Research ArticleDecay of Metastable State with Account of Agglomeration andRelaxation Processes

Victor Kurasov

Saint Petersburg State University Universitetskaya Embankment 79 Saint Petersburg 199034 Russia

Correspondence should be addressed to Victor Kurasov victor kurasovyahoocom

Received 27 July 2015 Revised 5 November 2015 Accepted 8 November 2015

Academic Editor Leonardo Palmisano

Copyright copy 2016 Victor Kurasov This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Theoretical description of the metastable phase decay kinetics in the presence of specific connections between the embryos of smallsizes has been given The theory of the decay kinetics in the presence of relaxation processes is constructed in analytical mannerThe119898-mers nucleation is investigated and the global kinetics of decay is also constructed in this case analytically

1 Introduction

The first-order phase transition kinetics is actively investi-gated since the publication of the pioneer papers by Wilson[1ndash3] concerning the famous chamber which became themain tool in investigations of the microworld The rangeof applications of the first-order phase transition kineticsinevitably grows since the phenomena concerning the self-organization [4 5] and aggregation [6] become the fieldof application of ideas lying in the base of the mentionedkineticsThe word ldquonucleationrdquo has nothing in common withnuclear phenomena but comes from the evident fact that theembryos of a new phase are rather compact objects and canbe treated as some nucleus of a new phase The main contentof the theory which predicts the rate of appearance of theembryos of a new phase was published in several papers inthe 1930s and 1940s which forms the classical theory of nucle-ationThe classical theory of nucleation [7 8] gives a completetheoretical description but the problem is that there existsno correspondence between the theoretical predictions andexperimental results

The first theoretical investigation on the problem of theembryos (droplet) formation belong to W Thomson (1870)who got the work of formation of the critical embryo and thesize of the critical embryo However Laplace and Young wereable to do this after their invention of the additional pressureunder the curved surface Certainly the true meaning of theformulas arises only with the theoretical investigations byGibbs (1877)

Historically the classical nucleation theory is associatedwith the following four papers [9ndash12] although many othercontributions (see eg [13] where for the first time wasstressed that the preexponential kinetic factor has to accountcollisions of a vapor molecule with cluster) are rather impor-tant in creation of the nucleation theory

After the creation of the classical nucleation theory themost radical reconsideration was proposed by Lothe andPound [14] who suggested including into the free energy ofa cluster the translational and rotational degrees of freedomThis point of view leads to the reconsideration of the rate ofnucleation in dozen of orders of magnitudes In some casesit brings the theory to the coincidence with experiment insome cases the situation is the opposite one The point ofview of Lothe and Pound was reconsidered in [15 16] andthe effect in comparison with the classical nucleation theorybecame radically smaller Here it is necessary to stress thatthe arguments of additional consideration of rotational andtranslational degrees of freedom for an embryo are ratherdoubtful

Other approaches to improve the classical nucleationtheory do not suggest such radical improvements One canmention here the attempts to account the partial pressure ofthe cluster which is treated as a molecule [17 18]The effect ofthis idea is not too big in comparison with the effect of Lothe-Poundsrsquo reconsideration

The next approach is the so-called kinetic theory ofnucleation suggested in [19 20]The long derivations in order

Hindawi Publishing CorporationAdvances in Physical ChemistryVolume 2016 Article ID 9468050 9 pageshttpdxdoiorg10115520169468050

2 Advances in Physical Chemistry

to solve the chain of kinetic equations finally led to thepractically known results except the requirement to take intoaccount the term corresponding to the surface tension of acluster with one molecule which can not be treated formallyin this sense but is certainly rather small correction factor

The Dillmann-Meier theory of nucleation is a rathermodel version which can be treated as some postulate onthe surface tension of a curved surface together with require-ments of thermodynamic consistency and some other minorimprovements [21 22] Certainly the theoretical ground ofthis approach is rather doubtful although the ideas arephysically attractive

The famous paper of Cahn andHillard [23] is very beauti-ful theoretically but the jumps of the density between phasescan be hardly described only by the first nontrivial term in thegradient decomposition This makes the range of applicationof this approach rather narrow namely in the vicinity of acritical point But in this vicinity one can hardly considerthe embryo as a closed object So the use of the mentionedapproach is rather restricted

One can not find any mathematical gaps in the classicaltheory of nucleation It means that some physical assump-tions of the model are not too appropriate since the math-ematical frames of theoretical construction are certainlycorrect

One of the possible sources of the deviation of the theoret-ical predictions from the experimental results is the existenceof the clusters (nonmonomers) with a relatively small freeenergy which actively changes the picture of the nucleationprocess Nowadays effect is known under the name of prenu-cleation [24] although some of such clusters can belong tothe blind channels which can not lead to the big supercriticalclusters [25] they are not nucleation centers Neverthe-less such terminology exists

The last example belongs to biological systems The com-plex behavior of the nucleation kinetics in biological systemsis not a surprising fact since the complex structure of biolog-ical molecules is well known More intriguing is the prenu-cleation importance in calcium containing systems [26]The further analysis of the prenucleation phenomena [27]shows the possibility ofmany paths through a nucleation bar-rier to a supercritical zoneThe nucleation processes throughthe different channels can even compete [28] Unfortunatelythe experimental observations are rarely accompanied by theweighted theoretical constructions

An evident intention to solve the appearing problems is touse computer calculations Unfortunately it is impossible tocheck all suppositions of the theory by computer simulationsince the nucleation process is the principally collective phe-nomenon and the true simulation has to consider billions ofparticles with precise numerical integration because the effectof nucleation is hidden in excesses from the phase coexistenceline

Ordinarily the attempts to improve the theory are aimedat getting the more reliable expression for the free energy ofthe critical embryoThere is no doubt that the critical embryois the central object of the nucleation theory considerationbut the constructions of the classical nucleation theoryapproach are valid only under the ldquogoodrdquo behavior of the free

energies of embryos of other sizes and kinetic coefficients ofejection and absorption of the molecules for the embryos farfrom the critical one

The ldquogoodrdquo behaviormeans that there are no peculiaritiesin behavior of kinetic coefficients and no local extremalpoints of the free energy of classically noncritical embryosMeanwhile there is absolutely no insurance that the men-tioned ldquogoodrdquo behavior really takes place The experimentalinformation is rather poor kinetic coefficients are mainlymeasured on the basis of the growth rate [29] and thischaracteristic is measured with such an accuracy that one canput a question about the excesses in the growth rate [30]Themeasurements of the growth rates can give only the differencebetween the adsorption and ejection coefficients but notexactly the values of these coefficients

One can note that in the presence of the prenucleationcenters the reconsideration of the rate of the stationary nucle-ation can be a nontrivial problem only in situations whenthe embryos have rather complex structure coming fromvarious structural components (see eg [25]) The methodsto construct the stationary rate of nucleation are ratherwell known although one can find many interesting subjectsin this question also

Herewewill focus on the kinetics of the global nucleationthat is on the description of the global evolution of the systemstarting from the appearance of metastability up to the end ofthe active formation of the embryos of a new phase whichgrow irreversibly until the Ostwald ripening The last stagesof the phase transition including the coagulation [31] andLifshic-Slezov asymptote [32 33] are not the subject of thisanalysis because we are interested in the total number ofsupercritical embryos and formation of their sizes spectrumThe total number of embryos and their size spectrum are themain results of the phase transformation and later the coagu-lation and mutual consumption (it is better to use namelythis terminology because the ordinary used word ldquocoales-cencerdquo implies absolutely another physical process) will onlydeform the size spectrum and the total number of supercrit-ical embryos

To describe the global nucleation kinetics it is necessary toput the initial conditionsThis requirement seems to be quitenatural We suppose that at the initial moment of time thereexists in the system a metastable phase without a new phaseand the process of nucleation beginsOtherwisewe can definethe initial moment as the last moment before appearance ofthe first traces of a new phase No further external action onthe system is implied This type of the initial and boundaryconditions is quite natural and has the name of the decay ofthe metastable state

The situation of decay without peculiarities in behaviorof the free energy of a small cluster was considered in [34]where the complete theory was constructed It is shownthat the evolution of the system can be described by thesubstance balance equation in some special form For ourpurposes to describe the kinetics in the situation where onesize of clusters is extracted by a very small free energy theanalogy with heterogeneous nucleation will be very useful Inheterogeneous nucleation the kinetics of the process is gov-erned not only by the fall of the supersaturation but also by

Advances in Physical Chemistry 3

the exhaustion of heterogeneous centers which become thecenters of the droplets Here the theory was given in 1989 [35]but the gap in the theory was filled only in 1993 [36]

The peculiarities in connections between the small clus-ters in a mother phase can cause very specific behavior of theglobal nucleation kinetics Although alreadymany importantcases were considered there exist enough important situa-tions necessary to be described theoretically Below we willgive the theoretical description of some of them

2 Weak Relaxation of Seeds inMonomer Nucleation

Thefirst important case is the situationwhen in themonomernucleation there exists a weak connection between the region(in the scale of sizes) of monomers and the region of119898-mersand it is necessary to take into account not only the consump-tion of 119898-mers by the supercritical embryos (every embryohas to consume one 119898-mer to start the life) but also theweak source of monomers being transformed into 119898-mersThis source changes the kinetics of the process

Here it is very useful to use the analogy with heteroge-neous nucleation Really the 119898-mers can be considered assome objects analogous to heterogeneous centers But thesituation with the relaxation of heterogeneous centers has notbeen considered So there exists a nontrivial mathematicalcontent of the constructions presented below

We introduce the value of 120579 as

120579 =

119899 [119898 119905]

119899 [119898 119905 = 0]

(1)

This value can be less or greater than 1 Here 119899[119898 119905] is thenumber of embryos with119898molecules at the time moment 119905

One can formulate the system of equations for unknownfunctions 119892 and 120579 and after some scaling get

119889120579

119889119905

= minus119895 (120579 minus 120579lim) minus 119879120579 [119905] exp (minus119892 [119905]) (2)

119892 [119905] = int

119905

0

(119905 minus 1199051015840)

3

120579 [1199051015840] exp (minus119892 [1199051015840]) 1198891199051015840 (3)

Here 119879 is some positive parameter which can not be avoidedThe positive parameter 119895 shows the intensity of the relaxationprocess The value 120579lim is the equilibrium value It is also aparameter

Now we will clarify the meaning of two unknown func-tions 119892 and 120579 of the time 119905 The function 119892 is the relativenumber of themolecules in a new phaseThe function 120579 is therelative number of the free (unoccupied by the supercriticalembryos) heterogeneous centers

Ordinarily the solution of the system of condensationequations (analogous to (2)-(3)) is given by iterations (see[37]) The behavior of the spectrum of sizes

119891 [119905] = 120579 [119905] exp (minus119892 [119905]) (4)

is rather complex even already in the first iterations whichare constructed in a way to simplify the form of the size

spectrumThis fact means that in our case one can not inventtoo ldquointelligentrdquo iterations Namely the presentation of (3) inthe already integrated form seems to produce too complexiteration procedure

In investigation of the nucleation process with theabsence of the relaxation process (see [37]) the followingchains of inequalities

120579(1)

lt 120579(3)

lt sdot sdot sdot lt 120579 lt sdot sdot sdot lt 120579(2)

119892(0)

lt 119892(2)

lt sdot sdot sdot lt 119892 lt sdot sdot sdot lt 119892(3)

lt 119892(1)

(5)

were formulated This chain is very productive for the accu-racy estimates which lead to the guaranteed accuracy of theobtained approximations

To conserve these chains one can propose the followingrecurrent scheme

119889120579(119894+1)

119889119905

= minus119895 (120579(119894)minus 120579lim) minus 119879120579

(119894)[119905] exp (minus119892

(119894)[119905])

119892(119894+1)

[119905] = int

119905

0

(119905 minus 1199051015840)

3

120579(119894)[1199051015840] exp (minus119892

(119894)[1199051015840]) 1198891199051015840

(6)

with initial condition

120579 [119905 = 0] = 1 (7)

and initial approximations

120579(0)

= 120579lim (8)

or

120579(0)

[119905] = 120579 [119905 = 0]

119892(0)

= 0

(9)

or the recurrent scheme

119889120579(119894+1)

119889119905

= minus119895 (120579(119894+1)

minus 120579lim)

minus 119879120579(119894+1)

[119905] exp (minus119892(119894)[119905])

119892(119894+1)

[119905] = int

119905

0

(119905 minus 1199051015840)

3

120579(119894)[1199051015840] exp (minus119892

(119894)[1199051015840]) 1198891199051015840

(10)

with the same initial condition and the same initial approxi-mations The third scheme is

119889120579(119894+1)

119889119905

= minus119895 (120579(119894)minus 120579lim) minus 119879120579

(119894+1)[119905] exp (minus119892

(119894)[119905])

119892(119894+1)

[119905] = int

119905

0

(119905 minus 1199051015840)

3

120579(119894)[1199051015840] exp (minus119892

(119894)[1199051015840]) 1198891199051015840

(11)

with the same initial approximations and the same initialcondition

We have to choose what type of iterations is the best oneThefirst procedure is absolutely correct and it can not lead

to divergence at some step of iterations The second and thethird schemes require additional regularization One has toput an additional restriction

4 Advances in Physical Chemistry

The Nucleation Process Stops When 120579 Attains Zero Thisrestriction is necessary to avoid the negative rate of nucleationand the divergence of iterations

The account of the relaxation process in the nucleationkinetics is a hard problem Already the first iteration fromthose where the effect of the relaxation term can be seenin the behavior of the supersaturation can not be calculatedanalytically

To fulfil analytical calculations we have to fulfil an addi-tional simplification based on the property of the avalancheconsumption

Thementioned property can be formulated as follows

Under Any Behavior of 120579 gt 0 the Relative Growth of 119892 OccursFasterThan (119905minus119905

0)3 One can prove this property analytically

Actually this property is evident since we take into accountthat 119892 = int

119905

0(119905 minus 1199051015840)3120579[1199051015840]exp(minus119892[1199051015840])1198891199051015840 and 120579 gt 0

On the basis of this property we see that the ldquopseudo-homogeneous spectrum of sizesrdquo (the spectrum of sizeswhich would be formed if there would be no action of thenumber of free seeds on the nucleation process)

119891hom sim exp (minus119892) (12)

will be truncated in a manner sharper than

exp(minus(

119905

119905cut)

3

) (13)

where 119905cut is the characteristic timeThen one can speak aboutthe cut-off type of the pseudo-homogeneous spectrum ofsizes

Now one can see the qualitative behavior of 120579 At first 120579relaxes to the value

120579119899=

119895

119895 + 119879

120579lim (14)

with characteristic time

119905rel 119899 = (119895 + 119879)minus1

(15)

This relaxation can lead to the increase of 120579 when

120579 [119905 = 0] lt 120579119899

(16)

or to the decrease of 120579 in the opposite situationAt the moment 119905cut this relaxation is changed by the

relaxation of 120579 to 120579lim with the characteristic time 119905rel = 119895minus1

The question which we are interested in is whether thechange of 120579 can destroy the cut-off type of the real spectrumof sizes 120579 exp(minus119892) (earlier the cut-off type was shown forexp(minus119892))

The destruction of the cut-off type of 120579 exp(minus119892) can takeplace only under the sharp relative increase of 120579 So we haveto estimate (119889120579119889119905)120579 Since

119889120579

119889119905

lt 119895120579lim

120579 gt

119895

119895 + 119879

120579lim

(17)

we see that119889120579119889119905

120579

lt 119895 + 119879 (18)

When 119879 ≫ 119895 one simply can not see the relaxationThe picture in this case is the picture described earlier (see[37]) plus some small perturbation Then having excludedthis situation we can rewrite the last inequality as

119889120579119889119905

120579

lt (3 divide 4) 119895 (19)

Now we see that the danger of destruction of the cut-offform of the size spectrum can appear only at big values of 119895Actually it has to be

119895 ≫ 119905minus1

cut (20)

If 119905cut ≪ 119879minus1 then there is no exhaustion of seeds So the

actual situation is 119905cut ge 119879minus1 and then we see that

119895 ≫ 119879 (21)

Here 120579lim is close to 120579119899and then 119905cut based on 120579lim is close

to 120579 based on 120579119899 This means that the cut-off behavior is not

broken here So the property of the cut-off takes place in allsituations

The property of the cut-off allows using the approxima-tion exp(minus119892) = 1 for 119905 lt 119905cut and exp(minus119892) = 0 for 119905 gt 119905cut Themoment 119905cut can be determined as the root of equation

119892 (119905cut) = 1 (22)

Then for 119905 lt 119905cut

119889120579

119889119905

= minus119895 (120579 minus 120579lim) minus 119879120579 [119905]

119892 [119905] = int

119905

0

(119905 minus 1199051015840)

3

120579 [1199051015840] 1198891199051015840

(23)

and one can explicitly calculate

120579 [119905] = 120579119899minus (120579119899minus 120579 [119905 = 0]) exp (minus (119895 + 119879) 119905) (24)

The value of 119892 can be presented as

119892 = int

119905

0

(119905 minus 1199051015840)

3

sdot (120579119899minus (120579119899minus 120579 [119905 = 0]) exp (minus (119895 + 119879) 119905

1015840)) 1198891199051015840

(25)

and can be easily found analytically which allows getting 119905cutas the solution of an ordinary algebraic equation

Equation (22) can be easily solved since we know theestimates 120579

11989911991144 and 120579[0]119911

44 for the behavior of 119892 which

leads to the estimates for the rootsWhat estimate is the aboveestimate andwhat the below estimate is are determined by thesign of 120579lim minus 120579[0]

The total number of droplets in the renormalized unitscan be found as

119873 = 119860int

119905cut

0

1 119889119905 = 119860119905cut (26)

Advances in Physical Chemistry 5

where 119860 is the initial amplitude (in renormalized values itequals 1) or as

119873 = 119860int

119905cut

0

exp (minus119892 [119905]) 120579 [119905] 119889119905 (27)

where for 119892 and for 120579 one can take the mentioned approx-imations The linearization of appearing exponents is alsoapproximately suitable and leads to the presence of the resultin elementary functions One has to note that the first recipeseems to bemore rough than the second one but it gives betterresults The reason is the effective compensation of decreaseof the intensity of appearance of new embryos before 119905cut bythe existence of the tail after 119905cut The refinement of equationfor 119905cut (it is better to use 119892[119905cut] = ln 2) can make the firstrecipe very accurate Certainly the further refinement withthe help of the perturbation technique can be fulfilled

Even the linear approximation

120579 = 120579 [0] + (119895 + 119879) (120579119899minus 120579 [0]) 119905 (28)

truncated at the crossing point with 120579119899 that is at (119895 + 119879)

minus1and prolonged further as 120579

119899gives a rather good approxima-

tion for the characteristics of the processThis approximationensures the pure algebraic equation of the fifth power for 119905cutIt is a really suitable approximation since this approximationhas themaximal deviation as (119895+119879)minus1 which is relatively small(simexp(minus1)) in comparison with initial deviation of 120579 from 120579

119899

Then the difference in integrals staying in (22) is less thantwo times which gives the difference for the duration of theintensive nucleation period in 2

14asymp 12 times (in reality the

difference in duration is much more small) The last approx-imation brings equation on root to the explicit formulas

This completes the analytical description of the situationof the decay in the system with the relaxation process for thecenters of nucleation

3 Accumulation of Several Seeds inthe Monomer Nucleation

The second important task is to consider the situation whenit is necessary to spend 119902 119898-mers to form the supercriticalembryo and the rate of nucleation is proportional to thenumber of119898-mers in power 119902

119869 sim 119899 [119898]119902 (29)

It is interesting to investigate this situation and to see thetransition to the case when 119902 is great enough Certainly theprobability

119875 sim 119899 [119898]119902 (30)

transfers into

119875 sim exp (minus120583119902) (31)

with a chemical potential

120583 = minus ln (119899 [119898]) (32)

corresponding to the chemical potential of ideal gasThe equation

119889119899 [119898]

119889119905

= 119899 [119898]119902119891 [119905] (33)

with some known function 119891[119905] can be written in thefollowing integral form

119899 [119898] =

119899 [119898 119905 = 0]

((119902 minus 1) int

119905

0119891 [1199051015840] 1198891199051015840+ 1)

1(119902minus1) (34)

Now we write the expression for the rate of nucleation inthe generalized Clausius-Clapeyron integrated form as

119869 = 1198690120579119902 exp(minus

Γ (120577 minus Φlowast)

Φlowast

) (35)

where

Γ =

Φlowast119889119865119888

119889120577

10038161003816100381610038161003816100381610038161003816120577=Φlowast

(36)

and Φlowastis the base of decomposition Here 119869

0is the initial

value of the rate of nucleation and 120579 is the relative numberof119898-mers referred to initial value

One can get

Γ = (

Φlowast

(Φlowast+ 1)

) ]1119888 (37)

where ]1119888is the number ofmonomers in critical embryoThis

number is connected with the total number of molecules ]119888

in the critical embryo as

]119888= ]1119888+ 119902119898 (38)

After the scaling the system of evolution equations is thefollowing one

119892 [119911] = int

119911

0

(119911 minus 119909)3120579119902[119909] exp (minus119892 [119909]) 119889119909

120579 [119911] = ((119902 minus 1)119860int

119911

0

exp (minus119892 [119909]) 119889119909 + 1)

minus1(119902minus1)

(39)

Here 119860 is the coefficient which remains after the scaling andcan not be canceled

We can formulate the iteration procedure as follows

119892(119894+1)

(119911) = int

119911

0

(119911 minus 119909)3120579119902

(119894)[119909] exp (minus119892

(119894)[119909]) 119889119909

120579(119894+1)

[119911]

= ((119902 minus 1)119860int

119911

0

exp (minus119892(119894)[119909]) 119889119909 + 1)

minus1(119902minus1)

(40)

6 Advances in Physical Chemistry

The separation of the spectrum in 119892 and 120579 which seemsto be natural from the physical point of view is quite unusualfrom the formal point of view when the unique equation

119892 [119911] = int

119911

0

(119911 minus 119909)3

sdot ((119902 minus 1)119860int

119909

0

exp (minus119892 [1199091015840]) 1198891199091015840 + 1)

minus119902(119902minus1)

sdot exp (minus119892 [119909]) 119889119909

(41)

is considered Then it is more natural to construct iterationsas

119892(119894+1)

[119911] = int

119911

0

(119911 minus 119909)3

sdot ((119902 minus 1)119860int

119909

0

exp (minus119892(119894)[1199091015840]) 1198891199091015840+ 1)

minus119902(119902minus1)

sdot exp (minus119892(119894)[119909]) 119889119909

(42)

but this way does not correspond to appearance of the aboveand below estimates although it is possible to prove that theseiterations with the initial approximation

119892(0)

= 0 (43)

converge to solutionThe procedure for 120579 and 119892 as the separate functions which

was formulated above will lead under the initial approxima-tions

120579(0)

= 1

119892(0)

= 0

(44)

to the chains of inequalities

119892(0)

lt 119892(2)

lt sdot sdot sdot lt 119892 lt sdot sdot sdot lt 119892(3)

lt 119892(1)

120579(1)

lt 120579(3)

lt sdot sdot sdot lt 120579 lt sdot sdot sdot lt 120579(2)

lt 120579(0)

(45)

Parameter 119860 is considered here as constant value for alliterations and the concrete value of 119860 will be determined onthe basis of the last (precise) iteration

It is useful to calculate the total number of supercriticalembryos as

119873tot sim int

infin

0

120579119902[119909] exp (minus119892 [119909]) 119889119909 (46)

On the basis of iteration procedure one can calculate

119873tot(119894+1) sim int

infin

0

120579119902

(119894)[119909] exp (minus119892

(119894)[119909]) 119889119909 (47)

The calculation of iterations gives

119892(0)

=

1199114

4

120579(1)

= ((119902 minus 1)119860119911 + 1)minus1(119902minus1)

(48)

Then

119873tot(2) = int

infin

0

((119902 minus 1)119860119911 + 1)minus119902(119902minus1) exp(minus119911

4

4

)119889119911 (49)

and this can be expressed through the hypergeometric func-tion

119873tot(2) = 2(minus92+2(119902(119902minus1)))

(119902 + 119860)minus119902(119902minus1)

(

1

3

sdot 2minus(52)(119902(119902minus1))+12

1205874(119902 + 119860)

3

(

1

119902 + 119860

)

minus119902(119902minus1)

119867([

3

4

+

1

4

119902

119902 minus 1

1 +

1

4

119902

119902 minus 1

5

4

+

1

4

119902

119902 minus 1

] [

3

2

7

4

5

4

]

minus

1

4

(119902 + 119860)4

)

sdot

sec ((14) 120587 (119902 (119902 minus 1))) Γ (3 + 119902 (119902 minus 1))

Γ (32 + (14) (119902 (119902 minus 1)))

minus 2(1minus(52)(119902(119902minus1)))

1205874(119902 + 119860)

2

(

1

119902 + 119860

)

minus119902(119902minus1)

119867([

3

4

+

1

4

119902

119902 minus 1

1 +

1

4

119902

119902 minus 1

1

2

+

1

4

119902

119902 minus 1

] [

3

2

3

4

5

4

]

minus

1

4

(119902 + 119860)4

)

sdot

csc ((14) 120587 + (14) 120587 (119902 (119902 minus 1))) Γ (2 + 119902 (119902 minus 1))

Γ (54 + (14) (119902 (119902 minus 1)))

+ 2(52minus(52)(119902(119902minus1)))

1205874(119902 + 119860)(

1

119902 + 119860

)

minus119902(119902minus1)

119867([

3

4

+

1

4

119902

119902 minus 1

1

4

+

1

4

119902

119902 minus 1

1

2

+

1

4

119902

119902 minus 1

] [

3

4

5

4

1

2

]

minus

1

4

(119902 + 119860)4

)

sdot

csc ((14) 120587 (119902 (119902 minus 1))) Γ (1 + 119902 (119902 minus 1))

Γ (1 + (14) (119902 (119902 minus 1)))

+ 2(3minus(52)(119902(119902minus1)))

1205874(

1

119902 + 119860

)

minus119902(119902minus1)

119867([

1

4

119902

119902 minus 1

1

4

+

1

4

119902

119902 minus 1

1

2

+

1

4

119902

119902 minus 1

] [

3

4

1

2

1

4

] minus

1

4

(119902 + 119860)4

)

sdot

sec ((14) 120587 + (14) 120587 (119902 (119902 minus 1))) Γ (119902 (119902 minus 1))

Γ (34 + (14) (119902 (119902 minus 1)))

+ 2(132minus2(119902(119902minus1)))

1205873(minus

1

2

+

1

4

119902

119902 minus 1

) (119902 + 119860)

Advances in Physical Chemistry 7

sdot 119867([1

3

4

1

2

1

4

] [

5

4

minus

1

4

119902

119902 minus 1

minus

1

4

119902

119902 minus 1

+

1

2

minus

1

4

119902

119902 minus 1

+

3

4

minus

1

4

119902

119902 minus 1

+ 1] minus

1

4

(119902 + 119860)4

)Γ(minus2

+

119902

119902 minus 1

))(1205873Γ(

119902

119902 minus 1

))

minus1

(50)

where Γ is the gamma-function and119867 is the hypergeometricfunction

It is possible to show analytically that this expression israther accurate The error is less than one percent Despitethe known form it is rather difficult to perform calculationsaccording to this expression and it is necessary to simplify thecalculations

Again we use the property of the avalanche consumptionHere one can show that 120579 is the decreasing (or at least notrapidly increasing) function of time (or of the size of theinitial embryo)Then one can see the cut-off type of the func-tion exp(minus119892) and the possibility of truncation of the function120579119902 exp(minus119892(119911)) at 119905cut The value of 119905cut can be chosen asthe root of equation 119892 = 1 Then the spectrum of sizes

119891 sim 120579119902[119909] exp (minus119892 [119909]) (51)

can be approximated as

119891 sim ((119902 minus 1)119860119911 + 1)minus119902(119902minus1) (52)

for 119911 lt 119911cut where 119911cut is the size corresponding to 119905cut Forother 119911 the spectrum is zero

The number of embryos can be easily calculated

119873tot = int

119911

0

((119902 minus 1)119860119909 + 1)minus119902(119902minus1)

119889119909

= minus

(119860119911119902 minus 119860119911 + 1)minus1(119902minus1)

minus 1

119860

(53)

This solves the formal problem of our investigationTo show the consistency of the presented approach we

have to investigate the inclusion of the119898-mers consumptionkinetics into the general nucleation Here one can find atleast two aspects of the problem The first aspect is the inputof 119898-mers in the overcoming of activation barrier and thecorresponding input in the stationary rate of nucleation Thesecond aspect is the consumption of the119898-mers by the near-critical and precritical embryos (this is the first type) andby the supercritical embryos (the second type) The rate ofconsumption will be different and their manifestation in theglobal kinetics 119904 will be described by different ways

To clarify the first aspect we rewrite the stationary rate ofnucleation as

119869 sim exp (minus119902 ln (119899 [119898])) exp (minus119865119888) (54)

One has to stress that here 119865119888is the free energy of the critical

embryo formation on 119902 119898-mers This implies that the work

of collecting of119898-mers is considered separatelyThis work inapproximation of ideal gas of119898-mers is

1198650= minus119902 ln (119899 [119898]) (55)

At the same time we have to note that in the embryocontaining ]

119888molecules there are only ]

119888minus 119902119898 molecules

coming from monomers Then 1198650has to be written as

1198650= minus119887 (]

119888minus 119902119898) + 120574119860 (56)

where 119887 is the excess of the chemical potential for monomers120574 is the renormalized surface tension and 119860 is the surfacearea of the embryo It is clear that in the determination of thesurface area it is necessary to account that the part of the vol-ume inside the embryos surface is occupied by the moleculescoming from119898-mersWhen the surface of the embryo simplysurrounds the embryo one has to write

119860 = 4120587(

3V119897(]1119888+ 119902119898)

4120587

)

23

(57)

where V119897is the volume per onemolecule in a liquid phase and

]1119888is the number of molecules coming from monomersIn the state of equilibrium the chemical potential of 119898-

mer has to be equal to 119898 chemical potentials of monomers(here we neglect the surface term or consider the deformedchemical potential) So

119899 [119898]

119899 [119898 119904 = 0]

= exp (119887119898) (58)

and we see that the total free energy is

119888= 119865119888minus 119902119887119898 = minus]

119888119887 + 120574119860 (59)

and the total free energy coincides with the classical expres-sion

One has to stress that the parameter Γ in the previoussimplification of the dependence of the free energy on super-saturation (or the dependence of the stationary rate of nucle-ation on the supersaturation) is the number of moleculescoming from monomers multiplied on Φ

lowast(Φlowast

+ 1)that is

Γ 997888rarr Γ1=

Φlowast]1119888

(Φlowast+ 1)

(60)

Γ1is Φlowast119889119865119888119889120577|120577=Φlowast

If we calculate the derivative of the total free energy

119888we

get

Φlowast

119889119888

119889120577

100381610038161003816100381610038161003816100381610038161003816120577=Φlowast

= Γ1+ 119902119898

119889119887

119889120577

(61)

or

Φlowast

119889119888

119889120577

100381610038161003816100381610038161003816100381610038161003816120577=Φlowast

= Γ1+ Γ119898 (62)

8 Advances in Physical Chemistry

where

Γ119898=

120577119902119898

(120577 + 1)

10038161003816100381610038161003816100381610038161003816120577=Φlowast

(63)

As a result

Φlowast

119889119888

119889120577

100381610038161003816100381610038161003816100381610038161003816120577=Φlowast

= Γ

Γ =

120577]119888

(120577 + 1)

10038161003816100381610038161003816100381610038161003816120577=Φlowast

(64)

We will call this property the separation of Γ It takes placeonly in the first derivatives of the free energies and onlybecause the surface term does not act on the derivative of thefree energy The surface term in the free energy only changesthe number of molecules in the critical embryo and has nodirect influence on derivative

So we see that in the free energy there is the directcorrespondence between the119898-mers picture and monomerspicture

One can propose another picture when the number of119898-mers necessary to form the critical embryo is proportionalto the number of molecules inside the critical embryo In theenormously big critical embryos namely thismechanismhasto be considered Then the picture is equivalent to the binarynucleation with two vapors (the vapor of monomers and thevapor of119898-mers)

The second aspect is connected with the mechanism ofconsumption Here one has to confess that there are two prin-cipal mechanisms of the substance consumption The firstmechanism is ldquoone embryo-fixed number of clusters (mono-mers or 119898-mers)rdquo The second mechanism is ldquoone embryo-growing number of clusters (monomers and 119898-mers) corre-sponding to the law of the regular growthrdquo

Rigorously speaking we have to write for each type ofactive monomers two types of consumption and the balanceequations With the help of exponential approximationsmentioned above one can unify these mechanisms in one lawof growth Ordinarily when the first mechanism is importantthe second is absent When the second mechanism acts itmeans that there is enough clusters of such type to neglect theexhaustion of seeds We will call this effect as the dominatingpreference of one mechanism

Despite the mentioned dominating preference propertythe simultaneous existence of several mechanisms can not beexcluded as a specific case But it is clear that with the helpof approaches developed here it is possible to construct thetheoretical description of any process with different mecha-nisms of formation different mechanisms of substance con-sumption and different relaxation processes correspondingto bounds or connections between the resources of clusters ofdifferent sizes

The different relations between the characteristic timesof the mentioned processes make the processes of nucleationrather different The size spectrums are very various also Todescribe these situations one has to demonstrate the powerof the approaches making the base of theoretical description

In every situation the leading idea will be the avalanche con-sumption of metastability in different manifestations of thisproperty

4 Main Results

This paper contains at least two important derivations con-cerning the kinetics of decay with account of the relaxationprocesses and the kinetics of decay when the119898-mers nucleationis important Both cases are important and one can see thataccount of these processes radically changes the numericalvalues of parameters of the nucleations process But one hasto confess that the qualitative picture remains similar to thetraditional case The last does not mean that nothing in theform of the particles size distribution is changed Contrarilythe form of the size spectrum will be absolutely another Butthe property of the avalanche cut-off of the pseudo-homo-geneous size spectrum remains The last property was veryproductive in construction of the above presented procedure

It would be interesting to analyze these situations in thecase of the smooth external influence on the system whichcauses nucleation that is in the case of dynamic conditionsAlso it would be interesting to see the influence of the relax-ation processes in the overcoming of the near-critical regionin the process of the binary nucleation Certainly these per-spectives make the subjects of separate publications

Beside the direct construction of the nucleation globalkinetics description we managed to extract some importantfeatures of the effect of119898-mers inclusion into the nucleationprocess One can mention here the property of separation ofΓ and the correspondence between the monomers picture and119898-mers picture These properties are important enough to bementioned in the general theory of nucleation

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] C T R Wilson ldquoCondensation of water vapour in the presenceof dust-free air and other gasesrdquo Philosophical Transactions ofthe Royal Society A Mathematical Physical and Engineering Sci-ences vol 189 pp 265ndash307 1897

[2] C T RWilson ldquoOn the condensation nuclei produced in gasesby the action of roentgen rays uranium rays ultraviolet rays andother agentsrdquoPhilosophical Transactions A vol 192 no 9 article403 1899

[3] C T RWilson ldquoOn the comparative efficiency as condensationnuclei of positively and negatively charged ionsrdquo PhilosophicalTransactions of the Royal Society A Containing Papers of aMath-ematical or Physical Character vol 193 pp 289ndash308 1900

[4] A Benmouna R Benmouna M R Bockstaller and I FHakem ldquoSelf-organization schemes towards thermodynamicstable bulk heterojunction morphologies a perspective onfuture fabrication strategies of polymer photovoltaic architec-turesrdquo Advances in Physical Chemistry vol 2013 Article ID948189 8 pages 2013

Advances in Physical Chemistry 9

[5] Y Kazzi H Awada and M Nardin ldquoAdhesion on nanoorga-nized multilayers surface thermodynamics and local energydissipationrdquo Advances in Physical Chemistry vol 2010 ArticleID 502709 11 pages 2010

[6] A I Burshtein ldquoContact and distant luminescence quenchingin solutionsrdquo Advances in Physical Chemistry vol 2009 ArticleID 214219 34 pages 2009

[7] A C Zettlemoyer Nucleation Dekker New York NY USA1969

[8] F F Abraham Homogeneous Nucleation Theory AcademicNew York NY USA 1974

[9] M Volmer and A Weber ldquoKeimbildung in ubersattigtenGebildenrdquo Zeitschrift fur Physikalische Chemie vol 119 pp 277ndash301 1925

[10] R Becker andW Doring ldquoKinetische behandlung der keimbil-dung in uberattigten dampfernrdquo Annals of Physics vol 24 pp719ndash752 1935

[11] J Frenkel ldquoA general theory of heterophase fluctuations andpretransition phenomenardquoThe Journal of Chemical Physics vol7 no 7 pp 538ndash547 1939

[12] J B Zeldovich ldquoOn the theory of new phase formation cavi-tationrdquo Acta Physicochimica USSR vol 18 pp 1ndash22 1943

[13] L Farkas ldquoThe velocity of nucleus formation in superheatedvaporsrdquo Zeitschrift fur Physikalische Chemie vol 125 pp 236ndash240 1927

[14] J Lothe and G M Pound ldquoReconsiderations of nucleationtheoryrdquo The Journal of Chemical Physics vol 36 no 8 article2080 1962

[15] H Reiss J L Katz and E R Cohen ldquoTranslation-rotation para-dox in the theory of nucleationrdquoThe Journal of Chemical Phys-ics vol 48 no 12 pp 5553ndash5560 1968

[16] H Reiss ldquoTreatment of droplike clusters by means of the clas-sical phase integral in nucleation theoryrdquo Journal of StatisticalPhysics vol 2 no 1 pp 83ndash104 1970

[17] W G Courtney ldquoRemarks on homogeneous nucleationrdquo TheJournal of Chemical Physics vol 35 no 6 pp 2249ndash2250 1961

[18] M Blander and J L Katz ldquoThe thermodynamics of cluster for-mation in nucleation theoryrdquo Journal of Statistical Physics vol 4no 1 pp 55ndash59 1972

[19] J L Katz and H Weidersich ldquoNucleation theory without Max-well Demonsrdquo Journal of Colloid and Interface Science vol 61no 2 pp 351ndash355 1977

[20] J L Katz and M D Donohue ldquoA kinetic approach to homo-geneous nucleation theoryrdquo in Advances in Chemical Physics IPrigogine and S A Rice Eds vol 40 chapter 3 pp 137ndash155Wiley 1979

[21] A Dillmann and G E A Meier ldquoHomogeneous nucleation ofsupersaturated vaporsrdquo Chemical Physics Letters vol 160 no 1pp 71ndash74 1989

[22] A Dillmann and G E A Meier ldquoA refined droplet approach tothe problem of homogeneous nucleation from the vapor phaserdquoThe Journal of Chemical Physics vol 94 no 5 pp 3872ndash38841991

[23] J W Cahn and J E Hillard ldquoFree energy of a nonuniformsystem III Nucleation in a two-component incompressiblefluidrdquoTheJournal of Chemical Physics vol 31 no 3 pp 688ndash6991959

[24] D Gebauer and H Colfen ldquoPrenucleation clusters and non-classical nucleationrdquo Nano Today vol 6 no 6 pp 564ndash5842011

[25] D Sept and J A McCammon ldquoThermodynamics and kineticsof actin filament nucleationrdquo Biophysical Journal vol 81 no 2pp 667ndash674 2001

[26] A S Myerson and B L Trout ldquoNucleation from solutionrdquo Sci-ence vol 341 no 6148 pp 855ndash856 2013

[27] D Zahn ldquoThermodynamics and kinetics of prenucleation clus-ters classical and non-classical nucleationrdquo ChemPhysChemvol 16 no 10 pp 2069ndash2075 2015

[28] H Deng S Wang X Wang et al ldquoTwo competitive nucle-ation mechanisms of calcium carbonate biomineralization inresponse to surface functionality in low calcium ion concentra-tion solutionrdquo Regenerative Biomaterials vol 2 no 3 pp 187ndash195 2015

[29] D Kozma and G Toth ldquoMicroscopic rate constants of crystalgrowth from molecular dynamic simulations combined withmetadynamicsrdquo Advances in Physical Chemistry vol 2012Article ID 135172 10 pages 2012

[30] S Fakruddin C Srinivasu B R Venkateswara Rao and KNarendra ldquoExcess transport properties of binary mixtures ofquinoline with xylenes at different temperaturesrdquo Advances inPhysical Chemistry vol 2012 Article ID 324098 9 pages 2012

[31] A A Lushnikov ldquoEvolution of coagulating systemsrdquo Journal ofColloid and Interface Science vol 45 no 3 pp 549ndash556 1973

[32] I M Lifshic and V V Slezov ldquoKinetics of diffusive decomposi-tion of supersaturated solid solutionsrdquo Soviet PhysicsmdashJournalof Experimental and Theoretical Physics vol 35 pp 331ndash3391959

[33] IM Lifshic andV V Slezov ldquoThe kinetics of precipitation fromsupersaturated solid solutionsrdquo Journal of Physics andChemistryof Solids vol 19 no 1-2 pp 35ndash50 1961

[34] F M Kuni and A P Grinin ldquoKinetics of homogeneous conden-sationat the stage of formation of the main quantity of a newphaserdquo Colloid Journal of the USSR vol 46 p 460 1984

[35] F M Kuni T Y Novozhilova and I A Terentiev ldquoKineticequation of heterogenous condensationrdquo Letters in Mathemati-cal Physics vol 14 no 2 pp 161ndash167 1987

[36] V Kurasov ldquoDecay of metastable multicomponentmixturerdquohttparxivorgabscond-mat9310005

[37] V B Kurasov ldquoKinetics of heterogeneous condensation underdynamic conditionsrdquo Physical Review E vol 49 no 5 pp 3948ndash3955 1994

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Inorganic ChemistryInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal ofPhotoenergy

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Carbohydrate Chemistry

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

Physical Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom

Analytical Methods in Chemistry

Journal of

Volume 2014

Bioinorganic Chemistry and ApplicationsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

SpectroscopyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Medicinal ChemistryInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chromatography Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied ChemistryJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Theoretical ChemistryJournal of

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Journal of

Spectroscopy

Analytical ChemistryInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Quantum Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Organic Chemistry International

ElectrochemistryInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CatalystsJournal of

2 Advances in Physical Chemistry

to solve the chain of kinetic equations finally led to thepractically known results except the requirement to take intoaccount the term corresponding to the surface tension of acluster with one molecule which can not be treated formallyin this sense but is certainly rather small correction factor

The Dillmann-Meier theory of nucleation is a rathermodel version which can be treated as some postulate onthe surface tension of a curved surface together with require-ments of thermodynamic consistency and some other minorimprovements [21 22] Certainly the theoretical ground ofthis approach is rather doubtful although the ideas arephysically attractive

The famous paper of Cahn andHillard [23] is very beauti-ful theoretically but the jumps of the density between phasescan be hardly described only by the first nontrivial term in thegradient decomposition This makes the range of applicationof this approach rather narrow namely in the vicinity of acritical point But in this vicinity one can hardly considerthe embryo as a closed object So the use of the mentionedapproach is rather restricted

One can not find any mathematical gaps in the classicaltheory of nucleation It means that some physical assump-tions of the model are not too appropriate since the math-ematical frames of theoretical construction are certainlycorrect

One of the possible sources of the deviation of the theoret-ical predictions from the experimental results is the existenceof the clusters (nonmonomers) with a relatively small freeenergy which actively changes the picture of the nucleationprocess Nowadays effect is known under the name of prenu-cleation [24] although some of such clusters can belong tothe blind channels which can not lead to the big supercriticalclusters [25] they are not nucleation centers Neverthe-less such terminology exists

The last example belongs to biological systems The com-plex behavior of the nucleation kinetics in biological systemsis not a surprising fact since the complex structure of biolog-ical molecules is well known More intriguing is the prenu-cleation importance in calcium containing systems [26]The further analysis of the prenucleation phenomena [27]shows the possibility ofmany paths through a nucleation bar-rier to a supercritical zoneThe nucleation processes throughthe different channels can even compete [28] Unfortunatelythe experimental observations are rarely accompanied by theweighted theoretical constructions

An evident intention to solve the appearing problems is touse computer calculations Unfortunately it is impossible tocheck all suppositions of the theory by computer simulationsince the nucleation process is the principally collective phe-nomenon and the true simulation has to consider billions ofparticles with precise numerical integration because the effectof nucleation is hidden in excesses from the phase coexistenceline

Ordinarily the attempts to improve the theory are aimedat getting the more reliable expression for the free energy ofthe critical embryoThere is no doubt that the critical embryois the central object of the nucleation theory considerationbut the constructions of the classical nucleation theoryapproach are valid only under the ldquogoodrdquo behavior of the free

energies of embryos of other sizes and kinetic coefficients ofejection and absorption of the molecules for the embryos farfrom the critical one

The ldquogoodrdquo behaviormeans that there are no peculiaritiesin behavior of kinetic coefficients and no local extremalpoints of the free energy of classically noncritical embryosMeanwhile there is absolutely no insurance that the men-tioned ldquogoodrdquo behavior really takes place The experimentalinformation is rather poor kinetic coefficients are mainlymeasured on the basis of the growth rate [29] and thischaracteristic is measured with such an accuracy that one canput a question about the excesses in the growth rate [30]Themeasurements of the growth rates can give only the differencebetween the adsorption and ejection coefficients but notexactly the values of these coefficients

One can note that in the presence of the prenucleationcenters the reconsideration of the rate of the stationary nucle-ation can be a nontrivial problem only in situations whenthe embryos have rather complex structure coming fromvarious structural components (see eg [25]) The methodsto construct the stationary rate of nucleation are ratherwell known although one can find many interesting subjectsin this question also

Herewewill focus on the kinetics of the global nucleationthat is on the description of the global evolution of the systemstarting from the appearance of metastability up to the end ofthe active formation of the embryos of a new phase whichgrow irreversibly until the Ostwald ripening The last stagesof the phase transition including the coagulation [31] andLifshic-Slezov asymptote [32 33] are not the subject of thisanalysis because we are interested in the total number ofsupercritical embryos and formation of their sizes spectrumThe total number of embryos and their size spectrum are themain results of the phase transformation and later the coagu-lation and mutual consumption (it is better to use namelythis terminology because the ordinary used word ldquocoales-cencerdquo implies absolutely another physical process) will onlydeform the size spectrum and the total number of supercrit-ical embryos

To describe the global nucleation kinetics it is necessary toput the initial conditionsThis requirement seems to be quitenatural We suppose that at the initial moment of time thereexists in the system a metastable phase without a new phaseand the process of nucleation beginsOtherwisewe can definethe initial moment as the last moment before appearance ofthe first traces of a new phase No further external action onthe system is implied This type of the initial and boundaryconditions is quite natural and has the name of the decay ofthe metastable state

The situation of decay without peculiarities in behaviorof the free energy of a small cluster was considered in [34]where the complete theory was constructed It is shownthat the evolution of the system can be described by thesubstance balance equation in some special form For ourpurposes to describe the kinetics in the situation where onesize of clusters is extracted by a very small free energy theanalogy with heterogeneous nucleation will be very useful Inheterogeneous nucleation the kinetics of the process is gov-erned not only by the fall of the supersaturation but also by

Advances in Physical Chemistry 3

the exhaustion of heterogeneous centers which become thecenters of the droplets Here the theory was given in 1989 [35]but the gap in the theory was filled only in 1993 [36]

The peculiarities in connections between the small clus-ters in a mother phase can cause very specific behavior of theglobal nucleation kinetics Although alreadymany importantcases were considered there exist enough important situa-tions necessary to be described theoretically Below we willgive the theoretical description of some of them

2 Weak Relaxation of Seeds inMonomer Nucleation

Thefirst important case is the situationwhen in themonomernucleation there exists a weak connection between the region(in the scale of sizes) of monomers and the region of119898-mersand it is necessary to take into account not only the consump-tion of 119898-mers by the supercritical embryos (every embryohas to consume one 119898-mer to start the life) but also theweak source of monomers being transformed into 119898-mersThis source changes the kinetics of the process

Here it is very useful to use the analogy with heteroge-neous nucleation Really the 119898-mers can be considered assome objects analogous to heterogeneous centers But thesituation with the relaxation of heterogeneous centers has notbeen considered So there exists a nontrivial mathematicalcontent of the constructions presented below

We introduce the value of 120579 as

120579 =

119899 [119898 119905]

119899 [119898 119905 = 0]

(1)

This value can be less or greater than 1 Here 119899[119898 119905] is thenumber of embryos with119898molecules at the time moment 119905

One can formulate the system of equations for unknownfunctions 119892 and 120579 and after some scaling get

119889120579

119889119905

= minus119895 (120579 minus 120579lim) minus 119879120579 [119905] exp (minus119892 [119905]) (2)

119892 [119905] = int

119905

0

(119905 minus 1199051015840)

3

120579 [1199051015840] exp (minus119892 [1199051015840]) 1198891199051015840 (3)

Here 119879 is some positive parameter which can not be avoidedThe positive parameter 119895 shows the intensity of the relaxationprocess The value 120579lim is the equilibrium value It is also aparameter

Now we will clarify the meaning of two unknown func-tions 119892 and 120579 of the time 119905 The function 119892 is the relativenumber of themolecules in a new phaseThe function 120579 is therelative number of the free (unoccupied by the supercriticalembryos) heterogeneous centers

Ordinarily the solution of the system of condensationequations (analogous to (2)-(3)) is given by iterations (see[37]) The behavior of the spectrum of sizes

119891 [119905] = 120579 [119905] exp (minus119892 [119905]) (4)

is rather complex even already in the first iterations whichare constructed in a way to simplify the form of the size

spectrumThis fact means that in our case one can not inventtoo ldquointelligentrdquo iterations Namely the presentation of (3) inthe already integrated form seems to produce too complexiteration procedure

In investigation of the nucleation process with theabsence of the relaxation process (see [37]) the followingchains of inequalities

120579(1)

lt 120579(3)

lt sdot sdot sdot lt 120579 lt sdot sdot sdot lt 120579(2)

119892(0)

lt 119892(2)

lt sdot sdot sdot lt 119892 lt sdot sdot sdot lt 119892(3)

lt 119892(1)

(5)

were formulated This chain is very productive for the accu-racy estimates which lead to the guaranteed accuracy of theobtained approximations

To conserve these chains one can propose the followingrecurrent scheme

119889120579(119894+1)

119889119905

= minus119895 (120579(119894)minus 120579lim) minus 119879120579

(119894)[119905] exp (minus119892

(119894)[119905])

119892(119894+1)

[119905] = int

119905

0

(119905 minus 1199051015840)

3

120579(119894)[1199051015840] exp (minus119892

(119894)[1199051015840]) 1198891199051015840

(6)

with initial condition

120579 [119905 = 0] = 1 (7)

and initial approximations

120579(0)

= 120579lim (8)

or

120579(0)

[119905] = 120579 [119905 = 0]

119892(0)

= 0

(9)

or the recurrent scheme

119889120579(119894+1)

119889119905

= minus119895 (120579(119894+1)

minus 120579lim)

minus 119879120579(119894+1)

[119905] exp (minus119892(119894)[119905])

119892(119894+1)

[119905] = int

119905

0

(119905 minus 1199051015840)

3

120579(119894)[1199051015840] exp (minus119892

(119894)[1199051015840]) 1198891199051015840

(10)

with the same initial condition and the same initial approxi-mations The third scheme is

119889120579(119894+1)

119889119905

= minus119895 (120579(119894)minus 120579lim) minus 119879120579

(119894+1)[119905] exp (minus119892

(119894)[119905])

119892(119894+1)

[119905] = int

119905

0

(119905 minus 1199051015840)

3

120579(119894)[1199051015840] exp (minus119892

(119894)[1199051015840]) 1198891199051015840

(11)

with the same initial approximations and the same initialcondition

We have to choose what type of iterations is the best oneThefirst procedure is absolutely correct and it can not lead

to divergence at some step of iterations The second and thethird schemes require additional regularization One has toput an additional restriction

4 Advances in Physical Chemistry

The Nucleation Process Stops When 120579 Attains Zero Thisrestriction is necessary to avoid the negative rate of nucleationand the divergence of iterations

The account of the relaxation process in the nucleationkinetics is a hard problem Already the first iteration fromthose where the effect of the relaxation term can be seenin the behavior of the supersaturation can not be calculatedanalytically

To fulfil analytical calculations we have to fulfil an addi-tional simplification based on the property of the avalancheconsumption

Thementioned property can be formulated as follows

Under Any Behavior of 120579 gt 0 the Relative Growth of 119892 OccursFasterThan (119905minus119905

0)3 One can prove this property analytically

Actually this property is evident since we take into accountthat 119892 = int

119905

0(119905 minus 1199051015840)3120579[1199051015840]exp(minus119892[1199051015840])1198891199051015840 and 120579 gt 0

On the basis of this property we see that the ldquopseudo-homogeneous spectrum of sizesrdquo (the spectrum of sizeswhich would be formed if there would be no action of thenumber of free seeds on the nucleation process)

119891hom sim exp (minus119892) (12)

will be truncated in a manner sharper than

exp(minus(

119905

119905cut)

3

) (13)

where 119905cut is the characteristic timeThen one can speak aboutthe cut-off type of the pseudo-homogeneous spectrum ofsizes

Now one can see the qualitative behavior of 120579 At first 120579relaxes to the value

120579119899=

119895

119895 + 119879

120579lim (14)

with characteristic time

119905rel 119899 = (119895 + 119879)minus1

(15)

This relaxation can lead to the increase of 120579 when

120579 [119905 = 0] lt 120579119899

(16)

or to the decrease of 120579 in the opposite situationAt the moment 119905cut this relaxation is changed by the

relaxation of 120579 to 120579lim with the characteristic time 119905rel = 119895minus1

The question which we are interested in is whether thechange of 120579 can destroy the cut-off type of the real spectrumof sizes 120579 exp(minus119892) (earlier the cut-off type was shown forexp(minus119892))

The destruction of the cut-off type of 120579 exp(minus119892) can takeplace only under the sharp relative increase of 120579 So we haveto estimate (119889120579119889119905)120579 Since

119889120579

119889119905

lt 119895120579lim

120579 gt

119895

119895 + 119879

120579lim

(17)

we see that119889120579119889119905

120579

lt 119895 + 119879 (18)

When 119879 ≫ 119895 one simply can not see the relaxationThe picture in this case is the picture described earlier (see[37]) plus some small perturbation Then having excludedthis situation we can rewrite the last inequality as

119889120579119889119905

120579

lt (3 divide 4) 119895 (19)

Now we see that the danger of destruction of the cut-offform of the size spectrum can appear only at big values of 119895Actually it has to be

119895 ≫ 119905minus1

cut (20)

If 119905cut ≪ 119879minus1 then there is no exhaustion of seeds So the

actual situation is 119905cut ge 119879minus1 and then we see that

119895 ≫ 119879 (21)

Here 120579lim is close to 120579119899and then 119905cut based on 120579lim is close

to 120579 based on 120579119899 This means that the cut-off behavior is not

broken here So the property of the cut-off takes place in allsituations

The property of the cut-off allows using the approxima-tion exp(minus119892) = 1 for 119905 lt 119905cut and exp(minus119892) = 0 for 119905 gt 119905cut Themoment 119905cut can be determined as the root of equation

119892 (119905cut) = 1 (22)

Then for 119905 lt 119905cut

119889120579

119889119905

= minus119895 (120579 minus 120579lim) minus 119879120579 [119905]

119892 [119905] = int

119905

0

(119905 minus 1199051015840)

3

120579 [1199051015840] 1198891199051015840

(23)

and one can explicitly calculate

120579 [119905] = 120579119899minus (120579119899minus 120579 [119905 = 0]) exp (minus (119895 + 119879) 119905) (24)

The value of 119892 can be presented as

119892 = int

119905

0

(119905 minus 1199051015840)

3

sdot (120579119899minus (120579119899minus 120579 [119905 = 0]) exp (minus (119895 + 119879) 119905

1015840)) 1198891199051015840

(25)

and can be easily found analytically which allows getting 119905cutas the solution of an ordinary algebraic equation

Equation (22) can be easily solved since we know theestimates 120579

11989911991144 and 120579[0]119911

44 for the behavior of 119892 which

leads to the estimates for the rootsWhat estimate is the aboveestimate andwhat the below estimate is are determined by thesign of 120579lim minus 120579[0]

The total number of droplets in the renormalized unitscan be found as

119873 = 119860int

119905cut

0

1 119889119905 = 119860119905cut (26)

Advances in Physical Chemistry 5

where 119860 is the initial amplitude (in renormalized values itequals 1) or as

119873 = 119860int

119905cut

0

exp (minus119892 [119905]) 120579 [119905] 119889119905 (27)

where for 119892 and for 120579 one can take the mentioned approx-imations The linearization of appearing exponents is alsoapproximately suitable and leads to the presence of the resultin elementary functions One has to note that the first recipeseems to bemore rough than the second one but it gives betterresults The reason is the effective compensation of decreaseof the intensity of appearance of new embryos before 119905cut bythe existence of the tail after 119905cut The refinement of equationfor 119905cut (it is better to use 119892[119905cut] = ln 2) can make the firstrecipe very accurate Certainly the further refinement withthe help of the perturbation technique can be fulfilled

Even the linear approximation

120579 = 120579 [0] + (119895 + 119879) (120579119899minus 120579 [0]) 119905 (28)

truncated at the crossing point with 120579119899 that is at (119895 + 119879)

minus1and prolonged further as 120579

119899gives a rather good approxima-

tion for the characteristics of the processThis approximationensures the pure algebraic equation of the fifth power for 119905cutIt is a really suitable approximation since this approximationhas themaximal deviation as (119895+119879)minus1 which is relatively small(simexp(minus1)) in comparison with initial deviation of 120579 from 120579

119899

Then the difference in integrals staying in (22) is less thantwo times which gives the difference for the duration of theintensive nucleation period in 2

14asymp 12 times (in reality the

difference in duration is much more small) The last approx-imation brings equation on root to the explicit formulas

This completes the analytical description of the situationof the decay in the system with the relaxation process for thecenters of nucleation

3 Accumulation of Several Seeds inthe Monomer Nucleation

The second important task is to consider the situation whenit is necessary to spend 119902 119898-mers to form the supercriticalembryo and the rate of nucleation is proportional to thenumber of119898-mers in power 119902

119869 sim 119899 [119898]119902 (29)

It is interesting to investigate this situation and to see thetransition to the case when 119902 is great enough Certainly theprobability

119875 sim 119899 [119898]119902 (30)

transfers into

119875 sim exp (minus120583119902) (31)

with a chemical potential

120583 = minus ln (119899 [119898]) (32)

corresponding to the chemical potential of ideal gasThe equation

119889119899 [119898]

119889119905

= 119899 [119898]119902119891 [119905] (33)

with some known function 119891[119905] can be written in thefollowing integral form

119899 [119898] =

119899 [119898 119905 = 0]

((119902 minus 1) int

119905

0119891 [1199051015840] 1198891199051015840+ 1)

1(119902minus1) (34)

Now we write the expression for the rate of nucleation inthe generalized Clausius-Clapeyron integrated form as

119869 = 1198690120579119902 exp(minus

Γ (120577 minus Φlowast)

Φlowast

) (35)

where

Γ =

Φlowast119889119865119888

119889120577

10038161003816100381610038161003816100381610038161003816120577=Φlowast

(36)

and Φlowastis the base of decomposition Here 119869

0is the initial

value of the rate of nucleation and 120579 is the relative numberof119898-mers referred to initial value

One can get

Γ = (

Φlowast

(Φlowast+ 1)

) ]1119888 (37)

where ]1119888is the number ofmonomers in critical embryoThis

number is connected with the total number of molecules ]119888

in the critical embryo as

]119888= ]1119888+ 119902119898 (38)

After the scaling the system of evolution equations is thefollowing one

119892 [119911] = int

119911

0

(119911 minus 119909)3120579119902[119909] exp (minus119892 [119909]) 119889119909

120579 [119911] = ((119902 minus 1)119860int

119911

0

exp (minus119892 [119909]) 119889119909 + 1)

minus1(119902minus1)

(39)

Here 119860 is the coefficient which remains after the scaling andcan not be canceled

We can formulate the iteration procedure as follows

119892(119894+1)

(119911) = int

119911

0

(119911 minus 119909)3120579119902

(119894)[119909] exp (minus119892

(119894)[119909]) 119889119909

120579(119894+1)

[119911]

= ((119902 minus 1)119860int

119911

0

exp (minus119892(119894)[119909]) 119889119909 + 1)

minus1(119902minus1)

(40)

6 Advances in Physical Chemistry

The separation of the spectrum in 119892 and 120579 which seemsto be natural from the physical point of view is quite unusualfrom the formal point of view when the unique equation

119892 [119911] = int

119911

0

(119911 minus 119909)3

sdot ((119902 minus 1)119860int

119909

0

exp (minus119892 [1199091015840]) 1198891199091015840 + 1)

minus119902(119902minus1)

sdot exp (minus119892 [119909]) 119889119909

(41)

is considered Then it is more natural to construct iterationsas

119892(119894+1)

[119911] = int

119911

0

(119911 minus 119909)3

sdot ((119902 minus 1)119860int

119909

0

exp (minus119892(119894)[1199091015840]) 1198891199091015840+ 1)

minus119902(119902minus1)

sdot exp (minus119892(119894)[119909]) 119889119909

(42)

but this way does not correspond to appearance of the aboveand below estimates although it is possible to prove that theseiterations with the initial approximation

119892(0)

= 0 (43)

converge to solutionThe procedure for 120579 and 119892 as the separate functions which

was formulated above will lead under the initial approxima-tions

120579(0)

= 1

119892(0)

= 0

(44)

to the chains of inequalities

119892(0)

lt 119892(2)

lt sdot sdot sdot lt 119892 lt sdot sdot sdot lt 119892(3)

lt 119892(1)

120579(1)

lt 120579(3)

lt sdot sdot sdot lt 120579 lt sdot sdot sdot lt 120579(2)

lt 120579(0)

(45)

Parameter 119860 is considered here as constant value for alliterations and the concrete value of 119860 will be determined onthe basis of the last (precise) iteration

It is useful to calculate the total number of supercriticalembryos as

119873tot sim int

infin

0

120579119902[119909] exp (minus119892 [119909]) 119889119909 (46)

On the basis of iteration procedure one can calculate

119873tot(119894+1) sim int

infin

0

120579119902

(119894)[119909] exp (minus119892

(119894)[119909]) 119889119909 (47)

The calculation of iterations gives

119892(0)

=

1199114

4

120579(1)

= ((119902 minus 1)119860119911 + 1)minus1(119902minus1)

(48)

Then

119873tot(2) = int

infin

0

((119902 minus 1)119860119911 + 1)minus119902(119902minus1) exp(minus119911

4

4

)119889119911 (49)

and this can be expressed through the hypergeometric func-tion

119873tot(2) = 2(minus92+2(119902(119902minus1)))

(119902 + 119860)minus119902(119902minus1)

(

1

3

sdot 2minus(52)(119902(119902minus1))+12

1205874(119902 + 119860)

3

(

1

119902 + 119860

)

minus119902(119902minus1)

119867([

3

4

+

1

4

119902

119902 minus 1

1 +

1

4

119902

119902 minus 1

5

4

+

1

4

119902

119902 minus 1

] [

3

2

7

4

5

4

]

minus

1

4

(119902 + 119860)4

)

sdot

sec ((14) 120587 (119902 (119902 minus 1))) Γ (3 + 119902 (119902 minus 1))

Γ (32 + (14) (119902 (119902 minus 1)))

minus 2(1minus(52)(119902(119902minus1)))

1205874(119902 + 119860)

2

(

1

119902 + 119860

)

minus119902(119902minus1)

119867([

3

4

+

1

4

119902

119902 minus 1

1 +

1

4

119902

119902 minus 1

1

2

+

1

4

119902

119902 minus 1

] [

3

2

3

4

5

4

]

minus

1

4

(119902 + 119860)4

)

sdot

csc ((14) 120587 + (14) 120587 (119902 (119902 minus 1))) Γ (2 + 119902 (119902 minus 1))

Γ (54 + (14) (119902 (119902 minus 1)))

+ 2(52minus(52)(119902(119902minus1)))

1205874(119902 + 119860)(

1

119902 + 119860

)

minus119902(119902minus1)

119867([

3

4

+

1

4

119902

119902 minus 1

1

4

+

1

4

119902

119902 minus 1

1

2

+

1

4

119902

119902 minus 1

] [

3

4

5

4

1

2

]

minus

1

4

(119902 + 119860)4

)

sdot

csc ((14) 120587 (119902 (119902 minus 1))) Γ (1 + 119902 (119902 minus 1))

Γ (1 + (14) (119902 (119902 minus 1)))

+ 2(3minus(52)(119902(119902minus1)))

1205874(

1

119902 + 119860

)

minus119902(119902minus1)

119867([

1

4

119902

119902 minus 1

1

4

+

1

4

119902

119902 minus 1

1

2

+

1

4

119902

119902 minus 1

] [

3

4

1

2

1

4

] minus

1

4

(119902 + 119860)4

)

sdot

sec ((14) 120587 + (14) 120587 (119902 (119902 minus 1))) Γ (119902 (119902 minus 1))

Γ (34 + (14) (119902 (119902 minus 1)))

+ 2(132minus2(119902(119902minus1)))

1205873(minus

1

2

+

1

4

119902

119902 minus 1

) (119902 + 119860)

Advances in Physical Chemistry 7

sdot 119867([1

3

4

1

2

1

4

] [

5

4

minus

1

4

119902

119902 minus 1

minus

1

4

119902

119902 minus 1

+

1

2

minus

1

4

119902

119902 minus 1

+

3

4

minus

1

4

119902

119902 minus 1

+ 1] minus

1

4

(119902 + 119860)4

)Γ(minus2

+

119902

119902 minus 1

))(1205873Γ(

119902

119902 minus 1

))

minus1

(50)

where Γ is the gamma-function and119867 is the hypergeometricfunction

It is possible to show analytically that this expression israther accurate The error is less than one percent Despitethe known form it is rather difficult to perform calculationsaccording to this expression and it is necessary to simplify thecalculations

Again we use the property of the avalanche consumptionHere one can show that 120579 is the decreasing (or at least notrapidly increasing) function of time (or of the size of theinitial embryo)Then one can see the cut-off type of the func-tion exp(minus119892) and the possibility of truncation of the function120579119902 exp(minus119892(119911)) at 119905cut The value of 119905cut can be chosen asthe root of equation 119892 = 1 Then the spectrum of sizes

119891 sim 120579119902[119909] exp (minus119892 [119909]) (51)

can be approximated as

119891 sim ((119902 minus 1)119860119911 + 1)minus119902(119902minus1) (52)

for 119911 lt 119911cut where 119911cut is the size corresponding to 119905cut Forother 119911 the spectrum is zero

The number of embryos can be easily calculated

119873tot = int

119911

0

((119902 minus 1)119860119909 + 1)minus119902(119902minus1)

119889119909

= minus

(119860119911119902 minus 119860119911 + 1)minus1(119902minus1)

minus 1

119860

(53)

This solves the formal problem of our investigationTo show the consistency of the presented approach we

have to investigate the inclusion of the119898-mers consumptionkinetics into the general nucleation Here one can find atleast two aspects of the problem The first aspect is the inputof 119898-mers in the overcoming of activation barrier and thecorresponding input in the stationary rate of nucleation Thesecond aspect is the consumption of the119898-mers by the near-critical and precritical embryos (this is the first type) andby the supercritical embryos (the second type) The rate ofconsumption will be different and their manifestation in theglobal kinetics 119904 will be described by different ways

To clarify the first aspect we rewrite the stationary rate ofnucleation as

119869 sim exp (minus119902 ln (119899 [119898])) exp (minus119865119888) (54)

One has to stress that here 119865119888is the free energy of the critical

embryo formation on 119902 119898-mers This implies that the work

of collecting of119898-mers is considered separatelyThis work inapproximation of ideal gas of119898-mers is

1198650= minus119902 ln (119899 [119898]) (55)

At the same time we have to note that in the embryocontaining ]

119888molecules there are only ]

119888minus 119902119898 molecules

coming from monomers Then 1198650has to be written as

1198650= minus119887 (]

119888minus 119902119898) + 120574119860 (56)

where 119887 is the excess of the chemical potential for monomers120574 is the renormalized surface tension and 119860 is the surfacearea of the embryo It is clear that in the determination of thesurface area it is necessary to account that the part of the vol-ume inside the embryos surface is occupied by the moleculescoming from119898-mersWhen the surface of the embryo simplysurrounds the embryo one has to write

119860 = 4120587(

3V119897(]1119888+ 119902119898)

4120587

)

23

(57)

where V119897is the volume per onemolecule in a liquid phase and

]1119888is the number of molecules coming from monomersIn the state of equilibrium the chemical potential of 119898-

mer has to be equal to 119898 chemical potentials of monomers(here we neglect the surface term or consider the deformedchemical potential) So

119899 [119898]

119899 [119898 119904 = 0]

= exp (119887119898) (58)

and we see that the total free energy is

119888= 119865119888minus 119902119887119898 = minus]

119888119887 + 120574119860 (59)

and the total free energy coincides with the classical expres-sion

One has to stress that the parameter Γ in the previoussimplification of the dependence of the free energy on super-saturation (or the dependence of the stationary rate of nucle-ation on the supersaturation) is the number of moleculescoming from monomers multiplied on Φ

lowast(Φlowast

+ 1)that is

Γ 997888rarr Γ1=

Φlowast]1119888

(Φlowast+ 1)

(60)

Γ1is Φlowast119889119865119888119889120577|120577=Φlowast

If we calculate the derivative of the total free energy

119888we

get

Φlowast

119889119888

119889120577

100381610038161003816100381610038161003816100381610038161003816120577=Φlowast

= Γ1+ 119902119898

119889119887

119889120577

(61)

or

Φlowast

119889119888

119889120577

100381610038161003816100381610038161003816100381610038161003816120577=Φlowast

= Γ1+ Γ119898 (62)

8 Advances in Physical Chemistry

where

Γ119898=

120577119902119898

(120577 + 1)

10038161003816100381610038161003816100381610038161003816120577=Φlowast

(63)

As a result

Φlowast

119889119888

119889120577

100381610038161003816100381610038161003816100381610038161003816120577=Φlowast

= Γ

Γ =

120577]119888

(120577 + 1)

10038161003816100381610038161003816100381610038161003816120577=Φlowast

(64)

We will call this property the separation of Γ It takes placeonly in the first derivatives of the free energies and onlybecause the surface term does not act on the derivative of thefree energy The surface term in the free energy only changesthe number of molecules in the critical embryo and has nodirect influence on derivative

So we see that in the free energy there is the directcorrespondence between the119898-mers picture and monomerspicture

One can propose another picture when the number of119898-mers necessary to form the critical embryo is proportionalto the number of molecules inside the critical embryo In theenormously big critical embryos namely thismechanismhasto be considered Then the picture is equivalent to the binarynucleation with two vapors (the vapor of monomers and thevapor of119898-mers)

The second aspect is connected with the mechanism ofconsumption Here one has to confess that there are two prin-cipal mechanisms of the substance consumption The firstmechanism is ldquoone embryo-fixed number of clusters (mono-mers or 119898-mers)rdquo The second mechanism is ldquoone embryo-growing number of clusters (monomers and 119898-mers) corre-sponding to the law of the regular growthrdquo

Rigorously speaking we have to write for each type ofactive monomers two types of consumption and the balanceequations With the help of exponential approximationsmentioned above one can unify these mechanisms in one lawof growth Ordinarily when the first mechanism is importantthe second is absent When the second mechanism acts itmeans that there is enough clusters of such type to neglect theexhaustion of seeds We will call this effect as the dominatingpreference of one mechanism

Despite the mentioned dominating preference propertythe simultaneous existence of several mechanisms can not beexcluded as a specific case But it is clear that with the helpof approaches developed here it is possible to construct thetheoretical description of any process with different mecha-nisms of formation different mechanisms of substance con-sumption and different relaxation processes correspondingto bounds or connections between the resources of clusters ofdifferent sizes

The different relations between the characteristic timesof the mentioned processes make the processes of nucleationrather different The size spectrums are very various also Todescribe these situations one has to demonstrate the powerof the approaches making the base of theoretical description

In every situation the leading idea will be the avalanche con-sumption of metastability in different manifestations of thisproperty

4 Main Results

This paper contains at least two important derivations con-cerning the kinetics of decay with account of the relaxationprocesses and the kinetics of decay when the119898-mers nucleationis important Both cases are important and one can see thataccount of these processes radically changes the numericalvalues of parameters of the nucleations process But one hasto confess that the qualitative picture remains similar to thetraditional case The last does not mean that nothing in theform of the particles size distribution is changed Contrarilythe form of the size spectrum will be absolutely another Butthe property of the avalanche cut-off of the pseudo-homo-geneous size spectrum remains The last property was veryproductive in construction of the above presented procedure

It would be interesting to analyze these situations in thecase of the smooth external influence on the system whichcauses nucleation that is in the case of dynamic conditionsAlso it would be interesting to see the influence of the relax-ation processes in the overcoming of the near-critical regionin the process of the binary nucleation Certainly these per-spectives make the subjects of separate publications

Beside the direct construction of the nucleation globalkinetics description we managed to extract some importantfeatures of the effect of119898-mers inclusion into the nucleationprocess One can mention here the property of separation ofΓ and the correspondence between the monomers picture and119898-mers picture These properties are important enough to bementioned in the general theory of nucleation

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] C T R Wilson ldquoCondensation of water vapour in the presenceof dust-free air and other gasesrdquo Philosophical Transactions ofthe Royal Society A Mathematical Physical and Engineering Sci-ences vol 189 pp 265ndash307 1897

[2] C T RWilson ldquoOn the condensation nuclei produced in gasesby the action of roentgen rays uranium rays ultraviolet rays andother agentsrdquoPhilosophical Transactions A vol 192 no 9 article403 1899

[3] C T RWilson ldquoOn the comparative efficiency as condensationnuclei of positively and negatively charged ionsrdquo PhilosophicalTransactions of the Royal Society A Containing Papers of aMath-ematical or Physical Character vol 193 pp 289ndash308 1900

[4] A Benmouna R Benmouna M R Bockstaller and I FHakem ldquoSelf-organization schemes towards thermodynamicstable bulk heterojunction morphologies a perspective onfuture fabrication strategies of polymer photovoltaic architec-turesrdquo Advances in Physical Chemistry vol 2013 Article ID948189 8 pages 2013

Advances in Physical Chemistry 9

[5] Y Kazzi H Awada and M Nardin ldquoAdhesion on nanoorga-nized multilayers surface thermodynamics and local energydissipationrdquo Advances in Physical Chemistry vol 2010 ArticleID 502709 11 pages 2010

[6] A I Burshtein ldquoContact and distant luminescence quenchingin solutionsrdquo Advances in Physical Chemistry vol 2009 ArticleID 214219 34 pages 2009

[7] A C Zettlemoyer Nucleation Dekker New York NY USA1969

[8] F F Abraham Homogeneous Nucleation Theory AcademicNew York NY USA 1974

[9] M Volmer and A Weber ldquoKeimbildung in ubersattigtenGebildenrdquo Zeitschrift fur Physikalische Chemie vol 119 pp 277ndash301 1925

[10] R Becker andW Doring ldquoKinetische behandlung der keimbil-dung in uberattigten dampfernrdquo Annals of Physics vol 24 pp719ndash752 1935

[11] J Frenkel ldquoA general theory of heterophase fluctuations andpretransition phenomenardquoThe Journal of Chemical Physics vol7 no 7 pp 538ndash547 1939

[12] J B Zeldovich ldquoOn the theory of new phase formation cavi-tationrdquo Acta Physicochimica USSR vol 18 pp 1ndash22 1943

[13] L Farkas ldquoThe velocity of nucleus formation in superheatedvaporsrdquo Zeitschrift fur Physikalische Chemie vol 125 pp 236ndash240 1927

[14] J Lothe and G M Pound ldquoReconsiderations of nucleationtheoryrdquo The Journal of Chemical Physics vol 36 no 8 article2080 1962

[15] H Reiss J L Katz and E R Cohen ldquoTranslation-rotation para-dox in the theory of nucleationrdquoThe Journal of Chemical Phys-ics vol 48 no 12 pp 5553ndash5560 1968

[16] H Reiss ldquoTreatment of droplike clusters by means of the clas-sical phase integral in nucleation theoryrdquo Journal of StatisticalPhysics vol 2 no 1 pp 83ndash104 1970

[17] W G Courtney ldquoRemarks on homogeneous nucleationrdquo TheJournal of Chemical Physics vol 35 no 6 pp 2249ndash2250 1961

[18] M Blander and J L Katz ldquoThe thermodynamics of cluster for-mation in nucleation theoryrdquo Journal of Statistical Physics vol 4no 1 pp 55ndash59 1972

[19] J L Katz and H Weidersich ldquoNucleation theory without Max-well Demonsrdquo Journal of Colloid and Interface Science vol 61no 2 pp 351ndash355 1977

[20] J L Katz and M D Donohue ldquoA kinetic approach to homo-geneous nucleation theoryrdquo in Advances in Chemical Physics IPrigogine and S A Rice Eds vol 40 chapter 3 pp 137ndash155Wiley 1979

[21] A Dillmann and G E A Meier ldquoHomogeneous nucleation ofsupersaturated vaporsrdquo Chemical Physics Letters vol 160 no 1pp 71ndash74 1989

[22] A Dillmann and G E A Meier ldquoA refined droplet approach tothe problem of homogeneous nucleation from the vapor phaserdquoThe Journal of Chemical Physics vol 94 no 5 pp 3872ndash38841991

[23] J W Cahn and J E Hillard ldquoFree energy of a nonuniformsystem III Nucleation in a two-component incompressiblefluidrdquoTheJournal of Chemical Physics vol 31 no 3 pp 688ndash6991959

[24] D Gebauer and H Colfen ldquoPrenucleation clusters and non-classical nucleationrdquo Nano Today vol 6 no 6 pp 564ndash5842011

[25] D Sept and J A McCammon ldquoThermodynamics and kineticsof actin filament nucleationrdquo Biophysical Journal vol 81 no 2pp 667ndash674 2001

[26] A S Myerson and B L Trout ldquoNucleation from solutionrdquo Sci-ence vol 341 no 6148 pp 855ndash856 2013

[27] D Zahn ldquoThermodynamics and kinetics of prenucleation clus-ters classical and non-classical nucleationrdquo ChemPhysChemvol 16 no 10 pp 2069ndash2075 2015

[28] H Deng S Wang X Wang et al ldquoTwo competitive nucle-ation mechanisms of calcium carbonate biomineralization inresponse to surface functionality in low calcium ion concentra-tion solutionrdquo Regenerative Biomaterials vol 2 no 3 pp 187ndash195 2015

[29] D Kozma and G Toth ldquoMicroscopic rate constants of crystalgrowth from molecular dynamic simulations combined withmetadynamicsrdquo Advances in Physical Chemistry vol 2012Article ID 135172 10 pages 2012

[30] S Fakruddin C Srinivasu B R Venkateswara Rao and KNarendra ldquoExcess transport properties of binary mixtures ofquinoline with xylenes at different temperaturesrdquo Advances inPhysical Chemistry vol 2012 Article ID 324098 9 pages 2012

[31] A A Lushnikov ldquoEvolution of coagulating systemsrdquo Journal ofColloid and Interface Science vol 45 no 3 pp 549ndash556 1973

[32] I M Lifshic and V V Slezov ldquoKinetics of diffusive decomposi-tion of supersaturated solid solutionsrdquo Soviet PhysicsmdashJournalof Experimental and Theoretical Physics vol 35 pp 331ndash3391959

[33] IM Lifshic andV V Slezov ldquoThe kinetics of precipitation fromsupersaturated solid solutionsrdquo Journal of Physics andChemistryof Solids vol 19 no 1-2 pp 35ndash50 1961

[34] F M Kuni and A P Grinin ldquoKinetics of homogeneous conden-sationat the stage of formation of the main quantity of a newphaserdquo Colloid Journal of the USSR vol 46 p 460 1984

[35] F M Kuni T Y Novozhilova and I A Terentiev ldquoKineticequation of heterogenous condensationrdquo Letters in Mathemati-cal Physics vol 14 no 2 pp 161ndash167 1987

[36] V Kurasov ldquoDecay of metastable multicomponentmixturerdquohttparxivorgabscond-mat9310005

[37] V B Kurasov ldquoKinetics of heterogeneous condensation underdynamic conditionsrdquo Physical Review E vol 49 no 5 pp 3948ndash3955 1994

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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CatalystsJournal of

Advances in Physical Chemistry 3

the exhaustion of heterogeneous centers which become thecenters of the droplets Here the theory was given in 1989 [35]but the gap in the theory was filled only in 1993 [36]

The peculiarities in connections between the small clus-ters in a mother phase can cause very specific behavior of theglobal nucleation kinetics Although alreadymany importantcases were considered there exist enough important situa-tions necessary to be described theoretically Below we willgive the theoretical description of some of them

2 Weak Relaxation of Seeds inMonomer Nucleation

Thefirst important case is the situationwhen in themonomernucleation there exists a weak connection between the region(in the scale of sizes) of monomers and the region of119898-mersand it is necessary to take into account not only the consump-tion of 119898-mers by the supercritical embryos (every embryohas to consume one 119898-mer to start the life) but also theweak source of monomers being transformed into 119898-mersThis source changes the kinetics of the process

Here it is very useful to use the analogy with heteroge-neous nucleation Really the 119898-mers can be considered assome objects analogous to heterogeneous centers But thesituation with the relaxation of heterogeneous centers has notbeen considered So there exists a nontrivial mathematicalcontent of the constructions presented below

We introduce the value of 120579 as

120579 =

119899 [119898 119905]

119899 [119898 119905 = 0]

(1)

This value can be less or greater than 1 Here 119899[119898 119905] is thenumber of embryos with119898molecules at the time moment 119905

One can formulate the system of equations for unknownfunctions 119892 and 120579 and after some scaling get

119889120579

119889119905

= minus119895 (120579 minus 120579lim) minus 119879120579 [119905] exp (minus119892 [119905]) (2)

119892 [119905] = int

119905

0

(119905 minus 1199051015840)

3

120579 [1199051015840] exp (minus119892 [1199051015840]) 1198891199051015840 (3)

Here 119879 is some positive parameter which can not be avoidedThe positive parameter 119895 shows the intensity of the relaxationprocess The value 120579lim is the equilibrium value It is also aparameter

Now we will clarify the meaning of two unknown func-tions 119892 and 120579 of the time 119905 The function 119892 is the relativenumber of themolecules in a new phaseThe function 120579 is therelative number of the free (unoccupied by the supercriticalembryos) heterogeneous centers

Ordinarily the solution of the system of condensationequations (analogous to (2)-(3)) is given by iterations (see[37]) The behavior of the spectrum of sizes

119891 [119905] = 120579 [119905] exp (minus119892 [119905]) (4)

is rather complex even already in the first iterations whichare constructed in a way to simplify the form of the size

spectrumThis fact means that in our case one can not inventtoo ldquointelligentrdquo iterations Namely the presentation of (3) inthe already integrated form seems to produce too complexiteration procedure

In investigation of the nucleation process with theabsence of the relaxation process (see [37]) the followingchains of inequalities

120579(1)

lt 120579(3)

lt sdot sdot sdot lt 120579 lt sdot sdot sdot lt 120579(2)

119892(0)

lt 119892(2)

lt sdot sdot sdot lt 119892 lt sdot sdot sdot lt 119892(3)

lt 119892(1)

(5)

were formulated This chain is very productive for the accu-racy estimates which lead to the guaranteed accuracy of theobtained approximations

To conserve these chains one can propose the followingrecurrent scheme

119889120579(119894+1)

119889119905

= minus119895 (120579(119894)minus 120579lim) minus 119879120579

(119894)[119905] exp (minus119892

(119894)[119905])

119892(119894+1)

[119905] = int

119905

0

(119905 minus 1199051015840)

3

120579(119894)[1199051015840] exp (minus119892

(119894)[1199051015840]) 1198891199051015840

(6)

with initial condition

120579 [119905 = 0] = 1 (7)

and initial approximations

120579(0)

= 120579lim (8)

or

120579(0)

[119905] = 120579 [119905 = 0]

119892(0)

= 0

(9)

or the recurrent scheme

119889120579(119894+1)

119889119905

= minus119895 (120579(119894+1)

minus 120579lim)

minus 119879120579(119894+1)

[119905] exp (minus119892(119894)[119905])

119892(119894+1)

[119905] = int

119905

0

(119905 minus 1199051015840)

3

120579(119894)[1199051015840] exp (minus119892

(119894)[1199051015840]) 1198891199051015840

(10)

with the same initial condition and the same initial approxi-mations The third scheme is

119889120579(119894+1)

119889119905

= minus119895 (120579(119894)minus 120579lim) minus 119879120579

(119894+1)[119905] exp (minus119892

(119894)[119905])

119892(119894+1)

[119905] = int

119905

0

(119905 minus 1199051015840)

3

120579(119894)[1199051015840] exp (minus119892

(119894)[1199051015840]) 1198891199051015840

(11)

with the same initial approximations and the same initialcondition

We have to choose what type of iterations is the best oneThefirst procedure is absolutely correct and it can not lead

to divergence at some step of iterations The second and thethird schemes require additional regularization One has toput an additional restriction

4 Advances in Physical Chemistry

The Nucleation Process Stops When 120579 Attains Zero Thisrestriction is necessary to avoid the negative rate of nucleationand the divergence of iterations

The account of the relaxation process in the nucleationkinetics is a hard problem Already the first iteration fromthose where the effect of the relaxation term can be seenin the behavior of the supersaturation can not be calculatedanalytically

To fulfil analytical calculations we have to fulfil an addi-tional simplification based on the property of the avalancheconsumption

Thementioned property can be formulated as follows

Under Any Behavior of 120579 gt 0 the Relative Growth of 119892 OccursFasterThan (119905minus119905

0)3 One can prove this property analytically

Actually this property is evident since we take into accountthat 119892 = int

119905

0(119905 minus 1199051015840)3120579[1199051015840]exp(minus119892[1199051015840])1198891199051015840 and 120579 gt 0

On the basis of this property we see that the ldquopseudo-homogeneous spectrum of sizesrdquo (the spectrum of sizeswhich would be formed if there would be no action of thenumber of free seeds on the nucleation process)

119891hom sim exp (minus119892) (12)

will be truncated in a manner sharper than

exp(minus(

119905

119905cut)

3

) (13)

where 119905cut is the characteristic timeThen one can speak aboutthe cut-off type of the pseudo-homogeneous spectrum ofsizes

Now one can see the qualitative behavior of 120579 At first 120579relaxes to the value

120579119899=

119895

119895 + 119879

120579lim (14)

with characteristic time

119905rel 119899 = (119895 + 119879)minus1

(15)

This relaxation can lead to the increase of 120579 when

120579 [119905 = 0] lt 120579119899

(16)

or to the decrease of 120579 in the opposite situationAt the moment 119905cut this relaxation is changed by the

relaxation of 120579 to 120579lim with the characteristic time 119905rel = 119895minus1

The question which we are interested in is whether thechange of 120579 can destroy the cut-off type of the real spectrumof sizes 120579 exp(minus119892) (earlier the cut-off type was shown forexp(minus119892))

The destruction of the cut-off type of 120579 exp(minus119892) can takeplace only under the sharp relative increase of 120579 So we haveto estimate (119889120579119889119905)120579 Since

119889120579

119889119905

lt 119895120579lim

120579 gt

119895

119895 + 119879

120579lim

(17)

we see that119889120579119889119905

120579

lt 119895 + 119879 (18)

When 119879 ≫ 119895 one simply can not see the relaxationThe picture in this case is the picture described earlier (see[37]) plus some small perturbation Then having excludedthis situation we can rewrite the last inequality as

119889120579119889119905

120579

lt (3 divide 4) 119895 (19)

Now we see that the danger of destruction of the cut-offform of the size spectrum can appear only at big values of 119895Actually it has to be

119895 ≫ 119905minus1

cut (20)

If 119905cut ≪ 119879minus1 then there is no exhaustion of seeds So the

actual situation is 119905cut ge 119879minus1 and then we see that

119895 ≫ 119879 (21)

Here 120579lim is close to 120579119899and then 119905cut based on 120579lim is close

to 120579 based on 120579119899 This means that the cut-off behavior is not

broken here So the property of the cut-off takes place in allsituations

The property of the cut-off allows using the approxima-tion exp(minus119892) = 1 for 119905 lt 119905cut and exp(minus119892) = 0 for 119905 gt 119905cut Themoment 119905cut can be determined as the root of equation

119892 (119905cut) = 1 (22)

Then for 119905 lt 119905cut

119889120579

119889119905

= minus119895 (120579 minus 120579lim) minus 119879120579 [119905]

119892 [119905] = int

119905

0

(119905 minus 1199051015840)

3

120579 [1199051015840] 1198891199051015840

(23)

and one can explicitly calculate

120579 [119905] = 120579119899minus (120579119899minus 120579 [119905 = 0]) exp (minus (119895 + 119879) 119905) (24)

The value of 119892 can be presented as

119892 = int

119905

0

(119905 minus 1199051015840)

3

sdot (120579119899minus (120579119899minus 120579 [119905 = 0]) exp (minus (119895 + 119879) 119905

1015840)) 1198891199051015840

(25)

and can be easily found analytically which allows getting 119905cutas the solution of an ordinary algebraic equation

Equation (22) can be easily solved since we know theestimates 120579

11989911991144 and 120579[0]119911

44 for the behavior of 119892 which

leads to the estimates for the rootsWhat estimate is the aboveestimate andwhat the below estimate is are determined by thesign of 120579lim minus 120579[0]

The total number of droplets in the renormalized unitscan be found as

119873 = 119860int

119905cut

0

1 119889119905 = 119860119905cut (26)

Advances in Physical Chemistry 5

where 119860 is the initial amplitude (in renormalized values itequals 1) or as

119873 = 119860int

119905cut

0

exp (minus119892 [119905]) 120579 [119905] 119889119905 (27)

where for 119892 and for 120579 one can take the mentioned approx-imations The linearization of appearing exponents is alsoapproximately suitable and leads to the presence of the resultin elementary functions One has to note that the first recipeseems to bemore rough than the second one but it gives betterresults The reason is the effective compensation of decreaseof the intensity of appearance of new embryos before 119905cut bythe existence of the tail after 119905cut The refinement of equationfor 119905cut (it is better to use 119892[119905cut] = ln 2) can make the firstrecipe very accurate Certainly the further refinement withthe help of the perturbation technique can be fulfilled

Even the linear approximation

120579 = 120579 [0] + (119895 + 119879) (120579119899minus 120579 [0]) 119905 (28)

truncated at the crossing point with 120579119899 that is at (119895 + 119879)

minus1and prolonged further as 120579

119899gives a rather good approxima-

tion for the characteristics of the processThis approximationensures the pure algebraic equation of the fifth power for 119905cutIt is a really suitable approximation since this approximationhas themaximal deviation as (119895+119879)minus1 which is relatively small(simexp(minus1)) in comparison with initial deviation of 120579 from 120579

119899

Then the difference in integrals staying in (22) is less thantwo times which gives the difference for the duration of theintensive nucleation period in 2

14asymp 12 times (in reality the

difference in duration is much more small) The last approx-imation brings equation on root to the explicit formulas

This completes the analytical description of the situationof the decay in the system with the relaxation process for thecenters of nucleation

3 Accumulation of Several Seeds inthe Monomer Nucleation

The second important task is to consider the situation whenit is necessary to spend 119902 119898-mers to form the supercriticalembryo and the rate of nucleation is proportional to thenumber of119898-mers in power 119902

119869 sim 119899 [119898]119902 (29)

It is interesting to investigate this situation and to see thetransition to the case when 119902 is great enough Certainly theprobability

119875 sim 119899 [119898]119902 (30)

transfers into

119875 sim exp (minus120583119902) (31)

with a chemical potential

120583 = minus ln (119899 [119898]) (32)

corresponding to the chemical potential of ideal gasThe equation

119889119899 [119898]

119889119905

= 119899 [119898]119902119891 [119905] (33)

with some known function 119891[119905] can be written in thefollowing integral form

119899 [119898] =

119899 [119898 119905 = 0]

((119902 minus 1) int

119905

0119891 [1199051015840] 1198891199051015840+ 1)

1(119902minus1) (34)

Now we write the expression for the rate of nucleation inthe generalized Clausius-Clapeyron integrated form as

119869 = 1198690120579119902 exp(minus

Γ (120577 minus Φlowast)

Φlowast

) (35)

where

Γ =

Φlowast119889119865119888

119889120577

10038161003816100381610038161003816100381610038161003816120577=Φlowast

(36)

and Φlowastis the base of decomposition Here 119869

0is the initial

value of the rate of nucleation and 120579 is the relative numberof119898-mers referred to initial value

One can get

Γ = (

Φlowast

(Φlowast+ 1)

) ]1119888 (37)

where ]1119888is the number ofmonomers in critical embryoThis

number is connected with the total number of molecules ]119888

in the critical embryo as

]119888= ]1119888+ 119902119898 (38)

After the scaling the system of evolution equations is thefollowing one

119892 [119911] = int

119911

0

(119911 minus 119909)3120579119902[119909] exp (minus119892 [119909]) 119889119909

120579 [119911] = ((119902 minus 1)119860int

119911

0

exp (minus119892 [119909]) 119889119909 + 1)

minus1(119902minus1)

(39)

Here 119860 is the coefficient which remains after the scaling andcan not be canceled

We can formulate the iteration procedure as follows

119892(119894+1)

(119911) = int

119911

0

(119911 minus 119909)3120579119902

(119894)[119909] exp (minus119892

(119894)[119909]) 119889119909

120579(119894+1)

[119911]

= ((119902 minus 1)119860int

119911

0

exp (minus119892(119894)[119909]) 119889119909 + 1)

minus1(119902minus1)

(40)

6 Advances in Physical Chemistry

The separation of the spectrum in 119892 and 120579 which seemsto be natural from the physical point of view is quite unusualfrom the formal point of view when the unique equation

119892 [119911] = int

119911

0

(119911 minus 119909)3

sdot ((119902 minus 1)119860int

119909

0

exp (minus119892 [1199091015840]) 1198891199091015840 + 1)

minus119902(119902minus1)

sdot exp (minus119892 [119909]) 119889119909

(41)

is considered Then it is more natural to construct iterationsas

119892(119894+1)

[119911] = int

119911

0

(119911 minus 119909)3

sdot ((119902 minus 1)119860int

119909

0

exp (minus119892(119894)[1199091015840]) 1198891199091015840+ 1)

minus119902(119902minus1)

sdot exp (minus119892(119894)[119909]) 119889119909

(42)

but this way does not correspond to appearance of the aboveand below estimates although it is possible to prove that theseiterations with the initial approximation

119892(0)

= 0 (43)

converge to solutionThe procedure for 120579 and 119892 as the separate functions which

was formulated above will lead under the initial approxima-tions

120579(0)

= 1

119892(0)

= 0

(44)

to the chains of inequalities

119892(0)

lt 119892(2)

lt sdot sdot sdot lt 119892 lt sdot sdot sdot lt 119892(3)

lt 119892(1)

120579(1)

lt 120579(3)

lt sdot sdot sdot lt 120579 lt sdot sdot sdot lt 120579(2)

lt 120579(0)

(45)

Parameter 119860 is considered here as constant value for alliterations and the concrete value of 119860 will be determined onthe basis of the last (precise) iteration

It is useful to calculate the total number of supercriticalembryos as

119873tot sim int

infin

0

120579119902[119909] exp (minus119892 [119909]) 119889119909 (46)

On the basis of iteration procedure one can calculate

119873tot(119894+1) sim int

infin

0

120579119902

(119894)[119909] exp (minus119892

(119894)[119909]) 119889119909 (47)

The calculation of iterations gives

119892(0)

=

1199114

4

120579(1)

= ((119902 minus 1)119860119911 + 1)minus1(119902minus1)

(48)

Then

119873tot(2) = int

infin

0

((119902 minus 1)119860119911 + 1)minus119902(119902minus1) exp(minus119911

4

4

)119889119911 (49)

and this can be expressed through the hypergeometric func-tion

119873tot(2) = 2(minus92+2(119902(119902minus1)))

(119902 + 119860)minus119902(119902minus1)

(

1

3

sdot 2minus(52)(119902(119902minus1))+12

1205874(119902 + 119860)

3

(

1

119902 + 119860

)

minus119902(119902minus1)

119867([

3

4

+

1

4

119902

119902 minus 1

1 +

1

4

119902

119902 minus 1

5

4

+

1

4

119902

119902 minus 1

] [

3

2

7

4

5

4

]

minus

1

4

(119902 + 119860)4

)

sdot

sec ((14) 120587 (119902 (119902 minus 1))) Γ (3 + 119902 (119902 minus 1))

Γ (32 + (14) (119902 (119902 minus 1)))

minus 2(1minus(52)(119902(119902minus1)))

1205874(119902 + 119860)

2

(

1

119902 + 119860

)

minus119902(119902minus1)

119867([

3

4

+

1

4

119902

119902 minus 1

1 +

1

4

119902

119902 minus 1

1

2

+

1

4

119902

119902 minus 1

] [

3

2

3

4

5

4

]

minus

1

4

(119902 + 119860)4

)

sdot

csc ((14) 120587 + (14) 120587 (119902 (119902 minus 1))) Γ (2 + 119902 (119902 minus 1))

Γ (54 + (14) (119902 (119902 minus 1)))

+ 2(52minus(52)(119902(119902minus1)))

1205874(119902 + 119860)(

1

119902 + 119860

)

minus119902(119902minus1)

119867([

3

4

+

1

4

119902

119902 minus 1

1

4

+

1

4

119902

119902 minus 1

1

2

+

1

4

119902

119902 minus 1

] [

3

4

5

4

1

2

]

minus

1

4

(119902 + 119860)4

)

sdot

csc ((14) 120587 (119902 (119902 minus 1))) Γ (1 + 119902 (119902 minus 1))

Γ (1 + (14) (119902 (119902 minus 1)))

+ 2(3minus(52)(119902(119902minus1)))

1205874(

1

119902 + 119860

)

minus119902(119902minus1)

119867([

1

4

119902

119902 minus 1

1

4

+

1

4

119902

119902 minus 1

1

2

+

1

4

119902

119902 minus 1

] [

3

4

1

2

1

4

] minus

1

4

(119902 + 119860)4

)

sdot

sec ((14) 120587 + (14) 120587 (119902 (119902 minus 1))) Γ (119902 (119902 minus 1))

Γ (34 + (14) (119902 (119902 minus 1)))

+ 2(132minus2(119902(119902minus1)))

1205873(minus

1

2

+

1

4

119902

119902 minus 1

) (119902 + 119860)

Advances in Physical Chemistry 7

sdot 119867([1

3

4

1

2

1

4

] [

5

4

minus

1

4

119902

119902 minus 1

minus

1

4

119902

119902 minus 1

+

1

2

minus

1

4

119902

119902 minus 1

+

3

4

minus

1

4

119902

119902 minus 1

+ 1] minus

1

4

(119902 + 119860)4

)Γ(minus2

+

119902

119902 minus 1

))(1205873Γ(

119902

119902 minus 1

))

minus1

(50)

where Γ is the gamma-function and119867 is the hypergeometricfunction

It is possible to show analytically that this expression israther accurate The error is less than one percent Despitethe known form it is rather difficult to perform calculationsaccording to this expression and it is necessary to simplify thecalculations

Again we use the property of the avalanche consumptionHere one can show that 120579 is the decreasing (or at least notrapidly increasing) function of time (or of the size of theinitial embryo)Then one can see the cut-off type of the func-tion exp(minus119892) and the possibility of truncation of the function120579119902 exp(minus119892(119911)) at 119905cut The value of 119905cut can be chosen asthe root of equation 119892 = 1 Then the spectrum of sizes

119891 sim 120579119902[119909] exp (minus119892 [119909]) (51)

can be approximated as

119891 sim ((119902 minus 1)119860119911 + 1)minus119902(119902minus1) (52)

for 119911 lt 119911cut where 119911cut is the size corresponding to 119905cut Forother 119911 the spectrum is zero

The number of embryos can be easily calculated

119873tot = int

119911

0

((119902 minus 1)119860119909 + 1)minus119902(119902minus1)

119889119909

= minus

(119860119911119902 minus 119860119911 + 1)minus1(119902minus1)

minus 1

119860

(53)

This solves the formal problem of our investigationTo show the consistency of the presented approach we

have to investigate the inclusion of the119898-mers consumptionkinetics into the general nucleation Here one can find atleast two aspects of the problem The first aspect is the inputof 119898-mers in the overcoming of activation barrier and thecorresponding input in the stationary rate of nucleation Thesecond aspect is the consumption of the119898-mers by the near-critical and precritical embryos (this is the first type) andby the supercritical embryos (the second type) The rate ofconsumption will be different and their manifestation in theglobal kinetics 119904 will be described by different ways

To clarify the first aspect we rewrite the stationary rate ofnucleation as

119869 sim exp (minus119902 ln (119899 [119898])) exp (minus119865119888) (54)

One has to stress that here 119865119888is the free energy of the critical

embryo formation on 119902 119898-mers This implies that the work

of collecting of119898-mers is considered separatelyThis work inapproximation of ideal gas of119898-mers is

1198650= minus119902 ln (119899 [119898]) (55)

At the same time we have to note that in the embryocontaining ]

119888molecules there are only ]

119888minus 119902119898 molecules

coming from monomers Then 1198650has to be written as

1198650= minus119887 (]

119888minus 119902119898) + 120574119860 (56)

where 119887 is the excess of the chemical potential for monomers120574 is the renormalized surface tension and 119860 is the surfacearea of the embryo It is clear that in the determination of thesurface area it is necessary to account that the part of the vol-ume inside the embryos surface is occupied by the moleculescoming from119898-mersWhen the surface of the embryo simplysurrounds the embryo one has to write

119860 = 4120587(

3V119897(]1119888+ 119902119898)

4120587

)

23

(57)

where V119897is the volume per onemolecule in a liquid phase and

]1119888is the number of molecules coming from monomersIn the state of equilibrium the chemical potential of 119898-

mer has to be equal to 119898 chemical potentials of monomers(here we neglect the surface term or consider the deformedchemical potential) So

119899 [119898]

119899 [119898 119904 = 0]

= exp (119887119898) (58)

and we see that the total free energy is

119888= 119865119888minus 119902119887119898 = minus]

119888119887 + 120574119860 (59)

and the total free energy coincides with the classical expres-sion

One has to stress that the parameter Γ in the previoussimplification of the dependence of the free energy on super-saturation (or the dependence of the stationary rate of nucle-ation on the supersaturation) is the number of moleculescoming from monomers multiplied on Φ

lowast(Φlowast

+ 1)that is

Γ 997888rarr Γ1=

Φlowast]1119888

(Φlowast+ 1)

(60)

Γ1is Φlowast119889119865119888119889120577|120577=Φlowast

If we calculate the derivative of the total free energy

119888we

get

Φlowast

119889119888

119889120577

100381610038161003816100381610038161003816100381610038161003816120577=Φlowast

= Γ1+ 119902119898

119889119887

119889120577

(61)

or

Φlowast

119889119888

119889120577

100381610038161003816100381610038161003816100381610038161003816120577=Φlowast

= Γ1+ Γ119898 (62)

8 Advances in Physical Chemistry

where

Γ119898=

120577119902119898

(120577 + 1)

10038161003816100381610038161003816100381610038161003816120577=Φlowast

(63)

As a result

Φlowast

119889119888

119889120577

100381610038161003816100381610038161003816100381610038161003816120577=Φlowast

= Γ

Γ =

120577]119888

(120577 + 1)

10038161003816100381610038161003816100381610038161003816120577=Φlowast

(64)

We will call this property the separation of Γ It takes placeonly in the first derivatives of the free energies and onlybecause the surface term does not act on the derivative of thefree energy The surface term in the free energy only changesthe number of molecules in the critical embryo and has nodirect influence on derivative

So we see that in the free energy there is the directcorrespondence between the119898-mers picture and monomerspicture

One can propose another picture when the number of119898-mers necessary to form the critical embryo is proportionalto the number of molecules inside the critical embryo In theenormously big critical embryos namely thismechanismhasto be considered Then the picture is equivalent to the binarynucleation with two vapors (the vapor of monomers and thevapor of119898-mers)

The second aspect is connected with the mechanism ofconsumption Here one has to confess that there are two prin-cipal mechanisms of the substance consumption The firstmechanism is ldquoone embryo-fixed number of clusters (mono-mers or 119898-mers)rdquo The second mechanism is ldquoone embryo-growing number of clusters (monomers and 119898-mers) corre-sponding to the law of the regular growthrdquo

Rigorously speaking we have to write for each type ofactive monomers two types of consumption and the balanceequations With the help of exponential approximationsmentioned above one can unify these mechanisms in one lawof growth Ordinarily when the first mechanism is importantthe second is absent When the second mechanism acts itmeans that there is enough clusters of such type to neglect theexhaustion of seeds We will call this effect as the dominatingpreference of one mechanism

Despite the mentioned dominating preference propertythe simultaneous existence of several mechanisms can not beexcluded as a specific case But it is clear that with the helpof approaches developed here it is possible to construct thetheoretical description of any process with different mecha-nisms of formation different mechanisms of substance con-sumption and different relaxation processes correspondingto bounds or connections between the resources of clusters ofdifferent sizes

The different relations between the characteristic timesof the mentioned processes make the processes of nucleationrather different The size spectrums are very various also Todescribe these situations one has to demonstrate the powerof the approaches making the base of theoretical description

In every situation the leading idea will be the avalanche con-sumption of metastability in different manifestations of thisproperty

4 Main Results

This paper contains at least two important derivations con-cerning the kinetics of decay with account of the relaxationprocesses and the kinetics of decay when the119898-mers nucleationis important Both cases are important and one can see thataccount of these processes radically changes the numericalvalues of parameters of the nucleations process But one hasto confess that the qualitative picture remains similar to thetraditional case The last does not mean that nothing in theform of the particles size distribution is changed Contrarilythe form of the size spectrum will be absolutely another Butthe property of the avalanche cut-off of the pseudo-homo-geneous size spectrum remains The last property was veryproductive in construction of the above presented procedure

It would be interesting to analyze these situations in thecase of the smooth external influence on the system whichcauses nucleation that is in the case of dynamic conditionsAlso it would be interesting to see the influence of the relax-ation processes in the overcoming of the near-critical regionin the process of the binary nucleation Certainly these per-spectives make the subjects of separate publications

Beside the direct construction of the nucleation globalkinetics description we managed to extract some importantfeatures of the effect of119898-mers inclusion into the nucleationprocess One can mention here the property of separation ofΓ and the correspondence between the monomers picture and119898-mers picture These properties are important enough to bementioned in the general theory of nucleation

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] C T R Wilson ldquoCondensation of water vapour in the presenceof dust-free air and other gasesrdquo Philosophical Transactions ofthe Royal Society A Mathematical Physical and Engineering Sci-ences vol 189 pp 265ndash307 1897

[2] C T RWilson ldquoOn the condensation nuclei produced in gasesby the action of roentgen rays uranium rays ultraviolet rays andother agentsrdquoPhilosophical Transactions A vol 192 no 9 article403 1899

[3] C T RWilson ldquoOn the comparative efficiency as condensationnuclei of positively and negatively charged ionsrdquo PhilosophicalTransactions of the Royal Society A Containing Papers of aMath-ematical or Physical Character vol 193 pp 289ndash308 1900

[4] A Benmouna R Benmouna M R Bockstaller and I FHakem ldquoSelf-organization schemes towards thermodynamicstable bulk heterojunction morphologies a perspective onfuture fabrication strategies of polymer photovoltaic architec-turesrdquo Advances in Physical Chemistry vol 2013 Article ID948189 8 pages 2013

Advances in Physical Chemistry 9

[5] Y Kazzi H Awada and M Nardin ldquoAdhesion on nanoorga-nized multilayers surface thermodynamics and local energydissipationrdquo Advances in Physical Chemistry vol 2010 ArticleID 502709 11 pages 2010

[6] A I Burshtein ldquoContact and distant luminescence quenchingin solutionsrdquo Advances in Physical Chemistry vol 2009 ArticleID 214219 34 pages 2009

[7] A C Zettlemoyer Nucleation Dekker New York NY USA1969

[8] F F Abraham Homogeneous Nucleation Theory AcademicNew York NY USA 1974

[9] M Volmer and A Weber ldquoKeimbildung in ubersattigtenGebildenrdquo Zeitschrift fur Physikalische Chemie vol 119 pp 277ndash301 1925

[10] R Becker andW Doring ldquoKinetische behandlung der keimbil-dung in uberattigten dampfernrdquo Annals of Physics vol 24 pp719ndash752 1935

[11] J Frenkel ldquoA general theory of heterophase fluctuations andpretransition phenomenardquoThe Journal of Chemical Physics vol7 no 7 pp 538ndash547 1939

[12] J B Zeldovich ldquoOn the theory of new phase formation cavi-tationrdquo Acta Physicochimica USSR vol 18 pp 1ndash22 1943

[13] L Farkas ldquoThe velocity of nucleus formation in superheatedvaporsrdquo Zeitschrift fur Physikalische Chemie vol 125 pp 236ndash240 1927

[14] J Lothe and G M Pound ldquoReconsiderations of nucleationtheoryrdquo The Journal of Chemical Physics vol 36 no 8 article2080 1962

[15] H Reiss J L Katz and E R Cohen ldquoTranslation-rotation para-dox in the theory of nucleationrdquoThe Journal of Chemical Phys-ics vol 48 no 12 pp 5553ndash5560 1968

[16] H Reiss ldquoTreatment of droplike clusters by means of the clas-sical phase integral in nucleation theoryrdquo Journal of StatisticalPhysics vol 2 no 1 pp 83ndash104 1970

[17] W G Courtney ldquoRemarks on homogeneous nucleationrdquo TheJournal of Chemical Physics vol 35 no 6 pp 2249ndash2250 1961

[18] M Blander and J L Katz ldquoThe thermodynamics of cluster for-mation in nucleation theoryrdquo Journal of Statistical Physics vol 4no 1 pp 55ndash59 1972

[19] J L Katz and H Weidersich ldquoNucleation theory without Max-well Demonsrdquo Journal of Colloid and Interface Science vol 61no 2 pp 351ndash355 1977

[20] J L Katz and M D Donohue ldquoA kinetic approach to homo-geneous nucleation theoryrdquo in Advances in Chemical Physics IPrigogine and S A Rice Eds vol 40 chapter 3 pp 137ndash155Wiley 1979

[21] A Dillmann and G E A Meier ldquoHomogeneous nucleation ofsupersaturated vaporsrdquo Chemical Physics Letters vol 160 no 1pp 71ndash74 1989

[22] A Dillmann and G E A Meier ldquoA refined droplet approach tothe problem of homogeneous nucleation from the vapor phaserdquoThe Journal of Chemical Physics vol 94 no 5 pp 3872ndash38841991

[23] J W Cahn and J E Hillard ldquoFree energy of a nonuniformsystem III Nucleation in a two-component incompressiblefluidrdquoTheJournal of Chemical Physics vol 31 no 3 pp 688ndash6991959

[24] D Gebauer and H Colfen ldquoPrenucleation clusters and non-classical nucleationrdquo Nano Today vol 6 no 6 pp 564ndash5842011

[25] D Sept and J A McCammon ldquoThermodynamics and kineticsof actin filament nucleationrdquo Biophysical Journal vol 81 no 2pp 667ndash674 2001

[26] A S Myerson and B L Trout ldquoNucleation from solutionrdquo Sci-ence vol 341 no 6148 pp 855ndash856 2013

[27] D Zahn ldquoThermodynamics and kinetics of prenucleation clus-ters classical and non-classical nucleationrdquo ChemPhysChemvol 16 no 10 pp 2069ndash2075 2015

[28] H Deng S Wang X Wang et al ldquoTwo competitive nucle-ation mechanisms of calcium carbonate biomineralization inresponse to surface functionality in low calcium ion concentra-tion solutionrdquo Regenerative Biomaterials vol 2 no 3 pp 187ndash195 2015

[29] D Kozma and G Toth ldquoMicroscopic rate constants of crystalgrowth from molecular dynamic simulations combined withmetadynamicsrdquo Advances in Physical Chemistry vol 2012Article ID 135172 10 pages 2012

[30] S Fakruddin C Srinivasu B R Venkateswara Rao and KNarendra ldquoExcess transport properties of binary mixtures ofquinoline with xylenes at different temperaturesrdquo Advances inPhysical Chemistry vol 2012 Article ID 324098 9 pages 2012

[31] A A Lushnikov ldquoEvolution of coagulating systemsrdquo Journal ofColloid and Interface Science vol 45 no 3 pp 549ndash556 1973

[32] I M Lifshic and V V Slezov ldquoKinetics of diffusive decomposi-tion of supersaturated solid solutionsrdquo Soviet PhysicsmdashJournalof Experimental and Theoretical Physics vol 35 pp 331ndash3391959

[33] IM Lifshic andV V Slezov ldquoThe kinetics of precipitation fromsupersaturated solid solutionsrdquo Journal of Physics andChemistryof Solids vol 19 no 1-2 pp 35ndash50 1961

[34] F M Kuni and A P Grinin ldquoKinetics of homogeneous conden-sationat the stage of formation of the main quantity of a newphaserdquo Colloid Journal of the USSR vol 46 p 460 1984

[35] F M Kuni T Y Novozhilova and I A Terentiev ldquoKineticequation of heterogenous condensationrdquo Letters in Mathemati-cal Physics vol 14 no 2 pp 161ndash167 1987

[36] V Kurasov ldquoDecay of metastable multicomponentmixturerdquohttparxivorgabscond-mat9310005

[37] V B Kurasov ldquoKinetics of heterogeneous condensation underdynamic conditionsrdquo Physical Review E vol 49 no 5 pp 3948ndash3955 1994

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Inorganic ChemistryInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal ofPhotoenergy

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Carbohydrate Chemistry

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

Physical Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom

Analytical Methods in Chemistry

Journal of

Volume 2014

Bioinorganic Chemistry and ApplicationsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

SpectroscopyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Medicinal ChemistryInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chromatography Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied ChemistryJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Theoretical ChemistryJournal of

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Journal of

Spectroscopy

Analytical ChemistryInternational Journal of

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Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Quantum Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Organic Chemistry International

ElectrochemistryInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CatalystsJournal of

4 Advances in Physical Chemistry

The Nucleation Process Stops When 120579 Attains Zero Thisrestriction is necessary to avoid the negative rate of nucleationand the divergence of iterations

The account of the relaxation process in the nucleationkinetics is a hard problem Already the first iteration fromthose where the effect of the relaxation term can be seenin the behavior of the supersaturation can not be calculatedanalytically

To fulfil analytical calculations we have to fulfil an addi-tional simplification based on the property of the avalancheconsumption

Thementioned property can be formulated as follows

Under Any Behavior of 120579 gt 0 the Relative Growth of 119892 OccursFasterThan (119905minus119905

0)3 One can prove this property analytically

Actually this property is evident since we take into accountthat 119892 = int

119905

0(119905 minus 1199051015840)3120579[1199051015840]exp(minus119892[1199051015840])1198891199051015840 and 120579 gt 0

On the basis of this property we see that the ldquopseudo-homogeneous spectrum of sizesrdquo (the spectrum of sizeswhich would be formed if there would be no action of thenumber of free seeds on the nucleation process)

119891hom sim exp (minus119892) (12)

will be truncated in a manner sharper than

exp(minus(

119905

119905cut)

3

) (13)

where 119905cut is the characteristic timeThen one can speak aboutthe cut-off type of the pseudo-homogeneous spectrum ofsizes

Now one can see the qualitative behavior of 120579 At first 120579relaxes to the value

120579119899=

119895

119895 + 119879

120579lim (14)

with characteristic time

119905rel 119899 = (119895 + 119879)minus1

(15)

This relaxation can lead to the increase of 120579 when

120579 [119905 = 0] lt 120579119899

(16)

or to the decrease of 120579 in the opposite situationAt the moment 119905cut this relaxation is changed by the

relaxation of 120579 to 120579lim with the characteristic time 119905rel = 119895minus1

The question which we are interested in is whether thechange of 120579 can destroy the cut-off type of the real spectrumof sizes 120579 exp(minus119892) (earlier the cut-off type was shown forexp(minus119892))

The destruction of the cut-off type of 120579 exp(minus119892) can takeplace only under the sharp relative increase of 120579 So we haveto estimate (119889120579119889119905)120579 Since

119889120579

119889119905

lt 119895120579lim

120579 gt

119895

119895 + 119879

120579lim

(17)

we see that119889120579119889119905

120579

lt 119895 + 119879 (18)

When 119879 ≫ 119895 one simply can not see the relaxationThe picture in this case is the picture described earlier (see[37]) plus some small perturbation Then having excludedthis situation we can rewrite the last inequality as

119889120579119889119905

120579

lt (3 divide 4) 119895 (19)

Now we see that the danger of destruction of the cut-offform of the size spectrum can appear only at big values of 119895Actually it has to be

119895 ≫ 119905minus1

cut (20)

If 119905cut ≪ 119879minus1 then there is no exhaustion of seeds So the

actual situation is 119905cut ge 119879minus1 and then we see that

119895 ≫ 119879 (21)

Here 120579lim is close to 120579119899and then 119905cut based on 120579lim is close

to 120579 based on 120579119899 This means that the cut-off behavior is not

broken here So the property of the cut-off takes place in allsituations

The property of the cut-off allows using the approxima-tion exp(minus119892) = 1 for 119905 lt 119905cut and exp(minus119892) = 0 for 119905 gt 119905cut Themoment 119905cut can be determined as the root of equation

119892 (119905cut) = 1 (22)

Then for 119905 lt 119905cut

119889120579

119889119905

= minus119895 (120579 minus 120579lim) minus 119879120579 [119905]

119892 [119905] = int

119905

0

(119905 minus 1199051015840)

3

120579 [1199051015840] 1198891199051015840

(23)

and one can explicitly calculate

120579 [119905] = 120579119899minus (120579119899minus 120579 [119905 = 0]) exp (minus (119895 + 119879) 119905) (24)

The value of 119892 can be presented as

119892 = int

119905

0

(119905 minus 1199051015840)

3

sdot (120579119899minus (120579119899minus 120579 [119905 = 0]) exp (minus (119895 + 119879) 119905

1015840)) 1198891199051015840

(25)

and can be easily found analytically which allows getting 119905cutas the solution of an ordinary algebraic equation

Equation (22) can be easily solved since we know theestimates 120579

11989911991144 and 120579[0]119911

44 for the behavior of 119892 which

leads to the estimates for the rootsWhat estimate is the aboveestimate andwhat the below estimate is are determined by thesign of 120579lim minus 120579[0]

The total number of droplets in the renormalized unitscan be found as

119873 = 119860int

119905cut

0

1 119889119905 = 119860119905cut (26)

Advances in Physical Chemistry 5

where 119860 is the initial amplitude (in renormalized values itequals 1) or as

119873 = 119860int

119905cut

0

exp (minus119892 [119905]) 120579 [119905] 119889119905 (27)

where for 119892 and for 120579 one can take the mentioned approx-imations The linearization of appearing exponents is alsoapproximately suitable and leads to the presence of the resultin elementary functions One has to note that the first recipeseems to bemore rough than the second one but it gives betterresults The reason is the effective compensation of decreaseof the intensity of appearance of new embryos before 119905cut bythe existence of the tail after 119905cut The refinement of equationfor 119905cut (it is better to use 119892[119905cut] = ln 2) can make the firstrecipe very accurate Certainly the further refinement withthe help of the perturbation technique can be fulfilled

Even the linear approximation

120579 = 120579 [0] + (119895 + 119879) (120579119899minus 120579 [0]) 119905 (28)

truncated at the crossing point with 120579119899 that is at (119895 + 119879)

minus1and prolonged further as 120579

119899gives a rather good approxima-

tion for the characteristics of the processThis approximationensures the pure algebraic equation of the fifth power for 119905cutIt is a really suitable approximation since this approximationhas themaximal deviation as (119895+119879)minus1 which is relatively small(simexp(minus1)) in comparison with initial deviation of 120579 from 120579

119899

Then the difference in integrals staying in (22) is less thantwo times which gives the difference for the duration of theintensive nucleation period in 2

14asymp 12 times (in reality the

difference in duration is much more small) The last approx-imation brings equation on root to the explicit formulas

This completes the analytical description of the situationof the decay in the system with the relaxation process for thecenters of nucleation

3 Accumulation of Several Seeds inthe Monomer Nucleation

The second important task is to consider the situation whenit is necessary to spend 119902 119898-mers to form the supercriticalembryo and the rate of nucleation is proportional to thenumber of119898-mers in power 119902

119869 sim 119899 [119898]119902 (29)

It is interesting to investigate this situation and to see thetransition to the case when 119902 is great enough Certainly theprobability

119875 sim 119899 [119898]119902 (30)

transfers into

119875 sim exp (minus120583119902) (31)

with a chemical potential

120583 = minus ln (119899 [119898]) (32)

corresponding to the chemical potential of ideal gasThe equation

119889119899 [119898]

119889119905

= 119899 [119898]119902119891 [119905] (33)

with some known function 119891[119905] can be written in thefollowing integral form

119899 [119898] =

119899 [119898 119905 = 0]

((119902 minus 1) int

119905

0119891 [1199051015840] 1198891199051015840+ 1)

1(119902minus1) (34)

Now we write the expression for the rate of nucleation inthe generalized Clausius-Clapeyron integrated form as

119869 = 1198690120579119902 exp(minus

Γ (120577 minus Φlowast)

Φlowast

) (35)

where

Γ =

Φlowast119889119865119888

119889120577

10038161003816100381610038161003816100381610038161003816120577=Φlowast

(36)

and Φlowastis the base of decomposition Here 119869

0is the initial

value of the rate of nucleation and 120579 is the relative numberof119898-mers referred to initial value

One can get

Γ = (

Φlowast

(Φlowast+ 1)

) ]1119888 (37)

where ]1119888is the number ofmonomers in critical embryoThis

number is connected with the total number of molecules ]119888

in the critical embryo as

]119888= ]1119888+ 119902119898 (38)

After the scaling the system of evolution equations is thefollowing one

119892 [119911] = int

119911

0

(119911 minus 119909)3120579119902[119909] exp (minus119892 [119909]) 119889119909

120579 [119911] = ((119902 minus 1)119860int

119911

0

exp (minus119892 [119909]) 119889119909 + 1)

minus1(119902minus1)

(39)

Here 119860 is the coefficient which remains after the scaling andcan not be canceled

We can formulate the iteration procedure as follows

119892(119894+1)

(119911) = int

119911

0

(119911 minus 119909)3120579119902

(119894)[119909] exp (minus119892

(119894)[119909]) 119889119909

120579(119894+1)

[119911]

= ((119902 minus 1)119860int

119911

0

exp (minus119892(119894)[119909]) 119889119909 + 1)

minus1(119902minus1)

(40)

6 Advances in Physical Chemistry

The separation of the spectrum in 119892 and 120579 which seemsto be natural from the physical point of view is quite unusualfrom the formal point of view when the unique equation

119892 [119911] = int

119911

0

(119911 minus 119909)3

sdot ((119902 minus 1)119860int

119909

0

exp (minus119892 [1199091015840]) 1198891199091015840 + 1)

minus119902(119902minus1)

sdot exp (minus119892 [119909]) 119889119909

(41)

is considered Then it is more natural to construct iterationsas

119892(119894+1)

[119911] = int

119911

0

(119911 minus 119909)3

sdot ((119902 minus 1)119860int

119909

0

exp (minus119892(119894)[1199091015840]) 1198891199091015840+ 1)

minus119902(119902minus1)

sdot exp (minus119892(119894)[119909]) 119889119909

(42)

but this way does not correspond to appearance of the aboveand below estimates although it is possible to prove that theseiterations with the initial approximation

119892(0)

= 0 (43)

converge to solutionThe procedure for 120579 and 119892 as the separate functions which

was formulated above will lead under the initial approxima-tions

120579(0)

= 1

119892(0)

= 0

(44)

to the chains of inequalities

119892(0)

lt 119892(2)

lt sdot sdot sdot lt 119892 lt sdot sdot sdot lt 119892(3)

lt 119892(1)

120579(1)

lt 120579(3)

lt sdot sdot sdot lt 120579 lt sdot sdot sdot lt 120579(2)

lt 120579(0)

(45)

Parameter 119860 is considered here as constant value for alliterations and the concrete value of 119860 will be determined onthe basis of the last (precise) iteration

It is useful to calculate the total number of supercriticalembryos as

119873tot sim int

infin

0

120579119902[119909] exp (minus119892 [119909]) 119889119909 (46)

On the basis of iteration procedure one can calculate

119873tot(119894+1) sim int

infin

0

120579119902

(119894)[119909] exp (minus119892

(119894)[119909]) 119889119909 (47)

The calculation of iterations gives

119892(0)

=

1199114

4

120579(1)

= ((119902 minus 1)119860119911 + 1)minus1(119902minus1)

(48)

Then

119873tot(2) = int

infin

0

((119902 minus 1)119860119911 + 1)minus119902(119902minus1) exp(minus119911

4

4

)119889119911 (49)

and this can be expressed through the hypergeometric func-tion

119873tot(2) = 2(minus92+2(119902(119902minus1)))

(119902 + 119860)minus119902(119902minus1)

(

1

3

sdot 2minus(52)(119902(119902minus1))+12

1205874(119902 + 119860)

3

(

1

119902 + 119860

)

minus119902(119902minus1)

119867([

3

4

+

1

4

119902

119902 minus 1

1 +

1

4

119902

119902 minus 1

5

4

+

1

4

119902

119902 minus 1

] [

3

2

7

4

5

4

]

minus

1

4

(119902 + 119860)4

)

sdot

sec ((14) 120587 (119902 (119902 minus 1))) Γ (3 + 119902 (119902 minus 1))

Γ (32 + (14) (119902 (119902 minus 1)))

minus 2(1minus(52)(119902(119902minus1)))

1205874(119902 + 119860)

2

(

1

119902 + 119860

)

minus119902(119902minus1)

119867([

3

4

+

1

4

119902

119902 minus 1

1 +

1

4

119902

119902 minus 1

1

2

+

1

4

119902

119902 minus 1

] [

3

2

3

4

5

4

]

minus

1

4

(119902 + 119860)4

)

sdot

csc ((14) 120587 + (14) 120587 (119902 (119902 minus 1))) Γ (2 + 119902 (119902 minus 1))

Γ (54 + (14) (119902 (119902 minus 1)))

+ 2(52minus(52)(119902(119902minus1)))

1205874(119902 + 119860)(

1

119902 + 119860

)

minus119902(119902minus1)

119867([

3

4

+

1

4

119902

119902 minus 1

1

4

+

1

4

119902

119902 minus 1

1

2

+

1

4

119902

119902 minus 1

] [

3

4

5

4

1

2

]

minus

1

4

(119902 + 119860)4

)

sdot

csc ((14) 120587 (119902 (119902 minus 1))) Γ (1 + 119902 (119902 minus 1))

Γ (1 + (14) (119902 (119902 minus 1)))

+ 2(3minus(52)(119902(119902minus1)))

1205874(

1

119902 + 119860

)

minus119902(119902minus1)

119867([

1

4

119902

119902 minus 1

1

4

+

1

4

119902

119902 minus 1

1

2

+

1

4

119902

119902 minus 1

] [

3

4

1

2

1

4

] minus

1

4

(119902 + 119860)4

)

sdot

sec ((14) 120587 + (14) 120587 (119902 (119902 minus 1))) Γ (119902 (119902 minus 1))

Γ (34 + (14) (119902 (119902 minus 1)))

+ 2(132minus2(119902(119902minus1)))

1205873(minus

1

2

+

1

4

119902

119902 minus 1

) (119902 + 119860)

Advances in Physical Chemistry 7

sdot 119867([1

3

4

1

2

1

4

] [

5

4

minus

1

4

119902

119902 minus 1

minus

1

4

119902

119902 minus 1

+

1

2

minus

1

4

119902

119902 minus 1

+

3

4

minus

1

4

119902

119902 minus 1

+ 1] minus

1

4

(119902 + 119860)4

)Γ(minus2

+

119902

119902 minus 1

))(1205873Γ(

119902

119902 minus 1

))

minus1

(50)

where Γ is the gamma-function and119867 is the hypergeometricfunction

It is possible to show analytically that this expression israther accurate The error is less than one percent Despitethe known form it is rather difficult to perform calculationsaccording to this expression and it is necessary to simplify thecalculations

Again we use the property of the avalanche consumptionHere one can show that 120579 is the decreasing (or at least notrapidly increasing) function of time (or of the size of theinitial embryo)Then one can see the cut-off type of the func-tion exp(minus119892) and the possibility of truncation of the function120579119902 exp(minus119892(119911)) at 119905cut The value of 119905cut can be chosen asthe root of equation 119892 = 1 Then the spectrum of sizes

119891 sim 120579119902[119909] exp (minus119892 [119909]) (51)

can be approximated as

119891 sim ((119902 minus 1)119860119911 + 1)minus119902(119902minus1) (52)

for 119911 lt 119911cut where 119911cut is the size corresponding to 119905cut Forother 119911 the spectrum is zero

The number of embryos can be easily calculated

119873tot = int

119911

0

((119902 minus 1)119860119909 + 1)minus119902(119902minus1)

119889119909

= minus

(119860119911119902 minus 119860119911 + 1)minus1(119902minus1)

minus 1

119860

(53)

This solves the formal problem of our investigationTo show the consistency of the presented approach we

have to investigate the inclusion of the119898-mers consumptionkinetics into the general nucleation Here one can find atleast two aspects of the problem The first aspect is the inputof 119898-mers in the overcoming of activation barrier and thecorresponding input in the stationary rate of nucleation Thesecond aspect is the consumption of the119898-mers by the near-critical and precritical embryos (this is the first type) andby the supercritical embryos (the second type) The rate ofconsumption will be different and their manifestation in theglobal kinetics 119904 will be described by different ways

To clarify the first aspect we rewrite the stationary rate ofnucleation as

119869 sim exp (minus119902 ln (119899 [119898])) exp (minus119865119888) (54)

One has to stress that here 119865119888is the free energy of the critical

embryo formation on 119902 119898-mers This implies that the work

of collecting of119898-mers is considered separatelyThis work inapproximation of ideal gas of119898-mers is

1198650= minus119902 ln (119899 [119898]) (55)

At the same time we have to note that in the embryocontaining ]

119888molecules there are only ]

119888minus 119902119898 molecules

coming from monomers Then 1198650has to be written as

1198650= minus119887 (]

119888minus 119902119898) + 120574119860 (56)

where 119887 is the excess of the chemical potential for monomers120574 is the renormalized surface tension and 119860 is the surfacearea of the embryo It is clear that in the determination of thesurface area it is necessary to account that the part of the vol-ume inside the embryos surface is occupied by the moleculescoming from119898-mersWhen the surface of the embryo simplysurrounds the embryo one has to write

119860 = 4120587(

3V119897(]1119888+ 119902119898)

4120587

)

23

(57)

where V119897is the volume per onemolecule in a liquid phase and

]1119888is the number of molecules coming from monomersIn the state of equilibrium the chemical potential of 119898-

mer has to be equal to 119898 chemical potentials of monomers(here we neglect the surface term or consider the deformedchemical potential) So

119899 [119898]

119899 [119898 119904 = 0]

= exp (119887119898) (58)

and we see that the total free energy is

119888= 119865119888minus 119902119887119898 = minus]

119888119887 + 120574119860 (59)

and the total free energy coincides with the classical expres-sion

One has to stress that the parameter Γ in the previoussimplification of the dependence of the free energy on super-saturation (or the dependence of the stationary rate of nucle-ation on the supersaturation) is the number of moleculescoming from monomers multiplied on Φ

lowast(Φlowast

+ 1)that is

Γ 997888rarr Γ1=

Φlowast]1119888

(Φlowast+ 1)

(60)

Γ1is Φlowast119889119865119888119889120577|120577=Φlowast

If we calculate the derivative of the total free energy

119888we

get

Φlowast

119889119888

119889120577

100381610038161003816100381610038161003816100381610038161003816120577=Φlowast

= Γ1+ 119902119898

119889119887

119889120577

(61)

or

Φlowast

119889119888

119889120577

100381610038161003816100381610038161003816100381610038161003816120577=Φlowast

= Γ1+ Γ119898 (62)

8 Advances in Physical Chemistry

where

Γ119898=

120577119902119898

(120577 + 1)

10038161003816100381610038161003816100381610038161003816120577=Φlowast

(63)

As a result

Φlowast

119889119888

119889120577

100381610038161003816100381610038161003816100381610038161003816120577=Φlowast

= Γ

Γ =

120577]119888

(120577 + 1)

10038161003816100381610038161003816100381610038161003816120577=Φlowast

(64)

We will call this property the separation of Γ It takes placeonly in the first derivatives of the free energies and onlybecause the surface term does not act on the derivative of thefree energy The surface term in the free energy only changesthe number of molecules in the critical embryo and has nodirect influence on derivative

So we see that in the free energy there is the directcorrespondence between the119898-mers picture and monomerspicture

One can propose another picture when the number of119898-mers necessary to form the critical embryo is proportionalto the number of molecules inside the critical embryo In theenormously big critical embryos namely thismechanismhasto be considered Then the picture is equivalent to the binarynucleation with two vapors (the vapor of monomers and thevapor of119898-mers)

The second aspect is connected with the mechanism ofconsumption Here one has to confess that there are two prin-cipal mechanisms of the substance consumption The firstmechanism is ldquoone embryo-fixed number of clusters (mono-mers or 119898-mers)rdquo The second mechanism is ldquoone embryo-growing number of clusters (monomers and 119898-mers) corre-sponding to the law of the regular growthrdquo

Rigorously speaking we have to write for each type ofactive monomers two types of consumption and the balanceequations With the help of exponential approximationsmentioned above one can unify these mechanisms in one lawof growth Ordinarily when the first mechanism is importantthe second is absent When the second mechanism acts itmeans that there is enough clusters of such type to neglect theexhaustion of seeds We will call this effect as the dominatingpreference of one mechanism

Despite the mentioned dominating preference propertythe simultaneous existence of several mechanisms can not beexcluded as a specific case But it is clear that with the helpof approaches developed here it is possible to construct thetheoretical description of any process with different mecha-nisms of formation different mechanisms of substance con-sumption and different relaxation processes correspondingto bounds or connections between the resources of clusters ofdifferent sizes

The different relations between the characteristic timesof the mentioned processes make the processes of nucleationrather different The size spectrums are very various also Todescribe these situations one has to demonstrate the powerof the approaches making the base of theoretical description

In every situation the leading idea will be the avalanche con-sumption of metastability in different manifestations of thisproperty

4 Main Results

This paper contains at least two important derivations con-cerning the kinetics of decay with account of the relaxationprocesses and the kinetics of decay when the119898-mers nucleationis important Both cases are important and one can see thataccount of these processes radically changes the numericalvalues of parameters of the nucleations process But one hasto confess that the qualitative picture remains similar to thetraditional case The last does not mean that nothing in theform of the particles size distribution is changed Contrarilythe form of the size spectrum will be absolutely another Butthe property of the avalanche cut-off of the pseudo-homo-geneous size spectrum remains The last property was veryproductive in construction of the above presented procedure

It would be interesting to analyze these situations in thecase of the smooth external influence on the system whichcauses nucleation that is in the case of dynamic conditionsAlso it would be interesting to see the influence of the relax-ation processes in the overcoming of the near-critical regionin the process of the binary nucleation Certainly these per-spectives make the subjects of separate publications

Beside the direct construction of the nucleation globalkinetics description we managed to extract some importantfeatures of the effect of119898-mers inclusion into the nucleationprocess One can mention here the property of separation ofΓ and the correspondence between the monomers picture and119898-mers picture These properties are important enough to bementioned in the general theory of nucleation

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] C T R Wilson ldquoCondensation of water vapour in the presenceof dust-free air and other gasesrdquo Philosophical Transactions ofthe Royal Society A Mathematical Physical and Engineering Sci-ences vol 189 pp 265ndash307 1897

[2] C T RWilson ldquoOn the condensation nuclei produced in gasesby the action of roentgen rays uranium rays ultraviolet rays andother agentsrdquoPhilosophical Transactions A vol 192 no 9 article403 1899

[3] C T RWilson ldquoOn the comparative efficiency as condensationnuclei of positively and negatively charged ionsrdquo PhilosophicalTransactions of the Royal Society A Containing Papers of aMath-ematical or Physical Character vol 193 pp 289ndash308 1900

[4] A Benmouna R Benmouna M R Bockstaller and I FHakem ldquoSelf-organization schemes towards thermodynamicstable bulk heterojunction morphologies a perspective onfuture fabrication strategies of polymer photovoltaic architec-turesrdquo Advances in Physical Chemistry vol 2013 Article ID948189 8 pages 2013

Advances in Physical Chemistry 9

[5] Y Kazzi H Awada and M Nardin ldquoAdhesion on nanoorga-nized multilayers surface thermodynamics and local energydissipationrdquo Advances in Physical Chemistry vol 2010 ArticleID 502709 11 pages 2010

[6] A I Burshtein ldquoContact and distant luminescence quenchingin solutionsrdquo Advances in Physical Chemistry vol 2009 ArticleID 214219 34 pages 2009

[7] A C Zettlemoyer Nucleation Dekker New York NY USA1969

[8] F F Abraham Homogeneous Nucleation Theory AcademicNew York NY USA 1974

[9] M Volmer and A Weber ldquoKeimbildung in ubersattigtenGebildenrdquo Zeitschrift fur Physikalische Chemie vol 119 pp 277ndash301 1925

[10] R Becker andW Doring ldquoKinetische behandlung der keimbil-dung in uberattigten dampfernrdquo Annals of Physics vol 24 pp719ndash752 1935

[11] J Frenkel ldquoA general theory of heterophase fluctuations andpretransition phenomenardquoThe Journal of Chemical Physics vol7 no 7 pp 538ndash547 1939

[12] J B Zeldovich ldquoOn the theory of new phase formation cavi-tationrdquo Acta Physicochimica USSR vol 18 pp 1ndash22 1943

[13] L Farkas ldquoThe velocity of nucleus formation in superheatedvaporsrdquo Zeitschrift fur Physikalische Chemie vol 125 pp 236ndash240 1927

[14] J Lothe and G M Pound ldquoReconsiderations of nucleationtheoryrdquo The Journal of Chemical Physics vol 36 no 8 article2080 1962

[15] H Reiss J L Katz and E R Cohen ldquoTranslation-rotation para-dox in the theory of nucleationrdquoThe Journal of Chemical Phys-ics vol 48 no 12 pp 5553ndash5560 1968

[16] H Reiss ldquoTreatment of droplike clusters by means of the clas-sical phase integral in nucleation theoryrdquo Journal of StatisticalPhysics vol 2 no 1 pp 83ndash104 1970

[17] W G Courtney ldquoRemarks on homogeneous nucleationrdquo TheJournal of Chemical Physics vol 35 no 6 pp 2249ndash2250 1961

[18] M Blander and J L Katz ldquoThe thermodynamics of cluster for-mation in nucleation theoryrdquo Journal of Statistical Physics vol 4no 1 pp 55ndash59 1972

[19] J L Katz and H Weidersich ldquoNucleation theory without Max-well Demonsrdquo Journal of Colloid and Interface Science vol 61no 2 pp 351ndash355 1977

[20] J L Katz and M D Donohue ldquoA kinetic approach to homo-geneous nucleation theoryrdquo in Advances in Chemical Physics IPrigogine and S A Rice Eds vol 40 chapter 3 pp 137ndash155Wiley 1979

[21] A Dillmann and G E A Meier ldquoHomogeneous nucleation ofsupersaturated vaporsrdquo Chemical Physics Letters vol 160 no 1pp 71ndash74 1989

[22] A Dillmann and G E A Meier ldquoA refined droplet approach tothe problem of homogeneous nucleation from the vapor phaserdquoThe Journal of Chemical Physics vol 94 no 5 pp 3872ndash38841991

[23] J W Cahn and J E Hillard ldquoFree energy of a nonuniformsystem III Nucleation in a two-component incompressiblefluidrdquoTheJournal of Chemical Physics vol 31 no 3 pp 688ndash6991959

[24] D Gebauer and H Colfen ldquoPrenucleation clusters and non-classical nucleationrdquo Nano Today vol 6 no 6 pp 564ndash5842011

[25] D Sept and J A McCammon ldquoThermodynamics and kineticsof actin filament nucleationrdquo Biophysical Journal vol 81 no 2pp 667ndash674 2001

[26] A S Myerson and B L Trout ldquoNucleation from solutionrdquo Sci-ence vol 341 no 6148 pp 855ndash856 2013

[27] D Zahn ldquoThermodynamics and kinetics of prenucleation clus-ters classical and non-classical nucleationrdquo ChemPhysChemvol 16 no 10 pp 2069ndash2075 2015

[28] H Deng S Wang X Wang et al ldquoTwo competitive nucle-ation mechanisms of calcium carbonate biomineralization inresponse to surface functionality in low calcium ion concentra-tion solutionrdquo Regenerative Biomaterials vol 2 no 3 pp 187ndash195 2015

[29] D Kozma and G Toth ldquoMicroscopic rate constants of crystalgrowth from molecular dynamic simulations combined withmetadynamicsrdquo Advances in Physical Chemistry vol 2012Article ID 135172 10 pages 2012

[30] S Fakruddin C Srinivasu B R Venkateswara Rao and KNarendra ldquoExcess transport properties of binary mixtures ofquinoline with xylenes at different temperaturesrdquo Advances inPhysical Chemistry vol 2012 Article ID 324098 9 pages 2012

[31] A A Lushnikov ldquoEvolution of coagulating systemsrdquo Journal ofColloid and Interface Science vol 45 no 3 pp 549ndash556 1973

[32] I M Lifshic and V V Slezov ldquoKinetics of diffusive decomposi-tion of supersaturated solid solutionsrdquo Soviet PhysicsmdashJournalof Experimental and Theoretical Physics vol 35 pp 331ndash3391959

[33] IM Lifshic andV V Slezov ldquoThe kinetics of precipitation fromsupersaturated solid solutionsrdquo Journal of Physics andChemistryof Solids vol 19 no 1-2 pp 35ndash50 1961

[34] F M Kuni and A P Grinin ldquoKinetics of homogeneous conden-sationat the stage of formation of the main quantity of a newphaserdquo Colloid Journal of the USSR vol 46 p 460 1984

[35] F M Kuni T Y Novozhilova and I A Terentiev ldquoKineticequation of heterogenous condensationrdquo Letters in Mathemati-cal Physics vol 14 no 2 pp 161ndash167 1987

[36] V Kurasov ldquoDecay of metastable multicomponentmixturerdquohttparxivorgabscond-mat9310005

[37] V B Kurasov ldquoKinetics of heterogeneous condensation underdynamic conditionsrdquo Physical Review E vol 49 no 5 pp 3948ndash3955 1994

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Inorganic ChemistryInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal ofPhotoenergy

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Carbohydrate Chemistry

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

Physical Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom

Analytical Methods in Chemistry

Journal of

Volume 2014

Bioinorganic Chemistry and ApplicationsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

SpectroscopyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Medicinal ChemistryInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chromatography Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied ChemistryJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Theoretical ChemistryJournal of

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Journal of

Spectroscopy

Analytical ChemistryInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Quantum Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Organic Chemistry International

ElectrochemistryInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CatalystsJournal of

Advances in Physical Chemistry 5

where 119860 is the initial amplitude (in renormalized values itequals 1) or as

119873 = 119860int

119905cut

0

exp (minus119892 [119905]) 120579 [119905] 119889119905 (27)

where for 119892 and for 120579 one can take the mentioned approx-imations The linearization of appearing exponents is alsoapproximately suitable and leads to the presence of the resultin elementary functions One has to note that the first recipeseems to bemore rough than the second one but it gives betterresults The reason is the effective compensation of decreaseof the intensity of appearance of new embryos before 119905cut bythe existence of the tail after 119905cut The refinement of equationfor 119905cut (it is better to use 119892[119905cut] = ln 2) can make the firstrecipe very accurate Certainly the further refinement withthe help of the perturbation technique can be fulfilled

Even the linear approximation

120579 = 120579 [0] + (119895 + 119879) (120579119899minus 120579 [0]) 119905 (28)

truncated at the crossing point with 120579119899 that is at (119895 + 119879)

minus1and prolonged further as 120579

119899gives a rather good approxima-

tion for the characteristics of the processThis approximationensures the pure algebraic equation of the fifth power for 119905cutIt is a really suitable approximation since this approximationhas themaximal deviation as (119895+119879)minus1 which is relatively small(simexp(minus1)) in comparison with initial deviation of 120579 from 120579

119899

Then the difference in integrals staying in (22) is less thantwo times which gives the difference for the duration of theintensive nucleation period in 2

14asymp 12 times (in reality the

difference in duration is much more small) The last approx-imation brings equation on root to the explicit formulas

This completes the analytical description of the situationof the decay in the system with the relaxation process for thecenters of nucleation

3 Accumulation of Several Seeds inthe Monomer Nucleation

The second important task is to consider the situation whenit is necessary to spend 119902 119898-mers to form the supercriticalembryo and the rate of nucleation is proportional to thenumber of119898-mers in power 119902

119869 sim 119899 [119898]119902 (29)

It is interesting to investigate this situation and to see thetransition to the case when 119902 is great enough Certainly theprobability

119875 sim 119899 [119898]119902 (30)

transfers into

119875 sim exp (minus120583119902) (31)

with a chemical potential

120583 = minus ln (119899 [119898]) (32)

corresponding to the chemical potential of ideal gasThe equation

119889119899 [119898]

119889119905

= 119899 [119898]119902119891 [119905] (33)

with some known function 119891[119905] can be written in thefollowing integral form

119899 [119898] =

119899 [119898 119905 = 0]

((119902 minus 1) int

119905

0119891 [1199051015840] 1198891199051015840+ 1)

1(119902minus1) (34)

Now we write the expression for the rate of nucleation inthe generalized Clausius-Clapeyron integrated form as

119869 = 1198690120579119902 exp(minus

Γ (120577 minus Φlowast)

Φlowast

) (35)

where

Γ =

Φlowast119889119865119888

119889120577

10038161003816100381610038161003816100381610038161003816120577=Φlowast

(36)

and Φlowastis the base of decomposition Here 119869

0is the initial

value of the rate of nucleation and 120579 is the relative numberof119898-mers referred to initial value

One can get

Γ = (

Φlowast

(Φlowast+ 1)

) ]1119888 (37)

where ]1119888is the number ofmonomers in critical embryoThis

number is connected with the total number of molecules ]119888

in the critical embryo as

]119888= ]1119888+ 119902119898 (38)

After the scaling the system of evolution equations is thefollowing one

119892 [119911] = int

119911

0

(119911 minus 119909)3120579119902[119909] exp (minus119892 [119909]) 119889119909

120579 [119911] = ((119902 minus 1)119860int

119911

0

exp (minus119892 [119909]) 119889119909 + 1)

minus1(119902minus1)

(39)

Here 119860 is the coefficient which remains after the scaling andcan not be canceled

We can formulate the iteration procedure as follows

119892(119894+1)

(119911) = int

119911

0

(119911 minus 119909)3120579119902

(119894)[119909] exp (minus119892

(119894)[119909]) 119889119909

120579(119894+1)

[119911]

= ((119902 minus 1)119860int

119911

0

exp (minus119892(119894)[119909]) 119889119909 + 1)

minus1(119902minus1)

(40)

6 Advances in Physical Chemistry

The separation of the spectrum in 119892 and 120579 which seemsto be natural from the physical point of view is quite unusualfrom the formal point of view when the unique equation

119892 [119911] = int

119911

0

(119911 minus 119909)3

sdot ((119902 minus 1)119860int

119909

0

exp (minus119892 [1199091015840]) 1198891199091015840 + 1)

minus119902(119902minus1)

sdot exp (minus119892 [119909]) 119889119909

(41)

is considered Then it is more natural to construct iterationsas

119892(119894+1)

[119911] = int

119911

0

(119911 minus 119909)3

sdot ((119902 minus 1)119860int

119909

0

exp (minus119892(119894)[1199091015840]) 1198891199091015840+ 1)

minus119902(119902minus1)

sdot exp (minus119892(119894)[119909]) 119889119909

(42)

but this way does not correspond to appearance of the aboveand below estimates although it is possible to prove that theseiterations with the initial approximation

119892(0)

= 0 (43)

converge to solutionThe procedure for 120579 and 119892 as the separate functions which

was formulated above will lead under the initial approxima-tions

120579(0)

= 1

119892(0)

= 0

(44)

to the chains of inequalities

119892(0)

lt 119892(2)

lt sdot sdot sdot lt 119892 lt sdot sdot sdot lt 119892(3)

lt 119892(1)

120579(1)

lt 120579(3)

lt sdot sdot sdot lt 120579 lt sdot sdot sdot lt 120579(2)

lt 120579(0)

(45)

Parameter 119860 is considered here as constant value for alliterations and the concrete value of 119860 will be determined onthe basis of the last (precise) iteration

It is useful to calculate the total number of supercriticalembryos as

119873tot sim int

infin

0

120579119902[119909] exp (minus119892 [119909]) 119889119909 (46)

On the basis of iteration procedure one can calculate

119873tot(119894+1) sim int

infin

0

120579119902

(119894)[119909] exp (minus119892

(119894)[119909]) 119889119909 (47)

The calculation of iterations gives

119892(0)

=

1199114

4

120579(1)

= ((119902 minus 1)119860119911 + 1)minus1(119902minus1)

(48)

Then

119873tot(2) = int

infin

0

((119902 minus 1)119860119911 + 1)minus119902(119902minus1) exp(minus119911

4

4

)119889119911 (49)

and this can be expressed through the hypergeometric func-tion

119873tot(2) = 2(minus92+2(119902(119902minus1)))

(119902 + 119860)minus119902(119902minus1)

(

1

3

sdot 2minus(52)(119902(119902minus1))+12

1205874(119902 + 119860)

3

(

1

119902 + 119860

)

minus119902(119902minus1)

119867([

3

4

+

1

4

119902

119902 minus 1

1 +

1

4

119902

119902 minus 1

5

4

+

1

4

119902

119902 minus 1

] [

3

2

7

4

5

4

]

minus

1

4

(119902 + 119860)4

)

sdot

sec ((14) 120587 (119902 (119902 minus 1))) Γ (3 + 119902 (119902 minus 1))

Γ (32 + (14) (119902 (119902 minus 1)))

minus 2(1minus(52)(119902(119902minus1)))

1205874(119902 + 119860)

2

(

1

119902 + 119860

)

minus119902(119902minus1)

119867([

3

4

+

1

4

119902

119902 minus 1

1 +

1

4

119902

119902 minus 1

1

2

+

1

4

119902

119902 minus 1

] [

3

2

3

4

5

4

]

minus

1

4

(119902 + 119860)4

)

sdot

csc ((14) 120587 + (14) 120587 (119902 (119902 minus 1))) Γ (2 + 119902 (119902 minus 1))

Γ (54 + (14) (119902 (119902 minus 1)))

+ 2(52minus(52)(119902(119902minus1)))

1205874(119902 + 119860)(

1

119902 + 119860

)

minus119902(119902minus1)

119867([

3

4

+

1

4

119902

119902 minus 1

1

4

+

1

4

119902

119902 minus 1

1

2

+

1

4

119902

119902 minus 1

] [

3

4

5

4

1

2

]

minus

1

4

(119902 + 119860)4

)

sdot

csc ((14) 120587 (119902 (119902 minus 1))) Γ (1 + 119902 (119902 minus 1))

Γ (1 + (14) (119902 (119902 minus 1)))

+ 2(3minus(52)(119902(119902minus1)))

1205874(

1

119902 + 119860

)

minus119902(119902minus1)

119867([

1

4

119902

119902 minus 1

1

4

+

1

4

119902

119902 minus 1

1

2

+

1

4

119902

119902 minus 1

] [

3

4

1

2

1

4

] minus

1

4

(119902 + 119860)4

)

sdot

sec ((14) 120587 + (14) 120587 (119902 (119902 minus 1))) Γ (119902 (119902 minus 1))

Γ (34 + (14) (119902 (119902 minus 1)))

+ 2(132minus2(119902(119902minus1)))

1205873(minus

1

2

+

1

4

119902

119902 minus 1

) (119902 + 119860)

Advances in Physical Chemistry 7

sdot 119867([1

3

4

1

2

1

4

] [

5

4

minus

1

4

119902

119902 minus 1

minus

1

4

119902

119902 minus 1

+

1

2

minus

1

4

119902

119902 minus 1

+

3

4

minus

1

4

119902

119902 minus 1

+ 1] minus

1

4

(119902 + 119860)4

)Γ(minus2

+

119902

119902 minus 1

))(1205873Γ(

119902

119902 minus 1

))

minus1

(50)

where Γ is the gamma-function and119867 is the hypergeometricfunction

It is possible to show analytically that this expression israther accurate The error is less than one percent Despitethe known form it is rather difficult to perform calculationsaccording to this expression and it is necessary to simplify thecalculations

Again we use the property of the avalanche consumptionHere one can show that 120579 is the decreasing (or at least notrapidly increasing) function of time (or of the size of theinitial embryo)Then one can see the cut-off type of the func-tion exp(minus119892) and the possibility of truncation of the function120579119902 exp(minus119892(119911)) at 119905cut The value of 119905cut can be chosen asthe root of equation 119892 = 1 Then the spectrum of sizes

119891 sim 120579119902[119909] exp (minus119892 [119909]) (51)

can be approximated as

119891 sim ((119902 minus 1)119860119911 + 1)minus119902(119902minus1) (52)

for 119911 lt 119911cut where 119911cut is the size corresponding to 119905cut Forother 119911 the spectrum is zero

The number of embryos can be easily calculated

119873tot = int

119911

0

((119902 minus 1)119860119909 + 1)minus119902(119902minus1)

119889119909

= minus

(119860119911119902 minus 119860119911 + 1)minus1(119902minus1)

minus 1

119860

(53)

This solves the formal problem of our investigationTo show the consistency of the presented approach we

have to investigate the inclusion of the119898-mers consumptionkinetics into the general nucleation Here one can find atleast two aspects of the problem The first aspect is the inputof 119898-mers in the overcoming of activation barrier and thecorresponding input in the stationary rate of nucleation Thesecond aspect is the consumption of the119898-mers by the near-critical and precritical embryos (this is the first type) andby the supercritical embryos (the second type) The rate ofconsumption will be different and their manifestation in theglobal kinetics 119904 will be described by different ways

To clarify the first aspect we rewrite the stationary rate ofnucleation as

119869 sim exp (minus119902 ln (119899 [119898])) exp (minus119865119888) (54)

One has to stress that here 119865119888is the free energy of the critical

embryo formation on 119902 119898-mers This implies that the work

of collecting of119898-mers is considered separatelyThis work inapproximation of ideal gas of119898-mers is

1198650= minus119902 ln (119899 [119898]) (55)

At the same time we have to note that in the embryocontaining ]

119888molecules there are only ]

119888minus 119902119898 molecules

coming from monomers Then 1198650has to be written as

1198650= minus119887 (]

119888minus 119902119898) + 120574119860 (56)

where 119887 is the excess of the chemical potential for monomers120574 is the renormalized surface tension and 119860 is the surfacearea of the embryo It is clear that in the determination of thesurface area it is necessary to account that the part of the vol-ume inside the embryos surface is occupied by the moleculescoming from119898-mersWhen the surface of the embryo simplysurrounds the embryo one has to write

119860 = 4120587(

3V119897(]1119888+ 119902119898)

4120587

)

23

(57)

where V119897is the volume per onemolecule in a liquid phase and

]1119888is the number of molecules coming from monomersIn the state of equilibrium the chemical potential of 119898-

mer has to be equal to 119898 chemical potentials of monomers(here we neglect the surface term or consider the deformedchemical potential) So

119899 [119898]

119899 [119898 119904 = 0]

= exp (119887119898) (58)

and we see that the total free energy is

119888= 119865119888minus 119902119887119898 = minus]

119888119887 + 120574119860 (59)

and the total free energy coincides with the classical expres-sion

One has to stress that the parameter Γ in the previoussimplification of the dependence of the free energy on super-saturation (or the dependence of the stationary rate of nucle-ation on the supersaturation) is the number of moleculescoming from monomers multiplied on Φ

lowast(Φlowast

+ 1)that is

Γ 997888rarr Γ1=

Φlowast]1119888

(Φlowast+ 1)

(60)

Γ1is Φlowast119889119865119888119889120577|120577=Φlowast

If we calculate the derivative of the total free energy

119888we

get

Φlowast

119889119888

119889120577

100381610038161003816100381610038161003816100381610038161003816120577=Φlowast

= Γ1+ 119902119898

119889119887

119889120577

(61)

or

Φlowast

119889119888

119889120577

100381610038161003816100381610038161003816100381610038161003816120577=Φlowast

= Γ1+ Γ119898 (62)

8 Advances in Physical Chemistry

where

Γ119898=

120577119902119898

(120577 + 1)

10038161003816100381610038161003816100381610038161003816120577=Φlowast

(63)

As a result

Φlowast

119889119888

119889120577

100381610038161003816100381610038161003816100381610038161003816120577=Φlowast

= Γ

Γ =

120577]119888

(120577 + 1)

10038161003816100381610038161003816100381610038161003816120577=Φlowast

(64)

We will call this property the separation of Γ It takes placeonly in the first derivatives of the free energies and onlybecause the surface term does not act on the derivative of thefree energy The surface term in the free energy only changesthe number of molecules in the critical embryo and has nodirect influence on derivative

So we see that in the free energy there is the directcorrespondence between the119898-mers picture and monomerspicture

One can propose another picture when the number of119898-mers necessary to form the critical embryo is proportionalto the number of molecules inside the critical embryo In theenormously big critical embryos namely thismechanismhasto be considered Then the picture is equivalent to the binarynucleation with two vapors (the vapor of monomers and thevapor of119898-mers)

The second aspect is connected with the mechanism ofconsumption Here one has to confess that there are two prin-cipal mechanisms of the substance consumption The firstmechanism is ldquoone embryo-fixed number of clusters (mono-mers or 119898-mers)rdquo The second mechanism is ldquoone embryo-growing number of clusters (monomers and 119898-mers) corre-sponding to the law of the regular growthrdquo

Rigorously speaking we have to write for each type ofactive monomers two types of consumption and the balanceequations With the help of exponential approximationsmentioned above one can unify these mechanisms in one lawof growth Ordinarily when the first mechanism is importantthe second is absent When the second mechanism acts itmeans that there is enough clusters of such type to neglect theexhaustion of seeds We will call this effect as the dominatingpreference of one mechanism

Despite the mentioned dominating preference propertythe simultaneous existence of several mechanisms can not beexcluded as a specific case But it is clear that with the helpof approaches developed here it is possible to construct thetheoretical description of any process with different mecha-nisms of formation different mechanisms of substance con-sumption and different relaxation processes correspondingto bounds or connections between the resources of clusters ofdifferent sizes

The different relations between the characteristic timesof the mentioned processes make the processes of nucleationrather different The size spectrums are very various also Todescribe these situations one has to demonstrate the powerof the approaches making the base of theoretical description

In every situation the leading idea will be the avalanche con-sumption of metastability in different manifestations of thisproperty

4 Main Results

This paper contains at least two important derivations con-cerning the kinetics of decay with account of the relaxationprocesses and the kinetics of decay when the119898-mers nucleationis important Both cases are important and one can see thataccount of these processes radically changes the numericalvalues of parameters of the nucleations process But one hasto confess that the qualitative picture remains similar to thetraditional case The last does not mean that nothing in theform of the particles size distribution is changed Contrarilythe form of the size spectrum will be absolutely another Butthe property of the avalanche cut-off of the pseudo-homo-geneous size spectrum remains The last property was veryproductive in construction of the above presented procedure

It would be interesting to analyze these situations in thecase of the smooth external influence on the system whichcauses nucleation that is in the case of dynamic conditionsAlso it would be interesting to see the influence of the relax-ation processes in the overcoming of the near-critical regionin the process of the binary nucleation Certainly these per-spectives make the subjects of separate publications

Beside the direct construction of the nucleation globalkinetics description we managed to extract some importantfeatures of the effect of119898-mers inclusion into the nucleationprocess One can mention here the property of separation ofΓ and the correspondence between the monomers picture and119898-mers picture These properties are important enough to bementioned in the general theory of nucleation

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] C T R Wilson ldquoCondensation of water vapour in the presenceof dust-free air and other gasesrdquo Philosophical Transactions ofthe Royal Society A Mathematical Physical and Engineering Sci-ences vol 189 pp 265ndash307 1897

[2] C T RWilson ldquoOn the condensation nuclei produced in gasesby the action of roentgen rays uranium rays ultraviolet rays andother agentsrdquoPhilosophical Transactions A vol 192 no 9 article403 1899

[3] C T RWilson ldquoOn the comparative efficiency as condensationnuclei of positively and negatively charged ionsrdquo PhilosophicalTransactions of the Royal Society A Containing Papers of aMath-ematical or Physical Character vol 193 pp 289ndash308 1900

[4] A Benmouna R Benmouna M R Bockstaller and I FHakem ldquoSelf-organization schemes towards thermodynamicstable bulk heterojunction morphologies a perspective onfuture fabrication strategies of polymer photovoltaic architec-turesrdquo Advances in Physical Chemistry vol 2013 Article ID948189 8 pages 2013

Advances in Physical Chemistry 9

[5] Y Kazzi H Awada and M Nardin ldquoAdhesion on nanoorga-nized multilayers surface thermodynamics and local energydissipationrdquo Advances in Physical Chemistry vol 2010 ArticleID 502709 11 pages 2010

[6] A I Burshtein ldquoContact and distant luminescence quenchingin solutionsrdquo Advances in Physical Chemistry vol 2009 ArticleID 214219 34 pages 2009

[7] A C Zettlemoyer Nucleation Dekker New York NY USA1969

[8] F F Abraham Homogeneous Nucleation Theory AcademicNew York NY USA 1974

[9] M Volmer and A Weber ldquoKeimbildung in ubersattigtenGebildenrdquo Zeitschrift fur Physikalische Chemie vol 119 pp 277ndash301 1925

[10] R Becker andW Doring ldquoKinetische behandlung der keimbil-dung in uberattigten dampfernrdquo Annals of Physics vol 24 pp719ndash752 1935

[11] J Frenkel ldquoA general theory of heterophase fluctuations andpretransition phenomenardquoThe Journal of Chemical Physics vol7 no 7 pp 538ndash547 1939

[12] J B Zeldovich ldquoOn the theory of new phase formation cavi-tationrdquo Acta Physicochimica USSR vol 18 pp 1ndash22 1943

[13] L Farkas ldquoThe velocity of nucleus formation in superheatedvaporsrdquo Zeitschrift fur Physikalische Chemie vol 125 pp 236ndash240 1927

[14] J Lothe and G M Pound ldquoReconsiderations of nucleationtheoryrdquo The Journal of Chemical Physics vol 36 no 8 article2080 1962

[15] H Reiss J L Katz and E R Cohen ldquoTranslation-rotation para-dox in the theory of nucleationrdquoThe Journal of Chemical Phys-ics vol 48 no 12 pp 5553ndash5560 1968

[16] H Reiss ldquoTreatment of droplike clusters by means of the clas-sical phase integral in nucleation theoryrdquo Journal of StatisticalPhysics vol 2 no 1 pp 83ndash104 1970

[17] W G Courtney ldquoRemarks on homogeneous nucleationrdquo TheJournal of Chemical Physics vol 35 no 6 pp 2249ndash2250 1961

[18] M Blander and J L Katz ldquoThe thermodynamics of cluster for-mation in nucleation theoryrdquo Journal of Statistical Physics vol 4no 1 pp 55ndash59 1972

[19] J L Katz and H Weidersich ldquoNucleation theory without Max-well Demonsrdquo Journal of Colloid and Interface Science vol 61no 2 pp 351ndash355 1977

[20] J L Katz and M D Donohue ldquoA kinetic approach to homo-geneous nucleation theoryrdquo in Advances in Chemical Physics IPrigogine and S A Rice Eds vol 40 chapter 3 pp 137ndash155Wiley 1979

[21] A Dillmann and G E A Meier ldquoHomogeneous nucleation ofsupersaturated vaporsrdquo Chemical Physics Letters vol 160 no 1pp 71ndash74 1989

[22] A Dillmann and G E A Meier ldquoA refined droplet approach tothe problem of homogeneous nucleation from the vapor phaserdquoThe Journal of Chemical Physics vol 94 no 5 pp 3872ndash38841991

[23] J W Cahn and J E Hillard ldquoFree energy of a nonuniformsystem III Nucleation in a two-component incompressiblefluidrdquoTheJournal of Chemical Physics vol 31 no 3 pp 688ndash6991959

[24] D Gebauer and H Colfen ldquoPrenucleation clusters and non-classical nucleationrdquo Nano Today vol 6 no 6 pp 564ndash5842011

[25] D Sept and J A McCammon ldquoThermodynamics and kineticsof actin filament nucleationrdquo Biophysical Journal vol 81 no 2pp 667ndash674 2001

[26] A S Myerson and B L Trout ldquoNucleation from solutionrdquo Sci-ence vol 341 no 6148 pp 855ndash856 2013

[27] D Zahn ldquoThermodynamics and kinetics of prenucleation clus-ters classical and non-classical nucleationrdquo ChemPhysChemvol 16 no 10 pp 2069ndash2075 2015

[28] H Deng S Wang X Wang et al ldquoTwo competitive nucle-ation mechanisms of calcium carbonate biomineralization inresponse to surface functionality in low calcium ion concentra-tion solutionrdquo Regenerative Biomaterials vol 2 no 3 pp 187ndash195 2015

[29] D Kozma and G Toth ldquoMicroscopic rate constants of crystalgrowth from molecular dynamic simulations combined withmetadynamicsrdquo Advances in Physical Chemistry vol 2012Article ID 135172 10 pages 2012

[30] S Fakruddin C Srinivasu B R Venkateswara Rao and KNarendra ldquoExcess transport properties of binary mixtures ofquinoline with xylenes at different temperaturesrdquo Advances inPhysical Chemistry vol 2012 Article ID 324098 9 pages 2012

[31] A A Lushnikov ldquoEvolution of coagulating systemsrdquo Journal ofColloid and Interface Science vol 45 no 3 pp 549ndash556 1973

[32] I M Lifshic and V V Slezov ldquoKinetics of diffusive decomposi-tion of supersaturated solid solutionsrdquo Soviet PhysicsmdashJournalof Experimental and Theoretical Physics vol 35 pp 331ndash3391959

[33] IM Lifshic andV V Slezov ldquoThe kinetics of precipitation fromsupersaturated solid solutionsrdquo Journal of Physics andChemistryof Solids vol 19 no 1-2 pp 35ndash50 1961

[34] F M Kuni and A P Grinin ldquoKinetics of homogeneous conden-sationat the stage of formation of the main quantity of a newphaserdquo Colloid Journal of the USSR vol 46 p 460 1984

[35] F M Kuni T Y Novozhilova and I A Terentiev ldquoKineticequation of heterogenous condensationrdquo Letters in Mathemati-cal Physics vol 14 no 2 pp 161ndash167 1987

[36] V Kurasov ldquoDecay of metastable multicomponentmixturerdquohttparxivorgabscond-mat9310005

[37] V B Kurasov ldquoKinetics of heterogeneous condensation underdynamic conditionsrdquo Physical Review E vol 49 no 5 pp 3948ndash3955 1994

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Inorganic ChemistryInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal ofPhotoenergy

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Carbohydrate Chemistry

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

Physical Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom

Analytical Methods in Chemistry

Journal of

Volume 2014

Bioinorganic Chemistry and ApplicationsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

SpectroscopyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Medicinal ChemistryInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chromatography Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied ChemistryJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Theoretical ChemistryJournal of

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Journal of

Spectroscopy

Analytical ChemistryInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Quantum Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Organic Chemistry International

ElectrochemistryInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CatalystsJournal of

6 Advances in Physical Chemistry

The separation of the spectrum in 119892 and 120579 which seemsto be natural from the physical point of view is quite unusualfrom the formal point of view when the unique equation

119892 [119911] = int

119911

0

(119911 minus 119909)3

sdot ((119902 minus 1)119860int

119909

0

exp (minus119892 [1199091015840]) 1198891199091015840 + 1)

minus119902(119902minus1)

sdot exp (minus119892 [119909]) 119889119909

(41)

is considered Then it is more natural to construct iterationsas

119892(119894+1)

[119911] = int

119911

0

(119911 minus 119909)3

sdot ((119902 minus 1)119860int

119909

0

exp (minus119892(119894)[1199091015840]) 1198891199091015840+ 1)

minus119902(119902minus1)

sdot exp (minus119892(119894)[119909]) 119889119909

(42)

but this way does not correspond to appearance of the aboveand below estimates although it is possible to prove that theseiterations with the initial approximation

119892(0)

= 0 (43)

converge to solutionThe procedure for 120579 and 119892 as the separate functions which

was formulated above will lead under the initial approxima-tions

120579(0)

= 1

119892(0)

= 0

(44)

to the chains of inequalities

119892(0)

lt 119892(2)

lt sdot sdot sdot lt 119892 lt sdot sdot sdot lt 119892(3)

lt 119892(1)

120579(1)

lt 120579(3)

lt sdot sdot sdot lt 120579 lt sdot sdot sdot lt 120579(2)

lt 120579(0)

(45)

Parameter 119860 is considered here as constant value for alliterations and the concrete value of 119860 will be determined onthe basis of the last (precise) iteration

It is useful to calculate the total number of supercriticalembryos as

119873tot sim int

infin

0

120579119902[119909] exp (minus119892 [119909]) 119889119909 (46)

On the basis of iteration procedure one can calculate

119873tot(119894+1) sim int

infin

0

120579119902

(119894)[119909] exp (minus119892

(119894)[119909]) 119889119909 (47)

The calculation of iterations gives

119892(0)

=

1199114

4

120579(1)

= ((119902 minus 1)119860119911 + 1)minus1(119902minus1)

(48)

Then

119873tot(2) = int

infin

0

((119902 minus 1)119860119911 + 1)minus119902(119902minus1) exp(minus119911

4

4

)119889119911 (49)

and this can be expressed through the hypergeometric func-tion

119873tot(2) = 2(minus92+2(119902(119902minus1)))

(119902 + 119860)minus119902(119902minus1)

(

1

3

sdot 2minus(52)(119902(119902minus1))+12

1205874(119902 + 119860)

3

(

1

119902 + 119860

)

minus119902(119902minus1)

119867([

3

4

+

1

4

119902

119902 minus 1

1 +

1

4

119902

119902 minus 1

5

4

+

1

4

119902

119902 minus 1

] [

3

2

7

4

5

4

]

minus

1

4

(119902 + 119860)4

)

sdot

sec ((14) 120587 (119902 (119902 minus 1))) Γ (3 + 119902 (119902 minus 1))

Γ (32 + (14) (119902 (119902 minus 1)))

minus 2(1minus(52)(119902(119902minus1)))

1205874(119902 + 119860)

2

(

1

119902 + 119860

)

minus119902(119902minus1)

119867([

3

4

+

1

4

119902

119902 minus 1

1 +

1

4

119902

119902 minus 1

1

2

+

1

4

119902

119902 minus 1

] [

3

2

3

4

5

4

]

minus

1

4

(119902 + 119860)4

)

sdot

csc ((14) 120587 + (14) 120587 (119902 (119902 minus 1))) Γ (2 + 119902 (119902 minus 1))

Γ (54 + (14) (119902 (119902 minus 1)))

+ 2(52minus(52)(119902(119902minus1)))

1205874(119902 + 119860)(

1

119902 + 119860

)

minus119902(119902minus1)

119867([

3

4

+

1

4

119902

119902 minus 1

1

4

+

1

4

119902

119902 minus 1

1

2

+

1

4

119902

119902 minus 1

] [

3

4

5

4

1

2

]

minus

1

4

(119902 + 119860)4

)

sdot

csc ((14) 120587 (119902 (119902 minus 1))) Γ (1 + 119902 (119902 minus 1))

Γ (1 + (14) (119902 (119902 minus 1)))

+ 2(3minus(52)(119902(119902minus1)))

1205874(

1

119902 + 119860

)

minus119902(119902minus1)

119867([

1

4

119902

119902 minus 1

1

4

+

1

4

119902

119902 minus 1

1

2

+

1

4

119902

119902 minus 1

] [

3

4

1

2

1

4

] minus

1

4

(119902 + 119860)4

)

sdot

sec ((14) 120587 + (14) 120587 (119902 (119902 minus 1))) Γ (119902 (119902 minus 1))

Γ (34 + (14) (119902 (119902 minus 1)))

+ 2(132minus2(119902(119902minus1)))

1205873(minus

1

2

+

1

4

119902

119902 minus 1

) (119902 + 119860)

Advances in Physical Chemistry 7

sdot 119867([1

3

4

1

2

1

4

] [

5

4

minus

1

4

119902

119902 minus 1

minus

1

4

119902

119902 minus 1

+

1

2

minus

1

4

119902

119902 minus 1

+

3

4

minus

1

4

119902

119902 minus 1

+ 1] minus

1

4

(119902 + 119860)4

)Γ(minus2

+

119902

119902 minus 1

))(1205873Γ(

119902

119902 minus 1

))

minus1

(50)

where Γ is the gamma-function and119867 is the hypergeometricfunction

It is possible to show analytically that this expression israther accurate The error is less than one percent Despitethe known form it is rather difficult to perform calculationsaccording to this expression and it is necessary to simplify thecalculations

Again we use the property of the avalanche consumptionHere one can show that 120579 is the decreasing (or at least notrapidly increasing) function of time (or of the size of theinitial embryo)Then one can see the cut-off type of the func-tion exp(minus119892) and the possibility of truncation of the function120579119902 exp(minus119892(119911)) at 119905cut The value of 119905cut can be chosen asthe root of equation 119892 = 1 Then the spectrum of sizes

119891 sim 120579119902[119909] exp (minus119892 [119909]) (51)

can be approximated as

119891 sim ((119902 minus 1)119860119911 + 1)minus119902(119902minus1) (52)

for 119911 lt 119911cut where 119911cut is the size corresponding to 119905cut Forother 119911 the spectrum is zero

The number of embryos can be easily calculated

119873tot = int

119911

0

((119902 minus 1)119860119909 + 1)minus119902(119902minus1)

119889119909

= minus

(119860119911119902 minus 119860119911 + 1)minus1(119902minus1)

minus 1

119860

(53)

This solves the formal problem of our investigationTo show the consistency of the presented approach we

have to investigate the inclusion of the119898-mers consumptionkinetics into the general nucleation Here one can find atleast two aspects of the problem The first aspect is the inputof 119898-mers in the overcoming of activation barrier and thecorresponding input in the stationary rate of nucleation Thesecond aspect is the consumption of the119898-mers by the near-critical and precritical embryos (this is the first type) andby the supercritical embryos (the second type) The rate ofconsumption will be different and their manifestation in theglobal kinetics 119904 will be described by different ways

To clarify the first aspect we rewrite the stationary rate ofnucleation as

119869 sim exp (minus119902 ln (119899 [119898])) exp (minus119865119888) (54)

One has to stress that here 119865119888is the free energy of the critical

embryo formation on 119902 119898-mers This implies that the work

of collecting of119898-mers is considered separatelyThis work inapproximation of ideal gas of119898-mers is

1198650= minus119902 ln (119899 [119898]) (55)

At the same time we have to note that in the embryocontaining ]

119888molecules there are only ]

119888minus 119902119898 molecules

coming from monomers Then 1198650has to be written as

1198650= minus119887 (]

119888minus 119902119898) + 120574119860 (56)

where 119887 is the excess of the chemical potential for monomers120574 is the renormalized surface tension and 119860 is the surfacearea of the embryo It is clear that in the determination of thesurface area it is necessary to account that the part of the vol-ume inside the embryos surface is occupied by the moleculescoming from119898-mersWhen the surface of the embryo simplysurrounds the embryo one has to write

119860 = 4120587(

3V119897(]1119888+ 119902119898)

4120587

)

23

(57)

where V119897is the volume per onemolecule in a liquid phase and

]1119888is the number of molecules coming from monomersIn the state of equilibrium the chemical potential of 119898-

mer has to be equal to 119898 chemical potentials of monomers(here we neglect the surface term or consider the deformedchemical potential) So

119899 [119898]

119899 [119898 119904 = 0]

= exp (119887119898) (58)

and we see that the total free energy is

119888= 119865119888minus 119902119887119898 = minus]

119888119887 + 120574119860 (59)

and the total free energy coincides with the classical expres-sion

One has to stress that the parameter Γ in the previoussimplification of the dependence of the free energy on super-saturation (or the dependence of the stationary rate of nucle-ation on the supersaturation) is the number of moleculescoming from monomers multiplied on Φ

lowast(Φlowast

+ 1)that is

Γ 997888rarr Γ1=

Φlowast]1119888

(Φlowast+ 1)

(60)

Γ1is Φlowast119889119865119888119889120577|120577=Φlowast

If we calculate the derivative of the total free energy

119888we

get

Φlowast

119889119888

119889120577

100381610038161003816100381610038161003816100381610038161003816120577=Φlowast

= Γ1+ 119902119898

119889119887

119889120577

(61)

or

Φlowast

119889119888

119889120577

100381610038161003816100381610038161003816100381610038161003816120577=Φlowast

= Γ1+ Γ119898 (62)

8 Advances in Physical Chemistry

where

Γ119898=

120577119902119898

(120577 + 1)

10038161003816100381610038161003816100381610038161003816120577=Φlowast

(63)

As a result

Φlowast

119889119888

119889120577

100381610038161003816100381610038161003816100381610038161003816120577=Φlowast

= Γ

Γ =

120577]119888

(120577 + 1)

10038161003816100381610038161003816100381610038161003816120577=Φlowast

(64)

We will call this property the separation of Γ It takes placeonly in the first derivatives of the free energies and onlybecause the surface term does not act on the derivative of thefree energy The surface term in the free energy only changesthe number of molecules in the critical embryo and has nodirect influence on derivative

So we see that in the free energy there is the directcorrespondence between the119898-mers picture and monomerspicture

One can propose another picture when the number of119898-mers necessary to form the critical embryo is proportionalto the number of molecules inside the critical embryo In theenormously big critical embryos namely thismechanismhasto be considered Then the picture is equivalent to the binarynucleation with two vapors (the vapor of monomers and thevapor of119898-mers)

The second aspect is connected with the mechanism ofconsumption Here one has to confess that there are two prin-cipal mechanisms of the substance consumption The firstmechanism is ldquoone embryo-fixed number of clusters (mono-mers or 119898-mers)rdquo The second mechanism is ldquoone embryo-growing number of clusters (monomers and 119898-mers) corre-sponding to the law of the regular growthrdquo

Rigorously speaking we have to write for each type ofactive monomers two types of consumption and the balanceequations With the help of exponential approximationsmentioned above one can unify these mechanisms in one lawof growth Ordinarily when the first mechanism is importantthe second is absent When the second mechanism acts itmeans that there is enough clusters of such type to neglect theexhaustion of seeds We will call this effect as the dominatingpreference of one mechanism

Despite the mentioned dominating preference propertythe simultaneous existence of several mechanisms can not beexcluded as a specific case But it is clear that with the helpof approaches developed here it is possible to construct thetheoretical description of any process with different mecha-nisms of formation different mechanisms of substance con-sumption and different relaxation processes correspondingto bounds or connections between the resources of clusters ofdifferent sizes

The different relations between the characteristic timesof the mentioned processes make the processes of nucleationrather different The size spectrums are very various also Todescribe these situations one has to demonstrate the powerof the approaches making the base of theoretical description

In every situation the leading idea will be the avalanche con-sumption of metastability in different manifestations of thisproperty

4 Main Results

This paper contains at least two important derivations con-cerning the kinetics of decay with account of the relaxationprocesses and the kinetics of decay when the119898-mers nucleationis important Both cases are important and one can see thataccount of these processes radically changes the numericalvalues of parameters of the nucleations process But one hasto confess that the qualitative picture remains similar to thetraditional case The last does not mean that nothing in theform of the particles size distribution is changed Contrarilythe form of the size spectrum will be absolutely another Butthe property of the avalanche cut-off of the pseudo-homo-geneous size spectrum remains The last property was veryproductive in construction of the above presented procedure

It would be interesting to analyze these situations in thecase of the smooth external influence on the system whichcauses nucleation that is in the case of dynamic conditionsAlso it would be interesting to see the influence of the relax-ation processes in the overcoming of the near-critical regionin the process of the binary nucleation Certainly these per-spectives make the subjects of separate publications

Beside the direct construction of the nucleation globalkinetics description we managed to extract some importantfeatures of the effect of119898-mers inclusion into the nucleationprocess One can mention here the property of separation ofΓ and the correspondence between the monomers picture and119898-mers picture These properties are important enough to bementioned in the general theory of nucleation

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] C T R Wilson ldquoCondensation of water vapour in the presenceof dust-free air and other gasesrdquo Philosophical Transactions ofthe Royal Society A Mathematical Physical and Engineering Sci-ences vol 189 pp 265ndash307 1897

[2] C T RWilson ldquoOn the condensation nuclei produced in gasesby the action of roentgen rays uranium rays ultraviolet rays andother agentsrdquoPhilosophical Transactions A vol 192 no 9 article403 1899

[3] C T RWilson ldquoOn the comparative efficiency as condensationnuclei of positively and negatively charged ionsrdquo PhilosophicalTransactions of the Royal Society A Containing Papers of aMath-ematical or Physical Character vol 193 pp 289ndash308 1900

[4] A Benmouna R Benmouna M R Bockstaller and I FHakem ldquoSelf-organization schemes towards thermodynamicstable bulk heterojunction morphologies a perspective onfuture fabrication strategies of polymer photovoltaic architec-turesrdquo Advances in Physical Chemistry vol 2013 Article ID948189 8 pages 2013

Advances in Physical Chemistry 9

[5] Y Kazzi H Awada and M Nardin ldquoAdhesion on nanoorga-nized multilayers surface thermodynamics and local energydissipationrdquo Advances in Physical Chemistry vol 2010 ArticleID 502709 11 pages 2010

[6] A I Burshtein ldquoContact and distant luminescence quenchingin solutionsrdquo Advances in Physical Chemistry vol 2009 ArticleID 214219 34 pages 2009

[7] A C Zettlemoyer Nucleation Dekker New York NY USA1969

[8] F F Abraham Homogeneous Nucleation Theory AcademicNew York NY USA 1974

[9] M Volmer and A Weber ldquoKeimbildung in ubersattigtenGebildenrdquo Zeitschrift fur Physikalische Chemie vol 119 pp 277ndash301 1925

[10] R Becker andW Doring ldquoKinetische behandlung der keimbil-dung in uberattigten dampfernrdquo Annals of Physics vol 24 pp719ndash752 1935

[11] J Frenkel ldquoA general theory of heterophase fluctuations andpretransition phenomenardquoThe Journal of Chemical Physics vol7 no 7 pp 538ndash547 1939

[12] J B Zeldovich ldquoOn the theory of new phase formation cavi-tationrdquo Acta Physicochimica USSR vol 18 pp 1ndash22 1943

[13] L Farkas ldquoThe velocity of nucleus formation in superheatedvaporsrdquo Zeitschrift fur Physikalische Chemie vol 125 pp 236ndash240 1927

[14] J Lothe and G M Pound ldquoReconsiderations of nucleationtheoryrdquo The Journal of Chemical Physics vol 36 no 8 article2080 1962

[15] H Reiss J L Katz and E R Cohen ldquoTranslation-rotation para-dox in the theory of nucleationrdquoThe Journal of Chemical Phys-ics vol 48 no 12 pp 5553ndash5560 1968

[16] H Reiss ldquoTreatment of droplike clusters by means of the clas-sical phase integral in nucleation theoryrdquo Journal of StatisticalPhysics vol 2 no 1 pp 83ndash104 1970

[17] W G Courtney ldquoRemarks on homogeneous nucleationrdquo TheJournal of Chemical Physics vol 35 no 6 pp 2249ndash2250 1961

[18] M Blander and J L Katz ldquoThe thermodynamics of cluster for-mation in nucleation theoryrdquo Journal of Statistical Physics vol 4no 1 pp 55ndash59 1972

[19] J L Katz and H Weidersich ldquoNucleation theory without Max-well Demonsrdquo Journal of Colloid and Interface Science vol 61no 2 pp 351ndash355 1977

[20] J L Katz and M D Donohue ldquoA kinetic approach to homo-geneous nucleation theoryrdquo in Advances in Chemical Physics IPrigogine and S A Rice Eds vol 40 chapter 3 pp 137ndash155Wiley 1979

[21] A Dillmann and G E A Meier ldquoHomogeneous nucleation ofsupersaturated vaporsrdquo Chemical Physics Letters vol 160 no 1pp 71ndash74 1989

[22] A Dillmann and G E A Meier ldquoA refined droplet approach tothe problem of homogeneous nucleation from the vapor phaserdquoThe Journal of Chemical Physics vol 94 no 5 pp 3872ndash38841991

[23] J W Cahn and J E Hillard ldquoFree energy of a nonuniformsystem III Nucleation in a two-component incompressiblefluidrdquoTheJournal of Chemical Physics vol 31 no 3 pp 688ndash6991959

[24] D Gebauer and H Colfen ldquoPrenucleation clusters and non-classical nucleationrdquo Nano Today vol 6 no 6 pp 564ndash5842011

[25] D Sept and J A McCammon ldquoThermodynamics and kineticsof actin filament nucleationrdquo Biophysical Journal vol 81 no 2pp 667ndash674 2001

[26] A S Myerson and B L Trout ldquoNucleation from solutionrdquo Sci-ence vol 341 no 6148 pp 855ndash856 2013

[27] D Zahn ldquoThermodynamics and kinetics of prenucleation clus-ters classical and non-classical nucleationrdquo ChemPhysChemvol 16 no 10 pp 2069ndash2075 2015

[28] H Deng S Wang X Wang et al ldquoTwo competitive nucle-ation mechanisms of calcium carbonate biomineralization inresponse to surface functionality in low calcium ion concentra-tion solutionrdquo Regenerative Biomaterials vol 2 no 3 pp 187ndash195 2015

[29] D Kozma and G Toth ldquoMicroscopic rate constants of crystalgrowth from molecular dynamic simulations combined withmetadynamicsrdquo Advances in Physical Chemistry vol 2012Article ID 135172 10 pages 2012

[30] S Fakruddin C Srinivasu B R Venkateswara Rao and KNarendra ldquoExcess transport properties of binary mixtures ofquinoline with xylenes at different temperaturesrdquo Advances inPhysical Chemistry vol 2012 Article ID 324098 9 pages 2012

[31] A A Lushnikov ldquoEvolution of coagulating systemsrdquo Journal ofColloid and Interface Science vol 45 no 3 pp 549ndash556 1973

[32] I M Lifshic and V V Slezov ldquoKinetics of diffusive decomposi-tion of supersaturated solid solutionsrdquo Soviet PhysicsmdashJournalof Experimental and Theoretical Physics vol 35 pp 331ndash3391959

[33] IM Lifshic andV V Slezov ldquoThe kinetics of precipitation fromsupersaturated solid solutionsrdquo Journal of Physics andChemistryof Solids vol 19 no 1-2 pp 35ndash50 1961

[34] F M Kuni and A P Grinin ldquoKinetics of homogeneous conden-sationat the stage of formation of the main quantity of a newphaserdquo Colloid Journal of the USSR vol 46 p 460 1984

[35] F M Kuni T Y Novozhilova and I A Terentiev ldquoKineticequation of heterogenous condensationrdquo Letters in Mathemati-cal Physics vol 14 no 2 pp 161ndash167 1987

[36] V Kurasov ldquoDecay of metastable multicomponentmixturerdquohttparxivorgabscond-mat9310005

[37] V B Kurasov ldquoKinetics of heterogeneous condensation underdynamic conditionsrdquo Physical Review E vol 49 no 5 pp 3948ndash3955 1994

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Inorganic ChemistryInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal ofPhotoenergy

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Carbohydrate Chemistry

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

Physical Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom

Analytical Methods in Chemistry

Journal of

Volume 2014

Bioinorganic Chemistry and ApplicationsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

SpectroscopyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Medicinal ChemistryInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chromatography Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied ChemistryJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Theoretical ChemistryJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Spectroscopy

Analytical ChemistryInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Quantum Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Organic Chemistry International

ElectrochemistryInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CatalystsJournal of

Advances in Physical Chemistry 7

sdot 119867([1

3

4

1

2

1

4

] [

5

4

minus

1

4

119902

119902 minus 1

minus

1

4

119902

119902 minus 1

+

1

2

minus

1

4

119902

119902 minus 1

+

3

4

minus

1

4

119902

119902 minus 1

+ 1] minus

1

4

(119902 + 119860)4

)Γ(minus2

+

119902

119902 minus 1

))(1205873Γ(

119902

119902 minus 1

))

minus1

(50)

where Γ is the gamma-function and119867 is the hypergeometricfunction

It is possible to show analytically that this expression israther accurate The error is less than one percent Despitethe known form it is rather difficult to perform calculationsaccording to this expression and it is necessary to simplify thecalculations

Again we use the property of the avalanche consumptionHere one can show that 120579 is the decreasing (or at least notrapidly increasing) function of time (or of the size of theinitial embryo)Then one can see the cut-off type of the func-tion exp(minus119892) and the possibility of truncation of the function120579119902 exp(minus119892(119911)) at 119905cut The value of 119905cut can be chosen asthe root of equation 119892 = 1 Then the spectrum of sizes

119891 sim 120579119902[119909] exp (minus119892 [119909]) (51)

can be approximated as

119891 sim ((119902 minus 1)119860119911 + 1)minus119902(119902minus1) (52)

for 119911 lt 119911cut where 119911cut is the size corresponding to 119905cut Forother 119911 the spectrum is zero

The number of embryos can be easily calculated

119873tot = int

119911

0

((119902 minus 1)119860119909 + 1)minus119902(119902minus1)

119889119909

= minus

(119860119911119902 minus 119860119911 + 1)minus1(119902minus1)

minus 1

119860

(53)

This solves the formal problem of our investigationTo show the consistency of the presented approach we

have to investigate the inclusion of the119898-mers consumptionkinetics into the general nucleation Here one can find atleast two aspects of the problem The first aspect is the inputof 119898-mers in the overcoming of activation barrier and thecorresponding input in the stationary rate of nucleation Thesecond aspect is the consumption of the119898-mers by the near-critical and precritical embryos (this is the first type) andby the supercritical embryos (the second type) The rate ofconsumption will be different and their manifestation in theglobal kinetics 119904 will be described by different ways

To clarify the first aspect we rewrite the stationary rate ofnucleation as

119869 sim exp (minus119902 ln (119899 [119898])) exp (minus119865119888) (54)

One has to stress that here 119865119888is the free energy of the critical

embryo formation on 119902 119898-mers This implies that the work

of collecting of119898-mers is considered separatelyThis work inapproximation of ideal gas of119898-mers is

1198650= minus119902 ln (119899 [119898]) (55)

At the same time we have to note that in the embryocontaining ]

119888molecules there are only ]

119888minus 119902119898 molecules

coming from monomers Then 1198650has to be written as

1198650= minus119887 (]

119888minus 119902119898) + 120574119860 (56)

where 119887 is the excess of the chemical potential for monomers120574 is the renormalized surface tension and 119860 is the surfacearea of the embryo It is clear that in the determination of thesurface area it is necessary to account that the part of the vol-ume inside the embryos surface is occupied by the moleculescoming from119898-mersWhen the surface of the embryo simplysurrounds the embryo one has to write

119860 = 4120587(

3V119897(]1119888+ 119902119898)

4120587

)

23

(57)

where V119897is the volume per onemolecule in a liquid phase and

]1119888is the number of molecules coming from monomersIn the state of equilibrium the chemical potential of 119898-

mer has to be equal to 119898 chemical potentials of monomers(here we neglect the surface term or consider the deformedchemical potential) So

119899 [119898]

119899 [119898 119904 = 0]

= exp (119887119898) (58)

and we see that the total free energy is

119888= 119865119888minus 119902119887119898 = minus]

119888119887 + 120574119860 (59)

and the total free energy coincides with the classical expres-sion

One has to stress that the parameter Γ in the previoussimplification of the dependence of the free energy on super-saturation (or the dependence of the stationary rate of nucle-ation on the supersaturation) is the number of moleculescoming from monomers multiplied on Φ

lowast(Φlowast

+ 1)that is

Γ 997888rarr Γ1=

Φlowast]1119888

(Φlowast+ 1)

(60)

Γ1is Φlowast119889119865119888119889120577|120577=Φlowast

If we calculate the derivative of the total free energy

119888we

get

Φlowast

119889119888

119889120577

100381610038161003816100381610038161003816100381610038161003816120577=Φlowast

= Γ1+ 119902119898

119889119887

119889120577

(61)

or

Φlowast

119889119888

119889120577

100381610038161003816100381610038161003816100381610038161003816120577=Φlowast

= Γ1+ Γ119898 (62)

8 Advances in Physical Chemistry

where

Γ119898=

120577119902119898

(120577 + 1)

10038161003816100381610038161003816100381610038161003816120577=Φlowast

(63)

As a result

Φlowast

119889119888

119889120577

100381610038161003816100381610038161003816100381610038161003816120577=Φlowast

= Γ

Γ =

120577]119888

(120577 + 1)

10038161003816100381610038161003816100381610038161003816120577=Φlowast

(64)

We will call this property the separation of Γ It takes placeonly in the first derivatives of the free energies and onlybecause the surface term does not act on the derivative of thefree energy The surface term in the free energy only changesthe number of molecules in the critical embryo and has nodirect influence on derivative

So we see that in the free energy there is the directcorrespondence between the119898-mers picture and monomerspicture

One can propose another picture when the number of119898-mers necessary to form the critical embryo is proportionalto the number of molecules inside the critical embryo In theenormously big critical embryos namely thismechanismhasto be considered Then the picture is equivalent to the binarynucleation with two vapors (the vapor of monomers and thevapor of119898-mers)

The second aspect is connected with the mechanism ofconsumption Here one has to confess that there are two prin-cipal mechanisms of the substance consumption The firstmechanism is ldquoone embryo-fixed number of clusters (mono-mers or 119898-mers)rdquo The second mechanism is ldquoone embryo-growing number of clusters (monomers and 119898-mers) corre-sponding to the law of the regular growthrdquo

Rigorously speaking we have to write for each type ofactive monomers two types of consumption and the balanceequations With the help of exponential approximationsmentioned above one can unify these mechanisms in one lawof growth Ordinarily when the first mechanism is importantthe second is absent When the second mechanism acts itmeans that there is enough clusters of such type to neglect theexhaustion of seeds We will call this effect as the dominatingpreference of one mechanism

Despite the mentioned dominating preference propertythe simultaneous existence of several mechanisms can not beexcluded as a specific case But it is clear that with the helpof approaches developed here it is possible to construct thetheoretical description of any process with different mecha-nisms of formation different mechanisms of substance con-sumption and different relaxation processes correspondingto bounds or connections between the resources of clusters ofdifferent sizes

The different relations between the characteristic timesof the mentioned processes make the processes of nucleationrather different The size spectrums are very various also Todescribe these situations one has to demonstrate the powerof the approaches making the base of theoretical description

In every situation the leading idea will be the avalanche con-sumption of metastability in different manifestations of thisproperty

4 Main Results

This paper contains at least two important derivations con-cerning the kinetics of decay with account of the relaxationprocesses and the kinetics of decay when the119898-mers nucleationis important Both cases are important and one can see thataccount of these processes radically changes the numericalvalues of parameters of the nucleations process But one hasto confess that the qualitative picture remains similar to thetraditional case The last does not mean that nothing in theform of the particles size distribution is changed Contrarilythe form of the size spectrum will be absolutely another Butthe property of the avalanche cut-off of the pseudo-homo-geneous size spectrum remains The last property was veryproductive in construction of the above presented procedure

It would be interesting to analyze these situations in thecase of the smooth external influence on the system whichcauses nucleation that is in the case of dynamic conditionsAlso it would be interesting to see the influence of the relax-ation processes in the overcoming of the near-critical regionin the process of the binary nucleation Certainly these per-spectives make the subjects of separate publications

Beside the direct construction of the nucleation globalkinetics description we managed to extract some importantfeatures of the effect of119898-mers inclusion into the nucleationprocess One can mention here the property of separation ofΓ and the correspondence between the monomers picture and119898-mers picture These properties are important enough to bementioned in the general theory of nucleation

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] C T R Wilson ldquoCondensation of water vapour in the presenceof dust-free air and other gasesrdquo Philosophical Transactions ofthe Royal Society A Mathematical Physical and Engineering Sci-ences vol 189 pp 265ndash307 1897

[2] C T RWilson ldquoOn the condensation nuclei produced in gasesby the action of roentgen rays uranium rays ultraviolet rays andother agentsrdquoPhilosophical Transactions A vol 192 no 9 article403 1899

[3] C T RWilson ldquoOn the comparative efficiency as condensationnuclei of positively and negatively charged ionsrdquo PhilosophicalTransactions of the Royal Society A Containing Papers of aMath-ematical or Physical Character vol 193 pp 289ndash308 1900

[4] A Benmouna R Benmouna M R Bockstaller and I FHakem ldquoSelf-organization schemes towards thermodynamicstable bulk heterojunction morphologies a perspective onfuture fabrication strategies of polymer photovoltaic architec-turesrdquo Advances in Physical Chemistry vol 2013 Article ID948189 8 pages 2013

Advances in Physical Chemistry 9

[5] Y Kazzi H Awada and M Nardin ldquoAdhesion on nanoorga-nized multilayers surface thermodynamics and local energydissipationrdquo Advances in Physical Chemistry vol 2010 ArticleID 502709 11 pages 2010

[6] A I Burshtein ldquoContact and distant luminescence quenchingin solutionsrdquo Advances in Physical Chemistry vol 2009 ArticleID 214219 34 pages 2009

[7] A C Zettlemoyer Nucleation Dekker New York NY USA1969

[8] F F Abraham Homogeneous Nucleation Theory AcademicNew York NY USA 1974

[9] M Volmer and A Weber ldquoKeimbildung in ubersattigtenGebildenrdquo Zeitschrift fur Physikalische Chemie vol 119 pp 277ndash301 1925

[10] R Becker andW Doring ldquoKinetische behandlung der keimbil-dung in uberattigten dampfernrdquo Annals of Physics vol 24 pp719ndash752 1935

[11] J Frenkel ldquoA general theory of heterophase fluctuations andpretransition phenomenardquoThe Journal of Chemical Physics vol7 no 7 pp 538ndash547 1939

[12] J B Zeldovich ldquoOn the theory of new phase formation cavi-tationrdquo Acta Physicochimica USSR vol 18 pp 1ndash22 1943

[13] L Farkas ldquoThe velocity of nucleus formation in superheatedvaporsrdquo Zeitschrift fur Physikalische Chemie vol 125 pp 236ndash240 1927

[14] J Lothe and G M Pound ldquoReconsiderations of nucleationtheoryrdquo The Journal of Chemical Physics vol 36 no 8 article2080 1962

[15] H Reiss J L Katz and E R Cohen ldquoTranslation-rotation para-dox in the theory of nucleationrdquoThe Journal of Chemical Phys-ics vol 48 no 12 pp 5553ndash5560 1968

[16] H Reiss ldquoTreatment of droplike clusters by means of the clas-sical phase integral in nucleation theoryrdquo Journal of StatisticalPhysics vol 2 no 1 pp 83ndash104 1970

[17] W G Courtney ldquoRemarks on homogeneous nucleationrdquo TheJournal of Chemical Physics vol 35 no 6 pp 2249ndash2250 1961

[18] M Blander and J L Katz ldquoThe thermodynamics of cluster for-mation in nucleation theoryrdquo Journal of Statistical Physics vol 4no 1 pp 55ndash59 1972

[19] J L Katz and H Weidersich ldquoNucleation theory without Max-well Demonsrdquo Journal of Colloid and Interface Science vol 61no 2 pp 351ndash355 1977

[20] J L Katz and M D Donohue ldquoA kinetic approach to homo-geneous nucleation theoryrdquo in Advances in Chemical Physics IPrigogine and S A Rice Eds vol 40 chapter 3 pp 137ndash155Wiley 1979

[21] A Dillmann and G E A Meier ldquoHomogeneous nucleation ofsupersaturated vaporsrdquo Chemical Physics Letters vol 160 no 1pp 71ndash74 1989

[22] A Dillmann and G E A Meier ldquoA refined droplet approach tothe problem of homogeneous nucleation from the vapor phaserdquoThe Journal of Chemical Physics vol 94 no 5 pp 3872ndash38841991

[23] J W Cahn and J E Hillard ldquoFree energy of a nonuniformsystem III Nucleation in a two-component incompressiblefluidrdquoTheJournal of Chemical Physics vol 31 no 3 pp 688ndash6991959

[24] D Gebauer and H Colfen ldquoPrenucleation clusters and non-classical nucleationrdquo Nano Today vol 6 no 6 pp 564ndash5842011

[25] D Sept and J A McCammon ldquoThermodynamics and kineticsof actin filament nucleationrdquo Biophysical Journal vol 81 no 2pp 667ndash674 2001

[26] A S Myerson and B L Trout ldquoNucleation from solutionrdquo Sci-ence vol 341 no 6148 pp 855ndash856 2013

[27] D Zahn ldquoThermodynamics and kinetics of prenucleation clus-ters classical and non-classical nucleationrdquo ChemPhysChemvol 16 no 10 pp 2069ndash2075 2015

[28] H Deng S Wang X Wang et al ldquoTwo competitive nucle-ation mechanisms of calcium carbonate biomineralization inresponse to surface functionality in low calcium ion concentra-tion solutionrdquo Regenerative Biomaterials vol 2 no 3 pp 187ndash195 2015

[29] D Kozma and G Toth ldquoMicroscopic rate constants of crystalgrowth from molecular dynamic simulations combined withmetadynamicsrdquo Advances in Physical Chemistry vol 2012Article ID 135172 10 pages 2012

[30] S Fakruddin C Srinivasu B R Venkateswara Rao and KNarendra ldquoExcess transport properties of binary mixtures ofquinoline with xylenes at different temperaturesrdquo Advances inPhysical Chemistry vol 2012 Article ID 324098 9 pages 2012

[31] A A Lushnikov ldquoEvolution of coagulating systemsrdquo Journal ofColloid and Interface Science vol 45 no 3 pp 549ndash556 1973

[32] I M Lifshic and V V Slezov ldquoKinetics of diffusive decomposi-tion of supersaturated solid solutionsrdquo Soviet PhysicsmdashJournalof Experimental and Theoretical Physics vol 35 pp 331ndash3391959

[33] IM Lifshic andV V Slezov ldquoThe kinetics of precipitation fromsupersaturated solid solutionsrdquo Journal of Physics andChemistryof Solids vol 19 no 1-2 pp 35ndash50 1961

[34] F M Kuni and A P Grinin ldquoKinetics of homogeneous conden-sationat the stage of formation of the main quantity of a newphaserdquo Colloid Journal of the USSR vol 46 p 460 1984

[35] F M Kuni T Y Novozhilova and I A Terentiev ldquoKineticequation of heterogenous condensationrdquo Letters in Mathemati-cal Physics vol 14 no 2 pp 161ndash167 1987

[36] V Kurasov ldquoDecay of metastable multicomponentmixturerdquohttparxivorgabscond-mat9310005

[37] V B Kurasov ldquoKinetics of heterogeneous condensation underdynamic conditionsrdquo Physical Review E vol 49 no 5 pp 3948ndash3955 1994

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Inorganic ChemistryInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal ofPhotoenergy

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Carbohydrate Chemistry

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

Physical Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom

Analytical Methods in Chemistry

Journal of

Volume 2014

Bioinorganic Chemistry and ApplicationsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

SpectroscopyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Medicinal ChemistryInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chromatography Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied ChemistryJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Theoretical ChemistryJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Spectroscopy

Analytical ChemistryInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Quantum Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Organic Chemistry International

ElectrochemistryInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CatalystsJournal of

8 Advances in Physical Chemistry

where

Γ119898=

120577119902119898

(120577 + 1)

10038161003816100381610038161003816100381610038161003816120577=Φlowast

(63)

As a result

Φlowast

119889119888

119889120577

100381610038161003816100381610038161003816100381610038161003816120577=Φlowast

= Γ

Γ =

120577]119888

(120577 + 1)

10038161003816100381610038161003816100381610038161003816120577=Φlowast

(64)

We will call this property the separation of Γ It takes placeonly in the first derivatives of the free energies and onlybecause the surface term does not act on the derivative of thefree energy The surface term in the free energy only changesthe number of molecules in the critical embryo and has nodirect influence on derivative

So we see that in the free energy there is the directcorrespondence between the119898-mers picture and monomerspicture

One can propose another picture when the number of119898-mers necessary to form the critical embryo is proportionalto the number of molecules inside the critical embryo In theenormously big critical embryos namely thismechanismhasto be considered Then the picture is equivalent to the binarynucleation with two vapors (the vapor of monomers and thevapor of119898-mers)

The second aspect is connected with the mechanism ofconsumption Here one has to confess that there are two prin-cipal mechanisms of the substance consumption The firstmechanism is ldquoone embryo-fixed number of clusters (mono-mers or 119898-mers)rdquo The second mechanism is ldquoone embryo-growing number of clusters (monomers and 119898-mers) corre-sponding to the law of the regular growthrdquo

Rigorously speaking we have to write for each type ofactive monomers two types of consumption and the balanceequations With the help of exponential approximationsmentioned above one can unify these mechanisms in one lawof growth Ordinarily when the first mechanism is importantthe second is absent When the second mechanism acts itmeans that there is enough clusters of such type to neglect theexhaustion of seeds We will call this effect as the dominatingpreference of one mechanism

Despite the mentioned dominating preference propertythe simultaneous existence of several mechanisms can not beexcluded as a specific case But it is clear that with the helpof approaches developed here it is possible to construct thetheoretical description of any process with different mecha-nisms of formation different mechanisms of substance con-sumption and different relaxation processes correspondingto bounds or connections between the resources of clusters ofdifferent sizes

The different relations between the characteristic timesof the mentioned processes make the processes of nucleationrather different The size spectrums are very various also Todescribe these situations one has to demonstrate the powerof the approaches making the base of theoretical description

In every situation the leading idea will be the avalanche con-sumption of metastability in different manifestations of thisproperty

4 Main Results

This paper contains at least two important derivations con-cerning the kinetics of decay with account of the relaxationprocesses and the kinetics of decay when the119898-mers nucleationis important Both cases are important and one can see thataccount of these processes radically changes the numericalvalues of parameters of the nucleations process But one hasto confess that the qualitative picture remains similar to thetraditional case The last does not mean that nothing in theform of the particles size distribution is changed Contrarilythe form of the size spectrum will be absolutely another Butthe property of the avalanche cut-off of the pseudo-homo-geneous size spectrum remains The last property was veryproductive in construction of the above presented procedure

It would be interesting to analyze these situations in thecase of the smooth external influence on the system whichcauses nucleation that is in the case of dynamic conditionsAlso it would be interesting to see the influence of the relax-ation processes in the overcoming of the near-critical regionin the process of the binary nucleation Certainly these per-spectives make the subjects of separate publications

Beside the direct construction of the nucleation globalkinetics description we managed to extract some importantfeatures of the effect of119898-mers inclusion into the nucleationprocess One can mention here the property of separation ofΓ and the correspondence between the monomers picture and119898-mers picture These properties are important enough to bementioned in the general theory of nucleation

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] C T R Wilson ldquoCondensation of water vapour in the presenceof dust-free air and other gasesrdquo Philosophical Transactions ofthe Royal Society A Mathematical Physical and Engineering Sci-ences vol 189 pp 265ndash307 1897

[2] C T RWilson ldquoOn the condensation nuclei produced in gasesby the action of roentgen rays uranium rays ultraviolet rays andother agentsrdquoPhilosophical Transactions A vol 192 no 9 article403 1899

[3] C T RWilson ldquoOn the comparative efficiency as condensationnuclei of positively and negatively charged ionsrdquo PhilosophicalTransactions of the Royal Society A Containing Papers of aMath-ematical or Physical Character vol 193 pp 289ndash308 1900

[4] A Benmouna R Benmouna M R Bockstaller and I FHakem ldquoSelf-organization schemes towards thermodynamicstable bulk heterojunction morphologies a perspective onfuture fabrication strategies of polymer photovoltaic architec-turesrdquo Advances in Physical Chemistry vol 2013 Article ID948189 8 pages 2013

Advances in Physical Chemistry 9

[5] Y Kazzi H Awada and M Nardin ldquoAdhesion on nanoorga-nized multilayers surface thermodynamics and local energydissipationrdquo Advances in Physical Chemistry vol 2010 ArticleID 502709 11 pages 2010

[6] A I Burshtein ldquoContact and distant luminescence quenchingin solutionsrdquo Advances in Physical Chemistry vol 2009 ArticleID 214219 34 pages 2009

[7] A C Zettlemoyer Nucleation Dekker New York NY USA1969

[8] F F Abraham Homogeneous Nucleation Theory AcademicNew York NY USA 1974

[9] M Volmer and A Weber ldquoKeimbildung in ubersattigtenGebildenrdquo Zeitschrift fur Physikalische Chemie vol 119 pp 277ndash301 1925

[10] R Becker andW Doring ldquoKinetische behandlung der keimbil-dung in uberattigten dampfernrdquo Annals of Physics vol 24 pp719ndash752 1935

[11] J Frenkel ldquoA general theory of heterophase fluctuations andpretransition phenomenardquoThe Journal of Chemical Physics vol7 no 7 pp 538ndash547 1939

[12] J B Zeldovich ldquoOn the theory of new phase formation cavi-tationrdquo Acta Physicochimica USSR vol 18 pp 1ndash22 1943

[13] L Farkas ldquoThe velocity of nucleus formation in superheatedvaporsrdquo Zeitschrift fur Physikalische Chemie vol 125 pp 236ndash240 1927

[14] J Lothe and G M Pound ldquoReconsiderations of nucleationtheoryrdquo The Journal of Chemical Physics vol 36 no 8 article2080 1962

[15] H Reiss J L Katz and E R Cohen ldquoTranslation-rotation para-dox in the theory of nucleationrdquoThe Journal of Chemical Phys-ics vol 48 no 12 pp 5553ndash5560 1968

[16] H Reiss ldquoTreatment of droplike clusters by means of the clas-sical phase integral in nucleation theoryrdquo Journal of StatisticalPhysics vol 2 no 1 pp 83ndash104 1970

[17] W G Courtney ldquoRemarks on homogeneous nucleationrdquo TheJournal of Chemical Physics vol 35 no 6 pp 2249ndash2250 1961

[18] M Blander and J L Katz ldquoThe thermodynamics of cluster for-mation in nucleation theoryrdquo Journal of Statistical Physics vol 4no 1 pp 55ndash59 1972

[19] J L Katz and H Weidersich ldquoNucleation theory without Max-well Demonsrdquo Journal of Colloid and Interface Science vol 61no 2 pp 351ndash355 1977

[20] J L Katz and M D Donohue ldquoA kinetic approach to homo-geneous nucleation theoryrdquo in Advances in Chemical Physics IPrigogine and S A Rice Eds vol 40 chapter 3 pp 137ndash155Wiley 1979

[21] A Dillmann and G E A Meier ldquoHomogeneous nucleation ofsupersaturated vaporsrdquo Chemical Physics Letters vol 160 no 1pp 71ndash74 1989

[22] A Dillmann and G E A Meier ldquoA refined droplet approach tothe problem of homogeneous nucleation from the vapor phaserdquoThe Journal of Chemical Physics vol 94 no 5 pp 3872ndash38841991

[23] J W Cahn and J E Hillard ldquoFree energy of a nonuniformsystem III Nucleation in a two-component incompressiblefluidrdquoTheJournal of Chemical Physics vol 31 no 3 pp 688ndash6991959

[24] D Gebauer and H Colfen ldquoPrenucleation clusters and non-classical nucleationrdquo Nano Today vol 6 no 6 pp 564ndash5842011

[25] D Sept and J A McCammon ldquoThermodynamics and kineticsof actin filament nucleationrdquo Biophysical Journal vol 81 no 2pp 667ndash674 2001

[26] A S Myerson and B L Trout ldquoNucleation from solutionrdquo Sci-ence vol 341 no 6148 pp 855ndash856 2013

[27] D Zahn ldquoThermodynamics and kinetics of prenucleation clus-ters classical and non-classical nucleationrdquo ChemPhysChemvol 16 no 10 pp 2069ndash2075 2015

[28] H Deng S Wang X Wang et al ldquoTwo competitive nucle-ation mechanisms of calcium carbonate biomineralization inresponse to surface functionality in low calcium ion concentra-tion solutionrdquo Regenerative Biomaterials vol 2 no 3 pp 187ndash195 2015

[29] D Kozma and G Toth ldquoMicroscopic rate constants of crystalgrowth from molecular dynamic simulations combined withmetadynamicsrdquo Advances in Physical Chemistry vol 2012Article ID 135172 10 pages 2012

[30] S Fakruddin C Srinivasu B R Venkateswara Rao and KNarendra ldquoExcess transport properties of binary mixtures ofquinoline with xylenes at different temperaturesrdquo Advances inPhysical Chemistry vol 2012 Article ID 324098 9 pages 2012

[31] A A Lushnikov ldquoEvolution of coagulating systemsrdquo Journal ofColloid and Interface Science vol 45 no 3 pp 549ndash556 1973

[32] I M Lifshic and V V Slezov ldquoKinetics of diffusive decomposi-tion of supersaturated solid solutionsrdquo Soviet PhysicsmdashJournalof Experimental and Theoretical Physics vol 35 pp 331ndash3391959

[33] IM Lifshic andV V Slezov ldquoThe kinetics of precipitation fromsupersaturated solid solutionsrdquo Journal of Physics andChemistryof Solids vol 19 no 1-2 pp 35ndash50 1961

[34] F M Kuni and A P Grinin ldquoKinetics of homogeneous conden-sationat the stage of formation of the main quantity of a newphaserdquo Colloid Journal of the USSR vol 46 p 460 1984

[35] F M Kuni T Y Novozhilova and I A Terentiev ldquoKineticequation of heterogenous condensationrdquo Letters in Mathemati-cal Physics vol 14 no 2 pp 161ndash167 1987

[36] V Kurasov ldquoDecay of metastable multicomponentmixturerdquohttparxivorgabscond-mat9310005

[37] V B Kurasov ldquoKinetics of heterogeneous condensation underdynamic conditionsrdquo Physical Review E vol 49 no 5 pp 3948ndash3955 1994

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Inorganic ChemistryInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal ofPhotoenergy

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Carbohydrate Chemistry

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

Physical Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom

Analytical Methods in Chemistry

Journal of

Volume 2014

Bioinorganic Chemistry and ApplicationsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

SpectroscopyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Medicinal ChemistryInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chromatography Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied ChemistryJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Theoretical ChemistryJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Spectroscopy

Analytical ChemistryInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Quantum Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Organic Chemistry International

ElectrochemistryInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CatalystsJournal of

Advances in Physical Chemistry 9

[5] Y Kazzi H Awada and M Nardin ldquoAdhesion on nanoorga-nized multilayers surface thermodynamics and local energydissipationrdquo Advances in Physical Chemistry vol 2010 ArticleID 502709 11 pages 2010

[6] A I Burshtein ldquoContact and distant luminescence quenchingin solutionsrdquo Advances in Physical Chemistry vol 2009 ArticleID 214219 34 pages 2009

[7] A C Zettlemoyer Nucleation Dekker New York NY USA1969

[8] F F Abraham Homogeneous Nucleation Theory AcademicNew York NY USA 1974

[9] M Volmer and A Weber ldquoKeimbildung in ubersattigtenGebildenrdquo Zeitschrift fur Physikalische Chemie vol 119 pp 277ndash301 1925

[10] R Becker andW Doring ldquoKinetische behandlung der keimbil-dung in uberattigten dampfernrdquo Annals of Physics vol 24 pp719ndash752 1935

[11] J Frenkel ldquoA general theory of heterophase fluctuations andpretransition phenomenardquoThe Journal of Chemical Physics vol7 no 7 pp 538ndash547 1939

[12] J B Zeldovich ldquoOn the theory of new phase formation cavi-tationrdquo Acta Physicochimica USSR vol 18 pp 1ndash22 1943

[13] L Farkas ldquoThe velocity of nucleus formation in superheatedvaporsrdquo Zeitschrift fur Physikalische Chemie vol 125 pp 236ndash240 1927

[14] J Lothe and G M Pound ldquoReconsiderations of nucleationtheoryrdquo The Journal of Chemical Physics vol 36 no 8 article2080 1962

[15] H Reiss J L Katz and E R Cohen ldquoTranslation-rotation para-dox in the theory of nucleationrdquoThe Journal of Chemical Phys-ics vol 48 no 12 pp 5553ndash5560 1968

[16] H Reiss ldquoTreatment of droplike clusters by means of the clas-sical phase integral in nucleation theoryrdquo Journal of StatisticalPhysics vol 2 no 1 pp 83ndash104 1970

[17] W G Courtney ldquoRemarks on homogeneous nucleationrdquo TheJournal of Chemical Physics vol 35 no 6 pp 2249ndash2250 1961

[18] M Blander and J L Katz ldquoThe thermodynamics of cluster for-mation in nucleation theoryrdquo Journal of Statistical Physics vol 4no 1 pp 55ndash59 1972

[19] J L Katz and H Weidersich ldquoNucleation theory without Max-well Demonsrdquo Journal of Colloid and Interface Science vol 61no 2 pp 351ndash355 1977

[20] J L Katz and M D Donohue ldquoA kinetic approach to homo-geneous nucleation theoryrdquo in Advances in Chemical Physics IPrigogine and S A Rice Eds vol 40 chapter 3 pp 137ndash155Wiley 1979

[21] A Dillmann and G E A Meier ldquoHomogeneous nucleation ofsupersaturated vaporsrdquo Chemical Physics Letters vol 160 no 1pp 71ndash74 1989

[22] A Dillmann and G E A Meier ldquoA refined droplet approach tothe problem of homogeneous nucleation from the vapor phaserdquoThe Journal of Chemical Physics vol 94 no 5 pp 3872ndash38841991

[23] J W Cahn and J E Hillard ldquoFree energy of a nonuniformsystem III Nucleation in a two-component incompressiblefluidrdquoTheJournal of Chemical Physics vol 31 no 3 pp 688ndash6991959

[24] D Gebauer and H Colfen ldquoPrenucleation clusters and non-classical nucleationrdquo Nano Today vol 6 no 6 pp 564ndash5842011

[25] D Sept and J A McCammon ldquoThermodynamics and kineticsof actin filament nucleationrdquo Biophysical Journal vol 81 no 2pp 667ndash674 2001

[26] A S Myerson and B L Trout ldquoNucleation from solutionrdquo Sci-ence vol 341 no 6148 pp 855ndash856 2013

[27] D Zahn ldquoThermodynamics and kinetics of prenucleation clus-ters classical and non-classical nucleationrdquo ChemPhysChemvol 16 no 10 pp 2069ndash2075 2015

[28] H Deng S Wang X Wang et al ldquoTwo competitive nucle-ation mechanisms of calcium carbonate biomineralization inresponse to surface functionality in low calcium ion concentra-tion solutionrdquo Regenerative Biomaterials vol 2 no 3 pp 187ndash195 2015

[29] D Kozma and G Toth ldquoMicroscopic rate constants of crystalgrowth from molecular dynamic simulations combined withmetadynamicsrdquo Advances in Physical Chemistry vol 2012Article ID 135172 10 pages 2012

[30] S Fakruddin C Srinivasu B R Venkateswara Rao and KNarendra ldquoExcess transport properties of binary mixtures ofquinoline with xylenes at different temperaturesrdquo Advances inPhysical Chemistry vol 2012 Article ID 324098 9 pages 2012

[31] A A Lushnikov ldquoEvolution of coagulating systemsrdquo Journal ofColloid and Interface Science vol 45 no 3 pp 549ndash556 1973

[32] I M Lifshic and V V Slezov ldquoKinetics of diffusive decomposi-tion of supersaturated solid solutionsrdquo Soviet PhysicsmdashJournalof Experimental and Theoretical Physics vol 35 pp 331ndash3391959

[33] IM Lifshic andV V Slezov ldquoThe kinetics of precipitation fromsupersaturated solid solutionsrdquo Journal of Physics andChemistryof Solids vol 19 no 1-2 pp 35ndash50 1961

[34] F M Kuni and A P Grinin ldquoKinetics of homogeneous conden-sationat the stage of formation of the main quantity of a newphaserdquo Colloid Journal of the USSR vol 46 p 460 1984

[35] F M Kuni T Y Novozhilova and I A Terentiev ldquoKineticequation of heterogenous condensationrdquo Letters in Mathemati-cal Physics vol 14 no 2 pp 161ndash167 1987

[36] V Kurasov ldquoDecay of metastable multicomponentmixturerdquohttparxivorgabscond-mat9310005

[37] V B Kurasov ldquoKinetics of heterogeneous condensation underdynamic conditionsrdquo Physical Review E vol 49 no 5 pp 3948ndash3955 1994

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Inorganic ChemistryInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal ofPhotoenergy

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Carbohydrate Chemistry

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

Physical Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom

Analytical Methods in Chemistry

Journal of

Volume 2014

Bioinorganic Chemistry and ApplicationsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

SpectroscopyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Medicinal ChemistryInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chromatography Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied ChemistryJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Theoretical ChemistryJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Spectroscopy

Analytical ChemistryInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Quantum Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Organic Chemistry International

ElectrochemistryInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CatalystsJournal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Inorganic ChemistryInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal ofPhotoenergy

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Carbohydrate Chemistry

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

Physical Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom

Analytical Methods in Chemistry

Journal of

Volume 2014

Bioinorganic Chemistry and ApplicationsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

SpectroscopyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Medicinal ChemistryInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chromatography Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied ChemistryJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Theoretical ChemistryJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Spectroscopy

Analytical ChemistryInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Quantum Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Organic Chemistry International

ElectrochemistryInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CatalystsJournal of