23 January 1998

Ž .Chemical Physics Letters 282 1998 403–408

Molar mass dependence of critical amplitudes for chain-moleculesolutions

Xueqin An, Fuguo Jiang, Chuanyin Chen, Weiguo Shen )

Department of Chemistry, Lanzhou UniÕersity, Lanzhou, Gansu 73000, China

Received 26 September 1997

Abstract

The power-law dependences of critical amplitudes on molar mass for coexistence curve, correlation length andsusceptibility have been derived from a Landau–Ginsburg–Wilson type model for chain-molecule solutions of both smallmolecules and polymers. A series of turbidity measurements and determinations of coexistence curves for solutions ofn-alkane in polar liquids and polymethylmethacrylate in 3-octanone have been conducted. Our experimental results and thoseof polystyrene solutions indicate that this model is satisfied within experimental errors. q 1998 Elsevier Science B.V.

1. Introduction

Recently, there has been considerable interest inthe dependence of the critical behaviours of polymersolutions on the number N of monomer units in thepolymer which is proportional to the molecular mass

w xM 1–9 . It appears that the critical amplitudes van-ish or diverge with M. These can be described as thegeneral forms:

< < b yb < < bDfsB t AM t , 1aŽ .

< <yn n < <ynjsj t AM t , 1bŽ .0

< <yg g < <ygxsx t AM t , 1cŽ .0

with

ts TyT rT , 2Ž . Ž .c c

where Df, j and x are difference of volume frac-

) Corresponding author.

tions of two coexisting phases, correlation length andŽsusceptibility or alternatively the osmatic compress-

.ibility , T is the critical temperature, b , n and g arec

t exponents and well established as universal indicesby experiment and theory. The coefficients B, j 0

Ž .and x in Eq. 1 are defined as the amplitudes0

which are M dependent. This M dependence may becharacterized by the M exponents b, n and g andthe values of the M exponents also seem to beuniversal. There is another exponent r that describesthe dependence of the critical volume fraction of thepolymer on M:

f AMyr . 3Ž .c

The value of r seems well established by experimen-w xtal studies of polymer solutions 5,6,10,11 , which is

0.38"0.01. The value of r calculated by Muthuku-w x w xmar is 1r3 8 . Recently, Wilding et al. 7 per-

formed computer simulations for the polymer-solvent

0009-2614r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved.Ž .PII S0009-2614 97 01291-8

( )X. An et al.rChemical Physics Letters 282 1998 403–408404

solutions with chain lengths of up to 60 monomersŽ . 0.37and found a relation of f r 1y1.1f AN . Thec c

most of experimental studies to determine b, n andg are limited to polystyrene solutions. The coexis-tence curves for solutions of polystyrene in methyl-cyclohexane over a wide range of molar masses have

w xbeen determined by Dobashi et al. 10 and the valuew xof b obtained was 0.23, while Sanchez 6 quoted

0.28 by re-analysing Dobashi’s data. Shinozaki et al.w x12 found bs0.34 from their own measurementswith full weight to their data for the highest molarmass, while their data for the lower molecular massalone were consistent with 0.23. The exponents nand g were determined by light scattering experi-ments for solutions of polystyrene in methylcyclo-

w xhexane 13,14 and the values of n and g wereŽfound to be 0.28 two experimental points with low

4 .molecular weights M -5.0=10 were deletedww xand 0.08, respectively 14 . When the correlation

length was plotted against molecular weight in aln–ln scale, an upwards curvature was observed andthe value of n was possibly as low as 0.10 for small

w xM 14,15 .Some theories such as Gennes approximation, the

Landau–Ginsburg–Wilson model, Muthukumar’sthree-body-interaction treatment and scaling lawshave been used to explain the experimental results.The values from the different theories varied from0.22 to 0.337 for b, 0.17 to 0.195 for n and y0.03to 0.025 for g. These differences are difficult todistinguish by available experimental results. Mostof the theories used to explore the exponents startedwith Flory–Huggins theory and took the limit oflarge M and f <1. The Flory–Huggins theory hasc

been widely used in predicting the properties ofmixtures of small molecules containing n-alkaneswith various chain lengths. These mixtures are incorresponding states as polymer solutions. Two ques-

Ž .tions then arise: 1 does a more universal M depen-dence of critical amplitudes exist in chain-molecule

Ž .solutions of both small molecules and polymers? 2Ž .Does the limitation of 1yf ™1 mask this uni-c

versality? In order to answer these two questions, wehave been intending to study the critical behavioursof a series of mixtures containing chain molecules.

Recently, we derived the relations of B and fc

dependent on molar mass from Flory–Huggins the-ory and the symmetric Landau-type expansion of the

Ž .free energy without the limitation of 1yf ™ 1cw x16 . They take the forms:

f r 1yf AMyr , 4Ž . Ž .c c

Br 1yf AMyb , 5Ž . Ž .c

with rs0.5 and bs0.25. Our measurements of thecritical volume fractions for mixtures of alkane in

w x w xN,N-dimethylacetamide 16 and in nitrobenzene 17yield the values of 0.41"0.01 and 0.40"0.01 forr. This consistent with that obtained from the simula-

w xtion study 7 for chain length up to 60 monomers,within computation errors. The value of b deducedfrom coexistence curves for mixtures of alkane in

w xN,N-dimethylacetamide is 0.06"0.04 16 , which issignificantly smaller than 0.25 from the classicaltheory and 0.28 from the experiments of polymersolutions. We also re-analysed the data of phase-sep-aration experiments of polystyrene in methylcyclo-

w xhexane 10 , polymethylmethacrylate in 3-octanonew x11 and observed that r is 0.402"0.008 and 0.417"0.006, respectively. The experimental values of rfor small chain-molecule mixtures are consistent withthose for polymer solutions, which implies that Eq.Ž . Ž .4 is more likely to be universal than Eq. 3 .

2. Modelling

By using Flory–Huggins theory and the assump-w xtion that xAarT 1,5 , the symmetric Landau-type

expansion of the free energy per lattice site aroundw xthe critical point takes the form 16 :

2 y20.5frk sa t fyf qa M 1yfŽ . Ž .B 1 c 2 2 c

=4

fyf , 6Ž . Ž .c

where k is Boltzmann’s constant, M is molarB 2Žmass of component 2 i.e. chain molecules in our

.experimental studies . The Hamiltonian for a Lan-dau–Ginsburg–Wilson model is obtained by adding

w Ž .x2 Ž .the gradient term k = f in Eq. 6 as follows:r

2 y20.5Hrk sa t fyf qa M 1yfŽ . Ž .B 1 c 2 2 c

=24

fyf qk = f , 7Ž . Ž . Ž .c r

( )X. An et al.rChemical Physics Letters 282 1998 403–408 405

where a and a are constants independent of M1 2

and t. The correlation length j from Landau–Gins-w xburg theory can be written as 18 :

0.5w xjs kr2 a t , T)T . 8Ž .1 c

Ž . Ž .Combining Eqs. 8 and 1b in the classical limityields:

kAj 2 . 9Ž .0

If the solution is so dilute that the interferenceamong different chains can be neglected, the follow-ing expression for the correlation length can be

w xobtained 19,20 :

y12 y1j Af 1yf . 10Ž . Ž .0 c c

Ž .In the polymer solutions of large M, 1yf ™ 1,c

then k is proportional to fy1, which is consistentcw xwith that suggested by de Gennes 21 and Sanchez

w x Ž . Ž .6 . Substituting Eq. 10 into Eq. 7 and ignoringall the constants independent of M and t gives:

2 y2 40.5Hs t fyf qM 1yf fyfŽ . Ž . Ž .c c c

2y1 y1q 1yf f = f . 11Ž . Ž . Ž .c c r

If the critical concentration of a solution is notŽsufficiently dilute i.e. the chain molecule is not

sufficiently long compared with the solvent.molecule , the segments in the different chains may

Ž . Ž .interfere with each other and then Eqs. 11 and 122 y1 Žare invalid. We simply assume j Af 1y0 c

.ySf for the chain-molecule solutions of both smallcŽ .molecules and polymers. Eq. 11 then become:

2 y2 40.5Hs t fyf qM 1yf fyfŽ . Ž . Ž .c c c

2yS y1q 1yf f = f . 12Ž . Ž . Ž .c c r

The direct renormalization group calculation of thew xcoexistence curve can be avoided 22 by introducing

the following transformation of variables:

y1 .5Sq1y1.5 y0.25cs fyf f M 1yf , 13aŽ . Ž . Ž .c c c

Ž .2 Sy10.5 2xsrM f 1yf , 13bŽ . Ž .c c

y3 Sq4y1 y3m s t M f 1yf , 13cŽ . Ž .0 c c

Ž .y6 Sy1y1.5 y6H sHM f 1yf , 13dŽ . Ž .x c c

Ž .and converting Eq. 12 to the standard form:

22 4H sm c qc q = c . 14Ž . Ž .x 0 x

Ž .The correlation length associated with Eq. 14 scalesŽ .ynas j A m . Using the relation xrj srrj , thex 0 x

Ž .correlation length j associated with Eq. 12 then isexpressed as:

Ž .3ny2Žny0.5.jAM f r 1yfŽ .c c

=wŽ . x3Sy1 ny2 S < <1yf t yn . 15Ž . Ž .c

Ž .The coexistence curve corresponding to Eq. 14 isŽ . bdescribed by cA ym . The difference of volume0

fractions of two existing phases is then expressed asfollows:

Ž .1.5y3bŽ0 .25yb .DfAM f r 1yfŽ .c c

=w Ž . x1.5Sq0.5q 1y3S b b< <1yf t . 16Ž . Ž .c

w xThe susceptibility can be expressed as 6 :

ygŽ . 2y6 Sy1y1.5 y6xAM 1yf f EfrEc mŽ . Ž . Ž .c c 0

Ž .3gy3Žgy1.sM f r 1yfŽ .c c

=Ž .Ž .3Sy1 gy1 yg< <1yf t . 17Ž . Ž .c

In the mean field limit, bs0.5, ns0.5 and gs1,then we have

y0 .25 < < 0.5Dfr 1yf AM t , 18aŽ . Ž .c

y0 .5 Ž .y0.5 Sq1 y0.5< <jA f r 1yf 1yf t ,Ž . Ž .c c c

18bŽ .

< <y1xA t . 18cŽ .

Ž .In the limit of 1yf ™1 for large M in polymercy0.5 Ž .solutions, taking f AM , Eqs. 18 become Eqs.c

Ž .1 with bsns0.25, gs0, bsns0.5 and gs1.ŽUsing bs0.327, ns0.63, gs1.24 and f r 1yc

. yr Ž . Ž .f AM , Eqs. 15 – 17 become:c

y0 .519Sy0.827 0.327yb < <Df 1yf AM t ,Ž .c

bs0.077q0.519r , 19aŽ .

( )X. An et al.rChemical Physics Letters 282 1998 403–408406

0.11Sq0.63 y0.63n < <j 1yf AM t ,Ž .c

ns0.13q0.11r , 19bŽ .Ž .y0.24 3Sy1 y1.24g < <x 1yf AM t ,Ž .c

gs0.24y0.72 r . 19cŽ .Ž . Ž .A combination of Eqs. 1 and 19 yields:

y0 .519Sy0.827 ybB 1yf AM ,Ž .c

bs0.077q0.519r , 20aŽ .0.11Sq0.63 nj 1yf AM ,Ž .0 c

ns0.13q0.11r , 20bŽ .Ž .y0.24 3Sy1 gx 1yf AM ,Ž .0 c

gs0.24y0.72 r . 20cŽ .

3. Experimental and discussions

Coexistence curves for n-alkane in dimethylac-etamide have been determined in the previous workw x16 . Taking the experimental value of rs0.41, a

Ž b. Ž .ln–ln plot of BM against 1yf yielded a valuec

of 2.04"0.3 for s. The plot is shown in Fig. 1. Wealso reanalysed the data of the coexistence curves forpolystyrene in methylcyclohexane reported by

w xDobashi et al. 10 and found ss1.99"0.15. Theln–ln plot is shown in Fig. 2. The values for twosystems are in good agreement and significantly

Ž b. Ž .Fig. 1. ln–ln plot of BM against 1yf for n-alkane incŽ . Ž .dimethylacetamide: v experimental values; — fit of Eq.

Ž .20a .

Ž b. Ž .Fig. 2. ln–ln plots of BM against 1yf for polystyrene incŽ . Ž .methylcyclohexane: v experimental values; — fit of Eq.

Ž .20a .

larger than the value of 1 suggested in the literature.Ž .Taking ss2 and rs0.41 the average value of r ,

Ž .Eqs. 20 then become:y1 .865 ybB 1yf AM , bs0.29, 21aŽ . Ž .c

0.85 nj 1yf AM , ns0.18, 21bŽ . Ž .0 c

y1 .20 gx 1yf AM , gsy0.06. 21cŽ . Ž .0 c

Ž .In order to test Eqs. 21 , a series of coexistence-curve determinations and turbidity measurements forvarious binary chain-molecule solutions of smallmolecules and polymers were conducted. The experi-mental details for measurements of coexistence

w xcurves have been described previously 16 . A ln–lnŽplot of the critical amplitude B against M see Eqs.

Ž .. w x1 for polymethylmethacrylate in 3-octanone 23 isŽ .shown by a in Fig. 3, in which one experimental

point with low molar mass is significantly off theŽ .straight regression line. However, as shown in b of

Fig. 3 all experimental points fall on the straight lineŽ .y1.865 Žin a ln–ln plot of B 1yf against M seec

Ž ..Eq. 21 with a value of bs0.26"0.02 as theslope. This value is slightly smaller than theoretical

Žprediction of 0.29. A ln–ln plot of critical B 1y.y1.865f against M for n-alkane in nitrobenzenec

yields a value of 0.28"0.02, which is in goodagreement with the prediction. The turbidity data forn-alkane in nitrobenzene at various wavelengths andtemperatures and of polymethylmethacrylate in 3-oc-

( )X. An et al.rChemical Physics Letters 282 1998 403–408 407

Ž .y1 .865Fig. 3. ln–ln plots of B 1yf and B against M forcŽ .polymethylmathacrylate in 3-octanone: v experimental values

w Ž .y1 .865 x Ž . Ž .of ln B 1yf ; B experimental values of ln B; — fitcŽ . ybof Eq. 21a and the equation Bs M .

tanone at fixed wavelength 632.8 nm and varioustemperatures were reduced to obtain the critical am-plitudes associated with the correlation length andsusceptibility. The samples of critical compositionsfor turbidity measurements were prepared in 10 mmpath-length optical cells. For small chain-moleculesolutions, the turbidities were measured by a spec-

Ž .trometer Beckman DU-7 . The sample cells were setin a metal block and the block was mounted in thespectrometer. The temperature in the block was con-stant within "0.003 K. For the polymer solutionsw x24 , the samples were placed in a water bath, where

Ž .0.85Fig. 4. ln–ln plot of j 1yf against M for polymethyl-0 cŽ . Ž .mathacrylate in 3-octanone: v experimental values; — fit of

Ž .Eq. 21b .

the temperature stability was better than "0.002 K.The intensities were measured by a detector assem-bly, consisting of a 632.8 nm interference filter, aphotodiode and an amplifier. The ln–ln plots ofŽ .0.85 Ž .y1.20j 1yf and x 1yf against M for0 c 0 c

polymethylmethacrylate in 3-octanone are shown inFigs. 4 and 5. We obtained ns0.18"0.03 andgsy0.10"0.04 for polymethylmethacrylate in 3-octanone and ns0.21"0.04 and gsy0.12"

0.09 for n-alkane in nitrobenzene, which are consis-tent with the theoretical predictions of 0.18 and

Ž .y0.06 within experimental uncertainties. Eq. 21bis also supported by results from reanalysing the data

w xof Shinozaki et al. 14 . As shown in Fig. 7 of Ref.w x14 a discontinuous step divided the six experimen-

Žtal points into two parts with different slopes i.e..with different value of n in a ln–ln plot of j 0

against M, one slope is 0.13 and the other is 0.17.This discontinuous step still exists in a ln–ln plot ofŽ .0.85j 1yf against M, but the values of n are0 c

0.18 and 0.19 and in good agreement with the theo-retical prediction of 0.18. It would be interesting toprecisely remeasure the correlation lengths and sus-ceptibilities for a series of mixtures of polystyrene inmethylcyclohexane with various molecular weightsof polymer.

Ž .In conclusion, Eqs. 21 derived from the LGWmodel, without the limitation of large M, seem to besatisfied for the chain-molecule solutions of both

Ž .y1 .20Fig. 5. ln–ln plot of x 1yf against M for polymethyl-0 cŽ . Ž .mathacrylate in 3-octanone: v experimental values; — fit of

Ž .Eq. 21c .

( )X. An et al.rChemical Physics Letters 282 1998 403–408408

small molecules and polymers. Further experimentalstudies on the critical behaviours of various systemsof chain-molecule solutions are being carried out totest this universality.

This work was supported by the National NaturalŽ .Science Foundation of China Project 29673019 , the

State Education Committee, and the Natural ScienceŽ .Foundation of Gansu province No. ZR-96-009 , P.R.

China.

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