Indian Journal of ChemistryVol. 34A, December 1995, pp. 931-937

Physicochemical studies on microemulsions IV-A comprehensive estimationof the energetics

Subinoy Paul & S P Moulik*Centre for Surface Science, Department-of Chemistry, Jadavpur University, Calcutta 700032, India

Received 3 October 1994; revised 9 June 1995; accepted 3 July 1995

A comprehensive presentation of the proposed thermodynamic theories on microemulsion forma-tion has been made. The thermodynamic functions (parameters) viz. van der Waals potential, configura-tional entropy, potential energy and droplet free energy in water-in-oil microemulsion stabilized by theamphiphiles aerosol OT, sodium dodecyl sulphate, cetyltrimethylammonium bromide, butanol and hex-ylamine have been evaluated. The estimated van der Waals potential has been found to be much lowerthan the critical phase separation value of 10.6. Inter droplet interaction leading to amphiphile penetra-tion has been considered to be responsible for this effect. It has been observed that the effective surfa-ctant chain length has significant effect on the droplet free energy and the potential energy. The hy-drodynamic radius of the microwater droplets has been found to be inversely proportional to the confi-gurational entropy.

Microemulsions are dispersions (size 5-50 nm) ofone liquid into another in the presnece of amphi-philes':". The preparations are normally isotropic,low viscous and thermodynamically stable!".Because of their wide spread applications in thearea of pharmacy, industry and versatile otherday-to-day uses, investigations on their prepara-tion, stability, physical properties and thermody-namics of formation, etc. are of contemporary in-terests. The studies of their phase behaviours" arealso of significant importance. There are severaltheories on the energetics and stability of rnicro-emulsions->". In this presentation our objective isto make a comprehensive presentation of thesetheories and estimate the energetics of the water-in-oil systems mainly on the basis of particle di-mension and their inherent tendency of interac-tion. The systems considered were studied in ourlaboratory and particle dimensions were evaluatedby the method of conductance in the range of per-colation'".

Theoretical considerationApparently, the thermodynamic stability of mic-

roemulsions appears to be contradictory for dis-persion of a bulk phase into drops increases theinterfacial area resulting in a positive change offree energy. To resolve this paradox, Bowcott andSchulman 13 have postulated that a negative initialinterfacial tension develops between oil and waterleading to spontaneous emulsification. Rehbinder!"

has recognized low interfacial tension so that thefree energy decreases due to the entropy of dis-persion which outweighs the increase in the inter-facial free energy due to dispersion. Ultralow va-lues of interfacial tensions at the oil-water inter-face of the order of 10 - 2-10 - 4 dyne/ em havebeen reported by several workers'<'" for systemshaving oil, water, salt, short chain alcohols andpetroleum sulphonates. Ruckenstein and Chi7 andReiss" have applied the theory of ultralow interfa-cial tension successfully to microemulsions andthis has been further developed by Ruckensteinand Krishnarr". Most of the thermodynamic ana-lyses consider micellar solutions consisting of largesubmicroscopic water-like and oil-like regions sep-arated by a surfactant rich interfacial region. Theshapes of these regions are taken to be spherical,polyhedron or planar. The drop sizes may vary insome cases, it may remain constant in other cases.In the present treatment, we have assumed iso-tropic solubilized micellar systems having sphericaldrops of radius r.

A dividing surface is considered within the oil!water interface so that the surface excesses rHO

. 2.and rOil of water and oil respectively are bothzero.

The total free energy of rnicroemulsion system+'expressed per unit volume is

y<p 2<pH[=-+-2-+ LCilli - Pr;+!1F.r r ... (1)

932 INDIAN 1 CHEM. SEe. A, DECEMBER 1995

where,fjJ = volume of the dispersed phase per unit

volume,= radius of drop,=concentration in the i-th phase,= chemical potential of the i-th phase,= interfacial tension,=exterior phase' pressure of the drop,= bending moment of the interface,

rCj

#jYPfJH·andI!!.Fc = droplet free energy.

There are three contributions to the free ener-gy.

1. The free energy of the interface.2. The free energy of the bulk phase, and3. The free energy of the droplets (I!!.Fe).

Several works24-2? on the contribution of oneand two appear in literature. For the free energyof the interface, the bending moment H plays animportant role. The contributions of Ruckensteinand Chi", Overbeek ", Miller29 and Huh30 are wor-thy of mention. Comparatively less effort has beenmade to calculate the magnitude of the dropletcontribution to the total free energy. We have hereattempted to estimate it considering the approach-of Huh31. I!!.Fe includes both interaction energyand entropic contribution. This effect entirelycomes from the entropic terms provided the dropsare non-interacting. The Il.Fe increases with the in-crease of droplet in fixed disperse phase volume.Huh has included the van der Waals' attractive in-teraction potential e liP.) with hard sphere modelwhich is more generalised and we have also usedthis in our calculation.

The following condition" prevails for thermod-ynamic eq.uilibrium at constant temperature (T),

.(O~- =0Or ..

C,.P~.J

The simplified thermodynamic equation for equi-librium is

Of = _ 3,y -.:.6'lH + OI!!.Fe=OOr r2 r: Or

... (2)

This equation is independent of components andthe nature of the surfactant.

Miller and Neogi' have proposed an equilibri-um condition,

where Vm is the total volume of the microemul-sion, N is the total number of drops, n~ is the to-tal amount of surfactant in ali the drops, H is thebending stress, n; is the total amount of water, n~is the total amount of oil and B2 is the second vir-ial coefficient.

The above equation differs from Eq. (2); it ismeant for dilute microemulsions.

The term I!!.Fe can be calculated from differentapproaches of which the lattice model consideredby .Ruckenstein and Chi? is the simplest. Thedrops are .considered non-interacting and the freeenergy is contributed by the entropy.

... (3)

where the new term Vc designates the volume of asingle continuous phase molecule.

The approach made by Overbeek" is based onhard sphere model. Miller29 has subsequently sug-gested that

This approach gives smaller value compared tonon-interacting droplets. Huh31 has consideredhard-spheres with van der Waals' interactions toexpress the free energy as

... (5)

where

br

b = half of minimum possible separation betweentwo drops,

PAUL et al.: PHYSICOCHEMICAL STUDIES ON MICROEMULSIONS 933

AIb =-forO/W system,K

b = A2a rm Ifor W/0 systen,AI and A2= constant,

1= Debye length,

K

ex: = Flory's expansion parameter,m = No. of segments in the lipophilic chain,I = segment length and1'/ =(1 + l)3 ~

when l and £11 (l) tend to zero, Eq. 3 reduces toEq. 4, i.e. the hard sphere model. The attractivevan der Waals' potential is defined as

£11 (l)= 3k~~~l)3 [~In (1+~) +(1+ l)

+(1+ l)3ln 2l + l2] ((l+lf ... 6)

where Ab is the. Hamaker constant and is definedas Ab = Aww+Aoo - 2Awo (A;j is the Hamakarconstant between the i and j phases).

Phase separationHuh31 has considered the possibility of separa-

tion of microemulsion into two phases with differ-ent number densities of the spheres. Such a separ-ation is possible with and without a polymer. Inthe absence of polymer, the phenomenon has beenshown to occur due to London-van der Waals' at-traction between the drops. The critical state forthe phenomenon is at 7]0=0.13044; the van derWaals' attraction energy is larger .than a critical va-lue i.e. e 11> £c = 10.6012.

According to Eq. (6), at constant I\b' van derWaals' potential can vary depending on A.= (b/a)for a given Ab; phase separation can occur if l issmall enough so that e \1 > e c. At high l, a micro-emulsion phase will be stable, e \1 may also in-crease if' the Hainakar constant Ab is higher. In thecase of polymer induced phase separation, theseparated phases are 'drop rich' and 'polymer rich'with enhanced repulsion between the microemul-sion drop and polymer molecule arising when apolymer molecule approaching or microemulsiondrop loosing its configurational freedom. In thepresent analysis the role of polymer in phase se-paration has not been considered.

Thermodynamic functionsVarious thermodynamic functions can be calcu-

lated adopting the approach of Miller et aL32 ThusU, the potential energy of micellar interaction perunit volume is given by the relation

... (7)

The micelles are taken as spheres (radius ao) in aclosed packed hexagonal array.

where,

(0.74)113D'=2a --

o ~

The configurational entropy per unit volume, Sis given by the relation,

3k~ [4.1la! [(0.74)1/3')1S=-- in--+31n - -14.1la~ 3 Vc ~

... (8)

The potential energy function U(h) for interac-tion between micelles consists of (i) the attractiveinteraction due to London-van der Waals' forcesaccording to the relation

.. , (9))

where,

and (ii) the repulsive force for oil continuous sys-tem given by the relation

R 2 2 - khU = !...fJ_£_a~o--,t/J'...:o:!...e_._

el h+ 2ao... (10)

Uel is, however, assumed to be zero.

Results and DiscussionThe Hamakar constant (Ah) was introduced by

H C Hamakar-" which gives a measure of interac-tion among particles of colloidal dimension. TheLondon-van der Waals' forces between two parti-cles of the same material embedded in a liquid is

934 INDIAN J CHEM. SEC. A, DECEMBER 1995

always attractive. If the particles are ot differentcompositions, the resultant force may be repulsive.For attractive interaction Ab is positive. The actualvalue of Ab varies from case to case, therefore, asingle value cannot be assigned to Ah• In general,Ah varies in the extreme limits of 10- 14 to 10- 11

erg. In most cases it lies between 10 -13 to 10 -12

erg. In the present calculation, it has been as-sumed to be 2 x 10- 12 erg, and is a little higherthan the general value but well within the rangeconsidered by Huh31 •

In the present study, twenty three WI 0 micro-emulsions previously prepared and studied by US12

have been considered. The surfactants used inthese systems have been cetyitrimethylammoniumbromide (Cf'Als), sodium bis 2-ethylbexyl sulpho-succinate (AOT) and sodium dodecylsulphate(SOS) and the co-surfactants have been butanol(Bu) and hexylamine (Ha). The various oils usedare heptane, decane, xylene and cholesterol in xy-lene. The conductance and percolation propertiesof the above systems have been studied for theevaluation of the particle size and their population.

The two studied experimental temperatures havebeen 293K and 303K at surfactant Ico-surfactantmass ratios of 0.33 and 0.50.

The van der Waals' interaction potential Ell (A.)has been estimated from Eq. (6). The parameterdefined by Huh31 A. = (b/ r), where b is the half ofminimum possible separation between the twodrops and is equal to A2a. [in I for WI 0 system.It is very difficult to find out the exact values ofthese parameters, we have reasonably assumedb = R; and r= R; + 112, where, Rw = radius of thewater droplet and .1=chain length of the surfact-ant. Therefore, in our calculation,

With the assumed values of the parameters, andtaking Ab = 2 x 10 - 12 erg, we have realized thatEu(A.) [Table 1] varies between the limits 0.14 and0.43. The lower values are characteristics for thesurfactant AOT and the values of E u (A.), on the

Table I-Calculated thermodynamic parameters of W/O microemulsions at 0.2 wt fraction of water"0

- I1Fe x 10-5 ~Ux 10-19System Temp/K S/CS{w/w) Re{R.,)/A EII{.t) S(erg. mr ') (erg. ml") (erg.ml=K'")

CTAB/Bu/Hp 293 0,33 4O.5{18.8) 0.43 8.09 1.46 433CTAB/Bu/Hp 293 0.50 45.2{23.5) 0.33 4.90 1.28 353CTAB/Bu/Hp 303 0.50 44.8{23.1) 0.34 4.69 1.29 356CTAB/Bu/Hp 293 0.50 44.5(22.8) 0.38 5.05 1.35 339SDS/Bu/Hp 293 0.33 52.5(35.5) 0.21 1.78 1.26 308SDS/Bu/Hp 293 0.50 69.0(52.0} 0.18 0.78 0.97 169SDS/Bu/Hp 303 0.50 65.5{48.5) 0.18 0.98 1.01 193SDS/Bu/Dc 293 0.50 74.8(57.8} 0.20 0.58 0.90 140SDS/Bu/Xy 293 1).50 77.5(60.5) 0.17 0.54 0:86 129

SDSlBu/30% (w/v) 293 0.50 75,0(58.0) 0.19 0.60 0.88 139Cholesterol + XyAOT/Bu/Hp 293 033 45.5(36.5) 0.16 2.10 1.43 409

AOT/Bu/Hp 293 0.50 58.0(49.0) 0.14 0.99 1.16 250

AOT/Bu/Hp 303 0.50 53.2(44.2) 0.15 1.3~ 1.23 297

AOT/Bu/Dc 293 0.50 56.8(47.8} 0.19 1.17 1.32 260

CTAB/Ha/Hp 293 0.50 53.0(31.3) 0.28 .2.62 1.32 303

CTAB/Ha/Hp 303 0.50 51.3{29.6) 0.28 3.20 1.28 322

CTAB/Ha/Dc 293 0.50 53.5(31.8) 0.26 2.44 1.24 296

SDS/Ha/Hp 293 0.50 64.0(47.0} 0.19 1.02 1.06 201

SDS/Ha/Hp 303 0.50 61.5(44.5} 0.19 1.34 1.09 220

SDS/Ha/Dc 293 0.50 63.0(46.0) 0.19 1.18 1.07 208

AOT/Ha/Hp 293 0.50 55.0(46.0) 0.18 1.18 1.24 285

AOT/Ha/Hp 303 0.50 50.5(41.5) 0.15 1.58 1.31 335

AOT/Ha/Dc 293 0.50 55.6{46.6) 0.18 1.11 1.23 277

a. All systems are taken from Ref. 12.

PAUL et al.: PHYSICOCHEMICAL STUDIES ON MICROEMULSIONS 935

other hand, either overlap or get closer for theamphiphiles AOT and SOS. It is noticed that thevalues of E 11 (,1.)for the amphiphiles follow the or-der CTAB > SDS > AOT. The co-surfactant Bugives higher result compared to the co-surfactantHa. Nevertlieless, the values of El1 (,1.)are signifi-cantly lower than the critical value of 10.6 forphase separation as defined by Huh3l. 'Accordingto this measure, the presently dealt systems are farfrom the phase separation state and are verystable. A comment on this will be made at the endof this section. The calculated value of E 11 (,1.)maybe increased by decreasing ,1.or by increasing Ah•

We have reasonably assumed the value of ,1.whichvaries between 0.6-0.9 for different systems andthere is not much possibility of lowering it to in-crease El1 (,1.).The other way of increasing Ell isby increasing the Hamakar constant (Ah). The as-sumed value is already on the higher side. Themaximum value of it can be assumed to be 10- II

erg. Even with such a high value of Ah, the esti-mated Ellis also significantly lower than 10.6.Calje et aL34 have also observed the necessity oftoo large a value of Ah for Eq. (6) to represent thephase separation behaviour. Experiments haveshown that for preparations, where E 11(,1.)is wellbelow 10.6; phase separation can occur. Thisanomaly may be a weakness of the main theory orneglect of an interaction which is fairly significant.The possiblity of such an interaction has been,therefore, proposed", The low values of E 11 sug-gests necessity of consideration of an effect thatleads to intermicellar interaction. According toHamakar-", intermicellar interaction is possiblesince the chemical nature of the aliphatic chains ofthe surfactant and of the alcohol are very similarto that of the organic solvent (oil). The order ofmagnitude of the attractive interactions is expect-ed to be appreciable. The contributions of the ali-phatic part including the overlapping effect andentropic contribution has been attempted byseveral workers=-". The entropic contributionarises from intermicellar penetration and is mainlydue to the change in configuration and conforma-tion during the overlap of the aliphatic interfaces.Brunettis et a£35 have found out the second virialcoefficient B ranging from - 27 to + 6 using lightscattering techniques for the systems SOS/ water /dodecane containing co-surfactant n-alkanols (C5-

C 7)' They have suggested that long range interac-tion forces can be operative in microemulsions byvarying the components and the chemical compo-sitions of the continuous and the disperised phase.The most sensitive component is the alkanol.Strong attractive interactions are found for the

microemulsions containing pentanol which is lessprominent for the system containing hexanol andheptanol. Also, the interactions are proportionalto the micellar radius. Attractive energy producespenetration of the aliphatic layer of the micellesand is proportional to the penetrated volume. Theextent of variation of this volume may result bythe change in the alkanol chain length or by thechange in the micellar size. In the present context,the micellar radius varied between 41-78A but thecalculated Ell has shown an inverse dependenceon the size; the chain length effect is, therefore, ofmore significance. Among the twoco-surfactants(butanol and hexylamine) used, the former hasshown greater E 11' Bothorel et al.35 have alsofound significant negative second virial coefficientfrom light scattering measurements on microemul-sions prepared with alkanols, which increased withthe decrease in the alkanol chain length. Accor-dingly, butanol is expected to show significant in-terdroplet interaction (penetration). A scope,therefore, remains to improve upon the theory byincorporating an appropriate correction term inthe matter of droplet interaction.

The droplet free energy ~ Fe has been calculat-ed from Eq. (5) which includes hard sphere plusvan der Waals' interaction. The values are in theorder of 105 erg.ml r 1 (Table 1) with a negativesign. The systems containing CTAB have givenlower values (higher negative values) than the

. other two surfactants AOT and SOS. Except for afew cases with AOT and SOS, it is found that thesurfactant dependence of ~Fe follows the orderSDS > AOT> CTAB. The co-surfactant Ha giveshigher result than Bu. Further, with increasing hy-drodynamic radius (Re), the droplet free energyincreases in identical fashion irrespective of thetype of surfactant (Fig. lA). The same trend isobserved for the potential energy (Fig. IB) calcu-lated from the Eq. (7) assuming ao = Re' But inthis case, the values are closer than ~Fe' So, it isclear that in the presence of co-surfactant, CTABis a better stabilizer for microemulsion formationthan AOT which is again better than SOS. Thetrend of the potential energy produced by the Sur-factants follows the order CTAB <AOT <SOS.The effective chain length of the surfactants mayalso support this view. The chain lengths" ofCTAB, AOT and SOS have been considered to be21.7,9.0 and 17A respectively. AOT has two non-polar tails, the effective chain length contributionis likely to be twofold i.e. 18A, the order of thechain length is thus CTAB > AOT> SDS. On thewhole, increasing surfactant chain length offers in-creasing stability to the microemulsions. The

936 INDIAN J CHEM. SEe. A, DECEMBER 1995

, I£• AOT

A

Go; 9

• eTAB

.,,- • 50S

'2".eo-<l ,

• 6--.0

.at .#

, ..~

B

i I.lt • AOT

~ " s.: , .' • eTAI

•00<,

1·20 .' ••

~O5

'g 1.101-

"•

~~'

I 1·00

••

0.90

70

• AOT•• eTAI

• 50S

1>0 c

"<,'.

1>0 70 80

R. It.

Fig. I-Dependence of free energy of droplets (curve A), pot-ential energy (curve B) and entropy (curve C) of W/O micro-emulsion on hydrodynamic radius of droplets. (Symbols: with-out prime - Hp; single prime - Dc; double prime - Xy; triple

prime - Cholesterol +Xy)

above trend also follows the order of the headgroup sizes of the surfactantsv-" CTAB (SIN),AOT (SON) and SDS (3SN). A correlation withthe charges of the head group is difficult to pro-pose at this stage.

The configurational entropy has been calculatedfrom the Eq. 8. The values vary from 129 to 433erg-ml : I.K - I for different surfactants (Table 1).The values decrease with increasing hydrodynamicradius of the droplet, R, (Fig. 1C) irrespective ofthe amphiphiles. It is quite obvious that with in-creasing radius the freedom of the droplets arerestricted, consequently the entropy decreases forlarger droplets.

The differences in the thermodynamic beha-

viours (Fig. 1) of the microemulsions resultedusing crAB, AOT and SDS tell about the role ofintrinsic nature of the amphiphiles, their chain.lengths and head groups including their sizes andcharges being modified in the presence of the co-surfactants Bu and Ha. The dispersed water glo-bules interacting with the oil via the surfactantmolecules are governed by the property of theamphiphiles. The droplet formation, is thus the ef-fect, the nature of the amphiphiles is the underly-ing cause .

The following conclusions can be drawn:(i) The surfactant dependent droplet free energy

and potential energy in microemulsion has beenwitnessed to follow the order CTAB < AOT <SDS which is considered to be a reflection of thesequence of the effective chain length of the am-phiphiles, CTAB > AOT > SDS.

(ii) It has been found that the critical van derWaals' potential for the phase separation of themicroemulsion is much less than the theoreticalprediction owing to inter droplet interaction lead-ing to mutual amphiphile interpenetration.

(iii) The configurational entropy has beenobserved to decrease with increasing hydrody-namic radius irrespective of the amphiphiles dueto decreasing randomness of larger particles.

AcknowledgementS. Paul highly appreciates the laboratory and

other facilities given to him by the JadavpurUniversity.

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