Microemulsions for dermal drug delivery studied by dynamic light scattering: Effect of interparticle...

Microemulsions for Dermal Drug Delivery Studied byDynamic Light Scattering: Effect of InterparticleInteractions in Oil-in-Water Microemulsions

ANUJ SHUKLA,1 MARTIN JANICH,2 KONSTANZE JAHN,3 REINHARD H.H. NEUBERT3

1Department of Physics, Optics Group, Martin-Luther-University Halle-Wittenberg, 06099 Halle/Saale, Germany

2Interdisciplinary Centre of Material Science, Martin-Luther-University Halle-Wittenberg, 06099 Halle/Saale, Germany

3Department of Pharmacy, Institute of Pharmaceutics and Biopharmaceutics, Martin-Luther-University Halle-Wittenberg,06120 Halle/Saale, Germany

Received 13 June 2002; revised 24 September 2002; accepted 16 October 2002

ABSTRACT: Dynamic light scattering (DLS) was used to study the droplet size and thedroplet interaction of o/w microemulsions (MEs) consisting of oils, a blend of a high and alow hydrophilic–lipophilic balance (HLB) surfactant, and a hydrophilic phase (propyleneglycol/water). Like many MEs, these systems could not be diluted to infinite dilutionwithout phase separation. Consequently, to allow a meaningful calculation of dropletdiameter from data obtained from DLS, it is necessary to correct scattering measure-ments in high concentration regions for the nonideality arising from interparticleinteraction. Scattering data were corrected for interparticle interaction using a suitableinteraction model proposed for our systems. From the total interparticle interactionenergy, coagulation time was calculated. The ratio between rapid and slow coagulationwas of the order of 10100, which is consistent with the observed stability of the MEsstudied. � 2003 Wiley-Liss, Inc. and the American Pharmaceutical Association J Pharm Sci

92:730–738, 2003

Keywords: o/w microemulsions; dynamic light scattering; interparticle interaction

INTRODUCTION

Microemulsions (ME), which are thermodynami-cally stable dispersions of oil and water that arestabilized by surfactants and in some cases addi-tionally by cosurfactants, 1,2 have attracted muchinterest in recent years because of their greatpractical importance in terms of their drugdelivery potential and their interesting physicalproperties.3 Extensive studies have been done onMEs using cosurfactants in the last few decades.A cosurfactant is usually a short-chain alcohol.

These alcohols are not considered edible and maybe irritating to the skin. In the present study,MEs were prepared using a blend of a high and alow hydrophilic–lipophilic balance (HLB) surfac-tant instead of a cosurfactant. For selecting asuitable ME system for drug delivery, it is im-portant to know something about the physico-chemical properties of the ME, such as drugsolubility, area of ME in the phase diagram, andthe resulting size of ME droplets. A particulardifficulty concerning the use of MEs as a modeldrug delivery system is the direct and unambig-uous determination of droplet radius, which isexpected to be a strong function of interparticleinteraction. Few methods are available to defi-nitively prove the existence of a ME; the mostwidely used are scattering techniques or nuc-lear magnetic resonance (NMR) self-diffusion

730 JOURNAL OF PHARMACEUTICAL SCIENCES, VOL. 92, NO. 4, APRIL 2003

Correspondence to: Reinhard H.H. Neubert (Telephone: 49-345-5525000; Fax: 49-345-5527292;E-mail: [email protected])

Journal of Pharmaceutical Sciences, Vol. 92, 730–738 (2003)� 2003 Wiley-Liss, Inc. and the American Pharmaceutical Association

measurements. Scattering techniques collectivelysuffer from the disadvantage that to obtain areliable estimation of particle size, measurementshould be made in the range of low disperse-phasevolume fractions and should be extrapolated toinfinite dilution to avoid the problems encoun-tered as a result of particle–particle interactions.Like many MEs, our systems could not be dilut-ed without phase separation. Consequently, toallow a meaningful calculation of droplet size atfinite droplet concentration, a scattering techni-que [e.g., static light scattering (SLS), dynamiclight scattering (DLS), and small-angle neu-tron scattering (SANS)] cannot be used withoutmaking an assumption about the nature of theinterparticle interaction. DLS studies on the con-centration dependence of the diffusion coefficientallow one to determine the virial coefficient ofdiffusion, kd, and the hydrodynamic radius ofME droplets. Several theories relate this virialcoefficient kd to interparticle interaction.4–6

In Part I7 of our publication series, we char-acterized the droplet size of oil-in-water (o/w)MEs, made with different kinds of pharmaceuticaloils, for dermal administration by DLS and SANS,and we modeled a ME droplet as a layered sphereconsisting of an oil core radius, Rcore, and a pene-trable shell of surfactants, Lshell (see Figure 1). Inthe region up to Rcore in Figure 1, the dark blackpart represents pure oil and the remainder indi-cates the penetration of the surfactant tail intothe oil. The region from Rcore to the position in thecontinuous aqueous phase forms a penetrableshell of surfactants, incorporating strongly bound-ed surfactants and a hydration shell. The hydro-dynamic radius, Rh, measured by DLS is supposed

to consist of the oil core and the strongly boundedsurfactant film and lies somewhere in between thecore radius and end of shell. In Part II,8 DLSmeasurements were used to characterize w/o MEsmade with different kinds of surfactants or dis-persed phase or a mixture of the two. As discussedin Parts I and II of this series, a hard sphere model,which prevents the overlap of different hardsphere droplets, is used to correct the results forparticle–particle interaction. In this work, a dilu-tion procedure was used in the region of the phasediagram where surfactant-covered oil dropletswere formed,9 which allowed us to deduce theconcentration dependence of the diffusion coeffi-cient from experimental data. The theory ofFelderhof,4 along with an appropriate model forinterparticle interaction, is used to extract theinformation about droplet size and interparticleinteraction from the concentration dependence ofthe diffusion coefficient. From the total interactionenergy, it is possible to derive a criterion for thestability of colloids.10

MATERIALS AND METHODS

Chemicals

Tween1 80, Tagat1 O2, Eutanol1 G, isopropylpalmitate (IPP), propylene glycol (PG), and oleicacid were purchased from Caesar& Loretz, Hil-den, Germany. Poloxamer 331 was kindly pro-vided by C.H. Erbsloeh, Krefeld, Germany. Thechemical structures of these compounds are pre-sented in Figure 2. Bidistilled water was used.

Sample Preparation and Experimental Set-upof DLS Experiments

The basic ME consists of a quaternary mixture ofoil (5 wt % Poloxamer 331) and either Tagat1

O2 or Tween1 80 at 20 wt % in a 3:2 ratio and PG/H2O at 75 wt % in a 2:1 ratio. This basic ME wasdiluted with the continuous phase, keeping theratio in eq. 1 constant and varying the oil weightpercentage (a) in the range 2<a<5, to preserve aconstant droplet radius.

m ¼ wt % of surfactants½ �½wt % of oil� ¼ 4 ð1Þ

The compositions of the MEs used are summar-ized in Table 1.

Prior to the measurements, dust particles wereeliminated by filtration with a 1.2-mm pore size

Figure 1. Different contributions to the effectiveinteraction potential. The meanings of the various radiiand interaction potential are explained in the text.

MICROEMULSION DROPLET SIZE AND INTERACTIONS DETERMINED BY DLS 731

JOURNAL OF PHARMACEUTICAL SCIENCES, VOL. 92, NO. 4, APRIL 2003

filter (Sartorius, Goettingen, Germany). Thevolume fraction of the ME droplets (f) is defined as

f ¼ vs þ vi

vs þ vi þ voð2Þ

where vs, vi, and vo are the volumes of surfactant,dispersed phase, and oil, respectively.

The light scattering hardware set-up consistsof commercially available equipment for simul-taneous static and dynamic experiments made byALV-Laser Vertriebsgesellschaft m.b.H. Langen,Germany. A green Nd:YAG DPSS-200 laser(532 nm) from Coherent (Auburn, CA) with anoutput of 200 mW was used. The thermostatedsample cell is placed on a motor-driven precisiongoniometer (� 0.018), which enables the photo-multiplier detector to be moved from a 208 to a 1508scattering angle. The intensity time–correlationfunctions (ITCF), g2(t), are recorded with an ALV-5000E multiple t digital correlator with fastoption. The minimal sampling time of this cor-relator is 12.5 ns. The cylindrical sample cellsare made of Suprasil1 quartz glass by Hellma,Muellheim, Germany, and have a diameter of10 mm.

Refractive Index

The refractive indices of all samples are measur-ed with an Abbe Refractometer (ABBEMAT,Dr. Kernchen GmbH, Seelze, Germany) at 25.0�0.28C.

Dynamic Viscosity

The viscosity of the external phase (PG/H2Omixture) was determined at different shearrates (0.2–200 s�1) using the Couette principle(RHEOMETRICS SCIENTIFIC, Bensheim,Germany) at 25.08C. Newtonian behavior wasobserved. These data are necessary to calculatethe hydrodynamic radius, Rh, from the diffusioncoefficient.

Dynamic Light Scattering

In DLS experiments, time-dependent fluctuationsof scattered light intensity due to Brownianmotion are measured. The normalized field auto-correlation function, g(1)(t) was derived from thescattered intensity with the Siegert relation:11

gð1ÞðtÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 �

I0s ðtÞIsðtþ tÞ

� �I2

s

� �s

ð3Þ

In very dilute monodisperse solutions,11 g(1)(t) is

gð1ÞðtÞ�� ¼ expð�GtÞ ð4Þ

In practice, the 0 delay (t¼ 0) intercept is ‘A1’,where A1� 1, and the base line is ‘A0’, where

Figure 2. Chemical structures of species used information of the o/w microemulsions.

Table 1. Compositions of the Microemulsions Studied

Microemulsion Composition m

1 Poloxamer 331/Tagat1 O2 (3:2) 4Eutanol1 GPG/water (2:1)

2 Poloxamer 331/Tagat1 O2 (3:2) 4IPPPG/water (2:1)

3 Poloxamer 331/Tween1 80 (3:2) 4IPPPG/water (2:1)

732 SHUKLA ET AL.


A0� 0. Thus, the following equation can bewritten:

gð1ÞðtÞ�� ¼ A0 þ A1 expð�GtÞ ð5Þ

where the relaxation rate, G, is connected with thetranslational diffusion coefficient, Dc, according to

G ¼ Dcq2 ð6Þ

where Dc is the translational diffusion coefficientof the molecules, y is the scattering wave vector,and q¼ (4pn/l)sin(y/2).

However, most systems display some poly-dispersity and should give rise to a sum ofexponentials:

gð1Þ tð Þ��

¼ A0 þ A1 exp �G1tþ1

2G2t2 � 1

6G3t3 þ � � �

� �ð7Þ

The correlation function of polydisperse solu-tions can be analyzed in terms of moments orcumulants.12 Logarithms of [ g1(t) � A0] are thenplotted as a function of the delay time, t, and fittedby a polynomial (A1 � G1t þ 1/2G2t

2 � 1/6G3t3),

from which the first cumulant (G1), secondcumulant (G2), and third cumulant (G3) wereextracted.

The apparent diffusion coefficient, Dapp, can beobtained from the first cumulant as follows:

<Dapp >¼ G1

q2ð8Þ

Another quantity, which is often used to specifythe polydispersity index Q, is the normalizedvariance:13

Q ¼ G2

G21

ð9Þ

The width of radius distribution, s, can be relatedto the experimentally determined polydispersityindex, Q, as follows:13

s ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR2

R2� 1

s¼

ffiffiffiffiQ

pð10Þ

The measured diffusion coefficients, Dapp, weretheoretically fitted, using a linear interactiontheory developed by Corti and Degiorgio,14 inwhich diffusion coefficients depend on the volumefraction (j) of the diffusing particles as follows:

Dapp ¼ D0 1 þ kdfð Þ ð11Þ

where D0 is the limit of D(j) as j! 0, andkd¼ ktþ kh is the linear correction to D(j).

Particle size, Rh (hydrodynamic radius), can bedetermined using the Stokes–Einstein equation,

Rh ¼ kBT

6phD0ð12Þ

where kB is the Boltzmann’s constant, T is theabsolute temperature, and h is the coefficient ofviscosity of the solvent (the continuous phase inthe case of a ME).

The perturbation coefficients kt and kh are dueto thermodynamic and hydrodynamic effects,respectively. Perturbation coefficients can be re-lated to the interaction potential of the particlesU(x), where x¼ (r� 2Rcore)/2Rcore and r is thedistance between the center of the two particles:15

Kt;h ¼ Khst;h þ

Z 1

0

dxGt;hðxÞ 1 � e�UðxÞ=kTh i

ð13Þ

where

Khst ¼ 8 ð14Þ

GtðxÞ ¼ 24ð1 þ xÞ2 ð15Þ

Khsh ¼ �6:44 ð16Þ

GhðxÞ ¼ �12ð1 þ xÞ þ 15

81 þ xð Þ�2

� 27

641 þ xð Þ�4� 75

64ð1 þ xÞ�5 ð17Þ

Knowledge of the origin of the potential U(x)interaction in the nonionic surfactant is verylimited. Because there are no charged groupspresent, one would expect that direct electrostaticinteractions do not play a major role, contrary tothe case of the ionic surfactant system. It is rathernatural to assume that the hydrogen bonding ofwater is responsible for interparticle interactionand, in fact, this view has been frequently usedin the literature.16–18 We too accept this viewand consider hydration force, which originatesfrom the increased structuring of the wateraround the head group of the surfactant mole-cules. Pair potential energy for hard sphere, vander Waals, and hydration forces can be written as(see Figure 1)

UðxÞ ¼ Ueff ðxÞ ¼ UbareðxÞ þUdirðxÞ þUindðxÞð18Þ



where bare Ubare represents a short-range strongrepulsive force that ensures the identity of the MEand prevents the overlap of different MEs. Thisforce is represented by the simple hard spherepotential.

Ubare ¼ Uh;s ¼1 for x < 00 for x < 0

�ð19Þ

The second term in eq. 18, Udir(r), represents thedirect interaction between droplets that does notdepend explicitly on the degrees of freedom ofwater. Dispersion forces give the dominant con-tribution to Udir in the case of nonionic micelles.This term gives rise to attractive forces that arestate independent. For our present purpose, we donot assume that the specific form of Udir(r) isessential. Rather, we assume simple van derWaals force, which is given as16

UdirðxÞ ¼ � A

12

�x2 þ 2x� �1þ x2 þ 2xþ 1

� �1

þ 2 lnx2 þ 2x

x2 þ 2xþ 1

��ð20Þ

where A is a Hamaker constant.Finally, the third term, Uind, should involve the

degree of freedom of water. This indirect interac-tion should correspond to repulsive force:16

Uind ¼ U0 exp �ðx=DÞ=ð1 þ xÞ½ � ð21Þ

where D, which is a measure of the width of theshell of structured water, decreases with increas-ing temperature so that attractive interactionbecomes more effective, and the possibility ofphase separation occurs at high temperature. Theparameter U0 is the change in energy due toremoval of H2O molecules (see Figure 3).

RESULTS AND DISCUSSION

It is known that the particle size distribution isone of the most important characteristics of MEsfor the evaluation of their stability and penetra-tion mechanism into the skin.19,20 The apparentdiffusion coefficients, Dapp, and polydispersityindex, Q, were determined for all ME series, withthe cumulant fitting procedures described ineqs. 7–10, from the normalized field auto correla-tion function g1(t). The second cumulant is verysensitive to the correct value of the baseline A0

because of the high correlation between these twoparameters. The value of s, calculated from thesecond cumulant, was 0.24–0.50 for all samples.As already pointed out by several authors,21–23

the second cumulant, G2 (representing only asmall correction to the shape of the correlationfunction), overestimates the polydispersity of theME droplets. Because of these uncertainties in theestimation of s, we are inclined to give less weightto this parameter. We know that our systems haves values of �0.20, as measured by SANS forsimilar systems.7 Furthermore, as already pre-dicted theoretically from the multiple chemicalequilibrium approach,24we assume that the sizepolydispersities (s) for a stable ME should be inthe range of 0.1 to 0.25.

The intercept and slope of the plots of Dapp

versus j (Figure 4) yield D0 and D0 � kd¼D0(ktþ kh), respectively, and these values arelisted in Table 2. The Rh, which is supposed torepresent the oil core surrounded by a surfactantfilm, was calculated using D0 from eq. 12 and theresults are listed in Table 2. It was observed thatthe kind of oil marginally influenced the droplet

Figure 3. Schematic representation of two overlap-ping droplets. The dashed region corresponds to theoverlap volume. The meanings of the various radii areexplained in the text.

Figure 4. Diffusion coefficient, Dapp� 3% error as afunction of the volume fraction for ME1, ME2, and ME3.

734 SHUKLA ET AL.


size, which is consistent with previously publishedresults.7 The head group of nonionic surfactants(Tween1 80 and Tagat1 O2) is attached to thehydrophobic part of the molecule by polymeriza-tion of ethylene oxide (EO), and the number ofmonomer units per molecule cannot be exactlycontrolled. For block copolymer (Poloxamer 331),there may also be a distribution in the hydrophobicdomain that involves polymerization of propyleneoxide (see Figure 2). The uncertainty in chainlength of each surfactant molecule due to theirsynthesis may be the reason that MEs withdifferent kinds of oil have marginally differentdroplet sizes.

If one assume that the suspended particles arehard spheres, one then obtains kd¼ ktþ kh¼ 1.56.This result is clearly not in accordance with theexperimental value. A possible explanation forthe observed greater value for kd is to assume thepresence of the effective interaction except thehard sphere force between suspended particles.The effects of interparticle interactions on observ-ed droplet diffusivities have been investigated byCorti and Degiorgio.14 Using the generalizedStokes–Einstein equation, with the hydrody-namic correction of Felderhof4 and a pair potentialfrom DLVO theory,10 they obtained quantitativefits to their measurements of sodium dodecyl sul-fate (SDS). Basically, the same procedure is usedin this work to correct the measured ME diffusioncoefficients for particle–particle interactions andhence to calculate the radius of the ME droplets. Inthis paper, the model effective potential shown inFigure 1 is used instead of pair potential usingDLVO theory. Because of phase separation at lowand at high volume concentration of dispersedphase, it was not possible to freely vary the volumefraction of the dispersed phase as was the casein previous work.14 All measured data are fittedusing eqs. 4–21. Both integrals in kt and kh cannotbe solved without a lower cutoff (xL> 0), becausethe attractive part of the potential diverges. Thegenerally accepted value xL¼ 0.04, which cor-responds to the expected Stern layer thickness,has been used.25,26

The unknown parameters for the least squarefitting procedure are the Hamaker constant A,the width of the layer of structured water D, andthe change in energy due to removal of watermolecules U0. Because of the functional depen-dence of the described theory, only one value forthe slope of the curve has to fit. The change inenergy U0 due to removal of water molecules fromthe overlapping volume is calculated (see Appen-dix I) for ME 1, ME 2 and ME 3 at volume fractionj� 0.12. These values are listed in Table 2. A com-plication arises in that the value of the Hamakerconstant is not known and cannot be easily esti-mated. Fortunately, the results are not stronglydependent on the value of the Hamaker constantbecause the repulsive part of the potential isdominant. Therefore, A¼ 25 kBT was used in thepresent fitting, which is a well-chosen value for thestability of colloids.10 An additional complicationarises in estimating the volume fraction, j, for theMEs. Partition coefficients for surfactants are notknown. It was assumed that the surfactant wascompletely incorporated into the droplet phase.For this assumption, the volume of the dispersedphase is just equal to the sum of the volumes of thesurfactants and the added oil. All experimentaldata were fitted by means of only one free para-meter that defines the interaction potential; thatis, width of structured water, D. The behavior ofthe potential for various values of D is given inFigure 5. The shell of structured water is almostthe same for all series of MEs. The combinationof repulsive forces and the attractive van derWaals interactions may produce a significantpotential energy barrier between the primarymaximum (Ueff)max at r¼ 2Rcore, and the shallowsecondary minimum (Ueff)min at a separationdistance r somewhat larger than the 2Rcore. Letus review the situations relevant to our experi-ment in terms of the parameters that characterizethe potential.

1. A large positive (Ueff)max and (Ueff)min

practically equal to zero. In this case, theenergy barrier prevents the particles from

Table 2. Dynamic Light Scattering Resultsa

ME Dapp �108 [cm2/s] Rh [nm] kapp U0 104 [kBT] D Ueff [kT] tf,Ueff¼0 10�5 [s] W

1 1.51 15.18 2.93 12.57 0.0191 76.12 9.8 10157

2 1.80 12.21 2.07 6.56 0.0193 193.66 5.10 1084

3 1.70 13.55 2.72 8.87 0.0194 266.43 7.01 10115

aErrors are smaller than �3% for radii and �5% for polydispersity index Q.



coming close enough and the colloid isstable.

2. A significant and positive (Ueff)max and a deepsecondary minimum (Ueff)min. In this case,when the particles reach separations corre-sponding to a separation minimum, theparticles stick together and flocculationoccurs.

3. A low value of (Ueff)max and a deep secondaryminimum (Ueff)min. This case corresponds toan intermediate situation, when the systemevolves towards an irreversible coagulationstarting from particles that colloid and sticktogether; that is, individual particles areadded one at a time to a growing cluster.

4. Finally, in the absence of repulsion, everycollision give rise to irreversible coagulationand phase separation will occur.

We observed (Ueff)max practically infinite atr¼ 2Rcore due to hard sphere potential (seeFigure 3). As shown in Figure 5, the repulsive partof (Ueff)min increases with increasing D value. Forfitted value of D (�0.02), (Ueff)min is practicallyzero, indicating that our systems follow case 1. Therapid coagulation time, tf,Ueff¼ 0, and ratio betweenrapid and slow coagulation time, W, which is themeasure of colloidal stability, were also calculatedusing interaction potential (see Appendix II). Theobserved effective potential diminishes the velo-city of coagulation by a factor of �10100. Thesestability results are consistent with observedinfinite stability (>3 years) of our MEs.

SUMMARY

Stable o/w MEs were studied with DLS measure-ments in the region of the phase diagram wheresurfactant-covered oil droplets were formed. Adilution procedure was used to determine in-formation about diffusion coefficient, droplet size,interparticle interaction, and polydispersitiesfrom experimental data. The second cumulantmay considerably overestimate the polydispersityat higher concentrations, where a high correlationbetween the second cumulant and base line isexpected. The stability and size of the ME dropletswere estimated. The treatment accounts for thehard sphere, van der Waals interaction potentialamong the droplets as well as local free energyassociated with a given configuration of the drop-lets. A local free energy associated with a givenconfiguration should correspond to a repulsiveforce. In fact, the head groups of surfactantmolecules can form hydrogen bonds with thewater molecules, and an increased structuring ofwater around each ME should be present. Thewidth of the layer of structured water wasestimated for our systems. The repulsive part ofthe interaction increases with D. The value of Dis consistent with the value necessary for stableMEs. When two ME droplets come close to eachother at a distance such that there is somesuperposition of the shells of the structured wateraround the two droplets, an effective repulsiveforce should arise because of the breaking ofhydrogen bonds due to removal of water mole-cules from overlapping volume. Change in energyU0 due to removal of water molecules fromoverlapping volume was estimated for our sys-tems. The observed particle–particle interactionfor MEs increases the coagulation time by thefactor W� 10100 in comparison with rapid coagu-lation, when there is no interaction between theparticles except a very steep attraction whenparticles touch each other. This result is consis-tent with the observed stability of our systems.Different kinds of oils were used in this study,which shows that the system has droplet size thatis independent of the kind of oil. It is clear that theinterpretation of scattering data on concentratedMEs is by no means simple. In our analysis, theuse of a suitable interparticle interaction modelprovides one possible approach to the interpreta-tion of data obtained from concentrated systems.In the absence of such a correction, the value ofdroplet size obtained is taken as indicative onlyfor the presence of ME droplets.

Figure 5. Behavior of the interaction potential as afunction of the width of the shell of structured water.

736 SHUKLA ET AL.


APPENDIX I. CHANGE IN ENERGY DUE TOREMOVAL OF WATER MOLECULE FROMSURFACTANT LAYER

Different contributions to effective interaction areshown in Figure 1. DLS yields the hydrodynamicradius Rh, which is supposed to consist of the oilcore and surfactant film containing some solventmolecules (i.e., the length of the surfactant film isRh�Ri). In this study, the oil/surfactant ratio waskept constant at 1/4 for all our samples, whichmeans

R3i

R3h � R3

i

¼ 1

4) Rh

Ri¼

ffiffiffi5

3p

ðA1Þ

A model of a penetrable droplet is shown inFigure 3, where 2r* is the distance between twodroplets of radius Rh, and Ri is the oil coreradius. One can determine r* using the followingassumption

r� ¼ RiþRh�Ri

2¼ Rh þ Ri

2ðA2Þ

Substituting the value of Ri from eq. A1 intoeq. A2, one gets

r� ¼ 0:7924Rh ðA3Þ

If two droplets of radius Rh are separated by adistance r (r< 2Rh), then the volume of over-lapping is

Voverlapping ¼ p6

2Rh � rð Þ2 2Rh þ r

2

�ðA4Þ

where r¼ 2r* in our case.If we assume that each surfactant molecule

tends to arrange itself in such a way that it sub-tends a constant volume, then the number of sur-factant molecules coating each droplet is given by

ns ¼fs

ns n

.ðA5Þ

where n¼ji þ js/(4/3)pRh3 is the total number of

droplets, ji is the volume occupied by oil, js is thevolume occupied by surfactant, ns ¼ Ms

Nadis the

volume of a surfactant molecule, and Ms, Na, andds are the molar mass of surfactant, Avogadronumber, and density of surfactant, respectively.

The number of water molecules bound per ethy-lene oxide (EO) group is 2, and the average asso-ciation energy per water molecule is 22 kJ/mol.27

The total number of EO groups associated witheach droplet, nEO, equals the number of EO groupsassociated with each surfactant molecule, ns*. The

total number of water molecules bound to eachdroplet, nw, equals 2*nEO.

The volume of surfactant shell incorporated ineach droplet is determined by

v1 ¼ 4

3p R3

h � r�ð Þ3n o

ðA6Þ

The number of water molecules in overlappingvolume is determined by

noverlapping ¼ nw � voverlapping

v1ðA7Þ

The change in energy due to removal of watermolecules in overlapping region Uo is

U0 ¼ 22

Na noverlappingkJ ðA8Þ

APPENDIX II. STABILITY OF COLLOIDS

The coagulation velocity when Ueff¼ 0 (i.e., whenthere is no interaction between the particles,except a very steep attraction when particlestouch each other) is equal to:10

GUeff¼ 8pRDappn ðA9Þ

where R, Dapp, and n are the particle radius,diffusion coefficient, and total number of particleper unit volume, respectively.

The time of coagulation10 when Ueff¼ 0 can bedetermined with eq. A8 using the Stokes–Einsteinrelation

tf ;Ueff ¼ 0 ¼ 3h

4kTnðA10Þ

where h � is the viscosity of the continuous phase.The interaction Ueff diminishes the velocity of

coagulation by a factor; that is, the coagulationtime increases by a factor,

W ¼Z 1

2

eUeff=kT

S2ds ðA11Þ

where the particle completely loses its velo-city after traveling over a distance. To get themagnitude of Ueff, Ueff is integrated from xL toinfinity.

The time of coagulation in the presence ofinteraction potential is

tf ¼ tf ;Ueff�0W ðA12Þ

A colloid may be termed stable when it does notflocculate, for example, in a week or a month,which means that the time of flocculation should



be >106 s. Consequently, the ratio between W ofrapid coagulation and that of slow coagulationmust surpass 105 for diluted and 109 for veryconcentrated colloids to give the colloids reason-able stability.

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