PHYSICAL REVIEW A VOLUME 39, NUMBER 8 APRIL 15, 1989

Dynamical properties of a potassium oleate microemulsion determinedby photon-correlation spectroscopy

G. Maisano and F. MallamaceIstituto di Fisica dell'UniUersita, Vill. S. Agata, Casella Postale 55, 98166 Messina, Italy

N. MicaliIstituto di Tecniche Spettroscopiche del Consiglio Nazionale delle Ricerche, Vill. S. Agata,

Casella Postale 55, 98166Messina, Italy(Received 4 November 1987; revised manuscript received 14 December 1988)

Light-scattering correlation spectroscopy measurements were made on a water-in-oil microemul-sion at a constant volume fraction of water P as a function of the temperature. The density correla-tion function shows nonexponential decay, which is analyzed in terms of cooperative or structuralmotion and of single-particle diffusional motion. The system shows a critical behavior at T =41'C.The fast relaxation rate (cooperative diffusion) gives information about the details of the criticaltransition via the hypothesis of droplet clustering. The slow relaxation rate is connected to the self-

tracer motion.

I. INTRODUCTION

Under suitable conditions a mixture of water and oil(hydrocarbon), together with other chemicals acting assurfacant and cosurfactant, becomes a thermodynamical-ly stable, macroscopically homogeneous, and opticallyclear system called a microemulsion. ' Its microscopicstructure depends on the thermodynamic variables(volume fraction P, temperature, pressure ) and chemi-cal nature of the interface component. At lower waterconcentrations, microemulsions consist of very small wa-ter droplets dispersed in oil, while at a higher water con-tent the situation is reversed and the system consists ofoil droplets dispersed in water. Between these twophases, the system shows an intermediate situation whichconsists of layers of surfactant separating alternate layersof water and oil, or other more complicated structures(bicontinuous structures).

A number of experimental and theoretical works (lightscattering, ' ultrasound, neutron scattering, ' dielec-tric properties, " electrical conductivity, ' and viscosityhave discussed the well-characterized structural proper-ties and thermodynamical behavior of microemulsions.The phase where the system consists of a water-in-oil mi-croemulsion is very different from the reversed one, be-cause the interaction among droplets, which is an attrac-tive interaction' in the first case, becomes ionic (Coulom-bic forces) in the second one; in particular, the water-in-oil microemulsion may be viewed as a colloidal suspen-sion and the small-angle neutron scattering (SANS) datahave been explained in terms of a phenomenological po-tential function having a hard core plus a Yukawa tailwhich represents the attractive interaction. ' The water-in-oil phase is a very interesting one, often showing a crit-ical behavior. By changing the thermodynamical param-eters such as pressure or temperature, ' molar ratio ofwater to surfactant, ' or the chemical nature of the com-ponents (e.g. , the alkyl-carbon chain length of the exter-

nal oil phase, ' the homogeneous system may in fact be-come unstable and separate into two stable phases. Thetemperature at which this separation occurs is called theupper cloud-point temperature. This transition showsthe typical characteristics of a critical transition, such asopalescence and the slowing down of concentration Auc-tuations, and for this reason the upper cloud temperatureis often called the critical temperature' and indicatedwith the symbol T, . A dynamical slowing down' and anonexponential decay of the density correlation functionhave recently been observed in an AOT (sodium di-2-ethylhexylsulfosuccinate) water-oil microemulsion. Thisbehavior has been presumed to be connected with a tran-sition in the structured water droplets similar to the glasstransition in supercooled liquids or in dense polymericsolutions. A nonexponential decay was also observed inwater-xylene-SDBSO (sodium dodecylbenzene sulfonate)-hexanol by Pusey. In a previous work we have investi-gated the phase diagram of a series of microemulsions ofwater in n-dodecane, with potassium oleate and hexanolas surfactant and cosurfactant; in this system the headgroups of the surfactant (—CO& K+ ) are partially disso-ciated in the water (internal phase) and the alkyl chainsextend into the continuous oil region. As the dropletsbear no net charge, the only possible interaction is attrac-tive. ' In our investigation, a miscibility gap was ob-served; across the boundary phenomena arise similar tothose observed in a phase transition.

Relatively few systems of microemulsions have been in-vestigated near the cloud point with the use of quasielas-tic light-scattering techniques; their analysis is complete-ly based in the idea that the entire contribution to thephoton-correlation function is due to critical scattering.The aim of this work is to take into account also the con-tribution of the single-particle motion to the light scatter-ing. This contribution due to the narrow-polydispersityeffects in a system of interactive particles can give furtheruseful information about the critical process.

39 4103 1989 The American Physical Society

4104 G. MAISANO, F. MALLAMACE, AND N. MICALI 39

II. EXPERIMENT

Quasielastic light-scattering measurements (by pho-ton-correlation spectroscopy) are performed on a singlephase of the previously described microemulsion at afixed volume fraction of water as a function of the tem-perature T. At the investigated concentration (/=0. 28)our microemulsion is of the water-in-oil type and itscloud-point temperature is T, =41'C. The sample wasprepared by adding n-dodecane to hexanol and high-purity potassium oleate in suitable proportions; waterv as added, drop by drop in the desired quantity; the mi-croemulsion was filtered into an optical cell. A 10-mW

0He-Ne laser operating at 6328 A, a 128-channel single-clipped homemade correlator and a temperature-stabilized spectrometer (temperature regulated within+0.01'C) were used for our experiment. Data were col-lected at the scattering angle of 60' that corresponds toan exchange wave vector k = 1.309 X 10 cm '. Thetransmitted light was not significantly attenuated, so mul-tiple scattering effects could be neglected.

Measurements were also made with the heterodynetechnique in order to test the Gaussianity of the scatteredfield; the results are in agreement with those using thehomodyne technique.

On the other hand, SANS (Ref. 10) and light-scattering' measurements on an AOT microemulsionand more recent' SANS measurements on a potassiumoleate microemulsion, very similar to the one investigatedhere, show that in these systems the droplet structurepersists in the region of the phase diagram for a volumefraction higher than our value. Therefore the so-calledbicontinuous structure apparently never set in.

The first-order correlation function of the dielectrictensor fluctuations,

& b e(~)AE(0) )& b, e'&

is connected with the experimental correlation of the in-tensity Iluctuations in the scattering light Cx-(w) by a sim-

ple form

where ~ is the delay time; k is the exchange wave vector( l. 309 X 10 cm '); K is the clipping level; Nfi=

& nx ) & n )N is the theoretical value of the correlationfunction at r~ ~; & nz ) and & n ) are, respectively, theaveraged values of the clipped signals and the signal; X isthe total number of the sample time; and f( 2, T) is afunction that depends on the coherence area 3 and thesample time T.

III. RESULTS AND DISCUSSION

Since the experimental value of the exchange wave vec-tor k (1.309X10 cm ') is smaller than the one corre-sponding to the possible peak in the structure factor, theself-correlation density function can be written as

g'"ccexp( —D, k r),

where D, is the cooperative diffusion coefficient,

D, = f dt —g &U, (r)u (0)),3S (k) o N, .

and u, is the velocity of the ith particle, and S(k) is thestructure factor.

In Fig. 1 we plot ln[g'' (k, r)] against w for differentvalues of temperature T, after subtraction of the uncorre-lated dc "backgrounds" and normalization for the valueof the correlation function at ~=0. As can be seen, a sin-gle exponential curve cannot fit the experimental resultsand the lower the temperature, the more marked is thedeparture from exponential behavior. This result is notsurprising, because as shown in dispersed systems of in-teracting particles' ' (with relatively high volume frac-tions and with fairly narrow size distributions), the light-scattering correlation function is, to a good approxima-tion, composed of two independent modes with well-separated decay times. The faster mode can be associatedwith the "average" collective coefficient and the slow onewith the average self-diffusion one. In our experimentaldata the deviation from a single exponential behavior ismore marked than that due to the polydispersity alone.The measured degree of polydispersion by SANS (Refs. 8and 17) in a potassium oleate microemulsion is 0.2 andthis value should show only a little deviation from a sin-gle exponential in the correlation function. Therefore theobserved long tail in the correlation function is essentiallydue to the narrow polydispersity in a system with inter-particle interactions, ' ' and it can be connected with anaverage self-diffusion coefficient, which determines therate at which matter is transported through the system.The self-diffusion coefficient is

D~= —,' dt U; tv;0

0

Therefore, in this frame, the experimental correlationfunction can be considered the sum of two contributions;the first one is cooperative in character and the secondone is due to a self-macroscopic motion.

We use the following two-exponential equation in or-der to fit our data:

g"'(k, r) =x exp( —I,r)+(1—x )exp( —I,r),where I,. =k D, and I",=k D, .

According to the above considerations for interactingand weakly polydisperse systems the diffusion coefficientscan be considered as average collective and self-diffusioncoefficients. The term x in Eq. (1) is connected with thepolydispersity; in fact, it is shown that' ' '

I & F(k) & I'

& IF(k) I'&

where F(k) is the form factor of the droplets and &. )

represents an average weighted by the distribution ofdroplet sizes. In general, x is a complicated function ofthe exchange wave vector k, the interaction, and the tem-perature, and we can write x =x(k, T).

In Fig. 1 the dots are the experimental data and thesolid line represents the best fit with Eq. (1). As can be

39 DYNAMICAL PROPERTIES OF A POTASSIUM OLEATE. . . 4105

-25 x]Q ~

—5xlp

20C~ ~ ~ ~ ~

OCl

(~oll

33 OC

10

r(sec ')

] x]P20

-- ~--).)0-530 7(oc) 40

I

0.001I

0.002t (sec)

I

0.003103

FIG. 1. Logarithm of the photon-correlation function atseveral temperatures; the nonexponential decay is clearlyshown. Dots are experimental data, solid lines are the best fitswith Eq. (1).

seen, it well reproduces the experimental data. Thereforewe analyze separately the two contributions to thedensity-Auctuation correlation function, namely, thecooperative and the self-contribution. The first one willbe connected to the critical transition in terms of a clus-tering of the particles with fixed form and polydispersity,the second one to a self-macroscopic motion. This pic-ture will be confirmed by analysis as a function of thetemperature of the weight x of the cooperative correla-tion function.

In Fig. 2 we report the results of this analysis, plottingI, and I", as a function of the temperature. As far as I,is concerned, two different behaviors are evident. In therange of temperatures far from the cloud-point tempera-ture (20—36'C) the microemulsion is stable and only asmall increase in the linewidth with temperature can beobserved. This small increase in I, is essentially due tothe temperature and viscosity effects. ' In the inset ofFig. 2 we report the temperature behavior of the relaxa-tion time r, = I/I „and the n-dodecane viscosity (con-tinuous oil phase). It is evident that in the temperaturerange where our microemulsion is stable, the behavior of

versus T is very close to the continuous oil-phaseviscosity behavior.

To describe the temperature dependence of I, close tothe cloud-point temperature I „we hypothesize the pres-ence of droplet clusters in the microemulsion. ' In fact,as shown in Ref. 24, in the critical region, the mean di-mension and the polydispersity of the droplets remainsunchanged. The dynamic and intensity light scatteringmeasure a quantity connected to the correlations amongthe droplets. Moreover, recently, ' a cluster percolationmodel in microemulsions was used to analyze data ofelectrical conductivity near the critical transition.

Therefore, near the critical point, we can use the well-

1020

I

30I

T ('C) I.O

FIG. 2. Linewidth values (dots) of single-particle I, and col-lective modes I, as a function of the temperature; solid lines areonly guides for the eye; in the inset are plotted the viscosity ofthe n-dodecane (dashed line) and ~, = 1/I, vs T (dots).

T —TC

(2)

For the exponent v the same theories predicted a value of3

In this frame we have analyzed the data concerningthe behavior of the cooperative part of the correlationfunction near the cloud-point temperature. We extractthe correlation length g in terms of the measured I",.The results obtained from this analysis are shown in Fig.

known ' results of the mode-mode coupling approach ofKawasaki in which the measured linewidth is the follow-ing power series of k:

I, -D, k (1+—',g k ),where g is the correlation length of concentration Auctua-tions of droplets' ' and the diffusion coeKcient has thesame value as in the Stokes-Einstein relation:

kbTD, =

6m.g*g

where g is the frequency-dependent shear viscosity. ' Ina microemulsion system the solvent shear viscosity g canbe used for g*.'

In the framework of the theories of critical phenome-na, ' the dependence of g on the reduced temperaturee=( T, —T ) /T, is given by the simple power law

4106 G. MAISANO, F. MALLAMACE, AND N. MICALI 39

3. We have an experimental value of v of 0.67+0.01,which is, within the accuracy of our data, in good agree-ment with the universal critical exponent value expectedfor a binary fluid.

For a complete analysis of the critical process it wouldbe useful also to have intensity measurements as a func-tion of the temperature and the exchange wave vector k.In such a way, it is possible to obtain a measure of thecorrelation length g in terms of the Ornstein-Zernike re-lation and the extrapolated forward-scattered intensityI(0). The correlation lengths g, at temperatures near T,obtained from the intensity data, are in agreement withthose obtained from the linewidth results (connected byus to the cooperative part of the photon-correlation func-tion). The analysis of the g values in terms of Eq. (2)gives, in such a case, v=0. 66+0.02. Further informationabout the critical behavior of our microemulsion systemcan be given in terms of the forward-scattered intensityI(0). This quantity can be obtained by an extrapolationof the I(k) data. The forward intensity is proportional tothe osmotic compressibility that diverges ' with the tem-perature at Kr-e ~ and therefore I(0)-E . We haveanalyzed this latter quantity and the value obtained for yis y = 1.21+0.05, in agreement with other measure-ments' in analogous systems. This last result is in agree-ment with our point of view that the cooperative part ofthe density correlation function takes into account thecritical process. Because the aim of this work is not acomplete study of the critical behavior [which is wellstudied in the literature in analogous systems by lightscattering and SANS (see, for example, Ref. 10, and refer-ences quoted therein)], but a study of the correlationfunction for density fluctuations, considering our mi-croemulsion as a system of interacting nearly poly-disperse particles, we do not detail further measurementsconnected to the critical phenomenon. In any case, wecan conclude, according to previous works, that thephase separation is driven, in analogy with binary fluids,by the concentration fluctuations of microemulsion drop-lets.

The linewidth I, is connected with the self-diffusionmotion coefficient of the droplets Dz, that gives informa-tion about the (true) motion of water in the continuous

oil phase. This diffusion coefficient should be the same asthe one obtained with a tracer diffusion experiment.

As far as the temperature dependence of I, is con-cerned, its variation with T (see Fig. 2) shows an increasefar from the cloud-point temperature. When the criticaltemperature is approached, I, decreases. This behavioris not surprising; in fact, measurements of viscosity per-formed in the same sample (/=0. 28) show a similartrend.

On the other hand, both the coefficients Dz and g, areconnected to a self-friction coefficient that takes the rela-tive motion of the particles into account.

In the plot of Fig. 4, we report the weight x of thecooperative correlation function versus the temperature.As can be seen, near the cloud point, the weight x in-creases sharply, showing that in this region the coopera-tive motion becomes dominant with respect to the self-motion contribution. This fact confirms our hypothesisthat the behavior of light-scattering intensity near thecloud point is essentially due to the correlated regions ofdroplets. In particular, the behavior of the parameterx(k, T) can be detailed if we consider it in the two dis-tinct regions in temperature of our microemulsion.

Since our system is composed of scattering poly-disperse homogeneous spheres, Refs. 10 and 24 show that

~(F(k))~ J IF(kR)I f(R)dRx(k, T)= (3)

f F(kR )f(R)dR0

where f(R)dR is the probability of a sphere having a ra-dius between R and R +dR. The above ratio is usuallycomputed assuming for f(R) the Schultz size distributionin which the width parameter z is connected to thepolydispersity index p =o.

R /R by

p =(z+1)where R is the mean value of the radius of the dropletsand o.z is the root-mean-square deviation from the meansize. For our microemulsion at /=0. 26 the polydispersi-ty obtained from the Schultz distribution by SANS (Ref.17) is -0.2 and the same value is obtained in anotherwork for concentrations near ours by a Gaussian distribu-tion' (for large-z values, weak polydispersity, the Schultz

10Tc —4,1.5 'C

0.5—

100

00

I l I I I I

100t I I I I I I II

c 1000

Tc T

Fits. 3. Correlation length g vs reduced temperature; thestraight line is the best power-law fit of the data.

020 30

T(~)

I

4,0

FIG. 4. Weight values x of the cooperative correlation func-tion [Eq. (I)] vs the temperature; the solid line is a guide for theeye.

39 DYNAMICAL PROPERTIES OF A POTASSIUM OLEATE. . . 4107

distribution tends to a Gaussian form).As shown by Chen and Kotlarchyk the temperature

dependence of the polydispersity p is weak; in fact, whilethe droplet size is a weakly decreasing function of thetemperature, the polydispersity weakly increases with Tin the entire range explored. This occurs also in theupper cloud-point region where the correlation length gdiverges. Therefore the factor x is nearly independent ofT and our results in the temperature region 20—36'C arein agreement with this behavior. The sharp increase of xnear the cloud point is due to the kR dependence in thescattered intensity.

According to Eq. (3) the factor x is a function of thewave vector k, but for our purpose it is important to dis-tinguish among the characteristic length of he scatterersof the system in the two temperature regions. In the re-gion far from the critical temperature, the sizes of mi-croemulsion droplets (in our case ' R -40 A) are al-ways much smaller than the wavelength A. (kR &&1), sothat the scattering amplitude and therefore F(kR) is in-dependent from the scattering vector k. In the critical re-gion the scattering process is dominated by the diver-gence in the concentration fluctuations and the correla-tion length g increases, approaching T„u tnil becomingcomparable with A, (kg-1). The increase in the factor xwith the temperature is then due to the increase in thecorrelation length g.

IU. CONCLUSIONS

Finally, our data analysis of the dynamic light scatter-ing in a microemulsion with relatively high volume frac-

tion indicates that interactions between the droplets arevery important for the system; the cooperative part of thelight-scattering correlation function and the measuredvalues of the factor x give us the indication that this in-teraction can originate clusters. SANS (Refs. 10 and 24)results confirm this picture; in fact, the mean radius ofthe droplets and the Schultz width parameter z remainnearly constant also in the critical region. At the mo-ment this clustering effect in microemulsions is the objectof research in our laboratory in order to better clarifysuch a picture.

In conclusion we have carried out a study of the densi-ty correlation function in a potassium oleate microemul-sion at a concentration where the system is in the water-oil phase, as a function of the temperature. We have alsostudied the behavior of such a microemulsion system atthe cloud points. The correlation function has been ana-lyzed in terms of a two-exponential function that takesinto account both the self-tracer motion and a coopera-tive process where aggregation phenomena are present.

ACKNOWLEDGMENTS

We acknowledge financial support from the GruppoNazionale Struttura della Materia del Consiglio Na-zionale d elle Ricerche, Italy, and from Centro In-teruniversitario di Struttura della Materia del Ministerodella Pubblica Istruzione, Italy.

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4108 G. MAISANO, F. MALLAMACE, AND N. MICALI 39

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