Experimental evidence for a continuous variation of effective critical exponents in a microemulsion system

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Experimental evidence for a continuous variation of effective critical exponents in a microemulsion system

A.M. Bellocq, P. Honorat, D. Roux

To cite this version:

A.M. Bellocq, P. Honorat, D. Roux. Experimental evidence for a continuous variation of effec- tive critical exponents in a microemulsion system. Journal de Physique, 1985, 46 (5), pp.743-748.

�10.1051/jphys:01985004605074300�. �jpa-00210016�

Experimental ^{evidence for a} continuous variation of effective critical exponents

in a microemulsion system

A. M. Bellocq, P. Honorat and D. Roux

Centre de Recherche Paul Pascal, ^CNRS (GRECO Microémulsions),

Domaine Universitaire, 33405 Talence Cedex, ^France (Reçu le 8 octobre 1984, accepté ^{le 3} janvier 1985)

Résumé.

Nous présentons des résultats obtenus par diffusion de la lumière en différents points ^d’une ligne ^cri- tique ^d’un système microémulsion quaternaire. ^Cette ligne débute par ^un point critique extreme. Les points ^cri- tiques ^{ont été} approchés, ^{soit en} faisant varier la température ^à composition ^constante (chemin I), ^{soit en faisant}

varier le rapport eau sur savon (X) ^à température ^constante (chemin II). ^Pour chaque point critique examiné, ^les

valeurs des exposants critiques ^{03BD et 03B3} mesurées le long ^{des deux chemins sont} proches. ^{Elles varient} continûment des valeurs d’Ising à des valeurs très petites quand ^le point critique ^considéré s’approche ^du point critique ^extreme.

Ce comportement critique complexe ^est analysé ^à l’aide des résultats du diagramme ^de phase.

Abstract

We present light scattering results obtained for a four-component microemulsion system ^{in several} distinct positions ^{of a} critical line. This line starts at a critical end point ^{The critical} points ^were approached ^either by raising ^the temperature ^{at fixed} composition (path I) ^or by increasing ^{the water over} surfactant ratio X at constant temperature (path II). ^{For each critical} point investigated, ^{the critical} exponents ^{03BD and 03B3} associated with the X variable (path II) ^{are close to} those obtained with the temperature (path ^I). They vary continuously ^{from the} Ising ^{values to} largely ^{smaller ones as the critical end} point ^is approached. ^This complex ^{critical behaviour is}

discussed in relation with the phase diagram findings.

Classification

Physics ^Abstracts

64.70J

78.35

82.70K

1. Introducfiom

Over the past ^few years ^a large number of papers have been devoted to the experimental study of critical

phenomena in mixtures involving surfactants. Two-, three-, four-, ^{and even} five-component systems ^were investigated [1-11]. ^{This current} experimental ^interest

in the critical behaviour of micellar and microemulsion solutions is due, ^{to a} great extent, ^{to the} very intricate results obtained so far. Indeed, ^{if some} experimental

results [1-8] ^{are in} agreement with those found in pure ^or usual binary ^fluids, other data indicate a more

complex ^behaviour [9-11]. ^In most cases, the critical

phenomenon is measured by its critical exponents ^v and _y which characterize, respectively, ^the divergences

of the correlation length and the osmotic compressibi- lity of the solution. The experimental values of v and _y for pure compounds ^or binary ^{mixtures are in} good agreement ^{with the} Ising ^values [12] (v

0.63,

y

1.24). ^{In some} surfactant solutions, the values measured correspond ^{to either} negative [10] ^or positive [11]. deviations from the Ising indices. This behaviour is far from being understood. All the critical

points investigated ^{so far are} lower critical points.

Several theoretical treatments for such points ⁱⁿ binary solutions of small molecules have been develop-

ed on the basis of decorated lattice models [13].

These models account for the strongly asymmetric

coexistence curves found for a wide variety of mixtures which present ^a lower critical point ^{and a} closed-loop

coexistence curve. They exhibit non-classical critical behaviour at both _upper and lower critical solution temperatures. These models also predict ^{that when}

the _upper and lower critical point ^{coalesce at a critical}

double point, ^the exponent fl ^{which describes the} shape of the coexistence curve in the vicinity ^of

the critical point is renormalized at the critical double point ^to the value fl*

2 p [14]. Recently,

doubled exponents (v ^and y) ^{have been} experimentally

found near the vanishing miscibility gap of the

pseudobinary ^mixture guaiacol ⁺ (glycerol/water) [15].

In 1970, Griffiths and Wheeler [16] pointed ^{out the} importance of the field variables in multicomponent systems. ^In contrast to densities, the fields have the property ^that they ^{take an} identical value in all the

phases ^{which are in} thermodynamic equilibrium.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01985004605074300

744

Temperature, pressure and chemical potential ^are fields, whereas concentration or refractive index are

densities. These authors also indicate that the direction

parallel ^{to the} coexisting surface in the _space of fields is singled ^out by ^the phase transition itself. Theoretical

predictions ^{also state that} along ^{such a} path ^the

critical behaviour is described by ^{the same} exponents whatever the field considered. The main difficulty ^{is to}

find an appropriate ^field variable, different from temperature ^or pressure, which _may be experimentally

controlled. In most of the surfactant systems investigat- ed, the critical point ^was approached by raising temperature. Recently, ^{two of us} [4] have found evidence that in the oil rich part of microemulsion

phase diagrams, ^{the water over} surfactant ratio X has the property ^{of a field} variable; ^for example,

it behaves as a chemical potential. Particularly, they

have shown that the variable X can be used similarly

as temperature ^to approach ^{a critical} point [4, 6].

In order to improve ^our understanding ^{of the}

critical behaviour in surfactant solutions we have undertaken a systematic light scattering investigation

of microemulsion systems [4, 6]. ^We present ^{in this}

paper results obtained for different points ^{of a critical}

line of a quaternary mixture. Each critical point ^has

been approached along ^{two different} paths, ^either by raising temperature ^{at fixed} composition (path I)

or by increasing X at constant temperature (path II).

The mixture investigated ^{contains water, dodecane,}

SDS and pentanol.

2. Phase diagram

We have recently reported preliminary ^{results on the} phase diagram ^{of the} quaternary system ^{made of} water, dodecane, ^sodium dodecyl sulfate and _pen- tanol [17]. ^The phase diagram ^is very complex. ^A

detailed description ^{will be} published ^elsewhere.

One of the important features found for this system,

at constant temperature, ^{is a} critical line which connects a critical end point P’ ^with probably ^a

critical point P’ located in the limiting ternary ^face of the system without oil. The critical end point PI

is located in the oil rich part ^{of the} diagram (critical

dodecane concentration = 80 %).

At fixed temperature and pressure, the phase diagram ^{can be} represented ⁱⁿ using ^three independent

variables. Figure ^la gives ^{a schematic} representation

of the phase diagram ^{of the} system investigated ^{in a}

mixed density-field space. Two of the variables are

densities, ^{the alcohol} (cA) ^{and oil} (co) concentrations,

while the third one is a field, ^{the water over surfactant}

ratio X.

Experimentally, ^{we have studied} pseudoternary diagrams at constant X. In the following, ^these

sections are referred to as an X plane. ^{The X values} (expressed ⁱⁿ weight) corresponding ^to pA ^and PB

are, respectively, 0.95 and 6.6. Therefore, any X

plane ^defined by ^{a X value between} 0.95 and 6.6

intercepts the critical line at one critical point Pc

Fig. ^1.

^Schematic representations ^{of the} phase diagram

of the pentanol-dodecane ^water-SDS system. la) ^{At fixed} temperature in the dodecane concentration-alcohol con-

centration-X space. lb) ^{At X constant in} the dodecane concentration-alcohol concentration-temperature space. Re-

gion ^{1 : :} microemulsion. Region ^{2 :} two-phase region.

Region ^{3 :} liquid crystalline phase. Region ^{4 :} polyphasic region.

called Pj. The critical line is located at the boundary

between a microemulsion domain (region 1) ^{and a} two-phase region ^{where two} microemulsions are in

equilibrium (region 2). ^Gas chromatography analysis

of the phases ⁱⁿ equilibrium has shown that the tie- lines corresponding ^{to the} two-phase region ^{2 are}

located in the X planes [18]. This result reflects the field character of the variable X. In a given ^X plane,

this property ^{holds even} far from the critical point.

In addition, ^{a smectic} liquid crystalline phase (region 3)

is found at a lower alcohol content than the micro- emulsion phase. ^This mesophase ^{can contain a very} high ^amount of alcohol plus ^oil up ^to 98 %. ^{The two} one-phase regions ^{1 and 3 are} separated by ^a complex polyphasic ^domain (region 4) ^which includes several two-, three-, ^{and even} four-phase equilibria. ^In particular, ^{in the} vicinity ^of PI ^a three-phase ^domain

is observed where the liquid crystalline phase ^is

in equilibrium ^{with two} microemulsions. The critical end point P’ ^{is the} point where the three-phase

domain disappears. ^{In this} part ^{of the} diagram,

the liquid crystalline phase ^{has a} composition very similar to that of the critical microemulsion phase.

Figure la shows that it is possible ^to approach

any critical point Peat ^fixed temperature by varying

the field X (path II). Along ^this path, measurements are made in the single phase ^{domain as a} function of X with a constant concentration for pentanol ^and dodecane; these latter are fixed at their critical values.

If one takes into account the temperature variation,

the critical line becomes a critical surface. The section at constant X for this surface gives ^a critical line.

Figure ^{1 b shows a schematic} phase diagram ^{in the} temperature, pentanol and dodecane concentrations space, for fixed ^{X. In} this space, any critical point Pr

can be approached by varying temperature (path I).

Along ^this path, measurements are made in the single phase microemulsion domain as a function of tempe-

rature with a constant overall composition ^{for all}

the components (pentanol, ^dodecane, X).

3. Light scattering ^results.

The phase diagram ^{of the} water-dodecane-pentanol-

sodium dodecyl-sulfate system ^{shows a} critical surface in the CA-co-X-temperature space. As mentioned above, any critical point ^{can be} approached ^{in the} single phase ^domain along ^{two different} paths,

either by varying temperature ^{at fixed X} (path I) ^or by varying ^{X at fixed} temperature (path II). ^The

critical points investigated ^are located in the oil rich

region ^of the microemulsion phase diagram. ^{In this} region, ^{the structure can} be described as a dispersion

of water in oil micelles [19]. ^The phase ^transition

studied has been previously interpreted ^{as a} liquid-gas

transition due to interactions between inverse micelles

[17]. Therefore, ^{the order} parameter of the transition is likely the volumic micellar concentration, ^{which is}

related to the concentration of the mixture water and surfactant.

In order to analyse whether the critical behaviour is the same at several distinct positions of the critical

surface, ^{we have} undertaken static light scattering

measurements for several critical samples.

The samples ^were prepared in sealed cylindrical glass cells. The cell temperature ^was controlled within 0.01 °C. The angular dissymmetry of the total scattered

intensity ^{was obtained} by measurements at five

angles ⁰ ranging ^{from 400 to 1400. The} intensity

and angular distribution of light ^{scattered were}

measured as a function of temperature ^{or X. The data} for each value of T or X can be described by ^the

Ornstein-Zernike relation, ^{which is} expected ^{to be a} good approximation ⁱⁿ the critical region : I(q)

I(0)1(1 + q2 ,2), where q ^{is the} scattering vector q

⁴ nnj  ^sin 0/2, n is the refractive index of the solution, A ^{the wave} length ^{of the} light ^{in vacuum} (5 145 Á), , ^is the correlation length ^{of the concen-}

tration fluctuations, ^and /(0) ^is proportional ^{to the}

osmotic compressibility of the solution.

3.1 CRITICAL BEHAVIOUR AT FIXED COMPOSITION. -

Six critical samples ^defined by ^X

1.034, 1.207, 1.372, 1.552, 3.448 and 5.172 have been prepared.

Both j ^and 1(0) ^{increase as the} temperature ^{is raised} and diverge ^at the critical point. Figure ^{2 shows} log-log plots ^of 1(0) and, ^versus the reduced tempe-

rature c

(Tc - T)jTc for the six critical mixtures.

The data are well described by ^the power laws

The t index indicates that the critical point ^is approached by varying temperature. The values obtained by ^a least square fitting procedure ^for Tc, yt, v, and Ço ^are given in table 1. The value T,

deduced from the fitting ^{of the} experimental ^{data is} slightly different from the experimentally ^found tempe-

rature Td ^at which the mixture separates. ^{The diffe-}

Fig. ^2.

Log-log plots of the correlation length ç and the total intensity 1(0) ^versus the reduced temperature for the six

following ^{critical mixtures. a :} X, = 1.034; ^{b :} Xr ^{= 1.207; c :} Xc = 1.372; ^{d :} X, = 1.552; ^{e :} Xc

3.448; ^{f :} X,

^5.172.

746

Table I.

Critical temperature Tc and critical ratio X, for ^{the six} points ^measured along paths ^{I and II. Values} of

the critical exponents vt, }’ t’ vx and yx ^and of ’the scale factors jo ^and Io.

(*) Arbitrary ^unit.

rence found between T c and Td (0.1 ^{°C-0. 1 5} °C) ^{is of}

the order of magnitude ^{of the} experimental ^uncer- tainty ^{on the} Td measurement. The values of the exponents ^{v and} y determined in using Td ^are very close to those obtained when temperature ^{is set free} in the fit of the experimental data. The uncertainties

given ^{in table I} correspond ^to the variations found when the critical temperature considered is either

T c ^or Td. ^The Y tl v ratio is found to be constant and close to 2. A continuous decrease of the exponents vt and yt is observed as the critical end point PI ^is approached. This decrease is accompanied by ^an

increase of Ço ^{from 30 A to more than 300 A and also}

an increase of Io. ^{The ratio} lolçõ ^{is not found constant.}

New experiments ^{are under} progress in order to obtain more accurate values of the constant Io.

In order to examine to which extent the critical exponents ^are depending ^on the determination of the critical composition, ^{we have studied two} samples d1

and d2 located in the plane ^X

^{1.55 near} the critical concentration (d). The dodecane concentration diffe-

rence between the samples di ^and d2 (1.8 %) ^is large

in comparison ^{with the} uncertainty ^{on the critical}

concentration (0.5 %). ^The exponent ^{v measured} for the three ^cases has the same value (d1 : ^v

^{0.55 ±} 0.03 ; ^{d : v}

^0.53 ± 0.03 ; d2 : v

^{0.54 ±} 0.03).

These results suggest that the observed behaviour cannot be related to errors in the determination of the critical point.

3.2 CRITICAL BEHAVIOUR AT FIXED TEMPERATURE.

Three of the six critical points investigated ^above

have also been approached ^{in the} single phase ^domain

at constant temperature by varying ^X (path II).

The experimental study of each critical point requires

the preparation of several samples. ^Each sample ^has

the critical alcohol and oil concentrations; they

differ by the value of X. Along this second path, /(0) and ç ^{are found to} follow the _power laws

The x index indicates that the critical point ^is approached by varying ^X.

ex is the reduced variable X : Ex

(Xc - X)/Xc

where Xc is the value of X at the critical point. Figure ³ gives log-log plots ^of 1(0) and, ^versus ex. The values

of vx and yx obtained by ^a least squares fit of the data for the three critical points ^studied (X c

3.424,

1.55 and 1.207) ^are given ^{in table I. As} X, ^decreases

the values of vx and yx ^{show a} trend similar to that found for vt and yt (Fig. 4). ^{For a} given ^critical point Px the values of v and _y obtained along ^{the two} paths ^{are very close.}

4. Discussion.

The continuous variation of the effective critical exponents ^{v and} y that we have evidenced is _very similar to that previously reported by Corti et al. [10]

for nonionic aqueous micellar solutions ^as the _sur- factant is changed. Indeed, the values of the critical exponents found for the _aqueous solutions of the

polyoxyethylene amphiphiles C6E3, C8E4, C12E8

and C,4E7 ^are ranging between 0.63 and 0.44 for v

and 1.25 and 0.87 for y, while the ’0 factor varies

between 3.4 and 17.5 A. The critical phenomena

Fig. ^3.

Log-log plots ^of the correlation length ( and the total intensity 1(0) ^{versus the reduced variable} ex

(Xc - X)j X c

for the three following ^critical points. ^{a :} X,

1.207; ^{b :} Xc

^{1.550; c :} X,

^3.424.

Fig. 4. - Variation of the critical exponents vh Î’ f’ V X’

yx and of the scale factor (o ^{versus X.}

found in these micellar solutions ^{are not} of the simple type ^described by ^the Ising ^model [9]. Corti et al.

have suggested ^{that a new} theoretical model is needed to describe the critical behaviour of micellar solu- tions [10]. ^However, it is worth mentioning ^{that in}

the ternary microemulsion system AOT-water-decane,

we do not observe such a variation in critical expo- nents [6]. ^This system presents ^{in the} temperature- composition ^{space a line of critical} points ^ended by ^a

critical end point. The critical exponents ^{v and} y measured in several distinct points of this line have

almost the same values, ^{even in the} vicinity ^{of the}

critical end point They ^{are found in} good agreement with the Ising ^indices [12, 18].

Although ^the complex behaviour found in the quaternary mixture under study ^{in this} paper is not

fully understood, ^several possible explanations ^{can be}

considered. First, ^we can assume that the temperature

or X range investigated (3 ^{x 10-4 e 3 x} 10-2)

is not sufficiently ^{close to} the critical point. ^{It is}

known that the range of the asymptotic simple ^power

law depends ^{on the} thermodynamic properties

considered [12]. Usually ^{for the} correlation length

and the osmotic compressibility, the correction terms turn out to be unimportant ^for binary mixtures.,

In order to clarify ^this point, ^further experimental

work in the _range E 10-4 is required.

Besides, it is known that in fluids (pure ^{or multi-} component mixtures) ^the exponents depend ^{on the} path followed in approaching the critical point [19].

This possibility complicates ^the analysis of the results.

However, the identical variation observed along ^the

two different paths (I ^and II) suggests that the observed behaviour is not only ^{due to} the choice of the specific path ^{followed. In both cases, the deviation from} Ising values increases significantly ^{in the} vicinity

of the critical end point

The exponent variation could be due to a crossover

between two sets of exponents, ^{one of them} being ^the Ising indices; these indices are found for the X,

^3.42

and 5.2 samples. ^{In this case,} the second set of expo- nents does not correspond ^{to the mean} field values since the exponents measured for the lowest X values

are much less than the ^mean field indices (v

0.5;

y

1). The second set of exponents ^{should be more} likely associated with a « special » critical end point

in relation with the existence of the liquid crystalline phase. ^With regard ^{to this} explanation is should be noted that we have observed in the vicinity ^{of the}

critical end point ^{an anomalous} behaviour of viscosity

748

similar to that previously reported ^for microemulsions

near a critical point [21]. ^{We have} measured, ^at fixed temperature, ^the viscosity of mixtures located along

the coexistence surface. Measurements have been done along paths ^defined by ^a fixed value of X ; the variable along ^each path is the dodecane and alcohol concentration. Figure ⁵ shows the variation of the viscosity along ^{the two} paths ^considered (X

1.02 and X

1.55). The data obtained for the X

1.02 plane ^{show an} anomalous behaviour in the oil concentration _range around the critical

point (volumic critical dodecane concentration 82 %).

This anomaly disappears ^{as X} increases. Indeed,

in the plane ^defined by ^X

^1.55, viscosity ^conti- nuously ^{decreases as} the oil concentration increases;

in this _case, no particular ^feature is observed at the critical point (volumic critical dodecane concen-

tration 77 %). Hence, the anomalous behaviour found for the X planes ^{close to} the critical end point PI (low ^{values of} X) ^{is not} only ^{due to} the presence of ^a critical point. This behaviour is more likely ^{related to}

the existence of the lamellar phase and could be due to precursor fluctuations of the smectic phase. ^Let

us recall that in the vicinity of the critical end point,

Fig. ^5.

Viscosity ^of microemulsions (21 OC) ^{versus the}

dodecane volume fraction for samples located in two X

planes : ^{+ X}

1.02; . ^X

^1.55.

the critical microemulsions and the birefringent phase

have _very similar compositions.

Acknowledgments.

The authors of this _paper are deeply ^{indebted to}

Professors B. Widom and V. Degiorgio, ^{and to}

Dr. J. Prost for valuable discussions and comments.

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