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Characterization of reverse micelles by dynamic light scattering
Colloids and Surfaces A: Physicochemical and Engineering Aspects 178 (2001) 313 – 323 www.elsevier.nl/locate/colsurfa
Characterization of reverse micelles by dynamic light scattering H.B. Bohidar a,b,*, M. Behboudnia b,1 a
Department of Chemistry, Indiana-Purdue Uni6ersity, 402 N Blackford Street, Indianapolis, IN 46202 -3274, USA b School of Physical Sciences, Jawaharlal Nehru Uni6ersity, New Delhi 110 067, India Received 24 February 2000; received in revised form 11 August 2000; accepted 21 August 2000
Abstract Reverse micelles of water–AOT–isooctane have been studied for the [Water]/[AOT]= W0 range varying from 1 to 50 by dynamic laser light scattering and capillary viscometry techniques with AOT concentrations chosen between 100 and 500 mM. The ratio of the rigidity of the liquid – liquid interface K to that of effective surface tension g, (K/g) defines a length scale j, (K/2g)= j 2 =RR0 where a reverse micelle of radius R is formed of surfactants having a spontaneous curvature 1/R0. We show that for W0 5 20, the reverse micelles can be modeled as spheres purely from geometrical considerations and for W0 \20, these conformationaly resemble deformed spheres. Correspondingly, the relative viscosity (to water) of the solution hr shows a micellar volume fraction dependence hr(f)= (1− f −lf 2)d, l being a fitting parameter. For low AOT concentration (100 mM), d= 2.5290.06 implying a near spherical shape (theoretical value = 2.5) for the micelles. This increased to d = 2.990.06 for AOT concentration 500 mM which we construe as micelles having deformed spherical conformation. The micellar stability has been deduced from free energy calculations and this yields an explicit expression for the pressure P(R) experienced inside the water core of a reverse micelle. The inter micellar interaction has been discussed within the framework of short-range interaction potential and perturbation potential r − 6. © 2001 Elsevier Science B.V. All rights reserved. Keywords: Reverse micelles; DLS; Stability; Conformation
1. Introduction Sodium bis (2-ethylhexyl) sulfosuccinate, i.e. AOT is a popular surfactant used for the formation of reverse micelles in a variety of organic * Corresponding author. E-mail address: [email protected] (H.B. Bohidar). 1 On leave from: Ardebil University, PO Box 179, Ardebil, Iran.
solvents [1–5]. One of the major advantages of AOT is its ability to create large water pools inside the reverse micellar core without necessitating the presence of a cosurfactant. The physical characterization of AOT mediated reverse micelles is mostly done through viscometry, light scattering, small angle neutron scattering, NMR, ultracentrifugation, ultrasonics and fluorescence spectroscopy [6–10]. The growth of microemulsion droplets has been modeled as chain-like ag-
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gregates purely from equilibrium thermodynamics, by Koper et al. [2]. In spite of the large body of literature available on this subject a complete physical understanding of the reverse micellar structure still remains unresolved. The AOT molecules remain locked at the oil– water interface with the polar head group partially immersed in the water pool and the non-polar tail extended in the oil medium. Assuming, the surfactant to be a diblock polymer with one polar side attached to a non-polar chain, Cantor [11] analyzed the stability of the interface. He assumed both the oil and water to be good solvents for their respective polymer segments, and using the theory of semi-dilute polymer solutions, described the interfacial forces in finer details. This was reviewed by de Gennes [12]. This model assumed a single saturated interface where the surfactant molecules experience a stretch perpendicular to the interface, thus generating a lateral force that opposes the usual surface tension g0 between the two ‘bare’ liquids. So, the effective surface tension experienced is g=g0 −gn where the polymer stretching at the interface contributes a ‘negative surface tension gn ’. A stable reverse micelles results when g 0. De Genne’s review was based on Schulman et al.’s [13] description of the saturated interface, Helfrich’s model [14] of interfacial rigidity and spontaneous curvature, and Cantor’s model [11] of diblock polymers entrapped at the oil–water interface. He defined a ‘persistence length jK ’ for a random interface that determines the interfacial curvature. For distances r B jK, the interface is assumed to be flat, otherwise it is curved. The interfacial rigidity K was related to jK at temperature T as [12] jK = a exp
2pK kB T
(1)
where a is a microscopic length related to the surfactant size and kB is Boltzmann constant. This concept of random surface was further extended to multi-surface problem. Consequently, a soft interface (defined by low K-values) was argued to give rise to a deformed droplet of effective radius R, having a characteristic relax-
ation time t in the dispersion medium of viscosity h given as [12] 1 K t 6phR 3
(2)
Such deformations have not been measured yet, though the occurrence of these appear very probable. The inter micellar interactions have been studied in the past by a variety of experimental techniques [15–18]. For a solution with micellar concentration f, the average intensity of the scattered light Is(f) shows a concentration dependence KI which gives the second virial coefficient of the osmotic pressure A2, which in turn is related to the concentration dependence [15– 18] KD of z-averaged diffusion coefficient, D(f)= D0(1+K fD) of the micelles as KD = KI − Kf where Kf, is the concentration dependence of translational frictional coefficient f(f)= f0(1+ Kff). This is accomplished through Einstein relation, D(f)= kB T/f(f) at temperature T with kB being Boltzmann constant. One major problem encountered here concerns the extrapolation of D(f) or Is to f 0. Determination of exact volume fraction of micelles is not trivial. This is attributed to the fact that the oil bound to the outer surface of these reverse micelles cannot be estimated to any accuracy. Hence, in most of the studies contribution of bound oil to the total micellar volume is neglected. The experimentally observed KD, Kf and KI have been interpreted through several model interaction potentials, but none of these models provide complete description of all the experimentally observed features. In this paper, experiments have been performed keeping the AOT concentration very high compared to the past studies. We argue that the stability can be visualized through the internal pressure prevailing inside reverse micelle. The micellar conformations have been verified from viscosity measurements. Finally, we provide a qualitative comparison of intermicellar interactions modeled through short range perturbation potential and perturbation potential of the form r − 6 in the context of our system.
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2. Pressure inside a reverse micelle Let us assume that the system has bulk oil (the dispersion medium), oil bound to the reverse micelles, bound and free water inside the reverse micelle and surfactant locked at the water–oil interface. Since, the mass of oil bound to the outer surface of the reverse micelle is very small compared to total mass of oil, we can neglect its contribution to the total free energy but shall account for it when for a given surfactant concentration, the micelle swells due to the addition of more water (high W0) thus forcing a larger amount of oil to get trapped in the new space created by the inter molecular separation of the surfactant molecules. Secondly, at this stage, we do not distinguish between bound and free water. Hence, the system has been simplified to comprise an oil body, water inside the micellar core and surfactants at the interface. If the mass of the total water, surfactant and oil in the system are MW, Ms and M0, respectively, and gW, gs and g0 are corresponding free energy per unit mass (same as the chemical potential of an infinite body of the substance), the total free energy can be written as [19] GTotal =M0g0 +Msgs +MWgW +4pNgR 2 NK (4pR) + 2pKN (3) R0 where there are N-droplets of reverse micelles of radius R each. The last three terms are the free energy of droplets as defined by Robbins [19]. Let us deduce the condition for stability for a given surfactant mass (concentration) by varying the water concentration in the system. Hence, the optimum condition is given by (GTotal =0 (4) (MW T Further as the droplet grows because of more water being added to the system, the separation between surfactant molecules attached to water core surface will increase creating bigger ‘lagoons’ (Fig. 1) of trapped oil. Hence, from the bulk a small fraction of oil dM0 =a% dMW gets attached to the droplets resulting in the loss of mass dM0 from the bulk. From Eqs. (3) and (4) one gets (under zeroth order approximation)
− a%g0 + gW +
(R (MW
8pRNg −
T
4pKN R0
315
=0 T
(5) Differentiating with respect to pressure P at constant temperature,
n
− a% +
(g0 (P
( (P
+
T
(gW (P
T
2Ng NK − rWR rWR0R 2
=0
(6)
T
where we used (R (MW
= T
1 4prWR 2
(7)
where rW is the mass density of water. Using Maxwell thermodynamic [20] relation ((g/(P)T = 1/r. Eq. (6) reduces to
a% 1 2gN 1 (R − =− r0 rW rW R 2 (P
2NK 1 (R + rWR0 R 3 (P
T
(8) T
with r0 being the mass density of oil and using
−
Fig. 1. A small reverse micelle (A) swells on addition of more water with fixed surfactant (AOT) concentration either to a spherical (B) or deformed spherical micelle (C), increasing the size of the ‘lagoons’ of oil attached to the outer surface (L). The volume of the locked oil increases as more water is added to cause swelling. A large water pool cannot be confined to a rigid spherical cell unless the rigidity parameter (K) is high. The micellar stability is decided by the condition imposed by Eq. (10). See text for details.
316
rs =
a% 1 − r0 rW
(9)
and integrating Eq. (8), the internal pressure experienced by each droplet P(R) at a given temperature T, will be P(R)=
K N 2g − rW rsR rsR0R 2
(10)
Here, the first term compresses the droplet due to surface tension and the second term which is a surface stiffness term counter balances this. A stable droplet is one where both these opposing forces are same in magnitude, implying P(R)=0. Hence, we get the stability condition as g=
K 2RR0
(11)
An identical relationship was established by Robbins [19], where he considered the free energy of the droplets only. Our approach is more complete since we account for the free energy of the whole system rather than of the droplet alone as done by Robbins [19] but we arrive at the same result. The parameter (K/g)1/2 is related to the geometric mean of R0 and R and can be defined as a length scale, say j, such that j = RR0 =
K 2g
1/2
(12)
Of the three parameters K, g, R0 and R, normally the first three are still not experimentally accessible. Nonetheless, it will be shown that from a reasonable estimate of R0 and measured value of R one can characterize the droplets to a good accuracy. From a purely geometrical consideration, the measured hydrodynamic radius R is (13)
R=d+l+rW
where d is the thickness of bound water layer, l is the length of surfactant molecule and rW is water core radius. It has been shown that for a micelle having n¯ number of surfactant molecules [21] rW =
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90W0n¯ 4p
Hence
1/3
(14)
R= d+l +
90W0n¯ 4p
1/3
(15)
A plot of measured R versus (90W0/4p)1/3 would yield a straight line with intercept (d +l) and slope n¯ 1/3, strictly for a spherical droplet. For a deformed droplet Eqs. (13)–(15) will have no well defined meaning.
3. Materials and methods The AOT material was bought from M/S BDH Laboratory Supplies, UK. Solvent used was double distilled de-ionized water and iso-octane had a nominal purity of 99.9% (Merck). These were used without further purification. The samples were ultracentrifuged at 8000 rpm for about 1 h, then directly loaded into optical quality borosilicate sample cells and sealed. Dynamic light scattering experiments were performed using a Brookhaven Instrument (Brookhaven Instruments, USA) 9000-AT, 1024 channel digital correlator. The details of the experimental set up is described elsewhere [22]. The correlation spectra were recorded at room temperature, T= (209 1)°C. The refractive index values needed for radius calculations were taken from CRC Handbook [23]. In all the experiments the difference between the measured and calculated baseline was not allowed to go beyond 9 0.5%. The data that showed excessive baseline difference were rejected. First the correlograms were analyzed through a non-linear Laplace inversion routine to check the relaxation time distributions. All the data exhibited a single narrow distribution. Once the unimodel particle size distribution was confirmed, the particle sizes were determined through the CONTIN software provided by the Brookhaven instruments company. These values are listed in Table 1. These radii values compare well with literature [21,24] data. The viscosity measurements were performed using a 0.25-mm quartz capillary Ostwald viscometer calibrated against glycerol. The minimum efflux time was ensured to be more than 50 s. The viscometer was mounted inside a thermostatic bath regulating temperature to an accuracy of 90.5°C of the set temperature. The temperature stability was in fact
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317
Table 1 Variation of hydrodynamic radius R of water–AOT–isooctane reverse micelles at 25°C for various AOT/H2O concentrations maximum polydispersity (R( 2/R( 2−1) observed was below 0.05 W0 = 1 5 10 20 30 40 50
[H2O] [AOT]
100 mM (nm) 29 0.2 2 90.2 39 0.2 4.59 0.4 6.59 0.5 10.090.5 13.09 1
200 mM (nm)
300 mM (nm)
400 mM (nm)
500 mM (nm)
29 0.2 2.5 90.3 39 0.2 4.59 0.4 6.09 0.5 12.09 0.6 15.09 2
2.59 0.3 3 9 0.3 3.59 0.2 4.0 90.3 10.0 90.8 17.0 90.8 19.0 92
2.5 9 0.3 3.5 9 0.3 3.59 0.2 4.0 9 0.3 17.5 9 0.8 21.0 9 0.8 –
2.5 90.3 3.5 90.3 4.0 90.3 4.5 90.4 21 9 1 – –
9 0.1°C over the duration of a set of three experimental runs carried out per sample. Double distilled water was used to flush the viscometer after use. The conversion of efflux time to viscosity requires the a priori knowledge of liquid density. This was measured by taking the weight of 100 ml of liquid (drawn using a micro-pipette) using a Mettler (Model: AJ 100) balance having a sensitivity of 0.1 mg. For calibration, we performed the density measurements on a standard liquid (acetic acid) and compared the values with the handbook data and the matching was excellent.
R0 =
3n¯V 4p
1/3
(16)
where V denotes the molecular volume of an AOT molecule. Assuming a conical shape for AOT and assigning head-group [25] cross-sectional area= 55 A, 2 and length 12 A, , one finds V: 220 A, 3. With the known values of n¯ and this value of V, it is possible to find a good estimate of R0 from Eq. (16), which is listed in Table 2. Consequently, the effective surface area S covered by each AOT molecule can be deduced from (listed in Table 2) S=
4pR 20 n¯
(17)
4. Results and discussion
4.1. Re6erse micellar size The plot of measured hydrodynamic radii R is plotted against (90W0/4p)1/3 in Fig. 2 for [AOT]=200 mM, which is a typical plot. The fitting of Eq. (15) to the first four points (W0 5 20) yielded (d +l) and n¯ values. These are listed in Table 2. It is generally assumed that l 1.1 nm [24,25]. This allows one to deduce the thickness of the bound water layer d. For the determination of j, one needs to know the values of R0 at different AOT concentrations. We propose that the curvature 1/R0 is not a constant but its value depends on the aggregation number n¯, the average number of surfactant molecules attached to a micellar interface. Hence,
Fig. 2. Variation of R with W0, scaling as (90W0/4p)1/3 for water – AOT – isooctane reverse micelles measured at 25°C. S. Data shown for only two selected AOT concentrations ( — 200 mM; and — 400 mM). Data are from Table 1.
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Table 2 Data obtained by fitting of Eq. (15) to the data reported in Table 1a [AOT] (mM)
(d+l) (nm)
d (nm)
n¯
R0 (nm)
S (nm)2
100 200 300 400 500
1.25 1.54 1.57 1.68 1.69
0.15 0.44 0.47 0.58 0.59
68 71 98 108 486
1.5 90.2 1.6 90.2 1.79 0.2 1.8 90.2 2.9 9 0.3
0.43 90.06 0.42 9 0.06 0.38 90.05 0.37 90.05 0.22 9 0.04
a The length of the AOT molecule was assumed to be l1.1 nm. The radius of spontaneous curvature R0 was calculated using Eq. (16) and the effective surface area per surfactant molecules S was deduced using Eq. (17). Fig. 4 shows a plot of S as function of AOT concentration.
In the event of assigning a cylindrical shape to AOT molecules, the calculated values of R0 will get multiplied by a constant factor (3)1/3 and S values will get multiplied approximately by a factor 2. However, we chose the cylindrical shape to be acceptable because the tabulated S values resemble those measured earlier by NMR technique [10]. The variation of surface area per molecule of the surfactant in contact with the water core surface S has been plotted as a function of AOT concentration in Fig. 3, along with the corresponding n¯ values. At low AOT concentration a larger portion of the surfactant head group is in contact with the water core. As AOT concentration increases, more AOT molecules get attached to the water core surface and there is an apparent internal competition among these molecules to offer maximum area of contact with the water core surface, thus reducing the effective contact area. We have coupled the two size parameters R and R0 to define j as the geometric mean. The values of j were deduced from known R and calculated R0 values taken from Tables 1 and 2. These are listed in Table 3 and plotted in Fig. 4. For W0 520, j5 (2.5 9 0.2) and above j\ (2.59 0.2) remained valid for all AOT concentrations except for 500 mM data. At this stage it is not possible to ascertain whether all the W0 5 20, points could be fitted to a master straight line. Looking at the scatter of data, we refrained from doing so. The distinction for higher W0 \ 20 values appears sharper for AOT concentrations higher than 200 mM. Assumption of a different molecular shape for AOT molecules will yield
curves qualitatively identical to Fig. 4 with similar built-in features. One is tempted to use the literature value [25] for l: 1.1 nm instead of R0 in Eq. (12) and evaluate j-values. It turns out that these two values differ typically by 10% for W0 520 and by more than 20% for W0 \ 20. In the absence of any definite method of evaluation of R0, it becomes difficult to assign any unique value to this parameter j.
4.2. Inter micellar interactions The pair wise interaction between reverse micelles can be modeled through the formulation developed by Cazabat and Langerin [26]. The volume fraction of the reverse micelle is defined as
Fig. 3. The effective surfactant surface area S (estimated from Eq. (17)) plotted as function of AOT concentration for water– AOT – isooctane reverse micelles measured at 25°C.
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319
Table 3 The calculated values of j using data of R and R0 from previous tables (these are plotted in Fig. 4) W0 =
[H2O] [AOT]
1 5 10 20 30 40 50
f=
100 mM (nm) 1.7 1.7 2.1 2.6 3.1 3.9 4.4
VW + Vs +Vbo VW +Vs +Vo
200 mM (nm)
300 mM (nm)
400 mM (nm)
500 mM (nm)
1.8 2.0 2.1 2.6 3.1 4.1 4.8
2.1 2.3 2.5 2.6 4.2 5.4 5.7
2.1 2.5 2.5 2.7 5.6 6.1 –
2.4 3.0 3.4 3.6 7.9 – –
(18)
where VW, Vs, Vo and Vbo stand for volume of water, surfactant, oil and oil-bound to the micelle respectively. We could not measure Vbo independently and hence to a first approximation its contribution is being neglected [26,27]. The translational diffusion coefficient D(f) depends on the volume fraction as D(f)=Do(1+ af)
(19)
V(r) has been approximated to be composed of a hard sphere part VHS(r) and an attractive part VA(r), treated as a perturbation term. Some of the representative diffusion coefficient values are plotted as function of f in Fig. 5. The D0, and a values were obtained from fitting of Eq. (19) to the measured data. Once a is known, one can find equivalent second virial coefficients from Eq. (21), for the case where the interaction potential is a short range potential A SR or for a perturbation
which, originates from the chemical potential m(f) as D(f)=
f 1 (m(f) (1− f) f(f) (f
(20)
PT
with a = aV +ao + aD +as +aA where D0, is the infinite dilution value of D(f), aV, is the osmotic virial contribution, ao is Oseen Tensor contribution, aD is the dipolar contribution, as, is short range interaction part and aA is a perturbation contribution to. A detailed analysis finally yields [26,27]
>
a= A 3A 1.56+ + ; 2 256 1.56+
5A 93A + ; 2 7168
(Short range perturbation potential) (perturbation potential r − 6)
Here, A is the second virial coefficient of the chemical potential m. The net interaction potential
Fig. 4. Variation of j with AOT concentration for water– AOT – isooctane reverse micelles. Note that up to W0 5 20 there is a clear linear variation which undergoes an abrupt change for W0 ]20. This is more pronounced for higher AOT concentrations. We propose deformed spherical conformation for the reverse micelles for W0 values above W0 =20. (AOT concentrations are — 100 mM; — 300 mM; and — 500 mM). See text for details.
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Fig. 5. Typical plot of volume fraction, F dependence of translational diffusion coefficient D at AOT concentration 300 mM. These have been fitted to Eq. (19) to deduce D0 and a. The solid line is a least squares fit.
Fig. 6. The fitted data obtained from Eq. (19) plotted as a function of AOT concentration. Note the sudden change in D0 and a values at [AOT] =300 mM. The values are tabulated in Table 4. See text for details.
potential A PR. These values are given in Table 4. The variations of D0 and a with AOT concentration are plotted in Fig. 6. Both D0 and a alter their trend at AOT concentration [AOT]= 300 mM. The exact cause for this is not known at present. However, we argue that the values of f used are underestimated by a factor Vbo which is unknown to us. This is crucially dependent on AOT concentration for a given W0 value. Hence, we refrain from making any definite comment on either D0 or a at this stage. An attempt was made to measure fm/I from static light scattering data but the data points were very scattered and not reproducible.
Nonetheless, one can resort to the following speculation based on the parameters a, A PR and A SR that give negative values at AOT concentration 100 mM implying strong inter micellar attraction. The magnitude of these reduce to a minimum at AOT concentration of 300 mM and then remains unchanged. The D0 values show similar feature. Hence, it can be argued that the strong attractive inter micellar interactions present below AOT concentration of 300 mM stabilize to a weaker attractive force as the surfactant concentration exceeds 300 mM. This could possibly happen because of the micellar to microemulsion transition. The radii values clearly indicate (Table 1) bigger micelles for AOT concentrations 300 mM and higher. In fact for W0 \ 30 and AOT concentration 400 and 500 mM, it was not possible to perform any measurement on the samples because these turned turbid implying some kind of transition.
Table 4 The infinite dilution translational diffusivity D0, and inter micellar interaction parameter a deduced from fitting of Eq. (19) to experimental dataa [AOT] (mM)
D0×107 (cm2/s)
a
A SR
A PR
100 200 300 400 500
3.069 0.6 2.679 0.5 2.379 0.5 3.029 0.6 3.7690.6
−8.49 −4.64 −3.56 −3.10 −2.95
−19.63 −12.11 −10.00 −9.10 −8.81
−3.99 −2.46 −2.03 −1.85 −1.79
a The contribution of short range potential parameter A SR and A PR to a has been deduced from Eq. (21).
4.3. Viscosity and density studies The measured solution viscosity (hsol) values were converted to relative viscosity (hr) with respect to water. Einstein relationship for spherical particles of volume fraction f, dispersed in a solvent is given by
H.B. Bohidar, M. Behboudnia / Colloids and Surfaces A: Physicochem. Eng. Aspects 178 (2001) 313–323
hr =1+ 2.5 f,
for f 1
(22)
At higher volume fractions Brinkham [28] proposed hr =(1 −f) − 2.5. However, for non-spherical or distorted spherical particles, those arising out of collision between spheres and for particles with weak interfacial layers (small K), one can use a more generalized expression for relative viscosity hr =(1−f− lf 2) − d
(23)
where l is fitting parameter (physically it is an interaction term), d is a shape dependent parameter with a value d = 2.5 for particles having spherical shape. Some typical plots of hr versus f are shown in Fig. 7. The values of l and d are listed in Table 5 (plotted in Fig. 8). The entire data set could be fitted to Eq. (23) reasonably well. The shape parameter d has a value 2.52 90.08 at 100 mM grows to d = 2.91 9 0.06 implying change in the three dimensional shape of the micelles as AOT concentration is raised. Our measured values follow the trend discussed by Dvolaitzky et al. [29] for microemulsions. For a sphere made of a fluid of viscosity h0 immiscible with continuous phase viscosity h1 can yield a relative viscosity given by [30]
Fig. 7. Variation of relative viscosity hr as function of volume fraction of the dispersed phase F plotted for two selected AOT concentrations, 100 mM () and 500 mM ( ). The data could be fitted to Eq. (23) quite well and the coefficients are plotted in Fig. 8 and tabulated in Table 5.
321
Table 5 Variation of l and d as function of micellar volume fraction f deduced from fitting of hr(f) to Eq. (23)a [AOT] (mM)
l
d
rsol (g/cc)
100 200 300 400 500
−0.74 −0.52 0.16 0.26 0.29
2.52 9 0.08 2.66 9 0.07 2.75 9 0.07 2.83 9 0.06 2.91 9 0.06
0.7669 0.008 0.784 9 0.008 0.806 9 0.008 0.823 9 0.009 0.847 90.009
The solution density values rsoln. did not show W0 dependence for a fixed AOT concentration within the cited experimental inaccuracy. a
hr = 1+
2.5h0 + h1 h0 + h1
(24)
Here, it is assumed that the velocity of the fluid must vanish at the surface of the sphere. Our data could not be fitted to such an expression, possibly because of the non-sphericity of our diffusing micellar particles. For low volume fractions 0.035 f 50.1, the coefficient l was deduced to be − 0.74 which slowly increased to 0.29 as f gradually increased to 0.3. At these low f values, the results could be fitted to hr = (1+ 2.5f), since the quadratic term is very small in Eq. (23). At these volume fractions our results qualitatively agree with those of Sein et al. [18]. At comparatively larger volume fractions our values are smaller than the literature [2] values by a factor 1/2
Fig. 8. Variations of the shape parameter d () and concentration dependence term l ( ) of Eq. (23) as a function of AOT concentration.
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which we could not explain. The shape dependent parameter d was found to be 2.529 0.06 for AOT concentration of 100 mM and this gradually increased to 2.919 0.06 for 500 mM AOT solutions. This clearly indicates the presence of spherical micelles at low surfactant concentrations. These get substantially deformed at surfactant concentrations more than 300 mM for W0 ] 30 as depicted in Fig. 1. Table 1 shows significantly bigger micelles in solutions when [AOT]] 300 mM. As these micelles grow bigger, the interfacial rigidity defined by K, becomes inadequate to provide sufficient strength to the interface to maintain a definite shape. Two observations have been made at this stage, one is, at low AOT concentrations for W0 50, deformations are observed in micellar shape and the second is, at higher surfactant concentrations this starts even at W0 ]30. Sein et al. [18] have seen interesting scattering features in AOT – CCl4 –H2O reverse micelles as the volume fraction of the dispersed phase is increased as in our case. For 0.055 f5 0.08, clear spherical micelles were seen which showed distinctive change at higher volume fractions and we, identify these as deformed spherical structures. The measured solution density rsol, values showed very little variation when AOT concentration was held constant and W0 was varied from 1 to 50. This is listed in Table 5. The measured data could not either be expressed as rsol = rwfw +rofo +rsfs or
conclusive. Zulauf and Eicke [24] have measured the molecule weight of reverse micelles and their corresponding hydrodynamic radii (R) values and made a plot of MW versus R that showed a scaling RM 0.30. An exponent of 0.33 would have indicated a perfect sphere. Hence, this possibly indicates the existence of non-spherical reverse micelles in the dispersion medium. We deduce an explicit expression for the pressure P(R) experienced by the water core of a reverse micelle of radius R. The micelle has been modeled to be stable when P(R)= 0. The free energy equation (Eq. (3)) has been written assuming all the water to be identical. In fact the bound water at the interface and the bulk at the core do exhibit different properties and are likely to contribute differently to the free energy equation. The same is true for the oil. This yielded the stability condition in terms of the surface rigidity K and effective interfacial tension g. Incorporating, the spontaneous curvature (1/R0) and R into a single parameter (their geometric mean) we find a length scale j which is more meaningful than R when it comes to conformational analysis. The transient features of spherical to deformed-spherical transitions in conformations remains to be probed.
Acknowledgements One of the authors (M. Behboudnia) acknowledges Ardebil University, Iran for providing the research fellowship and study leave.
rsol = rwxw + roxo +rsxs as is normally the case in miscible liquid mixtures, where f and x are volume and mole fraction of corresponding components.
5. Conclusions A distinctive feature of our studies is that we have taken higher AOT concentrations to form reverse micelles and varied W0 up to 50 and these has been characterized rigorously. The literature results on reverse micellar conformations are non-
References [1] J. Smeets, et al., J. Phys. Chem. 10 (1994) 1387. [2] G.J.M. Koper, et al., J. Phys. Chem. 99 (1995) 13291. [3] M.M. Kozlov, D. Andelman, Curr. Opin. Colloid Interface Sci. 1 (1996) 362. [4] H. Wennerstrom, Curr. Opin. Colloid Interface Sci. 1 (1996) 370. [5] M. Kotlarchyk, S.H. Chen, J.S. Huang, J. Phys. Chem. 86 (1982) 3273. [6] P.L. Luisi, M. Giomini, M.P. Pileni, B.H. Robinson, Biochem. Biophys. Acta 947 (1988) 209. [7] E. Gulari, B. Bedwell, S.J. Alkhafeji, Colloid Interface Sci. 77 (1980) 202.
H.B. Bohidar, M. Behboudnia / Colloids and Surfaces A: Physicochem. Eng. Aspects 178 (2001) 313–323 [8] R.A. Day, B.H. Robinson, J.H.R. Clarke, J.V. Doherty, J. Chem. Soc. Faraday Trans. 1 (75) (1979) 132. [9] G. D’Arrigo, A. Paparelli, J. Phys. Chem. 93 (1989) 8367. [10] N.J. Bridge, P.D.I. Fletcher, J. Chem. Soc. Faraday Trans. 1 (79) (1983) 2161. [11] R. Cantor, Macromolecules 14 (1987) 1186. [12] P.G. De Gennes, C. Taupin, J. Phys. Chem. 86 (1982) 2294. [13] J.H. Schulman, W. Stoecknius, L.M. Prince, J. Phys. Chem. 63 (1959) 167. [14] W.Z. Helfrich, Naturforsch. C, 28 (1973). [15] S. Brunetti, D. Roux, A.M. Bellocq, G. Fourche, P. Bothord, J. Phys. Chem. 87 (1983) 1028. [16] M. Corti, V. Degiorgio, J. Phys. Chem. 85 (1981) 711. [17] M.J. Hou, M. Kim, D.O.J. Shah, Colloid Intf. Sci. 123 (1988) 398. [18] E. Sein, J.R. Lalanee, J. Buchert, S.J. Kielich, Colloid Intf. Sci. 72 (1979) 363.
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[19] M.L. Robbins, in: K. Mittal (Ed.), Micellization, Solubilization and Microemulsion, vol. 2, Plenum, New York, 1977, p. 273. [20] K. Huang, Statistical Mechanics, Wiley, New York, 1987. [21] A. Maitra, J. Phys. Chem. 88 (1984) 5122. [22] S. Maity, H.B. Bohidar, Phys. Rev. E 58 (1988) 729. [23] CRC Handbook, CRC Press, Boca Raton, FL, 1989. [24] M. Zulauf, H.-F. Eicke, J. Phys. Chem. 83 (1979) 480. [25] H.F. Eicke, V.J. Arnold, Colloid Interface Sci. 46 (1974) 101. [26] A.M. Cazabat, D. Langerin, J. Chem. Phys. 74 (1981) 3148. [27] M. Hou, D.O. Shah, Langmuir 3 (1987) 1086. [28] H.C. Brinkham, J. Chem. Phys. 20 (1952) 571. [29] M. Dvolaitzky, M. Guyot, M. Lagues, J.P. Le Pesant, R. Ober, C. Sauterey, C. Taupin, J. Chem. Phys. 69 (1978) 3278. [30] G.I. Taylor, Proc. R. Soc. Lond. A 138 (1932) 41.
Characterization of reverse micelles by dynamic light scattering H.B. Bohidar a,b,*, M. Behboudnia b,1 a
Department of Chemistry, Indiana-Purdue Uni6ersity, 402 N Blackford Street, Indianapolis, IN 46202 -3274, USA b School of Physical Sciences, Jawaharlal Nehru Uni6ersity, New Delhi 110 067, India Received 24 February 2000; received in revised form 11 August 2000; accepted 21 August 2000
Abstract Reverse micelles of water–AOT–isooctane have been studied for the [Water]/[AOT]= W0 range varying from 1 to 50 by dynamic laser light scattering and capillary viscometry techniques with AOT concentrations chosen between 100 and 500 mM. The ratio of the rigidity of the liquid – liquid interface K to that of effective surface tension g, (K/g) defines a length scale j, (K/2g)= j 2 =RR0 where a reverse micelle of radius R is formed of surfactants having a spontaneous curvature 1/R0. We show that for W0 5 20, the reverse micelles can be modeled as spheres purely from geometrical considerations and for W0 \20, these conformationaly resemble deformed spheres. Correspondingly, the relative viscosity (to water) of the solution hr shows a micellar volume fraction dependence hr(f)= (1− f −lf 2)d, l being a fitting parameter. For low AOT concentration (100 mM), d= 2.5290.06 implying a near spherical shape (theoretical value = 2.5) for the micelles. This increased to d = 2.990.06 for AOT concentration 500 mM which we construe as micelles having deformed spherical conformation. The micellar stability has been deduced from free energy calculations and this yields an explicit expression for the pressure P(R) experienced inside the water core of a reverse micelle. The inter micellar interaction has been discussed within the framework of short-range interaction potential and perturbation potential r − 6. © 2001 Elsevier Science B.V. All rights reserved. Keywords: Reverse micelles; DLS; Stability; Conformation
1. Introduction Sodium bis (2-ethylhexyl) sulfosuccinate, i.e. AOT is a popular surfactant used for the formation of reverse micelles in a variety of organic * Corresponding author. E-mail address: [email protected] (H.B. Bohidar). 1 On leave from: Ardebil University, PO Box 179, Ardebil, Iran.
solvents [1–5]. One of the major advantages of AOT is its ability to create large water pools inside the reverse micellar core without necessitating the presence of a cosurfactant. The physical characterization of AOT mediated reverse micelles is mostly done through viscometry, light scattering, small angle neutron scattering, NMR, ultracentrifugation, ultrasonics and fluorescence spectroscopy [6–10]. The growth of microemulsion droplets has been modeled as chain-like ag-
0927-7757/01/$ - see front matter © 2001 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 7 - 7 7 5 7 ( 0 0 ) 0 0 7 3 6 - 6
314
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gregates purely from equilibrium thermodynamics, by Koper et al. [2]. In spite of the large body of literature available on this subject a complete physical understanding of the reverse micellar structure still remains unresolved. The AOT molecules remain locked at the oil– water interface with the polar head group partially immersed in the water pool and the non-polar tail extended in the oil medium. Assuming, the surfactant to be a diblock polymer with one polar side attached to a non-polar chain, Cantor [11] analyzed the stability of the interface. He assumed both the oil and water to be good solvents for their respective polymer segments, and using the theory of semi-dilute polymer solutions, described the interfacial forces in finer details. This was reviewed by de Gennes [12]. This model assumed a single saturated interface where the surfactant molecules experience a stretch perpendicular to the interface, thus generating a lateral force that opposes the usual surface tension g0 between the two ‘bare’ liquids. So, the effective surface tension experienced is g=g0 −gn where the polymer stretching at the interface contributes a ‘negative surface tension gn ’. A stable reverse micelles results when g 0. De Genne’s review was based on Schulman et al.’s [13] description of the saturated interface, Helfrich’s model [14] of interfacial rigidity and spontaneous curvature, and Cantor’s model [11] of diblock polymers entrapped at the oil–water interface. He defined a ‘persistence length jK ’ for a random interface that determines the interfacial curvature. For distances r B jK, the interface is assumed to be flat, otherwise it is curved. The interfacial rigidity K was related to jK at temperature T as [12] jK = a exp
2pK kB T
(1)
where a is a microscopic length related to the surfactant size and kB is Boltzmann constant. This concept of random surface was further extended to multi-surface problem. Consequently, a soft interface (defined by low K-values) was argued to give rise to a deformed droplet of effective radius R, having a characteristic relax-
ation time t in the dispersion medium of viscosity h given as [12] 1 K t 6phR 3
(2)
Such deformations have not been measured yet, though the occurrence of these appear very probable. The inter micellar interactions have been studied in the past by a variety of experimental techniques [15–18]. For a solution with micellar concentration f, the average intensity of the scattered light Is(f) shows a concentration dependence KI which gives the second virial coefficient of the osmotic pressure A2, which in turn is related to the concentration dependence [15– 18] KD of z-averaged diffusion coefficient, D(f)= D0(1+K fD) of the micelles as KD = KI − Kf where Kf, is the concentration dependence of translational frictional coefficient f(f)= f0(1+ Kff). This is accomplished through Einstein relation, D(f)= kB T/f(f) at temperature T with kB being Boltzmann constant. One major problem encountered here concerns the extrapolation of D(f) or Is to f 0. Determination of exact volume fraction of micelles is not trivial. This is attributed to the fact that the oil bound to the outer surface of these reverse micelles cannot be estimated to any accuracy. Hence, in most of the studies contribution of bound oil to the total micellar volume is neglected. The experimentally observed KD, Kf and KI have been interpreted through several model interaction potentials, but none of these models provide complete description of all the experimentally observed features. In this paper, experiments have been performed keeping the AOT concentration very high compared to the past studies. We argue that the stability can be visualized through the internal pressure prevailing inside reverse micelle. The micellar conformations have been verified from viscosity measurements. Finally, we provide a qualitative comparison of intermicellar interactions modeled through short range perturbation potential and perturbation potential of the form r − 6 in the context of our system.
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2. Pressure inside a reverse micelle Let us assume that the system has bulk oil (the dispersion medium), oil bound to the reverse micelles, bound and free water inside the reverse micelle and surfactant locked at the water–oil interface. Since, the mass of oil bound to the outer surface of the reverse micelle is very small compared to total mass of oil, we can neglect its contribution to the total free energy but shall account for it when for a given surfactant concentration, the micelle swells due to the addition of more water (high W0) thus forcing a larger amount of oil to get trapped in the new space created by the inter molecular separation of the surfactant molecules. Secondly, at this stage, we do not distinguish between bound and free water. Hence, the system has been simplified to comprise an oil body, water inside the micellar core and surfactants at the interface. If the mass of the total water, surfactant and oil in the system are MW, Ms and M0, respectively, and gW, gs and g0 are corresponding free energy per unit mass (same as the chemical potential of an infinite body of the substance), the total free energy can be written as [19] GTotal =M0g0 +Msgs +MWgW +4pNgR 2 NK (4pR) + 2pKN (3) R0 where there are N-droplets of reverse micelles of radius R each. The last three terms are the free energy of droplets as defined by Robbins [19]. Let us deduce the condition for stability for a given surfactant mass (concentration) by varying the water concentration in the system. Hence, the optimum condition is given by (GTotal =0 (4) (MW T Further as the droplet grows because of more water being added to the system, the separation between surfactant molecules attached to water core surface will increase creating bigger ‘lagoons’ (Fig. 1) of trapped oil. Hence, from the bulk a small fraction of oil dM0 =a% dMW gets attached to the droplets resulting in the loss of mass dM0 from the bulk. From Eqs. (3) and (4) one gets (under zeroth order approximation)
− a%g0 + gW +
(R (MW
8pRNg −
T
4pKN R0
315
=0 T
(5) Differentiating with respect to pressure P at constant temperature,
n
− a% +
(g0 (P
( (P
+
T
(gW (P
T
2Ng NK − rWR rWR0R 2
=0
(6)
T
where we used (R (MW
= T
1 4prWR 2
(7)
where rW is the mass density of water. Using Maxwell thermodynamic [20] relation ((g/(P)T = 1/r. Eq. (6) reduces to
a% 1 2gN 1 (R − =− r0 rW rW R 2 (P
2NK 1 (R + rWR0 R 3 (P
T
(8) T
with r0 being the mass density of oil and using
−
Fig. 1. A small reverse micelle (A) swells on addition of more water with fixed surfactant (AOT) concentration either to a spherical (B) or deformed spherical micelle (C), increasing the size of the ‘lagoons’ of oil attached to the outer surface (L). The volume of the locked oil increases as more water is added to cause swelling. A large water pool cannot be confined to a rigid spherical cell unless the rigidity parameter (K) is high. The micellar stability is decided by the condition imposed by Eq. (10). See text for details.
316
rs =
a% 1 − r0 rW
(9)
and integrating Eq. (8), the internal pressure experienced by each droplet P(R) at a given temperature T, will be P(R)=
K N 2g − rW rsR rsR0R 2
(10)
Here, the first term compresses the droplet due to surface tension and the second term which is a surface stiffness term counter balances this. A stable droplet is one where both these opposing forces are same in magnitude, implying P(R)=0. Hence, we get the stability condition as g=
K 2RR0
(11)
An identical relationship was established by Robbins [19], where he considered the free energy of the droplets only. Our approach is more complete since we account for the free energy of the whole system rather than of the droplet alone as done by Robbins [19] but we arrive at the same result. The parameter (K/g)1/2 is related to the geometric mean of R0 and R and can be defined as a length scale, say j, such that j = RR0 =
K 2g
1/2
(12)
Of the three parameters K, g, R0 and R, normally the first three are still not experimentally accessible. Nonetheless, it will be shown that from a reasonable estimate of R0 and measured value of R one can characterize the droplets to a good accuracy. From a purely geometrical consideration, the measured hydrodynamic radius R is (13)
R=d+l+rW
where d is the thickness of bound water layer, l is the length of surfactant molecule and rW is water core radius. It has been shown that for a micelle having n¯ number of surfactant molecules [21] rW =
H.B. Bohidar, M. Behboudnia / Colloids and Surfaces A: Physicochem. Eng. Aspects 178 (2001) 313–323
90W0n¯ 4p
Hence
1/3
(14)
R= d+l +
90W0n¯ 4p
1/3
(15)
A plot of measured R versus (90W0/4p)1/3 would yield a straight line with intercept (d +l) and slope n¯ 1/3, strictly for a spherical droplet. For a deformed droplet Eqs. (13)–(15) will have no well defined meaning.
3. Materials and methods The AOT material was bought from M/S BDH Laboratory Supplies, UK. Solvent used was double distilled de-ionized water and iso-octane had a nominal purity of 99.9% (Merck). These were used without further purification. The samples were ultracentrifuged at 8000 rpm for about 1 h, then directly loaded into optical quality borosilicate sample cells and sealed. Dynamic light scattering experiments were performed using a Brookhaven Instrument (Brookhaven Instruments, USA) 9000-AT, 1024 channel digital correlator. The details of the experimental set up is described elsewhere [22]. The correlation spectra were recorded at room temperature, T= (209 1)°C. The refractive index values needed for radius calculations were taken from CRC Handbook [23]. In all the experiments the difference between the measured and calculated baseline was not allowed to go beyond 9 0.5%. The data that showed excessive baseline difference were rejected. First the correlograms were analyzed through a non-linear Laplace inversion routine to check the relaxation time distributions. All the data exhibited a single narrow distribution. Once the unimodel particle size distribution was confirmed, the particle sizes were determined through the CONTIN software provided by the Brookhaven instruments company. These values are listed in Table 1. These radii values compare well with literature [21,24] data. The viscosity measurements were performed using a 0.25-mm quartz capillary Ostwald viscometer calibrated against glycerol. The minimum efflux time was ensured to be more than 50 s. The viscometer was mounted inside a thermostatic bath regulating temperature to an accuracy of 90.5°C of the set temperature. The temperature stability was in fact
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317
Table 1 Variation of hydrodynamic radius R of water–AOT–isooctane reverse micelles at 25°C for various AOT/H2O concentrations maximum polydispersity (R( 2/R( 2−1) observed was below 0.05 W0 = 1 5 10 20 30 40 50
[H2O] [AOT]
100 mM (nm) 29 0.2 2 90.2 39 0.2 4.59 0.4 6.59 0.5 10.090.5 13.09 1
200 mM (nm)
300 mM (nm)
400 mM (nm)
500 mM (nm)
29 0.2 2.5 90.3 39 0.2 4.59 0.4 6.09 0.5 12.09 0.6 15.09 2
2.59 0.3 3 9 0.3 3.59 0.2 4.0 90.3 10.0 90.8 17.0 90.8 19.0 92
2.5 9 0.3 3.5 9 0.3 3.59 0.2 4.0 9 0.3 17.5 9 0.8 21.0 9 0.8 –
2.5 90.3 3.5 90.3 4.0 90.3 4.5 90.4 21 9 1 – –
9 0.1°C over the duration of a set of three experimental runs carried out per sample. Double distilled water was used to flush the viscometer after use. The conversion of efflux time to viscosity requires the a priori knowledge of liquid density. This was measured by taking the weight of 100 ml of liquid (drawn using a micro-pipette) using a Mettler (Model: AJ 100) balance having a sensitivity of 0.1 mg. For calibration, we performed the density measurements on a standard liquid (acetic acid) and compared the values with the handbook data and the matching was excellent.
R0 =
3n¯V 4p
1/3
(16)
where V denotes the molecular volume of an AOT molecule. Assuming a conical shape for AOT and assigning head-group [25] cross-sectional area= 55 A, 2 and length 12 A, , one finds V: 220 A, 3. With the known values of n¯ and this value of V, it is possible to find a good estimate of R0 from Eq. (16), which is listed in Table 2. Consequently, the effective surface area S covered by each AOT molecule can be deduced from (listed in Table 2) S=
4pR 20 n¯
(17)
4. Results and discussion
4.1. Re6erse micellar size The plot of measured hydrodynamic radii R is plotted against (90W0/4p)1/3 in Fig. 2 for [AOT]=200 mM, which is a typical plot. The fitting of Eq. (15) to the first four points (W0 5 20) yielded (d +l) and n¯ values. These are listed in Table 2. It is generally assumed that l 1.1 nm [24,25]. This allows one to deduce the thickness of the bound water layer d. For the determination of j, one needs to know the values of R0 at different AOT concentrations. We propose that the curvature 1/R0 is not a constant but its value depends on the aggregation number n¯, the average number of surfactant molecules attached to a micellar interface. Hence,
Fig. 2. Variation of R with W0, scaling as (90W0/4p)1/3 for water – AOT – isooctane reverse micelles measured at 25°C. S. Data shown for only two selected AOT concentrations ( — 200 mM; and — 400 mM). Data are from Table 1.
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Table 2 Data obtained by fitting of Eq. (15) to the data reported in Table 1a [AOT] (mM)
(d+l) (nm)
d (nm)
n¯
R0 (nm)
S (nm)2
100 200 300 400 500
1.25 1.54 1.57 1.68 1.69
0.15 0.44 0.47 0.58 0.59
68 71 98 108 486
1.5 90.2 1.6 90.2 1.79 0.2 1.8 90.2 2.9 9 0.3
0.43 90.06 0.42 9 0.06 0.38 90.05 0.37 90.05 0.22 9 0.04
a The length of the AOT molecule was assumed to be l1.1 nm. The radius of spontaneous curvature R0 was calculated using Eq. (16) and the effective surface area per surfactant molecules S was deduced using Eq. (17). Fig. 4 shows a plot of S as function of AOT concentration.
In the event of assigning a cylindrical shape to AOT molecules, the calculated values of R0 will get multiplied by a constant factor (3)1/3 and S values will get multiplied approximately by a factor 2. However, we chose the cylindrical shape to be acceptable because the tabulated S values resemble those measured earlier by NMR technique [10]. The variation of surface area per molecule of the surfactant in contact with the water core surface S has been plotted as a function of AOT concentration in Fig. 3, along with the corresponding n¯ values. At low AOT concentration a larger portion of the surfactant head group is in contact with the water core. As AOT concentration increases, more AOT molecules get attached to the water core surface and there is an apparent internal competition among these molecules to offer maximum area of contact with the water core surface, thus reducing the effective contact area. We have coupled the two size parameters R and R0 to define j as the geometric mean. The values of j were deduced from known R and calculated R0 values taken from Tables 1 and 2. These are listed in Table 3 and plotted in Fig. 4. For W0 520, j5 (2.5 9 0.2) and above j\ (2.59 0.2) remained valid for all AOT concentrations except for 500 mM data. At this stage it is not possible to ascertain whether all the W0 5 20, points could be fitted to a master straight line. Looking at the scatter of data, we refrained from doing so. The distinction for higher W0 \ 20 values appears sharper for AOT concentrations higher than 200 mM. Assumption of a different molecular shape for AOT molecules will yield
curves qualitatively identical to Fig. 4 with similar built-in features. One is tempted to use the literature value [25] for l: 1.1 nm instead of R0 in Eq. (12) and evaluate j-values. It turns out that these two values differ typically by 10% for W0 520 and by more than 20% for W0 \ 20. In the absence of any definite method of evaluation of R0, it becomes difficult to assign any unique value to this parameter j.
4.2. Inter micellar interactions The pair wise interaction between reverse micelles can be modeled through the formulation developed by Cazabat and Langerin [26]. The volume fraction of the reverse micelle is defined as
Fig. 3. The effective surfactant surface area S (estimated from Eq. (17)) plotted as function of AOT concentration for water– AOT – isooctane reverse micelles measured at 25°C.
H.B. Bohidar, M. Behboudnia / Colloids and Surfaces A: Physicochem. Eng. Aspects 178 (2001) 313–323
319
Table 3 The calculated values of j using data of R and R0 from previous tables (these are plotted in Fig. 4) W0 =
[H2O] [AOT]
1 5 10 20 30 40 50
f=
100 mM (nm) 1.7 1.7 2.1 2.6 3.1 3.9 4.4
VW + Vs +Vbo VW +Vs +Vo
200 mM (nm)
300 mM (nm)
400 mM (nm)
500 mM (nm)
1.8 2.0 2.1 2.6 3.1 4.1 4.8
2.1 2.3 2.5 2.6 4.2 5.4 5.7
2.1 2.5 2.5 2.7 5.6 6.1 –
2.4 3.0 3.4 3.6 7.9 – –
(18)
where VW, Vs, Vo and Vbo stand for volume of water, surfactant, oil and oil-bound to the micelle respectively. We could not measure Vbo independently and hence to a first approximation its contribution is being neglected [26,27]. The translational diffusion coefficient D(f) depends on the volume fraction as D(f)=Do(1+ af)
(19)
V(r) has been approximated to be composed of a hard sphere part VHS(r) and an attractive part VA(r), treated as a perturbation term. Some of the representative diffusion coefficient values are plotted as function of f in Fig. 5. The D0, and a values were obtained from fitting of Eq. (19) to the measured data. Once a is known, one can find equivalent second virial coefficients from Eq. (21), for the case where the interaction potential is a short range potential A SR or for a perturbation
which, originates from the chemical potential m(f) as D(f)=
f 1 (m(f) (1− f) f(f) (f
(20)
PT
with a = aV +ao + aD +as +aA where D0, is the infinite dilution value of D(f), aV, is the osmotic virial contribution, ao is Oseen Tensor contribution, aD is the dipolar contribution, as, is short range interaction part and aA is a perturbation contribution to. A detailed analysis finally yields [26,27]
>
a= A 3A 1.56+ + ; 2 256 1.56+
5A 93A + ; 2 7168
(Short range perturbation potential) (perturbation potential r − 6)
Here, A is the second virial coefficient of the chemical potential m. The net interaction potential
Fig. 4. Variation of j with AOT concentration for water– AOT – isooctane reverse micelles. Note that up to W0 5 20 there is a clear linear variation which undergoes an abrupt change for W0 ]20. This is more pronounced for higher AOT concentrations. We propose deformed spherical conformation for the reverse micelles for W0 values above W0 =20. (AOT concentrations are — 100 mM; — 300 mM; and — 500 mM). See text for details.
320
H.B. Bohidar, M. Behboudnia / Colloids and Surfaces A: Physicochem. Eng. Aspects 178 (2001) 313–323
Fig. 5. Typical plot of volume fraction, F dependence of translational diffusion coefficient D at AOT concentration 300 mM. These have been fitted to Eq. (19) to deduce D0 and a. The solid line is a least squares fit.
Fig. 6. The fitted data obtained from Eq. (19) plotted as a function of AOT concentration. Note the sudden change in D0 and a values at [AOT] =300 mM. The values are tabulated in Table 4. See text for details.
potential A PR. These values are given in Table 4. The variations of D0 and a with AOT concentration are plotted in Fig. 6. Both D0 and a alter their trend at AOT concentration [AOT]= 300 mM. The exact cause for this is not known at present. However, we argue that the values of f used are underestimated by a factor Vbo which is unknown to us. This is crucially dependent on AOT concentration for a given W0 value. Hence, we refrain from making any definite comment on either D0 or a at this stage. An attempt was made to measure fm/I from static light scattering data but the data points were very scattered and not reproducible.
Nonetheless, one can resort to the following speculation based on the parameters a, A PR and A SR that give negative values at AOT concentration 100 mM implying strong inter micellar attraction. The magnitude of these reduce to a minimum at AOT concentration of 300 mM and then remains unchanged. The D0 values show similar feature. Hence, it can be argued that the strong attractive inter micellar interactions present below AOT concentration of 300 mM stabilize to a weaker attractive force as the surfactant concentration exceeds 300 mM. This could possibly happen because of the micellar to microemulsion transition. The radii values clearly indicate (Table 1) bigger micelles for AOT concentrations 300 mM and higher. In fact for W0 \ 30 and AOT concentration 400 and 500 mM, it was not possible to perform any measurement on the samples because these turned turbid implying some kind of transition.
Table 4 The infinite dilution translational diffusivity D0, and inter micellar interaction parameter a deduced from fitting of Eq. (19) to experimental dataa [AOT] (mM)
D0×107 (cm2/s)
a
A SR
A PR
100 200 300 400 500
3.069 0.6 2.679 0.5 2.379 0.5 3.029 0.6 3.7690.6
−8.49 −4.64 −3.56 −3.10 −2.95
−19.63 −12.11 −10.00 −9.10 −8.81
−3.99 −2.46 −2.03 −1.85 −1.79
a The contribution of short range potential parameter A SR and A PR to a has been deduced from Eq. (21).
4.3. Viscosity and density studies The measured solution viscosity (hsol) values were converted to relative viscosity (hr) with respect to water. Einstein relationship for spherical particles of volume fraction f, dispersed in a solvent is given by
H.B. Bohidar, M. Behboudnia / Colloids and Surfaces A: Physicochem. Eng. Aspects 178 (2001) 313–323
hr =1+ 2.5 f,
for f 1
(22)
At higher volume fractions Brinkham [28] proposed hr =(1 −f) − 2.5. However, for non-spherical or distorted spherical particles, those arising out of collision between spheres and for particles with weak interfacial layers (small K), one can use a more generalized expression for relative viscosity hr =(1−f− lf 2) − d
(23)
where l is fitting parameter (physically it is an interaction term), d is a shape dependent parameter with a value d = 2.5 for particles having spherical shape. Some typical plots of hr versus f are shown in Fig. 7. The values of l and d are listed in Table 5 (plotted in Fig. 8). The entire data set could be fitted to Eq. (23) reasonably well. The shape parameter d has a value 2.52 90.08 at 100 mM grows to d = 2.91 9 0.06 implying change in the three dimensional shape of the micelles as AOT concentration is raised. Our measured values follow the trend discussed by Dvolaitzky et al. [29] for microemulsions. For a sphere made of a fluid of viscosity h0 immiscible with continuous phase viscosity h1 can yield a relative viscosity given by [30]
Fig. 7. Variation of relative viscosity hr as function of volume fraction of the dispersed phase F plotted for two selected AOT concentrations, 100 mM () and 500 mM ( ). The data could be fitted to Eq. (23) quite well and the coefficients are plotted in Fig. 8 and tabulated in Table 5.
321
Table 5 Variation of l and d as function of micellar volume fraction f deduced from fitting of hr(f) to Eq. (23)a [AOT] (mM)
l
d
rsol (g/cc)
100 200 300 400 500
−0.74 −0.52 0.16 0.26 0.29
2.52 9 0.08 2.66 9 0.07 2.75 9 0.07 2.83 9 0.06 2.91 9 0.06
0.7669 0.008 0.784 9 0.008 0.806 9 0.008 0.823 9 0.009 0.847 90.009
The solution density values rsoln. did not show W0 dependence for a fixed AOT concentration within the cited experimental inaccuracy. a
hr = 1+
2.5h0 + h1 h0 + h1
(24)
Here, it is assumed that the velocity of the fluid must vanish at the surface of the sphere. Our data could not be fitted to such an expression, possibly because of the non-sphericity of our diffusing micellar particles. For low volume fractions 0.035 f 50.1, the coefficient l was deduced to be − 0.74 which slowly increased to 0.29 as f gradually increased to 0.3. At these low f values, the results could be fitted to hr = (1+ 2.5f), since the quadratic term is very small in Eq. (23). At these volume fractions our results qualitatively agree with those of Sein et al. [18]. At comparatively larger volume fractions our values are smaller than the literature [2] values by a factor 1/2
Fig. 8. Variations of the shape parameter d () and concentration dependence term l ( ) of Eq. (23) as a function of AOT concentration.
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which we could not explain. The shape dependent parameter d was found to be 2.529 0.06 for AOT concentration of 100 mM and this gradually increased to 2.919 0.06 for 500 mM AOT solutions. This clearly indicates the presence of spherical micelles at low surfactant concentrations. These get substantially deformed at surfactant concentrations more than 300 mM for W0 ] 30 as depicted in Fig. 1. Table 1 shows significantly bigger micelles in solutions when [AOT]] 300 mM. As these micelles grow bigger, the interfacial rigidity defined by K, becomes inadequate to provide sufficient strength to the interface to maintain a definite shape. Two observations have been made at this stage, one is, at low AOT concentrations for W0 50, deformations are observed in micellar shape and the second is, at higher surfactant concentrations this starts even at W0 ]30. Sein et al. [18] have seen interesting scattering features in AOT – CCl4 –H2O reverse micelles as the volume fraction of the dispersed phase is increased as in our case. For 0.055 f5 0.08, clear spherical micelles were seen which showed distinctive change at higher volume fractions and we, identify these as deformed spherical structures. The measured solution density rsol, values showed very little variation when AOT concentration was held constant and W0 was varied from 1 to 50. This is listed in Table 5. The measured data could not either be expressed as rsol = rwfw +rofo +rsfs or
conclusive. Zulauf and Eicke [24] have measured the molecule weight of reverse micelles and their corresponding hydrodynamic radii (R) values and made a plot of MW versus R that showed a scaling RM 0.30. An exponent of 0.33 would have indicated a perfect sphere. Hence, this possibly indicates the existence of non-spherical reverse micelles in the dispersion medium. We deduce an explicit expression for the pressure P(R) experienced by the water core of a reverse micelle of radius R. The micelle has been modeled to be stable when P(R)= 0. The free energy equation (Eq. (3)) has been written assuming all the water to be identical. In fact the bound water at the interface and the bulk at the core do exhibit different properties and are likely to contribute differently to the free energy equation. The same is true for the oil. This yielded the stability condition in terms of the surface rigidity K and effective interfacial tension g. Incorporating, the spontaneous curvature (1/R0) and R into a single parameter (their geometric mean) we find a length scale j which is more meaningful than R when it comes to conformational analysis. The transient features of spherical to deformed-spherical transitions in conformations remains to be probed.
Acknowledgements One of the authors (M. Behboudnia) acknowledges Ardebil University, Iran for providing the research fellowship and study leave.
rsol = rwxw + roxo +rsxs as is normally the case in miscible liquid mixtures, where f and x are volume and mole fraction of corresponding components.
5. Conclusions A distinctive feature of our studies is that we have taken higher AOT concentrations to form reverse micelles and varied W0 up to 50 and these has been characterized rigorously. The literature results on reverse micellar conformations are non-
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