Droplet Microemulsion

Dec 1, 1997 - a droplet microemulsion phase. We consider a homogeneous nucleation where aggregate growth occurs through addition of monomers. The nucl...
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Langmuir 1997, 13, 6972-6979

Homogeneous Nucleation in an Emulsion/Droplet Microemulsion System H. Wennerstro¨m,* J. Morris, and U. Olsson Division of Physical Chemistry 1, Center for Chemistry and Chemical Engineering, Lund University, P. O. Box 124, S-22100 Lund, Sweden Received July 11, 1997. In Final Form: October 10, 1997X We present a theoretical analysis of a model nucleation process where an oil phase separates out from a droplet microemulsion phase. We consider a homogeneous nucleation where aggregate growth occurs through addition of monomers. The nucleus is formed by the growth of an already existing microemulsion droplet. On the basis of previous equilibrium studies of the microemulsions of the same system we can be confident about the accuracy of the description of free energy changes during nucleation. Using the constraints of constant hydrocarbon volume and aggregate area, the change in curvature free energy is determined as an oil drop is nucleated rather than the change in surface free energy, as in a conventional nucleation theory. We obtain a simple analytical expression for the barrier which has the feature that it only exists in a finite parameter range. In the particular system that we have studied experimentally a two-phase system of microemulsion plus excess oil is reached through a temperature quench and a nucleation barrier is found for moderately deep quenches only. Having established an expression for the nucleation barrier, we analyze the kinetics and derive a diffusion equation in aggregate space, which considerably facilitates the calculation of the steady state rate for the formation of nuclei. Experiments confirm the existence of a nucleation barrier in the predicted range. They also show the concentration independence of the barrier and that experiments with different initial radii can be put on a common scale, as predicted. It is concluded that the system is very promising for fundamental studies of the dynamics of nucleation processes in liquids.

1. Introduction In microemulsions the preferred aggregate shape is controlled by the spontaneous curvature of the surfactant film.1 This is most easily demonstrated in ternary wateroil-nonionic surfactant systems. For alkyl oligo ethylene oxide surfactants, CnEm, the spontaneous curvature, H0, can be tuned by changing the temperature, and in this way one can explore a whole range of aggregate geometries. A particularly clear manifestation of the role of H0 is found for droplet microemulsions where in excess water the temperature is adjusted to give an H0-1 of the same magnitude as the radius of the microemulsion (oil/ surfactant) droplets. At a given oil to surfactant ratio the radius, R, of a spherical droplet is determined by the composition through the packing constraints of a fixed area to volume ratio. At an elevated temperature, corresponding to H0-1 > R, large nonspherical aggregates are formed,2,3 since they give a curvature closer to H0 than spheres, while at lower temperatures where H0-1 ≈ R there is a match between the optimal conditions for the surfactant film and the compositional constraints. At a still lower temperature H0-1 < R and the optimal spheres can no longer form in an homogeneous system. This leads to a separation into a microemulsion phase with R ≈ H0-1 and an excess pure oil phase.2,4,5 We term this boundary the solubilization phase boundary, occurring at TSPB. Recently6 we demonstrated experimentally that this type of system shows an interesting behavior when the * To whom correspondence should be addressed. X Abstract published in Advance ACS Abstracts, December 1, 1997. (1) Olsson, U.; Wennerstro¨m, H. Adv. Colloid Interface Sci. 1994, 49, 113. (2) Leaver, M. S.; Olsson, U.; Wennerstro¨m, H.; Strey, R. J. Phys. II 1994, 4, 515. (3) Leaver, M.; Furo´, I.; Olsson, U. Langmuir 1995, 11, 1524. (4) Turkevich, L. A.; Safran, S. A.; Pincus, P. A. Theory of Shape Transitions in Microemulsions. In Surfactants in Solution; Mittal, K. L., Bothorel, P., Eds.; Plenum Press: New York, 1986; Vol. 6, p 1177. (5) Safran, S. A. Statistical Thermodynamics of Surfaces, Interfaces, and Membranes; Addison-Wesley: Reading, MA, 1994. (6) Morris, J.; Olsson, U.; Wennerstro¨m, H. Langmuir 1997, 13, 606.

S0743-7463(97)00773-7 CCC: $14.00

droplet microemulsion was temperature quenched into a nonequilibrium state of two phases coexisting at equilibrium. The path to this equilibrium state can be very slow depending on the depth of the temperature quench. We qualitatively interpreted the experimental observations as being due to a homogeneous nucleation of the oil phase as small drops through an Ostwald ripening like process. A great advantage of the nonionic microemulsion systems in fundamental studies is that we have an unusually detailed understanding of the energetics of aggregate formation and also of the dynamics of the individual molecules. The aim of the present paper is to utilize this understanding to arrive at as detailed as possible a model description of the nucleation process leading to final equilibrium. We thus see the microemulsion as a very useful model system for fundamental studies of nucleation and growth in a multicomponent liquid. The paper is organized so that we separately consider two stages of the process. First we discuss the energetics of the formation of nuclei, and then we analyze the kinetics of the nucleation. The paper is concluded by a comparison with experimental findings on the phase separation kinetics. 2. Free Energy of Forming a Critical Nucleus Curvature Free Energy of Spherical Droplets. The free energy of a microemulsion droplet system can be described as a sum of the curvature energy and an entropy of mixing of droplets. The latter is small for large aggregates and high concentrations. Most nucleation processes in fluids are described in terms of a surface free energy with a size independent surface tension. In the present case the curvature free energy takes the role of a size dependent surface tension of which we have detailed knowledge from independent measurements.7 The curvature (free) energy is an integral over the area of the surfactant film of a curvature free energy (7) Rajagopalan, V.; Bagger-Jo¨rgensen, H.; Fukuda, K.; Olsson, U.; Jo¨nsson, B. Langmuir 1996, 12, 2939.

© 1997 American Chemical Society

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density, gc, so that

Gc )

∫dA gc

(1)

where gc in turn is usually developed to second order in the local principal curvatures c1 and c2. Normally one writes gc in the Helfrich form8

gc ) 2κ(H - H0)2 + κjK

(3)

where we call c0 the preferred curvature

(2κ2κ+ κj)

c0 ) H0

(4)

since it represents the optimal curvature of a symmetrically curved film. (H0 is the optimal curvature of a cylindrically curved film.) The bending modulus κ ′ is related to those of eq 2 by

κ ′ ) κ + κj/2

(5)

For spherical aggregates ∆c ) 0 and by rewriting the expression for the curvature free energy, we need only consider the first term of eq 3. The curvature free energy of a spherical droplet of radius R is thus

Gc ) 8πκ ′(1 - Rc0)2

()

Nf Ri ) Ni Rf

(2)

Here the two variables are the mean curvature H ) (c1 + c2)/2 and the Gaussian curvature K ) c1c2. The equation also contains three system specific parameters, where the significance of the spontaneous curvature H0 was already mentioned in the Introduction. The two modulii are called the bending rigidity κ and the saddle splay constant κj. It is particularly convenient to use eq 2 when comparing surfactant films of constant topology, since the gaussian curvature term then only contributes as a constant in eq 1 by virtue of the Gauss-Bonnet theorem. In the present paper we specifically choose to consider spherical aggregates only, where the topology of the system changes as the number of droplets changes. It is then more convenient to rewrite eq 2 and use instead of the Gaussian curvature the difference curvature, ∆c ) (c1 - c2)/2, as the second variable. This yields9

gc ) 2κ ′(H - c0)2 - κj(∆c)2

droplet radius R ) Rf ) 1/cf0 < Ri and where oil has been expelled from the microemulsion phase to the excess oil phase. By assumption the surfactant remains in the microemulsion phase conserving the total interfacial area. This implies that the number of droplets increases according to

(6)

which has its lowest value at Rc0 ) 1. This is our expression for the curvature free energy of a spherical microemulsion droplet, where we count curvature toward oil as positive. Equilibrium after a c0-Jump. Consider an oil-inwater droplet microemulsion coexisting with a reservoir of excess oil. For a given positive value of c0, which is not too close to zero (c0 typically has to be larger than 1/300 Å-1), the microemulsion phase contains at equilibrium spherical oil-in-water droplets of radius R ) 1/c0. In the literature this situation is often referred to as a Winsor I equilibrium or ‘emulsification failure’. Assume that for a time t < 0 we have such an initial equilibrium situation with c0 ) ci0 and correspondingly R ) Ri ) 1/ci0. At time t ) 0 we let c0 suddenly jump to a new value, cf0 > ci0. In the experimental system, as we will see below, this is achieved by a decrease in temperature. When sufficient time has elapsed, we reach a new equilibrium with a (8) Helfrich, W. Z. Naturforsch. 1973, 28c, 693. (9) Wennerstro¨m, H.; Anderson, D. M. In Statistical Mechanics and Differential Geometry of Micro-Structured Materials; Friedman, A., Nitsche, J. C. C., Davis, H. T., Eds.; Springer Verlag: Berlin, 1991.

2

(7)

where Ni and Nf are the number of droplets in the initial and final states, respectively. While area is being conserved all the way to the final state, the total volume of droplets decreases as oil is expelled from its initial value

4π Vi ) Ni R 3i 3

(8)

to become in the final state

Vf ) Nf

4π 3 4π R ) Ni Rf R2i 3 f 3

(9)

The path to this equilibrium state involves the nucleation of a macroscopic oil phase. In the final state the curvature energy has decreased by Ni8πκ ′(1 - Ri/Rf)2, but it is far from obvious that this process is downhill in curvature energy from the start. Nucleation of the Oil Phase. Geometrical Relations. After the jump of c0 we initially have a nonequilibrium state that could either be metastable or unstable. The further fate of the system depends on which molecular paths are available for approaching the equilibrium state. There are two main possibilities. Either aggregates grow in a stepwise way, adding molecules one at a time, or they grow by two aggregates merging to form a larger one (coalescence). For surfactant systems it is known that both modes can occur depending on the conditions. On the basis of our experimental findings6 we will below specifically analyze the stepwise growth case, which is for example also found for the formation of spherical micelles or in an Ostwald ripening process. Initially we have Ni droplets of radius Ri. The nucleation of an oil phase involves the growth of one (or a few) droplet to a radius Rbig, while all the others decrease infinitesimally from Ri to Ri + dR, where dR < 0. This adjustment of the other droplets also involves a change in their number to obey the area and volume constraints. With N ‘small’ droplets and one big drop of radius Rbig, conservation of area and volume implies 2 A ) N4π(Ri + dR)2 + 4πR big

(10)

4π 3 4π (R + dR)3 + R big 3 i 3

(11)

V)N

Here the number of small drops N adjusts as the big drop grows in size; otherwise it is impossible to simultaneously satisfy eqs 10 and 11 for arbitrary Rbig. If we keep terms up to linear in dR, we obtain

A 2 ) NR 2i + 2NRi dR + R big 4π

(12)

3V 3 ) NR 3i + 3NR 2i dR + R big 4π

(13)

From eqs 12 and 13 we can find dR and N, i.e. how the

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number of droplets and their size change with the size Rbig of the nucleating oil phase. A further simplification is obtained by writing

N ) Ni - 1 + dN

(14)

In the initial state of the nucleation process dN is small relative to Ni, so we can neglect terms of order dN dR. Then, combining eqs 12-14, we can solve for dR and dN

dN ) 2x3 - 3x2 + 1

(15)

Ni dR ) x2(1 - x)Ri

(16)

and

where we have introduced the reduced radius of the growing drop

x≡

Rbig Ri

(17)

From eq 15 it follows that the change in the number of small droplets, dN, is independent of the initial number of droplets, Ni. This might seem counterintuitive, but it is related to the result of eq 16, showing that the change in the radius of the small droplets, dR, is inversely proportional to Ni. Thus for a given value of the growth of the big drop, x, the smaller droplets decrease more in size the smaller the value of Ni, in such a way that the total change in number only depends on x. Equations 14-17 show how the number and radii of the many droplets vary when one drop grows in size to radius Rbig. They apply to the initial stage of the process when the material in the large drop is still negligible relative to the total amount. Knowing the radius and number of droplets, we can calculate how the curvature free energy varies during the nucleation of the oil phase. Curvature Energies during Nucleation. At equilibrium after the c0-jump the curvature energy is at its minimum value of zero with Rf ) (cf0)-1, neglecting the entropy for the mixing of droplets. In the initial state at t e 0 there are Ni droplets of radius Ri, and from eq 6 we find

Gc ) Ni8πκ ′(1 - y)2

Ri cf0 ) Ricf0 ) i Rf c

(19)

describing the c0-jump. As the nucleation process progresses, the curvature free energy changes, and with one big drop and many small ones

(20)

where the number of small droplets, N, is given by eqs 14 and 15. The change in curvature energy, ∆Gc(x,y) ) Gc(x,y) - Gc(1,y), is then given by

∆Gc(x,y) ) (x - 1)2(2x(1 - y) + 1) 8πκ ′

( )

∂ ∆Gc ) 2(x - 1)(2x(1 - y) + 1) + 2(x - 1)2(1 - y) ∂x 8πκ ′ (22) and determining its zeros, which occur at

x+ ) 1 x- )

0

Gc(x,y) ) (1 - xy)2 + N(1 - y - cf0 dR)2 8πκ ′

that ∆Gc is independent of N and thus not dependent on the initial droplet concentration. That ∆Gc does not depend on N has the same mechanistic origin as the independence of dN on Ni in eq 15. The bending elasticity enters as a simple proportionality parameter. The nature of ∆Gc depends crucially on y. This is illustrated in Figure 1, where we have plotted ∆Gc/κ′ as a function of x for some different values of y. For y ) 1, ∆Gc/8πκ′ ) (x - 1)2, a positive parabolic function with a minimum at x ) 1. For 1 < y < 3/2 the function still has a minimum at x ) 1 but displays a maximum for x > 1. The value at the maximum decreases in magnitude, and its position shifts to lower x as y increases from 1 to 3/2. At y ) 3/2 the maximum vanishes at x ) 1 as ∆Gc/8πκ′ is reduced to ∆Gc/8πκ′ ) (1 - x)3. For y > 3/2, ∆Gc decreases monotonically for x > 1. Nucleation Barrier. When there is a maximum in ∆Gc for x > 1, the initial state after the c0-jump is metastable. The maximum can also be found by evaluating the partial derivative of ∆Gc with respect to x

(18)

where we have introduced the reduced parameter

y≡

Figure 1. Variation of ∆Gc with the reduced radius of the growing drop, x ) Rbig/Ri, for four different values of the relative c0-jump y ) cf0/ci0.

(21)

to order Ni dR. Equation 21 gives the curvature energy as a function of x, describing the radius of the growing drop, and y, the reduced c0-jump. A striking feature is

y 3(y - 1)

For 1 < y < 3/2, the first root is consistent with the notion that the initial state with all drops of equal size is a local minimum. A possible barrier comes from the second root. Inserting this into the expression for the curvature energy of eq 21 yields the activation energy or energy barrier

2y 1- ) ( 3 ∆G* ) 8πκ ′

3

c

(1 - y)2

(23)

This activation energy is positive for 1 < y < 3/2. In fact the barrier diverges as y approaches 1, while it goes to zero as a triple root at y ) 3/2. Figure 2 shows how ∆G*c varies with y. Since κ′ is typically of order kBT, we can identify three regimes. For 1 < y < 1.2 the barrier is many kBT ’s high, sufficient to slow down the nucleation process significantly. There is a second regime for y > 1.2 where there still is a barrier, but it is questionable whether or not it is high enough to provide any measurable kinetic stability. Finally there is the unstable region for y > 3/2.

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the concentration of aggregate Aj is the flow Jj into j minus the flow Jj+1 out

d [A ] ) Jj - Jj+1 dt j

(25)

During the aggregation process each step will be rather close to local equilibrium, provided there are no exceedingly fast steps. We can then rewrite eq 24 as a transport equation. To this end we introduce the chemical potentials

µj ) µ°j + kBT ln[Aj]

(26)

in the expression for the flows Jj of eq 27 and expand the exponent to first order. Then Figure 2. Variation of ∆G* with the relative c0-jump y ) cf0/ci0.

The analysis of the free energy barrier provides a qualitative understanding of the nucleation process, but to get a more quantitative measure of the dynamics, it is necessary to analyze the kinetics of the growth process in detail. 3. Nucleation Kinetics Diffusion Equation of Stepwise Aggregation. In analyzing the free energy for nucleating an oil drop from the metastable microemulsion, we have considered that the drops grow in a stepwise continuous way. This implies that oil molecules are incorporated in the drop one at a time, and we could consider a general kinetic scheme kj+

M + Aj-1 {\ } Aj kj

all j

(I)

where in the present context M is an oil molecule and Aj is a drop with j oil molecules. By focusing on the process in scheme I as rate determining, we implicitly assume that other molecular events are faster. These involve the redistribution of surfactants, which should be fast on account of the higher aqueous solubility of the monomers, and a gradual change in the number of small droplets, which we also expect to be faster as long as the nucleation is a sluggish process. The kinetic scheme (eq I) is commonly occurring and has been extensively analyzed for a number of processes like homogeneous nucleation of liquids10-12 and precipitates13 and most recently prion aggregation.14 It also forms the basis of the Aniansson-Wall theory of micelle formation kinetics.15-17 In spite of the existence of this previous work, we chose to go through the kinetic consequences of the model to arrive at a novel formalism convenient for our purposes. Starting from scheme I, we can define a flow Jj of aggregates per unit time and volume in reaction step j as Jj ) k+ j [M][Aj-1] - kj [Aj]

(24)

where [ ] denotes concentration and J is counted as positive when aggregates grow in size. Note that the change in (10) Volmer, M.; Weber, A. Z. Phys. Chem. (Leipzig) 1926, 119, 227. (11) Becker, R.; Do¨ring, W. Ann. Phys. 1935, 24, 719. (12) Strey, R.; Wagner, P. E.; Viisanen, Y. J. Phys. Chem. 1994, 98, 7748. (13) Frenkel, J. Kinetic Theory of Liquids; Dover: New York, 1955. (14) Eigen, M. Biophys. Chem. 1996, 63, A1. (15) Aniansson, E. A. G.; Wall, S. N. J. Phys. Chem. 1974, 78, 1024. (16) Aniansson, E. A. G.; Wall, S. N. J. Phys. Chem. 1975, 79, 857. (17) Aniansson, E. A. G.; Wall, S. N.; Almgren, M.; Hoffmann, H.; Kielmann, I.; Ulbricht, W.; Zana, R.; Lang, J.; Tondre, C. J. Phys. Chem. 1976, 80, 905.

Jj ) -(kBT)-1 kj [Aj](µj - µj-1 - µM)

(27)

To obtain notational simplicity we introduce the compound chemical potential, related to the de Donder affinity,

µ′j ) µj - jµM

(28)

so that µ ′j ) 0 when aggregate Aj is in local equilibrium with the monomers. For an aggregation process to large j-values we can treat the aggregation number as a continuous variable and rewrite eq 27 as a differential equation in j-space

Jj ) -(kBT)-1kj [Aj]

dµ′ dj

(29)

which is formally analogous to the general form of Fick’s first law

J(x) ) -(kBT)-1D(x) c(x)

dµ dx

(30)

The equation of continuity, Fick’s second law, takes the form

{

}

1 d dµ′ d k [A ] [A ] ) dt j kBT dj j j dj d [M] ) dt

j>1

(31)

∫1∞ dj J(j)

where monomers act as a source. In their theory of micelle formation kinetics Aniansson and Wall18 discussed the similarity between aggregate growth in the presence of a barrier and diffusion through a confinement. Equations 29 and 30 make this analogy even more apparent, and the reverse rate constant takes the role of a diffusion constant, while the aggregate concentration takes that of the local concentration of the diffusing species. Apart from the conceptual advantage of this analogy, we can actually apply familiar methods for solving the diffusion equation also to the aggregate growth problem. Nucleation with Steady State Flow. After a ‘rapid’ c0-jump at time t ) 0, the initial, oligodisperse, droplets will rapidly respond to the change in c0 and expel some oil, giving a small but thermodynamically significant increase in the monomer concentration. After this first rapid adjustment, corresponding to the rapid relaxation process in micellar kinetics, a (quasi) steady state situation is established, where a few aggregates grow in size. In (18) Aniansson, E. A. G. On the Rate of Many-Step Processes. In Chemical and Biological Applications of Relaxation Spectrometry; WynJones, E., Ed.; D. Riedel Publishing Company: Dordrecht, 1975; p 245.

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analogy with solutions of the diffusion equation we can look for a constant flow Jss at this steady state condition. If we rearrange eq 29 and multiply with an integrating factor, we obtain

{ }

(kBT)-1 exp

µ′j dµ′ ) -Jss exp{µ′j/kBT}/kj [Aj] (32) kBT dj

Now we need two boundary conditions: one for the integration of the first order equation and one to determine Jss. Thus we integrate from j ) ji, corresponding to the most probable j-value of the microemulsion droplets, to j ) jm > jcrit on the far side of the critical nucleus size. The final expression for Jss should not depend on the detailed choice of jm. The integration of the left hand side of eq 32 is straightforward and

∫j j

m

i

dj

[Aji] [Ajm] exp{µ′/kBT} dµ′ = -1 (33) ) kBT dj [Aj ]0 [Aj ]0 m

i

where [Aj]0 denotes the concentration when aggregates Aj are in local equilibrium with the monomers so that µ ′j ) 0. The first equality follows from the definition of µ ′j in eq 28, and the second follows from the fact that the small droplets at j ) ji are in equilibrium with the monomers while the concentration of aggregates of sizes larger than the critical nucleus is negligible relative to [Aji] at the actual monomer concentration. We can also simplify the right hand side of eq 32 and obtain an expression for the steady state flow

∫j j dj (k-j [Aj]0)-1)-1

Jss ) (

m

i

where DM is the diffusion constant. The aggregate radius Rj is only weakly j-dependent, and for simplicity this dependence will be neglected below. The association and dissociation rate constants are interdependent through the equilibrium constant, so that eq + + kj [Aj]0 ) k [Aj]0/Kj ) k [M][Aj-1]0

(36)

In section 2 we calculated the curvature free energy ∆Gc associated with forming one drop of radius Rbig, implying aggregation number j, under the constraint of constant film area and enclosed volume. This ∆Gc contains all free energy contributions except the entropy of mixing. We should thus have a Boltzmann distribution (19) Kabalnov, A. S. Langmuir 1994, 10, 680.

Jss )

k+[M][Aji]

∫dj exp{∆Gc(j)/kBT}

(38)

In this form one can see the relation to the classical nucleation theory,10,11 which gives an analogous expression for the steady state nucleation flow with [Aji] replaced by the monomer concentration so that ji ) 1 accordingly. In that case the free energy barrier is due to a combination of a positive surface and a negative bulk free energy term. The concentration [Aji] of microemulsion droplets of the initial optimal value ji is proportional to the total concentration [M]tot to a good approximation. Deviations occur at very low concentrations due to the increased influence of the entropy of mixing. There is a size distribution of the microemulsion droplets which we assume to be Gaussian, with a standard deviation σ with respect to the distribution in aggregation numbers. The concentration of the most probable aggregate is then

[Aji] )

[M]tot jiσx2π

(39)

and we arrive at an expression for the rate of formation of nuclei in terms of accessible quantities

Jss )

(35)

(37)

of aggregate concentrations. Combining eqs 34, 36, and 37 and neglecting the difference in equilibrium concentration for j - 1 and j, we find for the steady state flow

(34)

Nucleation Rate with Diffusion-Controlled Addition of Monomers. The major unknown quantity in the integral of eq 34 is the rate constant and its j-dependence. At this stage it is necessary to be even more specific about the molecular processes that lead to aggregate growth according to scheme I. For the case of the analogous Ostwald ripening phenomenon there is the question of whether the oil molecules are transported to the growing drops by micelles or as monomers, and we have the same possibilities in the present case. For Ostwald ripening it seems that monomer diffusion is dominating,19 and we tentatively make the same assumption for the nucleation process. Addition of an uncharged solubilizate to a surfactant aggregate is a diffusioncontrolled process, and the forward rate constant is

k+ j ) 4πDMRj

[Aj]0 ) [Aji] exp{-∆Gc(j)/kBT}

k+[M]0[M]tot

∫j j dj exp{∆Gc(j)/kBT}

jiσx2π

m

(40)

i

Note that the integral will be dominated by the j-regime where ∆Gc has its maximum value, so that Jss is insensitive to the precise choice of jm provided there is a nucleation barrier. For the special case we are considering, eq 40 can be further simplified by using the explicit expression eq 21 for the curvature energy ∆Gc, after substituting the oil aggregation number j with the more suitable reduced radius, x, of the nucleating drop. To find the relation between j and x, we first need to define the aggregate radius R. We define this as the radius of the spherical interface enclosing the oil and a certain fraction, R, of the surfactant. We then have

jvo )

4πR3 - 4πR2Rls 3

(41)

where vo is the oil molecular volume and ls is the effective surfactant length defined as ls ≡ vs/as, where vs and as are the volume per surfactant molecule and the average area it occupies at the defined polar/apolar interface, respectively. From eq 41 we then obtain

dj )

4π 2 (R - 2RRls) dR vo

(42)

which in the reduced variable can be written

4πR3i 2 dj ) (x - 2xRls) dx vo and eq 43 is transformed to

(43)

Emulsion/Droplet Microemulsion System

Jss ) k+[M]0[M]totvo

∫1x dx m

(

⁄[

Langmuir, Vol. 13, No. 26, 1997 6977

jiσ(32π3)1/2R3i ×

) {

}]

ls 8πκ ′ x2 - xR exp (x - 1)2(2x(1 - y) + 1) Ri kBT

(44) which gives the initial nucleation rate in terms of the reduced c0-jump, y, describing the degree of supersaturation of oil. For a given value of κ′/kBT and y the integral is easily evaluated numerically. A convenient choice for the upper limit of integration is xm ) 2y/3(y - 1), corresponding to twice the value at the barrier maximum. Figure 3 shows the variation of Jss/[Aji] ) Jssjiσ x2π/[M]tot, which is the initial quasi steady state nucleation rate given as number of nuclei produced per second and per microemulsion droplet of optimal size, with the relative c0-jump height, y. In the calculations we have used [M]0 ) 2.2 × 1020 m-3,20 corresponding to the saturation solubility of decane in water, k+ ) 6 × 10-17 m3 s-1 calculated from eq 35, where DM ) 6 × 10-10 m2 s-1 was estimated from the Hayduk-Laudie equation,21 vo ) 323 Å3, R ) 75 Å, and κ′ ) 1.5 kBT. The parameters were chosen to correspond to the experimental system C12E5water-decane, which we will discuss below. When y ) 1, Jss ) 0, as expected. However Jss increases rapidly with increasing y as ∆G* decreases. At y ) 1.1 we obtain Jss/[Aji] ≈ 10-31 s-1, but for y ) 1.2 the value has already increased to 10-4 s-1. The calculated Jss represents the initial steady state rate. As nucleation progresses and the nuclei grow in size there will be a competition between nucleus formation and nucleus growth, where the latter process will ultimately dominate. The eq 44 only applies for early times where the nucleation is the main sink for the monomers. We postpone the experimental and theoretical study of the growth phase to a later occasion. 4. Comparison with Experiments Above we have made a detailed theoretical analysis of nucleation and growth in a nonequilibrium droplet microemulsion system. The basic motivation for this endeavor is that we have an experimentally realizable situation, with a nonionic surfactant-water-oil system, having unusually well-defined initial conditions and where both the energetics and the dynamic molecular processes can be given a quantitative description. We thus have a system where it is possible to obtain a critical test of the description of the fundamental processes nucleation22 and Ostwald ripening.23 We are presently working on experimental studies testing several aspects of the theoretical predictions presented above. There are many aspects of the theory, and it builds on a succession of assumptions, each of which might be inaccurate. Experiments have been performed on the well-studied C12E5-water-decane system. At the polar-apolar interface, defined as to divide oil and the surfactant alkyl chains from water and the polar ethylene oxide chains, the C12E5 surfactant has an essentially constant area, as ) 48 Å2, per molecule. Hence, this interface corresponds to the neutral surface of the surfactant film and is where we evaluate the film curvature. The dodecyl and pentaethylene oxide blocks of the surfactant have approxi(20) McAuliffe, C. Science 1969, 163, 478. (21) Hayduk, W.; Laudie, H. AIChE J. 1974, 20, 611. (22) Nucleation; Zettlemoyer, A. C., Ed.; Dekker: New York, 1969. (23) Kabalnov, A.; Shchukin, E. D. Adv. Colloid Interface Sci. 1992, 38, 69.

Figure 3. Variation of the quasi steady state nucleation rate Jss/[Aji] with the relative c0-jump y ) cf0/ci0. See text for the parameters used in the calculation.

mately the same volume, so that the apolar fraction R ) 0.5 (cf. eq 41). The radius of spherical oil-in-water droplets is given by the oil-to-surfactant ratio φo/φs (volume-toarea ratio). The stoichiometric hydrocarbon radius of a sphere enclosing the oil and the surfactant alkyl chains is given by

Rhc )

3φhcls φs

(45)

with φhc ) φo + Rφs. For C12E5 the surfactant length ls ) 14.5 Å.24 The preferred curvature of a film of a nonionic surfactant CnEm decreases with increasing temperature. Using a Taylor expansion to linear order, we write

c0(T) ) β(T - T0)

(46)

where T0 is the so-called balanced temperature where c0(T0) ) H0(T0) ) 0 and the film prefers a planar state. For the system C12E5-H2O-decane β ≈ -1 × 10-2 K-1 nm-1 and T0 ) 37.9 °C.7 The phase behavior is strongly dependent on c0, and for CnEm surfactants it is therefore strongly temperature dependent. Figure 4 shows a partial phase diagram of the C12E5-water-decane system, drawn as temperature versus φ ) φs + φo for a fixed ratio φo/φs ) 1.23. The microemulsion phase, L1, is stable in the temperature range 25-32 °C. At higher temperatures a lamellar (LR) and an L3 phase are formed. Near the lower phase boundary of the L1 phase (25 °C) the microemulsion consists of spherical oil-in-water droplets of radius Rhc ) 75 Å, which grow into a nonspherical shape if the temperature is increased away from the phase boundary. The lower phase boundary at 25 °C corresponds to the maximum oil solubility or ‘emulsification failure boundary’ discussed by Safran,4,5 where c0 ) 1/Rhc. We denote this the solubilization phase boundary with the corresponding temperature TSPB. When decreasing the temperature below TSPB, c0 becomes larger than 1/Rhc, resulting in a phase separation where the droplets respond to the increase in c0 by decreasing the radius, expelling oil to an excess oil phase (L1 + O equilibrium). A rapid c0-jump is in this experimental system achieved by a rapid drop or quench of the temperature. In particular, we consider a sample equilibrated at the initial temperature Ti ) TSPB ) 25 °C containing spherical droplets of radius Rhc ) Ri ) 1/ci0 and the evolution of the sample after a rapid quench of the temperature to a final (24) Bagger-Jo¨rgensen, H.; Olsson, U.; Mortensen, K. Langmuir 1997, 13, 1413.

6978 Langmuir, Vol. 13, No. 26, 1997

Wennerstro¨ m et al. Table 1. Experimental Data from the C12E5-Water-Decane System φo/φsa

Rib (Å)

φc

T0d (°C)

TSPBe (°C)

Tmf (°C)

ymg

1.23 1.23 1.23 2.26 2.26 2.26

75 75 75 120 120 120

0.059 0.12 0.23 0.067 0.13 0.26

37.9 37.9 37.9 36.2 36.2 36.2

25.0 25.0 25.1 29.8 29.6 29.3

22.5 21.9 21.6 27.5 28.0 27.7

1.19 1.24 1.27 1.36 1.24 1.23

a Oil-to-surfactant volume fraction ratio in the initial microemulsion. b Radius (Rhc, see eq 45) of the initial microemulsion droplets. For the φo/φs ) 1.23 system the radius has been measured by small angle neutron scattering,24 while the value for the φo/φs ) 2.26 system was calculated from eq 45. c Initial droplet volume fraction. d The temperature at which H0 ) c0 ) 0. e Temperature of the solubilization phase boundary (see text). f Temperature of the observed transition from metastability to instability. g The value of y ) cf0/ci0 at the metastability to instability transition calculated from T0, Tm, and TSPB according to eq 47.

Figure 4. Partial phase diagram of the C12E5-water-decane system for a constant oil-to-surfactant ratio φo/φs ) 1.23. The phase diagram is drawn as temperature versus φ ) φs + φo. L1 is the microemulsion phase, LR is a lamellar liquid crystalline phase, and L3 is another liquid phase, sometimes denoted the ‘sponge phase’. The lower phase boundary of the L1 phase represents the solubilization phase boundary (SPB) occurring at the temperature TSPB ) 25 °C.

value Tf < TSPB with the corresponding preferred curvature cf0 > ci0. With eq 46 we then have experimentally

y)

cf0 ci0

)

Tf - T0 Tf - T0 ) Ti - T0 TSPB - T0

(47)

On the basis of the curvature energy approach, we found in section 2 that a nucleation barrier exists for c0-jumps (temperature quenches) in a finite interval. We have recently verified this circumstance experimentally.6 In ref 6 we reported the results from time-resolved turbidity experiments performed on samples which after equilibration in the microemulsion phase, L1, were subject to a rapid drop in temperature into the two-phase area L1 + O. Three different droplet volume fractions were investigated, keeping φo/φs ) 1.23 constant, corresponding to the phase diagram of Figure 4. The variation of the turbidity with time was found to depend strongly on the quench depth. For shallow quenches with Tf > Tm ≈ 22 °C there was no detectable turbidity change within the time frame of 1 h. For deeper quenches with Tf < Tm an instantaneous turbidity increase was recorded as a function of time after the quench, where the (initial) slope increased with decreasing Tf. An important observation was that Tm is essentially independent of φ in this concentration range, which is consistent with the φ independence of TSPB. Tm and other data are summarized in Table 1. To summarise, it was found that the droplets are stable for T ) 25 °C and that there is an area of metastability between 22 and 25 °C followed by an area of instability for T < 22 °C. The observation of a metastable region is consistent with a nucleation barrier, ∆G*, as discussed and calculated in section 2. The temperature Tm ≈ 22 °C, where we observe a transition from metastability to instability, corresponds to a relative quench depth ym ≈ 1.25, as obtained by identifying Tf ) Tm in eq 50. For a more quantitative comparison with theory we consider eq 26 and the plotted function in Figure 2. While ∆G* does not vanish until y ) 3/2, it drops very fast initially when y is

Figure 5. Calculated initial quasi steady state nucleation rate as a function of the quench temperature Tf for the system φo/φs ) 1.23. The nucleation rate is expressed as the number of nuclei produced per total number of microemulsion droplets per second.

increased above 1 (cf. Figure 2) and has at y ) 1.25 already reached ∆G*/κ′ ≈ 1.5kBT. The value of the bending constant κ′ was recently estimated from an electrostatic stress experiment to have the value (1.5 ( 0.5)kBT.7 With this value we find ∆G* ≈ 2kBT at the operationally defined metastability to instability transition. It is also interesting to compare with the quasi steady state nucleation rate calculated above (eq 44, Figure 3). To obtain a more quantitative comparison, we have substituted the parameter y for the temperature Tf according to eq 47. Furthermore, while in Figure 3 we presented the nucleation rate as per number concentration of the most probable aggregate, [Aji], it is now more useful to calculate the rate per total number of droplets, i.e. Jss/ (N/V), where Ni/V ) σx2π[Aji] and where we recall that σ is the standard deviation of the distribution of oil aggregation numbers. The relative volume polydispersity of the initial droplets is typically of the order of 20-30%.24 Since the average oil aggregation number is approximately 4000, we then obtain σx2π ≈ 3000, assuming 30% polydispersity. The calculated quasi steady state nucleation rate, normalized with respect to N/V, is presented in Figure 5 as a function of Tf. Above 23 °C the nucleation rate is very low and decreases rapidly with increasing temperature. At the temperature, ≈22 °C, where we begin to record a turbidity increase in the experiments, the model calculation gives a nucleation rate of approximately one per million droplets a second. At present we cannot make a quantitative comparison between theory and experiments, but an estimate of the detection limit in the turbidity experiment is one big drop per 106 to 108 small ones. With the further uncertainty about the duration of

Emulsion/Droplet Microemulsion System

the nucleation phase, we estimate the operationally defined transition from metastability to instability to occur in the range 22-22.5 °C. This is a few tenths of a degree higher than that found experimentally. We find this relatively good agreement encouraging, and it provides some support for the basic assumptions that allowed us to derive eq 44. The model predicts ym to be essentially independent of φ, which was also observed experimentally. While Jss is proportional to the droplet concentration its variation with y and hence the temperature is much stronger. The factor of four increase in Jss, which is expected when going from our lowest to our highest concentration, may be compensated by a temperature decrease of less than 0.1 °C. Another prediction of the model is ym to be independent of Ri. To test this prediction, we have also performed experiments on samples with φo/φs ) 2.26, corresponding to Rhc ) 120 Å. Due to later use in small angle neutron scattering experiments, these samples were prepared with D2O instead of H2O, which has the effect that T0 is slightly lowered to T0 ≈ 36.2 °C.25 Results from the experiments are reported in Table 1. Similar ym values, around 1.25, as observed with the smaller droplets, were observed for three different concentrations with these bigger drops. Thus, in accordance with the model predictions, we do not find a significant droplet size dependence on ym in our experiments. To summarize, we have analyzed the nucleation of an oil phase in an oil-in-water droplet emulsion/microemulsion system. From curvature energy we find a nucleation barrier for shallow quenches into the two-phase area. This is consistent with the metastability observed experimentally with a nonionic surfactant-water-oil system. The calculated nucleation barrier decreases rapidly with increasing quench depth. This is also consistent with experiments where a sharp transition from metastability to instantaneous instability is observed. A quantitative comparison yields that this transition occurs when the barrier is reduced to about 2kBT. The driving force for the phase separation is the reduction in curvature energy when the droplets reduce their radius to match the new preferred curvature. It seems clear that the mechanism for removing oil is that a small fraction of the droplets are sacrificed and grow, thereby allowing the majority of droplets to decrease their size. Less clear at present, however, is the dominating mechanism behind the net flow of oil from the smaller droplets to the growing drops. Assuming this mechanism to be the transport of individual (25) Sottmann, T.; Strey, R.; Chen, S.-H. J. Chem. Phys. 1997, 106, 6483.

Langmuir, Vol. 13, No. 26, 1997 6979

oil molecules via the solvent water (stepwise aggregation), as in classical Ostwald ripening, we have also derived an expression for the quasi steady state nucleation rate valid for the initial state of the nucleation process. As mentioned, it is not yet clear if this is the dominating means of communication between droplets or if exchange through collisions, coalescence, and breakup or some other mechanism is more efficient. The dominating mechanism may also be different for different quench depths. For the stepwise aggregation mechanism the nucleation and growth rates should depend strongly on the oil solubility in water and therefore on the oil chain length. Such a strong oil chain length dependence was indeed observed by Nakajima,26,27 in similar quench experiments to ours. Taisne and Cabane,28 on the other hand, observed no significant difference between tetradecane and hexadecane systems, when following the evolution of the small angle neutron scattering after quenching from very high temperatures. Independently, Vollmer et al.29,30 have recently investigated the analogous phase separation of water from a water-in-oil microemulsion upon heating. Upon applying a constant heating rate, they were able to drive the system into a sequence of repeating phase separations as observed by an oscillating turbidity and heat capacity. The phase separation was analyzed using the same conceptual framework of curvature energy as used here. As a mechanism for the formation of large emulsion water drops Vollmer et al.29,30 propose and analyze the consequence of material exchange through binary collisions. This mechanism has also been discussed independently by Taisne and Cabane.28 In this paper we have addressed a different mechanism, assuming throughout that aggregate growth occurs by addition of monomers. Work is currently in progress to attempt to clarify the situation. Acknowledgment. This work was supported by the Swedish Natural Science Research Council (NFR) and the Go¨ran Gustafsson Foundation. LA970773+ (26) Nakajima, H.; Tomomasa, S.; Kouchi, M. J. Soc. Cosmet. Chem. Japan 1990, 23, 288. (27) Nakajima, H. Microemulsions in Cosmetics. In Industrial Applications of Microemulsions; Solans, C., Kunieda, H., Eds.; Marcel Dekker: New York, 1997; p 175. (28) Taisne, L. Echanges d'Huile entre Gouttes d'Emulsion, Theses l’Universite Paris VI, 1997. (29) Vollmer, D.; Strey, R.; Vollmer, J. J. Chem. Phys. 1997, 107, 3619. (30) Vollmer, J.; Vollmer, D.; Strey, R. J. Chem. Phys. 1997, 107, 3627.