Gelation and Coarsening in Dispersions of Highly Viscous Droplets

May 18, 2001 - Consolidation of foams deriving from emulsions by temperature cycling (“tempering”). E. Gravier , N. Drelon , L. Boisserie , A. Oma...
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Langmuir 2001, 17, 3545-3552

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Gelation and Coarsening in Dispersions of Highly Viscous Droplets J. Philip,‡ J. E. Poirier,§ J. Bibette,† and F. Leal-Calderon*,† Centre de Recherche Paul Pascal, CNRS, Avenue Albert Schweitzer, 33600 Pessac, France, Indira Gandhi Centre for Atomic Research, Kalpakkam 603 102, India, and COLAS, Laboratoire Central de Recherche, 4 rue Jean Mermoz, 78771 Magny-les-Hameaux, France Received September 29, 2000. In Final Form: March 26, 2001

We study the gelation and coarsening mechanisms in liquid-liquid dispersions made of highly viscous oils (viscosity about 105-106 Pa‚s). When a rupturing agent is added to the initially stable emulsion, a gel forms, which further contracts by preserving the geometry of the container. The microscopic observation of the gelation process and the characteristic time for gelation give qualitative insights into the growth mechanism which, in our experiments, is occurring via reaction-limited cluster aggregation (RLCA). The gelation is followed by a contraction process driven by surface tension, which is analogous to the sintering process occurring during the densification of solid powders. The initial stages of densification follow very well the “cylindrical model” for viscous sintering, but deviations are observed at the final stages of densification. We examine the influence of different factors (initial droplet volume fraction, droplet viscosity, rupturing agent nature, and concentration) on the gelation and contraction kinetics. Finally, we discuss the universality of the homothetic-sintering phenomenon.

I. Introduction Emulsions are dispersions of one liquid in another immiscible one, prepared by shearing the mixture of the two fluids in the presence of surface-active species (surfactants, polymers, etc.). The excess energy associated with large interfacial area created during shearing makes the emulsions metastable. The irreversible destruction of emulsions can be either due to Oswald ripening or coalescence. Ostwald ripening is due to the molecular diffusion of the dispersed phase trough the continuous phase.1 This mechanism does not involve film rupturing: instead, there is transfer of matter from the smaller to the larger dispersed droplets. When the two phases are poorly miscible, coarsening is essentially due to coalescence phenomena.2 Coalescence consists of the rupture of the thin liquid film in between two adjacent droplets through the nucleation of a small channel. This first nucleation step is followed by a shape relaxation driven by surface tension, which causes two droplets to fuse into a unique one. The characteristic time for shape relaxation is governed by the competition between surface tension and viscous dissipation and is given by Tr ∝ ηR/γ, where η is the viscosity of the droplets, R is their characteristic radius, and γ is their surface tension.3 When there is no energy barrier for coalescence, the droplets coalesce as soon as they collide. This nonactivated coarsening has been identified in the late stages of phase separations or in strongly unstable emulsions. The main limit, which was most extensively studied to date, corresponds to systems in which the characteristic shape relaxation time Tr is shorter compared to the time †

CNRS. Indira Gandhi Centre for Atomic Research. § COLAS. * Corresponding author: Tel 33 5 56 84 56 33; Fax 33 5 56 84 56 00; e-mail [email protected].

Tb separating two droplet collisions (under the effect of Brownian motion). In this limit, it was found both theoretically and experimentally that the average droplet size scales with time as t1/3 in 3-D systems.4 A very different scenario is observed in the limit where Tr is much larger than Tb. In this limit, the coarsening is limited by shape relaxation leading to very different structures and kinetics than in the previous case.5 This situation is frequently encountered in systems like emulsions of highly viscous substances (bitumen, colophon oil) or phase separations in binary mixtures of polymers. Here we aim to complete the description of coarsening mechanisms in systems where Tr . Tb. We use model emulsions of highly viscous oils (bitumen) which can be made suddenly unstable toward coalescence upon addition of a suitable chemical. Once the emulsion is made unstable, the droplet form a macroscopic gel made of an array of fused droplets. Then the gel continuously contracts with time in order to reduce its surface area.5 The contraction phenomenon in our case is reminiscent of the sintering process observed in ceramics and aerogels.6,7 It has been known for quite long time that powders made of fine packed particles can be sintered at temperatures well below the melting point of the same macroscopic solid phase. During sintering, the materials become tougher or denser by reducing the total surface area. In the case of gels and glasses, the sintering occurs due to viscous flow of matter inside the gel network formed by the interconnected solid particles. This phenomenon is widely used for the production of dense ceramics materials from powders, the manufacturing of refractory materials at low temperatures, the coating of electronic components and optics, etc. The first purpose of this paper is to propose a more detailed and precise presentation of the phenomena

‡

(1) Ostwald, W. Z. Phys. Chem. (Munich) 1901, 37, 385. (2) Deminie`re, B.; Colin, A.; Leal-Calderon, F.; Bibette, J. Phys. Rev. Lett. 1999, 82, 229. (3) Frenkel, J. J. Phys. (Moscow) 1945, 9, 385.

(4) Binder, K.; Stauffer, D. Phys. Rev. Lett. 1974, 33, 1006. (5) Philip, J.; Bonakdar, L.; Poulin, P.; Bibette, J.; Leal-Calderon, F. Phy. Rev. Lett. 2000, 84, 2018. (6) Mackenzie, J. K.; Shuttleworth, R. Proc. Phys. Soc. London 1949, 62, 833 (7) Scherer, G. W. J. Am. Ceram. Soc. 1977, 60, 236. Scherer, G. W. J. Am. Ceram. Soc. 1977, 60, 243.

10.1021/la001377l CCC: $20.00 © 2001 American Chemical Society Published on Web 05/18/2001

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that were first described in ref 5. We demonstrate that viscous sintering phenomena, which are well-known for glasses and gels, also occur in emulsions and can be observed on convenient time scales. In this way, it establishes a useful connection between the two fields of materials science. Our second purpose is to present some new results concerning the kinetics of the gelation process and of the macroscopic contraction as a function of different experimental parameters (initial droplet volume fraction, rupturing agent concentration and nature, bitumen viscosity, etc.). We discuss the basic features of the observed phenomena, and we compare our experimental results to some models from the literature. Finally, we discuss the universality of the observed phenomena. II. Experimental Section 1. Emulsification. Our emulsions are three-component suspensions consisting of dispersed bitumen oil, water as a continuous phase, and a surfactant partially adsorbed on the interface. The surfactant molecules play two roles in the emulsification process. First, they lower the interfacial tension and thereby make it easy to create the droplets. Second, they stabilize the dispersed droplets against coalescence after their formation. Tetradecyltrimethylammonium bromide (TTAB, critical micellar concentration cmc ) 3.5 × 10-3 M) purchased from Aldrich was used as surfactant. Mechanical energy is applied to the three-component system by using a colloidal mixer. A crude emulsion was prepared by gently shearing surfactant and bitumen at a temperature of 95 °C. The composition of the bitumen, TTAB, and water was 80:6:14 by weight. After emulsification, we obtained a polydisperse emulsion with a wide size distribution ranging from 0.1 to 10 µm. From the polydisperse emulsion, we get monodisperse fractions using a fractionation technique which is based on the liquid-solid phase transition induced by attractive depletion interactions.8 The fractionation method consists of introducing excess surfactant in the continuous phase. The large amount of surfactant micelles induces an attractive interaction (depletion forces) between the oil droplets, due to the micellar osmotic pressure.9 Since the depletion force is proportional to the size ratio between the droplets and the micelles, it becomes possible to separate droplets of different sizes using a fractionated crystallization technique. The final droplets size distribution was measured using a Malvern particle sizer (Mastersizer S). The polydispersity of our emulsions was around 15%. After the size selection process, the continuous phase contains excess surfactant. To set the surfactant concentration at a fixed value, we centrifuge the emulsion four times, and each time the continuous phase is replaced by a surfactant solution at a constant concentration. The final volume fraction of the stock emulsion after centrifugation was about 30%. The emulsions were found to be stable for several months. We use a phase contrast optical microscope equipped with a Normarsky contrast (Zeiss, Axiovert 100) for observation of the emulsions. 2. Viscosity Measurements. We used two different bitumens from NYNAS company with penetration grades 180/220 and 80/ 100. The penetration grade is an indication of the fluidity obtained from the sinking of a needle in bitumen in normalized conditions. The low shear viscosity of bitumen was obtained at different temperatures ranging from 20 to 90 °C using two different methods. At lower temperatures we measured the sedimentation velocity of a small iron sphere (radius ) 0.5 cm) immersed in the bitumen, and we used Stokes’ law to deduce the viscosity: η ) (2/9v)∆Fr2g where v is the sedimentation velocity, r is the radius of the metal ball, ∆F is the density difference between bitumen and water, and g is the acceleration due to gravity. A constant stain rheometer (Rheometrics) equipped with the cone and plate geometry was preferred at higher temperatures. The values obtained by these techniques were consistent with each other. The temperature-dependent viscosity of both bitumens is shown in Figure 1. (8) Bibette, J. J. Colloid Interface Sci. 1991, 147, 474. (9) Asakura, S.; Oosawa, J. J. Polym. Sci. 1958, 32, 183.

Philip et al.

Figure 1. Temperature-dependent viscosity of two different bitumens (80/100 and 180/220). 3. Contraction Experiments. To study the gelation and contraction phenomena, we introduce the emulsion of known initial volume fraction in a rectangular glass cell, and we add NaOH of known concentration. Since the emulsions are stabilized using a cationic surfactant (TTAB), the droplet interface is positively charged. Naturally, bitumen contains some acidic molecules which are also present at the bitumen-water interface. By adding NaOH in the continuous phase, the pH is raised, and the acidic dissociation gives rise to some negative charges that may partially or totally neutralize the interface. In the presence of low electrostatic repulsion, the droplets become unstable and coalesce. Right after the addition of NaOH, the emulsion is agitated for 2 s and stored at a given temperature. Initially, the system remains liquidlike, but after some time the emulsion does not flow any more. At this stage, microscopic observation reveals that the droplets stick together and form a threedimensional gel network.5 Once this network is formed, the gel starts to contract by reducing its surface area. In this process water is expelled from the space-filling network. The contraction remains remarkably homothetic, meaning that it preserves the geometry of the container. The contraction of the gel at different times (1, 9, and 15 min) after the addition of NaOH in a rectangular cell is shown in Figure 2a-c. The initial volume fraction of the emulsion was 16%. The initial droplet diameter was 1 µm, and the NaOH and TTAB concentrations were 1.5 M and 1 cmc, respectively. The contraction kinetics is followed by a digital camera. At regular intervals images are grabbed and stored in a computer, and later the exact dimension of the gel is measured using standard image processing software. By measuring the dimensions of the gel, we evaluate the volume fraction of bitumen inside the gel. To ensure that we measure the correct volume fraction, at the end of each experiment we remove the gel from the container and measure the final volume fraction by gravimetry. The two methods give identical results within 5%.

III. Results and Discussion 1. Gelation Process. a. Microscopic Observation of the Gelation Kinetics. The microscopic observations of an emulsion after the addition of NaOH at different time intervals are shown in Figure 3a-h. Here the bitumen used was 80/100 penetration grade, the droplet size was about 1.0 µm, and the TTAB concentration was equal to 1 cmc. Initial volume fraction of bitumen was 12%, and the NaOH concentration was 1 M. It can be seen that the growth process is occurring very slowly in this case. In the first slides (a-c) there are not many aggregates present in the system. However, at the end of the gelation process, all the droplets are connected and form a spacefilling interconnected network. Once the gelation process is complete, the gel begins to contract by reducing the surface area between bitumen and water phases, and during this process water is expelled from the gel meshes. For larger initial volume fractions and higher NaOH concentrations, the gelation process occurs more rapidly. Irreversible colloidal aggregation can occur via two distinct limiting regimes called diffusion-limited cluster aggregation (DLCA) or reaction-limited cluster aggregation (RLCA). The DLCA process occurs when there is

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Figure 2. Digital camera images of the gel at various time intervals after the addition of NaOH: (a) t ) 1 min; (b) t ) 9 min; (c) t ) 15 min.

at a time and that the diffusion coefficient of the clusters is given by Einstein’s equation for hard spheres:

negligible repulsive force between the colloidal particles so that the probability for sticking during a Brownian collision is equal to 1. In this case, the aggregation rate is limited by the time taken for the clusters to encounter each other by diffusion. In the RLCA process, the binding probability is less than unity. Here the aggregation occurs when there is a nonnegligible repulsive force between the particles, and the aggregation rate is limited by the time taken for two clusters to overcome this repulsive barrier by thermal activation. For a more detailed description about RLCA/DLCA processes, the reader is addressed to ref 10. So, we wanted to find out the limiting regime by which the aggregation phenomenon is occurring in our case as well as the scaling law of the gelation time with the initial drop volume fraction. b. Gelation Time. We define the gelation time Tg as the delay that has to be expected between the moment we introduce the rupturing agent (NaOH) and the moment at which the macroscopic contraction starts. Tg also coincides with the time at which the initially fluid emulsion does not flow any more. (Strictly speaking, the rheological elastic modulus becomes larger than the viscous loss modulus.) The gelation time is macroscopically obtained by regularly and slightly tilting the emulsion sample: when gelation occurs, the emulsion/air interface does not restore horizontality anymore due to the viscoelasticity of the gel network. The gelation time Tg as a function of initial volume fraction φ0 for a bitumen emulsion of grade 80/100for four different NaOH concentrations (2, 1.5, 1.0, and 0.5 M) are shown in Figure 4. In all these cases, Tg exhibits a power law behavior and scales as φ0x, where x value is around -3.0. We have also repeated the experiment by performing these measurements in a different emulsion made of bitumen 180/220 of droplet diameter 1 µm. Figure 5 shows the experimental values of Tg for different initial volume fractions and the best fits obtained for three different NaOH concentrations (0.8, 1.0, and 1.5 M). Again the power is around -3.0. At the microscopic level, Tg can be considered, as the time needed for all the growing clusters to stick and form a unique aggregate that spans over the whole volume. The main features of the gelation time may be obtained from a simplified version of the so-called Smoluchovski equation,11 assuming that only one cluster size is present

where df is the fractal dimension, which can be seen as a measure of the compactness of the clusters. To obtain the previous scaling law, it is admitted that the fractal dimension remains constant over the aggregation process, meaning that rearrangements in the growing clusters are totally inhibited. The fractal dimensions corresponding to the two limiting situations of the aggregation process DLCA and RLCA are reported to be 1.8 and 2.1, respectively.12 If we use these values for the evaluation of Tg as a function of φ0 , the expected values of the exponent are -2.5 and -3.3 respectively for the DLCA and RLCA regime. So, the exponent we have obtained was intermediate between the above two limits. Nevertheless, it must be emphasized that the experimental values for fractal dimensions (1.8 and 2.1 for DLCA and RLCA, respectively) have been obtained for very low volume fraction regimes (φ0 > 1%). Recent theoretical simulation studies13 predict that in the concentrated regime (higher volume fractions) the small clusters are already interpenetrated before sticking. As a result, when they stick, both the tips of the longer arms and the middle arms are in contact, leading to an increase in the compactness of the clusters with increasing concentrations and hence larger fractal dimensions. According to the above simulations, the fractal dimension should vary with initial droplet volume fraction as df ) 2.1 + 0.47φ00.66 for the RLCA regime and df ) 1.8 + 0.91φ00.51 for the DLCA regime. Figure 6 shows the evolution of Tg (expressed in arbitrary units) as a function of initial volume fraction, when the concentration dependent fractal dimension values for DLCA and RLCA regime are taken into consideration. The curves are roughly linear in a log-log plot, meaning that Tg obeys an effective power-law behavior. The values of the exponent are -2.16 and -3.09 for DLCA and RLCA regimes, respectively. Our experimental value of -3 is rather close to the theoretical value for RLCA aggregation. However, a direct comparison of our experimental data with the previous simplified model has to be considered

(10) Meakin, P. J. Sol-Gel Sci. Technol. 1999, 15, 97. Maekin, P. Physica D 1995, 86, 104. Mellema, M.; Van Opheusden, J. H. J.; Van Vliet, T. J. Chem. Phys. 1999, 111, 6129. (11) Von Smoluchovski, M. Phys. Z. 1916, 17, 585.

(12) Jullien, R.; Botet, R. In Aggregation and Fractal Aggregates; World Scientific: Singapore, 1987. (13) Lach-hab, M.; Gonzalez, A. E.; Blaisten-Barjos, E. Phys. Rev. E 1996, 54, 5456.

Tg ∝ φ03/(df-3)

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Figure 3. Microscopic images of the gel formation right after addition of NaOH. Initial emulsion volume fraction ) 12%. (a) t ) 0; (b) t ) 10 min; (c) t ) 20 min; (d) t ) 30 min; (d) t ) 40 min; (e) t ) 50 min; (f) t ) 60 min; (g) t ) 80 min; (h) t ) 100 min.

with some precautions. Indeed, it is clear that the measured values for the gelation times are quite long, and it is likely that some shape relaxation occurs during the aggregation stage. Relaxation decreases the effective size of the aggregates already formed, and this leads to an increase in df during aggregation. Therefore, strictly speaking, one should consider that the fractal dimension is not constant but increases in time. The previous model has been considered for the sake of simplicity and because it may provide some useful orders of magnitude. Indeed, in the case of DLCA process, we can evaluate the prefactor of the gelation time by means of the Smoluchovsky equation within the same approximations (one cluster size at a time and constant

fractal dimension):

Tg ) (πηc/kT)R3φ03/(df-3)

(1)

where ηc is the viscosity of the continuous phase, k is the Boltzmann constant, T is the absolute temperature, and R is the droplet radius. If we consider the concentrationdependent fractal dimension for the DLCA process (df ) 1.8 + 0.91φ00.51), an initial concentration of 10%, and droplet diameter of 1.0 µm, the estimated values of Tg at 293 K corresponds to 159 s. However, the experimentally observed value is much higher than the above value even with very high NaOH concentration. For 2 M NaOH, the Tg is about 600 s, and for lower salt concentrations the

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Figure 4. Gelation time Tg as a function of initial volume fraction for a bitumen emulsion of grade 80/100 for four different NaOH concentrations (2, 1.5, 1.0, and 0.5 M). The solid lines are the best fits to the experimental points.

Figure 5. Gelation time Tg as a function of initial volume fraction for a bitumen emulsion of grade 180/220 for three different NaOH concentrations (1.5, 1.0, and 0.8 M). The solid lines are the best fits to the experimental points.

Figure 6. A log-log plot of theoretical evolution of the gelation time as a function of initial droplet volume fraction (see text for details). The gelation time is expresses in arbitrary units.

experimental values are orders of magnitude larger than the expected values for the DLCA regime. This observation clearly suggests that the gelation process is not occurring via the DLCA process. Therefore, both the magnitude of Tg and the microscopic observations on the gelation process give convincing evidence that the gelation process is initiated via RLCA process. 2. Contraction Process. a. Effect of Initial Volume Fraction. Once the droplets form an interconnected network, the gel starts to contract by reducing its surface area. It may be appropriate to stress the point that the contraction process begins only after the formation of permanent irreversible network, and at this stage there is absolutely no free droplets remaining in the system. This is clearly evident from the transparent liquid, which separates from the contracted gel (see Figures 2 and 3). Figure 7 shows the evolution of the bitumen volume fraction in the gel as a function of time for different initial volume fractions (8, 10, 12, 14, 16, 18, and 20%) at room temperature (25 °C). For this set of experiments, we used the bitumen with penetration grade 180/220. In all the cases, the NaOH concentration was 1.5 M and TTAB was at cmc. The radius of the droplets in the emulsion was 0.5 µm. Under these conditions we probe the effect of initial volume fraction of the droplets on the contraction mech-

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Figure 7. Bitumen volume fraction in the gel as a function of time for emulsions of different initial volume fractions (8, 10, 12, 14, 16, 18, and 20%) for bitumen 180/220. NaOH and surfactant concentration in all the above cases were 1.5 M and cmc, respectively.

anism. As already mentioned, when NaOH is introduced in the emulsion, a delay Tg has to be expected before the gel starts to shrink. Once the contraction starts, we can clearly distinguish two different regimes. The rate of contraction at the initial stage is quite rapid and becomes much slower toward the final stages of contraction. This observed contraction phenomenon mimics the classical sintering in inorganic materials such as glasses and ceramics (where the materials tend to reduce their surface area at high temperatures). Frenkel3 has considered the sintering of a pair of spheres, which is representative of the sintering of a body of packed powder. According to Frenkel’s concept, the rate of viscous sintering can be calculated by equating the energy dissipated in viscous flow to the energy gained by reduction of surface area. As the centers of the spheres approach one another, the neck between them broadens. The change in distance between the centers of the two spheres is assumed to equal the linear contraction of a compact of such particles. The geometry is complex because the shape of the neck between the particles changes considerably by the time. The mathematical description of shape of the neck is therefore calculated with simplifying assumptions. Frenkel3 obtained a simple relationship for the growth in radius “x” of the neck between spheres of radius R as

(x/R)2 ) 3γt/2ηR

(2)

The above result has been experimentally confirmed in several studies. When the neck is relatively small, there is a simple geometrical relationship between x and R, which is used to derive an expression for the linear shrinkage:

L(t)/L(0) ) 1 - 3γt/8ηa

(3)

where L(t) corresponds to the center-to-center distance between the spheres at a time t. The above equation is used widely, and it works rather very well during the first few percentage of linear shrinkage. Using Frenkel’s concept, Mackenzie and Shuttleworth6 derived an expression for the shrinkage rate of a material containing closed pores. It is assumed that the deformation during sintering is due to surface tension and that the material is incompressible and has homogeneous mechanical properties. It is difficult to extend Frenkel’s concept to complex disordered systems due to geometrical constraints. To overcome these constraints, it has been shown that the sintering mechanism can be explained in terms of connected fractals aggregates or blobs that are

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formed by cluster-cluster aggregation14 This scaling approach employs only global symmetries of scale invariance and mass conservation. The scaling occurs in the range of length extending from the mean size “a” of the subunits up to the mean connecting distance ξ between the aggregates. As a result, the model ignores kinetic aspects and also the detailed characteristics of the materials such as surface energy, viscosity, etc. Scherer15 developed a simple and elegant approach to model the sintering mechanism in materials, especially glasses or ceramics, containing open pores, within the framework of Frenkel’s concept. We attempt now to explain the contraction mechanism of an emulsion using the socalled “cylindrical model” for sintering, which has been found to be sufficiently accurate for the intermediate stages of sintering. The model considers a cubic array formed by intersecting cylinders that are made up of strings of particles. The initial cylinder radius corresponds to the average radius of the particles in the material. Although the choice of cubic array is an idealized approximation compared to the complicated microstructures formed in actual situations, studies have shown that the geometry of the unit cell chosen (e.g., octahedral, tetrahedron, inverse tetrahedron) has very little influence on the kinetics.16,17 To reduce their surface area, the cylinders tend to become shorter and thicker. In these calculations, it is assumed that the geometry of the cell is preserved. A brief description of the calculations is as follows: If η is the viscosity of the dispersed phase and “l” and “a” are the length and radius of the cylinders, respectively, the energy dissipated in viscous flow (E˙ f) is given as

E˙ f ) (3πηa2/l)(dl/dt)2

(4)

The superscript dot indicates a derivative with respect to time. The energy change due to the reduction of surface area (Es) is given by

E˙ s ) (γ dSc/dt)

(5)

where γ is the interfacial tension and Sc is the surface area of a full cylinder. The energy balance requires the following condition:

E˙ f + E˙ s ) 0

(6)

From eqs 1 and 2, we get the rate of densification as

(γ/ηl0)(1/φ0)1/3(t - t0) )

∫0x2 dx/(3π - 8x2x)1/3x2/3

(7)

where x ) a/l. For a cubic cell, x is related to the cylinder volume fraction as

φ ) 3π(a/l)2 - 8x2(a/l)3

(8)

φ corresponds to the measured volume fraction of bitumen inside the gel. t0 is the fictitious time at which x ) 0. In eq 4, (γ/ηl0)(1/φ0)1/3 ) K is a constant for a given initial volume fraction φ0. Indeed, φ0 sets the initial cylinder height l0. When the ratio of cylinder radius to its height (14) Sempe´re, R.; Bourret, D.; Woignier, T.; Philippou, J.; Jullien, R. Phys. Rev. Lett. 1993, 71, 3307. (15) Scherer, G. W. J. Am. Ceram. Soc. 1991, 74, 1523. (16) Scherer, G. W.; Brinker, C. J.; Roth, P. E. J. Non-Cryst. Solids 1985, 72, 369. (17) Scherer, G. W. In Surface & Colloid Science; Matijevic, E., Ed.; Plenum Press: New Jersey, 1987; Vol. 14.

Figure 8. Gel volume fraction as a function of reduced time K(t - t0). Solid line corresponds to theory, and the dashed line has a slope of 1.

is equal to 1/2, the neighboring cylinders touch and the cell contains only closed pores. The corresponding theoretical density (volume fraction) of the sample would be 0.942. Therefore, the cylindrical model is not valid anymore for φ values larger than 94.2%. Figure 8 shows the evolution of the bitumen volume fraction φ inside the contracting gel as a function of reduced time K(t-t0). The solid line represents the theoretical curve obtained using eq 7. For φ values between 0.2 and 0.8, the theoretical curve is roughly linear with a slope of 1 (see dashed line). Equivalently, within the same φ range, the volume fraction should vary linearly with time with a slope equal to K. The experimental data from Figure 7 where recalculated in order to be plotted in reduced coordinates (φ ) f(K(t - t0)). For each initial volume fraction, K is deduced from the initial slope of the curve φ ) f(t) (for φ between roughly 0.2 and 0.6), and t0 is only a “translational” term which allows to translate the curves toward the same abscissa range. It should be noted that in all curves plotted here t corresponds to the time after gelation. On Figure 8, we observe that all the data lie within a unique curve, showing reasonable agreement with the theoretical one. However, the figure clearly shows that the initial stages of densification (up to 60%) correctly follow the cylindrical model, but a deviation appears at the final stages of densification. Let us remember that the cylindrical model assumes that the gel remains bicontinuous with an open pore structure for φ < 94.2%. We believe that the deviations are the consequence of formation of closed pores well below this limit. The pores could then entrap water and lower the contraction rate. This is confirmed by the fact that even after several days the gels never reach total compaction (φ ) 1), but instead they always retain small amount of water (between 5 and 10%) in volume. Here arises the question to know how the evolution of the gel topology takes place. This question will require further investigations, but it is likely that the kinetics of aggregation (DLCA or RLCA), which define the initial microscopic structure of the percolated gel, as well as the surface coalescence (the densification process can induce new contacts that may coalesce) may play an important role. b. Effect of Viscosity. According to Scherer’s model,15 the rate of contraction K should vary as the inverse viscosity. We have been able to explore the influence of viscosity on the rate of contraction by changing the temperature. We could measure it experimentally and varied over 3-4 orders of magnitude in the temperature range 20-90 °C (see Figure 1). For the 180/220 bitumen, the emulsion was the same as previously, and the experimental conditions used to obtain gelation and contraction were identical ([NaOH] ) 1.5 M and [TTAB]

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Figure 9. Slope K of densification curve as a function of viscosity for two different bitumens.

Figure 10. Bitumen volume fraction in the gel as a function of time for different NaOH concentrations (bitumen 180/220). The initial volume fraction was 10%

) 1 cmc). Concerning the 80/100 emulsion, the droplet radius was 0.25 µm; the NaOH and TTAB concentrations were 0.2 M and cmc/10, respectively. For both emulsions, the initial volume fraction was equal to 9%. Figure 9 shows the evolution of K as a function of viscosity. It can be noticed that lowering the viscosity has the effect to increase the rate of contraction K over several decades. As expected, the observed contraction is controlled by the viscous flow of bitumen through the gel network. The experimentally observed slopes around -0.85 for bitumen 80/100 and -0.92 for bitumen 180/220 are in reasonable agreement with the expected value, which is -1. This observation clearly demonstrates that the temperature-dependent densification is entirely governed by η(T) since the surface tension is weakly temperature dependent. This is the reason why the scaling essentially reflects the variation of viscosity.5 c. Effect of NaOH Concentration. Figure 10 shows the temporal evolution of the bitumen volume fraction in the gel for different NaOH concentrations (bitumen 180/ 220, [TTAB] ) 1 cmc, droplet diameter ) 1 µm). In all the curves the initial emulsion volume fraction was constant (10%). The contraction kinetics was similar to those observed earlier (two distinct regimes for the initial and final stages of contraction). However, the gelation time drastically changes when NaOH concentration is varied from 0.8 to 2.0 M, indicating that the repulsive barrier that controls the aggregation kinetics is strongly affected by the amount of sodium hydroxide. For 2 and 1.5 M NaOH, the contraction begins rapidly (within 10 min). In these cases, the gel volume fraction reaches 0.7 within half an hour. For salt concentration less than 1.5 M, the gelation occurs very slowly (within an hour), and we also observe

Figure 11. Digital camera images of the homothetic contraction of an esterified colophon oil emulsion after the addition of NaOH. The glass vessel shape is cylindrical: (a) t ) 1 min; (b) t ) 30 min.

that the rate of densification revealed by the initial slope of the curve φ ) f(t) (right after gelation) is slightly smaller. 4. Universality of the Homothetic Sintering Process. To see whether this contraction phenomenon is a universal one, we have prepared another oil-in-water emulsion using highly viscous colophon ester oil (105 Pa‚s at room temperature; this oil is obtained from pine resin after chemical modification). As in the case of bitumen, this oil also contains some residual acidic groups, and we could observe a similar contraction mechanism when the emulsion was allowed to break by adding NaOH (Figure 11). The same experiment was also reproduced with a silicone-in-water emulsion where the silicon droplets were acidified with small amounts of oleic acid. We have also probed the influence of different rupturing agents. For instance, adding small amounts of fused silica particles (below 50 nm in diameter) also produces a rapid gel formation and further homothetic contraction in bitumen

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emulsions. Everyday routine provides other examples for such phenomenon: after a period of storage of several days, some yogurts expel water, still keeping the shape of their container. In this latter case, it is likely that the contraction of the gel matrix is mainly due to rearrangements such as fat globule fusion. It is also interesting to note that the same type of phenomenology has been described for the so-called “macrosyneresis” process occurring during the aging of some protein gels.18 Therefore, we conclude that the contraction mechanism is a universal phenomenon in all emulsion systems made up of highly viscous oils. From all the experiments performed with different types of oils and rupturing agents, we may state that the gels contracts in an homothetic when drops viscosity exceeds around 100 Pa‚s. Below this limiting viscosity, gel relaxation is not homothetic anymore; we instead observe the formation of a structure that rapidly relaxes its shape to give a macroscopic spherical gel droplet. At this stage, we must emphasize the fact that our study refers to the “free contraction limit”, meaning that the gels do not stick neither on the walls of the container nor at the air/water interface. This is a necessary condition for the shape relaxation to be homothetic. Even partial sticking of the gel on one of its boundaries results in a nonhomothetic contraction. For example, with bitumen emulsions, we observe that the gel preferentially remains free in glass vessels but is strongly attached on the facets of some polymeric containers. In this latter case, we have observed that the gel does not macroscopically contract but develops numerous randomly distributed fractures. Using confocal microscopy and permeametry, Mellema et al. have studied the very first steps of the coarsening of an “attached gel” in the case of rennet-induced casein gels.19 They observe that during the coarsening process the average pore size of the gel increases (while in the “free contraction limit” the pore size decreases in time; see for example ref 17) and that the casein clusters become more compact. It was also noted that during aging strands (18) Walstra, P. In Cheese: Chemistry, Physics and Micrbiology 1; Fox, P. F., Ed.; Chapman & Hall: London, 1993. (19) Mellema, M.; Heesakkers, J. W. M.; van Opheusden, J. H. J.; van Vliet, T. Langmuir 2000, 16, 6847.

Philip et al.

connecting the clusters are stretched and become very thin and eventually break. These comments underline the importance of the boundary conditions when dealing with gel coarsening. IV. Conclusion In this paper, we have presented experimental evidence for viscous sintering phenomena in fluid-fluid dispersions. We have studied the gelation and coarsening mechanisms of different liquid-liquid dispersions made of highly viscous oils (viscosity about 105 Pa‚s). As a consequence of the high viscosity of the oil, when suddenly made unstable, the emulsions always form gels which further contracts in an homothetic way (gel contracts by preserving the geometry of the container). The observed power law scaling for the gelation time on the initial droplet volume fraction φ0 of the emulsion (Tg ∝ φ0x, where x ≈ -3.0) is in agreement with recent numerical simulations for RLCA aggregation. The sintering process may be of great technological importance in the field of emulsions since it allows transforming an initially liquid emulsion, into a dense and highly viscous material within a short period of time and at room temperature. Beside the field of application that may be very wide, the sintering process still raises fundamental questions that we aim to investigate. Among them, we can mention the evolution of the gel topology (formation of closed pores) and its influence on the rate of contraction. Another interesting question to be explored is the influence of the type of aggregation (RLCA or DLCA) on the rate of contraction as well as on the evolution of the gel topology. Indeed, we have identified some rupturing agents which can induce DLCA aggregation, and this limit is under current investigation. Acknowledgment. J.P. thanks Dr. Placid Rodriguez and Dr. Baldev Raj for constant encouragement and Dr. S. L. Mannan, Mr. P. Kalyanasundaram, and Dr. T. Jayakumar for advise. The COLAS Company has financially supported part of this work and is gratefully acknowledged. Authors are grateful to Prof. G. W. Scherer for many clarifications and fruitful suggestions. LA001377L