Langmuir 1998, 14, 2541-2547
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Structure Formation in Suspensions with a Liquid Crystalline Medium: Percolation Phenomena G. Pulla Reddy,† D. K. Chokappa,‡ V. M. Naik,‡ and D. V. Khakhar*,† Department of Chemical Engineering, Indian Institute of TechnologysBombay, Powai, Bombay 400076, India, and Hindustan Lever Research Centre, Chakala, Andheri (E), Bombay 400099, India Received May 28, 1997. In Final Form: January 13, 1998 Measurements of the storage modulus of suspensions of marble particles in a surfactant-based lyotropic liquid-crystalline medium (lamellar, smectic B) are reported. The suspension is a model system for several food and consumer products. The storage modulus is found to remain nearly constant with an increase in the particle volume fraction until a critical value is reached, beyond which a sharp increase in the storage modulus is obtained. The critical volume fraction is found to increase with an increase in particle size. The storage modulus of the suspensions exhibits a power law dependence on the oscillation frequency, and the power law exponent decreases sharply near the critical volume fraction. The experimental data indicate a transition from a multidomain lamellar liquid-crystalline structure to a structure with a rigid sample spanning network. A network model for the suspension is proposed in which a large number of parallel lamellae bridge randomly placed particles with surfaces closer than a specified distance, the maximum layer size (sm). The network formation is essentially a continuum percolation problem, for which the dimensionless critical number density is found by numerical simulations to be Rc ) F1/3 c Lm ≈ 0.87 where Lm is the maximum distance between the centers of bonded particles and Fc is the critical number density. The modulus (Gc) of the percolation network is obtained computationally as the stress required for a small tensile deformation of the network. The computed modulus is found to be of the form Gc ∝ F2/3(R - Rc)2.5, which is in good agreement with experimental data.
Introduction Consumer goods such as soap, detergent, and toothpaste, and processed foods such as ice cream, chocolate, and mango pulp all have a complex multiphase structure and can be broadly classified as composites of soft solids.1 In general, these materials are suspensions, in which the dispersed phase may be crystallized, or externally added solids, and in which the continuous phase is structured comprising lyotropic liquid-crystalline phases. The rheology of such materials is complicated because of the complex rheology of the medium and because of the interactions between the solids and the medium. A fundamental understanding of the rheology of such suspensions, particularly in relation to their microstructure, would be useful for the design and optimization of operations related to the industrial processing of soft solid composites. Only a few previous studies have focused on the rheology of suspensions with a structured continuous phase. Shouche et al.2 reported on a model system: marble particles of average diameter 12 µm dispersed in a potassium palmitate gel (40% w/w water). The lyotropic gel phase (also referred to as the Lβ phase or the smectic B phase) is a thermodynamically distinct phase, with a structure comprising surfactant layers with stiff alkyl chains, which can rotate about their axes, separated by water layers.3 The gel layers are, consequently, more rigid * To whom all correspondence may be addressed. † Indian Institute of TechnologysBombay. ‡ Hindustan Lever Research Centre. (1) Hermansson, A.-M. In Physical Chemistry of Foods; Jowitt, R., Escher, F., Hallstro¨m, B., Meffert, H. F. T., Spiess, W. E. L., Vos G., Eds.; Applied Science Publishers: New York, 1983; p 111. (2) Shouche, S. V.; Chokappa, D. K.; Naik, V. M.; Khakhar, D. V. J. Rheol. 1994, 38, 1871. (3) Adam, C. D.; Durrant, J. A.; Lowry, M. R.; Tiddy, G. J. T. J. Chem. Soc., Faraday Trans. 1984, 80, 789.
than those in lamellar liquid-crystalline phases (LR phases) in which the alkyl chains are in a liquid-like state. We briefly review the results obtained by Shouche et al.2 here. Suspensions with potassium palmitate gel as the continuous phase showed qualitatively different behavior compared to suspensions with a Newtonian medium (marble particles dispersed in silicone oil). The viscosity, obtained by capillary viscometry, increased more rapidly with the volume fraction of dispersed solids in the case of the gel medium, relative to the silicone oil based suspensions. The storage modulus in the linear viscoelastic regime increased sharply with the addition of solids for the suspension with a gel medium, in contrast to the Newtonian medium which showed a gradual rise. Both results were attributed to structure formation. In particular, the sharp increase in the storage modulus was suggested to be due to the formation of a sample spanning structure comprising particles bridged by surfactant gel lamellae. Kothari et al.4 in a study of the same system confirmed the formation of such large-scale structures by means of viscoelastic measurements at different temperatures, low-resolution NMR measurements, and differential scanning calorimetry. Structure formation in suspensions of colloidal particles has been the focus of several previous studies. Structure formation in such systems is either due to particle agglomeration caused by interparticle surface forces or due to bridging of the particles by polymers. In most cases, the evidence for structure formation is indirect and is inferred from the variation of the storage modulus with volume fraction, frequency, and temperature. Buscall et al.5 found a power law dependence of the storage modulus (4) Kothari, K.; Chokappa, D. K.; Naik, V. M.; Khakhar, D. V. J. Colloid Interface Sci. To be submitted for publication. (5) Buscall, R.; Mills, P. D. A.; Goodwin, J. W.; Lawson, D. W. J. Chem. Soc., Faraday Trans. 1988, 84, 4249.
S0743-7463(97)00558-1 CCC: $15.00 © 1998 American Chemical Society Published on Web 03/31/1998
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on the solids volume fraction (φ) for aggregated silica and polystyrene spheres in NaCl solutions of the form
G ) Kφn
(1)
where K is a constant and n ) 4 ( 0.5. Sohm and Tadros6 and Tadros and Hopkinson7 obtained similar scaling relationships for sodium montmorillonite suspensions and aggregated polystyrene lattices, respectively. Aranguren et al.8 studied fumed silica with different surface treatments in polydimethylsiloxane (PDMS) of different molecular weights. Increasing solid volume fractions gave a decreased dependence of G on the frequency in the linear viscoelastic regime, which is characteristic of an increasingly solid-like structure. Higher molecular weight PDMS suspensions gave higher values of G, and surface treatment which reduced the surface concentration of the highly reactive silanol groups on the particles gave lower values of G. The results were explained in terms of the bridging of particles by the polymer chains to form a network structure. Khan and Zoeller9 studied the effect of the interaction between the medium and the particle surface on structure formation using different media and particle surface treatments. Oscillatory rheometry and lightscattering measurements were carried out. Percolation refers to the formation of interconnectivity in disordered systems, in which a sample spanning network is formed due to the growth of isolated clusters.10 Percolation theory finds application in a wide variety of systems, and a detailed review of the applications is given by Sahimi.11 Percolation, in the context of structure formation in suspensions, was shown by Pouchelon and Vondrack12 in a study of colloidal silica particles (Aerosil 150) and carbon black particles in un-cross-linked silicone rubber. The storage modulus increased slowly with solid volume fractions until a critical value was reached, after which there was a sharp increase with a power law dependence of the form
G ∝ (φ - φc)t
(2)
The reported values of the exponent were close to 1 for both particles studied, with φc ) 0.09 and 0.14 for the silica particles and carbon black particles, respectively. The critical volume fraction for gelation (formation of a sample spanning cluster) was reported as φc ≈ 0.05 by Buscall et al.5 for silica and polystyrene particles in NaCl solutions. Several studies on silica aerogels have shown a dependence of the modulus as given in eq 2, with t ) 3.8.13 This is in good agreement with results of simulations based on three-dimensional elastic percolation networks14 in which the resistance to deformation is primarily due to rotation of bonds at joints. In this work, we consider the formation of sample spanning structures by marble particles suspended in potassium palmitate gel. The system is qualitatively (6) Sohm, R.; Tadros, T. F. J. Colloid Interface Sci. 1989, 132, 63. (7) Tadros, T. F.; Hopkinson, A. Faraday Discuss. Chem. Soc. 1990, 90, 41. (8) Aranguren, M. I.; Mora, E.; DeGroot, J. V.; Macosko, C. W. J. Rheol. 1992, 36, 1165. (9) Khan, S. A.; Zoeller, N. J. J. Rheol. 1993, 37, 1225-1235. (10) Stauffer, D.; Ahrony, A. Introduction to Percolation Theory; Taylor and Francis: London, 1992. (11) Sahimi, M. Applications of Percolation Theory; Taylor and Francis: London; 1994. (12) Pouchelon, A.; Vondrack, P. Rubber Chem. Technol. 1989, 62, 788. (13) Woignier, T.; Phalippou, J.; Sempere, R.; Pelons, J. J. Phys. France 1988, 49, 89. (14) Arbabi, S.; Sahimi, M. Phys. Rev. B 1988, 38, 7173.
Figure 1. Schematic view of bridging of particles by gel layers in the suspension. A bridge is formed when the interparticle spacing (s) is less than the maximum layer size (sm).
different from suspensions of colloidal particles discussed above: The particles in the gel-marble suspensions studied here are more than an order of magnitude larger than the colloidal particles used in previous studies. Consequently, interparticle spacing is of the order of 10 µm. Further, structure formation results from bridging by the gel, which is itself an aggregate of surfactant molecules. There is little previous work on such systems with mesoscopic bridging elements. The main objective of the work presented in this paper is to focus on the transition from a structure comprising isolated clusters to a sample spanning network, and thus to gain an insight into the interaction between the medium and the particles. Linear viscoelastic measurements and computations are used to investigate the behavior of the system. We first present a simple model for particle-induced structure formation in such suspension systems, followed by experimental details. The experimental and model results are discussed next, and the conclusions of the study are summarized in the final section. Model for Structure Formation in Suspensions We consider structure formation in the potassium palmitate gel-marble suspension resulting from bridging by elements of the medium. The potassium palmitate gel has a multidomain liquid-crystalline morphology, and the interactions between the particles and the gel domains are not well-understood. It is likely, however, that particles are excluded from the domains when the particle size is close to the domain size (10 µm). In such a situation, appropriately oriented gel domains, each comprising a large number of parallel surfactant layers, could form relatively rigid bridges between particle surfaces that are close enough. Here, we consider the gel lamellae to adsorb onto the surface of the particles and to form interparticle bonds as shown schematically in Figure 1. The primary assumption of the analysis is that the lamellae sizes are distributed in a range below a maximum value, sm. Thus interparticle bonds are formed whenever the distance between the surfaces of neighboring particles (s) is less than the maximum lamellae size, that is s < sm. The particles are assumed to be randomly distributed in the medium, and of uniform size, with radius R. The assumption of random distribution of particles is reasonable at low volume fractions of particles when a possible overlap of particles is negligible. Optical micrographs of the suspensions also do not show aggregation of particles or formation of particle chains as in the case of colloidal
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suspensions.15 The number density of particles is
F)
3φ 4πR3
(3)
where φ is the solid volume fraction. The network formation problem as defined above falls in the general category of continuum percolation and is referred to as the overlapping spheres model.16 Statistical properties of the model have been well-studied;16-19 however, most previous numerical studies of the modulus of percolation clusters have been carried out for regular lattices in two and three dimensions.20-22 The computed exponents are found to be sensitively dependent on the percolation threshold.21 Recent studies on two-dimensional generic networks (lattice-based networks with the nodal points randomly displaced) give results which are significantly different from the earlier reported results on regular lattices.23,24 The elastic modulus of continuous percolation networks has not been previously reported, and we focus on this problem here. We detail below the assumptions made in the model for the modulus of the suspension and briefly describe the computational procedure. The elastic modulus of the gel is relatively low because the gel layers form domains dispersed in an amorphous phase of relatively low strength. At low volume fractions of solids, the modulus remains low since the gel-particle clusters are isolated and do not contribute to reinforcement. However, the modulus of the suspension is significantly increased upon formation of sample spanning structures, if the layers (which are relatively stiff) are strongly adsorbed on the particles. We assume the latter to be valid, so that the modulus of the suspension depends on the modulus of the gel lamellae rather than the strength of adsorption of the lamellae on the particle surface. The relative modulus of the suspension, assuming that the modulus of the suspension is unaffected by the presence of the isolated clusters, is then
Gr )
G ) 1 + Gc G0
centers. The algorithm used in the calculations is briefly outlined below. A list of bonded neighbors for each particle (L < Lm) is first generated, for a given value of Lm. A cluster is grown from a seed particle chosen at random in a cubical region of side W/2, with the center of the region coinciding with the center of the domain. The clusters are started in the subdomain to minimize boundary effects. Nearest neighbors of the seed particles are assigned to the cluster first. Nearest neighbors of the neighbors are then assigned to the cluster if they do not already belong to it. This process is continued until no more bonds can be formed. Another seed particle from the subdomain, which does not already belong to a cluster, is then chosen and a cluster is grown. The above procedure is repeated with different seed particles until all seed particles in the subdomain are exhausted. Statistics of the clusters formed are then calculated which include the maximum cluster size (the number of particles in the largest cluster), the radius of gyration of the largest cluster (rgm), and the average number of bonds per particle (b h ). Sample spanning clusters are defined as those clusters which have grown to be of the size of the subdomain (rgm/W ) 1/4). The tensile force required to deform a sample spanning cluster for a given value of Lm is obtained by imposing a small deformation on the cluster, allowing the forces in the bonds to relax to their equilibrium values by moving the particles and calculating the total axial force on the bonds at the boundaries. All particles are first displaced according to
xi ) xi (1 + ∆) y i ) yi z i ) zi which corresponds to an affine tensile deformation of magnitude ∆. The dimensionless force in each bond joining particles i and j is calculated as
(4)
where Gc is the modulus of the sample spanning clusters, normalized by G0, the modulus of the gel. The clusters are considered to be a network of freely jointed Hookean springs, with the springs joined at the particle centers. A more detailed model for the system could be constucted in which the springs extend only between the particle surfaces. However, this would entail considerably greater computational complexity, and we limit the scope of this work to the simpler representation given above. The modulus of the network is obtained computationally from the force required to produce a small tensile deformation of the cluster (∆). A number of particles (N) are randomly distributed in a cubical region of side W. A bond is formed between any two particles that are closer than a maximum distance Lm ) (sm + 2R), where Lm is the distance between the particle (15) Kothari, K. M. Tech. Thesis, Indian Institute of Technologys Bombay, India, Bombay, 1995. (16) Quintanilla, J.; Torquato, S. Phys. Rev. E 1996, 54, 5331. (17) Haan, S. W.; Zwanzig, R. J. Phys. A 1977, 10, 1547. (18) Elam, W. T.; Kerstein, A. R.; Rher, J. J. Phys. Rev. Lett. 1984, 52, 1516. (19) Sevick, E. M.; Monson, P. A.; Ottino, J. M. J. Chem. Phys. 1988, 88, 1198. (20) Feng, S.; Sen, P. N. Phys. Rev. Lett. 1984, 52, 216. (21) Hansen, A.; Roux, S. Phys. Rev. B 1989, 40, 749. (22) Arbabi, S.; Sahimi, M. Phys. Rev. B 1993, 47, 695. (23) Jacob, D. J.; Thorpe, M. F. Phys. Rev. Lett. 1995, 75, 4051. (24) Moukarzel, C.; Duxbury, P. M. Phys. Rev. Lett. 1995, 75, 4055.
(5)
F h ij )
Fij Lij′ - Lij ) K Lij
(6)
where K is the effective spring constant, Lij is the initial bond length, and Lij′ is the bond length after the deformation. Particles which lie in the region xi ∈ (xm - W/4, xm + W/4) are allowed to move to relax the forces acting on them, while particles outside this region are held fixed, where xm is the x coordinate of the center of mass of the cluster. The motion of a particle is in the direction of the resultant force (F h i), and is given by
h iδ xi ) xi + F
(7)
where δ is a small displacement. The forces in the bonds are recalculated, and the particles moved until the maximum axial force on any particle that is free to move becomes small enough, that is
max|F h xi| < i
(8)
where is a small number. The total force acting on bonds joining the fixed and moving particles, in the direction of the displacement, is then obtained as Fc. On the basis of dimensional analysis, it is easily shown that the total force scales as
F h c ) F2/3W2f(R)
(9)
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Figure 2. Particle size distributions of marble particles used in the experiments. Table 1: Properties of Marble Particles, and Fitted Model Parameters for the Suspensions marble A marble B marble C
R (µm)
density (g/cm3)
sm (µm)
K (mN)
2.7 6.0 8.0
2.7 2.7 2.7
6.0 12.5 13.0
3.2 14.6 20.5
where R ) F1/3Lm. Finally, the modulus of the cluster is obtained as
Gc )
KF hc G0W2∆
)C
F2/3 f(R) ∆
(10)
where the constant C ) K/G0 is treated as a fitting parameter of the model. The entire procedure is repeated for increasing values of Lm, for the same particle configuration. Analogous calculations in two dimensions (N particles distributed in the square domain of side W) are also carried out to illustrate the cluster formation process and the stress relaxation process. Experimental Details Preparation of Suspensions. Potassium palmitate gel was prepared by reacting laboratory grade palmitic acid (98% assay, Loba Chemie Ltd.) with a stoichiometric amount of potassium hydroxide (85% assay, Ranbaxy Laboratories Ltd.) in a jacketed twin-blade sigma mixer at 85 °C. The moisture content was measured, and additional water was added to give a 40% w/w water content. Previous studies of the phase behavior of the potassium palmitate-water system have shown this composition to form a gel phase.25 Optical microscopy of the material prepared with crossed polarizers showed a texture which is typical of a gel. Marble particles of different sizes were used to prepare the suspensions. Physical properties of the particles are given in Table 1, and the particle size distributions obtained using a Coulter Counter are shown in Figure 2. The particles were washed with distilled water prior to mixing with the gel at 25 °C in an internal mixer with temperature control. Oscillatory Rheometry. Viscoelastic properties of the suspensions were studied using a controlled stress rheometer (Carrimed, model CSL100) with a parallel plate fixture (4 cm diameter). Measurements for the suspensions were carried out using tablets (4 cm diameter, 4 mm) made by compressing the suspension on an Instron Universal testing machine, while confining it in a ring of appropriate dimensions. The tablets were aged in sealed containers for 24 h for the internal structure (25) Small, D. Handbook of Lipids Research 4. Physical Chemistry of Lipids; Plenum: New York, 1985; p 285.
Figure 3. Variation of the maximum cluster size (Cm), the dimensionless radius of gyration of the largest cluster (rgm/W), and the average number of bonds per particle (b h ) with the dimensionless linear number density (R). The graphs are averaged results of 64 simulations and the error bars show the standard deviation for the largest system size (N ) 8000). to equilibrate. Care was taken while loading the tablets in the parallel plate fixture to prevent cracking. Measurements for the gel were made using a gap of 300 µm. The storage modulus in the linear viscoelastic region was obtained from the experiments.
Results and Discussion We present the computational results first. Figure 3 shows the variation of the maximum cluster size (Cm), the normalized radius of gyration of the largest cluster (rgm/ W), and the average number of bonds per particle (b h ) with the dimensionless linear number density R ) F1/3Lm. Results are presented for three different number densities (2000, 4000, and 8000 particles in a cubical domain of side W ) 1) for each of the parameters. The graphs show the averaged results of 64 simulations, and the error bars give the standard deviation of the computations for the largest system size (8000 particles). The maximum cluster size increases sharply with R beyond R ≈ 0.9, and the cluster size approaches the total number of particles at R ) 1.2. The rate of increase of the cluster size also becomes larger with the system size. The dimensionless radius of gyration of the largest cluster, which is of greater relevance to percolation, follows a trend similar to the cluster size and also increases sharply beyond R ≈ 0.85, to approach rgm/W ) 0.5 at R ) 1.2. The rate of increase of the radius of gyration increases with the system size similar to the cluster size. The average number of bonds
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Substituting for f(R) in eq 10 we finally obtain
Gc ) CR2(R - Rc)2.5
Figure 4. Variation of the standard deviation of the dimensionless radius of gyration of the largest cluster (∆rgm/W) with the dimensionless linear number density (R) for the results shown in Figure 3.
per particle (b h ) increases monotonically with dimensionless linear number density to reach a maximum value of about 7 at R ) 1.2. The computed results indicate that the number of bonds increases in proportion to the number density (F) (i.e., b h ∝ R3). An estimate of the percolation threshold obtained from the condition rgm ) W/4 is Rc ≈ 0.84. The average number of bonds per particle at the critical value of R is b h ≈ 2.4. The minimal condition for the formation of particle chains is that every particle (except the ones at the ends) must have at least two bonds. When the average number of neighbors is greater than 2, a significant fraction of the particles have more than two bonds which is the necessary condition for the formation of infinite networks. A more accurate estimate of the percolation threshold is obtained by considering the distribution of the radii of gyration obtained. Figure 4 shows the variation of the standard deviation of the dimensionless radius of gyration (∆rgm) with R. The maximum standard deviation is obtained at the percolation threshold Rc ) 0.87 ( 0.02, which is in reasonable agreement with previous estimates.19 Figure 5 illustrates the procedure used for calculating the modulus of a sample spanning cluster for a twodimensional system. The initial cluster (Figure 5a) is affinely deformed to give the starting configuration for the relaxation process (Figure 5b). The particles within the dashed lines are then moved iteratively until the equilibrium configuration (Figure 5c), corresponding to a vanishingly small total force on each particle that is free to move, is achieved. The total force is computed as the sum of the axial component of the force in each bond that crosses the dashed line. The variation of the calculated total force per unit deformation rescaled by the number density (f(R)/∆) with (R - Rc) is shown in Figure 6 for a fixed system size (W) and different values of the number density (F) in three dimensions. Results, averages of 64 runs, are presented for three different system sizes (N ) 500, 2000, and 4000). The error bars correspond to the computations with the largest system size. The scaling of the force with the number density is verified by the computations, since all the results collapse to a single curve. The computations show a power law dependence of the rescaled force only for (R - Rc) > 0.15, and the best fit line for the data in this region corresponds to (Figure 6)
f(R) ) 1.807(R - Rc)2.5 ∆
(11)
R > Rc
(12)
where C ) 1.807K/(G0Lm2). A somewhat smaller value of the exponent was obtained by Arbabi and Sahimi12 for the modulus of freely jointed, lattice-based networks (t ) 2.1). Such differences between regular lattice networks and random networks are not surprising in light of results recently obtained for generic networks.23,24 Close to the percolation threshold, there is a large deviation from a power law dependence, and the normalized force appears to approach an asymptotic value near the threshold. Reasonably large changes ((0.05) in the critical number density (Rc) do not affect this trend. Reasons for this behavior are not apparent, and computations with larger system sizes are required to confirm the results at small (R - Rc). We do not focus on this problem here since the differences between predictions of the power law equation and the computational data are small in absolute terms, particularly in comparison to the experimental errors in the data presented below. Figure 7 shows the experimental results for the variation of storage modulus of the gel-marble suspensions with solid volume fractions, for the different-sized marble particles. The experiments were repeated for a number of the suspensions and all data points are shown to indicate the experimental variation obtained, which was small. The data for all three sizes clearly shows the behavior typical of systems experiencing percolation: the modulus remains nearly unchanged until a critical volume fraction is reached, after which there is a sharp increase. Note the more than a 100-fold increase in the modulus with the addition of only 10% (vol) of the particles. Further, the critical volume fraction increases with particle size. Both the above features are in qualitative agreement with the proposed model. The variation of the elastic modulus with frequency of oscillation was found to be of the form26
Gr ) Aωp
(13)
The variation of the exponent (p) with solid volume fraction is shown in Figure 8. For the pure gel p ≈ 0.42; this is close to the value obtained for multidomain lamellar liquid crystalline systems (p ) 0.5).27,28 With increasing solid volume fractions there is a sharp decrease in the exponent near the percolation threshold. Low values of the exponent are indicative of a cross-linked network structure.8 The experimental data given in Figure 8 is thus consistent with the proposed transition of the morphology from a multidomain lamellar liquid-crystalline structure to a network-like structure. We consider a comparison between the experimental data and model predictions next. The experimental data are plotted in rescaled formsrelative storage modulus (Gr ) G/G0) versus Rsin Figure 9, along with the model predictions calculated from eq 12. Two parameters of the model (sm and C) are unknown and were used as fitting parameters. The scaling used for the experimental data requires only one of the parameters: the maximum layer size (sm), and this parameter was adjusted so that the dimensionless critical number density for the experimental (26) Pulla Reddy, G. M. Tech. Thesis, Indian Institute of Technologys Bombay, India, Bombay, 1996. (27) Bates, F. S. Macromolecules 1984, 17, 2607. (28) Kawasaki, K.; Onuki, A. Phys. Rev. A 1990, 42, 3664.
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Figure 5. Illustration of the procedure used to compute the modulus of the percolation network. (a) Initial cluster; dashed lines correspond to x ) xm ( W/4 where xm is the center of mass of the cluster. (b) Cluster after undergoing an affine deformation. (c) Equilibrium conformation of the cluster after relaxation. Particles within the dashed lines are permitted to move, whereas the ones outside are held fixed after the affine deformation.
data matched the computational value (Rc ) 0.87). The scaling for all three suspensions is very good and all the data collapse to a single curve for the fitted values of sm which are given in Table 1. The maximum layer size increases with particle size, and this could be a result of a larger number of layers in each bridge between the particles, because of the larger surface area available for adsorption for the larger particles. The values obtained for the maximum layer size are reasonable considering typical domain sizes of 10-15 µm in surfactant gels. Furthermore, there is very good agreement between the theoretical predictions and the experimental data for C ) 1745. The effective spring constant network elements is then K ) 966G0Lm2, and the calculated values are given
in Table 1. The increasing values of the spring constant with particle size are again due to a larger number of gel layers in the bridges formed between the larger particles due to the greater surface area available for adsorption. Since the gel layers only span the gaps between the particle surfaces, rather than the center-to-center distance as considered in the model, the spring constant of the bridging gel layers (KL) is lower than K. The spring constant of the layer may be estimated from
KL sj 2R ≈ ) 1K L L h
(
)
(14)
where the overbar denotes an average over all particles.
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Figure 6. Symbols give the variation of the rescaled force per unit strain (f(R)/∆) with (R - Rc) for Rc ) 0.87 obtained from computations for different number of particles (N) and ) 10-3. The data are averaged results of 64 simulations and the error bars show the standard deviation for the largest system size (N ) 4000). The solid line is fitted to the computational results for N)4000 for data points in the region (R - Rc) > 0.15.
Figure 9. Comparison of experimental results to model predictions. Filled symbols denote the rescaled experimental data, empty symbols denote the computational results, and the solid line denotes the power law equation (eq 12).
Thus KL is about 60% of K for the largest volume fractions considered (φ ) 0.1). A comparison of the computational data to the experimental results is also shown in Figure 9. While there is a clear difference between the computational results and the power law equation, the difference is not significant in comparion to the experimental errors, as mentioned earlier. Conclusions
Figure 7. Measured elastic modulus (G) for gel-marble suspensions with different marble particles and for varying solid volume fractions (φ). The measurements are made in the linear viscoelastic regime at a frequency of 1 Hz.
Figure 8. Variation of the power law exponent (p) of the dependence of the elastic modulus on frequency (ref. eq 13) for varying solid volume fractions (φ) of marble C.
Considering a cubic lattice, we have L ) F-1/3 and
[ (6φπ ) ]
KL ≈ K 1 -
1/3
(15)
Experimental evidence is presented for percolation in the context of structure formation in suspensions with a surfactant gel medium: the shear modulus increases sharply with volume fraction of the dispersed phase beyond a critical volume fraction. Experiments with particles for different sizes show that the threshold volume fraction increases with particle size. The power law exponent for the frequency dependence of the elastic modulus decreases sharply near the percolation threshold. The experimental data indicate a transition from a multidomain lamellar structure to a network structure. The behavior is modeled in terms of a simple network model in which gel layers form interparticle bridges whenever the distance between particle surfaces is less than the maximum layer size sm. At low-particle volume fractions when possible overlap between particles can be neglected, the problem reduces to a continuum percolation problem. The dimensionless critical number density of particles at percolation is obtained as Rc ) F1/3 c Lm ) 0.87, which is in agreement with previous computational results. The modulus of the network, calculated by imposing a small tensile deformation assuming the bonds to be freely jointed, is found to increase as Gc ∝ F2/3(R Rc)2.5 for (R - Rc) > 0.15. Near the threshold there is a significant deviation of the modulus from power law behavior. Computations with larger system sizes are required to confirm these results. The experimental data of the storage modulus for different particle sizes is rescaled to collapse onto a single curve for the maximum layer size in the range sm ) 6-13 µm. These values are reasonable considering typical domain sizes of 10-15 µm in the gel. The theoretical predictions of the model are in good agreement with experimental data. Values of the spring constant for the bonds back-calculated from the model are found to increase with increasing particle size. LA970558T
