Brownian Dynamics Study of Gel-Forming Colloidal Particles - The

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Brownian Dynamics Study of Gel-Forming Colloidal Particles P. H. S. Santos,*,† O. H. Campanella,† and M. A. Carignano‡ Department of Agricultural and Biological Engineering, Purdue UniVersity, 225 South UniVersity Street, West Lafayette, Indiana 47907, United States, and Department of Biomedical Engineering and Chemistry of Life Processes Institute, Northwestern UniVersity, 2145 Sheridan Road, EVanston, Illinois 60208, United States ReceiVed: June 21, 2010; ReVised Manuscript ReceiVed: September 9, 2010

Brownian dynamics simulations are used to study the formation process of colloidal gels and the effect of particle concentration on their rheological properties. To model the interaction between particles, we adopted an R-shifted 12-6 Lennard-Jones (LJ) potential, which allows independent control of the particle size and attractive range. For a short-ranged potential, the percolated network characteristic of a gel exhibited a viscoelastic behavior of weak gels. The storage modulus (G′) increased with the particle concentration increase. Moreover, the dependency of storage modulus (G′) on the particle concentration followed a power law function, which is commonly reported in the literature for experimental data. Simulating frequency sweep tests showed that the system behaved as a liquid-like or solid-like material, depending on the frequency applied. The crossover frequency, i.e., the frequency at which G′ and G′′ are equal, appears to shift slightly to lower values when the particle concentration increases, suggesting a more solid-like behavior for systems with higher particle concentration. During gelation, the storage modulus increases as a stretched exponential and reaches a constant value at long times. Introduction Gels formed by colloidal particles are soft materials rich in solvent with distinctive viscoelastic properties.1 Colloidal suspensions can be induced to aggregate by adding destabilizing agents.2 If the particle concentration is high enough, a destabilizing colloidal suspension will form a space-filling network rather than disconnected clusters. The resulting percolated cluster constitutes the skeleton of the gel. This skeleton has a characteristic internal dimension around 10 to 100 times the particle size, leaving large open regions that are occupied by the low molecular weight solvent. There are many examples of colloidal gels in many different areas. Yogurt and many dairy products are colloidal gels where the particles are large proteins or protein aggregates.2-5 In the pharmaceutical industry, colloidal topical formulations formed with liposomes are used as drug carriers.6 In more recent years, there has been a surge of interest in nanotechnology and the use of nanocomposite gels in the design of biocompatible complex materials for biomedical devices and drug delivery systems.7 Biodegradable and noncytotoxic three-dimensional (3-D) gels arise as an emergent alternative in tissue engineering and in cell culture.8-10 For other applications, the flow properties of some colloidal gels may also convey high performance and improved safety for rocket propellants, combining properties and advantages of solid and liquid fuels.11,12 The destabilized and aggregated colloidal network may be either permanent or transient, depending on the attractive forces between particles in the system.13 Very strong attractions or the formation of chemical bonds at relatively low volume fractions may result in irreversible gels by permanent fractal aggregation, resulting in a large interconnected network.14 Several experiments and computational simulations have been carried out in * To whom correspondence should be addressed. E-mail: psantos@ purdue.edu. † Purdue University. ‡ Northwestern University. E-mail: [email protected].

order to get a better understanding about irreversible colloid aggregation.15-18 Reversible gelation leads to the formation of transient gels. In this article we will only consider reversible gels: gels formed by particles that do not make permanent bonds. Namely, colloidal gels form under temperature and concentration conditions that will eventually lead to a phase separation of the system. Therefore, the gel is a transient state undergoing a very slow process that tends to evolve toward a heterogeneous system with a particle-rich phase in contact with a solvent-rich phase. Under certain conditions, this slow phase separation process may be completely arrested,1,19 and the system remains trapped in the gel state. In the cases where the phase separation continues, the time scale of this process is usually very long (days to years) compared with the time scale of the application of the gel. It is known that particle gels exhibit local ordering; however, they have no long-range correlations.20,21 Computer simulations of transient colloidal gels have been carried out for systems with short- and long-range attractive potentials.22-24 In this context, we refer to short- and long-range interactions as determined by the ratio between the effective attractive range and the size of the particle. A ratio that is larger or equal to one characterizes a long-range potential, as is the case for 12-6 Lennard-Jones. A ratio that is much smaller than 1 corresponds to a short-range potential, as is the case for 72-36 Lennard-Jones. Lodge and Heyes20,25-29 extensively studied systems interacting with Lennard-Jones 2n-n potentials. By taking different values for n, the ratio of range to particle size changes, affecting the gelation properties of the model. It is clear that the magnitude of the range of the interparticle attraction relative to their size has a profound effect on the system phase diagram.30 For example, Lennard-Jones particles have a well known phase diagram with the approximate shape of an inverted parabola that displays a gas/liquid coexistence curve characterized by a approximate critical point T* ∼ 1.31 and F ∼ 0.316.30 When the attractive range becomes

10.1021/jp105711y  2010 American Chemical Society Published on Web 09/28/2010

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extremely short, the liquid phase becomes metastable and there is a gas/solid phase transition for those systems.30-32 Weak gels are thermodynamically unstable systems in which the active gellant agent forms a fragile and space-filling network. Spherical particles may form weak gels if the range of the attraction between the particles is considerable smaller than the characteristic size of the particles. At sufficiently low temperature, the short-ranged potentials lead to the formation of gels, while longer range potentials (for example Lennard-Jones 12-6) tend to result in a faster phase separation. However, depending on the time scale studied, a transient gel structure can also be observed for long potential range.28 Performing Brownian dynamics simulations for a short-ranged potential, we show the formation of transient gels at low packing fractions for relatively long simulation times. Most of the efforts on colloidal systems focus on gelation routes of transient gels and on the development of a general theoretical framework that covers all concentration ranges, including high packing fractions. It is still a challenge to link macroscopic properties of materials with their microstructural properties and conformation. In practice, macroscopic properties of gels provide important and useful inputs to the design of new materials, and that is the reason why most fundamental research on gels describes their properties in terms of viscoelasticity. In this paper we use a rheological approach to study short-ranged transient gels and the effect of particle concentration on the gel formation and on their mechanical properties using computer simulations. The main objective is to get a better understanding about relations between microstructural and macroscopic (viscoelastic) properties of colloidal gels. Model and Simulation Details We performed Brownian Dynamics (BD) simulations in the overdamped limit. Namely, we numerically integrate the position Langevin equation:

dri 1 ) fi(t) + θi(t) dt γ

(1)

where γ is the friction constant, the same for all the particles, fi(t) is the force acting on particle i due to the interaction with the other colloidal particles, and θi(t) is a random noise satisfying 〈θi(t)θi(t)〉 ) 2δijδ(t)kBT/γ. Here kB is the Boltzmann constant and T the absolute temperature. The hydrodynamic interaction among the particles is beyond the scope of this work and is not considered. Within the Einstein-Smoluchowski theory of Brownian motion, the mean square displacement of a particle is given by:

〈r2〉 ) 6D0t

(2)

where D0 is the diffusion coefficient of the particles at infinite dilution. From here we can define the unit of time, as is usually done in BD simulations, as a2/D0, where a is the radius of the particles. This time unit is proportional to the time needed by an isolated particle to diffuse a distance proportional to its size. At this point and to complete the computational framework, we introduce Stokes law:

γ ) 6πηa

(3)

which relates the friction constant to the radius of the sphere, a, and the viscosity of the pure solvent, η. Finally, the Stokes-Einstein relation establishes the connection between the diffusion process and the thermal energy

D0 )

k BT kBT ) γ 6πηa

(4)

Considering that our model particles interact by a smooth continuous interaction potential, there is no clear definition of the particle radius and the range of attraction between the particles. We will define an effective diameter following the Weeks-Chandler-Andersen theory:33,34

σeff )

∫0∞ [1 - e-U

rep(r)/kBT

(5)

]dr

where Urep(r) is the repulsive part of the interparticle potential. Therefore, our time unit is expressed in terms of σeff instead of the radius of particle a. Then the time unit can be defined as 2 /4D0. σeff To model the interaction between colloidal particles, we adopted an R-shifted 12-6 Lennard-Jones potential:

{ [(

∞, U(r) ) 4ε

σ r - R0

) ( 12

σ r - R0

)] 6

r e R0

, r > R0

(6)

where r is the distance between the centers of the particles and R0 is the shifting distance. In all the simulations, a spherical cutoff was imposed at rc ) R0 + 2.5σ. This interaction potential allows for an independent control of the size of the particles through the shifting distance R0 and the range of attraction through Lennard-Jones parameter σ. Taking ε as the unit of energy, and σeff as the unit of length, we can rewrite the interaction potential in the reduced form:

U(r) ) ε ∞,

{ [(

1 4 r' - R0/σeff

) ( 12

1 r' - R0/σeff

r' e R0/σeff

)] 6

, r' > R0/σeff

(7)

It is also convenient to introduce the reduced temperature T* ) kBT/ε. All the simulations presented here are with N ) 2000 particles and constant volume V. In order to prevent particle overlap, a sufficiently small time step was used (8 × 10-6σ2eff/4D0) and no special provision was implemented to avoid the region of infinite potential energy. We use σeff as a nominal size (diameter) of the particle and define the volume fraction by

φ)

Nπ 3 σ V 6 eff

(8)

For particles interacting through a potential with R0 ) 9, σ ) 1, and T* ) 0.2, the nominal diameter or σeff corresponds to 10.075σ. The initial configuration for all simulations corresponds to a random configuration obtained by running a short molecular dynamics simulation at high temperature. The initial potential

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Figure 1. Potential energy as a function of time for a system interacting via standard Lennard-Jones 12-6 potential. The insert shows the initial 100 time units in which the particles form a percolated cluster. The drop in U between t ) 200 and t ) 300 corresponds to the breaking in the connectivity of the percolated cluster in one of the Cartesian directions, and the subsequent reorganization of the system’s structure.

energy is zero or a very small negative number for all the cases, since there is practically no contact between the particles. Results We started our work considering the pure Lennard-Jones system. This is a particular case of an interaction potential with R0 ) 0. Even though this system is not in our main interest, it represents a good test case for the simulation methodology due to the existence of previous works in similar systems. In Figure 1 we show the time evolution of the potential energy of the system, for a simulation corresponding to a volume fraction of φ ) 0.13 and a temperature T* ) 0.2, and in Figure 2 we show a series of snapshots characterizing the whole process. The potential energy shows a monotonic decreasing behavior as a function of time. At first the decrease is very sharp as the particles cluster themselves to form a gel-like structure with several branches. Figure 2A shows a snapshot of this transient gel that remains as such up to a time of t ≈ 200 (in reduced units). Then there is a major reaccommodation of the branches as they cluster together to form a single bulky structure, still percolated, but reducing its surface area as shown in Figure 2B which corresponds to t ) 500. The formation of the central bulky structure is reflected by a change in the slope of the potential energy curve between t ) 200 and t ) 300. The system keeps evolving and the connectivity across the box in one of the directions is lost, as shown in Figure 2C corresponding to t ) 1000. The breaking of the space-filling network is an

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Figure 3. Potential energy of a system with 2000 particles, interacting with the R-shifted potential with R0 ) 9 and σ ) 1. Different colors correspond to different volume fractions: 0.10 (black), 0.13 (red), 0.16 (green), and 0.20 (blue). For the intermediate volume fraction, we run two independent simulations starting from a different initial condition 2 2 /4D0 and (8 × 10-7)σeff / and with different time steps ((8 × 10-6)σeff 4D0). Longer simulation was performed for the system with volume fraction 0.13.

indication of the advance of the separation process. Figure 2D shows the final structure, at t ) 10 000, that has a cylindrical shape as the connectivity across the box is maintained only in one of three directions. Figure 2D also reveals the crystalline order in parts of the system. Lodge and Heyes20,25-29 previously reported the formation of transient gels using a Lennard-Jones 2n-n potential, volume fractions of 0.10, 0.16, and 0.20, and T* ) 0.3. They used 864 particles and a total simulation time of 81 time units. In their conclusion, they discussed the surprisingly gel-like behavior of the longest range potential system (n ) 6). Moreover, they commented that there were no signs of complete phase separation for the long-range system, which could be due to the time scale of their simulations, since those systems tend to phase separate more rapidly. We also observed visually a gel-like structure using Lennard-Jones potential (n ) 6), volume fraction of 0.13, and T* ) 0.2 at short simulation times (t ) 100). However, if the simulation goes on for longer times, the system no longer exhibits a percolated network characteristic of gel, which shows the important role that the potential range and time scale play in transient gels. We now consider particles interacting through a short-range potential, as is the case with R0 ) 9, and σ ) 1. In Figure 3 we show the time evolution of the potential energy for six trajectories corresponding to four different volume fractions. All the simulations were done at the same temperature T* )

Figure 2. Snapshots of the standard Lennard-Jones system at (from left to right) t ) 100, t ) 500, t ) 1000, and t ) 10 000. In the first two snapshots, the system forms a percolated cluster interconnected through all the faces of the simulation box. At t ) 1000 the connectivity in the vertical direction has been lost, and at t ) 10 000 the connectivity in a second direction is also broken and the cluster adopts a cylindrical shape.

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Figure 4. Snapshots of a system with volume fraction of 0.13 interacting with the R-shifted potential with R0 ) 9 and σ ) 1 at (from left to right) 2 t ) 100, t ) 500, t ) 1000, and t ) 10 000. The typical network of a gel remains up to 10 000σeff /4D0.

0.2, and two trajectories starting from different initial conditions 2 /4D0 and (8 × and using a different time step ((8 × 10-6)σeff 2 /4D0) have been obtained for the intermediate volume 10-7)σeff fractions. Judging by the potential energy and also visually inspecting the simulated trajectories, our results show no important differences between these two time steps. The volume fraction has some effect on the kinetics of gelation and on the final potential energy of the system. All four different volume fractions show qualitatively the same behavior. There is an initial fast (t < 10) clustering, followed a process of gel formation (10 < t < 100). For t > 100, the gel keeps evolving at a very low rate, and no kinetically arrested final state is ever reached. Note that Figure 3 shows the time evolution for one of the simulations corresponding to a volume fraction of 0.13 up to t ) 10 000, confirming that the system keeps decreasing its potential energy, indicating an accommodation of the particles in more energetically favorable conformations, and may be a result of aging process.35 Comparing the structures corresponding to t ) 1000 and t ) 10 000, we see a minor global reaccommodation, but a larger degree of local ordering is achieved at long times. As shown in Figure 4, the characteristic network of a gel remains for the longest simulation time (t ) 10 000). The times in which the snapshots were taken are equivalent to those presented in Figure 2 for the pure Lennard-Jones particles. In Figure 5 we present (A) the time evolution of the size of the largest cluster in the system, and (B) the time evolution of the average cluster size for the same systems considered in Figure 3. A particle is considered to be part of the cluster if the distance to any of the particles already in the cluster is smaller than a cutoff distance that corresponds to the first minimum in the particle-particle radial distribution function. The four systems display the same behavior: except for short time fluctuations, there is a monotonic increase in both the average and the largest cluster sizes. The time scale to reach total percolation is longer for the most diluted systems. This result is expected, since the average initial interparticle distance is smaller in the more concentrated system favoring the cluster development. However, it is interesting to note that the time to achieve total percolation for φ ) 0.10 is approximately 2 orders of magnitude longer than the corresponding time for φ ) 0.20. This strong dependency could be interpreted in terms of the slow diffusion of the intermediate clusters that need to travel a longer distance in the systems with low concentrations in order to aggregate into a single percolated structure. The overall structure of the systems at t ) 1000 corresponds to a collection of small clusters fused together. The local structure of the system is displayed in Figure 6 for the four studied systems. The number of first and second neighbors is 8.0 and 11.6, respectively, and the positions of the first and second peaks in the g(r) are approximately 1.0σeff and 1.4σeff,

Figure 5. (A) Size of the maximum cluster as a function of time for the system characterized by R0 ) 9 and σ ) 1, and different volume fractions. (B) Average cluster size as a function of the packing fraction. Different colors correspond to different volume fractions: 0.10 (black), 0.13 (red), 0.16 (green), and 0.20 (blue).

respectively. These peaks correspond to an fcc lattice. The reason why only eight first neighbors are found and not twelve, as is the optimum for hexagonal close packing, is that the system has a large exposed surface, as can be seen in Figure 4. The insert of Figure 6 shows a detail of the local crystalline structure corresponding to φ ) 0.13, showing sections of two crystalline grains in different relative orientations. From an experimental point of view, gels can be defined as a system with at least two components: (i) a solvent and (ii) a gellant that forms a macroscopic space-filling structure that provides mechanical rigidity.36 In terms of deformation and flow properties, complex materials such as gels do not follow Newton’s law of viscosity or even Hooke’s law of elasticity. Unsteady shear material functions are commonly used to characterize gel properties and address their applications. Small amplitude oscillatory shear (SAOS) strain is applied to the fluid, producing a shear stress wave of the same frequency but usually not in phase. Knowing the resulting shear stress and phase

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Figure 6. Cumulative g(r) for the system characterized by R0 ) 9 and σ ) 1 at different volume fractions: 0.10 (black), 0.13 (red), 0.16 (green), and 0.20 (blue). The insert shows the crystalline structure of two flocs in different relative orientation. The g(r) and the snapshot 2 correspond to t ) 1000 σeff /4D0.

difference between stress and strain waves, we can calculate two properties: storage modulus (G′) and loss modulus (G′′).37 The storage modulus (G′) describes the solid-like (elastic) characteristic of the system whereas the loss modulus (G′′) describes its liquid-like (viscous) behavior. These properties are often determined experimentally38-42 to characterize the rheological properties of gels since most of them are viscoelastic materials, i.e., they exhibit both viscous and elastic properties. In practice, they can describe the material in terms of strength and weakness, which are useful properties in designing materials and products. In addition, previous simulations also calculated the rheological properties of colloidal gels for Lennard-Jones potential and for systems with irreversible bonds.15,18,28,29,43,44 However, it is still a challenge in different areas such as material and food sciences to optimize the mechanical properties of systems and to obtain a better understanding of the relation between microscopy properties and rheology.2 In order to calculate the viscoelastic properties of the simulated gels, i.e., the storage modulus (G′) and loss modulus (G′′), we followed an approach similar to the experimental setup.15 Namely, we performed Brownian dynamics simulation on systems undergoing a small amplitude oscillatory shear. The simulation box was deformed by displacing the top wall normal to the z-axis following an oscillatory pattern with a fixed frequency ω. A general triclinic simulation box can be described by a 3 × 3 matrix. A cubic box is a particular case of a triclinic box, with all the diagonal elements equal to the length of the box’s edge, and the off diagonal elements equal to zero. The oscillatory shear strain that we apply to the simulation box corresponds to set the xy component of the matrix equal to

γxy(t) ) γ0 sin(ωt)

(9)

This oscillatory shear was applied for a very short time, as compared to the time required for the system to have a significant change in structure. To calculate the rheological properties of the system, we used the xy-component of the stress tensor, which follows an oscillatory function with the same frequency of the applied strain:

σxy(t) ) σ0 sin(ωt + δ)

(10)

The amplitude σ0 and the phase shift δ are obtained from a fit. Figure 7A shows the simulated sinusoidal strain applied to

Figure 7. Simulated small amplitude oscillatory shear (SAOS) for the 2 system R0 ) 9, σ ) 1, and φ ) 0.13, taken at t ) 1000σeff /4D0: (A) Simulated oscillatory strain (black), stress response (red), and phase delay (δ) at ω ) 100. (B) Stress response fitting for G′ and G′′ calculations (green). Lxy and Pxy are the xy components of stress and strain tensors.

the gel (t ) 1000), the shear stress response, and the difference in phase between them. Figure 7B shows the fitting of the shear stress response, and constants were used to compute the storage modulus (G′) and loss modulus (G′′):

G'(ω) )

σ0 cos δ γ0

(11)

G'
(ω) )

σ0 sin δ γ0

(12)

For a purely elastic material, the stress response is in phase with the strain whereas for a purely viscous material the angle phase between them corresponds to 90°. The phase angle δ for viscoelastic materials ranges from 0 to 90°. The first goal in this work was to determine the regime in which the stress is linearly related to strain, a standard test performed in experimental rheology to find the range of linear viscoelasticity of the sample. Strains of different magnitudes were applied to the gel, and the storage modulus (G′) was calculated at a constant frequency (ω ) 10 000 4D0/σ2eff). Figure 8 shows the strain dependence of G′ for gels of different packing fractions. For all particle concentrations, G′ does not vary for a range of small strains and exhibits significant reduction when a certain strain is achieved. This range characterizes the gel’s linear viscoelastic region, and the strain in which this relationship is no longer linear is defined as critical deformation value (γc). In addition, Figure 8 shows that the higher the particle

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Figure 8. Storage modulus (G′) dependency on maximum strain for the system R0 ) 9, σ ) 1, and different volume fractions: 0.10 (black), 0.13 (red), 0.16 (green), and 0.20 (blue). All calculations were performed at a frequency of ω ) 10 000. The calculation was performed 2 at t ) 1000σeff /4D0.

concentration, the higher the storage modulus (G′), for that particular frequency. Shih et al.45 developed the scaling fractal theory which considers that the gel structure is formed by clusters that aggregate with each other. This theory allows the collection of structural information on the gels from experimental rheological data. From experimental datan they showed a power law dependency of the storage modulus (G′ ∝ Øn) and of the critical deformation (γc ∝ Øn) on the particle concentration. Two different behaviors were found to describe the gels, and they were defined in terms of strong and weak links. In a weak-link regime, both G′ and γc increase with increasing particle concentration. In a strong-link regime, G′ also increases as concentration increases (higher dependence than in a weaklink gel); however, γc decreases as concentration increases. In addition, the interactions between the colloidal flocs (aggregates) are stronger than the interactions within flocs in the strong-link systems. In contrast, in a weak-link gel the interaction within flocs is stronger than that between aggregates. We also observe a power law dependency of G′ on particle concentration for our transient gels. The storage modulus (G′) satisfies a power law function with an index of 1.04. A weak dependence of storage modulus on the particle concentration is observed, which means that G′ increases slowly as packing fractions increases. The critical deformation (γc), which limits the linear dependency of G′ on strain, showed a weak dependence on particle concentration for the studied range. These results suggest that our system behaves as a weak-link colloidal gel. The characteristic behavior of a weak gel shows up also when simulated frequency sweep tests are performed in the linear region (γ ) 2.5%) for different volume fractions. This test is commonly used in experimental studies to investigate the frequency dependence of the storage modulus (G′) and loss modulus (G′′). From rheological point of view the flow properties of weak gels depend on the angular frequency (ω) of the applied stress.36 The storage (G′) and loss (G′′) modulus are frequency dependent, exhibiting solid-like or liquid-like properties depending on the frequency applied. Solid-like gels, i.e., strong gels, exhibit an elastic modulus much larger than the viscous modulus (G′ . G′′). Figure 9 shows the simulated frequency sweep tests for our transient gels. For all particle concentrations, the colloidal gels displayed a typical viscoelastic behavior of a weak gel. At low frequencies, the loss modulus (G′′) is the dominant response of the system, which means that for that frequency range, the system behaves as a liquid-like

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Figure 9. Storage (G′) and loss (G′′) moduli dependency on frequency for the system R0 ) 9, σ ) 1, and different volume fractions: 0.10 (black), 0.13 (red), 0.16 (green), and 0.20 (blue). All calculations were performed in the linear viscoelastic region (γ0 ) 2.5%) and at t ) 2 1000σeff /4D0. The insert shows G′ and G′′ as a function of volume fraction at fixed frequency ω ) 1000.

material. As the frequency increases, both of them (G′ and G′′) increase. When a certain frequency is achieved, the storage modulus (G′) exhibits higher values than those of the loss modulus (G′′). The frequency at which G′ and G′′ are the same is defined as the crossover frequency. At higher frequencies, the solid-like characteristic of the material is predominant. The effect of particle concentration on this solid/liquid like “behavior transition” can also be observed in Figure 9. As the particle concentration is increased, the crossover frequency shifted to lower frequency values, suggesting that higher concentration systems exhibit a more solid-like behavior. For the range studied, the higher the concentration, the higher the frequency range in which the material exhibits solid-like behavior. The variation of the crossover frequency is not large, 2 confirm this but the change in behavior at ω ) 1000 4D0/σeff trend. In the insert of Figure 9, we display G′ and G′′ at ω ) 2 1000 4D0/σeff as a function of the volume fraction φ, showing a larger slope for G′ than for G′′. This dependency implies that the transition to solid-like behavior happens at lower frequencies for higher particle concentration. All the macro properties presented so far were calculated for the gel network formed after a time of 1000σ2eff/4D0. In order to obtain some information on the rheology of the system during the gel formation, conformations of the system at different times were used to determine G′ during the aggregation process. In Figure 10, the calculated G′ for φ ) 0.10 and φ ) 0.20 packing fractions is plotted as function of time. This calculation was performed at a frequency in which the gels exhibited predominantly solidlike properties (ω ) 10 000). As the aggregation process goes on, the storage modulus increases, approaching an asymptotic value. The higher the particle concentration, the higher the asymptotic value, implying that solid-like behavior is enhanced by concentration. In addition, the G′ values during gel formation increase exponentially according to a stretched exponential of k the form G′ ) G∞ exp-(B/t) . Fitting the data to this function the constants G∞ ) 2.28, B ) 7.29, k ) 0.36, and G∞ ) 4.81, B ) 2.37, k ) 0.34, are obtained for φ ) 0.10 and φ ) 0.20, respectively. The stretching factor k is approximately the same for the two cases, but the time constant B is significant larger for the lower concentration. A stretched exponential time dependency has been observed in the other properties of the gels, such as the relaxation of incoherent scattering function.46

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Figure 10. Storage modulus calculation for the system R0 ) 9, σ ) 1, and volume fractions of 0.10 (squares) and 0.20 (circles) during gel formation. The calculations were performed in the linear viscoelastic 2 /4D0. region (γ0 ) 2.5%), at frequency ω ) 10 000 and t ) 1000σeff

Conclusions Brownian dynamics simulations were performed to study the relationship between microstructural and macroscopic properties of transient colloidal gels. We initially considered a standard Lennard-Jones 12-6 system corresponding to a volume fraction of φ ) 0.13 and a temperature T* ) 0.1. A transient gel is formed at relatively short times (t ≈ 100), but due to major reaccommodation of the branches, the system collapses to form a single bulky structure with some crystalline local order. To model particles interacting through a short-range potential, we adopted a R-shifted 12-6 Lennard-Jones model. Since the potential range plays an important role in the gel formation,24 the advantage of using this interaction potential is the independent control of the size of the particles and the range of attraction. In addition, we observed that for the case corresponding to R0 ) 9 and σ ) 1, a percolated network characteristic of a gel is formed for packing fractions varying from 0.10 to 0.20 and it remains for relatively long simulation 2 /4D0). Using a rheological approach, we times (up to 10 000σeff characterized those systems and determined their viscoelastic behavior to investigate the effect of particle concentration on their mechanical properties. Our simulation results using the R-shifted Lennard-Jones potential show similar trends and behavior to those observed experimentally for weak gels. First of all, the regime in which the stress is linearly related to strain was determined for all the considered concentrations. As particle concentration increased, the storage modulus (G′) in the linear viscoelastic region increased. Moreover, it satisfies a power law function with index of 1.04. The simulated frequency sweep tests showed that our colloidal gels behave as a weak gel. For all particle concentrations, the gels displayed a typical viscoelastic behavior. As the applied frequency is increased, the system undergoes a transition from a liquid-like to a solid-like material. The frequency at which G′ and G′′ are the same, which is defined as the crossover frequency, shifts slightly to lower values as the particle concentration increases, suggesting that higher concentration systems exhibit a more solid-like behavior. During the gelation process, the storage modulus (G′), which in practice characterizes the gel strength, increases as a stretched exponential as the aggregation goes on, reaching a constant value for very long times. Acknowledgment. This work was supported by the U.S. Army Research Office under the Multi-University Research Initiative (MURI) grant number W911NF-08-1-0171.

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