Yielding Behavior of Thermo-reversible Colloidal Gels - Langmuir

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Yielding Behavior of Thermo-reversible Colloidal Gels V. Gopalakrishnan and C. F. Zukoski* Department of Chemical and Biomolecular Engineering, UniVersity of Illinois, Urbana, Illinois 61801 ReceiVed July 18, 2006. In Final Form: February 4, 2007 The breakdown of structure in gelled suspensions due to the application of an external stress results in flow. Here we explore the onset of flow by investigating the onset of nonlinear behavior in the elastic moduli of a widely studied class of thermo-reversible gels over a range of volume fractions. We employ the system composed of octadecyl-coated silica particles (radius ) 24 nm) suspended in decalin that displays a transition from a liquid to a gel below a volume-fraction-dependent gel temperature, Tgel. The perturbative yield stress at which the gel modulus drops to 90% of its value in the linear viscoelastic limit is found to increase monotonically with volume fraction and decreasing temperature. The recently proposed activated barrier-hopping theory of Schweizer and co-workers1,2 presents a framework to capture the impact of external forces on the mechanical properties of structurally arrested systems. By characterizing particle interactions with a Yukawa potential and employing the resultant static structure factor as input into the activated barrier-hopping theory, we make predictions for how the elastic modulus evolves with the applied stress. Comparisons of these calculations with experiments reveal that the theory does an excellent job of quantitatively capturing the perturbative yield stresses over the entire range of volume fractions and temperatures explored in the study. The match of predictions with experimental results suggests that the theory not only captures particle localization but also how this localization is modulated in the presence of an external stress.

1. Introduction Thermo-reversible gels composed of silica particles coated with octadecyl chains suspended in a medium which is a good solvent of the chains at high temperature and a poor solvent at low temperatures are a model system for exploring the origins of gelation and the microstructure and mechanics of gels. These systems have been studied for over two decades,3-15 providing a test bed for ideas about the fractal nature of gels and the influence of weak interparticle attractions on microstructure and mechanics. Above a solvent-dependent critical temperature, suspensions of these octadecyl silica particles are model hard spheres. The van der Waals forces between particles are minimized and nearly eliminated by employing solvents such as toluene, aliphatic hydrocarbons, and decalin, resulting in very-close-to-pure volume exclusion interactions at high temperature, especially if the particles have a diameter less than 100 nm. Suspensions of these particles have been used to provide a detailed picture of the dynamics of hard-sphere systems. At a solvent-specific temperature, suspension viscosities rapidly increase above the hard* To whom correspondence [email protected].

should

be

addressed.

E-mail:

(1) Kobelev, V.; Schweizer, K. S. Phys. ReV. E 2005, 71, 041405. (2) Schweizer, K. S.; Saltzman, E. J. J. Chem. Phys. 2003, 119, 1181-1196. (3) Vermant, J.; Solomon, M. J. J. Phys.-Condens. Matter 2005, 17, R187R216. (4) Verduin, H.; deGans, B. J.; Dhont, J. K. G. Langmuir 1996, 12, 29472955. (5) Verduin, H.; Dhont, J. K. G. J. Colloid Interface Sci. 1995, 172, 425-437. (6) Chen, M.; Russel, W. B. J. Colloid Interface Sci. 1991, 141, 564-577. (7) Grant, M. C.; Russel, W. B. Phys. ReV. E 1993, 47, 2606-2614. (8) Jansen, J. W.; Dekruif, C. G.; Vrij, A. J. Colloid Interface Sci. 1986, 114, 471-480. (9) Jansen, J. W.; Dekruif, C. G.; Vrij, A. J. Colloid Interface Sci. 1986, 114, 481-491. (10) Jansen, J. W.; Dekruif, C. G.; Vrij, A. J. Colloid Interface Sci. 1986, 114, 492-500. (11) Jansen, J. W.; Dekruif, C. G.; Vrij, A. J. Colloid Interface Sci. 1986, 114, 501-504. (12) Ramakrishnan, S.; Gopalakrishnan, V.; Zukoski, C. F. Langmuir 2005, 21, 9917-9925. (13) Rouw, P. W.; Vrij, A.; Dekruif, C. G. Colloids Surf. 1988, 31, 299-309. (14) Rueb, C. J.; Zukoski, C. F. J. Rheol. 1997, 41, 197-218. (15) Rueb, C. J.; Zukoski, C. F. J. Rheol. 1998, 42, 1451-1476.

sphere value,15 and over a narrow temperature range, the suspensions become stiff, space-filling gels. The gel temperature is an increasing function of volume fraction. In dilute suspensions, gelation is associated with the formation of fractal aggregates16 while at volume fractions above about 0.15, gelation occurs with little observable change in microstructure and is associated with the absence of long-range particle diffusion. The modulus of the gel increases rapidly as temperature is lowered and typically has a power law dependence on volume fraction with a power law exponent of ∼5 independent of temperature.12,14 Several models have been developed to describe the mechanical properties of colloidal gels typically based on an assumed fractal microstructure and descriptions of bond bending moduli.17-19 These models can reproduce the mechanical properties but rarely are able to describe both the conditions when gels form and the mechanical properties of the resulting gels in a consistent framework. Recently, mode-coupling approaches20 have been applied to depletion gels (where non-adsorbing polymer is added to a suspension of hard spheres to induce attractions) with great success. These models accurately predict the effects of varying the volume fraction, polymer concentration, and polymer molecular weight on the linear viscoelastic modulus of dense depletion gels. Ramakrishnan and Zukoski21 extended modecoupling theory to provide descriptions for gelation and the elastic modulus in the linear viscoelastic regime of suspensions composed of particles interacting with Yukawa potentials. Applying this model to thermo-reversible gels composed of octadecyl-coated silica particles (radius, R ) 45 nm) suspended in decalin, they demonstrated that the model captures the location of the gel boundary and the volume fraction and temperature dependencies and magnitude of the gel elastic modulus. (16) Varadan, P.; Solomon, M. J. Langmuir 2001, 17, 2918-2929. (17) Potanin, A. A.; Derooij, R.; Vandenende, D.; Mellema, J. J. Chem. Phys. 1995, 102, 5845-5853. (18) Shih, W. H.; Shih, W. Y.; Kim, S. I.; Liu, J.; Aksay, I. A. Phys. ReV. A 1990, 42, 4772-4779. (19) Shih, W. Y.; Shih, W. H.; Aksay, I. A. J. Am. Ceram. Soc. 1999, 82, 616-624. (20) Chen, Y. L.; Schweizer, K. S. J. Chem. Phys. 2004, 120, 7212-7222. (21) Ramakrishnan, S.; Zukoski, C. F. Langmuir 2006, in press.

10.1021/la0620915 CCC: $37.00 © 2007 American Chemical Society Published on Web 06/13/2007

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Schweizer and Saltzman2,22 recently proposed the activated barrier-hopping theory which incorporates thermally activated processes into the framework of the naı¨ve mode coupling theory (NMCT), and Kobelev and Schweizer1 extended this theory to incorporate the effects of stress on the mechanical properties of structurally arrested systems. The model framework accounts for the ability of an applied stress to weaken interparticle bonds, thereby bringing about a reduction in elastic modulus and suspension viscosity. Experiments on concentrated suspensions23 containing particles that can be treated as effective hard spheres reveal that the perturbative yield strain predictions of the theory for hard sphere glasses are in agreement with the experiment results. Here perturbative yielding is defined as the regime indicating the onset of yielding, and the perturbative yield strain is defined as the strain imposed on the sample that causes the sample modulus to drop to 90% of its value at the low-stress limit (the corresponding stress is the perturbative stress). In this study, we are interested in testing the generality of the activated barrier-hopping framework by extending the test of the theory to attractive gel systems. Although we perform oscillatory stresssweep experiments to observe the complete yielding of the thermoreversible gel from a gel-state to a liquid-state, this study focuses on the onset of yielding. Using the characterization tools developed by Ramakrishnan and Zukoski, the octadecyl-silica-decalin thermo-reversible gels serve as an excellent experimental system to test the predictions of the theory for systems where structural arrest is driven by the formation of physical interparticle bonds. The methods of Ramakrishnan and Zukoski set all parameters required to characterize the system. Thus, a comparison of the predictions of the perturbative yield stress (henceforth referred to as the perturbative stress) using the activated barrier-hopping theory with that from experiment is a parameter-free test of the ability of this theory to predict the onset of nonlinear viscoelastic behavior in an attractive gel system. The paper is structured as follows. Section 2 details the experimental techniques. In Section 3, we detail our experimental results. We spend a significant portion of our discussion in Section 4 providing a concise description of the characterization technique of Ramakrishnan and Zukoski. Our success in using these tools with a particle size distinct from theirs provides evidence for the robustness of their technique. In addition, we briefly describe the activated barrier-hopping theory including the framework incorporating the impact of stress. In Section 5, we show that the theory predictions of the perturbative limit are in excellent quantitative agreement with experiment, and we discuss conclusions in Section 6. 2. Experimental Section Silica particles were synthesized via the Stober technique,24 which involves the base catalysis of tetraethyl ortho-silicate using ethanol as the solvent medium. An average particle radius, R ) 24 nm, was obtained with this technique. Once synthesized, the particles are coated with a hydrophobic layer of octadecyl chains that are chemically grafted onto the surface of the silica particles, as described by Van Helden et.al.25 The particles are washed in excess chloroform and then centrifuged several (3-6) times following the coating process to remove any traces of unreacted octadecanol. Then they are dried in a vacuum oven for ∼24 h before being used to prepare suspensions. The solvent used in our study is decalin, which is (22) Saltzman, E. J.; Schweizer, K. S. J. Chem. Phys. 2003, 119, 1197-1203. (23) Rao, R. B.; Kobelev, V. L.; Li, Q.; Lewis, J. A.; Schweizer, K. S. Langmuir 2006, 22, 2441-2443. (24) Stober, W.; Fink, A.; Bohn, E. J. Colloid Interface Sci. 1968, 26, 62. (25) Van Helden, A. K.; Jansen, J. W.; Vrij, A. J. Colloid Interface Sci. 1981, 81, 354-368.

Gopalakrishnan and Zukoski purchased from the Sigma Aldrich company. Being hydrophobic, the suspensions of these coated silica particles are stable in a hydrocarbon solvent like decalin at room temperature (25 °C). The near match of the refractive indices of silica and decalin minimize van der Waals interactions, and the residual attractions are eliminated by the steric barrier on the particles. This identical system was employed in a previous study26 where the zero-shear viscosity measurements of these suspensions at high volume fractions (φ ≈ 0.38-0.54) provides evidence for the near-hard-sphere nature of this experimental system at room temperature. In addition, Static Structure factor measurements from ultra small-angle X-ray scattering (USAXS) experiments are well captured by models26 for the average static structure factor of hard sphere suspensions with moderate polydispersity (∼10%). In our further discussion, we refer to all suspensions in this study at room temperature as hard-sphere suspensions. Sample suspensions are prepared by obtaining stock solutions of the hard-sphere suspensions and then mixing it with the appropriate quantities of decalin to obtain the desired volume fractions. The final volume fractions are verified by dry weight measurements, which are converted to a volume fraction by using a particle density of 1.6 g/mL. In our study, we worked with three volume fractions, φ ) 0.25, 0.30, and 0.38. As the sample temperature is decreased, suspensions retain fluidlike behavior until a critical temperature is reached. Below this temperature, Tgel, a gel phase is obtained. The gel phase results from attractions that are induced by the solvent decalin turning from a good solvent for the octadecyl hairs above the critical temperature to a poor solvent below the critical temperature. This transition is driven by enthalpic forces. This shift in solvent quality prompts the surface octadecyl hairs to interact with surface hairs on neighboring particles to minimize contact with the solvent, thereby minimizing the system free energy. The formation of the gel phase is reversible as, upon increasing the temperature, fluidlike behavior is restored.6,14 The gel boundary in our study has been determined rheologically from dynamic oscillation experiments. When going down in temperature, Tgel is defined as that temperature at which the suspension elastic modulus (G′) first exceeds the viscous modulus (G′′). In our study, we have made use of two rheometers. Experiments on the φ ) 0.38 sample were performed on a C-VOR Bohlin rheometer in the constant stress mode, whereas for the φ ) 0.25 and 0.30 samples, experiments were performed on the CS Bohlin rheometer in the constant stress mode. The only difference in the two rheometers is that the C-VOR rheometer has a greater degree of sensitivity. However, this is not an issue, as both the linear viscoelastic regime and the perturbative region where the modulus begins to drop (parameters of interest in our study) with increasing stress fall within the window of accurate measurability for both rheometers. A cupand-bob geometry is used for all rheological measurements. The bob diameter is 14 mm, and the gap is 0.7 mm. A roughened bob is employed for all measurements to eliminate slip at the bob-sample interface. By employing the technique of Walls et.al.,27 a plot of the elastic energy (G′γ) against the strain, γ, shows a single maximum, associated with samples that do not display wall slip. This reassures us that slip indeed does not play a role in our dynamic oscillatory measurements. For all measurements, the temperature in the sample is accurate to within (0.2 °C. Thus, we are able to probe the gel temperature to within an accuracy of 0.5 °C. Within the gel phase, the following protocol is followed to ensure a consistent means of measuring the sample modulus. To obtain data at a particular volume fraction and temperature, the sample is loaded into the rheometer cup as a liquid at room temperature. The sample temperature is then lowered to the desired value and kept constant for 45 min to ensure a uniform sample temperature. The sample is then presheared at a high shear rate (∼500 s-1) for 30 s, (26) Gopalakrishnan, V.; Zukoski, C. F. Ind. Eng. Chem. Res. 2006, 45, 69066914. (27) Walls, H. J.; Caines, S. B.; Sanchez, A. M.; Khan, S. A. J. Rheol. 2003, 47, 847-868.

Yielding BehaVior of Thermo-reVersible Colloidal Gels

Figure 1. Gel boundary of the 24 nm particles used in this study compared against that of the 45 nm particles used in a previous study.21 As can be seen, decreasing the particle radius shifts the gel boundary to higher temperatures. In addition, the gel boundary also becomes less sensitive to volume fraction. and then the elastic modulus is measured as a function of time to determine the recovery time for the sample. We find that for our samples, the recovery time is on the order of several hours. A steady state is defined as when the elastic modulus is finally independent of time. After sample recovery, elastic modulus is measured as a function of the oscillation frequency over 4 orders in magnitude (10-2-50 Hz) in the linear viscoelastic regime. Once the frequency dependence has been determined, a stress sweep experiment is performed to determine how the elastic modulus varies as a function of applied stress at a frequency of 1 Hz. In these experiments, the stress sweep is over 4 orders of magnitude going from 0.1-1000 Pa. Stress sweep experiments were also performed at 5, 6, 7 °C at φ ) 0.30 for different frequencies (0.1, 1 and 10 Hz) which reveal that the perturbative stress (stress where the modulus drops to 90% of its value in the linear viscoelastic regime) is independent of the applied frequency. Therefore, we only report data at 1 Hz. For experiments at any temperature, the sample is always reheated to room temperature before being recooled to the new temperature of interest.

3. Results In Figure 1, the volume-fraction dependence of the gel temperature for particles used in this study is compared with that of the 45 nm particles employed by Ramakrishnan and Zukoski.21 From the figure, it is apparent that at the same colloid volume fraction, decreasing the particle size shifts the gel boundary to higher temperatures. In addition, the gel-boundary temperature is less sensitive to φ at the smaller particle size over the range of volume fractions explored. Prior to any experiments, the gels are presheared to eliminate shear-history effects. After preshear, the modulus of the gel recovers over a period of time. For the sample displayed in Figure 2, recovery times are ∼8 h. Plotting the data on a log-log plot (not shown here) does indeed reveal a very slow increase in the elastic modulus even after a near-complete recovery indicating the possible presence of a slow aging process. However, over the time frame over which we performed our experiments after establishing a steady state, this slow aging may be relevant to long time relaxations of colloidal gels as described in the trap models of Sollich and co-workers28 but does not impact our observations (i.e., the perturbative stress is weakly dependent on time). The inset plot shows the results from the frequency sweeps at φ ) 0.30 for a range of temperatures. For all our samples, once in the gel phase, the modulus is a weak function of the stress frequency, as seen by the near-flat profiles in the inset plot of (28) Fielding, S. M.; Sollich, P.; Cates, M. E. J. Rheol. 2000, 44, 323-369.

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Figure 2. Elastic modulus, G′, as a function of the time elapsed after preshear. As can be seen, once the gel structure is disrupted by shear, it can take up to 6-10 h before a steady-state modulus is obtained. This was consistent for all the gel samples employed in the study. The inset plot shows the linear viscoelastic modulus, G′0, against applied frequency in a dynamic oscillatory measurement. The measurement is made after a steady-state modulus is achieved. Over ∼4 orders of magnitude in frequency, the modulus is relatively independent of the applied frequency.

Figure 3. Nondimensionalized linear viscoelastic modulus as a function of sample temperature, T, for the three volume fractions studied. An increase in modulus is brought about by either a decrease in T or an increase in φ.

Figure 2. In addition, decreasing the sample temperature shifts the moduli to higher values. This is expected considering that attractions get stronger with decreasing temperature. A summary of the volume fraction and temperature dependencies of the elastic modulus at 1 Hz is shown in Figure 3. The effect of increasing the stress at a frequency of 1 Hz for samples at φ ) 0.30 is shown in Figure 4a. At the lowest stresses, the modulus is independent of the applied stress. This is the linear viscoelastic regime. However, with increasing stress, the modulus decreases. Beyond the initial drop in the modulus, a relatively small increase in stress triggers a catastrophic drop in the elastic modulus. This is found to occur at all temperatures and at all volume fractions and can be understood by considering the large recovery times for these thermo-reversible gel systems. Once the microstructure has been broken by external stresses, a very large period of time is required to restore solidlike behavior (as seen in Figure 1). In a dynamic experiment where an oscillatory stress is applied continuously and gradually increased, we hypothesize that the system is driven sufficiently far from equilibrium beyond the perturbative stress limit that recovery is no longer possible and further oscillatory deformation promotes further breakage thereby leading to a catastrophic collapse in the modulus. In Figure 4b, the data are replotted by normalizing the modulus by the modulus in the linear viscoelastic limit where

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of magnitude. On the other hand, at a fixed temperature, σpert does not increase as strongly over the range of explored volume fractions. To analyze these results, we wish to use the activated barrierhopping theory of Schweizer and co-workers.1,2 Employing this approach requires that we know the structure factor of the suspension, which in turn requires that we characterize the interparticle attractions. We choose to assume that the particles interact via an attractive short-range Yukawa potential for which structure factors can be computed. To extract the temperaturedependent strength of attraction and range of the attraction, we follow the characterization methods of Ramakrishnan and Zukoski21. Below we describe how this is accomplished and also briefly describe the activated barrier-hopping theory and its ability to provide predictions for a stress dependent modulus. We then compare predictions of the theory with experimental data on the perturbative stress, σpert.

4. Theoretical Models

Figure 4. (a) Nondimensionalized modulus as a function of the nondimensionalized stress. As can be seen, decreasing the temperature increases the magnitude of the modulus in the linear viscoelastic regime. The stress at which the modulus drops from the plateau value increases with a decrease in temperature. Another interesting feature is that beyond the initial decrease in modulus, further increase in stress results in a catastrophic drop in the modulus. The lines are to guide the eye. (b) Elastic modulus normalized to the plateau value as a function of the nondimensionalized stress. The stress at which the modulus drops to 90% of its plateau value, defined as the perturbative stress, σpert, monotonically increases with a decrease in temperature. The lines are to guide the eye.

Figure 5. Nondimensionalized perturbative stress as a function of the sample temperature, T, for all three volume fractions employed in the study. The lines are drawn for the data at the lowest and highest volume fractions to guide the eye. As can be seen, σpert increases more strongly with a decrease in temperature than by increasing the volume fraction over the range studied here.

we find that the perturbative yield stress increases with a decrease in temperature. This feature is observed at all three volume fractions. A summary of the perturbative stresses, σpert, as a function of temperature and volume fraction is shown in Figure 5. The lines through the data sets of the lowest and highest φ show that, over the range of explored temperatures, σpert increases by an order

4.1. Naı1ve Mode Coupling Theory (NMCT) with the Yukawa Potential. NMCT20,29 is a simplified version of the ideal mode coupling approaches30 that aim to provide a selfconsistent picture of the dynamics of particles and how particle motion depends on correlations in density fluctuations on all length scales in the system. Building on the original concept of Kirkpatrick and Wolynes29 and using the equations of motion, NMCT describes conditions under which particles are localized and therefore unable to undergo long-range diffusion. Particles are gelled when they are localized within a cage defined by the number of nearest neighbors. They are free to diffuse within the cage giving rise to a localization length, rloc, calculated from

1 1 ) r2loc 9

∫0∞

4πq2 dq (2π)

3

q2FC2(q)S(q) e-(q rloc/6) (1+1/S(q)) (1) 2 2

where q is the wavevector, an inverse length scale in Fourier space, and S(q) is the static structure factor that represents the correlations in density fluctuations at length scales ∼1/q. C(q) is the direct correlation function that depends solely on S(q) and F ) (3φ)/(4πR3) is the number density. For a fixed strength of attraction at low enough volume fractions, 1/rloc is zero, indicating that particles diffuse freely. At a well-defined volume fraction, rloc jumps to a finite value, indicating that particles are localized and the gel boundary has been crossed. As volume fraction or strength of attraction are increased, rloc decreases, indicating that particles are more tightly held within their cages. From knowledge of rloc, the zero-frequency elastic modulus can be determined by the Green-Kubo relation and the Mode coupling theory factorization31-33 as

G′ )

k BT 60π

2

∫0∞ dqq4(

)

d ln S(q) 2 -(q2r2 /3) (1/S(q)) loc e dq

(2)

Bear in mind that this modulus is in the linear viscoelastic regime (limit of zero stress). To determine the location of the gel boundary and gel elastic modulus, S(q) is required as input. Numerous methods are available for calculating S(q) for attractive systems, but each requires that the nature of the particle pair potential be (29) Kirkpatrick, T. R.; Wolynes, P. G. Phys. ReV. A 1987, 35, 3072-3080. (30) Gotze, W. J. Phys. Condens. Matter. 1999, 11, A1-A45. (31) Geszti, T. J. Phys. C Solid State Phys. 1983, 16, 5805-5814. (32) Nagele, G.; Bergenholtz, J. J. Chem. Phys. 1998, 108, 9893-9904. (33) Verberg, R.; de Schepper, I. M.; Cohen, E. G. D. Phys. ReV. E 1997, 55, 3143-3158.

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defined. A common model for the pair interaction that captures many colloidal interactions is the two-parameter Yukawa potential that accounts for strength of attraction at contact and range of attraction:

{

∞ for r < 2R uYukawa(r)  ) e-κ(2R)(r/(2R)-1) for r > 2R kBT kBT r/(2R)

}

(3)

where κ-1 is the characteristic length scale of attraction potential and /kBT is the contact potential. r is the interparticle separation distance. S(q) for the Yukawa system has been calculated using the code of J. Bergenholtz, which uses the mean spherical approximation (MSA) closure.34 Using Tao and Raetto’s35 recommendations to improve the predictions of the MSA closure, Ramakrishnan and Zukoski showed that the simple relation, MSA/ kBT ) 0.69 e0.8/kBTcan be used as input information of the contact potential into the code. This transformation is required to link the structure factor calculated from MSA in a Yukawa system with the contact potential as the MSA closure overestimates the effects of attractions. To predict S(q), the parameters needed are /kBT, κ, and φ. Ramakrishnan and Zukoski21 were able to show that over a wide range of parameters, 11 e κ e 100 and 0.15 e φ e 0.381, the linear viscoelastic modulus predictions of the NMCT theory for different /kBT can be collapsed onto a single universal curve defined by 2

[

G′D3 0.58φ(κD) ) 2 exp 1.6χ(φ,κD) kBT F (φ,κD)

{( ) ( ) }]   kBT kBT

(4)

gel

where F(φ,κD) ) [(3.11 - 27.31φ + 106.37φ2 - 137.45φ3)‚ (κD)0.2]-1/2. Therefore, knowing φ, κD can be determined from knowledge of the prefactor, 0.58φ(κD)2/F2(φ,κD). χ(φ,κD) can be determined from knowledge of φ and κD provided in Table 2 in the original reference.21 In relating the interaction potential, /kBT to the sample temperature, T, we follow the suggestion of Jansen et.al.8

(

Tθ  )A -1 kBT T

)

for T e Tθ

(5)

where A is a factor proportional to the overlapping of the octadecyl chains and Tθ is the θ temperature for the solvent-polymer chain pair. Substituting eq 5 into eq 4 yields 2

[

(

Figure 6. Nondimensionalized linear viscoelastic modulus as a function of (1/T - 1/Tgel). The dotted line through the experimental data is the exponential fit, y ) 35.416 e55 501x. The fit is used to characterize the attractive interaction as a Yukawa potential as proposed by Ramakrishnan and Zukoski.21 The inset is the nondimensionalized version of Figure 3, and the dotted lines are predictions of the modulus from the NMCT model where the S(q) input is provided by characterizing the attractions as described above.

1 G′D3 0.58φ[κD] 1 exp 1.6χ(φ,κD)ATθ ) 2 kBT T T F (φ,κD) gel

)]

(6)

Thus, plotting the experimentally measured linear viscoelastic modulus, G′D3/kBT against (1/T - 1/Tgel) provides an exponential fit from which the parameters κD can be extracted from the intercept and the product ATθ can be extracted from the slope. To obtain values of A and Tθ, the experimental gel boundary is matched with calculations of the NMCT gel boundary for φ ) 0.25 in our study using eq 5. Figure 6 shows the plot of the non-dimensionalized experimental plateau moduli against (1/T - 1/Tgel). The data for all three different volume fractions collapse onto a single curve. The collapse of the data from samples made of the 24 nm particles in a similar fashion to that of the 45 nm particles as shown by Ramakrishnan and Zukoski21 provide evidence that this technique (34) Cummings, P. T.; Smith, E. R. Chem. Phys. 1979, 42, 241-247. (35) Tau, M.; Reatto, L. J. Chem. Phys. 1985, 83, 1921-1926.

Figure 7. Nonequilibrium free energy, in units of kBT, as a function of the particle displacement, normalized to the particle size. With increasing stress, the nonequilibrium free energy curve is modified. The values next to the curves display the corresponding stress in units of kBT/(8R3). With increasing stress, the barrier height drops until the absolute yield stress (24.5 for the above case) where the barrier is completely wiped out. With a reduction in barrier height, the position of the minimum, the localization length, rloc, shifts to larger values. This is better seen in the inset plot where the symbols represent calculations at specific stress values and the line is to guide the eye. An increasing rloc translates to a decrease in the elastic shear modulus.

employed to characterize the octadecyl-silica-decalin thermoreversible gels is independent of particle size. The dotted line is the exponential fit 8G′R3/kBT ) 35.416 exp[55 501(1/T 1/Tgel)] to the experimental data. From the fit, the extracted values of the parameters are as follows. κD ) 13, χ(φ,κD) ) 3.4, A ) 32.9 and Tθ ) 309.6 K. Using the parameters A and Tθ , we can convert the sample temperature to an interaction potential using eq 5. Equipped with κD, /kBT, and φ, we can now calculate S(q) to make predictions of the elastic modulus. The inset plot in Figure 6 shows the predictions from the NMCT model and compares it with the experimental data that were plotted in Figure 3. The match confirms that the parameters extracted do indeed characterize the thermo-reversible gel system and accurately predict the plateau elastic modulus. 4.2. Activated Barrier-Hopping Theory. As discussed earlier, the NMCT model20 makes predictions of the elastic modulus in the limit of zero external stress. In a manner consistent with eq 1, (i.e., the NMCT), Schweizer and Saltzman2 propose the

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activated barrier hopping theory which describes particle localization in terms of a nonequilibrium free energy function, F (defined in units of kBT) which is dependent on particle displacement, r. This nonequilibrium free energy is the effective interaction energy that would have to be imposed to account for the effects of correlated motions of particles that result in localization. As such, it does not alter the equilibrium microstructure but acts only to modulate particle dynamics. As the strength of attraction, /kBT, or the volume fraction, φ, is increased, F(r) evolves from a monotonically decreasing function to one with a point of inflection at critical values of /kBT and φ. This occurs at the point where rloc first takes on a finite value, thereby signifying particle localization. As either /kBT and φ are further increased, the inflection develops into a minimum, thus forming an energy well and localization length moves to smaller values signifying stronger localization. The nonequilibrium free energy is defined as resulting from an effective force arising from many particle interactions, thus providing a means to describe particle localization in the energy well and the resistance provided by this energy well (or barrier) to long-time diffusion in the event of thermally activated motion. F is defined as

3 F(R) ) ln(R) 2

dq b

∫ (2π)3 FC2(q)S(q)[1 + S-1(q)]-1 ×

(

q2 [1 + S-1(q)] (7a) 4R

R ) 3/2r2

(7b)

exp -

)

where

Although F(R) in eq 7a is in the limit of zero stress, this sets the framework to incorporate the effects of external stress as developed by Kobelev and Schweizer1 who argue that the macroscopic stresses applied on the system are transferred to the particles in such a manner that they distort the nonequilibrium free energy function as follows

3 F(R) ) ln(R) 2

(

Figure 8. Theoretical elastic modulus normalized to the value at zero stress as a function of the nondimensionalized stress. With increasing stress, the modulus decreases from the plateau value to register a drop that is qualitatively similar to experiment. As seen in the experiment, the stress at which this decrease in modulus is seen increases monotonically with decreasing temperature. The inset compares experimental observations (data points) at 5 and 9 °C at φ ) 0.30 with predictions from theory (lines).

dq b

∫ (2π)3FC2(q)S(q)[1 + S-1(q)]-1 ×

)

q2 σ exp - [1 + S-1(q)] - 2/3 x3/(2R(2R)2) (8) 4R φ where R is the particle radius and σ is the applied stress (in units of kBT/(8R3)) on the sample. As we have determined all the parameters needed to calculate S(q) for our experimental system, we can predict how F changes with applied stress. As shown in Figure 7 for φ ) 0.30 and T ) 6 °C, as the external stress is increased, the energy well which is defined as the barrier height decreases. In addition, the position of the minimum, defined as the localization length, is shifted to larger values. This is better illustrated in the inset plot where rloc is plotted as a function of the applied stress. As can be seen in this plot, at low stresses, rloc is virtually unchanged. As the applied stress increases, rloc increases much faster, finally reaching a point where the energy barrier, and therefore particle localization, is destroyed at the absolute yield stress. As seen from eq 2, rloc influences the modulus of the system. As rloc increases, the modulus as predicted using eq 2 drops. This is illustrated in Figure 8 where the modulus predictions of the activated barrier-hopping theory are plotted for φ ) 0.30 for a series of temperatures. As the applied stress increases, G′ drops from its value in the linear viscoelastic regime, G′0. Within the model, the strain on the system is then calculated in a consistent

fashion from knowledge of the applied stress and the resultant modulus as γ ) σ/G′. Although parameters for characterizing the attractive interactions are determined from experimental measurements of the linear viscoelastic modulus, the predictions of how this modulus evolves with the applied stress using the activated barrier-hopping theory have no inputs from experimental data. Thus, a comparison of the perturbative stress measurements from the stress sweep experiments with corresponding predictions from the theory with zero adjustable parameters provides an excellent test for how well the theory is capable of capturing the perturbative stresses of the thermo-reversible gels for a range of volume fractions and temperatures.

5. Discussion A comparison of Figures 4b (experiment) and 8 (model predictions) (also seen in the inset of Figure 8) for identical temperatures and φ ) 0.30 reveals that the model does not predict a drop in modulus that is as catastrophic as that observed in experiment. As mentioned earlier, the drop observed in experiment can be attributed to the inability of the thermo-reversible gels in our experimental system to recover at short times once the microstructure has been sufficiently perturbed. The theory is developed to account for small perturbations from equilibrium and thus cannot capture effects of these microstructural changes. Confocal microscopy studies36 on thermo-reversible gels also indicate that imposing large strains induces heterogeneities that are unaccounted for in the theory which assumes a uniform microstructure within the thermo-reversible gel. These heterogeneities at large strains could further weaken the gel, perpetuating a catastrophic drop in the elastic modulus. Therefore, in this study, we restrict ourselves to comparing theory predictions of the perturbative stress, σpert, with experiments. A comparison of experimental and theoretical predictions of σpert is shown in Figure 9. As seen in the experimental results, over the explored range of parameters, the model predicts a stronger dependence of the perturbative stress on sample temperature than the sample volume fraction, φ. Indeed, the main figure demonstrates that the model does an excellent job of capturing experimental perturbative stress values. The inset plot (36) Varadan, P.; Solomon, M. J. J. Rheol. 2003, 47, 943-968.

Yielding BehaVior of Thermo-reVersible Colloidal Gels

Figure 9. Nondimensionalized perturbative stress, σpert, as a function of the sample temperature for all three volume fractions. The data points are from experiments whereas the linesssolid line for 0.25, dashed line for 0.30, and short-dashed line for 0.38sare from the activated barrier-hopping theory calculations. The theory predictions quantitatively match experimental values for σpert without any adjustable parameters. The inset plot shows the perturbative strains from experiment and model predictions. Again, the theory predictions are of the same order of magnitude as that from experiments for the perturbative strain.

shows a comparison of the perturbative strains, defined as γpert ) σpert/G′, which demonstrates that the perturbative strain predictions are in the same order of magnitude as that seen in experiments with the largest differences seen at temperatures close to the gel boundary. The fact that perturbative strain values from theory predictions and experiments are off by a factor of ∼2-4 is due to small differences in predictions and experimental values of the elastic moduli and the perturbative stresses. We emphasize, however, that the agreement between experiments and the theory is very good given that we have used no adjustable parameters in predicting σpert or γpert. In previous work,23 the activated barrier-hopping theory has been shown to provide accurate predictions for perturbative strains for concentrated nanoparticle suspensions that behave as effective hard spheres. The values for the linear viscoelastic modulus and the perturbative stresses in the study differ from that predicted by the theory for hard spheres by an order in magnitude, and this is attributed to the presence of the soft steric barrier on the particles which is absent in the calculations of the theory. However, this difference is expected to have a negligible effect on the yield strains, and indeed, good agreement is observed between theory and experiment for the perturbative strains. In our study on thermoreversible gels, the interactions between the particles are characterized by a Yukawa potential by ensuring that the theory captures the linear viscoelastic modulus from the experiment. By doing this, we provide an accurate measure of the density fluctuations in the thermo-reversible gels (S(q)) as input into the activated barrier-hopping theory which then permits a more critical test of the theory predictions for the perturbative stresses and strains. The remarkable quantitative agreement for the perturbative stresses and a reasonably good agreement for the perturbative strains seen over a range of volume fractions and temperatures indicate that the theory captures conditions for the onset of nonlinear behavior, extending the applicability of the model beyond hard-sphere glasses. There has been a great deal of interest on shear induced mesoscopic structures and their influence on the mechanical properties of gels.12,37-39 In particular for depletion systems and (37) Varadan, P.; Solomon, M. J. Langmuir 2003, 19, 509-512. (38) Hoekstra, H.; Mewis, J.; Narayanan, T.; Vermant, J. Langmuir 2005, 21, 11017-11025. (39) Mohraz, A.; Solomon, M. J. J. Rheol. 2005, 49, 657-681.

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low volume fraction thermal gels, mesoscopic structures are observed and are expected to influence yielding and flow. Mode coupling theory is constructed to predict properties based on equilibrium microstructures and thus cannot account for the presence of these mesoscopic heterogeneities. However, for two experimental systems where interaction energies have been characterized, NMCT predictions agree with experiments on the volume fraction, polymer concentration, and size dependence of the elastic modulus of depletion systems40 and on the elastic modulus and perturbative stress of dense thermoreversible gels. In the case of depletion gel systems, there is strong evidence for mesoscopic structures, while for dense thermoreversible gels, on length scales measurable by USAXS the suspensions are uniform on length scales larger than a few particle diameters.12 The observations of structural heterogeneities on much larger length scales36,37 when contrasted to the agreement found here indicates the limited understanding that exists of the role of mesoscopic structures on the mechanics of gels.

6. Conclusion In this study, we test the predictions of the newly developed activated barrier-hopping theory that accounts for the implications of externally applied stress on the elastic modulus of attractive colloidal gels. Zero shear viscosity predictions of the theory have shown excellent agreement with corresponding experimental data on well-characterized hard-sphere systems.22 In addition, for depletion-gel systems, the theory predicts the right trends in the linear viscoelastic modulus with φ, polymer concentration, and radius of gyration.20,41 Accounting for large-scale structural features also enables a quantitative agreement. Thus, the framework developed in this approach has been successful in capturing the particle dynamics that influence mechanical properties in the limit of weak stresses that do not significantly distort the suspension microstructure. Kobelev and Schweizer’s1 extension of the theory to incorporate the impact of external forces has been tested on nanoparticles that mimic hard spheres in the glassy region. The perturbative strain predictions were found to capture quantitatively those from experiments.23 Thermoreversible gels present an experimentally convenient and tractable attractive-colloidal system since the strength of attractions can be controlled by merely changing sample temperature in a reversible manner. Using a recently developed technique21 for characterizing the parameters of the pair interaction potential from the linear viscoelastic modulus of the gels, we are able to predict the microstructure in the suspension, the S(q), as a function of φ and T, as is required to test the activated barrier-hopping theory. With no adjustable parameters, the model predictions of the perturbative stress are in excellent quantitative agreement with experiment. The perturbative strains too are within a factor of 2-4 of the experimental values. Acknowledgment. The authors thank Dr. J. Bergenholtz for access to the code for calculating S(q) for systems characterized by Yukawa potentials. We acknowledge Prof. S. Ramakrishnan for the gel boundary data for 45 nm particles and Prof. K. S. Schweizer for his insights and discussions. This work was supported by the Nanoscale Science and Engineering Initiative of the National Science Foundation under NSF Award No. DMR-0117792. LA0620915 (40) Ramakrishnan, S.; Chen, Y. L.; Schweizer, K. S.; Zukoski, C. F. Phys. ReV. E 2004, 70. (41) Shah, S. A.; Chen, Y. L.; Schweizer, K. S.; Zukoski, C. F. J. Chem. Phys. 2003, 119, 8747-8760.