Langmuir 2008, 24, 7565-7572
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A Metastable van der Waals Gel: Transitioning from Weak to Strong Attractions Ryan C. Kramb and Charles F. Zukoski* Department of Chemical and Biomolecular Engineering, UniVersity of Illinois, Urbana, Illinois 61801 ReceiVed January 3, 2008. ReVised Manuscript ReceiVed April 16, 2008 Here we describe a method to create gels where the gel point is decoupled from gel elastic properties. Working with charge stabilized polystyrene latex particles with diameters, D, of 508-625 nm at ionic strengths of 0.1-1 M, the gel volume fraction is varied from 0.10-0.35 through the addition of less than monolayer coverage of hexaethylene glycol monododecyl ether (C6E12). At each surfactant concentration, the gel volume fraction depends on the background ionic strength. The changes in gel point with surfactant concentration suggest the strength of interparticle attraction decreases with increasing surfactant concentration. These changes are not reflected in the gel moduli, which are independent of surfactant concentration and ionic strength. We propose a model to describe this behavior based on gelation due to localization in a shallow truncated van der Waals minimum produced by the surfactant acting as a steric stabilizing layer. The surfactant remains mobile on the surface. Below the gel volume fraction, the time particles spend in the truncated well are not sufficient for the surfactant to be displaced such that the particles can only sample the shallow well. Above the gel volume fraction, particles are localized in the truncated van der Waals minima for sufficient periods of time to displace the surfactant layers with the result being that the particles fall into a primary van der Waals minimum. The result is gel points sensitive to surfactant concentration but moduli that are independent of the gel volume fraction.
1. Introduction Colloidal gels are produced by increasing the strength of attraction between particles and elevating the suspension volume fraction. As soft solids, the gels have useful properties in a variety of products.1,2 Gels can be thought of as metastable states of matter where particles are localized in nonequilibrium configurations for long periods of time. Two important properties for these systems are (1) the conditions where there is a transition from a low viscosity fluid to a nonflowing solid-like state and (2) the strength and elasticity of the material after it has entered the solid state. We will use the volume fraction of particles at the gel transition, φgel, for the former, where we recognize that φgel depends on solution conditions, particle composition, and size. For the latter, we will use the dimensionless elastic modulus, G′D3/kT, where G′ is the gel elastic modulus, D is the particle diameter, and kT is the product of Boltzmann’s constant and absolute temperature. In most systems, these two characteristic properties have an inverse relationship. For example, when G′ ≈ Bφx, x is often found to be independent of φgel, but B is inversely dependent on φgel. Indeed, it is often found that B ∼ B′/φgelx, where B′ is independent of φgel.1–5 One reason for this dependence is that both properties are related to the strength of interparticle attractions. Strong attractions result in localization at lower volume fractions as well as elevated moduli.6–8 The close correlation between φgel and gel modulus can present a problem when the goal is to produce a gel that is both strong and high in solids content (pigment, adhesive, etc.). Weakly *
[email protected]. (1) Lewis, J. A. J. Am. Ceram. Soc. 2000, 83, 2341. (2) Russel, W. B. S., D. A.; Schowalter, W. R. Colloidal Dispersions; Cambridge University Press: New York, 1989. (3) Rueb, C. J.; Zukoski, C. F. J. Rheol. 1997, 41, 197–218. (4) Ramakrishnan, S.; Zukoski, C. F. Langmuir 2006, 22, 7833–7842. (5) Chen, L. B.; Ackerson, B. J.; Zukoski, C. F. J. Rheol. 1994, 38, 193. (6) Bergenholtz, J.; Fuchs, M. J. Phys.: Condens. Matter 1999, 11, 10171– 10182. (7) Bergenholtz, J.; Fuchs, M. Phys. ReV. E 1999, 59, 5706–5715. (8) Schweizer, K. S.; Saltzman, E. J. J. Chem. Phys. 2003, 119, 1181–1196.
attractive systems resulting in a low viscosity suspension at modest to high volume fractions do not produce gels with large moduli, while strongly attractive systems gel at low volume fractions and require some energy intensive process (e.g., pressure filtration) to increase the volume fraction significantly above φgel.9,10 Here we discuss a method for overcoming this dilemma. Our system gels at volume fractions above 0.30, but these gels have moduli with magnitudes equal to those that first gel at much lower volume fractions.11 Understanding of colloidal gels has been greatly advanced by the presence of model systems where the strength of attraction can be characterized. Two commonly used systems are depletion gels, where attractions are produced by adding a nonadsorbing polymer, and thermal gels, where attractions are produced by driving hydrocarbon chains covalently bound to a particle surface through their θ point by lowering the suspension temperature.4,12,13 These systems are often engineered such that, in the absence of polymer or at elevated temperatures, the particle interactions are those of hard spheres. As a result, the strength of attraction can be tuned from zero, where glasses are observed at volume fractions near 0.58, to very attractive, where gels are observed at volume fractions lower than 0.1. While of great value in understanding phase behavior and localization phenomena, these are not characteristic of a broad class of colloidal gels where the ubiquitous van der Waals interactions dominate the attractions and the range and strength of the attraction are set by material properties. van der Waals gels are difficult to study because, unlike depletion and thermal gels, where the range of attraction and (9) Landman, K. A.; Sirakoff, C.; White, L. R. Phys. Fluids A: Fluid Dynamics 1991, 3(6), 1495–1509. (10) Pham, K. N.; Petekidis, G.; Vlassopoulos, D.; Egelhaaf, S. U.; Pusey, P. N.; Poon, W. C. K. Europhys. Lett. 2006, 75(4), 624. (11) Buscall, R.; Mills, P. D. A.; Goodwin, J. W.; Lawson, D. W. J. Chem. Soc. Faraday Trans. 1 1988, 84(12), 4249–4260. (12) Gopalakrishnan, V.; Zukoski, C. F. Langmuir 2007, 23, 8187–8193. (13) Ramakrishnan, S.; Gopalakrishnan, V.; Zukoski, C. F. Langmuir 2005, 21, 9917–9925.
10.1021/la800021h CCC: $40.75 2008 American Chemical Society Published on Web 06/18/2008
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well depth are understood, the interactions of charged particles experiencing van der Waals forces are difficult to simplify due to the poorly defined strength of attraction at contact. Partridge et al. explored a system that was well characterized at high ionic strength by a van der Waals attraction truncated by adsorbed monodispersed nonionic surfactant.14 At high ionic strength, where the Debye-Huckel double layer thickness, κ-1, is much smaller than the particle diameter, van der Waals attractions will dominate attractions as long as surface to surface separation remains larger that ∼κ-1. The van der Waals attraction is well approximated by
{
uVDW(x) 1 1 A + + )kT 12kT x2 + 2x x2 + 2x + 1 x2 + 2x 2 ln 2 x + 2x + 1
(
)}
(1)
where A is the Hamaker coefficient for particles in a continuous phase, and x ) (r/D - 1), with r being the center to center separation. At complete coverage, the surfactant truncates the van der Waals attraction at a minimum separation of approximately twice the surfactant chain length. If we assume that the surfactant provides a hard repulsion when the particles are in contact, uVDW/kT ∼ -AD/24(2δ), where 2δ is twice the surfactant chain length. For the polystyrene latex studied here, A ∼ 3.2kT, and for the surfactant described below, 2δ ∼ 8 nm.15 As a result, for D ) 508-625 nm, the surfactant stabilized value of the van der Waals attraction minimum is -(6-9)kT. Thus, the truncated attraction based solely on van der Waals attractions is weak, suggesting that the particles will be stable at low volume fractions and will gel as the volume fraction is raised.4,6,13,16–18 When the bulk concentration of the surfactant is below that where the particle surfaces are saturated, the van der Waals attraction can be approximated as being truncated at a separation smaller than 2δ such that the attractive well depth is deeper. In addition, because the surface is not saturated, the surfactant may rearrange such that the particles come into closer contact. Here we argue that the gel volume fraction can be understood in terms of the truncated van der Waals attraction while the modulus of the resulting gels can be understood in terms of this surfactant rearrangement. Below in section 2, we detail experimental techniques used in particle synthesis and suspension preparation. Gel boundaries, flow properties, and microstructural properties of the suspensions are discussed in section 3, while in section 4, we discuss the mechanism of surfactant redistribution as the origin of how the system is able to achieve the unconventional high φgel, high G′ behavior we have observed, and finally, conclusions are discussed in section 5.
2. Experimental Section For this work, two sizes of spherical polystyrene particles were synthesized on the basis of the recipe of Goodwin, resulting in average diameters of 508 and 625 nm.19 The particles were created in a 5 L round-bottom flask immersed in a constant temperature water (14) Partridge, S. J. Rheology of CohesiVe Sediments; Bristol University: Bristol, 1985. (15) Romero-Cano, M. S.; Martin-Rodriguez, A.; Chauveteau, G.; de las Nieves, F. J. J. Colloid Interface Sci. 1998, 198(2), 273–281. (16) Bergenholtz, J.; Poon, W. C. K.; Fuchs, M. Langmuir 2003, 19(10), 4493– 4503. (17) Shah, S. A.; Chen, Y.-L.; Ramakrishnan, S.; Schweizer, K. S.; Zukoski, C. F. J. Phys.: Condens. Matter 2003, 15, 4751–4778. (18) Shah, S. A.; Chen, Y.-L.; Schweizer, K. S.; Zukoski, C. F. J. Chem. Phys. 2003, 119(16), 8747–8760. (19) Goodwin, J. W.; Hearn, J.; Ho, C. C.; Ottewill, R. H. Colloid Polym. Sci. 1974, 252, 464–471.
bath. A poly(tetraflouroethylene)-coated blade attached to a rotating glass bar was used for stirring. After allowing 3350 mL of deionized water to reach a steady-state temperature of 80 °C, 402 mL of styrene monomer (Sigma-Aldrich, 99% purity grade) was added. The solution was stirred at 80 °C for 1 h at a vigorous speed. A 2.985 g portion of potassium persulfate initiator (Fisher Scientific, 99.5% purity grade) dissolved in 560 mL of water was then added, and the reaction was allowed to proceed for 24 h. Although the same amounts of reactants were used to produce the 508 and 625 nm particles, slight differences in stirring speed likely contributed to the size variations. After completion of the reactions, all suspensions were dialyzed against a solution of polyethylene glycol (20 000 molecular weight, Sigma) in deionized water with approximately 30 g of PEG and 3 L of water. This process had two effects: to concentrate the particles by withdrawing water from inside the dialysis sack and reduce the ionic strength of the solution by allowing neutral salt to diffuse out of the sack. The PEG/water solution was changed daily until the particles reached an adequate volume fraction and ionic strength, as measured by a conductivity meter (YSI, Inc., model 34 conductance-resistance meter, YSI, Inc., model 3403 cell K ) 1.0/ cm). Ionic strength was determined to be sufficiently low when conductance matched that of a 10-5 M NaCl in water solution to the limits of the conductivity meter (∼2 µS). When iridescence was observed during dialysis, indicating an ordered structure, volume fractions were determined from wet and dry weights of samples taken from the dialysis sacks. It was found that ordering occurred at approximately φ ) 0.2 at ionic strengths of about 10-4 M. In order to work with the range of ionic strengths where gel transitions occur, the background ionic strength of the particle solutions was increased by the addition of sodium chloride (Fischer Scientific, crystalline) dissolved in deionized water. The final ionic strength was set by addition of necessary volumes of NaCl solutions at concentrations of 5, 1, 0.5, 0.3, and 0.2 M. C12E6 was also added to the NaCl solutions to maintain the desired surfactant concentrations. The particle sizes were determined to be 508 ( 12 and 625 ( 16 nm with scanning electron microscopy (SEM Hitachi S4700), by measuring ∼100 particle diameters at various positions on a sample grid (SPI supplies, Formvar coated). The accuracy of the SEM measurements was confirmed by dynamic light scattering using a Brookhaven Instruments fiber optic quasielastic light scattering (FOQELS) device. On the basis of the work of Partridge,14 a nonionic surfactant, hexaethylene monododecyl ether, C12E6 (Sigma) was chosen as the adsorbate to sterically stabilize the particles and truncate the van der Waals attraction. The length of this molecule, and therefore half the distance of closest particle separation at full coverage, was found to be 3.85 nm by Partridge. The critical micelle concentration of C12E6 in water is approximately 8 × 10-5 M.14,20,21 The structure of the surfactant molecule is shown in Figure 1. The extent of surfactant coverage was determined by measuring the surface tension of a suspension with φ ) 0.3 at 10-4 M ionic strength. Figure 2 shows the measured surface tension of both deionized water and the 508 nm particle solution as more surfactant is gradually added. The data from the water confirms the cmc found from in literature, 8 × 10-4 M. The surface tension data from the particle-containing solution shows that with φ ) 0.3 for the 508 nm particles, a minimum surfactant concentration of 10-2 M is needed to ensure full particle surface coverage. For higher volume fractions or smaller particle sizes, the concentration of surfactant must be increased to ensure full coverage. By fitting a line to the surface tension vs log [C12E6] plot, one can create a Langmuir isotherm to determine the number of surfactant molecules per surface area of particle. This resulted in a surface concentration of 1.7 × 1018 molecules m-2 at full coverage and an area per molecule of 0.62 nm2. This compares to 3.75 × 1018 (20) Corkhill, J. M.; Goodman, J. F.; Harrold, S. P. Trans. Faraday Soc. 1964, 60, 202. (21) Hough, D. B., Ph.D. Dissertation, Bristol University, 1973.
A Metastable Van der Waals Gel
Langmuir, Vol. 24, No. 14, 2008 7567 Table 1. Surface Potential Measurements of 508 nm Particles ionic strength (M) -4
10 0.1 0.2 0.3
Figure 1. Chemical structure of the surfactant molecule C12E6. The length of the molecule determines the distance of closest separation and therefore the truncation distance for van der Waals force calculations for fully covered particles.
Figure 2. Surface tension measurements of DI water as a function of C12E6 concentration (open squares), and a φ ) 0.3 sample of 508 nm particles in DI water (closed circles). The critical micelle concentration of C12E6 in water was found to be 8 × 10-4 M, in agreement with literature values. A concentration of 10-2M is needed to ensure monolayer coverage of the 508 nm φ ) 0.3 sample.
molecules m-2 and 0.27 nm2 found by Partridge14 and 0.60 nm2 by Lange22 and 0.55 nm2 by Corkill.23 At an ionic strength of 10-4 M, when dialyzed against a PEG solution, irreversible aggregation is observed at volume fractions greater than 0.45 in the absence of a stabilizing monolayer of surfactant. However, when the surface is saturated, at volume fractions greater than 0.55, the suspensions remain stable. All of the suspensions studied here had been exposed to surfactant at a bulk concentration of 10-4 M. Since we desired to have some samples with near zero surfactant coverage, a few suspensions were dialyzed exhaustively against an aqueous solution containing no surfactant to remove any excess surfactant. However, there is undoubtedly some surfactant left on the particle surface of these samples. The effects of this residual surfactant are observed in determining the gel point at the lowest surfactant concentrations. The surface charge of the particles in the presence of surfactant was measured by the ζ potential (Brookhaven Instruments phase analysis light scattering) at the edge of the surfactant monolayer, as seen in Table 1. Since aκ > 100, the Smoluchoski theory equation is used to convert electrophoretic mobility to ζ potential. Rheological measurements including viscosities and moduli were obtained using a Bohlin CS-10 rheometer with a cup and bob geometry. The bob is a made from roughened titanium with a diameter (22) Lange, H. Kolloid Z. 1965, 201, 131. (23) Corkhill, J. M.; Goodman, J. F.; Tate, J. R. Trans. Faraday Soc. 1966, 62, 979.
ζ potential measurements (mV)
Dκ
-31 -21 -11 -9
20 540 760 930
of 14 mm, a gap size of 0.7 mm, and a sample volume of 3 mL. Temperature was maintained with a constant temperature water bath at 25 ( 3 °C. The following protocol was used to obtain measurements from the rheometer. Various amounts of the concentrated suspension, surfactant (diluted in water if needed), and any additional water needed to achieve the desired volume fraction were mixed in a 20 mL scintillation vial. After stirring the solution in the vial, an aqueous sodium chloride solution at a known ionic strength was added and the mixture stirred. If the sample remained fluid-like, 3 mL was transferred to the rheometer with a pipet. If the sample gelled, approximately 3 mL had to be scooped out of the vial with a small metal spatula and smeared onto the inside wall of the rheometer cup. Samples were then presheared at a rate of 500 s-1 for 30 s. Moduli were measured at a frequency of 1 Hz as a function of time to determine recovery time. Strain amplitudes of 10-3 or smaller were maintained until a modulus plateau was reached. These terminal moduli are reported here. Viscosities and shear rates were measured during stress sweep experiments to determine zero shear viscosities and yield stresses. A solvent trap was used in conjunction with the cup and bob geometry to minimize evaporation. Gel points were determined with rheological data. We define the gel point by plotting the volume fraction on the horizontal axis with double vertical axes. The left vertical axis shows the zero shear viscosity and the right shows the elastic modulus. In these plots there is a zone of a few percent where, as we increase volume fraction, the viscosity rises rapidly to immeasurably high levels and the elasticity goes from zero to some higher value where it then becomes a power law in volume fraction outside of the region. The gel point is defined as the approximate midpoint of this zone, where an elastic modulus is first measurable. An example plot is shown in Figure 3.
3. Results The gel volume fraction was determined as the surfactant concentration is increased. The results of these experiments are shown in Figure 4. At a concentration of C12E6 ) 10-2 M, which was only tested for the 625 nm particles, gelation could not be achieved at 0.3 M ionic strength. Only at an ionic strength of 1.0 M was gelation observed at volume fractions of 0.5 or less. In determining the gel boundary at low surfactant concentrations experimentally, the particles were first exposed to 10-4 M C12E6 and then dialyzed exhaustively against a 10-4 M NaCl solution containing no C12E6. Thus the “zero” surfactant concentration (i.e., Deff kT r/Deff
Figure 9. van der Waals interaction with electrostatics for 508 nm particles at 0.1, 0.2, and 0.3 M ionic strength. Electrostatics weaken the energy minimum to 3-5kT by pushing distance of separation to 10-15 nm.
}
Figure 10. Yukawa fits to van der Waals interactions using parameters found in Table 2. Changes in the interaction energy due to changes in electrostatics or surfactant concentration can now be mimicked by adjusting the truncation point of the Yukawa calculations so that nMCT predictions of the elastic modulus can be performed.
ε/kT at twice the surfactant length in both the van der Waals calculation and the Yukawa fit, we adjust λDeff to match the slopes at the separation distance of 2δ. Using the same van der Waals calculations from Figure 8, we see the accuracy of these fits in Figure 10 and the Yukawa parameters in Table 2.
(3)
In this expression, ε/kT represents the strength of attraction at contact (i.e, when r ) Deff), and Deff is the effective diameter of the particle where we expect Deff to lie close to D + 2δ. The extent of the attraction is characterized by λ such that λDeff is the dimensionless inverse range of the attractive well. By fixing
Table 2. Parameters for Yukawa Fit of van der Waals Attraction (at r ) 2δ) particle size (nm)
ε/kT (at 2δ)
λDeff
508 625
-7.0 -8.9
80 80
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given range and strength of attraction.4 The calculations, based on λDeff ) 80, were reduced to expressions correlating calculated values such that
Another possible method for fitting the van der Waals interaction to a Yukawa expression is to equate the second virial coefficient for each interaction energy using the following equation.
B2 ) -2π
∫0
∞
-U⁄kT
(e
- 1)r dr 2
ε kT
[ ]
(4)
)
gel
ln(15.22φ-0.555) 0.8
(5)
Experimental gel volume fractions from Figure 4 are then compared with the curve in Figure 11 to extract strengths of attraction, and these are given in Table 3. As the surfactant concentration is increased, the strength of attraction decreases. Large changes in gel volume fraction are associated with small changes in effective well depth. While the volume fraction at gelation varies from 0.1 to 0.35, the predicted strength of attraction varies by only 1kT. In keeping with our model, this small change in strength of attraction corresponds to small changes in the location for the minimum in the potential energy. As expected, this location corresponds to a combined surfactant and electrostatic stabilizing layer thickness of 10-15 nm. These results suggest that even at sub-monolayer coverage, the surfactant stands up on the surface such that the minimum distance of separation remains approximately two surfactant chain thicknesses. The parameters required to predict the gel line for the van der Waals gels seen here are well inline with those expected on the basis of the physical chemistry of the polystyrene latex, C12E6, water system. Given a range and strength of attraction the nMCT correlations developed for Yukawa systems can be used predict the volume fraction dependence of the gel modulus. The elasticity of nMCT gels is determined by the localization length, rloc, which is associated with the range of motion of particles localized in a narrow well by the action of neighboring particles. The modulus can be written as G′D3/kT ∼ 0.58 < φ(D/rloc)2. At the gel point, rloc/D ∼ 0.8/(λD) and then decreases as φ increases above φgel. Previous studies have verified the predictions of nMCT for depletion and thermal gel systems.4,13 In studies of depletion gels, the volume fraction and polymer radius of gyration dependence of the modulus were captured by the model. However, the absolute magnitude of the experimental moduli were smaller than the predicted values by a constant factor of ∼100. This was attributed to the cluster microstructure observed in these gels.13 The magnitude and volume fraction dependencies for the thermal gels as well as the absolute magnitude were well described by the nMCT predictions.4 Here we can apply the correlations that were developed for rloc and G′D3/kT by detailed solutions of nMCT developed for attractive Yukawa particles to the van der Waals gels using the well depths shown in Table 2. We find that these correlations underpredict the moduli by several orders of magnitude (Figure 7). Thus, while the models of the interaction energy and application of nMCT agree in suggesting that the surfactantstabilized systems studied here gel due to the particles experi-
where B2 is the second virial coefficient and U/kT is the expression for either the van der Waals interaction from eq 1 or the Yukawa interaction from eq 3. ε/kT would again be fixed at a distance of 2δ, and λDeff would be adjusted until B2VDW ) B2Y. As an example, for the 508 nm particles, this method yields a B2 of -4.4 and a λDeff of 46. This results in a reduction of ε/kTgel of 0.3kT and a reduction in the predicted elastic modulus by a factor of ∼3 compared to the method above for obtaining λDeff. These quantitative differences may be important when undertaking detailed studies of nMCT predictions. Here we are instead interested in understanding why G′ in the gel is independent of φgel. With this purpose in mind, we focus attention on the ability of surfactant to be displaced between the gap as opposed to understanding the subtleties of connecting gel points using different interaction energies. For this reason, we continue to use the “slope” method to estimate the range of the effective Yukawa interaction potential. As we are working at high ionic strengths and we have not fully characterized the structure of the surfactant layer, we use the Yukawa attraction as a means of characterizing our system. In the following, we fix λDeff ) 80 and then fit experimental parameters by adjusting ε/kT. This allows us to mimic changes in the interaction energy due to ionic strength or surfactant concentration. Using detailed expressions for the equilibrium structure of weakly attractive Yukawa particles, nMCT has been applied to predict the volume fractions where gelation will occur for a
Figure 11. Calculated strength of attraction at the gel point for particles interacting in a Yukawa potential with κD ) 80 using nMCT predictions. Strengths of attraction between 5 and 4 kT should result in gel points between 0.1 and 0.4. Small changes in the interaction energy result in large changes in the gel points.
Table 3. Strength of Attraction at Gel Point Using NMCT Calculations 508 nm, 0.1 M
625 nm, 0.3 M, 1.0 M
[C12E6] (M)
φgel
stabilizing layer thickness (nm)
-U/kTmin
φgel
stabilizing layer thickness (nm)
-U/kTmin
