Langmuir 2000, 16, 10509-10514
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Prediction of the Interaction Potential of Microgel Particles from Rheometric Data. Comparison with Different Models Claudio L. A. Berli and Daniel Quemada* Laboratoire de Biorheologie et d’Hydrodynamique Physico-chimique (LBHP), Case 7056, Universite´ Paris VII, 2 Place Jussieu, 75251 Paris Cedex 05, France Received June 19, 2000. In Final Form: September 20, 2000 This work is concerned with the determination of the effective particle-particle interaction in colloidal suspensions from rheology. In particular, a concentrated microgel suspension is considered, in which particles interact through a repulsive polymer-polymer potential. The rheological modeling of this suspension was carried out by the authors in a recent work. Here, the interaction potential provided by that model is compared with the ones resulting from different models found in the literature. In this sense, a series of approaches relating the pair potential to rheological data such as dynamic modulus, viscosity, and yield stress is examined. Then the results of applying these models to experimental data are discussed, showing the advantages and limitations of each approach for the particular case of concentrated microgel suspensions. de Gennes’ theoretical pair potential for polymer-covered particles is also included for comparison. The proposed viscosity model appears rather general and it is expected to have success in determining the effective potential in colloidal systems with different particle-particle force laws.
Introduction The search for a direct connection between macroscopic rheology and microstructure constitutes an attractive problem currently discussed in the literature concerning colloidal suspensions.1-6 In fact, the possibility to relate the flow behavior of suspensions to the interaction between particles is a subject of interest for both the basic scientific problem and the useful industrial applications. In principle, a model which involves a correct description of the structure should allow one to predict the rheological functions starting from the knowledge of the potential energy between particles. The inverse problem is equally interesting: obtaining the interaction potential from a flow curve (viscosity) or a dynamic response (elastic modulus) and then using it to predict how particle size, concentration, or ionic strength influence the rheological functions. In this theoretical context, here we deal with a suspension of microgels, which are particles composed of a central zone of cross-linked polymer and an external layer of polymer chains.7-9 The suspension is thermodynamically stable since the thickness of the polymer layer is high enough to ensure that the van der Waals attraction potential is negligible in comparison with the Brownian thermal energy.1,10 Furthermore, the overlap of polymer * Corresponding author. E-mail:
[email protected]. (1) Russel, W. B.; Saville, D. A.; Schowalter, W. R. Colloidal Dispersions, 2nd ed.; Cambridge University Press: Cambridge, 1991. (2) Buscall, R. J. Chem. Soc., Faraday Trans. 1991, 87, 1365. (3) Buscall, R. Langmuir 1992, 8, 2077. (4) van der Vorst, B.; van den Ende, D.; Mellema, J. J. Rheology 1995, 39, 1183. (5) Ogawa, A.; Yamada, H.; Matsuda, S.; Okajima, K.; Doi, M. J. Rheology 1997, 4, 769. (6) Larson, R. G. The Structure and Rheology of Complex Fluids; Oxford University Press: New York, 1999. (7) Wolfe, M. S.; Scopazzi, C. J. Colloid Interface Sci. 1989, 133, 265. (8) Rodriguez, E.; Wolfe, M. S.; Fryd, M. Macromolecules 1994, 27, 6642. (9) Fridrikh, S.; Raquois, C.; Tassin, J. F.; Rezaiguia, S. J. Chim. Phys. 1996, 93, 941. (10) Israelachvili, J. Intermolecular and Surface Forces, 3rd ed.; Academic Press: London, 1997.
layers leads to a repulsive force between particles. Thus, important rheological changes arise in the suspension when the particle concentration is increased, namely, a liquid-solidlike transition. A rheological modeling for this suspension was carried out in a recent work.11 This modeling gave the possibility to determine an effective potential of interaction between particles directly from the rheometric data. In the present work, we compare the interaction energy provided by that model with the ones resulting from different models known in the literature, to (a) discuss the advantages of each approach for the particular case of microgel suspensions and (b) confirm and validate the prediction of our viscosity model. Fundamental calculations in suspension rheology, for instance, nonequilibrium statistical dynamics,12,13 clearly involve the pair potential. Nevertheless, these methodologies usually demand rather elaborate computations. In the present work, we are considering a group of models useful for practical purposes. In the following we present the main characteristics of the suspension considered and the particle-particle interaction involved. Then we list a series of models which relate the pair potential in colloidal suspensions to rheological data such as dynamic modulus, viscosity, and yield stress. Finally, the results obtained in applying these models to experimental data of microgel suspensions are discussed. Characteristics of the Microgel Suspension. The suspension consists of monodisperse spherical particles of cross-linked polymers (copolymer of styrene and acrylic monomers) dispersed in a good solvent (xylene). The physicochemical characterization of this colloidal suspension is well described elsewhere.9,14,15 Here it is important to mention that microgel particles present a radius a ) (11) Berli, C. L. A.; Quemada, D. Langmuir 2000, 16, 7968. (12) Russel, W. B.; Gast, A. P. J. Chem. Phys. 1986, 84, 1815. (13) Wagner, N. J.; Russel, W. B. Physica A 1989, 155, 475. (14) Raquois, C.; Tassin, J. F.; Rezaiguia, S.; Gindre, A. V. Prog. Org. Coatings 1995, 26, 239. (15) Raquois, C. Ph.D. Thesis, University of Maine, Le Mans, France, 1996.
10.1021/la0008481 CCC: $19.00 © 2000 American Chemical Society Published on Web 11/30/2000
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Figure 1. Theoretical interaction energy between polymercovered flat surfaces (eq 1) and schematic representation of a pair of microgel particles at different distances of approach. The dashed line, whose slope is -2π, represents the exponential decay described by Israelachvili.10
Figure 2. Dimensionless values of both high-frequency shear modulus G′∞ (open circles)15 and yield stress σY (solid circles)11 as a function of concentration φp for the suspension of microgels at 20 °C. The value of the ratio σY/G′∞ is close to C ) 0.03 (see text).
43.5 nm in latex phase and a hydrodynamic radius ah ) 117 nm when they are dispersed in the solvent. Throughout our analysis, these particles are assumed to have a simple core-shell configuration8,9 composed of a core of radius a and a polymer layer of thickness L ≈ ah - a at infinite dilution. The particle concentration equivalent to core volume fraction is designated φp. The critical concentration at which particles reach the random close packing is φ/p ) φRCP(a/ah)3 ) 3.3%, where φRCP ) 0.637. Beyond this critical concentration, polymer layers are progressively confined between the core surfaces as φp increases. The overlap of polymer layers reduces the volume available to each single chain and hence increases the free energy, producing a repulsive force between particles.1,10 This interaction has been studied by means of both experimental and theoretical methods.1,10,16-19 At the present, the main theory available to evaluate the pair interaction energy between polymer-covered flat surfaces is that of de Gennes,17 which takes into account the osmotic repulsion between polymer layers and the elastic energy of the chains. That is,
particle interaction in the microgel suspension is that given by the de Gennes theory. Figure 1 also shows schematically a pair of microgel particles at different distances of approach, which is equivalent to different concentrations in the suspension. The mean surface-to-surface distance D ) 2a[(φRCP/φp)1/3 - 1] becomes lower than 2L when φp > φ/p. This fact strongly affects the flow behavior of the suspension. At the lowest concentrations, rheology is governed mainly by Brownian motion and the suspension presents a liquidlike behavior. At concentrations above φ/p, rheology is governed by interparticle forces and the suspension presents a solidlike behavior involving elasticity and yield stress. In fact, Figure 2 shows dimensionless values of both the high-frequency shear modulus G′∞ and the apparent yield stress σY as function of concentration of microgel particles. The ratio σY/G′∞, corresponding to each concentration φp, presents slightly scattered values around the mean C ) 0.03, which is in perfect agreement with results reported in the literature for different colloidal systems (0.02 e σY/G′∞ e 0.04).20-22 Actually, Figure 2 constitutes a stress-concentration phase diagram in which the curve of rheometric data defines the transition between the zone of liquidlike behavior (left) and the zone of solidlike behavior (right). Theoretical Relations between the Interaction Energy and Rheological Functions. 1. Dynamic Rheometry. Applying small amplitude oscillatory shear perturbs slightly the microstructure and hence the measured shear modulus reflects both the interparticle energy and the structure. In particular, the high-frequency limit G′∞ is rather accessible to theoretical treatments and it has been the object of intensive research from several authors. Calculations of G′∞ usually lead to integrodifferential equations23 involving the interparticle energy U(R), the pair distribution function g(R), and hydrodynamic functions in the more rigorous approaches.24,25 When strongly repulsive forces are present, particles order in a
E(D) )
{( )
8LkBT 1 D 5 2L s3
-5/4
+
1 D 7 2L
7/4
( )
-
}
12 35
(1)
where D is the surface-to-surface distance, kB is the Boltzman constant, T is the absolute temperature, and s is the mean distance between the chain attachment points at the surface. The profile of the interaction predicted by eq 1 is presented in Figure 1. It is readily observed that, as described by Israelachvili,10 the repulsive interaction is roughly exponential (dashed line) except for the limits corresponding to very low and very high polymer layers overlapping. This feature is very useful from a practical point of view (see last section). Finally, since there is not electric double layer interaction and van der Waals forces are negligible, it may be assumed that the main particle(16) Markovic, I.; Ottewill, R. H.; Underwood, S. M.; Tadros, Th. F. Langmuir 1986, 2, 625. (17) de Gennes, P.-G. Adv. Colloid Interface Sci. 1987, 27, 189. (18) Milner, S. T.; Witten, T. A.; Cates, M. E. Macromolecules 1988, 21, 2610. (19) Costello, B. A.; Luckham, P. F.; Tadros, Th. F. Langmuir 1992, 8, 464.
(20) Buscall, R.; Goodwin, J. W.; Hawkins, M. W.; Ottewill, R. H. J. Chem. Soc., Faraday Trans. 1 1982, 78, 2873. (21) Chen, L.-B.; Zukoski C. F. J. Chem. Soc., Faraday Trans. 1990, 86, 2629. (22) Fagan, M. E.; Zukoski C. F. J. Rheology 1997, 41, 373. (23) Zwanzig, R.; Mountain, R. D. J. Chem. Phys. 1965, 43, 4464. (24) Wagner, N. J. J. Colloid Interface Sci. 1993, 161, 169. (25) Elliot, L.; Russel, W. B. J. Rheology 1998, 42, 361.
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crystalline array and the distribution function may be approximated by a δ function. This is a common starting point in the literature to obtain useful expressions allowing correlation of G′∞ data with the interparticle energy U(R).2,19,24,26,27 Here we use three different approximations of the expression of Zwanzig and Mountain23 proposed in the literature. 1.1. Equation derived by Evans and Lips.26
G′∞ ) nkBT +
(
)
Nφm 4 ∂U(R) ∂2U(R) + 5πR R ∂R ∂R2
(2)
where n ) 3φp/4πa3 is the particle density number, N is the number of nearest neighbors, φm is the maximum packing fraction, and R is the center-to-center particle distance. Equation 2 was initially proposed for microgel dispersions and then it was used for suspensions of polymerically stabilized particles at high concentrations.19 1.2. Equation derived by Buscall.2
Nφm ∂2U(R) G′∞ ) 5πR ∂R2
(3)
In particular, this equation is well-known in the literature since it is equivalent, except for numerical constants, to expressions derived previously by considering the restoring force of a crystalline array after a small shear deformation.28,29 1.3. Equation Derived by Wagner.24
G′∞ )
(
)
Nφm 4 ∂U(R) ∂2U(R) + 5πR R ∂R ∂R2
(4)
This expression was also obtained by van der Vorst et al.4 from the stress tensor due to electrostatic forces between particles in a crystalline array. 2. Steady Shear Flow Rheometry. In steady shear flow the structure is perturbed significantly more than in oscillatory shear. Hence, theoretical calculations of viscosity have been limited to low particle concentration and small perturbations from equilibrium.1 Concentrated systems at finite shear stress are usually treated in a more phenomenological context. Here we consider the following approaches relating suspension viscosity to interparticle forces. 2.1. Models Based on the Effective Particle Radius Derived from the Balance of Colloidal Forces. In stabilized suspensions, repulsive forces keep particles apart from one another. Thus, it is usually thought that electrostatic as well as polymer-polymer interactions increase the effective radius of particles.1,6,30 In particular, if the suspension is at rest, it may be assumed that there is an effective particle radius aeff satisfying U(R ) 2aeff) ≈ kBT. A detailed derivation of the electroviscous effects was made by Russel,31,32 by considering the effective radius at which the balance between electrostatic and hydrodynamic forces is attained. Based on these ideas, Buscall2,3,33 engaged in (26) Evans, D.; Lips, A. J. Chem. Soc., Faraday Trans. 1990, 86, 3413. (27) Mewis, J.; D’Haene, P. Macromol. Chem. Symp. 1993, 68, 213. (28) Buscall, R.; Goodwin, J. W.; Hawkins, M. W.; Ottewill, R. H. J. Chem. Soc., Faraday Trans. 1 1982, 78, 2889. (29) Goodwin, J. W.; Gregory, T.; Miles, J. A.; Warren, B. C. H. J. Colloid Interface Sci. 1984, 97, 488. (30) Goodwin, J. W.; Ottewill, R. H. J. Chem. Soc., Faraday Trans. 1991, 87, 357. (31) Russel, W. B. J. Fluid Mech. 1978, 85, 209. (32) Russel, W. B. J. Rheol. 1980, 24, 287. (33) Buscall, R. Colloids Surf., A 1994, 83, 33.
the problem of concentrated suspensions under finite shear rates. The author suggested that, when the viscous force acting on particles is considered, the force balance leads to the following expression:
(
U(R ) 2aeff) σa3eff ) k1 +1 kBT kBTK(φeff)
)
(5)
where σ is the shear stress and φeff is the effective volume fraction defined by
φeff ) φp(aeff/a)3
(6)
Also in eq 5, k1 ≈ 1/2 and K(φeff) ) 0.016 + 0.52φeff are phenomenological factors, the values of which correspond to hard sphere suspensions. In this model, the viscosity η relative to the fluid viscosity ηF is given by the equation of Krieger and Dougherty,
(
)
φeff η ) 1ηF φm
-(5/2)φm
(7)
Applying eqs 6 and 7 to data η(σ) gives a definition of aeff(σ,φp). Introducing then aeff(σ,φp) into eq 5 allows one to obtain an apparent interaction potential from any given viscosity curve. That is so because, for a fixed particle concentration, the potential U(R ) 2aeff) depends on shear stress σ by means of the effective radius aeff. In this sense, a test for the model is that potential values obtained from different volume fractions should lie on the same potential curve.3,33 It must be also mentioned that eq 5 is adequate for σa3eff/kBT . 1. In the low shear stress limit, the relationship between aeff and U(R) is better stated from the equation of Barker and Henderson.34 Indeed, Buscall2 included the viscous energy in this integral equation and then considered eq 5 to be the approximate solution for R ) 2aeff. 2.2. Models Derived from the Classical Theory of the Activation Process. The theory of rate processes35 has been recently used to derive theoretical relations between the suspension viscosity and the interparticle potential energy. In this theory, particles are assumed to move from their equilibrium positions to a vacant site in an ordered array, through a self-diffusion mechanism. In this context, Baxter-Drayton and Brady36 proposed a model for the viscosity of aggregated suspensions, by considering a shear stress induced variation of the activation energy barrier and hence the diffusion time of particles. The model consists of the equation of Ree and Eyring with a modified relaxation time containing the interaction energy. That is,
a2 exp (U/kBT) η - η∞ sinh-1(βγ˘ ) ; β ) A2 (8) ) η0 - η∞ βγ˘ DcU/kBT where η0 and η∞ represent the limiting viscosity values for the shear rate limits γ˘ f 0 and γ˘ f ∞, respectively. In the characteristic time β, Dc is the diffusion coefficient and A2 is a proportionality constant that depends on the network structure. In fact, the model concerns aggregated systems with an attractive potential. Nevertheless, it is interesting to observe that the interparticle energy arises through the critical shear rate γ˘ C ) β-1 (see next eq 11). (34) Barker, J. A.; Henderson, D. J. Chem. Phys. 1967, 47, 4714. (35) Glasstone, S.; Laidler, K. J.; Eyring, H. The Theory of the Rate Processes; McGraw-Hill: New York, 1941. (36) Baxter-Drayton, Y.; Brady, J. F. J. Rheol. 1996, 40, 1027.
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In a similar theoretical framework, Ogawa et al.5 developed a viscosity model for well-stabilized suspensions. By contrast, this approach considers the shear stress in colloidal suspensions as composed of two terms, the viscous stress σv and the particle stress σp. Consequently, the viscosity is written as the sum of two components,
(
)
c1d3σp ηv η U ) + c2φ exp ηF ηF kBT φkBT
(9)
where the viscous term ηv/ηF is given by eq 7 with φm ) 0.71, in agreement with the limit η(σ f ∞). In the interaction term, c1 (theoretically π/6) and c2 are numerical constants, d is the particle diameter, and φ is the volume fraction. It is appropriate to mention that this kind of modeling was carried out first by Goodwin et al.37 to relate the low shear limiting viscosity to the electrostatic potential of concentrated latices. 2.3. Model Proposed for Microgel Suspensions. Here we present briefly the viscosity model proposed for microgel suspensions, the detailed discussion of which is given in a previous work.11 In this model, it is assumed that particles have an equivalent hard sphere radius aHS which involves the core radius plus the hydrodynamic thickness of the polymer layer. Thus, aHS permits one to quantify the true volume faction φ ) φp(aHS/a)3 occupied by the particles. Variations in volume fraction, hence in viscosity, due to shear stress are described through a rate equation which accounts for the forces perturbing the microstructure and those restoring the equilibrium state. In this sense, it is assumed that the relaxation of particles after a shear deformation depends on interparticle repulsive forces (in addition to Brownian motion) which drive particles away one from another. The final form of the model involves the following explicit relationship between the suspension viscosity and shear stress,
(
)
1 + σ/σc χ + σ/σc
η(σ) ) η∞
2
(10)
which includes the interparticle potential through the critical shear stress,
σc )
(
kBT a3HS
1+
)
U(R) kBT
(11)
with R ) 2a(φRCP/φp)1/3. When repulsive forces among particles are negligible, eq 11 gives σc ) kBT/a3HS, which is the well-known scaling for colloidal suspensions of noninteracting particles.1,38 In eq 10, χ ) (1 - φ/φ0)/(1 φ/φ∞), where φ0 and φ∞ are the maximum packing fractions for the shear stress limits σ f 0 and σ f ∞, respectively. 3. Yield Stress Measurements. The apparent yield stress σY appearing in concentrated colloidal suspensions is also related to the interparticle potential. In fact, the following expressions were suggested in the literature. 3.1. From the Viscosity Model of Buscall.2 In this model the yield stress arises as the effective particle radius is high enough to produce a dense packing of particles. Therefore, the author suggests (37) Goodwin, J. W.; Gregory, T.; Stile, J. A. Adv. Colloid Interface Sci. 1982, 17, 185. (38) Krieger, I. M. Adv. Colloid Interface Sci. 1972, 3, 111.
σY ≈ K(φeff)
(
)
U(R ) 2am) - kBT a3m
(12)
where am is the maximum radius that particles can take because of the spatial constraints of the other particles. It must be noted here that, in comparison with eq 5, eq 12 misses the constant k1 (see also ref 6). 3.2. From the Viscosity Model of Ogawa et al.5 In this work it is indicated that, when the effect of repulsive interaction is strong, there is an apparent yield stress given by,
σY )
φU(R)
(13)
c1d3
Although we have not found how to deduce this relation from eq 9, eq 13 is equivalent to the expression of σY derived by Chen and Zukoski.21 In fact, by using the mean field potential proposed by these authors, eq 13 is easily obtained with c1 ) 1/3 instead of π/6. 3.3. From the Proposed Viscosity Model.11 In eq 10, the viscosity η(σ) diverges when the ratio χ f 0. Hence, negative values of χ are associated to the onset of the plastic behavior with a yield stress σY ) -χσc. Thus, the following expression can be written:
σY )
(
kBT a3HS
1+
)(
)
U φ/φ0 - 1 kBT 1 - φ/φ∞
(14)
which is only valid if φ > φ0 ≈ φRCP, i.e., when particles are densely packed and the suspension cannot flow. Results and Discussion 1. Pair Potential from the High-Frequency Shear Modulus. Here we use eqs 2-4 to determine the potential U(R) from the experimental values of G′∞ versus φp reported in Figure 2. In these equations, the maximum packing fraction φm corresponding to a face-centered cubic array will be considered, i.e., φFCC ) 0.74, and hence N ) 12. Also the interparticle distance is R ) 2a(φFCC/φp)1/3. Equations 2-4 involve the first and second derivatives of the interaction potential. Therefore, an analytical expression of U(R) is required beforehand in order to simplify the calculations. In this sense we introduced, as in ref 11,
U(R)/kBT ) A exp[-B(R - 2a)/2L]
(15)
according to the character of the interaction mentioned above (Figure 1). In addition, this function leads to expressions analytically simple for ∂U/∂R and ∂2U/∂R2, to be included in eqs 2-4. Thus, one may extract the interaction energy by fitting the data of G′∞ as function of R, where the unknown parameters are the constants A and B from eq 15. Results are shown in Figure 3, where the dimensionless potential U/kBT is plotted as a function of the relative distance between core surfaces D/2L, by using the geometrical relation D ) R - 2a. Clearly, after including the exponential form, the resulting potential curves are straight lines in the semilogarithmic plot. 2. Pair Potential from Viscosity Curves. The main feature of Buscall’s model is that a curve of potential U(R) can be obtained from a single curve η(σ). However, in the case of microgel suspensions, different flow curves,11,15 corresponding to different concentrations φp, do not lead to the same potential; that is, curves U(R) do not superimpose as should be expected (see Appendix, part 1). Equation 8 fits the viscosity data well in the concen-
Interaction Potential of Microgel Particles
Figure 3. Dimensionless interaction potential, as function of the relative surface-to-surface distance between particles, obtained from G′∞ data of the microgel suspension.
Figure 4. Dimensionless interaction potential, as function of the relative surface-to-surface distance between particles, obtained from σY data of the microgel suspension.
tration range of liquidlike behavior only (the model strictly predicts S-shaped viscosity curves). Even in this case, the predicted values of U(R) strongly depend on the unknown constant A2 (one must also take into account that the model was originally written for aggregated systems). Similar difficulties were found in applying the model of Ogawa et al. (eq 9) to suspension viscosity data reported in refs 11 and 15. In fact, this model is not able to predict viscosity curves showing a divergence at low shear stress, i.e., a plastic behavior. In addition, the predicted potential U(R) depends on the values taken for c2, which is equal to 1 according to the authors. Finally, one may conclude that the determination of the pair potential U(R) between microgel particles from the viscosity models listed in the previous section cannot be achieved directly, with the exception of eq 10.11 The main reason is that these models do not hold at very high concentrations, where suspensions present a plastic behavior. In addition, the equations η(σ) involve empirical constants which are not known a priori for microgel suspensions. 3. Pair Potential from the Yield Stress. Figure 4 shows the potential U/kBT predicted by eqs 12-14 from data of σY reported in Figure 2. In Buscall’s model (eq 12) the constant K ) 0.4 is included, corresponding to the maximum packing condition.33 Also for the microgel suspension the maximum radius of particles is am ) a(φm/ φp)1/3, with the maximum packing φm ) φFCC suggested by the author. In eq 13, c1 ) π/6, φ/d3 ) φp/(2a)3, R ) 2a(φm/
Langmuir, Vol. 16, No. 26, 2000 10513
Figure 5. Dimensionless interaction potential, as function of the relative surface-to-surface distance between particles, obtained from different rheometric data of the microgel suspension. Solid circles are the potential determined previously from viscosity curves.11 The full line is the theoretical prediction (eq A2 with s/a ) 0.358).
φp)1/3, and φm ) 0.71 were introduced. Calculations with eq 14 include the volume fractions reported previously.11 It is interesting to observe in Figure 4 that, although the models used come from different derivations, they predict almost the same potential curve. From a mathematical point of view, this is evident for U/kBT . 1, where eqs 12-14 differentiate each other by proportionality constants only. 4. Final Comparison. Figure 5 presents a comparison between the potential determined previously from the viscosity curves, by using eqs 10 and 11,11 and the potentials obtained here from different models and different experimental data of the microgel suspension (high-frequency shear modulus and yield stress). Since all curves in Figure 5 are rather coincident, one may say that the viscosity model in ref 11 predicts the correct form of the potential energy in microgel dispersions (it should be noted that a linear scale is used in Figure 5). The theoretical pair interaction of polymer-covered particles is also included here, which was derived from eq 1 by using the Derjaguin approximation (see Appendix, part 2). In fact, the full line in Figure 5 represents eq A2 with s/a ) 0.358. This value was calculated by fitting eq A2 to the data U/kBT determined from viscosity (solid circles). It should be noted that the resulting value of s/a lies into the range of values inferred in the Appendix, part 2. Concluding Remarks Two aspects are important to note here. First, the interaction potential determined from the viscosity model presents a remarkable agreement with (a) the potential determined from the high-frequency shear modulus, which is a rather accurate calculation, and (b) the theoretical pair potential for polymer-covered particles. Second, for the particular case of the microgel suspension considered here, this viscosity model improves on previous models found in the literature, from which the calculation of the interaction potential from viscosity data cannot be achieved directly. In addition, the proposed viscosity model appears rather robust (namely, it holds at high particle concentrations and describes the shear induced solid-liquid transition).11 Furthermore, as the specific characteristics of microgel particles do not strongly enter this model, one can expect it to succeed in determining the effective potential of colloidal systems with different particleparticle force laws.
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superimpose and coincide with the dotted line in Figure 6 as long as a decreasing function K(φeff) is included in eq 5 for each concentration. 2. Theoretical Potential between Particles. Equation 1 gives the interaction energy per unit area, E(D), between polymer-covered flat surfaces in a good solvent. Here we estimate the total interaction potential U(D) between spherical particles of radius a, by using the Derjaguin approximation,10
U(D) ) πa
Figure 6. Dimensionless interaction potential as function of the relative center-to-center distance between particles. Symbols are values obtained from viscosity curves of the microgel suspension11,15 by using eqs 5-7. The dotted line is the potential predicted from the high-frequency shear modulus by using eq 3.
(39) de Kruif, C. G.; van Iersel, E. M. F.; Vrij, A.; Russel, W. B. J. Chem. Phys. 1985, 83, 4717.
(A1)
For this approximation to be valid, all the particle-particle interactions should come from the region around the point of closest approach.40 The interaction law of de Gennes (Figure 1) decays rapidly with distance D and, in particular, it predicts E(D g 2L) ≡ 0. Thus, introducing 2L as the upper limit of the integral in eq A1, the following expression results,
U(D) ) 64πaL2kBT 1 D 5 2L s3
{( )
Appendix 1. Interaction Potential from Buscall’s Model. Figure 6 presents the potential U(R) obtained by applying eqs 5-7 to viscosity data11,15 of the microgel suspension considered here. It is readily observed that the potential curves, coming from flow curves η(σ) at different concentrations φp, do not superimpose. This limitation was also found by Buscall in a previous work,33 where the author argues that the origin of this problem is the variation of the particle radius with concentration. For comparison, the dotted line in Figure 6 is the potential obtained from G′∞ by using the equation also proposed by Buscall (eq 3) and already shown in Figure 3. Within the mathematical context of the viscosity model, a possible way to obtain superimposed potential curves would be by considering an appropriate empirical function K(φeff) for soft spheres. In fact, the function K(φeff) ) 0.016 + 0.52φeff was obtained by Buscall from data of the critical shear stress σ/c of hard spheres reported by de Kruif et al.,39 assuming that K(φeff) is equivalent to σ/c ) kBT/a3eff in eq 5. In the case of this microgel suspension, we found that the potential curves
∫D∞E(D′) dD′
-1/4
-
1 D 77 2L
( )
11/4
+
}
3 D 3 35 2L 11 (A2)
( )
Therefore, as a first approximation, here we take eq A2 to represent the pair interaction potential between microgel particles. In this equation, the only unknown is the value of the mean distance s between attachment points at the core surface. Nevertheless, by using geometrical scaling ratios, one may estimate an order of magnitude of the mean distance l between the cross-link points in the microgel particles. In fact, assuming that all the crosslinks are inside the core, and that each cross-link point involves a spherical volume of radius l/2, one may write M/Mc ∼ [a/(l/2)]3, where M ) 20.8 × 107 is the molecular weight of particles and Mc ) 17.3 × 104 is the mean molecular weight between cross-link points.15 Thus one has l/a ∼ 0.2. On the other extreme, if the cross-links are uniformly distributed in the whole particle volume, one may write M/Mc ∼ [(a + L)/(l/2)]3, which leads to l/a ∼ 0.5. Finally, assuming s ≈ l, the relative mean distance s/a may be expected to be in the range 0.2-0.5. Acknowledgment. C. L. A. B. acknowledges receipt of a postdoctoral fellowship from CONICET (Argentina) and educational leave of absence from UNL (Argentina). LA0008481 (40) White, L. R. J. Colloid Interface Sci. 1983, 95, 286.
