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Rheological Modeling of Microgel Suspensions Involving Solid-Liquid Transition 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 March 13, 2000. In Final Form: May 23, 2000 The rheological behavior of suspensions containing cross-linked polymer particles (microgels) was studied. Under flow, these particles behave as soft spheres due to the shear-induced deformability of their external polymeric layer and the suspensions are strongly shear-thinning. At concentrations where particles are densely packed, the suspensions present a solidlike behavior involving elasticity and yield stress. The rheological modeling of such suspensions was based on the concept of the effective volume fraction of particles, which depends on hydrodynamic forces. In this sense, microgel particles were assumed to have a simple core-shell configuration and to interact through a repulsive polymer-polymer potential. The viscosity equation resulting from the modeling allows: (a) appropriate correlation of the experimental data shear stress versus shear rate in steady shear flow and (b) accounting for the softness of particles, which changes with concentration and shear stress. Also the predictions of the model showed to be quite consistent with results obtained previously for the same suspensions through different techniques such as dynamic rheometry and rheoptics.
Introduction Most of complex fluids used for industrial applications have rheological properties that show strong similarities in comparison with those of concentrated dispersions. These rheological properties are due to both the existence of an internal structure (microstructure) and the shearinduced changes in the microstructure. Suspensions of microgels are a good example of such systems. Microgels are polymer particles composed of a central zone of crosslinked polymer and an external layer of polymer chains.1-3 Under flow, these particles behave as soft spheres due to the deformability of the external polymeric layer. At concentrations where particles are densely packed, the suspensions present a solidlike behavior involving elasticity and yield stress. Indeed, microgel suspensions present several rheological features appearing in suspensions of polymerically stabilized particles.4-6 The aim of the present work is to predict the rheological behavior of concentrated microgel suspensions on the basis of appropriate scaling principles. In this sense we make use of the effective volume fraction of particles, which has been shown to be an essential variable for rheological modeling of colloidal dispersions.7-10 In particular, a description of the flow-induced variation of the effective volume fraction is carried out here, to account for the softness of microgel particles. Also the influence of interparticle forces on the microstucture dynamics is considered in the modeling. The resulting viscosity equa* Corresponding author. E-mail:
[email protected]. (1) Wolfe, M. S.; Scopazzi, C. J. Colloid Interface Sci. 1989, 133, 265. (2) Rodriguez, E.; Wolfe, M. S.; Fryd, M. Macromolecules 1994, 27, 6642. (3) Fridrikh, S.; Raquois, C.; Tassin, J. F.; Rezaiguia, S. J. Chim. Phys. 1996, 93, 941. (4) Choi, G. N.; Krieger, I. M. J. Colloid Interface Sci. 1986, 113, 101. (5) Mewis, J.; Frith, W. J.; Strivens, T. A.; Russel, W. B. AIChE J. 1989, 35, 415. (6) Buscall, R. Colloid Surfaces A 1994, 83, 33. (7) Krieger, I. M. Adv. Colloid Interface Sci. 1972, 3, 111. (8) Russel, W. B.; Saville, D. A.; Schowalter, W. R. Colloidal Dispersions, 2nd ed.; Cambridge University Press: Cambridge, 1991. (9) Buscall, R. J. Chem. Soc., Faraday Trans. 1991, 87, 1365. (10) Quemada, D. Eur. Phys. J. AP 1998, 1, 119.
tion correlates appropriately experimental data in shear flow. Following we present the main characteristics of the microgel suspensions considered. Then we discuss some theoretical aspects and the viscosity model proposed. Finally, the results of modeling are discussed showing the goodness of the viscosity equation to predict the liquidsolid transition in microgel suspensions. Description of Experimental Data Sample Characterization. The suspensions here studied consist of monodisperse spherical particles of crosslinked polymers (copolymer of styrene and acrylic monomers) dispersed in a good solvent (xylene). These polymeric particles (microgels) were obtained by Tassin and coworkers11 using an emulsion polymerization process. The particle characterization was made by the same authors and it is well described elsewhere.3,11,12 In particular, for the suspension considered in this work, the main characteristics of particles are presented in Table 1.12 When dispersed in a good solvent, the microgel particles swell, increasing several times their volume. This is readily inferred from data in Table 1, by comparing the particle hydrodynamic radius (ah) to the particle radius in the latex phase (a). In fact, these kinds of particles are usually thought as composed of a hard core (central zone of crosslinked polymer chains) and an external layer (polymer chains grafted to the core surface).1-3 It may be also assumed that the swelling of the inner part is negligible in front of that of the external layer.3 Thus, as a first approximation, we shall consider that the particles have a core radius a and hence a polymer layer thickness at infinite dilution L ≈ ah - a (see Figure 1). Also throughout this work, the volume fraction corresponding to nonswelled polymer particles (cores) will be designated φp. The mean surface-to-surface distance between cores (Figure 1) is related to the core volume fraction by (11) Raquois, C.; Tassin, J. F.; Rezaiguia, S.; Gindre, A. V. Prog. Org. Coat. 1995, 26, 239. (12) Raquois, C. Ph.D. Thesis, University of Maine, Le Mans, France, 1996.
10.1021/la000365x CCC: $19.00 © 2000 American Chemical Society Published on Web 09/23/2000
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Table 1. Physicochemical Characteristics of the Cross-linked Polymer Particles (Microgels)12 name
nomenclature
value
molecular weight particle radius in latex phase hydrodynamic radius intrinsic viscosity
M a ah [η]
20.8 × 107 43.5 nm 117 nm 0.48 dl/g
Figure 1. Schematical representation of the cross-linked polymer particles (microgels).
D ) 2a(φm/φp)1/3 - 2a
(1)
where φm is the maximum packing fraction. Values of D were calculated for a dispersion of spherical particles in random close packing, i.e., putting φm ) φRCP ) 0.637 in eq 1. Since D decreases as φp increases, it exists a critical particle concentration φ*p above which the polymer layers overlap, i.e., D/2L < 1. The value φ*p may be estimated as the concentration at which particles of radius ah are packed; that is, φ*p ) φRCP(a/ah)3 ) 3.3%. As evidence, beyond this critical concentration polymer layers are compressed and particles progressively deswell as φp increases. Rheometry. In this section we present the experimental results of rheometry obtained by Raquois12 for the microgel suspensions presented above. Figure 2 shows typical curves of the relative viscosity ηr ) η/ηF, where η is the suspension viscosity and ηF is the suspending fluid viscosity, as a function of shear stress σ for different concentrations φp. It is observed that the rheological response of the suspension is highly dependent on the volume fraction of particles. At low concentrations, viscosity curves present the general form observed in colloidal suspensions, that is, a low shear Newtonian plateau followed by a shear-thinning region which ends in a second Newtonian plateau at high shear stress. Above a given concentration, the viscosity curves change in shape, showing a divergence at low shear stress and hence a strong shear-thinning. More precisely, as is observed in Figure 2, near φp ≈ 4% the low shear Newtonian plateau disappears and, instead of it, an apparent yield stress develops. This particular feature already described for suspensions containing polymerically stabilized particles4,5,9 was confirmed by means of a rheoptical technique3,12 (see next Figure 5 ). The high-frequency plateau modulus of the suspension measured at different concentrations φp also reveals the onset of a solidlike behavior in the suspension12 (see next Figure 7). This liquid-solid transition appearing in the microgel suspension will be further analyzed next in the context of a rheological model. It is appropriate to note here that the particle concentration at which the rheological transition arises is slightly higher than the critical value φp predicted above from geometrical considerations. This may be interpreted by taking into account that the osmotic pressure of the solution grows with particle concentration, against the internal osmotic pressure of particles,3,6 which reduces the particles swelling for concentrations below φ*p. Con-
Figure 2. Relative viscosity ηr ) η/ηF as a function of shear stress σ at different concentrations φp for the suspension of microgels at 20 °C.12
sequently, the values of φp at which particles could come “in contact” (and hence produce the rheological changes) are slightly higher than φ*p. Theoretical Concepts Interaction Energy between Particles. From a physicochemical point of view, the suspension is well stabilized. In fact, the polymer chains branched to the particle core constitute a polymer layer, the thickness of which is high enough to ensure that the van der Waals attraction potential is negligible in relation to the Brownian thermal energy. This mechanism is well-known as steric stabilization.8,13 As the concentration increases in the suspension, particles approach each other and, around the critical value φ*p, the polymer layers begin to overlap. Then, because of the spatial constraints at high concentrations, the polymer chains are compressed between the core surfaces, leading to a repulsive force between particles. This interaction has been studied by means of both experimental and theoretical methods.13-17 At the present time, the main theory available to evaluate the pair interaction energy between polymer-covered surfaces is that of de Gennes15 which takes into account the osmotic repulsion between polymer layers and the elastic energy of the chains. It must be also mentioned that the theory of Milner et al.16 comes from a more complex derivation, nevertheless resulting in an equation similar to that of de Gennes. For practical purposes one may also follow the suggestion of Israelachvili.13 In fact, for D < 2L the repulsive interaction energy is roughly exponential (see also the experimental results of Costello et al.)17 and may be written (13) Israelachvili, J. Intermolecular and Surface Forces, 3rd ed.; Academic Press: London, 1997. (14) Markovic, I.; Ottewill, R. H.; Underwood, S. M.; Tadros, Th. F. Langmuir 1986, 2, 625. (15) de Gennes, P.-G. Adv. Colloid Interface Sci. 1987, 27, 189. (16) Milner, S. T.; Witten, T. A.; Cates, M. E. Macromolecules 1988, 21, 2610. (17) Costello, B. A.; Luckham, P. F.; Tadros, Th. F. Langmuir 1992, 8, 464.
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U(D) ) AkBTe-πD/L
Berli and Quemada
(2)
dS -1 -1 ) -τ-1 hy (S - S∞) + τBr (S0 - S) + τre (S0 - S) (5) dt
where A is a constant related to the length L and the mean distance s between the chain attachment points at the core surface. Also in eq 2, kB is the Boltzman constant and T is the absolute temperature. Finally, since there is no electric double layer interaction and van der Waals forces are negligible, the main interaction governing the microstructure of the microgel suspension is that coming from the repulsion between polymer layers. Rheological Model for Steady Shear Flow. According to the experimental results described above, the rheological response of the suspension may be classified in two concentration regimes. In fact, if φp < φ*p, particles do not have spatial constraints, rheology is governed mainly by Brownian motion and shear forces on particles and the suspension presents a liquidlike behavior. In contrast, if φp > φ*p, particles are densely packed, rheology is governed by interparticle forces and the suspension presents a solidlike behavior. From a rheological point of view, the effective volume fraction φ occupied by the particles plays a crucial role. In this sense, here we develop a rheological modeling for this suspension in the framework of the generalized hard sphere model proposed by one of the authors.10 First, it is assumed that particles have an effective radius aHS which involves the core radius plus the hydrodynamic thickness of the polymer layer. Thus, aHS defines an equivalent hard sphere radius that permits to quantify φ ) φp(aHS/a)3. In addition, the radius of swelled particles decreases when φp > φ*p, hence aHS ) aHS(φp). Applying the effective volume fraction concept to a suspension of soft spheres requires to start from
where τhy, τBr, and τre are the characteristic relaxation times of hydrodynamic, Brownian, and repulsive interactions, respectively. Both first and second terms in the right hand side of eq 5 are usually considered to explain microstructural processes.20,21 The last term involving the particle-particle interaction is included here to extend the structural model to the case of suspensions containing highly repulsive particles. In fact, for flow to occur in these systems, particles must be forced to move against the force fields of the other particles, demanding an additional energy. Therefore, it may be assumed that the relaxation of particles after a shear deformation also depends on interparticle repulsive forces, which drive particles away one from another. At a given shear stress σ, the microstructure reaches a dynamical equilibrium. Consequently, dS/dt ) 0 in eq 5 and then the steady value of S is given by
(
)
φeff φm
ηr ) 1 -
-2
Br
τBr ∼
(4)
where φ is the volume fraction of the swelled polymer particles, C is a “compactness factor” and S is the structural variable.10 For well-stabilized suspensions in quiescent conditions, the value S ) S0 characterizes the equilibrium structure for which the statistical distribution of particles is governed by Brownian motion and repulsive particleparticle interactions. When a shear stress is applied to the suspension, hydrodynamic interactions perturb the equilibrium distribution of particles resulting in S < S0. In the case of high shear stress, hydrodynamic forces overcome both interparticle and Brownian randomizing forces, leading to a structure level S ) S∞ , S0. A rate equation for S can be written, which accounts for the forces perturbing the microstructure and those restoring the equilibrium state. In this sense, all the kinetic processes involved in the microstructure dynamics are supposed to be of relaxation type. Hence, for fixed values of concentration φp and temperature T, the structural variable is governed by (18) Quemada, D. In Lecture Notes in Physics; Casas-Vasquez, J., Lebon, G., Eds.; Springer-Verlag: Berlin, 1982; Vol. 164, p 210. (19) Brady, J. F. J. Chem. Phys. 1993, 99, 567.
(6)
re
where θ is a kinetic rate ratio. For dilute suspensions of hard spheres the Brownian time scale is 6πηFa3/kBT which corresponds to the diffusion time of a particle over a distance near its radius.8 In concentrated suspensions the diffusional movement of particles is strongly diminished. For these cases, the medium viscosity η instead ηF is currently considered.5,21 Moreover, for suspensions of particles containing a polymer layer, a must be taken as the effective particle radius aHS.4,8 Thus, the characteristic time τBr scales as
(3)
which generalizes a relationship between viscosity and volume fraction for concentrated colloidal dispersions.7,18,19 In eq 3, the effective volume fraction of the disperse phase is defined according to
φeff ) φ(1 + CS)
τ-1 S0 - S hy ) -1 ≡θ S - S∞ τ + τ-1
ηa3HS kBT
(7)
On the other hand, the characteristic time of the hydrodynamic interaction can be seen as the lifetime of a doublet rotating in a flow field of shear rate γ˘ , that is π/γ˘ .22 Therefore, one may write
τhy ∼ γ-1
(8)
Following it is necessary to infer the relaxation time due to the presence of repulsive forces in the suspension. For this purpose, the simplest assumption is to consider that each single particle stays in the minimum of the total potential energy UT generated by its neighbors. Under shear stress, the particle moves in the r direction over a distance ∆r, however remaining within a cage formed by the nearest neighbors (specially in concentrated systems, where particle displacements are very limited). Thus, the time required for the particle to reach its equilibrium position after suppressing the shear deformation must be proportional to f∆r/|∂U/∂r|, where f ) 6πηFa is the Stokes frictional coefficient and ∂U/∂r represents the restoring force arising from the potential energy field. Here we are supposing that the change of potential experimented by the particle is proportional to the pair potential U, which is a reasonable approximation for exponentially decaying potentials. Therefore, assuming that the movement of the (20) Cheng, D. C.-H.; Evans, F. British J. Appl. Phys. 1965, 16, 1599. (21) Quemada, D. Rheol. Acta 1978, 17, 643. (22) Goldsmith, H.; Mason, S. In Rheology; Eirich, F., Ed.; Academic Press: London, 1967; Vol. 4, Chapter 2.
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particle involves a distance around its radius, the characteristic time τre scales as
ηa3HS τre ∼ U
(9)
In this expression, both aHS and η were included instead of a and ηF respectively, to take into account the same effects considered above in deriving eq 7. Then, from eqs 7-9 the ratio θ results
θ∼
σa3HS ηa3HSγ˘ ) kBT + U kBT + U
(10)
It is then clear that θ expresses the balance of hydrodynamic interaction energy σa3HS against both Brownian thermal energy kBT and repulsive interaction energy U. Therefore, writing θ ) σ/σc, one may define the critical shear stress σc of the suspension as
σc )
kBT + U
(11)
a3HS
It must be observed that 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, where shear forces are in competition with Brownian motion only.7,8 The last step in modeling consists of writing an explicit relationship between viscosity and shear stress, which describes the non-Newtonian behavior of the suspension under steady conditions. Indeed, by introducing the steadystate solution of S (eq 6) into eq 4, the effective volume fraction results
φeff φ/φ0 + φ/φ∞θ ) φm 1+θ
(12)
where φ0 ) φm/(1 + CS0) and φ∞ ) φm/(1 + CS∞) are the effective maximum packing fractions corresponding to θ f 0 and θ f ∞, respectively.10 Then, using eq 12 in eq 3 leads to the following expression:18
(
η(σ) ) η∞
)
1 + σ/σc χ + σ/σc
2
(13)
In this equation, χ is the ratio
χ)
(
) ()
1 - φ/φ0 η∞ ≡ 1 - φ/φ∞ η0
1/2
(14)
where η0 and η∞ represent the limiting viscosity values corresponding to the shear stress limits σ f 0 and σ f ∞ respectively. Note that the second identity in eq 14 only works for χ g 0. In fact, these viscosities satisfy the following equations
ηr,0 ) (1 - φ/φ0)-2
(15a)
ηr,∞ ) (1 - φ/φ∞)-2
(15b)
where, for the case of hard sphere dispersions, the maximum packing fractions were reported φ0 ) 0.63 and φ∞ ) 0.71.23,24 Nevertheless, since we are dealing with (23) de Kruif, C. G.; van Iersel, E. M. F.; Vrij, A.; Russel, W. B. J. Chem. Phys. 1985, 83, 4717.
Table 2. Parameters of the Viscosity Model for the Microgel Suspension in Steady Shear Flow φp (%)
φ
φ0
φ∞
U/kBT
σCa3HSkBT
0.99 1.96 2.91 3.38 3.85 4.76 5.66 6.54 7.41 9.09 9.91 10.95
0.2284 0.4468 0.5856 0.6265 0.6299 0.6353 0.6352 0.6349 0.6353 0.6329 0.6322 0.6324
0.63 0.63 0.63 0.63 0.63 0.63 0.63 0.63 0.63 0.63 0.63 0.63
0.825 0.860 0.791 0.773 0.741 0.720 0.698 0.692 0.679 0.673 0.665
0.103 0.269 0.628 1.343 5.587 11.06 11.36 20.21 47.41 71.14 97.82
1.102 1.268 1.627 2.345 6.584 12.06 12.35 21.22 48.41 75.14 98.80
particles containing a soft polymeric layer, an additional shear stress dependence is expected in the maximum packing fraction. It was discussed above that as microgel concentration approaches the critical value φ*p, the suspension viscosity diverges and hence an apparent yield stress σY develops. In the mathematical context of the model, the viscosity η(σ) diverges when the ratio χ f 0. Hence, negative values of χ, which result from eq 14 if φ > φ0, can be associated with the onset of the plastic behavior with a yield stress σY ) -χσc. That is,
σY ) σc
(
)
φ/φ0 - 1 1 - φ/φ∞
(16)
which only exists if φ > φ0 (see also the work of Zhou et al.25). In other words, for φ < φ0 a shear-thinning behavior is observed without yield stress. As φ is increased, this response abruptly changes at φ ) φ0 (where σY ) 0) becoming a plastic behavior at φ > φ0, with a yield stress given by eq 16. Thus, for the concentration regime where φ > φ0 (equivalent to φp > φ*p in the microgel suspensions) eq 13 can be rewritten in the following form:10
(
η(σ) ) η∞
)
σ + σC 2 ; σ > σY σ - σY
(17)
Therefore, the viscosity model is able to describe the fluidsolid transition appearing in concentrated colloidal dispersions. In addition, it is worth noting that the viscosity equation suggested here incorporates the pair interaction energy U between particles. This physicochemical variable comes through the critical shear stress σC (eq 11) which characterizes the shear-thinning behavior of the suspension. In this way, one should be able to calculate the microscopic potential energy among particles from rheometric data of steady shear flow. Results and Discussion The viscosity model here proposed for suspensions of microgels involves three parameters in both cases, that is, η0, η∞, and σC for the liquidlike regime (eq 13) or σY, η∞, and σC for the solidlike regime (eq 17). These parameters are also related to the volume fractions φ, φ0, and φ∞ and the interparticle energy U in the case of σC. The procedure used to quantify them is described in the Appendix and the results are presented in Table 2 and throughout Figures 3-6.Figure 3 shows the curves of shear stress σ (24) Jones, D. A. R.; Leary, B.; Boger, D. V. J. Colloid Interface Sci. 1991, 147, 479. (25) Zhou, J. Z. K.; Fang, T.; Luo, G.; Uhlherr, P. H. T. Rheol. Acta 1995, 34, 544.
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Figure 3. Shear stress σ as a function of shear rate γ˘ at different concentrations φp for the suspension of microgels at 20 °C. Symbols are experimental data12 and full lines are the prediction of the viscosity model.
Berli and Quemada
Figure 5. Yield stress σY as a function of concentration φp. The circles are the prediction of the viscosity model and the squares are experimental data obtained through different techniques.3,12
Figure 6. Interaction potential energy U normalized with the thermal energy kBT as a function of the dimensionless surfaceto-surface distance D/2L for the microgel suspension. The symbols are the values obtained from viscosity data and the full line represents eq 2 with A ) 1497.
Figure 4. (a) Low (ηr,0) and high (ηr,∞) shear limits of the relative viscosity as a function of concentration φp and (b) low (φ/φ0) and high (φ/φ∞) shear limits of the relative volume fraction as a function of concentration φp.
as a function of shear rate γ˘ for different concentrations φp. Symbols are the experimental values measured by Raquois12 and full lines are the model curves. A good agreement between the model and rheometric data is observed in the full range of concentrations. It is relevant
to note here that the model incorporates the interparticle energy through the critical shear stress σC, which is a crucial parameter giving the right position of the flow curve in the σ-axis. In this sense, one may observe in Table 2 that the values of σCa3HS/kBT are near above 1 for the lowest concentrations, where the interparticle energy is weak, consistently with experimental and theoretical results for dilute colloidal suspensions.7,8 Nevertheless, as the particle concentration is increased, the values of σC differ in more than 1 order of magnitude from the scaling kBT/a3HS. This result reveals that the particle-particle interaction energy U plays a very important role in the critical shear stress and hence in the rheological response of microgel suspensions at φp > φ*p.
Rheological Modeling of Microgel Suspensions
Figure 4a presents the relative viscosities ηr,0 and ηr,∞ as a function of concentration φp for the microgel suspensions. It is observed that the low shear limiting viscosity varies around 7 orders of magnitude as φp grows from 0 to 4%, and clearly diverges after φp ≈ 4%, where particles “touch” each other. This is in agreement with a hard spherelike behavior of particles in the limit of σ f 0, where the effective radius is not affected appreciably by shear rate. In contrast, the high shear limiting viscosity increases almost linearly with concentration in the semilogarithmic plot (Figure 4a). This behavior already described for suspensions of soft particles1,5 shows the effects of hydrodynamic forces on the polymer layer of particles (see also the work of Goodwin and Ottewill).26 In fact, when σ f ∞ particles seem to diminish their effective radius, resulting in a suspension viscosity lower than that expected for an equivalent suspension of hard spheres. This interpretation of the rheological results is explained further in Figure 4b, where the ratios φ/φ0 and φ/φ∞ as a function of concentration are presented. For σ f 0, the relative volume fraction grows almost linearly as φp increases and reaches the limit 1 corresponding to maximum packing when φp ≈ 4%. For σ f ∞ however, the relative volume fraction as a function of φp presents a lower slope and reaches asymptotically the limit 1 at higher concentrations. Thus, one may observe that the shear stress increases the packing capability of particles (φeff/φm decreases with σ) and this is due to the combination of two effects: (a) the decrease in the effective radius of particles as a consequence of the influence of hydrodynamic forces on the polymer layers, and (b) the increase in the maximum packing fraction induced by shear, as it is wellknown for hard spheres. One may also observe that the difference φ/φ0 - φ/φ∞ (a measure of the packing capability of particles) decreases with φp. In fact, when particles are densely packed, the polymer layers are compressed and the possibility of an ulterior diminution of the effective particle radius by shear seems to be less feasible. Figure 4b also shows graphically how the yield stress appears in the model and indeed in the microgel suspension. Clearly if φ/φ0 g 1 particles are locked and the flow is not possible. Therefore, symbols over the dashed line in Figure 4b correspond to concentrations for which the suspension presents a solid-fluid transition induced by shear. That is, if the imposed shear stress is high enough, the effective volume fraction becomes lower than the maximum packing and the suspension flows. The yield stress σY is then the minimum value of shear stress required to get φeff/φm e 1. In Figure 5, the values of σY obtained from the model are compared with those measured previously through flow birefringence experiments and rheometry.3,12 The agreement is acceptable, taking in mind that measurements of each type were performed on different samples: it is well-known how much the conditions of sample preparation and setting in the apparatus affect drastically the measured values of yield stress. Figure 6 presents the pair interaction energy U/kBT obtained from suspension viscosity (symbols) as a function of the mean distance between core surfaces D/2L, where each value of D comes from a different concentration φp through eq 1. The full line in Figure 6 represents eq 2 with A ) 1497. This value was calculated by fitting the data U/kBT obtained from suspension viscosity, in the range D/2L e 1 where eq 2 is strictly valid. Since both curves in Figure 6 are rather coincident, one may say that the viscosity model predicts the correct form of the (26) Goodwin, J. W.; Ottewill, R. H. J. Chem. Soc., Faraday Trans. 1991, 87, 357.
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Figure 7. Dimensionless dynamic modulus G′∞a3/kBT as a function of concentration φp. The symbols are experimental values.12 The full and dashed lines represent the prediction of eqs 18 and 19, respectively, by using the pair interaction energy U(R) obtained from viscosity data (Figure 6).
potential interaction in microgel dispersions. Moreover, it should be stressed that changing the value of A leads to shift vertically the full line-curve, since the functionality with D/2L is fixed in eq 2. Figure 7 presents a cross-check for our modeling. In fact, we compare experimental values of the highfrequency shear modulus G′∞ with theoretical predictions involving the interaction energy obtained above from rheometric data in shear flow experiments. In this sense we use two approximations of the Zwanzig-Mountain expression27 proposed in the literature for suspensions of polymerically stabilized particles at high concentrations:
(a) the equation derived by Evans and Lips28 (see also,17): G′∞ ) nkBT +
(
)
Nφm 4 ∂U(R) ∂2U(R) + 5πR R ∂R ∂R2
(18)
where n ) 3φp/4πa3 is the particle density number, N is the number of nearest neighbors and R is the center-tocenter particle distance.
(b) the equation derived by Buscall29 (see also,9): G′∞ )
Nφm∂2U(R) 5πR ∂R2
(19)
In both equations, the maximum packing fraction φm corresponding to a face-centered cubic array is considered, i.e., φfcc ) 0.74 and hence N ) 12, according to the crystalline order showed by concentrated colloids in highfrequency experiments. Also the interparticle distance is taken R ) 2a(φfcc/φp)1/3. The symbols in Figure 7 correspond to the experimental values of G′∞ obtained by Raquois.12 (27) Zwanzig, R.; Mountain, R. D. J. Chem. Phys. 1965, 43, 4464. (28) Evans, D.; Lips, A. J. Chem. Soc., Faraday Trans. 1990, 86, 3413. (29) Mewis, J.; D’Haene, P. Macromol. Chem. Symp. 1993, 68, 213.
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The full and dashed lines represent the predictions of eqs 18 and 19, respectively. In these equations, the interparticle energy U(R) is included according to eq 2 using D ) R - 2a and the above obtained value A ) 1497 (Figure 6). The excellent agreement of both theoretical predictions with the experimental values of G′∞ indicates that the model suggested here to explain the rheological behavior of microgel suspensions in steady shear flow experiments is quite consistent with the results obtained through dynamic rheometry. Furthermore, one may say that a suitable prediction of U(R)/kBT was made through the viscosity model (it should be noted once again that the vertical position of the model curves in Figure 7 depends basically on A). Concluding Remarks and Perspectives The method used in deriving the viscosity model is phenomenological and based on scaling principles. In this sense, several improvements could be made, namely changing the characteristics of the structural kinetics. Nevertheless, the resulting viscosity equation gives a satisfactory description of the flow behavior of microgel suspensions in steady shear flow and, in particular, predicts successfully the liquid-solid transition. In addition, the viscosity model allows one to determine an effective potential of interaction directly from the rheometric data in shear flow. This important problem of suspension rheology has been discussed lately,6,9,30 however by means of different approaches. Therefore, a detailed comparison between the prediction of our model and that of different models known in the literature is required. This comparison is currently in progress and will be the subject of a future paper. Acknowledgment. One of the authors (C.L.A.B.) acknowledges receipt of a postdoctoral fellowship from CONICET (Argentina) and educational leave of absence from UNL (Argentina). D.Q. wishes to thank Professor F. Tassin and Dr. C. Raquois for kindly providing the microgel rheological data. Appendix Here we describe briefly the procedure used to quantify the parameters involved in the viscosity model proposed (30) Ogawa, A.; Yamada, H.; Matsuda, S.; Okajima, K.; Doi, M. J. Rheol. 1997, 4, 769.
Berli and Quemada
for microgel suspensions. By inverting eq 3 the following relationship is obtained
φeff ) 1 - η-1/2 r φm
(A1)
which permits the transformation of the data η versus σ presented in Figure 2 into different sets of data φeff/φm versus σ, corresponding to different concentrations φp. Then eq 12 may be fitted to these data to obtain the volume fraction of swelled particles φ, the low shear packing fraction φ0, the high shear packing fraction φ∞ and the shear stress σC as adjustable parameters. Furthermore, the critical shear stress σC can be written explicitly in terms of φ by using a3HS ) a3φ/φp. Thus, introducing eq 11 into eq 12 leads to
φeff φ/φ0 + φ/φ∞[σφβ(1 + U/kBT)] ) φm 1 + σφβ(1 + U/kBT)
(A2)
where β ) a3/φpkBT is a known factor. Consequently, the parameters to be calculated from rheometric data are the volume fractions φ, φ0, and φ∞, and the ratio U/kBT which represents the apparent interaction potential between particles. A standard fitting procedure was used for this purpose. In a first analysis, all the parameters were allowed to vary freely for each concentration φp, i.e., for each curve φeff/φm versus σ. The most relevant feature was that φ0 presented slightly scattered values around 0.63. This is consistent with theoretical expectation: for σ f 0 hydrodynamic forces do not affect the effective radius of particles and, in this particular situation, soft particles should behave as hard spheres. Then, the fitting procedure was performed imposing φ0 ) 0.63 for all the concentrations, hence reducing to three the number of free parameters. The results obtained in this way are presented in Table 2. The corresponding values of the critical shear stress σC (eq 11) are also reported in Table 2. The last step in calculations is straightforward: once the volume fractions φ, φ0, and φ∞ are known, the values of ηr,0 and ηr,∞ are obtained from eqs 15a and 15b, respectively. Results are presented in Figure 4. Finally, the yield stress σY is calculated from eq 16 and presented in Figure 5. LA000365X
