Ind. Eng. Chem. Res. 1994,33,2391-2397
2391
The Huggins Coefficient for the Square-Well Colloidal Fluid Johan Bergenholtz and Norman J. Wagner' Colburn Laboratory, Department of Chemical Engineering, University of Delaware, Newark, Delaware 19716
The Huggins coefficient, which characterizesthe exact, dilute limiting viscosity of colloidal suspensions with square-well interactions, is calculated. An overview of the method and numerical results are presented for both monodisperse and bidisperse suspensions of spherical colloids. The results demonstrate that attractions increase the viscosity much more than repulsions of similar magnitude. Further, the viscosity of polydisperse suspensions of these particles is very sensitive t o the smaller size fraction. These calculations complement other theoretical work on the equilibrium thermodynamic properties and colloidal diffusion t o provide a more complete reference state for analyzing the role of colloidal potentials on suspension properties.
Introduction The classic square-well interaction potential has frequently been invoked to successfullydescribe the cohesive nature of real, simple liquids. Its utility as a reference system lies in its simplicity while retaining both repulsive and attractive interactions. As for most molecular fluids and colloidal suspensions these interactions act over a relatively short range, the square-well potential is well suited for theoretical and numerical studies of how the interparticle potential influences specific material properties and molecular behavior. As a result, the square-well fluid has been extensively studied in the past by means of perturbation theory (Barker and Henderson, 1967), integral equations (Smith et al., 1974;Regnaut and Ravey, 1989),and computer simulations (Heyes, 1991;Heyes and Aston, 1992). Due to the analogy between colloidal systemsand simple liquid systems, the square-well potential has found use in the modeling of equilibrium structural and thermodynamic properties of systems in which macroparticles interact via short-ranged interactions, such as sterically stabilizedpoly(methylmethacrylate) (Rao and Debnath, 1990)and silica (Roux and de Kruif, 1988; Jansen et al. 1986) dispersions and inverse microemulsions (Huang, 1985). Indeed the thermodynamic and equilibrium structural properties of molecular and colloidal fluids that have the same interaction potential are the same. There is also a need for models of the transport properties of colloidal systems, such as diffusivities and viscosities, and how these properties depend on colloidal interactions. The transport properties of molecular square-welldense gases have been calculated (Hirschfelder et al. (1964),section 8.3). As is well-known, however, the presence of the solvent continuum for the colloidal square-well fluid distinguishesthe dynamical and transport properties from that of molecular fluids. Recently Cichocki and Felderhof (1991) have calculated the self-diffusioncoefficient for dilute suspensions of spherical, Brownian colloids interacting via the square-wellpotential, taking into proper account the presence of the solvent. The other important transport coefficient is the Huggins coefficient, which characterizes the exact, dilute limiting suspension viscosity. Measurement of the Huggins coefficient is often used for the determination of the interaction potential for colloidal fluids, in much the same manner as dilute gas viscosity measurements are used to determine intermolecular potential parameters. To date there have
* Author to whom correspondence should be sent. E-mail:
[email protected].
I
2
8r,b
Figure 1. Square-well potential.
been no calculations of the Huggins coefficient for squarewell fluids, which is the goal of this paper. In the following study we examine the effect of the square-well potential on the steady, low-shear viscosity of a system of Brownian spherical particles in the dilute, or pair interaction, limit. To calculate the dilute limiting viscosity, we first relate the potential to the excess stress created by the presence of suspended particles in the solvent. A calculation of this stress necessitates determining the suspensions's nonequilibrium microstructure under weak shearing. Finally, we will briefly examine the effect of polydispersity on the viscosity through model calculations for bidisperse suspensions. As many of the derivations and details of the calculations are contained in a previous paper concerning hard-sphere behavior (Wagner and Woutersen, 19941, we will only highlight the major steps and any differences with the approach contained therein.
The Square-Well Potential The square-well interaction potential (Figure 1) is defined in terms of three parameters: a well depth Ua,b characterizing the strength of interaction relative to kT, the thermal energy; a well width &,b defining the range of interaction between two spheres of radii a and b; and a hard-core radius ( ( a+ b)/2). Defining the dimensionless center-to-center separation distance ass = 2r/(a b) yields the following definition of the square-well potential (Barker and Henderson (1967)):
+
Although this interaction potential allows for size de-
Q888-5885/94/2633-2391$04.50/0@ 1994 American Chemical Society
2392 Ind. Eng. Chem. Res., Vol. 33, No. 10, 1994
pendent interaction parameters in polydisperse suspensions, we limit our calculation to systems of particles interacting via a fixed well depth u,,b/kT = uo/kT. Furthermore, we treat suspensions characterized by a constant absolute well width Aa,b = A. These limitations are justified for many colloidal systems as the interaction range is determined by steric effects, such as the interpenetration of chain segments of fixed length on neighboring particles (Huang et al., 1984; Huang, 1985). These restrictions for polydisperse suspensions retain the simplicity of necessitating only three parameters to define the potential, yet they can easily be relaxed if the physical situation warrants. An important, limiting case of the square-well potential is the sticky-sphere potential (Baxter, 1968), which is obtained by applying the limit
A+,
lim
A exp(-u,JkT)
~0-m
-
1
7
(2)
where 7 is a dimensionless temperature usually known as a stickiness parameter. In modeling the equilibrium properties it has been customary to map the potential parameters of the square-well onto the stickiness parameter. For monodisperse suspensions, equating the second virial coefficients of the respective interaction potentials one obtains (Cummings et al., 1976) 7-l
2 ~ + ~ 3
= 4(exp(-udkT) - l ) ( ( ~ - 1))
+ xbG(R-b)
Stresses The reduced suspension viscosity (7, normalized by the solvent viscosity ~ 0 can ) be expanded in powers of mass concentration c to give the followingdilute limit expansion: = 1 + [TI&
+ [11,2k~C' +
(5)
In this viral expansion Cqlo is defined as the intrinsic viscosity and is a function of the molecular weight of the individual particle. The quantity kH is known as the Huggins coefficient, which is characteristic of the suspended particles as it contains information about particle (pair) interactions analogously to the second virial coefficient (Hirschfelder et al. (1964), p 158). For suspensions of spherical particles an equivalent expansion is usually made in terms of 4, the particle volume fraction q,//.L, = 1
5 + c42 + ... + 54
(7)
where 2 is the total stress tensor, I.T. is an isotropic term, and E = (1/2)(Vv ( V V ) ~is) the strain rate tensor where Vv is the velocity gradient. The stress tensor is explicitly divided into contributions from the solvent and stresses arising from interparticle forces (I),Brownian motion (B), and hydrodynamic interactions (H). These contributions are given in eq 7 as ensemble averages ( S i ) ,that is weighted averages of the stress dipoles over all possible particle configurations. In what follows, we will consider dilute suspensions for which only pair interactions are important. In addition, we will work at asymptotically low shear rates, for which the condition
+
Pe =
+
b)~blEl sw) using the contact and far-field boundary conditions and the jump conditions for the outer boundary of the square well.
Results To calculate this nonequilibrium microstructure for evaluation of the particle stresses, eq 16 was discretized and written in matrix form. The hydrodynamic functions for pair interactions are known numerically as functions of the separation distance between the spheres for both monodisperse (Jeffrey and Onishi, 1984; Kim and Mifflin, 1985) and bidisperse suspensions (Wagner and Woutersen, 1994). In the near field, exploiting the behavior of the hydrodynamic function G(s) (G(s) 2(s-2) as s 2), provides a valid contact boundary condition in place of eq 17, giving f(s=2) in terms of df(s=2)/ds. At a separation of 25 diameters, a practical boundary condition was constructed to replace eq 18 by differentiating the analytic far field solution (see the Appendix). Imposing eq 19 at the well boundary results in two interdependent tridiagonal matrices of coefficients. These were inverted using a numerical back substitution algorithm (Press et al., 1986). The solution was iterated upon by adjusting the value for f(s=sW) until convergence against eq 20 was obtained. All calculations were iterated upon until the results were free of dependencies on the number of intervals and the range of discretization. Figure 2 shows the structural distortion due to shearing a monodisperse suspension of square-well spheres with a fixed well width of 10% of the radius for various well depths. The hard sphere structure agrees with that obtained previously by others (Batchelor, 1977). As a
-
-
2394 Ind. Eng. Chem. Res., Vol. 33, No. 10, 1994 2
\
.
25
Ala = 0 15 20
15
A l a = 0 10 A l a = 0 05
c
-
-
I 2
22
26
24
28
2
3
I
0
S
Figure 2. Nonequilibrium structure function f(s) as a function of radial distance s, plotted for varying amounts of attraction and repulsion, as labeled.
.10 -s
t
1
2
3
Figure 4. Interparticle contribution to the Huggins coefficient as a function of square-well potential parameters, as labeled.
A / a = 0 05 A l a = 0 10 -
-
A/. = 0 15
J
1
-12 2
4
UolkT
1
0
1
2
3
4
ugJkT
Figure 3. Brownian contribution to the Huggins coefficient as a function of square-well potential parameters, as labeled.
consequence of the singularity at the well boundary, the structures shown in Figure 2 show a discontinuity in the first derivative as soon as the square-well attraction or repulsion is introduced. For increasingly attractive wells f ( s ) becomes negative. This suggests that there is now an increase in particle density at angular positions where for hard spheres there is a depletion in density. This occurs because, for strongly attractive wells, closely neighboring particles act effectively as a dumbbell and rotate (by 90') in such a way as to decrease the friction being made with the surrounding medium. This type of behavior has been observed in spheres with surface adhesion, or sticky spheres (Russel, 1984; Cichocki and Felderhof, 1990). In the sticky limit (eq 2) permanent doublets are present which orient much the same way as the more strongly attractive pairs in this study. In approaching the sticky limit (i-e.,A/a = 0.0001 and uo/kT = -20) a value for f(s=sW) = f(s=2) of -2.919 was obtained, which agrees well with the sticky limit result of -2.963 calculated by Cichocki and Felderhof (1990).
This nonequilibrium structure is inserted into the stress equation for the Brownian forces,eq 12, yieldingthe results shown in Figure 3. As the Brownian force opposes the formation of pairs near contact, it tends to decrease the Huggins coefficient for increasingly attractive wells. Repulsive wells have little effect on this contribution to the Huggins coefficient. For the interparticle contribution to the Huggins coefficient, attractive wells strive to retain pairs near contact. The results of calculating eq 15 are presented in Figure 4, showing that this generally results in an increase to the Huggins coefficient. Again, repulsive wells yield only a relatively minor increase in comparison. Figure 5 demonstrates the important effect of attractive interactions on the hydrodynamic contribution to the Huggins coefficient. As for attractive wells particles are, on average, in closer contact; there is increased viscous
2
1
0
1
2
3
4
uolkT
Figure 5. Hydrodynamic contribution to the Huggins coefficient as a function of square-well potential parameters, as labeled.
Figure 6. Square-well Huggins coefficient k~ as a function of potential parameters: relative well depth (uo/kT) and relative well width (A/a).
dissipation resulting in an increased contribution to the Huggins coefficient over that of hard spheres. Again, repulsive interactions have the opposite effect, yet it is minor in comparison to strong attractions. Combining the various stress contributions yields the Huggins coefficient as
where the first term results directly from eq 11. Figure 6 shows the overall dependence of the Huggins coefficient on the square-well potential parameters. Figures 7 and 8 are projections of this surface onto planes of constant well width and well depth, respectively. It is seen that, in general, any amount of repulsion increases the Huggins coefficient over the hard sphere value (which is calculated to be 0.946 (C = 5.912) in this study). This agrees qualitatively with previous findings on the excluded shell potential (Russel, 1984). The increase in the Huggins coefficient due to repulsion is modest, however, in comparison to the rapid increase resulting from attractions.
Ind. Eng. Chem. Res., Vol. 33, No. 10, 1994 2395 16 14
--
A l a = 0.15 A l a = 0.10 A l a = 0.05
-
t
/i
'i
-*
2
25
3
35
4
45
5
S
UolkT
Figure 7. Projection of Huggins coefficientas a function of squarewell parameters, aa labeled. 31
I
uolkT = -3 uo/kT = - 2
uo/kT = - 1 uoJkT = 0
-
-
.-
-
15
1 -
0
002
004
006
008
01
014
012
Ala
Figure 8. Projection of Huggins coefficientaa a function of squarewell parameters, aa labeled. 1,
I '
A/o=O 098
kH
.
/
-
I
I
097 096
-
0
-02
-04
-0 6
08
-1
-1 2
-1 4
UolkT
Figure 9. Enhanced projection of Huggins coefficient aa a function of square-well parameters, aa labeled.
For instance, particles interacting with Ala = 0.10 and a 4kT repulsion have an increase of = 5 % in the Huggins coefficient over the hard sphere value, whereas a corresponding attraction of 4kT gives a full order of magnitude increase (=1000%) (see Figure 7). Figure 9, which is an enlargement about the hard sphere limit, reveals that weakly attractive wells yield a Huggins coefficient lower than that of hard spheres. This effect is of theoretical interest as it has been interpreted from previous calculations for adhesive spheres that hard sphere interactions define the minimum attainable viscosity for a given suspension. To prove that this observation is not a numerical artifact, a linear perturbation expansion in the parameter -uolkT is presented in the Appendix. The first-order correction to f(s) for hard spheres satisfies a microstructural equation similar to eq 16 with modified boundary conditions (see the Appendix and Figure 10). For all well widths, the solution is a negative perturbation to the hard sphere f(s) for attractive wells. Hence, weakly attractive square-well structures are perturbed less by shear than corresponding hard sphere structures. The effect of the attractive perturbation on the hydrodynamic
Figure 10. Order ( - 4 k T ) perturbationto nonequilibriumstructure function aa a function of relative well width from top to bottom Ala = 0.01, 0.1, 0.5, 1.0, respectively.
2396 Ind. Eng. Chem. Res., Vol. 33, No. 10, 1994 Table 1. Data for Square-Well Huggins Coefficient, k ~for , Monodisperse Spheres as a Function of Square-Well Parameters uo/kT 2
1 0
-1 -2 -3 -4 -5 -6
0.025 0.952 0.950 0.946 0.944 0.984 1.29 2.60 6.80 18.7
0.05 0.961 0.956 0.946 0.947 1.09 1.90 4.86 13.6 37.9
0.075 0.973 0.963 0.946 0.956 1.23 2.65 7.43 21.2 59.0
% 0.1 0.987 0.971 0.946 0.971 1.41 3.46 10.1 29.0 80.7
0.125 1.00
0.981 0.946 0.989 1.61 4.36 13.0 37.4 104
0.15 1.02 0.991 0.946 1.01 1.84 5.34 16.1 46.3 129
specific case of uo/kT = -2.8 and 2A/(a + b) = 0.063, which is representative of inverse microemulsions (Bergenholtz and Wagner, 1994). Hydrodynamic functions for unequal sized spheres were calculated in the same manner as described elsewhere (Wagner and Woutersen, 1994). The nonequilibrium structure was obtained as before, by solving eq 16, but now with the appropriate A-dependency in the hydrodynamic functions. The results for f ( s ) as a function of A are plotted in Figure 12, where it is seen that increased size asymmetry results in structures which are distorted less by the shearing flow in the near field. This is the same qualitative effect as seen in hard sphere mixtures (Wagner and Woutersen, 1994), and leads to a generalreduction in the Hugginscoefficientwith increasing polydispersity (size asymmetry). Figure 13 shows the Huggins coefficient as a function of mixing volume fraction for various values of size asymmetry. Clearly, the Huggins coefficient has a minimum near suspensions pure in the larger sized component and is greater for the suspensions rich in small particles. Also, increased size asymmetries result in increased Huggins coefficients for suspensions rich in the smaller sized component. This bias toward higher Huggins coefficients for the smaller sized particles can be readily understood; for a fixed absolute well width the relative well width of the smaller sized particles is increased with increased size asymmetry. The interactions between smaller particles is, therefore, stronger. Huggins coefficient measurements of colloidal suspensions with fixed square-well attractions are thus more sensitive to the smaller end of the size distribution. Previous results for the Huggins coefficient for hard spheres demonstrated that increasing asymmetry reduced the viscosity and that the coefficient was symmetric in volume mixing ratio ($Jsrnall/$Jtoa) (Wagner and Woutersen, 1994). Experimentally, concentrated bidisperse mixtures of spherical particles usually exhibit a minimum in viscosity that is toward the side rich in large particles, or in other words, the replacement of a few large particles by an equal volume of small particles results in large reductions in overall suspension viscosity (Goto and Kuno, 1984). One possible source of this asymmetry was thought to be interparticle interactions. Figure 13 shows similar behavior, suggesting that simple attractions of our postulated form could be responsible for the measured behavior of more concentrated, bidisperse suspensions.
0 -0 2
/
-
i-I
-1 2
22
2
26
21
3
28
S
Figure 12. Nonequilibrium structure functionf ( s ) as a function of radial distance s, plotted for varying amount of size asymmetry, as labeled, for the case of 2A/(a + b) = 0.063 and uo/kT = -2.8. 8
/ I
02
0
04
06
08
1
OsrnolilO*otol
Figure 13. Huggins coefficient for bidisperse mixture as a function of mixing volume ratio and size asymmetry for the case of 2A/(a + b) = 0.063 and uo/kT = -2.8 from top to bottom along the right axis X = 8, 4, 3, 1.5, 1, respectively.
Appendix Far-FieldSolution. Using the leading order, far-field forms for the hydrodynamic functions, a far-field solution to eq 16 can be constructed:
( + -SA "1
-3 3 s - 4
f(s) = c s-3
sw < s
(22)
where a = 4A/(1+ AI2 and the constant c depends on the potential parameters through the boundary conditions (eqs 17, 19, and 20). The outside solution (s > sw) can be differentiated and back-substituted to eliminate the constant c, yielding
r=-
3s4
+
+ 6as"
s - ~ (3a/2) s4
(23)
This condition was imposed in place of condition 18 in the numerical routine for the solution of eq 16. Perturbative Solution for Weak Interaction Strengths. Expanding the nonequilibrium structure as f(s) = p(s)+ (-m/kT)f'(s)for asymptotically weak interactions, to order (-uo/kT), and substituting into eq 16 yields the following for the perturbation to the nonequilibrium, hard-sphere structure
Acknowledgment We are pleased to submit this article in recognition of the achievements of our mentor and colleague A. B. Metzner. Support for this project is through the NSF (CTS-9158164) and the Kodak corporation.
The boundary conditions, to the same order, become, in the near and far field respectively
Ind. Eng. Chem. Res., Vol. 33, No. 10,1994 2397
and at the well boundary (s = SW)
For attractive interactions, the solution to the above set of equations shows that, as demonstrated in Figure 10, P(s) < 0 for all well widths, even as sw 2. As a consequence of this, to order (-udkT),the net thermodynamiccontribution to the Huggins coefficientis negative for attractive wells and positive for repulsive steps. As this outweighs the perturbation contribution to the hydrodynamic stress, which is always increased over the hard sphere value by attractions, the total Huggins coefficient is decreased below that of hard spheres for weak attractions. It should be noted that in this perturbation expansion as sw 2 the Huggins coefficient goes to its hard sphere value (from below for attractive wells). Hence, this is a fundamentally different limit than the sticky limit. The sticky limit produces doublets which always contribute positivelyto the Huggins coefficient (Russel, 1984;Cichocki and Felderhof, 1990).
-
-
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Cichocki, B.; Felderhof, B. U. Self-diffusion of Brownian particles with hydrodynamic interaction and square step or well potential. J. Chem. Phys. 1991,94,563-568. Cummings, P. T.; Perram, J. W.; Smith, E. R. Percus-Yevick Theory of Correlation Functions and Nucleation Effects in the Sticky Hard-Sphere Model. Mol. Phys. 1976,31 (2),535-548. Goto, H.; Kuno, H. Flow of suspensions containing particles of two different sizes through a capillary tube. ii. effect of the particle size ratio. J. Rheol. 1984,28,197. Heyes, D. M. Coordination Number and Equation of State of Squarewell and Square-shoulder Fluids: Simulation and Quasi-chemical Model. J. Chem. SOC.,Faraday Trans. 1991,87(20),3313-3317. Heyes, D. M.; Aston, P. J. Square-Well and Squareshoulder Fluids: Simulation and Equations of State. J. Chem. Phys. 1992,97(8), 5738-5148. Hirschfelder, J. 0.;Curties, C. F.; Bird, R. B. Molecular Theory of Gases and Liquids; John Wiley and Sons: New York, 1964. Huang, J. S.Surfactant Interactions in Oil Continuous Microemulsions. J. Chem. Phys. 1985,82(l),480-484. Huang, J. S.; Safran, S. A,; Kim, M. W.; Grest, G. S.; Kotlarchyk, M.; Quirke, N. Attractive Interactions in Micelles and Microemulsions. Phys. Rev. Lett. 1984,53(6),592-595. Jansen, W. J.; de Kruif,C. G.; Vrij, A. Attractions in Sterically Stabilized Silica Dispersions. J. Colloid Interface Sei. 1986,114, 411-480,481-491,492-500. Jeffrey, D. J.; Onishi, Y. Calculation of the Resistance and Mobility Functions for T w o Unequal Rigid Spheres in Low-ReynoldsNumber Flow. J. Fluid Mech. 1984,139,261-290. Kim, S.; Mifflin, R. T. The Resistance and Mobility Functions of Two Equal Spheres in Low-Reynolds-NumberFlow. Phys. Fluids 1985,28,2033-2045. Press, W. H.; Flannery, B. P.; Teukolaly, S. A.; Vetterling, W. T. Numerical Recipes; Cambridge University Press: 1986. Regnaut, C.; Ravey, J. C. Analysis of the adhesive sphere fluid as a reference model for colloidal suspensions. h o g . Colloid Polym. Sci. 1989,79,332-331. Rao, R. V. G.; Debnath, D. Structural, thermodynamic and light scattering properties of PMMA latex as a square well fluid in benzene. Colloid Polym. Sci. 1990,268,6044311. Row, P. W.; de Kruif, C. G. Adhesive hard sphere colloidaldispersions I. Diffusion coefficient as a function of well depth. J. Chem. Phys. 1988,88,7799-1186. Russel, W. B. The Huggins Coefficient aa a Means for Characterizing Suspended Particles. J. Chem. Soc., Faraday Trans. 2 1984,80, 31-41. Russel, W.B.; Saville,D. A.; Schowalter, W. R. Colloidal Dispersionu; Cambridge University Press: Cambridge, 1989. Smith, W. R.;Henderson, D.; Murphy, R. D. Percus-YevickEquation of State for the Square-Well Fluid at High Densities. J. Chem. P h p . 1974,61 (7),2911-2919. Wagner, N. J.; Woutersen, A. T. J. M. The Viscosity of Bimodal and Polydisperse Suspensions of Hard-Spheres in the Dilute Limit. J. Fluid Mech. 1994,in press. Woutersen, A. T. J. M.; de Kruif, C. G. The viscosity of semi-dilute bidisperse suspensions of hard spheres. J. Rheol. 1993,37,681.
Received for review January 14, 1994 Revised manuscript receiued April 25, 1994 Accepted May 10, 1994. e Abstract published in Advance ACS Abstracts, September 1, 1994.
