electrolyte solution interface

Monte Carlo Simulation of Small Particle Adsorption at the Solid/Electrolyte Solution Interface INTRODUCTION

0

The adsorption (and desorption) of colloidal-sized particles at various phase boundaries is of fundamental importance in many technologies, ranging from detergency to the stabilization of emulsions and foams. In a recent series of papers Vincent and co-workers (1-4) "have discussed the adsorption of small ( ~ 0 . 2 #m diameter), positively charged polystyrene latex particles at the negatively charged polystyrene/aqueous electrolyte solution interface. Larger (>2 #m diameter) polystyrene latex particles were in fact used as the adsorbent of high surface area for adsorption. In these experiments both sets of polystyrene particles carried a preadsorbed layer of poly(vinyl alcohol). The combination of electrostatic (plus van der Waals) attraction and steric repulsion implies that the net particle adsorption energy is controlled essentially by two main parameters: the electrolyte concentration (c) and the adsorbed polymer layer thickness (6). It was found (1, 2, 4) that for a given value of~ the small particle adsorption isotherm has a high-affinity character at low values of c, but a low-affinity character at high values of c. Moreover, there appears to be a critical value ofc (c*), or a very narrow range ofc values, over which the isotherm changes from the high-affinity to the low-affinity type; c* decreases as ~ increases. Two sets of small particle adsorption isotherms are shown in Fig. 1, corresponding to two sets of different sized particles (2, 4, 5). In both eases the adsorbed polymer layer was poly(vinyl alcohol) of molecular weight 24,000. Qualitatively, the same trends are observed with changing electrolyte concentrations. Quantitative agreement is only apparent at 10-~ mole dm -3 NaC1, however. At this relatively high ionic strength the electrical double layer interactions are small compared to the van der Waals interaction between the particles, and it is the latter which controls the adsorption behavior. Therefore, as the size of the small particles is similar in the two sets of experiments, one may expect the net adsorption energy to be similar in both cases. At lower ionic strengths, however, the electrostatic forces become increasingly more dominant, and the differences observed in the two sets of isotherms shown in Fig. 1 presumably reflect different electrostatic parameters operative in the two systems. In the work reported previously (1-4) it was tentatively assumed that the net particle adsorption energy consisted of two (additive) contributions: (i) the normal interaction (attraction) between a large particle and an

o

14 .

x

o b

0

f

o x

8 I

°0

--

0.2

0/.,

0.6

~)x103 08

FIG. l. Experimental adsorption isotherms for small, positive polystyrene latex particles adsorbed onto larger, negative polystyrene latex particles. ~7, 10-~ mole dm -3 NaCI; A, 10-2 mole dm -3 NaC1; (3, 10-3 mole dm -3 NaC1; ×, 10-4 mole dm -3 NaC1; 1:3, 10-5 mole dm -3 NaC1. (a) Radius of small particles, 0.2/~m; large particles, 3.0 #m (1, 5); (b) radius of small particles, 0.25 #m; large particles, 2.0 ~m (4).

adsorbed small particle; (ii) the lateral interaction (net repulsion at low ionic strengths: net attraction at high ionic strengths) between neighboring adsorbed particles. Clearly, the validity of this assumption is open to question, particularly at low ionic strengths, where the electrical double layers are long range, i.e., ra ~ or <1 (where ~ is the Debye-Hiickel reciprocal thickness of the electrical double layer, and a is the radius of the small particles). Strictly, one should consider the net interaction with all the other particles of the "adsorbate" rather than just those within the two-dimensional layer on the surface. In order to examine just how reasonable the assumptions referred to above are, a Monte Carlo computer simulation of an analogous system was attempted, i.e., where a dispersion of small particles contained within a rectangular prism is allowed to adsorb on the oppositely charged, end faces of that box.

262 0021-9797/83/030262-03503.00/0 Copyright © 1983 by Academic Press, Inc. All rights of reproduction in any form reserved.

Journal of Colloid and Interface Science, Vol. 92, No. I, March 1983

NOTES

,,,(,r5 t

t

I

zi

I

•

I

/

I

I FIG. 2. Model used in computer simulation. MONTE CARLO SIMULATION The model used is shown in Fig. 2. Both sets of pairwise interactions, i.e., V~p(Zi)between a particle (i) and an end-surface and V~,(h#) between two particles i and j, were taken to be the sum of the electrical double layer interaction (VE) and the van der Waals interaction (VA). The Hogg-Healy-Fuerstenau expression (6) was used for lie and the approximate Hamaker expression (7) for VA. Thus,

v~p(z~) = ~ , ~ ( ~ + ~P) ~ gFL~z~+ +ln(1-exp-2xz~ +

1 z + 26 + 2a

+ In

in (11+- expeXp- KZ~]KZi/

]} a[z' (

-6-

+26

z +2~

.~l

z + 26 + 2 a l l

[11

and V~(h~s) = lr~reoa~ In (1 + exp - xh0)

A F 2a 2 2a 2 [r~ - 4 a 2 \ l + ~ + In / "--~-~-----// 6 r~j - 4a 2 rij ~ rij /J

L

[21

where r~j = h~j + 2a. Note that the leading term on the right-hand side of Eq. [2] is the Derjaguin expression (8) for the electrostatic interaction between two identical spheres (Ka ,> 1 and ¢ < 25 mV). ~r is the relative permittivity of free space. A is the effective Hamaker constant for the polystyrene/water system (7.5 × 10-21 J) (9), and a the particle radius (0.2 ~m). 6, the thickness of the adsorbed polymer layer, was taken to be 32 nm for poly(vinyl alcohol) of molecular weight 24,000 on polystyrene latex (10). The values of Cs and Cp are difficult to estimate, but may, to a first approximation, be set equal to the appropriate zeta-potential (g'), if h and z are defined as indicated in Fig. 2. Even so some difficulty still exists in assigning values of ~'~and ~'p.For the purposes of these calculations the values shown in Table I were used, based on the experiments reported by Vincent et al. (1, 11) for

263

poly(vinyl alcohol), molecular weight 24,000 adsorbed on small, positive particles (~'p) and large, negative particles (~'J. The experiments are concerned with the adsorption of small particles onto much larger particles. It should, therefore, be reasonable to treat the surface of the large particle as planar. A computer simulation study of this colloidal problem can then proceed in exactly the same way as studies of the distribution of molecular fluids at plane solid surfaces (12). The system consists of an assembly of small particles constrained to a rectangular prism, of dimensions 30 × 10 × 10 d3; the largest of these dimensions represents the spacing between the plane surfaces representing the large particles; the usual periodic boundary conditions are invoked in the directions parallel to the solid surfaces. The very low particle number densities, expressed in units of d 3 in Table II, allow one to treat the colloidal "gas" of small particles as ideal. Thus the chemical potential # and the number density p are simply related by # = In p.

[3]

The configurational energy of the assembly of N small particlesis ~N = Z V~,(hi~) + Z [V~(z~) + V~(H - z~)l i
[41

i

where H(= 30d) is the spacing between the solid surfaces and the pair potential Vpp was truncated at 5 particle diameters. The scheme used to generate particle configurations and to determine the adsorption excess is the Grand Canonical Ensemble Monte Carlo method discussed in detail elsewhere (12). For those cases where ~'p is nonzero the potential V~ has a fairly deep and narrow minimum. Consequently, the system was slow to equilibrate; several million configurations were required before normal sampling, consisting of a further 2 to 3 million configurations, was commenced. RESULTS AND CONCLUSIONS The results of the computations are shown in Table II for the range of NaCI concentrations 10-4 to 10 -~ mole dm -3. Several trends may be observed which correlate well with the experimental results reported previously (2, 4) (Fig. 1): (i) For 10-4, 10-3, and 10 -2 mole dm -3 electrolyte high-affinity isotherms are found. TABLE I Values of ~'-Potentials Used in the Monte Carlo Simulations [NaCI] (mole d m -3)

10-'

10 -2

10-3

10 "4

~'p(mV)

0

0

+4

+21

~'~(mY)

-8

- 12

-25

-60

Journal o f Colloid a n d Interface Science, Vol. 92, No. l, March 1983

264

NOTES TABLE II Results o f Monte Carlo Simulation a*

[NaCI](moledm -J)

t~

o

Nb

N

AN

F

10-I

-6.2 -6.9 -7.5

2.03 × 10 -3 1.01 × 10 -3 5.53 X 10 -4

6.1 3.0 1.7

54.4 26.8 14.6

48.3 23.8 12.9

0.242 0.119 0.065

10 -2

-5.5 -6.2 -6.9 -7.5

4.09 2.03 1.01 5.5

10 -3 10 -3 10 -3 10 -4

12.3 6.1 3.0 1.7

65.0 56.4 58.7 59.2

52.7 50.3 55.7 57.5

0.26 0.25 0.28 0.29

10 -a

-5.5 -6.9 -7.5

3.9 × 10 -3 9.8 × 10 -4 5.5 X 10 -4

11.7 2.9 1.7

85.3 77.6 76.4

73.6 74.7 74.7

0.37 0.37 0.37

10 -4

-5.5 -6.2 -7.5

3.55 X 10-3 1.9 × 10 -3 5.5 × 10 -4

10.7 5.7 1.7

84.0 80.7 78.4

73.3 75.0 76.7

0.26 0.27 0.28

× × × ×

* u is the chemical potential, O the number density of small particles, N b the number of particles in the cell without adsorbing surfaces, N the ensemble average number of particles in the cell with adsorbing surfaces, AN = /V - Nb, and the adsorption excess I' = AN/200 ------0.

(ii) For this range o f electrolyte concentrations there appears to be a m a x i m u m in the plateau value o f the coverage (0) at ~ 10-3 mole din-3; the value o f the maxi m u m is 0 ~ 0.37. (iii) The plateau value of the coverage first increases with decreasing electrolyte concentration and then decreases. (iv) For 10-1 mole d m -3 electrolyte a tow-affinity isotherm results. Clearly, the exact values obtained for 0 will depend on the actual values chosen for ~'s and ~'p, but over the range o f reasonable values for these parameters, similar trends to those listed above can be expected. Also the actual form o f the potential energy function will affect 0. Hence, these adsorption data may prove useful for testing theories of interparticle forces. From the above and the fact that the Monte Carlo simulations show only monolayer adsorption, we cond u d e that the two-dimensional model used to interpret the earlier experimental results is reasonable. ACKNOWLEDGMENT This work was carried out while B.V. was a visiting fellow at the University o f Melbourne. REFERENCES 1. Vincent, B., Young, C. A., and Tadros, Th. F., Faraday Disc. Chem. Soc. 6 5 , 296 (1978). 2. Vincent, B., Young, C. A., and Tadros, Th. F., J. Chem. Soc. Faraday Trans. 76, 665 (1980).

JournalofColloidandInterfaceScience.Vol. 92. No. 1. March 1983

3. Vincent, B., Jafelicci, M., Luckham, P. F., and Tadros, Th. F., J. Chem. Soc. Faraday Trans. 7 6 , 674 (1980). 4. Luckham, P. F., Vincent, B., Hart, C. A., and Tadros, Th. F., Colloids Surf 1, 281 (1980). 5. Young, C. A., Ph.D. thesis, Bristol, 1978. 6. Hogg, R., Healy, T. W., and Fuerstenau, D. W., Trans. Faraday Soc. 62, 1638 (1966). 7. Hamaker, H. C., Physica (Utrecht) 4, 1058 (1937). 8. Derjaguin, B. V., Trans. Faraday Soc. 36, 203 (1950). 9. Vincent, B., J. ColloidlnterfaceSci. 42, 270 (1973). 10. van den Boomgaard, Th., King, T. A., Tadros, Th. F., Tang, H., and Vincent, B., J. Colloid Interface Sci. 66, 68 (1978). 11. Ref. (1), p. 334. 12. Snook, I. K., and van Megen, W., J. Chem. Phys. 72, 2907 (1980). W. VAN MEGEN* I. SNOOK* B. V1NCENTt"z

*Department of Applied Physics Royal Melbourne Institute of Technology Melbourne, Victoria, Australia tDepartment of Physical Chemistry University of Bristol Bristol BS8 1TS, England Received June 1, 1982; accepted June 7, 1982 t To whom correspondence should be addressed.