Langmuir 1992,8, 2880-2884
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Static Structure of Strongly Interacting Colloidal Particles I. K.Snook* Department of Applied Physics, Royal Melbourne Institute of Technology, Melbourne, Victoria 3001,Australia
J. B. Hayter Solid State Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831 Received April 9,1992 We show that in order to we or even to test the accuracy of rescaled mean spherical approximation (RMSA) model of the structure of colloidal dispersions, one muat take into account the penetrating background (PB)inherent in the we of the mean sphericalmodel applied to the one-componentmacrofluid model (OCM)of colloidaldispersions. This can be done analyticallyand no major modificationsof existing computer programsnor of analysesof experimentaldata need be made. All that is needed is areinterpretation of the parameters obtained from such studies. Furthermore, for typical systems of strongly interacting colloidal particles we show by comparison with machine-generatedvalues that the RMSA with PB gives extremely accurate results. Introduction One of the main sources of information about the structure and interparticle forces in systems of strongly interacting colloidalparticles is the static structure factor, S(q).lv2However, in order to use the informationcontained in S(q)about real space structure and interparticle forces, one must have a theory which relates the reciprocal space quantity S(q)to the radial distribution function g(r) and to the interparticle potentialenergy function, a. It should be noted that even thoughg(r) and S(q)are simply related by a Fourier transform, experimental data on S(q) are rarely extensive or accurate enough to enable the determination of g(r) by inverse Fourier transformation of s(@. Thus, we need accurate statistical mechanical models which relate g(r), S(q), and e. In order to describetypical stronglyinteracting colloidal particles, it is firstly convenient to think in terms of an effective pair potential, bij, i.e.
i uj. There is no reason to expect all members of a given class to display the samestructure, however. Indeed, when uj (or qj) is sufficientlylarge, we expect the structure to be dominated by the excluded volume correlations and,
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2882 Langmuir, Vol. 8, No. 12, 1992
hence, to be strongly dependent on volume fraction. Even for small coupling, however, the potential rises rapidly above kBT as uj becomes less than some value, u*. Then for all uj > 1 and the probability of real contact configurations becomes negligible; g(y I uj) = 0. In this regime, the finite size of the particle can no longer play a physical role and the structure is dominated by the Coulomb correlations, and hence by rk. We thus arrive at the followingscaling result for charged OCM systems: for a given volume fraction 7j such that U j 1 h(x) = g ( x ) - 1 = -1, x < 1 c ( x ) = -@V(x),x
(6a) (6b)
where c ( x ) is the direct correlation function and x = r/u. Condition 6b is equivalent to the hard-core condition, /3V(x kBT. At high density, however, the MSA works very well, and Hansen and Hayter12proposed using the scaling result of the last section to take advantage of this fact. Rather than attempting to compute the'structure of the given (dilute) system,eq 4 is used to choose a new, but entirelyequivalent system, at a density where the MSA is an accurate approximation. The fully analytic MSA is then used to calculatethe structure, and the results are triviallyrescaled back for the length scaleappropriate to the originalsystem. The density 71 at which the actual computation is performed is chosen by a straightforward application of the Gillan criterion>l g(y = u1+) = 0. For small to intermediate Coulombcoupling;the RMSA gives results which compare very well with other procedures, but at higher couplings Hansen and Hayter12(and later Svensson and Jonsson14)noted that both the RMSA and the hypernetted chain (HNC) approximations someQD,
(19)Hayter, J. B.; Penfold, J. Mol. Phy8. 1981,42, 109. (20)Roeenfeld, Y.; Ashcroft, N. W. Phys. Rev. A 1979,20, 1208. (21)Gillan, M.J. J. Phys. C 1974, 7, L1.
what underestimate the structure compared with Monte Carlo or modified HNC (HNC) calculations. In the next section we will show that this is not due to a breakdown in the accuracy of the RMSA procedure but rather to a failure to take into account the uniform penetrating background inherent in the one-component MSA. RMSA with Penetrating Background In a colloidal system, the small ions present maintain electroneutrality and screen the potential energy of interaction of the large, charged macroions. In the OCM model of a colloid, both of these functions are absorbed into a background which is assumed to be completely uniform, implying that a fraction 7 of this background penetrates the colloidal spheres reducing the charge by a factor (1 - 7).l8 In order to maintain effective charge on a given sphere, Ze-, at its nominal value in the presence of such a permeating background,the actual charge must therefore be increased to Ze*- = Ze-/(l- 7) (7) In any comparison of the penetrating background MSA with a Monte Carlo or primitive model, which excludes thisbackground,this charge renormalizationmust be taken into account; a formal argument for this being given by Hansen. In the original RMSA computations it was wrongly assumed that this effect was negligible for small values of 7. This is because the effect of background penetration must be included at the actual density at which the MSA is used; after rescaling, this density is often high enough to require a non-negligible compensation. With inclusion of the penetrating background the presence of screening is straightforward. If we wish the MSA to model a physical system characterized by the parameters 7 , y, and k , the MSA computation must be undertaken using the effective potential parameters y* and k*, where from (lb) and (7) VI2 (8) and from the requirement of constant Coulomb coupling Y* = ?,/(I-
k* = k - 2~ log (1- 7) (9) where u = 71/3. Thus, in using the RMSA scheme, eqs 8 and 9 must be applied to the rescaled parameters each time the MSA is evaluated. We call this the PBRMSA scheme. In order to demonstrate the accuracy of the RMSA with allowance for the effect of penetrating background, we will now compare the RMSA using eqs 8 and 9 with accurate machine-generated results for typical values of y and k. Results and Discussion The many experimental tests of the RMSA are based on comparison of theoretical and experimental structure factors, S(q),usually by fitting theory to experiment. In general, the fits are excellent and the impact of the penetrating background connection on such fits is small. However, the absence of exact a priori knowledge of the colloidal charge prevents an unambiguous test of the impact of eqs 8 and 9, although it does suggest the form of the RMSA S(q) is accurate and that it varies correctly with the bulk physical parameters characterising the system. An example of an analysis of experimental data is given in Table I, which comphes analysisof typicalexperimental
Langmuir, Vol. 8, No. 12, 1992 2883
Strongly Interacting Colloidal Particles Table 1. Charge (2) and Aggregation Number ( N )of Sodium Dodecyl Sulfate (SDS) Micelles Derived by Fitting the RMSA without and with the Background Correction to Experimental SANS Datala background ignored 2 N 22.8 77.1 23.3 98.1 119.6 26.9 26.8 138.6
[SDSI, [NaClI, m~l.dm-~ mol.dm3 0.0 0.016 0.0 0.1 0.4 0.0 0.1 0.4
background included 2 N 22.1 77.1 20.7 98.1 22.4 119.5 24.2 138.7
Table 11. $0,
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C,
mV system u, A m01.h-3 A 10-6 150 230 B 104 150 230 C 104 150 230 D 10-6 150 230 E 104 150 230 F 10-6 230 900 104 G 230 900 H 1o-s 230 900 I 40 314 10-3 10-3 J 40 314
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9 x 10-5 1.5 X lo4 3 x 10-4 4.4 x 104 4.4 x 104 8.4 x 10-5 1.2 x 104 2.3 X lo-" 8 x 10-2 1.3 X lo-'
figure simulation 2a MC 2b MC 3a MC 3b MCb 4 MCc BD 5 BD 6 7 BD BD 8 BD 9
X
Figure 2. g(r) versus 3c = r / u for systems (a) A and (b)B whose parameters are given in Table I 1 (-) PBRMSA;(- - -) MC results.
$0 is the surface potential, u the particle diameter, c the added electrolyte concentration, and 1) the particle volume fraction of the systems simulated by the MC and BD methods. N = 108 particles. N = 256 particles. I
S(q) SANS3 data without and with correction for the
penetrating background. Although the charges derived from the two methods are slightly different, either set of results is in reasonable agreementwith available estimates of the expected degree of i0nizati0n.l~ A much more stringent test is provided by comparing the RMSA g(x) with the accurate values calculated by machine simulations such as the Monte Carlo (MCI9 method and the Brownian dynamics (BD) method22of the same OCM model. Such simulations have been performed on hard particle^^^^^ interacting through a pair potentialof the form (1). Since no background is included in these simulations the parameters used in the potential are those of the physical system. However, published comparison^'^ of such data with the RMSA have not included the background correction in the analytic calculation. In the latter case, the RMSA leads to a consistently less structured g(x) than the machine results at moderate to high coupling, as would be expected. Thus, in order to provide a proper test of the RMSA, we have used some existing MC data@for some typical highly charged colloidal particles and generated some new data by the BD method on other typical colloidal systems. The data characterizing these systems are shown in Table I1 and is appropriate to those systems which have been studied by SANS and laser light scattering. In fact rather than merely trying an enormous range of arbitraryparameters, we have chosen values characteristic of typical systems of strongly interacting colloidal particles which have been studied by dynamic light scattering or small angle neutron scattering.*25 The MC and BD calculations were carried out with both N = 108 and 256 ~
(22) Snook,I. K.;vanMegen, W.; Gaylor,K.;Watte,R. 0.Adu. Colloid Interface Sci. 1982, 17, 33. (23) Brown, J. C.; Pusey, P. N.; Goodwin, J. W.; Ottewill, R. H. J . Phys. A 1975,8,664. (24) Pusey, P. N. Philos. Trans. R. SOC.London, Sect. A 1979, 293, 219. (25)Cebula, D. J.; Goodwin,J. W.; Jeffrey, G. C.; Ottewill, R. H.; Parentich, A.; Richardson, R. A. Faraday Discuss. Chem. SOC.1983,76, 37.
/
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Figure 3. As in Figure 1 but for systems (a) C and (b) D.
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Figure 4. As in Figure 1 but for system E (to be compared with Figure 3b in order to illustrate N dependence).
particles in order to test for size dependence of g(r). As discussed before it is sometimes necessary to use a large number of particles when very long ranged screened Coulomb interactions are used. This is illustrated in Figures 2b and 3; however, in all cases here N = 256 proved to be adequate. As can be seen from the comparisone of g(r) generated by machine simulationand the PBRMSA,see Figures 2-8, the accuracy of the PBRMSA method is very good. In the most extreme cases, i.e. systems A-D, there is a small discrepancy between the two results, but it is minor and this is the most extreme case as it involves very highly charged, small particles at very low electrolyte concentration. In all cases, however, the discrepancy between S ( 4 ) obtained from Fourier transforming the machinegenerated g(r) and from the PBRMSA model would not be discernible on figures of the scale used here and are, thus, not shown. We have also tested the PBRMSA resulta for some systems obtained by varying the parameters involved in systems I and J. These involved many combinationsof the parameters $0, C, and $ in the range 30-60 mV, lo-* to 5 X mol dmJ, and 1 to 13%
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2884 Langmuir, Vol. 8, No. 12, 1992 20
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Figure 5. g(r) for system F of Table I1 (0)BD results.
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Figure 8. As for Figure 6 but for system I. 20
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Figure 9. As for Figure 5 but for system J.
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Figure 6. As for Figure 6 but for system G. 2.5
g(x I
Conclusions The physical model upon which the MSA (or RMSA) method for the one-componentmodel of colloids is based includes a penetrating background. Thus, in order either to use the RMSA model to interpret experimental data or to compare with machine generated results, one must take this fact into account. Fortunately, this does not alter the form of the RMSA results forg(r)and S(q)but only means that one must relate the effective RMSA charge Z* and parameter y* to the actual values of Z and y appropriate to the real physical system being treated. Furthermore, there exist analytical relationships between Z* and 2and y* and y,and thus, previous results may be corrected (if necessary) and existing programs used to implement the RMSA need no major revision. Such corrections are illustrated here by reinterpreting SANS S(q) data for concentrated micelle solutions (see Table I). In fact, we have shown that the results of the RMSA model including the effect of the penetrating background (the PBRMSA model) for g(r) are in excellent agreement with those of machine-generated results. This was done for a wide range of physical parameters which are typical of those involved in systemsof strongly interactingcolloidal particles.
. i
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40
X
Figure 7. As for Figure 6 but for system H.
respectively. In no case could we find discrepanciesgreater than those shown in Figures 7 and 8 and, thus, we do not include these comparisons here. It may be possible to find some sets of parameters where the PBRMSA fails but we were unable to find them for the typical colloidal systems we studied here.
