J. Chem. SOC., Faraday Trans. 1, 1985, 81, 117-125 Studies of Electrical Double-layer Interactions in Concentrated Silica Sols by Small-angle Neutron Scattering BY JEFF PENFOLD AND JOHN D. F. RAMSAY* Neutron Division, Rutherford Appleton Laboratory, Chilton, Oxfordshire and Chemistry Division, AERE, Harwell, Oxfordshire Receiued 5th April, 1984 Effects of ionic strength, volume fraction and pH on the interparticle interactions in concentrated silica sols of different particle size have been investigated by small-angle neutron scattering. The results have been modelled using the solution of the mean spherical approximation (MSA) of Hayter and Penfold and Hansen and Hayter. In this model the pair potential comprises a hard core and a soft tail, due to the screened coulombic repulsion between the electrical double layers surrounding the spherical particles. Closed analytic solutions of the structure factor, S(Q), for this potential have been used in the fits to the experimental scattering data to obtain the surface charge and potential of the silica particles.Although the surface charge so obtained rises with increases in the pH of the sols, as expected, values are markedly less than the total surface charge density of silica as previously reported. There has recently been considerable interest in the determination of the effective pair potential resulting from the double-layer interaction between colloids in concen- trated dispersions. Particular progress has been made from the analysis of small-angle neutron scattering (SANS) from monodispersed latex particle^^-^ and surfactant micelles4 by modelling the behaviour of the dispersions as a one-component fluid of colloidal particle^.^ This approach, which has also been applied to dilute dispersions, has been based on established liquid-state models which involve either computer- simulation6? ’ or integral-equation methods.8> The latter approach, using the solution of the mean spherical approximation (MSA) as developed by Hayter et al.9*10 has recently been applied to experimental SANS data obtained with silica Results of the MSA analysis are described more fully in the present paper, which is an extension of our previous structural investigations12 and results obtained by modelling the systems using a hard-sphere (HS) p0tentia1.l~ An important feature which is demonstrated here is the effect of the pH of the silica sols on the small-angle scattering behaviour and the corresponding effective surface charge obtained using the MSA model.EXPERIMENTAL MATERIALS Concentrated silica sols of different particle size (Ludox SM, HS and TM) were obtained commercially (E. I. du Pont de Nemours & Co.) from designated sample batches (S2, S3 and S4) which have been studied previously.l29 l 3 These stock sols were dialysed repeatedly against dilute NaNO, solutions of different ionic strength and controlled pH as already described. To reduce the background incoherent neutron scattering from water, the sols were prepared in deuterium oxide ( > 99 D,O). Particle-size distributions, determined by transmission electron 117118 DOUBLE-LAYER INTERACTIONS IN SILICA SOLS microscopy,12 showed discrete, almost spherical, particles having mean diameters, d,, of 12, 17 and 30 nm for S2, S3 and S4, respectively (standard deviation o z 0.2).SMALL-ANGLE NEUTRON SCATTERING Measurements were made as previo~slyl~ at a wavelength, A, of 10 A on sol samples contained in silica cells (path length 2 mm) using a multidetector instrument14 installed in the PLUTO reactor at AERE, Harwell. Data were analysed using standard programs to normalise counter efficiencies and to correct for sample self-absorption and incoherent background. Absolute scattered intensities, expressed as macroscopic cross-sections, [dZ/di2],,,, were obtained using a light-water standard. MICROELECTROPHORESIS Measurements of electrophoretic mobility, u, were made on dilute (< g cmP3) S3 sols with a laser-electrophoresis instrument of similar design to that described15 previously.The corresponding zeta potentials, [, were calculated using the Huckel equation (viz. u = 3&[/2q; for KR < 1). DATA TREATMENT AND ANALYSIS NEUTRON SCATTERING The coherent macroscopic scattering cross-section for a concentrated colloidal dispersion of identical particles is given by16 where Q is the scattering vector, defined as for a scattering angle 28 and wavelength A, pp and ps are, respectively, the mean scattering length densities of the particles and solvent, & is the volume of each particle, np is the particle number density and P(Q) is the particle form factor, which for spheres of radius R is given by 3[sin (QR) - QR cos (QR)] Q3R3 P(Q) = ( (3) The structure factor, S(Q), is determined by the nature of the particle interaction potential; for non-interacting systems S(Q) = 1 .The spatial distribution of the particles as a function of the mean interparticle separation, r, is given by the particle pair-distribution function g(r) and is related to S(Q) by the Fourier transform as already illustrated12 for the silica sols considered here. DATA FITTING The MSA model gives a closed analytic solution to S(Q) as a function of several parameters of the system defining the interparticle potential : (volume fraction, d p , charge, 0, and screening, K ) . This function is calculated as has been described previo~sly,~~ lo and the product KP(Q)S(Q), where K contains the terms remaining in eqn (l), is fitted to the experimental data.The form of the potential, U(r), between the spherical particles is defined by a hard-J. PENFOLD AND J. D . F. RAMSAY 119 core potential and the standard Coulombic potential due to the mutual interaction of their electric double layers :17 U(r) = co, r < 2R U(r) = 4 n ~ , ER*I,Y; exp [ - ~ ( ~ - 2 R ) l / r , r > 2R where I,Y, is the surface potential, E is the dielectric constant of the solvent, E , is the permittivity of vacuum and IC is the Debye-Hiickel inverse screening length, which is given by where ci is the ionic strength of the solution and N is Avogadro’s number. The surface charge on the particle zp is related to yo by the approximation y / , = Zp/47tEEo R( 1 + KR).(8) By inputting K and &,, experimental data are fitted by least-squares refinement to an arbitrary intensity scale given by the calculated product of KP(Q)S(Q) to obtain 0, the surface charge density. The factor, YSCAL, by which the absolute theory must be multiplied to match the absolutely scaled data, is then calculated. In view of systematic errors and uncertainties in cross-sections, fits are considered satisfactory1* if this ratio is within 30% of unity. Here we have taken psio2 and pDZ0 as 3.49 x loplo cmP2 and 6.34 x cm-*, respectively, and [dC/dR],2, = 0.514 cm-l sr-l, and have obtained values of YSCAL which were consistently in the range ca. 0.8-> 0.9. RESULTS AND DISCUSSION EFFECTS OF SOL CONCENTRATION AND IONIC STRENGTH The dependence of scattered neutron intensity, I@), on momentum transfer, Q, for S3 silica sols, covering a range of concentration, which have been dialysed against electrolytes of two different ionic strengths (lop4 and 5 x lop3 mol dmP3) but similar pH (ca.8), are shown in fig. 1 and 2, respectively. The development of the maxima in I(Q) is caused by interference effects and indicates that the particles are not arranged at random but have some short-range ordering due to interparticle repulsion. Thus the movement of the maxima to higher values of Q with increasing sol concentration reflects a reduction in the equilibrium separation distance, rg(r)max, between the particles as already discussed.l* Values of rg(r)max in table 1 are those derived by Fourier-transforming [cf.eqn (4)] the fitted S(Q) obtained with the MSA model. It would seem from the similarities in scattering behaviour shown in fig. 1 and 2 that S(Q), and hence the interparticle potential, is almost unaffected by changes in the ionic strength of the dialysing electrolyte, cf. This at first sight would seem surprising because it is well known17 that the screening of the electrostatic interaction between colloidal particles is controlled by the ionic strength of the supporting electrolyte, ci. In the present situation, however, cf and ci may differ significantly due to the Donnan effect, which results from the establishment of osmotic equilibrium across the dialysing membrane, as has recently been discussed.3 Thus if we consider the colloidal particles as macroions of charge zp, then to preserve electroneutrality there will be an additional compensating counter-ion contribution to the bulk - electrolyte such that where zi is the counter-ion charge.Nc, zi 2 Ncf zi + np zp (9)120 DOUBLE-LAYER INTERACTIONS IN SILICA SOLS 100 3 I I Fig. 1. Small-angle neutron scattering of S3 sols of different concentrations dialysed against mol dm-3 sodium nitrate solution: 0, 0.081; 0, 0.175 and 0, 0.384 g ~ m - ~ ; continuous lines are the theoretical fits to the data. 0 0.025 0 0.025 0 0.025 0.05 Q1A-l Fig. 2. Small-angle neutron scattering of S3 sols of different concentrations dialysed against 5 x mol dm-3sodiumnitratesolution: .,0.137; .,0.266and +,0.550 g ~m-~;continuous lines are the theoretical fits to the data. It can be seen that the difference between ci and cf will increase as the particle number concentration, np, increases, and will hence be particularly marked where the particles are of small size and have a high volume fraction, as occurs here.Thus it can be shown that for a colloidal dispersion containing particles with radius R, volume fraction q$,J. PENFOLD AND J. D. F. RAMSAY 121 Table 1. Results from fitting the MSA model to SANS data obtained on silica sols of different concentrations, pH ca. 8 sol conc./g cmP3 cflmol dm-3 c;/mol dm-3 2R/nm a/pC cmP2 cy,/mV ~ ~ ( ~ ) ~ ~ ~ / n s 3 0.08 1 ca. loP4 5 x 20.0 0.346 28.2 45 s 3 0.175 ca. lop4 1.5 x 19.2 0.452 27.7 33 s 3 0.384 ca. lop4 4 . 6 ~ loP3 17.9 0.613 25.2 24 s 3 0.137 5 x 7 x 19.7 1.01 28.0 35 s 3 0.266 5 x 10-3 9 x lop3 18.6 0.765 26.0 27 s 3 0.550 5 x 1 .4 ~ 10P 16.7 0.662 18.5 21 with an effective surface density of charge, a, corresponding to the diffuse part of the electrical double layer, we have (10) where 4, = (4/3)nR"n,. Thus for a 1 : 1 electrolyte the effective counter-ion concentration, c; = [ci/( 1 -&)I, within the interstitial solvent volume between particles (rather than the overall dispersion volume) which will determine the screening parameter, IC, is then given by np zp = 3q4, a/ R c; = (cf + 3.12& all?)/( 1 - f$p) where, expressed in dimensionless units, we have c;/mol dmP3, R / A and a/pC cm-2. In fig. I and 2 the fits to the experimental data make allowance for an increase in K resulting from the additional counter-ion concentration, c;.Values of the fitted parameters are given in table 1. These show that R is slightly larger, although in satisfactory accord with that determined by electron microscopy, and that the effective surface potential, yo, is approximately half the zeta potential as measured by electrophoresis (see fig. 4 later) on much more dilute S3 sols under similar pH conditions. Values of the effective surface charge, a, are furthermore markedly lower than those obtained by conductimetric titration19 ( > 2 pC cm-2). It is also evident that values of c( for the two sets of sols, despite having been dialysed at markedly different values of ci, are similar. This would thus explain the similar scattering behaviour observed at comparable sol concentrations and the apparent insensitivity of the interaction potential to the ionic strength of the supporting electrolyte.The latter feature will be particularly evident as the particle size is decreased and thus may explain why maxima in S(Q) for silica and ceria sols, containing exceptionally small particles ( R < 100 A), do not increase markedly with &,13 as is observed2 with latex dispersions with a considerably greater particle diameter (2 300 A). Nevertheless there is evidence that S(Q) does indeed become less sensitive to K with increasing sol concentrations. This is demonstrated in fig. 3, which shows scattering data for a sol of high concentration (dialysed mol dmp3) and fits using the MSA model with fixed values of K corresponding to ionic strengths of lop4, and lop2 mol dm-3.Thus it is evident that all the fittings are similar in the region of the maximum in I(Q) and only show marked discrepancies in the low-Q limit, which is outside the range of measurement with the present instrument. Furthermore, using unrealistically high and low values of K , fits can be achieved by relatively marginal changes in the fitted surface charge. This would suggest that at high volume fractions the form of S(Q) is much more sensitive to a than K .122 DOUBLE-LAYER INTERACTIONS IN SILICA SOLS Fig. 3. Theoretical fits to scattering data I(Q) for silica sol S3 (0.38 g ~ r n - ~ ) dialysed against NaNO,; c; = and mol drn-3. K is fixed and corresponds to ionic strengths of mol drn-3 for (a), (6) and (c) giving a/pC cm-2 of 0.42, 0.44 and 0.83 respectively.Fig. 4. 50 - - I n - -80 E E - > E . - -40 I I I I 0 10 --. ; - 5 : 3 7 25 2 4 6 8 PH Dependence of electrophoretic mobility, u, and zeta potential, [, on pH (< g ~ r n - ~ ) silica sol S3 (“a+] z lop4 mol dmp3). I n I m --. 5 3 0. I lo 10 for a dilute EFFECTS OF pH It is well known20 that the surface of silica has a negative charge which increases gradually with rising pH, as is demonstrated in fig. 4 by the electrophoresis results obtained with dilute S3 sols. Changes in the scattering behaviour of concentrated S3 sols shown in fig. 5 reflect the difference in surface charge which results from dialysing at pH values of 7.2 and 4.5. Thus the sharper peak obtained at the higher pH indicates a stronger repulsive interaction, which arises from a greater effective surface charge, as is confirmed by the 0 values obtained from the fits given in table 2.The marked effect of surface charge on S(Q) is also demonstrated (fig. 6) by the scattering of S2 sols. Because these have a smaller particle size it is possible, with the same accessible range at small Q, to examine effects at lower volume fraction. It is then evident that differences in ionic strengths of the dialysing electrolyte produce significant changes in the S(Q) of sols of similar pH, which leads to a correspondingJ. PENFOLD AND J. D. F. RAMSAY I23 100 - I L. n ... I f 2 50 5 , w =3 ( b ) 1 I 0.03 0 0.03 0.06 QiA Fig. 5. I(Q) plotted against Q for S3 sols of similar concentrations (ca. 0.19 g cmP3) but different pH, ( a ) 7.2 and (b) 4.5, both dialysed against loP4 mol dmP3 sodium nitrate.-A 0.03 Fig. 6. Small-angle scattering of S2 sols of different pH dialysed against sodium nitrate of concentration (a) 5 x loP3 and (b) loP4 mol dmP3. Concentrations/g and pH of sols are, respectively: 0, 0.129 and 10.0, 0, 0.157 and 5.6, 17, 0.092 and 7.5 and ., 0.097 and 4.1.124 DOUBLE-LAYER INTERACTIONS IN SILICA SOLS Table 2. Results from MSA model fits to SANS data obtained on silica sols of different pH sol conc./g ci/mol dm-3 pH c;/mol dm-3 2R/nm o/pC cmP2 y/,/mV s3 0.197 ca. lop4 4.5 6 x 18.7 0.181 13.7 s 3 0.182 ca. 7.2 1 . 2 ~ 18.9 0.352 22.9 s 2 0.092 ca. 7.5 1 x 10-3 14.8 0.357 21.4 s2 0.097 ca. lop4 4.1 5~ 10-4 15.3 0.177 11.9 s 2 0.05 1 5~ 10-3 10.0 6 x 15.2 0.886 32.8 s2 0.129 5~ 10-3 10.0 9~ 10-3 13.9 1.32 40.9 s 4 0.314 ca.5.0 1 x 10-3 27.3 0.226 18.1 difference in the fitted 0 value (cf. table 2). However, the most striking feature is the complete absence of a maximum in I(Q) in fig. 6(a), which demonstrates that the interparticle potential is very weak at pH 5 5 owing to the low charge and short screening distance, K . CONCLUSION Measurements of interaction potential between silica-sol particles obtained from the MSA model reported here clearly reflect the effects of changing pH on the zeta potential as determined independently by laser electrophoresis. Zeta potentials are nevertheless significantly larger, a feature which may arise because measurements are of necessity restricted to much less concentrated sols. The considerably smaller effective surface charges obtained using the MSA model and the total surface charge as measured by conductimetric titration can be ascribed to the adsorption of counter-ions (Na+) at the silica surface.This feature, which has already been noted to a lesser extent with latex particles,l9 may arise because the silica particles have a diffuse surface with a high adsorption capacity for counter-ions.21* 22 This diffuseness may also be responsible for the slight, but significant, increase in the value of R, obtained from the MSA fits as the sol concentration is decreased compared with the particle size determined by electron microscopy. Another possible reason for the discrepancy between surface potentials obtained from model fits and other experimental measurements which cannot be overlooked concerns the assumptions inherent in the MSA model, in particular the use of the Debye-Huckel potential to describe double-layer interactions in concentrated dispersions.Consequently while the application of the Hansen-Hayter MSA form for S(Q) is a useful approach for displaying trends in potential parameters as shown here, it may have limitations in providing absolute values. Further verification of the model may be obtained in the future, if SANS measurements are extended to a much lower Q than accessible here, since it is now well known from liquid theory that details of the interaction potential are contained in S(Q) particularly in this range and that the form of S(Q) at higher Q represents the ‘softness’ of the repulsive core of the pair potential. We are indebted to Mr L.Benest for making electrophoretic mobility measurements and gratefully acknowledge helpful discussions with Prof. R. H. Ottewill and Dr J. B. Hayter.J. PENFOLD AND J. D. F. RAMSAY 125 K. Alexander, D. J. Cebula, J. W. Goodwin, R. H. Ottewill and A. Parentich, Colloids SurJ, 1983, 7, 233. D. J. Cebula, J. W. Goodwin, G. C. Jeffrey, R. H. Ottewill, A. Parentich and R. A. kchardson, Faraday Discuss. Chem. Soc., 1983, 76, 37. B. Beresford-Smith and D. Y. C . Chan, Faraday Discuss. Chem. Soc., 1983, 76, 65. J. B. Hayter, Faraday Discuss. Chem. Soc., 1983, 76, 7. W. van Megen and I. Snook, J. Chem. Phys., 1977,66, 813. ’ E. Dickinson, Faraday Discuss. Chem. Soc., 1978, 65, 127. D. W. Shaeffer, J. Chem. Ph-ys., 1977, 66, 3980. J. B. Hayter and J. Penfold, Mol. Phys., 1981, 42, 109. l o J. P. Hansen and J. B. Hayter, Mol. Phys., 1982, 46, 651. J. D. F. Ramsay, Faraday Discuss. Chem. Soc., 1983, 76, 108. l 2 J. D. F. Ramsay and B. 0. Booth, J . Chem. 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