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Assessment of Electrostatic Interactions in Dense Colloidal Suspensions with Multiply Scattered Light Y. Huang,† Z. G. Sun,†,‡ and E. M. Sevick-Muraca*,† Department of Chemical Engineering, Texas A&M University, College Station, Texas 77843-3122, and School of Chemical Engineering, Purdue University, West Lafayette, Indiana 47907-1283 Received September 14, 2001 The electrostatic repulsive force between particles impacts the static structure of colloidal systems and dramatically hinders visible light scattering. In this investigation, we employed frequency domain photon migration (FDPM) to measure the isotropic scattering coefficients of dialyzed polystyrene latex suspensions at 687 and 828 nm as a function of ionic strengths between 1 and 120 mM NaCl equivalents. Measured isotropic scattering coefficients increased with decreasing ionic strength of the suspensions, suggesting that changes in electrostatic interactions can be evaluated in ensemble measurements of visible light scattering. At 120 mM NaCl equivalents, the isotropic scattering coefficients were accurately predicted by Mie theory with the Percus-Yevick (PY) polydisperse model to account for hard-sphere (HS) interactions. At lower ionic strengths, the isotropic scattering coefficients at varying colloidal volume fractions were regressed to scattering theory which incorporated the mean spherical approximation (MSA) with the hard sphere Yukawa (HSY) interaction for monodisperse suspensions in order to yield an effective surface charge given the size and ionic strength. The estimates of surface charges obtained from scattering data at 687 and 828 nm were consistently similar but varied with ionic strength. A three-component MSA primary model (PM) was used in an attempt to account for sample polydispersity but yielded poor fit and estimates of effective surface charges which varied for data taken at the two wavelengths.
1.Introduction The interactions among colloidal particles decide the local structure, which can impact optical properties, rheology, colloidal stability, and therefore the final performance of the industrial colloidal products.1,2 Typically, the static structure of complex fluids can be obtained from small-angle light scattering, neutron scattering, and X-ray scattering measurements. However, concentrated interacting colloidal mixtures multiply scattered light, making small-angle light scattering measurements cumbersome unless refractive index matching is accomplished. Neutron and X-ray scattering require a reactor or a synchrotron source, which limits their practical and ubiquitous measurement of structure in complex fluids. In addition, these techniques are limited to nanometer colloids. Recent work to use light scattering involves monitoring structure from turbidity measurements3,4 or diffuse light measurements.5 Turbidity3,4 measurements are limited in that they assume no multiple scattering into the path length and are consequently limited to dilute colloidal suspensions, while continuous wave (CW) diffuse light measurements can be difficult to calibrate for the extraction of pertinent structure information. * To whom correspondence should be addressed. Phone: (979)458-3206. Fax: (979)-845-6446. E-mail:
[email protected]. † Texas A&M University. ‡ Purdue University. (1) Hiemenz, P. C.; Rajagopalan, R. Principles of Colloid and Surface Chemistry, 3rd ed.; Marcel & Dekker: 1997; Chapter 11. (2) Hunter, R. J. Foundations of Colloid Science; Oxford Science, 1987. (3) Afpel, U.; Grunder, R.; Ballauff, M. A turbidity study of particle interaction in latex suspensions. Colloid Polym. Sci. 1994, 272, 820829. (4) Afpel, U.; Grunder, R.; Ballauff, M. Precise analysis of the turbidity spectra of a concentrated latex. Langmuir 1995, 11, 3401-3407. (5) Kaplan, P. D.; Dinsmore, A. D.; Yodh, A. G.; Pine, D. J. Diffuse transmission spectroscopy - a structural probe of opaque colloidal mixtures. Phys. Rev. E 1994, 50, 4827-4835.
Nonetheless, the structure of dispersions has strong influence over light scattering efficiency and it not only should be taken into account when sizing particles using ensemble light scattering techniques, but also can be used to provide information about particle interactions within an ensemble. Recently, in our laboratory, we have developed frequency domain photon migration (FDPM) techniques for accurate and precise measurements of isotropic scattering coefficients, which are sensitive to microstructure.6,7 While this and other work8,9 successfully demonstrated the use of the monodisperse hard-sphere Percus-Yevick model to predict static structure and therefore light scattering in dense polystyrene suspensions, these studies have been limited to the structure predominantly set up by volumeexclusion effects. More recently, we have used the analytical solution for mono- and polydisperse hard-sphere Percus-Yevick fluids to demonstrate the ability to perform particle sizing in dense colloid suspensions.6,7 In this work, we seek to extend time-dependent multiple light scattering techniques to include more subtle, yet significant interactions owing to electrostatics. In the following, we provide a brief background to the static structure factor S(q,φ) of a suspension starting with the radial distribution function, g(r), and OrnsteinZernike (O-Z) integral equation for relating g(r) to S(q,φ) using phenomenological models for particle potential u(r) (6) Sun, Z. G.; Tomlin, C. D.; Sevick-Muraca, E. M. Investigation of particle interactions in dense colloidal suspensions using frequency domain photon migration: monodisperse system. J. Colloid Interface Sci. 2001, 245, 281-291. (7) Sun, Z. G.; Tomlin, C. D.; Sevick-Muraca, E. M. Particle sizing of colloidal suspension at high volume fractions using frequency domain photon migration technique. Langmuir 2001, 17, 6142-6147. (8) Fraden S.; Maret G. Multiple light scattering from concentrated, interacting suspensions. Phy. Rev. Lett. 1990, 65, 512-515. (9) Garg, R.; Prud’homme, R. K.; Aksay, I. A.; Liu, F.; Alfano, R. R. Optical transmission in highly concentrated dispersion. J. Opt. Soc, Am. A 1998, 15, 932-935.
10.1021/la011436a CCC: $22.00 © 2002 American Chemical Society Published on Web 02/08/2002
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and closure relations. We develop the theoretical basis for the experimentally measured changes in isotropic scattering and angle-integrated structure factor in dense polystyrene suspensions of varying ionic strengths, which is used to evaluate actual data obtained from FDPM measurements. 2. Background The structure of a colloidal dispersion is typically represented by the radial distribution function, g(r). The structure factor, S(q,φ), is the Fourier transform of radial distribution function and directly relates the scattering at the wave vector, q, from an individual particle to the scattering intensity of an ensemble of particles at volume fraction φ. The structure of a colloidal dispersion is usually investigated by using Monte Carlo simulation or perturbation theory or by solving the O-Z integral integration with the appropriate potential model of pairwise particle interaction. The O-Z equation for a mixture is written as10
hij(r) ) cij(r) +
∑l Fi∫clj(r′)hil(|r - r′|) dr′
(1)
where i and j represent the unlike components in the colloidal mixture, r is the center to center distance between particles of components i and j, cij(r) is direct correlation function, and hij(r) is the total correlation function between components i and j. The total correlation function, hij(r), is related to the radial distribution function by hij(r) ) gij(r) - 1, where gij(r) is the radial distribution function between components i and j and Fi is the number density of particles of component i. Owing to the two unknown functions, hij(r) and cij(r), an additional approximate relation, such as mean spherical approximation or the Percus-Yevick approximation is needed as closure to solve for the structure factor. The PY approximation closes the O-Z equation with the approximated relation
cij(r) ) [1 - exp(βuij(r))]gij(r)
(2)
and the MSA closes the O-Z with the relations
hij(r) ) -1, r < σij ) (σi + σj)/2
(3)
cij(r) ) - βuij(r), r > σij
(4)
where σi is the diameter of the particle component i, the term β is kBT, and uij(r) is the pairwise interaction potential between unlike particles of component i and j at the separation distance of r. The pairwise interaction potential can be predicted by one of several models, which, for this study of charged hard spheres, are chosen to be the hard-sphere Yukawa model and primary interaction model. We choose two analytical solutions employing the HSY potential model because analytical solutions to the O-Z have been previously developed. The mean spherical approximation using the hard-sphere Yukawa potential model11-18 is appropriate for prediction of structure in monodisperse (10) Hansen, J. P.; McDonald I. R. Theory of Simple Liquids, 2nd ed.; Academic Press: New York, 1986. (11) Hayter, J. B.; Penfold, J. An analytical structure factor for macroion solutions. Mol. Phys. 1981, 42 (1), 109-118. (12) Blum, L.; Hoye, J. S. Solution of the Ornstein-Zernike equation with Yukawa closure for a mixture. J. Stat. Phys. 1978, 19 (4), 317324. (13) Blum, L. Solution of the Ornstein-Zernike equation for a mixture of hard ions and Yukawa closure. J. Stat. Phys. 1980, 22 (6), 661-672.
suspensions, while the mean spherical approximation using the primary model (MSA-PM)19-22 can be used to discretize a polydisperse suspension into separate interacting populations of spheres for prediction of structure. For uncharged systems, analytical solution to the O-Z exists for Percus-Yevick approximation using the pure hard-sphere potential model for mono- and polydisperse suspensions.2,10,23,24 2.1. Hard-Sphere Interactions with Yukawa Tail for Monodisperse Systems. The potential interaction of a monodisperse colloidal system can be expressed as a hard-sphere interaction with a Yukawa tail:
u(r) )
{
∞ r σ
(5)
where e is electron charge, 0 is the electric permittivity of vacuum, and is dielectric constant of the suspending medium. The parameter σ is again the particle diameter, z is the effective particle surface charge, and κ is the inverse Debye screening length. The HSY potential model does not conserve neutrality, and charged particles are modeled as macro ions immersed in a neutral background, surrounded by the screened electric double layer of thickness κ-1, which decreases with increasing ionic strength. The charged particles repulsively interact through volume exclusion and double-layer overlapping. The HSY model is also usually termed as a one-component model (OCM).10 The solution of the O-Z equation using the MSA approximation with HSY potential is possible as shown by Blum and Hoye.12,13,25,26 Hayter and Penfold have also derived an analytical expression for the structure factor for monodisperse suspensions using MSA-HSY.11 In their solutions, multiple roots to a polynomial equation do not enable guaranteed uniqueness of their solution. Herrera et al.15 derived a simple explicit expression of structure factors for MSA-HSY, which we use in this study to fit our experimental data. More recently, Ginoza and Yasutomi17,18 also derived the analytical expressions for structure factors for HSY mixtures that exhibit both charge and size polydispersity. (14) D’Aguanno B.; Klein, R. Structural effects of polydispersity in charged colloidal dispersions. J. Chem. Soc., Faraday Trans. 1991, 87 (3), 379-390. (15) Herrera, J. N.; Cummings, P. T.; Ruiz-Estrada, H. R. Static structure factor for simple liquid metals. Mol. Phys. 1999, 5, 835-847. (16) Pastore, G. Uniqueness and the choice of the acceptable solutions of MSA. Mol. Phys. 1988, 63, 731. (17) Ginoza, M.; Yasutomi, M. Analytical model of the static structure factor of a colloidal dispersion: interaction polydispersity effect. Mol. Phys. 1998, 93 (3), 399-404. (18) Ginoza, M.; Yasutomi, M. Analytical structure factors for colloidal fluids with size and interaction polydispersities. Phys. Rev. E 1998, 58 (3), 3329-3333. (19) Blum L. Mean spherical approximation for asymmetric electrolytes I. Method of solution. Mol. Phys. 1975, 30 (5), 1529-1535. (20) Blum, L.; Hoye J. S. Mean spherical model for asymmetric electrolytes. 2. Thermodynamic properties and the pair correlation function. J. Phys. Chem. 1977, 81 (13), 1311-1316. (21) Hiroike, K. Ornstein-Zernike relation for a fluid mixture with direct correlation functions of finite range. J. Phys. Soc. Jpn. 1969, 27 (6), 1415-1421. (22) Hiroike, K. Supplement to Blum’s theory for asymmetric electrolytes. Mol. Phys. 1977, 33 (4), 1195-1198. (23) Baxter, R. J. Ornstein-Zernike relation for a disordered fluid. Aust. J. Phys. 1968, 21, 563-569. (24) Baxter, R. J. Ornstein-Zernike relation and Percus-Yevick approximation for fluid mixtures. J. Chem. Phys. 1970, 52 (9), 45594562. (25) Hoye, J. S.; Blum, L. Solution of the Yukawa closure of the Ornstein-Zernike equation. J. Stat. Phys. 1977, 16 (5), 399-412. (26) Blum, L.; Ubriaco, M. Analytical solution of the Yukawa closure of the Ornstein-Zernike equation IV: the general 1-component case. Mol. Phys. 2000, 98 (12), 829-835.
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2.2. Primary Model of Interaction Potential for Mixtures. In the primary model, the interaction potential between two charged particles is given by
uij(r) )
{
r e σij ∞ 2 e zizj/(4πor) r > σij
(6)
where ΣiFizi ) 0 and zi is the surface charge of component i in the colloidal mixture. Unlike the HSY interaction model, the electroneutrality of the dispersion is maintained, and the particles interact with each other through direct Columbic and volume exclusion interactions. Counterions are also considered as a component in the dispersions. Blum and Hoye19,20 solved the O-Z equation analytically by using MSA for the PM interaction model. Hiroike21,22 reorganized Blum’s solution and derived an explicit expression of direct correlation function, whose Fourier transform can be used to directly calculate partial structure factor Sij(q).2,27 Yet when dealing with dilute particle systems of strong electrostatic interactions, the MSA predicts negative probability density, g(r), at r ) σ. Hansen and Hayter28 rescaled the physical diameter of a monodisperse dispersion to a larger effective diameter σeff, which satisfies zero contact probability density, g(σeff) ) 0; the new model is called the rescaled mean-sphere model (RMSA). Finally, Ruiz-Estrada et al.29 extended the scaling technique to the colloidal mixtures using MSA-PM by requiring g(σeff,ii) ) 0, establishing the connection between MSA-HSY and MSA-PM for polydisperse suspensions. 2.3. Scattering Properties of Dense Colloidal Suspensions. For a well-characterized colloidal suspension, the scattering intensity, I(q), of the suspension, can be approximately predicted by30,31
I(q) )
∑i ∑j Fij(q,σi,σj)(FiFj)1/2Sij(q)
(7)
where Fij(q,σi,σj) is the binary form factor between the homogeneous spherical particles of size σi and σj at wave vector q, and binary form factor Fij can be calculated by
Fij ) Re(f1,if*1,j + f2,if*2,j)
(8)
where f1,i and f2,i are the scattering amplitudes into two orthogonal polarization states arising from a particle with size σi, which can be calculated by Mie scattering theory. f*1,j and f*2,j are the complex conjugates of f1,i and f2,i, respectively. For a monodisperse system, particles are of the same size and the form factor Fii can be simply calculated by
Fii ) f1,if*1,j + f2,if*2,j
(9)
It is noted that the calculation of the binary form factor with different sizes using Mie theory of light scattering is not the same as that for X-ray or neutron scattering. (27) Salgi, P.; Rajagopalan, R. polydispersity in colloids: implications to static structure and scattering. Adv. Colloidal Interface Sci. 1993, 43, 169-288. (28) Hansen, J. H.; Hayter, J. B. A rescaled MSA structure for dilute charged colloidal dispersion. Mol. Phys. 1982, 46 (3), 651-656. (29) Ruiz-Estrada, H.; Medina-Noyola, M.; Nagele, G. Rescaled mean spherical approximation for colloidal mixtures. Phys. A 1990, 168, 919941. (30) Griffith, W. L.; Triolo, R.; Compere, A. L. Analytical scattering function of a polydisperse Percus-Yevick fluid with Schulz - (Γ-) distributed diameters. Phys. Rev. A 1987, 35, 2200-2206. (31) Blum, L.; Stell, G. J. Polydisperse systems. I. Scattering function for polydisperse fluids of hard or permeable spheres. J. Chem. Phys. 1979, 71 (1), 42-46.
Because the wavelengths of X-rays or neutrons are much longer than the effective sizes of the nuclei and electrons, which scatter, the Rayleigh-Gans-Debye (RGD) approximation can be appropriately used and the scattering amplitudes are real. However, Mie scattering theory should be employed in this study because the wavelengths used in our experiments are comparable to the particle sizes of dispersions. In Mie theory, the scattering amplitudes are complex, so eq 8 should be used to calculate binary form factors for particles of different sizes. The partial structure factor Sij(q) indicates the averaged interference effects of scattered light resulting from component i and component j at wave vector q. In diluted suspensions (volume fractions 300. The mean size and ζ potential of the polystyrene was measured by dynamic light scattering (Zetasizer 3000HS, Malvern), after the sample was diluted in 0.1 M NaCl solution. The surface charge of the particle, z, can be approximately evaluated by z ) πσ(2 + κσ)φo.1 Here σ and φo are the particle diameter and the electrostatic potential at particle surface, respectively. ζ potential, the potential at the shear plane, can be measured and is usually considered close to the actual surface potential, which so far cannot be experimentally determined. By use of the ζ potential instead of φo, the surface charge can be estimated.
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Figure 1. Number density size distribution for the threecomponent discretization (solid bars), which has the same mean diameter and standard deviation with continuous Schulz distribution (line) with mean size determined by DLS and deviation determined by TEM. Table 1. Discretization of the Particle Size Distribution According to the Schulz Distribution12a particle diameter, nm no. fraction
1
component 2
3
102.3 0.255
136.4 0.643
177.2 0.102
a Volume-averaged diameter is 142 nm; number-averaged diameter is 132 nm; polydispersity index is 1.08.
4. Results and Discussion 4.1. Particle Characterization and Size Distribution Discretization. The volume-averaged diameter determined by DLS was found to be 143 ( 3 nm, and the standard deviation of the size distribution obtained by digitizing TEM images associated with Image Pro software is 22 ( 2 nm. The ζ potential of the polystyrene used is 82 ( 4 mV, and the corresponding average surface charge is calculated to be around 1000 e. To employ the MSA model for polydisperse mixtures, the particle size distribution (PSD) is histogrammatically represented by a three-component mixture,14,39 which is shown in Figure 1. The level of discretization is determined by requiring the equality of the first six moments of both the continuous PSD and the discretized three-component distributions. The result of the discretization is listed in the Table 1. The discretization using more components did not change the predicted isotropic scattering coefficient significantly, consistent with other reports.39 Our calculation shows that the difference between three- and fivecomponent discretization results in approximately 0.2% change in scattering, which is less than the precision of the FDPM measurement. 4.2. Isotropic Scattering versus Volume Fraction. Figure 2 shows the isotropic scattering coefficients [cm-1] at wavelengths of 687 and 828 nm as a function of volume (39) Nagele, G. On the dynamics and structure of charge-stabilized suspensions. Phys. Rep. 1996, 272, 215-372.
Figure 2. Isotropic scattering coefficients versus volume fraction obtained from FDPM measurements at two wavelengths and five ionic strengths in NaCl equivalents (solid circle, 120 mM, 687 nm; open circle, 120 nm, 828 nm; solid square, 60 mM, 687 nm; open square, 60 mM, 828 nm; solid triangle, 25 nm, 687 nm; open triangle, 25 nm, 828 nm; solid diamond, 5 mM, 687 nm; open diamond, 5 mM, 828 nm; star, 1 mM, 687 nm; +, 1 mM, 828 nm). Table 2. Fitted Effective Average Charge of MSA Models ionic strength, NaCl equiv, mM debye screening lenth, nm effective charge 687 nm MSA-HSY, e 828 nm effective charge 687 nm MSA-PM, e 828 nm
120 mM
60 mM
25 mM
5 mM
1 mM
0.88 1.9 0 0 0
1.24 1240 1005 7 3
1.93 836 1137 12 9
4.31 404 489 15 12
9.63 137 184 15 13
fraction and ionic strength (NaCl mM equiv). The debye lengths corresponding to these ionic strengths are listed in Table 2. The scattering coefficients of all samples at 828 nm are less than those at 687 nm at the same volume fraction, as expected for the 143-nm mean diameter. It is evident that the isotropic scattering coefficients, at both wavelengths, first increase with the volume fraction, achieve a maximum at volume fractions around 18%, and
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Figure 3. Comparison of the isotropic scattering coefficient obtained from FDPM measurements (solid circle, 687 nm; open circle, 828 nm) and predictions using Mie theory and the Percus-Yevick model (solid line, 687 nm; dash line, 828 nm).
then decrease. The plateau and the decrease of the isotropic scattering result from the interference of the scattered light owing to the structure or nonrandom orientation of scatters. The isotropic scattering coefficient also increases as the ionic strength increases or as the Debye screening length decreases. When the ionic strength decreases from 5 to 1 mM NaCl equiv (correspondingly, the debye length doubles from 4.3 to 9.6 nm), the scattering coefficient does not change significantly. Indeed, even when the ionic strength is as low as 0.1 mM NaCl equiv, the measured scattering coefficient does not further decrease (data not shown for brevity). This suggests that the polystyrene dispersions at experimental volume fraction are already well-structured at the ionic strength of 5 mM, especially at the high volume fractions. 4.3. Isotropic Scattering Coefficients Predicted by the PY-HS Model. Figure 3 shows the isotropic scattering coefficients obtained from FDPM measurements at 120 mM NaCl equiv and at 687 and 828 nm. Isotropic scattering coefficients calculated by Mie theory and the polydisperse PY-HS model with the mean size as the only fitting parameter are also showed in the figure. The fitted mean diameter obtained from the data taken separately at each wavelength was 142 nm and is comparable to the DLS measurement. It is noteworthy that the MSA-PM reduces to the PY-HS model when the particle charge approaches zero and also that the MSAPM model approaches the PY-HS model with increasing ionic strength. 4.4. Isotropic Scattering Coefficients Predicted by the MSA-HSY Model. Isotropic scattering coefficients as a function of volume fraction at ionic strengths of 60 mM, 25 and 5 mM equiv are fitted by using eqs 7 and 10 and Herrera’s analytic solution for structure factor of the MSA-HSY system.15 Here, the effective surface, z, is the only fitting parameter using the least squares method, and the experimental error is used as a weighting factor. The converge criteria were that the step size of surface charge was less than 1 e, and the relative decrease in the merit function, χ2, was less than 0.1. At same ionic strength, the effective charges are similar for data evaluated at the two different wavelengths as shown in Table 2. Figure 4 shows that the regressed curves predicted with MSA-HSY model match the FDPM experimental data well. The fitted effective charges change only with the ionic strength, increasing with ionic strength, from around 160 e at 1 mM NaCl equiv to 1000 e at 60 mM NaCl equiv. At 25 and 60 mM NaCl equiv, the fitted surface charge is approximately equal to the value estimated from the
Huang et al.
Figure 4. Isotropic scattering coefficients versus volume fraction obtained from FDPM measurements as a function of wavelength and ionic strength in NaCl equivalents (solid square, 60 mM, 687 nm; open square, 60 mM, 828 nm; solid triangle, 25 mM, 687 nm; open triangle, 25 nm, 828 nm; solid diamond, 5 mM, 687 nm; open diamond, 5 mM, 828 nm) and the corresponding MSA-HSY model predictions (solid line, 60 mM, 687 and 828 nm; dashed line, 25 mM, 687 and 828 nm; dotted line, 5 mM, 687 and 828 nm).
Figure 5. Isotropic scattering coefficients predicted from MSAHSY versus volume fraction as a function of surface charge at 25 mM NaCl equiv ionic strength at the wavelengths of 687 and 828 nm (solid line, z ) 400, 687, and 828 nm; dashed line, z ) 1000, 687, and 828 nm; dotted line, z ) 1400, 687, and 828 nm). Experimental values of isotropic scattering coefficients obtained from FDPM are represented as triangles (solid triangle, 687 nm; open triangle, 828 nm).
ζ potential measurement. To illustrate the sensitivity of MSA-HSY model to surface charge at each wavelength, Figure 5 shows the variation in the isotropic scattering coefficient with surface charge, z, at 25 mM. At higher ionic strengths, the change of isotropic scattering coefficient is less sensitive to the change of magnitude of the surface charges than at lower ionic strengths (not shown for brevity). 4.5. Isotropic Scattering Coefficients Predicted by the MSA-PM Model. Figure 6 shows experimental values of isotropic scattering as a function of volume fraction and ionic strength with the fitted prediction using MSA-PM mixture model. The only regression parameter of the MSA-PM model for the mixture is the surface charge density, which is assumed to be a constant for all the particles in the mixture. The number-averaged charge calculated from the fitted surface charge density employing the MSA-PM model are listed on the Table 2. The fitted surface charges using MSA-PM are 1-2 orders of magnitude less than those fitted by using MSA-HSY. The surface charge density regressed by MSA-PM increases as the ionic strength decreases from 60 to 1 mM NaCl equiv, while the average surface charge regressed by MSAHSY decreases as the ionic strength decreases from 25
Assessment of Electrostatic Interactions
Figure 6. Isotropic scattering coefficients versus volume fraction obtained from FDPM measurements as a function of ionic strength in NaCl equivalents and wavelength (square, 60 mM, 687 nm and 828 nm; triangle, 25 mM, 687 and 828 nm; diamond, 5 mM, 687 and 828 nm) and the MSA-PM model predictions (solid line, 60 mM, 687 nm and 828 nm; dashed line, 25 mM, 687 and 828 nm; dotted line, 5 mM, 687 and 828 nm).
mM to 1 mM NaCl equiv. The fitted surface charges using both MSA models at 828 nm are greater than those at 687 nm in most cases. To illustrate the sensitivity of the MSAPM model to surface charge, Figure 7 plots predicted data and illustrates the predicted decrease of scattering with increasing average particle charge at the wavelength of 687 nm. Figure 7 also shows that at volume fractions greater than 15%, the MSA-PM model tends to overpredict the scattering coefficient. Both the MSA-HSY and MSA-PM models predict hindered scattering with the increase in surface charge, and with the decrease of ionic strength, and could be used to regress experimental light scattering data. In our studies, the MSA-HSY match FDPM experimental data better than the MSA-PM mixture model. Yet the change in the parameter estimation of surface charges with ionic strength suggests that the physics are not correctly captured by the MSA model or the two potential models. 5. Conclusion This work shows that first principle models, PY-HS, MSA-HSY, or MSA-PM, can be used to explain the scattering of charged dispersions as measured by FDPM. At high ionic strength (>120 mM NaCl equiv), even when
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Figure 7. Isotropic scattering coefficients at 687 nm versus volume fraction predicted from MSA-PM at 687 nm and 25 mM NaCl equiv (solid line, z ) 6; dashed line, z ) 10; dotted line, z ) 16) and FDPM experimentally measured values at corresponding wavelength and ionic strength (symbols).
the particles are strongly charged, the PY-HS model predicts the isotropic scattering coefficient with accuracy. When the ionic strength is less than 60 mM NaCl equiv and the volume fraction is greater than 5%, the electrostatic interaction increases the structure of the charged colloid dispersion significantly and hindrance to scattering coefficients becomes greater than that measured at an ionic strength of 120 mM NaCl equiv or higher. The scattering coefficients at different volume fractions can be fitted with the surface charge as the single parameter estimate using MSA models. While surface charge should be independent of ionic strengths, we found that the fitted surface charge varied with the ionic strength of the dispersion. At some intermediate ionic strengths, around 25-60 mM NaCl equiv, the fitted effective charges using MSA-HSY are close to the surface charges estimated from ζ potential measurements. The fitted surface charges using MSA-PM are 1-2 orders of magnitude less than the actual charge. Using MSA-PM for a mixture with the actual polydisperisty index has the trend to overpredict the isotropic scattering coefficients at high volume fractions. Acknowledgment. This work is supported by the National Science Foundation (CTS -9876583). LA011436A
