16996
J. Phys. Chem. 1996, 100, 16996-17003
Dielectric Dispersion of Dilute Suspensions of Colloid Particles: Practical Applications Ali A. Garrouch,* Haitham M. S. Lababidi, and Ridha B. Gharbi Departments of Petroleum and Chemical Engineering, Kuwait UniVersity, P.O. Box 13060 Safat, Kuwait ReceiVed: May 16, 1996X
The two characteristic relaxation frequencies at which either the Maxwell-Wagner or the double layer polarization mechanism prevails have been defined in terms of characteristic parameters for colloidal suspensions subjected to an oscillating electric field. Both frequencies are found to increase with increasing ion diffusion coefficient, with decreasing particle radius, and with increasing salinity. For both thin and thick double layers, these frequencies appear to be invariant at low zeta potential (ζ) values. Correlations are developed between these characteristic relaxation frequencies and the colloid suspension parameters. The correlations constants appear to be related to the Boltzman constant, temperature, and electrostatic charge. These correlations can be useful for estimating particle size and counterion diffusion coefficient and can be used in developing dielectric dispersion models for colloid particles suspensions and for porous media. A rigorous dispersion model has been inverted to reciprocate simultaneously the average zeta potential (ζ), particle size, and particle volume fraction of the dispersed material from dielectric permittivity data at three distinct frequencies. Relaxation of colloidal suspensions appears to be an adequate alternative for quantifying these parameters as opposed to using intricate measurements of electrophoretic mobility.
Introduction The study of dielectric dispersion on simple and well-defined colloidal systems is important for understanding many complex phenomena like those occurring in biological systems or systems with selective membranes. Suspensions of charged particles in aqueous electrolyte solutions are known to display high dielectric constants in the presence of low-frequency alternating fields.1-5 The application of an external oscillating electric field disturbs the symmetry of bound charges on the particles by causing positive ions to move in the direction of the field and polarize at one extreme end of the particle, and negative ions to polarize at the other extreme end. This polarization enhances the dielectric constant because a large dipole is created as a result of the asymmetric distribution of charges. Ions from the bulk solution will diffuse to satisfy electroneutrality around the negatively charged surface as well as toward the new charge accumulations around the particles. The time required for diffusion processes to restore the equilibrium charge distribution around the colloid particles is called the characteristic relaxation time (τ). Various models have appeared over the past decades for predicting the dielectric enhancement of suspensions of charged particles.1-5 The development of some of these models requires an advance estimate of the double layer relaxation time. This, in return, influences the magnitude of the dielectric dispersion and the frequency range over which the double layer polarization dominates the Maxwell-Wagner polarization. Many of these models have been based on a simplistic estimate of double layer relaxation time:6 τ ) a2/2D. For illustrative purposes, we wish to shed some light on Shurr’s theory7 which is the basis for a few dielectric dispersion models for colloidal suspensions. An up-to-date and thorough review of all relevant theories is available in Lyklema.8 In Schurr’s theory, the ionic atmosphere around a charged sphere immersed in an aqueous electrolyte is divided into a thin layer of fixed counterions with density Fb and another as a diffuse layer with an ionic density Fd. Shurr assumes that (i) the fixed X
Abstract published in AdVance ACS Abstracts, September 1, 1996.
S0022-3654(96)01426-8 CCC: $12.00
Stern layer charges are only free to move tangentially, (ii) the diffuse layer charges exchange readily with the bulk electrolyte and this in return eliminates polarization in the diffuse atmosphere, and (iii) no ions are exchanged between the Stern and the diffuse layer. Under these assumptions Shurr’s solution for the complex conductivity σ j c* of a sphere and its associated electric double layer is given by
(
σ j c* ) λ +
) [
2λ0τ λ0ω2τ2 2 - iω ′c + 2 2 a 1+ω τ a(1 + ω2τ2)
]
(1)
Here, λ ) e0Fdµ1 and λ0 ) e0Fbµ1 are surface conductivities and τ ) a2/2D1 is the characteristic relaxation time of the charged particle. Parameters µ1 and D1 are the mobility and diffusion coefficient of the counterion in the bound layer, e0 is the charge of the counterion, and ′c is the real dielectric permittivity of the particle. The effective conductivity σc,ef and the effective permittivity c,ef which are real functions of frequency constitute the experimentally measured parameters and are, therefore, given by
c,ef ) ′c + σc,ef )
2λ0τ a(1 + ω2τ2)
2λ0ω2τ2 2λ + a a(1 + ω2τ2)
(2)
(3)
Equations 2 and 3 clearly illustrate the strong dependence of the permittivity and conductivity profiles on the characteristic time τ. An error in estimating τ causes an error in both the magnitude and profile of both permittivity and conductivity. Applying the Chain rule on (2), one can show that the relative error caused on the dielectric permittivity due to an error in the estimate of τ is given by
2λ0(1 - ω2τ2) ∆c,ef ∆τ ≈ c,ef (1 + ω2τ2)[a′c(1 + ω2τ2) + 2λ0τ] © 1996 American Chemical Society
(4)
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J. Phys. Chem., Vol. 100, No. 42, 1996 16997
As ω f 0, the relative error is given by
∆c,ef ∆τ 1 ≈ c,ef a′c τ +1 2λ0τ
(
)
(5)
For frequencies less than 10 MHz, the term a′c/2λ0τ is much less than unity, for dilute systems with reasonable values of bound layer ion concentration (Fb ) 1.33 × 1024 ions/m3) and mobility (µ1 ) 5.19 × 10-8 m2/(V s)). With these assumptions, the relative error simplifies to
∆c,ef ∆τ ≈ c,ef τ
(6)
At the other extreme (as ω f ∞), the relative error in the effective dielectric permittivity vanishes to zero. A similar analysis, starting from (3), shows that as ω f ∞, a relative error in the relaxation time causes a relative error on the effective conductivity approximated by
4λ0 ∆τ ∆σc,ef ≈ σc,ef [λ0 + λ] τ
(7)
For the case where Fb ) Fd/2, this relative error is approximately given by
∆σc,ef 4 ∆τ ≈ σc,ef 3 τ
(8)
These derivations present a vivid illustration of the relaxation time error effects on the permittivity and conductivity at both low and high frequencies. Perhaps the most commonly used estimate of the diffuse double layer relaxation time is that given by Delacey and White.2 The authors argue that as the field changes direction in the time 2π/ω, the double layer ions must travel a distance approximately equal to the diameter of the particle and its associated electric double layer length which is given by 2(a + κ-1). The time it takes for the counterions to diffuse this distance is therefore approximately equal to [2(a + κ-1)]2/6D, where D is the diffusion coefficient of the counterions. Delacey and White2 estimated heuristically that the relaxation of the double layer occurs at a characteristic time approximated by
τ ) 2π(a + κ-1)2/D
(9)
Lim and Franses9 measured the dielectric constants and conductivities of aqueous dispersions of monodisperse polymer microspheres for various sizes, weight fractions and NaCl concentrations in the frequency range 10-105 Hz. They estimated empirically the characteristic time to be proportional to (a + κ-1)n where n ) 1.5 ( 0.12. This exponent estimate differs from Delacey and White2 predicted value of 2.0. The rigorous model10 used in this study, which is a generalization of the Schwarz11 model, may still stand as one of the most comprehensive and realistic models for dielectric dispersion of colloid particles because it accounts for both Stern and diffuse layer polarization. Bound ions are allowed to exchange radially with the diffuse layer and to polarize tangentially as well, whereas diffuse layer ions are allowed to flow only radially. Because of the limited mobility of the Stern ions, at high frequencies, this model reduces to a diffuse layer polarization model (i.e., as if the Stern ions were immobile and behaved like bound charges). Here, the dielectric increment is related to the suspension properties as follows:
Figure 1. Dielectric increment versus angular frequency for particle volume fraction (p) ) 0.08; particle radius (a) ) 0.22 × 10-6 m; κa ) 300; zeta potential (ζ) ) 50 mV; counterion diffusivity coefficient (D) ) 2 × 10-9 m2/s.
(A1a2 - A2a1)(1 + W + W2) ∆(ω) 9 ) p(κa)2 (10) a 4 (A1 + A1W)2 + (A1W + A2W2)2 where W, A1, A2, a1, and a2 are intricate functions of the ζ potential, ion valence, particle size, the potential at the wall, counterion diffusion coefficient, and angular frequency. These functions are given in the Appendix. In this notation, p is the particle volume fraction of the dispersed material and a is the relative dielectric constant of the dispersion medium. The symbol ∆(ω), referred to as the dielectric increment, stands for the difference in relative dielectric constants between the colloidal suspension and the medium. Lyklema et al.10 asserted the dominance of the diffuse layer on polarization. They found the contribution of the bound counterions to be a maximum when both bound and diffuse layers surface charge densities are equal to half the colloid particle surface charge density. Polarization of the diffuse layer increased more with increasing surface charge than with polarization due to bound ions, for a fixed κa, where κ is the inverse Debye-Huckel length. Moreover, they showed that the influence of bound counterions becomes significant at moderate values of Stern plane potential φd e φcr d . The critical Stern plane potential is implicitly given by10
sinh(zFφcr d /2RT) = 0.03 κa
(11)
They have determined conditions for which diffuse layer polarization dominates bound counterion layer polarization. However, conditions for which the Maxwell-Wagner polarization prevails over double layer polarization are yet to be determined. Part of this endeavor aims at characterizing the distinction between these two mechanisms in terms of colloidal suspension parameters. The remainder of the paper is divided into two parts. In the first part, we solve for the inflection points of the dielectric increment-frequency profile given by (10). Two of these inflection points, ω1 and ω2, represent the characteristic relaxation frequencies marking the double layer and MaxwellWagner polarization mechanisms, respectively (Figure 1). The Lyklema et al. model,10 discussed next, is used to simulate profiles of dielectric permittivity increment versus angular frequency at various values of ζ potential, particle radius, particle volume fraction, salinity, and ion diffusivity. The simulation runs have been instrumental for correlating the characteristic relaxation frequencies, obtained from the inflection points, to
16998 J. Phys. Chem., Vol. 100, No. 42, 1996
Garrouch et al.
TABLE 1: Data Summary for Simulation Runs Generated in This Study a (µm) 0.22 0.22 0.22 0.22 0.22 0.22 0.22
p
κa
D‚109 (m2/s)
0.02 0.02 0.02 0.02 0.01, 0.02, 0.04, 0.08, 0.1 0.01, 0.02, 0.04, 0.08, 0.1 0.02
50 300 10, 20, 50, 100, 150, 180, 200 10, 20, 50, 100, 150, 180, 200 50 300 50
0.1, 0.2, 0.4, 0.6, 0.8, 1, 2 0.1, 0.2, 0.4, 0.6, 0.8, 1, 2 2 2 2 2 2
0.02
300
2
0.02 0.02
50 300
2 2
ζ potential (mV)
50 50 50 150 50 50 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 120, 130, 140 0.22 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 120, 130, 140 0.1, 0.2, 0.5, 1.0, 2.0, 5.0 50 0.1, 0.2, 0.5, 1.0, 2.0, 5.0 300
the suspension parameters. In the second part, we invert the dispersion model10 so that three measurements or readings of dielectric permittivity increments (∆1(ω1), ∆2(ω2), and ∆3(ω3)) reciprocate simultaneously three important suspension parameters a, p, and ζ. The latter parameter, traditionally estimated from electrophoretic measurements, is not estimated for its own sake, but rather to evaluate particle interaction as it is required in rheology and colloid stability. Here, we present dielectric measurements, simplified recently by the use of innovative four-electrode complex impedance setups,12 as an alternative for electrophoretic measurements. Discussion and Results 1. Inflection Points Analysis. The polarization phenomenon is typically characterized by a distribution of relaxation frequencies. Two of these characterize the polarization phenomenon of colloidal suspensions. The lower frequency corresponds to double layer polarization dominance, whereas the higher frequency corresponds to dominance of the Maxwell-Wagner polarization mechanism. These two characteristic frequencies are obtained by solving for the inflection points of the dielectric increment-frequency profile (eq 10). This is done by finding analytical expressions for the first and second derivatives of the dielectric increment with respect to frequency. Because of the high nonlinearity associated with the model, analytical solutions for these derivatives offer an advantage over numerical solutions plagued by approximation errors and instabilities. A symbolic Mathematica13 code has been written for this task. Another Mathematica code was also developed for generating data for the dielectric increment-frequency profiles, presented in this study. Seventy-two simulation runs have been conducted to generate dielectric increment-angular frequency profiles and to deduce the relaxation frequencies ω1 and ω2 (Table 1). These runs have been designed to incorporate a wide range of values for all independent variables; particle volume fraction (p) varied from 0.02 to 0.08, counterion diffusion coefficient (D) from 2 × 10-9 to 2 × 10-10 m2/s; zeta potential (ζ) from 30 to 300 mV, and particle radius (a) from 0.1 to 5.0 µm. Conditions of low and high suspension salinity have been simulated by varying the Debye-Huckel length (κ). These simulation runs were instrumental in revealing qualitative dependencies of ω1 and ω2 on different suspension parameters. These dependencies, discussed next, are used as a guide to develop nonlinear correlations between the characteristic frequencies and other parameters pertaining to the colloidal suspensions. Effect of Suspension Parameters on ω1 and ω2. Figure 2 shows that both ω1 and ω2 are insensitive to the particle volume fraction p for a saline environment. Similar results were also obtained for fresh environments simulated by low κa values. These results suggest that neither interparticle nor double-layer interactions exist when the colloidal dispersion is stable. This is also the case when the average interparticle distance (in the
Figure 2. Effect of particle volume fraction on the relaxation frequencies. Particle radius (a) ) 0.22 × 10-6 m; κa ) 300; zeta potential (ζ) ) 50 mV; counterion diffusivity coefficient (D) ) 2 × 10-9 m2/s.
Figure 3. Effect of particle radius (a) on relaxation frequencies. Counterion diffusion coefficient (D) ) 2 × 10-9 m2/s; particle volume fraction (p) ) 0.02; κa ) 30; zeta potential (ζ) ) 50 mV.
order of µm) is much larger than the Debye-Huckel length (in the order of 100th of a µm). These results suggest that these frequencies are a characteristic property of the particle size. As the particle radius increased, both ω1 and ω2, for a dilute suspension of charged particles, decreased (Figure 3). This is also in line with Grosse and Foster6 model as well as Delacey and White2 model that applies to a single charged particle suspended in an electrolyte solution and subjected to an oscillating electric field. Figure 4 shows that as κa increases (the diffuse layer becomes thinner), the relaxation frequency decreases. This shift in relaxation frequency to smaller values is expected if the
Dilute Suspensions of Colloid Particles
J. Phys. Chem., Vol. 100, No. 42, 1996 16999
Figure 4. Relaxation frequencies versus κa. Particle radius (a) ) 0.22 × 10-6 m; particle volume fraction (p) ) 0.02; zeta potential (ζ) ) 50 mV; counterion diffusivity coefficient (D) ) 2 × 10-9 m2/s.
Figure 5. Relaxation frequencies versus zeta potential (ζ). Particle radius (a) ) 2.2 × 10-6 m; particle volume fraction (p) ) 0.02; κa ) 100; counterion diffusivity coefficient (D) ) 2 × 10-9 m2/s.
characteristic frequency (ω1) is given by
ω1 = constant
2πDκ2 (1 + κa)2
(12)
Basically, as κ-1 decreases, the distance through which the double layer is displaced decreases and the corresponding diffusion time is reduced. Thus, the double layer continues to follow the field until higher frequencies are reached. Both ω1 and ω2 decrease with increasing values of κa. At very high values of κa, these frequencies become invariant. Both ω1 and ω2 increase with increasing ζ potential (Figure 5). The result shown in Figure 5 are for a κa value of 100 only. These trends have been observed for a variety of increasing κa values which indicate increasing salt concentrations. These results indicate that an increase in the ζ potential causes an increase in the space charge around the particle, which in turn increases the induced electric field due to concentration gradients. Therefore, the time required for diffusion processes to restore the equilibrium charge distribution around the particle is now reduced and the corresponding angular frequency is, therefore, increased. As shown in Figure 5 for saline environments (high κa values) ω1 and ω2 are invariant with low ζ potential values. Similar results were also obtained for κa values ranging from 50 to 300. Figure 6 shows that as the diffusion coefficient doubles, ω1 and ω2 also double. These results are for κa equals 30. A similar trend has been observed for κa values of up to 300.
Figure 6. Relaxation frequencies versus counterion diffusion coefficient (D). Particle radius (a) ) 2.2 × 10-6 m; particle volume fraction (p) ) 0.02; κa ) 30; zeta potential (ζ) ) 50 mV.
This is in line with simple models for ω1, of a single charged particle suspended in an electrolyte solution, suggested by Grosse and Foster6 or Delacey and White.2 DeVelopment of Correlations. Delacey and White model (eq 9), however simplistic, is probably the most commonly used expression for estimating values of ω1. The percent relative error between values of ω1 obtained from the inflection points solutions and values of ω1 computed using Delacey and White model was as high as 92% and had an average of 16% with a standard deviation of 16%. This high error calls for the development of a better model for estimating ω1 that incorporates parameters influencing both magnitude and shape of the dielectric dispersion profile. This study asserts a dependence of ω1 on counterion diffusion coefficient, ζ potential, particle radius, and salinity of the medium. We have included these variables in a nondimensional b;b B) as form to come up with a nonlinear correlation ω1 ) ω1(x a function of suspension parameters, lumped here in the vector b x, and fitting parameters given by the vector B b. This is obtained by minimizing a χ2 merit function approximated by its Taylor series expansion as14
1 χ2(b B) ≈ γ - B d ‚b B+ B b ‚D h ‚b B 2
(13)
In this notation, B b is a vector consisting of the set of M unknown d is an M vector given by fitting parameters bk, k ) 1, 2, ..., M; B B d ) ∇χ2|Bb. This gradient has components given by
∂χ2 ∂bk
N
) -2∑
[ω1i - ω1(xi;b B)]∂ω1(xi;b B) σi2
i)1
k ) 1, 2, ..., M
∂bk
(14)
D h is the second derivative matrix (Hessian) of the χ2 merit function, and its components are given by
[
N B) ∂ω1(xi;b B) 1 ∂ω1(xi;b ) 2∑ 2 ∂bk∂bl ∂bk ∂bl i)1 σ i
∂ 2 χ2
]
∂2ω1(xi;b B)
[ω1i - ω1(xi;b B)] The function γ is given by
∂bk∂bl
(15)
17000 J. Phys. Chem., Vol. 100, No. 42, 1996
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TABLE 2: Data Used To Validate the Proposed Models for ω1 and ω2 κa
ζ (mV)
a (µm)
D‚109 (m2/s)
estimated ω1
actual ω1
ω1 error (%)
estimated ω2
actual ω2
ω2 error (%)
120a 120 56 40 80 15.81b 19.91 25.46 23.64 12.1 14.9 16.3
37 36 34 12 33 23.5 24.2 25.0 22.7 26.4 27.5 28.2
2 2 1 1 1 0.222 0.222 0.222 0.222 0.222 0.222 0.222
1.57 1.57 1.57 1.57 1.57 1.54 1.54 1.54 1.54 1.37 1.37 1.37
51.4 51.2 211.6 187.1 206.7 4226.8 4225.7 4207.9 4165.3 4287.3 4319.5 4331.7
55 54 229 199 203 4050 4201 4129 4098 4146 4015 4158
7.04 5.51 7.42 6.02 1.95 4.36 0.59 1.91 1.64 3.41 7.59 4.18
659 655 1170 994 1134 60764 60743 60406 59601 61915 62530 62762
710 715 1276 1076 1126 61263 61488 60134 59820 61073 61156 62150
7.20 8.41 8.33 7.60 0.70 0.81 1.21 0.45 0.37 1.38 2.25 1.00
a
Beginning of Lim and Franses9 experimental data. b Beginning of Springer et al.17 experimental data. N
γ)∑ i)1
[
]
ω1i - ω1(xi;b B) σi
2
(16)
with σi being the standard deviation of the ith data point. Because of the nonlinear dependencies, the minimization proceeds iteratively. Given trial values for the parameters vector B bcur, the χ2 merit function is estimated and checked against the convergence limit. If this limit is not reached then we estimate the minimizing vector parameter using the gradient descent technique,14 namely
b cur - λ∇χ2(b Bcur) B b next ) B
(17)
The parameter λ is an arbitrary constant. The LevenbergMarquardt13,14 algorithm is implemented to linearize the system of equations given by (13) and to optimize λ value to be small enough not to bypass the global minimum of the merit function and to be large enough for fast convergence. The following expression for ω1 is found to mimic the data with reasonable degree of accuracy for moderate values of Stern plane potential φd e φcr d:
()
2πDκ2 φd ω1 ) R (1 + κa)2 φcr d
m
(18)
In the above notation R is a constant equal to 0.024 286 1 and m is also a constant equal to 0.136 106. The average relative error obtained was 4.1% with a standard deviation on the error of 2.4%. The average relative error is defined here as the absolute value of the difference between the estimate of ω1 based on inflection points solution and its estimate based on our proposed new model (eq 18), divided by the inflection point solution of ω1. It is interesting to notice at this point that the value of R obtained from this correlation is approximately equal to the numerical value of KT/e, where K is the Botlzman constant, T is the temperature, and e is the electrostatic charge. The term KT/e is an electric potential conventionally used to normalize colloid suspension potential solutions. In a sense this term is proportional to the probability that a counterion will be present at a particular position around the colloid particle, at equilibrium.15,16 This is a significant result since the constant R happens to be a carefully chosen number related to the Boltzman ion distribution in the presence of an electric double layer. For all frequencies obtained, we found that ω1 and ω2 scaled logarithmically as follows:
ω2 ≈ ω1/0.607 1
for ω1 < 100
(19a)
ω2 ≈ ω1/0.758 1
for ω1 g 100
(19b)
These results were inferred from plots of ω1 and ω2 presented earlier. These frequencies displayed parallel profiles when plotted in logarithmic scales as shown in Figures 2-6. Experimental data9 for polymer microspheres, and for homodisperse polystyrene latices,17 have been used to validate the two models for ω1 and ω2 proposed here. The experimental data provided by Springer et al.17 did not include values of ζ potential for the polystyrene latices used. Since these experimental data matched well simulation profiles of dielectric permittivity versus frequency, corresponding ζ potential were estimated by inverting Lyklema model. The inversion procedure is detailed in part 2 of this paper. Table 2 presents a summary of available data from the literature used for validating the proposed correlations. This table also shows the relative error between the proposed models estimates for ω1 and ω2 (eqs 18 and 19) with actual reported data in the literature.9,17 The average error between our estimate for ω1 and the reported data is approximately 4.3% with a standard deviation of 2.4%. The average error between our estimate for ω2 and the tested data is approximately 3.3% with a standard deviation of 4.4%. 2. Inversion Model. A great deal of research related to the dispersion of colloid suspensions in the presence of electric fields has been concerned primarily with the interpretation of electrophoretic mobility and electrical conductivity measurements.8 These quantities are measured using a steady electric field and are classified collectively under the electrokinetic phenomenon referred to customarily as electrophoresis. When the particles are charged, externally applied steady electric fields move them relative to the fluid. From this motion, particle ζ potential can be ascertained. Particle electrophoresis has been used in many practical areas of surface sciences such as water purification, detergency, and characterization of bacterial surfaces.18 Rozen and Saville19,20 presented ample evidence indicating that ζ potentials estimated from electrophoretic mobility measurements are systematically lower than corresponding potentials estimated from dielectric dispersion measurements. Recently, Kiljlstra et al.12 have indicated that this discrepancy is partly due to the fact that existing electrophoretic models do not account for bound ion layer polarization, or surface conduction. Unlike dielectric dispersion measurements, electrophoretic measurements are highly dependent on particle interaction. This means that unless the electrolyte concentration, solution pH and temperature are carefully monitored, particles might coagulate and settle on capillary walls making mobility measurements hard to make. In this part of the paper, we explore the use of dielectric dispersion, as an alternative to electrophoresis for estimating ζ potential as well as other parameters pertaining to colloidal suspensions. A Matlab code has been written to inverse the Lyklema et al. model for dielectric dispersion of dilute suspensions. The inversion model requires three readings of dielectric permittivity
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J. Phys. Chem., Vol. 100, No. 42, 1996 17001
TABLE 3: Inverse Model Validation Data (D ) 2 × 10-9 m2/s) ω1
ω2
ω3
∆1
∆2
∆3
κa
a (µm)
p
ζ (mV)
10 Hza 1 kHz 1 kHz 1 kHz 1 kHz 289 Hzb 207 Hz 206 Hz 196 Hz 415 Hz 108 Hz
100 Hz 10 KHz 10 KHz 10 KHz 10 KHz 2008 Hz 1864 Hz 1998 Hz 1030 Hz 1963 Hz 980 Hz
1000 Hz 100 KHz 100 KHz 100 KHz 100 KHz 9995 Hz 9604 Hz 9748 Hz 5831 Hz 6191 Hz 6068 Hz
699.6 637.0 53.1 2535.9 28.8 947.5 891.6 618.2 1807.6 611.6 108.3
683.5 521.3 42.8 2115.5 27.3 134.0 139.4 125.4 524.4 131.1 980.2
637.0 303.5 21.2 1516.3 23.4 17.9 14.7 12.6 76.5 12.7 6068.5
50 50 50 50 50 80 56 40 56 56 56
0.22 0.22 0.22 0.22 0.05 0.55 0.55 0.55 0.55 0.55 0.55
0.02 0.02 0.04 0.02 0.02 0.0107 0.0107 0.0107 0.0214 0.0107 0.00536
100 100 50 150 50 33 34 12 19 33 16
a Input data for inverse model was generated using Lyklema et al.10 model. b Input data for inverse model was generated from Lim and Franses9 experimental data.
increments (∆1(ω1), ∆2(ω2), and ∆3(ω3)) to reciprocate simultaneously three suspension parameters a, p, and ζ. This is accomplished by solving the following three nonlinear equations simultaneously
∆1(ω1) (A1a2 - A2a1)(1 + W1 + W12) 9 )0 p(κa)2 2 2 2 4 a (A1 + A1W1) + (A1W1 + A2W1 ) (20) ∆2(ω2) (A1a2 - A2a1)(1 + W2 + W22) 9 )0 p(κa)2 2 2 2 4 a (A1 + A1W2) + (A1W2 + A2W2 ) (21) ∆3(ω3) (A1a2 - A2a1)(1 + W3 + W32) 9 )0 p(κa)2 2 2 2 4 a (A1 + A1W3) + (A1W3 + A2W3 ) (22) Here, the function Wi is W (eq 10) evaluated at ωi with subscript (i) taking values of 1, 2, and 3, respectively. The three unknowns we are solving for (a, p, and ζ) are embedded into intricate expressions of A1, A2, a1, and a2, presented in the Appendix. Solving for the nonlinear system of equations given above is typically an iterative procedure that requires evaluation of the Jacobian (Jh), given by
[ ]
∂∆1 ∂a ∂∆2 hJ ) ∂a ∂∆3 ∂a
∂∆1 ∂p ∂∆2 ∂p ∂∆3 ∂p
∂∆1 ∂ζ ∂∆2 ∂ζ ∂∆3 ∂ζ
(23)
Analytical solutions have been found for all Jacobian elements by applying the chain rule on the associated functions presented in the Appendix. The job of evaluating these elements is both long and tedious. It is, however, advantageous since this exact solution for the Jacobian eliminates potential computational errors that would have accumulated, had the Jacobian been estimated numerically. Because nonlinear systems tend to converge to nonunique solutions, the following physical constraints were imposed
10 < κa e 300
(24a)
0 < p e 5%
(24b)
-200 e ζ e 200
(24c)
Since the direct model formulation stands for suspensions of κa values greater than 10 only, this variable was constrained in the inverse model to be greater than 10 and less than 300. The
upper boundary for κa is set to avoid particle flocculation induced by an increase in the solution electrolyte concentration. Particle interaction in dielectric dispersion is commonly ignored provided that the particle volume fraction is less than 5%.8 As a consequence, p was constrained in the inverse model to be less than 5%. The ζ potential was also constrained between -200 and 200 mV. These limits include a range of ζ potential values commonly encountered in practice. In what follows, we illustrate the procedure for validating the inverse model. The direct model has been run for several cases (Table 1) showing variations in ζ potential, diffusion coefficient, particle radius, Debye-Huckel length, and particle volume fraction. The output of each of these simulation runs is dielectric increment versus angular frequency. This data is used as an input in the inverse model. Specifically the inverse model requires input data of three distinct values of dielectric increment corresponding to three distinct values of angular frequencies. We have focused our study on two sets of angular frequencies at which we read the input dielectric increments. These are {10; 102; 103 Hz}, and {1; 10; 100 kHz}. The diffusivity coefficient was varied from 2 × 10-10 to 2 × 10-9 m2/s, and κa was varied from 50 to 300. We have examined cases for which the zeta potential varied from 50 to 100 then to 150 mV; the particle radius varied from 0.05 to 0.22 µm; and the particle volume fraction was fixed to either 0.02 or 0.04. Table 3 presents a sample of this data for a constant diffusion coefficient value of 2 × 10-9 m2/s. Predicted values, using the inverse model, of ζ potential, particle volume fraction (p), and particle radius (a) matched perfectly the corresponding input values used to generate the dielectric increment-frequency profiles when initial guesses for these variables were within (80% of the actual output values. Figure 7 shows perfect inverse model prediction of the three parameters (a, p, and ζ) for κa and ζ potential values of 50 and 50 mV, respectively. Similar perfect predictions were obtained for cases of high κa and ζ potential values and for different diffusion coefficient values mentioned earlier. Lim and Franses’ data9 were used to reciprocate simultaneously a, p, and ζ. The inverse model also converged to within 6.5% of these variables actual values when the initial guesses of a and p and ζ were within 35%. We have used this inverse algorithm to specifically generate zeta potential values which were not tabulated in Springer et al. data.17 These values were needed for the development of the relaxation frequencies models presented in part I. The inverse algorithm converged to exact Springer et al. values of a and p, indicating convergence to the initially unknown values of ζ potential (Table 2), since the solution is now unique. Conclusions This paper puts in perspective a couple of applications for dielectric dispersion models. These applications consist of
17002 J. Phys. Chem., Vol. 100, No. 42, 1996
Garrouch et al. + a1 ) a+(1 + Rc ) - R (1 + Rc ) + a2 ) R+(1 + R1 ) - R (1 + R1 ) + + A1 ) (z- - R-)(1 + R+ c ) + (z + R )(1 + Rc ) + + A2 ) (z- - R-)(1 + R+ 1 ) + (z + R )(1 + R1 )
R( )
( ( [3/2mI( 2 - z I1 ] κa
R( c )
( [3/2mI( 2c - I1 ] κa
R( 1 )
[ (
I( 1 ) -2 exp -
correlations development for characteristic relaxation frequencies and development of an algorithm for reciprocating colloid suspensions properties from dielectric dispersion data. Our choice of the Lyklema et al.10 model as a basis for this study stems from the fact that the model accounts for double layer structure and surface conductivity effects. The general analytical formulation of the model makes it also a very suitable starting point for this kind of analysis. The same analysis can be extended to other models as well. In the first part of the paper, we solve for the inflection points of the dielectric increment-frequency profile of the Lyklema et al.10 model using a symbolic code. This symbolic manipulation is an advantage, given the nonlinearity of the problem, since we get exact solutions for these inflection points, thus eliminating the errors that can accumulate when using numerical computations. A systematic study is performed showing the effects of particle volume fraction, medium salinity, ζ potential, particle radius, and diffusion coefficient on these characteristic frequencies pertinent to the inflection points. Correlations between the relaxation frequencies and the suspension properties are then developed. The correlations presented in this study constitute a significant improvement over many simplistic models currently available in the literature, by including the dependence of the relaxation frequencies on a dimensionless stern plane potential. They can also be used as an integral part of future models of dielectric dispersion of colloidal suspensions as well as porous media, or for estimating various suspension properties. In the second part of the paper, we demonstrated the use of dielectric dispersion data to estimate back simultaneously three fundamental suspension properties a, p, and ζ from three distinct readings of dielectric permittivity increments. In this lowfrequency range investigated (10 Hz-10 MHz) dielectric dispersion provides a lot more information than electrophoretic mobility measurements using static electric fields, used to estimate the ζ potential only. Acknowledgment. The authors express deep gratitude and thanks for Kuwait University for the financial support to this project (EPP001). Appendix Constants for Lyklema et al.10 Dielectric Dispersion Model.
I( 2c
)]
2Fζ 4 zFζ - 1 - exp RT z 2RT
[ ( )] zFζ 2Fζ 4 zFζ ) - 16 ln cosh( - [1 - exp() ] zRT z 2RT 4RT) I( 2 )
Figure 7. Inversion model solutions of zeta potential, particle radius and particle volume fraction versus initial guess vicinity. Actual values are ζ ) 50 mV; a ) 0.22 mm; p ) 0.02; D ) 1 × 10-9 m2/s, κa ) 50.
zFφd -1 RT
( z([I( 1 + I3 ] κa
2
3 cosh(zFφd/2RT) - exp(-zFφd/RT) - 2 I( 3 )cosh(zFφd/2RT) m ) 20a(RT/F)2/3ηD a R m κ T m z n* K R φd φcr d η F a 0 ζ p e0 σ0 λ0 τ D τ1 τ2 σm ω ω1 ω2
particle radius corelation constant ) 0.024 286 1 correlation constant ) 0.136 106 inverse of Debye-Huckel length (κ ) x(2n*e2z2/mKT) in S.I. units) temperature medium permittivity valence equilibrium ion concentration boltzman’s constant universal constant (8.3144 J/(g mol K)) potential at stern plane critical potential at the Stern plane medium viscosity Faraday’s number (9.65E4 coulomb/(g equiv) relative dielectric constant of the dispersion medium permittivity of free space potential at the slipping plane (zeta potential) volume fraction of the dispersed material electron charge surface charge density high-frequency limit of the frequency-dependent part of the surface conductivity relaxation time counterion diffusion coefficient relaxation time marking dominance of the double layer polarization mechanism relaxation time marking the Maxwell-Wagner polarization mechanism dominance medium conductivity angular frequency angular frequency marking dominance of the double layer polarization mechanism angular frequency marking the Maxwell-Wagner polarization mechanism dominance
Dilute Suspensions of Colloid Particles ∆(ω) J0 t F ψ χ2 b x B b B d D h hJ µ1 D1 σ j c* σc,ef c,ef ′c σi
dielectric increment current density time Net charge concentration electric potential merit function suspension parameters vector fitting parameters vector merit function gradient Hessian of merit function Jacobian matrix mobility of counterions in bound layer diffusion coefficient of counterions in bound layer complex conductivity of a sphere and its associated electric double layer effective conductivity of a sphere and its associated electric double layer effective permittivity of a sphere and its associated electric double layer particle real dielectric permittivity standard deviation of the ith data point
References and Notes (1) Chew, C. W.; Sen, P. N. J. Chem. Phys. 1982, 77, 4683-4693. (2) Delacey, E. H. B.; White, L. R. J. Chem. Soc., Faraday Trans. 2 1981, 77, 2007-2039.
J. Phys. Chem., Vol. 100, No. 42, 1996 17003 (3) Fixman, M. J. Chem. Phys. 1980, 72, 5177-5186. (4) Rozieres, J. D.; Middleton, M. A.; Schecter, R. S. J. Colloid Interface Sci. 1988, 124, 407-415. (5) Russel, W. B.; Saville, A. A.; Schowalter, W. R. Colloidal Dispersions; Cambridge University Press: London, 1989; pp 238-247. (6) Grosse, C.; Foster, K. R. J. Phys. Chem. 1987, 91, 3073-3076. (7) Schurr, J. M. J. Phys. Chem. 1964, 68, 2407-2413. (8) Lyklema, J. Fundamentals of interface and colloid science: Volume II: Solid-Liquid Interfaces; Academic Press: New York, 1995. (9) Lim, K.; Franses, E. I. J. Colloid Interface Sci. 1986, 110, 201213. (10) Lyklema, J.; Dukhin, S. S.; Shilov, V. N. J. Electroanal. Chem. 1983, 143, 1-21. (11) Schwarz, G. J. Phys. Chem. 1962, 66, 2636-2642. (12) Kijlstra, J.; van Leeuwen, H. P.; Lyklema, J. 1993, 9, 1625-1633. (13) Waltram, S. Mathematica, a system for doing mathematics by computer; Addison Wesley Publishing Co.: Reading, MA, 1991. (14) Press, W. H.; Teukolsky, S. A.; Vetterling, W. T.; Flannery, B. P. Numerical recipes in Fortran: The art of scientific computing; Cambridge University Press: London, 1992. (15) Adamson, A. W. Physical chemistry of surfaces; John Wiley and Sons, Inc.: New York, 1990. (16) Israelachvili, J. N. Intermolecular and surface forces; Academic Press: New York, 1992. (17) Springer, M. M.; Korteweg, A.; Lyklema, J. J. Electroanal. Chem. 1983, 153, 55-66. (18) Myers, D. Surfaces, interfaces, and colloids; VCH Publishers: New York, 1991. (19) Rozen, L. A.; Saville, D. A. Langmuir 1991, 7, 36-42. (20) Rozen, L. A.; Saville, D. A. J. Colloid Interface Sci. 1992, 149, 542-552.
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