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Approximations for Calculating van der Waals Interaction Energy between Spherical Particles—A Comparison
Journal of Colloid and Interface Science 243, 136–142 (2001) doi:10.1006/jcis.2001.7851, available online at http://www.idealibrary.com on
Approximations for Calculating van der Waals Interaction Energy between Spherical Particles—A Comparison Suresh N. Thennadil1 and Luis H. Garcia-Rubio2 Department of Chemical Engineering, University of South Florida, Tampa, Florida 33620 Received June 11, 2001; accepted July 14, 2001; published online September 24, 2001
The DLVO theory has been widely used to study the stability of colloidal systems with the van der Waals interactions modeled according to the Hamaker (microscopic) theory. The Hamaker theory, in addition to neglecting many-body interactions, does not account for retardation effects. Retardation effects can be included within the framework of the microscopic theory. For spherical particles, an “exact” expression has been derived by Clayfield et al. Lifshitz (macroscopic) theory, on the other hand, accounts for both manybody and retardation effects but is computationally intensive and requires dielectric data at different frequencies. For spherical particles, it is not practical to use the “exact” expression for van der Waals interaction derived by Langbein; therefore approximations have to be used. Approximate expressions to calculate the van der Waals interaction energy between spheres were considered in terms of accuracy, ease of computation, and required material parameters with the “exact” expression derived by Langbein used as the benchmark. It was found that the “exact” expression using the microscopic theory works as well as the best approximation to the macroscopic theory. °C 2001 Academic Press Key Words: colloidal dispersions; van der Waals interactions; retardation effects; Lifshitz theory; Hamaker theory; microscopic theory.
1. INTRODUCTION
The DLVO theory (1) has been widely used to study the stability of colloidal dispersions. According to this theory, the total interaction between charged particles dispersed in a solvent containing electrolyte is assumed to be the sum of van der Waals and electrostatic interactions. At moderate to high salt concentrations, which are the regime of interest when aggregation phenomena are investigated, the two forces compete. Under these conditions the electrostatic interactions are adequately described by the classical Poisson–Boltzmann equation. The van der Waals interactions are much more complicated, and an adequate
1 To whom correspondence should be addressed at present address: Instrumentation Metrics Inc., 7470 W. Chandler Blvd., Chandler, AZ 85226. Fax: (480) 755-9832. E-mail: [email protected]. 2 Present address: ENB 118, Department of Chemical Engineering, University of South Florida, 3202 E. Fowler Avenue Tampa, FL 33620. Fax: (813) 9743997. E-mail: [email protected].
0021-9797/01 $35.00
C 2001 by Academic Press Copyright ° All rights of reproduction in any form reserved.
description of these interactions is essential for the study of the structure of colloidal dispersions as well as aggregation phenomena in a quantitative manner. Usually, the van der Waals interaction energy is obtained by using Hamaker’s equation. This equation is based on the microscopic theory, which assumes that the interaction between two particles is just the sum of the interactions between each molecule in one particle and all of the molecules in the other particles. This assumption of pairwise additivity neglects many-body interactions. In addition, the Hamaker equation neglects retardation effects. Within the framework of the microscopic theory, retardation effects can be included by using the Casimir–Polder equation (2) as the starting point. A more rigorous theory called the macroscopic or Lifshitz theory, which takes into account many-body interactions, was developed by Lifshitz (3–6). According to this theory, the particles and the intervening fluids are treated as individual macroscopic phases, which are fully characterized by their dielectric permittivities. This theory can be applied to any body at any temperature and takes retardation effects into account. For the case of spherical particles, Langbein (7, 8) derived an “exact” expression within the framework of Lifshitz theory. This equation is computationally intensive, in addition to exhibiting problems of convergence at close separations, making it impractical to use. Several approximations to this expression have been proposed (9–11). On the other hand, the “exact” expression within the framework of the microscopic theory, derived by Clayfield et al. (12) though algebraically involved, is easy to compute. Simplified expressions that include retardation effects using the microscopic theory have been proposed (13–15). Ideally, we would like to use Langbein’s “exact” expression based on the Lifshitz theory to calculate the energy of interaction between two spheres. However, as mentioned earlier, this expression is computationally intensive and is beset with problems of convergence. Further, we require reliable dielectric data over a wide range of frequencies to be able to use the Lifshitz theory. Such data may not be available for materials under investigation. The general tendency has been to use Hamaker’s equation, although it has been shown to lead to significant deviations (16, 17) in the estimation of van der Waals interactions. The advantage of using the Hamaker equation is that, in addition to its simplicity, the Hamaker constant can be experimentally determined (18, 19) and these data are available for a
136
137
VAN DER WAALS ENERGY BETWEEN SPHERICAL PARTICLES
large number of materials. While several approximations to the “exact” macroscopic and microscopic theories exist, the availability of high-speed computers alleviates the need to oversimplify the equations just for the sake of reducing the computational intensity. In this manuscript, calculations will be performed for polystyrene spheres in water. These calculations will be used to evaluate the effects of retardation and the accuracy and practicality of using different approximations. 2. EXPRESSIONS FOR VAN DER WAALS INTERACTION ENERGY BETWEEN SPHERES
Retardation effects can be included by using the Casimir– Polder equation as the starting point. In this case, the method of pairwise summation for deriving the interaction energy between two spheres gives (12) Z 2a1 Aeff d x x(2a1 − x) (d + a1 + a2 ) 0 Z 2a2 y(2a2 − y) dy F(r ), × r5 0
V (d) = −
where d = r − a1 − a2 and r = d + x + y. F(r ) is given by
The “exact” expression for the interaction energy between two equal spheres is given by (8, 11)
V (d) = kB T
∞ X ∞ X
(2m + 1)(2n + 1)1m 1n
F(r ) = A − Br
r < 3λ/2π
F(R) = C/r − D/r 2 r > 3λ/2π, µm X µ=−µm
N =0 m,n=1
[3]
µ −µ Vmn (kr )Vnm (kr )
[4]
[1]
¡
¢ k 2 − k12 m jm (ak) jm (ak1 ) + kk1 [k1 jm (ak1 ) jm−1 (ak) − k jm (ak) jm−1 (ak1 )] ¢ k 2 − k12 m h m (ak) jm (ak1 ) + kk1 [k1 jm (ak1 )h m−1 (ak) − kh m (ak) jm−1 (ak1 )]
1m = ¡
µ µ Vmn (kr ) = Umn (kr ) +
n−µ+1 (n + µ) µ µ krUmn+1 (kr ) − krUmn−1 (kr ) (n + 1)(2n + 1) n(2n + 1)
(n − µ)!(m − µ)! µ V (kr ) (n + µ)!(m + µ)! mn µ ¶µ X νm 2 µ µν Umn (kr ) = Smn h m+n−µ−2ν (kr ) kr ν=0
−µ (kr ) = Vmn
µν Smn =
0(m − ν + 0.5)0(n − ν + 0.5)0(m + ν + 0.5)(m + n − ν)!(m + n − µ − 2ν + 0.5) , 0(m + n − µ − ν + 1.5)0(µ + 0.5)0(0.5)(m − µ − ν)!(n − µ − ν)!ν!
where k 2 = (ξN /c)2 ε3 (iξN ), k12 = (ξN /c)2 ε1 (iξN ), ξN = 2π N kT / h- , h- = h/2π, h is the Planck’s constant, µm = min(m, n), νm = min(m − µ, n − µ), r is the center-to-center distance, d =r − 2a is the surface to surface distance, a is the radius of the spheres, and jm and h m are modified spherical Bessel functions of order m of the first and third kind, respectively. According to the microscopic theory, the interaction between two spheres of radius a1 and a2 separated by a center-to-center distance r is obtained by pairwise summation of interactions of all molecules in sphere 1 with all molecules in sphere 2. When London’s equation is used as the starting point, pairwise summation leads to the Hamaker equation, which does not include retardation effects. The interaction energy for two spheres is then given by
where λ is the characteristic wavelength (typically set to 100 nm), A = 1.01, B = 0.14(2π λ), C = 2.45/(2π/λ), and D = 2.04/(2π/λ)2 . Clayfield et al. (12) obtained an analytical expression for Eq. [3], which is lengthy but computationally tractable, by dividing the separation distance into five regions. The analytical solution to [3] will henceforth be referred to as the pairwise additive (PA) approximation. Within the framework of the macroscopic theory, the interaction between two spheres could be computed by using the Derjaguin approximation,
· Aeff 2a1 a2 2a1 a2 + 2 V (r ) = − 2 2 6 r − (a1 + a2 ) r − (a1 − a2 )2 µ 2 ¶¸ r − (a1 + a2 )2 + ln 2 . r − (a1 − a2 )2
where Vf is the interaction energy between two semiinfinite slabs which, according to Liftshitz theory, is given by [2]
2πa1 a2 V (d) = a1 + a2
Vf (l) = −
Z
∞
Vf (l) dl,
[5]
h
Aeff , l2πl 2
[6]
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THENNADIL AND GARCIA-RUBIO
where l is the distance between the two slabs and Aeff is given by ∞ X 3 0 Aeff = − kB T I (ξ N , d) 2 N =0 µ ½ √ ¶ Z 2ξ N d ε3 2 ∞ ¯ 13 1 ¯ 23 p dp ln(1 − 1 I (ξ N , d) = c µ √ ¶ 2ξ N pd ε3 × exp − c µ µ √ ¶¶¾ 2ξ N pd ε3 + ln 1 − 113 123 exp − c
[7]
[8]
¯ j3 = s j ε3 − pε j 1 j3 = s j − p 1 s j ε3 + pε j sj + p q s j = p 2 − 1 + ε j /ε3 ,
spheres based on the interaction energy of two slabs. According to this approximation, the interaction energy between two spheres is given by
where ε = ε(iξN ) and j = 1, 2. Pailthorpe and Russel (11) evaluated an approximation suggested by Mahanty and Ninham (20) for the case of high ionic strengths when the microwave (N = 0) term is completely suppressed. They used the Hamaker equation [2] for two equal spheres with the Hamaker constant calculated using Eq. [10] with the summation carried out from N > 0. They found that this expression was superior to the Derjaguin approximation. For salt-free systems, the microwave term could be calculated using the zero-frequency term of the approximate expression for nonretarded van der Waals interaction energy derived by Kiefer et al. (10): V A = −0.5kB T g(z) ¶ 2ν µ 1 1 1 1 g(z) = + 2 2 8 sinh νθ ν cosh νθ
FIG. 1. Papadopoulous and Cheh (PC) method for computing the interaction energy of two spheres.
[9]
1 − ln{[1 + F(1)][1 + F(−1)]} 8 ∞ X sinh θ F(1) = 1m sinh (m + 1)θ m=1 µ ¶ ε−1 1= ζ (z) = 2[η(z) + ϑ(z)]0.5 − 1 ε + ζ (z) µ ¶ 1 3 2 1 + − 2 η(z) = 2 2 2 z −4 z z −1 µ µ ¶ ¶ 1 1 4 1 1 + + ln 1 − 2 ϑ(z) = 2 z2 − 4 z2 4 z µ ¶ ε1 (0) z r ; θ = cosh−1 . z= ; ε= a ε3 (0) 2 The Papadopoulous–Cheh (PC) approximation (21) has been proposed as an improvement over Derjaguin’s derivation for
Zθmax V (d) =
V f (s) sin θ dθ +
πa12
πa22
0
Zχmax sin χ V f (s) dχ , [10] 0
where θ and χ are angles shown in Fig. 1, s is the length of the arc connecting the two rings in each sphere, and θmax and χmax are defined by θmax + χmax = π. The accuracy of this approximation in the computation of van der Waals forces has not been evaluated. In this manuscript, the van der Waals interaction energy will be calculated using different approximations for polystyrene spheres in salt-free water. The “exact” expression of Langbein will be used as the benchmark for evaluating the efficacy of the various expressions. 3. RESULTS AND DISCUSSION
The calculation of van der Waals interactions using Lifshitz theory requires the construction of the quantity ε(iξ ), the dielectric response of the particles and the medium for imaginary frequencies. We can also compute the Hamaker constant given ε(iξ ) using Aeff =
∞ X ∞ 3kB T X (112 123 ) j 2 N =0 j=1 j3
1ik =
εi − εk . εi + εk
[11]
When data at major absorption frequencies are available, ε(iξN ) could be constructed by fitting the data to a damped linear oscillator model (4), ε(ξ ) = 1 +
X j
ω2j
f j − iξ h j , − ξ 2 − iξ g j
[12]
139
VAN DER WAALS ENERGY BETWEEN SPHERICAL PARTICLES
To evaluate Eq. [1], we first have to rewrite the sums on m and n P by makingPa change of variables, j = m + n, so that P ∞ ∞ ∞ P j−1 → m=1 n=1 j=2 m=1 . On plotting the terms in the sum over j, it appeared that the convergence was approximately geometric for j > 10. Therefore, an improved estimate of the jth partial sum S j when j > 10 can be obtained by (26, 27) S j = S j+1 −
(S j+1 − S j )2 . S j+1 − 2S j + S j−1
[13]
The improved estimates of the partial sums converge much faster. The same technique can be used on the outer sum over N . As an example, consider the case of two polystyrene spheres of radius 100 nm separated by a distance of 20 nm in water. By directly computing the sum over j, to achieve convergence with
FIG. 2. Comparison of the macroscopic and microscopic theories for polystyrene spheres in water. (a) Interaction energy using Langbein’s and Clayfield’s (PA approximation) expressions for two spheres of 100 nm radius. (b) Percentage difference for different sizes.
where the parameters f j correspond to oscillator strengths, ω j to resonance frequencies, and g j to the bandwidth of the response. This approach has been used (16, 22, 23) to calculate the Hamaker constants of several materials. Dielectric data for polystyrene and water at 20◦ C were constructed using the estimates given by Parsegian and Weiss (24) for the parameters in Eq. [10]. The convergence of Langbein’s “exact” expression (Eq. [1]) is very slow. Pailthorpe and Russel (11) evaluated [1] by computing term by term the difference between the retarded and nonretarded interactions, which converges much faster. They then evaluated the nonretarded interaction by using Love’s technique (25), which is faster than evaluating Langbein’s nonretarded expression, and added this to the difference. In this study, a different approach was used to accelerate the convergence.
FIG. 3. Comparison of the interaction energy computed using the MNK approximation with the exact expression. (a) Interaction energy for two spheres of 100 nm radius. (b) Percentage difference for different sizes.
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THENNADIL AND GARCIA-RUBIO
error when the MNK approximation is used are similar to those of the “exact” microscopic theory. Figure 4 shows the percent deviation from the “exact” Langbein’s expression of the Derjaguin and the PC approximations. It is seen that, while the PC approximation is more accurate than the Derjaguin approximation, it still is much worse than the previous two approximations. Since the PA approximation is an analytical expression requiring negligible computing time and gives the same order of error as the MNK approximation, it would be the approximation of choice. However, in systems containing electrolyte, the screening of van der Waals interactions has to be taken into account (20, 28–31). Only the microwave term is affected by the presence of electrolyte. Within the framework of the macroscopic theory, this effect can be taken into account by treating the N = 0
FIG. 4. Comparison of Derjaguin’s and the PC approximation with Langbein’s expression. (a) Percentage deviation of Derjaguin’s approximation for different sizes. (b) Percentage deviation of the PC approximation for different sizes.
a tolerance of 10−4 , 43–61 terms were needed for each N , and for the convergence of the outer sum over N , 109 terms were needed. By using Eq. [11] for the same tolerance levels, the j terms needed ranged from 21–45 and number of terms needed for the sum over N was only 59. Thus, the computation time was greatly reduced. Figure 2 compares the PA approximation with the “exact” macroscopic theory given by Eq. [1]. It can be seen that the PA approximation overestimates the interaction energy at small separations and underestimates it at large separations. As the particle size increases, the separation at which it starts to underestimate becomes smaller. Figure 3 compares the approximation using Eqs. [8] and [9], which will henceforth be referred to as the MNK approximation. We observe that, for the case of polystyrene in water, the magnitude and characteristics of the
FIG. 5. Comparison of the modified PA and MN approximation with Langbein’s expression at high electrolyte concentration. (a) Percentage deviation of the MN approximation for different sizes. (b) Percentage deviation of the modified PA approximation for different sizes.
VAN DER WAALS ENERGY BETWEEN SPHERICAL PARTICLES
term separately. In the case of the microscopic theory, a plausible way to include this screening effect would be to evaluate Eq. [3] using the value of Aeff computed by suppressing the N = 0 term in Eq. [11]. This value would give the interaction energy contribution from the higher frequencies, which are not affected by the presence of electrolyte. The microwave term can then be treated separately to account for the effect of electrolyte. To test the feasibility of this approach, consider the case where the salt concentration is sufficiently high to completely screen the microwave term. Figure 5a shows the percent deviation from the “exact” expression when the approximation of Mahanty and Ninham (MN) is used. Figure 5b shows the deviation when the PA approximation is used with Aeff computed by suppressing the N = 0 term. It can be seen that the MN approximation is slightly superior to the PA approximation. Nevertheless, the ease of computing the PA approximation makes it an attractive choice. Further, in this case, only two parameters are required to use this modified PA approximation, viz., the relative dielectric constant (for the microwave term) and the Hamaker constant at high salt concentration, which can be obtained experimentally. On the other hand, the MN approximation requires dielectric data over a wide range of frequencies, which are not available for many materials. 4. SUMMARY AND CONCLUSIONS
In this manuscript, van der Waals interaction energies for polystyrene spheres in pure water were computed using approximations whose performances were compared using the “exact” expression derived by Langbein as the benchmark. Langbein’s expression was evaluated by using Aitken’s method to speed up convergence of the infinite series. It was found that the accuracy of the PA approximation is comparable to that of the MNK approximation. The PC approximation showed significant improvement over the Derjaguin approximation, but the errors were much higher than in the PA and MNK approximations. The PA approximation requires negligible computation time and requires only the value of the effective Hamaker constant, which is available for a large number of materials. On the other hand, the MNK approximation is computationally more intensive and requires dielectric data over a wide frequency range, which is available only for a limited number of materials. In situations where the van der Waals interaction energy has to be evaluated several times for different particle sizes and distances, the MNK approximation will be much more cumbersome to use than the PA approximation. The study of aggregation during a polymerization reaction (32) will be one such example where the simplicity of the PA approximation will be a tremendous advantage. Another example where this simplicity helps is in the study of the equilibrium structure of colloidal dispersions using Monte Carlo simulations (33). In light of these considerations, the PA approximation would be the method of choice for calculating the van der Waals interactions between spherical particles.
141
An apparent disadvantage in using the PA approximation is that it does not include the effect of screening in systems containing electrolyte. In this manuscript, a modification to the PA approximation was proposed to overcome this disadvantage. This modification consisted of suppressing the zero-frequency term in the computation of the Hamaker constant, so that the value calculated by the PA approximation was the contribution from frequencies greater than zero. The zero-frequency term can then be calculated by using the macroscopic theory. Calculations in the limit of very high electrolyte concentrations suggest that this approach is feasible. Currently, an expression for calculating the zero-frequency term for spherical particles in solvent containing electrolyte is not available. The screening effect is calculated by applying the Derjaguin approximation to two parallel semi-infinite slabs in electrolyte. In this manuscript, it is shown that the PC approximation is more accurate than the Derjaguin approximation and thus could be used for such computations. In conclusion, properties of dispersions of interacting particles in moderate to high salt concentrations are heavily influenced by van der Waals interactions. In quantitative investigations of the stability and structure of colloidal dispersions, a practical and accurate means of calculating the van der Waals interactions is necessary. Analysis reported in this manuscript indicates that the PA approximation is the best alternative in terms of accuracy and computational considerations. REFERENCES 1. Verwey, E. J. W., and Overbeek, J. Th. G., “Theory of the Stability of Lyophobic Colloids.” Elsevier, 1948. 2. Casimir, H. B. G., and Polder, D., Phys. Rev. 73, 360 (1948). 3. Lifshitz, E. M., Soviet Phys. JETP 2, 73 (1956). 4. Parsegian, V. A., in “Physical Chemistry: Enriching Topics from Colloid and Surface Science” (van Olphen and K. J. Mysels, Eds.), Ch. 4. Theorex, La Jolla, CA, 1975. 5. van Kampen, N. G., Nijboer, B. R. A., and Schram, K., Phys. Lett. A 26, 307 (1968). 6. Ninham, B. W., Parsegian, V. A., and Weiss, G. H., J. Stat. Phys. 2, 323 (1970). 7. Langbein, D., Phys. Rev. B 2, 3371 (1970). 8. Langbein, D., “Theory of van der Waals Attraction,” Springer Tracts in Modern Physics, Vol. 72. Springer-Verlag, Berlin, New York, 1974. 9. Mitchell, D. J., and Ninham, B. W., J. Chem. Phys. 56, 1117 (1972). 10. Kiefer, J. E., Parsegian, V. A., and Weiss, G. H., J. Colloid Interface Sci. 57, 580 (1976). 11. Pailthorpe, B. A., and Russel, W. B., J. Colloid Interface Sci. 89, 563 (1982). 12. Clayfield, E. J., Lumb, E. C., and Mackey, P. H., J. Colloid Interface Sci. 37, 382 (1971). 13. Schenkel, J. H., and Kitchener, J. A., Trans. Faraday Soc. 56, 161 (1960). 14. Ho, N. F. H., and Higuchi, W. I., J. Pharm. Sci. 57, 436 (1968). 15. Gregory, J., J. Colloid Interface Sci. 83, 138 (1981). 16. Parsegian, V. A., and Ninham, B. W., J. Colloid Interface Sci. 37, 332 (1971). 17. Smith, E. R., Mitchell, D. J., and Ninham, B. W., J. Colloid Interface Sci. 45, 55 (1973). 18. Visser, J., Adv. Colloid Interface Sci. 3, 331 (1972). 19. Israelachvili, J., “Intermolecular and surface forces.” Academic Press, New York, 1991.
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20. Mahanty, J., and Ninham, B. W., “Dispersion Forces.” Academic Press, New York, 1976. 21. Papadopoulous, K. D., and Cheh, H. Y., AICHE J. 30, 7 (1984). 22. Prieve, D. C., and Russel, W. B., J. Colloid Interface Sci. 125, 1 (1988). 23. Gregory, J., Adv. Colloid Interface Sci. 2, 396 (1981). 24. Parsegian, V. A., and Weiss, G. H., J. Colloid Interface Sci. 81, 285 (1981). 25. Love, J. D., J. Chem. Soc., Faraday Trans. II 73, 669 (1977). 26. Hamming, R. W., “Numerical Methods for Scientists and Engineers.” Dover, 1986. 27. Press, W. H., Teukolsky, S. A., Vettering, W. T., and Flannery, B. P., “Numerical Recipes in Fortran—The Art of Scientific Computing.” Cambridge University Press, 1994.
28. Davies, B., and Ninham, B. W., J. Chem. Phys. 56, 5797 (1972). 29. Barouch, E., Perram, J. W., and Smith, E. R., Chem. Phys. Lett. 19, 131 (1973). 30. Mitchell, D. J., and Richmond, P., J. Colloid Interface Sci. 46, 118 (1974). 31. Barnes, C. J., and Davies, B., J. Chem. Soc. Faraday Trans. II 71, 1667 (1975). 32. Marquez, E., “Mathematical Modelling and Simulation of Agglomeration Phenomena in Mass and Suspension PVC,” Ph.D. Dissertation. University of South Florida, 1993. 33. Thennadil, S. N., “Investigation of Structure of Concentrated Colloidal Dispersions and Its Effect on Light Scattering Using Monte Carlo Simulations,” Ph.D. Dissertation. University of South Florida, 2001.
Approximations for Calculating van der Waals Interaction Energy between Spherical Particles—A Comparison Suresh N. Thennadil1 and Luis H. Garcia-Rubio2 Department of Chemical Engineering, University of South Florida, Tampa, Florida 33620 Received June 11, 2001; accepted July 14, 2001; published online September 24, 2001
The DLVO theory has been widely used to study the stability of colloidal systems with the van der Waals interactions modeled according to the Hamaker (microscopic) theory. The Hamaker theory, in addition to neglecting many-body interactions, does not account for retardation effects. Retardation effects can be included within the framework of the microscopic theory. For spherical particles, an “exact” expression has been derived by Clayfield et al. Lifshitz (macroscopic) theory, on the other hand, accounts for both manybody and retardation effects but is computationally intensive and requires dielectric data at different frequencies. For spherical particles, it is not practical to use the “exact” expression for van der Waals interaction derived by Langbein; therefore approximations have to be used. Approximate expressions to calculate the van der Waals interaction energy between spheres were considered in terms of accuracy, ease of computation, and required material parameters with the “exact” expression derived by Langbein used as the benchmark. It was found that the “exact” expression using the microscopic theory works as well as the best approximation to the macroscopic theory. °C 2001 Academic Press Key Words: colloidal dispersions; van der Waals interactions; retardation effects; Lifshitz theory; Hamaker theory; microscopic theory.
1. INTRODUCTION
The DLVO theory (1) has been widely used to study the stability of colloidal dispersions. According to this theory, the total interaction between charged particles dispersed in a solvent containing electrolyte is assumed to be the sum of van der Waals and electrostatic interactions. At moderate to high salt concentrations, which are the regime of interest when aggregation phenomena are investigated, the two forces compete. Under these conditions the electrostatic interactions are adequately described by the classical Poisson–Boltzmann equation. The van der Waals interactions are much more complicated, and an adequate
1 To whom correspondence should be addressed at present address: Instrumentation Metrics Inc., 7470 W. Chandler Blvd., Chandler, AZ 85226. Fax: (480) 755-9832. E-mail: [email protected]. 2 Present address: ENB 118, Department of Chemical Engineering, University of South Florida, 3202 E. Fowler Avenue Tampa, FL 33620. Fax: (813) 9743997. E-mail: [email protected].
0021-9797/01 $35.00
C 2001 by Academic Press Copyright ° All rights of reproduction in any form reserved.
description of these interactions is essential for the study of the structure of colloidal dispersions as well as aggregation phenomena in a quantitative manner. Usually, the van der Waals interaction energy is obtained by using Hamaker’s equation. This equation is based on the microscopic theory, which assumes that the interaction between two particles is just the sum of the interactions between each molecule in one particle and all of the molecules in the other particles. This assumption of pairwise additivity neglects many-body interactions. In addition, the Hamaker equation neglects retardation effects. Within the framework of the microscopic theory, retardation effects can be included by using the Casimir–Polder equation (2) as the starting point. A more rigorous theory called the macroscopic or Lifshitz theory, which takes into account many-body interactions, was developed by Lifshitz (3–6). According to this theory, the particles and the intervening fluids are treated as individual macroscopic phases, which are fully characterized by their dielectric permittivities. This theory can be applied to any body at any temperature and takes retardation effects into account. For the case of spherical particles, Langbein (7, 8) derived an “exact” expression within the framework of Lifshitz theory. This equation is computationally intensive, in addition to exhibiting problems of convergence at close separations, making it impractical to use. Several approximations to this expression have been proposed (9–11). On the other hand, the “exact” expression within the framework of the microscopic theory, derived by Clayfield et al. (12) though algebraically involved, is easy to compute. Simplified expressions that include retardation effects using the microscopic theory have been proposed (13–15). Ideally, we would like to use Langbein’s “exact” expression based on the Lifshitz theory to calculate the energy of interaction between two spheres. However, as mentioned earlier, this expression is computationally intensive and is beset with problems of convergence. Further, we require reliable dielectric data over a wide range of frequencies to be able to use the Lifshitz theory. Such data may not be available for materials under investigation. The general tendency has been to use Hamaker’s equation, although it has been shown to lead to significant deviations (16, 17) in the estimation of van der Waals interactions. The advantage of using the Hamaker equation is that, in addition to its simplicity, the Hamaker constant can be experimentally determined (18, 19) and these data are available for a
136
137
VAN DER WAALS ENERGY BETWEEN SPHERICAL PARTICLES
large number of materials. While several approximations to the “exact” macroscopic and microscopic theories exist, the availability of high-speed computers alleviates the need to oversimplify the equations just for the sake of reducing the computational intensity. In this manuscript, calculations will be performed for polystyrene spheres in water. These calculations will be used to evaluate the effects of retardation and the accuracy and practicality of using different approximations. 2. EXPRESSIONS FOR VAN DER WAALS INTERACTION ENERGY BETWEEN SPHERES
Retardation effects can be included by using the Casimir– Polder equation as the starting point. In this case, the method of pairwise summation for deriving the interaction energy between two spheres gives (12) Z 2a1 Aeff d x x(2a1 − x) (d + a1 + a2 ) 0 Z 2a2 y(2a2 − y) dy F(r ), × r5 0
V (d) = −
where d = r − a1 − a2 and r = d + x + y. F(r ) is given by
The “exact” expression for the interaction energy between two equal spheres is given by (8, 11)
V (d) = kB T
∞ X ∞ X
(2m + 1)(2n + 1)1m 1n
F(r ) = A − Br
r < 3λ/2π
F(R) = C/r − D/r 2 r > 3λ/2π, µm X µ=−µm
N =0 m,n=1
[3]
µ −µ Vmn (kr )Vnm (kr )
[4]
[1]
¡
¢ k 2 − k12 m jm (ak) jm (ak1 ) + kk1 [k1 jm (ak1 ) jm−1 (ak) − k jm (ak) jm−1 (ak1 )] ¢ k 2 − k12 m h m (ak) jm (ak1 ) + kk1 [k1 jm (ak1 )h m−1 (ak) − kh m (ak) jm−1 (ak1 )]
1m = ¡
µ µ Vmn (kr ) = Umn (kr ) +
n−µ+1 (n + µ) µ µ krUmn+1 (kr ) − krUmn−1 (kr ) (n + 1)(2n + 1) n(2n + 1)
(n − µ)!(m − µ)! µ V (kr ) (n + µ)!(m + µ)! mn µ ¶µ X νm 2 µ µν Umn (kr ) = Smn h m+n−µ−2ν (kr ) kr ν=0
−µ (kr ) = Vmn
µν Smn =
0(m − ν + 0.5)0(n − ν + 0.5)0(m + ν + 0.5)(m + n − ν)!(m + n − µ − 2ν + 0.5) , 0(m + n − µ − ν + 1.5)0(µ + 0.5)0(0.5)(m − µ − ν)!(n − µ − ν)!ν!
where k 2 = (ξN /c)2 ε3 (iξN ), k12 = (ξN /c)2 ε1 (iξN ), ξN = 2π N kT / h- , h- = h/2π, h is the Planck’s constant, µm = min(m, n), νm = min(m − µ, n − µ), r is the center-to-center distance, d =r − 2a is the surface to surface distance, a is the radius of the spheres, and jm and h m are modified spherical Bessel functions of order m of the first and third kind, respectively. According to the microscopic theory, the interaction between two spheres of radius a1 and a2 separated by a center-to-center distance r is obtained by pairwise summation of interactions of all molecules in sphere 1 with all molecules in sphere 2. When London’s equation is used as the starting point, pairwise summation leads to the Hamaker equation, which does not include retardation effects. The interaction energy for two spheres is then given by
where λ is the characteristic wavelength (typically set to 100 nm), A = 1.01, B = 0.14(2π λ), C = 2.45/(2π/λ), and D = 2.04/(2π/λ)2 . Clayfield et al. (12) obtained an analytical expression for Eq. [3], which is lengthy but computationally tractable, by dividing the separation distance into five regions. The analytical solution to [3] will henceforth be referred to as the pairwise additive (PA) approximation. Within the framework of the macroscopic theory, the interaction between two spheres could be computed by using the Derjaguin approximation,
· Aeff 2a1 a2 2a1 a2 + 2 V (r ) = − 2 2 6 r − (a1 + a2 ) r − (a1 − a2 )2 µ 2 ¶¸ r − (a1 + a2 )2 + ln 2 . r − (a1 − a2 )2
where Vf is the interaction energy between two semiinfinite slabs which, according to Liftshitz theory, is given by [2]
2πa1 a2 V (d) = a1 + a2
Vf (l) = −
Z
∞
Vf (l) dl,
[5]
h
Aeff , l2πl 2
[6]
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THENNADIL AND GARCIA-RUBIO
where l is the distance between the two slabs and Aeff is given by ∞ X 3 0 Aeff = − kB T I (ξ N , d) 2 N =0 µ ½ √ ¶ Z 2ξ N d ε3 2 ∞ ¯ 13 1 ¯ 23 p dp ln(1 − 1 I (ξ N , d) = c µ √ ¶ 2ξ N pd ε3 × exp − c µ µ √ ¶¶¾ 2ξ N pd ε3 + ln 1 − 113 123 exp − c
[7]
[8]
¯ j3 = s j ε3 − pε j 1 j3 = s j − p 1 s j ε3 + pε j sj + p q s j = p 2 − 1 + ε j /ε3 ,
spheres based on the interaction energy of two slabs. According to this approximation, the interaction energy between two spheres is given by
where ε = ε(iξN ) and j = 1, 2. Pailthorpe and Russel (11) evaluated an approximation suggested by Mahanty and Ninham (20) for the case of high ionic strengths when the microwave (N = 0) term is completely suppressed. They used the Hamaker equation [2] for two equal spheres with the Hamaker constant calculated using Eq. [10] with the summation carried out from N > 0. They found that this expression was superior to the Derjaguin approximation. For salt-free systems, the microwave term could be calculated using the zero-frequency term of the approximate expression for nonretarded van der Waals interaction energy derived by Kiefer et al. (10): V A = −0.5kB T g(z) ¶ 2ν µ 1 1 1 1 g(z) = + 2 2 8 sinh νθ ν cosh νθ
FIG. 1. Papadopoulous and Cheh (PC) method for computing the interaction energy of two spheres.
[9]
1 − ln{[1 + F(1)][1 + F(−1)]} 8 ∞ X sinh θ F(1) = 1m sinh (m + 1)θ m=1 µ ¶ ε−1 1= ζ (z) = 2[η(z) + ϑ(z)]0.5 − 1 ε + ζ (z) µ ¶ 1 3 2 1 + − 2 η(z) = 2 2 2 z −4 z z −1 µ µ ¶ ¶ 1 1 4 1 1 + + ln 1 − 2 ϑ(z) = 2 z2 − 4 z2 4 z µ ¶ ε1 (0) z r ; θ = cosh−1 . z= ; ε= a ε3 (0) 2 The Papadopoulous–Cheh (PC) approximation (21) has been proposed as an improvement over Derjaguin’s derivation for
Zθmax V (d) =
V f (s) sin θ dθ +
πa12
πa22
0
Zχmax sin χ V f (s) dχ , [10] 0
where θ and χ are angles shown in Fig. 1, s is the length of the arc connecting the two rings in each sphere, and θmax and χmax are defined by θmax + χmax = π. The accuracy of this approximation in the computation of van der Waals forces has not been evaluated. In this manuscript, the van der Waals interaction energy will be calculated using different approximations for polystyrene spheres in salt-free water. The “exact” expression of Langbein will be used as the benchmark for evaluating the efficacy of the various expressions. 3. RESULTS AND DISCUSSION
The calculation of van der Waals interactions using Lifshitz theory requires the construction of the quantity ε(iξ ), the dielectric response of the particles and the medium for imaginary frequencies. We can also compute the Hamaker constant given ε(iξ ) using Aeff =
∞ X ∞ 3kB T X (112 123 ) j 2 N =0 j=1 j3
1ik =
εi − εk . εi + εk
[11]
When data at major absorption frequencies are available, ε(iξN ) could be constructed by fitting the data to a damped linear oscillator model (4), ε(ξ ) = 1 +
X j
ω2j
f j − iξ h j , − ξ 2 − iξ g j
[12]
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VAN DER WAALS ENERGY BETWEEN SPHERICAL PARTICLES
To evaluate Eq. [1], we first have to rewrite the sums on m and n P by makingPa change of variables, j = m + n, so that P ∞ ∞ ∞ P j−1 → m=1 n=1 j=2 m=1 . On plotting the terms in the sum over j, it appeared that the convergence was approximately geometric for j > 10. Therefore, an improved estimate of the jth partial sum S j when j > 10 can be obtained by (26, 27) S j = S j+1 −
(S j+1 − S j )2 . S j+1 − 2S j + S j−1
[13]
The improved estimates of the partial sums converge much faster. The same technique can be used on the outer sum over N . As an example, consider the case of two polystyrene spheres of radius 100 nm separated by a distance of 20 nm in water. By directly computing the sum over j, to achieve convergence with
FIG. 2. Comparison of the macroscopic and microscopic theories for polystyrene spheres in water. (a) Interaction energy using Langbein’s and Clayfield’s (PA approximation) expressions for two spheres of 100 nm radius. (b) Percentage difference for different sizes.
where the parameters f j correspond to oscillator strengths, ω j to resonance frequencies, and g j to the bandwidth of the response. This approach has been used (16, 22, 23) to calculate the Hamaker constants of several materials. Dielectric data for polystyrene and water at 20◦ C were constructed using the estimates given by Parsegian and Weiss (24) for the parameters in Eq. [10]. The convergence of Langbein’s “exact” expression (Eq. [1]) is very slow. Pailthorpe and Russel (11) evaluated [1] by computing term by term the difference between the retarded and nonretarded interactions, which converges much faster. They then evaluated the nonretarded interaction by using Love’s technique (25), which is faster than evaluating Langbein’s nonretarded expression, and added this to the difference. In this study, a different approach was used to accelerate the convergence.
FIG. 3. Comparison of the interaction energy computed using the MNK approximation with the exact expression. (a) Interaction energy for two spheres of 100 nm radius. (b) Percentage difference for different sizes.
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THENNADIL AND GARCIA-RUBIO
error when the MNK approximation is used are similar to those of the “exact” microscopic theory. Figure 4 shows the percent deviation from the “exact” Langbein’s expression of the Derjaguin and the PC approximations. It is seen that, while the PC approximation is more accurate than the Derjaguin approximation, it still is much worse than the previous two approximations. Since the PA approximation is an analytical expression requiring negligible computing time and gives the same order of error as the MNK approximation, it would be the approximation of choice. However, in systems containing electrolyte, the screening of van der Waals interactions has to be taken into account (20, 28–31). Only the microwave term is affected by the presence of electrolyte. Within the framework of the macroscopic theory, this effect can be taken into account by treating the N = 0
FIG. 4. Comparison of Derjaguin’s and the PC approximation with Langbein’s expression. (a) Percentage deviation of Derjaguin’s approximation for different sizes. (b) Percentage deviation of the PC approximation for different sizes.
a tolerance of 10−4 , 43–61 terms were needed for each N , and for the convergence of the outer sum over N , 109 terms were needed. By using Eq. [11] for the same tolerance levels, the j terms needed ranged from 21–45 and number of terms needed for the sum over N was only 59. Thus, the computation time was greatly reduced. Figure 2 compares the PA approximation with the “exact” macroscopic theory given by Eq. [1]. It can be seen that the PA approximation overestimates the interaction energy at small separations and underestimates it at large separations. As the particle size increases, the separation at which it starts to underestimate becomes smaller. Figure 3 compares the approximation using Eqs. [8] and [9], which will henceforth be referred to as the MNK approximation. We observe that, for the case of polystyrene in water, the magnitude and characteristics of the
FIG. 5. Comparison of the modified PA and MN approximation with Langbein’s expression at high electrolyte concentration. (a) Percentage deviation of the MN approximation for different sizes. (b) Percentage deviation of the modified PA approximation for different sizes.
VAN DER WAALS ENERGY BETWEEN SPHERICAL PARTICLES
term separately. In the case of the microscopic theory, a plausible way to include this screening effect would be to evaluate Eq. [3] using the value of Aeff computed by suppressing the N = 0 term in Eq. [11]. This value would give the interaction energy contribution from the higher frequencies, which are not affected by the presence of electrolyte. The microwave term can then be treated separately to account for the effect of electrolyte. To test the feasibility of this approach, consider the case where the salt concentration is sufficiently high to completely screen the microwave term. Figure 5a shows the percent deviation from the “exact” expression when the approximation of Mahanty and Ninham (MN) is used. Figure 5b shows the deviation when the PA approximation is used with Aeff computed by suppressing the N = 0 term. It can be seen that the MN approximation is slightly superior to the PA approximation. Nevertheless, the ease of computing the PA approximation makes it an attractive choice. Further, in this case, only two parameters are required to use this modified PA approximation, viz., the relative dielectric constant (for the microwave term) and the Hamaker constant at high salt concentration, which can be obtained experimentally. On the other hand, the MN approximation requires dielectric data over a wide range of frequencies, which are not available for many materials. 4. SUMMARY AND CONCLUSIONS
In this manuscript, van der Waals interaction energies for polystyrene spheres in pure water were computed using approximations whose performances were compared using the “exact” expression derived by Langbein as the benchmark. Langbein’s expression was evaluated by using Aitken’s method to speed up convergence of the infinite series. It was found that the accuracy of the PA approximation is comparable to that of the MNK approximation. The PC approximation showed significant improvement over the Derjaguin approximation, but the errors were much higher than in the PA and MNK approximations. The PA approximation requires negligible computation time and requires only the value of the effective Hamaker constant, which is available for a large number of materials. On the other hand, the MNK approximation is computationally more intensive and requires dielectric data over a wide frequency range, which is available only for a limited number of materials. In situations where the van der Waals interaction energy has to be evaluated several times for different particle sizes and distances, the MNK approximation will be much more cumbersome to use than the PA approximation. The study of aggregation during a polymerization reaction (32) will be one such example where the simplicity of the PA approximation will be a tremendous advantage. Another example where this simplicity helps is in the study of the equilibrium structure of colloidal dispersions using Monte Carlo simulations (33). In light of these considerations, the PA approximation would be the method of choice for calculating the van der Waals interactions between spherical particles.
141
An apparent disadvantage in using the PA approximation is that it does not include the effect of screening in systems containing electrolyte. In this manuscript, a modification to the PA approximation was proposed to overcome this disadvantage. This modification consisted of suppressing the zero-frequency term in the computation of the Hamaker constant, so that the value calculated by the PA approximation was the contribution from frequencies greater than zero. The zero-frequency term can then be calculated by using the macroscopic theory. Calculations in the limit of very high electrolyte concentrations suggest that this approach is feasible. Currently, an expression for calculating the zero-frequency term for spherical particles in solvent containing electrolyte is not available. The screening effect is calculated by applying the Derjaguin approximation to two parallel semi-infinite slabs in electrolyte. In this manuscript, it is shown that the PC approximation is more accurate than the Derjaguin approximation and thus could be used for such computations. In conclusion, properties of dispersions of interacting particles in moderate to high salt concentrations are heavily influenced by van der Waals interactions. In quantitative investigations of the stability and structure of colloidal dispersions, a practical and accurate means of calculating the van der Waals interactions is necessary. Analysis reported in this manuscript indicates that the PA approximation is the best alternative in terms of accuracy and computational considerations. REFERENCES 1. Verwey, E. J. W., and Overbeek, J. Th. G., “Theory of the Stability of Lyophobic Colloids.” Elsevier, 1948. 2. Casimir, H. B. G., and Polder, D., Phys. Rev. 73, 360 (1948). 3. Lifshitz, E. M., Soviet Phys. JETP 2, 73 (1956). 4. Parsegian, V. A., in “Physical Chemistry: Enriching Topics from Colloid and Surface Science” (van Olphen and K. J. Mysels, Eds.), Ch. 4. Theorex, La Jolla, CA, 1975. 5. van Kampen, N. G., Nijboer, B. R. A., and Schram, K., Phys. Lett. A 26, 307 (1968). 6. Ninham, B. W., Parsegian, V. A., and Weiss, G. H., J. Stat. Phys. 2, 323 (1970). 7. Langbein, D., Phys. Rev. B 2, 3371 (1970). 8. Langbein, D., “Theory of van der Waals Attraction,” Springer Tracts in Modern Physics, Vol. 72. Springer-Verlag, Berlin, New York, 1974. 9. Mitchell, D. J., and Ninham, B. W., J. Chem. Phys. 56, 1117 (1972). 10. Kiefer, J. E., Parsegian, V. A., and Weiss, G. H., J. Colloid Interface Sci. 57, 580 (1976). 11. Pailthorpe, B. A., and Russel, W. B., J. Colloid Interface Sci. 89, 563 (1982). 12. Clayfield, E. J., Lumb, E. C., and Mackey, P. H., J. Colloid Interface Sci. 37, 382 (1971). 13. Schenkel, J. H., and Kitchener, J. A., Trans. Faraday Soc. 56, 161 (1960). 14. Ho, N. F. H., and Higuchi, W. I., J. Pharm. Sci. 57, 436 (1968). 15. Gregory, J., J. Colloid Interface Sci. 83, 138 (1981). 16. Parsegian, V. A., and Ninham, B. W., J. Colloid Interface Sci. 37, 332 (1971). 17. Smith, E. R., Mitchell, D. J., and Ninham, B. W., J. Colloid Interface Sci. 45, 55 (1973). 18. Visser, J., Adv. Colloid Interface Sci. 3, 331 (1972). 19. Israelachvili, J., “Intermolecular and surface forces.” Academic Press, New York, 1991.
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20. Mahanty, J., and Ninham, B. W., “Dispersion Forces.” Academic Press, New York, 1976. 21. Papadopoulous, K. D., and Cheh, H. Y., AICHE J. 30, 7 (1984). 22. Prieve, D. C., and Russel, W. B., J. Colloid Interface Sci. 125, 1 (1988). 23. Gregory, J., Adv. Colloid Interface Sci. 2, 396 (1981). 24. Parsegian, V. A., and Weiss, G. H., J. Colloid Interface Sci. 81, 285 (1981). 25. Love, J. D., J. Chem. Soc., Faraday Trans. II 73, 669 (1977). 26. Hamming, R. W., “Numerical Methods for Scientists and Engineers.” Dover, 1986. 27. Press, W. H., Teukolsky, S. A., Vettering, W. T., and Flannery, B. P., “Numerical Recipes in Fortran—The Art of Scientific Computing.” Cambridge University Press, 1994.
28. Davies, B., and Ninham, B. W., J. Chem. Phys. 56, 5797 (1972). 29. Barouch, E., Perram, J. W., and Smith, E. R., Chem. Phys. Lett. 19, 131 (1973). 30. Mitchell, D. J., and Richmond, P., J. Colloid Interface Sci. 46, 118 (1974). 31. Barnes, C. J., and Davies, B., J. Chem. Soc. Faraday Trans. II 71, 1667 (1975). 32. Marquez, E., “Mathematical Modelling and Simulation of Agglomeration Phenomena in Mass and Suspension PVC,” Ph.D. Dissertation. University of South Florida, 1993. 33. Thennadil, S. N., “Investigation of Structure of Concentrated Colloidal Dispersions and Its Effect on Light Scattering Using Monte Carlo Simulations,” Ph.D. Dissertation. University of South Florida, 2001.