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Limitations of the Derjaguin Approximation and the Lorentz-Berthelot Mixing Rule Jan Forsman*,† and Clifford E. Woodward‡ †
Theoretical Chemistry, Chemical Centre, P.O. Box 124, S-221 00 Lund, Sweden, and ‡School of Physical, Environmental and Mathematical Sciences, University College, University of New South Wales, ADFA Canberra ACT 2600, Australia Received July 27, 2009. Revised Manuscript Received February 12, 2010
We investigate the Derjaguin approximation by explicitly determining the interactions between two spherical colloids using density functional theory solved in cylindrical coordinates. The colloids are composed of close-packed LennardJones particles. The solvent particles are also modeled via Lennard-Jones interactions. Cross interactions are assumed to follow the commonly used Lorentz-Berthelot (LB) mixing rule. We demonstrate that this system may display a net repulsive interaction across a substantial separation range. This contradicts the Hamaker-Lifshitz theory, which predicts attractions between identical polarizable particles immersed in a polarizable medium. The source of this repulsion is traced to the LB mixing rule. Surprisingly, we also observe nonmonotonic convergences to the Derjaguin limit. This behavior is best understood by decomposing the total interaction between the colloids into separate contributions. With increasing colloid size, each of these contributions approach the Derjaguin limit in a monotonic manner, but their different rates of convergence mean that their sum may display nonmonotonic behavior.
I. Introduction The Derjaguin approximation (DA) 1 is a simple relation that is often used to estimate forces between spherical colloidal particles by using the interaction between flat surfaces. It is particularly useful in generalizing theoretical results for planar geometries and is often explicitly (or implicitly) used when interpreting measurements with the surface force apparatus (SFA).2 At first glance, the DA would appear to be valid as long as the colloids are larger than the range of the intermolecular interactions. However, in this work, we will show that convergence to the Derjaguin limit may be problematic in those cases where a balance of opposing forces exists. We shall use density functional theory (DFT) to study the interaction between two neutral spherical colloids consisting of a collection of Lennard-Jones (L-J) particles. These colloids are immersed in a solvent, also consisting of L-J particles. A similar investigation has also been carried out by Amokrane et al.3 using different levels of theory. In that study, however, a truncated L-J potential was used. Because we wish to model dispersion interactions, we will use the full L-J interaction. The particles in the colloids are generally different than those of the solvent, thus the Lorentz-Berthelot (LB) mixing rule is used to obtain cross interactions. An unexpected and interesting outcome of our investigations has been to uncover problems with the use of these rules for colloidal interactions. Indeed, our results call into question their use in macromolecular systems in general. The LB rules have been criticized in previous work,4-10 particularly in connection with interactions in rare gases. The (1) Derjaguin, B. V. Kolloid-Z. 1934, 69, 155. (2) Rentsch, S.; Pericet-Camara, R.; Papastavrou, G.; Borkovec, M. Phys. Chem. Chem. Phys. 2006, 8, 2531. (3) Amokrane, S.; Malherbe, J. G. J. Phys.: Condens. Matter 2001, 13, 7199. (4) Kong, C. H. J. Chem. Phys. 1973, 59, 2464. (5) Tang, K.; Toennies, J. P. Z. Phys. D 1986, 1, 91. (6) Waldman, M.; Hagler, A. T. J. Comput. Chem. 1993, 14, 1077. (7) Duh, D.-M.; Henderson, D.; Rowley, R. Mol. Phys. 1997, 91, 1143. (8) Delhommelle, J.; Milli, P. Mol. Phys. 2001, 99, 619. (9) Boda, D.; Henderson, D. Mol. Phys. 2008, 106, 2367. (10) Rouha, M.; Nezbeda, I. Fluid Phase Equilib. 2009, 277, 42.
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LB rules are commonly used in colloid and biophysical applications and are standard in most major molecular simulation packages (force fields). Thus, they would typically be used in full and simplified atomistic treatments of proteins, for example. As far as we know, the issues that we identify regarding the LB rules do not seem to have been appreciated in the literature. We should emphasize that these issues pertain to fundamental inconsistencies with the use of the LB rules in obtaining the potential of mean force between colloids, rather than just their accurate representation of cross interactions.
II. Model and Theory We consider the potential of mean force between two identical spherical colloids, with a radius R, immersed in a solvent. The colloids are assumed to be made of discrete particles, and all particle interactions, ΦRβ, are of the L-J type: " 6 # σ Rβ 12 σRβ ΦRβ ðrÞ ¼ 4ERβ r r
ð1Þ
The particles that make up the colloids (labeled “c”) are assumed to be at a uniform density, nc. This density is chosen to have the close-packed value, ncσcc3 = (2)1/2. All of the particles in our model system interact with the same L-J energy parameter (i.e., ɛcc = ɛss = ɛsc = 1.1 β-1. Here, β is the inverse thermal energy and the subscript “s” refers to the solvent. The last equality follows from the Berthelot combination rule, ɛsc2 = ɛssɛcc. The only interaction parameter that we vary is the ratio between particle sizes σcc/σss. We will investigate two cases, σcc/σss = 1.5 and 2. Cross diameters are obtained from the Lorentz rule, 2σsc = σcc þ σss. Note that σcc is the size of the particles that constitute the colloid and is not the colloidal diameter. The latter is given by 2R. We define a size asymmetry parameter, R = 2R/σss.
Published on Web 02/24/2010
DOI: 10.1021/la904769x
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The free energy of interaction between the colloids is determined using a nonlocal DFT.11 The grand free energy, Ω, is formulated as a function of the solvent density, ns(r), according to Z βΩ ¼ Z þβ
Z ns ðrÞðln½ns ðrÞ -1Þ dr þ β
ns ðrÞf ex ½ns ðrÞ dr
Z Z β ns ðrÞ ns ðrÞΦss ðjr0 -rjÞ dr0 ðVext -μs Þns ðrÞ dr þ 2 Z dr þ βnc χcc ðjr -rB jÞ dr ð2Þ r∈A
R Here, n is a weighted density and n(r) = 3/(4πσss3) |r-r0 | 10. In this work, we use a brute-force DFT approach. Here, the free energy is a function of the solvent density only and is solved in the full asymmetric potential set up by both colloidal particles. Repeating this procedure, for an increasing asymmetry parameter (R), will provide a consistent framework within which to evaluate the convergence to the Derjaguin limit. We anticipate that the brute-force method provides a more well-defined approach to the Derjaguin limit than the test particle methods mentioned earlier. In the former case, R affects only the external potential, which acts on the pure solvent. The R-dependent direct AB interaction does not affect the intrinsic free energy of the solvent. Therefore, in effect, we are investigating the response of the solvent to the application of varying applied potentials whereas the fluid correlations are treated in a consistent way that is intrinsic to the functional (eq 2). However, in the test-particle approach, as R varies, both the external potential (of the single colloid) and the correlations in the dilute fluid mixture change. This will sorely test the accuracy of any DFT, in particular, its ability to reflect changes accurately in the correlations in a fluid (11) Nordholm, S.; Johnson, M.; Freasier, B. C. Aust. J. Chem. 1980, 33, 2139. (12) Carnahan, N. F.; Starling, K. E. J. Chem. Phys. 1969, 51, 635. (13) Oetell, N. Phys. Rev. E 2004, 69, 041404.
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mixture, as the interactions are dramatically altered. It also makes the interpretation of the approach to the Derjaguin limit more difficult because of the influence of R on two different aspects of the physics. The solution of the functional (eq 2) is straightforward for planar surfaces. However, the potential generated by two spherical colloids has cylindrical symmetry. We solved this latter case in cylindrical coordinates, with the z axis along the line joining the centers of the colloids. For planar surfaces, we denote the interaction free energy per unit area as g(D) = Ωeq(D)/A þ PbD, with D being the surface separation. Here, Ωeq is the equilibrium grand potential, A is the surface area, and Pb is the bulk pressure. The quantity gs(D) = g(D) - g(D = ¥) is thus the net interaction free energy per unit area. For two spheres, it is useful to define an excess free energy W = Ωeq - Ωref, where Ωref is the equilibrium grand potential for the bulk solvent.14 The potential of mean force is then obtained as Ws = W(D) - W(¥). In addition to the full DFT solutions, we also consider the case where the solvent is assumed to adopt a uniform density about the colloidal particles. We denote this the uniform density approximation (UDA). We note that the UDA is more consistent with the usual Hamaker(-Lifshitz) approach to the interaction between polarizable bodies in an intervening polarizable medium. Hamaker’s classical study dealt particularly with the attractive dispersive component of the particle interactions. Here, we will also consider the repulsive component of the interactions, as modeled by the Lennard-Jones interactions. In our approach, the interaction between the solvent and the particles that constitute the colloids carries the L-J parameters determined by the LB rules. We shall see below that this can lead to unphysical predictions of colloidal interactions. The difference between the full DFT solution and the UDA result is due to the structuring of the solvent. This effect has been investigated in previous work15 for the planar case. Here, we shall consider the influence of solvent structuring on the approach to the Derjaguin limit.
III. Results The DA can be obtained by several routes.13,16 R In our nomenclature, it can be expressed as Ws(D) ≈ -πR ¥ D dx gs(x) where gs(x) is the interaction free energy per unit area between two planar surfaces at separation x. The force, F, acting between the colloids is FðDÞ πRgs ðDÞ
ð4Þ
In Figure 1, we show forces and interaction free energies, as obtained for several colloidal diameters. We chose two different values for σcc/σss: 1.5 and 2. First, we note that, for both cases, there is quite a slow convergence to the Derjaguin limit. Even for a colloidal diameter of 40σss there is a notable discrepancy between the spherical and planar results. Second, despite the small difference in the diameter of particles that constitute the colloids, the resulting interactions are remarkably different. For σcc/σss = 1.5, we observe an attraction for small colloidal radii that turns weakly repulsive for intermediate sizes and then attractive again when the colloids are sufficiently large. The Derjaguin limit is attractive. However, with σcc/σsc = 2, the DFT solution predicts that small colloids attract whereas larger ones repel, at least at short and intermediate separations. The Derjaguin limit is predominately repulsive, with very weak attraction at large (14) Forsman, J.; Woodward, C. E. J. Chem. Phys. 2009, 131, 044903. (15) Forsman, J.; J€onsson, B.; Woodward, C. E.; Wennerstr€om, H. J. Phys. Chem. B 1997, 101, 4253. (16) Henderson, J. R. Physica A 2002, 313, 321.
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Figure 2. Evaluations of the Derjaguin approximation, using the UDA approximation, for L-J colloids immersed in an L-J solvent. Black lines are the limiting flat surface results: βπgsσss2. (a) σcc = 1.5σss and (b) σcc = 2σss.
Figure 1. Evaluations of the Derjaguin approximation using full DFT calculations for L-J colloids of various radii, R, immersed in an L-J solvent. The colloid sizes are expressed in terms of the solvent diameter: R = 2R/σss. The thick black lines in graphs a and c are the limiting flat surface results: βπgsσss2. (a) Forces, with σcc = 1.5σss. (b) Free energies, with σcc = 1.5σss. (c) Forces, with σcc = 2σss. The dashed lines with square symbols show forces obtained from δWs/ δD for R = 5 and 40. (d) Free energies, with σcc = 2σss.
separations. The dashed lines with square symbols in Figure 1c are obtained as discrete derivatives of the potential of mean forces, Ws (for R = 5 and 40), and we see that they agree with the corresponding results obtained by summing up forces between the colloids and the solvent (our “default” method). This is an important consistency check for any nonlocal DFT. The reason for the slow convergence can be ascertained from Figure 2, which shows results from the UDA. Even though the solvent density is constrained to be constant, the Derjaguin limit is only slowly approached. This is due to the slow convergence of the attractive dispersion component. Dispersion interactions between colloids, Wdisp, are commonly described by the Hamaker expression17
Wdisp
2 !3 A4 1 1 xðx þ 2Þ 5 ¼ þ 2 ln þ 12 xðx þ 2Þ ðxþ1Þ2 ðxþ1Þ2
ð5Þ
where A is the Hamaker constant and x = D/R. Equation 5 approaches the planar limit as 1/R ln(R), which is rather slow. For colloids in a solvent, there are three distinct contributions to the total interaction with the form of eq 5,17 (also see below). Each has a slightly different distance dependence (due to the differing radii), leading to a more complex approach to the Derjaguin limit. Even subtracting the UDA contribution from the full DFT results would produce a rather slow convergence because of the spatially varying solvent density. The slow convergence to the Derjaguin limit in the presence of dispersion interactions can be compared with the case where the interactions between all species contain only the repulsive (r-12) component of the L-J interaction. Such systems have been investigated by Henderson16 in the depletion regime (small separations). We shall not consider depletion but focus on those separations considered above. In Figure 3, we show the full DFT (17) Hamaker, H. Physica A 1937, 10, 1058.
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Figure 3. Evaluations of the Derjaguin approximation for repulsive L-J colloids (no r -6 term) with two different radii, R, immersed in a repulsive L-J solvent. σcc = 2σss.
prediction for forces between two colloids with sizes of 2R/σss = 10 and 20 for the σcc/σss = 2 case. Clearly, the lack of the longranged dispersion contribution to the intermolecular forces means that the system rapidly converges to the Derjaguin limit. Hamaker theory predicts that the asymptotic interaction between two similar polarizable spheres in a polarizable medium will be attractive. In our model, the UDA predicts a predominately attractive force for σcc/σsc = 1.5 and a repulsive force for σcc/σsc = 2. The magnitudes of the interactions that have been obtained are, for large colloids, about 2 to 3 times larger than typical hydrocarbon-water-hydrocarbon dispersion interactions. The presence of net repulsive interactions in these cases is surprising because the UDA is very close in spirit to the original Hamaker derivation. Furthermore, we see that a small change in the diameter of the particles that make up the colloids is enough to trigger a qualitative change in the predicted surface interactions. The full DFT solutions show similar trends. The source behind our anomalous results can be traced to our use of the LB mixing rule for mixed interactions. This is easily demonstrated for planar surfaces using the UDA approach. The free energy can then be divided into three contributions: Ws ¼ Δucc þ Δusc þ Δuss
ð6Þ
These contributions arise when one envisages a process of changing the colloid separation by the appropriate removal of solvent to create space for the new colloidal positions and the subsequent filling with solvent of the cavities left behind. We will use only the attractive component of the Lennard-Jones interaction to evaluate these contributions and thus probe the dispersion attractions only. The first term on the RHS of eq 6 is a direct colloid-colloid interaction and is given by βΔucc = -C(nc*)2/D2, where C is a DOI: 10.1021/la904769x
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positive constant and nc* = ncσcc3. The second term is a (repulsive, “de-solvation”) colloid-solvent contribution, βΔusc = 2Cnc*nsσsc6/(σcc3D2), and the final term is a solvent-solvent contribution given by βΔuss = -C(ns*)2/D2. When the Lorentz rule (arithmetic mean) is used for σsc, the repulsive component, Δusc, may dominate the other terms, resulting in the repulsions that we have observed. Hamaker-Lifshitz theory is more consistent with the geometric mean (i.e., σsc2 = σssσcc). This reflects the product of polarizabilities used for mixed dispersive forces in that approach and gives the following wholly attractive expression for the total interaction: βWs = -C(nc* - ns*)2/D2. The geometic mean is thus an appropriate combination rule for particle diameters in the sense that one obtains qualitatively correct attractive dispersion interactions. However, whereas a geometric mean will give the appropriate behavior at long range, it may underestimate the repulsion at short range. A solution might be to abandon the L-J form for cross interactions, as suggested by several authors, in connection with studies of rare gas mixtures.4-6,8 A pragmatic solution would be to use the Lorentz combination rule for the short-ranged part of the L-J interaction and the geometric combination rule for the long-ranged part. In the Supporting Information, we have explicitly verified that such a mixing rule generates physically reasonable net attractive colloid-colloid interactions.
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IV. Conclusions There are two main messages contained in this work. First, we have demonstrated that the Derjaguin approximation may converge slowly in the presence of dispersion forces and when the net interaction contains opposing contributions it may converge over different ranges. This result is certainly of significance in polydisperse colloidal mixtures, where both attractive and repulsive interactions may be at play, depending upon the radii of the colloidal particles involved. An important example of opposing contributions to colloid interactions arises when these are charged, as manifested by DLVO theory. Second, we note that the Lorentz-Berthelot rules can lead to qualitatively erroneous predictions for dispersion forces and should be replaced by a combination rule that is more consistent with the HamakerLifshitz theory. Acknowledgment. J.F. is supported by the Swedish Research Council. Supporting Information Available: Verification of the net interactions between L-J colloids in an L-J solvent. This material is available free of charge via the Internet at http:// pubs.acs.org.
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