pubs.acs.org/Langmuir © 2010 American Chemical Society
Depletion Force between Anisometric Colloidal Particles† S. Kr€uger, H.-J. M€ogel, M. Wahab,* and P. Schiller Department of Physical Chemistry, TU Bergakademie Freiberg, Leipziger Str. 29, 09599 Freiberg, Germany Received October 14, 2010. Revised Manuscript Received November 23, 2010 A simple mathematical model for the depletion force between two arbitrarily shaped large convex colloidal particles immersed in a suspension of small spherical particles is proposed. Using differential geometry, the interaction potential is expressed in terms of the mean and Gaussian curvature of the particle surfaces. The accuracy of theoretical results is tested by Monte Carlo simulations for parallel and nonparallel circular cylinders. The agreement between theoretical results and simulated data is very good if the density of the depletion agent is not too high.
1. Introduction In colloidal suspensions containing large and small particles a peculiar attraction force appears, which can lead to the aggregation of large particles. Especially, if a nonadsorbing polymer is introduced into a colloidal system flocculation has often been observed.1 The interaction range of the attractive force is approximately equal to the diameter of the polymer coils. Considering large colloidal particles as large spheres and polymer coils as small spheres, Asakura and Oosawa2,3 found a reasonable explanation for the attractive interaction. If the gap between adjacent large particles is small, no any polymer coil can reside inside the gap. Thus, it appears an osmotic pressure between the suspension inside and outside the gap region. For a dilute polymer solution the osmotic pressure may be expressed as P = FkT, where F is the number density of polymer coils, k Boltzmann’s constant, and T the temperature. The force between adjacent colloidal particles is simply -P multiplied with the cross-sectional area of the gap region which is inaccessible for polymer coils. This model has been widely used to describe flocculation in various experimental systems which contain a mixture of small and large colloidal particles.4-9 In most cases the smaller particles in the colloidal mixture are assumed to have a spherical shape, but there are also theoretical models focused on systems with particles possessing nonspherical shapes.10-12 Furthermore, as an extension of Onsager’s model13 for the isotropic-nematic phase transition, the depletion interaction between hard spherocylinders due to the addition of spherical polymer coils has been taken into account.14 Arranging anisometric particles into ordered assemblies is of great interest in technology. Tuning the interaction potential by depletion effects offers an additional possibility to facilitate the † Dedicated to Erich Miersemann on the occasion of his 65th bithday. *To whom correspondence should be addressed.
(1) Napper, D. H. Polymeric Stabilization of Colloidal Dispersions; Academic Press: London, 1983. (2) Asakura, S.; Oosawa, F. J. Polym. Sci. 1958, 33, 183. (3) Asakura, S.; Oosawa, F. J. Chem. Phys. 1954, 22, 1255. (4) Jenkins, P.; Snowden, M. Adv. Colloid Interface Sci. 1996, 68, 57. (5) de Hek, H.; Vrij, A. J. Colloid Interface Sci. 1981, 84, 409. (6) Sperry, P. R. J. Colloid Interface Sci. 1982, 87, 375. (7) Gast, A. P.; Hall, C. K.; Russel, W. B. J. Colloid Interface Sci. 1983, 96, 251. (8) Gast, A. P.; Hall, C. K.; Russel, W. B. J. Colloid Interface Sci. 1986, 109, 161. (9) Pate1, P. D.; Russel, W. B. Colloids Surf. 1988, 31, 355. (10) Yaman, K.; Jeppesen, C.; Marques, C. M. Europhys. Lett. 1989, 42, 221. (11) Piech, M.; Walz, J. Y. J. Colloid Interface Sci. 2000, 232, 86. (12) Oversteegen, S. M.; Lekkerkerker, H. N. W. Phys. Rev. E 2003, 68, 021404. (13) Onsager, L. Ann. N.Y. Acad. Sci. 1949, 51, 627. (14) Lekkerkerker, H. N. W.; Stroobants, A. Nuovo Cimento D 1994, 16, 949.
646 DOI: 10.1021/la104141w
self-organization of suspensions into liquid-crystalline or crystalline ordering.15 A proper way to derive approximate expressions for evaluating the depletion force between spheres and cylinders is Derjaguin’s method.16 This approximation provides good results if the size of the interacting particles is substantially larger than their interaction range. At low number densities F of the depletion agent and weak curvatures of particle surfaces, the interaction potential can be obtained by combining the approach of Asakura-Oosawa2 and Derjaguin’s approximation.16 Let R be the distance between the gravity centers of two equal spheres with radius r and d = R - 2r the minimum of the gap width between them. If all particles, the large spheres and the smaller suspended particles of the depletion agent, are considered as hard impenetrable bodies, the depletion interaction potential has the simple form3 π UðdÞ ¼ - FkTrðσ - dÞ2 2
ð1Þ
where σ is the diameter of the smaller spherical particles. Equation 1 has been found to be surprisingly accurate, even if the ratio r/σ is not very large.12 However, in the case of clays particles resemble disks rather than spheres. For evaluating depletion forces for clays and many other suspensions containing anisometric particles, eq 1 should be generalized since the depletion force depends on the mutual orientation of the particles. A possible way to generalize eq 1 is based on a method proposed by White,17 which leads to interaction potentials in dependence on the curvature radii of the particle surfaces. However, for practical applications it is more convenient to modify this approach by expressing the interaction potential in terms of mean and Gaussian curvatures, which can easily be evaluated by standard procedures for many anisometric particle shapes. In this paper we apply differential geometry and Derjaguin’s approximation to evaluate an expression for the depletion interaction potential of two large nonspherical convex bodies in a suspension containing small hard spherical particles. The physical model is based on the theory of Asakura and Oosawa.2 This concept will be tested by Monte Carlo simulations for two parallel and nonparallel circular cylinders. The model is useful to evaluate (15) Baranov, D.; Fiore, A.; van Huis, M.; Giannini, C.; Falqui, A.; Lafont, U.; Zandbergen, H.; Zanella, M.; Cingolani, R.; Manna, L. Nano Lett. 2010, 10, 743. (16) Israelachvili, J. N. Intermolecular and Surface Forces; Academic Press: London, 1992. (17) White, L. R. J. Colloid Interface Sci. 1983, 95, 286.
Published on Web 12/20/2010
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symbol O(3) indicates that terms of higher magnitude like x3 or x2 y are neglected. Such a truncation of the Taylor series is justified if the interaction range σ is much smaller than the size of the considered large colloidal particles a and b. Taking into account that the axes x and x^ enclose the angle φ, the coordinate transformation x^ ¼ x cos φ þ y sin φ and y^ ¼ - x sin φ þ y cos φ leads to σ ¼ dþ
x2 a y2 ðK1 þ Kb1 cos φ2 þ Kb2 sin φ2 Þ þ ðKa2 þ Kb1 sin φ2 2 2
þ Kb2 cos φ2 Þ þ xyðKb1 - Kb2 Þ sin φ cos φ þ Oð3Þ Figure 1. Adjacent anisometric particles.
the depletion interaction potential between smooth anisometric particles like uniaxial or biaxial ellipsoids.
2. Depletion Force between Two Convex Particles Let us consider two adjacent surfaces a and b of convex colloidal particles (Figure 1) with size much larger than their interaction range. In the present case this range is equal to the diameter of polymer coils that are depleted in a gap between both colloidal convex bodies. For evaluating depletion forces, the polymer coil is considered as a hard sphere with diameter σ. In the region where the distance between the surfaces a and b is smaller than the sphere diameter σ, an osmotic pressure P is maintained between the regions inside and outside the gap. Hence, the depletion force between the convex bodies is obtained from F = -PAc, where Ac denotes the cross-sectional area of the depleted region. There is a simple way to evaluate Ac. Let d be the shortest distance between the surfaces a and b, which is the connecting line between the surface points Ta and Tb in Figure 1. Obviously, the tangent planes at the points Ta and Tb are parallel to each other. Now we introduce a Cartesian coordinate system with origin at point Ta, so that d coincides with a part of the z-axis, while the axes x and y define the tangent plane. This plane contains the tangents to the lines of curvature at point Ta. At each nonumbilic surface point two lines of curvature that are parallel to the directions of largest and lowest curvature meet. As known from differential geometry, both lines of curvature cross at right angles.18 The x-axis may be chosen in such a way that its direction is parallel to a line of curvature passing point Ta. Hence, the y-axis is parallel to the second line of curvature at point Ta. In a similar way, a Cartesian x^-^ y-^ z coordinate system with origin at point Tb is defined. The directions of the axes x^ and y^ are chosen to be parallel to the lines of curvature at point Tb. The z^-axis is parallel to the z-axis, but the coordinate axes x^ and y^ will generally be rotated, so that the x-axis and the x^-axis enclose an angle φ. The curve surrounding the area Ac, where the gap between the surfaces is narrower than σ, can be evaluated by σ ¼ dþ
x2 y2 x^2 y^2 þ þ þ þ Oð3Þ 2ra1 2ra2 2rb1 2rb2
ð2Þ
where rai (i = 1, 2) denotes the principal radius of curvature at point Ta and rbi the principal radius of curvature at point Tb. The (18) Struik, D. J. Lectures on Classical Differential Geometry; Dover Publications: New York, 1988.
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ð3Þ
b -1 where κai = r-1 ai and κi = rbi (i = 1, 2) are the principal curvatures. The depleted area with boundaries described by eq 3 is an ellipse. Its area can be evaluated by using the relation Ac = πAB, where A and B are the half-axes of the ellipse obtained from the quadratic form eq 3. This form is rewritten as σ = d þ 1 /2mBTCmB, where mBT = (x, y) is the transposed vector of the column vector mB and C is a symmetric matrix. After a certain rotation of the coordinate system the quadratic form is transformed into σ = d þ 1/2(λ1X2 þ λ2Y2), where λ1 and λ2 are the eigenvalues of matrix C, and the axes of the Cartesian coordinates (X, Y) are parallel to the principal axes of the ellipse. Denoting by E the unit matrix, the eigenvalues result from the secular equation det(C - λE) = 0. Without loss of generality, we can allocate the coordinate axes in such a way that either the conditions κa1 e κa2 and κb1 e κb2 or κa1 g κa2 and κb1 g κb2 are satisfied. Then the secular equation
λ2 - 2ðHa þ Hb Þλ þ Ka þ Kb þ 2Ha Hb qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi - 2 ðHa 2 - Ka ÞðHb 2 - Kb Þ cos 2φ ¼ 0
ð4Þ
results, where Ha = (κa1 þ κa2)/2, Hb = (κb1 þ κb2)/2 and Ga = (κa1κa2), Gb = (κb1κb2) are the mean and Gaussian curvatures for the particle surfaces at the points Ta and Tb. Using the solutions λ1 and λ2 of eq 4, the area of the ellipse depleted from particles is Ac = 2π (σ - d)/(λ1λ2)1/2. This relation can be written as 2πðσ - dÞ Ac ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Ka þ Kb þ 2Ha Hb - 2Sab cos 2φ where we introduce the notation qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Sab ¼ ðHa 2 - Ka ÞðHb 2 - Kb Þ
ð5Þ
ð6Þ
Finally, recalling that the attractive force between the surfaces a and b is F = -PAc and F = -∂U/∂d, the interaction potential - πðσ - dÞ2 Pθðσ - dÞ Uðd, φÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Ka þ Kb þ 2Ha Hb - 2Sab cos 2φ
ð7Þ
results, where Heaviside’s function θ(σ - d ) is equal to one for σ > d and zero for σ < d. The interaction between the surfaces a and b involves an angular momentum Ld = -∂U/∂φ, which may be evaluated by Ld ðd, φÞ ¼
- 2πðσ - dÞ2 PSab θðσ - dÞ sin 2φ ðKa þKb þ2Ha Hb - 2Sab cos 2φÞ3=2 DOI: 10.1021/la104141w
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Table 1. Interaction Potentials for Spheres with Radius r and Circular Cylinders with Radius r and Length La Ha
Ka
Hb
Kb
U(d,φ)
equal spheres 1/r 1/r 1/r 1/r - /2π(σ - d )2rP plane and sphere 0 0 1/r 1/r2 -π(σ√- d )2rP -(1/ 2)π(σ - d)2rP sphere and cylinder 1/r 1/r2 1/2r 0 (nonparallel) cylinders 1/2r 0 1/2r 0 -π(σ - d )2rP|sin φ|-1 parallel cylinders 1/2r 0 1/2r 0 -4/3L(σ - d)3/2r1/2P a The Heaviside function θ(σ - d ), which appears as a factor in eqs 7 and 9, has been omitted. 2
2
1
where the index d indicates that the vector of angular momentum is parallel to the connection between the points of shortest distance Ta and Tb. It should be noted that eqs 7 and 8 could also be derived by using a slightly different method published by White.17 Equation 7 is not applicable to parallel cylindrical bodies, since in this case one of the eigenvalues of matrix C becomes zero. Since lines of curvature on a cylinder surface are parallel to the cylinder axis, φ is identified as the angle between the cylinder axes. For parallel cylinders the depletion interaction potential is defined as a potential per unit length U (d )/L, where L is the cylinder length. An appropriate modified expansion, obtained by inserting ra2 = ¥ and rb2 = ¥ into eq 2, leads to UðdÞ - 4ðσ - dÞ3=2 Pθðσ - dÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ L 3 Ha þ Hb
ð9Þ
and Lz(d,φ) = 0. Table 1 shows some examples of interaction potentials for spheres and circular cylinders with radius r. Some results in Table 1 agree with corresponding formulas published by Israelachvili.16 The osmotic pressure P appearing as factor in eqs 7 and 9 and Table 1 is equal to FkT for a dilute suspension. If the density is higher, however, the interactions between hard spherical particles lead to deviations from the simple perfect gas equation of state used to determine the osmotic pressure. F is the number density for the small spherical colloidal particles in the accessible volume which is not occupied by the large colloidal particles of the sample. When the condition for the volume density η = πσ3F/6 , 1 is violated, it seems reasonable to replace the equation of state for a perfect gas by an improved equation which accounts for the excluded volume interaction between spheres. Thus, for evaluating the depletion force at higher densities, we can use the Carnahan-Starling equation for a hard sphere gas Z ¼
1 þ η þ η2 - η3 ð1 - ηÞ3
ð10Þ
where Z = P/FkT is the compressibility factor.
3. Comparison with Monte Carlo Simulations The accuracy of approximate expressions for the pair potential of depletion forces can be tested by Monte Carlo simulations. We have done simulations (N-V-T ensemble) for a 50σ 50σ 50σ box containing one or two large circular cylinders, which penetrate the whole system (Figure 3), and N = 10 000 hard spheres with diameter σ. This particle number corresponds to a volume density η = πσ3F/6 = 0.0579, where F is the particle number density of the spheres in the part of box volume, which is not excluded by the cylinders. To reduce boundary effects, periodic boundary conditions were used. In most simulations the cylinder 648 DOI: 10.1021/la104141w
Figure 2. Lines of curvature of a prolate ellipsoid.
Figure 3. Simulation box with two circular cylinders.
radius r = 10.5σ was chosen. The interaction potential between a cylinder and a sphere is defined as Uðri Þ ¼
¥ for ri < r þ σ=2 0 for ri g r þ σ=2
ð11Þ
where ri is the radial distance between the cylinder axis and the center of gravity of a sphere i (i = 1, 2, .., N). In a similar way the pair potential for hard spheres is defined. It becomes infinity if the distance between the gravity centers of the spheres is smaller than σ and zero for larger distances. For determining the depletion force of nonparallel cylinders, one cylinder is turned by an angle φ around the axis d in Figure 3. Every simulation comprises 106 Monte Carlo steps, where 20% of them were used for equilibration. In a first step, we check if the Carnahan-Starling equation is reproduced in our simulations. It is appropriate to insert only one cylinder into the simulation box. Clearly, in this case averaged interaction forces between the cylinder and the spherical particles are zero (ÆFxæ = ÆFyæ = ÆFzæ = 0). However, there is a pressure exerted on the cylinder surface, which can be evaluated by using a modified version of the contact theorem for hard-sphere gases19 P ¼ kTFðr þ σ=2Þ
ð12Þ
(19) Henderson, D. Statistical Mechanical Sum Rules. In Fundamentals of Inhomogeneous Fluids; Henderson, D., Ed.; Marcel Dekker: New York, 1992; p 61.
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Figure 4. Pressure evaluated by Monte Carlo simulation for a hard-sphere gas in comparison with the Carnahan-Starling equation: (Δ) Monte Carlo simulation, (---) perfect gas equation of state, (;) theory (eq 10).
where F(r þ σ/2) is the averaged particle number density of the spheres at a distance where the cylinder surface and the surrounding spheres may touch each other. If the volume density η is not small, F(r þ σ/2) is higher than the average particle number density F in the bulk. F(R þ σ/2) is obtained by counting the gravity centers of spheres located within an annulus surrounding the cylinder surface. The volume of an annulus with a small width ε is Vε = 2πε(r þ σ/2)L, where L denotes the length of the cylinder. If Nε centers of gravity are localized inside the annulus, the approximate relation F(r þ σ/2) = Nε/Vε holds. Many simulation snapshots are needed to reduce statistical errors. Furthermore, the width ε should be chosen in such a way that statistical and systematic errors remain small. In our simulation we have chosen the value ε = 0.1σ and in some cases ε = 0.05σ. Figure 4 illustrates the results of the simulation in comparison to the perfect gas equation of state and the Carnahan-Starling equation (eq 10). Although we only consider dilute systems (η = 0.0579), the Carnahan-Starling equation is expected to give better results. When two parallel cylinders are placed into the box (Figure 3), a force ÆFxæ between them may appear, whereas the averaged total forces for the directions y and z should be equal to zero. A symmetry consideration reveals that even if the left cylinder in Figure 3 is rotated through an angle φ around axis d, which is perpendicular to the cylinder axes, the averaged forces ÆFxæ and ÆFyæ exerted on the right cylinder remain equal to zero. In a similar way as for a sphere,20 the averaged force exerted on a cylinder can be obtained from an integral of the local particle number density over the cylinder surface. Introducing the polar angle R (Figure 3), the force exerted on a cylinder by hard spherical particles with diameter σ is expressed as Z
2π
ÆFx æ ¼ rLkT
Fðr þ σ=2, RÞ cos R dR
ð13Þ
0
For utilizing simulations, which produce snapshots of configurations, eq 13 is replaced by the sum ÆFx æ ¼
Nε kT X cos Rj æC Æ ε j¼1
ð14Þ
In this relation, Nε is equal to the number of spherical particles localized in the region r þ σ/2 < rj e r þ σ/2 þ ε, where rj is the radial distance between the cylinder axis and a particle j ( j = 1, 2, .., Nε). The brackets Æ...æC indicate statistical averaging. (20) Dickmann, R.; Attard, P.; Simonian, V. J. Chem. Phys. 1997, 107, 205.
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Figure 5. Depletion force fx versus the reduced gap width d/σ for parallel circular cylinders: (Δ) Monte Carlo simulation, (---) theory (Z = 1), and (;) theory (Z = 1.269).
Figure 6. Depletion force fx versus the reduced diameter 2r/σ for parallel circular cylinders, which are in contact (d = 0): (Δ) Monte Carlo simulation, (---) theory (Z = 1), and (;) theory (Z = 1.269).
For parallel cylinders the potential listed in Table 1 is accompanied by the force Fx = -2Lr1/2(σ - d)1/2 P. Introducing the notation for a reduced force fx ¼
Fx σ 2 LkT
ð15Þ
we get the equation pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi fx ¼ - 2Fσ2 rðσ - dÞZθðσ - dÞ
ð16Þ
where Z = P/FkT. The compressibility factor Z is equal to one, if the excluded volume interaction between the hard spheres is neglected. If for an improved description the Carnahan-Starling equation (eq 10) is used, we obtain the compressibility factor Z = 1.269 for η = 0.0579. If the density of the depletion agent is as low as in our simulations, the interaction range is not larger than the diameter σ of a sphere. For gaps broader than σ the interaction potential of the cylinders can be considered as negligibly small. Figure 5 depicts a plot of fx versus the reduced gap width d/σ. Obviously, the agreement between simulated data and the theoretical curve based on the Carnahan-Starling equation is very good. The simpler model based on the perfect gas equation of state for the spheres underestimates the depletion force. It is interesting to elucidate if the theory is still applicable to smaller values of diameter ratios 2r/σ, although the spirit of Derjaguin’s approximation requires large values of 2r/σ. In Figure 6 the reduced depletion force fx is plotted versus 2r/σ for d = 0. Again, a good agreement with simulated data is found, when the osmotic pressure of the hard spheres is calculated by using eq 10. The theoretical predictions for the depletion force are still accurate for moderate diameter ratios (2r/σ = 4). DOI: 10.1021/la104141w
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Figure 9. Depletion force fx versus the volume density for parallel cylinders, which are in contact (d = 0): (Δ) Monte Carlo simulation, (---) theory (Z = 1), and (;) theory (eq 10).
of the simulated values. Finally, we have elucidated how accurate is the theory at higher volume densities η. Figure 9 demonstrates that the evaluation of the depletion force is rather accurate as long as the value of the volume density η is lower than 0.15.
4. Summary Figure 7. Depletion force f^x versus the reduced gap width d/σ for tilted cylinders: (a) φ = π/4; (b) φ = π/2. (Δ) Monte Carlo simulation, (---) theory (Z = 1), and (;) theory (Z = 1.269).
Figure 8. Depletion force f^x versus the tilt angle φ for cylinders, which are in contact (d = 0): (Δ) Monte Carlo simulation, (---) theory (Z = 1), and (;) theory (Z = 1.269).
According to Table 1, the scaling of the force for parallel and tilted cylinders should be chosen differently. In the case of tilted cylinders (φ 6¼ 0) the reduced force ^fx ¼ Fx σ kT
ð17Þ
^fx ¼ - 2πFσrðσ - dÞZθðσ - dÞ jsin φj
ð18Þ
satisfies the equation
Figure 7 illustrates the dependence of the depletion force f^x on the gap width d for the tilt angles φ = π/4 (a) and φ = π/2 (b). If the Carnahan-Starling equation is used for the hard spheres, the simulated data are described fairly well by eq 18. In Figure 8 the force f^x is plotted versus the tilt angle φ. The approach based on the Carnahan-Starling equation leads to a proper reproduction
650 DOI: 10.1021/la104141w
Depletion forces produced by polymer coils or small colloidal particles may influence the structure of sols and gels. Most models for explaining these forces are based on excluded-volume interactions between hard impenetrable particles. We have combined the Asakura-Oosawa theory2 with Derjaguin’s approximation16 to derive a pair interaction potential for the depletion force between large anisometric particles, which are immersed in a suspension containing smaller spherical particles. The potential is expressed in terms of mean and Gaussian curvatures at the points of shortest distance between large anisometric particles. This formulation is suitable to evaluate anisotropic depletion forces between particles, which may have a more complex geometry than spheres or cylinders. The depletion force between prolate and oblate ellipsoids will be considered in a forthcoming paper. Monte Carlo simulations are suitable to test eq 7 for the depletion potential U(d,φ). Instead of the potential itself, it is more convenient to compare the force F = -∂U(d,φ)/∂d with simulated data. The original Asakura-Oosawa theory uses the equation of state for noninteracting particles to evaluate the osmotic pressure of the depletion agent. In this case the theory is restricted to rather low osmotic pressures. Monte Carlo simulations demonstrate that an accurate theoretical description for higher osmotic pressures up to moderate volume densities (η = 0.15) requires a proper equation of state for the hard-sphere particles of the depletion agent. The extended theory is suitable to evaluate orientation-dependent depletion forces between anisometric particles. However, at higher volume densities structural aspects of hard-sphere fluids near walls become important. In this case some intricate theoretical problems are unsolved yet.21,22 Nevertheless, models for small densities may be useful, since in many practical cases depletion agents have low or moderate volume densities. In conclusion, the anisotropic depletion force between colloidal particles can be evaluated in terms of mean and Gaussian curvatures of the particle surfaces. This differential geometric approach offers a convenient possibility to evaluate pair potentials for anisometric particles. (21) Oettel, M. Phys. Rev. E 2004, 69, 041404. (22) Henderson, J. R. Physica A 2002, 313, 321.
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