Derivation of the Derjaguin approximation for the case of inhomogeneous solvents

The Derjaguin approximation (DA) relates the force F(h) between two curved surfaces to the interaction free energy per unit area $Ap(h)$ between two planar surfaces of the same materials^1–3 and is valid in the limit where the surface-to-surface separation h is much smaller than the radius (or radii) of curvature of the surfaces. When properly applied, it yields significant computational simplifications and has played a major role in our understanding of the fundamental forces acting in both biological and synthetic colloidal systems, for example, via the celebrated Derjaguin–Landau–Verwey–Overbeek (DLVO) theory of colloidal stability.^3–6 Another important practical application of DA is the surface force apparatus technique,⁷ which relies on a mapping between the force measured between two crossed cylinders to the interaction free energy between two planar surfaces.

The original statement of the approximation due to Derjaguin¹ is that the interaction force F(h) between two spheres of radii R₁ and R₂ can be expressed as follows:

where h is the distance of closest approach between the spheres. The relation for two spheres is easily generalized to other geometries like a sphere and a plane, a plane and a parallel cylinder, or two crossed cylinders; the only resulting modification of Eq. (1) is the geometrically determined prefactor, while the physical effects are contained in $Ap$⁠. In Derjaguin’s original derivation, as well as in many textbook derivations, $Ap$ is obtained from the explicit, pairwise summation of a direct interaction potential u(r) acting between the constituent atoms or molecules of the two bodies. Assuming that the entropic contribution to the direct interaction is negligible, this yields an expression for $Ap(h)$⁠. Equation (1) then holds under the two conditions that (i) u(r) is short-ranged relative to R₁ and R₂ and (ii) h is small relative to R₁ and R₂. More recently, the validity limits of DA have been experimentally verified,^8–11 and generalizations of DA to more complex geometries, such as anisotropic particles and rough surfaces, have been derived.^12,13

For the specific case of van der Waals interactions, u(r) decays as r⁻⁶ and DA is often derived based on a Hamaker description, where u(r) is explicitly summed to yield $Ap(h)$ as follows:^2,3

where H₁₂ is the Hamaker constant describing the interaction between materials 1 and 2. A more sophisticated description of the same physical effect can be obtained using Lifshitz theory,^14,15 which avoids the assumption of pairwise additivity between interparticle interactions that is particularly poor for the non-dispersion (classical) contributions to the van der Waals interaction. Within the Lifshitz formalism, the two interacting surfaces are treated as continuous materials described by their frequency-dependent dielectric responses. This treatment yields the same distance dependence of the interaction as the Hamaker treatment, although with a modified value of the Hamaker constant H₁₂. Furthermore, the effect of a medium between the bodies can be readily included into Lifshitz theory: this medium is also characterized by its bulk dielectric properties, which are assumed independent of the separation between the bodies. In reality, however, the density of the intervening fluid medium will be position-dependent in a way that depends on h. For nearly incompressible pure liquids, this is a negligible effect, while for a compressible liquid or for media containing more than one component, substantial changes of density or composition can occur as the two bodies approach; an extreme case occurs when a new phase forms between the bodies, as in the case of capillary condensation.¹⁶ The standard geometrical derivations of DA are not capable of treating such effects of inhomogeneity, which instead require an explicit statistical–mechanical treatment of the molecular solvent.

The application of DA to situations with a medium between the surfaces has previously been discussed for a range of specific interactions. Oversteegen and Lekkerkerker^10,11 studied DA in the context of the depletion force between hard spherical bodies. Schnitzer and Morozov¹⁷ derived a generalized version of DA valid at any separation for the specific case of electrostatic double layer interactions, while Forsman and Woodward analyzed the validity of DA for two spherical particles in a Lennard-Jones fluid¹⁸ or in a polymer solution.¹⁹ In this paper, we provide a complement to these studies, covering the general case of the interaction between curved bodies immersed in a molecular solvent. We present a straightforward, statistical–mechanical derivation of (i) an exact expression for the interaction free energy between two infinite planes and (ii) DA for a sphere and a plane, both expressed in terms of the position-dependent and separation-dependent solvent density. Our results show that DA remains valid for the case of a molecular solvent, with two additional constraints compared to the Lifshitz treatment, namely, that (i) the surface–molecule interactions are of a much shorter range than the radius R of the sphere and (ii) the density correlation length in the solvent is smaller than R. We then extend our analysis to the case where a phase transition occurs between the surfaces, which cannot be straightforwardly covered using a statistical–mechanical formalism due to the discontinuous change in the density of the medium. Thus, we instead take a continuum thermodynamics approach and show that the phase transformation induces an attractive force between the bodies, whose value between curved surfaces can be related to the corresponding free energy for a planar system in accordance with DA.

We start by considering the interaction between a homogeneous planar wall placed at z = 0 and a sphere of radius R made from the same material as the wall, centered at r_s = (0, 0, z_s), immersed in a molecular solvent containing N molecules in equilibrium with a large reservoir. Molecule i interacts with the wall and the sphere with potentials $uiw$(z_i) and u_is(r_is), respectively, where r_is is the distance between the molecule i and the center of the sphere; see Fig. 1. In addition, all molecular pairs i, j interact through a pairwise potential u_ij(r_ij). For simplicity, we consider interactions independent of the molecular orientation; while it is straightforward to generalize the formalism to orientation-dependent interactions, it would lead to a more extensive notation, but adding only marginally to the general understanding. We furthermore ignore the direct interaction between the two bodies, which is already covered by the standard derivations^2,3 and does not influence the molecular degrees of freedom that are our focus here. The configuration integral Z_N of the solvent particles is formally expressed as follows:

where

is the total energy of the system and $β=(kBT)−1$ is the inverse thermal energy. The excess free energy of the system is given by A(h) = −k_BT ln Z_N(h), and the force between the plane and the sphere is

If we vary h by displacing the sphere while keeping the wall fixed, the only term in Eq. (4) that changes is the one containing u_is:

where the second equality follows from the identity of all solvent molecules. The integral over ${ri}2N$ can be identified as Z_Nρ(r₁), where ρ(r₁) is the single-particle density.²⁰ We furthermore write $r1s=[x12+y12+(z1−R−h)2]1/2$⁠, so that

We now change the coordinate system to the center of the sphere, i.e., z = z₁ − R − h, which enables us to rewrite Eq. (6) as follows:

where we have dropped the subscript “1” from the variables. Changing to a spherical coordinate system, performing the trivial integration over φ and using Eq. (5) now yields the following exact expression for the solvent contribution to the force F(h) between a sphere and a plane:

Using the fact that, as h → ∞, ρ becomes independent of θ, we can define

Since the force on a free spherical particle in the solution vanishes, the integral in Eq. (9) becomes zero in this limit, and we can replace ρ(r, θ) in Eq. (9) by Δρ(r, θ) ≡ ρ(r, θ) − ρ_∞(r) to yield

Note that, while seemingly independent of the wall–molecule interaction u_w(z), the force in Eq. (11) depends implicitly on u_w through its dependence on Δρ. A fully analogous derivation of the force per unit area $Fp(h)$ between two half-planes (corresponding to R → ∞), furthermore, yields

where Δρ_p(z) is the corresponding density difference between the two planes. Importantly, the integral in Eq. (12) runs from z = −∞ rather than z = 0: This is because, even though ρ_p = 0 for z < 0, Δρ_p is not, since the left wall replaces the solvent as it is brought from infinite separation. This yields a nonzero contribution to the force even for an incompressible solvent between the planes for which Δρ_p = 0 in the gap. For the case of van der Waals interactions, this contribution corresponds to the solvent-induced part of the force from the Lifshitz treatment. Equations (11) and (12) are exact and will in the following be used to derive DA through a series of approximations.

We now divide the solvent volume into two parts, as illustrated in Fig. 2(b): region A directly between the sphere and the wall and the remainder, region B, outside the projection of the sphere onto the plane. Assuming that the presence of the wall does not affect the density in region B, we can set Δρ = 0 here. In region A, we replace θ by the local perpendicular distance h′ between the wall and the sphere surface [see Fig. 2(b)]. Using the chord theorem, these variables can be related by

which, together with Eq. (11), yields

Under the additional approximations that (i) h ≪ R and (ii) Δρ is nonzero only in the region where h′ ≪ R, i.e., that the region where h′ ≈ R does not contribute significantly to the force, we can neglect h and h′ in the geometrical factor of Eq. (14), which simplifies to

In Eq. (15), we have extended the upper limit in the outer integral from R + h to ∞, again based on the assumption that Δρ is negligible in this region. In order to map the force between a plane and a sphere to that between two planes, we now assume that density correlations in the solvent are of a significantly shorter range than R, implying that the local density difference Δρ(r, h′) between a sphere and a plane is the same as that between two planes at separation h′. In other words, we assume that the radius of curvature is large enough that the surface can be regarded as locally flat. Thus, the density variations are unaffected by the local curvature of the surface as long as the density correlation length is smaller than the radius of curvature. Furthermore, we interpret the molecule-sphere force $−dusdr$ as the limiting value when R → ∞, so that $dusdr=duwdz$ Thus, Eq. (15) can be expressed solely in terms of parameters for the planar system:

From Eq. (12), we can now identify the inner integral as the expression for the force between two planes at separation h′, so that

We can now readily identify the integral of $Fp(h)$ as the free energy $Ap(h)$⁠, which, finally, leads us to DA,

In addition to the usual restrictions on h and the range of the direct interaction relative to R, we have employed the constraints that (i) the range of the solvent–sphere interactions is small relative to R and (ii) the density correlations in the solvent are short-ranged compared to R. As discussed above, this derivation assumes that the interaction between the solvent molecules and the bodies is independent of molecular orientation; including orientational degrees of freedom is straightforward and would lead to position-dependent densities and orientation distributions. This would, in turn, impose further constraints on the decay of orientational correlations, which need to be short-ranged relative to R, leading to complications for a medium close to the isotropic–nematic transition.

Above, we have shown that DA remains valid also for the case when the there are substantial density variations in the medium induced by the surface–surface interactions, under the tacit assumption that the solvent density varies smoothly in space. While all the arguments given above are based on a formally exact expression for the configurational integral, Eq. (3), complications arise when treating the case of a phase transition taking place between the surfaces, as it is notoriously difficult to treat a first order phase transition using a configurational integral approach. The most important such case is when a capillary-induced phase separation occurs as the surfaces approach each other, resulting in an attractive force between both planar and curved surfaces. This practically important case is thus not covered by the formalism of Secs. II and III, and it is not a priori obvious whether DA can be used also in this case. One specific complication not present in the case treated above is that a new phase formed between two planar surfaces has an infinite extent in the xy direction, while for curved surfaces, the newly formed phase has a finite volume, which makes a direct application of DA non-trivial. Since we do not have the tools to treat this case using a basic statistical–mechanical formalism, we, instead, make use of a continuum thermodynamic approach.

Following along the lines of previous derivations,^2,16 we consider an emerging phase β that forms a cylindrical lens of radius R_m between a sphere and a plane separated by a minimum distance h immersed in the bulk phase α (see Fig. 3). Note that we consider the force between two bodies joined by a medium in equilibrium with an infinite reservoir, so that changes in separation occur at constant chemical potential of the medium, while the case of a condensed phase of constant volume is mathematically much more demanding.²¹ The driving force behind the phase transformation is a reduction in the surface free energy, quantified by Δγ ≡ γ_sβ − γ_sα < 0, where γ_sα/β is the surface free energy between the solid phase and phase α/β. The decrease in surface free energy equals $As(h)=2πRm2Δγ$⁠, while the phase transformation is associated with an increase in bulk free energy, proportional to the volume of the lens and the difference in bulk free energy density Δf = f_β − f_α > 0, and an increase due to the surface energy γ_αβ between the two liquid phases. The volume of the lens can be straightforwardly derived by taking into account the volume of the cylinder and that of the spherical cap to second order in R_m, yielding $Vlens=πRm2(h+Rm2/4R)$⁠. To this order, the total free energy change upon forming the new phase is

Neglecting, for now, the contribution from the αβ interface (i.e., putting γ_αβ = 0), the equilibrium lens radius can be obtained by minimizing Eq. (19) with respect to R_m, yielding

where

is the so-called Kelvin radius. The maximum separation h_c where phase separation occurs can then be calculated by inserting Eq. (20) into Eq. (19) and setting A(h) = 0, yielding

where

This shows that the capillary phase separation between a sphere and a plane occurs at a somewhat smaller separation than between two planes, where h_c = R_K, due to the surface energy between the two liquid phases. The force F(h) due to the new phase can now be obtained by differentiating Eq. (19),

showing that the force is linear and attractive, since h < R_K.For two parallel planes, the formed β phase is infinite, and the free energy per unit area $Ap(h)$ is readily obtained in an analogous way as Eq. (19),

A direct comparison with Eq. (24) shows that

i.e., that DA holds even in the case of a phase transformation taking place in the solvent, with the additional constraint that h ≪ R_m ≪ R, which was implicitly applied when truncating Eq. (19) at second order in R_m.

The Derjaguin approximation is a very versatile tool for both the direct interpretation of surface force measurements and for the more conceptual understanding of surface forces. In contrast to the standard derivations, based on a direct interaction between two bodies separated by a gap, the derivations presented here consider the validity of DA in the case when the interaction force involves an active participation of a molecular medium surrounding the bodies. Our results show that DA remains generally valid, as is tacitly assumed in many applications, but with two new parameter limitations: First, in analogy with the constraint on the direct interaction between the bodies, the interaction between the solvent molecules and the constituents of the bodies should have a range much shorter than the radius of curvature R of the bodies. The second, somewhat less obvious, constraint is that the density correlation length in the solvent is small relative to R, which can be an important limitation in the vicinity of critical points, such as when studying critical Casimir forces.²² We also showed that DA remains valid even in the case of a new phase forming in the gap, such as for capillary-induced condensation or evaporation, with the additional constraint that the radius R_m of the lens formed by the new phase is smaller than R, but larger than h. As presented, our derivation covers the case of a single component medium between the bodies. Generalizing our derivation to the case of a two-component system, such as the case of a molecular solute present in the solvent, where density inhomogeneities are usually more significant, is an interesting route for future work.

Data sharing is not applicable to this article as no new data were created or analyzed in this study.

J.S. acknowledges funding from the Swedish Research Council (Grant Nos. 2015-05449 and 2019-03718).