Interfacial free energy of a hard-sphere fluid in contact with curved hard surfaces

Abstract

Using molecular-dynamics simulation, we have calculated the interfacial free energy γ between a hard-sphere fluid and hard spherical and cylindrical colloidal particles, as functions of the particle radius R and the fluid packing fraction η = ρσ 3 /6, where ρ and σ are the number density and hard-sphere diameter, respectively. These results verify that Hadwiger's theorem from integral geometry, which predicts that γ for a fluid at a surface, with certain restrictions, should be a linear combination of the average mean and Gaussian surface curvatures, is valid within the precision of the calculation for spherical and cylindrical surfaces up to η ≈ 0.42. In addition, earlier results for γ for this system [Bryk et al., Phys. Rev. E 68, 031602 (2003)] using a geometrically based classical density functional theory are in excellent agreement with the current simulation results for packing fractions in the range where Hadwiger's theorem is valid. However, above η ≈ 0.42, γ (R) shows significant deviations from the Hadwiger form indicating limitations to its use for high-density hard-sphere fluids. Using the results of this study together with Hadwiger's theorem allows one, in principle, to determine γ for any sufficiently smooth surface immersed in a hard-sphere fluid. The solid-liquid interfacial free energy γ is a central property governing a wide variety of technologically important phenomena from crystal nucleation and growth to wetting. Because accurate and reliable experimental measurements of γ are rare, much effort has been devoted in recent years to the development of atomistic simulation methods to determine this quantity for interfaces between coexisting solid and fluid phases [1][2][3][4][5] and for systems in which the solid is modeled by a static wall [6][7][8][9][10][11]. These efforts have thus far been primarily restricted to planar interfaces; however, there are many physically relevant systems in which interfacial curvature is relevant, for example, in the formation of critical nuclei in nucleation [12][13][14] or the solvation or wetting of hydrophobic nanoscale particles [15][16][17]. There have been a number of previous simulation studies that examine the effect of curvature in liquid-vapor interfaces [18][19][20][21], but direct simulation studies on solid-liquid interfaces are lacking. In this Rapid Communication, we examine the dependence of γ on the surface curvature for a hard-sphere fluid in contact with curved hard surfaces, specifically at spherical and cylindrical colloidal particles.König et al. [22] have recently shown that Hadwiger's theorem [23] from integral geometry puts severe restrictions on the shape (curvature) dependence of the interfacial free energy. In their analysis, the interfacial free energy of an object with a surface S is given bywhere h and κ are constants depending upon the thermodynamic state, but independent of the specific surface S. HereH andK are the averaged mean and Gaussian curvatures of S, * Author to whom correspondence should be addressed: blaird@ku.edu d...

Communication: Non-Hadwiger terms in morphological thermodynamics of fluids

“…We introduce an extended fundamental measure theory (EFMT) for the bulk fluid, from which we obtain these deviations in leading order for arbitrary density of the fluid. The result is in excellent agreement with recent numerical simulations by Laird et al 12 The smallness of the corrections to Eq. (1) for packing fractions η 0.4 is encouraging regarding the validity of previous results obtained within morphological thermodynamics.…”

“…(15) 24 The remaining degree of freedom can be eliminated by prescribing ∂ ∂ξ 1 . By fitting to recent results from simulations 12 we obtain Fig. 2.…”

“…3 for the corresponding EFMT result. 12 The black curve is the analytical EFMT result for κ 3 , Eq. (21).…”

Hard rectangles near curved hard walls: Tuning the sign of the Tolman length

“…2 in Ref. [28]. As we do not see a change of sign of the Tolman length for disks, we conclude that the change of sign of the Tolman length is caused by the anisotropy of the particle shape, and not with various radii of curvature R including a flat wall as limiting case (green, R = ∞).…”

“…The interfacial tension between a planar hard wall and a fluid hard-sphere bulk phase has been explored by computer simulations [16][17][18][19][20] and provides an ideal testing ground for the performance of approximations in classical density functional theory (DFT) of inhomogeneous fluids [21][22][23][24][25][26]. Subsequent analytic calculations [27], simulations [28], and DFT calculations [29][30][31] have considered a curved wall exposed to a hard-sphere fluid and found a negative sign of the Tolman length for hard spheres around a spherical obstacle. Moreover, the Tolman length has been accessed for other interactions such as (modified) Lennard-Jones potentials [20,[32][33][34][35][36][37][38] or Yukawa potentials [32,39], at phase boundaries [37,40], and in lattice models [41] [42].…”