Gradient theory computation of the radius-dependent surface tension and nucleation rate for n-nonane clusters

Abstract

The Van der Waals-Cahn-Hilliard gradient theory ͑GT͒ is applied to determine the structure and the work of formation of clusters in supersaturated n-nonane vapor. The results are analyzed as functions of the difference of pressures of the liquid phase and vapor phase in chemical equilibrium, which is a measure for the supersaturation. The surface tension as a function of pressure difference shows first a weak maximum and then decreases monotonically. The computed Tolman length is in agreement with earlier results ͓L. Gránásy, J. Chem. Phys. 109, 9660 ͑1998͔͒ obtained with a different equation of state. A method based on the Gibbs adsorption equation is developed to check the consistency of GT results ͑or other simulation techniques providing the work of formation and excess number of molecules͒, and to enable an efficient interpolation. A cluster model is devised based on the density profile of the planar phase interface. Using this model we analyze the dependency of the surface tension on the pressure difference. We find three major contributions: ͑i͒ the effect of asymmetry of the density profile resulting into a linear increase of the surface tension, ͑ii͒ the effect of finite thickness of the phase interface resulting into a negative quadratic term, and ͑iii͒ the effect of buildup of a low-density tail of the density profile, also contributing as a negative quadratic term. Contributions ͑i͒-͑iii͒ fully explain the dependency of the surface tension on the pressure difference, including the range relevant to nucleation experiments. Contributions ͑i͒ and ͑ii͒ can be predicted from the planar density profile. The work of formation of noncritical clusters is derived and the nucleation rate is computed. The computed nucleation rates are closer to the experimental nucleation rate results than the classical Becker-Döring theory, and also the dependence on supersaturation is better predicted.

Thermodynamic stability of nanosized multicomponent bubbles/droplets: The square gradient theory and the capillary approach

“…24,34,43 Since thermodynamic equilibrium in the canonical ensemble is characterized by a minimum in the Helmholtz energy, we find the molar density distributions, c i , which give a stationary state of the Helmholtz energy functional…”

Homogeneous droplet nucleation modeled using the gradient theory combined with the PC-SAFT equation of state

“…Owing to much better approach in this work compared to the previous work [2], computations could be performed in routine to get density profiles corresponding to many temperatures. Values corresponding to the experimental temperatures by Rudek et al [17], Hung et al [18], Luijten [19], Viisaanen et al [20] and Wagner and Strey [21] were used:…”

“…The whole number of molecules N in the critical droplet is the sum of the bulk volume number and the excess number [2],…”

“…This causes very significant cumulation of numerical errors. In previous work [2], this BVP was solved using the matlab procedure BPV4 [14]. This procedure uses the finite difference scheme: it devides the original interval of the problem into subintervals, where the right-hand side is integrated.…”

“…On the other hand, the PC-SAFT EoS combines different intermolecular forces, e.g., hydrogen bonding, covalent bonding, Coulombic forces which makes it more accurate in predicting of the physical variables. We continued in our previous works [2,3] by solving the boundary value problem which arose by mathematical solution of the DGT formulation and including the boundary conditions. Achieving the numerical solution was rather tricky; this study describes some of the crucial developments that helped us to overcome the partial problems.…”