## Equilibrium and Nonequilibrium Thermodynamics of Planar and Curved Interfaces

##### Doctoral thesis

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##### Date

2015##### Metadata

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- Institutt for kjemi [1129]

##### Abstract

Interfaces can be found everywhere and their properties are of major importance
to a wide variety of processes, ranging from DNA replication to weather forecasts.
This work is a fundamental study of systems with planar and curved interfaces at
equilibrium and nonequilibrium.
First, we studied the thermodynamic stability of bubbles and droplets in systems
with both finite volume and number of particles. We showed that the macroscopic
capillary approach predicted thermodynamic stability limits nearly identical to
those from square gradient theory, a more rigorous mesoscopic theory. Both approaches
gave a stable and unstable branch of solutions, representing bubbles and
droplets in the canonical ensemble. They also predicted a minimum threshold size
for stability. This minimum threshold size was used to explain how fluids can be
superstabilized in nanocontainers and to derive formulas to estimate the container
volume below which no droplet or bubble can form. Finite-size effects also influence
nucleation processes. A comprehensive theoretical analysis of cavitation and
droplet formation in finite systems was presented. Using general thermodynamic
arguments, we derived simple, yet accurate formulas to calculate the finite-size corrections
for the critical size, the nucleation barrier, and the nucleation rates in the
canonical ensemble. These formulas can be used to select the appropriate system
size for simulations or to obtain a precise evaluation of nucleation rates of complex
substances at atmospheric conditions by using a small number of molecules and
correcting for finite-size effects.
Next, the curvature dependence of the surface tension was examined by evaluating
the leading terms of an expansion in the total and Gaussian curvatures. The
first-order curvature correction is the Tolman length and the second-order corrections
are the rigidity constants. By combining the square gradient theory with an
accurate equation of state, we reproduced the surface tension and Tolman length
for the shifted and truncated Lennard-Jones fluid from molecular simulations. The
curvature expansion accurately described the change in surface tension, until the
radius of the bubbles and droplets became similar to the interfacial width of the
planar interface at coexistence. For metastablities even closer to the spinodal limits,
the curvature remained nearly constant, but the surface tension decreased due
to a decreasing difference between the densities of the bubble/droplet interior and
exterior. The size of droplets and bubbles in the region of constant curvature
was found to be proportional to the correlation length of fluctuations in the liquid
phase. The discovery of a region with constant curvature offers a new perspective
on how nucleation should be described at high metastabilities. We next estimated
the Tolman length and rigidity constants of water. The Tolman length of water was
negative and weakly temperature dependent, having the relatively small value of
about −0.05 nm, which is in agreement with previous estimates in the literature.
Using the leading terms of the curvature expansion, we incorporated the curvature
dependence of the surface tension into the framework of classical nucleation
theory. The modification corrected the temperature dependence of the nucleation
rates given by the classical theory, thereby improving the agreement between theoretical
and experimental results. Thus, the procedure offered a promising way
to alleviate the problems of classical nucleation theory and obtain quantitatively
accurate predictions, which hopefully is possible also for other substances.
We next examined the inherent properties of the temperature across interfaces.
Several microscopic expressions to calculate temperature in molecular simulations
were evaluated, where some included configurational contributions. We investigated
their accuracy and usefulness in high- and low-density bulk systems and
across vapor-liquid interfaces both at equilibrium and with a temperature gradient.
We showed that the configurational temperature was equivalent to the kinetic
temperature in steady-state molecular dynamics simulations. This agreement however,
was obtained only if the discontinuity in the derivatives of the interaction
potential was handled properly by using a sufficiently long truncation distance
or tail-corrections. If the entropy density is included as a variable in the square
gradient theory framework, the theory suggests that the temperature has different
contributions from directions parallel and perpendicular to the interface at
equilibrium. We found a similar anisotropy in simulations by examining the configurational
temperature in molecular dynamics. The results from the simulations
agreed qualitatively with the theory. According to the theory, the temperature
anisotropy provides evidence for new entropic contributions to the tension tensor,
responsible for ∼20% of the surface tension of argon.
Eventually, we studied transport of heat and mass across planar and curved interfaces.
The transport equations were formulated with the framework of nonequilibrium
thermodynamics by using a set of interface transfer coefficients. We started
by investigating the interface transfer coefficients of bubbles and droplets in both
single- and two-component systems. We verified that the coefficients obtained by
combining the integral relations with results from the equilibrium square gradient
model were the same as those obtained by using the nonequilibrium square gradient
model, in which gradients in temperature, pressure and composition were imposed
across the interface. The local heat transfer characteristics of the interfacial region
(local resistivity) were found to have a large influence on the predicted curvature
dependence of the interface transfer coefficients.
We obtained the general curvature dependence of the interface transfer coefficients
by expanding the coefficients in the total and Gaussian curvatures. This provided
an accurate description of heat and mass transfer across interfaces of complex nanogeometries.
A method to obtain the leading terms in the curvature expansion was
presented. The method was demonstrated for an oblate spheroidal droplet (Mentos
candy), a prolate spheroidal bubble (rugby ball), and a toroidal bubble (donut).
Depending on the sign and magnitude of the total and Gaussian curvatures, the
interface transfer could increase or decrease significantly. We next derived analytical
expressions for the terms in the curvature expansion of the interface transfer
coefficients, allowing us to calculate them with higher accuracy. The magnitude
and temperature-dependence of the interface transfer coefficients and the terms
in the curvature expansion were discussed in detail for the truncated and shifted
Lennard-Jones fluid.
The tools developed were next used to describe transfer across planar and curved
water interfaces. For the complicated case of water, it was necessary to use a
temperature-dependent influence parameter in the square gradient model. We
showed how this extension modified the nonequilibrium square gradient model,
and lead to thermodynamic quantities that depended on temperature gradients.
The modified formulation was found to be thermodynamically consistent and to
give an interface at local equilibrium, in agreement with previous work. We then
calculated the interface transfer coefficients and the corresponding curvature corrections
for the vapor-liquid interface of water from 260 K to 550 K. A novel
approach was used; this approach took advantage of low-temperature water evaporation
experiments, nonequilibrium molecular dynamics with the TIP4P/2005
model at high temperatures and square gradient theory. We used the framework
to obtain new understanding of transport across the interface of coalescing nanosized
water droplets, in which the resistance to heat and mass transfer was found
to be significantly increased in the junction between the droplets.
This work has provided new insight into topics heavily debated in the literature,
such as the sign and relevance of the Tolman length and the rigidity constants, and
investigated previously unaddressed topics, such as the inherent properties of the
temperature across interfaces and the curvature dependence of the interface transfer
coefficients. Several tools and techniques developed in this work can be taken
advantage of to study the properties of highly metastable fluids, the magnitude
and curvature dependence of interface transfer coefficients, improve the predictions
from nucleation theory and design future nano devices with enhanced properties.