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Prediction of liquid properties

See also: Computational materials science

Methods for predicting liquid properties can be organized by their "scale" of description, that is, the length scales and time scales over which they apply.

• Macroscopic methods use equations that directly model the large-scale behavior of liquids, such as their thermodynamic properties and flow behavior.
• Microscopic methods use equations that model the dynamics of individual molecules.
• Mesoscopic methods fall in between, combining elements of both continuum and particle-based models.

Macroscopic

Empirical correlations

Empirical correlations are simple mathematical expressions intended to approximate a liquid's properties over a range of experimental conditions, such as varying temperature and pressure. They are constructed by fitting simple functional forms to experimental data. For example, the temperature-dependence of liquid viscosity is sometimes approximated by the function $\eta (T)=Ae^{B/T}$, where $A$ and $B$ are fitting constants. Empirical correlations allow for extremely efficient estimates of physical properties, which can be useful in thermophysical simulations. However, they require high quality experimental data to obtain a good fit and cannot reliably extrapolate beyond the conditions covered by experiments.

Thermodynamic potentials

See also: Equation of state

Thermodynamic potentials are functions that characterize the equilibrium state of a substance. An example is the Gibbs free energy $G(p,T)$, which is a function of pressure and temperature. Knowing any one thermodynamic potential ${\mathcal {F}}$ is sufficient to compute all equilibrium properties of a substance, often simply by taking derivatives of ${\mathcal {F}}$. Thus, a single correlation for ${\mathcal {F}}$ can replace separate correlations for individual properties. Conversely, a variety of experimental measurements (e.g., density, heat capacity, vapor pressure) can be incorporated into the same fit; in principle, this would allow one to predict hard-to-measure properties like heat capacity in terms of other, more readily available measurements (e.g., vapor pressure).

Hydrodynamics

Main articles: Fluid dynamics and Computational fluid dynamics

Hydrodynamic theories describe liquids in terms of space- and time-dependent macroscopic fields, such as density, velocity, and temperature. These fields obey partial differential equations, which can be linear or nonlinear. Hydrodynamic theories are more general than equilibrium thermodynamic descriptions, which assume that liquids are approximately homogeneous and time-independent. The Navier-Stokes equations are a well-known example: they are partial differential equations giving the time evolution of density, velocity, and temperature of a viscous fluid. There are numerous methods for numerically solving the Navier-Stokes equations and its variants.

Mesoscopic

See also: Mesoscopic physics

Mesoscopic methods operate on length and time scales between the particle and continuum levels. For this reason, they combine elements of particle-based dynamics and continuum hydrodynamics.

An example is the lattice Boltzmann method, which models a fluid as a collection of fictitious particles that exist on a lattice. The particles evolve in time through streaming (straight-line motion) and collisions. Conceptually, it is based on the Boltzmann equation for dilute gases, where the dynamics of a molecule consists of free motion interrupted by discrete binary collisions, but it is also applied to liquids. Despite the analogy with individual molecular trajectories, it is a coarse-grained description that typically operates on length and time scales larger than those of true molecular dynamics (hence the notion of "fictitious" particles).

Other methods that combine elements of continuum and particle-level dynamics include smoothed-particle hydrodynamics, dissipative particle dynamics, and multiparticle collision dynamics.

Microscopic

Microscopic simulation methods work directly with the equations of motion (classical or quantum) of the constituent molecules.

Classical molecular dynamics

Main articles: Molecular dynamics and Molecular mechanics

Classical molecular dynamics (MD) simulates liquids using Newton's law of motion; from Newton's second law ($F=m{\ddot {x}}$), the trajectories of molecules can be traced out explicitly and used to compute macroscopic liquid properties like density or viscosity. However, classical MD requires expressions for the intermolecular forces ("F" in Newton's second law). Usually, these must be approximated using experimental data or some other input.

Ab initio (quantum) molecular dynamics

See also: Car–Parrinello molecular dynamics

Ab initio quantum mechanical methods simulate liquids using only the laws of quantum mechanics and fundamental atomic constants. In contrast with classical molecular dynamics, the intermolecular force fields are an output of the calculation, rather than an input based on experimental measurements or other considerations. In principle, ab initio methods can simulate the properties of a given liquid without any prior experimental data. However, they are very expensive computationally, especially for large molecules with internal structure.