Constant pressure ensemble

In the preceding section we have set up the canonical ensemble partition function (independent variables N, V, T). This is a necessary step whether one decides to use the canonical ensemble itself or some other ensemble such as the grand canonical ensemble (p, V, T), the constant pressure canonical ensemble (N, P, T), the generalized ensemble of Hill33 (p, P, T), or some form of constant pressure ensemble like those described by Hill34 in which either a system of the ensemble is open with respect to some but not all of the chemical components or the system is open with respect to all components but the total number of atoms is specified as constant for each system of the ensemble. We now consider briefly the selection of the most convenient formalism for the present problem. 

G. J. M. Koper and H. Reiss. Length scale for the constant pressure ensemble application to small systems and relation to Einstein fluctuation theory. J. Phys. Chem., 100 422 132, 1996. 

Simulations essentially extend possibility to study supercooled liquid water, as crystallization may be suppressed. However, there is no water model, which adequately reproduces phase diagram of water and its properties even in the thermodynamic region, where experimental data are available. In such situation, only comparative analysis of the results, obtained for various water models, can give information, relevant for the behaviour of real water in supercooled region. Additional complication appears due to the necessity to use sophisticated simulation methods, appropriate for the studies of the phase transitions, such as Monte Carlo simulations in the grand canonical or in the Gibbs ensemble (see Refs.7,16 for more details). Note, that simulations in the simple constant-volume or constant-pressure ensembles, widely used in the studies of supercooled water (see, for example Refs. 17,18), are not appropriate for the location of the phase transitions. 

The constant-temperature, constant-pressure ensemble (NPT) allows control over both the temperature and pressure. The unit cell vectors are allowed to change, and the pressure is adjusted by adjusting the volume. This is the ensemble of choice when the correct pressure, volume, and densities are important in the simulation. This ensemble can also be used during equilibration to achieve the desired temperature and pressure before changing to the constant-volume or constant-energy ensemble when data collection starts. 

The constant-temperature, constant-stress ensemble (NST) is an extension of the constant-pressure ensemble. In addition to the hydrostatic pressure that is applied isotropically, constant-stress ensemble allows you to control the xx, yy, zz, xy, yz, and zx components of the stress tensor (sometimes also known as the pressure tensor). This ensemble is particularly useful if one wants to study the stress-strain relationship in polymeric or metallic materials. 

One can now also investigate other ensembles for the single vesicle adhered to a substrate such as the ensemble in which besides the volume also the surface area A is kept constant. As we discussed previously, this is the ensemble for which Seifert and Lipowsky constructed their phase diagrams of the vesicle unbinding transition. Another ensemble one might wish to consider is the constant pressure ensemble, for which Q is the appropriate free energy, in which instead of the vesicle volume the Laplace pressure difference Ap is... 

The previous discussions translate directly over into pressure-volume variables, if we compare the constant-NVT and constant-A PJ ensembles. Double-peaked distributions of volumes are seen near a transition at constant pressure. 

For a multicomponent system, it is possible to simulate at constant pressure rather than constant volume, as separation into phases of different compositions is still allowed. The method allows one to study straightforwardly phase equilibria in confined systems such as pores [166]. Configuration-biased MC methods can be used in combination with the Gibbs ensemble. An impressive demonstration of this has been the detennination by Siepmaim et al [167] and Smit et al [168] of liquid-vapour coexistence curves for n-alkane chain molecules as long as 48 atoms. 

However, it is common practice to sample an isothermal isobaric ensemble NPT, constant pressure and constant temperature), which normally reflects standard laboratory conditions well. Similarly to temperature control, the system is coupled to an external bath with the desired target pressure Pq. By rescaling the dimensions of the periodic box and the atomic coordinates by the factor // at each integration step At according to Eq. (46), the volume of the box and the forces of the solvent molecules acting on the box walls are adjusted. 

Just as one may wish to specify the temperature in a molecular dynamics simulation, so may be desired to maintain the system at a constant pressure. This enables the behavior of the system to be explored as a function of the pressure, enabling one to study phenomer such as the onset of pressure-induced phase transitions. Many experimental measuremen are made under conditions of constant temperature and pressure, and so simulations in tl isothermal-isobaric ensemble are most directly relevant to experimental data. Certai structural rearrangements may be achieved more easily in an isobaric simulation than i a simulation at constant volume. Constant pressure conditions may also be importai when the number of particles in the system changes (as in some of the test particle methoc for calculating free energies and chemical potentials see Section 8.9). 

A macroscopic system maintains constant pressure by changing its volume. A simulation i the isothermal-isobaric ensemble also maintains constant pressure by changing the volurr... 

Thermal averages in the ensemble with constant pressure p are given via the corresponding partition function dVexp[—/3pF]Z(A, F, T). 

These equations of motion can be integrated by many standard ensembles constant energy, constant volume, constant temperature and constant pressure. More complex forms of the extended Lagrangian are possible and readers are referred to Ref. [17] for a Lagrangian that allows intermolecular charge transfer. 

Most methods for the determination of phase equilibria by simulation rely on particle insertions to equilibrate or determine the chemical potentials of the components. Methods that rely on insertions experience severe difficulties for dense or highly structured phases. If a point on the coexistence curve is known (e.g., from Gibbs ensemble simulations), the remarkable method of Kofke [32, 33] enables the calculation of a complete phase diagram from a series of constant-pressure, NPT, simulations that do not involve any transfers of particles. For one-component systems, the method is based on integration of the Clausius-Clapeyron equation over temperature,... 

The Car-Parrinello method is similar in spirit to the extended system methods [37] for constant temperature [38, 39] or constant pressure dynamics [40], Extensions of the original scheme to the canonical NVT-ensemble, the NPT-ensemble, or to variable cell constant-pressure dynamics [41] are hence in principle straightforward [42, 43]. The treatment of quantum effects on the ionic motion is also easily included in the framework of a path-integral formalism [44-47]. 

Similar schemes to the above can be used in molecular dynamics simulations in other ensembles such as those at constant temperature or constant pressure (see Frenkel and Smit, and Allen and Tildesley (Further reading)). A molecular dynamics simulation is computationally much more intensive than an energy minimization. Typically with modern computers the real time sampled in a simulation run for large cells is of the order of nanoseconds (106 time steps). Dynamical processes operating on longer time-scales will thus not be revealed. 

To show this, we will present and discuss some typical results readily available in the literature. Here we will select systems simulated in the constant-pressure constant-surface-tension ensemble, i.e. with a temperature, pressure and surface tension control. 

Venable, R. M., Brooks, B. R. and Pastor, R. W. (2000). Molecular dynamics simulations of gel phase lipid bilayers in constant pressure and constant surface area ensembles, J. Chem. Phys., 112, 4822-4832. 

Second virial coefficients represent the first approximation to the system equation of state. Yethiraj and Hall [148] obtained the compressibility factor, i.e., pV/kgTn, for small stars. They found no significant differences with respect to the linear chains in the pressure vs volume behavior. Escobedo and de Pablo [149] performed simulations in the NPT ensemble (constant pressure) with an extended continuum configurational bias algorithm to determine volumetric properties of small branched chains with a squared-well attractive potential... 

While Eqs. (9)-(16) provide a recipe for evaluating Sc T) for constant volume (V) systems (i.e., constant cj)), they can easily be applied to constant pressure systems by computing cj) for a given pressure P and temperature T from the equation of state. For internal consistency, this equation of state must be derived from the free energy T expression appropriate to the /-ensemble. 

However, these results do not immediately carry over to the problems of interest here where (while PBCs are the norm) the ensembles are frequently open or constant pressure, and the systems do not fit in to the lattice model framework. Even in the apparently simple case of crystalline solids in NVT, the free translation of the center of mass introduces /-dependent phase space factors in the configurational integral which manifest themselves as additional finite-size corrections to the free energy these may not yet be fully understood [58, 97]. If one adopts the traditional stance, then, one is typically faced with having to make extrapolations of the free-energy densities in each of the two phases, without a secure understanding of the underlying form (jf . ..) of the corrections involved. 

In what follows, the two fluctuating interfaces will be replaced by many small, independent surfaces of area S, separated by a distance z (see Figure 4). The (metastable) distribution of the distances between the surfaces, in an ensemble subjected to a constant pressure, will be assumed Boltzmannian. It will be also assumed that the fluctuating interfaces have constant total areas and an elastic bending modulus Kq. Let us first consider that the interfaces interact per unit area through a harmonic potential... 

There are many excellent reviews on the standard molecular dynamics method dealing with calculations in the microcanonical ensemble as well as on the Monte Carlo method involving calculations in the canonical, isothermal isobaric, and grand canonical ensemble (< ). In the present article, we shall limit ourselves exclusively to those developments that have taken place since the work of Andersen (4). In the molecular dynamics method, the developments are the constant-pressure, constant-temperature, constant-temperature-constant-pressure, variable shape simulation cell MD, and isostress calculations in the Monte Carlo method, it is the variable shape simulation cell calculation. 

The ab initio molecular simulation was carried out in the NPT ensemble with version 3.9.1 of CPMD [29]. In the calculation, 20 water and one acetic acid molecules were simulated in a periodic cubic cell under an ambient condition of a constant pressure 1 bar, a constant temperature 300 K, and density of 1.0 gem-3. The starting structure was constructed in a simple flexible cubic cell with a length of 8.8794 A, by randomly adding 11 water molecules to an optimized hydration compound composed of a single acetic acid molecule and 9 water molecules. 

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