Constant-pressure simulation

Constant-temperature and constant-pressure simulations—Section VII.C, this chapter. 

The question arises as to how useful atomistic models may be in predicting the phase behaviour of real liquid crystal molecules. There is some evidence that atomistic models may be quite promising in this respect. For instance, in constant pressure simulations of CCH5 [25, 26] stable nematic and isotropic phases are seen at the right temperatures, even though the simulations of up to 700 ps are too short to observe spontaneous formation of the nematic phase from the isotropic liquid. However, at the present time one must conclude that atomistic models can only be expected to provide qualitative data about individual systems rather than quantitative predictions of phase transition temperatures. Such predictions must await simulations on larger systems, where the system size dependency has been eliminated, and where constant... 

Constant Pressure MD. The conventional MD technique uses a fixed size for the simulation box, that is, the calculation is performed under constant volume conditions. Using methods developed by Parrinello and Rahman and by Berendsen and coworkers, it is now possible to undertake constant pressure simulations by allowing cell dimensions to vary dnring the simulation. Detailed discussions are given in Referenced . The most obvious field of application of this technique is to the study of phase transitions, and useful applications have been reported to the study of melting and glass formation as discussed below. 

R. H. Boyd, Macromolecules, 22, 2477 (1989). An Off-Lattice Constant-Pressure Simulation of Liquid Polymethylene. 

Moaronak, R., Donadio, D., and Parrinello, M. Polyamorphism of ice at low temperatures from constant-pressure simulations, Phys. Rev. Lett., 92, 225702, 2004. 

Solid-state phase transitions often involve deformation of the unit cell along with rearrangements of the molecules inside the unit cell. Therefore a simple constant-pressure simulation which allows only isotropic expansion or contraction of the unit cell may not be able to reproduce all of the aspects of a solid-state phase transition. The technique of Parrinello and Rahman [160] introduces a time-dependent metric tensor in the Lagrangian of the system, which allows changes of both volume and shape of the xmit cells. As such, simple solid-state phase transitions can be directly simulated with this technique. However, this method cannot be used for solid phases with very different unit cells.[145] The orientational order-disorder transitions in the solid state in some cases occur with little change in the unit cell parameters or molecular rearrangements. These orientational transitions are suitable for the Parrinello-Rahman technique.[161]... 

A volume change in an isobaric simulation can be achieved by changing the volume in all directions, or in just one direction. It is instructive to consider the range of volume changes that one might expect to observe in a constant pressure simulation of a typical system. The isothermal compressibility is related to the mean square volume displacement by ... 

As with a molecular dynamics simulation, a Monte Carlo simulation comprises an equilibration phase followed by a production phase During equilibration, appropriate thermodynamic and structural quantities such as the total energy (and the partitioning of the energy among the various components), mean square displacement and order parameters (as appropriate) are monitored until they achieve stable values, whereupon the production phase can commence. In a Monte Carlo simulation from the canonical ensemble, the temperature and volume are, of course, fixed. In a constant pressure simulation the volume will change and should therefore also be monitored to ensure that a stable system density is achieved. 

A related observation can be found in Table 1 The higher of the two values for the diffusion coefficient for Oj in PIB corresponds to a constant-volume simulation at the experimental density of PIB, the lower to a constant-pressure simulation at a slightly higher (3%) density. The authors believed that the difference between the two simulation results stems largely from the increase in density when the constant-volume constraint is relaxed. 

Figure 11. Equation of state from NPT MD and NVT MD simulations using the SW potential. Nine isotherms at temperatures above and below the critical temperature of the liquid-liquid transition are shown. The open symbols represent data from NPT MD simulations and the opaque symbols represent data from NVT MD simulation. The solid lines are polynomial fits to the data points, (s The isotherms above T = 1133K are monotonic and continuous and below T = 1133K show a jump in density for small change in pressure in constant pressure simulations, (b) Constant volume (NVT) MD simulation data for T < 1133K show nonmonotonic behavior indicating a first-order phase transition.

Most methods for 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 [41,42] enables the calculation of a complete phase diagram from a series of constant-pressure 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. 

Trofimov, S. Y., Nies, E. L. R, and Michels, M. A. J. 2005. Constant pressure simulations with dissipative particle dynamics. J. Chem. Phys. 123 144102. 

The system was studied as a function of external load on the surfaces. As molecules which detached from the droplet were removed from the simulation, the system was not in equilibrium with the vapor phase. The solid substrates were modeled after crystalline solids with both weak attraction (ej = 1) and strong attraction (es = 3) between the surface atoms and the polymer segments, which were treated as UA monomers here es is the well-depth of the surface-monomer potential and is measured in units of the well-depth of the monomer-monomer potential. Bond angles were constrained in the simulation. In Ref. 32, constant pressure simulations of liquid tridecane were performed for a system periodic in the x and y directions. The surface structure was that of the (111) face of an fee crystalline solid. Here, an explicit atom representation of the alkane chains was used. Results are presented for surface atom-polymer atom interactions equal to those of the carbon-carbon and carbon-hydrogen interactions for carbon and hydrogen atoms, respectively (a weakly attractive surface) for films nominally 4nm thick at 450 K. 

Constant pressure simulations of tiidecane confined between atomistic, weakly attractive parallel plates using a calibrated EA force field have been performed for films of nominal 4 nm thickness as 450 The most interesting difference between these and earlier UA simulations is the extension of ordering effects near the surfaces farther into the bulk, similar to the effects seen in the previous constant volume EA simulations (see Fig. 8.17). The monomer density profile and order parameter for the constant pressure EA simulations at 450 K are very close to those for the constant volume EA simulations. The apparent diffusivity perpendicular to the surfaces shows the characteristic decrease as one approaches the surfaces, while the overall mobility is essentially unaffected. This is consistent with previous results obtained for surfaces which are not strongly attractive. The bulk value of the diffusion constant of 3.3 x 10 cm /s, obtained from the diffusivity at the center of the 4nm film, agrees reasonably well with the experimental value of 5.0 x 10 cm /s. This is in contrast to some UA... 

The shape, as well as the size of the simulation cell may vary during the simulation. In order to keep the simulation cell a square, one can also apply the average osmotic pressure Va = Vi - -Vi)/2 in both directions. One advantage of the constant pressure simulation method is that the grafting density p can be changed dynamically by simply varying H,. 

Constant temperature and pressure were maintained by so called weak coupling in the lac repressor, homeodomain, endonuclease, and TATA-box binding protein (TBP) (only temperature) systems. The glucocorticoid receptor DNA-binding domain (GRDBD) simulation with a shell of water should also be considered as a constant pressure simulation, and the rest are constant volume and constant potential energy simulations. 

More recent constant pressure simulations generally agree with results from Smith and co-workers, however, su esting that pressure effects do not change the qualitative conclusions from entropy of association calculations. 

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