Isothermal-isobaric MD simulations

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. 

To perform the multibaric-multithermal MD simulation, we just solve the above equations of motion (4.29)-(4.34) for the regular isobaric-isothermal ensemble (with arbitrary reference temperature T = To and reference pressure V = Vo), where the enthalpy H is replaced by the multibaric-multithermal enthalpy 7mbt in (4.30) and (4.34) [63]. 

We now present an example of MREM. We consider an isobaric-isothermal ensemble and exchange not only the temperature but also the pressure values of pairs of replicas during a MC or MD simulation [94]. Namely, suppose we have M replicas with M different values of temperature and pressure (Tm,Vm). We are setting Eo = E, V = V, and Am = Vm in (4.58). We exchange replicas i and j which are at (Tm,Vm) and (Tn,Vn), respectively. The transition probability of this replica-exchange process is then given by (4.48), where (4.60) now reads [3,80,96]... 

Figures 4.5a-c show the time series of volume V obtained by the conventional isobaric-isothermal MD simulations. The volume fluctuates in narrow ranges. The MUBATH MD simulation, on the other hand, performs a random walk that covers a range of V = 1.8 3.5 nm3, as shown in Fig. 4.5d, which is 3-5 times wider than that by the isobaric-isothermal MD simulations.

Fig. 4.4. Time series of potential energy E from (a) the conventional isobaric-isothermal MD simulation at To = 240 K and Vo = 0.1 MPa (b) the conventional isobaric-isothermal MD simulation at To = 298 K and Vo = 0.1 MPa (c) the conventional isobaric-isothermal MD simulation at To = 298 K and Vo = 300 MPa and (d) the multibaric-multithermal MD simulation...

MD simulations can also be used for the pmf calculations with umbrella sampling or the perturbation method. An effective procedure for implementing isothermal-isobaric conditions is to couple the system to a constant temperature and pressure bath. A leap-frog algorithm is then preferred for integrating Newton s equations. " In addition, the SHAKE procedure can be applied to constrain intramolecular degrees of freedom, This is often done to eliminate bond vibrations which permits use of a larger time step, about 1-2 fs. 

In 2002, Morrow and Maginn presented an all-atom force field for [C4mim][PF6] using a combination of DFT calculations (B3LYP/6-311+G ) and CHARMM 22 parameter values [13]. MD simulations were carried out in the isothermal-isobaric ensemble at three different temperatures. The calculated properties contained infrared frequencies, molar volumes, volume expansivities, isothermal compressibilities, self-diffusivities, cation-anion exchange rates, rotational dynamics, and radial distribution functions. These thermodynamic properties were found to be in good agreement with available experimental values [13]. 

Generally, MD is performed to simulate a continuous phase trajectory in the microcanonical N,E,V) ensemble, while in MC method, individual phase points of an (N,V,T) ensemble are simulated. As far as the equilibrium properties are concerned, the trajectory average of MD and the configurational average of MC are equivalent (Allen and Tildesley, 1987). Recently, MD simulation of other types of ensembles also have been achieved. In the N,E,V) simulations, volume and total energy are held constant and the temperature and pressure are allowed to fluctuate. Andersen (1980) suggested methods for simulation of isobaric-isoenthalpic and isobaric-isothermal (A, P,7) ensembles. A... 

Cool down the system stepwise from a high temperature to low temperature at a constant interval. At each temperature, MD simulations perform in the canonical (NVT) ensemble and isothermal-isobaric (NPT). 

The MD simulations can be perfomed in maity different ensembles, such as grand canonical (pVT), microcanonical (NVE), canonical (NVT) and isothermal-isobaric (NPT). The constant temperature and pressure can be controlled by adding an appropriate thermostat (e g., Berendsen, Nose, Nose-Hoover, and Nose-Poincare) and barostat (e.g., Andersen, Hoover, and Berendsen), respectively. Applying MD into polymer composites allows us to investigate into the effects of fillers on polymer stracture and dynamics in the vicinity of polymer-filler interfaee and also to probe the effects of polymer-filler interactions on the materials properties. 

There are several modifications of the MD algorithm, allowing one to carry out the simulations in the canonical NVT) or isothermal-isobaric (NPT) ensembles. Relationships similar to Equations (l)-(3) and many others can be systematically derived for these ensembles, as well (Allen and Tildesley 1987 Frenkel and Smit 1996). 

Figure 17. Isothermal compressibility against temperature for diffaent isobars from MD simulations using the SW potential. The iocation of the minima along the isobars defines the TMinC line.

FIGURE 9.5 Comparison of the equation of state (reduced axial pressure versus reduced numerical density) of Square-Well molecules of = 1.5 confined in cylindrical hard pore with diameter, D/a = 2.2, obtained by isobaric-isothermal Monte Carlo (NPT MC) and molecular dynamic (MD) simulations. Here, squares indicate NPT MC result and circles the MD result. The solid line indicates an analytical fit of the result at the fluid branch, and the dash line is the second order polynomial fit to the solid branch. Error bars are the standard deviation of five independent runs. (From Huang, H. C., J. Chem. Phys., 132, 224504, 2010. With permission. Copyright 2010, American Institute of Physics.)... 

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