NEMD

Theories and simulation of the operation of AFM in liquid have been attempted [102-104], In principle, molecular dynamics or NEMD may be a suitable method to mimic the operation of a scanning tip. The time scale, however, precludes simulating a long-enough scan to see a complete atom. Most studies, therefore, were made with equilibrium conditions and a fixed position of the AFM tip. Explicit consideration of electrolytes and electrostatic effects has not been modeled. 

FIG. 18 Setup of the cylindrical simulation cell with periodic boundary condition in the axial (z) direction. In the NEMD simulation, a constant gradient in the potential 4>(z) is applied, and concentration c(z) is maintained constant by recycling ions. 

One attraction of MD simulation is the possibility of computer animation. The mobility of ions inside a charged cylindrical pore can be visualized. Some movie clips of EMD and NEMD are downloadable at http //chem.hku.hk/ kyc/movies/. mpg. Some features that escape statistical averages can be learned in watching the animation. While the coions are present mainly in the center of the pore, occasional collisions with the wall do occur, as observed in the movie. The time scale of a coion staying near the wall is of the order of 1 ps, compared to 10 ps for the counterion. While the averaged equilibrium distributions indicate an infinitesimal concentration of coion at the wall, reaction of coion with the wall can occur within a time scale of 1 ps. From the video, it can also be observed that the radial mobility of the counterion is more significant compared to the coion s and compared to the axial mobility. It is consistent with the statistical results. 

Evans and Baranyai [51, 52] have explored what they describe as a nonlinear generalization of Prigogine s principle of minimum entropy production. In their theory the rate of (first) entropy production is equated to the rate of phase space compression. Since phase space is incompressible under Hamilton s equations of motion, which all real systems obey, the compression of phase space that occurs in nonequilibrium molecular dynamics (NEMD) simulations is purely an artifact of the non-Hamiltonian equations of motion that arise in implementing the Evans-Hoover thermostat [53, 54]. (See Section VIIIC for a critical discussion of the NEMD method.) While the NEMD method is a valid simulation approach in the linear regime, the phase space compression induced by the thermostat awaits physical interpretation even if it does turn out to be related to the rate of first entropy production, then the hurdle posed by Question (3) remains to be surmounted. 

Perhaps the most common computer simulation method for nonequilibrium systems is the nonequilibrium molecular dynamics (NEMD) method [53, 88]. This typically consists of Hamilton s equations of motion augmented with an artificial force designed to mimic particular nonequilibrium fluxes, and a constraint force or thermostat designed to keep the kinetic energy or temperature constant. Here is given a brief derivation and critique of the main elements of that method. 

Depending on the point of view, it is either a strength or a weakness of the NEMD method that it gives a uniform structure for the nonequilibrium system... 

In the practical implementation of the NEMD method, it is usual to set the momentum derivative of the nonequilibrium potential to zero, Ap,a = 0 [53, 89]. Presumably the reason for imposing this condition is that it preserves the classical relationship between velocity and momentum, q,.-, = p,rjJm. In view of this condition, the rate of change of the nonequilibrium potential reduces to... 

Figure 7. Nonequilibrium Monte Carlo results for the thermal conductivity (To = 2). The circles and squares are the present steady-state results for bulk and inhomogeneous systems, respectively (horizontally offset by 0.015 for clarity), and the triangles are NEMD results [89, 91]. (From Ref. 5.)...

Nonequilibrium molecular dynamics (NEMD) Monte Carlo heat flow simulation, 71-74 theoretical background, 6 Nonequilibrium probability, time-dependent mechanical work, 51-53 Nonequilibrium quantum statistical mechanics, 57-58... 

To bridge the time-scale gap between microscopic and macroscopic scales and accurately capture dynamic phenomena on the coarsegrained level, systematic time-scale-bridging molecular dynamics was recently introduced by using an alternative MC-MD iteration scheme, which also shows higher calculation efficiency than standard NEMD (Ilg et al, 2009). 

Recently, Miiller-Plathe suggested a NEMD method [51] for calculation of thermal conductivity in atomic fluids that was subsequently adapted and applied... 

Using the imposed heat flux method, we carried out NEMD simulations for the HMX melt at six temperatures (550 K - 800 K, in 50 K intervals) and atmospheric pressure. The simulation methodology was similar to the one described above for equilibrium MD simulations with a few exceptions. Each system contained 100 HMX molecules. The orthorhombic simulation box, extended in the z direction, was subdivided into 10 equal slabs with width of about 5.0 A and cross-sectional area of about 625.0 A2. The molecular center-of-mass velocities were exchanged every 500 fs (W=0.002 fs 1) for pairs of molecules belonging to the cold and hot slabs. This choice of the W was based on our previous experience with simulations of liquid n-butane and water [52],... 

In Fig. 8 we show a comparison of the thermal conductivity for liquid HMX obtained from our NEMD simulations with measured values for crystalline HMX [54] as well as values used in combustion models for HMX [55]. Despite being weak, the temperature dependence of the thermal conductivity of liquid HMX is not featureless. The thermal conductivity exhibits a sharp drop in the temperature interval from the melting point (550 K) up to 650 K. At higher temperatures the thermal conductivity exhibits almost no temperature dependence. The predicted value at 550 K is consistent with the HMX crystal data [54]. The thermal conductivity used in some combustion models [55] agrees to within about 25% with our NEMD predictions over the entire temperature interval. 

It should be mentionned that the result of q-dependent chain orientation in shear was also found by nonequilibrium molecular dynamics (NEMD) simulations by Kroger et al [32], The increasing power of computational techniques will surely result in increased accuracy and usefulness of this type of numerical simulation. 

Figure 1. Variation of transport coefficient with density. Inset depicts the comparison of a purely no-slip viscous theoiy with the NEMD results at l.SO K.

A large number of DCV-GCMC runs were also conducted at 150 K, for various pair of density values in the two end sections, and the effective Fickian transport coefficient, obtained. A similar coefficient is also estimated from the EMD and NEMD values of Du, that correspond to a chemical potential gradient driving force, following... 

Figure 2. Comparison of measured transport coefficients, obtained using DCV-GCMD, with those predicted based on coefficients obtained using EMD ( ) and NEMD (O). Inset shows that equilibrium density profiles are attained in EMD and NEMD.

Transport coefficients of molecular model systems can be calculated by two methods [8] Equilibrium Green-Kubo (GK) methods where one evaluates the GK-relation for the transport coefficient in question by performing an equilibrium molecular dynamics (EMD) simulation and Nonequilibrium molecular dynamics (NEMD) methods. In the latter case one couples the system to a fictitious mechanical field. The algebraical expression for the field is chosen in such a way that the currents driven by the field are the same as the currents driven by real Navier-Stokes forces such as temperature gradients, chemical potential gradients or velocity gradients. By applying linear response theory one can prove that the zero field limit of the ratio of the current and the field is equal to the transport coefficient in question. 

The topic of this article is the study of transport properties of liquid crystal model systems by various molecular dynamics simulations techniques. It will be shown how GK relations and NEMD algorithms for isotropic liquids can be generalised to liquid crystals. It is intended as a complement to the texts on transport theory such as the monograph "Statistical Mechanics of Nonequilibrium liquids [8] by Evans and Morriss and "Recent Developments in Non-Newtonian Molecular Dynamics [9] by Sarman, Evans and Cummings and textbooks on liquid crystals such as "The physics of liquid crystals" [2] by de-Gennes and Frost and "Liquid Crystals" [3] by Chandrasehkar. 

The article is organised as follows In Section 2 we review the basic theory, in Section 3 we describe NEMD-algorithms for the evaluation of the thermal conductivity and the viscosity, in Section 4 we discuss flow properties of liquid crystals, in Section 5 we present results of flow simulations of liquid crystals and finally in section 6 there is a conclusion. 

In a liquid crystal most properties are best expressed relative to a director based coordinate system. This is not a problem in a macroscopic system where the director is virtually fixed. However, it can be a problem in a small system such as a simulation cell where the director is constantly diffusing on the unit sphere. Thus a director based frame is not an inertial frame. Correction terms should therefore be added to time dependent properties. Time correlation functions with slowly decaying tails might also be affected by the director reorientation. Transport coefficient obtained from them will consequently be incorrect. When NEMD-simulation algorithms are applied, the fictitious external field exerts a torque that constantly twists the director, which could make it impossible to reach a steady state. 

Comparison of the thermal conductivities of prolate (p) and oblate (o) nematic liquid crystals. The entries for zero field have been obtained by using the Green-Kubo relation (3.3). The entries for finite field have been obtained by applying the heat flow algorithm (3.5). Note that the EMD GK estimates and the NEMD estimates agree within the statistical error. 

We have presented EMD and NEMD simulation algorithms for the study of transport properties of liquid crystals. Their transport properties are richer than those of isotropic fluids. For example, in a uniaxially symmetric nematic liquid crystal the thermal conductivity has two independent components and the viscosity has seven. So far the different algorithms have been applied to various variants of the Gay-Beme fluid. This is a very simple model but the qualitative features resembles those of real liquid crystals and it is useful for the development of molecular dynamics algorithms for transport coefficients. These algorithms are completely general and can be applied to more realistic model systems. If the speed of electronic computers continues to increase at the present rate it will become possible to study such systems and to obtain agreement with experimental measurements in the near future. 

One easy extension of equilibrium molecular dynamics is the computation of the properties of condensed matter systems in the presence of external fields. Many experiments are done by applying an external field, whether it be an electric field, heat gradient, or some type of flow. Usually, the experimentalist waits some time for the system under investigation to reach a steady state in the presence of the applied field. Measurements are then performed to deduce structural or dynamical information. Performing this task on a computer is what is commonly known as nonequilibrium molecular dynamics (NEMD). It is nonequilibrium in the sense that, in the presence of an external field, the system at the steady state will be in a state of lower entropy. Upon removing the external field, the system will return to the state of maximum entropy or the equilibrium state. 

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