# Molecular simulation

### Literature Review

### Ⅰ Introduction

### A. General aspects of molecular simulation

Before a fundamentally new invention becomes a widespread application, it usually takes decades rather than years. However, when computer simulation started as a tool to utilize electronic computing machines which had been developed after World WarⅡ, the story is different. In the early 1950s, computers could be utilized as an available tool for non-military use and this was the beginning of computer simulation [1]. In order to assess the potential use of computers and their logical structure, scientists began to broaden their range of capabilities, in this period.

It was a fact that it was very difficult to predict the properties of a molecular substance before the use of computer simulation emerged; this fact affected the development of related subjects severely. However, scientists could make use of theories which provied an approximate description of the material. These theories will evaluate the properties of interest with enough information about intermolecular interaction; unfortunately, our knowledge of the simplest molecules is quite limited. We can not determine the intermolecular interactions of all molecules exactly. This leads to a problem if we try to test the validity of a particular theory by comparing to experimental data directly. If we find that the theory and the experiment data is in disagreement, it may mean that our theory is wrong, or that we have an incorrect estimate of the intermolecular interaction, or both.

Obviously, it would be ideal if we can obtain exact results for a given model system without approximate theories. Computer simulation allows us to do that easily and precisely. On the one hand, we can now compare the calculated properties of a model system with those of an experimental system: if these two disagree, our model is incorrect, therefore we have to do some improvement on our estimation of the intermolecular interactions. On the other hand, we can compare the result of a simulation of a given model system with the predictions of an approximate analytical theory which can be applied to the same model. If we find that the theory and the simulation is in disagreement, the theory is flawed [2]. As we can see, the computer simulation is a complementary technique to experiments and to theory, which can play the role of the experiment designed to test the theory. However, we have to note that the computer just offers us numbers, and in the numbers there exists statistical errors. As a result, we can not get a theoretical relation directly from simulation. We still need to extract the useful information from the numbers, much like we do in a real experiment.

Within the broad range of applications of computer simulation, the most common is to predict the properties of materials. The need for computer simulation may not be immediately obvious. After all, it is much easier to scale the freezing point of water than to measure it from a computer simulation. The point is that it is easy to measure the freezing point of water at low atmosphere. However, for very high pressures, the situation is different. It is often very difficult and expensive to measure the properties of real materials at high pressures or temperatures. It makes no difference to a computer if the pressures and temperatures are high or low, to the computer it is just a number to input into a calculation; it is not on fire when you do a simulation with a specified system at 10,000 K. In addition, more significantly, we can use computer simulation to predict the properties of materials which have not yet been synthesized. With the development of science, computer simulations are widely used in data analysis. For example, a very efficient technique for obtaining structural information about macromolecules from 2D-NMR is to feed the experiment data into a Molecular Dynamics simulation and let the computer find the structure which is both energetically favorable and compatible with the available NMR data.[3]

### B. Overview of molecular simulation methods

Computer simulation allows scientists to study properties of multi-particle systems; however, not all of the properties can be directly measured in a simulation. Conversely, most of the quantities that can be calculated in a simulation process do not link to properties of the system directly. This is in contrast to real experiments, in which you can measure and compare data directly with their corresponding properties. For instance, in an MD simulation of water, we can acquire data of the instantaneous positions and velocities of all molecules in the liquid. However, this kind of information cannot be compared to experimental data, because no real experiment can provide us with such information. The real experiment can always measure the data averaged over a large number of particles and also averaged over the time of the measurement. If we wish to utilize computer simulation as a numerical counterpart to experiments, we have to know what kind of average data we should aim to compute. In order to explain this, we require another language, which is statistical mechanics. However, it is not enough to do the calculation of molecular simulation solely with statistical mechanics; we also need some specific methods to perform this. Clearly, there are two main methods to connect the statistical mechanics with computer simulation, the MC (Monte Carlo) and the MD (Molecular Dynamics) methods. So following part will introduce the MC method and the MD method briefly.

Monte Carlo methods are a class of computational algorithms that depend on repeated random sampling to work out their results. Because of their reliance on repeated computation of random or pseudo-random numbers, these methods are most suited to calculation by computers and tend to be employed when it is infeasible or impossible to calculate a precise result with a settled algorithm.[4] For example, the Metropolis-sampling algorithm is described by the sequence of operations given in 1.1[5].

Monte Carlo simulation methods are especially useful in analyzing systems with a large number of coupled degrees of freedom, such as fluids, disordered materials, strongly coupled solids, and cellular structures. The term "Monte Carlo method" was coined in the 1940s by physicists including Metropolis and his co-workers who were working on nuclear weaponry projects in the Los Alamos National Laboratory. [6]

Molecular dynamics is a specialized subject of molecular modeling and computer simulation based on statistical mechanics; the main justification of the MD method is that statistical ensemble averages are equal to time averages of the system, known as the ergodic hypothesis. MD has also been termed "statistical mechanics by numbers" and "Laplace's vision of Newtonian mechanics" of predicting the future by animating nature's forces [7, 8] and allowing insight into molecular motion on an atomic scale. However, long MD simulations are mathematically ill-conditioned, generating accumulative errors in numerical integration that can be minimized with appropriate selection of algorithms and parameters, but cannot be obviated completely. In addition, current potential functions are, in many cases, not sufficiently accurate to reproduce the dynamics of molecular systems, so the much more computationally demanding Ab Initio Molecular Dynamics method must be used. Nevertheless, molecular dynamics techniques allow detailed time and space resolution into representative behavior in phase space for carefully selected systems. The best introduction to molecular Dynamics simulation is to consider the structure of a simple program. The program we point out is kept as simple as possible to illustrate the core features of Molecular Dynamics simulations.

### The program can be constructed as follows:

1. We read in the parameters that specify the conditions of the run (e.g., initial T, number of particles, density, time step).

2. We initialize the system (i.e., we select initial positions and velocities).

3. We compute the forces on all particles.

4. We integrate Newton's equations of motion. This step and the previous one make up the core of the simulation. They are repeated until we have computed the time evolution of the system for the desired length of time.

5. After completion of the central loop, we compute and print the averages of the measured quantities, and stop.

Because molecular systems generally comprise of a large number of particles, in general it is hard to estimate the properties of such complex system analytically. When the number of particles interacting is higher than two, the result is disorderly motion. MD simulation circumvents the analytical intractability by using numerical methods. It acts as an interface between laboratory experiments and theory, and can be considered as a "virtual experiment". MD examines the relationship between molecular structure, movement and function. Molecular dynamics is a multidisciplinary method; its laws and theories stem from mathematics, physics, and chemistry, and it uses algorithms from computer science and information theory. It was originally conceived within theoretical physics in the late 1950s [9] and the early 1960s [10], but is applied today mostly in materials science and computer simulation.

### Ⅱ Intermolecular potentials for simple fluids

### A. General potentials

There are many suitable choices for intermolecular potential energy functions. Although for the attractive contribution, there is theoretical justification for a r-6 dependence. [5] The repulsive term is not as readily characterized. Qualitatively, as molecules come near each other, strong repulsions will occur when negatively charged electron clouds overlap. These repulsions are estimated in terms of an empirical function with a high order dependence on r, typically.

Short range (r≤σ): Overlapping electron clouds leading intensive repulsive interactions

Intermediate range (σ＜r≤r0):”Residual valence” interactions.

Long range (r≥r0): There are 4 main long range effects, fluctuation or dispersion energies (London forces), polarization or induction energies, direct electrostatic energies, coulombic energies. And the details about these effects will not discuss here. [5]

As stated earlier, we cannot out the exact intermolecular potential energy function with mathematics form used in computer simulation. So in the computer simulation area, the physicists create some kinds of intermolecular potential which can use in the simulation process. For instance, the Lennard- Jones 12-6 potential is the most widely used form for predicting properties of simple fluids in statistical physics and related scientiﬁc domains. [12]

Though Lennard- Jones 12-6 potential which has its own flaw cannot describe the interaction between molecules ideally, it stands the test of time and is used widely. The situations for hard-sphere, square-well, Kihara, Mie (n-m) and other potentials are nearly the same. In the following part, I will introduce the potentials separately.

### 1. Hard-sphere potential (HS)

Hard sphere is widely used in computer simulations for fluid and solid. They are defined simply as impenetrable spheres that cannot overlap in space. They mimic the extremely strong repulsion that atoms and spherical molecules experience at very close distances. [13]

Hard sphere of diameter σ is particles with the following pairwise interaction potential:

Where r1 and r2 are the positions of the two particles.

From the previous definition, you can tell that hard spheres potential, which are quite easy to handle, received particular attention. [14]However, the hard-sphere potential does not contain an attractive term it is, of course, a poor description of the force between actual molecules. [15]

### 2. Square-well potential (SW)

The square-well potential is identical to the Hard Sphere Potential with the addition of an attractive well of depth − ε. Mathematically, it is defined as follows:

Where σ is the diameter of the particles and rc is the cutoff. So this potential has 3 adjustable parameters.

By its definition, the force between square-well molecules is impulsive, i.e., any collision occurs instantly. This is one of its advantages, which simplify the interaction of particles. However, in real liquids or dense fluids, a molecule is in continuous interaction with many molecules in the surrounding medium. [16] As we can see, this disagreement limits the application of this kind of potential.

And also, the square-well potential is often thought of as a coarse-grained treatment of The Lennard-Jones Potential because it also takes into account both excluded volume interaction and attractive interactions. See 2.2, the comparison between Square-Well Potential and Lennard-Jones Potential.

### 3. Lennard- Jones n-m potential

It is relatively common to encounter potential functions given by:

\Phi_{12}(r) = c_{n,m} \epsilon \left[ \left( \frac{ \sigma }{r } \right)^n - \left( \frac{\sigma}{r} \right)^m \right].

Where:

Φ12(r) is the intermolecular pair potential between two particles or sites

σ is the diameter (length), i.e. the value of r where Φ12(r) = 0

ε is the well depth (energy)

n and m are positive integers and n > m. cn,m is chosen such that the minimum value of Φ12(r) being Φmin = − ε. Such forms are usually referred to as n-m Lennard-Jones Potential. There is another form which is widely used in computer simulation, the12- 6 Lennard-Jones potential, where m is fixed at 6, and n is 12. The 12-6 Lennard-Jones potential is given by

\Phi_{12}(r) = 4 \epsilon \left[ \left(\frac{\sigma}{r} \right)^{12}- \left( \frac{\sigma}{r}\right)^6 \right]

The Lennard-Jones intermolecular pair potential is a special case of the Mie potential and takes its name from Sir John Edward Lennard-Jones. [18, 19] The Lennard-Jones model comprises of two parts; a steep repulsive term m, and smoother attractive term n, representing the London dispersion forces.

This potential indicts most of the features found in experiment for simple fluid. However, from a strict theoretical point of view, it is well known that the LJ potential cannot be a true representation of even two-body interactions between argon atoms. In particular, we have to note that it represents the decay of repulsive interaction by a 12–power dependence on r, which was chosen mainly for mathematical convenience (from fitting to actual data, e.g. Second virial coefficient or viscosity) and has no physical soundness. [20] This flaw will lead big disagreement in some specified cases, which limit the application of LJ potential.

The following 2.3 is a plot of the Lennard-Jones (12-6) model for the parameters \epsilon/k_B \approx120 K and \sigma \approx0.34 nm.

### 4. Kihara potential

Here the finite size of the molecule is taken into consideration by including a core diameter a. [21, 22] Note that if a=0, this potential will turn to Lennard-Jones (12-6).

The advantage of this potential is that it can describe interactions simply even in the complex cases. Also, the theory of Kihara molecule fluid is developed well. However, this kind of potential makes use of surface-to-surface distance instead of the site –site distance of two molecules. Clearly, the previous one is more complex than the latter one, so this is the main disadvantage of the potential. [23] Several simulation studies of the Kihara model [24-26] have shown, but, that the Kihara potential constitutes an effective pair-potential for modeling the behavior of substances such as nitrogen or propane; hence, the interest in developing theories computationally is less demanding than the simulation studies for this kind of fluids. [27].This is the main reason why this potential is not used widely.

### 5. Stockmayer potential

The Stockmayer potential consists of the LJ 12-6 potential with an enclosed point dipole. Thus the Stockmayer potential becomes:

\Phi_{12}(r, \theta_1, \theta_2, \phi) = 4 \epsilon \left[ \left(\frac{\sigma}{r} \right)^{12}- \left( \frac{\sigma}{r}\right)^6 \right] - \frac{\mu_1 \mu_2}{4\pi \epsilon_0 r^3} \left(2 \cos \theta_1 \cos \theta_2 - \sin \theta_1 \sin \theta_2 \cos \phi\right)

Where:

Φ12(r) is the intermolecular pair potential between two particles at a distance r;

σ is the diameter (length), i.e. the value of r at Φ(r) = 0;

ε: well depth (energy) , ε0 is the permittivity of the vacuum

μ is the dipole moment, ϕ is the azimuth angle between the two dipole moments

θ1, θ2 is the inclination of the two dipole axes with respect to the intermolecular axis.

The Stockmayer potential is one of three potentials used extensively to model dipolar fluids. [28] More recently the complex phase behaviour of dipolar liquids has become the focus of attention, because of interesting for researchers working on polymers and (lyotropic) liquid crystals. [29] This recent attention will board the application of Stockmayer potential. However, we have to note that the simulation of the Stockmayer molecule is relatively complicated, because the long-range corrections should be treated carefully. [30]

### 6. Buckingham exponential-6 potential

Where ε and σ are the Lennard-Jones energy and size parameters respectively and r is the separation distance between groups. [31] rm is the distance at which the potential has a minimum. The characteristic size parameter σ for this potential is defined as the distance for which U(r) =0 and is acquired numerically. A hard core is introduced at the distance of the (unphysical) maximum of the potential at short distances. [32] The reason a cutoff distance is required is that at very short distances, the original Buckingham exponential-6potential becomes negative. [33]Though this potential has 4 parameters which are more than LJ potential, it gives us more choices to describe the behaviour of real fluid. For instance, from Ref. 34 we can find that the accuracy of representation is significantly higher in comparison to the LJ potential, at the cost of introducing one extra parameter in the potential.

### 7. Sutherland potential

The Sutherland potential, consists of a hard-sphere repulsive core plus an attractive inverse power tail, that is in which σ is the diameter of the particles, -ε is the maximum potential depth, and γ is a parameter that determines the effective range of the potential. [35]

Relatively little attention has been paid to the study of systems with molecules interacting by Sutherland potential from computer simulation [36-38]. However, this potential model might be useful because of its resemblance with other intermolecular potentials. With the variation of γ, this potential can represent variety kinds of fluids both for long-range and short-range interactions. For ionic fluid, the main interactions are those between ion pairs. After averaging their orientations, the force behaves asymptotically like the Sutherland potential with γ=4; [39] thus, the Sutherland potential resembles the Lennard–Jones potential, for γ = 6, and therefore would be useful for studying simple fluids, and the Mie potential, widely used for modeling complex molecular fluids [40-42], for other values of γ. Furthermore, quite recently, the Sutherland potential has been used to modelize different kinds of interactions in complex fluids and mixtures [25, 43, 44].

### 8. Mie (n-m) potential

The details about Mie potential will be discussed in the next section.

With the development of computer simulation, the potential becomes the limiting factor in the whole process. Early methods for describing phase equilibrium needed the formulation of an equation of state from simulation at different state points. [45–48]However, the emergence of the Gibbs ensemble technique [49-51] simplified this procedure greatly by determining the properties of two coexisting phases at an exact temperature through computer simulation directly. This method is more efficient than the previous one, except near critical points. More recently, histogram reweighting grand canonical Monte Carlo simulations, [52, 53] combined with mixed-field finite-size scaling concepts, [54, 55] have been used to specify critical parameters and phase behaviour for different systems unprecedented accurately.

These methods have brought computer simulation to the point where it can be utilized to predict the thermophysical properties of complex, industrially important fluids reliably. As we can see, the limiting factor is the lack of accurate intermolecular potential for real fluids. However, the determination of these potential inevitably involves the fitting process of parameters to experimental data [56-58], which can be impractical if numerous state points and parameters are needed. All in all, the development of potential cannot satisfy the requirement of computer simulation, which this fact will need scientists to broad the options of potential.

### Ⅲ The Mie n-m potential

### A. Historical background

Gustav Mie (1903) [59] suggested to describe interaction in a diatomic molecule with the help of potential similar to

Here A, B > 0, n > m >0. This function has a minimum at

In the early twentieth century the Mie potential was widely used for calculation of physical properties of gases and solids. [60] In 1924, Lennard-Jones [61] first used such potentials in studying gases. London (1930) [62] proved by the quantum theory that at large distance the interaction energy is proportional to –r-6. It is therefore natural to use the value m=6. Lennard-Jones (1931) [63] refitted the experimental data with m=6 and n equal to 9 or 10 or 12. In the 1960s, Guggenheim & McGlashan [62] showed that the particular choice n = 12, m = 6, leading to the modern "12-6 potential" by fitting the experiment data.

### B. Current simulation of mie potential for vapor-liquid equilibrium

Phase equilibria of real fluids are very interested by scientists and engineers. Not long time ago, the investigation of phase equilibria for the simple system was still a hard task. [34] During the last two decades, great progress has been achieved in investigating and predicting phase diagrams of molecular systems. [65] The development of statistical mechanics methods and potential with the increased use of computer simulation have provided an in-depth understanding of the phase behavior of systems as complex as associating substances, molecules that exhibit liquid crystalline phases, charged molecules, and molecular chains. [66] With given intermolecular interactions, faster computers and improved algorithms such as the Gibbs ensemble [49-51] and extended ensembles [70-72] make the procedure of the determination of phase coexistence properties for fluids easily.

Vapor-liquid equilibrium is a condition where its liquid phase and its gas phase are in equilibrium with each other, and also a condition where the rate of evaporation equals to the rate of condensation on a molecular level. Although equilibrium takes forever to reach in theory, equilibrium can be reached in a relatively closed location if liquid and vapor are allowed to contact with each other long enough without interference from the outside. In thermodynamics, we can use temperature, (partial) pressure and Gibbs free energy to describe the vapor-liquid equilibrium. Because, for pure systems, the temperature, partial pressure, Gibbs free energy of the two phases is the same at equilibrium. [73] For multicomponent systems where the liquid and vapor consist of more than one type of molecule, the situation is quite same; expect that the partial molar Gibbs free energy of each phase have to be equal. [73]

As we can see, if the data of temperature, pressure or Gibbs free energy can be calculated in the computer simulation, we can determine the phase equilibrium. The fact is the pioneer scientists have made possible the direct evaluation of phase coexistence by MD and MC. For instance, several methods have been developed to draw the coexistence curve as theoretical studies. [80] Following are some examples: (1) Moller and Fischer proposed the NpT plus test particle method. [78] This method determines the equilibrium vapor pressure by computing the chemical potential to estimate ρv and ρL . (2) Panagiotopoulos introduced the Gibbs ensemble Monte Carlo technique. [49-51]The method is based on executing a simulation in two regions, each with separate periodic boundary conditions. The thermodynamic conditions of phase coexistence are satisfied in a statistical sense by particle displacements, volume changes and particle transfers, respectively ensuring internal equilibration, pressure and chemical potential equality.[55] (3) The Maxwell construction method is also used to determine the coexistence curve. [79] Pressures can be calculated for various volumes and coexistence properties are determined by the Maxwell equal-area rule. However, in this method, it is necessary to perform simulations in an unstable region for estimating the van der Waals loop. The curve in the unstable region of the van der Waals loop is not well defined in molecular simulations. Therefore, the result of this method is questionable.

More recently, histogram reweighting grand canonical Monte Carlo simulations, [52, 53] combined with mixed-field finite-size scaling concepts, [54, 55] have been used to specify critical parameters and phase behaviour for different systems unprecedented accurately.

For Mie (n-6) potential, Kenji Kiyohara et al. [84] applied thermodynamic scaling Gibbs ensemble (TSGE) method which is a combination of thermodynamic scaling and Gibbs ensemble concepts to a test case, phase coexistence of the (n-6) family of Mie fluid. In his study, a temperature range of T*(= k T /ε) = 0.7-1.5 and three different Hamiltonians, n = 12, 11, and 10 for a Mie (n -6) potential, were covered. The results for coexistence densities are shown in 3.1 with a temperature interval of ΔT* = 0.05.

3.1 Coexistence densities for the liquid and gas phase for three Hamiltonians calculated by TSGE. Open circles, diamonds, and triangles are for n = 12, 11, and 10, respectively. Solid curves are fits to the law of rectilinear diameters and the scaling law for the density with Ising exponent (β= 0.326). The system size effect on the exponent β (see [85]) and the crossover to classical values near the critical point are not discussed in this paper. Filled circle, diamond, and triangle are the estimates of the critical points for each Hamiltonian by the fitting.

From the work of Hisashi Okumura and Fumiko Yonezawa [86], the liquid–vapor coexistence curves in the T – ρ plane are shown in Figs. 3.2–3.4 for mie n-6 (n=7, 8, 9, 10, 12, 15, 18, 20, 32) potential.

In this case, the coexistence curves of the mie n-6 potential were drawn by the NpT plus test particle method.

Furthermore, the coexistence curves studied for potentials as Mie n-6 (n=10–22) potential are also presented in Ref 87.

Except mie n-6 potential, mie 2n-n potential was also studied widely by scientists. From the work of M. Hasegawa [88], he was used mie 2n-n potential to compare with hard-sphere attractive Yukawa (HSAY) potential. See Fig 3.5, the phase diagram for n=10, 12 and 14.

In M. Hasegawa's work [88], the disappearance of a stable liquid phase and the occurrence of an isostructural solid–solid transition (Fig 3.6) are concerned with primarily in his case. In addition, in his case, n equal to 200 was covered.

### C. Mie data for other thermodynamics properties

Both design and efficient cost running of industrial processes involving fluids depend on the possibility of determining an excellent concurrent description of phase equilibria and thermodynamic properties of the pure fluids and mixtures involved. For instance, in the oil and gas industries, special attention is devoted to the recovery of the heavy crude oil. In this particular case, the characteristic parameters that must be fitted in each situation are usually tuned almost exclusively to phase equilibrium data in order to obtain the best representation possible for describing vapor-liquid equilibrium. But these fitting parameters usually yield poor estimations of other properties. Moreover, if the parameters from fitting experiment data are used to extend estimations of second-derivative properties, deviations from real behavior reach limits that make results useless. The contrary also applies: estimations obtained from parameters fitted on thermophysical properties show poor agreement with experimental phase equilibrium. Related to this fact, it is widely agreed that estimation of second-derivative properties is one of the most demanding tests to check the performance limits for a thermodynamic model or equation.

Thermodynamic properties of a substance include temperature T, pressure P, isochoric heat capacity CV, isobaric heat capacity CP, isothermal compressibility βT, isentropic compressibility βS, Joule-Thomson coefficient μJT, velocity of sound u and isothermal pressure coefficient γV. Expect T and P, other properties are second-derivative which quantities are related to second-order derivatives of the entropy.

### 1.1 Isochoric heat capacity CV and isobaric heat capacity CP

CV and CP are the specific heat of substances typically measured under constant volume or constant pressure [91], respectively. They can be described by the equation (Eq 3.1 and 3.2) as follow:

In addition, the relationship between CV and CP is as Eq 3.3 and Eq 3.4.

where, α is the coefficient of thermal expansion, βT is the isothermal compressibility, and ρ is density.

γ is ratio of specific heats.

### 1.2 Isothermal compressibility βT and isentropic compressibility βS

Compressibility is a measure of the relative volume change of a fluid or solid as a response to a pressure (or mean stress) change. But, for any object or system the magnitude of the compressibility depends strongly on whether the process is adiabatic or isothermal. Accordingly we define the isothermal compressibility [92] as Eq 3.5:

And adiabatic compressibility [92] as Eq 3.6:

Moreover, the isothermal compressibility is related to the isentropic (or adiabatic) compressibility by the relation as Eq3.7:

via Maxwell's relations. More simply stated,

Eq 3.8

From Eq 3.4, we can rearrange Eq 3.8 to Eq 3.9

Eq 3.9

### 1.3 Joule–Thomson coefficient μJT

In thermodynamics, the Joule–Thomson effect or Joule–Kelvin effect or Kelvin–Joule effect describes the temperature change of a gas or liquid when it is forced through a valve or porous plug while kept insulated so that no heat is exchanged with the environment. [93-95]

The rate of change of temperature T with respect to pressure P in a Joule–Thomson process (that is, at constant enthalpy H) is the Joule–Thomson coefficient μJT. This coefficient can be expressed in terms of the gas's volume V, its heat capacity at constant pressure Cp, and its coefficient of thermal expansion α as: [93, 95and 96]

\mu_{\mathrm{JT}} \equiv \left( {\partial T \over \partial P} \right)_H = \frac{V}{C_{\mathrm{p}}}\left(\alpha T - 1\right)\,

### 1.4 Speed of sound u

The speed of sound is the rate of travel of a sound wave through an elastic medium. The speed of sound in a fluid is given by [97]

Where K is the bulk modulus of the fluid, the bulk modulus K can be formally defined by the equation: K=-V\frac{\partial P}{\partial V}

So

### 1.5 Isothermal pressure coefficient γV

Isothermal pressure coefficient http://www.imathas.com/cgi-bin/mimetex.cgi?%5Cdisplaystyle%5CgammaV is defined as

However, from Ref .98 we can get the relationship between γV and βT as:

### 2.1 Calculate thermodynamics properties by MD

By the method of MD, thermodynamic properties above can be calculated in the classical molecular dynamics ensemble, at least in principle. [99-101] The classical molecular dynamics ensemble (NVEP) [100] with fixed number of particles N, volume V, total energy E, and total linear momentum P is closest to the microcanonical ensemble (NVE) , the natural starting point of statistical mechanics. Consider an isolated system of N rigid nonspherical particles in a volume V with total energy E, constrained to a subspace of the microcanonical ensemble by a total linear momentum. The classical phase space integrals are [103-105] the phase space volume:

The phase space density:

and the ensemble average of any dynamical property:

In classical statistical mechanics it is shown that in the thermodynamic limit both the phase space volume and the phase space density are proper analogs to the entropy [103] through

with the property that[106]

Through R. Lustig's work, the basic phase space function can be determined as follow.

Where K is the kinetic and U the potential energy of the system and the angular brackets denote an ensemble average. The symbol (x), denotes the Pochhammer polynomial x(x+1)…(x+n-1) with (x)0l and S stands for the Kronecker symbol. F, M are the total number of degrees of freedom and the total mass of the system, respectively. The cilk are combinatorial numbers. The functions , which are products of certain derivatives of the potential energy of the system with respect to the volume, are of the form to be dealt with below. Details in Ref. [100]

If the phase space function can be determined by MD, thermodynamic properties can be calculated through the relation in Table 3.1. [107]

Table 3.1Relations of important thermodynamic state variables in terms of phase-space functions

The general first order volume derivative of the potential energy can be written as [108]

With the assumption of a cubic container holding the system, so the equation above can be rearranged to Eq

where

with pairwise additivity

Extending the above calculation to the general nth order derivative leads to

where the superscript (n) means that the operator (rij -) is applied to U n times.

Expanding Eq 3.11 and using Eq 3.12 one finally has Eq 3.13

The general mie (n-m) potential is used for the site-site interaction.

So one obtains an explicit expression for the volume derivatives of the potential energy as follow:

where (x)k is the Pochhammer symbol. The combinatorial numbers apk and bpk are given in the Ref [101]. The only assumptions involved in equation above are (i) cubic container; (ii) pairwise additivity; and (iii) the multicenter n/m-Mie potential.

As we can see, the consequences of using a general multicenter Mie n-m potential have been explored. More importantly, it is actually possible to generate from MD a terms of the basic phase space functions variety of thermodynamic data for simple real molecular substances over the whole fluid region of the phase diagrams with technical accuracy using mie potentials.

2.2 Calculate thermodynamic properties by MC

By the method of MC, thermodynamic properties can be determined by using the microcanonical ensemble (NVE) with fixed number of particles N, volume V, and total energy E. The procedure of calculating is quite similar with MD method with classical molecular dynamics ensemble (NVEP).

The phase space volume can be determined as follow equation:

the phase space density

and the ensemble average of any dynamical property of the type

To factorize Equations into momentum and configurational parts the very same Laplace transformation technique as outlined in Refs. 104, 105 are used here. The final results are

Where is the total number of degrees of freedom of the system, Г is the gamma function, and the constants C are of no interest here. These results could have been anticipated from Ref. 101 without calculation. The only difference from the NVEP ensemble is that whenever F occurs there, it is replaced by F+3. The excess are the three components of the total linear momentum P.

As shown in Ref. 101 the complete thermodynamics can conveniently be constructed from the so-called basic phase space functions as defined by

The next step will be the identification of the Ωmn's with the ensemble average of a configurational property of the system. Following Ref.101 the result is as same as what we have mentioned in MD method. In addition, relations of important thermodynamic state variables in terms of phase-space functions are the same as MD method, too, see Table 3.1.

A general analytic expression for the volume derivatives of the potential energy has been found as follow:

The coefficients are related to Pochhammer polynomials and are given by

and the combinatorial numbers apk and bpk are also given in Appendix B of Ref 111. This equation is evaluated in the innermost loop of a MC code and is thus time determining.

As we can see, through MC method with the microcanonical ensemble (NVE), thermodynamic properties can be calculated as MD method with the classical molecular dynamics ensemble (NVEP).

### D. Applications of Mie potentials in real fluid.

In the work of Jeffrey J. Potoff and Damien A. Bernard-Brunel [112], mie n - 6 potentials are applied to normal alkanes and perfluorocarbons. Vapor-liquid coexistence curves, vapor pressures, heats of vaporization and critical points are calculated from histogram-reweighting Monte Carlo in the grand canonical ensemble. By varying the repulsive term of potential, potential parameters can be optimized such that saturated liquid densities and vapor pressures are predicted with great accuracy, without the need for alteration of the experimental bond lengths. The vapor-liquid coexistence curves for n-alkanes and n-perfluoroalkane are shown; see 3.9 and 3.10

3.9 Vapor-liquid coexistence curves for n-alkanes. Line represents experimental data [113, 114]; open circles denote the predictions of MC simulations. Stars and filled circles correspond to critical points determined from experiment and simulation, respectively

3.210 Vapor-liquid coexistence curves for n-perfluoroalkane. Line represents experimental data [114]; open circles denote the predictions of MC simulations. Stars and filled circles correspond to critical points determined from experiment and simulation, respectively

As we can see from these two s, calculations of vapor-liquid coexistence curves for n-alkanes and n-perfluoroalkanes show good agreement with experimental data

In the work of Nasrabad [115], the Mie (14-7) potential was used in computer simulations to determine thermodynamic and structural properties of the fluid phases. The GEMC simulation technique was utilized to compute the fluid phase envelope and the NVT-MC simulation method was applied to determine the equation of state and the pair correlation functions. A test of the results of his work for argon fluid phase envelope shows excellent agreement (see 3.11) with empirical data and also shows that the results of Mie 14-7 potential predictions are better than that of the predictions made by the LJ 12-6 potential. Based on the results of this work and those from the literature [116], it is expected that the Mie 14-7 potential predictions for the phase equilibria of simple fluids are better than those from the mie n-6 potentials (n=10~20).

3.11 Fluid phase envelope for argon. The empirical data (Ref. 117) (dash Line) and the simulation results using the M14-7 model (○), the LJ (12, 6) model (Ref. 118) (+), and the ab initio potential (Ref. 119) (◇) are shown. The experimental critical point (Ref. 120) (□), those predicted by the M14-7 potential (X), the LJ (12, 6) model (Ref. 121) (△), and the ab initio potential (Ref. 119) () are also plotted.

### Ⅳ Project proposal

In this project, I will focus studying second-derivative thermodynamics properties by changing with the exponent n of the mie potential. The potential used in this project is the form of mie n-6 potential, and the MD method will be used.

The plan of this project is listed below.

Firstly, how the phase diagram change with the exponent will be studied in details using MD method.

Secondly, I will try to study how the second-derivative thermodynamics properties, like Cp CV and u, changing with the exponent n.

Finally, summarize the results of above works and finish the thesis.

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