What Kinds of Molecular-Microsimulation Methods are Useful for Colloidal Dispersions ?

A. Satoh    Faculty of System Science and Technology, Akita Prefectural University, Japan

A colloidal dispersion is a suspension of fine particles (dispersoid) which are stably dispersed in a base liquid (dispersion medium). The dimensions of the dispersed particles are generally within the range of 1 μm to 10 pm. If the particles are smaller than about 1 nm, the dispersion behaves like a true solution, and at the other end of the scale the particles tend to sink due to the force of gravity. Hence a dispersion of particles outside this range is not regarded as being in the colloidal state. The main objective of the present book is to discuss molecular-microsimulation methods for colloidal dispersions, whereas general matters concerning colloid science should be referred to in other textbooks [1–3]. In this book, we restrict our attention to solid particles as colloidal ones which are not distorted in shape in a flow field.

One possible technique for generating functional or intelligent materials is to systematically combine known materials to make a new material with the desired functional properties. Such functional or intelligent materials are expected to exhibit highly attractive properties under certain circumstances. This method of generating new materials is similar to the concept of developing micromachines in the mechanical engineering field. Such intelligent materials may be called “artificially-controlled materials.” According to this concept, new functional colloids such as ferrofluids, ER fluids, and MR suspensions have been generated in the colloid physics-engineering field. These functional fluids are generated by making functional particles stably dispersed in a base liquid. They behave as if the liquid itself had functional properties in an external field such as an applied magnetic or electric field. Another example of expanding the concept of the synthesis of functional colloids in the materials field of the surface quality change leads to the possibility of developing higher quality magnetic recording materials (tapes).

In order to investigate a physical phenomenon theoretically, the governing or basic equations are first constructed, and then solved analytically or numerically. Where possible, the theoretical or simulation results are compared with experimental observations in order to deepen our understanding of the physical phenomena. When we construct governing equations for a physical phenomenon, we don’t usually need the information of the microstructure of molecules or atoms which constitute the system, rather we generally stand on a larger scale. For example, let us consider the flow problem of water around a circular cylinder under the condition of an ordinary pressure. The physical phenomenon in this case is governed by the Navier-Stokes equation which is well known as a basic equation for flow problems in the fluid mechanics field. In the derivation of the Navier-Stokes equation, we never need the detailed information concerning a water molecule which is composed of one oxygen and two hydrogen atoms. Rather, water is regarded as a continuum, and the Navier-Stokes equation is derived on such a continuum scale.

How does the molecular simulation or microsimulation deal with physical phenomena? In molecular-microsimulations, we stand on the microscopic level of the constituents of a system, such as atoms, molecules or sometimes ultrafine particles, and follow the motion of the constituents to evaluate microscopic or macroscopic physical quantities. Hence, in investigating the above-mentioned flow problem around a cylinder, the equations for the translational and rotational motion of water molecules have to be solved, and the desired macroscopic quantities are evaluated from an averaging procedure over the corresponding microscopic ones, which may be a function of molecular positions and velocities. In this example, the flow field is evaluated by averaging the velocity of each molecule and illustrates the fact that we cannot simulate a physical phenomenon unless the molecular structure and interaction energies between molecules are known. It is, therefore, clear that molecular- microsimulation methods are completely different from numerical analysis methods such as finite-difference or finite-element methods. The former methods stand on the microscopic level of molecules or fine particles, and the latter methods on the macroscopic level of a continuum, in which the governing equations are discretized and these algebraic equations are solved numerically.

Molecular-microsimulations for colloidal dispersions may be classified mainly into four groups based on the simulation method. Monte Carlo methods are highly useful for thermodynamic equilibrium in a quiescent flow field. If the particle Brownian motion is negligible and a dispersion is dilute, molecular dynamics methods are applicable. If the particle Brownian motion can be neglected and a dispersion is not dilute, we have to take into account hydrodynamic interactions among particles. In this case, Stokesian dynamics methods have to be used instead of molecular dynamics methods. Finally, if colloidal particles are not sufficiently smaller than micron-order and the particle Brownian motion has to be taken into account, then Brownian dynamics methods are indispensable. In the following sections we just survey the essence of each simulation method.

1.1 Monte Carlo Methods

The Monte Carlo method is named after the famous gambling town, Monte Carlo, which is the Principality of Monaco. In this method, a series of microscopic states is generated successively through the judgment of whether or not a new state is acceptable using random numbers. This procedure is similar to a gambling game where success or failure is determined by dice. Monte Carlo methods are used for simulating physical phenomena for a system in thermodynamic equilibrium, and are therefore based on the theory of equilibrium statistical mechanics.

We now consider a canonical ensemble of the particle number N, system volume V, and temperature T. For this statistical ensemble, the probability density function, that is, the canonical distribution p(r) is written as

r=1Qexp−1kTUr,

in which the abbreviation r is used for r1, r2, …, rN for simplicity. With this distribution, the statistical average of a certain quality A is expressed as

=∫Arρrdr,

in which k is Boltzmann’s constant, dr = dx1dy1dz1··· dxNdyNdzN , Q is the configuration integral, and U is the total interaction energy of a system. The expressions for Q and U are written as

=∫exp−1kTUrdr,

=∑i=1N∑j=1Nuijj>i

In Eq. (1.4), uij is the interaction energy between particles i and j, and the contribution of three-body interactions to the system energy has been neglected.

Since the configuration integral Q is 3 N-fold for a three-dimensional system, analytical evaluation is generally impracticable and numerical integration methods such as the Simpson method are also quite unsuitable for multi-fold integrals. However, if it is taken into account that the integrand is an exponential function, we can expect that certain particle configurations make a further more important contribution to the integral than others. Thus, if such important microscopic states can be sampled with priority in a numerical procedure, then it may be possible to evaluate the ensemble average, Eq. (1.2), in a reasonable time. This is exactly the concept of importance sampling. It is the Monte Carlo method that generates microscopic states successively with the concept of important sampling. As will be shown in Sec. 3.3, Metropolis’ transition probability is almost universally used for the transition of one microscopic state to the next. With this transition probability, the probability of realizing a microscopic state is determined by the canonical distribution for the case of the canonical ensemble. Hence Eq. (1.2) can be simplified as

≈∑n=1MArn/M

in which M is the total sampling number, and rn is the n-th sampled microscopic state.

On the other hand, the quantity ¯, which is measured in the laboratory, is the time average of A and expressed as

¯=t→∞lim1t−t0∫t0tAτdτ.

The relationship between the statistical ensemble and the time average ¯ will be discussed later in Sec. 2.1.

1.2 Molecular Dynamics Methods

In this book, we consider molecular dynamics methods from the viewpoint of microsimulation for colloidal dispersions. Thus general matters of molecular dynamics methods should be referred to...