Waterlike particles in two dimensions

The study of waterlike particles in two dimensions has two important merits. First, it can be viewed as a prelude to the study of the more difficult three-dimensional cases. Second, the study of nonsimple particles by the available statistical mechanical methods is interesting in its own right. For 

A system of nonspherical particles in two dimensions is, from the computational point of view, an intermediate case between spherical particles (in either two or three dimensions) and nonspherical particles in three dimensions. The pair potential in this case depends on three coordinates (see below), compared with six in the three-dimensional case. Some very useful information on the numerical procedure, on the problem of convergence, and so forth, can thus be gained in a system which is relatively simpler than the three-dimensional case. We shall also present some results on the generalized molecular distribution function, which thus far are available only in two dimensions, yet are of relevance to the case of real liquid water. 

The particles comprising our system are basically spherical disks. The intermolecular potential function operating between two particles is a superposition of two functions a Lennard-Jones type, depending on the distance only, and a hydrogen-bond-like potential, which depends on the distance as well as on the relative orientation of the pair of particles. The construction of the latter potential is, with some simplifications, similar to the construction of the potential function given in (6.16). 

The above qualitative description of the bonding between the particles is now translated into more precise language. Let Xi, Ti be the Cartesian coordinates of the center of the ith particle. Let (A = 1, 2, 3) be the kth unit vector of the ith particle. The angle between the unit vectors is In 13. The orientation of the ith particle is given by the angle between the vector ii and the positive direction of the x axis. This coordinate system is shown in Fig. 6.19. We denote by the vector Xj = Tj, the full configuration 

As a measure of the structure of the system, we may use a quantity defined in a similar manner to the quantity A hb in Section 6.10. First, we define the set of configurations 

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A. Ben-Naim, Statistical mechanics of waterlike particles in two dimensions. I. Physical model and application of the Percus-Yevick equation. J. Chem. Phys., 54 (1971), 3682-95. 

Fig. 6.18. A sample of waterlike particles in two dimensions. The circles indicate the Lennard-Jones diameter of the particles. The arrows attached to each particle are unit vectors along which a hydrogen bond may be formed. Particles 1 and 2 are considered to be bonded. Particles 2 and 10 are correctly oriented but too far to form a bond. Particles 8 and 9 are almost at the correct configuration for bond formation. Particles 3, 4, and 5 are connected successively by hydrogen bonds.

We present here an example of complementary information on the system of waterlike particles in two dimensions, obtained by the standard Monte Carlo method. The model is the same as above, but we focus our attention mainly on the singlet generalized molecular distribution functions (Chapter 5). Figure 6.24 shows a sample of 36 waterlike particles. The molecular parameters chosen for this particular illustration are... 

Figure 7.10 depicts a possible form of the function x iv) for water and a simple fluid. (While the details of the form of these functions are hypothetical the drawings have been influenced by the knowledge of the form of this function for spherical and waterlike particles in two dimensions, as presented in Chapter 6.) In a simple fluid, the distribution Xb v) is expected to be concentrated under one narrow peak. Hence, choice of a cutoff point at, say, will produce two dissimilar components. However, one of the components will have a very low concentration, thus fulfilling the first of the above conditions. On the other hand, choice of a cutoff point at, say, will produce two components with almost equal concentrations. These will be quite similar, hence fulfilling the second condition. In both cases, we must end up with a small stabilization effect. 

We present here a few results calculated for a system of waterlike (BN2D) particles in two dimensions, obtained by the Monte Carlo method. The molecular parameters chosen for this particular illustration are... 

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