Waterlike particles

This relationship was first used in the context of computer calculations by Rahman and Stillinger in their MD simulation of waterlike particles, and it has since been applied in MC calculations for simple dipolar systems.Equation 3.36 is of a form usually associated with spherical samples, and this has led to some discussion of whether, and if so... 

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. 2.5 The conditional average binding energy of particles having a fixed coordination number Kina two-dimensional system of (a) spherical particles for different LJ parameter s (as indicated next to each curve) and (b) of waterlike particles for different hydrogen bond energies shb (Sec. 2.6).

Figure 2.39 shows the radial distribution function or the angular average of the pair correlation function for the waterlike particles (BN2D) calculated by the PY equation with the... 

In this section, we present some of the characteristics of an effective pair potential that may be used for simulating the properties of water. Because of the rather crude and preliminary stage of this subject, it is more appropriate to speak of waterlike particles that are presumed to interact according to some specified pair potential. 

Fig. 6.11. Two waterlike particles in a favorable orientation to form a hydrogen bond.

This function is not yet known for liquid water. Therefore, the features of the function depicted in Fig. 6.17 are purely hypothetical. A similar function for a two-dimensional system of waterlike particles is discussed in Section 6.11. 

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... 

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.

Fig. 6.21. The radial distribution function g R) for the waterlike particles with the parameters listed in (6.126). The value of the HB energy parameter nlkT is indicated next to each curve. The density for all the curves is = 0.6.

Figure 6.21 shows the angle-average pair correlation function for a system of waterlike particles with the following parameters ... 

Fig. 6.23. Various ideal geometries of nearest neighbors for waterlike particles. The separation between particles 1 and 2 is (a) R =...

There is one important conclusion that can be drawn from the study of the pair correlation function for two-dimensional waterlike particles which is relevant to the study of liquid water. If strong directional forces, or bonds, are operative at some selected directions, then the correlation between the positions of two particles is propagated mainly through a chain of bonds, and less by the filling of space —a characteristic feature of the mode of packing of simple fluids. 

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 6.25 shows the radial distribution function for our waterlike particles with different HB energies as in (6.132). The peak at about R = 0.8 is the normal peak one would expect for a system of spherical particles without the HB potential. As we increase the HB energy, the peak at... 

Fig. 6.24. A sample of 36 waterlike particles with parameters given in (6.132). The diameter of the particles is Ow — the HB length is and the first coordination radius is Rc = X. iaw Note...

As the HB energy becomes very large s jkT = —8.0), most of the waterlike particles tend to engage in three hydrogen bonds hence we get a strong peak at about vjkT —24, with small peaks corresponding to particles with two, one, and zero bonds. 

Another feature of the mode of packing of waterlike particles akin to the behavior of liquid water is demonstrated by the joint singlet generalized molecular distribution function, constructed by combining the binding energy and coordination number (Fig. 6.28). The values of K) Av... 

Fig. 6.28. Schematic illustration of the function Xn ciy,K)Av for waterlike particles, with parameters given in (6.132), but with HB energy e IkT = —8.0. Note the difference between this figure and the corresponding figure for spherical particles (Fig. 5.8).

Simulation of the behavior of water by waterlike particles in three dimensions has all the merits discussed in the previous section. In addition, this type of computation, which may be referred to as the ab initio approach to liquid water, is of importance in establishing the most appropriate effective pair potential for water molecules. On the other hand, simulations in the three-dimensional case vastly increase the computer time required to execute a typical computation. In particular, because of the strong attractive forces operating among water molecules, the convergence of the numerical methods is usually slower than in the case of particles with relatively weak attractive forces. This aspect was discussed in the previous section, but it pertains to the three-dimensional case equally well. 

Three well-established techniques in the theory of simple fluids have been adapted recently to the study of waterlike particles in three dimensions. The effective pair potential which has been tried in all of the methods is essentially the one based on the Bjerrum model described in Section 6.4. 

Barker and Watts (1969) published a preliminary report on the computations of energy, heat capacity, and the radial distribution function for waterlike particles. The potential function used for these calculations is similar to the one discussed in Section 6.4 however, instead of a smooth switching function, they used a hard-sphere cutoff at 2 A so that the point charges could not approach each other to zero separation. 

The most successful and complete study of the behavior of a sample of waterlike particles in three dimensions has been carried out by Rahman... 

There is a wealth of other information, on both equilibrium and kinetic properties, of this sample of waterlike particles obtained by the molecular dynamics computations which are not reproduced here. The interested reader is referred to the original articles for further details. 

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 one result obtained by this method. The system is two dimensional, similar to the system described in Section 6.11. Here, we have waterlike particles at a number density of = 0.65 and a simple solute with a number density of qs = 0 05. The other molecular parameters for this particular computation are... 

A. Geiger, H. E. Stanley, Tests of universality of percolation exponents for a three-dimensional continuum system of interacting waterlike particles, Phys. Rev. Lett. 49 (1982) 1895-1898. 

---