Water-like particles

J. P. Valleau and A. A. Gardner,/. Chem. Phys., 86, 4162 (1987). Water-like Particles at Surfaces. I. The Uncharged, Unpolarized Surface. 

Recently, Yamamoto and Hyodo have employed the DPD method for studying Nafion membranes [20]. The systems considered in this study were built using two distinct molecular species, denoted comb-shaped polymer ip) and water (w). The polymer was presented as a branched sequence of beads. It consisted of a main chain (backbone) of iV = 20 effective monomer units (-CF2CF2CF2CF2 ) linked with rig = 5 short side chains of = 2 units [-0CF2C(CF3)F0 and F2CF2S03H] the total number of interaction sites in the macromolecule was Np= N/, + n xn = 30. A water-like particle was modeled as the same size as the units of the Nafion fragment (<7 = 6.1 A) and represented four water molecules. The x parameters were found using an atomistic calculation. The DPD simulation was performed for water volume... 

One stringent test of this principle was to apply it to a onedimensional model of water-like particles and see if one can obtain the anomalous properties of water while bypassing the concept of structure. Indeed, the model described in Sec. 2.S utilizes nothing but the principle. To my astonishment, it has worked very well in one- and two-dimensional systems. 

Chapters 3 and 4). In Sec. 2.6, we discuss a 2-D model of water-like particles first constructed in 1971 and 1972, and later extended to the study of aqueous solutions of both ionic and non-ionic solutes. 

There is one very important reason for performing simulations of water-like particles. This is when we study some problems of aqueous solutions that are not accessible to experiments. Examples are the hydrophobic or hydrophilic interactions discussed in Chapter 4, or the understanding of the role of water in biological systems. 

Thus, although the model pair potential might not be realistic, it can still be useful in application to complex systems. Sometimes, the water-like particles are non-realistic from the outset (such as 1-D or 2-D models see Secs. 2.4 and 2.5). Nevertheless, the study of such systems can teach us what features of the molecular model are essential for the manifestation of some macroscopic properties of the system. From such studies, one can draw similar conclusions about the features of the models in 3-D that account for the specific and outstanding macroscopic properties of real liquid water. 

Fig. 2.4 The distribution of the combined properties of coordination number (CN) and binding energy BE) (a) for spherical particles and (b) for water-like particles in two dimensions (Sec. 2.6). The number densities are shown next to each curve.

Fig. 2.8 The correlation between local density and binding energy in a onedimensional model for (a) normal fluid and (b) water-like particles.

Consider a system consisting of N water-like particles denoted w, in one dimension (Fig. 2.9a). We take only nearest-neighbor interactions between pairs of consecutive particles into account. The potential function is described in Fig. 2.10d and is defined as... 

Fig. 2.9 (a) The primitive model and (b) the primitive cluster model for water-like particles in one dimension. In the latter, each sequence of bonded molecules is viewed as a different component. ... 

Fig. 2.15 The isotope effect on the pressure dependence of the volume of the water-like particles near the phase transition. ...

HBing are very close to each other, and therefore the g R) does not resolve into two peaks. We shall see similar phenomena for water-like particles in Sec. 2.6. 

The physical model of water-like particles in two dimensions... 

Fig. 2.36 A sample of water-like 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. See Sec. 2,6.2.

Fig. 2. 39 The spatial pair correlation function for water-like particles in two dimensions. (The hydrogen bond energies she are indicated near each curve.)...

In contrast to the spherical, or the normal case, we find that for the water-like particles the height of the first peak decreases as we increase the density up to about p = 0.5, then starts to increase (Fig. 2.40a). Also, the second peak is at about R = VSRh- This corresponds to the distance of the second nearest neighbors of HBod particles (see Fig. 2.41b). This is the analog of the distance R = 4.5A in liquid water (Fig. 1.29). The angular dependence of the RDF was also calculated. Since the RDF was calculated only for five values of the orientation angles a = 0°, 30°, 60°, 90°, and 120°, we can write the RDF... 

There is one important conclusion that can be drawn from the study of the pair correlation function for 2-D water-like 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 two 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. 

Fig. 2.42 The spatial pair correlation function for a system of water-like particles, with parameters given in (2.6.26). The HB energies shb/ bT are indicated next to each curve. The locations of the various maxima are indicated on the abscissa.

As the HB energy becomes very large (suB/kBT = —8.0), most of the water-like particles tend to engage in three hydrogen bonds hence, we get a strong peak at about v/k T —24, with small peaks corresponding to particles with two, one, and zero bonds. 

Fig. 2.46 (a) A linear hydrogen bond formed by two water-like particles... 

The three terms in (2.7.6) correspond to the short-range, long-range and intermediate-range interactions between the water-like particles. 

As I have discussed in Sec. 2.7.1, a model that produces better agreement between calculated and experimental results does not necessarily mean that it is closer to describing real water molecules. All one can say is that a system of water-like particles show the characteristic behavior of real liquid water and nothing more. 

The first simulation of liquid water was reported by Barker and Watts (1969). The potential function they used was that used earlier by Rowlinson (1951). The Monte Carlo method was applied to 64 water-like particles at 25°C. 

This method was successfully applied for simple liquids. It was applied to liquid water only after the development of the BNS pair potential. Rahman and Stillinger carried out an extensive MD simulation of water-like particles based on the BNS potential.They applied the molecular dynamics method for a system of 216 particles interacting via the effective pair... 

In this section, we extend the application of the primitive cluster model discussed in Sec. 2.5.4 to examine the solvation thermodynamics of simple solutes in the water-like solvent. The model is schematically described in Fig. 3.21b. The only new feature that is added here compared with the model discussed in Sec. 2.5.4 is that clusters of water-like particles contain holes in which solute molecules can be accommodated. This feature is the analog of the cavities formed by the network of hydrogen-bonded molecules in real water. 

Silverstein etal. (1998,1999) performed a Monte Carlo simulation of a system of pure water as well as for aqueous solutions of L/ particles. They examined the angular distribution of a water-like particle near the solute. As expected, the water molecules straddle the solute to avoid the loss of hydrogen bonds. A very detailed study of the solvation of simple solutes in the 2-D model was undertaken by Silverstein et al. (1999). A more recent investigation of this model was carried out by Southall and Dill (2000,2002), Southall etal, (2002), and more recently by Urbic et al. (2002, 2003, 2004, 2007). The application of a 2-D model of water was extended to solvation of polymers. 

The first study of the PMF of two solutes in water-like particles by simulation methods was published by Geiger et al. (1979). A molecular dynamic simulation of 216 water-like particles and two neon -like solutes was carried out. Initially, the two solutes were placed at contact distance. During the evolution of the system, the two solute particles separated to a configuration of two cages, which is essentially a configuration such that the structure of water is not disrupted by the inclusion of the solute particles. The orientation of the water molecules around the solute was found to be such that no hydrogen bonding capability was lost. This is similar to the situation described in Fig. 4.41. 

An extension of the molecular dynamics simulation to study aggregation of H0O solutes in water was carried out by Wallqvist (1991a,b). Rappaport and Scheraga (1982) found no evidence of aggregration of non-polar solutes in water-like solvent. Recently, Paschek (2004) carried out extensive molecular dynamics simulations of aqueous solutions using various models for water-like particles. 

Pangali et al. (1979a,b) calculated the PMF between two LJ solute particles (with ctss = 4.12 A and Sss/k = 170.1 K) in water-like particles by Monte Carlo simulation. They found a... 

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