Liquid-like pair correlation function

As noted earlier, the diffraction of X-rays, unlike the diffraction of neutrons, is primarily sensitive to the distribution of 00 separations. Although many of the early studies 9> of amorphous solid water included electron or X-ray diffraction measurements, the nature of the samples prepared and the restricted angular range of the measurements reported combine to prevent extraction of detailed structural information. The most complete of the early X-ray studies is by Bon-dot 26>. Only scanty description is given of the conditions of deposition but it appears likely his sample of amorphous solid water had little or no contamination with crystalline ice. He found a liquid-like distribution of 00 separations at 83 K, with the first neighbor peak centered at 2.77 A. If the pair correlation function is decomposed into a superposition of Gaussian peaks, the area of the near neighbor peak is found to correspond to 4.23 molecules, and to have a root mean square width of 0.50 A. 

The liquid-like order present in the pair correlation function manifests itself as a peak in the static structure factor (S(q)). The scaling of the position qm of this maximum with the density has attracted much attention in the literature [40, 51-53]. Scaling arguments suggest [35, 42, 49, 51] that qm obeys the relation qm p1/3 for dilute solutions and qm pv/(3v 1) for semidilute solutions. Here v is the scaling exponent for the end-to-end distance, i.e., RE hT. The overlap threshold concentration is estimated as p N1 3v. As a conse-... 

LDA and HDA were interpreted to be similar to two limiting structural states of supercooled liquid water up to pressures of 0.6 GPa and down to 208 K. In this interpretation, the liquid structure at high pressure is nearly independent of temperature, and it is remarkably similar to the known structure of HDA. At a low pressure, the liquid structure approaches the structure of LDA as temperature decreases [180-182]. The hydrogen bond network in HDA is deformed strongly in a manner analogous to that found in water at high temperatures, whereas the pair correlation function of LDA is closer to that of supercooled water [183], At ambient conditions, water was suggested to be a mixture of HDA-like and LDA-like states in an approximate proportion 2 3 [184-186],... 

Amorphous ice has been studied in some detail by both X-ray and neutron diffraction [738-740]. The O- -O pair-correlation functions are similar to those of liquid water, except that on condensing on very cold surfaces, i.e., 10 K, there is an extra sharp peak at 3.3 A. This indicates some interpenetration of the tetrahedral disordered ice-like short-range structures. It appears that none of the many proposed atom-atom potential energy functions can simulate a structure for liquid water that predicts pair-correlation functions which are a satisfactory fit to the experimental data [741, 742]. Opinions seem to differ as to whether the discrepancy is in the theory or the experiments. 

Figures 2a and 2b show how the predicted solvent-solute pair correlation function, gAB varies with density. At the highest density (p = 0.6) the structure is liquid-like with hrst, second, third,. .. maxima/minima oscillating about 1.0. The size of the solvent-solute cluster for this state was calculated to be about —1 solvent molecule the presence of one solute molecule at this state excludes about one solvent molecule.

At analytic theory based on these ideas, which yields Ti in terms of molecular parameters like masses and bondlengths and also equilibrium pair correlation functions of the liquid solution, has been described elsewhere [19,24]. 

In this section the results given by the various theories are described and compared, insofar as is possible, with MD or MC calculations. Also a qualitative comparison with experimental data for real liquids is made. The computer simulations do not provide as clear an evaluation of the different approximate theories as one would like, since for the reasons discussed in Section III.C, totally convincing estimates of e have not been obtained. Therefore, to get some idea of the accuracy of the different approximations and to illustrate several of the points made in Section III.C, it is useful to begin by examining the pair correlation function. 

In Fig. 2.26 we show the pair correlation function g(R) for the square-well potential (Fig. 2.10b) and for the primitive model (Fig. 2.10d). These two should represent the normal liquid and the water-like liquid. Note that in the case of the normal liquid (Fig. 2.26a) the height of the first peak of (R) increases as we increase the density of the liquid. We have seen this behavior in Sec. 1.4, Fig. 1.28. 

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. 

The formalism described above applies equally to any mixtures of simple liquids. The form of the decay is very similar to that obtained above, except that the prefactor A, and the correlation length are complicated expressions. The important point is the following while the prefactor depends on the species pairs, the correlation length is the same for all species pairs. In other words, all pair correlation functions decay in an identical fashion. We will demonstrate this in a very general case in Section 7.3. Here, we examine the case of isotropic interactions and binary mixtures. If we label the two species as 1 and 2, we have 3 independent interactions M (r) and U22(r) between like species 1 and 2, respectively, and u 2(f) = U2 r) between unlike species. Similarly, there will be three RDFs gu(/), 822(6, and gi2(r ), and three DCFs Cnfr), C22(r), and Ci2(r). All these functions are related to each other through the two same exact equations, the OZ equation written in the Fourier space as... 

The molecular structure of a solution, like that of a pure liquid, can be described in terms of the pair correlation function. We may specify, for example, a function b a( ) that describes the pair correlation for a solute molecule B surrounded by molecules of the solvent A. As in a pure liquid, the amount of order seen in the function depends on the strength and directionality of the binding forces involved. Some qualitative conclusions can often be drawn from the nature... 

We have tried to characterize the structure by any statistic information, and found that the pair correlation function can be extracted layer-by-layer, even for this kind of strongly anisotropic structure of molecules. Figure 4 shows the in-plane pair correlation function for the contacting layer, which demonstrates that the structure at the higher temperature is rather liquid-like with random nature. On the other hand a decrease in teinierature down to 185 K brings almost perfect hexagonal order, which is typically demonstrated by the first minimum reaching down to zero, and by the sprit of the second peak. 

Ordering of molecules in a monolayer phase can be characterized by the in-plane pair correlation functions go-oir) (Fig. 87). Upon heating, the tetrahedral arrangement of water molecules, represented by the maximum of go-oir) at about 4.5 A, strongly diminishes both in the bulk liquid water (right panel in Fig. 87) and in the surface water layer (left panel in Fig. 87). However, this is not the case for the maximum of go-oir) at about 5.5 A, which reflects chain-like ordering of... 

There have been several liquid-solid interface simulations on the LJ system. These are reviewed in some detail in Ref. 3. Of these, by far the most extensive are those of Broughton and Gilmer. These studies of the structure and thermodynamics of fee [100], [110] and [111] LJ crystal-liquid interfaces were part of a six-part series on the bulk and surface properties of the LJ system. Like most of the earlier simulations, these were done under triple-point conditions. The numbers of particles for the [111], [100] and [110] simulations were 1790, 1598 and 1674, respectively. Analysis of diffusion profiles, various layer-dependent trajectory plots, pair correlation functions, nearest-neighbor fractions and angular correlations yield a width of about three atomic diameters for all three interfaces. The density profiles indicate an interface width that is larger... 

This strategy is common in atomic fluid theory at low to moderate densities, and for Coulombic systems, and corresponds to the reference idea ubiquitous in liquid state theory [5], The purely hard core problem is treated using the accurate [27] PY closure. (2) The construction of a closure approximation for the tail part of the potential is subject to the constraint of exactly describing the weak coupling limit. In physical terms, for fractal-like interpenetrating molecules these indirect processes may strongly couple the direct correlation functions associated with those pairs of sites which are in simultaneous contact The number of such two-molecule pair contacts. Me. scales with N as [23] ... 

The relation between collective and self-motion in simple monoatomic liquids was theoretically deduced by de Gennes [233] applying the second sum rule to a simple diffusive process. Phenomenological approaches like those proposed by Vineyard [ 194] and Skbld [234] also relate pair and single particle motions and may be applied to non-exponential functions. The first clearly fails to describe the PIB results since it considers the same time dependence for both correlators. Taking into account the stretched exponential forms for Spair(Q.t) (Eq. 4.21) and Sseif(Q>0 (Eq 4.9), the Skold approximation ... 

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