Interstitial molecules

The model consists of dividing the fluid into N cells with index i e [1,. . . , N], each with volume Vq, and occupation variable rij = 0 (for a cell with gas-like density) or = 1 (for a cell with liquid-like density). Each cell is assumed in contact with 4 nearest neighbor (n.n.) cells, mimicking the first shell of liquid water, in the simplified assumption of no interstitial molecules. 

The coefficient of thermal expansion can be determined in two different ways. The first, and most direct, is simply to take a single crystal of ice and measure its dimensions as a function of temperature. From the symmetry of the crystal the results can be expressed in terms of two linear expansion coefficients a = l dljdT) in directions parallel to the r-axis and to an a-axis respectively. Alternatively X-ray diffraction methods can be used to measure the c and a dimensions of the unit cell as a function of temperature. These two methods do not, in fact, measure exactly the same thing, since in a real crystal there will be an equilibrium concentration of vacancies and interstitial molecules, and these concentrations will change with temperature. As we shall see, however, the experimental results for ice are not sufficiently accurate to enable a meaningful distinction to be drawn. 

Because of the open, four-coordinated structure of ordinary ice, the energy involved in the creation of an interstitial molecule is not extreme. This is doubly clear from the existence of Ice VII, in which the energy penalty for the creation of a whole network of interstitials in a self-clathrate structure is more than balanced by the energy gained from the decrease in volume once the pressure exceeds 22 kbar (see chapter 3), though the phase transition I VII... 

X traps are localised at a particular point in the lattice, without extending over larger distances in one or more dimensions. They thus belong to the class of point defects. Other point defects can be caused by missing molecules in the lattice, i.e. lattice vacancies or so-called Schottky defects, or by interstitial molecules, i.e. Frenkel defects, or finally by improperly oriented molecules on regular lattice sites. Point defects can usually be observed only indirectly, through the effects they have on other physical properties of the crystal. 

Frenkel defects in general do not play an important role in molecular crystals. The asymmetric shape of the molecules and the steeply increasing repulsive potential between molecules at short intermolecular distances make the occurrence of interstitial molecules in molecular crystals thermodynamically improbable. The... 

The majority of the fluid that is filtered from the microcirculation into the interstitial space is carried out of the tissue via the lymphatic network. This unidirectional transport system originates with a set of blind channels in distal regions of the microcirculation. It carries a variety of interstitial molecules, proteins, metabolites, colloids, and even cells along channels deeply embedded in the tissue parenchyma toward a set of sequential lymph nodes and eventually back into the venous system via the right and left thoracic ducts. The lymphatics are the pathways for immune surveillance by the lymphocytes and thus, they are one of the important pathways of the immune system [Wei et al, 2003]. 

Rull (2002) recently provided Raman spectroscopic evidence supporting the mixture model, a major fraction consisting of domains with linear HBs in a tetrahedral like configuration, the other of interstitial molecules, with either bifurcated or else weak or no HBs. Soper (2010) commented on the two-state model that the different domains must be very short lived, in view of the rapid diffusion of the water molecules, one of them moving over 150 molecular diameters away in 1 ms. 

The remaining Nh = Nu, — Ny molecules are assumed to occupy the holes and may be referred to as interstitial molecules.A schematic illustration of such a model in two dimensions is shown in Fig. 2.6. 

Clearly, the condition Nh < NqNl must be satisfied. Alternatively, the mole fraction of interstitial molecules is restricted to vary between the limiting values ... 

For instance, in the model depicted in Fig. 2.6, No = j, i.e. each lattice particle belongs to three holes, but each hole is built up by six particles, so that each particle contributes half a hole. Thus, the mole fraction of interstitial molecules cannot exceed j. 

Exercise E.2.7 Explore numerically the temperature and the pressure dependence of the molar volume and the heat capacity in this model. Choose values of El < 0 and Eh < 0 and Vl > 0 as the molecular parameters. Follow the mole fractions xl, xh as a function of T and F. Suppose that El — Eh < 0, i.e. the energy per lattice molecule is lower than the interstitial molecule. In this case, we always find that xp decreases with increase in temperature. Explain why. 

To keep the complexity of the model at a minimum, we require the following simplifying assumptions (1) All the holes have the same structure. (2) A hole can accommodate at most one water molecule in such a way so as not to distort the lattice structure to a significant extent. (3) The interstitial molecules do not see each other, i.e., there is no direct interaction between interstitial molecules in adjacent holes. Hence, occupancy of a certain hole does not affect the chances of an adjacent hole being empty or filled. (4) The lattice molecules are assumed to hold the equilibrium lattice points, and vibrational excitation is negligible. (5) An interstitial molecule is assumed to be situated in a fixed position in the hole. 

Frank and Quist (1961) considered two kinds of holes in the lattice, only one of which was occupied by interstitial molecules. 

The first interstitial molecule can be put in any of the NqNi, holes, the second in any of NqNl — 1 holes, etc. The total number of such arrangements is (NqNlXNqNl — 1) (NqNl — iVn + 1). However, because of the equivalence of the molecules, we must divide this number by to obtain the number of distinguishable arrangements, given in (6.29). 

Let us now consider some general features of the present model. At the very outset, we emphasize that the basic variables of our system are r, P, and i.e., we have a one-component system, and there is no trace of a mixture-model approach. The variables and in (6.32) play the role of convenient intermediary variables. Once we have carried out the summation in (6.32), the dependence of the partition function on disappears. Nevertheless, the nature of the model suggests a new way of looking at this one-component system namely, we decide to refer to a lattice molecule as an L-cule and to an interstitial molecule as an LT-cule (the letters L and H in the present context are chosen to remind us of water in lattice and water in holes. The same notation will be used in Section 6.8 to indicate the type of environment, that of low local density and high local density, respectively.) Once we have made this classifica-... 

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