Empty lattice sites

Let us consider a simple self-avoiding walk (SAW) on a lattice. The net interaction of solvent-solvent, chain-solvent and chain-chain is summarized in the excluded volume between the monomers. The empty lattice sites then represent the solvent. In order to fulfill the excluded volume requirement each lattice site can be occupied only once. Since this is the only requirement, each available conformation of an A-step walk has the same probability. If we fix the first step, then each new step is taken with probability q— 1), where q is the coordination number of the lattice ( = 4 for a square lattice, = 6 for a simple cubic lattice, etc.). 

Fig. 6.93. A typical equilibrium configuration for a model of bisulfate adsorption on Rh(111), generated by Monte Carlo methods in the ordered (-

With the introduction of the lattice structure and electroneutrality condition, one has to define two elementary SE units which do not refer to chemical species. These elementary units are l) the empty lattice site (vacancy) and 2) the elementary electrical charge. Both are definite (statistical) entities of their own in the lattice reference system and have to be taken into account in constructing the partition function of the crystal. Structure elements do not exist outside the crystal and thus do not have real chemical potentials. For example, vacancies do not possess a vapor pressure. Nevertheless, vacancies and other SE s of a crystal can, in principle, be seen , for example, as color centers through spectroscopic observations or otherwise. The electrical charges can be detected by electrical conductivity. 

However, in order to perform intracrystalline chemical reactions that affect the components and not just the SE s, empty lattice sites (vacancies) constituting a complete... 

Consider now the fluctuations of the order parameter in the system possessing the chemical reaction this problem could be perfectly illustrated by computer simulations on lattices. We start with the bimolecular A + B -y 0 reaction discussed above, and first of all froze particle diffusion. Let the recombination event happen instantly when a pair AB of dissimilar particles occupies the nearest lattice sites (assume lattice to be squared). Immobile particles enter into reaction as a result of their creation with the equal probabilities in empty lattice sites from time to time a newly created particle A(B) finds itself nearby pre-created B(A) and they recombine. (Since this recombination event is instant, the creation rate is of no importance.) This model describes, in particular, Frenkel defect accumulation in solids under... 

Within the frameworks of the lattice gas model it is reasonable to classify the elementary processes by the number of sites m, which a given process occurs on, i.e., one- and two-site cases. In the first case the changing parameter is the occupancy state of one site. The processes such as these include isomerization associated with changes in the internal degrees of freedom of the adspecies (ZA- ZB, i.e., transition of the adspecies from state A to state B), adsorption-desorption of the atoms and nondissociating molecules (A + Z- ZA), reaction according to the collision mechanism (A + ZB ->ZD + C, Eley-Rideal s-type mechanism). It should be remembered that ZA, Z and A denote adspecies A, empty lattice site and species A in the gaseous phase, respectively. 

How many ways are there to put the first molecule on the lattice That s right, nQ. The first molecule can go in any one of the empty lattice sites (Figure 11-5). (Again, there is a nice animation of this process on our Polymer Science CD that you might wish to view.)... 

The analysis depends on evaluation of the number Vj of situations (i.e., appropriate sequences of empty lattice sites) accessible to molecule j with orientation yj when j — 1 molecules have been added previously to the space, or lattice, in which the mixture is confined. This quantity can be formulated as the product of the number of lattice sites available to the initial segment of the chain and the probabilities that the sites required for each successive segment of the cham are unoccupied and, hence, accessible. The number of eligible locations for the initial segment of the chain is just the total number Hq — x(j — 1) of vacant sites, where no is the total number of lattice sites. In formulating the probability that the site required by the second seg-... 

There are two types of lattice defects that occur in all real crystals and at very high concentration in irradiated crystals. These are known as point defects and line defects. Point defects occur as the result of displacements of atoms from their normal lattice sites. The displaced atoms usually occupy sites that are not in the lattice framework they are then known as interstitials. The empty lattice site left behind by the interstitial is called a vacancy. Avacancy produced by displacement of an anion or cation, along with its interstitial ion, is called a Frenkel pair, or simply a... 

V can be varied by varying the number of empty lattice sites in the system. It is easy to show (9-10,19,21-23,27) that the equations that minimize the configurational Helmholtz free energy Ac with respect to the order variables at constant V and T also minimize the configurational Gibbs free energy Gc with respect to the order variables at constant P and T. The most stable state is the state of lowest free energy or lowest chemical potential at constant P and... 

The lattice fluid equation-of-state theory for polymers, polymer solutions, and polymer mixtures is a useful tool which can provide information on equa-tion-of-state properties, and also allows prediction of surface tension of polymers, phase stability of polymer blends, etc. [17-20]. The theory uses empty lattice sites to account for free volume, and therefore one may treat volume changes upon mixing, which are not possible in the Flory-Huggins theory. As a result, lower critical solution temperature (LCST) behaviors can, in principle, be described in polymer systems which interact chiefly through dispersion forces [17]. The equation-of-state theory involves characteristic parameters, p, v, and T, which have to be determined from experimental data. The least-squares fitting of density data as a function of temperature and pressure yields a set of parameters which best represent the data over the temperature and pressure ranges considered [21]. The method,however,requires tedious experiments to deter-... 

Conduction can occur by ions either moving to an empty lattice site or moving to an interstitial site. An interstitial site is a gap in the lattice... 

Neutral compounds the interactions between the layer are primarily of the van der Waals type, and the interlayer space is an array of empty lattice sites. 

Suppose there were Nx molecules and N lattice sites. For any distribution of the A, molecules among the N lattice sites there will be Nx lattice sites with molecules, and N2 = N - Nx empty lattice sites. Thus we can consider the lattice gas to be a mixture of Nx molecules and N2 holes,... 

For Si, there are three types of native defects the vacancy, the interstitial, and the interstitialcy. The vacancy, V, is an empty lattice site. Depending on the configuration of the unsatisfied bonds due to the missing atom, a vacancy in Si can be either neutral, negatively or positively charged. A vacancy is also referred to as a Schottky defect. A Si atom residing in the interstices of the Si lattice is defined as a self-interstitial. A Frenkel pair is a vacancy-interstitial pair formed when an atom is displaced from a lattice site to an interstitial site. An interstitialcy... 

Materials for cathodes and anodes (insertion electrodes) for rechargeable lithium batteries are called intercalation compounds and constimte a special class of electroactive material [122]. The intercalation refers to the reversible insertion of mobile guest species into a crystalline host lattice, which contains an interconnected system of empty lattice sites of appropriate size, while the structural integrity of the host lattice is formally conserved [122]. 

Figure 6 Diagram illustrating kth step of the construction of a square king lattice of L X L spins with the stochastic models (SM) method solid circles denote lattice sites already filled with spins ( 1) in preceding steps of the process open circles denote the still empty lattice sites. The linear nature of the buildup construction is achieved by using spiral boundary conditions (i.e., the first spin in a row interacts with the last spin of the preceding row). Whereas all the L uncovered spins (at sites k - L,k — L + 1,..., k - ) determine the transition probability for selecting spin k, the spins in close proximity to k k - 1, k - L, etc.) have the largest effect. The local states method is based on the SM construction. Thus, the transition probabilities for spin k are obtained from a Metropolis Monte Carlo sample by calculating the number of occurrences of the various local states, (a, a) = n k-v k-2 k-L k-L v k-L 2 k-L 3 l- transition probability is jS(cT d ) = (cr, ff)/[ (a = 1,ct) -I- n(a = These transition...

Electrons and holes can be produced by thermal motion or the absorption of light. Excitons are electron/hole pairs. Excitons are produced when an electron takes up energy, but not a sufficient amount to escape the hole produced. Consequently, the electronic charge of an exciton is zero. The exciton can transport energy but cannot conduct an electric current. Empty lattice sites are called site defects or vacancies. Atoms residing in sites between lattice ponts are called interstitial defects. 

The interactions are described by contact energies and the electrostatic energy. For contact energies we used the common matrix of interaction parameters in which the reference interactions ( i.e., those where solvent is involved) are zero. The optimized parameters in units of kT are s-s = 0, pma-s = 0, c-s = 0, fipMA-PMA = -0.27, pMA-c = 0.8, and c-c = 0.8, where S, C and PMA stand for solvent (i.e., empty lattice site, occupied implicitly by solvent), C core (lattice point at the surface of tlie core or an attached hydrophobic pendant group), and the PMA bead (i.e., the Kuhn lattice segment, irrespectively of the ionization). 

For polymer systems, a lot of EOS have been developed (e.g., Flory-Orwoll-Vrij EOS [10], Sanchez-Lacombe EOS (11), Panayiotou-Vera EOS [12], lattice gas EOS [13], group-contribution lattice fluid EOS [14]). In all these equations, the pressure is introduced via empty lattice sites allowing the compressibility of the lattice. 

Kirkendall Effect The Kirkendall effect is a phenomenon observed frequently in solid materials [38]. It refers to a vacancy counter diffusion process through an interface of two solid materials, metals in particular, to compensate the unequal material flow formation at the interface [38a]. In metals and metallic alloys, the vacancy is atomic defect, that is, empty lattice site. Combination of excess vacancies can lead to the formation of void within the fast-diffusion side of the interface [39]. While this phenomenon has been known for a very long time, synthesis of hollow nanostructures based on Kirkendall effect was realized fairly recently [40]. Ym studied the time evolution in the formation of hollow nanospheres and found that Kirkendall diffusion followed the Tick s law [41]. This means that the diffusion of atoms and vacancies is driven by the difference in atom concentration. Wu et al. synthesized hollow nanostructures of CoCuPt alloy catalyst by using Co nanoparticles as the sacrificial templates. For this trimetallic system, Co atoms diffused faster than those of Pt or Cu to form core-shell like Co CuPt hollow nanoparticles and then the CoCuPt hollow spheres (Fig. 2.10) [42]. 

The term intercalation is used by chemists to describe the insertion of mobile guest species (atoms, molecules, or ions) into a crystalline host lattice containing an interconnected system of empty lattice sites ( ) of appropriate size, according to the general equation (see Reference 1)... 

Frenkle defects result when an atom moves away from a lattice site and assumes a position between lattice sites (i.e., takes an interstitial position), as shown in Fig. 2.17(b). The Frenkle defect, therefore, corresponds to a pair of defects, namely, the empty lattice site and the extra interstitiaUy positioned atom. The activation... 

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