Lattice Coordination Number Research Articles

Polymer Plasticization Using Supercritical Carbon Dioxide: Experiment and Modeling

The most important effect of sorption of compressed gases and supercritical fluids into glassy polymers is the reduction of the glass transition temperature (Tg). This plasticization effect causes changes in mechanical and thermophysical properties of the polymers. In this work, a thermodynamic study based on experimental and theoretical results is addressed. New data were carried out for poly(2,6-dimethylphenylene oxide) (PPO), poly(acrylic acid) (PAA), and the copolymer vinylpyrrolidone−vinyl acetate [P(VP−VA)] using an inverse gas chromatographic technique. To model the Tg behavior of diluent−polymer systems, a model that couples the lattice-fluid equation of state and the Gibbs−DiMarzio criterion was used. A parametric study of the influence of the physical properties (lattice coordination number, molecular weight, and binary interaction parameter) on the Tg behavior is presented. The thermodynamic model fairly describes the experimental data measured in this work and gives a phenomenological representation of the retrograde vitrification for the systems PPO−CO2, PVP−CO2, and P(VP−VA)−CO2.

Underpotential Deposition of Metals from Thermodynamic and Structural Considerations. 2. Nonaqueous Solvents

The underpotential deposition (UPD) shifts of different metallic couples in acetonitrile and propylene carbonate are estimated using work function differences, lattice coordination numbers, and ads...

Fractal structure in adsorbates in a catalysed reaction on a surface

The influence of interaction and lattice coordination number on the reactivity and structure of adsorbed particles on a surface has been analysed for a reaction of a monomer and a dimer that dissociates, both of which are adsorbed on a surface and react. For the case without interaction there is a narrow range for a control parameter in which reaction is effective. Depending on the interaction, this range can grow very large, spanning almost all the range of the control parameter or, on the contrary, the reaction can disappear. Aggregates of adsorbed particles with fractal structure have been observed. Fractal dimension has been determined for different values of a control parameter for square and triangular lattices.

Underpotential Deposition of Metals: Structural and Thermodynamic Considerations

The dependence of underpotential deposition (UPD) shift on work function differences, lattice coordination numbers, solvent desorption energies, and surface coverages is analyzed. The transport processes that govern monolayer formation and bulk deposition are incorporated. The parameteric dependence of the UPD shift on different thermodynamic quantities and adsorbate charge densities is reported. The validity of the formalism is demonstrated by comparison with the experimental data.

The Reaction �A2 + BA ? A2B on Square and Triangular Lattices

We performed Monte Carlo simulations to study catalytic surface reaction of the type ½A2 + BA → A2B. We chose two different types of catalytic surfaces in our studies: one represented by a square lattice (coordination number four) and the other represented by a triangular lattice (coordination number six). The whole process can be described by a single parameter, which is taken as the relative adsorption of BA molecules. In our model, we always need two nearest neighbor sites to the adsorption of both molecules. After deposition, the A2 molecules dissociate occupying two nearest neighbor sites, while the BA molecules stay bound. The reaction occurs only when a free A atom is nearest neighbor of a B atom of the BA molecule. In spite of each reaction freeing three empty sites on the lattice, this model does not exhibit any reactive window for both lattices. Only when we introduce processes like the diffusion of BA molecules or free A atoms, or desorption of BA molecules, the catalytic surface can become active for some values of deposition rate yBA. As to be expected, the reactivity over the triangular lattice is higher than that observed on the square lattice.

The modified quasi-chemical model: Part II. Multicomponent solutions

Further improvements to the modified quasi-chemical model in the pair approximation for shortrange ordering (SRO) in liquids are extended to multicomponent solutions. The energy of pair formation may be expanded in terms of the pair fractions or in terms of the component fractions, and coordination numbers are permitted to vary with composition. The model permits complete freedom of choice to treat any ternary subsystem with a symmetric or an asymmetric model. An improved general functional form for “ternary terms” in the excess Gibbs energy expression is proposed. These terms are related to the effect of a third component upon the binary pair interaction energies. It is shown how binary subsystems that have been optimized with the quasi-chemical model can be combined in the same multicomponent Gibbs energy equation with binary subsystems that have been optimized with a random-mixing Bragg-Williams model and a polynomial expression for the excess Gibbs energy. This is of much practical importance in the development of large databases for multicomponent solutions. The model also applies to SRO in solid solutions as a special case, when the number of lattice sites and coordination numbers are constant.

The modified quasi-chemical model: Part III. Two sublattices

The modified quasi-chemical model in the pair approximation for short-range ordering (SRO) in liquids is extended to solutions with two sublattices. Short-range ordering of nearest-neighbor pairs is treated, and the effect of second-nearest-neighbor (SNN) interactions upon this ordering is taken into account. The model also applies to solid solutions, if the number of lattice sites and coordination numbers are held constant. It may be combined with the compound-energy formalism to treat a wide variety of solution types. A significant computational simplification is achieved by formally treating the nearest-neighbor pairs as the “components” of the solution. The model is applied to an evaluation/optimization of the phase diagram of the Li,Na,K/F,Cl,SO4 system.

Spanning trees on graphs and lattices inddimensions

The problem of enumerating spanning trees on graphs and lattices is considered. We obtain bounds on the number of spanning trees NST and establish inequalities relating the numbers of spanning trees of different graphs or lattices. A general formulation is presented for the enumeration of spanning trees on lattices in d⩾2 dimensions, and is applied to the hypercubic, body-centred cubic, face-centred cubic and specific planar lattices including the kagomé, diced, 4-8-8 (bathroom-tile), Union Jack and 3-12-12 lattices. This leads to closed-form expressions for NST for these lattices of finite sizes. We prove a theorem concerning the classes of graphs and lattices with the property that NST~exp (nz) as the number of vertices n→∞, where z is a finite non-zero constant. This includes the bulk limit of lattices in any spatial dimension, and also sections of lattices whose lengths in some dimensions go to infinity while others are finite. We evaluate z exactly for the lattices we consider, and discuss the dependence of z on d and the lattice coordination number. We also establish a relation connecting z to the free energy of the critical Ising model for planar lattices.

Semiflexible polymer on an anisotropic Bethe lattice

The mean-square end-to-end distance of an N-step polymer on a Bethe lattice is calculated. We consider semiflexible polymers placed on isotropic and anisotropic lattices. The distance on the Cayley tree is defined by embedding the tree on a sufficiently high-dimensional Euclidean space, considering that every bend of the polymer defines a direction orthogonal to all the previous ones. In the isotropic case, the result obtained for the mean-square end-to-end distance turns out to be identical to the one obtained for ideal chains without immediate returns on an hypercubic lattice with the same coordination number of the Bethe lattice. For the general case, we obtain asymptotic behavior in both the semiflexible and almost rigid limits.

Electron theory related to mechanical properties of condensed phases

In this article, some recent work is surveyed on mechanical properties of both metals and semiconducting silicon. In quantum chemical language, such properties as (1) cleavage forces and (2) tribological properties involve bond stretching, and here it is argued that electron correlation must play an important role. In the context of long-range dispersion forces, this is already evident in first-principles work of Lifshitz, which, particularly in simple metals, leads to an asymptotic form of cleavage force F(z), with z the interplanar separation beyond equilibrium spacing on cleavage, falling off as z{sup {minus}3}, the magnitude of this term being expressible in terms of the electron plasma frequency. Here, it is argued, from quantum chemistry, that in covalently binded systems, such as semiconducting Si, the elasticity of a covalent bond in the crystal is quite comparable with that of the H{sub 2} molecule bonding in free space. The transition from delocalized molecular-orbital theory to Heitler-London or valence bond regimes comes about as the bonds are stretched to breaking point. This situation with covalent bonds is compared and contrasted with that of a largely nondirectionally bonded metal such as Cu, in which there is a closed d shell, and a more complicated body-centered-cubicmore » Fe, in which the charge density can be demonstrated experimentally, through overlapping reflections in X-ray diffraction, to have marked directionally. Even for simple sp metals, it is shown that some of the main features of the heavier alkalis, and in particular K, Rb, and Cs, can be explained by viewing the dimer, say K{sub 2}, as the basic building block in representing the energy of the crystal of K as a function of lattice spacing and local coordination number. Finally, a brief discussion of liquid alkalis is given, mechanical properties now embracing viscosities.« less

Monolayer adsorption of nonrandom mixtures

A model for monomers on a lattice is presented based on local density calculations that were first proposed by Ono and Kondo in 1947 and recently generalized by Aranovich, Donohue, and co-workers. The model allows one to describe the adsorption behavior of molecules at a surface (or interface), and the phase behavior of adsorbed molecules, as well as of molecules in the bulk on the basis of short-range ordering in two and three dimensions. While there are prior lattice theories that predict nonrandom behavior for arbitrary lattice coordination numbers, the derivation of adsorption models from these theories is usually based on ideal fluid behavior in the bulk. However, the new adsorption model presented here is consistent in that molecular behavior in the bulk as well as in the adsorbed surface layer is based on identical assumptions. This is accomplished by calculating the total free energy of the system; the corresponding adsorption model follows through minimization of the free energy. This procedure is also used for deriving a new adsorption equation based on the quasi-chemical approximation to the Ising problem. Results from this equation are very similar to those obtained from the equation based on Ono–Kondo theory. When compared with lattice Monte Carlo computer simulations, the new adsorption models based on nonrandom mixing consistently show better agreement than those based on random behavior. For simplicity, the discussion of results is restricted to single-component systems. However, the new adsorption model based on Ono–Kondo theory is applicable to systems of arbitrary numbers of components without introducing any further assumptions.

The ferrimagnetic mixed spin [formula omitted] and spin 1 Ising system

A cluster variational theory, within pair approximation, of the mixed spin 1 2 –spin 1 ferrimagnet with single-ion anisotropy, is presented. General expressions for the Curie temperature and the tricritical point are obtained. The first-order transitions are located from a free energy analysis. The thermal dependence of the magnetization is studied, and the existence and dependence of a compensation point on single-ion anisotropy strength is investigated for different lattice coordination numbers. Results of the theory are compared with those of other approximate methods, and with the exact results that are available in the case of the honeycomb lattice.

Environmental Effects on Anion Polarizability: Variation with Lattice Parameter and Coordination Number

The results of ab initio calculations of in-crystal ionic polarizabilities, α, over a wide range of lattice parameters, R, are presented for LiF, NaF, KF, LiCl, NaCl, KCl, LiBr, NaBr, KCl, and MgO. The derivatives of the mean polarizability with respect to lattice parameters are compared with experimental values obtained from the variation of the refractive index with pressure. The environmental effects on the polarizability of an anion may be viewed as the consequence of imposition of a confining potential on its electron density whose origin includes Coulombic interactions and the exclusion of these electrons from the region occupied by the electron density of the first-neighbor shell of cations. This model suggests scaling relationships, between values of α(R) obtained at different levels of calculation and for a given anion with different cations, which are shown to be semiquantitative. These findings lead to the proposal of a universal representation of the polarizability of a given anion, which predicts the dependence on lattice parameter and crystal form and transfers from one substance to another.

Phase Equilibria of Lattice Polymers from Histogram Reweighting Monte Carlo Simulations

Histogram-reweighting Monte Carlo simulations were used to obtain polymer / solvent phase diagrams for lattice homopolymers of chain lengths up to r=1000 monomers. The simulation technique was based on performing a series of grand canonical Monte Carlo calculations for a small number of state points and combining the results to obtain the phase behavior of a system over a range of temperatures and densities. Critical parameters were determined from mixed-field finite-size scaling concepts by matching the order parameter distribution near the critical point to the distribution for the three-dimensional Ising universality class. Calculations for the simple cubic lattice (coordination number z=6) and for a high coordination number version of the same lattice (z=26) were performed for chain lengths significantly longer than in previous simulation studies. The critical temperature was found to scale with chain length following the Flory-Huggins functional form. For the z=6 lattice, the extrapolated infinite chain length critical temperature is 3.70+-0.01, in excellent agreement with previous calculations of the temperature at which the osmotic second virial coefficient is zero and the mean end-to-end distance proportional to the number of bonds. This confirms that the three alternative definitions of the Theta-temperature are equivalent in the limit of long chains. The critical volume fraction scales with chain length with an exponent equal to 0.38+-0.01, in agreement with experimental data but in disagreement with polymer solution theories. The width of the coexistence curve prefactor was tentatively found to scale with chain length with an exponent of 0.20+-0.03 for z = 6 and 0.22+-0.03 for z = 26. These values are near the lower range of values obtained from experimental data.

Lower bounds and series for the ground-state entropy of the Potts antiferromagnet on Archimedean lattices and their duals

We prove a general rigorous lower bound for $W(\ensuremath{\Lambda},q)=\mathrm{exp}[{S}_{0}(\ensuremath{\Lambda}{,q)/k}_{B}]$, the exponent of the ground-state entropy of the $q$-state Potts antiferromagnet, on an arbitrary Archimedean lattice $\ensuremath{\Lambda}$. We calculate large-$q$ series expansions for the exact ${W}_{r}(\ensuremath{\Lambda}{,q)=q}^{\ensuremath{-}1}W(\ensuremath{\Lambda},q)$ and compare these with our lower bounds on this function on the various Archimedean lattices. It is shown that the lower bounds coincide with a number of terms in the large-$q$ expansions and hence serve not just as bounds but also as very good approximations to the respective exact functions ${W}_{r}(\ensuremath{\Lambda},q)$ for large $q$ on the various lattices $\ensuremath{\Lambda}$. Plots of ${W}_{r}(\ensuremath{\Lambda},q)$ are given and the general dependence on lattice coordination number is noted. Lower bounds and series are also presented for the duals of Archimedean lattices. As part of the study, the chromatic number is determined for all Archimedean lattices and their duals. Finally, we report calculations of chromatic zeros for several lattices; these provide further support for our earlier conjecture that a sufficient condition for ${W}_{r}(\ensuremath{\Lambda},q)$ to be analytic at $1/q=0$ is that $\ensuremath{\Lambda}$ is a regular lattice.

Application of graph theory to the statistical thermodynamics of lattice polymers. I. Elements of theory and test for dimers

The lattice cluster theory (LCT) is extended to enable inclusion of longer range correlation contributions to the partition function of lattice model polymers in the athermal limit. A diagrammatic technique represents the expansion of the partition function in powers of the inverse lattice coordination number. Graph theory is applied to sort, classify, and evaluate the numerous diagrams appearing in higher orders. New general theorems are proven that provide a significant reduction in the computational labor required to evaluate the contributions from higher order correlations. The new algorithm efficiently generates the correction to the Flory mean field approximation from as many as eight sterically interacting bonds. While the new results contain the essential ingredients for treating a system of flexible chains with arbitrary lengths and concentrations, the complexity of our new algorithm motivates us to test the theory here for the simplest case of a system of lattice dimers by comparison to the dimer packing entropies from the work of Gaunt. This comparison demonstrates that the eight bond LCT is exact through order φ5 for dimers in one through three dimensions, where φ is the volume fraction of dimers. A subsequent work will use the contracted diagrams, derived and tested here, to treat the packing entropy for a system of flexible N-mers at a volume fraction of φ on hypercubic lattices.

An effective field theory for dilute anisotropic Heisenberg ferromagnets

Dilute anisotropic Heisenberg ferromagnets have been studied by an effective field theory based on a two-site cluster approximation with an Ising character assumed for the surrounding spins. The results for the Curie temperature, T C , and the critical concentration for magnetic ordering, are improvements on those of a single-site approximation where the exchange anisotropy cannot be accounted for. The initial reduction in T C with dilution is confirmed to decrease with increases in both the lattice coordination number and exchange anisotropy. Finally, by utilising its connection with the zero-temperature magnetization for the Ising system, the percolation probability is examined.

Flory and Guggenheim lattice statistics reinterpreted and extended to include molecular clustering and chemical dissociation

A new method is presented for counting the number of distinguishable arrangements available to a mixture of molecules in a lattice model. Chain molecules are used and these are allowed to undergo molecular clustering and chemical dissociation. The resulting equation for the configurational degeneracy is shown to reduce to the equations of Flory and Guggenheim in the appropriate limits. The Guggenheim equation is shown to be exact in one dimension; extension to higher dimensions requires allowing the lattice coordination number to increase beyond 2. The Flory equation is shown to be the low volume fraction approximation of the Guggenheim equation. The counting method developed can be used to model entropy effects associated with chemical reactions. The equation of state derived using the new configurational degeneracy term is shown to reduce to a number of well known equations. Comparison to van der Waals equation of state shows that the improved ability of the new equation to model observed behavior, results from a redefinition of the van der Waals free volume.

Monte Carlo simulations of chain molecules in confined environments

The conformational and thermodynamic properties of lattice chain molecules in confined environments have been studied by Monte Carlo simulations. In the case of homopolymers with purely repulsive interactions with the walls the scaling laws proposed by Daoud and deGennes for the free energy of confinement, from three dimensions to one dimension and for two dimensions to one dimension, have been confirmed. The number of self-avoiding walks (SAW) of a chain with n segments for each film thickness d has been found to follow an effective relationship of the form NSAW(d)∝zeffn (d)nγ (d)−1, where zeff (d) is a thickness dependent effective lattice coordination number and γ (d) is the d dependent enhancement exponent. In the case of triblock copolymers in which the segments of the end blocks have attractions with the walls of the confined media it was found that for strong enough attractive interactions there is a minimum in the effective interactions between parallel walls. The distance at which the interaction potential has its minimum corresponds to the bulk end-to-end distance of the middle (nonattractive) block. This turns out to also be the distance at which the chains have the maximum number of bridging configurations between the two surfaces.

Self-avoiding walks including next-nearest-neighbor steps.

A self-avoiding-walk series on lattices of high effective coordination number in two dimensions are obtained by allowing next-nearest-neighbor steps in addition to nearest-neighbor steps. The dependence of the connective constant of self-avoiding walks on the lattice coordination number is studied by means of standard methods of series analysis.