Static equilibrium properties

The ability to generate an equilibrium ensemble from a dynamical trajectory has a number of useful features. One can obtain not only ordinary static equilibrium properties from Eq. [25], but also dynamical information. In fact, dynamical information is available on two levels. On the one hand, equilibrium time correlation functions can be calculated, leading to the prediction of vibrational spectra, transport coefficients and so on. On the other, the trajectory allows access to the microscopic detailed motion of individual atoms. Therefore, one can, in a sense, visualize at an atomistic level the dynamical behavior of the system as a function of time, which can lead to valuable insights about chemical reaction mechanics, structural rearrangements, and other details of the system that can be captured only by visualization at this level of detail. 

X HE CONFIGURATIONAL PROPERTIES OF WATER-SOLUBLE POLYMERS and their interactions with other polymers and with colloidal particles are relevant to understanding the intermolecular interactions in associative thickeners and steric stabilization of colloidal particles. We used fluorescence spectroscopy to study both the dynamic and the static equilibrium properties... 

The work described in this chapter is concerned with static (equilibrium) properties of polymers in random media. There is a lot of theoretical work stiU to be done related to the dynamics, and especially nonequilibrium properties of polymers in random media. This is also of practical importance, for example for the separation of chain of different length or mass like DNA molecules under the effect of an applied force when embedded in a random medium like a gel [38]. 

The minor chain (MC) model of reptating chains as shown in Figure 1 was proposed by Kim and Wool to analyze interdiffusion in polymer melts. Only those parts of the chains which have escaped by reptation from their initial tubes (the minor chains) at the time of contact can contribute to interdiffusion. Using this model, the average molecular properties of the interface were derived and are summarized in Table 1. The molecular properties have a common scaling law which relates the dynamic properties, H t)y to the static equilibrium properties, //, via the reduced time, t/T, by t ... 

Thermal phase transitions show that dynamic properties like transport coefficients or relatmtion times may have di ient exponents for different materials and models even if the static equilibrium properties have the same exponents. Thus the static univeisality classes are split into smaUer dynamic universality groups. Conversely, certain etqxinent ratios like y/v or filv may remain constant even ifp,y, and v are a hmetion of a parameter . Nothing seems to be known yet about whether or not gelation and percolation exhibit simitar effects. 

Given the character of the water-water interaction, particularly its strength, directionality and saturability, it is tempting to formulate a lattice model, or a cell model, of the liquid. In such models, local structure is the most important of the factors determining equilibrium properties. This structure appears when the molecular motion is defined relative to the vertices of a virtual lattice that spans the volume occupied by the liquid. In general, the translational motion of a molecule is either suppressed completely (static lattice model), or confined to the interior of a small region defined by repulsive interactions with surrounding molecules (cell model). Clearly, the nature of these models is such that they describe best those properties which are structure determined, and describe poorly those properties which, in some sense, depend on the breakdown of positional and orientational correlations between molecules. 

In Fig. 3.16 dynamic structure factor data from a A =36 kg/mol PE melt are displayed as a function of the Rouse variable VWt (Eq. 3.25) [4]. In Fig. 3.6 the scaled data followed a common master curve but here they spht into different branches which come close together only at small values of the scahng variable. This splitting is a consequence of the existing dynamic length scale, which invalidates the Rouse scaling properties. We note that this length is of purely dynamic character and cannot be observed in static equilibrium experiments. 

We will introduce basic kinetic concepts that are frequently used and illustrate them with pertinent examples. One of those concepts is the idea of dynamic equilibrium, as opposed to static (mechanical) equilibrium. Dynamic equilibrium at a phase boundary, for example, means that equal fluxes of particles are continuously crossing the boundary in both directions so that the (macroscopic) net flux is always zero. This concept enables us to understand the non-equilibrium state of a system as a monotonic deviation from the equilibrium state. Driven by the deviations from equilibrium of certain functions of state, a change in time for such a system can then be understood as the return to equilibrium. We can select these functions of state according to the imposed constraints. If the deviations from equilibrium are sufficiently small, the result falls within a linear theory of process rates. As long as the kinetic coefficients can be explained in terms of the dynamic equilibrium properties, the reaction rates are directly proportional to the deviations. The thermodynamic equilibrium state is chosen as the reference state in which the driving forces X, vanish, but not the random thermal motions of structure elements i. Therefore, systems which we wish to study kinetically must first be understood at equilibrium, where the SE fluxes vanish individually both in the interior of all phases and across phase boundaries. This concept will be worked out in Section 4.2.1 after fluxes of matter, charge, etc. have been introduced through the formalism of irreversible thermodynamics. 

All the preceding particulate handling steps are affected by the unique properties of all particulates, including polymeric particulates while they may behave in a fluidlike fashion when they are dry, fluidized and above 100 pm, they also exhibit solidlike behavior, because of the solid-solid interparticle and particle-vessel friction coefficients. The simplest and most common example of the hermaphroditic solid/ fluidlike nature of particulates is the pouring of particulates out of a container (fluidlike behavior) onto a flat surface, whereupon they assume a stable-mount, solidlike behavior, shown in Fig. 4.2. This particulate mount supports shear stresses without flowing and, thus by definition, it is a solid. The angle of repose, shown below, reflects the static equilibrium between unconfined loose particulates. 

A molecule contains a nuclear distribution and an electronic distribution there is nothing else in a molecule. The nuclear arrangement is fully reflected in the electronic density distribution, consequently, the electronic density and its changes are sufficient to derive all information on all molecular properties. Molecular bodies are the fuzzy bodies of electronic charge density distributions consequently, the shape and shape changes of these fuzzy bodies potentially describe all molecular properties. Modern computational methods of quantum chemistry provide practical means to describe molecular electron distributions, and sufficiently accurate quantum chemical representations of the fuzzy molecular bodies are of importance for many reasons. A detailed analysis and understanding of "static" molecular properties such as "equilibrium" structure, and the more important dynamic properties such as vibrations, conformational changes and chemical reactions are hardly possible without a description of the molecule itself that implies a description of molecular bodies. 

Attention should be drawn to the fact that there has been a degree of inconsistency in the treatments of ionic clouds (Chapter 3) and the elementary theory of ionic drift (Section 4.4.2). When the ion atmosphere was described, the central ion was considered—from a time-averaged point of view—at rest. To the extent that one seeks to interpret the equilibrium properties of electrolytic solutions, this picture of a static central ion is quite reasonable. This is because in the absence of a spatially directed field acting on the ions, the only ionic motion to be considered is random walk, the characteristic of which is that the mean distance traveled by an ion (not the mean square distance see Section 4.2.5) is zero. The central ion can therefore be considered to remain where it is, i.e., to be at rest. 

We shall consider, in detail, studies of MgO lattice properties using the potential-induced breathing model and three complementary band-theoretical studies of electron distribution. Early studies using the modified electron-gas method (Cohen and Gordon, 1976) gave reasonably good agreement with experiment for equilibrium static lattice properties and... 

Phase equilibrium is a dynamic process that is quite different from the static equilibrium achieved as a marble rolls to a stop after being spun into a bowl. In the equilibrium between liquid water and water vapor, the partial pressure levels off, not because evaporation and condensation stop, but because at equilibrium their rates become the same. The properties of a system at equilibrium are independent of the direction from which equilibrium is approached, a conclusion that can be drawn by observing the behavior of the liquid-vapor system. If we inject enough water vapor into the empty flask so that initially the pressure of the vapor is above the vapor pressure of liquid water, PvaplHiO)) then liquid water will condense until the same equilibrium vapor pressure is achieved (0.03126 atm at 25°C). Of course, if we do not use enough water vapor to exceed a pressure of 0.03126 atm, all the water will remain in the vapor phase and two-phase equilibrium will not be reached. 

When a solid is subjected to a shearing force, the solid (simultaneously with the application of force) deforms, and internal stresses develop until a condition of static equilibrium is reached. Within the elastic limit of a substance, these internal stresses are proportional to the induced shearing strains (deformation). The ability of a material to reach static equilibrium, rather than deform continuously, is due to a property called shear strength. 

Since the equilibrium properties of the medium are involved in determining these effects they are called static medium effects. However, there is another type of relevant phenomenon which is related to the dynamic properties of a condensed phase. Since the movement of molecules with respect to one another is required for the reaction to take place, the local viscosity of the system can also influence the rate of reaction. This property is also related to local intermolecular forces. Effects which depend on local viscosity have also been studied experimentally and are known as dynamic medium effects. 

By virtue of its yield stress, a viscoplastic material in an unsheared state will support an immersed particle for an indefinite period of time. In recent years, this property has been successfiilly exploited in the design of slurry pipelines, as briefly discussed in section 4.3. Before undertaking an examination of the drag force on a spherical particle in a viscoplastic medium, the question of static equilibrium will be discussed and a criterion will be developed to delineate the conditions under which a sphere will either settle or be held stationary in a liquid exhibiting a yield stress. 

The nuclear relaxation (NR) contributions were computed using a finite field approach [73,74]. In this approach one first optimizes the geometry in the presence of a static electric field, maintaining the Eckart conditions. The difference in the static electric properties induced by the field can then be expanded as a power series in the field. Each coefficient in this series is the sum of a static electronic (hyper) polarizability at the equilibrium geometry and a nuclear relaxation term. The terms evaluated in Ref. [61] were the change of the dipole moment up to the third power of the field, and that of the linear polarizability up to the first power ... 

At the end of this subsection it should be noted, that having a set of parameters Bpq attached to the crystallographic coordinate system one gets the opportunity (1) to describe the magnetic properties of a crystal at any appropriate value and direction of applied magnetic fields (2) to obtain reasonable estimates of different effects due to electrons phonon interaction with only those model parameters which have been introduced to describe energy spectra in the static equilibrium lattice. 

Both the Anderson and the Kondo (or Coqblin-Schrieffer) model have been solved exactly for thermodynamic properties such as the 4f-electron valence, specific heat, static magnetic and charge susceptibilities, and the magnetization as a function of temperature and magnetic field B by means of the Bethe ansatz (see Schlottmann 1989, and references therein). This method also allows one to calculate the zero-temperature resistivity as a function of B. Non-equilibrium properties, such as the finite temperature resistivity, thermopower, heat conductivity or dynamic susceptibility, could be calculated in a self-consistent approximation (the non-crossing approximation), which works well and is based on an /N expansion where N is the degeneracy of the 4f level. 

The fundamentals ofj 2 he DF method are discussed in detail elsewhere in this volume the present lecture notes start where those of R. M. Martin ended the method provides us with 1) energy of the unit cell 2) forces on atoms and 3) stress over a unit volume. Only those details of the method that are specific to our present applications are summarized in Section 2. The successive steps leading to dynamical properties - static equilibrium, frozen phonon method - are then explained in Sections 3. and 4 the topic of frozen phonons is treated only briefly in these notes, because an adequate text already exists detailed material completing Section 4 is to be found in Ref. 13. 

The critical temperatures of most metals can only be estimated and most lie well above those of mercury and the alkali metals (7. > 3000 K). Conventional measurements under static, equilibrium conditions are nearly impossible at such extreme temperatures and pressures. Transient methods such as shock waves, exploding wires, and laser heating have been developed to study at least a few properties of the high critical-point metals. These include the equation of state and the velocity of sound. But these techniques are less accurate than static measurements and it has not been possible to obtain measurements very close to the critical point, or indeed, even to determine with any reasonable level of precision the location of the critical points of most of the metallic elements. 

Recent important progresses have been made in this direction, for the theory of static and equilibrium properties of polyelectrolytes by Barbosa and Levin. The interested reader will find all required details which are out the scope of this chapter in the original papers. [2] [3] [4] [5] [6]. 

Extensions of the QC Method Because of its versatility, the QC method has been widely applied and, naturally, extended as well. While its original formulation was for zero-temperature static problems only, several groups have modified it to allow for finite-temperature investigations of equilibrium properties as well. A detailed discussion of some of these methodologies is presented in the discussion of finite-temperature methods below. Also, Dupoy et al. have extended it to include a finite-temperature alternative to molecular dynamics (see below). Lastly, the quasi-continuum method has also been coupled to a DFT description of the system in the OFDFT-QC (orbital-free DFT-QC) methodology discussed below. 

Many studies point to the significance of conformational factors in influencing solubility, partitioning, and even bioavailability of flexible com-pounds. 0 54,55,158-160 Because the above-described log P calculation methods are based on a static view of molecular structure, they will fail to model adequately dynamic equilibrium properties such as lipophilicity and biodistribution. There are at least two distinct ways to approach the lipophilic behavior of flexible compounds ... 

Among all possible dynamical and kinetic applications of the cluster densities analysis, it should be noticed that cluster approach can also be fruitful for the study of static and equilibrium properties of electrolyte solutions. 

---