Liquids solid-like models

Of course, the effect of excluded volume is opposite and greatly exceeds that shown in Fig. 1.10, which is produced by uncorrelated collective interaction. Unfortunately, neither of them results in sign-alternating behaviour of angular or translational momentum correlation functions. This does not have a simple explanation either in gas-like or solid-like models of liquids. As is clearly seen from MD calculations, even in... 

As the density of a gas increases, free rotation of the molecules is gradually transformed into rotational diffusion of the molecular orientation. After unfreezing , rotational motion in molecular crystals also transforms into rotational diffusion. Although a phenomenological description of rotational diffusion with the Debye theory [1] is universal, the gas-like and solid-like mechanisms are different in essence. In a dense gas the change of molecular orientation results from a sequence of short free rotations interrupted by collisions [2], In contrast, reorientation in solids results from jumps between various directions defined by a crystal structure, and in these orientational sites libration occurs during intervals between jumps. We consider these mechanisms to be competing models of molecular rotation in liquids. The only way to discriminate between them is to compare the theory with experiment, which is mainly spectroscopic. 

A more rigorous free-volume treatment is due to Cohen and Grest (CG) [34,35], according to which the material is comprised of liquid and solid-like cells. The former have free volume, but mobility requires continuity of the local empty space. The temperature dependence of the relaxation times according to the CG model is... 

Semibatch reactors are often used to mn highly exothermic reactions isothermally, to run gas-liquid(-solid) processes isobarically, and to prevent dangerous accumulation of some reactants in the reaction mixture. Contrary to batch of)eration, temperature and pressure in semibatch reactors can be varied independently. The liquid reaction mixture can be considered as ideally mixed, while it is assumed that the introduced gas flows up like a piston (certainly this is not entirely true). Kinetic modelling of semibatch experiments is as difficult as that of batch, non-isotherma experiments. 

It appears like a miracle how aliphatic chains (mainly olefins and paraffins) are formed from a mixture of CO and H2. But miracle means only high complexity of unknown order (Figure 9.1). Problems in FT synthesis research include the visualization of a multistep reaction scheme where adsorbed intermediates are not easily identified. Kinetic constants of the elemental reactions are not directly accessible. Models and assumptions are needed. The steady state develops slowly. The true catalyst is assembled under reaction conditions. Difficulties with product analysis result from the presence of hundreds of compounds (gases, liquids, solids) and from changes of composition with time. 

Complementing the equilibrium measurements will be a series of time resolved studies. Dynamics experiments will measure solvent relaxation rates around chromophores adsorbed to different solid-liquid interfaces. Interfacial solvation dynamics will be compared to their bulk solution limits, and efforts to correlate the polar order found at liquid surfaces with interfacial mobility will be made. Experiments will test existing theories about surface solvation at hydrophobic and hydrophilic boundaries as well as recent models of dielectric friction at interfaces. Of particular interest is whether or not strong dipole-dipole forces at surfaces induce solid-like structure in an adjacent solvent. If so, then these interactions will have profound effects on interpretations of interfacial surface chemistry and relaxation. 

Prior to gelation, two types of radicals with solid- and liquid-like mobility are present they are possibly located in microgels (solid-like mobility) and in monomers (liquid-like mobility). The concentration of free radicals increases continuously, so that the pseudo steady-state assumption cannot be applied to model the reaction kinetics. 

In fact, extremum tendencies expressing the dominant mechanisms in systems like turbulent pipe flow (Li et al, 1999), gas-liquid-solid flow (Liu et al, 2001), granular flow, emulsions, foam drainages, and multiphase micro-/nanoflows also follow similar scenarios of compromising as in gas-solid and gas-liquid systems (Ge et al., 2007), and therefore, stability conditions established on this basis also lead to reasonable descriptions of the meso-scale structures in these systems. We believe that such an EMMS-based methodology accords with the structure of the problems being solved, and hence realize the similarity of the structures between the physical model and the problems. That is the fundamental reason why the EMMS-based multi-scale CFD improves the... 

If a sample shows elastic, solid-like deformation below a certain shear stress ay and starts flowing above this value, ay is called a yield stress value. This phenomenon can occur even in solutions with quite low viscosity. A practical indication for the existence of a yield stress value is the trapping of bubbles in the liquid Small air bubbles that are shaken into the sample do not rise for a long time whereas they climb up to the surface sooner or later in a liquid without yield stress even if their viscosity is much higher. A simple model for the description of a liquid with a yield stress is called Bingham s solid ... 

If you step back and think about it, the mechanical and rheological properties of many solids and liquids can be modeled fairly well by just two simple laws, Hooke s law and Newton s law. Both of these are what we call linear models, the stress is proportional to the strain or rate of strain. If we examine viscoelastic properties like creep, the variation of strain with time appears decidedly non-linear (see Figure 13-75). Nevertheless, it is possible to model this non-linear time dependence by the assumption of a linear relationship between stress and strain. By this we mean that if, for example, we measure the strain as a function of time in a creep experiment, then for a given time period (say 1 hour) the strain measured when the applied stress is 2o would be twice the strain measured when the stress was o. 

Pharmaceutical materials are rarely described by simple mechanical equivalents such as the Kelvin (solid-like behavior) or the Maxwell (liquid-like behavior) model. 

Methanol molecules confined in micropores can form the close packed structure in larger micropores, though they cannot form in narrower micropores because of the misfit space size for formation of the solid-like structure of the high packing density. On the other hand, the close packed structure like bulk liquid can be formed in short distance still in narrower micropores. Ethanol molecules ad.sorbed in carbon micropores can form the solid-like ordered structure in wider micropores but cannot form in narrower micropores. Ethanol molecules should be oriented parallel to the pore walls in wider microporcs. In narrower micropores, ethanol molecules form a specific ordered structure different from bulk solid. The model having a flat orientation for the surface of narrower micropores can support the results of adsorbed density and ERDF of adsorbed ethanol on P5. 

Two different modeling approaches are used for simulated moving bed reactors. The first approach combines the model of several batch columns with the mass balances for the external inlet and outlet streams. By periodically changing the boundary conditions the transient behavior of the process is taken into account. The model is based on the SMB model introduced in Chapter 6 and is, therefore, referred to as the SMBR model. The second approach assumes a true counter-current flow of the solid and the liquid phase like the TMBR. Therefore, this approach is called the TMBR model. 

The first indications that certain systems might violate the phase rule came from computer simulations of small clusters of atoms. A number of studies revealed clearly defined solid-like and liquid-like forms [5-14]. These embraced both molecular dynamics and Monte Carlo simulations, and explored a variety of clusters. These included several based on atomic models with interparticle Lennard-Jones forces, which mimic rare gas clusters rather well. There were also models of alkali halide clusters. Hence, the existence of solid and liquid forms for such small systems seemed not only plausible but general, not restricted to any one kind of system. Shortly after these studies appeared, another, of a 55-atom cluster with Lennard-Jones interparticle forces, showed not only solid and liquid forms but also a form in which the surface of the cluster (with icosahedral structure) is liquid... 

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