Volume, excluded hydrodynamic

A porous particle contains many interior voids known as open or closed pores. A pore is characterized as open when it is connected to the exterior surface of the particle, whereas a pore is closed (or blind) when it is inaccessible from the surface. So, a fluid flowing around a particle can see an open pore, but not a closed one. There are several densities used in the literature and therefore one has to know which density is being referred to (Table 3.15). True density may be defined as the mass of a powder or particle divided by its volume excluding all pores and voids. True density is also referred to as absolute density or crystalline density in the case of pure compounds. However, this density is very difficult to be determined and can be calculated only through X-ray or neutron diffraction analysis of single-crystal samples. Particle density is defined as the mass of a particle divided by its hydrodynamic volume. The hydrodynamic volume includes the volume of all the open and closed pores. Practically, the hydrodynamic volume is identified with the volume included by the outer surface of the particle. The particle density is also called apparent or envelope density. The term skeletal density is also used. The skeletal density of a porous particle is higher than the particle one, since it is the mass of the particle divided by the volume of solid material making up the particle. In this volume, the closed pores volume is included. The interrelationship between these two types of density is as follows (ASTM, 1994 BSI, 1991) ... 

As a contribution to the study of these problems, stochastic models are here developed for two cases a freely-jointed chain in any number of dimensions, and a one-dimensional chain with nearest-neighbor correlations. Our work has been directly inspired by two different sources the Monte Carlo studies by Verdier23,24 of the dynamics of chains confined to simple cubic lattices, and the analytical treatment by Glauber25 of the dynamics of linear Ising models. No attempt is made in this work to introduce the effects of excluded volume or hydrodynamic interactions. 

Leonov AI (1994) On a self-consistent molecular modelling of linear relaxation phenomena in polymer melts and concentrated solutions. J Rheol 38( 1) 1—11 Liu B, Diinweg B (2003) Translational diffusion of polymer chains with excluded volume and hydrodynamic interactions by Brownian dynamics simulation. J Chem Phys 118(17) 8061-8072... 

This proportionality makes sense in both limits. In the melt (= 1), both excluded volume and hydrodynamic interactions are fully screened to the level of individual monomers, so b. At the overlap concentration... 

On length scales larger than the screening length the dynamics are many-chain-like, with both excluded volume and hydrodynamic interactions... 

Hydrodynamic interactions of the solute with the gel were not considered by Ogston. In general, the diffusion of probe molecules through a gel can be described as depending primarily on steric factors, i.e., on the volume excluded by the fiber matrix of the gel ... 

An effective medium approach, which uses hydraulic permeabilities to define the resistance of the fiber network to diffusion, has been used to estimate reduced diffusion coefficients in gels [77]. For a particle diffusing within a fiber matrix, the rate of particle diffusion is influenced by steric effects (due to the volume excluded by the fibers in the gel, which is inaccessible to the diffusing particle) and hydrodynamic effects (due to increased hydrodynamic drag on the diffusing particle caused by the presence of fibers). Recently, it was proposed that these two effects are multiplicative [78], so that the diffusion coefficient observed for particles in a fiber mesh can be predicted from ... 

Rouse Model. In 1953 P. E. Rouse proposed a simple model to describe the dynamics of a polymer chain in dilute solution (21,59). The model considers the chain as a sequence of Brownian particles which are connected by harmonic springs. Being immersed in a (structureless) solvent the chain experiences a random force by the incessant collisions with the (infinitesimally small) solvent particles. The random force is assumed to act on each monomer separately and to create a monomeric friction coefficient. The model therefore contains chain connectivity, a local firiction and a local random force. All non-local interactions between monomers distant along the backbone of the chain, such as excluded-volume or hydrodynamic in-... 

First, the fundamental forces between neutral colloidal spheres are the same as the fundamental forces between neutral polymers. Polymers and colloids are equally subject to excluded-volume forces, hydrodynamic forces, van der Waals interactions, and to the random thermal forces that drive Browiuan motion. 

In 2-butanone, an intermediate behavior is observed. There is a pronounced decrease in Ie/Im at low c, followed by a plateau region at intermediate c. At high polystyrene concentrations Ie/1m decreases once more, with values superimposing on those in toluene at similar polystyrene concentrations. One has rather detailed insights into the interplay of excluded volume and hydrodynamic interactions on internal chain dynamics, and the screening effects of added unlabelled polymer. 

Equation (23) predicts a dependence of xR on M2. Experimentally, it was found that the relaxation time for flexible polymer chains in dilute solutions obeys a different scaling law, i.e. t M3/2. The Rouse model does not consider excluded volume effects or polymer-solvent interactions, it assumes a Gaussian behavior for the chain conformation even when distorted by the flow. Its domain of validity is therefore limited to modest deformations under 0-conditions. The weakest point, however, was neglecting hydrodynamic interaction which will now be discussed. 

Arai, T, Sawatari, N., Yoshizaki, T., Einaga, Y. and Yamakawa, H. (1996) Excluded-volume effects on the hydrodynamic radius of atactic and isotactic oligo- and poly(methylmethacrylate)s in dilute solution. Macromolecules, 29, 2309-2314. 

The range of semi-dilute network solutions is characterised by (1) polymer-polymer interactions which lead to a coil shrinkage (2) each blob acts as individual unit with both hydrodynamic and excluded volume effects and (3) for blobs in the same chain all interactions are screened out (the word blob denotes the portion of chain between two entanglements points). In this concentration range the flow characteristics and therefore also the relaxation time behaviour are not solely governed by the molar mass of the sample and its concentration, but also by the thermodynamic quality of the solvent. This leads to a shift factor, hm°d, is a function of the molar mass, concentration and solvent power. 

However, this is true only for good solvent conditions, where (c) is also the correlation length, beyond which both the excluded volume and the hydrodynamic interaction are screened and self-entanglements (intramolecular... 

Under -conditions the situation is more complex. On one side the excluded volume interactions are canceled and E,(c) is only related to the screening length of the hydrodynamic interactions. In addition, there is a finite probability for the occurrence of self-entanglements which are separated by the average distance E,i(c) = ( (c)/)1/2. As a consequence the single chain dynamics as typical for dilute -conditions will be restricted to length scales r < (c) [155,156],... 

It is generally accepted that in semi-dilute solutions under good solvent conditions both the excluded volume interactions and the hydrodynamic interactions are screened owing to the presence of other chains [4,5,103], With respect to the correlation lengths (c) and H(c) there is no consensus as to whether these quantities have to be equal [11] or in general would be different [160],... 

Muthukumar and Winter [42] investigated the behavior of monodisperse polymeric fractals following Rouse chain dynamics, i.e. Gaussian chains (excluded volume fully screened) with fully screened hydrodynamic interactions. They predicted that n and d (the fractal dimension of the polymer if the excluded volume effect is fully screened) are related by... 

There are numerous equations in the literature describing the concentration dependence of the viscosity of dispersions. Some are from curve fitting whilst others are based on a model of the flow. A common theme is to start with a dilute dispersion, for which we may define the viscosity from the hydrodynamic analysis, and then to consider what occurs when more particles are added to replace some of the continuous phase. The best analysis of this situation is due to Dougherty and Krieger18 and the analysis presented here, due to Ball and Richmond,19 is particularly transparent and emphasises the problem of excluded volume. The starting point is the differentiation of Equation (3.42) to give the initial rate of change of viscosity with concentration ... 

Table 2). All the radii have a certain molar mass dependence. The magnitudes of these radii, however, can deviate strongly from each other. These differences result from the fact that they are physically differently defined. The radius of gyration, R, is solely geometrically defined the thermodynamically equivalent sphere radius, R-p is defined by the domains of interaction between two macromolecules, or in other words, on the excluded volume. The two hydrodynamic radii R and R result from the interaction of the macromolecule with the solvent (where the latter differs from R by the fact that in viscometry the particle is exposed to a shear gradient field).

Through control of the amount of cross-linking, nature of the packing material, and specific processing procedures, spheres of widely varying porosity are available. The motion in and out of the stationary phase depends on a number of factors including Brownian motion, chain size, and conformation. The latter two are related to the polymer chain s hydrodynamic volume—the real, excluded volume occupied by the polymer chain. Since smaller chains preferentially permeate the gel particles, the largest chains are eluted first. As noted above, the fractions are separated on the basis of size. 

FIGURE 16.3 Dependences of the polymer retention volume on the logarithm of its molar mass M or hydrodynamic volume log M [T ] (Section 16.2.2). (a) Idealized dependence with a long linear part in absence of enthalpic interactions. Vq is the interstitial volume in the column packed with porous particles, is the total volume of liquid in the column and is the excluded molar mass, (b) log M vs. dependences for the polymer HPLC systems, in which the enthalpic interaction between macromolecules and column packing exceed entropic (exclusion) effects (1-3). Fully retained polymer molar masses are marked with an empty circle. For comparison, the ideal SEC dependence (Figure 16.3a) is shown (4). (c) log M vs. dependences for the polymer HPLC systems, in which the enthalpic interactions are present but the exclusion effects dominate (1), or in which the full (2) or partial (3,4) compensation of enthalpy and entropy appears. For comparison, the ideal SEC dependence (Figure 16.3a) is shown (5). (d) log M vs. dependences for the polymer HPLC systems, in which the enthalpic interactions affect the exclusion based courses. This leads to the enthalpy assisted SEC behavior especially in the vicinity of For comparison, the ideal SEC dependence (Eigure 16.3a) is shown (4). 

Although shear rate effects are more pronounced in good solvents, the intrinsic viscosity decreases with shear rate even in 0-solvents, where excluded volume is zero (317,318). The Zimm model employs the hydrodynamic interaction coefficients in the mean equilibrium configuration for all shear rates, despite the fact that the mean segment spacings change with coil deformation. Fixman has allowed the interaction matrix to vary in an appropriate way with coil deformation (334). The initial departure from [ ]0 was calculated by a perturbation scheme, and a decrease with increasing shear rate in 0-systems was predicted to take place in the vicinity of / = 1. 

All these potential mechanisms for shear rate dependence—changes in excluded volume, changes in hydrodynamic interaction, finite extensibility and... 

Changes in excluded volume and in intramolecular hydrodynamic interaction appear at the present time to be the only acceptable explanations for the onset of shear rate dependence in systems without appreciable intermolecular interactions. It seems likely that both internal viscosity and finite extensibility would assume importance only at much higher shear rates. 

Intcrmolecular Contributions. Increasing concentration reduces the effects of excluded volume and intramolecular, hydrodynamic on viscoelastic properties (Section 5). Internal viscosity and finite extensibilty have already been eliminated as primary causes of shear rate dependence in the viscosity. Thus, none of the intramolecular mechanisms, even abetted by an increased effective viscosity in the molecular environment, can account for the increase in shear rate dependence with concentration, e.g., the dependence of power-law exponent on coil overlap c[r/] (Fig. 8.9). Changes in intermolecular interaction with increased shear rate seems to be the only reasonable source of enhanced shear rate dependence, at least with respect to the early deviations from Newtonian behavior and through a substantial portion of the power law regime. 

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