Viscosity free volume models

The free-volume model was originally derived to explain the temperature dependence of the viscosity. We have shown that it has a much broader application and can explain many of the outstanding experimental observations. This includes the existence of an entropy catastrophe at 7 and the approximate equality of Tj, and 7], first observed by Angell and coworkers.The relation between ln and 7, measured by Moynihan et al., also follows naturally and quantitatively from the notion that the liquidlike cell fraction p is the important variable that ceases to reach equilibrium when the relaxation rates become longer than the time scale for the measurement. 

Time-temperature superpositioning was originally derived from free volume models, which assume that the rates of molecular motions are governed by the available unoccupied space. Early studies of molecular liquids led to the Doolittle equation, relating the viscosity to the fractional free volume, f =V /(V - Vo), where V is the specific volume and Vo is the occupied volume normalized by the mass) (Doolittle and Doolittle, 1957 Cohen and Turnbull,... 

The constants correspond to coo = C, B = yVflEf, and Tq = T o (7b is the Vogel temperature). With Cohen and Turnbull delivered this free-volume model, a theoretical justification of the empirical VFTH equation and the equivalent Williams-Landel-Ferry (WLF) [Williams et al., 1955] equation as well as of the empirical free-volume models of viscosity [Fox and Flory, 1950 Doolittle, 1951]. 

YASOTUMI S..BAIR S. and WINER W.O. "An Application of a Free-Volume Model to Lubricant Rheology (I) Dependence of Viscosity on Temperature and PressureClI) Variation in Viscosity of Binary Blended Lubricants",Trans A.S.M.E.,Journal of Tribology,vol 106,p291-312,1984. 

In spite of this success, there is still some argument about the role of free volume or fractional free-volume in determining the mobility of liquids. Some fluids can apparently be described by the WLF-type equations over reasonably large ranges of temperature and pressure, but in other cases the description results in unreasonable values for the free-volume parameters.Also, under conditions where the free-volume fraction is held constant by increasing pressure when the temperature is increased, the viscosity of the fluid is not constant.Thus it is argued that an energy of activation is required in the free-volume model. 

Loutfy and coworkers [29, 30] assumed a different mechanism of interaction between the molecular rotor molecule and the surrounding solvent. The basic assumption was a proportionality of the diffusion constant D of the rotor in a solvent and the rotational reorientation rate kOI. Deviations from the Debye-Stokes-Einstein hydrodynamic model were observed, and Loutfy and Arnold [29] found that the reorientation rate followed a behavior analogous to the Gierer-Wirtz model [31]. The Gierer-Wirtz model considers molecular free volume and leads to a power-law relationship between the reorientation rate and viscosity. The molecular free volume can be envisioned as the void space between the packed solvent molecules, and Doolittle found an empirical relationship between free volume and viscosity [32] (6),... 

With further understanding how molecular rotors interact with their environment and with application-specific chemical modifications, a more widespread use of molecular rotors in biological and chemical studies can be expected. Ratiometric dyes and lifetime imaging will enable accurate viscosity measurements in cells where concentration gradients exist. The examination of polymerization dynamics benefits from the use of molecular rotors because of their real-time response rates. Presently, the reaction may force the reporters into specific areas of the polymer matrix, for example, into water pockets, but targeted molecular rotors that integrate with the matrix could prevent this behavior. With their relationship to free volume, the field of fluid dynamics can benefit from molecular rotors, because the applicability of viscosity models (DSE, Gierer-Wirtz, free volume, and WLF models) can be elucidated. Lastly, an important field of development is the surface-immobilization of molecular rotors, which promises new solid-state sensors for microviscosity [145]. 

This is a theoretical equation that was derived from free volume theory. If extruding materials at lower than normal temperatures, the higher sensitivity of the viscosity to temperature is an issue that needs to be considered. The engineering-based viscosity equation developed by Adams and Campbell [18] has been shown to hold for all nominal processing temperatures, from within a few degrees of Tg [26, 27] to conventional extruder melt temperatures. The Adams-Campbell model limiting shear temperature dependence is ... 

A modified version of the free-volume theory is used to calculate the viscoelastic scaling factor or the Newtonian viscosity reduction where the fractional free volumes of pure polymer and polymer-SCF mixtures are determined from thermodynamic data and equation-of-state models. The significance of the combined EOS and free-volume theory is that the viscoelastic scaling factor can be predicted accurately without requiring any mixture rheological data. 

The hole model for molecular liquids was elaborated by Furth [12], who supposed that the free volume of a liquid is not distributed uniformly between its molecules like in crystals, but is concentrated like some holes which can disappear in one place and appear in another place. These holes are in permanent motion, so that the situation is different from the jumps of the holes in a crystal. The appearance and disappearance of the holes in a liquid are a result of the fluctuations connected with thermal movements. These holes in liquids have no definite shape and size they can increase or decrease spontaneously. Furth [12] tried to calculate a large number of properties of the liquids viscosity, compressibility, thermal expansion, thermal conductivity, but the results were not successful. However, Furth obtained a precise result of the calculation of the volume change by melting and the entropy of melting. 

Analysis of the dependence of viscosity on the concentration of disperse systems and on the free volume of condensed liquid systems shows that there is a considerable similarity between the concepts based on the description of the properties of these systems. This is evidently explained by the similarity of geometric models describing the behavior of these systems based on the description of the... 

On the other hand, some phenomenological distributions of relaxation times, such as the well known Williams-Watts distribution (see Table 1, WW) provided a rather good description of dielectric relaxation experiments in polymer melts, but they are not of considerable help in understanding molecular phenomena since they are not associated with a molecular model. In the same way, the glass transition theories account well for macroscopic properties such as viscosity, but they are based on general thermodynamic concepts as the free volume or the configurational entropy and they completely ignore the nature of molecular motions. 

Measurements of molecular density and intrinsic viscosity of PAMAM dendrimers indicate an unusual variation with dendrimer generation. Minimum density and maximum intrinsic viscosity were observed at around G4, which suggests that the fully developed dendrimers have a high accessible internal surface area in a solvent-filled intramolecular free volume that may consist of internal cavities and channels. Similar findings were reported by Mourey et al. for PBE monodendrons (based on dihydroxybenzyl alcohol) and tridendrons produced by coupling these dendrons to a trifunctional core, l,l,l-tris(4 -hydro-xyphenyl)ethane prepared by the convergent method. A maximum intrinsic viscosity occurred at G3 for tridendrons and G5 for monodendrons, consistent with the model developed by Lescanec and Muthukumar... 

Structural studies in fused salts by means of careful and thorough high-temperature measurements of electrical conductivity, density, viscosity, and laser- Raman spectroscopy have been reviewed. Four problem areas are discussed (1) melting mechanisms of ionic compounds with large polyatomic cations, (2) salts as ultra-concentrated electrolyte solutions, (3) structural aspects and Raman spectroscopy, and (4) electrolysis of molten carbonates. The results in these areas are summarized and significant contributions to new experimental techniques for molten-salt studies are discussed.275 The physical properties and structure of molten salts have also been reviewed in terms of operational (hole, free volume, partly disordered crystal) and a priori (intermolecular potential) models.276 Electrochemistry... 

Viscosity can be considered as a measure of the ease of movement of molecules in a liquid undergoing shear. Several factors may influence this ease of movement including molecule size and intermolecular attraction, but a major factor is the amount of space available between the molecules, hence, the variety of models incorporating a free volume term. 

A different approach was made by Dyre et al. (1996) to account for the experimental viscosity variations with temperature as an alternative to VTF and AG models. They considered the flow in viscous liquids to arise from sudden events involving motion and reorganization of several molecules. From the viewpoint of mechanism, the energy required for such flow is minimized if the surrounding liquid is shoved aside to create the necessary volume for rearrangement. This volume is fundamentally different from the volume of the free volume theory and is, in principle, an activation volume. The free energy involved may be written as... 

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