Gibbs model

Adam-Gibbs Model for the Supercooled Dynamics in the Ortho-Terphenyl Ortho-Phenylphenol Mixture. 

While the combination of the LCT for polymer glasses and the postulates of the Adam-Gibbs model for relaxation in cooled liquids captures many aspects of... 

Equations D3.5.30 and D3.5.32 are both very valuable. They state that the rate of adsorption can be obtained from plots of the interfacial tension versus either tA- (for t—>0) or lth (for the long-term solution f— >). With these two equations the tool to extract the adsorption rate from experimentally obtained surface tension-time curves is at hand. It should be noted that instead of the Gibbs model, one could use one of the previously mentioned adsorption isotherms such as the Langmuir adsorption isotherm to convert interfacial tension to interfacial coverage data. The adsorption isotherms may be obtained by fitting equilibrium surface tension data versus surfactant concentration. 

In the Gibbs model the interface is ideally thin (Va = 0) and the total volume is... 

In the Gibbs model of an ideal interface there is one problem where precisely do we position the ideal interface Let us therefore look at a liquid-vapor interface of a pure liquid more closely. The density decreases continuously from the high density of the bulk liquid to the low density of the bulk vapor (see Fig. 3.2). There could even be a density maximum in between since it should in principle be possible to have an increased density at the interface. It is natural to place the ideal interface in the middle of the interfacial region so that T = 0. In this case the two dotted regions, left and right from the ideal interface, are equal in size. If the ideal interface is placed more into the vapor phase the total number of molecules extrapolated from the bulk densities is higher than the real number of molecules, N < caVa + c V13. Therefore the surface excess is negative. Vice versa if the ideal interface is placed more into the liquid phase, the total number of molecules extrapolated from the bulk densities is lower than the real number of molecules, N > caVa + surface excess is positive. 

In the Gibbs model all volume terms disappear. In the Guggenheim model one must also keep P constant during the integration. 

The number of moles of i must be the same in the actual and the Gibbs model system ... 

To proceed further, we make use of the Adam-Gibbs model for the temperature dependence of relaxation time x(T) of cooperative rearranging regions in glassforming liquids [41]... 

From the Adam-Gibbs model, rnmm can be written as... 

In contrast, the barrier for cooperative rearrangements in the Adam-Gibbs model is inversely proportional to configurational entropy density [41]... 

One way to improve the Adam-Gibbs model is to include details of the structure of the interface between the various aperiodic minima [39]. Near the Kauzmann temperature, the interface broadens, and correct scaling laws are obtained by wetting the droplet surface [39]. In this case, the surface tension of the entropic droplet is a function of its radius and can be obtained by renormalization group arguments. Analysis reveals that the activation barrier to configuration rearrangement is [39]... 

This equation was derived by Hildebrand (1936) and is based on the simple Gibbs model of a dividing surface. It should be emphasized that the interaction parameter B is related to the interaction of components in the bulk liquid and therefore, the surface tension is influenced by the behavior of the bulk. 

Arrhenius plots demonstrating the effect of temperature on lettuce seed aging rate (squares) and on molecular mobility calculated using the Adam-Gibbs model and heat capacity measurements (open circles) and from rotational motion using electron spin resonance measurements. Data are from Walters et al., (2004) (aging rate), Walters, (2004) (glass relaxation rates) and Buitink et al., (2000) (ESR measurements). 

By using the Gibbs model, it is possible to obtain a definition of the siuface or interfacial tension y, starting from the Gibbs-Duhem equation [2], that is... 

Using the Gibbs model, it is possible to obtain a definition of the surface or interfacial tension y. The surface free energy dG comprises three components (i) an entropy term S dT (ii) an interfacial energy term Ady, and (iii) a composition term S d/t (where W is the number of moles of component i with chemical potential nf. The Gibbs-Duhem equation is,... 

Figure 1. Comparison of the reduced CXC-densities pi,g/pc for real substances Ar C2H, (m/u), CO2 ( /<>), H2O c/o) with the van der Waals-Maxwell-Gibbs model s predictions ) based on the disorder parameter x the respective rectilinear diameter (pi+pg)/2pc as a function of x (—).

Hierarchical surfaces may also provide a realisation of the Adam-Gibbs model [178,179], which assumes that relaxation is due to cooperatively rearranging regions whose size increases with decreasing temperature. On a hierarchical landscape, structural change involves longer pathways with higher barriers as the temperature decreases. 

It is important to realise that the Gibbs model does not imply that the surface excess is concentrated at the Gibbs dividing surface this is clearly physically impossible since molecules have a finite size and cannot occupy a mathematical surface. What the Gibbs method docs is to recognise the existence of concentration profiles such as those shown in Figure 5.3, whose exact form cannot yet be measured experimentally, and to provide a method of expressing the observable consequences of their existence. 

Cp° is the conhgurational heat capacity, i.e. the difference in Cp between the liquid (glass) and crystal states, and T2 is the temperature at which the conhgurational entropy reaches zero, T k- A modihed Adam-Gibbs model describes non-equilibrium relaxation in the glassy state. A hctive temperature is introduced, which represents the... 

This notion is complex and a more detailed presentation would need to go beyond the scope of this document. In particular, in order to describe the thermodynamics of interfaces one would need to refer to the notion of surface excess as defined by the Gibbs model. Ne can still say that TJ is the integral of the volume concentration over a distance equal to the thickness of the interfacial zone... 

Fig. 12.23 Relaxation time versus temperature for the blend system PS/PoClS open circles, pure PS (molecular weight 700 g/mol) triangles, 25 % PoClS squares, 50 % PoClS stars, 75 % PoClS open squares, pure PoClS. The solid lines are fits of the combined Adam/Gibbs - selfconcentration approach as described in the text. The dashed lines are fits of the Adam and Gibbs model to the data of the pure components. For details see reference (Data were taken fi om reference Cangialosi et al. (2005))...

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