Universal values

A useful observation is that E is very nearly proportional to Tm over the range usually employed. Use Eq. XVIII-3 to demonstrate this. Suggestion Find Tm for various x,x = EjRTm, and thence E for various Tm- One can conclude from this an approximate universal value of x. Explain and give this value. 

The situation becomes most complicated in multicomponent systems, for example, if we speak about filling of plasticized polymers and solutions. The viscosity of a dispersion medium may vary here due to different reasons, namely a change in the nature of the solvent, concentration of the solution, molecular weight of the polymer. Naturally, here the interaction between the liquid and the filler changes, for one, a distinct adsorption layer, which modifies the surface and hence the activity (net-formation ability) of the filler, arises. Therefore in such multicomponent systems in the general case we can hardly expect universal values of yield stress, depending only on the concentration of the filler. Experimental data also confirm this conclusion [13],... 

It seems however that this problem is not fully cleared up. Thus, it was stated in paper [19] on the basis of experimental data obtained earlier that Ye increased with a filler s concentration in proportion to cp4J. Such a law for the concentration dependence of yield stress at the shear Y(from different ways of analytical approximation of particular experimental data. The only unquestionable fact is that Ye as well as Y grow very sharply with an increase in concentration. 

Now calculations of Ta and Tk are plagued by the usual difficulties of liquid state structure theory and the accuracy of approximations, some of which are hard to control. Still, even in the face of such approximations, such microscopic considerations lead us to expect a universal value of y/Tg at Tg as we shall discuss next. 

Two other remarkable universalities emerge from the value of y. First, at a reference laboratory time scale of 1 h IO tq, we have a universal value of Sc O.Sfe. This implies Sc Tg)/sc Tm) — 0.7, where Sc Tm) is, of course, also the fusion entropy. This relation is independent of the precise identity of the moving subunit and holds very well. A second important universal feature emerges from the universal value of y/Tgi the cooperative size at Tg is nearly universal. 

Precise knowledge of the critical point is not required to determine k by this method because the scaling relation holds over a finite range of p at intermediate frequency. The exponent k has been evaluated for each of the experiments of Scanlan and Winter [122]. Within the limits of experimental error, the experiments indicate that k takes on a universal value. The average value from 30 experiments on the PDMS system with various stoichiometry, chain length, and concentration is k = 0.214 + 0.017. Exponent k has a value of about 0.2 for all the systems which we have studied so far. Colby et al. [38] reported a value of 0.24 for their polyester system. It seems to be insensitive to molecular detail. We expect the dynamic critical exponent k to be related to the other critical exponents. The frequency range of the above observations has to be explored further. 

Difficult to estimate reliably because C and Ci uncertain based on universal values. 

Since co2 =K/m, the mean potential and kinetic energy terms are equal and the total energy of the linear oscillator is twice its mean kinetic energy. Since there are three oscillators per atom, for a monoatomic crystal U m =3RT and Cy m =3R = 2494 J K-1 mol-1. This first useful model for the heat capacity of crystals (solids), proposed by Dulong and Petit in 1819, states that the molar heat capacity has a universal value for all chemical elements independent of the atomic mass and crystal structure and furthermore independent of temperature. Dulong-Petit s law works well at high temperatures, but fails at lower temperatures where the heat capacity decreases and approaches zero at 0 K. More thorough models are thus needed for the lattice heat capacity of crystals. 

Parameter values considered universal for all 29 events were determined for the six events that were selected for calibration. These universal values are shown in Table 7.3. The remaining parameters—dependent on the actual event — were determined separately. 

In a superconducting system, when one increases the temperature at a given chemical potential, thermal motion will eventually break up the quark Cooper pairs. In the weakly interacting Bardeen-Copper-Schrieffer (BCS) theory, the transition between the superconducting and normal phases is usually of second order. The ratio of the critical temperature TcBCS to the zero temperature value of the gap AbGS is a universal value [18]... 

The optimum UNIQUAC interaction parameters u, between methylcyclohexane, methanol, and ethylbenzene were determined using the observed liquid-liquid data, where the interaction parameters describe the interaction energy between molecules i and j or between each pair of compounds. Table 4 show the calculated value of the UNIQUAC binary interaction parameters for the mixture methanol + ethylbenzene using universal values for the UNIQUAC structural parameters. The equilibrium model was optimized using an objective function, which was developed by Sorensen [15],... 

The Saybolt universal viscosity equivalent to a given kinematic viscosity varies slightly with the temperature at which the determination is made because the temperature of the calibrated receiving flask used in the Saybolt method is not the same as that of the oil. Conversion factors are used to convert kinematic viscosity from 2 to 70 cSt at 38°C (100°F) and 99°C (210°F) to equivalent Saybolt universal viscosity in seconds. Appropriate multipliers are listed to convert kinematic viscosity over 70 cSt. For a kinematic viscosity determined at any other temperature, the equivalent Saybolt universal value is calculated by use of the Saybolt equivalent at 38°C (100°F) and a multiplier that varies with the temperature ... 

Indicator parameters can be valid only for a certain time period, in a limited geographic area, and for a particular environmental system. On the other hand, a parameter such as drinking water taste will have more universal value as an indicative parameter (10). 

It is apparent that only the average value corresponds to the universal value in WLF theory, and that the deviations from this value can be very considerable. In spite of this, we may believe that free-volume is determined mainly by the hole volume. At the same time it follows from Eq. (63) that the SB constant is a very complex value and a function of p and a. Plotting the experimental data in coordinates... 

The idea that the fractional free-volume at glass temperature as found experimentally depends on the mode of molecular motions was put forward in 196746 47 as a result of calculating/g from data obtained from isothermal volume relaxation for some polymer systems. By estimating average relaxation time at different temperatures it was possible to find the fractional free-volume/g at Te according to WLF theory. If we accept the validity of the theory as regards the universal dependence of the reduction factor aT on (T - Tg), then on the basis of data on Aa and theoretical values aT calculated from universal values of the coefficients C and C, it is possible to make an estimate of/g. In this case the value found corresponds to the universal one. If, however, we use the experimental values aT, the fractional free-... 

It was found that Afi Tg and Aa Tg are not constant and therefore the SB equation has limited applicability. Hie results indicate an increase in Aa Te with increasing Tg. Therefore it is inadmissible to use the product A a Tg as a universal value in any theoretical discussion of the glass-transition phenomenon. At the same time, this conclusion in no way excludes the free-volume theory and the role of free-volume in the transition from the glassy to the liquid or rubberlike state. 

In some reports83,84) the change in the fractional free-volume was calculated at temperatures above Tg for epoxy resin filled with polystyrene particles on the basis of the experimental value of the reduction factor aT and the universal value fg according to the equation... 

Table 2. Expansion coefficients and universal values for some filled systems...

Values 0 are given in Table 4 for the universal value of the SB constant and for the values found experimentally for a given system. It is interesting to note that these data conflict with the results of calculating for the same samples according to WLF theory. 

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