Equilibrium degree

Whereas only a few atomie jumps may be neeessary to enable ehanges in the equilibrium degree of SRO, without atomie movement over long distanees ehanges in LRO may not be suffieient to reaeh equilibrium. This ean lead to a eompetition between inerease of SRO in the matrix and formation of the new LR0-phase when lowering the temperature below an order/disorder phase boundary. In those cases, thermal and/or meehanieal pretreatment of the sample is of erueial importanee for what is observed in the sample. 

As compared to ECC produced under equilibrium conditions, ECC formed af a considerable supercooling are at thermodynamic equilibrium only from the standpoint of thermokinetics60). Indeed, under chosen conditions (fi and crystallization temperatures), these crystals exhibit some equilibrium degree of crystallinity at which a minimum free energy of the system is attained compared to all other possible states. In this sense, the system is in a state of thermodynamic equilibrium and is stable, i.e. it will maintain this state for any period of time after the field is removed. However, with respect to crystals with completely extended chains obtained under equilibrium conditions, this system corresponds only to a relative minimum of free energy, i.e. its state is metastable from the standpoint of equilibrium thermodynamics60,61). 

The degree of dissociation a is the equilibrium degree of conversion, i.e. the fraction of the number of molecules originally present that dissociated at the given concentration. The degree of dissociation depends directly on the given dissociation constant. Obviously a = [Bz+]/v+c = [Az ]/v c, [Bv+Av ] = c(l - a) and the dissociation constant is then given as... 

T Canal, NA Peppas. Correlation between mesh size and equilibrium degree of swelling of polymeric networks. J Biomed Mater Res 23 1183-1193, 1989. 

Gel type Reaction time (hr) Equilibrium degree of swelling Shear modulus (g/cm2) Effective cross-link density (10 5 g/cm3)... 

The first step in the analysis is to determine if the chemical equations A to C are independent by applying the test described above. When one does this one finds that only two of the reactions are independent. We will choose the first two for use in subsequent calculations. Let the variables a and B represent the equilibrium degrees of advancement of reactions A and B, respectively. A mole table indicating the mole numbers of the various species present at equilibrium may be prepared using the following form of equation 1.1.6. 

Equating the chemical potential to zero gives a relationship between the equilibrium degree of swelling and the molecular weight Mc. The relation for Mc ph is obtained for a tetrafunctional phantom network model as... 

The equilibrium degree of crystallization, (0, can be determined from the condition 3AFc/3u) 0, or... 

Both Qv(x) and Q°(x) decrease as the polymerization proceeds and, after a definite conversion Qv(x) may reach the value of Qx(x). Since the dilution of a gel cannot be greater than its equilibrium degree of swelling, the excess of solvent should separate from the gel phase resulting in the syneresis, i.e. in phase separation. The condition for incipient phase separation during copolymerization of divinyl/vinyl monomers is given by [107]... 

In homogeneous networks such as that of crosslinked rubber, the equilibrium degree of swelling qm depends on the molecular weight of the network chain Mc, satisfying the following relation for a given rubber network-solvent system [56],... 

The measured a at room temperature of gel cylinders with various diameters are shown in Fig. 9. It is seen that a increases with the diameter of cylinders, and also a measured on plates and cubes are larger than that of cylinders. This result is surprising because it shows quite unambiguously that the equilibrium degree of swelling depends on the shape of samples. The transition temperature also depends strongly on the shape of samples, as shown in Fig. 10. The difference between the T0 of the thinnest (0.2 mm) and of the thickest (5 mm) cylinders measured in this study differed by more than 8 °C. 

Gel type Approximate gelation time (min) Shear modulus (gcm 2) Equilibrium degree of swelling Effective crosslinking density (mol cm-3) Crosslinking efficiency (%)... 

Fig. 5. Equilibrium degree of swelling of radiation-crosslinked PVME gel (Trial 4) at different temperatures. The line added is to guide the eye. Reprinted from Polymer (1991) 33 990 by permission of the publishers, Butterworth Heinemann [46]...

Fig. 9. Plot of normalized approach to equilibrium mass against the square root of time for a temperature-sensitive 10 x 4 PNIPAAm gel sheet swelling and shrinking between 10 and 25 °C-Shown are the curve fits to the kinetic data of theory developed from Fick s law of diffusion in a polymer-fixed reference frame [149]. The equilibrium degree of swelling is 17.0 at 10 °C and 11.1 at 25 °C the diffusion coefficients obtained from the curve fits are 2.3 x 10 7 cm2/s for swelling and 3.6 x 10 7 cm2/s for shrinking [121]...

Figure 2.20. Temperature dependence of the equilibrium degree of crystallinity for polycaproamide.

By integrating Eq. (2.50) for different cooling rates, i.e., for different functions T(t), it is possible to find the time dependence of crystallinity and the rate of the crystallization process. It is also necessary to bear in mind the temperature dependence of the equilibrium degree of crystallinity, a x (T). As an example, this dependence is shown for polycaproamide in Fig. 2.20.97 It is evident from Eq. (2.53) that the functiona(T) must have a maximum whose location on the temperature axis depends on the cooling rate. This is illustrated in Fig. 2.21, where values of the rate of heat output dQ/dt, proportional to da/dt, and degree of crystallinity a are shown as functions of temperature. It is worth mentioning that all the curves in this figure are adequately described by Eq. (2.52). 

This equation can be easily modified if the equilibrium degree of conversion at T = Tiisequal, notto 1, but to some value dp < 1. 

The values of the modulus of the end product depend on temperature therefore it is necessary to introduce the equilibrium degree of conversion as a reference point ... 

Fig. 8 Polymer volume fraction (j) = ho//t,w, where ho is the thickness of the polymer film after spin-coating and hsw is the thickness of a swollen film, measured by in situ spectroscopic ellip-sometry as a function of (a) the relative solvent vapor pressure for thin films of homopolymers PS, PB, and SBS block copolymer. Reprinted from [49], with permission. Copyright 2004 American Institute of Physics, (b) Polymer volume fraction as a function of the swelling time for PS- >-P2VP (SV) block copolymer and for homopolymers PS and P2VP at p/po = 1.0 [118]. The equilibrium degree of swelling indicates that toluene is a selective solvent for the PS block, and that SV block copolymer shows asymmetric swelling under toluene vapor. Reproduced by permission of The Royal Society of Chemistry (RSC) [118]...

The intensity of the decrease of the equilibrium degree of adsorption (coverage) with rising temperature as well as the point of approximate saturation, however, are unknown for the problem at hand. But since, as discussed before, all reactions under consideration require aqueous solutions anyway, adsorptions on solid, i.e., dry surfaces are... 

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