Couchman equation

Cottrell equation Cottrell unit Couchman equation Couette flow Couette viscometers Cough drops Coughlozenges... 

By developing a model based on the arguments given by Goldstein [171], Painter et al. [172] modified the Couchman equation [173] as follows ... 

The foam processing window of the blend systems is also controlled by the glass transition temperature of the blend phases. With regard to the neat blend system, the addition of PS continuously lowers the glass transition temperature of the PPE/PS phase, as predicted by the Couchman equation [77] (Fig. 25). For the carbon dioxide-laden case, the plastifying effect needs to be taken into account, which lowers the glass transition temperature of both PPE/PS and SAN. 

For the PPE/PS phase, the previously described Chow equation can be combined with the Couchman equation to estimate the Tg as a function of the blend composition. The results are highlighted in Fig. 25. For the prediction, the heat capacity and the molar mass of the repeat unit of the PPE/PS blends is regarded to scale linearly with the mass content of the blend partners, and a constant lattice coordination number of z = 2 is used [75]. While the addition of PS to PPE allows one to reduce continuously the Tg in presence of carbon dioxide, the plasticization effect is less pronounced, mainly driven by the decreasing solubility via addition of PS. 

Fig. 25 Glass transition temperature of PPE/PS blends in absence and presence of carbon dioxide, as calculated by the Couchman equation [77] and by the Chow equation [75]...

A )> PpCIS/PPO, low temperature transition (each point dropped 7 C on the ordinate for purpose of comparison with the lower Tg copol3miers), Curve 1 was drawn from values of Tg s calculated by means of the Couchman equation (Eq. 1) for the compatible Copolymer B/PPO blends. 

Plotting the glass transition temperatures of Fig. 7.73 as a function of concentration, yields Fig. 7.74. Only the Gordon-Taylor equation with a fitted constant represents the data. Similarly it is possible to fit with the Schneider equation with its two constants. Two additional equations, not in Fig. 7.69, are compared in Fig. 7.74 to the data one, is the Couchman equation, based on additivity of the products of ACp with the logarithm of T, the other uses a molar additivity of the logarithn of T. All equations without adjustable parameters do not fit the experimental data (o). 

Mixed system glass transition temperature, Tg is defined by the requirement that entropy for the glassy state is identical to that for the rubbery state [5]. A revisit of the Couchman equation without neglecting the differences of entropy of mixing in the glassy and rubbery state is used to develop a mathematical framewoilc to represent multiple glass transition tanperatures in partially miscible copolymers in blends. 

A variation of the Couchman equation was proposed assuming Tgi ACpi = constant [5]... 

The equations of electrocapillarity become complicated in the case of the solid metal-electrolyte interface. The problem is that the work spent in a differential stretching of the interface is not equal to that in forming an infinitesimal amount of new surface, if the surface is under elastic strain. Couchman and co-workers [142, 143] and Mobliner and Beck [144] have, among others, discussed the thermodynamics of the situation, including some of the problems of terminology. 

In this case, the constants are related to the heat capacity changes at the glass transition of the homopolymers. While the above three equations have rather different origins and appear to be quite different, over the range of many measurements, the results do not differ significantly. Couchman (26) has... 

Table 1 Glass transition behavior /(, and T% width (ATg)] measured by DSC and DMA as well as theoretical values predicted by Fox [53] and Couchman [54] equations for the neat materials and PPE/PS blends...

The effect of molecular weight of PMMA on the miscibility of PMMA/PS blends was examined by studying the specific heat increment ACp at Tg [Bums and Kim, 1988]. Using Couch-man s equation, C for PMMA was calculated and was found to decrease with the composition of PS [Couchman, 1978]. The C for PS similarly decreased with PMMA composition. From these results the authors inferred that some of the PMMA dissolved in the PS phase and vice versa. Thus, the blends were found to be partially miscible. This result was found to be consistent with the polymer-polymer interaction parameter values. The authors also studied the PC/SAN blends miscibility by the thermal analysis [Kim and Burns, 1988]. The values of the specific heat increment AC at T for PC and SAN in PC/SAN... 

The classical Gibbs-Duhem equation must be modified to describe theimodynamic properties of solids. If temperature and pressure are kept constant, one obtains by the function of Everett and Couchman ... 

FIGURE 13.1 Experimental T values resulting from the antiplasticization effect of PVP90 (o), PVP12 (o), or MD simulated values for PVP (>t) in amorphous mixtures with indomethacin. r predicted is based on the Gordon-Taylor equation with the value of K estimated using the Couchman and Karasz relationship. 

One of the properties that is affected by the production of miscible blends is the glass transition temperature or Tg. The Tg of miscible blends has been modeled using several different approaches in the literature (Wood 1958 Couchman and Karasz 1978). The simplest of these is the Flory-Fox equation ... 

Because of the complexity of our blend system which contains two crystallizable polymers the more general equation of Couchman is difficult to apply and we decided to plot theoretical curves of Tg versus molar percent PCL based on equations (II), (III) and (IV). These plots are illustrated in figure 6 and represent a lenticular shaped... 

Many polymer blends that are used in industrial practice have been found to be partially miscible. Examples are PVC/SAN, PC/SAN at certain AN compositions, PET/PHB, etc. The entropic difference model was developed by taking into account the change in entropy of mixing at glass transition in the Couchman model. Equation (6.15) quadratic expression for blend TgS is obtained upon Taylor approximation of the relation obtained by equating the entropy of the blend in glassy phase with the entropy of blend in rubbery phase at glass transition of the blend. 

Figure 11.6 Schematic representation of relationships between water activity, water content, Tg and viscosity for wheat gluten-based films. Calculated values were obtained using the (GAB) equation [176], Couchman and Karasz equation (CK) [171], and Williams Landel and Ferry equation (WLF) [153]. The critical water activity (aw) and Me are indicated when Tg is equal to the ambient temperature...

The empirical DiBenedetto equation was developed in the late 1960s to mathematically relate Tg and conversion (Nielson 1969 DiBenedetto 1987). Excellent theoretical treatises on the Tg-conversion relationship can be found in Pascault and Williams (1990), Hale et al. (1991), and Venditti and Gillham (1997). Venditti and Gillham (1997) developed an equation based on thermodynamic considerations put forth by Couchman and Karasz (1978) to predict Tg versus mole fraction of constituents of a linear copolymer ... 

The T of polymer blends can be related to the blend composition by different equations, as shown in Table 12.2. The T of miscible blends of poly(p-dioxa-none) with poly(vinyl phenol) (PPDO/PVPh) [74], as studied using the Fox [75], Gordon-Taylor [76], Couchman-Karasz [77] and Kwei [78] models, showed that the experimental data lay below the Fox equation, suggesting that the free volume of the blends was larger than predicted, assuming free volume additivity. On the other hand, the Gordon-Taylor and the simplified Kwei equations fitted the experimental T values appropriately. Other studies have shown that the Ta of SAN/PMMA blends was also effectively approached by the Gordon-Taylor relationship [79,80]. 

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