Fox equation

Changes in heat capacity and measurement of T for blends have been used to determine components of copolymers and blends (126—129), although dynamic mechanical analysis has been found to give better resolution. Equations relating T of miscible blends and ratios of components have been developed from dsc techniques, eg, the Fox equation (eq. 1), where f the blend, or is the weight fraction of component 1 or 2,... 

Wi is the weight fraction of the elastomer, W2 the tackifier, W3 a further compatible additive, such as an oil, and so forth, for the remaining components in the formulated PSA. Application of the Fox equation to the poly (/-butylstyrene) tackified natural rubber adhesive (cited above) gives a value of —11°C, in good agreement with the interpolated value of — 13°C. 

Mixtures of monomers can be used to balance properties. This is possible due to the ease of copolymer formation via free-radical polymerization. The glass transition temperature of acrylic copolymers can be predicted from the weight fraction of the component monomers and the glass transition temperatures of the respective homopolymers [20]. Eq. 3 (commonly known as the Fox equation) is reported ... 

Monthly evaporation estimates can be made using the Kohler-Nordenson-Fox equation with a pan coefficient of 0.7. The Kohler-Nordenson-Fox equation describes evaporation as the combination of water loss due to radiation heat energy and the aerodynamic removal of water vapor from a saturated surface. The general form for the combination equation is ... 

These data showed that within the range of 0.5 < TPU/PVC < 1.075, the Tg value of the blends is very close to the theoretical value calculated from the Fox equation. This suggests that 892024TPU is compatible with PVC within this range. 

The main function of a plasticizer is to increase the mobility of the polymer chain, thus decreasing its Tg. The Tg of the PVC/TPU system is mainly determined by the Tg of TPU. It could be calculated by the Fox equation ... 

For example, the T values of copolymers with dilactide increase in proportion to the dilactide content according to the Fox equation ... 

Fig. 10 Phase diagrams of dPS/PnPMA blends, o UCST at ambient pressure, LCST at ambient pressure. Closed-loop phase behaviours are observed at higher pressure A 97 bar 117 bar 138 bar 166 bar 186 bar. Dashed line Prediction of Tg, blend by Fox equation at ambient pressure. From [59]. Copyright 2004 American Chemical Society... 

Figure 2. Glass transition temperature of PU/PMMA SINs as a function of composition —, Pochan equation -----, Fox equation.

It is known that the T of binary blends made miscible by hydrogen bonding shows a composition dependence significantly deviated from the weight-average law or the Fox equation given by ... 

Table 10 summarizes the glass transition behavior of these polyimide blends and demonstrates that there is only one Tg for each blend. Similar results have been confirmed by Koros of the University of Texas [29], This data confirms that as long as the dianhydride is the same in the composition, the change of 6F diamine from 3,3 to 4,4 does not alter solubility significantly, and the pairs are miscible. The relationship between the Tg and the Fox equation is discussed by MacKnight et al. work [17]. 

Tg of individual polymer in a blend b Measured Tg of a blend c Calculated Tg from the Fox Equation d Difference between the measured Tg and the calculated Based on temperature difference... 

Equation (3.60) has just the form of the well-known Flory-Fox equation (98). As is well-known, this equation holds for sufficiently high molecular weights. With decreasing molecular weights shielding is expected to become less effective [cf. also Kirkwood and Riseman ( 8)]. Apparently, the Flory-Fox parameter of eq. (3.60) reads... 

In GPC, the product [77] M, (or the hydrodynamic radius Re) has been widely accepted as a universal calibration parameter. In the Ptitsyn-Eizner modification of the Flory-Fox equation the quantity 4>, which relates the dimensional parameters to the above product, is taken as a variable. The value of < depends upon molecular expansion in solution as represented by a function f(e). Because of this dependence polymeric species having the same [77] M value cannot have the same statistical dimensions (radius of gyration or end-to-end distance) unless they have the same e value. Thus, if [77] M is a universal calibration parameter, the statistical parameters cannot be used as such. A method is presented for obtaining the Mw/Mn ratio from GPC data even though universal calibration is used. 

The ratio (Re/R)3 = (I)3 is thus implicit in the value of in the Flory-Fox equation and has a value of 0.49, corresponding to the Flory-Fox value of 2.1 X 1023. It is clear from Equations 1, 2, and 3 that [77] M cannot be related to the statistical polymer dimensions h and R without a knowledge of , i.e., < , which varies with solvent for a given polymer. It follows, that if all species having the same [77] M elute together from the GPC columns, then only Re can be the universal parameter, since will not be the same for all solute-solvent pairs and h and R will not be equally correct for universal calibration. 

At least two different glass transition temperatures have been reported for PVdF homopolymer. Owing to the large proportion of crystalline structure in this polymer and the rapid crystallization which occurs while heating quenched amorphous samples, it is difficult experimentally to obtain an unambiguous, well-defined second-order transition. Mandel-kem, Martin, and Quinn (16) reported a value below — 40°C based upon an extrapolation of the Tg data for vinylidene fluoride-chlorotri-fluoroethylene copolymers in accordance with the Fox equation (6),... 

Using this experimental value for PVdF and the value of 95°C for PMMA, and plotting the experimental T0 data for the blends vs. wt % composition, the observed values show large deviation from the behavior which would be predicted by the Fox equation (Figure 2). It appeared... 

Figure 2. Comparison of experimental Tg values of PMMA-PVdF blends with theoretical curve based on Fox equation...

Thus, Tg should be a linear function of the volume fraction of each component rather than the weight fraction as implied by the Fox equation. When the Tg data are plotted vs. volume fraction, a reasonable fit is obtained for the PMMA-PVdF polyblends (Figure 3) and excellent agreement for PEMA-PVdF blends (Figure 4). [For these calculations,... 

Table 10.7 Component values of some important groups for the calculation of Tgj from the Fox equation. Remark (a) these component values are only applicable to the bridges between two aromatic nuclei. 

Shen et al. determined the BD/IP copolymerization parameters for the polymerization with the ternary catalyst system NdN/TIBA/EASC at 50 °C ted = 1.4 and np = 0.6 [92]. Over a wide range of BD/IP copolymer compositions the experimentally determined Tg values significantly deviate from the theoretical curve which was calculated by the Fox equation for random copolymers. Only for IP-contents < lOwt. % does the experimentally determined data coincide with the theoretical curve. Shen et al. also succeeded to synthesize block copolymers comprising poly(butadiene) and poly(isoprene) building blocks [92]. 

It is well established that the excluded volume effect vanishes under a special condition of temperature or solvent, which is usually known as the Flory theta temperature or solvent. Thus, light scattering measurements performed on solutions under theta conditions can furnish direct knowledge of the unperturbed dimensions [see, for example, Outer, Carr and Zimm (207) Shultz (233) and Notley and Debye (207)]. Viscosity measurements, though less directly, can also furnish similar knowledge with the aid of the Flory-Fox equation (103,109), which may be written... 

For comparison, the radius of gyration can also be calculated from the viscosity and the molecular weight of the polymer by use of the Flory-Fox equation (11) ... 

Tg of poly( 1,4 phenylene terephthaloyl) has been reported as 540 K (41). All of the points could not be fitted by the general Tg/composition formula by Wood (27), even when the Tg of poly (oxy terephthaloyl) was allowed to float. If the point for 3IGT was omitted, the rest of the points were fitted by the Fox equation (40), shown as a dashed line in Figure 12. 

Compare a plot of T versus composition using this data to that using the Fox equation (Equation 10-53). Explain any deviations you find in the plots. (Hint you won t find the answer in the text, you ll have to dig )... 

Differential Scanning Calorimetry. Thermal analysis of PVME/PS blends and IPNs by DSC indicates only one glass transition temperature, which is located between the homopolymer T s. The position of Tg is dependent upon composition although it does not follow the prediction of the Fox equation (21), Table II. The breadth of the transition also increases significantly with increased PS content for the blends and IPNs. The broad transition might result from either the clustering of like mers near... 

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