Effect of Molecular Weight on Tg

In a polymer sample of density p and molecular weight Af , the number of chains per unit volume is given by pNAv/ n- where Nav is Avogadro s number, and so the number of chain ends per unit volume is 2pNAv/Mn- If 9 is the contribution of one chain end to the free volume then the total fractional free volume due to chain ends fc is given by 

It can be argued that if a polymer with this value of fc has a glass transition temperature of Tg then fc will be equivalent to the increase in the volume on expanding the polymer thermally between Tg and Tg oo- This means that 

Problem 2.28 A polydisperse polystyrene sample was fractionated into four components of various molecular distributions and Tg of each fraction was measured  

Studies of the increase in Tg with increasing polymer molecular weight date back to the works of Ueberreiter in the 1930s (105). The theoretical analysis of Fox and Flory (74) (see Section 8.6.1.1) indicated that the general relationship between Tg at a molecular weight M was related to the glass temperature at infinite molecular weight, Tg.., by 

The molecular weight in equation (8.63) is for fractionated polystrene. For slow experiments, these equations suggest a 6 C increase in Tg at infinite molecular weight. 

According to equation (8.61) the glass transition depends on the molecular weight. What happens during an isothermal polymerization When the polymerization begins, the monomers are always in the liquid state. Sometimes, however, the system may go through Tg and the polymer may vitrify as the reaction proceeds. Since molecular motion is much reduced when the system is below Tg, the reaction substantially stops. 

Two conditions can be distinguished. First, during a chain polymerization, the monomer effectively acts hke a plasticizer for the nascent polymer. An example relates to the emulsion polymerization of polystyrene, often carried out at about 80°C. The reaction will not proceed quite to 1(X)% conversion, because the system vitrifies. 

Gillham (107-112) pointed out the need to postcure the polymer above T oo, the glass transition temperature of the fully cured system. He developed a time-temperature-transformation (TTT) reaction diagram that may be used 

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Branches present in small numbers on a polymer chain are known to decrease Tg. This effect also can be explained using. the free volume concept. Since branches give rise to chain ends, the above arrSfysis of the effect of molecular weight on Tg can be extended to branching. Thus, if the total number ends per chain is y, then by the analysis as given above one may write... 

Figure 1.13 Effect of molecular weight on Tg of interactive excipient.

Fig. 14. Effect of molecular weight on Tg of linear and branched polystyrenes. Symbols as in Fig. 8, Ref (penetration method)...

As with resoles, the central issue in design of novolacs is molecular weight. The effects of formaldehyde-to-phenol molar ratio and formaldehyde conversion on molecular weight of novolacs has been well studied and reported [192,193]. The effects of molecular weight on most of the important properties are also available [193]. These include Tg, melt viscosity, gel time, hot-plate flow, glass-plate flow. 

Fig. 4.41 The effect of molecular weight on the elastic modulus of an amorphous thermoplastic polymer above the Tg.

Figure 2.31a represents the variation of Tg vs. M for polystyrene. The slope of this straight line can be used to deduce the value of 0, since a, and p are known. It is obtained that 0 = 80 A, a value approximately equal to half the size of the styrene monomer unit. Figure 2.31b shows a plot of Tg vs. M as can be seen, the effect of the molecular weight on Tg has very little importance when the molecular weight is high (M > 10 ). 

The effect of cross-linking on Tg can be treated along similar lines to the effects of molecular weight and branching. When two polymer chains are connected by introducing a crosslink at intermediate points, the chains are pulled closer together at these points and the free volume is decreased. This reduction in the free volume raises Tg. 

A new correlation will be developed in Section 6.B, to enable the prediction of reasonable values of Tg without requiring and thus being limited by the availability of group contributions. This correlation will account for the effect of the composition and the structure of a polymer, as reflected in the chain stiffness and the cohesive forces between different chains, on Tg. A new correlation will be presented for the effects of the average molecular weight on Tg in Section 6.C. The effects of plasticization on Tg will be discussed in Section 6.D. A new correlation will be presented for the effects of crosslinking on Tg in Section 6.E. The effects of tacticity on Tg will be discussed in Section 6.F. Secondary relaxations will be discussed in Section 6.G. The crystalline melting temperature will be discussed in Section 6.H. Finally, the ratio Tm/Tg... 

The methods described in Chapter 6 can be used to estimate Tg, which is the most important input parameter. For polymers of low Mn (Mn Mcr), it is important to apply the correction described in Section 6.C for the effects of the molecular weight on Tg. 

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... 

Figure 13.20 Schematic representation of the effect of moiecuiar weight on shear modutus-temperature curve. Tg is the gtass transition temperature white Tg is the ftow temperature. Tg, Tg , Tg- represent low- medium-, and high-molecular-weight materials, respectively.

In Section 8.9 some effects of crystallinity and hydrogen bonding on Tg were considered. The effect of molecular weight was discussed in Section 8.7, and the effect of copolymerization was discussed in Section 8.8. This section discusses the effect of chemical structure in homopolymers. 

Noting the instruments mentioned in Section 8.3, what instrument would you most like to have in your laboratory if you were testing (a) each of the three theories of Tg, (b) the molecular weight dependence of Tg, (c) the effect of cross-linking on Tg. Defend your choice. 

The effect of the molecular weight on Tg is often ascribed to the small entropy of the polymer terminal and is expressed [45] as... 

To illustrate the effect of temperature on mechanical properties, it is sometimes preferable to plot the property vs. temperature for constant values of time. For example, data of the type shown in Fig. 18.21 may be cross-plotted as (10) (the 10-second relaxation modulus) vs. T, Such a plot is given in Fig. 18.23 for several polystyrene samples," The five regions of viscoelastic behavior are evident in the linear, amorphous (atactic) samples (A) and (C) along with the effect of molecular weight in the flow region. The drop in modulus in the vicinity of Tg (100°C) is dearly seen. The crystalline (isotactic) sample maintains a fairly high modulus all the way up to (a 235 "C). Given values of one can convert data in the form vs, t at constant T (a master curve) to vs. T at constant t and vice versa. 

Fig. 21 The effect of film thickness and molecular weight on Tg of a thin polystyrene film on silicon. (From Ref. 68.)...

Kumler, P. L., S. E. Keinath, and R. F Boyer, ESR Studies of Polymer Transitions III. Effect of Molecular Weight Distribution on Tg Values of Polystyrene as Determined by ESR Spin-Probe Studies, J. Macromol. Sci. Phys., 13, 631-646, 1977. 

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