The glass transition temperature

The estimation /(oo) for the number of polymers using literary data [11] shows that the value of /( x ) is approximately constant at [7]. This fact allows a conclnsion to be made that at some critical value of /(°°)(t / ) freezing of the formation of local order domains (clusters), i.e., the physical macromolecular entanglements clnster network, is impossible because of the high thermal mobility of macromolecules (the indicated network density at is equal to zero [8]). As a result. Equation 1.37 can be rewritten as follows [7]  

To express analytically the influence of on / is difficult, however it can be supposed that the restrictions for / are intensified with the growth in v, i.e., becomes smaller. The authors [7] carried out the allowance of these restrictions empirically (as was fulfilled by Flory [12]), namely by introduction of the the factor Cv , where C and n are constants, in the right-hand part of Equation 6.1. In this case Equation 6.1 for crosslinked systems has the form [7]  

Proceeding from the fact that approximately the same value of / for all polymers was accepted in paper [7], it was calculated according to the known data for polycarbonate [11]. Since, according to the fulfilled estimations, n 1/2 gives the best correspondence with the experiment, this magnitude of n was used in paper [7]. The value of C can be determined, using the value for one of the epoxy polymers studied in [5]. Finally Equation 6.2 is transformed into the form [7]  

the availability of a chemical crosslinking network not only influences the local order level (see, for example, Eigure 5.33) that was noted earlier [13], but it also restricts thermal fluctuations of segments in clusters. This effect defines to a considerable extent the properties of crosslinked systems and, in particular, it allows the antibate change of elasticity modulus and glass transition temperature to be explained [7]. 

The authors of paper [14] used fractal analysis methods for the description of the glass transition process of crosslinked polymers. For this purpose they used the expression for the estimation of beginning time T. of the accumulation of avalanche-type defects obtained in paper [15]  

The relative size of the spherulitecan be predicted from what is known of the crystallization of monatomic solids. Low-temperature solidification leads to small spherulites because the nucleation rate is high and the growth rate low. Conversely, high-temperature solidification leads to large spherulites because the nucleation rate is low relative to the growth rate. 

The glass transition temperature is usually obtained from a volume-temperature plot of observations taken on cooling. The precise value of Tg depends slightly on the rate of cooling, being lower for lower rates of cooling. The usual cooling rate in observations of is 1 C per minute. 

The amorphous fraction of a crystalline polymer exhibits a glass transition. The ratio of the glass transition temperature to the melting point is observed empirically to be of the order of 

At temperatures between and the solid consists of rigid crystals and an amorphous fraction of low modulus, so that the solid is flexible and tough (see 

Chapter 3). Below the solid consists of rigid crystals and a glassy, rigid amorphous fraction. Crystalline polymers find extensive application at temperatures above the of the amorphous fraction it is when the amorphous pads between the crystals are rubbery that the polymeric solid exhibits that highly valued toughness typical of crystalline polymers. 

How can Tg be determined In principle this can be achieved in various ways. However, two of the methods are of special importance and are used in the majority of cases. These are temperature-dependent measurements of the expansion coefficient or the heat capacity of a sample, carried out dm -ing heating or cooling rrnis. They need only small amoimts of material, and standard equipment is commercially available. 

In view of the broadening of the step and the rate effects, it does not seem appropriate to introduce a sharply defined Tg. For practical use as material parameter and for comparisons it is sufficient to conduct the measurements with a standard heating or cooling rate ( T = (10 — l)Ks ) and to pick out some temperature near the center of the step, for example, that associated with the maximum slope. The thus obtained values of Tg have a tolerance of some degrees, but this must be accepted regarding the physical nature of the phenomenon. 

The cause for the occurrence of the steps in the heat capacity and the expansion coefficient is easily seen. Cooling a sample below Tg results in a freezing of the o-modes. The observations tell us that the a-modes affect not only the shape of a sample, but also its volume and its enthalpy. This is not at all surprising. If segments move, they produce an additional volmne in their neighborhoods. In the literature, this is often called a free volume in order to stress that it is not occupied by the hard cores of the monomers. The free volume increases with temperature because motions intensify, that 

That a corresponding behavior is found for the enthalpy and the heat capacity is conceivable. As the free volume incorporates energy, changes in the volume and in the enthalpy are interrelated and this results in simultaneous steps in the expansion coefficient and the heat capacity. 

In an analysis, one has to consider the time dependence of relaxation processes under non-isothermal conditions as imposed during a cooling or heating run. Observations suggest that we represent the sample volume as a sum of two contributions 

At temperatures in excess of in Fig. 8.2, the not very well-defined melting temperature, these polymers are completely amorphous and show many of the features characteristic of liquids, the condition being termed viscofluid. Rapid cooling of the polymer from state A (Fig. 8.2) causes it to follow the path ABC, whether it it cross-linked or not. The polymer is amorphous at all stages along ABC, shows little or no change in specific volume at and when the temperature is below it is essentially glassy. 

Polymers that show a glass transition are called thermoplastics. At temperatures above Tg they extend readily under tension, the main effect being a straightening of the polymer chains. They retain this different shape when the load is removed, though the original shape can be recovered simply by raising the temperature sufficiently. 

Linear and branched polymers that have a few strong crosslinks do not show a glass transition. They can undergo large extensions under tension, but regain their original dimensions completely when the load is removed. Such materials are known 

When a polymer is heavily cross-linked it forms a rigid three-dimensional arrangement that deforms very little under load. As these materials polymerise, a process accelerated by raising the temperature, the monomers group themselves into a rigid framework that is not softened when the temperature is raised again. Materials of this sort, for example, urea formaldehyde, are known as thermosetting plastics. 

Different assumptions for s were proposed in the literature. Barton and Lee (1968) suggested s to be equal to the weight- or mole fraction of the relevant group in relation to the structural unit. Weyland et al. (1970) put s equal to Z , the number of backbone atoms of the contributing group. Becker (1978) and Kreibich and Batzer (1979) identified s, with the number of freely and independently oscillating elements in the backbone of the structural unit. In general s is held as a kind of entropy of transition . 

With regard to the sum iSiTgi most authors see it as proportional to the cohesion energy, so, e.g. Hayes (1961), Wolstenholme (1968) and Kreibich and Batzer (1979,1982). 

It should be annotated that the form of the aforementioned equation is the same as the well-known thermodynamic expression for phase transitions of the first order  

A serious objection against such a thermodynamic analogy, however, for identifying Lj SjTgj with the molar cohesion energy is twofold  

A quite different method for calculating Tg was proposed by Marcincin and Romanov (1975). They developed the formula (also based on cohesive energy)  

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Polymers will be elastic at temperatures that are above the glass-transition temperature and below the liquiflcation temperature. Elasticity is generally improved by the light cross linking of chains. This increases the liquiflcation temperature. It also keeps the material from being permanently deformed when stretched, which is due to chains sliding past one another. Computational techniques can be used to predict the glass-transition and liquiflcation temperatures as described below. 

Many polymers expand with increasing temperature. This can be predicted with simple analytic equations relating the volume at a given temperature V T) to the van der Waals volume F and the glass transition temperature, such as... 

Note that subtracting an amount log a from the coordinate values along the abscissa is equivalent to dividing each of the t s by the appropriate a-p value. This means that times are represented by the reduced variable t/a in which t is expressed as a multiple or fraction of a-p which is called the shift factor. The temperature at which the master curve is constructed is an arbitrary choice, although the glass transition temperature is widely used. When some value other than Tg is used as a reference temperature, we shall designate it by the symbol To. 

In addition to thermodynamic appUcations, 62 values have also been related to the glass transition temperature of a polymer, and the difference 62-61 to the viscosity of polymer solutions. The best values of 6 have been analyzed into group contributions, the sum of which can be used to estimate 62 for polymers which have not been characterized experimentally. 

Polymeric materials are unique owing to the presence of a glass-transition temperature. At the glass-transition temperatures, the specific volume of the material and its rate of change changes, thus, affecting a multitude of physical properties. Numerous types of devices could be developed based on this type of stimuli—response behavior however, this technology is beyond the scope of this article. 

Glass Transition. The glass-transition temperature T reflects the mechanical properties of polymers over a specified temperature range. 

Molecular Weight. The values of the mechanical properties of polymers increase as the molecular weight increases. However, beyond some critical molecular weight, often about 100,000 to 200,000 for amorphous polymers, the increase in property values is slight and levels off asymptotically. As an example, the glass-transition temperature of a polymer usually follows the relationship... 

Elastomeric Modified Adhesives. The major characteristic of the resins discussed above is that after cure, or after polymerization, they are extremely brittie. Thus, the utility of unmodified common resins as stmctural adhesives would be very limited. Eor highly cross-linked resin systems to be usehil stmctural adhesives, they have to be modified to ensure fracture resistance. Modification can be effected by the addition of an elastomer which is soluble within the cross-linked resin. Modification of a cross-linked resin in this fashion generally decreases the glass-transition temperature but increases the resin dexibiUty, and thus increases the fracture resistance of the cured adhesive. Recendy, stmctural adhesives have been modified by elastomers which are soluble within the uncured stmctural adhesive, but then phase separate during the cure to form a two-phase system. The matrix properties are mosdy retained the glass-transition temperature is only moderately affected by the presence of the elastomer, yet the fracture resistance is substantially improved. 

This type of adhesive is generally useful in the temperature range where the material is either leathery or mbbery, ie, between the glass-transition temperature and the melt temperature. Hot-melt adhesives are based on thermoplastic polymers that may be compounded or uncompounded ethylene—vinyl acetate copolymers, paraffin waxes, polypropylene, phenoxy resins, styrene—butadiene copolymers, ethylene—ethyl acrylate copolymers, and low, and low density polypropylene are used in the compounded state polyesters, polyamides, and polyurethanes are used in the mosdy uncompounded state. 

In the area of moleculady designed hot-melt adhesives, the most widely used resins are the polyamides (qv), formed upon reaction of a diamine and a dimer acid. Dimer acids (qv) are obtained from the Diels-Alder reaction of unsaturated fatty acids. Linoleic acid is an example. Judicious selection of diamine and diacid leads to a wide range of adhesive properties. Typical shear characteristics are in the range of thousands of kilopascals and are dependent upon temperature. Although hot-melt adhesives normally become quite brittle below the glass-transition temperature, these materials can often attain physical properties that approach those of a stmctural adhesive. These properties severely degrade as the material becomes Hquid above the melt temperature. 

Improved Hot—Wet Properties. Acryhc fibers tend to lose modulus under hot—wet conditions. Knits and woven fabrics tend to lose their bulk and shape in dyeing and, to a more limited extent, in washing and drying cycles as well as in high humidity weather. Moisture lowers the glass-transition temperature T of acrylonitrile copolymers and, therefore, crimp is lost when the yam is exposed to conditions requited for dyeing and laundering. 

The glass-transition temperature, T, of dry polyester is approximately 70°C and is slightly reduced ia water. The glass-transitioa temperatures of copolyesters are affected by both the amouat and chemical nature of the comonomer (32,47). Other thermal properties, including heat capacity and thermal conductivity, depend on the state of the polymer and are summarized ia Table 2. 

The porous electrodes in PEFCs are bonded to the surface of the ion-exchange membranes which are 0.12- to 0.25-mm thick by pressure and at a temperature usually between the glass-transition temperature and the thermal degradation temperature of the membrane. These conditions provide the necessary environment to produce an intimate contact between the electrocatalyst and the membrane surface. The early PEFCs contained Nafton membranes and about 4 mg/cm of Pt black in both the cathode and anode. Such electrode/membrane combinations, using the appropriate current coUectors and supporting stmcture in PEFCs and water electrolysis ceUs, are capable of operating at pressures up to 20.7 MPa (3000 psi), differential pressures up to 3.5 MPa (500 psi), and current densities of 2000 m A/cm. ... 

In methacrylic ester polymers, the glass-transition temperature, is influenced primarily by the nature of the alcohol group as can be seen in Table 1. Below the the polymers are hard, brittle, and glass-like above the they are relatively soft, flexible, and mbbery. At even higher temperatures, depending on molecular weight, they flow and are tacky. Table 1 also contains typical values for the density, solubiHty parameter, and refractive index for various methacrylic homopolymers. 

T is the glass-transition temperature at infinite molecular weight and is the number average molecular weight. The value of k for poly(methyl methacrylate) is about 2 x 10 the value for acrylate polymers is approximately the same (9). A detailed discussion on the effect of molecular weight on the properties of a polymer may be found in Reference 17. 

Determination of the glass-transition temperature, T, for HDPE is not straightforward due to its high crystallinity (16—18). The glass point is usually associated with one of the relaxation processes in HDPE, the y-relaxation, which occurs at a temperature between —100 and —140° C. The brittle point of HDPE is also close to its y-transition. 

The value of the glass-transition temperature, T, is dependent on the stereoregularity of the polymer, its molecular weight, and the measurement techniques used. Transition temperatures from —13 to 0°C ate reported for isotactic polypropylene, and —18 to 5°C for atactic (39,40). 

A plasticizer is a substance the addition of which to another material makes that material softer and more flexible. This broad definition encompasses the use of water to plasticize clay for the production of pottery, and oils to plasticize pitch for caulking boats. A more precise definition of plasticizers is that they are materials which, when added to a polymer, cause an increase in the flexibiUty and workabiUty, brought about by a decrease in the glass-transition temperature, T, of the polymer. The most widely plasticized polymer is poly(vinyl chloride) (PVC) due to its excellent plasticizer compatibility characteristics, and the development of plasticizers closely follows the development of this commodity polymer. However, plasticizers have also been used and remain in use with other polymer types. 

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