Glass transitions in polymers

Glass transitions in polymers

9 GLASS TRANSITIONS IN P O L Y M E R S MITCHEL C. SHEN North American Aviation ScienceCenter, Thousand Oaks, California and ADI EISENBERG Departmen...

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MITCHEL C. SHEN North American Aviation ScienceCenter, Thousand Oaks, California

and ADI EISENBERG Department of Chemistry, University of California, Los Angeles,California

I. I N T R O D U C T I O N

Although synthetic glasses have been known for several millennia, and naturally occurring glasses (for instance Obsidian) even longer, the phenomenology of the vitrification process was not explored until c a . 1920.") Thus, the initial study of glass transition phenomena is more recent than the discovery of radioactivity, the formulation of the early theories of wave mechanics, and the publication of the special theory of relatively, all thoroughly modern events in the physical sciences. One excellent indication of this novelty of the field is the fact that the article by Kauzmannt2) on the nature of the glassy state lists only three references prior to 1921, none of which are directly concerned with the glass transition. Another is the fact that only rather recently have comprehensive theories of the glass transition begun to appear, and these still view the phenomenon from widely differing viewpoints. These are discussed in the original articles, as well as in a series of reviews to which the interested reader is referred."- 13) The reviews which were mentioned before usually emphasize only a limited aspect of the vitrification process, and then, usually, from a rather specialized point of view. Some textbookst14- is) do discuss the phenomenon, but due to the nature of the treatment, only a limited amount of space can be devoted to it. The need has therefore arisen for a general introductory treatment of the glass transition which will deal with the fundamentals of both the phenomenological and theoretical aspects of the vitrification process. The present article is directed primarily at those who have a fair background in the physical sciences, but no direct experience with the glass transition phenomenon and those workers who, while familiar with some aspects, might wish to broaden their familiarity with the vitrification process. 407



While early treatments of the glassy state mention synthetic organic polymers only cursorily, it is precisely this class of materials which has given a great impetus to the study of the glass transition, mainly because the uses to which polymers are put are dictated by the onset of vitrification. It should be remembered, however, that these are only one of many classes of materials which can yield non-crystalline solids.t It is therefore fitting that a review of this type should include a brief discussion of the materials which exhibit this phenomenon. Of paramount importance to the formation of a glass is the existence of some barrier to crystallization. This barrier may be due to the existence of sufficient asymmetry in the structure of the material which would preclude the existence of a crystalline state, to the absence of sufficient thermal energy (provided the material can be cooled to a low enough temperature without crystallizing) to permit the material to reorganize to a crystal, or to a combination of the above. The materials are roughly classified by decreasing degree of importance of the covalent bonds to glass formation. (1) The network glasses, ¢8'9'19'20) for instance SiO2, B 2 0 3 , P 2 0 5 , A s 2 0 3 , are those in which the bonds themselves must be reorganizing in order to permit structural reorganization to proceed at a high enough rate for the glass transition to manifest itself. (2) If network modifiers,ts' 9,19, 2o) for instance Na20 are added to the above, the network tends to break down. For instance in the case of P2Os, this leads to the existence of linear NaPO 3 segments of lengths depending on the Na :P ratio to be located between O~P(O)a crosslinks, with other types of structures occurring in the other network formers. In these materials, the mechanism of the glass transition may be either diffusional motion of linear segments of the chains o r bond interchange. (3) The linear or branched polymers are the next group, t14- is) Here belong the myriad of synthetic organic materials, for instance polystyrene, as well as the elemental glass formers sulfur and selenium. The glass transition in these materials does not depend on interchange of primary bonds, but rather on diffusional motion of chain segments, which leaves the backbone intact. (4) Next is the wide range of the hydrogen-bonded glasses t2' 9, 21, 22) in which the transition may depend on the interchange of hydrogen bonds. Examples of this group are glycerine, glucose, mixtures of low alcohols, etc. (5) Finally, the low molecular weight materialst2' 22) such as 2-methylpentane should be mentioned, which possess enough asymmetry to prevent crystallization so that they can be supercooled until the glass transition is reached. In these cases, the transition probably is dependent upon motions of the molecule t It is perhaps preferrable to refer to non-crystalline polymeric materials as "non-crystallineI' rather than "amorphous." In polymers a certain degree of order exists, particularly with resp~t to internuclear distances along the chain, even if no crystallinity is present.

Glass transitionsin polymers


as a whole rather than segmental motion, with van der Waals' bonds being involved rather than hydrogen bonds or covalent bonds. There are three additional groups of glass-formers which do not fit conveniently into any of the above categories. These are : (6) Ionic or partially ionic salts or salt mixtures, t9'23'24) examples being ZnC12, BeF2, SbFs, K2CO 3 + MgCO3 and K N O 3 + Ca(NO3) 2. (7) Solutions of electrolytes, particularly at the eutectic point, t25) for instance HC1 + H 2 0 or H2SO 4 + H20. (8) Metals. These have been deposited as amorphous thin layers ~26) and also by admixtures of small amounts of non-metals, for instance phosphorus in nickel. (27) The range of materials covered in the above eight categories is truly enormous. The glass transition range in temperature from c a . 1300°C for SiO2 to c a . - 200°C for the low molecular weight hydrocarbons. The Appendix lists the glass transition temperatures of some representative materials. While it is true that some materials have not as yet been (or perhaps never will be) made into glasses (for instance the so-called noble gases), it is also true that some materials cannot be obtained as crystals, for instance the vinyl polymers with sufficiently large pendant groups. While the number of materials which can only exist as crystals (below their melting point) is greater than that of the materials which cannot be crystallized, it is important that both types of materials exist, and this in turn lends support to the view of glasses as a fourth state of matter. It is truly amazing that in spite of the differences in the mechanism of vitrification, the same phenomena are observed over almost the entire range of materials, and that with proper techniques the same methods can be employed to detect the transition. While this review will deal primarily with the glass transition in polymeric materials, the other materials will be referred to when advisable. It should be stressed that this work is not intended as a critical review of the field; no attempt is made to provide new insights into the subject; but rather to present a unified introduction to some of the current views and the underlying experimental facts. No claim is made for a comprehensive coverage; only representative experimental data and theories are given. The presentation can be logically divided into five parts: in the first the phenomenology of the glass transition will be discussed. The second will be devoted to several representative theoretical interpretations of the nature of these phenomena. The third section will present the molecular parameters affecting the glass transition, such as chain topology and intermolecular forces, and the fourth will discuss the various controllable factors which influence the transition, among them hydrostatic pressure, diluent concentration, molecular weight, etc. Finally, the multiplicity of transitions in polymers will be reviewed.




In the glass transition range, many physical properties of the material undergo a more or less drastic change. The most familiar, and perhaps technologically most important of these is the change from a viscous or flexible (rubbery) material above the transition to a hard and brittle solid below. However, there are a number of changes in other characteristics that are perhaps even more significant from a theoretical point of view. Whereas it is quite possible that these changes are all manifestations of the same mechanism, there is as yet no general concensus as to what exactl3~ this mechanism is; it is therefore of importance to have a clear perspective of the manifestations of these phenomena. This topic will constitute the subject matter of the present section. For the sake of convenience, we divide the phenomenology into four parts, i.e. volumetric, thermodynamic, mechanical and electromagnetic. The treatment centers mainly on the glass transition temperature. Other phenomena in the vitrification range will be discussed in Section VI. A. Volumetric Properties Volume dilatometry is perhaps the most widely accepted means for locating the glass transition temperature. Generally, the specific volume of the material is determined as a function of temperature, the slope of such a curve being the thermal expansion coefficient. At the glass transition temperature there is a sudden change in this coefficient, hence one observes two approximately straight lines, with different slopes, that intersect at a certain point. The temperature at which this intersection occurs is defined as To. Actual experimental data, however, do not often yield a sharp inflection point. It is, therefore, customary to extrapolate the linear portions, and take the intersection of these extrapolated lines as Tg. This is illustrated in Fig. 1 for polyvinyl acetate. (2s) One notes in Fig. 1 that the To obtained by rapid cooling (0.02 hr) is higher than the one by slow cooling (100 hr.). Apparently, the vitrification process is a rate-dependent phenomenon. In later sections we shall elaborate on the significance of this observation. For the moment, however, we shall concern ourselves only with its phenomenology. The data in Fig. 1 were obtained by first allowing the sample to equilibrate at a temperature far above To, and then quenching it rapidly to the temperature of interest. The specific volume, V, is measured as a function of elapsed time at that temperature. At short times V(t) decreases steadily, but eventually reaches an equilibrium value V(~). Figure 2 shows a plot of V(t) - V ( ~ ) as a function of time for a series of temperatures to which the sample was quenched. Clearly, as temperature approaches the transition, increasingly long times are required to. reach the equilibrium specific volume V(~). By choosing a certain value of time, Fig. 1 can be constructed. Thus, Tg's defined by fast cooling measurements represent nonequilibrated values. In principle, then, one must wait an infinite time in order


Glass transitions in polymers

to obtain equilibrated values. For practical purposes, however, it is desirable to define To for a convenient time scale. Kovacs (28) recommended 3 min. Since increasing this time by a factor of ten only changes TO by about 3 degrees, longer times would be experimentally infeasible.




I /IB x





] I



/ t


// I 0




T~ rg

I 25 Temperature,. C

I 50

FIG. 1. Volume-temperature curves of polyvinyl acetate measured after quick quenching from well above To. Closed circles: equilibrium values; open circles: values measured at cooling rates as indicated. (After Kovacs. (2s))






FIG. 2. Volume-time isotherms of polyvinyl acetate, measured after quick quenching from well above To to temperatures indicated. (After Kovacs. (zs))



Another important condition for dilatometric determination of Tg is to make measurements over a wide temperature range. Mandelkem et al. t29) first warned that there are in fact a series of breaks in the V - T curve of polystyrene rather than one, and this is representative of a wide range of polymers. As we shall see in Section VI, each of these breaks is associated with a certain molecular process. The primary glass transition, TO, is the one that has the most pronounced change. As a consequence, a sufficiently wide temperature range is necessary for the purpose of identifying To. An additional important criterion for locating the primary glass transition is the fact that rubber-glass transition occurs at TO. The best method for observing this manifestation is to study the modulus-temperature curve, which we shall describe in a later part of this section. The most often used dilatometric technique is the displacement method, in which the thermal expansion coefficient of the material in question is found by measuring the quantity of the confining fluid it has displaced as a function of temperature. Mercury is the most convenient confining fluid, (2s'29) since polymers tend to swell in ordinary liquids. The limitation of this method is that the temperature range must be above the solidification point of the fluid, but below its vaporization or decomposition temperature. In order to circumvent this difficulty, air has been used as the confining fluid. (30) Volume change of the sample in an air dilatometer is found by noting the slight differential pressure change with changing temperatures. In addition to volume dilatometry, the linear thermal expansion coefficient measurements are often utilized.~31'32) Other volume-dependent variables have also been employed, such as refractive index,133- 35) X-ray absorption, t36) and strain-gauge methods,taT) The last of these can be used to measure TO for liquid as well as bulk polymer samples.

B. Thermodynamic Properties One of the earliest specific heat (Cp) measurements of glass was performed at constant pressure on selenium. Regnault~38)showed that at low temperatures the Cp's of the crystalline and glassy selenium are of comparable magnitude, but above a certain temperature the glass possesses a higher value. In 1928, the increase in Cv around the transition point was also found in a polymeric material: natural rubber. ~39) Subsequent investigations led to the conclusion that the temperature at which the sudden change in Cp occurs is the glass transition temperature. Although this phenomenon resembles a thermodynamic second-order transition, some important differences exist, as we shall see in a later section. Usually the observed change in specific heat, Cp, at the glass transition temperature varies in magnitude from material to material. However, an interesting correlation has recently been found. Wunderlich~4°) collected the specific heat data of some forty glasses, and recalculated them on the basis

Glass transitionsin polymers


of one mole of fundamental units of the substances. He defined the simplest molecular unit as a "bead" of the polymer chain, such a s - - C H 2--, - - C H ( C H 3 }--o r a similar unit of the monomeric materials. Thus, polystyrene has an "average bead molecular weight" of 52.07, and polyvinyl chloride 31.25. Wunderlich's rule of constant Cp then states that the rise in specific heat during the glass transition is 2.7 _ 0.5 cal per mole of "beads". This value was confirmed by computations on the basis of hole theory of liquids. ~4°) Another important property of glasses is that it apparently does not seem to obey the Third Law of Thermodynamics: the entropy of the glass was found not to vanish at absolute zero. This "residual" entropy can be obtained from the following relation :(41,42) S~ = S°obs -- S~bs


where S°obs and S~,bs are the integrals of Cp/T for the glass//crystal respectively. The enthalpy and free energy can similarly be evaluated. The residual entropy for a number of glass-forming organic liquids is 2.5 e.u./mole; and the residual enthalpy 0.6-3.5 kcal/mole. There is some ambiguity in these determinations, however, since it is always possible that these supercooled liquids may have crystallized. This difficulty is circumvented if one utilizes a polymer of different tacticity. Passaglia and Kevorkian t421 made such measurements on atactic and isotactic polypropylenes. The former does not crystallize appreciably, while the latter is 64.9 per cent crystalline. After correcting for the degree of crystallinity, these authors found that the residual entropy for polypropylene is 0.62 _ 0.2 e.u./mole, and residual enthalpy 500 + 60 cal/mole. Heat capacity measurements are rate dependent much in the same way as the volumetric properties. Figures 3 and 4 respectively, show the schematic diagrams, due to Jones, ca) of the temperature dependence of specific volume and enthalpy, and also of their derivatives, i.e. the thermal expansion coefficient (fl) and heat capacity. In addition to the fact that value of To is affected by the rate of temperature variation, there is also a "hysteresis" effect in these properties. This effect is usually more noticeable in the derivative properties. In these figures, curve 1 illustrates the behavior of glass being heated at a rate faster than its initial cooling rate, while curve 3 shows the reverse case. Curve 2 is obtained at identical cooling and heating rates. Experimentally no hysteresis effect is clearly discernible here. In a detailed theoretical analysis, however, Volkenshtein and Ptitsyn ~43) concluded that even in this case hysteresis occurs, albeit it is very small. Experimentally, besides the conventional calorimetric measurements, a convenient technique for approximate heat capacity determinations is the differential thermal analysis (DTA). ~44) This method essentially consists of the measurement of the temperature differential, AT, between the sample and an inert reference material when they are heated simultaneously at a uniform rate. The virtue of DTA is that it is simple, rapid and amenable to automatic



recording. Tgis the temperature at which a shift in base line occurs, corresponding to a sudden change in specific heat. Another useful thermodynamic parameter that undergoes transition at To is the internal pressure : (2)

P i = ( ~ E I t 3 V ) T = T(t~SIt~V),r - P

Experimentally, it is obtained from the pressure-temperature coefficient at constant volume : Pi = T(dP/dT)v






t.0 "r






- - - Normal Heating


- ' - Equilibrium



FIG. 3. Schematic diagram of volume or enthalpy dependence on temperature of a glassforming material. (1) Slow cooling; (2) normal cooling; (3) fast cooling. (After Jones.(s))

- - - Normal Heating - - - Equilibrium



i 6



/ 13)/'~ • Temperature

- -

FIG. 4. Schematic diagram of thermal expansion coefficient and specific heat dependence on temperature of a glass-forming material. (1) Slow cooling; (2) normal cooling; (3) fast cooling. (After Jones.(s))

Glass transitions in polymers


where x is isothermal compressibility. The atmospheric pressure is usually negligible in comparison with the internal pressure. By using eqn. (3b), Dole ~45~ showed that internal pressure for polyisobutylene increases from 3940 to 5910 atm (or 100 to 140 cal/cc) as the polymer passes from the glassy to the rubbery state. Allen et al. (46) made direct pressure-temperature coefficient measurements and found Pl to increase sharply at the glass transition temperature. C. Mechanical Properties In a strict sense, the volumetric properties treated in Section II A should be regarded as part of the mechanical or the thermodynamic manifestation of glass transitions. However, due to the fact that it is a more or less established tradition to accept the inflection in the V - T curve as the definition of TO, we have chosen to discuss it separately so as to emphasize its importance. In this section, we shall treat other changes in mechanical properties during vitrification. The study of viscosity-temperature relationships is one of the oldest methods employed for determining the glass transition region. Generally, viscosity increases with falling temperature. As TO is approached, the increase becomes steeper. Figure 5 illustrates this phenomenon for a soda-lime-silica glass. 22

.-_ 18

£ ~


, 0



400 800 TEMPERATURE (=C)

, 1200

FIG. 5. Viscosity-temperaturerelation of a soda-lime-silicaglass. (AfterJones.~8)) Early in 1929, Parks and Gilky t47) summarized a large amount of viscositytemperature data of glass-forming materials, and suggested To as the temperature at which the viscosity becomes 1013. Subsequent workers t48' 49) concurred. In fact, the glass transition has been regarded as an iso-viscous state. For polymers, the study of viscoelastic properties as a function of temperature is even more important. In general, there are five regions of viscoelastic behavior for an amorphous polymer t~4) (Fig. 6). The region of interest is the temperature range where the tensile or shear modulus of the polymer experiences a catastrophic drop from more than 10 l° dynes/cm 2 to about 107 dynes/cm 2.



The temperature at which the modulus becomes 109 dynes/cm 2 was arbitrarily definedt51,52) as the inflection temperature, T~. It has been shown 151, 53) that T~ is closely related to Tg. Tobolsky 114) suggested that in this temperature region polymer segments undergo short-range diffusional motions in response to the imposed strain at a rate that is comparable to the experimental time. Ten seconds is often arbitrarily chosen as the reference time. t14)



~8 /

•,~ 6

RUBBERY PLATEAU ly crosslinked)





FIG. 6. Schematic diagram of the modulus-temperature curve of a high polymer. (After Tobolsky and ShentS°)). It is important to note that the experimental time scale we are referring to here is not the rate of heating or cooling, but rather the rate of imposing an external mechanical perturbation onto the system at a certain temperature. In the case of simple unidirectional constant-strain experiment for a real substance, the observed modulus is a time-dependent quantity, thus the relaxation modulus is E,(t) = f ( t ) / s


wheref(t) is the stress and s the strain. In most modulus-temperature studies, it is necessary to adopt the value of E, after an arbitrary reference time has elapsed since the instant of deformation, such as 10 sec. In fact the complete relaxation spectrum for a viscoelastic material is itself a very important property. Often the complete range of the viscoelastic behavior at a given temperature covers as much as 10 to 20 decades of time, which is experimentally inaccessible. A method of extrapolation is usually used, which is known as "the Time-Temperature Superposition Principle"314'54) This principle asserts that the effect of temperature on linear viscoelastic properties is equivalent to multiplying the time scale by a constant factor at each temperature. When a series of isothermal modulus-time curves is constructed at various


Glass transitions in polymers

temperatures, a smooth "master curve" can be obtained by horizontal translations along the logarithmic time axis (Fig. 7). In other words,

Er, T(Kt ) = Er, To(t)


where To is an arbitrarily chosen reference temperature, and K is the a m o u n t


6 ~ -4



















Relaxational master curve of polystyrene. (After Takahashi e t al. ~52>)

of shift necessary to accomplish the superposition. If one assigns to K a simple Arrhenius-type temperature dependence: K = A exp (-nact/kT)


then Hac t


2.303 R d log K/d(1/T)


where R is the gas constant. However, the activation energy, Hact, was found to vary with temperature rather than being constant. Figure 8 shows that Hae t exhibits a maximum at a certain temperature. The temperature at which this maximum occurs has been found to correspond to the glass transition temperature. 200



150 . . . . PPMA








2E ¸









Viscoelastic activation energy of polymethyl methacrylate and poly-n-propyl methacrylate as a function of temperature. (After Saito e t al.{1%



The physical significance of the appearance of this maximum can be understood by considering the meaning of the apparent activation energy. Essentially Hact expresses the manner in which the rate of stress-relaxation varies with temperature. At temperatures above and below the transition region, a change of temperature affects the relaxation rate only slightly. In the transition region, however, a slight temperature shift will cause drastic changes in the stressrelaxation curve. The temperature at which this change is greatest, therefore, corresponds to the maximum in the activation energy. Another form of temperature dependence of the shift factor has been suggested by Williams, Landel and Ferryt59' 60) as follows : log K ( T ) = log aT = -- 17.44(T-- TO) K(TO) 51.6 + ( T - To)


This equation is found to describe adequately a large number of relaxation and retardation data of glass-forming materials. In the next section we shall discuss the WLF equation in considerable detail. When the viscoelastic property is determined by dynamic loading patterns, the modulus is often expressed as the sum of the storage modulus, G', and the loss modulus, G". Thus, ~5°' s4) G* = G' + iG"


G* is called the complex dynamic modulus. G' is that part of G* where the strain is in phase with the stress, and G" out of phase. The ratio tan 6 = G"/G'


in the loss tangent, which can be used to measure the dissipation of stored energy into heat. Similar definitions hold for the Young's moduli. A number of experimental methods are available for dynamic mechanical measurements,tS4) In free decay experiments,~61-63) the sample is given an initial strain; upon release of the strain the sample vibrates, and the amplitude of vibration is measured. The observed amplitude decays with time. The logarithmic decrement A is then defined as the log of two successive amplitude maxima. It is related to the loss tangent by the following approximate expression. A = ~ tan 6


Two types of forced dynamic tests are available. The first of these is a nonresonance type of apparatus, which directly measures the stress-strain relations to determine their complex moduli, such as the Fitzgerald transducer,~64) the Maxwell rotating-beam tester, ¢65) etc. The other is the resonance technique. By subjecting the sample to a sinusoidally varying force, its resonance frequency can be determined.~66- 69) The internal friction is then Q- 1 = A f /~/3f



Glass transitions in polymers

where fo is the resonant frequency, and Af is the half-width in the amplitudefrequency plot. Internal friction is equivalent to the loss tangent. As pointed out by Ferry, t~s) the loss tangent and associated quantities convey no physical magnitude, but are a measure of the ratio of energy lost to energy stored in a cyclic deformation. The loss tangent is a useful parameter, however, since each maximum is associated with some energy absorption process in the sample. Study of the loss tangent as a function of temperature, in fact, is sometimes referred to as mechanical spectroscopy. Figure 9 shows









o.i __ \







I 20



x~T-"" 40


F16. 9. Storage modulus (G'), loss modulus (G") and logarithmic decrement (A) of a styrene-butadiene copolymer as a function of temperature. (After Nielsen. (16))

that around To, a large absorption peak in A is observed along with concomitant changes in moduli. (16) This is attributed to the onset of relaxation processes whose time scale corresponds to the measurement frequency. The position of the peak shifts to higher temperatures with increasing measurement frequency (Section V).

D. Electromagnetic Properties The study of dielectric behavior of materials at various frequencies is one of the important tools in vitrification research. Of interest is the dielectric relaxation due to the lag in the response of a system to a change in the electrical forces to which it is subjected. A complex dielectric constant can be defined as

~* =





and the loss tangent tan 6 = e"/e'




These quantities are analogous to those in dynamic mechanical property studies. Plots of dielectric loss, along with the mechanical loss, for polyoxymethylene(7°) at 100 c.p.s, are shown in Fig. 10. A number of investigations (T1- 73) have been carried out in which the dielectric and dynamical mechanical properties were studied simultaneously for some 0.10

Mechanical 0.025




f 0.010



I t I -lOO - 8 0 - 6 0 - 4 0








Temperature, eC

FIG. 10. Dielectric and mechanicallosses of polyoxymethyleneat 100 cps as a function of temperature. (AfterIshida.(7°)) amorphous polymers. Similar loss behavior has been found for electrical and mechanical losses, although the relative magnitudes are quite different (Figs. 9, 10). It is important to note that loss maxima are found at a comparable temperature and frequency range for a given polymer for either measurement. Generally two to three maxima are present throughout the temperature range studied. The largest peak, the so-called ct peak, has been associated with the glass transition temperature, presumably due to the onset of motions of the main polymer chains. A very useful quantity that can be derived from dielectric studies is the dielectric relaxation time. It is defined as the time required for the polarization in a dielectric to decrease to 1/e of its original value, where e is the natural logarithmic base. The polar molecules in an alternating electric field rotate to achieve a dynamic equilibrium distribution in molecular orientations with a corresponding dielectric polarization. (74) This relaxation process is structure dependent, and the relaxation time is a direct reflection of the rotatory motion of the molecules for structural rearrangement. Kauzmann (2) summarized the available dielectric data for a number of glass-forming materials and compared them with their respective glass transition temperatures. As indicated in Fig. 11, extrapolation of these curves yields values of the dielectric relaxation times in the range of minutes in the temperature region in which the glass transition is normally encountered. This is the time scale at which the Tg's were determined.


Glass transitions in polymers

From this evidence he concluded that vitrification is a relaxation process in which molecules jump from one equilibrium position to another owing to thermal agitation. Dielectric measurements, of course, require the presence of polar groups. For materials that contain these groups dielectric measurement is often an advantage in that it facilitates the identification of the type of motion responsible for the observed dielectric dispersion. This is a very useful tool for complementing dynamic mechanical studies, where such identification is difficult.





' J!JJ!J!

~ I0 ° I1: ~o 102I--

~ I04i ,o~




4~o '



FIG. 11. Correlation of Tgwith dielectricrelaxation times, the range of the glass transition temperature is indicated by arrows. (AfterKauzmann.~2)) Another valuable method for studying motions in solid materials is nuclear magnetic resonance. N M R is particularly amenable to studying materials that contain protons, which is usually true for organic polymers. A nuclear magnet is known to be strongly affected by its local environment as well as by its own motion and that of its neighbors. The motion has the effect of averaging out the local field variation, and thereby sharpen the resonance line. This phenomenon is known as "motional narrowing". (75) The line narrowing occurs when frequency of motions becomes equal to frequency of the line width itself. Thus, below the glass transition temperature, protons in the glass-forming material give rise to a broad N M R line since their motions are relatively sluggish. As temperature increases, increased motion narrows the spectrum until Tgis reached when a pronounced narrowing is observed. Figure 12 shows the plot of line width as a function of temperature for polyvinyl chloride. (76) The transition region is seen to be in good accord with the mechanical loss maximum and shear modulus data. In this section, we have presented the manifestation of glass transition in a number of properties of the glass-forming materials. These observed phenomena



have been instrumental in providing insight into the nature of vitrification. However, there are a number of other important property changes associated with glass transition that we have chosen to omit from the present discussion; these include electrical conductivity, (77) gas permeability, (79-81) chemical luminescence,(S 2) fluorescence,(S a. s4) etc. 9.0



"'.~H 8.0



I0 I










o n


3.O 2.0






0.5 ,



200 250 300 TEMPERATURE °K



FIG. 12. N M R line width (AH), shear modulus (G) and logarithmic decrement (A) of polyvinyl chloride as a function of temperature. (After Sauer and Woodward. (76))



As is evident from the preceding phenomenological description, in the glass transformation range the sudden changes in the thermodynamic properties of glass-forming materials are accompanied by concomitant rate effects. Inasmuch as both of these observations are equally characteristic of this behavior, this unique feature has caused much confusion concerning the origin of the glass transitions. Over the years a number of theories have been proposed, most of which, however, tended to emphasize only one or the other aspect. Considerable controversy ensued, and workers in the field were polarized into proponents of either equilibrium or relaxational theories. The former asserted that the glass transition represents a true equilibrium thermodynamic second order transition, while the latter regarded it an artifact due to the experimental impossibility of performing infinitely slow measurements. With the accumulation of both experimental and theoretical evidence, however, it has become increasingly clear that these seemingly contradictory viewpoints are actually

Glass transitions in polymers


explaining different aspects of the same phenomenon. They are in fact compatible with rather than contradictory to each other. In the following discussion we shall attempt to show that the observed glass transition is most probably a result of the kinetic manifestations of the approach to an equilibrium thermodynamic real transition. The fact that this "real" transition may never be observed empirically does not preclude the possibility of its existence. In this section, we shall examine various aspects of the glass transition while leading finally toward this conclusion. For the sake of simplicity, we shall only present representative theories in some detail rather than treating all of the existing ones. Pertinent references, however, will be made whenever appropriate. A. The Free Volume Theories

Before embarking on our discussion of the kinetics and thermodynamics of vitrification, we first introduce theories of glass transitions based on the free volume model. First developed by Eyring and others,t85-87~ this model is intuitively simple to grasp. Fox and Floryt88~ in 1950 used it to explain the glass transitions phenomena. Although the free volume remains operationally ill-defined, it has since been successful in illuminating many facets of the phenomena of vitrification. According to Fox and Flory, glass transition occurs when the free volume reaches some constant value, which does not change any further as temperature decreases below Tg. They separated the specific volume, V, into the sum of the free volume, VI, and the occupied volume, Vo. The latter is defined by Vo = V' + flgT


where V' is the hypothetical volume of the glass at absolute zero, and fig the thermal expansion coefficient of the glass. As we shall subsequently see, VI can be determined by the WLF equation as a universal constant for all glass-forming materials. Recently, however, Simha and Boyer(89) suggested another method for calculating free volume, primarily for polymers, at Tg. Supported by considerations of the principle of corresponding states,(9°- 93) these authors found T0(fir - fig) = 0.113


In eqn. (16) fir is the thermal expansion coefficient in the rubber state. This finding implies that at the glass transition temperature, the free volume constitutes approximately 11.3 per cent of the total volume of the material. This value is much higher than 2.5 per cent predicted by Williams, Landel and Ferry. This difference is due to the difference in their definitions of free volume, as illustrated in Fig. 13. A more detailed, microscopic definition of free volume has been offered by Turnbull and Cohen. ~94)The molecule, or the monomeric segment of a polymer, is considered to be in a cage formed by its nearest neighbors. The work, U(R),



required to remove it from the center of the cage to vacuum at 0°K is assumed to vary with the cage radius R-in the manner described by a Lennard-Jones potential function. U(R) reaches its minimum value at some value of the radius Ro. As temperature rises, the specific volume per molecule ~ increases. This increase should be proportional to the increase in the average c a g e radii, R. Thus,

(~ -

Vo)/V o =





where v0 is the volume of a molecule, and A a constant. The quantity (~ - v0) is considered to be the "excess volume", A~.

"~ ~ "~"~"-.~...--~ o







FIG. 13. Schematic diagram of free volume as defined by Fox and Flory(as) or Williams, Landel and Ferry(s°) (WLF), Simha and Boyer(sg) (S-B) and Turnbull and Cohen(94) if-C). At Ro, the potential energy is at a minimum when the excess volume is distributed uniformly a m o n g the cages. Non-uniform redistribution will increase the potential energy. This increase is greatest near Ro, but decreases as /~ becomes larger than Ro, and approaches zero when R is in the linear region of U(_R). Turnbull and Cohen defined that part of the excess volume which can be redistributed without energy change as the free volume, vy'. Thus, A~ = v~ + Ave


At low temperatures, there is no v) and the glass expands thermally like a crystalline solid, the expansion Ave being due to the anharmonicity of the vibrations of the molecules. When the excess volume increases to some criticai

Glass transitions in polymers


value A~o, corresponding to R in the linear range of U(R), the free volume v'r begins to be added and the material experiences a glass transition. Apparently, the Turnbull-Cohen definition of the excess volume, Ate, is analogous to the free volume, vI, according to Fox and Flory or Williams, Landel and Ferry. The free volume, v~, is related to vI via eqn. (18). The Turnbull-Cohen (T-C) free volume is illustrated by the shaded area in Fig. 13. Although the precise definition of free volume is as yet not generally agreed upon by the various authors, there is actually no conflict between the existing ones. The basic concept is the same for all of them, and each definition is closely related to the others. Since most of the workers to date have been following the W L F definition, we shall base our discussion on this free volume model simply for the sake of convenience. The role of free volume in the vitrification process lies mainly in its relation with molecular mobility. This idea was first used by Batchinski ~95) in describing liquid viscosity. More recently Doolittle ~96) found an empirical equation for viscosity based on this concept : ~1 = A exp (BVo/Vs)


where A and B are empirical constants, and Vo and VI are the occupied and free volume per gram, respectively. The Doolittle equation can be derived by considering the molecular transport as occurring in a liquid consisting of hard spheres)94, 97) These molecules are confined in cages. Fluctuations in density of the liquid cause redistribution in free volume, and as a consequence a hole may be opened up that permits the molecule to undergo diffusive motion with the gas kinetics velocity, u. The diffusion coefficient of this molecule is then :

D(v) = g a(v) u


where v is the actual free volume associated with the molecule, 9, a geometric factor and a(v) the diameter of the cage. A critical free volume v* is required for this motion, below which the molecule must remain in the cage. The average diffusion coefficient is then,

D = ~ D(v) p(v) dv



In eqn. (21), p(v) is the probability of finding the free volume between v and v + dr. Cohen and Turnbull ~94) found that this probability can be expressed by,

p(v) = (7/vs) exp ( - 7v/vl)


where 7 is a numerical factor (½ < 7 < 1) introduced to correct for overlap of free volume, vI is the average free volume. D(v) is a slowly varying function of v, hence it is set to equal D(v*) and

D = D(v*)p(v*)




where p(v*) is the total probability : p(v*) = ~ p(v) dv = exp (-TV*/vl) The diffusion coefficient for the liquid is therefore D = 9 a* u exp ( - 7v*/vf)


Since 7v* has the approximate magnitude of Vo, and that ~l = bT/D


where b is a constant, an expression of the form of Doolittle equation can thus readily be obtained. An important consequence of the Doolittle equation is that it provides a theoretical basis for the very important W L F equation. The W L F equation es9.60) is perhaps the most successful single effort in describing the glass transition as an iso-free volume state. While originally derived on purely empirical grounds, it bears a strong functional resemblance to the previously discussed Doolittle equation. ¢961 It will be presented here as its logical sequence, as was done toward the end of the original publication. 16°) Due to its considerable success, the derivation will be given here in some detail since it will also be needed later in the description of the variation of the glass transition with various parameters. The Doolittle equation t96) can be rewritten as In r / = In A + BVo/VI


It should be recalled here that, as pointed .out by Fox and Flory,taa) the free volume decreases drastically when the glass transition is approached. We can therefore replace, without appreciable loss of accuracy, the free volume by the fractional volume, f, defined as VI/(Vs + Vo). Choosing the glass transition temperature as the reference point and substituting Vo/VI ,.~ 1/f, we get In ar = In r/T/r/Tg = B(1/f -- 1/fo)


Several authors t17'88) have taken the free volume above To to be related to the free volume at To by the relation f = fo + flI( T -



where fll is the difference in expansion coefficients of the material above and below T0. Inserting eqn. (27) into eqn. (26) and rearranging, we get




log aT = 2.303L fo//~I + T - TO


Glass transitions in polymers


which is identical in form to the WLF equation. For To as the reference temperature eqn. (28) can be written as: log ar = - 17.44

T - To 51.6 + T - To


Doolittle found that B was close tO unity. The existence of a "universal" function (eqn. (29)) implies that B, to a good approximation, must be the same for all materials; therefore, it was taken ~59' 6o) as unity. With this in mind, one can compare eqns. (28) and (29) with the result that

fo ~

0.025, fly ~ 4.8 x 10 -4 ~C-1


The value of fly is indeed close to a universal constant, as has been confirmed for a wide range of materials (see, for instance, Simha and Boyer, ~s9) Table 1). The value offo, however, is only one of many which have been found by various techniques as we have already seen. Excellent discussions of this are found in the papers by Boyer ~11) and Bondi.! 9s) Over the past 10 years the WLF equation has been used very widely and successfully. Among many other examples it was extended by Bestul t99) to inorganic glasses, who showed that for those materials (B/2.303fg) had a range of 13 to 20 while (fg/fls) ranged from 30 to 380. It should be mentioned that the shape of the curve is not extremely sensitive to small variations in the value of the constant. Among the subsequent developments of the W L F approach must be mentioned the treatment of polystyrene by Williams ~1°°) who found the constant B in the Doolittle equation to be 0.91, independent of temperature and molecular weight; while the constant A is strongly molecular weight dependent, showing an inflection point on a log-log plot at M ~ 35,000, the slope above that point being 3.4. Vo was found to be only slightly dependent on the molecular weight. The latest work which bears a strong relation to be W L F treatment is that of Adam and Gibbs, ~1°~) which we shall discuss at the end of this section. It is important to point out that since below To there is presumably no further collapse of the free volume, the W L F equation is not applicable. On the other hand at high temperatures this equation reduces to the Arrhenius form, because in this temperature range the relaxation processes are probably governed by a number of other factors as well. Ferry tl 5) suggested that application of the W L F equation should be limited to the range from TOto TO + 100°. However, in a recent treatment of the viscosity of polymers, Fox and Allen ~1°21 extended the applicability of the W L F equation to higher temperatures by the addition of an Arrhenius term to the W L F term in the computation of the monomeric friction coefficient. Although the iso-free volume concept has been eminently successful in interpretating the glass transition phenomenon, recent evidence t 11, lo3 - l O 5 seems to indicate that the critical free volume may not be a "universal" value.



Miller t1°3~ in applying a modified Arrhenius treatment to the viscosities of polystyrene and polyisobutylene, showed that the free volume at TOincreases slightly with molecular weight, as found by Williams31° 1~A similar observation was also made in a recent theory proposed by Kanig31°4J In a recent review, Boyer~1~ noted that in addition to its variation with molecular weight, the fractional free volume at TOalso varies with degree of crosslinking, copolymer composition and other structional details such as bulkiness of side-group and chain stiffness, etc. Thus the iso-free volume concept must be regarded as approximate, and its quantitative application is meaningful only under restricted conditions. B. Kinetic Theories

In the preceding part of this section, the glass transition was represented essentially as an iso-free volume state of a glass forming material. The free volumes themselves, however, were obtained as a result of some type of a rate process, for instance viscosity or expansion coefficients. The latter of these is made a rate process by the finite rate of heating or cooling of the specimen. There are several lines of evidence which tend to indicate that the glass transition itself is a rate phenomenon. This implies that the free volume approach works either because the free volume itself is kinetically determined or that it does not vary greatly with rates of experimentation, the former of which is more probable. The best known experimental evidence in support of the kinetic nature of the glass transition is embodied in the graph shown in Fig. 1, which indicates that if one determines the glass transition temperature in a cooling run, the glass transition decreases with a decrease in the cooling rate3 TM Upon heating of samples cooled at different rates, one observes hysteresis phenomena which can only be explained on the basis of a kinetic glass transition. These are shown in Fig. 3, taken from the work of Jones3 s) Curve 3 (solid part) is the result of rapid cooling, indicating a high value of Tg; a "normal" heating run (dashed line) indicates that the volume approaches its equilibrium value sooner than might be expected, i.e. that it "thaws" (if one regards for the moment T~ as a freezing-in phenomenon) at a lower temperature. Conversely, curve 1 indicates that a "slow" cooling run gives a lower glass transition, with a "normal" heating run (dashed line) indicating that the structure remains "frozen in" beyond the equilibrium point. Another line of evidence is based on dynamic results, either mechanical or dielectric. It was indicated before in correlating the loss tangent maximum with the glass transition that if the frequency of measurement was decreased, the peak moved to lower temperatures. Still another line of evidence is based on precise volumetric results in the vicinity of the glass transition. Figure 2 shows that if a sample is quenched from way above the glass transition temperature to a temperature in the vicinity of the glass transition, the volume may take

Glass transitionsin polymers


quite some time to reach its equilibrium value, the length of time being dependent on the temperature. These phenomena can be explained only if the experimentally observed To is taken to be a kinetic phenomenon. Starting with a vitrifiable sample at a high temperature, we can be sure that the volume of that sample is in internal equilibrium with the temperature. If we now cool this sample to another temperature, it will take some time before the volume has readjusted, quite aside of the finite rate of heat transfer within the sample (for the sake of simplicity we will assume for the purposes of this discussion that either the rate of heat transfer is infinite, i.e. that heat is transferred instantaneously, or that it is very much faster than the volume relaxation time). These volume readjustments may be thought as being due to the "squeezing out of free volume" resulting from increased packing efficiency due to decreased thermal motion. This process clearly takes time, with the length of time required to achieve a certain percentage of the contraction increasing with decrease in temperature. As we keep on cooling, either stepwise or continuously, we eventually get to the point where the rate of volume readjustment (as measured, for example, by a fixed percentage of volume contraction), becomes of the order of seconds, minutes, hours,, days, etc. In a continuous cooling experiment, when the rate of cooling reaches the same order of magnitude as the rate of volume contraction, a discontinuity of some type is observed because as we continue cooling the volume is not given enough time to reach its equilibrium value. This inflection point is taken as the glass transition temperature for a particular cooling rate, but it is evident that a slower or faster cooling rate would give lower or higher values, respectively. The same argument can be cast in terms of the enthalpyt~O6, ~o7) or any of the other parameters, or for that matter, in terms of frequency in dielectric, dynamic-mechanical or acoustic experiments (see Section V F). To explain the relaxation feature from a different point of view, Alfrey, Goldfinger and Mark t87) first suggested the existence of two mechanisms occurring simultaneously. One mechanism responds instantaneously, and is due to changes in amplitude of anharmonic thermal vibrations as in a crystal lattice. The other is time-dependent. It involves molecular diffusion to effect configurational rearrangements so as to achieve an equilibrium structure with the temperature in question. This idea can be illustrated more clearly by the phenomenological model of Voigt for viscoelastic materials (1o,~7,1OP.lO9) (Fig. 14). Consider the volume change, AV, to correspond to strain which was brought about by the "stress": the temperature change, AT The "modulus" for the "instantaneous" mechanism is

, - Vol


where fll is the "instantaneous" thermal expansion coefficient, and V0 the specific



volume at 0°K. The time-dependent part of this model is then given by the differential equation: dAV A T = (Vofl2)AV + z(Vofl2) d t


where flz is the time-dependent thermal expansion coefficient, and x the relaxation timel Solution of eqn. (32) describes the thermal expansion behavior of a AT




FIG. 14. The Voigt model for the volume-temperature-time relationship of polymers.

material possessing a single relaxation time. For real polymers, a distribUtion of relaxation times is usually needed, in which case a series of properly constructed Voigt models must be used. Perhaps the most extensive series of investigations of volume re-equilibration in polymers was performed by Kovacs. t2s) His results are of particular significance in that they provide not only a measure of the magnitude of the volume relaxation time, but also a theoretical framework. Kovacs started with the viscosity formula originally proposed by Dienes tllo) Eh


In tt = A" + - ~ f + R ( T - Too)


where t/is the viscosity, A" is a constant, the term E ~ / R T corresponds to an activation process analogous to that of Arrhenius, and Ua/R(T - Too) introduces a temperature limit T~obelow which cohesive forces prevent any configurational rearrangement. Equation (33) is then cast in an equivalent form analogous to

Glass transitions in polymers


that of the Doolittle equation, i.e.

Eh b lnq = a' + ~ + ~


which differs from the Doolittle equation only by the introduction of an activation term. Regarding q as the bulk volume viscosity (equal to the product of the relaxation time z and the bulk modulus), rewriting eqn. (34) for z rather than r/and expressing the relaxation time (which below TOdepends not only on the temperature T but also on the time t during which the sample had been at that temperature) in its free volume dependent form, he obtains zr,, = z 0 e x P L ~ - \ ~ - - ~

+ b



where the dependence o f f on T and t is given by

f = fT + Aft = fg + fl:(T - To) + (V - Voo)/Vo~


wherefT is the equilibrium free volume fraction at T and Af is the relative free volume excess (or deficiency) resulting from the non-equilibrium state of the system at volume V,, the equilibrium volume being Voo. Rearranging the free volume expression in (35) as fT - - f + fo --fT




fo -




(flz/fo)(T - To) fo -+- flo (T -- TO)

-- fT~f


he obtains the differential equation of volume stabilization

[ b(V-Voo)Voo ] dv exp - ( v _ Vo)(Voo - vo) " v ~


at aTzg



~T ar -

tEn To - T

% -


( b / f ° ) ( T - 5)-

LX TO T (fo/[3:) +


,~ _ ToJ


which depends only on T. Equation (39) resembles very strongly the WLF equation except for the correction term involving the activation energy. Limiting oneself to the temperature range where V~ > Vo, one can rewrite eqn. (38) as exp ( - S) ds S "i - (fr/b)S

dt at%



S = (b/fr) - (b/f), or S=

bV~ (V - V~) (v-

Vo)(V~ - Vo)

= - In ( z r t ' ~




Equation (40) can be integrated to give

E,( -

S,) - E , ( - S) - ~

[exp ( - S,) - exp ( - S)] -







aTZ o

where x

Ei(x) = ~ e - " (du/u), oo

and the subscripts i refer to initial conditions. The solid lines in Fig. 15 have been calculated from this equation with f35 = 0.0250, flf = 3.6 x 10-4°C -1 Ta = (0.015 + 0.005) hr and t~ = 0.01 hr In general, eqn. (42) has been found to reproduce experimental values to better than 5 per cent in the range where Is, I < 10, in other words for relative changes in volume of ca. 4 x 10- 3









FIG. 15. Volume-time isotherms of glucose, measure after quick quenching from 40°C to temperaturesas indicated.Solidcurveswerecalculatedfromeqn.(42).(AfterKovacs/X2~)

This theory has been cast in terms of a single relaxation time. An extensive discussion of the implications of the one-relaxation (or retardation) time versus multiple relaxation time theories has been given by Goldstein (13) to which the interested reader is referred. A number of microscopic relaxation theories have been published. Huggins (111) first carried out some semi-quantitative calculations. It was assumed that two energy states, separated by a gap, are available to the vitrifying material. The freezing-in process is then a consequence of the insufficiency of the thermal energy to surmount the energy gap. Later a detailed relaxation theory of vitrification was proposed by Volkenshtein and Ptitsyn/43) More recently, Wunderlich ~1°7) discussed it in light of the Hirai-Eyring Illz- 1~4) hole theory.

Glass transitions in polymers


The latter two theories regarded the vitrification process as a chemical reaction involving the passage of kinetic particles from one energy, state to another. For the sake of simplicity, only the first order reaction will be considered here. The transition from the unexcited state (1) to the excited state (2) will thus involve only one kinetic particle. Their energy relationship is illustrated in Fig. 16. The quantity AE is the activation energy. If the number of excited





FIG. 16. Energy diagram of a vitrification process. (After Volkenshtein and Ptitsyn.(43)) kinetic particles, or the high energy conformations, is N out of the total Nt particles, then dN dt




t -




where w12 and w21 are the transition probabilities of 1 ~ 2 and 2 ~ 1. They are functions of temperature and pressure. At steady ;state, eqn. (43) is equal to zero, and the equilibrium number of high energy conformations is Ne


g/(1 + g)


where g -- w12/w 21. Ne may also be described by a Boltzmann distribution: N e = N, exp ( - A E / k T )


Now defining relaxation time z as


z -


WI2 + W21

eqn. (43) becomes dN dt


1 (N z




If the temperature is varied at the rate of q2F

dT dt




then the change of N with temperature is dN



d T = qr ( N e - N)

The solution of eqn. (49) is

fqeXp T

N = No exp [ - ~(T)] + exp [ - ~ ( T ) ]




where T ¸

to The subscript zero here refers to the quantities in question at time zero. One notes that No can be used to describe the various frozen-in states when different rates of cooling is adopted. In addition, the "f]ctive temperature" of Tool t115) Ts can be found by the eqn. (45). N O = N, exp (-AE/RTI)


A detailed analysis of the solution is mathematically quite involved, and hence will be omitted here. Suffice it to say that by adopting this approach, the relation between Tg and the rate of temperature variation can be predicted in a form resembling Fig. 3. Volkenshtein and Ptitsyn ~4a) further concluded that, as we have already mentioned earlier, hysteresis invariably occurs upon cooling and infrequent heating, even if they were conducted at identical rates. These predictions were later confirmed experimentally by the calorimetric data of Volkenshtein and Sharonov. ~ 16) C. Thermodynamic Theories

The first significant thermodynamic treatment of the vitreous state perhaps belongs to Lewis and Gibson, t117~ who in 1920 predicted that supercooled liquids would not obey the Nernst Heat Theorem. This prediction was soon confirmed by Simon tl is) in his measurement of the heat capacity of glycerol in the liquid and crystalline states down to 10°K. Using the crystalline specific heat as a reference level, Simon showed that the entropy of glassy glycerol at absolute zero would be 5 cal/mole deg rather than zero. It was l~ointed out, however, that although the crystal is in its equilibrium state, the glass is at best metastable. Parks, Hoffman and Cattoir, "19) for instance, found their glucose glass to crystallize after long standing. If the experiment is performed slowly to allow the supercooled liquid to equilibrate, the "liquidus" curve can be extended as in Fig. 26. This experiment was performed by Oblad and Newton. t12°) These workers showed that by slow measurements (,~ one week),


Glass transitions in polymers

TOcan be reduced by 15 ° and the entropy anomaly as a consequence is slightly reduced. These observations, of course, reflect the kinetic phenomena of glass transitions as we have just discussed. We shall now proceed to examine their thermodynamic implications. According to Simon, as a material is cooled through its transition temperature, its diffusional motion on a molecular scale is slowed down. As a consequence, the configuration cannot change sufficiently rapidly to reach equilibrium, and a "degree of order", Z, is frozen in. Cessation of this configurational contribution was held responsible for the observed discontinuties in heat capacity. As time elapses, however, a new value of Z will be reached


-"- 4 0

// e/



"5 2 0

r, J i


200 Temperoture (*K)


FIG. 17. Enthalpy-temperature curves of glycerol in liquid (a), supercooled liquid (b), glass (c), and crystalline states (d). Curve (e) is the expected curve if equilibrium condition prevailed for the supercooled liquid. (After Jones. (s))

which confers a different value on the associated property in question. At temperatures well below T0, changes in Z become so slow that it can be regarded as being fixed. Later Davies and Jones, (4' 121) based on the idea of Simon, defined Z as a measure of the configurational order which is continuous, and remains constant if pressure and temperature are changed rapidly. In order to assess the physical significance of the ordering parameter, Z, these authors considered the X-ray diffraction studies of some liquids. They pointed out that the liquid alkali metals, for instance, achieve an increased order with falling temperatures, as evidenced by the study of Gingrich (lz2) on the X-ray radial



distribution functions of these liquids. Unfortunately, no such investigation was conducted for the more complicated glass-forming materials. According to the order parameter theory, for a certain rate of temperature or pressure change, the existence of the discontinuity at Tg is entirely a result of the inability of the configurational contribution to take effect within the experimental times. (This concept is effectively the same as the "fictive temperature" idea of Tool. t115)) For sufficiently long times, then, the equilibrium curve would be expected to obey the Nernst Heat Theorem. The implication that the entropy of an equilibrated glass vanishes at absolute zero immediately 1.0




/ / //

........ ,/ ...." .. .-'/ GLft..CEROL."~,../ ....................... .................. / GLUCQSE'~':'"/ 1 / LACTICACID~ ../ / .................................. .'i 0.2


/ /






, i 0.6

, OB



FIG. 18. Differences in entropy expressed as fraction of the entropy of fusion between the supercooled liquid and crystalline phases as a function of reduced temperature for some organic liquids. (After Kauzmann. (2~)

raises the question whether the glass should have the same structure as the crystal. If so, one would then see an amorphous material continuously change into a crystalline one. This would be highly unlikely. To look at this somewhat differently, Kauzrnann t2) plotted the differences in entropy between the supercooled liquid and crystalline phases as a function of reduced temperatures, T/T=, where T,, is the melting temperature. Figure !8 shows that by extrapolating the equilibrium curve to low temperatures, one is led to an embarrassing conclusion that the entropy becomes negative when absolute zero is approached. To resolve this paradox, Kauzmann noted that the free energy barrier hindering crystal nucleation decreases with decreasing temperature, whereas the barrier for molecular flow has the opposite temperature dependence. At a certain "pseudo-critical" temperature, then, these barriers would become equal, and the liquid will crystallize first. It is, therefore,

Glass transitionsin polymers


not possible to obtain an equilibrated supercooled liquid below this temperature, and the question becomes "operationally meaningless".(2) If this ingenious argument has turned the problem into a metaphysical one for most glass-forming materials, it faces some difficulty in the case of polymers. For many atactic polymers, the topological configurations are such that it is virtually impossible to pack them into a regular array. As a consequence the prior crystallization concept cannot be employed here to resolve "the Kauzmann paradox". The question now arises, even though the observed glass transition is almost certainly a rate phenomenon, whether there may be a true thermodynamic transition underlying the kinetic transition. We recall from the discussion in the previous section that at the transition temperature, the enthalpy H = F -



F being the Gibbs free energy, is continuous, while the specific heat C p = (c3H/t?r)p = -



undergoes a discontinuous change. Several workers, TM x23.124) had suggested, on the basis of this observation, that the vitrification process is a second order transition in the sense of Ehrenfest.t125) Although these discontinuities are quite similar to those in the order-disorder transition, the latter are in true thermodynamic equilibrium. The rate of cooling affects only the shape of the curve as, for instance, in the case of the ), transition in ammonium chloride. Inasmuch as the glass transitions as observed are not at equilibrium, such identification does not seem desirable. Recently, Gibbs and DiMarzio t126-13o) in a series of papers made a renewed effort to resolve this apparent impasse. These authors argued that even though the observed glass transitions are indeed kinetic phenomena, the underlying true transitions can nevertheless possess equilibrium properties, albeit they are difficult to realize. At infinitely long times, a second order transition can be achieved when the material reaches equilibrium. They predicted that in an infinitely slow experiment, one could eventually obtain a glassy phase whose entropy becomes zero. Figure 19 shows the plot of the theoretical curve of the happens is designated as T2 which, as we shall see later, lies some 50°C below the observed To. , Gibbs and DiMarzio envisage the glass transition process as solely a consequence of the change in configurational entropy with temperature. As the temperature is lowered, the number of states available to the material decreases, causing the observed kinetic sluggishness as the transition temperature is approached. At the transition temperature, the equilibrium configurational entropy becomes zero. Figure 19 shows the plot of the theoretical curve of the configurational entropy as a function of temperature. The dotted line represents the counterpart of Kauzmann's extrapolation. Since the configurational



entropy becomes zero at the transition temperature T2, the negative entropy paradox is therefore removed. This theory is in some sense similar to the order parameter theory, if the configurational entropy is regarded as the ordering parameter. In this part we shall discuss the Gibbs-DiMarzio theory (hereafter referred to as the G - D theory) in some detail, since it is perhaps the most quantitative

o t-

Z 0


,7 / s











FiG. 19. Schematic diagram of the configurational entropy of a polymer as a function of temperature according to the G - D theory. (After Gibbs and DiMarzio. "28~)

molecular theory to date which is particularly applicable to polymers. The central problem of the theory is to find the configurational partition function from which the expression for the configurational entropy can be calculated. The word configuration refers to "the specification of the state of the whole system", t129) It includes the conformation of all the polymer molecules as well as their locations and orientations in the system. It is assumed that the vibrational and electronic degrees of freedom do not interact significantly with the external translational and rotational states of the molecules, so that the configurational partition function can be factored out of the total partition function.t The quasi-lattice model aa2~ of Meyer-Flory-Huggins was utilized. The problem is now to place nx linear polymer molecules, each of which consists of x monomeric segments, and no holes into the quasi-lattice. Conformations of these polymer chains are dependent on the rotation around the single bonds along the backbone of the polymer chains. These hindered rotations are subject to a potential energy such as the one shown in Fig. 20. The energy, el, is associated with one of these orientations, and another energy, e2, is associated with the remaining z-2 orientations, z being the coordination number (e.g. for organic t Some recent evidence,tlal~ however, seems to suggest that the electronic degree of freedom may play a more significant role than previously suspected.

Glass transitions in polymers


polymers with carbon atoms in the backbone of the chain, z = 4). These energies represent the intramolecular contribution; the intermolecular energy is given by the hole energy, ~. The sum of these then constitute the total energy of the system. The configurational partition function should thus be of the form:

9. =

Y~ • • • f i n x . . . ,

w(f,n~... ,Ln~..., no) no

exp [ - E(flnx...Jinx..., no)/kT]


wheref~nx is the number of molecules in conformation i. W is the total number of ways that the nx polymer molecules can be packed into Xnx + no sites on the




FIG. 20. Schematic diagram of the potential energy of a polymer as a function of angle of rotation around the chemical bond. (After Gibbs." 29))

quasi-lattice. An expression for W was previously given by Flory (133) for nx polymers chains and no solvent molecules, which was used by Gibbs and DiMarzio in their calculations. Since the energy of the system is known, the configurational partition function can thus be formulated. With the knowledge of Q, and that from statistical mechanics,

S = k T f\~ l ~T n Q ~Jv, N + k l n Q


one in principle can perform all the necessary calculations. The detailed mathematical procedures will be omitted here. After some labor, G - D theory arrives at the following expressions for T2 when the configurational entropy becomes zero.t t The assumption of zero configuration entropy is a convenient, but not necessarycondition for the second order transition to occur. The "ground state" for amorphous packing may be degenerate, in which case the entropy becomes a constant. This approximation, however, does not affect the essential features of the theory.



Soo.~(T~) -


n~k T2






In {[(z - 2)x + 2] (z - 1)12}


Iv, \:12- ~ Vo / Vo\ = ln/'°] + ~-. In I -~- I


where ~o



Vx \So}

and 2 (~_~2) _ X - x 3 { ln[1 + ( z - 2 ) e x p ( - e / k T ) ] e (z - 2 ) e x p ( - e / k T ) + kT 1 + ~-- ~exp~-T)J


In the preceding equation, Vo is the fraction of unoccupied sites : Vo = no/(no + x nx) S° = V° Sx= 1-So;

- z~ 1 V~= 1 -

(59) ,




Equation (56) is a simplified expression suggested by Moacanin and Simha. (134) Clearly, the G - D theory predicts that T2 is a function of intramolecular energy e(= e2 - el), the hole energy, a, and the degree of polymerization x for a given polymer with fixed coordination number z. As we have mentioned earlier, T2 is not equal to T0, hence a direct experimental comparison is not possible. However, since they are considered to be closely related, variation of Tz'with some experimental parameter should parallel that of TO. Therefore, by setting T2 = TO in eqn. (56), ~ can be chosen for a family of curves with different values of ~ that displays the variation of T2 with degrees of polymerization. It turns out that these curves are quite insensitive to ~, hence for all practical purposes a can be set to be infinity. The agreement has been found quite satisfactory. The insensitivity of T2 to at merits further consideration. Moacanin and Simha" 34) noted that for large values of x, the condition for T2 can be satisfied by the relation -q~ = 2 (eqn. (56)). Figure 21 is a plot of values e/kT2 vs. a/kT2 that satisfy the preceding relation for a series of values of z. First one notes that since e/kTz has a finite value, as e ~ 0, T2 must also vanish. This implies that for polymers having zero chain stiffness, the transition temperature would be absolute zero. In addition, Fig. 21 shows that for a certain polymer with a fixed coordination number, above a certain value ~ exerts no influence on T2. In most polymers, the intermolecular energy is quite high, e.g. c~/kT2 --- 1.19

Glass transitions in polymers


for polystyrene. As a consequence, G - D theory predicts that T2 is independent of intermolecular interaction. In the next section, we shall see that this prediction was not borne out by experimental findings. Moacanin and Simha further showed that for ~ - , oo, Vo ---, 0 at all temperatures and lim (fir - fig) Tz = 0 vo ~o


For finite values of T2, ~r - ~o is zero at the limit, which means that no break will be found in the volume-temperature curves for polymers possessing infinite intermolecular attraction. Despite the preceding cited inconsistencies, the G - D theory has been successful in explaining many other aspects of polymer vitrification, e.g. effects of plasticizer,(128. 129) copolymer units, (135) and degree of crosslinking, (13°) etc.








16 0

I 0.4

I 0.8

[ 12

I L6








a /8T 2

FIG. 21. Relationship between T2 and the energy parameters e and a according to the G D theory. (After Moacanin and Simha."34))

In later sections we shall have occasion to discuss some of them. We also note that these inconsistencies appear to relate more to the molecular model utilized in actual calculations rather than the basic philosophy of the theory itself. However, it is difficult to assess just what approximations are introduced by committing to a definite physical model, or for that matter, whether these inconsistencies can be removed by choosing a model that better represents the system.

D. A Unifying Treatment We see in our discussion thus far that the relaxation behavior of the glass



transition phenomena can be adequately described by pure kinetic considerations. In particular, the development of the WLF equation must be regarded to be a major triumph. On the other hand, equilibrium theories, notably the G - D theory, have been successful in explaining the thermodynamic aspect of vitrification. They have, however, all tended to focus their attention on one major aspect of the phenomenon. The former emphasizes the rate effect of the observed glass transitions, while the latter concentrated on the equilibrium behavior of the hypothetical second-order transitions. As we have already noted earlier, they are in fact compatible with each other. Nevertheless, it would be desirable to seek a unified treatment for both aspects of the transition. Such an attempt was recently proposed by Adam and Gibbs ~1°2) in their molecular kinetic theory. The gist of this theory is to relate the relaxation behavior with the quasistatic properties at T2. To accomplish this purpose, Adam and Gibbs proposed the concept of a "cooperatively rearranging region", which is defined as the smallest region capable of configurational change without a concomitant change outside its own region. By definition, then, this region is equal to the size of the sample at T2, at which only one configuration is available. The central problem now becomes the evaluation of the temperature dependence of these cooperatively rearranging regions. Assume these regions as weakly interacting subsystems each of which consist of z molecules (or monomeric segments). These equivalent and distinguishable subsystems form an isobaric, isothermal ensemble whose partition function may be given by:

Q(z,p,T) = ~ w(z,E,V)exp [ - ( E + pV)/kT]


E, V

If out of the total N subsystems, n of them do allow such rearrangement, a partition function Q'(z,p,T) can then be written for them. The cooperative transition probability can now be expressed as :

W(T) = n/N = Q'/Q


Since Q is related to Gibbs free energy: Q = exp ( - G/kT) = exp (-Zl~/kT)


W(T) --- A exp ( - zAI2/k T)



In eqn. (66), A/~ is the potential energy barrier for the cooperative rearrangement per molecule. Adam and Gibbs then proceeded to show that the average transition probability for all nonvanishing W(T) is : W(T) = A exp ( - z*Al~/kT)


Glass transitions in polymers


where A is frequently factor negligibly dependent on temperature, and z* is the smallest size of subsystems undergoing rearrangement. In order to evaluate the temperature dependence of z*, one recalls that the subsystems are independent and equivalent. The macroscopic configurational entropy, So, is therefore the sum of the configuration entropy of the subsystem, Sc;

Sc=klnWc = N sc


where W~ is number of configurations of the maximal term of the partition function. For one mole of molecules, then: Sc = k In (W~ IN) = k In (W~/u~'v)


where N A y is Avogadro's number. The smallest size of the system is t h u s z* = N A v S c*//Sc


Substituting it into eqn. (67), one obtains: W (T) = A exp (-S*A#/kTS~)


Replace S~ by So(T2) - S¢(T,) = ACp In (T2/T~) (72)


Sc(T~) = ACr In (T~/T:) where T~ is the characteristic temperature of Williams, Landel and Ferry. Since the transition probability is just the reciprocal of the relaxation time, the W L F shift factor can then be expressed by : - l o g aT = -- log [W(T~)/W(T)]

= al(T - T~)


a 2 + (T - T~) where a 1 = 2.303 S*A#/kACpT~ In (T~/Tz)


a2 = T~In (TJT2)/[1 + In (T~/T2)]



We see that, based on equilibrium thermodynamic considerations, Adam and Gibbs now were able to arrive at an expression that is similar to the well-known W L F equation governing the relaxational behavior of vitrifying materials. An experimental verification of eqn. (73) was also afforded by these authors. Based on the data collected by Ferry, C15) the calculated W L F curve was shown to agree with the empirical "universal" curve (Fig. 22). An even more important



conclusion is that they found for the materials examined : (76)

TdT~ = 1.30

and Tg -


T2 = 5 5 ° C

which were found to be in good agreement with the calometric dataJ 42' 136) 0



log aT -6

-8 I -I0

-12 0

I 20

I 40

I 60

I 80

Curve 2 I I00



FIG. 22. WLF shift factor as a function of temperature (T-To).Curve 1: experimental. Curve 2: calculated from eqn, (73).(AfterAdam and Gibbs31°2~)

IV. MOLECULAR PARAMETERS AFFECTING GLASS TRANSITIONS The glass transition temperatures of polymers are profoundly influenced by their structural factors. For a given polymer, its To can be varied by changing its molecular weight, degree of crosslinking, diluent concentration, etc. These factors are usually controllable so that "tailor-made" Tg's can be obtained. However, there are a number of molecular parameters that are inherent in the structural units of the polymer itself. These parameters are, so to say, "the nature of the beast" and cannot be changed at will. In this section we shall concern ourselves with these "uncontrollable" molecular parameters that influence the glass transition temperature of a polymer. The "controllable" parameters will be discussed in the next section.

A. The Effect of Chain Stiffness At the glass transition temperature, there is sufficient thermal energy to surmount the rotational energy barrier due to the neighboring atoms along the polymer chain so that greater configurational degrees of freedom may be


Glass transitions in polymers

achieved. It is, therefore, not surprising that the chain stiffness factor plays a very important role in determining To. Two types of factors influence the chain stiffness. The first of these may be called the internal mobility, which reflects the ease of rotational motion of the main chain backbone itself. To wit, the rotational energy barrier for polydimethyl siloxane CH3




CH3 is very low and hence the chain is highly mobile. Its To is -123°C. ('4) Polyethylene (--CH/--CH2--) has a higher energy barrier (3.3 kcal/mole), hence TABLE 1. GLASS TRANSITION TEMPERATURES OF SOME AROMATIC VINYL POLYMERS



Structural formula



() Poly-o-methyl styrene

~-CH 2 ( : H I


Poly-ct-vinyl naphthalene



Poly-vinyl biphenyl

[CH:--( :H-}-


<) <) CH 3


Poly-ct-methyl styrene






higher TO(-90°C). (t 1, 14) Polytetrafluoroethylene's (--CF2--CF2--) barrier is still higher (4.7 kcal/mole), and its To is -50°C.t Addition of side groups onto a given chain backbone increases chain stiffhess, which may be called geometrical stiffness. A case in point is the substitution of aromatic groups for hydrogen atoms on a hydrocarbon chain. A familiar example would be polystyrene, which has phenyl groups on alternate carbon atoms along the main chains. The glass transition temperature of this polymer is 100°C, almost 200°C higher than that of an unsubstituted hydrocarbon chain, polyethylene. An increase of the bulkiness of the side group by ring substitution increases the glass transition temperature further (see Table 1). Adding a methyl group onto the a-carbon produces a much higher rotational energy barrier, hence Tg of poly-~-methyl styrene is 175°C. For polyacenaphthalene the mobility of the chain is so severely inhibited that it vitrifies at 264°C, over 300°C higher than polyethylene. A very valuable method of continuously varying molecular parameters is to copolymerize two monomers with differing structures. For instance, methyl methacrylate CH3


(CH2~-~-C~COOCH3) has a methyl group on the s-position. It is stiffer than methyl acrylate (CH2~---CHCOOCHa), hence the To of its polymer (105°C) is nearly 100°C higher than that of the latter (9°C). By synthesizing copolymers with increasing amounts of methyl methacrylate, the TO's increase continuously. (14°) The same can be said of the styrene-acenaphthalene copolymers, (1as) as illustrated in Fig. 23. It must be emphasized, however, that polymers consisting of disubstituted units do not always possess higher TO's. Consider, for example, polyvinylidene chloride (--CH2--CC12-- , -17°C) vs. polyvinyl chloride (--CH2--CHCI--, 87°C), and polyisobutylene CH3


( - - C H 2 - - C - - , - 65°C)


CHa vS. polypropylene (-~CH2--CH-- , - 10°C).O 6)


CH3 t There is somedispute that the Tg'sof polyethyleneand polytetrafluoroethylenemay be higher; the figuresare given only for illustrative purposes.


Glass transitions in polymers

In these cases the disubstituted polymers have lower TO's than their monosubstituted analogs. This phenomenon has often been attributed to symmetry effects. Gibbs and DiMarzid 135) pointed out that although introduction of the second pendant group in the monomeric segment of the polymer main chain increases the absolute energies of bond rotation, the flex energy being the difference e2 - el, is now lowered. According to their theory, the effect of the latter would be a reduction of To as observed. 30C

/ F


T~ {*C)




FIG. 23. Effect of chain stiffness on the glass transition temperature of polymers. Monomer 2 represents units that constitute stiffer chains. (Data of Wood (14°) and Schaffhauser, Shen and Tobolsky. (138))

Since the advent of Ziegler-Natta catalysts, stereoregularity in polymers is becoming increasingly important. The effect of tacticity on glass transition temperatures seems to be negligible in most polymers.(141' 14.2) However, for polymethacrylates, the situation is different. Syndiotactic polymethyl methacrylate vitrifies at 115°C in contrast to 45°C for isotactic polymethyl methacrylate.(143) Shetter tx44) showed that as the ester side-chain lengthens, this difference seems to disappear. However, no systematic change was found for polyacrylates.



B. The Internal Diluent Effect It must be emphasized that the presence of side chains does not always increase the glass transition temperature. Its effect depends heavily on the flexibility of the particular side group in question. Table 2 illustrates a particularly apt example of the effect of side chain flexibility on To. The relatively rigid tertiary butyl groups are seen to yield polymers of highest TO's.The more flexible secondary butyl group lowers TO; while the very flexible normal butyl group depresses it still further. For all the polymers listed, there is a reduction TABLE 2. GLASS TRANSITION TEMPERATURES(of) OF SOME BUTYL-SIDE CHAIN POLYMERS



--C~CH 3

CH 3



















CH3 ~-CH25H-] Bu

~-CH2~H-3l COOBu CH3-







COOBu {-CH2--CH-] -


in To of the order of magnitude of 100°C when the pendant group changes from t-butyl to n-butyl. The geometric stiffness brought about by the presence of bulky, rigid side groups on the main chain can be alleviated by inserting flexible chain units between the side group and the chain backbone. Dunham et al. (145) studied the softening temperatures of a series of poly-~-olefins of the following structure :

(CH2).R where R represents sec-butyl, tert-butyl cyclohexyl and phenyl groups. It was found that as the number of the intervening methylene groups, n, increases,

Glass transitions in polymers


the transition temperature decreases. Poly-5-phenyl-pentene-1, for instance, has a T9 more than 120°C lower than that of polystyrene.~" A direct demonstration of the effect of flexible pendant groups on the glass temperature suppression is given in Fig. 24. Here Tg data (145-148) for four series of polymers are plotted as a function of the number of carbon atoms in the side chains. In the case of polymethacrylates a lowering of 200°C can be achieved by increasing the length of the side chains. It is believed that long flexible side chains are essentially acting as "internal diluents". The presence of these "diluents" lowers the frictional interaction between chains. As a consequence, the backbone chain rotations can commence with less thermal 120 CiH3

lOft .




60 40



~ '°


~(°clo i,-\

,, N~,



!;, -20


-40 -60 -80


-IOC -12C

I 5

I 5

I 7

I 9


I 15

/ 15

I 17



FIG. 24. Effect of side chain lengths on the glass transition temperatures of polymethacrylates(146) (C)), poly-p-alkyl styrenes(147) (O), poly~-°lefins(t45, 14s) (A), and polyacrylates(144)(A).

energy, which in effect facilitates the occurrence of the glass transition at a lower temperature. It must be pointed out that for polymers containing still longer side chains, a reverse trend may be observed. Above certain lengths the side chain crystallization sets in, which impedes chain motions and thus increases the glass transition temperatures. 1"The softening temperatures measured by Dunham determined by differential thermal analysis. 2G

et al.

were shown to be close to the T~s



The preceding interpretation is essentially based on the free-volume concept. The increased mobility is due to the addition of excess free volume of the side chains. On the other hand, DiMarzio and Gibbs ~35) based on their statistical thermodynamic theory regarded the side chain polymer as a special case of a copolymer. These authors considered the poly-n-alkyl methacrylates to be a polymethyl methacrylate onto which segments of polyethylene have been grafted. On this basis, T~s of higher polymethacrylates were calculated by their copolymer equation (seeSection VD) with very good agreement. C. Effect of lntermolecular Forces

A well-known quantity for describing energetics of interaction in liquid solubility is the cohesive energy density (CED), which is defined by the following equation :~49) CED = Evap/V


where Evap is the molar energy of vaporization and V the molar volume. Often the square root of this quantity is referred to as the solubility parameter (6p). The latter quantity can be determined experimentally for polymers by swelling measurements. ~15°) It has been noted c~L ~4) that T0's in general increase with increasing 6p. Apparently, the higher the intermolecular attraction, the more thermal energy is required to attain chain motions responsible for the transition. Boyer ~1~) found that T0's for a variety of polymers seem to be linearly dependent on their values of 6p, although considerable scatter is observed. In addition, he noted that T0's of polymers containing stiff chains seem to increase more rapidly than those with flexible chains. An essentially empirical equation was proposed t~Sa) to relate the glass transition temperature of polymers with molar cohesive energy CED = 0 . 5 m R T o - 25m


where m is a number analogous to the degrees of freedom in expressions of kinetic energy. Hayes gave several rules for determining m, which are all concerned with the ability of the atoms or groups to rotate. It was noted ~5~) that this equation holds for most of the polymers for which pertinent data were available, except for the stereoisomers of the polymethacrylates. In most instances it is difficult to isolate the effect of intermolecular forces from other structural factors. Thus in eqn. (79), it is necessary to introduce the adjustable parameter m, the presence of which limits its usefulness. In addition, we note that the solubility parameters for polyvinyl acetate and polyvinyl chloride are 9.4 and 9.5, respectively,~4) yet their glass transition temperatures differ by 53°C (PVC, 82°C; PVAc, 29°C). Recently, Tobolsky and Shen ~152) investigated the viscoelastic properties of the homopolymer of 2-hydroxyethyl methacrylate (HEMA) and its copolymers with ethyl methacrylate, n-propyl "methacrylate and methoxyethyl methacrylate (Fig. 25).


Glass transitions in polymers

In these cases the resultant polymers all have comparable chain stiffness and side chain lengths, but differ in energetics of interaction due to the hydrogen bonding capability of 2-hydroxyethyl methacrylate units. The inflection temperatures (T/), which were shown to be closely related to Tg's, of these polymers increase with increasing amounts of HEMA in the chain. The T~of pure poly-2hydroxyethyl methacrylate, for instance, is more than 50°C higher than that of the n-propyl methacrylate homopolymer. An even stronger intermolecular force than hydrogen bonds is the ionic interaction. Fitzgerald and Nielsen ~153) in studying the viscoelastic properties 120
















FIG. 25. Effect of hydrogen-bonds on the glass transition temperature of polymethacrylates: The comonomers are: 2-hydroxyethyl methacrylate (HEMA); ethyl methacrylate (EMA); n-propyl methacrylate (PMA); and methoxyethyl methacrylate (MEMA). Data of Tobolsky and Shen. t*52)

of the salts of some polymeric acids noted that Tg's increase with the addition of metallic ions. Increasing the valency of the incorporated ions, such as Na +, Ba + +, A1+ ÷ +, etc., pronouncedly enhances the values of To. In the case of the trivalent aluminium salts, an infusion mass was obtained even for polymers that have only a small amount of acid contentl A comprehensive investigation by Eisenberg, Farb, and Cool ~154) on the effect of the counter-ion on the glass transition of phosphate chains, revealed that a substitution ofNa + for hydrogen increases the glass transition from - 1 0 ° C to +280°C, while Cu ++ raises it still further to c a . 500°C. Figure 26 gives the T0-composition plot for copolymers of H P O 3 - - N a + P O 3 . The presence of ionic forces in place of hydrogen bonds



is thus seen to increase the intermolecular attraction to raise the glass transition temperatures. As we have mentioned in the previous section, the theory of Gibbs and DiMarzio (135) predicted that intermolecular forces exert little influence on the glass transition temperature of polymers. Evidence presented in this part is obviously not in agreement with their prediction. However, it should

300 Tg vs Copolymer concentrotion




Tg (*C)








% No PO3

FIG. 26. Effect of ionic interaction on the glass transition temperature of polyphosphoric acid. Data of Eisenberg, Farb and Cool. (~s4)

be pointed out that e and a are not necessarily independent of each other. Furthermore, it has been shown that e, calculated on the basis of the rotational isomeric model, depends not only on the nearest neighbors, but also on longer range interactions. Thus, even if the rotational energy barrier for nearest neighbors is zero, e still has a finite value for the polymer as a whole. Despite these relatively minor difficulties, however, the basic philosophy embodied in the G D theory offers perhaps the best current molecular interpretation of the glass transition in high polymers. V. C O N T R O L L A B L E



In the preceding section the effect of uncontrollable parameters on the glass transition was discussed, i.e. of parameters which could not be altered without changing the chemical nature of the polymer. In this section we shall attempt to answer the following question: "Given a specified polymer, say polystyrene, how can we change its glass transition in a continuous and, in

Glass transitionsin polymers


principle, controllable (if not always reversible) manner?" In other words, we will describe the effects of such parameters as pressure, diluent concentration, molecular weight, crosslink density, comonomer concentration and rate or frequency of measurement on the glass transition. This discussion will bring us to the concept of the multidimensionality of the glass transition, which we shall review very briefly. As we discuss the effect of each of the parameters mentioned before, we shall not only present typical experimental data to illustrate the phenomenon, but also discuss it briefly from a theoretical point of view. Due to the large number of available theories, however, we shall limit ourselves only to two, i.e. the simple free volume treatment and the GibbsDiMarzio theory. Wherever advisable, we shall mention other theories, but will usually limit ourselves to a very brief discussion or even just a literature reference. A. Pressure

Several investigations of vitrification under pressure have been undertaken in the past. ~155-16o~ One of the earliest was a study of selenium by Tamman and Jellinghans, t155~ and since the phenomenological results of this work do not differ from those of subsequent studies, they will be reviewed here in some detail. No matter which theory of the glass transition one accepts (see Section III), it seems reasonable that a decrease in temperature should have a qualitatively similar effect on the amorphous samples as an increase in pressure. Taking a purely kinetic approach and starting with the sample above its glass transition temperature, an increase in pressure would tend to decrease the mobility of the segments. The effect of pressure on viscosity is well documented. ~161~ Similar arguments can be made for the free-volume approach, an increase in pressure tends to "squeeze out" free volume. In other words, it has an effect similar to a temperature decrease, and thus tends to bring the glass transition closer at constant temperature. The figures below illustrate the behavior of an amorphous sample (selenium) in the vicinity of the glass transition. Figure 27 is a plot of the volume (in arbitrary units) as a function of temperature and pressure. ~155~ Planes parallel to the V,,T axes through this figure (at constant pressure) illustrate the wellknown volume-temperature behavior, the glass transition temperature being located at the inflection point. Similar sections at constant temperature are shown in Figs. 28 and 29. It is clear from Fig. 28 that unlike the V-T plots, the V-P plots are not even linear to a first approximation, but that the slopes above and below the glass transition do differ. Figure 28 also shows the V-P plot for a temperature at which the sample vitrifies as the pressure is increased. An intersection of two lines of different slope and curvature is clearly evident, the intersection point giving the glass transition pressure Pg in analogy to the glass transition temperature, Tg. It should be recalled that in both cases we are



actually speaking of ranges rather than points; i.e. glass transition temperature range and similarly the glass transition pressure range, the intersection point being a rather arbitrary definition, as was pointed out in Section II. One additional point is worth mentioning. As can be seen in Fig. 27, the V - P - T plot defines two curved surfaces, the intersection of which gives a Volume,



I ,.°'


/"2000 ,~/'2S00

Pressure, bars

FIG. 27. Volume of selenium as a function of temperature and pressure. (After Tamman and Jellinghaus31s 5~) 1.02 1.01 I.z LO0

----- .99

w .9B IE

~ 97 966










I 12 I0 - 3





I ~-. 18

L 20

FIG. 28. Volume of selenium as a function of pressure in the glass transitionregion,40°C. (After Tamman and Jellinghaus.~155)) smooth curved line. The projection of this line on the P - T plane gives simultaneously the variation of the glass transition temperature with pressure as well as the variation of the glass transition pressure with temperature. The projection is shown in Fig. 29, along with other similar plots resulting from more recent studies.~ 16t)

Glass transitions in polymers


Several studies of dynamic-mechanical ultrasonic or dielectric properties under pressure have been performed. Among these may be listed as examples the work of O'Reilly t159) on the dielectric properties of polyvinylacetate; the study of McKinney, Belcher and Marvin I~sS) on the dynamic compressibility of rubber. In all these cases it is found that the peak (for instance in E', the complex part of the dielectric constant in dielectric measurements at constant frequency), which can be correlated with the glass transition, always moves to higher temperatures with increasing pressure.

i30 120 I10


Phenolhthalein .











60 50 f ~ sQhcln S e l e n i u m



I __ 2000 I __ I000 PRESSURE,BARS

I 3600 3000

FIG. 29. Glass transition vs. pressure for various substances. (After Eisenberg.(162~)

It is quite easy to interpret the results of the above experiments in terms of the free volume model. In developing the W L F equation, we have seen that the free volume fraction at any temperature above To w a s f = f0 + /3I (T - TO). If we let the free volume compressibility be z r, then the above equation becomes fr, P = fo + / ~ I [ r - TO(0)] - z i P


where To(0) refers to the glass transition temperature at "zero" pressure. At the glass transition temperature (at whatever pressure),fr, p = f.o, and the equation becomes

fir [To - To(0)] = z I P




and, by differentiation,

Op j l



This relation was proposed in several publications, among them those by Ferry and Stratton, ~x63) O'Reilly, (a 59) McKinney, Belcher and Marvin, ° 58) and Goldstein, ~x64) and is strongly reminiscent of the relation of Ehrenfest (~2s) for the change of a second order transition temperature with pressure. It should be recalled, however, that the glass transition as observed experimentally is not a thermodynamic second order transition in the Ehrenfest sense. F o r a second order transition, the following relation holds:

ArlAfl = T V AfllACp


where AC e is the difference in heat capacities of the material above and below the transition. However, as Davies and Jones O21) point out, if a distribution of ordering parameters exist, the above relation becomes an inequality; i.e. Ax/Afl >_- T V Aft/AC e and this is indeed found for the glass transition. (~64) However, Bianchi t16°) recently found that, for a number of polymers, the inequality is much less conclusive than originally anticipated. TABLE3. PRESSURECOEFFICIENTSOFTHEGLASSTRANSITIONTEMPERATURESFORSELECTED MATERIALS Material Natural rubber Polyisobutylene Polyvinyl acetate Rosin Selenium Salicin Phenolphthalein Polyvinyl chloride Polystyrene Polymethyl methacrylate Boron trioxide




-72 -70 25 30 30 46 78 87 100 105 260

0.024 0.024 0.022 0.019 0.015-0.004t 0.005 0.019 0.016 0.031 0.020-0.023 0.020

159 159 159 162 162 159 162 159 162 160 159

t The variation is probably due to the different compressibilitiesof ring and chain material. F o r the sake of illustration of the orders of magnitude involved, several representative (cgTo/OP) values are presented in Table 3. They were taken from two recent summaries. (159, 162) In the development of the G - D theory, pressure does not enter explicitly into the equation, except in terms of the number of "holes" in the system;


Glass transitions in polymers

therefore no description of the effect of pressure within the framework of that theory will be given here. B. Diluent Concentration Due to their technological importance, the effect of diluents on polymers has been studied widely in the past. One of the extensive reviews of vitrification of polymer-diluent systems is that of Jenckel,t~65~ who treated the subject up to 1956. The incorporation of a low molecular weight material (the To of which would be quite low, if it can be measured at all) into a high polymer would be expected to depress the glass transition of the system below that of the pure polymer. A typical curve, taken from the work of Kelly and Buechet166~ on the polymethyl methacrylate-diethyl phthalate system, is shown in Fig. 30.

? 50

0 0.7



FIG. 30. Effect of diluent on the glass transition temperature. (After Kelley and Bueche.(16%

It should be noted that a "diluent" could conceivably have a higher glass transition temperature, in which case the Tg of the system would be higher than that of the pure polymer, but cases of this type are highly unusual.~154~ In terms of the simplest free volume treatment, the effect of diluents on the glass transition is quite straightforward. The derivation of Kelly and Bueche(166~ is given below, although with slightly changed symbols to conform to previous derivations. Starting with the free volume expression for the pure polymer, i.e. f = 0.025 + fly (T - TOp)


where 0.025 = fg and Tgp refers to the pure polymers, and assuming that the free volumes of the polymer and diluent are additive, we get for the total fractional free volume of the system f = 0.025 + tip (T - TOp)V, + /3d (T - TOe)Vd




where the subscripts d and p refer to diluent and poiymer, resPectively, and V is the volume fraction. At the glass transition temperature of the system, f becomes 0.025 and T becomes T0. Rearranging, we obtain To = [flpVpTop + [3d (1 - Vp) Toa]/[[3pVp + fld (1 -- Vp)]


where 1 - Vp was substituted for Vd. Values of TOeand fin are not usually available and may be quite difficult to determine. Using the "universal" value of fld = 10-3 °C-1, as suggested by the authors, and finding the value of Ton approximately from the glass transition of one polymer-diluent combination as -65°C, the solid line in Fig. 30 was obtained. The agreement is thus seen to be good if one treats fld and/or Toe as an adjustable parameter. A range of other equations has been derived; among these should be mentioned the one due to Jenckel and Heusch, (167) i.e. Ak. To = Top Cp + TOn Cd -- Cp Cd Aft


where the concentrations are given by weight, where Ak. is an expression incorporating the derivations from additivity of the indices of refraction of the pure materials and Aft is a similar expression for the temperature coefficient of the refractive index. Experimentally, it was found that (87)

Ak,/Afl ~ TOd- To,

With this substitution the equation can be regarded as an equation without adjustable parameters which fits a wide range of data quite satisfactorily. DiMarzio and Gibbs ~120) adapted their theory to polymer-diluent systems by regarding the diluent as a low molecular weight polymer (of degree of polymerization rB) of variable chain stiffness. The configurational partition function was written for a mixture of long-chain polymers and short-chain diluents, using a zeroth approximation according to which there is a completely random arrangement of the molecules. Holes were neglected in this calculation. Their result for the configurational entropy of the system is : S (z-2) [!z - 2)r8 + 2v] krgN A - 2(1 - - v) In zr B + r B (1 - v) ln +


v (r B - 3 )


v) (ln [1 + (z - 2)exp { - A e B / k T } ] + fBA~QkT) + In [1 + (z - 2) exp { - A e ~ j k T } ] + fAAeA/kT

where f/=

(z -- 2)exp {-Aei/kT } 1 + (z - 2) exp { - A e i / k T } '

i = A,B


Glass transitions in polymers


and NA, NR are the total numbers of A molecules and B molecules, respectively; v is the volume fraction of diluent and other symbols have their usual connotations. In their original publication~129~ they treated the system polystyrene-styrene as a no-parameter system, i.e. they regarded styrene as a polymer of a degree of polymerization of one and used the same intermolecular parameters as those developed for polystyrene, with excellent agreement. The latter development~129) allows wider flexibility in that more complex diluents of varying stiffness can be treated. Due to the complexity of the equations a computer program is vital, however. It should be mentioned that a range of semi-empirical or purely empirical equations has been given in the past for the glass transition behavior of polymer diluent systems, and these are reviewed by Shen and TobolskyJ 168)

C. Molecular Weight Although the first extensive study of the variation of the glass transition temperature with molecular weight was performed in 1950 (88'169'17°~ for polystyrene, only several additional systems have been studied to date. These include polyisobutylene,a69) polymethyl methacrylate,(lw) polyacrylonitrile,~ 72) polypropylene~lV 3) and polymeric sodium phosphate, ~32) an inorganic ionic polymer. From an elementary kinetic point of view, it is quite easy to predict the behavior of a glass transition as a function of the molecular weight. Since each chain middle is restrained at both ends by other repeat units, whereas the chain end, by definition, is restrained at only one end, the mobility of a chain end will be greater than that of a chain middle at the same temperature. Thus, the more chain ends we have in a particular polymer sample, the further we will have to cool the material to reach the point at which the relaxation time (for instance for volume viscosity) is the same as that of a sample containing no chain ends. In other words, the glass transition will decrease with the molecular weight. A similar argument can be made in terms of the free volume; each chain end has a higher free volume associated with it than a chain middle, again as a result of a decrease in constraints. Therefore, the sample containing more chain ends will have to be cooled further than a sample containing fewer chain ends to achieve the same free volume; therefore, again, the glass transition will decrease with the molecular weight. It is interesting to note that, by regarding chain ends as "diluents", equivalent expressions can be derived for effects of molecular weight and diluent concentration on TO.~168) One of the earliest general theories which correlated the glass transition of a polymer with the molecular weight was that Of Fox and Loshaek./174) It was known from previous experimental data that a plot of the specific volume at the glass transition temperature vs. To was linear, a fact which was subsequently confirmed by the G-D theory. With this as a starting point, the authors developed a theory which required only a knowledge of the volume-temperature



behavior of the infinite polymer and of the monomer, the glass transition temperature of the infinite polymer, and the slope of the specific volume vs. the glass transition temperature for the prediction of the Tg vs. molecular weight curve. The detailed development of the above theory is somewhat too involved to be presented here. We will reproduce, however, the development of the simplest free volume approach to the glass transition-molecular weight problem, as given, for instance, by Bueche. (17) If 0 is the %~xcess free volume" contributed by each chain end to the polymer, then 20 is the contribution pek chain, 20NAy (where NAy is Avogadro's number) is the contribution per mole of chains, 20Nav/M is the free volume contributed by the chain ends per gram of polymer, and 2pONAv/M per cm 3 (where p is the density). This excess free volume will cause a decrease in the glass transition from To(oo) to T0. If we assume that the free volume is constant at the glass transition and independent of the molecular weight, then the excess free volume introduced by the chain ends must be exactly compensated by the free volume contraction due to cooling from To(oo) to TO. Thus, if Afls is "free volume expansion coefficient", we obtain

2pNavO M - Aflf [Tg(oo) - TO],or 2pNAvO AflfM

TO = To(oo)


= TO(oo)



a very simple relationship which allows us to calculate the approximate excess free volume per chain and if K and p are known from experiment and Afly is taken to be the difference between the liquid and glassy cubic expansion coefficients. An example of the applicability of eqn. (89) is shown in Fig. 31. For polystyrene, Beevers and White "7~) found 0 = 40 A 3, in approximate agreement with the calculated value; while for the sodium polyphosphates it is O


0.5 A 3 for ~ OP(O-Na+)2 and O


ca. 30 A 3 for ~ O P - - O H .(32)


O-Na + The difference between these two will be discussed later.

Glass transitions in polymers


It should be recalled that this treatment is only approximate in that the free volume at the glass transition temperature is not completely independent of the molecular weight, tI°3'134) and the free volume expansion coefficient is not identical with the difference between the glassy and liquid expansion coefficients. The approach is, however, very useful as a first approximation, and the figures obtained from it appear to be reasonable estimates. 310



Gibbs ~¢ DiMorzio, Z = 4


Gibbs ~, OiMarzio, Z = 8

290 280 270 260 Tg

(°C) 250 240 250 220 •


-- OH


o -- O~Na ÷ terminated

2O0 0.00







0 05

I /Pr

FIG. 31. Effect of molecular weight on the glass transition in polymeric sodium phosphate. Pr is degree of polymerization. [ ] (After Eisenberg, Farb and Coo13154~)

As can be seen from eqn. (56), the G - D theory was developed explicitly in terms of the chain length, so a calculation of the transition temperature as a function of the molecular weight, while somewhat lengthy, presents no further problems. In their original publication t~zT) the authors give a graph of Tg vs. degree of polymerization for polystyrene showing very good agreement with experiment. Beevers and White ~171) applied a modified G - D treatment to their data on polymethyl methacrylate again with good results. Eisenberg and Sasada ~32~ applied the theory directly to the polyphosphates; their results are shown in Fig. 31. This figure illustrates the fact that in this connection the G - D theory applies to polymers in which the terminal groups do not differ from the repeat units by more than a proton, an assumption incidentally, which is tacitly made in all the other derivations. If this is not the case, it can be seen from Fig. 31 that the behavior of the polymer is quite different, and, if --~O-Na + is the terminal group rather than ,-~OH, the Tg changes only



vary slightly with the degree of polymerization. The very small value of the excess free volume per chain end (,~ 0.5 A 3) indicates that ~ O - N a ÷ possesses some crosslink character. Figure 31 also illustrates the fact that a change in the lattice coordination number does not affect the Tg vs. 1/P curve drastically. It is of interest to note that if bulky chain ends are utilized, the glass transition temperature can even increase with the molecular weight, t175) In addition, Fauche¢ ~76) found that for polypropylene and polybutylene oxides, T0's are independent of molecular weight.

D. Copolymer Composition As in the case of plasticized systems, the enormous industrial importance of copolymers prompted extensive studies of the glass transition as a function of composition. In 1958, W o o d (177) summarized the current theories and presented them in one unified treatment which will be summarized here. In the simplest instance, the glass transition of a copolymer should equal some type of a weighted average of the individual glass transitions of the homopolymers Tg, and Tg~.The weighting factor was made to include the weight concentration and a constant A which is as yet unspecified. For a binary copolymer the equation thus becomes A1CI(Tg - Tg,) + A2C2(Tg - Tg2) = 0


Several authors (x35' ~78-x80) used various theoretical approaches to arrive at the above equation. One of these will be presented below. Before that, however, it should be noted that if one defines k as A 2 / A , one obtains r. =

+ (k r . - r.,) 1 -

(1 -



which is the equation of Gordon and Taylor. "Ts) Alternately, if one defines R as T~2A2/Tg~ AI one obtains from eqn. (90) the relation




-~a = C1 + R C 2


+ Tg2 .3

which was first proposed by Mandelkern, Martin and Quinn. °79) Going one step further, and setting R = 1, one obtains the relation proposed by Fox, (iS°) i.e. 1 -- =







In keeping with the emphasis of the preceding sections, it is perhaps most suitable to present the derivation of eqn. (90) in terms of free volume, as given by Kelley and BuecheJ 166) Starting with eqn. (85b), which was originally developed for a polymer-diluent system, replacing the subscripts p and d with


Glass transitions in polymers

1 and 2 respectively, and setting Aflg2/Aflg1 equal to K', we obtain Tg = [Tgl + (K'Tg2 - Tg,) V2]/[1 + (K' - 1) V2~


which is identical with eqn. (91) except that here the volume fraction rather than the weight fraction is utilized and K' has the physical significance of the ratio of two "free volume expansion coefficients". Actually, if K' is regarded as an adjustable parameter, a much better fit may be expected. Among the subsequent developments in this field may be mentioned the work of Braun and Kovacs,(181) who include an interaction term to take into account non-ideality in the "mixing" of the two systems and are able to predict and explain the occurrence of discontinuties in some binary systems. Finally DiMarzio and Gibbs t129) applied their theory to copolymer systems. Starting out with the assumption of an "average stiffness energy" of the form 8 = BA8A q- BB%


where BA and BB are the fractions of rotatable bonds of type A and B in the copolymer and eA and es are the corresponding stiffness energies. The entropies at the thermodynamic glass transition are zero, i.e.

S(eA/k T2A) = S(el3/k Tzs ) = S(BAe A + na~B)/k T 2 = 0


for the homopolymers and the copolymer. Since the entropy is a monotonic function of temperature, the argument of the above equation can be equated, i.e. '~A 13B - BABA + BB£B (97) kT2g - kT2B kT2 Eliminating eA and ~a from this equation, and expressing it in terms of weight fractions rather than bond fractions, one gets MA (0~-A~ (Z2 -- T2A)- ~- MB ( ~ ) ( Z

2 - T2B) = 0


where MA is the weight fraction of comonomer A, czA is the number of flexible bonds per repeat unit and WA is the molecular weight of the repeat unit. This equation is identical in form to the Gordon-Taylor equation t178) a~ad the general equation 0fW0od t177) except that here the constants have a completely different meaning. The assumptions introduced for the sake Of simplicity in the derivation of the G-D theory, as applied to copolymers, make this only an approximation relation; however, the authors show that it is applicable with a deviation of only a few percent to a wide range of copolymers. E. Crosslinking Very little theoretical work has been done on the correlation of glass transition temperature with the degree of crosslinking in spite of the enormous



technological importance of crosslinks in polymers, for instance in rubber. In this section two treatments will be discussed; one of these is due to Fox and Loshaek (174) and the other to DiMarzio. "29) From the most elementary kinetic considerations we can deduce that the introduction of a crosslink would tend to lower the mobility of a polymer segment and thus bring the sample containing these crosslinks closer to the glass transition than an uncrosslinked sample would be. This has indeed been observed in a wide range of materials, an example of which is shown in Fig. 32, taken from the work of Martin and Mandelkern3182)





\ I







FIG. 32. Effect of degree of crosslinking (expressed as percent of bound sulfur) on the glass transition temperatures of natural rubber. Experimental points were taken from data of Martin and Mandelkern.~182) The solid curve was that calculated by DiMarzio~t29) on the basis of eqn. (100).

The theory of Fox and Loshack again is based on the assumption of a linear relationship between the specific volume)at the glass transition and the glass transition temperature itself. By allowing for a volume shrinkage (V~) and resulting from the introduction of a crosslink, the authors derived the equation Tg = To(~ ) - K / M + K x p


where p is the number of crosslinks per gram, Kx a constant, and the other expressions have been defined before (eqn. (89)). This equation is valid only for high M and low p; the precise equation is considerably more complicated, and can be found in the original work. DiMarzio ~129~ has adapted the G - D theory to take into account the effect of crosslinking. Since under normal circumstances the introduction of crosslinks decreases the configurational entropy below what it would be if there were no crosslinks, and the glass transition occurs at a constant value of the configurational entropy (zero in the case of the "thermodynamic" glass transition), the glass transition would be expected to rise with increasing crosslinked concentration. The configurational entropy for the crosslinked system (So) consists of two parts, i.e. the entropy which the system would have in the absence of crosslinks

Glass transitionsin polyrners


(So) plus the change in configurational entropy due to the crosslinks (AS0. So can be calculated directly from the original G - D theory, which leaves the problem of calculating AS1. The number of ways of arranging a polymer molecule whose ends are tied down (for instance by crosslinks) can be calculated, t1321 even if the crosslinks do not remain in one place but move about their most probable locations. AS1 can thus be calculated as a function of the chain length between crosslinks. Making a number of simplifying approximations and assumptions, DiMarzio obtains the relation T(X)- T(O) T(O) -

K"MX/7 1 - K"MX/7


where X is the number of crosslinks per gram, M the molecular weight, 7 the number of flexible bonds per residue, K" is a universal constant, and the T's are the glass transition temperatures. The theory was applied to experimental data for sulfur-cured natural rubber, polystyrene crosslinked with divinyl benzene and to polymethyl methacrylate crosslinked with ethylene glycol dimethacrylate. The values of K" obtained for these polymers are respectively, 1.30, 1.20, and 1.38 x 1023, showing very good agreement. (Fig. 32). F. Miscellaneous Factors There exist additional factors which influence the glass transition of any polymer, but which were not discussed implicitly or explicitly in the preceding sections. Among these may be included the following : 1. The effect of strain, t129' 163.18a) 2. Thermal history. 3. Crystal!!nity,"84) etc. Since these factors are not as yet so well characterized as those that have been discussed in previous sections, they will not be discussed further here. G. Rate or Frequency Effects Up to this point, we have tacitly ignored the effect of rates of heating or cooling or of the frequency of experiments on the value of the glass transition. It was pointed out in Section II that rates do affect this value, and in Section III the kinetic aspect of the glass transition was discussed somewhat more extensively. In this section we shall investigate the effect of rates on the glass transition further. The first extensive study of the effect of rates on the position of the glass transition temperature was performed by Tool; ~185) it was he who showed that the glass transition temperature is a function of the cooling rate in a manner depicted in Fig. 3. From the work of Kovacs, an example of which was shown in Fig. 15, and which was discussed in Section III, it is possible to construct curves (Fig. 1) quite analogous to the one obtained by Tool, except that 2H



here the points are taken at a constant time after being kept at the temperature shown on the ordinate. In other words, the points are obtained by quenching the sample from a high temperature to the temperature on the ordinate and keeping it at that temperature for a specified length of time. By this technique it is possible to define TOvalues for any arbitrary cooling rate and to see how TO changes with "cooling rate", although the experiments are not done at varying cooling rates, but at various quench temperatures. Incidentally, a rough quantitative idea of the change in TO with rate can be obtained by differentiating the W L F equation, i.e. dATO d log a T

C 2 3o -- ~



as reported by Ferry, (15) indicating that the glass transition would change roughly by 3 ° if the cooling rate (log aT) is changed by a factor of 10.







570 580 TEMPERATURE (*K)

I 590


FIG. 33. Effectof coolingrate on the glass transition in D.T.A. experiments.(Afterwunderlich, Bodilyand Kaplan.(1°7))Top line: fast cooling,bottom line: slow cooling;peaks can be correlated with Tg. In a completely different type of experiment, Wunderlich, Bodily and Kaplan t1°7) in a study by differential thermal analysis of polystyrene at varying heating rates showed that the apparent heat capacity shows a maximum in the glass transition region, and that this maximum shifts with heating rate. A plot of the maxima vs. the heating rate from their data is shown i n Fig. 33.

Glass transitions in polymers


If this m a x i m u m is identified as the glass transition, the curve can be taken as the variation of the glass transition with rate. In a review of the dynamic glass transition, t186' lsv) i.e. the temperature of the loss peak m a x i m u m on a frequency-temperature curve in dynamic measurements, A. F. Lewis and O. G. Lewis found several correlations. One of these showed that the dynamic glass transition temperature Td measured at a frequency of 10 -1'5 c.p.s, could be correlated with the "static" TO value by an equation of the type Td(--1.5) = aTO + b


where a and b are 1.074 and - 2 2 for sterically restricted polymers such as polystyrene, and 0.87 and 34 for sterically unrestricted polymers such as polyethylene or polyformaldehyde. Furthermore, they observed that log v d, the frequency at which the peak m a x i m u m or Td was observed could be correlated by a linear plot of log v a versus 1/Td, i.e. by an Arrhenius type relation. It was found that plots for the sterically restricted polymers tended to converge at a point vd = l0 s c.p.s, and Td = 458°K, while for the unrestricted polymers the values are v d -- 1018 c.p.s, and Td ---- 562°K. Thus the empirical relation was obtained log (valve) = ( - AHa/4.57 ) [(1/Td) -- C]


where vc = l08 and C = 2.2 x l0 -a for the sterically restricted polymers and vc -- 101 s and C -- 1.78 x 10-3 for the unrestricted polymers. Combining eqns. (102) and (103), recalling that Vd = 10-1.5 at To (see eqn. (102)) and solving for AHa, they obtain AH a

46.4 TO - 955 1.048 - 2.35 x 10-3

(for restricted polymers)


(for unrestricted polymers)



and AH~ =

77.6 To + 3030 0 . 9 4 - 1.55 x 10-3TO

Thus, an empirical bridge between the dynamic glass transition and the static transition has been established. H. M u h i d i m e n s i o n a l i t y o f the Glass Transition (162)

Throughout the preceding discussion we have always referred to glass transition as a glass transition temperature. The only indication that a glass transition can occur under isothermal conditions was given in Fig. 29, where it was shown that in certain temperature regions an increase in pressure could bring about vitrification. This was called the glass transition pressure. It is clear that the variation in the glass transition pressure with temperature is given by the same graph as the variation of the glass transition temperature with pressure, except that the axes should be relabeled Po for P and T for To



for the sake of clarity. Since the concept of the glass transition pressure has been discussed already, it will not be mentioned further. The other possible glass transitions, however, will be discussed. The first of these is the glass transition concentration, the evidence of which can be seen most clearly from an experiment of the type first performed by Lewis and Tobin. (t88) These authors investigated the dynamic-mechanical behavior of a series of polymers as a function of diluent concentration, all at room temperature, and at a certain concentration they observed an inflection point in the modulus-temperature curve and a maximum in the loss tangenttemperature curve, i.e. behavior characteristic of a glass transition. The concentration of polymer at that point was called the glass transition concentration, in analogy to the glass transition temperature. The glass transition concentration would naturally vary with temperature, and that variation, just as in the case of pressure, is given by the plot of Tg versus diluent concentration.


V375.K vs M



® •

856 mE E





I I0

lIM - 20

l 30


4l 0

I 50

I 60

I 70

I 80

M X 10-3 UNITS

FIG. 34. Specificvolume of polymethyl-methacrylateat 375°K vs. the molecularweight. (From data of Beeversand White(171) ; after Eisenberg.(162)) A somewhat more difficult concept is that of the glass transition molecular weight, Mg. It is true that we cannot vary the molecular weight continuously, but with modern techniques we have excellent control over the resulting molecular weights in polymerization reactions. To illustrate the validity of the Mg concept, a plot of the specific volume at 375°K vs. the molecular weight is shown in Fig. 34, the point representing interpolated values from the data of Beevers and White. (171) A clear inflection is observed at M ~ 15 x 103, quite characteristic of a glass transition. It should be noted that points to the right of that value represent glassy samples, while to the left they represent liquid or rubbery samples. Again, the variation of the glass transition molecular weight with temperature is given by the Tg vs. M plot.

Glass transitions in polymers


Several other glass transitions could be defined, among them the glass transition composition of copolymers; also, phenomenogical equations have been suggested °sg) correlating these various transitions with each other. It should also be recalled that, in dynamic measurement, the temperature at which the loss maximum occurs is shifted to higher values with increasing frequency. Thus, a glass transition frequency could also be defined. ~187) In summary, it can be stated that the glass transition is not a phenomenon which occurs in one dimension only; i.e. temperature, but that it is rather a multidimensional phenomenon, the dimensions being composition, molecular weight, pressure, etc., i.e. any parameter which influences the glass transition temperature if that is the one that is being investigated. VI. M U L T I P L I C I T Y


Up to this point we have focused our attention on the primary glass transition phenomena of glass-forming materials. While for many of these materials this is the only significant transition observed, in their solid and liquid states, the same does not in general hold true for high polymers. Due to their more complicated molecular structure, polymers have been known to undergo several "transitions" throughout the temperature scale upwards from 4.2°K. These transitions are not all identical or even similar to the primary glass transitions. However, they are associated with some energy-absorbing molecular processes that should therefore be properly reckoned. Although the existence of these transitions in polymers has been known for some time, it has received relatively little attention only until recently. For a comprehensive review on this subject, readers are referred to the article by Boyer.(tl) Consider as a typical example one of the most familiar polymers: polystyrene. It is generally accepted that the primary glass transition, Tg, occurs around 100°C, at which there is a precipitous drop in the modulus. However, in their dilatometric study of polystyrene as a function of molecular weight, Fox and Flory ~ss) found two breaks in the volume-temperature curves. One break is located at 100°C, the other at 160°C. Subsequently Krimm and Tobolsky ~a89) investigated the X-ray diffraction pattern of polystyrene as a function of temperature. These authors noted that the relative peak intensities of the amorphous halos show two abrupt changes in the region of 100°C and 160°C (Fig. 35), indicating some fairly definite changes in the material. Such observations are, in fact, not limited to polystyrene. A number of cellulose esters, for example, exhibit similar multiple transition behavior. " 9 ° - 193) As we have already mentioned in Section II, the thermal expansion of polymers in the glassy state is due to the increased amplitude of vibration of monomeric segments in the quasi-lattice. This is the behavior of polystyrene below To (100°C). Above this temperature, according to Krimm and Tobolsky,~189) the enhanced rotation about bonds produces a longer range of order



over the length of neighboring chains. This change in chain configuration is not accompanied by an increase in intermolecular distances until the temperature reaches 160°C. At this temperature the thermal expansion begins to reflect the contribution of intermolecular spacings. In the terms of Boyer,(11) 160°C is Tf,t for polystyrene at which there is a change in liquid structure (the liquid 1 ~ liquid 2 transition). 2.2 2£ I,E [o / I i

1.6 1.4 1.2 20

I 40

I 60

I 80

I I00

I I 120 140 Temperoture * C

I 160

I 180

I 200


FIG. 35. Relative peak intensities of the X-ray amorphous halos of polystyrene. (After Krimm and Tobolsky.(189))

T~,t is the highest transition temperature for noncrystalline polymers. For crystalline materials there is still a melting temperature, on which we shall not elaborate here. Below the primary glass transition, however, there are a number of glass-like transitions. Wunderlich and Bodily(194) recently found by differential thermal analysis that there is a secondary freezing process in the range of 350 ° to 230°K. The transition point lies at 325 + 4°K, which was called the 50°C or B-transition for polystyrene. It is known that nuclear second moment determined by NMR as a function of temperature" 95) shows a change at ,-~310°K; while dynamic mechanical data (196) (10 c.p.s.) exhibit a small maximum also at ~ 320°K. In addition, a maximum of the Bragg distances of phenyl groups was found by X-ray studies(19T) to lie around 320°K. From this evidence it was concluded(194- ~9~) that the transition of polystyrene at 50°C represents the onset of torsional vibrations of the phenyl groups. At still lower temperatures, several authors (a96' 198) noted a small maximum in mechanical loss at ~ 130°K (10 c.p.s.) for linear polystyrene. Illers and Jenckel (198) called it the y-transition, and attributed it to the irregularities in polymer structure. During polymerization process, head-to-head and tail-totail additions may occur, which produce sequences containing two methylene groups. Due to their relatively small steric hindrance, these "weak points" undergo motions that are similar to those occurring in polyethylene, the polyamides, etc. These authors found that "/-peaks are dependent on the condition of polymerization, which is consistent with their hypothesis. This may perhaps explain the fact that the "/-transition was not observed in similar experiments on polystyrene performed by several other workers. (x99' zoo)


Glass transitions in polymers

It is of interest to note that the molecular mechanism for the y-transition in polystyrene as visualized by Illers and Jenckel seems, in some sense, to anticipate the "crankshaft transition" theory proposed by Schatzki (2°t~ to interpret the y-peak in polyethylene. He suggested that the ~ 150°K transition in polyethylene is due to the motion of four methylene groups, as illustrated in Fig. 36. It is necessary that two carbon-carbon bonds are collinear so that the four intervening carbon atoms can undergo crankshaft-like motion. This theory is in agreement with the observation that the y-transition in polyethylene occurs whenever the --(CH2)4-- moiety exists in the polymer chain,t2°2) However, it is known ~2°3) that the y-transition for polypropylene shifts to ~200°K, presumably due to the inhibition by the pendant methyl group. It would thus be difficult to reconcile the lowering of the y-transition temperature in polystyrene whose phenyl groups are, of course, considerably bulkier. There is, at 1


~.r ) x. ~




FIG. 36. Molecular model for the crankshaft transition in polyethylene. (After Schatzki. ~z°~)

present, no evidence whether the y-dispersion in polystyrene is associated with the crankshaft transition. Recently, dynamic mechanical studies of polymers have been extended to cryogenic temperatures. Sinottt2°4) reported a loss maximum at 38°K for polystyrene determined at 5.6 c.p.s. Crissman and McCammont68) found this peak shifted to 48°K at a frequency of 6290 c.p.s. The mechanism for this particular relaxation process was attributed to either oscillation or wagging of the pendant phenyl groups, t2°4~ Crissman, Woodward, and Sauert2°5) showed that this transition is unaffected by the presence of crystallinity, although the peak is lower for the amorphous sample. A summary of the multiple transition behavior of polystyrene is given in Table 4. The mechanisms suggested must be regarded as tentative, since to date there are insufficient data to draw definitive conclusions. The number of transitions that a given polymer may have is strongly dependent on its molecular structure. Poly-~-methylstyrene,t2°5) for example, exhibits only a weak shoulder in its loss curve at 10° to 50°K but an additional peak at ,-~ 140°K (6731 c.p.s.). The latter was assignedt2°5~ to phenyl motions coupled with methyl group reorientations. The 10° to 50°K peak is somewhat similar to the one observed in polypropylene,t2°6) and could be due to hindered



rotations of methyl groups. On the other hand, polyethylene having no substituted pendant groups, shows no more loss peaks below the crankshaft transition of the main chain. An additional interesting example of the side-chain transition of polymers at low temperature is that of polyvinyl cyclohexane. <2°7) Besides the primary glass transition at 415°K, a loss peak was also found at 155°K. It was suggested that this transition reflects the motion of the cyclohexyl group in its boat-chair conformation conversions.



Suggested Mechanisms


433°K(160°C) 373°K(100°C) 325 ± 4°K(~50°C)


Liquid1 ~ Liquid2 Long-range chain motions Torsional vibrations of phenyl groups Motions due to -~7CH2-~ moieties Oscillation or wagging of phenyl groups

11,189 11,189 194,197

130°K 38~8°K

196,198 204, 205

So far most of the sub-Tg transitions were found by observing the response of the polymer to an externally imposed dynamic perturbation. As such, they are generally regarded to be due to relaxational processes. Some evidence for the transition was found in volume-temperature curves at low temperature for polystyrene,~146) and polyethylene-polypropylene blends, ~2°8) although the data were not sufficient, to make any definitive conclusions at this time. O'Reilly eta/. (2°9) accurately measured specific heats of both atactic and isotactic polystyrenes from 20°K to 350°K. No evidence of any transition was found. However, these do not necessarily imply that there is no change in thermodynamic functions associated with the loss peaks in mechanical spectroscopy. It is possible that their very small magnitudes are not detectable by these techniques. In any case, the phenomenon of multiple transitions is still in the stage of early development. Conclusive interpretations regarding their mechanism cannot be drawn until sufficient positive evidence has been accumulated. VII. S U M M A R Y


We hope that the preceding presentation has made it clear that the glass transition is not a simple phenomenon, but a very complicated process which is still not understood in all of its ramifications. The nature of the glass transition temperature (Tg), to begin with, has been the source of considerable controversy

Glass transitionsin polymers


for many years. The observation of its rate effects, on the other hand, and its similarity to second-order transition, on the other, has caused much confusion. It is not until recently that there is promise for reconciliation. It appears at present that To is what one experimentally observes when the material is on its "slow boat" to T2. In other words, Tg is a kinetic manifestation of a thermodynamic process (T2). There is, in fact, no inherent controversy between these two views, since each explains one aspect of the "many-faced" transition. The solution of this problem was actually alluded to in the classical paper of Kauzmann, although it was not recognized as such. The credit of explicit clarification goes to Gibbs and DiMarzio, who first argued the case in quantitative terms. Many unanswered questions remain, however. Some of these are : 1. If Tz indeed exists, is the configurational entropy t h e underlying parameter or are there any others? The postulate appears reasonable for polymers, but does it hold equally well for ordinary liquids and other glass-forming materials? 2. If T2 could ever be reached, what would one expect the detailed molecular structure to be like at that temperature? How would it differ from crystals at zero configurational entropy? 3. Pure kinetic theories have been successful in explaining the observed rate effects. Can one derive such equations from considerations of the GibbsDiMarzio theory? On the phenomenological side, there is even wide disagreement in defining the glass transition. We have seen that the glass transition manifests itself as a rather abrupt change in a number of different properties. Principally, they are volumetric, thermodynamic, mechanical and electromagnetic properties, some of which we have discussed in detail. There are a number of other property changes that are also associated with Tg. All of these changes have been used to determine the glass transition temperature. At Iast count, there are forty or fifty different methods that have actually been used in reporting Tg's in the published literature. These different methods necessarily produce different values. In addition, ~'s are profoundly affected by many external factors, such as pressure, diluent concentration, degree of crosslinking, molecular weight, etc. Values of T~ thus vary from sample to sample. It is therefore not surprising to see the lack of agreement in many reported T~'s of one certain material. We would like to echo the suggestion of KovacsC2lo) by asking : 4. Would it be practical to adopt an arbitrarily selected standard for measuring Tg's so that they may be meaningfully compared with each other? Besides these external, controllable factors that affect T o, the molecular parameters of the monomeric segments of a polymer also exert considerable influence. Particularly important are the effects of chain stiffness and intermolecular interaction. Although their importance has long since been recognized, these two effects have never been completely isolated from each other to provide quantitative data. According to the G-D theory, only chain stiffness is the



prime determinate of Tg and intermolecular interaction has very small effect. We thus pose the questions : 5. Can the effect of chain stiffness and intermolecular interaction be quantitatively measured? Can G - D theory be modified to account for the latter effect? We have seen that the glass transition is not just a temperature phenomenon, although it is often treated as such. Looking at it in a different perspective, it is more accurately called a state arrived at by varying a number of environmental parameters. Once a critical condition is reached, we have in addition to a glass transition temperature, a glass transition pressure, a glass transition frequency, a glass transition concentration, etc. In other words, glass transition is a multidimensional phenomenon. Now to broaden our view even further, there are more than just one glass transition, but rather a number of such glass-like transitions below and above Tg. The multiple transition phenomena are receiving more attention only of late. It is a field that boasts a great deal more questions than answers. To illustrate: 6. Is the nature of these transitions indeed "glass-like"? 7. Are we to wage another thermodynamic vs. kinetic war over these transitions? 8. How do these transitions correlate with molecular structure? and many more. During the course of writing this article, these authors have been often haunted by the story of the elephant and the three blind men. Fortunately, we are beginning to see the overall perspective. However, considering the stupendous nature of the problem, it has thus far received less attention than it deserves, particularly in areas other than polymers. As we recall from the Introduction, polymers are only one of the eight classes of glass-forming materials. It is to be hoped that more efforts will be directed to the study of the vitrification process fo r the sake of understanding it as a general phenomenon for all these materials. ACKNOWLEDGMENTS

It is a pleasure to acknowledge the assistance of R. J. Schaffhauser in supplying some of the references, J. Moacanin in reading part of the manuscript, and A. V. Tobolsky, R. Simha and T. F. Schatzki in supplying information prior to publication. One of us (A. E.) acknowledges the financial assistance of tile Office of Naval Research.


Glass transitions in polymers


Glass Transition Temperaturesfor Selected Materials Material



Polyacenaphthalene Polyvinyl pyrrolidone Poly-o-vinyl benzyl alcohol Poly-p-vinyl benzyl alcohol Polymethacrylonitrile Polyacrylic acid Polymethyl methacrylate Polyvinyl formal Polystyrene Polyacrylonitrile Polyvinyl chloride Polyvinyl alcohol Polyvinyl acetal Polyvinyl propional Polyethylene terephthalate Polyvinyl isobutyral P01yc~tprolactam (nylon 6) Polyhexamethylene adipamide (nylon 6,6) Polyvinyl butyral Polychlorotrifluorethylene Ethyl cellulose Polyhexamethylene sebacamide (nylon 6,10) Polyvinyl acetate Polyperfluoropropylene Polymethyl acrylate Polyvinylidene chloride Polyvinyl fluoride Poly- 1-butene Polyvinylidene fluoride Poly- 1-hexene Polychloroprene Polyvinyl-n-butyl ether Polytetramethylene sebacate Polybutylene oxide Polypropylene oxide Poly~ 1-octene Polyethylene adipate Polyisobutylene Natural rubber Polyisoprene Polybutadiene Polydimethyl siloxane

264 175 160 140 120 106 105 105 100 96.5 87 85 82 72 69 56 50 50 49 45 43 40 29 11 9 -17 -20 -25 -39 -50 -50 -52 -57 60 60 -65 70 -70 -72 -73 -85 123

138 16 16 16 16 14 14 16 14 211 16 16 16 16 16 16 16 16 16 14 16 16 14 14 14 16 16 16 14 16 16 16 16 176 176 16 16 14 14 16 14 14

Organic Polymers






MITCrlEL C. SHEr~ and AI)I EISENBERG Material Inorganic Polymers Silicon dioxide Polycalcium phosphate Polysodium phosphate Boron trioxide Arsenic trisulfide Arsenic trioxide Zinc chloride Sulfur Selenium Polyphosphoric acid Glass-forming Liquids Phenolphthalein Sucrose Salicin Glucose Diphenyl phthalate 13-naphthyl salicylate Dicyclohexyl phthalate Na2S203'5 H20 Phenyl salicylate Dibenzyl succinate Tricresyl phosphate Diisodecyl phthalate Di-2-ethylhexyl phthalate Diethyl phthalate Di-n-octyl phthalate d-l-Lactic acid Glycerol Propylene glycol Toluene sec-Butyl alcohol Ethanol 1-Propanol 3-Methyl hexane 2,3-Dimethylpentane



1200-1700 525 285 200-260 195 160 100 75 30 -10

2 154 154 2 20 20 23 212 213 154

78 67 46 7-27 -15

156 2 155 2 214 167 214 2 167 214 167 214 214 214 214 2 2 2 9 2 2 2 2 2

- 29.5


-33 -42 -56 -58 -64.5 -77 -82 -85 -87 -88 -67 -93- -83 123-- 108 -

















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