The glass transition

Result

The glass transition

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Statement

The freezing of an amorphous polymer.

Why it matters

degree-of-polymerisation and step-vs-chain-growth describe how chains are built and how long they are; the glass transition describes what happens to those chains once assembled, as an amorphous (non-crystalline) polymer is cooled. It is the single most practically important thermal property of most commercial plastics, elastomers, and adhesives, since it marks the boundary between a hard, glassy, brittle material and a soft, flexible, rubbery one made of the identical chemical repeat units — the same polystyrene that is rigid and glassy at room temperature becomes rubbery well above its glass transition temperature \(T_g\), and understanding what drives \(T_g\) is what lets a chemist tune a polymer's mechanical behaviour by design.

Hypotheses

The polymer is amorphous, or at least contains a significant amorphous fraction, in the temperature range of interest.A fully crystalline solid melts at a sharp, first-order melting point \(T_m\) instead; the glass transition is specifically a phenomenon of the disordered, amorphous regions of a polymer, and semicrystalline polymers show both a \(T_g\) (for their amorphous fraction) and a distinct, generally higher \(T_m\) (for their crystalline fraction). Below \(T_g\), large-scale segmental motion of the polymer backbone is effectively frozen out on the experimental timescale, even though the material remains a disordered, liquid-like structure at the molecular level.This is why the glass transition is classified as a kinetic phenomenon rather than a true thermodynamic phase transition in the strict sense: its observed temperature depends measurably on cooling rate (a more slowly cooled sample has more time to relax toward a lower-volume state before segmental motion freezes, giving a slightly lower apparent \(T_g\)), unlike a true first-order transition such as melting or freezing.

Proof

V_f(T) = V(T) - V_{\text{occupied}}(T)

Define the free volume \(V_f\) as the difference between a polymer sample's total specific volume and the volume genuinely occupied by its molecules; free-volume theory holds that segmental motion (chain segments rotating and translating past their neighbours) requires a minimum amount of locally available free volume to occur at all. A

\text{On cooling, } V_f \text{ decreases roughly linearly with } T \text{ until it reaches a critical minimum value at } T=T_g.

Above \(T_g\), thermal expansion in the melt/rubbery state contracts free volume steadily as temperature falls; once free volume drops to the minimum needed to sustain segmental motion, that motion effectively stops on experimental timescales, defining the glass transition temperature by this kinetic freezing-out criterion. A

\left(\frac{\partial V}{\partial T}\right)_{TT_g}, \qquad \left(\frac{\partial C_p}{\partial T}\right)\text{ discontinuous at } T_g

Because segmental motion, and the additional configurational degrees of freedom it provides, is frozen out below \(T_g\), the thermal expansion coefficient and heat capacity both drop to smaller (glassy-state) values below \(T_g\) than above it, giving a change in slope of volume versus temperature and a step in heat capacity — the standard experimental signatures (by dilatometry or differential scanning calorimetry) used to locate \(T_g\), in contrast to the discontinuity in volume itself seen at a true first-order melting transition. A

\log a_T = \frac{-C_1(T-T_g)}{C_2+(T-T_g)}

The Williams–Landel–Ferry (WLF) equation, an empirical but widely validated relation built directly on free-volume reasoning, describes how a polymer's relaxation timescale (and hence properties like viscosity) shift with temperature above \(T_g\), with \(C_1,C_2\) near-universal constants for many amorphous polymers when referenced to their own \(T_g\); it is the standard tool for predicting viscoelastic behaviour across the temperature range above the glass transition. B

Result

T < T_g:\ \text{glassy, rigid, low free volume} \qquad T > T_g:\ \text{rubbery/liquid, mobile, higher free volume}

Reading. \(T_g\) marks the kinetic boundary, driven by free volume dropping below the threshold needed for segmental motion, between a glassy amorphous polymer and its rubbery or liquid state, detected experimentally as a step in heat capacity and a change in thermal expansion coefficient rather than a sharp, first-order discontinuity in volume.

Scope. Applies to the amorphous fraction of a polymer; a fully crystalline sample shows no glass transition at all, and a semicrystalline sample shows a genuine \(T_g\) for its amorphous regions in addition to a separate melting point \(T_m\) for its crystalline regions.

Corollaries & converses

Anything that increases chain flexibility or free volume at fixed temperature lowers \(T_g\): plasticiser addition, lower molar mass (more chain-end free volume per unit mass), and flexible backbone or side-group chemistry all systematically lower \(T_g\); conversely, bulky rigid side groups, hydrogen bonding, and crosslinking (which directly restricts segmental motion) all raise it.
flory-huggins' interaction parameter \(\chi\) governs whether a small-molecule plasticiser mixes favourably with a polymer at all; only a miscible plasticiser can lower \(T_g\) by the free-volume mechanism described here, since an immiscible additive simply phase-separates rather than increasing the polymer's own segmental mobility.
Converse: measuring \(T_g\) as a function of added plasticiser concentration, or as a function of molar mass via degree-of-polymerisation, is a standard indirect probe of how strongly a given structural feature affects chain segmental mobility.

Fails without

Apply the concept to a fully crystalline polymer sample: since the underlying mechanism (Step 1–2) is specifically about free volume and segmental motion in a disordered, amorphous structure, a fully crystalline solid shows no glass transition at all, only a sharp, first-order melting point (Hypotheses).
Treat a measured \(T_g\) as a fixed, cooling-rate-independent thermodynamic constant: since \(T_g\) marks a kinetic freezing-out of segmental motion rather than a true equilibrium phase transition (Hypotheses), the same sample measured under different cooling or heating rates will show a measurably different apparent \(T_g\) (Problems, Q1).

Common errors

Confusing the glass transition temperature \(T_g\) with the melting point \(T_m\); they describe different phenomena (amorphous softening versus crystalline melting) and, in a semicrystalline polymer, are two distinct, generally well-separated temperatures.
Treating \(T_g\) as a sharply defined, cooling-rate-independent thermodynamic transition temperature like a true melting point, rather than the kinetic, somewhat rate-dependent quantity described in the Hypotheses.
Assuming higher molar mass always raises \(T_g\) without bound; in reality \(T_g\) rises with molar mass only up to a plateau, because the free-volume contribution of chain ends (Corollaries) becomes negligible once chains are already long relative to typical commercial polymer molar masses.
Forgetting that a fully amorphous polymer well above its \(T_g\) has no distinct melting point at all, since there is no crystalline order present to melt.

Discussion

Free-volume theory, and the WLF equation built on it, were developed through the mid-20th century as polymer science matured from a largely empirical, industrial discipline into one with a quantitative statistical-thermodynamic foundation; Malcolm Williams, Robert Landel, and John Ferry published their equation in 1955, finding remarkably similar constants \(C_1,C_2\) across a wide range of chemically different amorphous polymers when referenced to each polymer's own \(T_g\), a striking degree of universality for an empirical relation.

Whether the glass transition is best understood as a purely kinetic phenomenon, or whether it reflects an underlying, experimentally inaccessible true thermodynamic transition at some lower "ideal" glass transition temperature \(T_2\) (the basis of the Adam–Gibbs and related theories, connecting configurational entropy to relaxation time), remains an active and genuinely unresolved question in condensed-matter and polymer physics, well beyond the scope of the free-volume picture developed here.

Common misconception: that a polymer "melts" at its glass transition. Below \(T_g\) an amorphous polymer is a rigid, disordered solid; above \(T_g\) it is a soft, disordered rubber or viscous liquid, but at no point does a genuine crystalline lattice break down, since none was present — "melting," in the strict sense, applies only to the separate, crystalline-fraction transition at \(T_m\) in a semicrystalline sample.

Worked examples

\text{Polystyrene, } T_g\approx100^\circ\text{C}; \quad \text{natural rubber (polyisoprene), } T_g\approx-70^\circ\text{C}

Polystyrene's bulky, rigid phenyl side group severely restricts backbone rotation, giving a high \(T_g\) well above room temperature (hence its familiar rigid, glassy behaviour at ordinary conditions); polyisoprene's flexible backbone with few bulky substituents gives a very low \(T_g\), so it is already well into its rubbery regime at room temperature (hence its familiar elastic behaviour). A

\text{Plasticised PVC: } T_g(\text{unplasticised})\approx80^\circ\text{C} \ \longrightarrow\ T_g(\text{plasticised})\approx-20\ \text{to}\ 0^\circ\text{C (typical range)}

Adding a compatible small-molecule plasticiser increases free volume between chains and screens some chain–chain interactions, lowering \(T_g\) substantially (per Step 2's free-volume mechanism) — the standard industrial route by which rigid PVC pipe and flexible PVC (e.g. cabling, vinyl sheeting) are made from essentially the same base polymer. A

\text{Side-group bulk} \uparrow T_g;\quad \text{chain flexibility, plasticiser} \downarrow T_g

Reading. The same underlying free-volume mechanism explains both why chemically different polymers span an enormous range of room-temperature mechanical behaviour, and why a single polymer's behaviour can be deliberately tuned by formulation.

Scope. The qualitative direction of each structural effect (bulk raises \(T_g\), flexibility and plasticiser lower it) is reliable across essentially all amorphous polymers; precise \(T_g\) values still require direct measurement for a given formulation.

Problems

Explain, using free-volume theory (Step 1–2), why a polymer sample cooled very slowly through its glass transition typically shows a slightly lower measured \(T_g\) than the same sample cooled rapidly.
Solution
Slow cooling gives chain segments more time to relax and pack into a lower-free-volume configuration at each temperature before segmental motion becomes too slow to continue; the sample can therefore reach a smaller free volume, and hence remain mobile down to a lower temperature, before finally falling below the critical free-volume threshold of Step 2. Rapid cooling does not allow this relaxation, freezing the sample at a comparatively higher free volume (and hence, apparently, a higher \(T_g\)) — direct evidence that \(T_g\) is a kinetic, rate-dependent quantity rather than a sharp thermodynamic transition point.
A polymer chemist wants to raise the \(T_g\) of a flexible-backbone elastomer without changing its basic chemical repeat unit. Suggest one structural modification, referencing the Corollaries, and explain why it would work.
Solution
Introducing crosslinks between chains (vulcanisation-style covalent links) directly restricts the large-scale segmental motion that free-volume theory identifies as the mechanism of the rubbery state (Step 1–2); more crosslinking reduces the chains' effective mobility at any given temperature, raising the temperature needed before segmental motion can occur, and hence raising \(T_g\) (Corollaries, first bullet).
A semicrystalline polymer sample shows two distinct thermal features on heating: a step change in heat capacity near \(60^\circ\text{C}\) and a sharp endothermic peak near \(180^\circ\text{C}\). Identify which feature is \(T_g\) and which is \(T_m\), and justify using Step 3 and the Hypotheses.
Solution
The step change in heat capacity near \(60^\circ\text{C}\), with no accompanying sharp, discrete latent-heat peak, is the glass transition \(T_g\), consistent with Step 3's description of \(T_g\) as a change in slope (a second-order-like signature) rather than a first-order discontinuity. The sharp endothermic peak near \(180^\circ\text{C}\), representing a genuine latent heat absorbed at a comparatively well-defined temperature, is the crystalline melting point \(T_m\), consistent with the Hypotheses' distinction between the amorphous \(T_g\) and the separate, generally higher, crystalline \(T_m\) of a semicrystalline sample.