chemistry2u
Depth
Result

The glass transition

T-125Home CU-404Threads structure
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
1
\[ V_f(T) = V(T) - V_{\text{occupied}}(T) \]
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
2
\[ \text{On cooling, } V_f \text{ decreases roughly linearly with } T \text{ until it reaches a critical minimum value at } T=T_g. \]
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
3
\[ \left(\frac{\partial V}{\partial T}\right)_{T<T_g} \neq \left(\frac{\partial V}{\partial T}\right)_{T>T_g}, \qquad \left(\frac{\partial C_p}{\partial T}\right)\text{ discontinuous at } T_g \]
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
4
\[ \log a_T = \frac{-C_1(T-T_g)}{C_2+(T-T_g)} \]
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
1
\[ \text{Polystyrene, } T_g\approx100^\circ\text{C}; \quad \text{natural rubber (polyisoprene), } T_g\approx-70^\circ\text{C} \]
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
2
\[ \text{Plasticised PVC: } T_g(\text{unplasticised})\approx80^\circ\text{C} \ \longrightarrow\ T_g(\text{plasticised})\approx-20\ \text{to}\ 0^\circ\text{C (typical range)} \]
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.

Further example. A polymer chemist is formulating a flexible PVC cable jacket. The unplasticised PVC has a glass transition temperature \(T_g = 80^\circ\text{C}\). The target \(T_g\) for the final product is \(-20^\circ\text{C}\). Using the Fox equation for polymer/plasticiser blends, calculate the required mass fraction of a plasticiser with \(T_g = -80^\circ\text{C}\). Assume the system behaves ideally.

1
\[ T_g(\text{blend}) = -20^\circ\text{C} \implies T_{g,\text{K}} = 253.15\ \text{K} \]
The Fox equation is an empirical relation derived for thermodynamic mixing where temperatures must be in an absolute scale (Kelvin). Converting the target temperature from Celsius to Kelvin is the necessary first step. A
2
\[ T_{g,1} = 80 + 273.15 = 353.15\ \text{K}; \quad T_{g,2} = -80 + 273.15 = 193.15\ \text{K} \]
Convert the pure component glass transition temperatures to Kelvin to maintain unit consistency with the target temperature. A
3
\[ \frac{1}{T_{g,\text{mix}}} = \frac{w_1}{T_{g,1}} + \frac{w_2}{T_{g,2}} \]
Apply the Fox equation, where \(w_1\) and \(w_2\) are the mass fractions of the polymer and plasticiser, respectively. A
4
\[ \frac{1}{253.15} = \frac{w_1}{353.15} + \frac{1-w_1}{193.15} \]
Substitute the known temperatures and the constraint \(w_1 + w_2 = 1\) (so \(w_2 = 1 - w_1\)) into the equation. A
5
\[ 0.003950 = 0.002832 w_1 + 0.005177 (1 - w_1) \]
Calculate the reciprocal of each temperature term. \(1/253.15 \approx 0.003950\), \(1/353.15 \approx 0.002832\), and \(1/193.15 \approx 0.005177\). A
6
\[ 0.003950 = 0.002832 w_1 + 0.005177 - 0.005177 w_1 \implies 0.002345 w_1 = 0.001227 \implies w_1 \approx 0.523 \]
Rearrange to solve for \(w_1\). The mass fraction of the polymer is approximately 0.523. Consequently, the mass fraction of the plasticiser is \(w_2 = 1 - 0.523 = 0.477\). A
\[ w_{\text{plasticiser}} \approx 0.48 \]

Reading. To lower the \(T_g\) of PVC from \(80^\circ\text{C}\) to \(-20^\circ\text{C}\), nearly half the mass of the formulation must be plasticiser. This highlights the significant amount of additive required to achieve a large shift in thermal properties.

Scope. The Fox equation assumes ideal mixing and is most accurate for systems where the polymer and plasticiser are miscible and have similar molar volumes. It is a standard engineering approximation for PVC/plasticiser systems.

Problems
  1. 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.
    SolutionSlow 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.
  2. 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.
    SolutionIntroducing 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).
  3. 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.
    SolutionThe 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.
  4. A polymer sample is cooled from the melt at a rate of \(10\ \text{K/min}\) and a glass transition is observed at \(T_g = 100^\circ\text{C}\). The sample is then reheated at a rate of \(1\ \text{K/min}\). Using the empirical rule that \(T_g\) shifts by approximately \(2\ \text{K}\) for every order of magnitude change in cooling/heating rate, estimate the new \(T_g\) observed during the slower heating.
    Solution The initial rate is \(10\ \text{K/min}\) and the final rate is \(1\ \text{K/min}\). The ratio of rates is \(\frac{10}{1} = 10\), which is one order of magnitude. Since the rate is decreasing (slower heating), the material has more time to relax, allowing it to remain mobile to a lower temperature. Thus, \(T_g\) decreases. The shift is \(2\ \text{K}\) per order of magnitude. \(\Delta T_g = -2\ \text{K}\). New \(T_g = 100^\circ\text{C} - 2\ \text{K} = 98^\circ\text{C}\).
  5. A semicrystalline polymer has a glass transition temperature \(T_g = 60^\circ\text{C}\) and a melting point \(T_m = 180^\circ\text{C}\). The polymer is used in an application where it must be rigid at \(25^\circ\text{C}\) but deformable at \(100^\circ\text{C}\). Based on the thermal transitions, explain why this polymer is suitable for the application.
    Solution At \(25^\circ\text{C}\), the temperature is below \(T_g\) (\(25 < 60\)). In this glassy state, the amorphous chains are frozen, and the crystalline domains are intact, providing high rigidity and strength. At \(100^\circ\text{C}\), the temperature is above \(T_g\) (\(100 > 60\)) but below \(T_m\) (\(100 < 180\)). In this rubbery state, the amorphous chains have gained sufficient segmental mobility to allow deformation, while the crystalline domains remain intact to provide structural integrity and prevent flow. Thus, the polymer transitions from a rigid glass to a deformable rubber within the required temperature range.