Polymer Rheology, Viscoelasticity, Glass Transition ($T_g$) & Mechanical Properties

The mechanical response of solid polymers spans an extraordinary continuum from hard, glass-like brittleness to ductile necking and high-extensibility rubber elasticity, dictated fundamentally by temperature relative to the glass transition ($T_g$) and melting temperature ($T_m$).

Engineering Stress-Strain Definitions

In a uniaxial tensile test, a specimen of initial cross-sectional area $A_0$ and gauge length $L_0$ is pulled at constant deformation rate:

1. Engineering Stress: $\sigma = F / A_0$ (Pascals or $\text{MPa}$).

2. Engineering Strain: $\epsilon = (L - L_0) / L_0 = \Delta L / L_0$ (dimensionless or $\%\text{ strain}$).

3. True Stress: $\sigma_{\text{true}} = F / A = \sigma (1 + \epsilon)$ (assuming constant volume during deformation).

4. True Strain: $\epsilon_{\text{true}} = \ln(L / L_0) = \ln(1 + \epsilon)$.

5. Young's Modulus: $E = \lim_{\epsilon \to 0} (d\sigma / d\epsilon)$ (initial linear elastic slope).

Characteristic Deformation Regimes

1. Brittle Polymeric Glasses ($T \ll T_g$):

• Examples: Atactic polystyrene (PS), poly(methyl methacrylate) (PMMA) at room temperature.
• High Young's modulus ($E = 3.0 - 3.5\text{ GPa}$).
• Fracture occurs prematurely at small strains ($\epsilon_b < 2 - 3\%$) with zero macroscopic plastic yield. Failure is governed by crazing (microvoid formation bridged by drawn polymer fibrils) followed by brittle fracture.

2. Ductile Polymers with Yield and Cold-Drawing ($T < T_g$ but near $T_g$, or semi-crystalline $T_g < T < T_m$):

• Examples: Polycarbonate (PC), high-density polyethylene (HDPE), Nylon 6,6.
• Exhibits an initial linear elastic Hookean regime followed by a prominent yield point (yield stress $\sigma_y$).
• Beyond yield, necking (localized reduction in cross-sectional area) occurs with an engineering stress drop.
• As the neck propagates stably along the gauge length (cold-drawing), polymer chains undergo extensive uncoiling and orientation alignment parallel to the draw axis, resulting in strain hardening until rupture at $\epsilon_b = 50 - 500\%$.

3. Elastomers ($T \gg T_g$, lightly cross-linked):

• Examples: Vulcanized natural rubber, polybutadiene, PDMS silicone.
• Low initial modulus ($E \sim 1 - 10\text{ MPa}$).
• Reversible non-linear deformation up to colossal strains ($\epsilon_b = 500 - 1,000\%$) with instantaneous recovery upon release. Driven purely by conformational entropy.

Unlike purely elastic Hookean solids (where stress is instantaneous and proportional to strain: $\sigma = E \epsilon$) or purely viscous Newtonian fluids (where stress is proportional to strain rate: $\sigma = \eta \dot{\epsilon}$), polymers exhibit viscoelasticity—a time- and rate-dependent hybrid behavior combining elastic energy storage with viscous energy dissipation.

1. Creep and Creep Compliance $J(t)$

In a creep test, a constant instantaneous engineering stress $\sigma_0$ is applied at $t = 0$ and maintained:

where $H(t)$ is the Heaviside step function. The resulting strain $\epsilon(t)$ increases continuously over time:

• An instantaneous elastic response $\epsilon_0$.
• A time-dependent delayed elastic (anelastic) retarding strain.
• For uncrosslinked polymers, a continuous steady viscous flow.

The creep compliance $J(t)$ is defined as the time-dependent strain per unit applied stress:

(Units: $\text{Pa}^{-1}$ or $\text{MPa}^{-1}$).

2. Stress Relaxation and Relaxation Modulus $E(t)$

In a stress relaxation test, a constant instantaneous strain $\epsilon_0$ is applied at $t = 0$ and held constant:

To maintain this constant strain, the stress required to hold the sample decays continuously over time as macromolecular chains undergo conformational rearrangements, reptation, and disentanglement:

The stress relaxation modulus $E(t)$ is defined as:

(Units: $\text{Pa}$ or $\text{MPa}$). In linear viscoelasticity (Boltzmann superposition principle), $E(t)$ and $J(t)$ are interrelated via the convolution integral:

To model viscoelastic constitutive behavior mathematically, physical networks of ideal Hookean springs (elastic storage, modulus $E$) and Newtonian dashpots (viscous dissipation, viscosity $\eta$) are constructed.

1. The Maxwell Model (Viscoelastic Fluid)

A Hookean spring and a Newtonian dashpot connected in series:

• Total strain is the sum of spring and dashpot strains: $\epsilon = \epsilon_s + \epsilon_d$.
• Both elements experience identical stress: $\sigma = \sigma_s = \sigma_d$.

Differentiating with respect to time:

Multiplying by $E$ yields the Maxwell constitutive differential equation:

where $\tau_R \equiv \eta / E$ is the relaxation time. In stress relaxation at constant strain ($\epsilon = \epsilon_0, d\epsilon/dt = 0$):

The stress relaxation modulus is:

The Maxwell model describes uncrosslinked polymer melts: it exhibits instantaneous elasticity and exponential stress relaxation to zero at long times. However, it fails to predict primary creep recovery.

2. The Voigt-Kelvin Model (Viscoelastic Solid)

A Hookean spring and a Newtonian dashpot connected in parallel:

• Both elements experience identical strain: $\epsilon = \epsilon_s = \epsilon_d$.
• Total stress is the sum of spring and dashpot stresses: $\sigma = \sigma_s + \sigma_d = E \epsilon + \eta \frac{d\epsilon}{dt}$.

Rearranging:

where $\tau_C \equiv \eta / E$ is the retardation time. Under constant stress $\sigma_0$ (creep test):

The creep compliance is:

The Voigt-Kelvin model accurately predicts delayed elastic deformation and complete strain recovery upon load removal, but fails in stress relaxation (predicts infinite instantaneous stress).

3. The Standard Linear Solid (Zener Model)

Combines a Maxwell arm in parallel with an equilibrium Hookean spring ($E_2$):

Provides both an instantaneous glassy modulus $E_g = E_1 + E_2$ at $t = 0$, an exponential relaxation decay, and an equilibrium plateau modulus $E_e = E_2$ at $t \to \infty$, accurately capturing crosslinked elastomer behavior.

Dynamic Mechanical Analysis (DMA) is the premier analytical technique for probing the viscoelastic spectrum of polymers across wide frequency and temperature domains.

Oscillatory Harmonic Deformation

A sinusoidal oscillating strain of angular frequency $\omega$ (in $\text{rad/s}$) is applied to the specimen:

Because polymers are viscoelastic, the resulting steady-state stress oscillates at the identical frequency $\omega$ but leads the strain by a phase angle $\delta$ ($0^\circ \le \delta \le 90^\circ$):

Expanding using trigonometric angle addition:

Dividing by maximum strain $\epsilon_0$:

Complex Modulus Components

1. Storage Modulus ($E'$, In-Phase Component):

Represents the elastic energy stored and reversibly recovered per cycle of deformation. Directly proportional to specimen stiffness.

2. Loss Modulus ($E''$, Out-of-Phase Component):

Represents the viscous energy dissipated as internal heat per cycle through molecular friction.

3. Complex Modulus Magnitude ($|E^*|$):

4. Loss Factor (Damping Factor, $\tan \delta$):

The ratio of dissipated energy to stored elastic energy:

• Pure elastic Hookean solid: $\delta = 0^\circ \implies E'' = 0, \tan \delta = 0$.
• Pure viscous Newtonian liquid: $\delta = 90^\circ \implies E' = 0, \tan \delta = \infty$.
• Viscoelastic polymer: $0 < \tan \delta < \infty$.

In a DMA temperature sweep at constant frequency (e.g., $1\text{ Hz}$), the glass transition temperature $T_g$ is detected with unmatched sensitivity as a dramatic multi-decade plunge in storage modulus $E'$ (by 3 orders of magnitude from $\sim 3\text{ GPa}$ to $\sim 1\text{ MPa}$) accompanied by a pronounced peak in $\tan \delta$.

The glass transition temperature ($T_g$) is the temperature boundary where an amorphous polymer transitions from a hard, brittle, glassy state ($T < T_g$) into a soft, flexible rubbery or leathery state ($T > T_g$). Unlike crystalline melting ($T_m$), which is a true first-order thermodynamic phase transition with discontinuous changes in enthalpy and volume ($\Delta H_m > 0, \Delta V_m > 0$), the glass transition is a kinetic and pseudo-second-order transition characterized by a step change in heat capacity ($\Delta C_p$) and thermal expansion coefficient ($\Delta \alpha$).

Fox-Flory Free Volume Theory

According to the free volume model formulated by Thomas Fox and Paul Flory: The total macroscopic volume $V$ of a polymer solid consists of:

1. Occupied Volume ($V_{\text{occ}}$): The van der Waals volume occupied by the constituent atoms plus their core vibrational exclusion shells.

2. Free Volume ($V_f$): The interstitial unoccupied empty space between macromolecular chains resulting from packing inefficiencies.

Free volume fraction is defined as:

• Above $T_g$ ($T > T_g$): Thermal energy expands the free volume at thermal expansion rate $\alpha_f$:

where $\alpha_f = \alpha_r - \alpha_g \approx 4.8 \times 10^{-4}\text{ K}^{-1}$. Because $f$ is large ($f > 0.025$), polymer chain segments (typically $20 - 50$ backbone carbon atoms) have ample space to undergo coordinated rotational jumps and crankshaft motions, resulting in rubbery flexibility.

• Cooling toward $T_g$: As temperature decreases, free volume contracts.
• At $T_g$: The free volume fraction drops to a universal critical percolation threshold:

• Below $T_g$ ($T < T_g$): The free volume is too constricted to accommodate coordinated backbone bond rotations. Large-scale segmental motion freezes out. The chains are trapped in a non-equilibrium disordered glassy configuration. The only motions permitted are localized small-amplitude bond vibrations and minor side-group rotations ($eta$- and $\gamma$-relaxations).

The glass transition temperature of a polymer is governed directly by its chemical constitution and chain architecture:

1. Backbone Chain Flexibility:

• Flexible backbones containing low rotational barrier bonds ($-C-O-C-$, $-Si-O-Si-$, $-C-C-$) exhibit extremely low $T_g$:
• Poly(dimethylsiloxane) (PDMS): $T_g = -123^\circ\text{C}$
• Polyethylene (PE): $T_g \approx -120^\circ\text{C}$ to $-80^\circ\text{C}$
• Poly(ethylene oxide) (PEO): $T_g = -67^\circ\text{C}$
• Rigid backbones containing bulky aromatic rings, heterocyclic units, or conjugated double bonds possess high torsional barriers, elevating $T_g$:
• Poly(ethylene terephthalate) (PET): $T_g = +78^\circ\text{C}$
• Polycarbonate (PC): $T_g = +150^\circ\text{C}$
• Polyetheretherketone (PEEK): $T_g = +143^\circ\text{C}$
• Polyimides (Kapton): $T_g > +380^\circ\text{C}$

2. Steric Bulk of Pendant Substituents:

• Bulky side groups hinder backbone bond rotation, restricting segmental motion and driving $T_g$ upward:
• Polypropylene ($-\text{CH}_3$): $T_g = -10^\circ\text{C}$
• Polystyrene ($-\text{C}_6\text{H}_5$): $T_g = +100^\circ\text{C}$
• Poly(1-vinylnaphthalene): $T_g = +150^\circ\text{C}$
• Conversely, flexible aliphatic side chains act as internal plasticizers, pushing chains apart and increasing free volume (internal plasticization):
• Poly(methyl methacrylate): $T_g = +105^\circ\text{C}$
• Poly(ethyl methacrylate): $T_g = +65^\circ\text{C}$
• Poly(n-butyl methacrylate): $T_g = +20^\circ\text{C}$
• Poly(n-octyl methacrylate): $T_g = -20^\circ\text{C}$

3. Intermolecular Forces (Dipole-Dipole & Hydrogen Bonding):

Strong polar interactions physically anchor neighboring chains, requiring higher thermal energy to unlock segmental motion:

• Polypropylene (non-polar): $T_g = -10^\circ\text{C}$
• Poly(vinyl chloride) (polar $C-Cl$ dipoles): $T_g = +82^\circ\text{C}$
• Polyacrylonitrile (strong $-C\equiv N$ dipoles): $T_g = +105^\circ\text{C}$
• Poly(vinyl alcohol) (interchain hydrogen bonds): $T_g = +85^\circ\text{C}$
• Nylon 6,6 (dense amide hydrogen bond lattice): $T_g = +50^\circ\text{C}$

4. Cross-Link Density:

Covalent cross-links introduce topological constraints that severely restrict chain mobility. As cross-link density increases, $T_g$ increases according to the DiMarzio equation:

Highly cross-linked epoxy or phenolic resins exhibit $T_g > 180 - 250^\circ\text{C}$.

The Fox Equation for Random Copolymers and Miscible Blends

In 1956, Thomas G. Fox derived the fundamental thermodynamic equation predicting the glass transition temperature of a homogeneous, single-phase random copolymer or miscible polymer blend:

where:

• $w_1$ and $w_2$ are the mass fractions of components 1 and 2 ($w_1 + w_2 = 1.0$).
• $T_{g, 1}$ and $T_{g, 2}$ are the glass transition temperatures of the corresponding pure homopolymers (expressed strictly in Kelvin).
• $T_g$ is the resulting intermediate glass transition temperature of the copolymer (in Kelvin).

For multicomponent systems containing $N$ comonomers:

Physical Derivation from Free Volume Additivity

Fox derived this relationship by assuming that the free volume of the copolymer mixture at its glass transition ($f_g = 0.025$) is the weighted sum of the free volume contributions of its constituent segments:

Linear expansion above each component's $T_g$ yields the Fox equation.

• Diagnostic Criteria for Phase Homogeneity:
• A single-phase homogeneous random copolymer or completely miscible blend exhibits a single intermediate $T_g$ that obeys the Fox equation.
• In contrast, an immiscible, phase-separated blend (such as HIPS or block copolymers) exhibits two distinct, separate $T_g$ values corresponding to each independent microphase domain.

Plasticization Mechanics

Plasticization is the intentional reduction of polymer $T_g$ through the addition of a compatible, low-volatility small molecule diluent (plasticizer). Because small plasticizer molecules possess vast free volume and low glass transitions ($T_{g, \text{plast}} \approx 120 - 180\text{ K}$), applying the Fox equation demonstrates dramatic $T_g$ suppression:

Adding $30\text{ wt}\%$ DOP ($T_g = 190\text{ K}$) to rigid PVC ($T_g = 355\text{ K}$) drops the glass transition of the plasticized compound to $280\text{ K}$ ($+7^\circ\text{C}$), transforming an unyielding pipe resin into soft, flexible film!

Viscoelastic phenomena are governed by molecular relaxation processes that occur over time scales ranging from microseconds to decades. Measuring relaxation experimentally over 10 decades of time at a single temperature is physically impossible. However, because temperature and time exert mathematically equivalent effects on molecular mobility, Time-Temperature Superposition (TTS) allows short-term measurements taken across multiple temperatures to be shifted into a single universal master curve spanning enormous spans of frequency or time.

The Shift Factor $a_T$

To construct a master curve at an arbitrary reference temperature $T_{\text{ref}}$, relaxation modulus curves $E(t)$ measured at temperature $T$ are shifted horizontally along the logarithmic time axis by a dimensionless shift factor $a_T$:

• If $T > T_{\text{ref}}$: Molecular motions are faster; relaxation occurs at shorter times. The curve must be shifted to the right: $a_T < 1 \implies \log a_T < 0$.
• If $T < T_{\text{ref}}$: Molecular motions are sluggish; relaxation is delayed. The curve must be shifted to the left: $a_T > 1 \implies \log a_T > 0$.
• At $T = T_{\text{ref}}$: $a_T = 1.0 \implies \log a_T = 0$.

The Williams-Landel-Ferry (WLF) Equation

In 1955, Malcolm Williams, Robert Landel, and John D. Ferry demonstrated that for all amorphous polymers in the temperature range from $T_g$ to $T_g + 100^\circ\text{C}$, the shift factor $\log a_T$ obeys a universal empirical equation:

When the polymer's glass transition temperature is chosen as the reference temperature ($T_{\text{ref}} = T_g$), the empirical parameters take on universal values across virtually all amorphous polymers:

where $C_1 = 17.44$ and $C_2 = 51.6\text{ K}$.

Theoretical Derivation from Free Volume Theory

According to the Doolittle viscosity equation, molecular friction factor $\zeta$ depends exponentially on the reciprocal free volume fraction $1/f$:

Substituting the linear expansion $f(T) = f_g + \alpha_f (T - T_g)$:

Dividing numerator and denominator by $\alpha_f$ and converting to base-10 logarithm:

Comparing directly with the WLF equation yields the fundamental physical definitions:

Setting $B \approx 1.0$, $C_1 = 17.44$ yields the universal free volume at $T_g$:

and $C_2 = 51.6\text{ K}$ yields the thermal expansion coefficient of free volume:

This landmark derivation unites empirical polymer rheology with the statistical thermodynamics of free volume!