4.7 Crystallinity and glass transition

Crystallinity and glass transition are two of the most important concepts for understanding why polymers behave the way they do. Together, they explain differences in strength, transparency, flexibility, and thermal behavior. By controlling factors like cooling rate, chain regularity, and molecular weight, polymer chemists can tune these properties for specific applications.

Crystallinity in polymers

Crystallinity refers to the degree of structural order in a polymer, where chains align into organized regions. Most polymers aren't fully crystalline or fully amorphous; they're somewhere in between (semi-crystalline). The amount and type of crystallinity directly affects strength, transparency, melting point, and chemical resistance.

Types of polymer crystals

Several models describe how crystalline regions form in polymers:

• Fringed micelle model: Small crystalline regions are connected by amorphous chain segments. This was the earliest model and works well for low-crystallinity polymers.
• Folded chain model: Polymer chains fold back and forth on themselves to form thin, plate-like structures called lamellae. This is the most widely accepted model for polymer crystallization.
• Extended chain crystals: Chains align parallel to each other without folding. These are rare and typically only form under high pressure or in very stiff polymers.
• Spherulites: Three-dimensional structures made of lamellae radiating outward from a central nucleation point, resembling a sphere. These are the most common large-scale crystalline structures you'll see in semi-crystalline polymers.

Degree of crystallinity

The degree of crystallinity quantifies what fraction of a polymer sample is in an ordered, crystalline state. It typically ranges from 0% (fully amorphous, like atactic polystyrene) to around 90% (highly crystalline, like HDPE).

$\text{Degree of Crystallinity} = \frac{\text{Mass of Crystalline Regions}}{\text{Total Mass of Polymer}} \times 100\%$

Common measurement techniques include:

• X-ray diffraction (XRD): Compares crystalline peak areas to amorphous scattering
• Density measurements: Crystalline regions pack more efficiently and have higher density
• Differential scanning calorimetry (DSC): Compares the measured heat of fusion to that of a 100% crystalline reference

Higher crystallinity generally means greater tensile strength, increased opacity, and improved chemical resistance.

Factors affecting crystallinity

• Chain regularity: Regular, stereoregular chains pack more easily. Isotactic polypropylene crystallizes readily, while atactic polypropylene stays largely amorphous.
• Molecular weight: Higher molecular weight increases chain entanglement, which can slow crystallization but also affects the final degree of order.
• Cooling rate: Slow cooling gives chains more time to organize into crystals. Fast cooling (quenching) can trap chains in a disordered, amorphous state.
• Nucleating agents: Additives that provide surfaces for crystal growth, increasing the number of nucleation sites and speeding up crystallization.
• Intermolecular forces: Stronger interactions (hydrogen bonding in nylon, for example) promote tighter chain packing and higher crystallinity.

Crystallization process

Crystallization happens in stages:

1. Nucleation: Small clusters of aligned chains form. This can be homogeneous (spontaneous, within the melt) or heterogeneous (starting at a surface, impurity, or nucleating agent).
2. Primary crystallization: Rapid growth of crystallites as more chains join the ordered regions. Spherulites expand outward during this stage.
3. Secondary crystallization: A slower process where existing crystals thicken and become more perfect over time.

The kinetics of isothermal crystallization are described by the Avrami equation:

$X_t = 1 - e^{-kt^n}$

• $X_t$ = fraction crystallized at time $t$
• $k$ = crystallization rate constant (depends on temperature)
• $n$ = Avrami exponent, which reflects the geometry of crystal growth and the type of nucleation (values typically range from 1 to 4)

The linearized form is useful for experimental analysis:

$\ln[-\ln(1-X_t)] = \ln k + n \ln t$

Spherulites and lamellae

Spherulites are the dominant morphological feature in many semi-crystalline polymers. They consist of lamellae (thin, folded-chain crystals) that radiate outward from a central nucleus.

• Spherulite size affects optical properties: smaller spherulites scatter less light, producing more transparent materials.
• Lamellar thickness influences the melting temperature, as described by the Gibbs-Thomson equation. Thicker lamellae melt at higher temperatures.
• Spherulite growth can be directly observed using polarized optical microscopy, where they appear as characteristic "Maltese cross" patterns.

Glass transition temperature

The glass transition temperature ($T_g$) is the temperature at which an amorphous polymer (or the amorphous regions of a semi-crystalline polymer) transitions from a hard, glassy state to a soft, rubbery state. This isn't a sharp melting point; it occurs over a range of temperatures and is a kinetic phenomenon, not a true thermodynamic phase transition.

A useful rule of thumb: for many polymers, $T_g$ is roughly $\frac{2}{3}$ of the melting temperature $T_m$ (both in Kelvin).

Factors influencing Tg

• Chain flexibility: Flexible backbones (like polysiloxanes) have low $T_g$ values. Rigid backbones with aromatic rings (like polycarbonate) have high $T_g$ values.
• Intermolecular forces: Stronger forces (hydrogen bonding, dipole-dipole) restrict chain motion and raise $T_g$.
• Molecular weight: Higher molecular weight generally increases $T_g$ because there's less free volume from chain ends.
• Plasticizers: Small molecules that insert between chains, increasing free volume and lowering $T_g$. PVC is a classic example where plasticizer content determines whether you get rigid pipe or flexible tubing.
• Crosslinking: Restricts chain mobility and raises $T_g$.
• Tacticity: Isotactic polymers often have higher $T_g$ than their atactic counterparts due to more efficient packing.

Measurement techniques for Tg

• Differential Scanning Calorimetry (DSC): Detects the step change in heat capacity at $T_g$. Most common method.
• Dynamic Mechanical Analysis (DMA): Measures changes in storage modulus and loss modulus. The peak in tan delta (or loss modulus) identifies $T_g$. Often considered the most sensitive technique.
• Dilatometry: Tracks the change in thermal expansion coefficient at $T_g$.
• Thermomechanical Analysis (TMA): Monitors dimensional changes with temperature.
• Dielectric analysis: Measures changes in electrical properties across the transition.

Because $T_g$ is a kinetic phenomenon, the measured value depends on heating rate. Faster heating rates shift the apparent $T_g$ higher.

Importance in polymer applications

$T_g$ determines the usable temperature range for amorphous polymers. Below $T_g$, the material is rigid and glassy; above it, the material becomes rubbery and flexible.

• Processing: Extrusion and injection molding temperatures must account for $T_g$ (and $T_m$ for semi-crystalline polymers).
• Mechanical performance: Modulus, strength, and impact resistance all change dramatically around $T_g$.
• Blend compatibility: $T_g$ values of individual components affect miscibility and phase behavior in polymer blends.
• Applications: Food packaging, automotive parts, and aerospace components all require careful $T_g$ selection to ensure performance across expected temperature ranges.

Amorphous vs crystalline regions

Most commercial polymers are semi-crystalline, containing both amorphous and crystalline domains. The balance between these regions controls the overall properties of the material.

Structural differences

• Amorphous regions have randomly oriented, disordered chains. They undergo the glass transition.
• Crystalline regions have chains arranged in regular, ordered patterns. They exhibit a distinct melting temperature.
• Tie molecules bridge between crystalline regions through the amorphous phase, providing mechanical connectivity.
• Interfacial regions exist at the boundary between amorphous and crystalline domains, with intermediate levels of order.

Property comparisons

Influence on polymer behavior

Crystalline regions provide strength and stiffness, while amorphous regions contribute flexibility and toughness. The ratio between them determines the overall balance of properties.

• Amorphous regions undergo the glass transition; crystalline regions melt.
• Stress-induced crystallization can occur when amorphous regions are stretched (natural rubber is a classic example).
• Crystalline regions act as physical crosslinks, anchoring the amorphous chains and influencing viscoelastic behavior.
• Processing history (cooling rate, orientation, annealing) directly controls the development of both types of domains.

Thermal transitions in polymers

Polymers exhibit several thermal transitions, with the glass transition ($T_g$) and melting temperature ($T_m$) being the most important. Understanding these transitions is essential for selecting materials and setting processing conditions.

Melting temperature vs Tg

For many polymers, the ratio $T_m / T_g$ (both in Kelvin) is approximately 1.5. This is a useful approximation, not a strict rule.

The temperature gap between $T_m$ and $T_g$ defines the window where cold crystallization can occur: amorphous regions that were quenched too quickly can crystallize when reheated into this range.

Heat capacity changes

At $T_g$, you see a step change in heat capacity on a DSC trace. The size of this step reflects how much molecular mobility increases above $T_g$. At $T_m$, you see a sharp endothermic peak as crystalline regions absorb energy to melt.

The degree of crystallinity can be calculated from DSC data by comparing the measured heat of fusion to the theoretical heat of fusion for a 100% crystalline sample.

Thermal expansion coefficients

• Amorphous polymers show a distinct increase in thermal expansion coefficient at $T_g$.
• Crystalline regions have lower thermal expansion than amorphous regions.
• In oriented polymers, thermal expansion can be anisotropic (different in different directions). Some highly oriented fibers even show negative thermal expansion along the fiber axis.
• Mismatches in thermal expansion between amorphous and crystalline regions can create internal stresses during temperature cycling.

Crystallinity effects on properties

Mechanical properties

Increasing crystallinity generally improves tensile strength, modulus (stiffness), and creep resistance. However, there's a trade-off: highly crystalline polymers tend to be more brittle, with lower elongation at break and reduced impact strength.

• Yield stress increases with crystallinity.
• Fatigue performance depends on crystalline morphology and orientation.
• The relationship between crystallinity and strength is not perfectly linear; crystal size, perfection, and the number of tie molecules all play a role.

Optical properties

Crystalline regions scatter light because they have a different refractive index than the surrounding amorphous material. As crystallinity increases, transparency decreases and haze increases.

• Smaller spherulites scatter less light, so controlling nucleation to produce many small spherulites (rather than few large ones) improves clarity.
• Birefringence appears in oriented crystalline regions.
• Some crystalline structures can even produce iridescence.

Thermal properties

• Melting temperature increases with greater lamellar thickness and crystal perfection.
• Heat deflection temperature rises with crystallinity.
• Thermal conductivity improves with crystallinity (ordered regions conduct heat more efficiently).
• Thermal expansion coefficient decreases with higher crystallinity.

Barrier properties

Crystalline regions are essentially impermeable to gas and solvent molecules. Increasing crystallinity forces diffusing molecules to take longer, more tortuous paths through the amorphous regions.

• Gas permeability and moisture vapor transmission both decrease with crystallinity.
• Solvent resistance improves.
• The orientation of crystallites affects directional barrier properties.
• Tie molecules between crystalline regions influence overall permeability.

Glass transition effects on properties

Viscoelastic behavior

The most dramatic property changes in amorphous polymers occur around $T_g$. In DMA:

• Storage modulus drops by several orders of magnitude (often from ~$10^9$ Pa to ~$10^6$ Pa).
• Loss modulus shows a peak near $T_g$.
• Tan delta (the ratio of loss to storage modulus) also peaks, and this peak is often used to define $T_g$ in DMA experiments.

The time-temperature superposition principle applies in the viscoelastic region: the effect of increasing temperature is equivalent to increasing the time scale of observation. This allows construction of master curves that predict long-term behavior from short-term tests.

Mechanical properties

• Below $T_g$: the polymer is glassy, stiff, and brittle.
• Above $T_g$: the polymer becomes rubbery, flexible, and ductile.
• Impact strength generally improves above $T_g$.
• Yield stress drops significantly.
• Crazing behavior (formation of micro-voids under stress) changes across the glass transition.

Processing considerations

• $T_g$ sets the lower bound for processing temperatures of amorphous polymers.
• Mold temperature relative to $T_g$ affects surface quality and dimensional stability.
• Annealing near $T_g$ relieves internal stresses from processing.
• Physical aging occurs below $T_g$: the polymer slowly densifies over time, which can change mechanical properties (increased stiffness, decreased impact strength).
• Plasticizers lower $T_g$, making processing easier and the final product more flexible.

Crystallization kinetics

Nucleation and growth

Crystallization rate depends strongly on temperature. The maximum crystallization rate occurs at a temperature roughly halfway between $T_g$ and $T_m$. Below this optimum, chains lack mobility; above it, thermal energy disrupts crystal formation.

• Homogeneous nucleation: Spontaneous formation of nuclei within the melt. Requires significant supercooling.
• Heterogeneous nucleation: Starts at surfaces, impurities, or added nucleating agents. More common in practice and requires less supercooling.
• Secondary nucleation: Growth on existing crystal surfaces, which drives spherulite expansion.

Avrami equation

The Avrami equation describes isothermal crystallization:

$X_t = 1 - e^{-kt^n}$

The Avrami exponent $n$ provides information about the crystallization mechanism:

Cooling rate effects

• Fast cooling produces many small crystallites (or can suppress crystallization entirely if fast enough).
• Slow cooling allows fewer, larger, and more perfect crystals to form.
• Quenching below $T_g$ can freeze the polymer in a fully amorphous state. Reheating above $T_g$ then allows cold crystallization.
• The critical cooling rate is the minimum rate needed to prevent crystallization entirely. Polymers with slow crystallization kinetics (like PET) are easier to quench into an amorphous state than fast-crystallizing polymers (like polyethylene).
• Non-isothermal crystallization (the real-world case) is more complex and requires modified kinetic models.

Characterization techniques

X-ray diffraction

• Wide-angle X-ray scattering (WAXS): Reveals crystal structure and allows calculation of a crystallinity index from the ratio of crystalline peak areas to total scattering.
• Small-angle X-ray scattering (SAXS): Probes larger-scale morphology, including lamellar spacing (long period).
• Unit cell parameters come from peak positions; crystallite size is estimated using the Scherrer equation.
• Pole figure analysis assesses crystallite orientation.

Differential scanning calorimetry

DSC is the workhorse technique for thermal analysis of polymers. In a typical heating scan:

1. $T_g$ appears as a step change in the baseline (change in heat capacity).
2. Cold crystallization (if present) appears as an exothermic peak.
3. Melting appears as an endothermic peak.

The degree of crystallinity is calculated by comparing the measured heat of fusion (area under the melting peak) to the theoretical value for 100% crystallinity. Running multiple heating/cooling cycles reveals thermal history effects.

Dynamic mechanical analysis

DMA measures the storage modulus (elastic response), loss modulus (viscous response), and tan delta as functions of temperature, time, or frequency.

• $T_g$ is identified by the sharp drop in storage modulus or the peak in tan delta.
• Secondary relaxations (beta, gamma transitions) appear as smaller features at lower temperatures.
• Time-temperature superposition generates master curves for predicting long-term mechanical behavior.
• Crystallinity affects the magnitude of the modulus drop at $T_g$: more crystalline samples show a smaller drop because the crystalline regions maintain stiffness above $T_g$.

Structure-property relationships

Crystallinity vs strength

Higher crystallinity generally increases tensile strength and modulus, but the relationship isn't perfectly simple. Crystal size, perfection, and the density of tie molecules all matter.

• At very high crystallinities, brittleness can become a problem, and ultimate strength may actually decrease.
• Oriented semi-crystalline polymers show anisotropic properties: much stronger along the orientation direction.
• Tie molecules connecting crystalline regions through the amorphous phase are critical for transferring stress between crystallites.

Tg vs flexibility

A polymer's flexibility at a given use temperature depends heavily on where that temperature falls relative to $T_g$.

• Well above $T_g$: rubbery and flexible (think silicone rubber, $T_g$ ≈ -125°C).
• Well below $T_g$: stiff and glassy (think polystyrene at room temperature, $T_g$ ≈ 100°C).
• Plasticizers shift $T_g$ lower, increasing flexibility at a given temperature.
• Copolymerization can be used to fine-tune $T_g$ between the values of the two homopolymers.

Morphology vs performance

• Spherulite size affects both optical clarity (smaller = more transparent) and mechanical toughness.
• Lamellar thickness controls melting temperature and contributes to mechanical strength.
• Orientation creates anisotropic behavior, which can be advantageous in fibers and films.
• Strain-induced crystallization (as in natural rubber during stretching) can dramatically improve strength during deformation.
• Transcrystallinity at fiber-matrix interfaces in composites enhances load transfer.

Industrial applications

Polymer selection criteria

Selecting the right polymer requires matching $T_g$, $T_m$, and crystallinity to the application:

• The operating temperature range must fall in the right zone relative to $T_g$ and $T_m$.
• Crystallinity must be balanced for the needed combination of strength, transparency, and barrier performance.
• Processing requirements (melt viscosity, crystallization rate) must be practical for the manufacturing method.
• Long-term stability, chemical resistance, and aging behavior all connect back to $T_g$ and crystallinity.

Processing considerations

• Cooling rate control in injection molding and extrusion directly sets crystallinity levels.
• Annealing after processing can increase crystallinity and relieve residual stresses.
• Nucleating agents speed up crystallization and produce finer spherulitic textures.
• Processing temperatures are chosen based on $T_g$ (for amorphous polymers) or $T_m$ (for semi-crystalline polymers).
• In fiber spinning and film drawing, orientation and crystallinity develop simultaneously and must be carefully managed.

Product design implications

• Crystallization shrinkage causes dimensional changes that must be accounted for in mold design.
• Differential thermal expansion between amorphous and crystalline regions can cause warping or internal stresses.
• Oriented semi-crystalline polymers offer high directional strength for structural applications.
• Wall thickness affects local cooling rates, which in turn affect crystallinity. Thicker sections cool more slowly and may develop different crystallinity than thin sections.
• High-stress applications should account for the possibility of stress-induced crystallization during service.