Glass transition
The glass transition is the nonequilibrium process by which an equilibrium supercooled liquid transforms into a nonequilibrium amorphous solid (glass) upon cooling, or vice versa upon heating, marked by a dramatic slowdown in structural relaxation times without any change in the average atomic or molecular structure.[1] This transition occurs in amorphous materials, including polymers, metallic glasses, and inorganic oxides, and is characterized by a gradual shift from a rigid, brittle glassy state below the glass transition temperature (Tg) to a more viscous, rubbery state above it.[2] Unlike crystalline melting, the glass transition involves no latent heat or abrupt volume change, but rather a kinetic arrest where the material's viscosity reaches approximately 1012 Pa·s (or 1013 poise).[3] This viscosity-based definition is phenomenological, and its relationship to a true thermodynamic glass transition—if such a thing even exists—remains unclear, as the glass transition is debated between kinetic and thermodynamic interpretations.[4] The underlying physics of the glass transition remains a central puzzle in condensed matter science, with two competing theoretical frameworks: thermodynamic approaches, which propose an "ideal" transition driven by a rapid loss of configurational entropy leading to a Kauzmann entropy crisis, and kinetic models, which emphasize dynamic heterogeneity and the loss of ergodicity due to increasingly cooperative relaxation processes.[2] Phenomenological theories, such as the Vogel-Fulcher-Tammann equation for viscosity and mode-coupling theory for the dynamics near Tg, capture universal features like the super-Arrhenius increase in relaxation times and the emergence of a dynamic crossover temperature Td above Tg.[5] These aspects highlight the transition's universality across disparate glass-forming systems, from fragile organic liquids to strong network formers like silica.[6] In practice, Tg governs the mechanical and thermal properties of amorphous materials, setting limits on their processing temperatures, dimensional stability, and performance in applications ranging from structural polymers to pharmaceutical formulations.[7] For instance, in polymers, Tg influences ductility and toughness, with values typically ranging from -100°C for flexible elastomers to over 300°C for high-performance engineering plastics, modulated by factors like chain stiffness, intermolecular interactions, and cooling rate.[8] The transition's sensitivity to thermal history—manifesting as structural relaxation and aging effects—further complicates material design, underscoring the need for precise control in manufacturing.[9]Fundamentals
Characteristics
The glass transition represents a gradual and reversible transformation in amorphous materials, shifting from a hard, brittle glassy state to a viscous supercooled liquid state upon heating. This process occurs without the absorption or release of latent heat and lacks an abrupt change in volume, distinguishing it as a kinetic phenomenon driven by the freezing of molecular configurations rather than a thermodynamic equilibrium shift.[10][11] Across the transition, mechanical properties undergo a marked change: below the glass transition temperature T_g, the material displays high rigidity and elastic modulus, typically on the order of gigapascals, rendering it brittle under stress. Above T_g, it adopts a rubbery or viscous character with significantly reduced stiffness, often dropping to below 1 MPa in storage modulus, enabling greater flexibility and deformation.[12][11] The kinetic essence of the glass transition manifests in its sensitivity to experimental conditions, particularly the cooling or heating rate, which influences the point at which structural relaxation halts or resumes. Standard measurements, such as those via differential scanning calorimetry, employ rates of 10 K/min to capture this behavior reliably.[13][14] Unlike melting, which disrupts long-range crystalline order and involves a discontinuous volume expansion with latent heat, the glass transition pertains exclusively to amorphous structures devoid of such order, emphasizing a slowdown in dynamics over structural reorganization.[10] Early observations of softening behavior in undercooled liquids date to the work of Gustav Tammann in the early 20th century, particularly his 1903 publication on states of aggregation. The concept was formalized in 1930 by Fritz Simon, who analogized it to a second-order phase transition due to its continuous nature in thermodynamic properties.[11]Formal Definitions
The glass transition is formally defined as the temperature regime in which the relaxation times associated with molecular or structural rearrangements in a supercooled liquid become comparable to the duration of the experimental observation, typically on the order of 100 seconds.[6] This kinetic criterion highlights the nonequilibrium nature of the process, where the system's inability to equilibrate on accessible timescales leads to the formation of a structurally arrested, amorphous solid.[15] Operationally, the glass transition temperature T_g is often specified as the isotherm at which the shear viscosity \eta attains $10^{12} Pa·s, marking the onset of solid-like behavior in the material. Equivalently, it corresponds to the point where the structural relaxation time \tau reaches approximately 100 s, or where the instantaneous shear modulus transitions to a value around $10^9 Pa, reflecting the dominance of elastic over viscous responses.[16] From a thermodynamic viewpoint, the glass transition manifests as a pseudo-second-order transition, featuring a step-like discontinuity in the specific heat capacity C_p at T_g, while the first derivatives of the Gibbs free energy remain continuous; this implies a discontinuous second derivative of the free energy, akin to classical second-order phase changes but without true equilibrium singularity.[17] The underlying configurational entropy S_c plays a central role, as the transition arises when cooperative molecular rearrangements become too sluggish to allow the system to access its full entropy landscape, effectively freezing in a disordered configuration and driving the material out of thermodynamic equilibrium. This entropic arrest, formalized in the Adam-Gibbs framework, links the exponential divergence of relaxation times to the diminishing S_c near T_g, emphasizing the role of structural disorder in vitrification. The concept of fictive temperature T_f provides a means to quantify this nonequilibrium state, defined as the hypothetical temperature of an equilibrium supercooled liquid that would possess the same enthalpy and entropy as the actual glass at its formation temperature. Thus, T_f serves as a structural parameter that evolves with thermal history, bridging the glassy state's frozen properties to an equivalent equilibrium counterpart.Transition Temperature
Definition and Measurement
The glass transition temperature, denoted as T_g, is operationally defined using differential scanning calorimetry (DSC) as the midpoint of the step change in heat capacity, determined as the intersection of the tangent to the curve at the point of inflection with the extrapolated pre- and post-transition baselines, marking the point where the material's thermodynamic response shifts from rigid to more fluid-like behavior.[18] Alternatively, T_g can be identified from the change in the rate of volume expansion, where the coefficient of thermal expansion transitions from the lower value characteristic of the glass (\alpha_g) to the higher value of the supercooled liquid (\alpha_l).[19] This definition aligns with operational standards for assigning T_g in amorphous materials, emphasizing the kinetic nature of the transition rather than a strict thermodynamic equilibrium.[18] Differential scanning calorimetry (DSC) is a primary method for measuring T_g, where the transition appears as a step-like increase in heat capacity, manifesting as an endothermic baseline shift in the heat flow curve during heating.[20] The T_g is typically determined from the inflection point of this step, often calculated as the midpoint between the extrapolated onset and end of the transition, following standardized procedures such as ASTM E1356.[18] For precise measurement, DSC instruments are calibrated using high-purity indium as a reference standard, which has a well-defined melting transition at 156.5985 °C and enthalpy of fusion of 28.58 J/g, ensuring accurate temperature and heat flow scaling.[21] Dilatometry measures T_g by tracking dimensional changes with temperature, revealing a characteristic kink in the thermal expansion curve where the expansion rate accelerates due to the onset of structural relaxation. Below T_g, the relative length change follows \Delta L / L_0 = \alpha_g (T - T_g), while above it, \Delta L / L_0 = \alpha_l (T - T_g), with linear fits to the glassy and liquid regimes used to locate the intersection defining T_g.[22] This technique, often implemented via thermomechanical analysis (TMA), is particularly sensitive for materials where volume changes dominate the transition signature, as per ISO 11359-2 guidelines.[23] Viscometry defines T_g operationally as the temperature where the shear viscosity reaches $10^{12} Pa·s (log \eta = 12), corresponding to a relaxation time of approximately 100 seconds, measured using techniques like fiber elongation or beam bending for high-viscosity regimes.[24] Complementary to this, dynamic mechanical analysis (DMA) identifies T_g from the peak in the loss tangent (tan \delta) or the maximum in the loss modulus during oscillatory testing, reflecting the temperature where mechanical relaxation broadens significantly, as standardized in ASTM E1640. These rheological methods are essential for capturing the dynamic aspects of the transition in viscoelastic materials.[25] The measured T_g exhibits dependence on the heating or cooling rate q, with faster rates yielding higher T_g values due to the kinetic lag in structural equilibration; an empirical relation approximates this as T_g \approx T_{g0} + C \log q, where T_{g0} is the reference T_g at a standard rate (e.g., 10 K/min), and C is a material-specific constant typically ranging from 2 to 5 K per decade of rate.[26] This rate sensitivity underscores the nonequilibrium nature of the glass transition and necessitates consistent experimental conditions for comparability across studies.[27]Factors Influencing Tg
The glass transition temperature (Tg) is profoundly influenced by molecular architecture, particularly chain stiffness and intermolecular forces. In polymers, increased chain stiffness, such as from rigid aromatic groups or bulky side chains, restricts segmental mobility and raises Tg by reducing the entropy of the system.[28] Stronger intermolecular forces, like hydrogen bonding, further elevate Tg by enhancing cohesive interactions that hinder cooperative rearrangements during cooling.[28] Free volume theory posits that Tg arises when the available free volume falls below a critical threshold, limiting molecular motion; materials with lower inherent free volume, due to compact packing from stiff chains or strong attractions, exhibit higher Tg.[29] Cooling rate significantly modulates the apparent Tg through kinetic effects on structural relaxation. Faster cooling rates increase Tg because the system has insufficient time for complete relaxation, trapping it in a higher-energy, less equilibrated state with reduced free volume.[30] This kinetic origin is evident in techniques like differential scanning calorimetry (DSC), where Tg shifts by approximately 3–5 K per decade change in cooling rate.[31] Pressure dependence of Tg stems from its compression of free volume and alteration of relaxation dynamics, typically following a Clausius-Clapeyron-like relation derived from thermodynamic changes at the transition. The coefficient dTg/dP ranges from 0.1 to 0.3 K/MPa across various glass-formers, reflecting how elevated pressure slows dynamics by densifying the structure.[32] Additives and plasticizers lower Tg by introducing excess free volume and enhancing chain mobility. Low-molecular-weight solvents or plasticizers, such as dioctyl phthalate in polymers, disrupt intermolecular forces and increase the fractional free volume, depressing Tg proportionally to their concentration.[28] In nanoscale confinement, such as thin films or pores, Tg often decreases by 10–50 K compared to the bulk due to enhanced surface interactions that accelerate surface-layer dynamics and reduce overall cooperativity.[33] This effect is prominent in supported polymer films, where free surfaces dominate, leading to a gradient in mobility.[33] For mixed systems, universal mixing rules predict Tg based on component weight fractions (w_i) and pure-component Tg values. The Fox-Flory equation, applicable to miscible polymer blends, assumes ideal volume additivity of free volume contributions: \frac{1}{T_g} = \frac{w_1}{T_{g1}} + \frac{w_2}{T_{g2}} This linear reciprocal form captures deviations from additivity in copolymers. The more general Gordon-Taylor equation accounts for differing thermal expansivities via a fitting parameter k (often k ≈ ρ1 Cp1 / ρ2 Cp2, where ρ is density and Cp is heat capacity): T_g = \frac{w_1 T_{g1} + k w_2 T_{g2}}{w_1 + k w_2} This relation effectively models Tg in amorphous mixtures, including pharmaceuticals and composites.[34]Thermodynamic Aspects
Heat Capacity Changes
The glass transition is characterized by a discontinuous increase in the isobaric heat capacity C_p at the transition temperature T_g, typically on the order of \Delta C_p \approx 0.1 to $0.5 J/g·K for many amorphous materials, which signifies the onset of configurational degrees of freedom as the material shifts from a rigid glass to a more mobile supercooled liquid state. This jump arises because, below T_g, molecular rearrangements are frozen, limiting contributions to heat capacity primarily to vibrational modes, whereas above T_g, additional anharmonic and cooperative configurational excitations become active, enhancing the material's ability to absorb heat.[35] In metallic glasses, for instance, \Delta C_p values cluster around 13-14 J/mol·K, often approximating $3R/2 (where R is the gas constant), underscoring a universal scaling tied to atomic-scale freedoms.[36] Heat capacity in glasses can be modeled using temperature-dependent expressions that distinguish vibrational from configurational contributions. Below T_g, the heat capacity of the glass follows a predominantly vibrational form, approximated as C_{p,\text{glass}} \approx a + bT, where a and b are material-specific constants reflecting lattice vibrations akin to those in crystals, with minimal anharmonicity.[37] Above T_g, the supercooled liquid's heat capacity incorporates an additional configurational term, often modeled as C_{p,\text{liquid}} \approx a + bT + c/T, where the c/T contribution accounts for relaxational processes and structural rearrangements that scale inversely with temperature.[38] These models highlight how the configurational entropy, frozen in the glass, unfreezes upon heating through T_g, leading to a step-like enhancement in thermal response without a latent heat, distinguishing the transition from a first-order phase change.[39] The enthalpy and entropy changes associated with this \Delta C_p have significant thermodynamic implications for the supercooled liquid. The excess enthalpy relative to the glass is given by \Delta H = \int_{T_g}^{T} \Delta C_p \, dT', representing the energy stored in configurational states as temperature rises above T_g.[37] Consequently, the excess entropy in the supercooled liquid accumulates as \Delta S = \int_{T_g}^{T} (\Delta C_p / T') \, dT', reflecting the increased number of accessible microstates due to molecular mobility, which drives the material's viscous flow behavior.[40] This excess entropy persists in the supercooled regime, influencing stability and relaxation kinetics. Experimentally, heat capacity changes are observed in differential scanning calorimetry (DSC) as a sigmoidal step in the heat flow curve over a temperature range of about 10-20 K, centered at T_g, where the baseline shifts upward due to the \Delta C_p jump.[41] Upon reheating an aged glass sample, an overshoot often appears in the DSC trace near T_g, attributed to enthalpy recovery as structural relaxation releases stored energy from the nonequilibrium glassy state.[42] These features provide a direct probe of the transition's kinetic and thermodynamic nature. Below T_g, the vibrational density of states in glasses exhibits an excess contribution known as the boson peak, typically observed in the 1-10 THz range via inelastic neutron scattering or Raman spectroscopy, which correlates with an anomalous rise in low-temperature heat capacity beyond the Debye T^3 prediction.[43] This peak arises from quasilocalized vibrational modes in the disordered structure, contributing to the vibrational heat capacity and distinguishing glassy dynamics from crystalline ones, with its intensity linked to structural heterogeneity.[44]Kauzmann's Paradox
Kauzmann's paradox refers to a thermodynamic inconsistency that emerges when extrapolating the properties of supercooled liquids below the glass transition temperature T_g. Proposed by Walter Kauzmann in his seminal 1948 review, the paradox highlights an apparent "entropy catastrophe" where the extrapolated configurational entropy of the supercooled liquid would become negative at a finite temperature, violating fundamental thermodynamic principles.[45] The thermodynamic basis of the paradox lies in the decomposition of the liquid's entropy relative to the crystalline state. The entropy of the supercooled liquid is given byS_\text{liquid} = S_\text{crystal} + \Delta S_\text{vib} + \Delta S_\text{conf},
where \Delta S_\text{vib} represents the difference in vibrational entropy (typically small and positive) and \Delta S_\text{conf} is the configurational entropy difference arising from the multitude of accessible molecular arrangements in the liquid. Upon cooling below the melting point, \Delta S_\text{conf} decreases more rapidly than \Delta S_\text{vib} because the heat capacity at constant pressure C_p of the liquid exceeds that of the crystal, leading to a steeper decline in entropy with decreasing temperature.[45][10] This behavior originates from the observed discontinuity in heat capacity at T_g, where C_{p,\text{liquid}} > C_{p,\text{crystal}}, prompting a linear extrapolation of the liquid's entropy using the glassy state's properties below T_g. The extrapolation predicts that the excess entropy \Delta S = S_\text{liquid} - S_\text{crystal} reaches zero at the Kauzmann temperature T_K < T_g, implying \Delta S_\text{conf} < 0 for T < T_K, an unphysical state where the disordered liquid would possess less entropy than the ordered crystal.[45][10] The implications of this paradox challenge the validity of equilibrium thermodynamics for deeply supercooled liquids, as it suggests a breakdown where the liquid's disorder fails to maintain its expected entropic advantage, potentially contravening the third law of thermodynamics. In practice, glass formation circumvents the catastrophe by kinetically arresting the system at T_g, rendering the glass non-ergodic and frozen in a metastable, out-of-equilibrium configuration that does not follow the extrapolated equilibrium path.[45] Graphically, the paradox is depicted in a plot of entropy S versus inverse temperature $1/T, where the supercooled liquid's entropy curve, characterized by a larger slope (dS/d(1/T) = -C_p/T) due to its higher C_p, intersects the shallower crystal curve at $1/T_K, visually underscoring the impending entropy crossover.[45][10]