12.3 Phase transitions and critical phenomena

Phase transitions are fascinating phenomena that occur when a substance changes from one state to another. They're crucial in understanding how matter behaves under different conditions, from everyday occurrences like boiling water to complex systems in physics and chemistry.

In thermodynamics, phase transitions reveal the intricate balance between energy and entropy. We'll explore how molecular interactions drive these changes, and how critical points mark the boundary between distinct phases, offering insights into the nature of matter at its most fundamental level.

Phase transition classification

Ehrenfest classification scheme

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• Phase transitions are classified by the Ehrenfest classification scheme based on the lowest derivative of the free energy that is discontinuous at the transition
• First-order phase transitions exhibit a discontinuity in the first derivative of the free energy with respect to a thermodynamic variable
  • Examples include solid-liquid and liquid-gas transitions
• Second-order phase transitions have a continuous first derivative but a discontinuous second derivative of the free energy
  • Examples include the ferromagnetic transition and the superfluid transition in liquid helium

Molecular characteristics and latent heat

• The order of a phase transition is related to the latent heat
  • First-order transitions involve latent heat, while second-order transitions do not
• Molecular characteristics, such as the degree of molecular ordering and symmetry, change abruptly at first-order transitions and continuously at second-order transitions
• The Landau theory of phase transitions describes the behavior of the order parameter, which quantifies the degree of ordering in a system, across different types of phase transitions
  • The order parameter changes discontinuously at first-order transitions and continuously at second-order transitions
  • The free energy can be expanded in terms of the order parameter to study the behavior near the phase transition

Intermolecular forces and phase behavior

Critical point and intermolecular forces

• Intermolecular forces, including van der Waals forces, hydrogen bonding, and electrostatic interactions, play a crucial role in determining the phase behavior of molecular systems
• The critical point represents the end of the liquid-gas coexistence curve, beyond which the distinct liquid and gas phases no longer exist
  • It is characterized by the critical temperature (Tc), critical pressure (Pc), and critical volume (Vc)
• The strength and nature of intermolecular forces influence the location of the critical point
  • Stronger intermolecular attractions result in higher critical temperatures and pressures
  • For example, water has a higher critical point than carbon dioxide due to the presence of hydrogen bonding

Critical phenomena and fluctuations

• Near the critical point, the differences in density and other properties between the liquid and gas phases become negligible, leading to critical opalescence and enhanced fluctuations
• The compressibility of a system diverges at the critical point, indicating increased susceptibility to density fluctuations
• The correlation length, which characterizes the spatial extent of fluctuations, diverges at the critical point, leading to long-range correlations in the system
  • This results in the formation of large-scale structures and the slowing down of dynamics near the critical point
  • Critical opalescence, the scattering of light due to these large-scale fluctuations, is observed in fluids near their critical points

Universality and scaling in critical phenomena

Universality and critical exponents

• Universality refers to the observation that many systems exhibit similar critical behavior near phase transitions, regardless of their microscopic details
• Systems belonging to the same universality class share the same critical exponents, which describe the power-law dependence of various physical quantities on the reduced temperature (t = (T - Tc) / Tc) near the critical point
  • For example, the critical exponent β describes the behavior of the order parameter (magnetization for ferromagnets, density difference for fluids) as t → 0
• The critical exponents are universal and depend only on the dimensionality of the system and the symmetry of the order parameter
  • The Ising universality class, which includes ferromagnets and binary fluids, has specific values for critical exponents that are different from those of the Heisenberg universality class, which includes isotropic magnets

Scaling laws and renormalization group theory

• Scaling laws describe the relationship between different physical quantities near the critical point
  • For example, the Widom scaling law relates the critical exponents for the correlation length (ν), susceptibility (γ), and specific heat (α): γ = (2 - η)ν and α = 2 - dν, where d is the dimensionality and η is the anomalous dimension
• The scaling hypothesis states that the singular part of the free energy near the critical point is a homogeneous function of the reduced temperature and the ordering field (e.g., magnetic field for ferromagnets)
  • This leads to the collapse of data onto universal scaling functions when physical quantities are plotted in terms of scaled variables
• Renormalization group theory provides a framework for understanding the origin of universality and calculating critical exponents by systematically coarse-graining the system and studying the flow of coupling constants under scale transformations
  • The fixed points of the renormalization group flow correspond to the different universality classes, and the critical exponents are determined by the eigenvalues of the linearized flow near the fixed points