♨️Thermodynamics of Fluids Unit 15 – Critical Phenomena in Supercritical Fluids

Key Concepts and Definitions

• Critical point represents the highest temperature and pressure at which a substance can exist as a vapor and liquid in equilibrium
• Supercritical fluid is a state of matter that occurs when a substance is heated and compressed above its critical point
• Critical temperature ($T_c$) is the temperature above which vapor-liquid phases cannot exist
• Critical pressure ($P_c$) is the minimum pressure required to liquefy a gas at its critical temperature
• Critical density ($\rho_c$) is the density of a fluid at its critical point
• Compressibility factor ($Z$) measures the deviation of a fluid's behavior from that of an ideal gas, equals 1 at the critical point
• Critical opalescence is the phenomenon of increased light scattering and opacity near the critical point due to density fluctuations
  • Caused by the correlation length of density fluctuations becoming comparable to the wavelength of visible light

Supercritical Fluid Basics

• Supercritical fluids exhibit properties intermediate between those of gases and liquids
• Have gas-like diffusivity and viscosity, enabling rapid mass transfer and penetration into porous materials
• Possess liquid-like density, allowing them to act as effective solvents
• Lack a distinct phase boundary, transitioning smoothly from liquid-like to gas-like properties as temperature and pressure are varied
• Exhibit enhanced solubility for many substances compared to their solubility in the fluid's gas or liquid state
• Solvent power can be tuned by adjusting temperature and pressure, making them versatile for selective extraction and separation processes
• Commonly used supercritical fluids include carbon dioxide ($CO_2$), water ($H_2O$), and ethane ($C_2H_6$)
  • $CO_2$ is particularly popular due to its moderate critical temperature (31.1°C), non-toxicity, and low cost

Phase Diagrams and Critical Points

• Phase diagrams map the regions of pressure and temperature where distinct phases (solid, liquid, gas) exist in equilibrium
• Critical point is located at the end of the vapor-liquid equilibrium curve (coexistence curve) on a phase diagram
• Vapor-liquid equilibrium curve represents the conditions where liquid and vapor phases coexist
• Supercritical region lies beyond the critical point, where no phase boundaries exist
• Triple point is the unique combination of pressure and temperature where all three phases (solid, liquid, gas) can coexist
• Sublimation curve separates the solid and gas regions, indicating the conditions where solid transitions directly to gas (sublimation)
• Melting curve (fusion curve) separates the solid and liquid regions, representing the melting point at various pressures
• Critical opalescence occurs near the critical point due to increased density fluctuations and correlation length

Thermodynamic Properties Near the Critical Point

• Thermodynamic properties exhibit anomalous behavior and divergences near the critical point
• Isothermal compressibility ($\kappa_T$) diverges at the critical point, indicating enhanced density fluctuations
  • Defined as $\kappa_T = -\frac{1}{V} \left(\frac{\partial V}{\partial P}\right)_T$
• Isobaric heat capacity ($C_P$) also diverges at the critical point, reflecting the system's increased heat capacity
  • Defined as $C_P = T \left(\frac{\partial S}{\partial T}\right)_P$
• Thermal expansion coefficient ($\alpha$) diverges at the critical point, indicating enhanced thermal expansivity
  • Defined as $\alpha = \frac{1}{V} \left(\frac{\partial V}{\partial T}\right)_P$
• Density fluctuations become long-ranged near the critical point, with a correlation length ($\xi$) that diverges as $\xi \propto |T - T_c|^{-\nu}$, where $\nu$ is a critical exponent
• Transport properties such as viscosity and thermal conductivity also exhibit anomalous behavior near the critical point
• Universal scaling laws govern the behavior of thermodynamic properties near the critical point, with critical exponents that are independent of the specific substance

Experimental Techniques and Observations

• Light scattering techniques (e.g., Rayleigh scattering, Brillouin scattering) probe density fluctuations and correlation lengths near the critical point
• Small-angle X-ray scattering (SAXS) and small-angle neutron scattering (SANS) provide information on the structure and size of density fluctuations
• Interferometry and refractive index measurements detect changes in the fluid's optical properties near the critical point
• Calorimetry measures heat capacity and latent heat associated with phase transitions and critical phenomena
• Density measurements using vibrating tube densimeters or magnetic suspension balances track density changes and anomalies near the critical point
• Acoustic measurements (e.g., sound velocity, attenuation) probe the fluid's compressibility and relaxation processes
• Visual observations of critical opalescence, turbidity, and light scattering patterns provide qualitative evidence of critical phenomena
• Microgravity experiments (e.g., on the International Space Station) eliminate gravity-driven effects and allow for more precise studies of critical behavior

Applications in Industry and Research

• Supercritical fluid extraction (SFE) is used for selective extraction of valuable compounds from natural materials (e.g., caffeine from coffee beans, essential oils from plants)
• Supercritical fluid chromatography (SFC) employs supercritical fluids as mobile phases for efficient and selective separation of complex mixtures
• Supercritical water oxidation (SCWO) is a waste treatment technology that uses supercritical water to oxidize and destroy hazardous organic compounds
• Supercritical fluid synthesis is used to produce nanoparticles, aerogels, and other advanced materials with controlled size, morphology, and properties
• Enhanced oil recovery (EOR) techniques utilize supercritical $CO_2$ to extract residual oil from depleted reservoirs
• Supercritical fluids serve as reaction media for chemical synthesis, enabling improved selectivity, yield, and environmental sustainability compared to conventional solvents
• Fundamental research on critical phenomena advances our understanding of phase transitions, universality, and the behavior of matter under extreme conditions
• Supercritical fluid technology is being explored for applications in carbon capture and storage (CCS), hydrogen storage, and renewable energy systems

Mathematical Models and Equations

• Van der Waals equation of state captures the qualitative behavior of fluids near the critical point, including the existence of a critical point and vapor-liquid equilibrium
  • $\left(P + \frac{a}{V_m^2}\right)\left(V_m - b\right) = RT$, where $a$ and $b$ are substance-specific constants
• Renormalization group theory provides a framework for understanding the universal scaling behavior near the critical point
• Scaling laws relate the divergence of thermodynamic properties to the reduced temperature $\epsilon = \frac{T - T_c}{T_c}$ and critical exponents (e.g., $\alpha$, $\beta$, $\gamma$, $\delta$, $\nu$)
  • Example: $C_P \propto \epsilon^{-\alpha}$, $\rho_l - \rho_v \propto \epsilon^\beta$, $\kappa_T \propto \epsilon^{-\gamma}$
• Universality hypothesis states that critical exponents are determined by the system's dimensionality and symmetry, rather than the specific details of the intermolecular interactions
• Fluctuation-dissipation theorem relates the response of a system to external perturbations (e.g., compressibility) to the equilibrium fluctuations in the corresponding variable (e.g., density)
• Integral equations (e.g., Ornstein-Zernike equation) and density functional theory (DFT) provide microscopic descriptions of the structure and thermodynamics of fluids near the critical point
• Molecular dynamics (MD) and Monte Carlo (MC) simulations offer insights into the molecular-level behavior and dynamics of supercritical fluids

Challenges and Future Directions

• Developing more accurate and predictive equations of state for supercritical fluids, particularly in the near-critical region where classical models break down
• Improving the understanding of the microscopic origins of critical phenomena and the relationship between intermolecular interactions and macroscopic behavior
• Advancing experimental techniques for in situ characterization of supercritical fluids under high-pressure and high-temperature conditions
• Designing novel supercritical fluid-based processes for sustainable chemical synthesis, materials processing, and energy applications
• Exploring the use of supercritical fluids in emerging fields such as biotechnology, nanomedicine, and green chemistry
• Investigating the behavior of complex fluids (e.g., polymers, surfactants, ionic liquids) in supercritical media and their potential applications
• Harnessing the unique properties of supercritical fluids for the development of innovative technologies in carbon capture, hydrogen storage, and renewable energy systems
• Expanding the fundamental understanding of critical phenomena in multicomponent systems, confinement, and out-of-equilibrium conditions