Desorption Isotherm Modeling: A Review of Current Approaches and Challenges

1. Introduction

2. Current Approaches to Desorption Isotherm Modeling

2.1 Empirical and Semi-Empirical Models

• Modified BET Model:​ While the Brunauer-Emmett-Teller (BET) model is primarily for multilayer adsorption, modified versions attempt to fit desorption branches, particularly for high-moisture content regions. However, they often fail to capture the hysteresis loop adequately.
• GAB Model:​ The Guggenheim-Anderson-de Boer (GAB) model is a successful extension of BET for food applications, describing water sorption well over a wide range of moisture contents. Its application to desorption requires careful parameter fitting but is widely used for practical predictions.
• Halsey and Harkins-Jura Models:​ These are simple exponential models that provide reasonable fits for specific segments of the desorption curve but lack universality.
• Iglesias-Chirife Model:​ Specifically developed for food dehydration, this model accounts for the "constant rate period" and the falling rate period, linking desorption to heat and mass transfer phenomena.

2.2 Thermodynamic and Mechanistic Models

• Capillary Condensation Theory (Kelvin Equation):​ This classical approach explains the sharp increase in uptake at high relative pressures by invoking the condensation of vapor in cylindrical capillaries. The desorption branch is modeled by considering the evaporation from menisci with a different radius than during condensation. This theory successfully predicts the onset of hysteresis but fails to close the loop quantitatively due to oversimplified pore geometry assumptions.
• Independent Domain Theory (Potential Theory):​ Proposed by Everett, this theory suggests that the adsorbate occupies independent sites with a distribution of energies. Hysteresis arises from the difference in energy required for adsorption (nucleation) versus desorption (evaporation). It provides a qualitative framework but is mathematically complex for practical use.
• Pore Network Models (PNMs):​ These models represent the porous medium as a network of interconnected pores of various shapes (cylinders, slits, ink-bottle). The dynamics of liquid/vapor interfaces through this network are simulated using Young-Laplace equation. PNMs can reproduce realistic hysteresis loops and are powerful for studying the impact of pore structure. However, they are computationally intensive and require detailed knowledge of the microstructure.
• Density Functional Theory (DFT) for Pores:​ DFT, originally a quantum mechanical method, has been adapted for fluids in nanoporous materials. It calculates the local density profile of the fluid within a pore of a specific geometry, providing a molecular-level explanation for capillary condensation and hysteresis. It is highly accurate for model pores but becomes prohibitively expensive for complex, disordered real materials.

2.3 Modern Computational & Hybrid Approaches

• Molecular Dynamics (MD) Simulations:​ MD tracks the motion of individual atoms/molecules over time. It can simulate the entire desorption process in a small material volume, capturing both thermodynamic and kinetic aspects. It reveals atomic-scale mechanisms but is limited to very small length and time scales.
• Monte Carlo (MC) Simulations:​ Particularly Grand Canonical Monte Carlo (GCMC), MC simulations determine equilibrium configurations by sampling states according to statistical weights. It is excellent for calculating adsorption isotherms in model geometries and has been extended to study desorption pathways. Like MD, it is limited by computational cost.
• Machine Learning (ML) Models:​ Artificial Neural Networks (ANNs), Support Vector Machines (SVMs), and Gaussian Process Regression are being trained on large databases of experimental isotherms. ML models can predict desorption behavior with high accuracy if sufficient data is available. Their major drawback is being "black boxes" with poor interpretability and generalizability outside the training domain.
• Hybrid Multi-Scale Models:​ The frontier of research involves coupling models across scales. For example:
  • Using DFT to calculate adsorption energies for specific pore types, which then inform a larger-scale Pore Network Model.
  • Using experimental data to train an ML model that acts as a surrogate for a computationally expensive MD simulation.

3. Key Challenges in Desorption Isotherm Modeling

1. Accurate Representation of Hysteresis:​ No single model perfectly closes the hysteresis loop for all materials and conditions. Capturing the "main" loop and nested scanning curves remains a formidable task. The choice of model often depends on the material's porosity (microporous vs. mesoporous).
2. Multi-Component Systems:​ Most models are developed for single-component desorption (e.g., pure water vapor). Real-world systems involve mixtures (e.g., air-water-vapor, organic solvents). Competitive adsorption and co-desorption introduce immense complexity, and reliable models are scarce.
3. Temperature Dependence:​ Desorption is strongly temperature-dependent. While some models incorporate thermal effects via the Clausius-Clapeyron relation, accurately predicting desorption behavior across a wide temperature range for complex materials is challenging. Thermal gradients during actual desorption processes add another layer of difficulty.
4. Material Heterogeneity and Structural Complexity:​ Real materials have irregular, hierarchical pore structures (macro-, meso-, micro-pores) and surface chemical heterogeneity. Capturing this complexity in a tractable model is extremely difficult. Most models rely on idealized geometric representations.
5. Kinetics vs. Equilibrium:​ Traditional isotherm models describe equilibrium states. In practice, desorption is a dynamic, non-equilibrium process influenced by heat and mass transfer resistances. Bridging the gap between equilibrium isotherm models and transient kinetic models is crucial for industrial applications like drying.
6. Parameter Identification and Uniqueness:​ Many mechanistic models have multiple adjustable parameters. Fitting these parameters to experimental data can lead to non-unique solutions, making physical interpretation ambiguous. Robust inverse modeling techniques are needed.
7. Data Scarcity and Quality:​ High-quality, reproducible experimental desorption data, especially for novel materials or under extreme conditions, is often lacking. This limits the development and validation of new models, particularly for ML approaches.

4. Future Perspectives

• Integrated Multi-Scale Frameworks:​ The development of seamless workflows that link atomistic simulations (DFT, MD) to mesoscopic models (PNM) and finally to continuum-scale engineering models will be paramount.
• Advanced Machine Learning:​ Moving beyond simple regression, the use of physics-informed neural networks (PINNs) that embed known physical laws (e.g., mass conservation, thermodynamics) directly into the loss function could lead to more robust and interpretable AI models.
• Focus on Complex Fluids and Mixtures:​ There is a pressing need to develop theories and models for desorption of supercritical fluids, ionic liquids, and complex mixtures relevant to carbon capture, hydrogen storage, and pharmaceutical manufacturing.
• Dynamic and Non-Equilibrium Modeling:​ Coupling isotherm models with computational fluid dynamics (CFD) to simulate real-world desorption processes in reactors, dryers, and geological formations will enhance design and optimization.
• Standardization of Protocols:​ Establishing standard protocols for measuring and reporting desorption data would greatly facilitate model comparison and validation.

5. Conclusion