CahnHilliardligningen

CahnHilliardligningen is a theoretical framework that extends the classical Cahn–Hilliard approach to describe phase separation and pattern formation in lignin-containing polymer systems. It combines diffuse-interface theory with lignin-specific thermodynamics to model how lignin-rich and lignin-poor domains develop and evolve within biopolymer matrices.

In its typical formulation, the concentration of lignin (or an equivalent order parameter) obeys a Cahn–Hilliard-type equation: ∂c/∂t = ∇·(M ∇ μ), where μ = ∂f/∂c − κ ∇^2 c. Here f is a free-energy density with a double-well component representing two preferred lignin configurations, M is a mobility that may depend on c, and κ is a gradient-energy coefficient that governs interface width. The model may be augmented by coupling to additional fields such as solvent fraction, temperature, or crosslink density to reflect processing conditions in biomass materials. This allows the simulation of how processing steps like solvent exchange or thermal treatment influence microstructure.

Applications focus on lignin-containing materials, including lignin–cellulose composites, thermoplastic lignin blends, and biomaterial aging. The framework helps researchers explore how microstructure affects mechanical properties, diffusion, and stability during processing and use. Benefits include a tractable, continuum description of demixing; challenges involve parameterizing the heterogeneous lignin structure and extending the model to three-dimensional, highly crosslinked systems.

See also: Cahn–Hilliard equation; phase-field method; lignin; biomass materials; polymer blends.