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International Journal of Bioprinting Swelling–shrinking behavior of hydrogel

Figure 2. Finite element method modeling for investigating the humidity-driven swelling–shrinking behavior of hydrogel filaments. (A) Schematic
diagram showing the extrusion process of a hydrogel filament. (B) Simplified model and computational mesh of the printed filament.

of analyses were conducted across various diameters and where ϕ is the dimensionless phase field variable, which
relative humidity (RH) values. takes a value of 1 or −1 to distinguish one fluid phase from

2.2. The boundary conditions of the finite element another; u is the velocity vector; γ is the mobility rate,
method modeling which determines the time scale of the Cahn–Hilliard
diffusion; λ and ω are the mixing energy density and the
2.2.1. Two-phase flow modeling of the interface thickness parameter, respectively, which are both
printed filament highly related to the materials; and Ψ is the phase field
To simulate the swelling–shrinking behavior of hydrogel auxiliary variable. These parameters were calculated in the
filaments under varying ambient RH conditions, the two-
phase flow method was employed in the FEM model. A COMSOL software without artificial settings.
phase field method is suitable for modeling two immiscible 2.2.2. Flow condition of the ambient air
fluids with stable interfaces. It enables a sharp transition
between the two fluid phases, resulting in a high-resolution To conduct further analysis based on the two-phase flow
representation of the fluid–fluid interface. The dynamics model, the flow conditions within the ambient air were also
of two-phase flow is governed by a widely adopted considered. During the evaporation process of the printed
Cahn–Hilliard theory : filament, water contained within the hydrogel filament
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diffuses into the ambient air and transitions into vapor.
Conversely, water vapor from the ambient air is absorbed

u
(I)
t 2 into the hydrogel during the swelling process. This transfer
of water vapor results in a slight flow of moist air within
the ambient air domain. The Reynolds number (R e) was
2 ( 2 1) (II) employed to determine this type of flow condition :
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Volume 11 Issue 4 (2025) 413 doi: 10.36922/IJB025220222