Numerical Simulation of The Time-Dependent Permeation Through Barrier Structures
Flexible electronic devices are very sensitive against the ingress of oxygen and water vapor from the atmosphere and therefore have to be protected by encapsulation with high barrier films. Such films contain several polymeric or inorganic layers which are deposited from the liquid or gas phase on top of a flexible substrate.
Two important concepts of barrier structures will be discussed: 1) Alternating inorganic and polymeric layers; 2) Polymeric layers with dispersed inorganic particles. Theoretical modelling and numerical simulation of the transport processes give insight into the barrier mechanisms of these structures and allow to calculate steady-state and transient permeation rates dependent on geometrical and material parameters.
Sorption and diffusion of substances in polymeric layers are described by Henry’s and Fick’s laws, while the matrix of inorganic materials is assumed to be impermeable as a first approximation. However, the permeation through inorganic layers is possible within defects which are caused by the surface roughness of the substrate during deposition.
The resulting diffusion equation within the polymeric regions and the zero normal flux condition at the interfaces between these regions and the impermeable material of inorganic layers or particles govern the transport through single layers and multilayer structures. The equations were numerically solved by the finite element method using the General Form PDE interface of COMSOL Multiphysics®.
The considered model was applied to the transient permeation through two inorganic layers containing defects and an intermediate polymeric layer. According to the numerical results such a structure might exhibit a low steady-state permeability and, in addition, a long transient period during which the permeation rate approaches the steady-state value. This behavior correlates with the calculated evolution of the concentration profile over time. An explanation for the slow increase of the concentration within the intermediate layer is given by the large capacity of the layer for the permeating substance in contrast to the low permeation fluxes through the small defects of the surrounding inorganic layers. Based on this observation, a quasi-steady-state approach for the permeation through structures of alternating layers was developed.
Consequently, the barrier performance of a polymeric layer can be significantly improved by its combination with inorganic layers. Another concept to reduce its permeability consists in the integration of inorganic particles such as sheet silicates and graphene into the matrix of the polymeric layer.
Numerical simulations show that the period of transient permeation through a polymeric layer is extended if it contains impermeable particles. Increasing their volume fraction or aspect ratio results in an enhanced effect on the corresponding lag time. In the case of two-dimensional systems channels exist between the particles where the concentration gradient is nearly parallel to their direction. Therefore, the time-dependent permeation rate can be calculated approximately under the assumption of one-dimensional diffusion within the channels.
