Finite difference algorithm for solving lubrication equations with solute diffusion
AssigneeSeiko Epson Corporation
InventorsJie Zhang, Jiun-Der Yu, Shinri Sakai
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Abstract
A computer implemented method for simulating a final pattern of a droplet of a fluid having a plurality of fluid properties is disclosed. The method includes using lubrication equations to represent solute flow, diffusion and evaporation of a droplet on a substrate. The method further includes solving the lubrication equations through temporal discretization and spatial discretization; and deriving the final pattern of the droplet from results of the solving. The final pattern is stored on a computer readable medium.
Claims14
A computer-readable medium encoded with instructions for a processor to perform a method for simulating a final pattern of a droplet of a fluid, comprising:
- instructions for using lubrication equations that describe behavior of the droplet on a non-flat substrate;
- instructions for using the lubrication equations to represent diffusion in the droplet and evaporation of the droplet;
- instructions for solving the lubrication equations through temporal discretization and spatial discretization;
- instructions for deriving the final pattern of the droplet from results from solving the lubrication equations;
- and
- instructions for storing the final pattern,
- wherein evaporation in the droplet is calculated using a first high order differential function that is representative of the behavior of a height of the droplet over time, wherein a first function is equated to a second function, the first function being representative of a temporal derivative of a height (H) of the droplet above a plane, and the second function including a first term that is a function of the height (H-f) of the droplet above the substrate (f);
- and a Laplacian of the height (∇2H) of the droplet above the plane.
A computer-readable medium encoded with instructions for a processor to perform a method for simulating a final pattern of a droplet of a fluid, comprising:
- instructions for using lubrication equations that describe behavior of the droplet on a non-flat substrate;
- instructions for using the lubrication equations to represent diffusion in the droplet and evaporation of the droplet;
- instructions for solving the lubrication equations through temporal discretization and spatial discretization instructions for deriving the final pattern of the droplet from results from solving the lubrication equations;
- and
- instructions for storing the final pattern,
- wherein the diffusion in the droplet is calculated using a second high order differential function that is representative of the behavior of a concentration (C) of solute in the droplet, wherein
- a temporal derivative of the product of a height of the droplet above the substrate and the concentration ((H-f)C) is equated to a sum of a third function and a fourth function;
- the third function is a differential function of a product of:
- the concentration (C);
- the height of the droplet above the substrate (H-f);
- and
- a third high order differential function of the height of the droplet (H) above a plane;
- and
- the fourth function is a differential function of a product of:
- a differential function of the concentration (C);
- and
- the height of the droplet above the substrate (H-f).
A computer-readable medium encoded with instructions for a processor to perform a method for simulating the evolution of a height of an evaporating droplet comprising:
- generating a height function that is representative of the height (H) of the droplet above a plane at a first point in time at a plurality of points in a simulation space;
- generating a first differential function that describes a proportional relationship between an intermediate variable and a Laplacian of the height function (∇2H);
- generating a second differential function that comprises:
- a first term that is a partial derivative of the height (H) function with respect to time;
- a second term that is proportional to the evaporation rate (J) of the droplet;
- and
- a third term that is a third function of the height function (H), a height of a non-flat substrate (f) on which the droplet is located, and the intermediate variable;
- and
- determining the height function at a second point in time by finding an approximate solution that satisfies both the first differential function and the second differential function.
Dependencies are inferred from the claim text.