Abstract
We present the final film thickness' expression of spin-coated photoresist on a spherical substrate. Firstly, some reasonable assumptions are put forward for a concise derivation process. Then, on the basis of the motion equation of spin-coated photoresist on a plane, considering the spherical surface shape, we put forward the motion equation of spin-coated photoresist on a spherical substrate. So two evolution equations of film thickness and radial position are derived, and the expression of initial film thickness evolution in a radial position is also gained. Finally, considering some effects of solvent volatilization, we gain the expression of final film thickness. The experiment result indicates that the expression is accurate.
©2005 Optical Society of America
1. Introduction
There are many applications for concave gratings and diffractive optical elements (DOE), including light splitting, imaging, and focusing [1]. These optical elements are usually fabricated by the photolithography technology [2]. Thus, the problem of coating uniform photoresist film on a curved substrate is brought forward. However, few papers analyze it in detail until now [3]. Its main reason should be the complexity of curved equation derivation and the particularity of application.
The paper focuses on the evolution of film thickness of spin-coated photoresist on a spherical substrate and shows the derivation course in detail. Experiment results can affirm the accuracy of these equations and the method. Thus, these equations may be used for predicting the film thickness of spin-coated photoresist on a spherical substrate, and the derivation method can also be applied for other curved surfaces. Furthermore, the theoretical results can also guide the development of new equipment.
2. Flow mechanism
The flow mechanism of spin-coated photoresist on a spherical substrate relates to the centrifugal force, the coriolis force, the viscous force, the surface tension, the volatilization of solvent, and so on. Thus, following assumptions are required
- Dilute photoresist is Newtonian fluid approximately
- A spherical surface is an axial symmetry shape, so the coriolis force may be ignored [4]
- Because the film thickness is quite small in comparison with the characteristic dimension of substrate, the surface tension may also be ignored
- The shear force that arose from wind is also ignored.
The motion equation of spin-coated photoresist on a plane is given by [5]
In Eq. (1), ρω 2 r is the centrifugal force,μ is the kinematic viscosity of fluid, u is the radial velocity in z direction, z is the height in the film thickness direction, ρ is the density of fluid, ω is the angular velocity, and r is the radial position.
Obviously, on a spherical substrate, there is (see Fig. 1)
In Eq. (2), θ is the pitch angular of spherical tangent line in a radial position (obviously, θ ≺ 90°),R is the spherical radius. Considering the effect of gravity, we may derive the motion equation of spin-coated photoresist on a spherical substrate
In Eq. (3), g is the gravity force of infinitesimal fluid, and it relates to the density of fluid. Because the lubrication approximation of fluid is satisfied, the following boundary conditions can be gained.
On a substrate (z equal to zero), there is
The velocity gradient of a free surface is zero, and there is
Thus, the integral expression of Eq. (3) can be derived
The mass continuity equation of incompressible fluid is
According to Eq. (6) and Eq. (7), there is
The differential expression of the optional position's film thickness is
Compare Eq. (8) with Eq. (9), two evolution equations of photoresist thickness and radial position can be gained
Eq. (10) indicates that the change speed rate of film thickness disaccords among radial positions, and this is also different of the sphere’ and the planar’. As the radial position is determined, the integral expression of Eq. (10) is
In Eq. (12), h 0 is the initial film thickness of a radial position. Eq. (12) describes the variety that the initial film thickness (h 0) relates to the radial position (r).
According to Eq. (3), flow-out necessary condition of fluid film is
The solution of Eq. (13) is
While r = 0 , the film thickness can’t be calculated in Eq. (12) (see Eq. (7)). Actually, the film thickness of revolving spindle is non-uniform forever [6].
So, there is
According to Eq. (12), while r ~ 0.816R, there is
Provided that ≻ 0.816 R , the position of r ≈ 0.816R is a key point according to Eq. (12). While 0 ≺ r ≺ 0.816R , the fluid film thins gradually by the centrifugal force, and h ≺ h 0. However, while 0.816 R ≺ r ≺ , the fluid film thickens gradually by the gravity force, and h ≻ h 0. Thus, the spin-coated form should not be used provided that the caliber of a spherical substrate is larger than 1.632R (see Fig. 1), or else the uniform film can’t be gained forever.
While R ≻≻ r or high speed spin-coated, the effect of the gravity force may be ignored, the variety that the film thickness can be gained according to Eq. (12)
Obviously, Eq. (17) is the case of spin-coated fluid film on a plane (See Ref. [5]).
In order to gain the solid film, the volatilization of solvent must be considered. So more assumptions are necessary, including
- The initial concentration of solvent is uniform, and the concentration of solute (c) is changed uniformly by the volatilization of solvent, and c is independent of r
- The change of c in the direction of z may be ignored
- The density of solvent and solute is uniform, and the capacity of solution equals to the sum of the solvent capacity (L) and the solute capacity (S).
The actual fluid disaccords with these assumptions, but the difference has no effect on the results [7].
According to these assumptions, there is
The change of viscosity arises from the change of concentration, and it has an effect on the dynamics characteristic of fluid. Following equations are the change rate of S and L owing to the flow-out of fluid and volatilization of solvent
In Eq. (21), e is the speed rate of solvent volatilization, and hf is final film thickness, and Sf is final solute thickness, obviously, hf = Sf.
With the volatilization of solvent, the concentration of fluid increases. The viscosity of fluid increases, also. So the rheological change of fluid should be expressed by the function of power-law
The volatilization of solvent may be ignored while the film thickness is thinner than the critical point h 1/3 (it relates to the initial condition). While the film thickness is h 1/3, the volatilization of solvent and the radial flow have same effect on the variety of film thickness [8]. Moreover, e relates to the angular velocity of spin-coated (See Ref. [6]).
In Eq. (24), ω is the angular velocity of spin-coated, and C is the coefficient of laboratory and device.
Thus, the final film thickness may be attained
In Eq. (25), c 0 is the density of nonvolatile matter in the photoresist.
Commonly, as the photoresist is spun coating on a spherical substrate, the assumption of Newtonian fluids and laminar airflow (i.e., some assumptions of solvent volatilization) can be satisfied, the results of Eq. (25) are reasonable. As the spin-coated speed or the caliber of substrate is quite big, Sukanek' work must be considered seriously [9].
3. Experiment and results
The positive photoresist BP212-6 is diluted with the thinner, and is spun coating on a concave spherical substrate. Afterwards, the photoresist is exposed by a laser direct writing’ equipment, and a generating line figure is generated by concentric optical scan. After developing, in some radial positions, the film thickness is tested by an atomic force microscope (See Fig. 2). Experiment shows that the mathematical model is accurate.
4. Conclusions
Based on some assumptions, we present the evolution equation of spin-coated photoresist on a spherical substrate. Considering the effect of solvent volatilization, we derive the final film thickness of spin-coated photoresist on a spherical substrate. Moreover, two inferences may be gained
- In Eq. (25), the final photoresist thickness is independent of initial film thickness, so it is independent of the initial material amounts. While the radius of a spherical substrate and the angular speed are confirmed, the final film thickness relates only to the radial position, and the film thickness gets thick along with the increased radial dimension. Thus, the form of spin-coated must be altered to gain the even film thickness while r/R ≻ 0.816. For example, the opening of spherical substrate turns towards flank when the photoresist is spun coating.
- Obviously, these conclusions may be also applied to other similar fluid.
Acknowledgments
This study is supported by the National Defence Science Advanced Foundation No.10.4.2.ZK1001).
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