Spin casting of polymers is a simple technique, which is widely used for producing very thin organic films by applying centrifugal force, linear shear stress, and uniform evaporation of a polymer solution on a rotating disk or a substrate [Meyerhofer, 1978; Hall et al., 1998]. In the spin casting process, a solution of polymer is first deposited on the substrate, and the substrate is then accelerated rapidly to the desired rotation rate. During rotation, the polymer solution flows radially outward owing to the action of centrifugal force, its thickness being changed. Final film thickness is based on the initial properties of the polymer, solvent, and spin speed; it is independent of the size of the substrate. The film continues to thin slowly until the film reaches an equilibrium thickness. The final thinning of the film is then due solely to solvent evaporation.

Spin casting (coating) seems a simple and successful way of producing a very thin uniform film when the casting or coating liquid is Newtonian and not too volatile, and the substrate to be coated is smooth. But in many applications the casting liquid contains not only highly volatile solvent but also a solvent of lower volatility as well as dissolved polymers or suspended solids, or both. In those cases, the success of spin casting may require the proper combination of casting formulation, design, and sequence of processing conditions.

Spin casting (coating) is a batch process, which means a combination of different processes (Figure 1.1). The final result of spin casting is the application of a uniform thin film of some desired casting liquid (or liquid like slurry) to a rigid flat disk or substrate or other shape. The liquid film may or may not solidify as part of the process, i.e. during the spinning operation. Uniform films with thicknesses on the order of micrometers are readily produced industrially. There are two main applications for this process: (1) the production of ultrathin photoresist films (with thickness less than 200 nm) necessary for the manufacturing of microelectronic devices and (2) the deposition of magnetic dispersion on aluminum substrate during the production of magnetic memory devices.

The basic variables that come into play during the casting cycle are (1) speed or rpm; (2) casting cycle time; (3) direction of spin; (4) speed of pouring of polymer and (5) substrate cleanliness. The spin casting process can be divided into four major stages: deposition, spin-up, spin-off, and evaporation of solvents. These stages generally overlap. The first three can be sequential process steps during which the casting (coating) liquid flows over and off the rigid substrate due to the action of centrifugal force, which increases with spinning speeds. A short description of each stage is as follows:

This is the first stage of the spin casting process for delivering an excess of the liquid polymer to the surface of the substrate. During this process, the arriving liquid displaces all of the air or gas that initially covers the substrate’s surface. Liquid can be deposited in different ways: (a) as heavy rain that inundates or covers the entire disk; (b) as the continuos delivery at the center and around the perimeter of the entire substrate, the liquid then spreading over the rest of the substrate while spinning; or (c) as a continuous stream from an elevated port that moves radially over the entire substrate or disk, so that a rotating disk receives a spiral ridge of liquid that can subsequently flow over the rest of the disk, during spinning.

 



Figure 1.1.1. Different Process Involved in Spin Casting

    

1.2 Spin - up

This is the second stage of spin casting, for covering or wetting the entire surface of the substrate with excess casting liquid. The liquid flows radially at all locations under the centrifugal force generated by the spinning disk. Ordinarily this is a transient stage although it may include a pre-spin interval of constant, low angular velocity of the disk. The mechanics of liquid flow during this process can be quite complicated.

This is the third stage of spin casting, for removing the excess liquid from the entire surface of the substrate. From rotational speeds that may be in the range of from a few tens to a few thousand rpm, the liquid flies off the edge or boundary of the spinning substrate, breaking up into droplets as it does so. A film of nearly uniform thickness tends to become even more uniform as it thins further, the films continuing to thin ever more slowly, after an equilibrium thickness is achieved.

The tendency of the liquid film toward uniformity as it is thinned by the centrifugally-driven radial flow lies at the heart of the spin casting (coating) process. This tendency of uniformity can be distressed or disturbed by non-Newtonian rheology, by misdesigned continuous delivery of casting liquid during spin-off, and by nonuniform evaporation of solvents, among other things.

This is the fourth or last stage of the spin casting process, where the liquid solidifies due to the solvent evaporation. Solvent evaporation is the process of concentrating the suspended or slurried solids, dissolved polymer, and other solutes of low volatility in the material or solution that is to remain as the coated film. The rate of evaporation of a solvent from a casting or coating liquid depends mainly on two factors. First is the difference or variation in partial pressure of each solvent type between the free surface of the liquid layer and the bulk of the gas flowing nearby. Second is the area of convective and diffusive transport of vaporized solvents through the gas between the free surface and the bulk gas flowing nearby. The convective diffusion area or regime depends on many factors but mainly on how the gas flows above the rotational disk that is whether the gas flow is turbulent or laminar in nature. Plainly the evaporation of solvents can be quite complex as it overlaps the first three stages.

As the film grows thinner during spinning operation with time, the local concentrations and concentration gradients are more sharply altered by departure of a unit mass of solvent. But as evaporation proceeds, its rate tends to fall or reduced because the remaining solvents of casting or coating liquid encounter greater resistance to reach the free surface of the film and exerting their partial pressure there.

Moreover, mass transfer by evaporation has the ability to affect the flow and film thickness of the casting or coating liquid. During the solvent evaporation, the viscosity of the polymer solution increases and its tendency to radial flow reduces. Only this last stage is modeled in the present work, assuming some solidification has occurred.

Few experimental studies have been done for the modeling of spin casting of polymer films [Bornside et al., 1987; Extrand, 1994; Hall et al., 1998; Jenekhe, 1983-84; Meyerhofer, 1978; Warren et al., 1983]. With approximately Newtonian liquids, e.g., linseed oil, varnish, and certain enamels, it has been found that a uniform final thickness as a result of spin casting is independent of the amount of liquid dispensed and the liquid profile at the start of rotation, as long as that amount was sufficient to cover the rotating disk or substrate. The thickness of casting liquid, of course, decreases continuously with time as material is spun away. For non-Newtonian liquids, the film no longer approaches a uniform thickness condition. Different attempts have been made to correlate the thickness of more or less uniform polymer films with concentration of solvents initially in the coating liquid, viscosity of the coating liquid, angular velocity of the disk, duration of spinning, etc. An important finding [Bornside el al., 1987] was that film thickness late in the spin casting or coating process is affected very little or not at all by the deposition and spin-up stages. Because in standard practice, both are completed early, and together they take but a minor fraction of the time that is consumed by the succeeding spin-off and solvent evaporation stages. Another finding was that, in the absence of solvent evaporation, the casting or coating liquid continues to thin without changing concentration as the spin-off stage proceeds. In fact, the apparent final film thickness h_f achieved from the spin casting process can be related with the angular velocity of spinning w , roughly as h_f w ^-l , where .

Because of the widespread use of the spinning technique, numerous measurements of profile and film thickness have been reported. Damon [1967] studied Kodak photoresists and found the following equation which describes the final film thickness h on spinning rotation rate f and on initial solids concentration c_o as

                                                                                (2.1)

Here the constant k is a function of the polymer solvent system.

Meyerhofer [1978] did numerical calculation for a Newtonian fluid whose viscosity depends on its evaporation and concentration. He found that, initially, outward flow of casting liquid dominates the desired thickness without much change in concentration, and then evaporation takes over. Jenekhe [1983-84] introduced a general expression of a time-dependent film thickness of a solution flowing on a rotating disk accompanied by solvent evaporation and change in viscosity. Considering the relative importance of solvent evaporation to be small, he concluded the casting process is dominated by the changing of the rheological properties of the fluid.

For a spin casting process, Lai [1978] observed the dry film thickness h in the range of 0.3 to 2 m m depended on rotation rate w of the rigid substrate and on initial concentration c of solids as

                                                                                 (2.2)

which is similar to the expression of Damon [1967] for photoresist solution. Malangone and Needham [1982] studied an acrylate copolymer dissolved in 2-methoxyethyl ether at concentrations of 6 to 17 wt% at rates from 2000 to 6000 rpm and found

                                                                     (2.3)

for films thickness h ranging from 0.3 to 100 m m, where r is the density of the solution and m is its initial viscosity, w is the spinning speed of substrate, and c is the concentration of solids in casting liquid.

Jenekhe [1983-84] spun-cast films of polyamic acid and polyvinyle accetate where the casting solutions contained about 20% solids at spinning rates of 300 to 600 rpm, and the desired film thickness ranged from 2 to 65 m m. He compared his findings with the following analytic solution

                                 (2.4)

Where t is the spin time, h_o is the initial solution thickness, a is related to the concentration dependence of viscosity, Y is an adjustable "wet film contraction factor" that is equal to unity if all the solvent evaporates during spinning, and m is the initial viscosity of casting liquid.

Starting with the result and analysis of Esmile et al. [1958], they derived an expression to correlate experimental findings among different parameters. Considering the isothermal laminar flow of a layer of dilute solution with assumed constant density r and initial viscosity m across a disc spinning at w , the desired thickness h of the casting solution decreases with time t as:

                                                     (2.5)

where h_o is the initial thickness of the solution layer. Here it is assumed that no evaporation occurs until time t, when all the solvent instantaneously flashes off, leaving behind only the polymer, and reaches to the free air on the rigid substrate. The dry thickness h* of the remaining layer is related to the initial volume concentration c of the solution by

                                                                                     (2.6)

Thus,

                                             (2.7)

For sufficiently large spin rate (w ) and time (t), and sufficiently small viscosity (i.e. 1<< 4), Equation 2.7 simplifies to

                                                     (2.8)

From the studies of Meyerhofer [1978] it has been found that, at the start of spinning process, the outflow of casting liquid dominates the desired film thickness and the concentration does not change much. When the final film thickness h has dropped to about , where h_o is the initial thickness, evaporation takes over and the volume of the solids rapidly reaches its final value. At this time, the flow has ceased completely due to the high viscosity and the reduced liquid thickness and only evaporation continues.

The topography of the surface of the solution was observed [Meyerhofer, 1978] visually during spinning. After the first fraction of a second, the surface of the casting liquid appeared completely flat and remained so. As the layer thinned out during time, interference colors were observed to change uniformly through the various orders. The uniformity of the solid film is usually established by determining the thickness at a number of different points on the film, where the thickness is measured by using reflection spectrum over the visible range.

Among the four stages of spin casting of polymer liquid, spin-off and solvent evaporation stages are more potential compared to others, as they have significant effect on the desired film thickness. One of the most important findings of the spin casting is the tendency of the film to become uniform in thickness as it thins, and to remain perfectly uniform condition during thinning operation. All of the spin casting models that incorporate the solvent evaporation stage presume that the thinning film is perfectly uniform and remains so during time. The major assumptions that become common for spin casting modeling are summarized below:

1. Casting or coating fluid flow is symmetric according to the rotational axis.
2. Casting or coating fluid flow is considered to the both radial and vertical direction.
3. Surface tension forces are neglected which affects the validity of the model in the immediate vicinity of a thin ring surrounding the edge of the substrate.
4. Density of the casting or coating fluid stays constant which simplifies the mathematical calculation.
5. The casting or coating fluid temperature and partial pressure of the solvent in the spinning chamber remain constant. The fluid film to be isothermal throughout the casting process.
6. The rate of solvent evaporation from the liquid surface is uniform which is independent of the location of spinning substrate.

If the polymer solution has approximately Newtonian character, viscous and uniform thickness over the spinning disk, the flow in the spin-off stage is radial, rectilinear and slowly varying.

Figure 3.1.1. Modeling of Spin–off for Spin Casting

The hydrodynamics of polymer liquid are governed by the following equations:

Force balance:

                                                                                 (3.1.1)

Continuity equation:

                                                                                 (3.1.2)

Here, r is the fluid density, assumed to be constant here, m is the fluid viscosity which is markedly sensitive to the fluid composition, w is the angular velocity = 2p f, r is the radial distance from the axis of rotation, and z is the distance above the surface of the substrate or disk, h is the height of free surface of polymer solution, t is the spinning time and q is the volumetric flow of polymer solution per unit circumference. At the free surface of the liquid, the polymer solution ordinarily exerts negligible shear stress and so

at  z = h(r, t). With this boundary condition, the velocity distribution that balances the centrifugal and viscous forces is

                                                                (3.1.3)

The corresponding volumetric flow per unit circumference locally within the liquid layer is

                                                                                     (3.1.4)

The solution of these equations when there is no evaporation, i.e., no change of viscosity, is that the profile remains uniform and that thickness h decreases according to Emslile et al. [1958] as

                                                             (3.1.5)

Where h_o is the initial uniform thickness of the polymer film. Thus, an initially uniform film would remain uniform as it thins, if the liquid were Newtonian and its viscosity remained uniform.

In order to produce a solid film, the solvent must be evaporated. At the start of the spinning, the concentration of the solute is uniform. The solvent evaporates over the entire surface area causing the concentration c to increase. If the height of free polymer surface h and the solvent evaporation are independent of radial distance r, then the solvent concentration c will also be independent of radial distance r, while there will be some change of c with z (neglected in thin film).

As the casting liquid accumulates near the edge of the spinning substrate, the profile becomes more unstable under the high centrifugal force and ejects droplets until the arrival of liquid slows enough that an effective stable profile can persist. In practice, the film at the edge often ends up locally thicker than elsewhere unless the edge of the substrate is especially beveled or chamfered. It has been found that a uniform thickness would never be established near the axis or rotation when the casting polymer is non-Newtonian in behavior. This is done, as the viscosity of the non-Newtonian liquid would increase without bound at the axis.

Solvent evaporates by convection and diffusion process from the polymer solution to the gas affects the final thickness achieved by spin casting process. During the spin casting process, both the liquid surface and gas overhead offer mass transfer resistance to solvent evaporation. When solvent evaporates or leaves, polymer solution becomes more concentrated in solids and its viscosity is increased. In that case, polymer solution or casting (coating) fluid losses the tendency to flow radially and its thinning operation is ceased by radial depletion. The convective-diffusion process through which solvent evaporates from the liquid layer is

                                         (3.2.1)

Where, is the local rate of depletion of solvent A, D is the binary mass diffusivity, which is a strong function of mass concentration of solvent , v_r and v_z are the radial and vertical flow of polymer solution respectively. Vertical component of polymer flow, v_z is small and inconsequent during the spin-off process, but strongly affects the rate of solvent evaporation. In case of uniform viscosity and film thickness, the vertical component of polymer flow

                                                                                     (3.2.2)

The rate of solvent evaporation of total mass, m to the rate at which the receding surface overtakes the moving liquid is expressed as follows:

                                                                            (3.2.3)

The local rate of evaporation of a solvent A, from free surface of casting (coating) liquid depends on the followings:

1. the difference in partial pressure of the solvent in the gas next to the free surface, and in the gas supplied to swipe the free surface,
2. the process of convection and diffusion as that gas flows over the free surface.

It is important to be mentioned here that the partial pressure of solvent A, can be controlled and is considered as a parameter to be specified in any model of solvent evaporation of spin casting process. On the other hand, the partial pressure in the gas tied with polymer surface can be approximately considered as the value in equilibrium with the solvent concentration in the surface of the liquid layer. The solvent concentration in the remaining polymer solution depends on the convection and diffusion of solvent evaporation from liquid to gas.

A solution of convective-diffusion equation is required to express the local rate of evaporation of a solvent A, m_A(r, t) from the polymer solution, which in fact has to describe the actual nature of gas flow over the polymer surface. That is, the evaporation of a solvent from a polymer solution also depends whether the gas flow over the polymer surface is laminar or turbulent in pattern. Simply, the mass transfer of a solvent A can be described as

                                                                                 (3.2.4)

or,                                                                           (3.2.5)

Where, k is called mass transfer coefficient and is the solvent concentration of polymer liquid.

In spin casting modeling, Mayerhofer considered the uniform thickness and uniform solvent concentration of polymer solution, but both change with time. He also assumed a constant and uniform solvent evaporation, which was independent of solvent concentration but sensitive to the speed at which the substrate was rotating. He described the viscosity of polymer solution as follows:

                                                                                 (3.2.6)

Where, and are the adjustable parameters. Here the viscosity of the casting or coating liquid is increasing with solvent evaporation with time.

It was found in his modeling that, at the beginning of spin casting process the outflow, or, drainage of polymer solution dominates the film thickness and the concentration of solvent did not change much. But with passing time, the film thins at a certain level when the evaporation acts as main mechanism for further film thinning as flow of polymer solution ceases for extremely high viscosity. Mark tried to correlated the viscosity with solvent concentration as follows:

                                                             (3.2.7)

Where n is the kinematic viscosity and c is the concentration of solids in casting or coating solution.

Sukanek [1985] introduced a mass transfer coefficient k in his spin casting modeling to account roughly the gas-phase resistant offered to solvent evaporation. For convenience, he assumed that the gas supplied contained no vapor and considered the viscosity to depend on solvent concentration according to either

                                                                 (3.2.8) or

                                                                             (3.2.9)

Where m _A is the viscosity of pure solvent, m _O is the initial viscosity of the casting or coating liquid, r _A⁰ is the initial solvent density and h ,g and b are the empirical constants. He came to a similar conclusion of Meyerhofer [1978] that the film thins initially due to radial convective outflow and after a certain time, evaporation comes to dominate the thinning operation. He also mentioned that the initial solvent concentration plays an important role in final film thickness of a spin casting process.

It has been found that the rate of solvent evaporation, e depends very strongly on how fast the vapor phase above the casting liquid is removed. That is the rate of solvent evaporation during a spinning operation must be related to the airflow over the spinning disk. For the simplest model, the rate of solvent evaporation is proportional to the rate of the airflow over the free surface of polymer solution. The solvent evaporation, e can be related with the spinning speed, f of the substrate as

                                                                                                                 (3.2.10)

Finite element method is a numerical method for solving the problems of engineering and mathematics physics. The finite element formulation of the problem results in a system of simultaneous algebraic equations for solution, rather than requiring the solution of differential equations. These numerical methods yield approximate values of the unknowns at discrete numbers of points in the continuum. Hence this process of modeling a body by dividing it into an equivalent system of smaller bodies or units interconnected at points common to two or more elements (nodal points or nodes) and or boundary lines and or surface is called discretization. In the finite element method, instead of solving the problem for the entire body in one operation, we formulate the equations for each finite element and combine them to obtain the solution of the whole body.

An axisymmetric flat "polymer" slab of 1 m radius and 20 mm thickness has been deposited on a rigid substrate, which is capable of spinning at different speeds. The total thickness of the polymer consists of four layers of elements, 5 mm thick each. The interface between slab and substrate is defined by nodes 1-101, while the outer surface is defined by nodes 809-909. Each layer contains 50 elements in the radial direction, resulting in 200 total elements. Interior nodal rows and columns are shown in Figure 4.1.1.

Figure 4.1.1. Uniform spin casting model definition for finite element analysis.

The purpose of this investigation is to develop a model to investigate spin casting of thin polymer films. One specific task in the model development is to determine the maximum possible radial (in local r-direction) and vertical (in local z-direction) displacements of the polymer slab, considered as either elastic or viscoelastic, when it is subjected to different spinning speeds under the following boundary condition (m = friction co-efficient):

1. When the bottom layer of the polymer mold is fully unconstrained for radial displacement, i.e. a roller interface between the polymer and rigid substrate (m = 0).
2. When the bottom layer of the polymer mold is partially constrained for radial displacement, i.e. a friction interface between polymer and rigid substrate (0<m<¥ ).
3. When the bottom layer of the polymer mold is fully constrained for radial displacement, i.e. a fixed interface between the polymer and rigid substrate (m = ¥ ).

A finite element analysis of the spin casting process was conducted using the nonlinear code ABAQUS. The following characteristics define the polymer slab for ABAQUS analysis:

• Initial radius of the axisymmetric polymer slab: R_o = 1 m

• Total initial thickness of slab: H_o = 20 mm
• Total number of element layers along thickness = 4
• Thickness of each layer = 5 mm
• Total number of elements per layer along radial direction = 50

• Element type: 8 node axisymmetric continuum elements, with reduced integration where non-linear geometry effects have been considered.

• Interface between polymer slab and rigid substrate is either roller (friction coefficient m = 0) or friction (m = 0.4)

• Density of polymer = 1500 kg/m³
• Poisson’s ratio of polymer = 0.4

It is assumed that, after some solidification, the polymer solution can be modeled as a viscoelastic solid (Section 6).

• Both centrifugal and gravity loading are applied to the polymer for finding its maximum radial and vertical displacements.

When a roller interface exists between the polymer and the rigid substrate, the interface is fully unconstrained in the radial direction, and no friction force acts against radial displacement. Different values of mold elastic modulus and spin speeds have been applied to the polymer in order to find the maximum radial and vertical displacements. It has been found that the maximum radial displacement occurred at the interface and the maximum vertical displacement occurred at the surface.

For getting significant displacement in radial and vertical dimensions, the modulus of elasticity of polymer is kept constant at 10 MPa (1.456 ksi). The maximum radial and vertical displacements of the polymer, while spinning speed is increased from 1 Hz to 13 Hz is shown at table 4.4.1.

Table 4.4.1: Maximum radial and vertical displacement, and strain, for different spinning speeds.

In the case of the roller interface between the polymer and rigid substrate, the maximum radial displacement, u_r is 270.6 mm, which is found along the polymer interface at initial radial position r/R_o = 0.86 (Figure 4.4.1). Radial displacement of different columns (at r = constant planes) is nearly uniform (Figure 4.4.2). The maximum vertical displacement, u_z is - 8.810 mm, which is found along the polymer surface at initial radial position r/R_o = 0 (Figure 4.4.3).

Figure 4.4.1. Radial displacement of polymer interface for roller interface case.

Figure 4.4.2. Radial displacement of various columns (r = constant planes) for roller interface.

Figure 4.4.3. Vertical displacement of different polymer layer (z = constant planes) for roller interface.

When the roller interface exists between the polymer and the rigid substrate, the maximum spin speed that the polymer can withstand is 13 Hz, provided that nonlinear geometry command and reduced integration have been considered for analysis, modulus of elasticity of polymer E10 MPa, density = 1500 kg/m³, and Poisson’s ratio = 0.4. Spin speed was stopped at 13 Hz because of convergence problem arising in the ABAQUS analysis. When f > 13 Hz, some elements of the model are distorted so much that they turn inside out and some node points at the interface become open.

In this case, the maximum radial displacement, u_r is 207.6 mm and the average radial strain, <e_r> is 17.19 %; so the maximum radial displacement is ~ 0.2 times the original radial dimension, R_o. The maximum vertical displacement, u_z is 8.81 mm and the average vertical strain, <e_z > is 30.00 %; so the maximum vertical displacement is ~ 0.5 times the original vertical (z) dimension, H_o.

When friction exits between the polymer and the rigid substrate, the interface between the polymer and rigid substrate is partially constrained for radial displacement. Similar analysis to the roller interface case has been done for friction interface case, to determine the maximum radial and vertical displacement of the polymer when it is subjected to different spin speeds. In this case, the friction coefficient m = 0.4 has been considered. (No significant difference has been found in radial and vertical displacement for m = 0.4 to m = 0.1)

For the friction interface case, the maximum radial displacement, u_r is 200.8 mm at initial radial position, r/R_o = 0.87 of the polymer interface (Figure 4.5.1) and the maximum vertical displacement, u_z is – 6.782 mm at initial radial position, r/R_o = 0 of the polymer surface (Figure 4.5.2).

For the friction interface between the polymer and the rigid substrate, the maximum spin speed that can achieve convergence is 12 Hz, provided that the polymer slab has the same mechanical properties discussed for the roller interface case.

In this case, the maximum radial displacement, u_r is 200.8 mm, and the average radial strain is, <e_r> is 17.17 %; so the maximum radial displacement is ~ 0.2 times the original radial dimension, R_o. The maximum vertical displacement, u_z is 6.782 mm, and the average vertical strain, <e_z> is 29.44 %; so the maximum vertical strain is ~ 0.4 times the original vertical (z) dimension, H_o.

For spin speed f > 12 Hz, the same error has been found as for roller interface, i.e., some of the model elements are distorted so much that they turn inside out and some of node points at the interface become open.

Figure 4.5.1. Radial displacement of polymer interface for friction interface case (friction coefficient m = 0.4).

Figure 4.5.2. Vertical displacement of different rows (z = constant planes) for friction interface case (friction coefficient m = 0.4).

When the interface is fixed to the rigid substrate (m = ¥ ), its node points are fully constrained for radial and vertical displacement. Similar analysis, as in the roller and friction interface cases, has been done for the fixed interface case to determine the maximum radial and vertical displacement of the polymer when it is subjected to different spin speeds.

For the fixed interface case, the maximum radial displacement, u_r is 16.34 mm at initial radial position, r/R_o = 0.99 of the surface (Figure 4.6.1) and the minimum vertical displacement, u_z is -8.402 mm at initial radial position r/R_o = 1 of the polymer surface (Figure 4.6.2).

When the interface of the polymer is fixed to the rigid substrate, the maximum spin speed that the model can withstand is 58 Hz, provided that the polymer has the same mechanical properties discussed above. Although the polymer can withstand a higher spin speed (f = 58 Hz) for this case, the maximum radial and vertical displacements are less compared to the roller and friction interface cases. The maximum radial displacement, u_r is 163.4 mm and average radial strain, <e_r> is 0.3428 %; so the maximum radial displacement is ~ 0.17 times the original radial dimension, R_o. The maximum vertical displacement, u_z is 8.402 mm and the average vertical strain, <e_z> is – 2.707 %; so the maximum vertical displacement is ~ 0.42 times the original vertical (z) dimension, H_o. For spin speed f > 58 Hz, the same convergence error has been found as in the roller and friction interface cases.

Figure 4.6.1. Radial displacement of different polymer rows (z = constant rows) for fixed interface case.

Figure 4.6.2. Vertical displacement of polymer surface for fixed interface case.

able 4.7.1. Maximum spin speed, displacement and strain, for different interface conditions with polymer modulus of elasticity 10 MPa.

Another purpose of this investigation is to investigate nonuniformity by using different elasticity or viscoelastic data for different regions in the model. Nonuniform mass diffusion or rigid elements at different locations has also created nonuniformity in the model.

In order to represent as much viscoelastic flow as possible in the model, a minimum value of elastic modulus E_min that still achieved convergence was determined.

At first, nonuniformity was created by assigning different value of elasticity to the polymer solution, confined in a certain region of the model (Figure 5.1.1). The minimum value of uniform and nonuniform elasticity of the polymer solution, E_min [uniform model] and E^*_min [nonuniform model], respectively, that allowed convergence were determined when the model was spinning at frequency f = 1 Hz. This determines how compliant the spin casting model could be before convergence problems arise in ABAQUS analysis. [For manufacturing large polymer mirrors by the spin casting process, the spin speed of the mold is often kept constant near 1 Hz]. The minimum value of uniform elasticity was determined first, then the minimum value of nonuniform elasticity E^*_min was found by decreasing its value gradually until getting a convergence error, while the value of uniform elasticity has been kept constant. For this case, both elastic and viscoelastic analyses have been performed for roller and friction interface cases. The maximum values of radial and vertical displacement of polymer solution have also been determined for the above cases. Data for the various cases is given in Table 5.1.1.

Figure 5.1.1. FEM definition for nonuniform model with different elasticity. The nonuniformity represents about 4.3 % of the total volume.

Table 5.1.1. Minimum modulus of elasticity for uniform and nonuniform model for elastic and viscoelastic analysis

A spin casting model containing nonuniform elasticity E*_min, confined at a particular region, has been found to offer different radial and vertical displacements compared to the uniform model consisting of uniform elasticity E_min. The radial displacement at the polymer interface has been changed at the boundary of nonuniform elasticity region, and remains changed beyond the nonuniformity, provided that E*_min < E_min (Figure 5.2.1).

From an elastic analysis, the minimum value of uniform elasticity, E_min, of the model has been found to be 77 kPa and 69 kPa for roller and friction interface cases respectively, when the model is spinning at f = 1 Hz [Table 5.1.1]. For the same operating conditions, these values become 58 kPa and 68 kPa, respectively, for the nonuniform model, when the nonuniformity is confined at radial dimension 0.460-0.505 m [Table 5.1.1]. The minimum value of elasticity of the uniform and nonuniform model has been found to be unchanged for the viscoelastic analysis [Table 5.1.1].

For the roller interface case, the difference in radial displacement between the uniform and nonuniform models has been found to increase by 2.8% and 5.4% for the elastic and viscoelastic analysis, respectively, while it was unchanged for the friction interface case [Table 5.2.1].

Figure 5.2.1. Radial displacement of the polymer interface with nonuniform elasticity and roller interface having E_min = 77 kPa and E*_min = 58 kPa. The 3 nonuniformities are considered separately.

Table 5.2.1. Maximum radial (u_rmax) and vertical (u_zmax) displacement for uniform and nonuniform model, for elastic and viscoelastic analysis, and radial deviation [(u^*_rmax- u_rmax)/u_rmax] x100.

In the vertical displacement case, the effect of nonuniformity has been found to be locally confined to its original position for both elastic and viscoelastic analyses. This means that the vertical displacement of the polymer surface of the nonuniform model will be the same as the uniform model except in the region of the nonuniformity. In the region of nonuniformity, the vertical displacement has been found to be more than the uniform model, provided that E*_min < E_min (Figure 5.2.2).

Figure 5.2.2: Vertical displacement of the polymer surface with nonuniform elasticity and roller interface having E_min = 77 kPa and E*_min = 58 kPa. Here, the 3 nonuniformities are considered separately.

Again, the radial displacement at the polymer interface of a spin casting model increases with increasing number of nonuniformities (Figure 5.2.3). On the other hand, the shape of vertical displacement was found to be the same when, 3 nonunifomities were considered together (Figure 5.2.2 and 5.2.4). The maximum value of vertical displacement of the polymer surface has been found to increase if the nonuniformity, E*_min, has been shifted towards the center of the model. It does not depend on the number of nonuniformities, but on the initial radial position of the nonuniformity in the model (Figure 5.2.2 and 5.2.4).

Figure 5.2.3.Radial displacement at the polymer interface of nonuniform model with multiple nonuniformity with E_min = 77 kPa and E*_min = 58 kPa. The 3 nonuniformities are considered simultaneously.

It has been found that, when the uniform and nonuniform regions have the same viscosity or creep compliance, the final radial dimension (or maximum radial displacement) at the polymer interface does not depend significantly on the initial radial position of the nonuniformity in the model. The final radial dimension of the polymer interface after a certain period of time has been found to be almost the same, when the region of nonuniform elasticity is located at different locations in the model (Figure 5.2.1). However, if the model is less viscous in the nonuniform region compared to the uniform region, maximum radial displacement has been found to occur when the area of nonuniformity is located near the middle of the model (Figure 5.2.5).

Figure 5.2.5. Radial displacement of the polymer interface with nonuniform elasticity and viscosity. Nonuniform region has been assumed two times less viscous than uniform region.

In this case, also the vertical displacement of polymer surface has been found to increase when the nonuniformity is shifted towards the center of the model (Figure 5.2.6). The vertical displacement follows the same pattern as the model of nonuniform elasticity.

Figure 5.2.6. Vertical displacement of the polymer surface with nonuniform elasticity and viscosity. Nonuniform region has been assumed two times less viscous than uniform region

ABAQUS analysis of the spin casting model proves that the final radial and vertical displacement of polymer solution is model dependent, e.g., changing the polymer density will change the radial and vertical displacement (Figure 5.2.5 and 5.2.6). The radial displacement does not depend on the initial thickness of the polymer solution, but the vertical displacement does (Figure 5.2.7 and 5.2.8). The effect of Poisson's ratio has been found significant, for example, the maximum radial displacement to be occurred at the nondimensional initial radial position r/R₀ = 0.86 for n = 0.4, not at the edge of the model, which is supported well by theory. The same Poisson's effect also causes the maximum vertical displacement to occur at the center. Only Poisson's ratio of 0.00 has the maximum radial and vertical displacement at the edge of the model (Figure 5.2.9 and 5.2.10).

Table 5.2.2. Position for maximum radial and vertical displacement for different Poisson's ratio.

Figure 5.2.7. Radial displacement of the polymer interface for different polymer density.

Figure 5.2.8. Vertical displacement of the polymer surface for different polymer density.

Figure 5.2.9. Radial displacement of the polymer interface for different initial thickness

Figure 5.2.10. Vertical displacement of the polymer interface for different initial thickness.

Figure 5.2.11. Radial displacement of the polymer interface for Poisson's ratio equal to 0.00

Figure 5.2.12. Vertical displacement of the polymer surface for Poisson's ratio equal to 0.00.

Solvent evaporation plays an important role in the spin casting process. When the solvent evaporates during spinning, the polymer solution becomes increasingly viscous, and its tendency to radial flow reduces. This solvent evaporation may or may not be uniform throughout the substrate.

To model the solvent evaporation process, a mass diffusion analysis was performed using ABAQUS. Different viscosities were then generated at different locations to create the nonuniformity in the model. For simplicity and to avoid the convergence problems in ABAQUS analysis, the following model has been considered to visualize the effect the nonuniform viscosity.

A finite element analysis of the spin casting process was conducted using the Nonlinear code ABAQUS. The following characteristics define the polymer substrate for ABAQUS analysis:

• Initial radius of the axisymmetric polymer substrate: R_o = 1 m

• Total initial thickness of substrate: H_o = 1 mm
• Total number of element layers along thickness = 1
• Total number of elements along radial direction = 20

• Element type: 4 node axisymmetric continuum elements, with reduced integration where non-linear geometry effects have been considered.

• Density of polymer = 1500 kg/m³
• Poisson’s ratio of polymer = 0.4

• Elasticity of the model = 77 kPa (minimum value of modulus of elasticity for uniform model while spinning at 1 Hz.)

• Two sets of creep test data have been generated for different kinematic viscosity of polymer solution confined at different element sets. The nonuiform mass diffusion has been confined at the element number 10 (at radial dimension r = 0.45 m to 0.50 m).

• Both centrifugal and gravity loading (1 Hz and 9.81 m/s², respectively) were applied to the polymer solution for finding the maximum radial and vertical displacement. Effects of nonuniform mass diffusion has also been investigated.

Figure 6.1.1. Finite element model for uniform mass diffusion analysis.

Figure 6.1.2. Finite element model for nonuniform mass diffusion analysis at radial dimension r/R₀ = 0.45 – 0.50.

The viscosity of the polymer solution of a spin casting model increases due to gradual decrease of solvent concentration with time [Table 6.2.1]. For simplification of analysis, the solvent concentration of polymer solution has been considered step-wise continuous over the spinning process. Initially, the solvent concentration was considered equal to 80% and the kinemetic viscosity of the polymer solution was considered equal to 1 kPa-s.

A rule-of-mixtures procedure has been used for determining the kinematic viscosity of the polymer solution. The initial assumptions are:

Initial solvent concentration, X_o = 80 %

Initial kinematic viscosity of solvent, η_solvent = 0.001 kPa-s

Initial kinematic viscosity of polymer solution, η_o = 1 kPa-s

By way of example, the calculation of polymer kinematic viscosity is given for through the first two seconds:

η_o = (1-X₀) η_polymer + X₀* η_solvent (6.2.1)

• 1 = (1-0.8) η_polymer + 0.80*0.001
• η_polymer = 4.996 kPa-s

η₁ = (1-X₁) * η_polymer + X₁* η_solvent (6.2.2)

= (1-0.54) * 4.996 + 0.54 * 0.001

= 2.2987 kPa-s

h₂ = (1-X₂) * η_polymer + X₂* η_solvent (6.2.3)

= (1-0.37) * 4.996 + 0.37 * 0.001

= 3.1479 kPa-s

The kinematic viscosity of the polymer solution for the rest of the solvent concentration, in the uniform and nonuniform mass diffusion region, has also been determined in the same way as discussed above.

The following table depicts results form ABAQUS for the change of solvent concentration with time in the above spin casting model, using a mass diffusion analysis (see Appendix).

Table 6.2.1. Change of solvent concentration and kinemetic viscosity of the polymer solution with time.

The solvent concentration over time during the spinning process depends on the solubility and the diffusivity of the polymer solution and air; it also depends on its initial concentration. In this analysis, the absolute value of diffusivity and solubility have been considered very small, which is reasonable, and consequently did not change significantly over time. Nonuniform mass diffusion has been applied at the middle of the model, at element 10 (radial dimension of 0.45 to 0.50 m). The change in solvent concentration and kinematic viscosity, both in the uniform and nonuniform mass diffusion region, are shown in Figures 6.2.1 and 6.2.2, respectively, for the time period of 10 seconds.

Figure 6.2.1. Change in solvent concentration of polymer solution with time.

Figure 6.2.2. Change in kinematic viscosity of polymer solution with time from change in the solvent concentration.

For viscoelastic analysis, it is required to assign creep test data in an input file, namely as a normalized compliance with time. The "Standard Linear Solid" model (Figure 6.3.1), consisting of two springs and one dashpot, was used for determining the creep test data or creep compliance for ABAQUS analysis. In this model, the total strain e (t) as a function of time is given by:

                                             (6.3.1)

Figure 6.3.1. Standard linear solid model with spring and dashpot (Three Element Model).

The creep compliance J(t) is found by normalizing the total strain by the applied constant stress s _{0 :}

                         (6.3.2)

In this three element solid model, E₂ is the initial value of modulus of elasticity (E_min = 77 kPa is used) of the spin casting model, and h indicates its viscosity as a function of time. The value of E₁, used in this case, was kept very low (1 kPa) in order to minimize the "solid" nature of the analysis. Due to this minimal value of E₁, the three element model described above, in fact approaches a two element visco-fluid model which can be expressed as follows:

Figure 6.3.2. Two element visco-fluid model with spring and dashpot

Equation (6.3.2) has been used for determining the creep compliance of the uniform and nonuniform mass diffusion region, again by a step-wise approximation. Table 6.3.1 and Figure 6.3.1 show the creep compliance of the uniform and nonuniform mass diffusion region.

Table 6.3.1. Creep compliance of uniform and nonuniform mass diffusion region. This data is related to the kinematic viscosity data in Table 6.2.1 by use of Equation (6.3.2).

As the viscosity of the polymer solution increases with time due to solvent evaporation, it consequently decreases the compliance of the spin casting model. The nonuniform mass diffusion region is more compliant compared to the uniform mass diffusion region due to less change in viscosity.

Figure 6.3.3. Change in creep compliance for uniform and nonuniform mass diffusion region with time.

From ABAQUS analysis, the model is less compliant when its viscosity is increasing with time due to solvent evaporation and mass diffusion. Both radial and vertical displacements have been found to decrease when uniform mass diffusion occurs throughout the model (Figure 6.4.1 and Figure 6.4.2).

Figure 6.4.1. Effect of mass diffusion in radial displacement of polymer interface with roller interface (Friction Coefficient, m = 0).

Figure 6.4.2. Effect of mass diffusion in vertical displacement of the polymer surface with roller interface (Friction Coefficient m = 0).

The nonuniform mass diffusion in a typical spin casting model has an effect both on the radial and vertical displacements. With nonuniform mass diffusion (resulting in a lower viscosity confined to a particular region), more radial displacement is seen compared to the uniform mass diffusion model. The radial displacement increases at the boundary of the nonuniformity and remains increased beyond that location (Figure 6.4.3).

Figure 6.4.3. Effect of nonuniform mass diffusion in radial displacement of polymer interface for roller interface (Friction Coefficient m = 0).

For the same nonuniform mass diffusion spin casting model, additional vertical displacement has been found in the region of nonuniformity (Figure 6.4.4). Here, the effect of nonuniformity has been found to be locally confined. The vertical displacement of the polymer surface is the same as the uniform model, expect in the nonuniform mass diffusion region.

Figure 6.4.4. Effect of nonuniform mass diffusion in polymer surface with roller interface (Friction Coefficient m = 0)

The position of the nonuniform mass diffusion region in a spin casting model has been found to offer almost no effect on the final radial displacement of the polymer interface, provided that both uniform and nonuniform mass diffusion regions contain the same modulus of elasticity. This implies that the final radial displacement of the polymer interface will be the same, wherever the nonuniform mass diffusion region is confined in the model (Figure 6.4.5).

Figure 6.4.5. Radial displacement of polymer interface for different locations of nonuniform mass diffusion.

On the other hand, the maximum vertical displacement has been found to increase if the region of nonuniform mass diffusion has been shifted towards the center of the model (Figure 6.4.6).

Figure 6.4.6. Vertical displacement of polymer surface for different locations of nonuniform mass diffusion.

The polymer solution used for spin casting may contain some rigid inclusions, like sand or dust particles; these could affect the final radial and vertical displacement of the polymer interface and surface, respectively. This generates the interest for analyzing the spin casting model of a polymer solution with the presence of rigid elements at the different locations. Rigid elements have been assigned near the center, middle or at edge of the model, being located at the surface of the model and confined by an element

                       7.1 Model Geometry, Material Properties, and Loading

The model geometry, modulus of elasticity, density, Poisson’s ratio and loading have been kept the same as in the uniform elasticity model (chapter 4). Finite element software ABAQUS offers the analysis of rigid element by using the key word *RIGID BODY.

Figure 7.1.1. Finite element model with rigid element at the top of the surface. rigid element 1 is located at 0.20-0.22 m, rigid element 2 is located at 0.48-0.50 m and rigid element 3 is located at 0.80-0.82 m.

A mild difference has been found in the radial and vertical displacement cases of the nonuniform model with rigid element, attached at the surface, compared to the uniform model. The nonuniform model with rigid element offers the same radial and vertical displacement as the uniform model up to the beginning of the nonuniform region (Figure 7.2.1 and 7.2.2). As expected, no increase in radial displacement occurs across the nonuniform (rigid element) region, since it moves as a rigid body in translation (Figure 7.2.2). Radial displacement increases again from the end point of the rigid element and remains increasing with the radial dimension.

Vertical displacement has also been found to be similiar in the uniform and nonuniform model, except in the region of nonuniformity. In the nonuniform region, less vertical displacement was found. The difference in vertical displacement between the uniform and nonuniform model decreases when the location of rigid element moves towards the edge of the model (Figure 7.2.3). Vertical displacement has also been found to be similiar in the uniform and nonuniform model, except in the region of nonuniformity. In the nonuniform region, less vertical displacement was found. The difference in vertical displacement between the uniform and nonuniform model decreases when the location of rigid element moves towards the edge of the model (Figure 7.2.3).

Figure 7.2.1.Radial displacement of the polymer interface for a rigid element at the surface, but different radial locations (roller interface case).

Figure 7.2.2.Radial displacement of the polymer surface for a rigid element at the surface, but different radial locations (roller interface case).

Figure 7.2.3.Vertical displacement of the polymer surface for a rigid element at the surface, but different radial locations (roller interface case).

In this investigation, various effects of nonuniformity in the spin casting model have been studied, in particular whether this effect is locally confined or not. In precision applications, the uniformity of the polymer surface and its thickness becomes important. This work has investigated the effects of localized nonuniformites in the spin casting model. Of central importance is the fact that localized nonuniformities in elasticity or viscosity create local nonuniformities in the polymer surface. Other effects from the local nonuniformities have also been observed in model results. Nonuniform regions with rigid elements, attached at the surface of the spin casting model, move as a rigid body in translation, as expected, and consequently offers very slide difference in the radial and vertical displacement.

The radial and vertical displacement of the spin casting model was found to be depended on the model parameters, discussed in chapter 4. The maximum radial displacement was found not to be depended on the initial thickness of the polymer solution, but the vertical displacement was. The Poisson's ratio had been found to be significant for the position of maximum radial and vertical displacement of the spin casting model. Only the Poisson's ratio 0.00 offered the maximum radial and vertical displacement to be occurred at the edge.

A spin casting model consists of four steps has been described in introduction. Only the last step, evaporation, is modeled in this reported investigation, by using nonlinear code ABAQUS, assuming some solidification has occurred. Future work should incorporate further analysis of fluid behavior of spin casting model to construct a strong bridge between the fluid and solid model.

In the analysis of nonuniform elasticity of spin casting model, the viscosity of the nonuniform region has been considered the same as the uniform region. In reality, these two regions should have different viscosity. Proper mathematical formulation should be investigated for determining the viscosity of the nonuniform region, having different elasticity compared to the uniform region. An opposite investigation should also be carried out for the nonuniform mass diffusion analysis, where the nonuniform mass diffusion region should have different elasticity, compared to the uniform mass diffusion region. In the mass diffusion analysis, the properties of the polymer solution has been considered constant with time. In fact, fluid properties will be changed for changing the viscosity with time.

There might be some chance for solvent absorption into a flowing film of polymer solution on a flat rotating disk. So the combined effect of solvent evaporation and solvent absorption should be considered for further investigation.