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Spin coating of Blu-Ray disks: modeling, experiments, limitations, and manipulation.

Abstract A major challenge in the production of Blu-Ray Disks (BDs) is making the cover layer over the information area. The layer has to be both thick enough to protect the information and even enough for the information to be read through it optically. Furthermore, it is preferred not to cover the hole in the center of the disk. Spin coating is a candidate method for the production of these layers in a rapid reproduction process. When dispensing is performed off-center (in order not to cover the hole), a new complication appears, namely the formation of a slope toward the inner rim of the liquid film. Here, fundamental limitations for achieving even films in this system and ways to overcome the difficulties by manipulation of the process are studied. A mathematical model for this particular case of spin coating is obtained and validated by comparison with experiments made in industrial equipment aiming at producing BDs. The model agrees well with the experimental data. The model is then used to show that cover layers that fulfil the Blu-Ray specification are very difficult to produce with the spin-coating technique. Manipulation by inline curing and surface shear is added to the model and the results show that it is considerably easier to meet the BD specification when utilizing the manipulation.

Keywords Spin coating, Blu-Ray disk


The spin-coating process and previous work

Spin coating is used to apply thin liquid layers on the surface of objects. During the spin-coating process, the liquid is first dispensed on the surface. The liquid is then spread over the surface by the centrifugal force (Fig. 1). Emslie et al. (1) made a theoretical study of the development of a film of a Newtonian liquid flowing on a rotating disk. Under the assumptions that the liquid film is thin and that the effects of fluid acceleration, gravity, Coriolis force, and surface tension are negligible compared to the centrifugal force, the thickness of the film is described by the equation

[[[partial derivative][H.sup.*]]/[[partial derivative][t.sup.*]]] = - [1/[R.sup.*]] [[partial derivative]/[[partial derivative][R.sup.*]]] ([[w.sup.*2]/[3[v.sup.*]]] [R.sup.*2] [H.sup.*3]), (1)

where H is the thickness of the liquid film, R is the radial coordinate from the center of the disk, [omega] is the angular velocity, v is the kinematic viscosity of the liquid, and * denotes dimensional quantities.


To give a hint on the applicability of the model above in different spin-coating situations, we briefly introduce the key assumptions behind it. First, the initial liquid layer has to be even enough in the azimuthal direction. The liquid has to be viscous enough so that inertia can be neglected, which can (after the initial acceleration of the disk) be quantified as a low value of the Reynolds number:

Re = [[U.sup.*][H.sup.*]/[v.sup.*]], (2)

where [U.sup.*] is a characteristic velocity of the liquid, and can be approximated by the liquid surface velocity given by the model: [U.sup.*] = [[omega].sup.*2][R.sup.*][H.sup.*]/[2v.sup.*]. The layer is also assumed to be thin enough that there is no shear in vertical planes, so that the flow can be considered one dimensional. Following reference 1, the condition for neglecting gravity becomes:

[1/[R.sup.*]] [[[partial derivative][H.sup.*]]/[[partial derivative][R.sup.*]]] [much less than] [[[omega].sup.*2]/g], (3)

where g = 9.81 [m/s.sup.2], and for neglecting Coriolis force:

[v.sup.*] [much greater than] [[omega].sup.*] [H.sup.*]. (4)

In practice, the set of conditions above means that the equation applies to thin layers of viscous fluid, with height variations being small compared to the centrifugal force potential, which is a fair approximation for coating of Blu-Ray Disks (BDs). * Also, the spinning speed in this application is typically high enough to constrain the effect of surface tension near the rims of the lacquer layer.

Spin coating has been studied extensively and a review is found in reference 2. Summarizing (with example references), effects of surface roughness, (3) evaporation, (4) non-Newtonian rheology, (5) hydrodynamic stability, (6), (7) and inertia and shear on the surface (8) have been investigated. Later, the initial spreading of the liquid and the related contact line problem has also grasped attention. (9), (10) More applied studies aim at controlling the process to achieve even films for production of microelectronics by different means. (11-13) The particular case of spin coaling of BDs has also been studied (14) and this work will be returned to later on.

Modeling a liquid film starting at a nonzero radius

First, the problem is nondimensionalized. The coordinates are [R.sup.*] for the radial direction and [H.sup.*] for the thickness of the liquid layer. We introduce the following scaling:

H = [[H.sup.*]/[R.sub.j.sup.*]] R = [[R.sup.*]/[R.sub.j.sup.*]] t = [[[R.sub.i.sup.*2][[omega].sup.*2]]/[[3v.sup.*]]][t.sup.*], (5)

where [R.sub.i.sup.*] is the radius at which the liquid film starts, [[omega].sup.*] is the angular velocity of the disk, and [v.sup.*] is the kinematical viscosity of the liquid. The numerical factor 3 in the time scaling is introduced for later convenience. In this scaling, equation (1) gives

[[[partial derivative]H]/[[partial derivative]t]] = -[1/R] [[partial derivative]/[[partial derivative]R]] ([R.sup.2] [H.sup.3]). (6)

Emslie et al. (1) investigated the spreading of a liquid film starting from R = 0. This is generally not true for spin coating of optical disks, since the hole in the middle is not covered.

A liquid layer that docs not cover the center (R = 0), but instead begins at R = 1, is here modeled with the boundary condition:

H (R [less than or equal to] 1) = 0. (7)

This means that the inner contact line is assumed to be steady at R = 1. The geometrical development of the liquid film is thus completely determined by the initial condition H(R, t = 0), The parameters [R.sub.i.sub.*], [v.sup.*], and [[omega].sup.*] only scale the length and time.

The presence of the inner contact line implies that the slope has to be small also in this region for the model to be strictly valid (see the conditions in the previous section). However, it will be seen later that the slope tends to decrease fast while spinning, and more importantly, that the model agrees with experiments.

In nondimensional form, the BD criterion mentioned in the abstract (H.sup.*] = 100 [+ or -] 3 [micro]m for 24 mm [less than or equal to] [R.sup.*] [less than or equal to] 58 mm and [R.sub.i.sup.*] = 7.5 mm) becomes:

H = (9.09 [+ or -] 0.27) x [10.sup.-3] at 3.2 [less than or equal to] R [less than or equal to] 7.7 (8)

Equation (8) is obtained with the assumption that the lacquer starts at the rim of the hole, the configuration that minimizes the gradient al the information area.

Equation (6) can be solved by the method of characteristics (1) (the same solution can be reached via asymptotic expansions (15). First, take a set of radial positions [R.sub.0.sup.m] where m is an index over increasing radial positions. The corresponding film thicknesses at these positions are denoted [H.sub.0.sup.m]. The thickness distribution at time t is then obtained at the new radial positions [R.sub.t.sup.m]:

[R.sub.t.sup.m] = [R.sub.0.sup.m] [[1 + 4[([H.sub.0.sup.m]).sup.2]t].sup.0.75], (9)

where it is:

[H.sub.t.sup.m] = [[H.sub.0.sup.m]/[[1 + 4 [([H.sub.0.sup.m]).sup.2]t].sup.0.5]]], (10)

where [H.sub.t.sup.m] is the thickness at the radial position [R.sub.t.sup.m]. The boundary condition (7) is fulfilled at all times if H(t = 0, R = 1) = 0 since [R.sub.t.sup.m] = [R.sub.0.sup.m] and [H.sub.t.sup.m] = [H.sub.0.sup.m] if [H.sub.0.sup.m] - 0. This solution is valid only as long as the ordering sequence of [R.sub.t.sup.m] is intact, i.e., as long as

[R.sub.t.sup.m-1] < [R.sub.t.sup.m]. (11)

This last condition is equivalent to stating that no shocks are allowed to form on the surface. Note also that the points [R.sub.t.sup.m] are immaterial.

The problem stated by equations (6) and (7) can also be solved by integrating the equations numerically. This can be done by most standard numerical packages. Numerical integration is used (i) to compare with experiments in "Experiments and verification" and (ii) to evaluate the manipulation strategies in "Numerical evaluation and manipulation by surface shear and weak curing during spinning". Equations (9) and (10) are used in "Limitations and possibilities" to derive fundamental limitations of the process as such.

Present work

The background for this study is the production of BDs, the latest generation of optical disk formats. These disks have to be covered with a protective layer. The demands on the layer are stringent and the thickness of the layer is to be 100 [+ or -] 3 [micro]m over the information area that begins at a radius of 24 mm and ends at 58 mm. The radius of the hole is 7.5 mm. Spin coating seems to be an attractive method to achieve this in a rapid production process. In this spin-coating process (see Fig. 1), lacquer is initially dispensed at a finite radius because of the hole in the center of the disk. The dispensing is followed by spinning, and finally, the lacquer is cured.

With off-center dispensing it has been observed that the cover layers had an unacceptable gradient toward the inner radius. The lacquer was thinning toward the edge of the hole and this slope (or gradient) extended into the information area. The off-center dispensing was identified as the reason for the unacceptable thickness gradient. This study was initiated to understand and develop the spin-coating process, so that BD cover layers can be produced. The work deals with the physical origin of the gradient at the inner radius in terms of experimental verification of the model given by equations (6) and (7). It will also be shown that BD cover layers are not possible to produce by pure spin coating of a Newtonian fluid. Therefore, also possible manipulations of the process will be modeled, in the form of shear on the surface and/or inline curing of the lacquer. It will be demonstrated that spin coating with manipulation can be used to produce BD cover layers.

In "Experiments and verification", it is verified experimentally that the model described by equations (6) and (7) is sufficient to describe spin coating of BD cover layers. In "Limitations and possibilities", it is shown that fundamental limits can be derived from the model and that a thickness of 100 [+ or -] 3 [micro]m is practically impossible to achieve for the described spin-coating process. Thus, some kind of manipulation is necessary. Two ways of manipulation (shear on the surface and inline curing) will be introduced and studied in "Numerical evaluation and manipulation by surface shear and weak curing during spinning".

Experiments and verification

Experimental setup and methods

The objective of the experiments was to validate the model presented in the previous section. The experiments were made in a production line for DVDs (Alphabonder at AlphaSweden AB) specially adapted for studies of the BD spin-coating process. The liquid film was formed on CD substrates with a thickness of 1.1 mm. The substrate was placed on a chuck driven by an electrical motor with computer-controlled speed, and the liquid was placed from a dispense needle fed by a volumetric pump. The needle could be moved in the radial direction during dispensing, so that a larger portion of the substrate could be covered. The viscosity of the lacquer was 2000 cSt.

The dispensing process was computer controlled. For each experiment, 3 mL of lacquer was placed on the substrate while it was rotating at 2.5 rev/s. When the dispensing started, the needle was at [R.sup.*] = 11 mm and during the dispensing it moved outwards at constant speed (2.5 mm/s) to [R.sup.*] = 16 mm. Each dispensing process took 2 s.

After dispensing, the disk was accelerated to the spinning speed, 23.3 rev/s. After a certain time, the rotation was stopped and the lacquer was cured by UV radiation. After curing, the lacquer thickness was measured by two independent systems designed for measurements of BD cover layer thickness (Argus Universal Measurement System from Dr. Schwab gmbh and ProMeteus MT-200 from Dr. Schenk Inspection Systems).

The accuracy of the experimental procedure was tested in two ways. First, the variation of the thickness of the cured layer in the azimuthal direction at a given radius was measured and found to be within [+ or -]2.8 [micro]m. Henceforth, all experimental data are taken as the average in the azimuthal direction. The second test was the variation from one realization to the other. Ten cover layers were produced with nominally the same procedure. At a given radius, the layer thickness (i.e., the average in the azimuthal direction) was within [+ or -]2.2 [micro]m over the ten disks. Since the nominal thickness was 100 [micro]m, these variations were considered to be acceptable.

To compare equation (6) and the boundary condition (7) with the experiments, an initial distribution was obtained by spinning at the higher speed for the time [t.sub.ref] The initial thickness profile [H.sub.0.sup.*]([R.sup.*]) was then measured after curing. For the following disks, the dispensing sequence was identical and the spinning time at high rotation speed was increased. The time of spinning from the initial profile is then [t.sub.eff] = [t.sub.spin] - [t.sub.ref] and this is the time used in the simulations. Here, [t.sub.ref.sup.*] = 4 s and the maximum value of [t.sub.eff.sup.*] is 60 s.

Results and verification

The measured thickness as a function of R at different [t.sub.eff] is compared with calculated results from equation (6) and shown in Fig. 2. In (a), the complete profiles at t = 0 and five later time instants are shown. The experimental data covers R - 1.8-4.6 and the simulated R = 1-4.6. For the initial condition of the simulations, cubical splines are used to connect the boundary condition H = 0 at R = 1 with the first experimental data at R = 1.8. In Fig. 2a, some examples of complete profiles are shown and it is seen how the film gels thinner as time increases. The difference between the experiments and the simulations, [DELTA](R) = ([H.sub.sim](R) - [H.sub.exp](R))/[H.sub.exp], at different times is shown in Fig. 2b. For each instant, the maximum (*) and mean (*) deviations over all R are shown. The thickness of the film decreases by a factor of five and the agreement between measured and simulated values is within 5% at all times. In (c), the film thicknesses at two radial positions, R = 1.8 and R = 4.6, are shown as functions of time.


Limitations and possibilities

Assume that a film with a minimum thickness [H.sub.0] and a maximum thickness (1 + [k.sub.0])[H.sub.0] in a region of length [L.sub.0] starting at radius [R.sub.0] is to be produced. Below, it will be derived that for certain values of [R.sub.0], [L.sub.0], and [K.sub.0], the homogeneity of the initial layer can be arbitrary, as long as the layer is thick and long enough! Since equations (6) and (7) and the underlying assumptions now constitute an experimentally verified model of the thickness development, we can use them to study the inverse problem: which final conditions result in valuable properties for the initial layer. Turn to the analytical description of the thickness as a function of time in equations (9) and (10). Note that the temporal change of [H.sub.t.sup.m] in equation (10) is independent of R. Consequently, horizontal surfaces will remain horizontal during spinning. So, two horizontal lines limiting the final layer correspond to two equally horizontal lines at earlier times (see Fig. 3, which will be described in detail below). Henceforth, these lines will be referred to as the upper and the lower limit, respectively. In particular, a very special time instant will be studied, namely the lime instant at which the region relevant for the final layer starts at the rim of the hole. This means that the inner endpoint of the lower limit has moved to R = 1.


Figure 3 shows (i) the final region (thin lines, red online only) defined by the BD specification (equation 8) and (ii) the corresponding region (thick lines, blue online only) where the initial layer must be. The thick lines are obtained by equations (9) and (10) with negative values of t. The time of spinning (|t|) has been increased until the inner endpoint of the lower limit (thick, solid blue line online only) has reached the rim of the hole, i.e., [R.sub.t.sup.-] = 1. Longer spin times than these are not allowed, since the dispensed layer cannot extend over the hole. This is a decent first approximation of the innermost position achievable (surface tension and contact angle will of course have an impact, but since the thickness of the layer is only a few percent of the radial extension, these effects are of second order in this analysis). In Fig. 3, we define a number of quantities: the radial positions [R.sub.t.sup.+], [R.sub.t.sup.-], [R.sub.0], the lengths of the limits [L.sub.t.sup.+], [L.sub.t.sup.-], [L.sub.0], and finally [H.sub.0], [k.sub.0].

Two observations are made. First (as mentioned), the maximum spin time is given by the value of [([H.sub.0.sup.-]).sup.2] t giving [R.sub.t.sup.-] = 1, since longer spin times would mean that the initial layer would have to lean in over the hole. Second, at this spin time, the upper limit has moved further in than the lower one. If the outer edge of the upper limit at this time is at R [less than or equal to] 1, the final layer can be produced by applying a thick enough liquid layer from R = 1 to R = 1 + [L.sub.t.sup.-] and spin. In such cases, there is no upper bound on the initial layer. Mathematically, the criteria on the upper limit translates to [R.sub.t.sup.+] + [L.sub.t.sup.+] [less than or equal to] 1. Thus, if

[R.sub.t.sup.+] + [L.sub.t.sup.+] [less than or equal to] 1 (12)


[R.sub.t.sup.-] = 1, (13)

this situation is at hand. As mentioned, this means that a layer starting at the rim of the hole has no upper limit above the lower limit, since the whole upper limit has moved in over the hole. As a consequence, there is no condition on the thickness variations of the initial layer. The only remaining condition is that it has to have a thickness of at least [H.sub.t.sup.-] from [R.sub.t.sup.-] to [R.sub.t.sup.-] + [L.sub.t.sup.-]. Since there is no upper limit, this should be possible to achieve by adding enough lacquer. In our opinion, this should result in a very robust process.

Equations (9) and (10) can be used to investigate if a final layer specification fulfils this condition (a property which should result in a robust process) and, perhaps even more important, it will turn out that a BD layer does not.

First, equations (9) and (13) give the maximum spinning time as (with [H.sub.0.sup.m] = [H.sub.0], [R.sub.0.sup.m] = [R.sub.0], and [R.sub.t.sup.m] = 1):

4[H.sub.0.sup.2][t.sub.lim] = [(1/[R.sub.0]).sup.4/3] -1 = [R.sub.0.sup.-4/3] - 1. (14)

Note that the numerical value of [t.sub.lim] is negative since the inverse problem is studied. Nevertheless, the term "maximum spinning time" is used for this quantity due to its descriptive value.

The radial position of the outer edge of the upper limit at this time [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is obtained by equation (9) with [H.sub.0.sup.m] = (1 + [k.sub.0]) [H.sub.0]:


Equations (12-15) can be combined to give us the final tolerance [k.sub.0] for which the positive production circumstances are obtained:

[k.sub.0] [greater than or equal to] [square root of ([[[([R.sub.0] + [L.sub.0]).sup.-4/3] - 1]/[[R.sub.0.sup.-4/3] - 1]])] - 1. (16)

This result is shown in Fig. 4. In this figure, the lowest value of [k.sub.0] that obeys equation (13) is shown as a function of [R.sub.0] and [L.sub.0]. The gray area shows the values of [R.sub.0] and [L.sub.0] which are possible within the BD specification. Furthermore, the thick contour (red online only) shows [k.sub.0] = 0.06, the thickness variation allowed by the BD specification. The dashed and dashed-dotted lines will be explained later. As can be seen, the BD specification does not fulfill the condition for a robust process. The lowest thickness variation for which this condition is fulfilled for a BD-like layer accordingly is [k.sub.0] = 0.09. A layer that is to meet the BD specification would have to be dispensed very carefully, with a precision that might be hard to achieve in practice. If this is to be attempted, analysis of positions of the upper and lower limits for different spin times provides an exact quantification of the needed homogeneity. The actual homogeneity varies with surface tension and contact angle. For the BD case, the homogeneity is seen in Fig. 3 and is 98 [micro]m for the region where the upper limit overlaps the lower limit. This region has a radial extension of 3.6 mm.


One remedy to this problem would be to change the BD specification. With constant size of the disk, this can be done in three ways:

1. increase the allowed variation of the cover layer,

2. decrease the radius of the hole, or

3. increase the radius at which the information layer starts.

It is directly seen from Fig. 4 that point (i) implies [k.sub.0] = 0.09, i.e., a final layer definition of 100 [+ or -] 4.5 [micro]m. Furthermore, the straight lines show that the present condition, 100 [+ or -] 3 [micro]m ([k.sub.0] = 0.06), can be met if the hole diameter is decreased so that [R.sub.0] = 4.1 (the crossing of the dashed-solid line and the thick contour). This possibility is further illustrated in Fig. 5. In the latter figure, it is seen that either changing the hole radius to 5.8 mm instead of the present 7.5 mm or moving the inner limit of the information to 28.5 mm instead of 24 mm extends the available parameter region to the contour [k.sub.0] = 0.06. The latter suggestions imply a reduction of the information area of around 10%.


Numerical evaluation of manipulation by surface shear and weak curing during spinning


As known from the previous section, the BD condition cannot be met without some manipulation of the spin-coating process. Two means of manipulation are to apply shear on the film surface during spinning (e.g., by air streaming over the disk) and to cure (i.e., increase the viscosity of) the film during spinning. Below, mathematical models of such manipulation are formulated and a first evaluation of their potential is performed.

It will be demonstrated that such manipulation makes it possible to produce cover layers that fulfill the BD specification (equation 8). The ability is demonstrated numerically with an initial profile that is inspired by experimental data.

Demonstration case

The result of the manipulation methods is evaluated by its ability to produce lacquer layers meeting the specifications from a given initial thickness distribution inspired by the experimental distributions. The ability of the manipulation is measured by the quantity [k.sub.min], defined as




Thus, k(t) is the maximum instantaneous relative deviation from the desired cover layer thickness. The minimum value of this quantity over time is then the best quality that can be achieved, [k.sub.min]. The time at which this minimum is obtained is denoted [t.sub.min], and [k.sub.min] [<] 0.03 implies that the BD criterion has been met.

With surface shear or inline curing, the radial distribution and amplitude of surface shear or curing intensity enters the problem. The possibilities of surface shear and inline curing will not be considered in depth, and the results will be restricted to a demonstration of how the manipulations can increase the operating windows of the spin-coating process when making BDs.

Improvement by shear on the surface

Background and governing equations

One way to manipulate the temporal evolution of the film thickness is to apply a shear stress at the film surface. This can be simulated by equations obtained by applying the condition [partial derivative][U.sup.*]/[partial derivative][y.sup.*] = [[sigma].sup.*] ([R.sup.*])/[[mu].sup.*], where [U.sup.*] is the radial velocity of the liquid, [y.sup.*] is the disk-normal coordinate, [[sigma].sup.*] is the radial shear stress on the surface of the film, and [[mu].sup.*] is the dynamic viscosity. Again, superscript * denotes dimensional quantities. The analysis of reference 1 is readily adjusted and after nondimensionalization, the equation governing the evolution of the film thickness reads

[[[partial derivative]H]/[[partial derivative]t]] = -[1/R] [[partial derivative]/[[partial derivative]R]]([R.sup.2][H.sup.3] + [SIGMA](R)[RH.sup.2]), (18)

where the nondimensional shear stress on the surface is

[SIGMA](R) = [[3[v.sup.*]]/[2[[micro].sup.*] [[omega].sup.*2] [R.sub.i.sup.*2]]] [[sigma].sup.*] ([RR.sub.i.sup.*]). (19)

For a given initial distribution of the film thickness and a (possibly time-dependent) distribution of [SIGMA](R), the above equations can be integrated in time to give the film thickness as a function of time.

Remark: To be able to neglect surface tension near R = 1, and for the boundary condition (7) to be valid, no arbitrarily large values of shear are allowed in this model. However, this should not be a problem as long as only a necessary amount of shear is used to compensate for the slope formation.

Results with shear on the surface

Figure 6 shows the effects of shear on the surface. The initial profile, taken from an experimental measurement, is shown in Fig. 6a, and in (b) the chosen distribution of radial shear is shown.


Thickness distributions at the time [t.sub.min] (i.e., the time corresponding to [k.sub.min]) for different values of shear amplitude are shown in Fig. 6c. For low shear, the decrease of the thickness toward the inner radius is spread over a long radial distance, but with increasing shear the slope becomes steeper and steeper, giving a more even thickness distribution over a large portion of the surface. The efficiency of the manipulation is shown in Fig. 6d where [k.sub.min] is shown as a function of [[SIGMA].min]. The minimum value of [k.sub.min] is practically zero, indicating that completely even layers can be achieved with proper distributions of surface shear.

Control by weak curing during spinning

Curing during spinning is inspired by Heinz et al. (14) They imposed a temperature gradient from the inner radius and out to manipulate the viscosity. Here, a different model for the curing manipulation is used, namely that the viscosity, as a material parameter, is changed by the curing process.

Governing equations

Thus, we describe inline curing as an increase of viscosity in time. The relative viscosity increase is defined as

[bar.v](T) = [[[v.sup.*](T)]/[v.sub.0.sup.*]], (20)

where [v.sub.0.sup.*] is the initial viscosity.

To model inline curing, e.g., by a UV lamp, the material derivative of this quantity is assumed to be a function of the radial position:

[[D[bar.v]]/[Dt]] = [[[partial derivative][bar.v]]/[[partial derivative]t]] + [bar U][[[partial derivative][bar.v]]/[[partial derivative]R]] = If(R), (21)

where [bar.U] is the mean (in the vertical direction) of the velocity, I is a curing intensity, and f(R) is the radial distribution of this intensity. This model for the effect of curing assumes that the lacquer is permanently hardened when exposed to the UV radiation and the increase of viscosity is moved with the fluid in the radial direction. However, the fact that the liquid closest to the disk travels slower in the radial direction than liquid closer to the free surface is not taken into account. Together with boundary and initial conditions, equations (1) and (21) can be solved and possibilities of inline curing studied.

The possibilities of inline curing are illustrated in Fig. 7. The two intensity distributions in Fig. 7a are compared. The initial profile is the same as before (see Fig. 6a). Figure 7b shows [k.sub.min] as a function of maximum curing intensity I for the two distributions. The smooth distribution (the solid curve) is shown to give values of [k.sub.min] less than 2%, whereas the steep distribution (dash-dot line) does not manage to reach values below 3%. It should be noted that the smooth distribution gives a rather large range of I (200-400) for which the BD specification (3%) is satisfied.


The thicknesses as a function of radius at the time [t.sub.min] are shown in Fig. 7c and d for the smooth and steep intensity distribution, respectively. In each graph, the different curves show increasing curing intensities as indicated by the arrows. For high values of I, the viscosity becomes large at small radii and consequently there is a build-up of liquid there (especially with the steep cure-intensity distribution), giving large values of [k.sub.min]. For the most even profiles, this effect is moderate and the method might be possible to use in a real process.


Spin coating of disks has been studied aiming at producing the cover layer of BDs. A first-order model of spin coating (1) has been expanded to cases where coating is starting off-center. The expanded model has been verified against experiments. Thus, the assumptions behind the model are valid in this case. Fundamental limitations regarding the thickness homogeneity as a function of the radius at which the lacquer layer starts have been derived and it has been shown that the layer called upon by the BD specification is difficult to produce (due to the strong limitations it puts on the initial layer) without manipulation of the process. The limitations are not restricted to spin coating of BDs, but equation 16 can be applied to any other off-center coating situations where the model of reference 1 is suitable. Two methods by which these fundamental limitations can be overcome have been proposed. The first is to apply shear at the surface of the liquid film and the second is to cure the liquid while spinning. These two methods have been incorporated in the model and the results show that they can be used to improve the quality of the final cover layer.

The model for applying shear on the surface includes no additional simplifications compared to reference (1) and it can thus be assumed that the promising results in Fig. 6 are possible to achieve in a physical setup. The model for inline curing is more rudimental since viscosity and its convection speed are assumed to be constant through the film in the vertical direction. Thus, only qualitative similarity should be expected if the method is implemented in a physical setup. Nevertheless, each model should be possible to use both for further development and control of the process.

Acknowledgments This study was performed in cooperation with Alpha Sweden AB and their assistance, especially when it comes to the experimental part, is gratefully acknowledged. FL has been financed by the Swedish Research Council and OT by Linne FLOW Centre.


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O. Tammisola, F. Lundell (*), G. Hellstrom

Linne FLOW Centre, KTH Mechanics, 100 44 Stockholm, Sweden


T. Lagerstedt

Torgny Lagerstedt AB, Dobelnsgatan 89, 113 52 Stockholm, Sweden

* After the short initial acceleration and thinning phase.

J. Coat. Technol. Res., 7 (3) 315-323, 2010

DOI 10.1007/s11998-009-9204-2
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Author:Tammisola, Outi; Lundell, Fredrik; Hellstrom, Georg; Lagerstedt, Torgny
Publication:JCT Research
Date:May 1, 2010
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