Spin coating of Blu-Ray disks:

modelling, experiments, limitations and

manipulation

Outi Tammisola1, Fredrik Lundell1, Georg Hellstrom1 & Torgny Lagerstedt2

1Linne FLOW Centre, KTH Mechanics, S-100 44 Stockholm, Sweden2Torgny Lagerstedt AB, Dobelnsgatan 89, S-113 52 Stockholm, Sweden

July 9, 2009

Abstract

A major challenge in the production of Blu-Ray Disks (BDs) is to makethe cover layer over the information area. The layer has to be both thickenough to protect the information and even enough for the informationto be read through it optically. Furthermore, it is preferred not to coverthe hole in the centre of the disk. Spin coating is a candidate methodfor the production of these layers in a rapid reproduction process. Whendispensing is performed off-centre (in order not to cover the hole), a newcomplication appears, namely the formation of a slope towards the innerrim of the liquid film. Here, fundamental limitations for achieving evenfilms in this system and ways to overcome the difficulties by manipulationof the process are studied. A mathematical model for this particular caseof spin coating is obtained and validated by comparison with experimentsmade in industrial equipment aiming at producing Blu-Ray Disks. Themodel agrees well with the experimental data. The model is then usedto show that cover layers that fulfill the Blu-Ray (BD) specification arevery difficult to produce with the spin-coating technique. Manipulation byinline curing and surface shear is added to the model and the results showthat it is considerable easier to meet the BD specification when utilizingthe manipulation.

1 Introduction

1.1 The spin coating process and previous work

Spin coating is used to apply thin liquid layers on the surface of objects. Duringthe spin coating process the liquid is first dispensed on the surface. The liquidis then spread over the surface by the centrifugal force, figure 1. Emslie et

al.[1] made a theoretical study of the development of a film of a Newtonianliquid flowing on a rotating disk. Under the assumptions that the liquid film is

1

thin and that the effects of fluid acceleration, gravity, Coriolis force and surfacetension are negligible compared to the centrifugal force, the thickness of the filmis described by the equation

∂H∗

∂t∗= −

1

R∗

∂

∂R∗

(

ω∗2

3ν∗R∗2H∗3

)

, (1)

where H is the thickness of the liquid film, R is the radial coordinate from thecentre of the disk, ω is the angular velocity, ν is the kinematic viscosity of theliquid and ∗ denotes dimensional quantities.

To give a hint on the applicability of the model above in different spincoating situations, we shortly introduce the key assumptions behind it. Firstly,the initial liquid layer has to be even enough in the azimuthal direction. Theliquid has to be viscous enough so that inertia can be neglected, which can (afterthe initial acceleration of the disk) be quantified as a low value of the Reynolds

number :

Re =U∗H∗

ν∗, (2)

where U∗ is a characteristic velocity of the liquid, and can be approximated bythe liquid surface velocity given by the model: U∗ = ω∗2R∗H∗/2ν∗. The layeris also assumed to be thin enough that there is no shear in vertical planes, sothat the flow can be considered one-dimensional. Following [1], the conditionfor neglecting gravity becomes:

1

R∗

∂H∗

∂R∗<<

ω∗2

g, (3)

where g = 9.81 m/s2, and for neglecting Coriolis force:

ν∗ >> ω∗H∗. (4)

In practice, the set of conditions above means that the equation appliesto thin layers of viscous fluid, with height variations small compared to thecentrifugal force potential, which is a fair approximation for coating of BluRay1.Also, the spinning speed in this application is typically high enough to constrainthe effect of surface tension near the rims of the lacquer layer.

Spin coating has been studied extensively and a review is found in [2]. Sum-marizing (with example references), effects of surface roughness [3] , evaporation[4], non-Newtonian rheology [5], hydrodynamic stability [6, 7] and inertia andshear on the surface [8] have been investigated. Later, the initial spreadingof the liquid and the related contact line problem has also grasped attention[9, 10]. More applied studies aim at controlling the process in order to achieveeven films for production of microelectronics by different means [11, 12, 13].The particular case of spin coating of Blu-Ray disks has also been studied [14]and this work will be returned to later on.

1after the short initial acceleration and thinning phase.

2

1.2 Modeling a liquid film starting at a non-zero radius

First the problem is non-dimensionalized. The coordinates are R∗ for the radialdirection and H∗ for the thickness of the liquid layer. We introduce the followingscaling:

H =H∗

R∗

i

R =R∗

R∗

i

t =R∗2

i ω∗2

3ν∗t∗, (5)

where R∗

i is the radius at which the liquid film starts, ω∗ is the angular velocityof the disk and ν∗ is the kinematical viscosity of the liquid. The numericalfactor 3 in the time scaling is introduced for later convenience. In this scaling,equation (1) gives

∂H

∂t= −

1

R

∂

∂R

(

R2H3)

. (6)

Emslie et al.[1] investigated the spreading of a liquid film starting fromR = 0. This is generally not true for spin coating of optical disks, since the holein the middle is not covered.

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

H(R ≤ 1) = 0. (7)

This means that the inner contact line is assumed to be steady at R = 1. Thegeometrical development of the liquid film is thus completely determined by theinitial condition H(R, t = 0). The parameters R∗

i , ν∗ and ω∗ only scale thelength and time.

The presence of the inner contact line implies that the slope has to be smallalso in this region for the model to be strictly valid (see the conditions in theprevious section). However, it will be seen later that the slope tends to de-crease fast while spinning, and more importantly, that the model agrees withexperiments.

In nondimensional form, the BD criterion mentioned in the abstract (H∗ =100 ± 3 µm for 24 mm≤ R∗ ≤58 mm and R∗

i = 7.5 mm) becomes:

H = (9.09 ± 0.27)× 10−3 .at 3.2 ≤ R ≤ 7.7 (8)

Equation (8) is obtained with the assumption that the lacquer starts at the rimof the hole, the configuration that minimizes the gradient at the informationarea.

Equation (6) can be solved by the method of characteristics[1] (the samesolution can be reached at via asymptotic expansions[15]). First, take a setof radial positions Rm

0 , where m is an index over increasing radial positions.The corresponding film thicknesses at these positions are denoted Hm

0 . Thethickness distribution at time t is then obtained at the new radial positions Rm

t :

Rmt = Rm

0

[

1 + 4(Hm0 )2t

]0.75, (9)

3

where it is:

Hmt =

Hm0

[1 + 4(Hm0 )2t]0.5 , (10)

where Hmt is the thickness at the radial position Rm

t . The boundary condition(7) is fulfilled at all times if H(t = 0, R = 1) = 0 since Rm

t = Rm0 and Hm

t = Hm0

if Hm0 = 0. This solution is valid only as long as the ordering sequence of Rm

t

is intact, i.e. as long asRm−1

t < Rmt . (11)

This last condition is equivalent to stating that no shocks are allowed to formon the surface. Note also that the points Rm

t are immaterial.The problem stated by equations (6,7) 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 section 3 and(ii) to evaluate the manipulation strategies in section 5. Equations (9,10) areused in section 4 to derive fundamental limitations of the process as such.

1.3 Present work

The background for the present study is the production of Blu-Ray Disks (BDs),the latest generation of optical disk formats. These disks have to be covered witha protective layer. The demands on the layer are stringent and the thicknessof the layer is to be 100 ± 3 µm over the information area that begins at aradius of 24 mm and ends at 58 mm. The radius of the hole is 7.5 mm. Spincoating seems to be an attractive method to achieve this in a rapid productionprocess. In this spin-coating process (see figure 1), lacquer is initially dispensedat a finite radius because of the hole in the centre of the disk. The dispensingis followed by spinning and finally the lacquer is cured.

With off-centre dispensing it has been observed[14] that the cover layers hadan unacceptable gradient towards the inner radius. The lacquer was thinningtowards the edge of the hole and this slope (or gradient) extended into theinformation area. The off-centre dispensing was identified as the reason for theunacceptable thickness gradient. The present work was initiated in order tounderstand and develop the spin-coating process so that BD cover layers can beproduced. The work deals with the physical origin of the gradient at the innerradius in terms of experimental verification of the model given by equations(6,7). It will also be shown that BD cover layers are not possible to produce bypure spin coating of a Newtonian fluid. Therefore, also possible manipulationsof the process will be modeled, in the form of shear on the surface and/orinline curing of the lacquer. It will be demonstrated that spin coating withmanipulation can be used to produce BD cover layers.

In section 2, it is verified experimentally that the model described by equa-tions (6,7) is sufficient to describe spin coating of BD cover layers. In section3 it is shown that fundamental limits can be derived from the model and thata thickness of 100 ± 3 µm is practically impossible to achieve for the describedspin-coating process. Thus, some kind of manipulation is necessary. Two ways

4

dispensing

spinning

curing

UV light

Figure 1: (Color online) Schematic of the spin coating process in the manufac-turing of Blu Ray disks.

of manipulation (shear on the surface and inline curing) will be introduced andstudied in section 5. The conclusions are given in section 6.

2 Experiments and verification

2.1 Experimental setup and methods

The objective of the experiments was to validate the model presented in theprevious section. The experiments were made in a production line for DVD(Alphabonder at AlphaSweden AB) specially adapted for studies of the BD spin-coating process. The liquid film was formed on CD substrates with a thicknessof 1.1 mm. The substrate was placed on a chuck driven by an electrical motorwith computer controlled speed and the liquid was placed from a dispense needlefeed by a volumetric pump. The needle could be moved in the radial directionduring 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 mlof lacquer was placed on the substrate while it was rotating at 2.5 rev/s. Whenthe dispensing started, the needle was at R∗=11 mm and during the dispensing,it was moved outwards at constant speed (2.5 mm/s) to R∗=16 mm. Eachdispensing 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 UVradiation. After curing, the lacquer thickness was measured by two independentsystems 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 ata given radius was measured and found to be within ±2.8 µm. From now on,all experimental data are taken as the average in the azimuthal direction. The

5

1 2 3 40

0.005

0.01

R

0 0.02 0.04-10

0

10

t

0 0.02 0.040

0.005

0.01

t

HΔ [%]

(a)

(b) (c)

H

t

R

Figure 2: (Color online) Comparison between experimental and simulated re-sults. (a): Calculated (—, blue) and measured (· · ·, red) film thickness atincreasing times. The first experimental contour, also used as initial profile forthe calculations, is drawn with a thick line. (b): Difference between the mea-surements and calculations, ∗: maximum, ◦: mean value in R = 1.8—4.6. (c):Calculated (∗) and measured (◦) film thickness as a function of time at R = 1.8(—) and 4.6 (- -).

second test was the variation from one realization to the other. Ten cover layerswere produced with nominally the same procedure. At a given radius, the layerthickness (i.e. the average in the azimuthal direction) was within ±2.2 µm overthe ten disks. Since the nominal thickness was 100 µm, these variations wereconsidered to be acceptable.

In order to compare equation (6) and the boundary condition (7) with ex-periments, an initial distribution was obtained by spinning at the higher speedfor the time tref . The initial thickness profile H∗

0 (R∗) was then measured aftercuring. For the following disks, the dispensing sequence was identical and thespinning time at high rotation speed was increased. The time of spinning fromthe initial profile is then teff = tspin − tref and this is the time used in thesimulations. Here, t∗ref = 4 s and the maximum value of t∗eff is 60 s.

6

2.2 Results and verification

The measured thickness as a function of R at different teff is compared withcalculated results from equation (9) and shown in figure 2. In (a), the completeprofiles at t = 0 and five later time instants are shown. The experimental datacovers R = 1.8—4.6 and the simulated R = 1—4.6. For the initial conditionof the simulations, cubical splines are used to connect the boundary conditionH = 0 at R = 1 with the first experimental data at R = 1.8. In figure 2(a), some examples of complete profiles are shown and it is seen how the filmgets thinner as time increases. The difference between the experiments and thesimulations, ∆(R) = (Hsim(R)−Hexp(R))/Hexp, at different times is shown infigure 2 (b). For each instant, the maximum (∗) and mean (◦) deviations overall R are shown. The thickness of the film decreases by a factor of five and theagreement 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, areshown as functions of time.

3 Limitations and possibilities

Assume that a film with a minimum thickness H0 and a maximum thickness(1 + k0)H0 in a region of length L0 starting at radius R0 is to be produced.Below, it will be derived that for certain values of R0, L0 and k0, the homo-geneity of the initial layer can be arbitrary, as long as the layer is thick and longenough! Since equations (6, 7) and the underlying assumptions now constitutean experimentally verified model of the thickness development, we can use themto study the inverse problem: which final conditions result in valuable proper-ties for the initial layer. Turn to the analytical description of the thickness asa function of time in equations (9,10). Note that the temporal change of Hm

t

in equation (10) is independent of R. Consequently, horizontal surfaces will re-main horizontal during spinning. So, two horizontal lines limiting the final layercorrespond to two, equally horizontal lines at earlier times (see figure 3, whichwill be described in detail below). From now on, these lines will be referred toas the upper and the lower limit, respectively. In particular, a very special timeinstant will be studied, namely the time instant at which the region relevant forthe final layer starts at the rim of the hole. This means that the inner endpointof the lower limit has moved to R = 1.

Figure 3 shows (i) the final region (thin lines, red online) defined by theBD specification, equation (8) and (ii) the corresponding region (thick lines,blue online) where the initial layer must be. The thick lines are obtained byequations (9 & 10) with negative values of t. The time of spinning (|t|) has beenincreased until the inner endpoint of the lower limit (thick, blue online, solidline) has reached the rim of the hole, i.e R−

t = 1. Longer spin times than theseare not allowed, since the dispensed layer cannot extend over the hole. This is adecent first approximation of the innermost position achievable (surface tensionand contact angle will of course have an impact, but since the thickness of the

7

0 2 4 6 80

0.01

0.02

0.03

0.04

0.05

R

H

R 0

L0

k H00

R t

-

L t

-

R t

+

L t

+

disc

Figure 3: (Color online) Upper (- -) and lower (—) limit of the region where theliquid surface has to be before spinning (thick, blue online) in order to end up inthe BD region (thin, red online) defined by equation (8). Note that there mustbe fluid outside of the initial region so that condition (11) is fulfilled during thespinning process.

layer is only a few percent of the radial extension, these effects are of secondorder in this analysis). In figure 3 we define a number of quantities: the radialpositions R+

t , R−

t , R0, the lengths of the limits L+t , L−

t , L0 and finally H0, k0.Two observations are made. Firstly (as mentioned), the maximum spin time

is given by the value of (H−

0 )2t giving R−

t = 1, since longer spin times wouldmean that the initial layer would have to lean in over the hole. Secondly, at thisspin 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 ≤ 1, the final layer can be producedby applying a thick enough liquid layer from R = 1 to R = 1 + L−

t and spin.In such cases, there is no upper bound on the initial layer. Mathematically, thecriteria on the upper limit translates to R+

t + L+t ≤ 1. Thus, if

R+t + L+

t ≤ 1 (12)

whenR−

t = 1, (13)

this situation is at hand. As mentioned, this means that a layer starting at therim of the hole has no upper limit above the lower limit, since the whole upperlimit has moved in over the hole. As a consequence, there is no condition on thethickness variations of the initial layer. The only remaining condition is thatit has to have a thickness of at least H−

t from R−

t to R−

t + L−

t . Since there isno upper limit, this should be possible to achieve by adding enough lacquer. Inour opinion, this should result in a very robust process.

8

Equations (9) and (10) can be used to investigate if a final layer specificationfulfils 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 (withHm

0 = H0, Rm0 = R0 and Rm

t = 1):

4H20 tlim =

(

1

R0

)4/3

− 1 = R−4/3

0 − 1. (14)

Note that the numerical value of tlim is negative since the inverse problemis studied. Nevertheless, the term ”maximum spinning time” is used for thisquantity due to its descriptive value.

The radial position of the outer edge of the upper limit at this time (R+

tlim+

L+

tlim) is obtained by equation (9) with Hm

0 = (1 + k0)H0:

R+tlim

+ L+tlim

= (R0 + L0)[

1 + 4(1 + k0)2H2

0 tlim]0.75

. (15)

Equations (12–15) can be combined to give us the final tolerance k0 for whichthe positive production circumstances are obtained:

k0 ≥

√

(R0 + L0)−4/3 − 1

R−4/3

0 − 1− 1. (16)

This result is shown in figure 4. In this figure, the lowest value of k0 thatobeys equation (13) is shown as a function of R0 and L0. The grey area showsthe values of R0 and L0 which are possible within the BD-specification. Further-more, the thick contour (red online) shows k0 = 0.06, the thickness variationallowed by the BD-specification. The dashed and dashed-dotted lines will be ex-plained later. As can be seen, the BD specification does not fulfill the conditionfor a robust process. The lowest thickness variation for which this condition isfulfilled for a BD-like layer accordingly is k0 = 0.09. A layer that is to meetthe BD specification would have to be dispensed very carefully, with a precisionthat might be hard to achieve in practice. If this is to be attempted, analysis ofpositions of the upper and lower limits for different spin times provides an exactquantification of the needed homogeneity. The actual homogeneity varies withsurface tension and contact angle. For the BD case the homogeneity is seen infigure 3 and is 98 µm for the region where the upper limit overlaps the lowerlimit. This region has a radial extension of 3.6 mm.

One remedy to this problem would be to change the BD specification. Withconstant 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.

9

2 4 6 8 100

5

10

15

20

25

R0

L0

k0=0.09

0.06

0.1

0.0450.03

0.017

0.01

0.003

0.075

0.3

0.17

Figure 4: (Color online) Contours of the thickness variation of the final layerk0 for which the outer edge of the upper limit is inside the rim of the hole.The variation is a function of radial position at which the specification of thefinal layer start (R0) and its length (L0). The grey area shows the region thatcan be reached under the BD specification of hole diameter and informationregion. The thick (red online) contour shows the BD condition (k0 = 0.06)and the dashed and dash-dotted lines show the effect of decreasing the holediameter (dashed) and moving the beginning of the information area outwards(dash-dot).

It is directly seen from figure 4 that point (i) implies k0 = 0.09, i.e. a finallayer definition of 100 ± 4.5 µm. Furthermore, the straight lines show that thepresent condition, 100 ± 3 µm (k0 = 0.06) can be met if the hole diameter isdecreased so that R0 = 4.1 (the crossing of the dashed-solid line and the thickcontour). This possibility is further illustrated in figure 5. In the latter figure,it is seen that either changing the hole radius to 5.8 mm instead of the present7.5 mm or moving the inner limit of the information to 28.5 mm instead of24 mm extends the available parameter region to the contour k0 = 0.06. Thelatter suggestions imply a reduction of the information area of around 10%.

4 Numerical evaluation of manipulation by sur-

face shear and weak curing during spinning

4.1 Motivation

As follows from the previous section, the BD-condition cannot be met withoutsome manipulation of the spin coating process. Two means of manipulation are

10

1 2 3 4 50

2

4

6

8

10

R0

L0

BD spec.

Ri

*=28.5 mm

rh

*=5.8 mm

k0=0.06

Figure 5: (Color online) Illustration of regions in the R0, L0 plane that can bereached under the present BD-specification (black area), with a hole diameter of5.8 mm, and inner limit of information area of 28.5 mm (gray areas as indicated).The evenness condition for a BD cover layer, k0 = 0.06, is drawn with a curve(red online).

to apply shear on the film surface during spinning (e.g. by air streaming overthe disk) and to cure (i.e. increase the viscosity of) the film during spinning.Below, mathematical models of such manipulation are formulated and a firstevaluation of their potential is performed.

It will be demonstrated that such manipulation makes it possible to pro-duce cover layers that fulfill the BD specification, equation (8). The ability isdemonstrated numerically with an initial profile that is inspired by experimentaldata.

4.2 Demonstration case

The result of the manipulation methods is evaluated by its ability to producelacquer layers meeting the specifications from a given initial thickness distribu-tion inspired by the experimental distributions. The ability of the manipulationis measured by the quantity kmin, defined as

kmin = mint

k(t), (17)

where

k(t) = maxR

|H(R) − Hspec|

Hspec.

11

2 40

0.01

0.02

0.03

(a)

R

H

2 4

-1

0

R

-Σ(R

)/Σ

min

(b)

2 40

0.005

0.01

0.015

(c)

R

H

0 0.9 1.8

x 10-2

0

0.02

0.06

0.08

- Σmin

kmin

8.8

9.4

x 10 -3

(d)

incr. shear

Figure 6: (Color online) Evenness control by shear. (a): Initial profile calculatedfrom an experimentally obtained dispensing profile. (b): Variation of the radialshear at the surface. (c): Film thickness for different Σmin at tmin; (−·, blue):H(R, tmin, (—, red): BD criterion). In the insert the curves are: (—,blue):H(R, tmin) and (· · ·, red): BD criterion. (d): kmin as a function of Σmin; (—,blue): kmin, (- -, red): BD criterion.

Thus, k(t) is the maximum instantaneous relative deviation from the desiredcover layer thickness. The minimum value of this quantity over time is thenthe best quality that can be achieved, kmin. The time at which this minimumis obtained is denoted tmin and kmin < 0.03 implies that the BD criterion hasbeen met.

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

4.3 Improvement by shear on the surface

4.3.1 Background and governing equations

One way to manipulate the temporal evolution of the film thickness is to applya shear stress at the film surface. This can be simulated by equations obtainedby applying the condition ∂U∗/∂y∗ = σ∗(R∗)/µ∗, where U is the radial velocityof the liquid, y is the disk-normal coordinate, σ∗ is the radial shear stress on the

12

surface of the film and µ∗ is the dynamic viscosity. Again, superscript * denotesdimensional quantities. The analysis of [1] is readily adjusted and after non-dimensionalization, the equation governing the evolution of the film thicknessreads

∂H

∂t= −

1

R

∂

∂R

(

R2H3 + Σ(R)RH2)

, (18)

where the non-dimensional shear stress on the surface is

Σ(R) =3ν∗

2µ∗ω∗2R∗2i

σ∗(RR∗

i ). (19)

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

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

4.3.2 Results with shear on the surface

Figure 6 shows the effects of shear on the surface. The initial profile, taken froman experimental measurement, is shown in figure 6 (a), and in (b) the chosendistribution of radial shear is shown.

Thickness distributions at the time tmin (i.e. the time corresponding tokmin) for different values of shear amplitude are shown in figure 6 (c). For lowshear, the decrease of the thickness towards the inner radius is spread over along radial distance, but with increasing shear the slope becomes steeper andsteeper, giving a more even thickness distribution over a large portion of thesurface. The efficiency of the manipulation is shown in figure 6 (d) where kmin

is shown as a function of Σmin. The minimum value of kmin is practically zero,indicating that completely even layers can be achieved with proper distributionsof surface shear.

4.4 Control by weak curing during spinning

Curing during spinning is inspired by Heinz et al. (2006)[14]. They imposed atemperature gradient from the inner radius and out to manipulate the viscosity.Here, a different model for the curing manipulation is used, namely that theviscosity, as a material parameter, is changed by the curing process.

4.4.1 Governing equations

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

ν(T ) =ν∗(T )

ν∗

0

, (20)

13

2 4

0

0.5

1

R

f

(a)

0 5000

0.02

0.04

0.06

0.08

I

min

(kτ)

(b)

2 40

0.005

0.01

R

H

(c)

8.8

9.4

x 10-3

2 40

0.005

0.01(d)

R

H

8.8

9.4

x 10-3

1

1 1

I I

Figure 7: Evenness control by inline curing. (a): The two demonstration dis-tributions of the cure-intensity in equation (4.4.1), smooth (—) and steep (–·).(b): kmin defined in equation (17) for the two cure-intensity distributions in(a). (c) and (d): Film thickness for different I at tmin for the smooth and steepcure-intensity distribution in (c) and (d), respectively. Curve styles and colorsin (c) and (d) are (−·, blue): H(R, tmin, (—, red): BD criterion). In the insertsthe curves are: (—,blue): H(R, tmin) and (· · ·, red): BD criterion.

where ν∗

0 is the initial viscosity.In order 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:

Dν

Dt=

∂ν

∂t+ U

∂ν

∂R= If(R), (21)

where U is the mean (in the vertical direction) of the velocity, I is a curingintensity and f(R) the radial distribution of this intensity. This model for theeffect of curing assumes that the lacquer is permanently hardened when exposedto the UV radiation and the increase of viscosity is moved with the fluid in theradial direction. However, the fact that the liquid closest to the disk travelsslower in the radial direction than liquid closer to the free surface is not takeninto account. Together with boundary and initial conditions, equations (1,4.4.1)can be solved and possibilities of inline curing studied.

The possibilities of inline curing are illustrated in figure 7. The two intensitydistributions in figure 7 (a) are compared. The initial profile is the same as

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before (see figure 6(a)). Figure 7 (b) shows kmin as a function of maximumcuring intensity I for the two distributions. The smooth distribution (the solidcurve) is shown to give values of kmin less than 2% whereas the steep distribution(dash-dot line) does not manage to reach values below 3%. It should be notedthat the smooth distribution gives a rather large range of I (200–400) for whichthe BD specification (3%) is satisfied.

The thicknesses as a function of radius at the time tmin are shown in figure7 (c) and (d) for the smooth and steep intensity distribution, respectively. Ineach graph, the different curves show increasing curing intensities as indicatedby the arrows. For high values of I the viscosity becomes large at small radii andconsequently there is a build-up of liquid there (especially with the step cure-intensity distribution) giving large values of kmin. For the most even profiles,this effect is moderate and the method might be possible to use in a real process.

5 Conclusions

Spin coating of disks has been studied aiming at producing the cover layer ofBluRay disks. A first order model of spin coating[1] has been expanded to caseswhere coating is starting off-centre. The expanded model has been verifiedagainst experiments. Thus, the assumptions behind the model are valid in thepresent case. Fundamental limitations regarding the thickness homogeneity asa function of the radius at which the lacquer layer starts have been derived andit has been shown that the layer called upon by the BD specification is difficultto produce (due to the strong limitations it puts on the initial layer) withoutmanipulation of the process. The limitations are not restricted to spin coatingof BluRay disks, but eq. 16 can be applied to any other off-centre coatingsituations, where the model of [1] is suitable. Two methods by which thesefundamental limitations can be overcome have been proposed. The first is toapply shear at the surface of the liquid film and the second is to cure the liquidwhile spinning. These two methods have been incorporated in the model andthe results show that they can be used to improve the quality of the final coverlayer.

The model for applying shear on the surface includes no additional simplifi-cations compared to [1] and it can thus be assumed that the promising resultsin figure 6 are possible to achieve in a physical setup. The model for inlinecuring is more rudimental since viscosity and its convection speed are assumedto be constant through the film in the vertical direction. Thus only qualitativesimilarity should be expected if the method is implemented in a physical setup.Nevertheless, each model should be possible to use both for further developmentand control of the process.

This work has been performed in cooperation with Alpha Sweden AB andtheir assistance, especially when it comes to the experimental part, is gratefullyacknowledged. FL has been financed by the Swedish Research Council and OTby Linne FLOW Centre.

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References

[1] A. Emslie, F. Bonner, and L. Peck, J. Appl. Phys. 29 (1958).

[2] S. Kistler and P. Schweizer, Liquid Film Coating (Chapman and Hall,1997).

[3] L. E. Stillwagon and R. G. Larson, Phys. Fluids A. 2 (1990).

[4] C. J. Lawrence, Phys. Fluids A. 2 (1990).

[5] C. J. Lawrence and W. Zhou, J. Non-Newt. Fluid Mech. 39 (1991).

[6] B. Resifeld, S. G. Bankoff, and S. Davies, J. Appl. Phys. 70 (1991a).

[7] B. Resifeld, S. G. Bankoff, and S. Davies, J. Appl. Phys. 70 (1991b).

[8] T. J. Rehg and B. G. Higgins, Phys. Fluids 31 (1988).

[9] S. Wilson, R. Hunt, and B. Duffy, J. Fluid Mech. 413 (1999).

[10] E. Rame, S. Garoff, and K. Willson, Physical Review E 70 (2004).

[11] Y.-K. Kuo and C.-G. Chao, Microelectronics Journal 37, 759 (2006).

[12] C.-M. Cheng and R.-H. Chen, Japanese Journal of Applied Physics, Part1 (Regular Papers, Short Notes & Review Papers) 43, 8028 (2004).

[13] F.-C. Chou and P.-Y. Wu, Journal of the Electrochemical Society 147, 699(2000).

[14] B. Heinz, T. Eisenhammer, M. Dubs, C. Yavaser, and T. Pfaff, OpticalData Storage Topical Meeting, 2006 pp. 117–119 (2006).

[15] B. Dandapat, P. Daripa, and P. Ray, J. Appl. Phys. 94, 4144 (2003).

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