Applied Mathematics and Computation 223 (2013) 76–87

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Applied Mathematics and Computation

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Extended asymptotic solutions to the spin-coating modelwith small evaporation

0096-3003/$ - see front matter � 2013 Elsevier Inc. All rights reserved.http://dx.doi.org/10.1016/j.amc.2013.07.071

⇑ Corresponding author.E-mail address: vincent.cregan@ul.ie (V. Cregan).

Vincent Cregan ⇑, Stephen B.G. O’BrienMACSI, Department of Mathematics & Statistics, University of Limerick, Limerick, Ireland

a r t i c l e i n f o

Keywords:Spin-coatingEvaporationAsymptoticsMatched asymptotic expansions

a b s t r a c t

We obtain improved asymptotic solutions to the spin-coating with small evaporationmodel. In particular, the spinning of a solute-free fluid, and a two-component solvent–sol-ute solution are studied. In addition, the non-negligible effect of the rise of solute concen-tration on the solution viscosity at the end of the spin-coating process is considered.

� 2013 Elsevier Inc. All rights reserved.

1. Introduction

Thin film technology is used extensively in the production of a diverse range of electronic and optical devices, includingintegrated circuitry [6,9], semiconductor wafers [23,28], thin film transistors [19], compact discs (CDS), digital versatile discs(DVDS) and hard disk magnetic coatings [17,24]. Thin films are materials ranging from fractions of nanometres to severalmicrometres in thickness. A number of industrial techniques have been developed for the production of thin films includingelectrodeposition and spray-coating [17].

Of the thin film synthesis methods available, spin-coating is the most conventional, commonly used procedure, and it isused widely in the production of flat wafers for integrated circuitry. The basic spin-coating process is as follows. A thin filmsuspension, consisting of solid particles suspended in a solvent, is placed on a planar substrate. Rotation of the substrate en-sues, and it is accelerated up to its final, desired rotation speed, which typically lies in the range 2000–8000 rpm [5,23]. Thefilm thins via the centrifugal drainage of the liquid film and the concentration-driven evaporation of the solvent, as shown inthe schematic in Fig. 1. Initially, the fluid suspension height decreases rapidly due to spin forces. Evaporation occurs contin-uously throughout this initial spin-off stage but its contribution to liquid thinning is negligible relative to the centrifugalforce. The duration of this initial rapid film thinning stage is short relative to the duration of the entire operation. As the pro-cess continues, viscous forces and surface tension cause a residual layer to remain on the substrate. As the layer thins, evap-oration of the solvent increases the solute concentration of the liquid leading to increased solution viscosity. This increase inviscosity causes the fluid flow to slow down which in turns leads to evaporation playing a more significant role with respectto film thinning. When the solvent is completely evaporated off, a solid thin film remains. Several authors have given de-tailed descriptions of the spin-coating process [5,27].

Various aspects of the spin-coating process have been modelled in the literature. The first notable theoretical descriptionof the flow of a nonvolatile, Newtonian, viscous liquid on a rotating, planar surface was by Emslie et al. [14]. Their key resultwas to demonstrate that if the initial film profile is uniform then the resulting evolving profile is spatially independent.Meyerhofer extended the spin-coating model to allow for the spin-coating of a two-component volatile fluid consisting ofa solvent and solute [22]. Meyerhofer’s novel approach was to theorise that the process could be divided into two time inter-vals whose durations were approximately equal. In the initial interval, he assumed that film thinning was dominated by

CircularSubstrate

LiquidFilm

Spin rate ω

l*

LiquidEvaporation

Fig. 1. Spin-coating model schematic.

Asymptotic (Leading Order)Asymptotic (Higher Order)Numerical (Exact)

0

0.2

0.4

0.6

0.8

1

Nor

mal

ised

Liq

uid

Thi

ckne

ss

10 20 30 40 50Time t* (s)

Fig. 2. Exact and asymptotic solutions for pure liquid thickness. e = 0.1.

V. Cregan, S.B.G. O’Brien / Applied Mathematics and Computation 223 (2013) 76–87 77

radial outflow considerations, and evaporation was negligible. For the second interval Meyerhofer assumed that fluid flowhas effectively ceased, and the final solid layer is obtained solely via evaporation. Meyerhofer’s approach led to good agree-ment between the approximate final film thickness and corresponding experimental findings from the spinning of photore-sists solutions. Several authors have used asymptotic methods to describe various aspects of the spin-coating process, suchas fluid flow for low and high Reynolds number [15], and the relative importance of inertial, gravitational and surface tensionforces on film thinning and planarization [19,26,31]. Other authors have considered the spin-coating model subject to sub-strate topography allowing for surface tension [29], radial dependent surface roughness [16,21], and air shear at the fluidsurface [17]. The additional complexity of non-Newtonian fluids has also been studied. One significant difference betweenNewtonian and non-Newtonian liquids is that the latter typically exhibit nonuniform liquid configurations [1,10,18].

Walker et al.’s study of the production of thin layers of paint in the 1920’s was amongst the first experimental treatment’sof the spin-coating process [30]. Three decades later, Kleinschmidt documented obtaining uniform films of asphalt as thin as12lm via careful control of the fluid viscosity and spinning at rates of up to 4000 rpm [20]. Numerous authors have reportedthat the final solid film thickness decreases as the substrate angular velocity increases [5,7,22]. Daughton and Givens dem-onstrated experimentally that the initial dispense volume, dispense speed and spin acceleration do not have an influence onthe final film thickness, and that the final film characteristics depend strongly on fluid viscosity, final spin speed and finalspin time [12]. The experimental analysis of Franke and Birnie on composite solutions containing fibres, with typical lengthsand diameters of 0.5 mm and 10 lm, respectively, showed that the fibres orientate themselves with a general radial direc-tion during the spin-coating process [3]. In conjunction with optical interferometry and simple linear regression Birnie andManley estimated flow and evaporation rates for a range of solvents [4]. Their results were in accordance with previousapproximations for the flow and evaporation rates used in the literature. Britten and Thomas highlighted the radial nonuni-formity which occurs during the spinning of a shear thinning alumina sol gel [8].

Asymptotic (Leading Order)Asymptotic (Higher Order)Numerical (Exact)

0

0.2

0.4

0.6

0.8

1

Nor

mal

ised

Sol

vent

Thi

ckne

ss

10 20 30 40 50Time t*

Asymptotic (Leading Order)Asymptotic (Higher Order)Numerical (Exact)

0.5

0.6

0.7

0.8

0.9

1

Nor

mal

ised

Sol

ute

Thi

ckne

ss

0 10 20 30 40 50Time t*

Fig. 3. Exact and asymptotic solutions for (a) solvent and (b) solute thicknesses. e = 0.1, d = 0.01, a = 1 and d � 0.05.

78 V. Cregan, S.B.G. O’Brien / Applied Mathematics and Computation 223 (2013) 76–87

The importance of thin solid films is illustrated by their widespread application in the electronics industry. Hence, accu-rate expressions for the evolution of a thin film, and the final film thicknesses are extremely desirable. We previously used aformal asymptotic approach to derive leading order time-dependent expressions for the solute and solvent thicknesses [11].The smallness of evaporation relative to centrifugal-viscous effects was the basis of the perturbation approach. The analysisdemonstrated that the process could be represented by two stages; an initial, relatively short ‘‘inner’’ stage where flow ef-fects dominate, and a relatively long ‘‘outer’’ stage where flow and evaporation are equal. Matched asymptotic expansionswas used to combine the inner and outer solutions, and these solutions improved on Meyerhofer’s seminal spin-coatingapproximations [22]. Critically, the effect of the solute concentration dependent suspension viscosity was not included atleading order.

In this paper we seek an improvement on our original asymptotic results. The extended asymptotic solutions now allowfor the solute concentration dependent suspension viscosity. We also investigate how the rise in viscosity, with increasedsolute concentration, at the end of the process affects the spin-coating process. In particular, we consider its impact onthe asymptotic approach. In Sections 2 and 3, a solute-free liquid and a solution, respectively are considered. Section 4 out-lines the effect of the viscosity increase near the end of the process.

2. Solute-free liquid with evaporation

From Meyerhofer [22], the governing spatially independent model for the spinning of a solute-free liquid is

V. Cregan, S.B.G. O’Brien / Applied Mathematics and Computation 223 (2013) 76–87 79

dl�

dt�¼ �al�3 � b; ð2:1Þ

l� ðt� ¼ 0Þ ¼ LI; ð2:2Þ

where l�ðt�Þ is the liquid thickness, t� is time, a � 2x2=3m � 109 m�2 s�1 measures the relative importance of spin effects toviscosity, x is the substrate angular velocity, m is the liquid kinematic viscosity, b � 10�7 m s�1 is the evaporation rate, andLI � 10�5 m is the initial, uniform film thickness [6,25]. In practice, the initial profile may not be uniform. However, for New-tonian liquids, regardless of the initial liquid configuration, once spinning commences the profile quickly attains a uniformthickness [14,27,28]. For the case of non-Newtonian fluids, uniformity is more difficult to achieve [1]. From the experimentalliterature the evaporation rate for solvents such as methanol, ethanol and toluene is typically of the order 0.3–5 lm s�1

[4,22,25]. The approach will also work for highly volatile solvents such as acetone, assuming that the asymptotic conditiondiscussed in Section 2.1 is satisfied (i.e. e� 1).

2.1. Nondimensionalisation

From [11], substituting the outer rescalings

l� ¼ ba

� �1=3

l; t� ¼ 1

ðab2 Þ1=3 t; ð2:3Þ

into (2.2) gives the outer equation

dldt¼ � l3 � 1; ð2:4Þ

l ð0Þ ¼ 1e1=3 ; ð2:5Þ

where e � b=ðaL3I Þ � 1. Eq. (2.5) represents a physical balance between flow and evaporation. Rescaling (2.5) via the inner

variables

l ¼ 1e1=3 L; t ¼ e2=3T; ð2:6Þ

yields the inner equation

dLdT¼ �L3 � e; ð2:7Þ

L ð0Þ ¼ 1; ð2:8Þ

which represents the initial dominance of flow relative to evaporation.

2.2. Solutions

From [11], the OðeÞ inner solution to (2.8) is

L ¼ 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2T þ 1p þ e �1

5ð2T þ 1 Þ þ 1

51

ð2T þ 1 Þ3=2

!: ð2:9Þ

Substituting the outer expansion l � l0 þ e2=3 l1 into (2.5) yields the leading order outer equation

dl0

dt¼ �l3

0 � 1; ð2:10Þ

whose solution is given in the implicit form

lnð l0 þ 1 Þ1=3

ð l20 � l0 þ 1 Þ

1=6 þ1ffiffiffi3p arctan

1ffiffiffi3p ð2 l0 � 1Þ þ t þ C0 ¼ 0; ð2:11Þ

where C0 ¼ �p=ð2ffiffiffi3pÞ was found using asymptotic matching. The O ðe2=3Þ outer equation is

dl1

dt¼ �3 l2

0 l1;

80 V. Cregan, S.B.G. O’Brien / Applied Mathematics and Computation 223 (2013) 76–87

which upon integrating yields

l1 ¼ C1 exp �3Z t

l0ðuÞ2 du� �

: ð2:12Þ

Differentiating (2.10) with respect to t and substituting the resulting expression into (2.12) leads to

l1 ¼ C1 expZ t ðl0Þuu

ðl0Þudu

� �¼ C1 exp

Z t 1ðl0Þu

dðl0Þu� �

¼ C1 ðl0Þt ¼ C1 ð l30 þ 1 Þ; ð2:13Þ

where C1 is an integration constant.The matching procedure is complicated by the fact that the leading order outer solution is only known implicitly. The

simplest approach is to develop an explicit expression for l0 ðtÞ for t � 1. Rewriting (2.9) in terms of the outer variables,the leading order behaviour is l0 ðtÞ � 1=

ffiffiffiffiffiffi2 tp

. Substituting this expression into (2.11) and using successive small t expan-sions we find that

l � 1ffiffiffiffiffiffi2 tp � 2 t

5þ e2=3

ffiffiffi2p

C1

4 t3=2 þ2C1

5

!þ . . . ð t ! 0 Þ: ð2:14Þ

Restating (2.9) in terms of the outer variables and expanding gives

l � 1ffiffiffiffiffiffi2 tp � 2 t

5� e2=3

ffiffiffi2p

8 t3=2 þ15

!: ð2:15Þ

Comparing (2.14) and (2.15), it follows that C1 ¼ �1=2, and hence, the Oðe2=3Þ outer solution is

l1 ¼ �ðl30 þ 1Þ=2: ð2:16Þ

Combining the inner and outer solutions, and subtracting the common part yields the composite solution

lcomp ¼ l0 �e2=3

2ðl3

0 þ 1Þ þ 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 t þ e2=3p þ e2=3 � 2t

5e2=3 �15þ 1

ð2e�2=3t þ 1Þ3=2

!

� 1ffiffiffiffiffiffi2 tp � 2 t

5� e2=3

ffiffiffi2p

8 t3=2 þ15

" # !: ð2:17Þ

The leading order time of drying when the liquid has been removed from the substrate is found by letting l0 ¼ 0 in (2.11)to give tf0 ¼ ð2pÞ=ð3

ffiffiffi3pÞ. We improve on this result by going to the next order. Progress is assisted if explicit expressions for

l0ðtÞ and l1ðtÞ near the end of process are known. We seek asymptotic expressions when t ¼ tf0 þ T; ðT � 1Þ. The expansionfor l0 near the end of the process is

l0ðtf0þ TÞ ¼ l0ðtf0 Þ þ T l00ðtf0

Þ þ T2

2l000ðtf0 Þ þ O ðT3Þ;

where l0ðtf0 Þ ¼ 0 is exploited to give

l0ðtf0 þ TÞ � �T þ T4

4þ O ðT6Þ ðT � 1Þ:

The related expansion for l1ðtÞ is

l1 ð tf0þ T Þ � �1

2þ T3

2þ O ðT6Þ ðT � 1Þ:

Replacing T with e2=3 tf1 in the outer solution near the end of the process yields

l0 tf0þ e2=3 tf1

� �þ e2=3 l1 ð tf0

þ e2=3 tf1Þ ¼ e2=3 tf1

þ ðe2=3 tf1 Þ

4

4þ e2=3 �1

2þ ðe

2=3 tf1 Þ3

2

!:

When the liquid is removed from the substrate the above expression must equal zero to give

0 ¼ e2=3 �tf1� 1

2

� �þ oðe2=3Þ;

and it follows immediately that tf1 ¼ �1=2. Thus, for the current parameters, the dimensionless time of drying is

tf �2p

3ffiffiffi3p � e2=3

2� 1:101; ð2:18Þ

which is identical to the numerical (exact) time of drying. We note that the leading order drying time is 1.21.

V. Cregan, S.B.G. O’Brien / Applied Mathematics and Computation 223 (2013) 76–87 81

3. Spin-coating of a solution

In practical situations, spin-coating typically involves the spinning of a solution which comprises a solvent (volatile) andsolute (nonvolatile) component with the goal being to apply a uniform solid coating to the planar substrate. The solute ismixed with a solvent as it is easier to control the final thickness of the solid film coating. As in the case of a solute-free liquid,once spinning ensues, the solution thickness quickly levels out. Therefore the overall process is only weakly dependent onthe initial solution thickness [14]. When the optimal configuration is obtained, spinning is stopped and the remaining sol-vent evaporates off. When all the solvent has been completely removed a solid film remains.

The presence of solute in the solvent complicates the physics of the spin-coating model. Initially, the solvent concentra-tion is much greater than the solute concentration. As the process progresses, the solution is removed via flow, while thesolvent is removed from the solution by evaporation. Near the end of the process most of the solvent has been evaporatedoff. Consequently, the solute concentration increases significantly which affects the solution viscosity and hence, the rate offluid thinning [9].

Meyerhofer’s classical solution spin-coating model [22] is

dL�

dt�¼ � a

1 þ N S�

S� þ L�

� � L� S� þ L�ð Þ2 � b; L� ð0Þ ¼ LI; ð3:1Þ

dS�

dt�¼ � a

1 þ N S�

S� þ L�

� � S� S� þ L�ð Þ2; S� ð0Þ ¼ SI; ð3:2Þ

where L� is the solvent thickness, S� is the solute thickness, and LI � 10�5m and SI � 10�7m are the initial solvent and solutethicknesses, respectively. The kinematic viscosity, m� ¼ m� ðcÞ, is an increasing function of the solute concentration c. Assum-ing that S� � L� (which is true for the majority of the spin-coating process), the solution kinematic viscosity is written as

m� ðcÞ ¼ ml 1 þ NS�

S� þ L�

� �� �; � ml 1 þ a

S�

L�þ O

S�

L�

� �2 ! !

; ð3:3Þ

where ml is the viscosity of the solution at the beginning of the process, N is an arbitrary function of the solute concentration,and a is an Oð1Þ constant whose value depends on the solution. The viscosity ml is often taken to be the viscosity of the sol-vent in the absence of the solute. We assume that the viscosity of the solution follows [13] where Einstein deduced that theviscosity of a dilute suspension of spheres, of solids concentration c, could be expressed as m� ¼ mlð1 þ 2:5cÞ. Einstein’s resultstrictly only applies to a suspension. However, the transition from a suspension to a solution is a continuous one and Einsteinhimself did not hesitate to apply his results to solutions of cane sugar in water. The results presented here are valid for anysolution in which the solution viscosity can be written as a Taylor series in terms of the concentration provided the seriescoefficients do not interfere with the asymptotic structure.

3.1. Nondimensionalisation

From [11] the outer variables are

L� ¼ ba

� �1=3

l; S� ¼ dba

� �1=3

s; t� ¼ 1

ðab2Þ1=3 t; ð3:4Þ

where d ¼ SI=LI � 1. Substituting (3.4) into (3.1) and (3.2) leads to

dldt¼ � 1

1 þ N d sd sþ l

� � l ðds þ l Þ2 � 1; l ð0Þ ¼ 1e1=3 ; ð3:5Þ

dsdt¼ � 1

1 þ N d sd sþ l

� � s ðds þ l Þ2; s ð0Þ ¼ 1e1=3 ; ð3:6Þ

where e ¼ b=ðaL3I Þ � 1. The asymptotic approach is simplified by formulating a relationship between e2=3 and d which reflects

the numerical relationship between the physical parameters. This approach permits us to reduce the two parameter problem toa one parameter model. The exact nature of the relationship between e2=3 and d depends on the particular spin-coating solutionunder consideration. We choose the distinguished limit d � de2=3 (where d ¼ Oð1Þ) which ensures that inclusion of the smallinitial concentration is included in the model ahead of small evaporation effects. Hence, (3.5) and (3.6) become

dldt¼ �l3 � ð2 � a Þds l2 e2=3 � 1 þ O ðe4=3Þ; l ð0Þ ¼ 1

e1=3 ; ð3:7Þ

dsdt¼ �s l2 � ð2 � aÞds2 le2=3 þ O ðe4=3Þ; s ð0Þ ¼ 1

e1=3 : ð3:8Þ

82 V. Cregan, S.B.G. O’Brien / Applied Mathematics and Computation 223 (2013) 76–87

From an asymptotic standpoint the outer rescaling is not an appropriate representation of the initial conditions. As in thecase of the solute-free liquid, we anticipate that the first two terms in each of the above equations dominate near the start ofthe process.

To reflect the initial dominance of flow effects relative to evaporation, we rescale (3.5) and (3.6) via the inner variables

l ¼ 1e1=3 L; s ¼ 1

e1=3 S; t ¼ e2=3 T; ð3:9Þ

to give

dLdT¼ �L3 � ð2 � a ÞdSL2 e2=3 � e þ O ðe4=3Þ; L ð0Þ ¼ 1; ð3:10Þ

dSdT¼ �SL2 � ð2 � a ÞdS2 Le2=3 þ O ðe4=3Þ; S ð0Þ ¼ 1: ð3:11Þ

3.2. Solutions

We seek inner expansions for the solvent and solute thicknesses of the form

L � L0 þ e2=3 L1 þ eL2; S � S0 þ e2=3 S1 þ eS2 :

The leading order problems are

dL0

dT¼ �L3

0; L0 ð0Þ ¼ 1; ð3:12Þ

dS0

dT¼ �S0 L2

0; S0 ð0Þ ¼ 1; ð3:13Þ

with solutions

L0 ¼1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2T þ 1p ; S0 ¼

1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2T þ 1p : ð3:14Þ

The Oðe2=3Þ systems are

dL1

dTþ 3L2

0 L1 ¼ �dð2 � a ÞS0 L20; L1 ð0Þ ¼ 0; ð3:15Þ

dS1

dTþ S1 L2

0 ¼ �2S0 L0 L1 � dð2 � aÞS20 L0; S1 ð0Þ ¼ 0; ð3:16Þ

have solutions

L1 ¼d ða � 2 ÞTð2T þ 1 Þ3=2 ; S1 ¼

d ða � 2 ÞTð2T þ 1 Þ3=2 : ð3:17Þ

Finally, the OðeÞ problems

dL2

dT¼ �3L2

0 L2 � 1; L2 ð0Þ ¼ 0; ð3:18Þ

dS2

dT¼ �2S0 L0 L2 � S2 L2

0; S2 ð0Þ ¼ 0; ð3:19Þ

have solutions

L2 ¼ �2T5� 1

5þ 1

5 ð2T þ 1 Þ3=2 ; S2 ¼4T15þ 2

15� 1

3 ð2T þ 1 Þ1=2 þ1

5 ð2T þ 1 Þ3=2 : ð3:20Þ

We define the outer asymptotic expansions for the solvent and solute thicknesses to be

l � l0 þ e2=3 l1; s � s0 þ e2=3 s1:

The leading order outer equations are

dl0

dt¼ �l3

0 � 1;ds0

dt¼ �s0 l2

0;

with solutions

lnðl0 þ 1 Þ1=3

ðl20 � l0 þ 1Þ1=6

0@

1Aþ 1ffiffiffi

3p arctan

1ffiffiffi3p ð2 l0 � 1Þ � t þ C0 ¼ 0; s0 ¼ C1 ð l3

0 þ 1 Þ1=3; ð3:21Þ

V. Cregan, S.B.G. O’Brien / Applied Mathematics and Computation 223 (2013) 76–87 83

where C0 ¼ �p=ð2ffiffiffi3pÞ and C1 ¼ 1. The Oðe2=3Þ problems

dl1dtþ 3 l2

0 l1 ¼ �C1 d ð2 � a Þ ð l30 þ 1 Þ

1=3l20; ð3:22Þ

ds1

dtþ s1l20 ¼ �2C1ðl3

0 þ 1Þ1=3

l0 �C1dð2� aÞðl3

0 þ 1Þ1=3

2þ C2ðl3

0 þ 1Þ !

� C21dð2� aÞl0ðl3

0 þ 1Þ2=3; ð3:23Þ

have solutions

l1 ¼C1 � d ð2 � aÞ ð l3

0 þ 1 Þ1=3

2þ C2 ð l3

0 þ 1 Þ; s1 ¼ ðC1 C2 l20 þ C3 Þ ð l30 þ 1 Þ1=3; ð3:24Þ

where C2 and C3 are integration constants.

3.3. Asymptotic matching

To proceed to higher orders in e, the easiest approach again is to obtain explicit expressions for l0 ðtÞ and s0 ðtÞ for t � 1.Using (2.14) and (3.21) we find

s0 ðtÞ �1ffiffiffiffiffiffi2 tp þ 4 t

15þ 13

ffiffiffi2p

t5=3

450þ . . . ðt ! 0Þ: ð3:25Þ

Substituting the expressions for l0 and s0 into (3.24) leads to

l1 ðtÞ �C2

2ffiffiffi2p

t3=2� dffiffiffiffiffiffi

2 tp þ da

2ffiffiffiffiffiffi2 tp þ 2C2

5þ . . . ðt ! 0Þ;

s1 ðtÞ �C2

2ffiffiffi2p

t3=2� C3ffiffiffiffiffiffi

2 tp � 4C2

15þ . . . ðt ! 0Þ:

Thus, the full closed form outer expressions for t � 1 are

l � 1ffiffiffiffiffiffi2 tp � 2 t

5� 3t5=3

25ffiffiffi2p þ � � � þ e2=3 C2

2ffiffiffi2p

t3=2þ dð2� aÞ

2ffiffiffiffiffi2tp � 1

5

� �; ð3:26Þ

s � 1ffiffiffiffiffi2tp þ 4t

15þ 13

ffiffiffi2p

t5=3

450þ � � � þ e2=3 C2

2ffiffiffi2p

t3=2þ C3ffiffiffiffiffi

2tp � 4C2

15

� �: ð3:27Þ

To employ the Van Dyke matching scheme the full inner solutions are restated in terms of the outer variables, and uponexpanding for small e we obtain

l ¼ 1ffiffiffiffiffiffi2 tp � 2 t

5þ e2=3 � 1

4ffiffiffi2p

t3=2þ d ða � 2 Þ

2ffiffiffiffiffiffi2 tp � 1

5

� �; ð3:28Þ

s ¼ 1ffiffiffiffiffiffi2 tp þ 4 t

15þ e2=3 � 1

4ffiffiffi2p

t3=2þ d ða � 2 Þ

2ffiffiffiffiffiffi2 tp þ 2

15

� �: ð3:29Þ

Comparing (3.26) and (3.27), with (3.28) and (3.29), respectively, we require C2 ¼ �1=2 and C3 ¼ d ða � 2 Þ=2. In the usualway composite solutions are formed by combining the respective inner and outer solutions, and subtracting the relevantcommon parts to give

lcomp ¼ l0ðtÞ þ1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2t þ e2=3p � 1ffiffiffiffiffi

2tp

þ e2=3 �12

dð2� aÞðl0ðtÞ3 þ 1 Þ1=3� ðl0ðtÞ3 þ 1Þ

2þ dða� 2Þtð2t þ e2=3Þ3=2 þ 1

4ffiffiffi2p

t3=2� dða� 2 Þ

2ffiffiffiffiffi2tp

" #þ OðeÞ; ð3:30Þ

scomp ¼ ðl0ðtÞ3 þ 1Þ1=3þ 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2t þ e2=3p � 1ffiffiffiffiffi

2tp

þ e2=3 ðl0ðtÞ3 þ 1Þ1=3 d

2ða� 2Þ � l0ðtÞ2

2

!þ dða� 2Þtð2t þ e2=3Þ3=2 þ 1

4ffiffiffi2p

t3=2� dða� 2Þ

2ffiffiffiffiffi2tp

" #þ OðeÞ: ð3:31Þ

A method analogous to that in Section 2 is used to show that the dimensionless time when the solvent is removed from thesubstrate is

Fig. 4. Comparisons of exact and higher order asymptotic solute solutions for e = 0.1, a = 1, and for varying d and d. (a) d = 0.02, d � 0.1, and (b) d = 0.005,d � 0.02.

1

1.5

2

2.5

3

3.5

Nor

mal

ised

Vis

cosi

ty

0.2 0.4 0.6 0.8 1

Time t* (s)

Fig. 5. Dimensionless O (d) solute concentration dependent viscosity (4.1) for a = 1 and d = 0.01.

84 V. Cregan, S.B.G. O’Brien / Applied Mathematics and Computation 223 (2013) 76–87

V. Cregan, S.B.G. O’Brien / Applied Mathematics and Computation 223 (2013) 76–87 85

tf ¼2p

3ffiffiffi3p þ e2=3 �1

2� d ð2 � aÞ

2

� �� 1:1: ð3:32Þ

We note that the corresponding numerical (exact) and leading order drying times are 1.1 and 1.21, respectively.A brief numerical study of the solvent–solute system for varying d is given in Fig. 4. The higher order asymptotic solute

solution is in excellent agreement with the corresponding numerical solution, thus justifying the use of the distinguishedlimit d � de2=3 to simplify the analysis. This approach is applicable for all two-component solutions whose parameter valuesreflect this relationship in a numerical sense.

4. Solution with varying viscosity

Thus far the effects of variation in viscosity can be neglected despite the inclusion of a variable viscosity term in (3.1). It isphysically clear that if the process commences with a dilute solution, the initial solution viscosity will be similar to the vis-cosity of the associated solute-free liquid. Near the end of the process, the solute concentration becomes large and as a resultthe solution becomes thick and highly viscous [2,9,22]. Fig. 5.

4.1. Einstein formula for viscosity

For small identical rigid spheres suspended in a Newtonian liquid, the effective dynamic viscosity for small concentra-tions is given by the Einstein viscosity formula [13]

m ðcÞ ¼ ml ð1 þ 2:5cÞ:

In terms of the outer variables, the solution kinematic viscosity is

ml 1 þ Nds

ds þ l

� �� �¼ ml 1 þ ad

slþ O ðd2Þ

� �; ð4:1Þ

where a ¼ O ð1Þ (or at least ad� 1). Hence, at leading order the viscosity is constant at its initial value which in practice maybe the liquid viscosity or the initial viscosity of the solution.

To study the importance of the increase in viscosity near the end of the process (3.5) and (3.6) are rescaled via the endlayer variables t ¼ tf � ds; l ¼ dk; s ¼ r to obtain

dkds¼ 1

1 þ N rrþ k

� � dk ðdr þ dk Þ2 þ 1;drds¼ 1

1 þ N rrþ k

� � dr ðdr þ dk Þ2: ð4:2Þ

Given that the end layer viscosity is

ml 1 þ adr

dr þ dk

� �� �¼ ml 1 þ a

rr þ k

� �; ð4:3Þ

it would appear that viscosity is not constant at the end of process. However, from (4.2), in the end layer no flow takes placeat leading order so the increase in viscosity has no impact on fluid flow. Hence, to leading order the increase in viscosity withincreased solute concentration in the end layer may be ignored. This is an important consideration from a practical perspec-tive and in general is not recognised by the coating community.

4.2. Special case of polymer solutions ða 1Þ

There exists a class of coating liquids for which a 1, and for such solutions (depending on the value of a), the constantviscosity approximation may not be valid. Suppose the viscosity of such a solution is given in terms of the outer variables by

ml 1 þ ads

ds þ l

� �b !

; ð4:4Þ

where a 1 [22]. For instance, if a ¼ oðd�bÞ, the above approach is still valid in that ml � 1 until the end layer has beenreached though the error in the approximation must be investigated in each case. A worst case scenario occurs whena ¼ jd�b and j; b ¼ O ð1Þ. The behaviour of (4.4) in the asymptotic regions is now considered.

4.2.1. Inner layerExploiting SI � LI the viscosity at the beginning of the process is

ml 1 þ jd�b SI

SI þ LI

� �b !

� ml 1 þ jd�b SI

LI

� �b !

¼ ml ð1 þ j Þ;

86 V. Cregan, S.B.G. O’Brien / Applied Mathematics and Computation 223 (2013) 76–87

which implies that the value of j has a significant effect on the solution viscosity. In the inner layer at leading orderSðTÞ ¼ LðTÞ, and thus the viscosity becomes

Fig. 6.Parame

ml 1 þ jd�b dSdS þ L

� �b !

� ml 1 þ jSL

� �b !

¼ ml ð1 þ j Þ: ð4:5Þ

Hence, in the inner layer the solution viscosity stays approximately at its initial value despite variation in S and L. Signifi-cantly, even with variation in viscosity, the inner equations can still be solved as in Section 2.

4.2.2. Outer layerIn the outer layer the solution viscosity is

ml 1 þ jd�b dsd s þ l

� �b !

� ml 1 þ jsl

� �b� �

ð4:6Þ

and thus it is clear that the viscosity depends strongly on s=l which increases from its initial value of one to a value of O ðd�1Þ.Hence, a knowledge of the variation with viscosity with concentration is required to model the flow. The presence of the s=lterm in (4.6) makes the outer equations much more difficult to solve. When (4.6) is incorporated into the outer equations,non-linear terms in l and s arise and an appropriate numerical scheme is required to solve the resulting equations. Finally, inthe end layer the variation of viscosity with concentration has no impact on the flow since there is no flow term at leadingorder.

5. Summary

The asymptotic analysis used in [11] was extended to construct higher order solutions to the spin-coating with smallevaporation model. The concentration dependent viscosity at the end of process, and its impact on the asymptotic approachwas investigated.

For the spinning of a solute-free liquid, with initial thickness LI , the inclusion of the higher order correction term in theouter asymptotic expansion improves on the original asymptotic result. Fig. 2 demonstrates that the numerical (exact) andextended asymptotic solutions are indistinguishable. In addition, the leading order asymptotic solution overestimated thetime of drying, and this is now corrected by the inclusion of the higher order term. The process concludes when all the liquidhas been expelled from the rotating substrate at

t�f ¼2p

3ffiffiffi3p � e2=3

2

� �1

a1=3 b2=3 � 51s;

where we recall that e � b=ðaL3I Þ ¼ 0:1;a � 2x2=3m � 109m�2s�1, and b � 10�7ms�1 is the evaporation rate. The inclusion of

the O ðe2=3Þ term leads to a reduction in the leading order time of drying which is to be expected as evaporation is not presentin the leading order problem. We observe that care must be taken with the singular limit e ¼ 0 when the model scales areinvalid. Physically, this limit corresponds to evaporation being equal to zero and in which case the time to drying is infinite.

Solute-free higher order asymptotic solvent expression compared to experimental results of Mokarian–Tabari et al. for spin-coating of toluene [25].ters values are a � 4:5 1010m�2 s�1, b � 0:5 10�6 m s�1; L0 � 5 10�6 m, and e � 0:05.

V. Cregan, S.B.G. O’Brien / Applied Mathematics and Computation 223 (2013) 76–87 87

Fig. 6 shows good agreement between the higher order asymptotic solvent expression and the experimental findings of Mok-arian-Tabari et al. for the spin-coating of toluene [25].

For the spinning of a solution where the initial solvent and solute heights are LI and SI both extended asymptotic solutionsimproved on the previous results. The numerical (exact) and extended asymptotic solutions in Fig. 3 are in excellent agree-ment. The extended asymptotic solutions accurately predict the time of drying and, critically, the final solid film thickness.The dimensional time of drying is

t�f ¼2p

3ffiffiffi3p þ e2=3 �1

2� d ð2 � a Þ

2

� �1

a1=3 b2=3 � 51 s;

where d � SI=LI ¼ 0:01; d � d=e2=3 � 0:05 and a is an Oð1Þ parameter resulting from the expansion of the solute concentrationdependent viscosity. Of practical interest is the final, dimensional, solute thickness which is given by

S�f ¼ 1 þ e2=3 d ða� 2Þ2

� � SI

LI

ba

� �1=3

� 0:46 10�7 m: ð5:1Þ

Inspection of (5.1) suggests that the final dimensional solute thickness is proportional to the product of the initial solute sol-vent ratio and the intrinsic length scale ðb=aÞ1=3.

Acknowledgments

We gratefully acknowledge the financial support of the Mathematics Applications Consortium for Science and Industry(MACSI, http://www.macsi.ul.ie) supported by a Science Foundation Ireland mathematics Investigator Award 12/IA/1683.

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