2016
J. Phys. Chem. B 2008, 112, 2016-2023
Dynamic Instability of a Sol-Gel-Derived Thin Film S. Joon Kwon* and Jae-Gwan Park Nano Science and Technology DiVision, Korea Institute of Science and Technology, P.O. Box 131, Cheongryang, Seoul, 130-650, Korea ReceiVed: July 10, 2007; In Final Form: NoVember 9, 2007
We present a study on the dynamic instability of a sol-gel-derived (SG) thin film on a nonwettable substrate. Because of the structural instability accompanied by syneresis stress in a film deposited on the substrate, there exists a regular distribution of dewetting patterns required to relieve the in-plane stress, such as holes in the earlier stages, and droplets accompanying a regular polygonal distribution in the later stages of the dynamic instability. The characteristic length scales in each stage scaled linearly with the film thickness during the duration of dewetting. For the formation of holes during the earlier stages of rupture of the film, the dewetting velocity was analyzed with a viscous sintering theory of a SG thin film. In the earlier stages of the dynamic instability, the dewetting velocity decreases with increasing dewetting time and increases with increasing the initial film thickness, which indicates that the SG thin film behaves partially like a slipping polymer thin film. In the final times of the film rupture, the radius of the hole has a linear relationship with the film thickness, and the growth rate of the hole (dewetting velocity) is nearly constant, regardless of the film thickness. These dewetting behaviors indicate that the SG thin film in the final times of the rupture is somewhat similar to nonslipping film. From these observations, we found that the dewetting behavior of a SG thin film has ambivalent dewetting characteristics of slipping and nonslipping films and that a SG thin film is not a purely viscous film.
I. Introduction The dynamics of a thin film mainly concerns its morphological instability given by dynamic instability in the film. Among a variety of morphological instabilities, dewetting has been the subject of particular interest in recent decades.1-13 With regard to the characteristic period of the dewetting pattern, the concept of spinodal decomposition and nucleation followed by the growth of holes8 was found to explain well the scaling behavior of the wavelength with the initial thickness of the film (h0).3,4 To determine the velocity of the dewetting of polymer thin films, much research has concentrated on the dewetting rates of various kinds of experimental cases, such as highly elastic polymers,9,11 polymer thin films near the glass transition temperature,12 glassy polymers,13 viscoelastic polymers,14 solid monolayers,15 etc. Particularly, the dewetting dynamics of polymer thin films due to several kinds of structural instabilities involving Rayleigh, fingering, rim instability, etc., have been studied. Recently, researchers reported and explained that some kind of dewetting instability and its dynamics of thin films involving bilayers can originate from long-range van der Waals force,16 solid-liquid friction,17 molecular recoiling stress,18 thermal noise,19 film thickness ratio-dependent long-range dispersion force,20 and so on. Theoretically, Eggers suggested that a hydrodynamic framework can be applied to explain the dewetting of a liquid thin film when it is withdrawn from a liquid bath.21 Although both theoretical and experimental interest is still concentrated on the dewetting of the thin films, there have not been any notable studies on the dynamic instability of other kinds of thin films, such as sol-gel-derived (SG) thin films.22-24 * To whom correspondence should be addressed. E-mail: cheme@ kist.re.kr. Phone: +82-2-958-5504. Fax: +82-2-958-5489.
Recently,24 we found that the dewetting behavior of a SG thin film is similar to that of a relatively thick (h0 > 100 nm) polymer film, given the scaling properties of the wavelength with the film thickness with a power index of 1.5,6,9 The characteristic wavelength of the dewetting was also found to govern the periods of various kinds of patterns in the earlier and later stages of the dewetting. In order to conduct a more perspective analysis of the dewetting nature of a SG thin film, however, an understanding of the dynamic properties of the film is required. In this study, a theoretical analysis of the dewetting dynamics of a SG thin film on a nonwettable substrate was presented and compared with the experimental results. We found that the dewetting velocity of the hole formation resulting from the retraction of the film in the earlier stages is not closely related to the film thickness, although the SG thin film has one of dewetting properties of a nonslipping polymer thin film, in that the overall dewetting velocity decreases with increasing dewetting time. Also, the linearly scaling behavior of the final value of the radius of the holes with the film thickness could be theoretically predicted. By studying the dewetting dynamics, a better understanding of the structural and dynamic instability of a SG thin film can be obtained. II. Experimental Section In our experiments we used ZnO sol solution to fabricate the SG film. Because the synthetic process of the ZnO sol solution in a stable phase is well-known, we employed the ZnO sol solution as a model material.22-25 This ZnO sol solution was prepared according to a synthetic method using ZnO precursors in alcoholic solution, i.e., zinc acetate dehydrate (ZAD, (CH3COO)Zn‚2H2O, Aldrich) in methanol.22-24 The ZAD was provided in a state of solid powder, and the average size of the particles was 220 µm with the size distribution of
10.1021/jp0753629 CCC: $40.75 © 2008 American Chemical Society Published on Web 01/23/2008
Dynamic Instability of a Sol-Gel-Derived Thin Film 220 ( 32 µm (measured by particle size analyzer (High Performance Particle Sizer, Malvern Instrument)). Experimentally, there were no distinctive effects on the film properties given by the particle size or its distribution of the ZAD powder. The ZAD was rigorously dissolved in an anhydrous methanol at ambient temperature for 4 h, and the resulting concentration of the sol solution was 0.1-1.5 M. To stabilize the phase of the solution, we added a phase stabilizer such as monoethanolamine (MEA, C2H7NO, Aldrich), and the concentration ratio between the ZAD and MEA in the sol solution was 2-5. The as-obtained sol solution was spin coated onto a flat substrate at 3000 rpm for 30 s in order to obtain 15-250 nm thick films. The thickness was measured using field-emission scanning electron microscopy (FESEM, Hitachi S-4100, 40 kV). The substrates were prepared by spin casting a toluene solution of polystyrene (PS, Aldrich, MW ) 290 000, MW/MN < 1.09, Tg ≈ 100 °C) onto a Si (001) substrate to form 200-300 nm thick films (measured by ellipsometer (Uvisel, Horiba Jobin Yvon, Japan)). Being wettable on the surface of the substrate without PS layer coating, we adapted the PS layer in order to observe the spontaneous dewetting instability of the ZnO sol solution. Experimentally, there were no distinctive effects on the dynamic instability of the ZnO thin film given by the variation in the film thickness of the PS layer in the range of 200-300 nm. However, one can expect that there would be a notable effect that can participate in the dewetting of the SG thin film when the PS thickness is sufficiently thin in which van der Waals interaction has an important role in the dynamic instability. On the PS layer coated substrate, ZnO sol solution was nonwettable, and the spin-coated ZnO film was annealed at 90 °C for 2 h to remove the residual solvent in the PS layer. The as-prepared samples were annealed at 150 °C in a constant temperature box furnace for 1-60 min. Although the entire state of the system could not be observed in situ, the gel state of the film was confirmed by observing the cross-sectional FESEM image of the film (not shown). This image showed networks of agglomerated particulates, which also indicates the homogeneity of the gel in the direction normal to the substrate (z direction) without geometric anisotropy. The resulting structure was examined by optical microscopy, to measure the average number density and dewetting velocity of the holes and polygons observed in the film, and by scanning electron microscope (SEM, Hitachi SE-3000, 20 kV), to determine the characteristic wavelength of the resulting surface patterns and dewetting velocity. III. Results and Discussions 1. Dewetting of a SG Thin Film. Shown in parts a and b of Figure 1are the dewetting patterns in the earlier (holes) and later (droplets with polygonal distribution) stages of the dynamic instability, respectively. The dewetting begins on the whole surface of the film, and the dewetting patterns in the earlier and later stages are quite regular. This regularity indicates that the dewetting was not caused by anomalous surface defects or the field gradient of the substrate. Although the hole distribution in the earlier stages of the instability is not as perfectly uniform as that associated with the spinodal dewetting of a polymer thin film, it shows a considerable periodicity and a lack of directional order. Therefore, the concept of the critical wavelength is valid enough to explain the existence of characteristic length scales in the patterns observed during the dewetting of the SG thin film. To understand the dewetting dynamics, the characteristic wavelength of the dewetting patterns should be determined prior to the analysis of the dewetting velocity.
J. Phys. Chem. B, Vol. 112, No. 7, 2008 2017
Figure 1. (a) Dewetting patterns of a ZnO SG thin film (h0 ) 40 nm) accompanying regularly distributed holes on the PS-coated Si substrate in the earlier stages (t ) 1 min) and (b) Dewetting patterns accompanying regularly distributed droplets in conjunction with polygonal distribution on the PS-coated Si substrate in the later stage of the dynamic instability (t ) 20 min) after annealing at 150 °C. (c) Scaling relationship between the characteristic periods of the dewetting patterns such as hole-to-hole distance (λC), size of the droplets (2l), and the average length of the polygon sides (DP) and the film thickness (h0). Symbols are used to indicate the experimental data (9 for λC, ( for DP, and 2 for 2l) and the three different lines corresponding to the fitting are identified by the slopes. Error bars in each data point indicate the standard deviations.
A lateral flow in a fluidic film is described using the continuity equation7,9,24
∂h(x,t) ∂2 ) C 2P(x,t) ∂t ∂x
(1)
where h(x,t) is the local film thickness, the shape of the flow profile and the shear viscosity of the film η are absorbed into the constant C, which is a constant related to the fluidic properties of the film, and P(x,t) is the pressure across the film. In the case of a SG film, an additional factor should be considered in P(x,t), namely, the stress caused by the evaporation of the remaining solvent during the drying process. The solvent is drawn from the film, since the film is exposed to the solidvapor interface, where the interface energy is larger than that at the solid-liquid interface.22,26 Also, the film adheres to the substrate during the evaporation of the solvent. Therefore, the drying process gives rise to internal stress, σ(z). This stress in the direction of the z axis is normal to the interface (along the normal to the plane of the film) between the film and the substrate at the top of the film, which has a destabilizing effect on the layer, due to the lateral constraint imposed by the substrate. σ(z) is described in terms of the effective syneresis stress by which the spontaneous network shrinkage takes places. σ(z) can be expressed as follows24,26
2018 J. Phys. Chem. B, Vol. 112, No. 7, 2008
[ ]
[ ]
κz VE κ cosh h ηh2 σ(z) ) CNHG ,κ) h sinh(κ) DHG
( )
1/2
(2)
where CN ) (1 - 2σm)/(1 - σm), HG ) (η(1 - σm))/((1 2σm)(1 + σm)), σm is the Poisson’s ratio, VE is the volumetric evaporation rate of the solvent per unit area of the film, D is the permeability of the film, and the relaxation time, τp, is given by τp ) η/Em, where Em is the Young’s modulus of the film. In here, the relaxation time τp is for the transition of the film from an elastic to a viscous state. Therefore, one can obtain following form of σ(z)26
[ ( )]
κz τpEmVE κ cosh h ,κ) σ(z) ) (1 + σm)h sinh(κ)
[
] []
ηh2(1 - 2σm)(1 + σm) DτpEm(1 - σm)
1/2
)h
2 3D
1/2
(3)
where we have utilized typical relationship of τp ) η/Em with σm ≈ 1/3. The pressure in the film can be written as follows7,9,24
P(x,t) )
Aeff
∂2h - γP 2 + 〈σ(z)〉 6πh ∂x
(4)
3
where Aeff is the effective Hamaker constant for the van der Waals interaction, γP is the surface energy of the SG film, and the second term is responsible for the Laplace pressure. It should be noted that the average pressure resulting from the drying stress over the film, 〈σ(z)〉, was used in eq 4. This means that the stress component that is responsible for 〈σ(z)〉 is σzz because the other stress components such as σxz and σyz are not considered for the lateral flow. In here, axes x and y are for the axes parallel to the plane (normal to the axis z) [refer to Figure 2c]. 〈σ(z)〉 can be calculated using eq 3 such that24
〈σ(z)〉 )
3τpEmVE
∫0h
{( ) } 4τ E V 2 cosh{z( ) } dz ) 3D 3h
2 1/2 2h(6D)1/2 sinh h 3D
1/2
p m E
(5)
As in the case of the dewetting of polymer thin films, it can be considered that the dynamic instability is initiated by the formation of an unstable small modulation of the surface, which is given by following wave profile2-4,7,9
h(x,t) ) h0 + cos(kx) exp
(τt )
(6)
where is the amplitude, which is assumed to be much smaller than h0, k is the wave number, and τ is the characteristic time corresponding to the exponential growth of the modulation when τ > 0. Solving eq 1 with eqs 4, 5, and 6 yields the following relationship24
(Cτ)-1 ≈
(
Aeff
2πh0
4
+
)
4τPEmVE Λ 2 k - γPk4, Λ ) 2 3 h0
(7)
where the linearized form of eq 7 is based on the assumption that , h0 and ∂h/ ∂x , 1. The minimization of (Cτ)-1 with respect to k gives the critical wave number
kC )
[
]
(Aeff + 2πΛh02) 4πγPh04
Kwon and Park
1/2
(8)
for the fastest growing mode. For metalorganic SG thin films, the effect of the drying stress that determines kC is much larger than that of the van der Waals interaction, since Λ/h02 exceeds Aeff/h04 by 2 or 3 orders of magnitude.24 Then, kC ≈ (Λ/ 2γPh02)1/2, and therefore, the characteristic wavelength for the hole-to-hole distance is proportional to h0, since λC ) 2π/kC ∝ h0. Since SG films are stiff and strongly confined by the substrate due to the drying stress, we expected that their dewetting behavior would be analogous to that of high molecular weight viscoelastic polymer films, for which the relationship is λC ∝ h0n, where n is lower than 2.5,6,9 Shown in Figure 1c are the scaling behaviors of the dewetted patterns with h0 in the earlier and later stages of the dynamic instability. As shown in the double logarithmic plots, the theoretical predictions based on our theoretical analysis for the slopes of the fitting line are in accordance with the experimental results for λC (n ) 0.98 ( 0.07). The scaling behaviors of the average length of the polygon sides (DP, n ) 0.95 ( 0.07) and the droplets (2l, n ) 0.81 ( 0.06) in the later stages of the dynamic instability also show linearly scaling behaviors with h0.24 Particularly, the dependences of DP on h0 can be readily predicted using the relationship such that DP ≈ λC/x3, which is responsible for the predicted relationships DP ∝ h0, although the experimental values were somewhat greater or smaller than the predicted values of DP. For 2l, the maximum diameter of the resulting vertex drop, 2lm, is calculated as 2lm ) 2[(x3 λC2h0 sin θ)/(4(2 + cos θ)(1 cos θ)2)]1/3, which leads to the relationship: 2lm ∝ h0. Experimentally, the dewetted droplets showed a weaker dependence of the size on h0 than the predicted dependence, such that 2l ∝ h00.81(0.06, which is due to the Rayleigh instability and the existence of a series of smaller droplets between the two vertex drops. After the completion of the surface modulation, the trough in the modulation acts as a starting point for the homogeneous nucleation of the hole.3,8 Since the SG film can be assumed to be incompressible on the basis that the remaining solvent amount in the film after the spin-coating process is sufficiently small to neglect the density gradient during the dewetting, we can think that the film retracted from the nucleation point is piled up along the perimeter of the hole to form a rim. The holes meet each other when the summation of the radius (a) and the rim width (2b) of the two holes is equal to the hole-to-hole distance, λC. The resulting boundaries of the three rims leave triangular vertices. Figure 2a shows the morphological evolution of the dewetted SG thin film. By the experimental observation and the comparison with the theoretical analysis of the characteristic lengths of the dewetting patterns, we found that the distribution of the dewetted patterns was regularly distributed and that the size variation in the patterns was sufficiently small. We also found that the dewetting patterns occur on the entire surface of the SG thin film and not randomly on the surface of the substrate, since no extrinsic surface defects were included during the film preparation. Additionally, intrinsic surface defects which are dependent on atmospheric conditions can be ignored in our case due to the intrinsic surface defects are peculiar to the morphology of polymer thin film.27 The SG thin film is sufficiently compact and has very low sensitivity to the atmospheric conditions, opposed to the polymer thin films. Moreover, the SG thin film does not contain any kind of chain, and therefore, thickness-dependent nonequilibrium conformational stress by recoiling in the molecule such as polymer chain
Dynamic Instability of a Sol-Gel-Derived Thin Film
J. Phys. Chem. B, Vol. 112, No. 7, 2008 2019
Figure 2. (a) Morphological evolution of the dewetted pattern of the SG thin film (h0 ) 40 nm) resulting from annealing at 150 °C for 0-20 min. Plane SEM images were taken at different annealing times. (b) Relationship between the radius of the growing hole (a) and the width of the rim (2b) for different initial film thicknesses. The contact angle of the dewetted film on the substrate θ was assumed to be 30°. (c) Geometry of the dewetted SG film with a circular rim.
can be excluded in our analysis.18 Therefore, nucleation given by extrinsic or intrinsic defects was not considered in our analysis. If the film is sufficiently fluidic before the completion of the sol-gel conversion, the remaining rim is decayed into a truncated sphere-shaped droplet (namely, the vertex drop) that is placed on the vertex of the three holes, since it has a thermodynamically stable shape, as shown in the rightmost SEM image in Figure 2a. This evolution is similar to that of the dewetting of low molecular weight polymer thin films.2 During the growth of the holes, we could not find any rim instability that would cause rim fluctuation accompanied by irregular hole growth and, therefore, it follows that an increase in a results in an increase in 2b. To a first approximation without consideration of the asymmetry of the rim and the slippage of the film on the underlying PS film,28,29 we assumed a circular profile for the shape of the rim resulting from the force balance relation between the capillary force and the viscous force.30 The volume conservation relationship gives the following equation
2b2(a + b)
(
)
θ - cot θ ) h0(a + 2b)2 sin2 θ
(9)
where θ is the contact angle of the SG film on the substrate. As is apparent from Figure 2b, increases in h0 and a lead to an increase in 2b. For the sake of simplicity, we also approximated the shape of the hole to be that of a cylinder that has an average height of h1, which can be written such that h1 ) b/2(θ/(sin2 θ) - cot θ) [refer to Figure 2c]. 2. Dewetting Velocity of a SG Thin Film. During the growth of the holes, the dewetting velocity can be analyzed by considering the sintering process in which the densification of the SG thin film occurs.26 In the case of viscous sintering, the growth of the hole is accompanied by temporal variations in two different free energies, one of which corresponds to the lateral motion of the fluidic film over the surface of the substrate and the other to the increase in the interface energy. Because of the friction force, dissipation energy is expended for the
lateral motion of the film. According to Frenkel’s theory for the simple cylindrical model,26,31 the rate of change of the dissipation energy of the dewetted part of the film during the film flow (∂Ef/∂t) can be written as follows
(
)
(
)
∂Ef ∂r ∂z ∂r ∂z ) V(t) 2σr + σz + σz ) πa2h1 2σr ∂t ∂t ∂t ∂t ∂t
(10)
where σr and σz are the stress and r and z are the strain in the radial and z directions, respectively. It should be noted that we used the volume of the cylinder V(t) such that πa2h1 in order to consider the dissipation energy expense of the dewetted material. Because the film is incompressible, the rate of change of the volume, ∂V/∂t ) πa2h1(2∂r/∂t + ∂z/∂t) is equal to zero, which leads to following relationship
(
)
( )
∂z ∂z ∂Ef ) πa2h1 (σz - σr) ) 3πa2h1η ∂t ∂t ∂t
2
(11)
along with the strain-stress relation 3η∂z/∂t ) σz - σr. The rate of change of the interface energy (∂ES/∂t) can be written as follows
[
]
∂h1 ∂ES ∂a ∂a - 2π|S|a , |S| ) ) 2πγP 2a + (2h1 - h0) ∂t ∂t ∂t ∂t |γS - γSP - γP| (12) where |S| denotes the spreading coefficient of the SG film on the substrate, γSP is the interface energy between the film and the substrate, and γS is the surface energy of the substrate. At equilibrium, the rate of change of the total energy, ∂Ef/∂t + ∂ES/∂t, should be equal to zero13,26,31 due to energy balance, which gives the following equation 2
(∂a∂t ) + 2aγ
6h1η
P
∂h1 ∂a ∂a + (2h1 - h0)γP - |S|a ) 0 ∂t ∂t ∂t
(13)
2020 J. Phys. Chem. B, Vol. 112, No. 7, 2008
Figure 3. Time-dependent behaviors of the holes radius of the ZnO SG thin film on the PS-coated Si substrate after annealing 150 °C with different values of h0. The dotted line for each data indicates the state in which no variation in the holes radius was observed. The final values of the holes radius in the earlier stages of the dynamic instability, af, and the latest time of rupture, tf, were defined in this state.
where the rate of change of a was assumed to be solely dependent on the strain rate in the radial direction, such that ∂a/∂t 1/a ) - ∂r/∂t ) 1/2 ∂z/∂t. From eq 13 with eq 9, one can obtain the dewetting velocity, da/dt, although obtaining an analytic solution of eq 13 is difficult. By assumption that the dewetting velocity can be represented by the scaling behaviors of the displacement of the front, rear, and width of the rim in the case of polymer thin films on an adsorbed polymer or endgrafted polymer,10 we tried to determine whether there exists a distinctive time dependence such as a scaling relationship of the dewetting velocity in the case presented herein. Certainly, power laws based on the scaling relation cannot always govern the dewetting velocity of a thin film, and it was observed that there exists a logarithmic dependence in the dewetting of a viscoelastic polymer thin film.14 Although not shown here, we found that the variation in the power for the scaling relationship of each experimental data is not sufficiently small, which means the power law is not strong to be the governing time dependence for the dewetting velocity of each data. In Figure 3, time-dependent behaviors of the holes radius of the ZnO SG thin film on the PS-coated Si substrate after annealing at 150 °C with different values of h0 were shown. It should be noted that the very initial time for the holes formation (time when the holes start to emerge) for each cases with different film thicknesses were not measured and therefore were not plotted in Figure 3. Considering the relatively fast dewetting velocity of the SG thin film, time dependence analysis of the dewetting velocity for the entire dewetting duration would be varied providing that the initial holes formation time is involved. In the case of glassy polymer thin films,13 exceptionally, similar behavior of the dewetting velocity accompanying the asymptotic value of the hole radius (similar to the final value of the hole radius in our case, af) was found when the exponent, m, in the relationship, stress ∼ (strain rate)m, is smaller than 1/3. Assuredly, this similarity is limited in that the dewetting rate of the hole in glassy polymer thin films satisfies a power law. However, we could not find any distinctive scaling behavior represented by a scaling relationship in which the displacement (a) was proportional to tn in conjunction with a constant value of n. The values of power (n) for the scaling relationship for different thicknesses showed a considerable deviation from the average value, although the dewetting velocity of the films with different thicknesses
Kwon and Park commonly decreased with increasing dewetting time. This decreasing behavior of the dewetting velocity is one of dewetting characteristics of a slipping film; however, it does not prove the similarity of the dewetting nature of the SG thin film to that of a slipping film. Indeed, the nonexistence of a distinctive common time dependence (n) of a led us to conclude that the dewetting dynamics is not solely determined by the viscous dissipation energy. Moreover, the observation that the thicker the film the faster the dewetting velocity (the greater the rate of increase of the radius of the holes) in the initial times of the film rupture contrasts with the dewetting behavior of a slipping polymer thin film on an endgrafted substrate, in which the friction force is proportional to the average width of the rim.10 The increasing dewetting velocity with increasing film thickness would originate from the decreasing viscosity in the initial times of the dewetting with increasing film thickness. This decrease in the viscosity of the film would hinge on an increase in the volume fraction of the solvent vapor in the thicker film prior to the gelation of the film. Further challenge to the dewetting behavior of a SG thin film in the initial times of the dynamic instability is necessary. From these observations, one can find that a SG thin film is not a totally slipping film and that there exists an additional factor determining the dewetting dynamics of a SG thin film, apart from the friction and capillary forces during the viscous sintering process. It should also be noted that the latest time of the film rupture, tf, and final value of the holes radius, af, were indicated in Figure 3. We defined tf as the time when there is no variation in the holes radius as were shown in Figure 3 (dotted lines). In this meaning, tf can be considered as the time when the morphological transformation of the merged rim into droplets starts. At this moment, we can define af as a(tf) ) af as well as b(tf) ) bf and h1(tf) ) h1f, respectively. To determine the dewetting rate during the rupture of the film, we considered the rate in two extreme cases: the initial (short-time approximation) and the final (long-time approximation) times of the film rupture. In the initial times, the dewetting velocity da/dt can be written as follows
∂a ≈ ∂t
2h11/2(4h1 - 3h0) ∂h1 θ ∂t - cot θ h0(h1 - h0)1/2 sin2 θ
(
)
(14)
In eq 14, we utilized the volume conservation relationship given in eq 9 which can be rewritten as follows
a
(
)
θ - cot θ ) sin2 θ
[( ) ]
4h13/2(h1 - h0)1/2 1 h0
h0 h1
1/2
+1
4h13/2(h1 - h0)1/2 ≈ (15) h0
since the value of (h1 - h0)1/2 is sufficiently small. Then, eq 13 reads as follows
∂h1 + (h1 - h0)1/2(4h1 - 3h0) × ∂t h0 θ - cot θ - 2 S h13/2(h1 - h0)1/2 + h1 - h0γp 2 2 sin θ θ - cotθ ) 0 (16) 4γph1h0(h1 - h0)3/2 sin2 θ
12ηh13/2(4h1 - 3h0)2
[( ) (
) ||
(
)
]
However, we could not find an exact temporal profile of η,
Dynamic Instability of a Sol-Gel-Derived Thin Film
J. Phys. Chem. B, Vol. 112, No. 7, 2008 2021
since η varies rapidly in the initial times. Therefore, no analytic form of the dewetting velocity in the initial times could be obtained. In the final times of the rupture, the volume conservation relationship can be rewritten as follows
(
a
)
θ - cot θ ) sin2 θ 4h12 h0 h0 1- + 1h0 h1 h1
[
( )] 1/2
≈
2h1 (4h1 - 3h0) (17) h0
since h1 is much greater than h0. Therefore, da/dt can be written as follows
∂a ≈ ∂t
16h1 - 6h0 16h1 ∂h1 ∂h1 ≈ θ ∂t θ ∂t - cot θ - cot θ h0 h0 2 2 sin θ sin θ (18)
(
)
(
)
Then eq 13 can be written as follows
∂h1 ≈ |S| ∂t
12η
(19)
and the simplified form of dh1/dt in eq 19 leads to h1f - h1(t) ) |S|/12η(tf - t) with a boundary condition such that h1(tf) ) h1f. When t ) tf, we could find the relationship 2(af + bf) ) λC, which leads to the following relationship
af )
[(
λC λC
)
)} ]
{ ( )
1/2 θ θ - cot θ - 2h0 - h0λC - cot θ 2 2 sin θ sin θ θ 2λC - cot θ - 2h0 sin2 θ λC (20) ≈ 2
(
It should be noted that the parameter af was responsible for the average value of radius of different holes, which meet together at one value of tf. Other holes radius which did not meet together (or did not form a polygonal boundary) at tf were not involved in the measurement in our experiment. Solving eq 18 with eq 19 gives the temporal profile of a in the final times of the rupture of the film, as follows
af - a(t) )
4|S|h1f (tf - t) θ cot θ 3ηh0 sin2 θ
(
)
(21)
where h1f can be calculated using the relationship of h1f ) bf/ 2(θ/(sin2 θ) - cot θ) and eq 21. Interestingly, the slope of eq 21, 4|S|h1f/3ηh0(θ/(sin2 θ) - cot θ), is independent of h0 since h1f ∝ h0 based on the fact bf ∝ h0 given by following relationship
bf )
)} ] {( ) } ( )[ {( )( ) } ] ( ){ ( ) }
λC - 2af ) 2
[ { (
3λC h0 + h0λC
θ - cot θ sin2 θ
θ - cot θ - h0 2 λC sin2 θ 2γp 1/2 2γp 1/2 θ 3 + 3π cot θ 2π Λ Λ sin2 θ 1/2 2γp θ - cot θ 4π -3 2 Λ sin θ
1/2
)
1/2
h0 (22)
This independence indicates that the dewetting velocity in the final times of the rupture of the film represented by the hole growth rate is independent of h0. In the final times of the rupture of the film, eq 20 gives rise to a linear relationship between af and h0 such that af ≈ π(2γp/Λ)1/2h0. As was shown in Figure 4, this linear relationship indicates that the hole radius in the final times is governed by the linear relationship λC ∝ h0. Although not theoretically analyzed here, the value of tf also seems to scale linearly with h0, as is apparent from the inset plot of Figure 4. To confirm the theoretical prediction that the growth rate of the hole in the final times of the rupture is independent of the film thickness (eq 20), we plotted the normalized experimental data between af - a(t) and tf - t. As shown in Figure 5, the dewetting velocity gradually commonly decreases with different film thicknesses as the film rupture proceeds (as the value of tf - t approaches to zero). A tangential line was obtained from linear fitting of the last data points for the last 30 s near the last moment of the rupture (5 data points for each film with different film thickness, near tf - t ) 0), and the slopes of the lines for each data give an average value of 0.144 ( 0.0077 µm/s. Although the standard deviation of the average value of the slopes is relatively large (∼5%), we could find that the hole growth rate in the final times of the rupture is nearly constant, which is strongly independent to the film thickness. It also indicates that the dewetting velocity in the final times is determined solely by physical properties such as the interfacial and surface free energy and viscosity. Indeed, we found that η ≈ 3.5 × 108 N s m-2 using the value of the slope of the linear fitting, and the fact that |S| ≈ 2|γp - (γpγs)1/2|, using γS ≈ 3 × 10-2 Nm-1 and γP ≈ 5 × 10-1 Nm-1 assuming θ ≈ 30°. This value of η is of the same order of magnitude as the typical value of η for a metalorganic SG thin film after its gelation (η≈ τpEm ≈ 108 N s m-2),24,26 and therefore, our analysis can certainly capture the critical characteristics of the dynamics of a SG thin film. The difference in the exact values of η could originate from the estimation errors of the physical properties of the film.24 Interestingly, the time dependence of the dewetting velocity, which is asymptotic to linear in the final times of the rupture, is similar to the dewetting dynamics of a polymer thin film on an endgrafted substrate.10 The dewetting dynamics of a polymer thin film on such an endgrafted substrate was reported to satisfy a relationship such that the dewetting velocity is proportional to t1.05(0.05 in the later stages of dewetting.10 This property of the dewetting velocity is one of dewetting characteristics of a nonslipping film; however, it does not guarantee that the dewetting nature of the SG thin film is very similar to that of a nonslipping film. The near constancy of the hole growth rate regardless of h0 supports the observation that the dewetting velocity is not affected by the rim width that is proportional to the friction force, thereby causing a decrease in the dewetting velocity, due to the increase in the value of h0. In the case of the dewetting of thin water films on a highly hydrophobic polymer layer, it has been reported that the dewetting velocity
2022 J. Phys. Chem. B, Vol. 112, No. 7, 2008
Kwon and Park velocity in the final times of the rupture of the film also indicates that a SG thin film is not a purely viscous film. At present, the underlying physics cannot explain why a SG thin film has ambivalent dewetting behaviors. Further studies considering the spreading coefficient of the SG thin film on different kinds of underlying substrates and under different annealing temperatures are required. IV. Summary
Figure 4. Relationship of af and h0, which shows a linear scaling behavior. The right inset is for a relationship between tf and h0.
In summary, the dewetting dynamics of a SG thin film was studied. The linearly scaling behavior of the characteristic wavelength of the dewetting patterns was observed and theoretically analyzed. This wavelength governs periods of dewetting patterns in the earlier and later stages of the dynamic morphological instability. The dewetting velocity was analyzed by considering the viscous sintering process in which the SG thin film’s densification occurs. In the final times of the film’s rupture, we found that the value of the radius of the holes scales linearly with the initial film thickness. The dewetting velocity was found to decrease as the dewetting proceeds, which is one of dewetting properties of the slipping dewetting mode. We also observed that the dewetting rate of the holes due to film rupture is nearly constant, which is independent of the film thickness, in the final times of the rupture of the film. The dewetting behaviors were observed to have ambivalent properties. On the one hand, a decreasing behavior of the dewetting velocity was observed with increasing the dewetting time, which is one of the dewetting characteristics of a slipping film. On the other hand, the near constant growth rate (dewetting velocity) of the holes in the final times of the rupture of the film, regardless of the initial film thickness, together with the observation that the thicker the film, the faster the dewetting velocity (the greater increasing rate of the radius of holes) in the initial times of the film’s rupture, which is one of dewetting characteristics of a slipping film. These two contrasting dewetting behaviors show that a SG thin film is not a purely viscous film but has twofaced dewetting properties of the slipping and nonslipping dewetting modes. References and Notes
Figure 5. Relationship between the value of af - a(t) and tf - t with different values of h0 after annealing 150 °C. The dotted line indicates an average fitting line with the average slope of 0.144 ( 0.0077 µm/s for the growth rate of a(t) in the final times of the film rupture. Upper left inset is the expanded part of the relationship between af - a(t) and tf - t showing the linear fitting of the last data points for the last 30 s near the latest time of film rupture (near tf - t ) 0). Error bars in each data point indicate the standard deviations.
of the water film is constant due to flows in the substrate (lower polymer layer) rather than due to viscous losses and wave resistance.32 In contrast to the water-polymer bilayer, the SG thin film is strongly confined by the solid substrate, and therefore, the frictional dissipation energy (∂Ef) gives the critical effect in the determination of the dewetting velocity. By consideration of the decreasing behavior of the dewetting velocity together with the observation that the thicker the film, the faster the dewetting velocity (the greater the rate of increase of the radius of the holes) in the initial times of the film rupture, we can draw a conclusion that the dewetting of a SG thin film has two-faced dewetting properties of the slipping and nonslipping dewetting modes. The near constant value of the dewetting
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