APPLIED PHYSICS LETTERS
VOLUME 83, NUMBER 17
27 OCTOBER 2003
Instability and dynamics of thin slipping films Ashutosh Sharmaa) Department of Chemical Engineering, Indian Institute of Technology, Kanpur-208016, India
Kajari Kargupta Department of Chemical Engineering, Jadavpur University, Kolkata-700032, India
共Received 16 May 2003; accepted 22 August 2003兲 The linear stability analysis of the full Navier–Stokes equations shows that the surface instability and dynamics of thin liquid films are profoundly altered by the presence of slippage on the substrate. For example, the exponents for the length scale ( m ⬀h n0 ; h 0 is film thickness兲 and time scale of instability (t r ⬀h m 0 ) change nonmonotonically with slippage 关for van der Waals force induced instability, n苸(1.25,2), m苸(3,6)]. Slippage always encourages faster rupture and can greatly reduce the number density of holes for moderate to strong slip. Thus, any interpretation of thin film experiments, including determination of intermolecular forces from the length and time scales, needs to account for the possibility of slippage. © 2003 American Institute of Physics. 关DOI: 10.1063/1.1618376兴
Although ultrathin 共⬍100 nm兲 liquid films on solid substrates have attracted widespread interest in recent years, the current theoretical understanding of thin film instability, dynamics, dewetting is limited largely to the films that satisfy the Newtonian condition of no-slip on their substrates.1–3 However, slippage or weak adherence is evidenced in a variety of settings: in chain entangled high molecular weight polymer melts,4 –14 on autophobic grafted chains,8 –10,14 in superfluids,15 in confined films,16 on smooth nonwettable and hydrophobic substrates17–20 and in the presence of nanobubbles21 on substrates. The important role of slippage in the macroscopic contact line movement is already well known,4,6,8,9,10,14,22 but the behavior of highly confined 共⬍100 nm兲 films should be even more dramatically affected by slippage since the slip length, b, can be easily much larger than the film thickness.10,12,13,17–20 Experimental values of slip length vary from a microscopic length 共weak slip兲 comparable to film thickness to a macroscopic slip (b ⬃100– 1500 m or larger兲.12,13 Recent experiments with thin films of polymers23–25 show a substantially different dependence of the lengthscale of instability on the film thickness ( m ⬀h n with n⬃1 to 1.6兲 compared to the prediction for the nonslipping films (n ⫽2). The only existing theoretical studies7,26 of slippage in thin films concluded that the instability lengthscale is completely unaffected by slippage. Thus, the thin film experiments are invariably interpreted within the framework of noslip hydrodynamics, and it becomes tempting to assign the deviations from the theory to the ill-defined factors such as complex intermolecular forces, rheology, and heterogeneities. However, the invariance of instability length scale with slip, if it was true, would be at odds with the well known results27–30 on the stability of ‘‘free’’ foam like films that are hydrodynamically equivalent to the case of infinite slip due to zero shear at their midplane, but show a very different wavelength of instability. a兲
Author to whom correspondence should be addressed; electronic mail:
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In what follows, we consider the effects of slippage on the stability, dynamics, and morphology of thin films in the entire spectrum from the no-slip (b⫽0) to moderate slip (bⰇh), and on to the case of strong slip (b→⬁). Each regime shows different dependence of length scale and time scale of instability on the thin film potential and film thickness. For a Newtonian film on a solid substrate (x – y plane兲, the motion of fluid is described by the continuity and Navier–Stokes equations: ⵜ• ⫽0; ( t ⫹ •ⵜ )⫽ ⵜ 2 ⫺ⵜp⫺ⵜ ; where subscripts denote differentiation, (u,w) is the velocity vector, p is pressure and ⫺ (h) denotes the disjoining pressure engendered by the long-range intermolecular interactions. , , and ␥ are the viscosity, density, and interfacial tension, respectively. For a slipping film, the boundary conditions are: impermeability and shear stress balance at the solid boundary,29 z⫽0: w⫽0; u z ⫺b ⫺1 u⫽0. The slip length, b can also be interpreted in terms of a surface friction coefficient, s as, b⫽ / s . 4 –7 At the liquidgas interface, z⫽h(x,t), the zero shear stress and pressure jump conditions are given, respectively, by:2,26 –30 (u z ⫹w x ) ⫻(1⫺h 2x )⫹2h x (w z ⫺u x )⫽0 and 2 关 (1⫺h 2x )w z ⫺h x (u z ⫹w x ) 兴 (1⫹h 2x ) ⫺1 ⫺ p⫺ ␥ h xx (1⫹h x ) ⫺3/2⫽0. The kinematic condition at the free film surface, z⫽h(x,t) is: h t ⫹uh x ⫽w. Linear stability analysis for this system is carried out by introducing infinitesimal perturbations around the base state (u⫽w⫽0,h⫽h 0 ,p⫽ p 0 , 兩 h⫽h0 ⫽ 0 ): 关 u,w,p⫺p 0 ,h ˆ (z),pˆ (z),, h0 兴 exp(kx⫹t). The ⫺h 0 , ⫺ 0 兴 ⫽ 关 uˆ (z),w resulting dispersion relation is given by: S ⫽ 关共 h0 ⫹ ␥ k 2 兲 / 共 k 兲兴共 1⫺q 2 兲关 sinh  cosh ␣ ⫺q cosh  sinh ␣ ⫹kb 共 1⫺q 2 兲 sinh  sinh ␣ 兴 , S⫽ 兵 4q 共 q 2 ⫹1 兲 ⫹ 关共 q 2 ⫹1 兲 2 ⫹4q 2 兴 sinh ␣ sinh  ⫺q 关共 q 2 ⫹1 兲 2 ⫹4 兴 cosh  cosh ␣ ⫹kb 共 1⫺q 2 兲 ⫻ 关共 q 2 ⫹1 兲 2 sinh  cosh ␣ ⫺4q cosh  sinh ␣ 兴 其 , 共1兲
0003-6951/2003/83(17)/3549/3/$20.00 3549 © 2003 American Institute of Physics Downloaded 21 Oct 2003 to 203.197.196.1. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/aplo/aplcr.jsp
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Appl. Phys. Lett., Vol. 83, No. 17, 27 October 2003
A. Sharma and K. Kargupta
where is the growth coefficient 共⬎0 for instability兲, h 0 is the mean thickness, h0 ⫽( / h) 兩 h⫽h0 ; k is the wave number; ␣ ⫽kh 0 ,  ⫽kqh 0 and q⫽(1⫹ / k 2 ) 1/2. Simplification of this relation for two asymptotic limit, b⫽0 and b ⫽⬁, using the long wave approximation and expanding q for / k 2 Ⰶ1, gives consistently the results obtained previously for nonslipping films and free foam films, respectively.2,31 Although the full form of the resulting dispersion relation is rather complicated, one can anticipate and directly verify that the resulting instability has a long wavelength (kh 0 Ⰶ1), so that the dispersion relation can be simplified to the following in the dominant order
bh 0 2 ⫹ 关 4bk 2 h 0 ⫹1 兴 ⫹ 关 h0 k 2 h 20 ⫹ ␥ k 4 h 20 兴共 h 0 /3⫹b 兲 ⫽0.
共2兲
When the conditions, h0 ⬍0 and k 2 ⬍⫺ h0 / ␥ , are met, the film is unstable ( ⬎0) regardless of the slip length. The dominant wave number, k m 共dominant wavelength, m ⫽2 /k m ) and the corresponding maximum growth factor ( m ), are obtained numerically using Eq. 共2兲. The linear theory rupture time is given by, t r ⫽ 关 1/ m 兴 ln(h0 /). Slippage at the solid-fluid boundary effectively decreases the viscous dissipation and thus accelerates the growth of instability. For small to moderate slip lengths, the growth rate is slow, and the unsteady term, bh 0 2 , in Eq. 共2兲 is negligible, giving 2 m ⫽16 2 bh 0 /
m ⫽⫺ 共 h0 ⫹ ␥ /4bh 0 兲 h 20 共 h 0 /3⫹b 兲 / 关 4 bh 0 共 1⫹1/ 兲兴 , 共3兲 where ⫽⫺1⫹(1⫹4bh 0 兩 h0 兩 / ␥ ) 1/2⫽⫺1⫹(1⫹4B g ) 1/2. B g is the nondimensional slip length for a general intermolecular potential. In our framework, Eq. 共3兲 describes the no-slip, weak slip as well as the moderate slip regimes depending on the slip length. For no-slip (b⫽0), Eq. 共3兲 recovers the well-known results 2 2 3 m ⫽⫺8 2 ␥ / h0 ; m ⫽ h0 h 0 /12␥ .
共4兲
The weak slip regime of Eq. 共3兲 is characterized by the 2 2 h 0 )⫽( m /4 h 0 ) 2 i.e., condition: 0⬍b/h 0 Ⰶ1/(4k m 2 2 16 bh 0 / m ⫽ Ⰶ1, implying, (4bh 0 兩 h0 兩 / ␥ )Ⰶ1. Thus, in the weak slip regime 共to the leading order兲 2 2 2 m ⫽⫺8 2 ␥ / h0 and m ⫽ h0 h 0 共 h 0 ⫹3b 兲 /12␥ .
共5兲
Thus the length scale remains unaffected by a weak slip, but the time scale can be decreased for bⰇh 0 . The weak slip behavior is displayed in the regime, 0⬍ ⬍ T1 Ⰶ1 (0⬍b ⬍b T1 ), where T1 , b T1 denote the first transition from weak to moderate slip regime. ⬃1 gives the critical slip length, b C . From Eq. 共3兲: b C ⫽ ␥ /2h 0 兩 h0 兩 .b T1 Ⰶb C . The moderate-slip is found when T1 ⬍ ⬍ T2 (b c Ⰶb T2 ⬍b⬍b T1 Ⰶb c ), where T2 , b T2 denote the second transition from moderate to strong slip regime. The upper bound of moderate 共intermediate兲 slip regime is obtained with different scaling, when, b/h 0 Ⰷ( m /4 h 0 ) 2 , i.e., ( Ⰷ1), but bh 0 2 is still negligible, which simplifies Eq. 共3兲 to 4 ⫽⫺ h0 h 0 ⫺ ␥ k 2 h 0 ⫹ h0 共 4bk 2 兲 ⫺1 ⫹ ␥ /4b,
FIG. 1. Variation of the dominant wavelength 关Fig. 1共a兲兴 and rupture time (1/ m ) 关Fig. 1共b兲兴 with the thickness of the film, for different slip lengths. Squares and filled circles represent the no slip (b⫽0) and free slip (b ⫽⬁) conditions, respectively. Curves from bottom to top in Fig. 1共a兲 are for b⫽0.5, 25, 250 m, 1, 102 , 104 , and 106 cm, respectively. Curves from top to bottom in Fig. 1共b兲 are for b⫽0.001, 0.1, 1, 10, 102 m, 1, and 100 cm, respectively. 4 m ⫽⫺64 4 b ␥ h 0 / h0 ;
m ⫽ 共 ␥ ⫺4bh 0 h0 兲 /16 b.
共6兲
Thus, the moderate slip regime is an intermediate regime connecting the weak and strong slip regimes for which the nondimensional parameter varies from Ⰶ1 to Ⰷ1. At the asymptotic limit, where the length scale, m , predicted by Eqs. 共5兲 and 共6兲 are same, B g ⫽1 and ⫽1.236. However, the transition values T1 and T2 can only be predicted numerically solving the general Eq. 共2兲. Finally, a strong slip regime is obtained when Ⰷ1 ( ⭓ T2 ), but bh 0 2 in Eq. 共2兲 can no longer be neglected since becomes high for very high slip length or very thin films. The asymptotic case of a ‘‘free’’ film 共e.g., a foam film with zero stress at its midplane兲27–31 is also correctly recovered in case of infinite slip (b→⬁, →⬁):
2 h 0 ⫹4k 2 h 0 ⫹ 共 h0 k 2 h 20 ⫹ ␥ k 4 h 20 兲 ⫽0, 2 ⫽4 2 关 2 ␥ 兵 1⫺4 2 / h 0 ␥ 其 兴 m
⫻ 兵 h0 关 ⫺1⫹ 共 4 2 / h 0 ␥ 兲 1/2兴 其 ⫺1 ,
共7兲
m ⫽⫺ 共 h0 h 0 /4 兲 兵 1⫹2 ␥ 关 1⫺ 共 4 2 / h 0 ␥ 兲 1/2兴 ⫻ 关 2 ␥ 共 1⫺4 2 / h 0 ␥ 兲兴 ⫺1 其 . Equations 共3兲–共7兲 clearly show that the length scale and time scale of instability and their dependence on the mean film thickness depend on slip regime and the form of the intermolecular potential ( h0 ). For the illustration purpose, we employ the most commonly used van der Waals potential ⌽⫽A/6 h 3 .
共8兲
A is the Hamaker constant. The parameter values used for the analysis are: A⫽10⫺20 J, ␥ ⫽20 mJ/m2 , ⫽1 kg/共ms兲, ⫽1000 kg/m3 . Figures 1共a兲 and 1共b兲 summarize the effect of slippage on the length scale and time scale of instability for the van der Waals potential. Strong, shear free slippage (b→⬁) increases the length scale of instability by two orders of magnitude 关Fig. 1共a兲兴, and thus reduces greatly the initial number ⫺2 ) compared to the density of holes/depressions (N d ⬀ m nonslipping film. The length scale, m of films thicker than a transition thickness 共in the weak slip兲 is independent of sliplength 共e.g., h 0 ⬎100 nm for b⫽250 m). The transition thickness, h 0T increases with the slip length 关Fig. 1共a兲兴. In other words, for a film of given thickness, there exists a transition slip length (b T1 ) below which the length scale of
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Appl. Phys. Lett., Vol. 83, No. 17, 27 October 2003
FIG. 2. Variations of exponents n 关 m ⬀h n0 ; Fig. 2共a兲兴 and m 关 t r ⬀h m0 ; Fig. 2共b兲兴 for the van der Waals potential. Curves 1, 2, and 3 are for mean film thickness, h 0 ⫽10, 50, 100 nm, respectively.
instability remains unaffected by slippage. The invariance of length scale of instability with slip length below a transition slip length 关 b T1 ; T1 Ⰶ1, Fig. 1共a兲兴, obtained from the general dispersion relation 关Eq. 共2兲兴, is also consistent with an earlier theoretical study.26 The scaling used in Ref. 26 was, however, inspired by the no-slip case, and are thus valid only for the weak slip regime.26 In contrast, any amount of slip reduces the rupture time compared to that of a nonslipping film 关by about 3 to 8 共!兲 orders of magnitude in the range, 10 nm⬍h⬍100 nm; Fig. 1共b兲兴. The breakup of films becomes significantly faster even for a slip length as small as 10% of the film thickness 关Eq. 共5兲兴. For the van der Waals potential, length scale, m and the rupture time, t r can be related to the film thickness by the power law relationships: m ⬀h n0 and t r ⬀h m 0 . The different regimes of slip can be identified from the nonmonotonic variation of the exponents, m and n 关Figs. 2共a兲 and 2共b兲兴 with slip length. For van der Waals potential, the time scale exponent, m⫽5 关Eq. 共4兲兴 for a nonslipping flow, but increases to a maximum of 6 关Eq. 共5兲兴 in the weak slip regime 关Fig. 2共a兲兴, and then declines to 3 in the moderate and strong slip 关Eqs. 共6兲, 共7兲, Fig. 2共a兲兴. The length scale exponent, n⫽2 for nonslipping and weakly slipping flows 关Eq. 共5兲, Ⰶ1, b⬍b T1 Ⰶb C or h 0 ⬎h 0T ]. For example, for the system considered here, b⬍0.1 m, exponent for a 10 nm film 关Fig. 2共b兲; n ⫽2] is unaffected by the slip length. In the moderate slip regime (b T1 ⭐b⬍b T2 ), the exponent, n decreases from 2 to a minimum of 1.25 关Eq. 共6兲兴. According to Eq. 共7兲, in the strong slip regime (b T2 ⭐b⭐⬁), dominance of unsteady term raises n to about 1.75 when 4 2 / h 0 ␥ Ⰶ1 关Fig. 2共b兲兴. Decrease in the film thickness increases the effect of slippage and results in the transition of slip regime either from weak →moderate, or from moderate→strong, depending on the slip length. Accordingly, the exponents, m and n also vary. For example, if b⫽1 m, films 10 nm and 50 nm thick are in the moderate slip and weak slip regimes, respectively 关Fig. 2共a兲兴. The transitions of weak→moderate→strong slippage regimes crucially depend on the magnitude of the nondimensional parameter, 关Eq. 共3兲兴. It is found that for a van der Waals potential, the first transition from the weak to moderate slip regime occurs for T1 ⬃0.01 and the second transition from the moderate to strong slippage occurs for T2 ⬃500. For a given system, the values of at the transitions are found to be almost independent of film thickness. Thus, is the most useful key parameter to identify the slip regime that can be directly related to the nondimensional slip length, B g 关Eq. 共3兲兴. In summary, slippage can have profound influence on the
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length scale and time scale of instability in thin films. Three distinct flow regimes with slippage are identified: 共1兲 weakly slipping flow (b⭐b T1 ,0⬍ ⭐0.01); 共2兲 moderately slipping flow (b T1 ⬍b⬍b T2 ;0.01⬍ ⭐500); and 共3兲 strongly slipping flow (b T2 ⬍b⭐⬁, Ⰷ500). The flow regimes depend on the slip length and the mean film thickness. In the weakly slipping flow, the length scale of instability is not affected by slippage, but only the time scale is altered. However, moderate or strong slippage increases the length scale of instability, m , and engenders a weaker dependence of m on the film thickness. For a van der Waals potential, moderate slippage reduces the exponent, n, ( m ⬀h n0 ) from 2 to 1.25; whereas strong slippage reduces it to 1.75. Increase of slip length from 0 to ⬁ decreases the breakup time by several order of magnitudes. Weak slippage increases the exponent, m (t r ⬀h m 0 ) to 6 compared to a nonslipping flow (m⫽5), whereas strong slippage reduces it to 3. A decrease in the film thickness changes the slip regime from weak to moderate to strong. The results should aid in characterization of thin film potential based on the experimentally observed length and time scales.
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