- Email: [email protected]
Rupture of thin stagnant films on a solid surface due to random thermal and mechanical perturbations
Journal of Colloid and Interface Science 287 (2005) 624–633 www.elsevier.com/locate/jcis
Rupture of thin stagnant films on a solid surface due to random thermal and mechanical perturbations Ganesan Narsimhan ∗ Biochemical and Food Process Engineering, Department of Agricultural and Biological Engineering, Purdue University, West Lafayette, IN 47907, USA Received 11 August 2004; accepted 11 February 2005 Available online 18 March 2005
Abstract A generalized formalism for the rupture of a nondraining thin film on a solid support due to imposed random thermal and mechanical perturbations, modeled as a Gaussian white noise, is presented. The evolution of amplitude of perturbation is described by a stochastic differential equation. The average film rupture time is the average time for the amplitude of perturbation to equal to the film thickness and is calculated by employing a first passage time analysis for different amplitudes of imposed perturbations, wavenumbers, film thickness, van der Waals and electrostatic interactions and surface tensions. The results indicate the existence of an optimum wavenumber at which the rupture time is minimum. A critical film thickness is identified based on the sign of the disjoining pressure gradient, below which the film is unstable in that the rupture time is very small. The calculated values of rupture time as well as the optimum wavenumber in the present analysis agree well with the results of linear stability analysis for immobile as well as completely mobile gas–liquid film interfaces. For stable films, the rupture time is found to increase dramatically with film thickness near the critical film thickness. As expected, the average rupture time was found to be higher for smaller amplitudes of imposed perturbations, larger surface potentials, larger surface tensions and smaller Hamaker constants. 2005 Elsevier Inc. All rights reserved. Keywords: Thin film rupture; Thermal fluctuations; Pressure fluctuations; First passage time; Van der Waals forces; Electrostatic forces; Disjoining pressure; Stagnant thin film
1. Introduction Thin liquid films bounded by two fluids or by a fluid and a solid surface are encountered in many colloidal systems. Coalescence of bubbles and droplets in foams and emulsions depends on the rupture of the intervening thin liquid films [1]. The rupture of thin film on a solid surface is important in flotation of solid particles [2], microelectronics [3] and corneal tear film [4,5]. The thin film drains because of capillary forces. When the film thickness becomes of the order of the range of intermolecular van der Waals, electrostatic, steric, depletion and hydration forces, these forces will influence the film drainage and stability [1]. Extensive investigations of the stability of thin films have been car* Fax: +1 765 496 3816.
E-mail address: [email protected]. 0021-9797/$ – see front matter 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.jcis.2005.02.029
ried out and a comprehensive review is given by Malderelli and Jain [6]. The spontaneous rupture of ultrathin films on a solid surface due to imposed perturbations has been treated by Ruckenstein and Jain [7] by introducing a body force into the equations of motion to account for the surface effects. Jain and Ruckenstein [8] have also investigated the stability of a stagnant film on a solid surface and established the conditions of instability and further calculated the rupture time due to the growth of imposed thermal perturbations as a result of van der Waals interactions. Several other investigators [7,9–13] have examined the conditions for instability and the time of rupture of equilibrium thin films by linear stability analysis. The nonlinear effects on film rupture have also been investigated [14–16]. A graphical technique has been developed [9] for the evaluation of critical thickness of rupture for films with immobile interfaces. Ivanov [17] established the critical thickness of rupture for a thinning
G. Narsimhan / Journal of Colloid and Interface Science 287 (2005) 624–633
film with high concentration of surfactant and the critical thickness of rupture with mobile interfaces. The effects of surface viscosity [18] and hydrophobic interactions [19] on the drainage and rupture of thin films have been investigated. The effect of density variation on dewetting of thin film on a solid surface has been examined by Sharma and Mittal [20]. Comprehensive reviews of drainage and stability of thin films can be found elsewhere [21,22]. Traditionally, the stability analysis of thin film involves the investigation of the growth of imposed perturbations. If the imposed perturbations grow, the film is deemed unstable. On the other hand, if the imposed perturbations are attenuated, the film is deemed stable. The surface tension and repulsive interactions between the two faces of the film tend to attenuate the perturbation by restoring the film shape to its original plane parallel shape whereas the attractive van der Waals and depletion forces tend to result in the growth of the perturbation. Depending on the shape of the disjoining pressure curve, there is a critical film thickness below which the film is considered unstable as per the criterion described above. The traditional stability analysis considers the response to a single imposed perturbation. However, when the film is subjected to random mechanical perturbation, the film may be exposed to a series of perturbations of different amplitudes and frequencies at different times. Even though the film may be stable to a single perturbation, the cumulative effect of these perturbations may result in the rupture of the film. In this paper, we present an analysis of the response of film thickness to an imposed Gaussian white noise. The evolution of film thickness is governed by a stochastic differential equation because of the random nature of imposed perturbations. By a first passage time analysis, the sample paths of film thickness are calculated to obtain the mean rupture time. Interestingly, the results of the present analysis agree well with the rupture time predicted by the conventional stability analysis for unstable films. The advantage of the proposed analysis is the prediction of rupture time of stable films as a result of the cumulative effect of imposed disturbances. The effects of the amplitude of the imposed disturbance as well as the nature of van der Waals and electrostatic interactions on the rupture time are investigated.
2. Model for rupture of thin liquid film on a solid substrate Consider a nondraining thin liquid film formed on a planar solid interface. For simplicity, the film is assumed to be square of area A. The schematic of the film with coordinate system is shown in Fig. 1. The film thickness is h. Since the film is nondraining, we have p = pg − Π(h),
(1)
where p and pg refer to the pressure in the film and gas phase respectively and Π(h) is the disjoining pressure. The rupture of thin film occurs due to thermal and mechanical
625
Fig. 1. Schematic of thin film on a solid support.
perturbations which result in pressure fluctuations that are imposed on the film surface. The imposed pressure fluctuation pg (x, t) can be expressed as a superposition of disturbances of waves of different wavenumbers and amplitudes. In the following, we will investigate the effect of such an imposed disturbance of a certain wavenumber kn on the shape of the film interface. This analysis is strictly valid only for film of radius small compared to the wavelength of imposed perturbation. For films of larger radii, one needs to consider the effect of many such perturbations on film leading to hole formation as discussed by Tsekov and Radoev [23]. The imposed pressure disturbance therefore satisfies (x, t) = ∇ 2 pg,n
(x, t) ∂ 2 pg,n
= −kn2 pg,n (x, t). (2) ∂x 2 The imposed random pressure fluctuation is assumed to be a Gaussian process [24], i.e., (x, t) = An (x)σn Tf pg,n
1/2
ξ(t),
(3)
where σn is the amplitude of the imposed pressure fluctuation, Tf is the timescale of the fluctuation and ξ(t) is white noise process of mean 0 and standard deviation 1, i.e., ξ(t) = 0, ξ(t)ξ(t ) = δ(t − t ). (4) refers to ensemble average and δ(t) is Dirac delta function. Therefore, pg,n (x, t) = 0, (x, t ) = An (x)2 σn2 Tf δ(t − t ). pg,n (x, t)pg,n (5) From (2) and (3), one can see that d2 An (x) (6) = −kn2 An (x). dx 2 The solution of the above equation with the boundary conditions, An (0) = 0,
An (L) = 0,
(7)
L being the length of the film, is given by nπx nπ An (x) = sin (8) and kn = . L L Therefore, the imposed random pressure disturbance is given by nπx 1/2 σn Tf ξ(t). (x, t) = sin pg,n (9) L
626
G. Narsimhan / Journal of Colloid and Interface Science 287 (2005) 624–633
The resulting perturbation ζn (x, t) of the film interface will also be periodic and satisfies ∂ 2 ζn (x, t) (10) = −kn2 ζn (x, t). ∂x 2 The imposed disturbance will result in a flow within the thin film. The equations of continuity and motion for the resulting flow are given by [8]
∇ 2 ζn (x, t) =
∂vy ∂vx + = 0, ∂x ∂y ∂p ∂ 2 vx =µ 2 , ∂x ∂y ∂p = 0. ∂y
(11)
Eq. (15a) can be recast as nπx ∗ 1/2 γ kn2 σ T ξ(t) + − 1 ζn∗ (x, t). L n f Π (h) (17) From Eqs. (12) and (15a), we get
pn∗ (x, t) = sin
∂ 2 vx,n nπ nπx 1/2 = cos σn Tf ξ(t) µL L ∂y 2 ∂ζn (x, t) 1 2 γ kn − Π (h) + . µ ∂x
(18)
(12)
For an immobile gas–liquid interface, the boundary conditions become
(13)
y = 0,
vx,n = 0,
(19)
The normal stress boundary condition at the gas–liquid interface yields the following for the pressure in the thin film [18],
y = h,
vx,n = 0.
(20)
∂ 2 ζn (x, t) ∂x 2 − Π(h) − Π (h)ζn (x, t),
(x, t) − γ p + pn (x, t) = pg + pg,n
(14)
pn (x, t)
where p is the mean pressure in the film, is the fluctuating pressure because of imposed pressure fluctuation, γ is the surface tension. In the above equation, the first term on the right-hand side is the mean pressure, the second term is the applied random pressure fluctuation, and the third term is the difference in pressure due to curvature of the film as a result of imposed perturbations as given by Laplace equation. The last term in the above equation is the change in the disjoining pressure because of imposed disturbance and is written as the linear term of Taylor series expansion around the equilibrium film thickness. As will be discussed later, this approximation is indeed reasonable when the maximum in disjoining pressure that has to be overcome for film rupture is indeed much larger than the disjoining pressure value for equilibrium film thickness. It is possible to include second- and higher-order terms in the expansion. This will, however, make the problem quite intractable. From Eqs. (1) and (14), we have ∂ 2 ζn (x, t) − Π (h)ζn (x, t). (14a) ∂x 2 (x, t) from Eq. (9), we get Substituting for pg,n
(x, t) − γ pn (x, t) = pg,n
nπx ∂ 2 ζn (x, t) 1/2 σn Tf ξ(t) − γ L ∂x 2 − Π (h)ζn (x, t).
pn (x, t) = sin
(15)
Using Eq. (2), we obtain nπx 1/2 σn Tf ξ(t) + γ kn2 − Π (h) ζn (x, t). pn (x, t) = sin L (15a) Defining the following dimensionless parameters, pn∗
p = n , Π (h)h
σn∗
σn , = Π (h)h
ζn∗
ζn = , h
(16)
Integrating Eq. (18) twice with the boundary conditions (19) and (20), we get nπx nπ 1/2 cos σn Tf ξ(t)(hy − y 2 ) vx,n = − 2µL L ∂ζn (x, t) 1 2 γ kn − Π (h) (hy − y 2 ). − (21) 2µ ∂x From Eqs. (11) and (21) and using the boundary condition, y = 0,
vy = 0,
(22)
we get 2 kn2 hy y3 1/2 sin kn xσn Tf,n ξ(t) − µ 4 6 2 2 y3 k hy − . − n γ kn2 − Π (h) ζn (x, t) µ 4 6
vy = −
(23)
Therefore, the velocity of the gas–liquid interface due to imposed perturbations is given by ∂ζn (x, t) k 2 h3 1/2 = vζ = vy |y=h = − n sin kn xσn Tf,n ξ(t) ∂t 12µ + γ kn2 − Π (h) ζn (x, t) (24) or kn2 h3 2 γ kn − Π (h) ζn (x, t) dt 12µ k 2 h3 1/2 − n sin kn xσn Tf,n dW (t), 12µ
dζn (x, t) = −
(25)
where dW (t) is a Wiener process [24] defined as dW (t) = ξ(t) dt, 2 dW (t) = 0, dW (t) = dt.
(26) (27)
Defining an average perturbation as n ζ¯n (t) = L
L/n ζn (x, t) dx, 0
(28)
G. Narsimhan / Journal of Colloid and Interface Science 287 (2005) 624–633
we have from Eq. (25) kn2 h3 2 γ kn − Π (h) ζ¯n (t) dt 12µ k 2 h3 1/2 − n σn Tf,n dW (t). 6πµ
equilibrium film thickness will quickly die. Consequently, the equilibrium film thickness can be considered as a reflecting boundary. In other words,
dζ¯n (t) = −
(29)
For completely mobile film interface, no-slip boundary condition (20) is to be replaced with zero shear stress boundary condition dvx,n = 0, y = h, (30) dy and the corresponding stochastic differential equation for the perturbation is kn2 h3 2 γ kn − Π (h) ζ¯n (t) dt 3µ 2k 2 h3 1/2 − n σn Tf,n dW (t). 3πµ
dζ¯n (t) = −
(31)
Equation (29) is a stochastic differential equation describing the evolution of ζ¯n (t). This equation is written in Stratanovich sense [24]. Converting the above equation to its equivalent Ito form [24], we get dζ¯n = f (ζ¯n ) dt + g(ζ¯n ) dW (t). γ kn2 − Π (h) ζ¯n , 12µ h3 kn2 1/2 σn Tf,n , g(ζ¯n ) = − 6πµ
f (ζ¯n ) = −
(33) (34)
and for completely mobile gas–liquid interface, h3 kn2 2 γ kn − Π (h) ζ¯n , 3µ 2h3 kn2 1/2 σn Tf,n . g(ζ¯n ) = − 3πµ
f (ζ¯n ) = −
(35) (36)
Since ζ¯n (t) is a stochastic process, the evolution of mean disturbance on the film surface is random and is usually referred to as a sample path of the process. For an ensemble of large number of films with the same equilibrium thickness, the evolution of the disturbance on the film surface gives an ensemble of sample paths. From this ensemble, one can then define the probability that the mean disturbance is x at time t. One can then define p(x , t|x, 0) as the conditional probability that ζ¯n = x at time t with the initial condition x. The film will be stable if the perturbation is less than the equilibrium film thickness. We postulate, therefore, that the film ruptures when the average disturbance is the equilibrium film thickness. Therefore, p(h, t|x, 0) = 0.
dp(0, t|x, 0) = 0. (38) dx As pointed out earlier, we are interested in the average film rupture time. Consequently, the final point in the sample path is h. Therefore, we are interested in the evolution of this conditional probability backward in time. This is given by the backward Fokker–Planck equation [24] ∂p(x , t|x, 0) ∂p(x , t|x, 0) = f (x) ∂t ∂x 1 ∂ 2 p(x , t|x, 0) + g 2 (x) (39) , 2 ∂x 2 where f (x) and g(x) are given by Eqs. (29), (31) and (30), (32), respectively. One can define the probability G(x, t) that at time t after the imposition of disturbance the film is stable when the initial mean disturbance is x as h G(x, t) =
p(x , t|x, 0) dx .
(40)
0
(32)
For immobile gas–liquid interface, h3 kn2
627
(37)
Because of the restoring force of surface tension, any disturbance that tends to increase the film thickness above the
Equation (39) can then be integrated to yield ∂G(x, t) 1 2 ∂ 2 G(x, t) ∂G(x, t) = f (x) + g (x) . (41) ∂t ∂x 2 ∂x 2 First passage time is defined as the time after the imposition of disturbance at which the film ruptures. Since the imposed disturbance is random, the first passage time is also random which can be described by a distribution. The mean first passage time T (x) when the initial average disturbance is x is given by ∞ T (x) = −
∂G(x, t) dt = t ∂t
0
∞ G(x, t) dt.
(42)
0
Integrating Eq. (41) from 0 to ∞, we get ∞
∂G(x, t) dt = G(x, ∞) − G(x, 0) = −1 ∂t
0
dT (x) 1 2 d2 T (x) (43) + g (x) dx 2 dx 2 with the boundary conditions (obtained by integrating the boundary conditions (37) and (38)) = f (x)
x = 0, x = h,
dT (x) = 0, dx T (x) = 0.
(44) (45)
Note that boundary condition (44) is the reflecting boundary condition and (45) is the absorbing boundary condition. The solution of Eq. (43) with the boundary conditions (44) and
628
G. Narsimhan / Journal of Colloid and Interface Science 287 (2005) 624–633
(45) is given by [24] x T (x) = 2
dy ψ(y)
0
ψ(z) dz, g 2 (z)
(46)
y
h
where
x ψ(x) = exp
2 dx 2f (x )/g(x ) .
(47)
h
Since we are interested in calculating the average rupture time of an equilibrium film exposed to a periodic disturbance of a certain wavenumber, the mean rupture time T¯n is given by T¯n = T (0) = 2
0 h
dy ψ(y)
0
ψ(z) dz, g 2 (z)
(48)
y
where ψ is defined in terms of functions f and g written for wavenumber kn . The imposed pressure fluctuation of a certain wavenumber pn (t) is given by ∂vz,n pn (t) = 2µ (49) + γ kn2 − Π (h) ζn . ∂z z=h Neglecting the first term on the right-hand side, we get pn (t) ≈ γ kn2 − Π (h) ζn . (50) For thermal perturbation, the amplitude of perturbation is given by [25] 1/2 21/2 (kT )1/2 2 −1/2 γ kn − Π (h) ζn = ζn2 (51) = L with the corresponding amplitude of pressure fluctuation given by 1/2 21/2 (kT )1/2 2 γ kn − Π (h) . (52) L In the long wavelength limit, the amplitude of the squeezing mode of imposed perturbation relaxes as exp(−|t|/Tf ). Vrij [9] has treated this for immobile gas–liquid interface and has shown from the conservation of film volume that the timescale Tf is given by [25]
σn =
Tf =
24µ . kn2 h3 (γ kn2 − Π (h))
(53)
For completely mobile gas–liquid interface, we have [8] Tf =
6µ . kn2 h3 (γ kn2 − Π (h))
(54)
The disjoining pressure is given by [8,26] zeψ2 zeψ1 A tanh + 64n0 kT tanh Π(h) = − kT kT 6πh3 × exp(−κh), (55)
where A is the Hamaker constant, n0 is the number concentration of electrolytes, k is the Boltzmann’s constant, T is the temperature, κ is the Debye–Huckel parameter, and ψ1 and ψ2 are the surface potential at the gas–liquid interface and solid surface, respectively. Typical plot of disjoining pressure versus film thickness is shown in Fig. 2. As expected, the electrostatic interactions predominate over intermediate distances and van der Waals attraction predominates for very thin and thick films so that there is a maximum in the disjoining pressure at a critical film thickness hcrit . When h < hcrit , the imposed perturbations grow and hence the film is unstable. On the other hand, when h > hcrit , the imposed perturbations attenuate and conventional stability analysis predicts the film to be stable. Since the change in the disjoining pressure due to perturbation of the interface is expressed as Taylor series expansion around the equilibrium thickness retaining only the linear terms (see Eq. (14)), an average value of Π (h) was used for the unstable and stable branches of the disjoining pressure in the subsequent calculations.
3. Results and discussion When h < hcrit , the imposed perturbations grow with time thus rendering the film unstable. The average rupture time of an unstable film can be estimated from the evaluation of growth rate of perturbation via linear stability analysis. In the following, we will compare this time with the average rupture time estimated by the current analysis as given by Eq. (48). When h > hcrit , the imposed perturbations attenuate with time thus rendering the film stable. Existing stability analyses predict that the film is stable to all imposed perturbations. However, the current analysis can predict an average rupture time for this film as given by Eq. (48), this time being dependent on the amplitude of imposed perturbations. 3.1. Case I: unstable film The film is unstable when Π (h) > 0 (see Fig. 2). Linear stability analysis of growth of thermal perturbations yields the following expression for the growth coefficient for immobile gas–liquid interface [8,26]: β=
h3 (∂Π/∂h)2 . 48µγ
(56)
For completely mobile gas–liquid interface [8,26], βmobile =
h3 (∂Π/∂h)2 12µγ
with the dominant wavenumber kd given by [8,26]
1 ∂Π . kd ≈ 2γ ∂h
(57)
(58)
G. Narsimhan / Journal of Colloid and Interface Science 287 (2005) 624–633
629
Fig. 2. Plot of disjoining pressure and disjoining pressure gradient versus film thickness for the following parameters: A = 5 × 10−20 J, eψ1 /kT = 0.8, eψ2 /kT = 0.1, I = 0.01. Critical film thickness hcrit is the thickness at which disjoining pressure is maximum. The stable and unstable regions are shown in the figure. bn,mobile
The rupture time tr is given by h = h0,d exp(βtr ),
(59)
where h0,d is the amplitude of thermal fluctuation at the dominant wavenumber kd as given by Eq. (51). For an immobile gas–liquid interface, from Eqs. (56)–(59) we get
Lh Π (h) 1/2 48µγ ln , tr = 3 (60) 2 kT h (∂Π/∂h)2 and for completely mobile gas–liquid interface, we get
12µγ tr Lh Π (h) 1/2 ln . (61) tr,mobile = = 3 4 h (∂Π/∂h)2 2 kT
exp(x 2 ) erf(bn ) − erf(x) dx,
×
(64)
0
where the upper limit bn,mobile is given by bn,mobile =
(Π (h) − γ kn2 )1/2 31/2 πµ1/2 1/2
2kn h1/2 σn Tf,n
,
(65)
Tf,n being given by Eq. (54). It can be seen from Eqs. (62) and (64) that Tn,mobile = (1/4)Tn . A typical plot of the average rupture time for the current model as given by Eq. (62) for immobile gas–liquid interface
For the current analysis, for an immobile gas–liquid interface, Eq. (48) reduces to 12π 1/2 µ Tn = 2 3 kn h (Π (h) − γ kn2 )
bn
exp(x 2 ) erf(bn ) − erf(x) dx,
0
(62)
where the upper limit bn is given by bn =
(Π (h) − γ kn2 )1/2 31/2 πµ1/2 1/2
kn h1/2 σn Tf,n
,
(63)
Tf,n being given by Eq. (53). For thermal perturbation, the amplitude of pressure fluctuation σn is given by Eq. (52). For completely mobile gas–liquid interface, the current analysis gives Tn,mobile =
3π 1/2 µ kn2 h3 (Π (h) − γ kn2 )
Fig. 3. Plot of average rupture time versus wavenumber for an unstable film exposed to thermal perturbations. A = 5 × 10−20 J, eψ1 /kT = 0.8, eψ2 /kT = 0.1, I = 0.01, γ = 50 mN/m, L = 0.01 m, µ = 10−3 Pa s, h = 10−9 m, T = 298 K.
630
G. Narsimhan / Journal of Colloid and Interface Science 287 (2005) 624–633
Fig. 4. Plot of average rupture time versus film thickness for different Hamaker constants for an unstable film subjected to thermal perturbations. The other parameters are the same as in Fig. 3.
Fig. 5. Plot of average rupture time versus film thickness for different surface potential of air–liquid interface ψ1∗ = eψ1 /kT for an unstable film subjected to thermal perturbations. All the other parameters are the same as in Fig. 3.
for different wavenumbers is shown in Fig. 3. The rupture time increases dramatically at the critical wavenumber. The dominant wavenumber for minimum rupture time (maxi√ mum growth rate) is about 1/ 2 times the critical wavenumber, consistent with the results of linear stability analysis. This minimum rupture time at the dominant wavenumber is reported in subsequent calculations as the average rupture time. The average rupture time for the current model is compared with that predicted by linear stability analysis (Eq. (60)) for different parameters in Figs. 4–7. As expected,
Fig. 6. Plot of average rupture time versus film thickness for different surface tensions for an unstable film subjected to thermal perturbations. All the other parameters are the same as in Fig. 3.
Fig. 7. Plot of average rupture time versus film thickness for different viscosities for an unstable film subjected to thermal perturbations. All the other parameters are the same as in Fig. 3.
an increase in Hamaker constant reduces the rupture time, this effect being more pronounced for thicker films. Also, the rupture time is found to increase dramatically (see Fig. 4) near the critical film thickness. The effect of surface potential of the gas–liquid interface on the average rupture time is shown in Fig. 5. As expected, the rupture time increases with the surface potential. Also, the rupture time increases dramatically near the critical film thickness, the value of which decreases at higher surface potential. Interestingly, in the absence of electrostatic interactions (only van der Waals), the rupture time increases less rapidly with film thickness with the absence of a critical film thickness. The effects of surface
G. Narsimhan / Journal of Colloid and Interface Science 287 (2005) 624–633
631
tension and viscosity on the average rupture time are shown in Figs. 6 and 7, respectively. Since the ratio of the rupture times for immobile and completely mobile gas–liquid interfaces is 4 for both linear stability analysis and the current analysis, the comparison of rupture times for both analyses for the case of completely mobile gas–liquid interface is not shown here. 3.2. Case II: stable film The film is stable when Π (h) < 0 (see Fig. 2). For the current analysis, for a stable film, for immobile gas–liquid interface, Eq. (46) reduces to 12π 1/2 µ Tn = 2 3 kn h (γ kn2 − Π (h))
bn 2
exp(x ) erf(x) dx,
(66)
0
where the upper limit bn is given by hcrit (γ kn2 − Π (h))1/2 31/2 πµ1/2 1− , bn = 1/2 h kn h1/2 σn Tf,n
Fig. 8. Plot of average rupture time versus wavenumber for an unstable film subjected to pressure fluctuations. The parameters are A = 10−20 J, σ = 1000 Pa, h = 5 × 10−9 m, eψ1 /kT = 0.5, eψ2 /kT = 0.1, I = 0.01, µ = 10−3 Pa s, γ = 50 mN/m, L = 0.01 m.
(67)
Tf,n being given by Eq. (53). For thermal perturbation, the amplitude of pressure fluctuation σn is given by Eq. (52). The factor (1 − hcrit / h) is introduced in the upper limit in order to account for the fact that it is sufficient for the amplitude of perturbations to grow to (h − hcrit ) in order for the film to rupture. For completely mobile gas–liquid interface, we get 3π 1/2 µ Tn,mobile = 2 3 kn h (γ kn2 − Π (h))
bn,mobile
exp(x 2 ) erf(x) dx, 0
(68)
where the upper limit bn,mobile is given by hcrit (γ kn2 − Π (h))1/2 31/2 πµ1/2 bn,mobile = 1 − , (69) 1/2 h 2kn h1/2 σn Tf,n Tf,n being given by Eq. (54). Comparing Eqs. (66) and (68), it can be seen that Tn,mobile = (1/4)Tn . When bn 1, Eq. (66) can be approximated as Tn
12π 1/2 µ(1 − hcrit / h) kn2 h2 23/2 σn cn Π (h)h(1 − hcrit / h) 2 × exp , σn
(70)
where cn = π(1 + γ kn2 /Π (h))/23/2 . It can be seen that the exponential term in the above equation is the pressure barrier since the maximum disjoining pressure is at the film thickness hcrit . Typical plot of average rupture time versus wavenumber for a stable film (evaluated from Eq. (66)) as shown in Fig. 8 indicates the existence of an optimum wavenumber at which
Fig. 9. Plot of average rupture time versus film thickness for different amplitudes of imposed pressure fluctuations for a stable film. All the other parameter values are the same as in Fig. 8.
the rupture time is minimum. This is reported as the average rupture time. The variation of average rupture time with film thickness for different amplitudes of pressure fluctuation (including thermal perturbation) is shown in Fig. 9. The rupture time is indeed extremely large for thermal perturbations, decreasing considerably as the amplitude of pressure fluctuation increases. It is to be noted that the rupture time decreases dramatically as the film thickness approaches the critical film thickness. Fig. 10 shows that the average rupture time is sensitive to the surface potential increasing with the potential. The effect of pressure amplitude on the average rupture time is shown in Figs. 11a, 12a and 13 for different surface potentials, Hamaker constants and surface tension values, respectively. As expected, the average rupture time decreases with pressure amplitude. Also, larger Hamaker constants, smaller surface potentials and smaller surface tensions tend to decrease the average rupture time.
632
G. Narsimhan / Journal of Colloid and Interface Science 287 (2005) 624–633
Fig. 10. Plot of average rupture time versus film thickness for different surface potential of air–liquid interface ψ1∗ = eψ1 /kT for a stable film. All the other parameter values are the same as in Fig. 8.
(a)
(b)
(a)
Fig. 12. Plots of (a) average rupture time versus amplitude of imposed pressure fluctuation for different Hamaker constants and (b) average rupture time versus amplitude of imposed dimensionless pressure fluctuation for different Hamaker constants for a stable film. h = 10−8 m. All the other parameter values are the same as in Fig. 8.
(b) Fig. 11. Plots of (a) average rupture time versus amplitude of imposed pressure fluctuation for different surface potential of air–liquid interface and (b) average rupture time versus dimensionless amplitude of imposed pressure fluctuation for different surface potential of air–liquid interface. ψ1∗ = eψ1 /kT for a stable film. All the other parameter values are the same as in Fig. 8.
Fig. 13. Plot of average rupture time versus amplitude of imposed pressure fluctuation for different surface tensions for a stable film. All the other parameter values are the same as in Fig. 8.
G. Narsimhan / Journal of Colloid and Interface Science 287 (2005) 624–633
Fig. 14. Plot of ln Tn versus pressure barrier {Π (h)h(1 − hcrit / h)/σn }2 for different amplitudes of pressure fluctuation for a stable film. All the other parameter values are the same as in Fig. 8.
The effects of dimensionless pressure amplitude (as defined by Eq. (16)) on the average rupture time for different surface potentials and Hamaker constants are shown in Figs. 11b and 12b, respectively. Fig. 14 shows that the plot of ln Tn versus pressure barrier {Π (h)h(1 − hcrit / h)/σn }2 for different pressure amplitudes do fall approximately into a single straight line. Since the rupture time for completely mobile gas–liquid interface is 1/4 of that for immobile film interface provided the disjoining pressure is the same, the results of variation of rupture time for completely mobile interface are not shown here.
4. Conclusions The growth of perturbation of a thin film on a solid surface when subjected to random mechanical perturbation as described by a Gaussian white noise is analyzed. The growth of amplitude of perturbation is described by a stochastic differential equation. The sample paths of evolution of perturbation amplitude is analyzed by a first passage time analysis. The average rupture time (the average time taken for the amplitude to become equal to film thickness) is evaluated for different amplitudes and wavenumbers of imposed random mechanical perturbation. The results indicate the existence of an optimum wavenumber at which the rupture time is minimum. A critical film thickness is identified based on the sign of the gradient of disjoining pressure, below which the film is unstable in that the rupture time is very small. The
633
calculated values of rupture time from the present analysis were found to agree very well with the results of linear stability analysis. In addition, the optimum wavenumber at which the rupture time is minimum (or, equivalently, the growth coefficient is maximum) agreed with that given by linear stability analysis. The rupture time is also calculated for stable films for different amplitudes of imposed perturbation, film thickness and physical properties of the system. The rupture time was found to increase dramatically near the critical film thickness, this increase being more pronounced for films with electrostatic repulsions. The average rupture time for completely mobile gas–liquid interface is shown to be 1/4 of that for an immobile interface. As expected, the rupture time was found to be larger for smaller amplitude of imposed perturbation, larger film thickness, larger surface potentials, smaller Hamaker constants and larger surface tensions. The proposed analysis is extremely useful in the prediction of rupture of thin films when exposed to random mechanical perturbations.
References [1] I.B. Ivanov (Ed.), Thin Liquid Films: Fundametals and Applications, Dekker, 1988. [2] A. Scheludko, Kolloid Z. 191 (1963) 52. [3] E. Ruckenstein, C.S. Dunn, Thin Solid Films 51 (1978) 43. [4] A. Sharma, E. Ruckenstein, Am. J. Optom. Physiol. Opt. 62 (1985) 246. [5] A. Sharma, E. Ruckenstein, J. Colloid Interface Sci. 111 (1986) 8. [6] C. Malderelli, R. Jain, in: I.B. Ivanov (Ed.), Thin Liquid Films: Fundamentals and Applications, Dekker, 1988, pp. 497–568. [7] E. Ruckenstein, R. Jain, J. Chem. Soc. Faraday II 70 (1974) 132. [8] R. Jain, E. Ruckenstein, J. Colloid Interface Sci. 54 (1976) 108–116. [9] A. Vrij, Discuss. Faraday Soc. 42 (1966) 23. [10] A. Scheludko, Adv. Colloid Interface Sci. 1 (1967) 391–464. [11] B. Radoev, A. Scheludko, E. Manev, J. Colloid Interface Sci. 95 (1983) 254. [12] C. Malderelli, R. Jain, J. Colloid Interface Sci. 90 (1982) 263. [13] A. Vrij, F.Th. Hesselink, J. Lucassen, M. van den Temple, Proc. K. Ned. Akad. Wet. Ser. B 73 (1970) 124. [14] M.B. Williams, S.H. Davis, J. Colloid Interface Sci. 90 (1982) 220. [15] A. Sharma, E. Ruckenstein, J. Colloid Interface Sci. 113 (1986) 456. [16] A. Sharma, E. Ruckenstein, Langmuir 2 (1986) 480. [17] I.B. Ivanov, Pure Appl. Chem. 52 (1980) 1241. [18] I.B. Ivanov, D.S. Dimitrov, Colloid Polym. Sci. 252 (1974) 982–990. [19] A. Sharma, J. Colloid Interface Sci. 199 (1998) 212–214. [20] A. Sharma, J. Mittal, Phys. Rev. Lett. 89 (2002) 186101. [21] D.S. Dimitrov, Prog. Surf. Sci. 14 (1983) 295. [22] J.E. Coons, P.J. Halley, S.A. McGlashan, T. Tran-Cong, Adv. Colloid Interface Sci. 105 (2003) 3–62. [23] R. Tsekov, B. Radoev, Adv. Colloid Interface Sci. 38 (1992) 353–369. [24] C.W. Gardiner, Handbook of Stochastic Methods for Physics, Chemistry and Natural Sciences, Springer-Verlag, 1983. [25] J.G.H. Joosten, in: I.B. Ivanov (Ed.), Thin Liquid Films: Fundamentals and Applications, Dekker, 1988, pp. 569–662. [26] D.A. Edwards, H. Brenner, D.T. Wasan, Interfacial Transport Processes and Rheology, Butterworth–Heinemann, 1991.
Rupture of thin stagnant films on a solid surface due to random thermal and mechanical perturbations Ganesan Narsimhan ∗ Biochemical and Food Process Engineering, Department of Agricultural and Biological Engineering, Purdue University, West Lafayette, IN 47907, USA Received 11 August 2004; accepted 11 February 2005 Available online 18 March 2005
Abstract A generalized formalism for the rupture of a nondraining thin film on a solid support due to imposed random thermal and mechanical perturbations, modeled as a Gaussian white noise, is presented. The evolution of amplitude of perturbation is described by a stochastic differential equation. The average film rupture time is the average time for the amplitude of perturbation to equal to the film thickness and is calculated by employing a first passage time analysis for different amplitudes of imposed perturbations, wavenumbers, film thickness, van der Waals and electrostatic interactions and surface tensions. The results indicate the existence of an optimum wavenumber at which the rupture time is minimum. A critical film thickness is identified based on the sign of the disjoining pressure gradient, below which the film is unstable in that the rupture time is very small. The calculated values of rupture time as well as the optimum wavenumber in the present analysis agree well with the results of linear stability analysis for immobile as well as completely mobile gas–liquid film interfaces. For stable films, the rupture time is found to increase dramatically with film thickness near the critical film thickness. As expected, the average rupture time was found to be higher for smaller amplitudes of imposed perturbations, larger surface potentials, larger surface tensions and smaller Hamaker constants. 2005 Elsevier Inc. All rights reserved. Keywords: Thin film rupture; Thermal fluctuations; Pressure fluctuations; First passage time; Van der Waals forces; Electrostatic forces; Disjoining pressure; Stagnant thin film
1. Introduction Thin liquid films bounded by two fluids or by a fluid and a solid surface are encountered in many colloidal systems. Coalescence of bubbles and droplets in foams and emulsions depends on the rupture of the intervening thin liquid films [1]. The rupture of thin film on a solid surface is important in flotation of solid particles [2], microelectronics [3] and corneal tear film [4,5]. The thin film drains because of capillary forces. When the film thickness becomes of the order of the range of intermolecular van der Waals, electrostatic, steric, depletion and hydration forces, these forces will influence the film drainage and stability [1]. Extensive investigations of the stability of thin films have been car* Fax: +1 765 496 3816.
E-mail address: [email protected]. 0021-9797/$ – see front matter 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.jcis.2005.02.029
ried out and a comprehensive review is given by Malderelli and Jain [6]. The spontaneous rupture of ultrathin films on a solid surface due to imposed perturbations has been treated by Ruckenstein and Jain [7] by introducing a body force into the equations of motion to account for the surface effects. Jain and Ruckenstein [8] have also investigated the stability of a stagnant film on a solid surface and established the conditions of instability and further calculated the rupture time due to the growth of imposed thermal perturbations as a result of van der Waals interactions. Several other investigators [7,9–13] have examined the conditions for instability and the time of rupture of equilibrium thin films by linear stability analysis. The nonlinear effects on film rupture have also been investigated [14–16]. A graphical technique has been developed [9] for the evaluation of critical thickness of rupture for films with immobile interfaces. Ivanov [17] established the critical thickness of rupture for a thinning
G. Narsimhan / Journal of Colloid and Interface Science 287 (2005) 624–633
film with high concentration of surfactant and the critical thickness of rupture with mobile interfaces. The effects of surface viscosity [18] and hydrophobic interactions [19] on the drainage and rupture of thin films have been investigated. The effect of density variation on dewetting of thin film on a solid surface has been examined by Sharma and Mittal [20]. Comprehensive reviews of drainage and stability of thin films can be found elsewhere [21,22]. Traditionally, the stability analysis of thin film involves the investigation of the growth of imposed perturbations. If the imposed perturbations grow, the film is deemed unstable. On the other hand, if the imposed perturbations are attenuated, the film is deemed stable. The surface tension and repulsive interactions between the two faces of the film tend to attenuate the perturbation by restoring the film shape to its original plane parallel shape whereas the attractive van der Waals and depletion forces tend to result in the growth of the perturbation. Depending on the shape of the disjoining pressure curve, there is a critical film thickness below which the film is considered unstable as per the criterion described above. The traditional stability analysis considers the response to a single imposed perturbation. However, when the film is subjected to random mechanical perturbation, the film may be exposed to a series of perturbations of different amplitudes and frequencies at different times. Even though the film may be stable to a single perturbation, the cumulative effect of these perturbations may result in the rupture of the film. In this paper, we present an analysis of the response of film thickness to an imposed Gaussian white noise. The evolution of film thickness is governed by a stochastic differential equation because of the random nature of imposed perturbations. By a first passage time analysis, the sample paths of film thickness are calculated to obtain the mean rupture time. Interestingly, the results of the present analysis agree well with the rupture time predicted by the conventional stability analysis for unstable films. The advantage of the proposed analysis is the prediction of rupture time of stable films as a result of the cumulative effect of imposed disturbances. The effects of the amplitude of the imposed disturbance as well as the nature of van der Waals and electrostatic interactions on the rupture time are investigated.
2. Model for rupture of thin liquid film on a solid substrate Consider a nondraining thin liquid film formed on a planar solid interface. For simplicity, the film is assumed to be square of area A. The schematic of the film with coordinate system is shown in Fig. 1. The film thickness is h. Since the film is nondraining, we have p = pg − Π(h),
(1)
where p and pg refer to the pressure in the film and gas phase respectively and Π(h) is the disjoining pressure. The rupture of thin film occurs due to thermal and mechanical
625
Fig. 1. Schematic of thin film on a solid support.
perturbations which result in pressure fluctuations that are imposed on the film surface. The imposed pressure fluctuation pg (x, t) can be expressed as a superposition of disturbances of waves of different wavenumbers and amplitudes. In the following, we will investigate the effect of such an imposed disturbance of a certain wavenumber kn on the shape of the film interface. This analysis is strictly valid only for film of radius small compared to the wavelength of imposed perturbation. For films of larger radii, one needs to consider the effect of many such perturbations on film leading to hole formation as discussed by Tsekov and Radoev [23]. The imposed pressure disturbance therefore satisfies (x, t) = ∇ 2 pg,n
(x, t) ∂ 2 pg,n
= −kn2 pg,n (x, t). (2) ∂x 2 The imposed random pressure fluctuation is assumed to be a Gaussian process [24], i.e., (x, t) = An (x)σn Tf pg,n
1/2
ξ(t),
(3)
where σn is the amplitude of the imposed pressure fluctuation, Tf is the timescale of the fluctuation and ξ(t) is white noise process of mean 0 and standard deviation 1, i.e., ξ(t) = 0, ξ(t)ξ(t ) = δ(t − t ). (4) refers to ensemble average and δ(t) is Dirac delta function. Therefore, pg,n (x, t) = 0, (x, t ) = An (x)2 σn2 Tf δ(t − t ). pg,n (x, t)pg,n (5) From (2) and (3), one can see that d2 An (x) (6) = −kn2 An (x). dx 2 The solution of the above equation with the boundary conditions, An (0) = 0,
An (L) = 0,
(7)
L being the length of the film, is given by nπx nπ An (x) = sin (8) and kn = . L L Therefore, the imposed random pressure disturbance is given by nπx 1/2 σn Tf ξ(t). (x, t) = sin pg,n (9) L
626
G. Narsimhan / Journal of Colloid and Interface Science 287 (2005) 624–633
The resulting perturbation ζn (x, t) of the film interface will also be periodic and satisfies ∂ 2 ζn (x, t) (10) = −kn2 ζn (x, t). ∂x 2 The imposed disturbance will result in a flow within the thin film. The equations of continuity and motion for the resulting flow are given by [8]
∇ 2 ζn (x, t) =
∂vy ∂vx + = 0, ∂x ∂y ∂p ∂ 2 vx =µ 2 , ∂x ∂y ∂p = 0. ∂y
(11)
Eq. (15a) can be recast as nπx ∗ 1/2 γ kn2 σ T ξ(t) + − 1 ζn∗ (x, t). L n f Π (h) (17) From Eqs. (12) and (15a), we get
pn∗ (x, t) = sin
∂ 2 vx,n nπ nπx 1/2 = cos σn Tf ξ(t) µL L ∂y 2 ∂ζn (x, t) 1 2 γ kn − Π (h) + . µ ∂x
(18)
(12)
For an immobile gas–liquid interface, the boundary conditions become
(13)
y = 0,
vx,n = 0,
(19)
The normal stress boundary condition at the gas–liquid interface yields the following for the pressure in the thin film [18],
y = h,
vx,n = 0.
(20)
∂ 2 ζn (x, t) ∂x 2 − Π(h) − Π (h)ζn (x, t),
(x, t) − γ p + pn (x, t) = pg + pg,n
(14)
pn (x, t)
where p is the mean pressure in the film, is the fluctuating pressure because of imposed pressure fluctuation, γ is the surface tension. In the above equation, the first term on the right-hand side is the mean pressure, the second term is the applied random pressure fluctuation, and the third term is the difference in pressure due to curvature of the film as a result of imposed perturbations as given by Laplace equation. The last term in the above equation is the change in the disjoining pressure because of imposed disturbance and is written as the linear term of Taylor series expansion around the equilibrium film thickness. As will be discussed later, this approximation is indeed reasonable when the maximum in disjoining pressure that has to be overcome for film rupture is indeed much larger than the disjoining pressure value for equilibrium film thickness. It is possible to include second- and higher-order terms in the expansion. This will, however, make the problem quite intractable. From Eqs. (1) and (14), we have ∂ 2 ζn (x, t) − Π (h)ζn (x, t). (14a) ∂x 2 (x, t) from Eq. (9), we get Substituting for pg,n
(x, t) − γ pn (x, t) = pg,n
nπx ∂ 2 ζn (x, t) 1/2 σn Tf ξ(t) − γ L ∂x 2 − Π (h)ζn (x, t).
pn (x, t) = sin
(15)
Using Eq. (2), we obtain nπx 1/2 σn Tf ξ(t) + γ kn2 − Π (h) ζn (x, t). pn (x, t) = sin L (15a) Defining the following dimensionless parameters, pn∗
p = n , Π (h)h
σn∗
σn , = Π (h)h
ζn∗
ζn = , h
(16)
Integrating Eq. (18) twice with the boundary conditions (19) and (20), we get nπx nπ 1/2 cos σn Tf ξ(t)(hy − y 2 ) vx,n = − 2µL L ∂ζn (x, t) 1 2 γ kn − Π (h) (hy − y 2 ). − (21) 2µ ∂x From Eqs. (11) and (21) and using the boundary condition, y = 0,
vy = 0,
(22)
we get 2 kn2 hy y3 1/2 sin kn xσn Tf,n ξ(t) − µ 4 6 2 2 y3 k hy − . − n γ kn2 − Π (h) ζn (x, t) µ 4 6
vy = −
(23)
Therefore, the velocity of the gas–liquid interface due to imposed perturbations is given by ∂ζn (x, t) k 2 h3 1/2 = vζ = vy |y=h = − n sin kn xσn Tf,n ξ(t) ∂t 12µ + γ kn2 − Π (h) ζn (x, t) (24) or kn2 h3 2 γ kn − Π (h) ζn (x, t) dt 12µ k 2 h3 1/2 − n sin kn xσn Tf,n dW (t), 12µ
dζn (x, t) = −
(25)
where dW (t) is a Wiener process [24] defined as dW (t) = ξ(t) dt, 2 dW (t) = 0, dW (t) = dt.
(26) (27)
Defining an average perturbation as n ζ¯n (t) = L
L/n ζn (x, t) dx, 0
(28)
G. Narsimhan / Journal of Colloid and Interface Science 287 (2005) 624–633
we have from Eq. (25) kn2 h3 2 γ kn − Π (h) ζ¯n (t) dt 12µ k 2 h3 1/2 − n σn Tf,n dW (t). 6πµ
equilibrium film thickness will quickly die. Consequently, the equilibrium film thickness can be considered as a reflecting boundary. In other words,
dζ¯n (t) = −
(29)
For completely mobile film interface, no-slip boundary condition (20) is to be replaced with zero shear stress boundary condition dvx,n = 0, y = h, (30) dy and the corresponding stochastic differential equation for the perturbation is kn2 h3 2 γ kn − Π (h) ζ¯n (t) dt 3µ 2k 2 h3 1/2 − n σn Tf,n dW (t). 3πµ
dζ¯n (t) = −
(31)
Equation (29) is a stochastic differential equation describing the evolution of ζ¯n (t). This equation is written in Stratanovich sense [24]. Converting the above equation to its equivalent Ito form [24], we get dζ¯n = f (ζ¯n ) dt + g(ζ¯n ) dW (t). γ kn2 − Π (h) ζ¯n , 12µ h3 kn2 1/2 σn Tf,n , g(ζ¯n ) = − 6πµ
f (ζ¯n ) = −
(33) (34)
and for completely mobile gas–liquid interface, h3 kn2 2 γ kn − Π (h) ζ¯n , 3µ 2h3 kn2 1/2 σn Tf,n . g(ζ¯n ) = − 3πµ
f (ζ¯n ) = −
(35) (36)
Since ζ¯n (t) is a stochastic process, the evolution of mean disturbance on the film surface is random and is usually referred to as a sample path of the process. For an ensemble of large number of films with the same equilibrium thickness, the evolution of the disturbance on the film surface gives an ensemble of sample paths. From this ensemble, one can then define the probability that the mean disturbance is x at time t. One can then define p(x , t|x, 0) as the conditional probability that ζ¯n = x at time t with the initial condition x. The film will be stable if the perturbation is less than the equilibrium film thickness. We postulate, therefore, that the film ruptures when the average disturbance is the equilibrium film thickness. Therefore, p(h, t|x, 0) = 0.
dp(0, t|x, 0) = 0. (38) dx As pointed out earlier, we are interested in the average film rupture time. Consequently, the final point in the sample path is h. Therefore, we are interested in the evolution of this conditional probability backward in time. This is given by the backward Fokker–Planck equation [24] ∂p(x , t|x, 0) ∂p(x , t|x, 0) = f (x) ∂t ∂x 1 ∂ 2 p(x , t|x, 0) + g 2 (x) (39) , 2 ∂x 2 where f (x) and g(x) are given by Eqs. (29), (31) and (30), (32), respectively. One can define the probability G(x, t) that at time t after the imposition of disturbance the film is stable when the initial mean disturbance is x as h G(x, t) =
p(x , t|x, 0) dx .
(40)
0
(32)
For immobile gas–liquid interface, h3 kn2
627
(37)
Because of the restoring force of surface tension, any disturbance that tends to increase the film thickness above the
Equation (39) can then be integrated to yield ∂G(x, t) 1 2 ∂ 2 G(x, t) ∂G(x, t) = f (x) + g (x) . (41) ∂t ∂x 2 ∂x 2 First passage time is defined as the time after the imposition of disturbance at which the film ruptures. Since the imposed disturbance is random, the first passage time is also random which can be described by a distribution. The mean first passage time T (x) when the initial average disturbance is x is given by ∞ T (x) = −
∂G(x, t) dt = t ∂t
0
∞ G(x, t) dt.
(42)
0
Integrating Eq. (41) from 0 to ∞, we get ∞
∂G(x, t) dt = G(x, ∞) − G(x, 0) = −1 ∂t
0
dT (x) 1 2 d2 T (x) (43) + g (x) dx 2 dx 2 with the boundary conditions (obtained by integrating the boundary conditions (37) and (38)) = f (x)
x = 0, x = h,
dT (x) = 0, dx T (x) = 0.
(44) (45)
Note that boundary condition (44) is the reflecting boundary condition and (45) is the absorbing boundary condition. The solution of Eq. (43) with the boundary conditions (44) and
628
G. Narsimhan / Journal of Colloid and Interface Science 287 (2005) 624–633
(45) is given by [24] x T (x) = 2
dy ψ(y)
0
ψ(z) dz, g 2 (z)
(46)
y
h
where
x ψ(x) = exp
2 dx 2f (x )/g(x ) .
(47)
h
Since we are interested in calculating the average rupture time of an equilibrium film exposed to a periodic disturbance of a certain wavenumber, the mean rupture time T¯n is given by T¯n = T (0) = 2
0 h
dy ψ(y)
0
ψ(z) dz, g 2 (z)
(48)
y
where ψ is defined in terms of functions f and g written for wavenumber kn . The imposed pressure fluctuation of a certain wavenumber pn (t) is given by ∂vz,n pn (t) = 2µ (49) + γ kn2 − Π (h) ζn . ∂z z=h Neglecting the first term on the right-hand side, we get pn (t) ≈ γ kn2 − Π (h) ζn . (50) For thermal perturbation, the amplitude of perturbation is given by [25] 1/2 21/2 (kT )1/2 2 −1/2 γ kn − Π (h) ζn = ζn2 (51) = L with the corresponding amplitude of pressure fluctuation given by 1/2 21/2 (kT )1/2 2 γ kn − Π (h) . (52) L In the long wavelength limit, the amplitude of the squeezing mode of imposed perturbation relaxes as exp(−|t|/Tf ). Vrij [9] has treated this for immobile gas–liquid interface and has shown from the conservation of film volume that the timescale Tf is given by [25]
σn =
Tf =
24µ . kn2 h3 (γ kn2 − Π (h))
(53)
For completely mobile gas–liquid interface, we have [8] Tf =
6µ . kn2 h3 (γ kn2 − Π (h))
(54)
The disjoining pressure is given by [8,26] zeψ2 zeψ1 A tanh + 64n0 kT tanh Π(h) = − kT kT 6πh3 × exp(−κh), (55)
where A is the Hamaker constant, n0 is the number concentration of electrolytes, k is the Boltzmann’s constant, T is the temperature, κ is the Debye–Huckel parameter, and ψ1 and ψ2 are the surface potential at the gas–liquid interface and solid surface, respectively. Typical plot of disjoining pressure versus film thickness is shown in Fig. 2. As expected, the electrostatic interactions predominate over intermediate distances and van der Waals attraction predominates for very thin and thick films so that there is a maximum in the disjoining pressure at a critical film thickness hcrit . When h < hcrit , the imposed perturbations grow and hence the film is unstable. On the other hand, when h > hcrit , the imposed perturbations attenuate and conventional stability analysis predicts the film to be stable. Since the change in the disjoining pressure due to perturbation of the interface is expressed as Taylor series expansion around the equilibrium thickness retaining only the linear terms (see Eq. (14)), an average value of Π (h) was used for the unstable and stable branches of the disjoining pressure in the subsequent calculations.
3. Results and discussion When h < hcrit , the imposed perturbations grow with time thus rendering the film unstable. The average rupture time of an unstable film can be estimated from the evaluation of growth rate of perturbation via linear stability analysis. In the following, we will compare this time with the average rupture time estimated by the current analysis as given by Eq. (48). When h > hcrit , the imposed perturbations attenuate with time thus rendering the film stable. Existing stability analyses predict that the film is stable to all imposed perturbations. However, the current analysis can predict an average rupture time for this film as given by Eq. (48), this time being dependent on the amplitude of imposed perturbations. 3.1. Case I: unstable film The film is unstable when Π (h) > 0 (see Fig. 2). Linear stability analysis of growth of thermal perturbations yields the following expression for the growth coefficient for immobile gas–liquid interface [8,26]: β=
h3 (∂Π/∂h)2 . 48µγ
(56)
For completely mobile gas–liquid interface [8,26], βmobile =
h3 (∂Π/∂h)2 12µγ
with the dominant wavenumber kd given by [8,26]
1 ∂Π . kd ≈ 2γ ∂h
(57)
(58)
G. Narsimhan / Journal of Colloid and Interface Science 287 (2005) 624–633
629
Fig. 2. Plot of disjoining pressure and disjoining pressure gradient versus film thickness for the following parameters: A = 5 × 10−20 J, eψ1 /kT = 0.8, eψ2 /kT = 0.1, I = 0.01. Critical film thickness hcrit is the thickness at which disjoining pressure is maximum. The stable and unstable regions are shown in the figure. bn,mobile
The rupture time tr is given by h = h0,d exp(βtr ),
(59)
where h0,d is the amplitude of thermal fluctuation at the dominant wavenumber kd as given by Eq. (51). For an immobile gas–liquid interface, from Eqs. (56)–(59) we get
Lh Π (h) 1/2 48µγ ln , tr = 3 (60) 2 kT h (∂Π/∂h)2 and for completely mobile gas–liquid interface, we get
12µγ tr Lh Π (h) 1/2 ln . (61) tr,mobile = = 3 4 h (∂Π/∂h)2 2 kT
exp(x 2 ) erf(bn ) − erf(x) dx,
×
(64)
0
where the upper limit bn,mobile is given by bn,mobile =
(Π (h) − γ kn2 )1/2 31/2 πµ1/2 1/2
2kn h1/2 σn Tf,n
,
(65)
Tf,n being given by Eq. (54). It can be seen from Eqs. (62) and (64) that Tn,mobile = (1/4)Tn . A typical plot of the average rupture time for the current model as given by Eq. (62) for immobile gas–liquid interface
For the current analysis, for an immobile gas–liquid interface, Eq. (48) reduces to 12π 1/2 µ Tn = 2 3 kn h (Π (h) − γ kn2 )
bn
exp(x 2 ) erf(bn ) − erf(x) dx,
0
(62)
where the upper limit bn is given by bn =
(Π (h) − γ kn2 )1/2 31/2 πµ1/2 1/2
kn h1/2 σn Tf,n
,
(63)
Tf,n being given by Eq. (53). For thermal perturbation, the amplitude of pressure fluctuation σn is given by Eq. (52). For completely mobile gas–liquid interface, the current analysis gives Tn,mobile =
3π 1/2 µ kn2 h3 (Π (h) − γ kn2 )
Fig. 3. Plot of average rupture time versus wavenumber for an unstable film exposed to thermal perturbations. A = 5 × 10−20 J, eψ1 /kT = 0.8, eψ2 /kT = 0.1, I = 0.01, γ = 50 mN/m, L = 0.01 m, µ = 10−3 Pa s, h = 10−9 m, T = 298 K.
630
G. Narsimhan / Journal of Colloid and Interface Science 287 (2005) 624–633
Fig. 4. Plot of average rupture time versus film thickness for different Hamaker constants for an unstable film subjected to thermal perturbations. The other parameters are the same as in Fig. 3.
Fig. 5. Plot of average rupture time versus film thickness for different surface potential of air–liquid interface ψ1∗ = eψ1 /kT for an unstable film subjected to thermal perturbations. All the other parameters are the same as in Fig. 3.
for different wavenumbers is shown in Fig. 3. The rupture time increases dramatically at the critical wavenumber. The dominant wavenumber for minimum rupture time (maxi√ mum growth rate) is about 1/ 2 times the critical wavenumber, consistent with the results of linear stability analysis. This minimum rupture time at the dominant wavenumber is reported in subsequent calculations as the average rupture time. The average rupture time for the current model is compared with that predicted by linear stability analysis (Eq. (60)) for different parameters in Figs. 4–7. As expected,
Fig. 6. Plot of average rupture time versus film thickness for different surface tensions for an unstable film subjected to thermal perturbations. All the other parameters are the same as in Fig. 3.
Fig. 7. Plot of average rupture time versus film thickness for different viscosities for an unstable film subjected to thermal perturbations. All the other parameters are the same as in Fig. 3.
an increase in Hamaker constant reduces the rupture time, this effect being more pronounced for thicker films. Also, the rupture time is found to increase dramatically (see Fig. 4) near the critical film thickness. The effect of surface potential of the gas–liquid interface on the average rupture time is shown in Fig. 5. As expected, the rupture time increases with the surface potential. Also, the rupture time increases dramatically near the critical film thickness, the value of which decreases at higher surface potential. Interestingly, in the absence of electrostatic interactions (only van der Waals), the rupture time increases less rapidly with film thickness with the absence of a critical film thickness. The effects of surface
G. Narsimhan / Journal of Colloid and Interface Science 287 (2005) 624–633
631
tension and viscosity on the average rupture time are shown in Figs. 6 and 7, respectively. Since the ratio of the rupture times for immobile and completely mobile gas–liquid interfaces is 4 for both linear stability analysis and the current analysis, the comparison of rupture times for both analyses for the case of completely mobile gas–liquid interface is not shown here. 3.2. Case II: stable film The film is stable when Π (h) < 0 (see Fig. 2). For the current analysis, for a stable film, for immobile gas–liquid interface, Eq. (46) reduces to 12π 1/2 µ Tn = 2 3 kn h (γ kn2 − Π (h))
bn 2
exp(x ) erf(x) dx,
(66)
0
where the upper limit bn is given by hcrit (γ kn2 − Π (h))1/2 31/2 πµ1/2 1− , bn = 1/2 h kn h1/2 σn Tf,n
Fig. 8. Plot of average rupture time versus wavenumber for an unstable film subjected to pressure fluctuations. The parameters are A = 10−20 J, σ = 1000 Pa, h = 5 × 10−9 m, eψ1 /kT = 0.5, eψ2 /kT = 0.1, I = 0.01, µ = 10−3 Pa s, γ = 50 mN/m, L = 0.01 m.
(67)
Tf,n being given by Eq. (53). For thermal perturbation, the amplitude of pressure fluctuation σn is given by Eq. (52). The factor (1 − hcrit / h) is introduced in the upper limit in order to account for the fact that it is sufficient for the amplitude of perturbations to grow to (h − hcrit ) in order for the film to rupture. For completely mobile gas–liquid interface, we get 3π 1/2 µ Tn,mobile = 2 3 kn h (γ kn2 − Π (h))
bn,mobile
exp(x 2 ) erf(x) dx, 0
(68)
where the upper limit bn,mobile is given by hcrit (γ kn2 − Π (h))1/2 31/2 πµ1/2 bn,mobile = 1 − , (69) 1/2 h 2kn h1/2 σn Tf,n Tf,n being given by Eq. (54). Comparing Eqs. (66) and (68), it can be seen that Tn,mobile = (1/4)Tn . When bn 1, Eq. (66) can be approximated as Tn
12π 1/2 µ(1 − hcrit / h) kn2 h2 23/2 σn cn Π (h)h(1 − hcrit / h) 2 × exp , σn
(70)
where cn = π(1 + γ kn2 /Π (h))/23/2 . It can be seen that the exponential term in the above equation is the pressure barrier since the maximum disjoining pressure is at the film thickness hcrit . Typical plot of average rupture time versus wavenumber for a stable film (evaluated from Eq. (66)) as shown in Fig. 8 indicates the existence of an optimum wavenumber at which
Fig. 9. Plot of average rupture time versus film thickness for different amplitudes of imposed pressure fluctuations for a stable film. All the other parameter values are the same as in Fig. 8.
the rupture time is minimum. This is reported as the average rupture time. The variation of average rupture time with film thickness for different amplitudes of pressure fluctuation (including thermal perturbation) is shown in Fig. 9. The rupture time is indeed extremely large for thermal perturbations, decreasing considerably as the amplitude of pressure fluctuation increases. It is to be noted that the rupture time decreases dramatically as the film thickness approaches the critical film thickness. Fig. 10 shows that the average rupture time is sensitive to the surface potential increasing with the potential. The effect of pressure amplitude on the average rupture time is shown in Figs. 11a, 12a and 13 for different surface potentials, Hamaker constants and surface tension values, respectively. As expected, the average rupture time decreases with pressure amplitude. Also, larger Hamaker constants, smaller surface potentials and smaller surface tensions tend to decrease the average rupture time.
632
G. Narsimhan / Journal of Colloid and Interface Science 287 (2005) 624–633
Fig. 10. Plot of average rupture time versus film thickness for different surface potential of air–liquid interface ψ1∗ = eψ1 /kT for a stable film. All the other parameter values are the same as in Fig. 8.
(a)
(b)
(a)
Fig. 12. Plots of (a) average rupture time versus amplitude of imposed pressure fluctuation for different Hamaker constants and (b) average rupture time versus amplitude of imposed dimensionless pressure fluctuation for different Hamaker constants for a stable film. h = 10−8 m. All the other parameter values are the same as in Fig. 8.
(b) Fig. 11. Plots of (a) average rupture time versus amplitude of imposed pressure fluctuation for different surface potential of air–liquid interface and (b) average rupture time versus dimensionless amplitude of imposed pressure fluctuation for different surface potential of air–liquid interface. ψ1∗ = eψ1 /kT for a stable film. All the other parameter values are the same as in Fig. 8.
Fig. 13. Plot of average rupture time versus amplitude of imposed pressure fluctuation for different surface tensions for a stable film. All the other parameter values are the same as in Fig. 8.
G. Narsimhan / Journal of Colloid and Interface Science 287 (2005) 624–633
Fig. 14. Plot of ln Tn versus pressure barrier {Π (h)h(1 − hcrit / h)/σn }2 for different amplitudes of pressure fluctuation for a stable film. All the other parameter values are the same as in Fig. 8.
The effects of dimensionless pressure amplitude (as defined by Eq. (16)) on the average rupture time for different surface potentials and Hamaker constants are shown in Figs. 11b and 12b, respectively. Fig. 14 shows that the plot of ln Tn versus pressure barrier {Π (h)h(1 − hcrit / h)/σn }2 for different pressure amplitudes do fall approximately into a single straight line. Since the rupture time for completely mobile gas–liquid interface is 1/4 of that for immobile film interface provided the disjoining pressure is the same, the results of variation of rupture time for completely mobile interface are not shown here.
4. Conclusions The growth of perturbation of a thin film on a solid surface when subjected to random mechanical perturbation as described by a Gaussian white noise is analyzed. The growth of amplitude of perturbation is described by a stochastic differential equation. The sample paths of evolution of perturbation amplitude is analyzed by a first passage time analysis. The average rupture time (the average time taken for the amplitude to become equal to film thickness) is evaluated for different amplitudes and wavenumbers of imposed random mechanical perturbation. The results indicate the existence of an optimum wavenumber at which the rupture time is minimum. A critical film thickness is identified based on the sign of the gradient of disjoining pressure, below which the film is unstable in that the rupture time is very small. The
633
calculated values of rupture time from the present analysis were found to agree very well with the results of linear stability analysis. In addition, the optimum wavenumber at which the rupture time is minimum (or, equivalently, the growth coefficient is maximum) agreed with that given by linear stability analysis. The rupture time is also calculated for stable films for different amplitudes of imposed perturbation, film thickness and physical properties of the system. The rupture time was found to increase dramatically near the critical film thickness, this increase being more pronounced for films with electrostatic repulsions. The average rupture time for completely mobile gas–liquid interface is shown to be 1/4 of that for an immobile interface. As expected, the rupture time was found to be larger for smaller amplitude of imposed perturbation, larger film thickness, larger surface potentials, smaller Hamaker constants and larger surface tensions. The proposed analysis is extremely useful in the prediction of rupture of thin films when exposed to random mechanical perturbations.
References [1] I.B. Ivanov (Ed.), Thin Liquid Films: Fundametals and Applications, Dekker, 1988. [2] A. Scheludko, Kolloid Z. 191 (1963) 52. [3] E. Ruckenstein, C.S. Dunn, Thin Solid Films 51 (1978) 43. [4] A. Sharma, E. Ruckenstein, Am. J. Optom. Physiol. Opt. 62 (1985) 246. [5] A. Sharma, E. Ruckenstein, J. Colloid Interface Sci. 111 (1986) 8. [6] C. Malderelli, R. Jain, in: I.B. Ivanov (Ed.), Thin Liquid Films: Fundamentals and Applications, Dekker, 1988, pp. 497–568. [7] E. Ruckenstein, R. Jain, J. Chem. Soc. Faraday II 70 (1974) 132. [8] R. Jain, E. Ruckenstein, J. Colloid Interface Sci. 54 (1976) 108–116. [9] A. Vrij, Discuss. Faraday Soc. 42 (1966) 23. [10] A. Scheludko, Adv. Colloid Interface Sci. 1 (1967) 391–464. [11] B. Radoev, A. Scheludko, E. Manev, J. Colloid Interface Sci. 95 (1983) 254. [12] C. Malderelli, R. Jain, J. Colloid Interface Sci. 90 (1982) 263. [13] A. Vrij, F.Th. Hesselink, J. Lucassen, M. van den Temple, Proc. K. Ned. Akad. Wet. Ser. B 73 (1970) 124. [14] M.B. Williams, S.H. Davis, J. Colloid Interface Sci. 90 (1982) 220. [15] A. Sharma, E. Ruckenstein, J. Colloid Interface Sci. 113 (1986) 456. [16] A. Sharma, E. Ruckenstein, Langmuir 2 (1986) 480. [17] I.B. Ivanov, Pure Appl. Chem. 52 (1980) 1241. [18] I.B. Ivanov, D.S. Dimitrov, Colloid Polym. Sci. 252 (1974) 982–990. [19] A. Sharma, J. Colloid Interface Sci. 199 (1998) 212–214. [20] A. Sharma, J. Mittal, Phys. Rev. Lett. 89 (2002) 186101. [21] D.S. Dimitrov, Prog. Surf. Sci. 14 (1983) 295. [22] J.E. Coons, P.J. Halley, S.A. McGlashan, T. Tran-Cong, Adv. Colloid Interface Sci. 105 (2003) 3–62. [23] R. Tsekov, B. Radoev, Adv. Colloid Interface Sci. 38 (1992) 353–369. [24] C.W. Gardiner, Handbook of Stochastic Methods for Physics, Chemistry and Natural Sciences, Springer-Verlag, 1983. [25] J.G.H. Joosten, in: I.B. Ivanov (Ed.), Thin Liquid Films: Fundamentals and Applications, Dekker, 1988, pp. 569–662. [26] D.A. Edwards, H. Brenner, D.T. Wasan, Interfacial Transport Processes and Rheology, Butterworth–Heinemann, 1991.