Langmuir 2007, 23, 2437-2443
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Rupture of Draining Foam Films Due to Random Pressure Fluctuations Zebin Wang and Ganesan Narsimhan* Biochemical and Food Process Engineering, Department of Agricultural and Biological Engineering, Purdue UniVersity, West Lafayette, Indiana 47907 ReceiVed May 30, 2006. In Final Form: December 5, 2006 A generalized formalism for the rupture of a draining foam film due to imposed random pressure fluctuations, modeled as a Gaussian white noise, is presented in which the flow inside the film is decomposed into a flow due to film drainage and a flow due to imposed perturbation. The evolution of the amplitude of perturbation is described by a stochastic differential equation. The rupture time distribution is calculated from the sample paths of perturbation amplitude as the time for this amplitude to equal one-half the film thickness and is calculated for different amplitudes of imposed perturbations, film thicknesses, electrostatic interactions, viscosities, and interfacial mobilities. The probability of film rupture is high for thicker films, especially at smaller times, as a result of faster growth of perturbations in a thick film due to a smaller disjoining pressure gradient. Larger viscosity, larger surface viscosity, higher Marangoni number, and smaller imposed pressure fluctuation result in slower growth of perturbation of a draining film, thus leading to larger rupture time. It is shown that a composite rupture time distribution combining short time simulation results with equilibrium distribution is a good approximation.
Introduction A foam is a high volume fraction dispersion of gas in a liquid in which the gas bubbles are distorted in the form of polyhedra and are separated by thin films. Three adjacent thin films intersect in a Plateau border, and the continuous phase is interconnected through a network of Plateau borders. Foams are encountered in a variety of applications such as detergency, personal care products, food products, fire fighting, enhanced oil recovery, textiles, froth flotation, and foam fractionation and are usually formed by incorporating a high volume fraction of gas in a surfactant solution, which stabilizes the bubbles. The pressure in the films is less than the pressure in Plateau borders because of the radius of curvature of a Plateau border.1 This Plateau border suction leads to drainage of liquid from thin films to the neighboring Plateau border. This is counteracted by disjoining pressure caused by van der Waals, electrostatic, and steric interactions between two approaching faces of a draining film.2 The draining thin film may reach an equilibrium at which the Plateau border suction is counterbalanced by the disjoining pressure. The draining film may rupture due to the growth of imposed thermal and mechanical perturbations, thus leading to the coalescence of neighboring bubbles. Therefore, the stability of thin films greatly influences foam stability. The film will attain an equilibrium thickness if the time scale of film drainage is much smaller than the film rupture time. The rupture time of equilibrium films due to thermal perturbations have been calculated using linear3-7 and nonlinear8-10 stability analysis in * To whom correspondence should be addressed. Phone: (765) 4941199; e-mail:
[email protected]. (1) Bikerman, J. J. Foams; Springer: New York, 1973. (2) Ivanov, I. B. Thin Liquid Films: Fundamentals and Applications; MarcelDekker: New York, 1988. (3) Ruckenstein, E.; Jain, R. Spontaneous Rupture of Thin Liquid-Films. J. Chem. Soc., Faraday Trans. 2 1974, 70, 132. (4) Vrij, A.; Hesselink, F. T.; Lucassen, J.; van den Temple, M. Waves in Thin Liquid Films. 2. Symmetrical Modes in Very Thin Films and Film Rupture. Proc. K. Ned. Akad. Wet., Ser. B 1970, 73, 124. (5) Scheludko, A. Thin Liquid Films. AdV. Colloid Interface Sci. 1967, 1, 391-464. (6) Radoev, B.; Scheludko, A.; Manev, E. Critical Thickness of Thin Liquid FilmssTheory and Experiment. J. Colloid Interface Sci. 1983, 95, 254.
which the growth of an imposed thermal perturbation of different wavenumbers was investigated. The surface tension and repulsive interactions between the two faces of the film will attenuate the imposed perturbations, whereas the attractive van der Waals interactions will result in their growth. The film is deemed stable if the imposed perturbation decays and is considered to be unstable if the imposed perturbation grows. It has been shown that an imposed perturbation grows whenever the disjoining pressure gradient is positive, and decays whenever it is negative.11 For equilibrium films, the rupture time is considered the time needed for the amplitude of imposed perturbation of optimum wavenumber (of maximum growth rate) to attain one-half the film thickness. If the time scale of film drainage is comparable to the film rupture time, however, the growth of perturbation during film drainage will influence film rupture.5,12 In a draining unstable film, an imposed perturbation grows while the film thickness decreases with time. For unstable films, the stability analysis indicates the existence of a transition film thickness at which the perturbation grows.12 Consequently, the critical thickness of film rupture is the film thickness at which the amplitude of imposed perturbation of optimum wavenumber (of maximum growth rate) becomes equal to one-half the film thickness.2,12 The effect of local variations in film thickness on critical thickness13 is discussed, and scaling laws in terms of physical properties of the film have been proposed.14,15 According to the conventional stability analysis, a draining foam film stabilized by repulsive (7) Malderelli, C.; Jain, R. The Hydrodynamic Stability of Thin Films. In Thin Liquid Films: Fundamentals and Applications; Ivanov, I. B., Ed.; Marcel Dekker: New York, 1988; Vol. 29, pp 497-568. (8) Williams, M. B.; Davis, S. H. Non-Linear Theory of Film Rupture. J. Colloid Interface Sci. 1982, 90, 220. (9) Sharma, A.; Ruckenstein, E. An Analytical Nonlinear Theory of ThinFilm Rupture and its Application to Wetting Films. J. Colloid Interface Sci. 1986, 113, 456. (10) Sharma, A.; Ruckenstein, E. Finite-Amplitude Instability of Thin Free and Wetting FilmssPrediction of Lifetimes. Langmuir 1986, 2, 480. (11) Jain, R.; Ruckenstein, E. Stability of Stagnant Viscous Film on a Solid Surface. J. Colloid Interface Sci. 1976, 54, 108-116. (12) Ivanov, I. B.; Dimitrov, D. S. Hydrodynamics of Thin Liquid Films. Effect of Surface Viscosity on Thinning and Rupture of Foam Films. Colloid Polym. Sci. 1974, 252 (11), 982-990. (13) Manev, E. D.; Nguyen, A. V. Critical Thickness of Microscopic Thin Liquid Films. AdV. Colloid Interface Sci. 2005, 114-115, 133-146.
10.1021/la061536m CCC: $37.00 © 2007 American Chemical Society Published on Web 01/23/2007
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interactions (electrostatic and/or steric) is always stable to an imposed disturbance since the film thickness always lies on the branch of disjoining pressure curve for which the disjoining pressure gradient is negative and therefore cannot fully explain instability in such systems. In real systems, the foam is usually exposed to random mechanical perturbations, and therefore the film may be subjected to a series of perturbations of different amplitudes and frequencies at different times. Even though the film is stable to a single imposed perturbation, the cumulative effect of these random perturbations may result in its rupture. In recent publications,16,17 we have presented a methodology for the evaluation of mean rupture time and the rupture time distribution of equilibrium nondraining films when exposed to random pressure fluctuations modeled as Gaussian white noise using first passage time analysis. The results of this analysis for thermal perturbations has been shown to agree with conventional stability analysis for unstable film on a solid substrate16 as well as for an unstable foam film.17 In this paper, we present the analysis for the calculation of rupture time distribution of a draining foam film due to imposed pressure fluctuation modeled as Gaussian white noise. The flow inside the draining film is decomposed into a flow due to film drainage and a flow due to imposed perturbations. The effects of the surface tension gradient as well as surface viscosity on film drainage and the growth of imposed perturbations are considered in the analysis. The growth of perturbations is expressed in terms of a stochastic differential equation, which is solved along with a film drainage equation to obtain sample paths of amplitude of perturbations. These sample paths are then analyzed to obtain the rupture time distribution of the film. The current analysis can explain the observed initial rapid coarsening of bubbles in a standing foam18-20 in that thicker draining foam films tend to be less stable because of the tendency for faster growth of imposed random perturbations.
Model for the Growth of Perturbation in a Thinning Film Consider a circular thin liquid film between two foam bubbles. Liquid in the film drains into the neighboring Plateau border because of Plateau border suction. The rupture of thin film occurs as a result of thermal and mechanical perturbations, which result in pressure fluctuations that are imposed on the film surface at different times. Even though the draining film may be stable to a single perturbation, the cumulative effect of these perturbations may result in film rupture (Figure 1). The imposed pressure fluctuation p′g(r,t) can be expressed as a superposition of disturbances of waves of different wavenumbers and amplitudes. We shall only consider the case of asymmetrical disturbances (Figure 1) in the squeezing mode that are known to be most destabilizing for film rupture.12 The effect of such imposed random (14) Coons, J. E.; Halley, P. J.; McGlashan, S. A.; Tran-Cong, T. Scaling Laws for the Critical Rupture Thickness of Common Thin Films. Colloids Surf., A: Physicochem. Eng. Aspects 2005, 263 (1-3 SPEC ISS), 258-266. (15) Coons, J.; Halley, P.; McGlashan, S.; Tran-Cong, T. Bounding the Stability and Rupture Condition of Emulsion and Foam Films. Chem. Eng. Res. Des. 2005, 83 (7A), 915-925. (16) Narsimhan, G. Rupture of Thin Stagnant Films on a Solid Surface Due to Random Thermal and Mechanical Perturbations. J. Colloid Interface Sci. 2005, 287, 624-633. (17) Narsimhan, G.; Wang, Z. Rupture of Equilibrium Foam Films Due to Random Thermal and Mechanical Perturbations. Colloids Surf., A: Physicochem. Eng. Aspects 2006, 282, 24-36. (18) Wang, Z.; Narsimhan, G. Evolution of Liquid Holdup Profile in a Standing Protein Stabilized Foam. J. Colloid Interface Sci. 2004, 280 (1), 224-233. (19) Wang, Z.; Narsimhan, G. Model for Plateau Border Drainage of Power Law Fluid with Mobile Interface and Its Application to Foam Drainage. J. Colloid Interface Sci. 2006, 300 (1), 327-337. (20) Du, L.; Prokop, A.; Tanner, R. D. Variation of Bubble Size Distribution in a Protein Foam Fractionation Column Measured Using a Capillary Probe with Photoelectric Sensors. J. Colloid Interface Sci. 2003, 259 (1), 180-185.
Wang and Narsimhan
disturbances of a certain wavenumber kn on the shape of the film surface is investigated below. The imposed pressure disturbance therefore satisfies
(
)
1 d dp′g,n(r,t) r ) -kn2p′g,n(r,t) r dr dr
(1)
and is assumed to be a Gaussian process,16,17 that is,
p′g,n(r,t) ) An(r)σnTf1/2ξ(t)
(2)
where σn is the amplitude of the imposed pressure fluctuation, Tf is the time scale of the fluctuation and ξ(t) is the white noise process, that is, 〈ξ(t)〉 ) 0; 〈ξ(t)ξ(t′)〉 ) δ(t - t′), where 〈 〉 refers to the ensemble average, and δ(t) is the Dirac delta function. The solution of eq 1 with the boundary condition, pg,n(R) ) 0 yields
p′g,n(r,t) ) J0(knr)σnTf1/2ξ(t)
(3)
where J0(x) is the Bessel function of zero order, and kn ) λn/R, with λn being the zero of J0(x). The resulting perturbation ζn(r,t) of the film interface will also be periodic and satisfies
∇2ζn(r,t) )
(
)
1 d dζn(r,t) r ) -kn2ζn(r,t) r dr dr
(4)
The imposed pressure disturbance will augment the flow caused by film drainage. The equations of continuity and motion for the flow in the film are given by
∂Vz 1 ∂ + (rV ) ) 0 ∂z r ∂r r ∂2Vr ∂z
2
)
1 ∂p µ ∂r
∂p )0 ∂z
(5)
(6) (7)
where r and z are the radial and axial coordinates, respectively (z ) 0 refers to the plane of symmetry), p is the pressure, Vz and Vr are the two velocity components, and µ is the viscosity of the liquid. The validity of quasi steady-state and lubrication approximations for main flow due to film drainage12 as well as flow due to imposed disturbance21 has been discussed. The equation of continuity for the surfactant for the quasi steady state yields
∂2c 1 ∂ ∂c + r )0 ∂z2 r ∂r ∂r
( )
(8)
with c being the surfactant concentration, with the boundary conditions
c is finite r ) 0
(9)
c ) c 0, r ) R
(10)
∂c ) 0, z ) 0 ∂z
(11)
1 ∂ ∂c (rV Γ) -D |z)h/2 ) ∂z r ∂r r,s
(12)
where D and Γ are the diffusion coefficient and surface concentration of surfactant, respectively, and Vr,s is the radial
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Figure 1. Rupture of a draining film. (A) Nonequilibrium film with an initial thickness of h0. (B) Growth of surface perturbation and decrease in film thickness. (C) Rupture of the film. (D) Disjoining pressure profile. p′g: pressure fluctuation; ξ(t): surface perturbation at time t; h(t): film thickness at time t; heq: equilibrium film thickness; htr: transition film thickness at which dΠ/dh ) 0; pc: capillary pressure.
velocity at the interface. In writing the above equation, it is assumed that the diffusion of surfactant from the bulk solution to the subsurface is slower than the transfer of surfactant from the subsurface to the surface.22 In addition, if Γ ) Γ0 + Γ1, where Γ0 is the equilibrium surface concentration and Γ1 is the deviation from the equilibrium due to interfacial mobility, we assume that Γ1 , Γ012,22 so that eq 12 can be approximated as
1 ∂ ∂c (rV ) -D |z)h/2 ) Γ0 ∂z r ∂r r,s
(13)
The normal stress boundary condition at the gas-liquid interface yields the following for the pressure in the thin film: 12,16,17
p + p′n(r,t) ) pg + p′g,n(r,t) - γ∇2ζn(r,t) - Π(h) 2Π′(h)ζn(r,t) (14) where p is the pressure in the film, γ is the surface tension, pg is the mean pressure in the gas phase, and Π(h) is the disjoining pressure for a draining film of mean thickness h(t). 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. The last term is the change in the disjoining pressure due to imposed perturbation. The shear stress balance at the gas-liquid interface is given by
∂γ ∂c ∂ 1 ∂ h ∂Vr,n + µs z) ,µ ) (rV ) 2 ∂z ∂c ∂r ∂r r ∂r r,s
( )( )
the film interface ζn(r,t) can be expressed in terms of as
ζn(r,t) ) ξn(r,t)
Substituting all the variables in terms of , using eq 16, into eqs 5-15 and equating the terms of zero order in , we get identical equations of continuity, motion, surfactant balance, and boundary conditions as given by eqs 5-15, except that one cannot write the normal stress boundary condition for a plane parallel film but has to impose an overall force balance.2 The solution of these equations for the velocity of film drainage V is given by2,12
V)-
dh
)
dt h3∆p 24µR2
(
In the above equation, µs is the surface viscosity, and the first term on the right-hand side is the Marangoni stress. The flow field can be subdivided into main flow due to film drainage and perturbation to the main flow due to the superposition of random pressure fluctuation, that is,
Vz ) V0z + V1z ; Vr ) V0r + V1r ; p ) p0 + p1; c ) c0 + c1 (16) where the perturbation parameter ) σn/pg. The perturbation of (21) Narsimhan, G.; Wang, Z. Effect of Interfacial Mobility on Rupture of Stagnant Films on a Solid Surface Due to Random Mechanical Perturbations. J. Colloid Interface Sci. 2006, 298 (1), 491-496. (22) Valkovska, D. S.; Danov, K. D.; Ivanov, I. B. Stability of Draining PlaneParallel Films Containing Surfactants. AdV. Colloid Interface Sci. 2002, 96 (13), 101-29.
∞
6µ + µskn2RhJ2(λn)
n)1
(6µ + 6µR + µskn2Rh)λn3J1(λn)
∑
)
-1
(18)
where R is the film radius, ∆p is the pressure difference causing the drainage, and R is a dimensionless number characterizing the Marangoni effect defined as
R)-
3Dµ Γ0(∂γ0/∂c0)
(19)
The disjoining pressure can be calculated by17
Π(h) ) (15)
(17)
zeψ A + 64n0kT tanh2 exp(-κh) (20) 3 kT 6πh
( )
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 ψ is the surface potential at the air-liquid interface. It is to be noted that ∆p ) pc - Π(h), where pc is the capillary pressure. First order: Substituting all the variables in terms of , using eq 16, into eqs 5-15 and equating the terms of first order in , we get
∂V1z 1 ∂ 1 + (rV ) ) 0 ∂z r ∂r r ∂2V1r ∂z2
)
1 ∂p1 µ ∂r
∂p1 )0 ∂z
(21) (22) (23)
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Wang and Narsimhan
( )
∂2c1 1 ∂ ∂c1 + r )0 r ∂r ∂r ∂z2
(24)
c1 is finite r ) 0
(25)
c1 ) 0, r ) R
(26)
Tf,n(h) )
{[
(27)
1 ∂ 1 ∂c1 (rV ) -D |z)h/2 ) Γ0 ∂z r ∂r r,s
(28)
p1(r,t) ) J0(knr)pgTf,n1/2ξ(t) - γ∇2ξn(r,t) - 2Π′(h)ξn(r,t) (29)
( )( )
1
h ∂Vr,n ∂γ ∂c1 ∂ 1 ∂ 1 + µs z) ,µ ) (rV ) 2 ∂z ∂c ∂r ∂r r ∂r r,s
(30)
The solution of eq 24 with boundary conditions 25-27 gives
c1(r,z) ) CnJ0(knr) cosh(knz)
∂V1z )0 z ) 0, ∂z
(32)
we get 1 )Vr,n
kn 1 p T 1/2J (k r)ξ(t)z2 + (γkn2 - 2Π′ 2µ g f,n 1 n 2µ ∂ξn 2 (h)) z + C6(r) (33) ∂r
Using shear stress balance (eq 30), recasting in terms of perturbation ζn(r,t) using eq 17, and defining
ζhn(t) )
∫0R rζn(r,t)dr
2 R2
(34)
we get
[
Mγ cosh
( )
k nh 2
}
( )]
knh knh + Mµ sinh knh 2 2 (37)
From eq 35, we see that the variance 〈dζn(t)dζn(t)〉, up to linear terms in dt, is given by
kn2h(t)3 1 J12(λn)σn2 〈dζn(t)dζn(t)〉 ) 6µ λ 2(γk 2 - 2Π′(h))
{
n
n
( ) ( ) sinh
1+
knh(t) 2
(
)}
knh(t) knh(t) Mγknh(t) cosh + Mµknh(t) sinh 2 2
dt
(38)
(31)
Substituting eq 29 into eq 22 and integrating using the boundary condition
( )
sinh
kn2h3(γkn2 - 2Π′(h)) 1 +
1
∂c ) 0, z ) 0 ∂z
24µ
Since Π′(h) < 0 for a draining film, the above equation indicates that the growth rate of variance of surface fluctuation is greater for larger amplitudes of imposed pressure disturbance and is strongly influenced (square dependence) by it. Also, because of the increase in the magnitude of Π′(h) for a smaller film thickness for a draining film, the growth rate is smaller for smaller film thicknesses. In addition, the growth rate decreases for larger viscosity and larger values of Mµ and Mγ. As will be shown in the next section, the results of rupture time distribution obtained from the sample path calculations are consistent with the above observations.
Results Equation 35 is a stochastic differential equation for the average amplitude of perturbation ζhn. The film thickness h is to be evaluated at different times using eq 18. As can be seen from eq 35, the deterministic term on the right-hand side has a stabilizing influence on imposed perturbations whenever Π′(h) < 0, that is, whenever h > htr (htr, the transition thickness, is the film thickness at which the disjoining pressure is maximum; see
kn2h(t)3 2 dζhn(t) ) J (λ )σ T 1/2dW(t) + 24µ λn 1 n n f,n
{
1+
]
(γkn2 - 2Π′(h(t)))ζhn(t)dt ×
( ) ( ) sinh
knh(t) 2
( )
knh(t) knh(t) Mγknh(t) cosh + Mµknh(t) sinh 2 2
}
(35)
where dW(t) is a Weiner process, that is, 〈dW(t)〉 ) 0, 〈dW2(t)〉 ) dt, and the dimensionless numbers Mµ and Mγ are defined as
Mµ )
µskn ∂γ Γ0 ; Mγ ) µ ∂c Dµ
( )
(36)
In the above equation, Mµ is the dimensionless surface viscosity, and Mγ is the Marangoni number. It is to be noted that the parameter R as defined in eq 19 is equal to 3/Mγ. The time scale of pressure fluctuation is given by17
Figure 2. Sample paths of a foam film with an initial thickness of 25 nm. Other parameters are ψ* ) 0.3, µ ) 1 cP, A ) 10-20 J, γ ) 50 mN/m, R ) 0.1 mm, I ) 0.01, Mγ ) 104, µs ) 1 mN‚s/m,R ) 3 × 10-4, pc ) 8 kPa, and σn ) 20 kPa. The rupture time of the draining film is considered the time at which the average amplitude of an asymmetric perturbation in the squeezing model is (h - htr)/2 since the thinnest part of the film will rupture once its thickness reaches htr.
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Figure 3. Mean rupture time (normalized to its value at the dimensionless time interval of 0.0001) versus dimensionless time interval employed for the solution of eq 35 for different film thicknesses. Other parameters are ψ* ) 0.3, µ ) 1 cP, A ) 10-20 J, γ ) 50 mN/m, R ) 0.1 mm, and I ) 0.01; the film is immobile.
Figure 4. Comparison of mean rupture time obtained by current simulation with those obtained by the Fokker-Planck equation for different dimensionless surface potentials (φ* ) zeψ/kT). Other parameters are µ ) 1 cP, A ) 10-20 J, γ ) 50 mN/m, R ) 0.1 mm, and I ) 0.01; the gas-liquid interface of the film is immobile.
Figure 5. Rupture probability density of a foam film with an initial thickness of 20 nm. Other parameters are the same as those in Figure 4.
Figure 1D), the film is stable to a single imposed perturbation, as given by conventional stability analysis. However, the film may still rupture because of the cumulative effect of imposed random perturbations as dictated by the first random term. For a draining film, h > htr. Consequently, Π′(h) < 0. The draining film may reach mechanical equilibrium when the capillary pressure is counterbalanced by the disjoining pressure, provided the film does not rupture during drainage. Consistent with earlier observations for an equilibrium film,17 the growth of perturbation is found to be maximum for the first wavenumber k1 for a draining film. Consequently, the reported results are for the calculations
Figure 6. Cumulative rupture time distributions of draining films with different initial film thicknesses at (A) long time range and (B) short time range. Lines A-E are cumulative rupture time distributions for films with initial thicknesses of 25, 23, 20, 15, and 8.236 (equilibrium thickness) nm, respectively. Lines a-e are film thickness variations for films with corresponding initial film thicknesses. Other parameters are the same as those in Figure 4.
Figure 7. Constructed and simulated cumulative rupture time distribution of a thinning film with different initial film thicknesses. Other parameters are the same as those in Figure 4.
for wavenumber k1. Equation 35 was solved along with eq 18 to obtain a sample path for the evolution of ζhn using a standard Euler scheme.23 Since the linearization of the disjoining pressure is indeed valid only for sufficiently small perturbations, in the numerical simulation of the sample paths, the disjoining pressure gradient Π′(h) is replaced by the approximation Π′(h,ξh) ) {Π(h + ξh) - Π(h)}/ξh and is evaluated for each step. This approximation implies that Π′(h,ξh) has to be updated at each step. As can be (23) Milstein, G. N. Numerical Integration of Stochastic Differential Equations; Kluwer Academic Publishers: Norwell, MA, 1988.
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Figure 8. Cumulative rupture time distribution (thick lines) and film thickness (thin line) of a thinning film for different amplitudes σ of imposed pressure fluctuations: A - 20 kPa; B - 15 kPa; C - 10 kPa. The film thickness is 25 nm. The other parameters are the same as those in Figure 4.
seen, this approximation does indeed approach Π′(h) when the disturbance amplitude ξh is small. Typical sample paths are shown in Figure 2 along with the evolution of thickness of a draining film. Since the film becomes unstable when h e htr, leading to its spontaneous rupture, it is sufficient for the amplitude of perturbation to grow to (h/2)(1 - (htr/h)) in order for the film to rupture. Consequently, the rupture time of the film is taken as the time at which the amplitude is equal to (h/2)(1 - (htr/h)). The calculations of several (10 000) sample paths resulted in the evaluation of rupture time distribution and mean rupture time. Evaluation of mean rupture time for different dimensionless time steps () ∆t/Tf,n) indicated that the mean rupture time converged when the dimensionless time step was 0.001 or smaller, even though the results fluctuated for larger time steps, as shown in Figure 3. As a result, all calculations were carried out using a dimensionless time step of 0.001. The validity of the calculation scheme was verified by comparing the results of mean rupture time for a nondraining equilibrium film with the analytical results obtained using first passage time analysis.17 The results of the current calculations compared well with the analytical results (Figure 4). In this paper, we report only those results of the calculation that demonstrate the effect of a film thinning on its stability. The specific values of parameters that were employed in these reported calculations are typical values we have encountered in protein stabilized foam. Calculations were also performed for different dimensionless amplitudes of pressure disturbance, dimensionless surface potentials, surface tensions, and so forth. The qualitative nature of the effect of these parameters is similar to that reported in an earlier paper for equilibrium film.17 Therefore, these results are not reported here. The rupture time distribution of a draining film is shown in Figure 5. The distribution is unimodal and exhibits a large peak at a very short time followed by a long tail, that is, the probability of film rupture is large at very small times. However, because of the large tail, the mean rupture time is large. For a thick film, |Π′(h)| is small.17 As a result, the attenuation of perturbations is small, thus leading to faster growth of perturbations. In addition, the film thickness decreases due to thinning. As a result, the probability of film rupture is large. As the film thickness decreases, however, |Π′(h)| increases,17 thus leading to much slower growth of perturbations. This explains the large tail in the rupture time
distribution. In other words, a draining film has a higher probability of rupture than an equilibrium film of much smaller thickness. The cumulative rupture time distribution of films of different initial thicknesses at small and large times are shown in Figure 6a,b, respectively. Because of faster growth of imposed perturbations due to a smaller disjoining pressure gradient as explained above, the probability of rupture of thicker films was found to be larger at smaller times (see Figure 6b). At larger times, however, as expected, the cumulative rupture distribution increased faster for thinner films (see Figure 6a). It is interesting to note that the results indicate a higher probability of rupture for thicker films at smaller times as a result of imposed Gaussian pressure perturbations. In order to reduce the computation time for the evaluation of rupture time distribution, the distribution was constructed as the sum of two conditional probability distributions as given by
P(t) ) fP1(t|t e T0) + (1 - f)P2(t|t > T0) (39)
(39)
where P(t) is the probability distribution of film rupture, f is the fraction of films of rupture time eT0, P1(t|t e T0) is the conditional probability distribution of film rupture for films of rupture time e T0, and P2(t|t > T0) is the conditional probability distribution of film rupture for films of rupture time > T0. T0 is chosen such that (h(T0) - heq)/heq e δ, with δ being a small number. P1(t|t e T0) and f are evaluated by the solution of eqs 35 and 18 to generate sample paths up to t ) T0, and P2(t|t > T0) is evaluated from the first passage time analysis of film rupture for an equilibrium film17 as given in our earlier paper. The rupture time distributions as obtained from eq 39 for different initial film thicknesses were found to agree very well (Figure 7) with the distributions obtained by the generation of sample paths for sufficiently long times. The effect of the amplitude of imposed pressure fluctuation on the cumulative rupture time distribution of a draining film is shown in Figure 8. As expected, the probability of film rupture increases with an increase in the amplitude of imposed pressure fluctuation. Also, the probability of film rupture at small times is found to be larger for higher pressure amplitudes. The effect of viscosity on the thickness of a draining film as well as the cumulative film rupture time distribution are shown in Figure 9a. As expected, the film drains
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Figure 9. Cumulative rupture time distribution (thick lines) and film thickness (thin line) of a thinning film for different bulk viscosities in (A) actual time and (B) dimensionless time defined in eq 40. Initial film thickness is 25 nm. Other parameters are the same as those in Figure 4.
slower for higher viscosity, with the corresponding probability of rupture being smaller. It is to be noted that the equilibrium film thickness for a draining film is independent of viscosity, as can be seen from Figure 9a. The cumulative rupture distribution was nondimensionalized with respect to dimensionless time t* defined as
t* )
t Tf(heq)
(40)
with Tf(heq)being given by eq 37. Interestingly, the film thickness as well as the dimensionless cumulative rupture time distributions for different viscosities collapse into a single curve, as shown in Figure 9b. The effects of surface viscosity and the Marangoni effect on the rupture time distribution at short times are shown as plots of cumulative rupture time distribution for different values of µs and Mγ in Figure 10a,b, respectively. Higher values of the surface viscosity and Marangoni number result in slower growth of perturbations as well as slower film thinning, thus providing more stability. Results of the rupture time distribution at longer times (not shown here) also indicate that draining films of higher surface viscosity and Marangoni number are more stable.
Figure 10. Cumulative rupture time distributions of a foam film with an initial thickness of 25 nm with (A) different µs and (B) different Mγ. In panel A, Mγ ) 0.1, and lines A-C are for µs values of 0.01, 0.1, and 1 mN‚s/m, respectively. In panel B, µs ) 0, and lines A-C are for Mγ values of 0.1, 1, and 1000, respectively.
Conclusions The growth of perturbation due to random pressure fluctuations modeled as Gaussian white noise in a draining film is expressed as a stochastic differential equation, which was solved to obtain sample paths of perturbation amplitude. The film rupture time is defined as the time at which this amplitude becomes equal to one-half the thickness of a draining film. The sample paths were analyzed to obtain the rupture time distribution. The results indicate that the probability of rupture of a draining film is high at short times as a result of the faster growth of perturbations in a thick film due to a smaller disjoining pressure gradient. The rupture time distribution is unimodal with a sharp peak at small times and a long tail, thus resulting in a large mean rupture time. Also, the probability of rupture is larger at smaller times for thicker draining films. Larger viscosity, larger surface viscosity, higher Marangoni number, and smaller imposed pressure amplitude result in the slower growth of perturbation of a draining film, thus leading to a larger rupture time. It was shown that composite rupture time distribution combining short time simulation results with analytical equilibrium rupture time distribution agrees with the simulation results for the entire time range. LA061536M
