Stability of thin stagnant film on a solid surface with a viscoelastic air–liquid interface

Journal of Colloid and Interface Science 291 (2005) 296–302 www.elsevier.com/locate/jcis

Note

Stability of thin stagnant film on a solid surface with a viscoelastic air–liquid interface Ganesan Narsimhan ∗ , Zebin Wang Biochemical and Food Process Engineering, Department of Agricultural and Biological Engineering, Purdue University, West Lafayette, IN 47907, USA Received 11 March 2005; accepted 28 April 2005 Available online 31 May 2005

Abstract Linear stability analysis for a film on a solid surface with a viscoelastic air–liquid interface is presented. The interfacial dilatational and shear viscoelastic properties were described by Maxwell models. Dilatational and shear interfacial elasticity and viscosity were shown to improve film stability. When the interfacial rheological properties are extremely large or small, the maximum perturbation growth coefficient is shown to reduce to those for immobile and mobile interfaces respectively. Calculated values of maximum growth coefficient for thin film stabilized by 0.5% β-lactoglobulin approached those of mobile films for thick (>2000 nm) and those for immobile films for thin (<100 nm) films respectively with the values lying between the two limits for intermediate film thicknesses.  2005 Elsevier Inc. All rights reserved. Keywords: Film stability; Viscoelastic interface; Surface dilatational viscosity; Surface dilatational elasticity; Surface shear viscosity; Surface shear elasticity; Perturbation growth coefficient; Thin film on solid; Maxwell model; Viscoelasticity; Linear stability

1. Introduction Thin liquid films bounded by two fluids or by a fluid and a solid are encountered in many colloidal systems such as foams, emulsions, flotation, etc. Drainage of thin film in such systems occurs because of capillary forces. When the film thickness becomes of the order of nanometers, intermolecular interactions between the two faces of the film due to van der Waals, electrostatic, steric, hydration and depletion forces influence further film drainage and stability. In some cases, the film reaches an equilibrium thickness at which the disjoining pressure counterbalances the capillary forces. Several investigators [1–8] have carried out linear stability analysis of equilibrium thin film accounting for intermolecular forces to calculate the growth rate of perturbations and timescale of film rupture. These intermolecular forces have been accounted for either as a body force [2] or as a disjoining pressure in the normal stress boundary condition [9]. * Corresponding author. Fax: +1 (765) 496 3816.

E-mail address: [email protected] (G. Narsimhan). 0021-9797/$ – see front matter  2005 Elsevier Inc. All rights reserved. doi:10.1016/j.jcis.2005.04.093

In addition, critical film thickness of a draining film with mobile interfaces have been evaluated [10,11]. The effects of hydrophobic interactions [12], density variations [13] and bulk viscoelasticity [14] on film stability have been investigated. Influence of interfacial rheology on foam and emulsion stability has been discussed [15]. Different aspects of stability of thin films has been reviewed recently [16,17]. The analysis has been extended to include non-linear effects on thin film stability [6,18–20]. Rupture of non-Newtonian liquid thin films has been investigated [14,21,22]. Experimental observations have clearly established the role of interfacial elasticity on emulsion and foam stability in case of protein stabilized systems [23–25]. Factors affecting the interfacial rheology of adsorbed protein layers have been discussed [23,26]. High interfacial shear rheological properties of globular protein such as whey proteins, ovalbumin and soy glycinin enable them to provide superior emulsion and foam stability compared to casein [23,27]. In addition, the ability of β-lactoglobulin to provide superior dilatational rheological properties to sucrose monoesters is attributed to better foaming properties [28]. Experimental techniques for the characterization of interfacial shear and dilatational rhe-

G. Narsimhan, Z. Wang / Journal of Colloid and Interface Science 291 (2005) 296–302

ological properties have been described [26,29–33]. In this paper, a linear stability analysis of a thin film on a solid surface with a viscoelastic interfacial layer is formulated. The dilatational and shear viscoelasticity of the interface are expressed in terms of Maxwell models. The problem formulation is discussed in the next section followed by the discussion of the effect of different parameters on film stability.

Consider a thin film of thickness h on a solid surface. The film can be considered semi-infinite in x and z directions. The solid surface is the origin of y direction. At t = 0, a periodic perturbation f0 sin(kx) of wave number k is imposed on the film. If the perturbation grows, the film will eventually rupture; otherwise the film will be stable. The continuity equation is ∂vy ∂vx + = 0. ∂x ∂y

Equation of continuity for the surfactant for quasi-steady state yields ∂ 2c ∂ 2c + =0 ∂x 2 ∂y 2

(1)

Assuming quasi-steady state, the equations of motion are 
2 ∂p ∂ vx ∂ 2 vx = 0, − (2) +µ + ∂x ∂x 2 ∂y 2
2  ∂ 2 vy ∂ vy ∂p +µ = 0. − (3) + ∂y ∂x 2 ∂y 2

(11)

with the boundary conditions x = 0,

c = c0 ,

x = L,

c = c0 , ∂c = 0. ∂y

y = 0,

2. Governing equations

297

(12) (13)

Surfactant balance at the air–liquid interface neglecting surface diffusion yields  ∂c  ∂  0  v Γ . −D  (14) = ∂y y=h ∂x x Neglection of surface diffusion is a reasonable assumption for films stabilized by relatively large molecular weight surfactants. In writing the above equation, it is assumed that the diffusion of surfactant from the bulk solution to subsurface is slower than the transfer of surfactant from the subsurface to the surface [17]. 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  Γ0 [11,17] so that Eq. (14) can be approximated as  ∂  0 ∂c  v . −D  (15) = Γ0 ∂y ∂x x y=h

Since the motion in the liquid film is due to the imposed perturbation, it is reasonable to assume that the film thickness h(x, t), pressure p(x, y, t), velocity in x direction vx (x, y, t), and velocity in y direction vy (x, y, t) also have oscillations of the same frequency as the imposed perturbation, i.e.,

Assuming the concentration profile of the form

f (x, t) = f0 exp(ikx + βt),

(4)

p(x, y, t) = pss + p  (y) exp(ikx + βt),

(5)

Γ0 vx (h)k 2 ∂c = cosh(ky) exp(ikx + βt). ∂x Dk sinh(kh)

vx (x, y, t) = vx (y) exp(ikx + βt),

(6)

For a Langmuir adsorption isotherm of surfactant, we have

vy (x, y, t) = vy (y) exp(ikx + βt),

(7)

where pss is the pressure at solid surface and β is the growth coefficient. If β is positive, then the amplitude of perturbation will keep increasing resulting in rupture of the film, otherwise the perturbation decreases and disappears eventually. No-slip boundary condition at the solid surface gives vx (x, 0, t) = 0,

(8)

vy (x, 0, t) = 0.

(9)

At the air–liquid interface, the kinematic condition gives ∂f vy (x, h, t) = . ∂t

(10)

c (x, y) =

c(x, y) − c0 = C exp(ikx) cosh(ky), c0

(16)

Eq. (11) can be solved with the boundary conditions (12) and (13) to yield (17)

Γm Kc , (18) 1 + Kc where K and Γm are constants. Using Gibbs adsorption equation, Γ0 =

dσ (19) , dc where R is gas constant, T is temperature. At the air–liquid interface, normal force balance results in normal stress boundary conditions as given by [29] Γ0 = −RT c

−n·P ·nn ˜

˜˜

˜ ˜

t = 2H σ n + ˜

−∞

Gs (t − t  ) n(b −2Hs I s ) : ∇s v 0 dt  ˜ ˜

˜˜

˜

298

G. Narsimhan, Z. Wang / Journal of Colloid and Interface Science 291 (2005) 296–302

t +

Gd (t − t  )2H n ∇s · v 0 dt  ˜

−∞

t +

˜

Gs (t − t  )2H n ∇s · v 0 dt  , ˜

−∞

(20)

˜

where P refers to the difference in the pressure tensor across ˜˜ the interface, Gs (t) and Gd (t) refer to the surface shear and surface dilatational relaxation modulus, respectively, H is the curvature of the interface, σ is the surface tension, n is the unit normal, v 0 is the surface velocity vector, b is ˜the ˜ ˜ surface curvature dyadic defined as b = −∇s n, ∇s being the ˜ ˜ v0 0 surface gradient operator. Since I s : ∇s = ∇s · v , we have t − n · P · n n = 2H σ n + ˜

˜˜

˜ ˜

˜

t +

˜

˜˜

˜

−∞

˜

−∞

˜

˜

˜˜

˜˜

Gd (t − t  )∇s ∇s · v 0 dt  ˜

−∞

t

(21)

Gs (t − t  )∇s ∇s · v 0 dt 

+

˜

−∞

t

For a planar air–liquid interface, the above normal stress boundary condition becomes ∂vy p(x, h, t) − p0 + Π(h) + Π (h)f − 2µ ∂y  t
 ∂ 1 = h1 Gd (t − t  ) + Gs (t − t  ) σ+ ∂y h1 −∞


 ∂ 0 ∂ 1 0 × h1 vx + vy h1 dt  , ∂x ∂y h1 (22)

p(x, h, t) − p0 + Π(h) + Π  (h)f − 2µ

∂vy ∂y

 t   (d2 f/dx 2 ) =− Gd (t − t  ) + Gs (t − t  ) σ+ 2 1 + (df/dx) −∞ 

 vy0 (d2 f/dx 2 ) ∂vx0 2 dt  . − 1 + (df/dx) × ∂x 1 + (df/dx)2 (23) Retaining only linear terms, we get
2  ∂vy ∂ f  = −σ p(x, h, t) − p0 + Π(h) + Π (h)f − 2µ ∂y ∂x 2 (24)

    Gs (t − t  ) n ×∇s ∇s × v 0 · n

+

˜

−∞



where p(x, t) is the pressure inside the film, Π(h) is the van der Waals disjoining pressure, Π  (h) is the gradient of the disjoining pressure, µ is the bulk viscosity, σ is the surface tension and h1 is the surface metric (the other metric h2 being unity). Recognizing that h1 = dl/dx =

1 + (df/dx)2 , we get

(26)

For viscoelastic surface, shear stress balance can be expressed as [29] − n · P  · I s = ∇s σ +

˜

Gd (t − t  )2H n ∇s · v 0 dt  .

From Eqs. (4), (5) and (7), Eq. (25) reduces to 
 dvy (h)   p (h) + Π (h)f0 − 2µ = σf0 k 2 . dy

t

Gs (t − t  ) n b : ∇s v 0 dt  ˜ ˜

in which p0 is the atmosphere pressure. Since pss − p0 + Π(h) = 0 for an equilibrium film, we have 
∂vy (h)   p (h) exp(ikx + βt) + Π (h)f − 2µ ∂y
2  ∂ f . = −σ (25) ∂x 2

˜

  + 2(b −2H I s ) · ∇s v 0 · n dt  . ˜˜

˜˜

˜

˜

˜

(27)

For a planar gas–liquid interface, the above shear stress boundary condition becomes 
∂vy ∂vx + µ ∂x ∂y y=h t   ∂σ + Gd (t − t  ) + Gs (t − t  ) = h1 ∂x −∞ 
 ∂ ∂vx0 ∂ 0 ∂ 1 + h1 vy × h1 h1 dt  , ∂x ∂x ∂x ∂y h1

(28)

which can be simplified to 
∂vy ∂vx + µ ∂x ∂y y=h =



1 + f 2 

×

∂σ + ∂x

t

  Gd (t − t  ) + Gs (t − t  )

−∞

 ∂ 2 vx0  1 + f 2 ∂x 2



+



1 + f 2

∂vx0 f  f 

∂x 1 + f  2

f  ∂x 1 + f  2     2 f   f 4f 0 − 1 + f  2 vy − dt  . 1 + f  2 (1 + f  2 )2

−

1 + f 2

∂vy0

(29)

G. Narsimhan, Z. Wang / Journal of Colloid and Interface Science 291 (2005) 296–302

Retaining only the linear terms, the above equation reduces to 
∂vy ∂vx µ + ∂x ∂y y=h =

∂σ + ∂x

t

−∞

  ∂ 2 vx0  Gd (t − t  ) + Gs (t − t  ) dt , ∂x 2

(30)

where vx0 is the velocity at the air–liquid interface. The first term on the right-hand side in Eq. (30) is the Marangoni effect. Equation (30) can be rewritten as
 ∂vy ∂vx µ + ∂x ∂y y=h ∂σ ∂c + = ∂c ∂x

t

−∞

  ∂ 2 vx0  Gd (t − t  ) + Gs (t − t  ) dt . (31) ∂x 2

∂σ Γ0 vx (h)k 2 cosh(ky) exp(ikx + βt) ∂c Dk sinh(kh) t   ∂ 2 vx0  Gd (t − t  ) + Gs (t − t  ) dt . + ∂x 2

(32)

−∞

For a Maxwell model, 
κ s , Gd (s) = exp − λd λd 
µs s , Gs (s) = exp − λs λs


β=

Π  (h) − σ k 2 2µk




 M(kh)4 + 2(kh)3 . 6Mkh + 3

(36a)

If M is large, which implies that the surface viscosities are large while the bulk viscosity is small, Eq. (36) reduces to 

 sinh2 (kh) − k 2 h2 Π (h) − σ k 2 . β= (39) 2µk sinh(kh) cosh(kh) + kh For kh  1, Eqs. (38) and (39) become

(33a)

β=

(Π  (h) − σ k 2 )(kh)3 , 3µk

(38a)

β=

(Π  (h) − σ k 2 )(kh)3 . 12µk

(39a)

(33b)

where κ and µs are dilatational and shear viscosities respectively. λd and λs are relaxation times defined as κ λd = , gd µs , λs = gs

The growth coefficient β was obtained as 
 Π (h) − σ k 2 β= 2µk 
M(sinh2 (kh) − k 2 h2 ) + (sinh(kh) cosh(kh) − kh) , × M(sinh(kh) cosh(kh) + kh) + (cosh2 (kh) + k 2 h2 ) (36) A dΠ(h) Π  (h) = (37) = , dh 2πh4 where A is the Hamaker constant. For kh  1, Eqs. (35) and (36) reduce to 

∂σ Γ0 1 κ µs k − + M= , (35a) 2µ 1 + βλd 1 + βλs ∂c Dµ kh

If M is small, which implies that the surface viscosities are small while the bulk viscosity is large, Eq. (36) reduces to
  
Π (h) − σ k 2 sinh(kh) cosh(kh) − kh β= (38) . 2µk cosh2 (kh) + k 2 h2

Using Eqs. (17)–(19), Eq. (31) can be recast as
 ∂vy ∂vx µ + ∂x ∂y y=h =

299

(34a)

Equations (38), (39) are the same as those given by Jain and Ruckenstein [1] obtained for mobile and immobile air– liquid interface respectively.

(34b)

where gd and gs are dilatational and shear elasticity, respectively.

3. Perturbation growth coefficient Equations (1)–(3) were solved analytically with the assumptions Eqs. (4)–(6), the boundary conditions Eqs. (8)– (10), (26), (32), with the Maxwell model (33), (34). Define 

k ∂σ Γ0 κ µs M= + − cotanh(kh). 2µ 1 + βλd 1 + βλs ∂c Dµ (35)

4. Results In Eq. (36), the second term is always positive. Therefore, the sign of β is determined by the first term. As a result, when Π  (h) is larger than σ k 2 , β is positive, implying an unstable case; otherwise when Π  (h) is smaller than σ k 2 , β is negative, implying a stable case. The critical wave number is determined by Π  (h) = σ k 2 . That is, 1 kc = 2 h



A . 2πσ

(40)

(41)

300

G. Narsimhan, Z. Wang / Journal of Colloid and Interface Science 291 (2005) 296–302

Fig. 1. Growth coefficient versus wave number at a given film thickness. A–D are for mobile, viscoelastic (κ = 0.1 (N s)/m), viscoelastic (κ = 1 (N s)/m) and immobile interface respectively. Parameters for viscoelastic interface are: µs = 0.4 (N s)/m, λd = 10 s and λs = 4 s; other parameters are µ = 5 Pa s, σ = 50 mN/m, A = 10−20 J and h = 10−7 m.

For k < kc the film is unstable. The dominant wave number kd for which the growth coefficient is maximum is determined from the condition that dβ/dk = 0. Eqs. (35)–(37) are numerically solved to obtain kd and βmax . Typical plots of β versus k are shown in Fig. 1. βmax for a mobile interface is the largest, while that for an immobile interface is the smallest. βmax increases with decreasing surface rheological properties. This shows that surface rheological properties increased the film stability. For a given h, kc is independent of surface rheological properties, which can also be seen from Eq. √ (41). kd for a mobile and immobile interface is about kc / 2, which is consistent with the results obtained by Jain and Ruckenstein [1]. When the surface is between mobile and immobile, kd shifts a little to a lower wave number. βmax of a mobile interface is about 4 times that of an immobile interface, which is also consistent with the results obtained by Jain and Ruckenstein [1]. βmax for different film thickness is shown in Fig. 2. From the plots, film thickness has a dramatic effect on the stability of films. When the film thickness decreases from 8 × 10−7 to 10−7 m, βmax increases by a factor of 105 . βmax for a mobile interface is always about four times that of an immobile interface. βmax for a viscoelastic interface is always between the two. The viscoelastic interface behaves as a mobile interface at large film thickness and as an immobile interface at low film thickness. The limiting film thicknesses for asymptotic mobile film behavior as well as immobile film behavior are higher for larger viscoelastic rheological properties. kc is strongly dependent on h (see Eq. (41)). When h is small, kc is large, correspondingly kd is also large, as a result M is large. As discussed earlier, large value of M implies that the interface is immobile. Similarly, a large h implies small kc and kd , resulting in a small M, which implies a mobile interface behavior. Both the surface dilatational and shear rheological properties affect the stability of film. Effect of κ is shown in Fig. 1. µs has symmetric position in the equations and therefore the effect of µs is the same as that of κ.

Fig. 2. βmax for different film thickness. Parameters for A–D are the same as given in Fig. 1.

Fig. 3. βmax versus λd for different κ and µs . Parameters are µ = 5 Pa s, σ = 50 mN/m, A = 10−20 J and h = 10−7 m. κ and µs values for different curves are A: 0 and 0; B: 1 × 10−3 and 4 × 10−4 (N s)/m; C: 1 × 10−2 and 4 × 10−3 (N s)/m; D: 1 × 10−1 and 4 × 10−2 (N s)/m and E: 1 and 0.4 (N s)/m, respectively.

Effects of λd and λs are shown in Fig. 3. λd and λs also have symmetric position in equations, therefore only one of them (denoted as λ) is varied here. From Fig. 3, for any specified µs and κ, when λ is large enough, the film behaves as that with a mobile interface. For intermediate range of λ, βmax decreases with the decrease of λ. When λ decreases to a small enough value, βmax decreases to a constant value and is no longer dependent on λ. This intermediate range shifts to smaller λ when κ and µs decrease. This could be explained from Eq. (35). When λ becomes large, M becomes small, resulting in the behavior of mobile interface. When λ becomes small enough, M becomes independent of λ. In order to ascertain the importance of surface shear and dilatational rheological properties on stability of protein stabilized film, calculations of βmax for different film thickness was carried out for a film stabilized by β-lactoglobulin at pH 7 and ionic strength of 0.02. The surface dilatational and shear elasticity as well as viscosity were measured for different frequencies. The experimental methods for these measurements are given elsewhere [34,35]. The experimen-

G. Narsimhan, Z. Wang / Journal of Colloid and Interface Science 291 (2005) 296–302

301

properties. It is shown that both dilatational and shear interfacial rheological properties improved the film stability. When the interfacial rheological properties are extremely large or small, the maximum perturbation growth coefficient reduces to those for immobile or mobile interfaces respectively. Experimental values of surface shear and dilatational viscosities for 0.5% β-lactoglobulin were incorporated in maximum growth coefficient calculations for different film thicknesses. Calculated maximum growth coefficient values were found to lie between the limiting values for mobile and immobile interfaces. This clearly indicated that surface rheological properties are important for stability of protein stabilized thin films. Also, the calculated values of maximum growth coefficient approached those for mobile films for thick (>2000 nm) films and approached those for immobile films for thin (<100 nm) films.

References [1] [2] [3] [4]

Fig. 4. (A) Actual and (B) relative maximum growth coefficient (normalized by maximum growth coefficient for β-lactoglobulin) of a 0.5% β-lactoglobulin stabilized thin film at a solid surface. A = 10−20 J, σ = 50 (mN)/m. The viscoelastic properties of the interface are: κs = 12.9 (mN s)/m, gs = 18.8 mN/m, µs = 3.9 (mN s)/m, gd = 103 mN/m.

tal data of surface dilatational viscosity and elasticity values are reported elsewhere [34]. Based on high frequency limits of these surface shear and dilatational rheological properties and under the assumption of negligible Marangoni effect, βmax was calculated for different film thicknesses and compared with the corresponding values for mobile and immobile films (see Fig. 4). Fig. 4 also gives the relative values of βmax for different film thickness. It is interesting to note that βmax values lie between the mobile and immobile limits for film thickness range of 100 to 2000 nm thereby indicating that the effect of surface rheological properties on film stability is indeed important. Also, for very large film thickness (>2000 nm), the film can be considered to be mobile, whereas for very thin films (<100 nm), the film can be considered to be immobile.

5. Conclusions Linear stability analysis for a film on a solid surface with viscoelastic air–liquid interface is presented. The interfacial dilatational and shear viscoelastic properties were described by a Maxwell model. Perturbation growth coefficient is calculated in terms of film thickness and surface rheological

[5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28]

R. Jain, E. Ruckenstein, J. Colloid Interface Sci. 54 (1976) 108–116. E. Ruckenstein, R. Jain, J. Chem. Soc. Faraday II 70 (1974) 132. E. Ruckenstein, C.S. Dunn, Thin Solid Films 51 (1978) 43. C. Malderelli, R. Jain, The Hydrodynamic Stability of Thin Films, in: I.B. Ivanov (Ed.), Thin Liquid Films: Fundamentals and Applications, Dekker, New York, 1988, pp. 497–568. A. Sharma, E. Ruckenstein, Am. J. Optom. Physiol. Opt. 62 (1985) 246. K. Savettaseranee, D.T. Papageorgiou, P.G. Petropoulus, B.S. Tilley, Phys. Fluids 15 (2003) 641–652. L.Y. Yeo, O.K. Matar, J. Colloid Interface Sci. 261 (2) (2003) 575– 579. A. Sharma, K. Kargupta, App. Phys. Lett. 83 (2003) 3549–3551. I.B. Ivanov, Thin Liquid Films: Fundamentals and Applications, Dekker, New York, 1988. I.B. Ivanov, Pure Appl. Chem. 52 (1980) 1241. I.B. Ivanov, D.S. Dimitrov, Colloid Polym. Sci. 252 (1974) 982–990. A. Sharma, J. Colloid Interface Sci. 199 (2) (1998) 212–214. A. Sharma, J. Mittal, Phys. Rev. Lett. 89 (18) (2002) 186101. S.A. Safran, J. Klein, J. Phys. II (France) 3 (1993) 749–757. D. Langevin, Adv. Colloid Interface Sci. 88 (1–2) (2000) 209–222. J.E. Coons, P.M. Halley, S.A. McGlashan, T. Tran-Cong, Adv. Colloid Interface Sci. 105 (2003) 3–62. D.S. Valkovska, K.D. Danov, I.B. Ivanov, Adv. Colloid Interface Sci. 96 (1–3) (2002) 101–129. M.B. Williams, S.H. Davis, J. Colloid Interface Sci. 90 (1982) 220. A. Sharma, E. Ruckenstein, J. Colloid Interface Sci. 113 (1986) 456. C.K. Lin, C.C. Hwang, W.Y. Uen, J. Colloid Interface Sci. 231 (2) (2000) 379–393. A. Sathyagal, G. Narsimhan, Chem. Eng. Comm. 111 (1992) 161–175. Y.L. Zhang, O.K. Matar, R.V. Craster, J. Colloid Interface Sci. 262 (2003) 130–148. E. Dickinson, Colloids Surf. B Biointerfaces 20 (3) (2001) 197–210. L.A. Lobo, D.T. Wasan, AIChE Symp. Ser. 86 (277) (1990) 25–34. M.A. Bos, T. van Vliet, Adv. Colloid Interface Sci. 91 (3) (2001) 437– 471. B.S. Murray, E. Dickinson, Food Sci. Technol. Int. 2 (1996) 131–145. A.H. Martin, K. Grolle, M.A. Bos, M.A.C. Stuart, T. van Vliet, J. Colloid Interface Sci. 254 (1) (2002) 175–183. G. Garofalakis, B.S. Murray, Colloids Surf. B Biointerfaces 21 (1–3) (2001) 3–17.

302

G. Narsimhan, Z. Wang / Journal of Colloid and Interface Science 291 (2005) 296–302

[29] D.A. Edwards, H. Brenner, D.T. Wasan, Interfacial Transport Processes and Rheology, Butterworth–Heinemann, 1991. [30] E.H. Lucassen-Reynders, D.T. Wasan, Food Struct. 12 (1993) 1–12. [31] P. Joos, M. Vanuffelen, J. Colloid Interface Sci. 171 (2) (1995) 297– 305. [32] J. Benjamins, A. Cagna, E.H. Lucassen-Reynders, Colloids Surf. A Physicochem. Eng. Aspects 114 (1996) 245–254.

[33] R. Miller, R. Wüstneck, J. Kräger, G. Kretzschmar, Colloids Surf. A Physicochem. Eng. Aspects 111 (1–2) (1996) 75–118. [34] Z. Wang, Drainage and Stability of Protein Stabilized Standing Foam, Ph.D. thesis, Dec. 2004, Department of Agricultural and Biological Engineering, Purdue University, West Lafayette, IN 47907. [35] M. Cornec, D. Kim, G. Narsimhan, Food Hydrocol. 15 (2001) 303– 313.