Hydrodynamics of thin liquid films: Retrospective and perspectives.

CIS-01464; No of Pages 15 Advances in Colloid and Interface Science xxx (2014) xxx–xxx

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Hydrodynamics of thin liquid films: Retrospective and perspectives Stoyan I. Karakashev ⁎, Emil D. Manev Department of Physical Chemistry, Sofia University, James Bourchier Blvd, 1164 Sofia, Bulgaria

a r t i c l e

i n f o

Available online xxxx Keywords: Thin liquid film drainage Hydrodynamic theory Fractals Dimples

a b s t r a c t This review presents a summary of the results in the domain of microscopic liquid film hydrodynamics for several decades of experimental and theoretical research. It mainly focuses on the validation, application and further development of the Stefan–Reynolds theory on the liquid drainage, based on the accumulated knowledge of surface forces, surface tension caused by the surfactant adsorption, and diffusion of surfactants. Liquid films are of primary significance for colloidal disperse systems, and diverse industrial processes. The transient stability of the froth phase and the froth drainage is a function of the drainage and rupture of liquid films between air bubbles. In flotation, the bubble–particle attachment is controlled by the thinning and rupture of the intervening liquid film between an air bubble and a mineral particle. Both the experimental and theoretical results are mostly related to the foam liquid films between two bubbles, but can be principally generalized for emulsion films, formed in another liquid, as well as wetting films between a bubble and a solid surface. © 2014 Elsevier B.V. All rights reserved.

Contents 1. 2.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . Thin liquid films — basic assumptions and concepts . . . . . . . 2.1. Role of surfactants in the formation of liquid films . . . . 2.2. Surface forces in thin films . . . . . . . . . . . . . . . 2.3. Experimental techniques . . . . . . . . . . . . . . . . 3. Investigations of film thinning . . . . . . . . . . . . . . . . . 3.1. Drainage of planar films stabilized by non-ionic surfactants 3.2. Drainage of planar films stabilized by ionic surfactants . . 3.3. Effect of film thickness non-homogeneity . . . . . . . . 3.4. The model of Ruckenstein and Sharma (RSh) . . . . . . . 3.5. The model of Manev, Tsekov and Radoev (MTR) . . . . . 3.6. Fractal model [73,74] . . . . . . . . . . . . . . . . . 4. Perspectives of thin liquid drainage theory . . . . . . . . . . . 5. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1. Introduction Thin liquid films (TLFs), formed by the continuous phase (the dispersion medium), are central to colloid science and have vast practical (technological and environmental) importance. They are the fundamental ⁎ Corresponding author at: Department of Physical Chemistry, 1 James Bourchier Blvd, Sofia 1164, Bulgaria. Tel.: +359 28161283; fax: +359 29625438. E-mail address: [email protected]fia.bg (S.I. Karakashev).

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0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

structural element of various disperse systems widely spread in nature, including foam and emulsion [1,2]. The thin films separating the particles, gas bubbles and droplets of the disperse phase play a decisive role in a variety of chemical and biological applications and processes. Examples range from emulsion systems of food or pharmaceutical products to gas dispersion systems found in chemical reactors or fire foams [3]. Both foam and wetting films are vitally important to the froth flotation separation process widely used in the recovery of coal and valuable minerals from the rock, in the treatment and purification of wastewater, and in

http://dx.doi.org/10.1016/j.cis.2014.07.010 0001-8686/© 2014 Elsevier B.V. All rights reserved.

Please cite this article as: Karakashev SI, Manev ED, Hydrodynamics of thin liquid films: Retrospective and perspectives, Adv Colloid Interface Sci (2014), http://dx.doi.org/10.1016/j.cis.2014.07.010

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the de-inking of wastepaper [4–6]. The quality of the products and efficiency of the processes are chiefly determined by the stability of the intervening liquid films. An understanding of how the film thinning and rupture occur, leading to the stability or instability, is critical to the technologically significant processes. The aim of the present collection of our investigations is to systematize the original results which contain a number of factors governing the behavior of TLFs. Evaluation of the range of applicability and explanation of the deviations from the theoretical predictions for the effects of various parameters, such as the film size, the concentration of surfactants and electrolytes, on the film properties are pursued. The main research directions reported in this paper comprise studies of the applicability of the theoretical models of evolution of microscopic horizontal circular liquid films. The classic description of the hydrodynamic behavior of the microscopic liquid films was put forward by Scheludko [7,8] and is based on the Stefan–Reynolds equation [9–11]. Despite the limitations imposed by the basic requirements of the Stefan–Reynolds equation, its merits are perpetuated in both historical and applied terms. This is the reason why one of the main targets of the studies compiled here is to establish the scope of applicability of this simple theory. The main contributions of summarizing these works are the experimentally established significant effects of the surfactant's bulk and surface diffusion and the deformability of the fluid film interfaces in the process of the film thinning and the quantitative estimation of the deviations from the classical models for a planar film as described by the basic. New theoretical and empirical expressions developed to describe the real hydrodynamic behavior of the films (accelerated drainage) are also put forward and are subjected to the experimental verification. The attention of researchers in the field has been persistently focused on effects due to the presence of surfactants that build the adsorbed layers at the film surfaces. First of all, the presence of surfactant is the necessary condition for the formation of a foam and emulsion, which do not decompose instantly. Surfactant adsorption provides the films with the ability to resist the local thinning in the process of draining. Specific effects, related to the individual structure of adsorbed layers from different surfactants, are also observed. The characteristics of the surfactants strongly affect the behavior of the intervening films and the entire disperse system. The focus of both the experimental and theoretical studies is predominantly on the foam films, that is, a single free microscopic film with liquid–gas interfaces. Results of investigations discussed here favor the concept that in many respects (formation, thinning and rupture) the observed effects correspond to other types of liquid films and permit respective generalization. These are, first of all, the other type of symmetrical films with fluid interfaces: the emulsion films. 2. Thin liquid films — basic assumptions and concepts

They are approximated with a circular shape and axisymmetric cylindrical profile (Fig. 2). The influence of hydrostatic pressure on the drainage of horizontal films is insignificant and is not considered in the modeling. The diameter of the films investigated is usually in the range between 0.1 and 1.0 mm, and its largest thickness (at the formation) is, as a rule, less than 10 μm which is always much smaller than the film radius (h b b r, where r is the film radius). When the film is formed, its internal pressure initially equals the pressure in the concave meniscus with which it is in contact [1]. Because of the pressure jump on the curved liquid–gas interface, there exists a force due to the (Laplace) capillary pressure, Pσ, acting upon the flat film surface in direction of thinning. The pressure balance gives (subscripts g, l and f refer to the gas, liquid and film phases, respectively): P σ ¼ P g −P l

ð1Þ

P f ¼ Pl

ð2Þ

and the driving pressure, ΔP, for the film thinning is determined by: ΔP ≡ P g −P f ¼ P σ −Π;

ð3Þ

where Π is the disjoining pressure at the appropriate thickness. If at some thickness the disjoining pressure, Π, is equal to the capillary pressure, Pσ, the driving force ΔР is equal to zero, leading to the formation of an equilibrium film. 2.1. Role of surfactants in the formation of liquid films Adsorption of surfactants plays a major role for the behavior of liquid films. The quantitative relation between the adsorption density Γ i [mol/m2 ] of a certain (ith) component of the system and the surface tension σ is given by the Gibbs adsorption isotherm by X dσ ¼ − Γ i dμ i

ð4Þ

where μ i is the chemical potential of the component. For foam films the presence of a surface-active component is the necessary condition, e.g. Ref. [2]. First of all, the visco-elastic effect of the surfactant is necessary for the dynamic stabilization of the liquid film, in order that it can form and drain, while withstanding tangential stresses due to the outflow on its surfaces, and reaching the ‘critical thickness’ eventually. In the presence of surfactants the drainage at the film surfaces may be substantially reduced by an opposing gradient of the surface tension, the socalled ‘dynamic elasticity’ or ‘Marangoni effect’ [3].

The experimentally investigated here microscopic horizontal films are formed in the meniscus of a double-concave liquid drop (Fig. 1).

Z

r h/2

Fig. 1. Schematic of microscopic horizontal liquid films formed by two concave gas–liquid interfaces in the capillary of the Scheludko cell.

Fig. 2. Representation and notation of microscopic horizontal liquid films formed in the Scheludko cell.



3

It was found in our first investigation of the effect of surface diffusion on film thinning [83] that in order to provide the formation and further existence of a thinning film, the following condition has to be satisfied:

pressure, ΠvdW, as a function of the film thickness, h, combining the two approaches gives [18]:

h o r σ −σ ¼ P −P 2

Π vdW ¼ −

o

r

ð5Þ

where Po − P r is the pressure difference causing the drainage; the superscripts o and r refer to the locations at the center and periphery of the film, respectively. Scheludko [4] was the first to advance Eq. (5), but as a condition for zero surface velocity with insoluble surfactants. Therefore, a film can be formed from an aqueous solution even at very small reduction in the surface tension (Δσ ≅ 0.1 mN/m). The initial thickness of formation also depends on the surfactant content: It diminishes significantly at low concentrations. This effect has been put forward by Exerowa [5] as a criterion for the water purity. Foam films from nonaqueous liquids have smaller initial thickness (b300 nm), as compared to the “typical” initial thickness (N1 μm) of an aqueous film [4,5]. Surfactant adsorption affects the dynamic properties of the liquid surfaces, which govern the resistance of the film to the changes in the shape and size. No direct correlation is proved to exist between the absolute value of the surface tension (which depends on the surfactant type and content) and foaming [6], but the theoretical investigations indicate that there exist relationships between these two quantities through the elasticity and surface viscosity of the film [7]. The elasticity, defined by Gibbs [8], expresses the resistance of the film to local deformations. Any increase in the film area is accompanied by a local increase in surface tension and, consequently, a tendency to contraction. The tangential force per unit area towards restoring the initial un-deformed state is the Gibbs elasticity, Е, described by Ε ¼ 2S

dσ dσ ¼− dS d ln Γ

ð6Þ

where S is the area of the fluid interface. Eq. (6) predicts a steady increase in elasticity upon thinning, leading to the increase in the film stability. This, however, does not match reality; Gibbs himself understood that elasticity alone is not sufficient to prevent the film from rupture and has admitted the existence of another “still unknown force”. Plateau [9] assumed that surfactants increase the surface viscosity, thus reducing the drainage and enhancing the foam film stability. Boussinesq [10] developed a quantitative theory on the effect of surfactants on the hydrodynamic behavior of the disperse systems, describing the surface as a 2D fluid with two surface viscosities: shear viscosity and dilatational viscosity. Although certain correlations can be established, it is generally accepted that surface viscosity alone cannot explain dispersion stability. In many cases surface viscosity effect on stability is insignificant and not necessarily related to the latter [11] Of course, its effect on the film lifetime and critical thickness need not be excluded [7].

Aðh; T; κ Þ 6πh3

ð7Þ

where А(h,T,κ) is the so-called Hamaker–Lifshitz “constant” [18] which is in fact a function of temperature T and the Debye constant, κ, as well as a weak function of the film thickness, h, due to the “electromagnetic retardation effect” that follows from the limited velocity of propagation of the dispersion interactions [22]. In foam and emulsion films ΠvdW is always negative (i.e. attraction between the film surfaces). In asymmetrical (e.g. wetting) films it can become positive if attraction to the substrate prevails. Negative ΠvdW is responsible for the instability and rupture of the films at the critical thickness. The electrostatic disjoining pressure Πel arises in thin films from dilute electrolyte solutions and is due to the overlapping of the diffuse electric layers on the two film surfaces at small separation distances. For symmetrical electrolytes the equation derived from the Poisson– Boltzmann equation for Πel gives [18,23] zeψ −1 Π el ðhÞ ¼ 2cel kB T cosh kB T

where z is the valence of the binary electrolyte, ψ is the potential in the film at the position of the zero potential gradient (e.g. at the mid-plane for the symmetrical foam films), e is the charge of proton, cel is the electrolyte bulk concentration in the solution from which the film is formed, kBT is the molecular thermal energy. In foam (and emulsion) films Πel is always positive (repulsion). With wetting films the situation is more complex and in some cases electrostatic interactions can even lead to attraction of the film surfaces [18,24]. In addition to the long-range forces of attraction and repulsion, stability of films and the respective disperse system is also dependent on the short-range interactions in the adsorption layers [5,24,25], whose characteristics are an important factor in the formation of the stable ‘black’ films, but have little effect on film thinning. The calculation of the disjoining pressure as a function of the film thickness is critical to validation of the drainage theories. Approximate equations are not universal. For this reason advanced models are needed as described below [26]. The electrostatic disjoining pressure Πel can be obtained by numerical solution of the Poisson–Boltzmann equation employing appropriate boundary conditions at the film surfaces. Under the condition of constant surface potential, the numerical solution of the non-linear Poisson– Boltzmann equation can be semi-analytically represented as [27] 2

2.2. Surface forces in thin films The understanding of the properties of the thin liquid film requires knowledge of the fundamental forces acting in them. The TLF is a quasi-two-dimensional phase, in which the two interfacial layers overlap to form a unified non-homogeneous structure of specific properties. The film thickness is a fundamental quantitative characteristic of the deviations in the properties of the thin film from those of the bulk phase. Such deviations are adequately expressed by the disjoining pressure, introduced by B. V. Derjaguin in the early 1930s [12,13] and is a fundamental element in the DLVO theory of stability of lyophobic colloids [14, 15]. The classical DLVO theory incorporates the van-der-Waals and electrostatic double-layer interactions. The van der Waals interaction between the thin liquid film surfaces is described comprehensively in many books, including Refs. [16,17]. The interaction can be determined using either the Hamaker ‘microscopic’ approach [18], or the Lifshitz (‘macroscopic’) approach [18–21]. For the van der Waals disjoining

ð8Þ

Π el ðhÞ ¼ 32cel Rg T tanh

y 1 2y 0 þ f ðy0 Þ sinh 0 exp½−f ðy0 Þkh 1 þ coshκh 4 4

ð9Þ where cel is the molar concentration of electrolytes in the solution. The Debye constant for a binary electrolyte of valence z is defined as κ = {2celF2z2/(εε0RgT)}1/2, where ε0 is the dielectric permittivity of the vacuum. The normalized surface potential is defined as y0 = zFφs/(RgT), where F is the Faraday constant and ψs is the surface potential. For |y0| ≤ 7, the function f(y0) is defined as f(y0) = 2 cosh(0.332|y0| − 0.779). Eq. (9) is valid until 120 mV value of the surface potential. Under the condition of constant surface charge density, the exact numerical solution to the Poisson–Boltzmann equation gives [27]:

Π el ðhÞ ¼

e 2cel Rg T A rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : e e2 coth κhC=2 ½ coshðκhC Þ−1 1 þ B

ð10Þ


4


The model constants in Eq. (10) are functions of the surface potential, y0, at infinite separation (i.e., the single air–water surface) which are e¼B e sinh 1:854jy j−0:585jy j2 þ eC described for |y0| ≤ 5 as follows: A 0 0 e ¼ 0:571jy j exp −0:095jy j1:857 and 0:1127jy0 j3 −0:00815jy0 j4 Þ, B 0 0 e ¼ 1−0:00848jy j. C 0 Eq. (10) is valid until 186 mV value of the surface potential. It should be noted here that the regime of constant surface potential should be the most probable regime for TLF with fluid interfaces (foam and emulsion films), while the regime of the constant surface charge density is most probable for TLF with solid surfaces (e.g. in the suspensions). We must acknowledge that the electro-static disjoining pressure in the case of ionic surfactants is affected by the outflows in the course of the thin liquid film drainage [28]. For example, the surface-active ions are displaced in the periphery of the film, thus reducing the electrostatic repulsion in its center. Moreover, the shift of the electrical double layer (EDL) towards to periphery of the film creates streaming potential between the center and film's rim [29]. The latter causes the emergence of reverted fluxes decreasing the speed of film drainage. The van der Waals disjoining pressure, ΠvdW, as a function of the film thickness, h, for both the non-retarded and retarded regimes can be described as [30] Π vdW ¼ −

Aðh; κ Þ 1 dAðh; κ Þ þ 6πh3 12πh2 dh

ð11Þ

where А(h,κ) is the Hamaker–Lifshitz function, which depends on the film thickness and the Debye constant, κ, due to the electromagnetic retardation effect and is described as

−2κh

Aðh; κ Þ ¼ ð1 þ 2κhÞe

2 2 2 q −1=q 2 j ∞ 3kB T X 3ℏω n1 −n2 h −3 ε1 −ε2 j þ pffiffiffi 3=2 1 þ e 2 2 ε1 þ ε2 4 j¼1 16 2 n þ n λ 1

2

ð12Þ where ε1 is the static dielectric permittivity of the dispersion phase (2.379 for toluene at 298.15 K), ε2 is the static dielectric permittivity of the disperse medium (80 for water), ℏ = 1.055 × 10− 34 Js/rad is the Planck constant (divided by 2π), ω is the absorption frequency in the UV region — typically around 2.068 × 1016 rad/s for water, n1 and n2 are the characteristic refractive indices of the dispersion phase and the medium: n21 = 1 for air and n22 = 1.887 for water, and q = 1.185. qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi e ¼ 2v 2=n2 n2 þ n2 = The characteristic wavelength is defined as λ 2 1 2 2 π ω , where v is the speed of light. Eq. (11) is more precise than Eq. (7). 2.3. Experimental techniques The first investigations of the thin (foam) film thickness belong to Perrin [84]. The optical experimental setups, based on Perrin's ideas and applied by Wells [85], have preserved their methodology value up to the present day. The thickness is determined from the intensity of the light reflected by the film, visually comparing it to the (maximum) brightness of the ‘white film’ from the same monochromatic source. The accuracy of the averaged value is ca. 0.5 nm, which is remarkable for the technical level of that time. Three decades later, Derjaguin and Titijevskaya [31] in Russia and Scheludko [32] in Bulgaria have independently started their investigations almost simultaneously. Together with their collaborators they have developed new experimental techniques for thin liquid films. Scheludko [33] has applied soap bubble expansion to estimate the film thickness from the volume of the gas bubble and the liquid forming it. Again Scheludko [34] has used conductometry (longitudinal resistance) measurements with foam films. For measuring the thickness of a film from non-conducting liquid between two conducting droplets

Gas Phase

Plateau Border

Film

Glass cell

Water Phase

Fig. 3. The Scheludko cell.

(mercury) Sonntag [35] and Hanai, Haydon and Taylor [86] have applied capacitance techniques. Numerous precise measurements of different film parameters have been conducted using specialized techniques and setups. A good example is the widely used microinterferometric method, comprising specially designed cells and automatic recording of the film thickness evolution with time, which is one of the most significant developments in the field since the middle of the last century and is of a particular interest to the studies presented here. The methods have been developed and refined during the years by Scheludko [36], together with Exerowa [5] and many other researchers of the Bulgarian school of surface science. This method has been successfully tested and applied in a vast number of studies, including the pioneer studies on thinning and surface forces in the liquid films. The method is suitable for studying under defined conditions equilibrium and thinning films of different types: free (foam), emulsion, polymolecular films on solid/liquid substrate. It allows the experiment to cover the entire range of stability of films of vastly varying lifetime and is extensively exploited in our investigations. The set-up for interferometric measurements of the film thickness comprises two basic units: 1. Measuring cell (‘the Scheludko cell’, Fig. 3) in which the film is formed, and 2. Optico-electronic system for monitoring the film and registration of its thickness (Fig. 4). Several innovations have been introduced in the experimental technique. The most significant of them is the oscillating photometric probe [37]. Measuring the thickness on a small portion of the film surface (e.g. b5% of total area) has allowed in this case a very fine registration of deviations in the local film thickness from the variations in reflected light intensity (Fig. 4). These effects as a function of time were used in the investigation of the film non-homogeneity and critical thickness. An example of interferogram of planar foam film and its corresponding film thickness is presented in Fig. 5 [26]. The measuring technique with the oscillating photometric probe has further permitted the quantitative estimate of the magnitude of the film non-homogeneity, its dependence on film radius and the size of the (thin and thick) domains over the film area. The obtained data thus have been used to establish the laws of the detailed evolution of film thickness with time. More advanced techniques for studying the film thickness inhomogeneity is the line-scan camera, which is scanning the foam film during its drainage in line with thickness of about 0.5 μm with a given frequency [38]. This allows one to investigate precisely the profile of the thin



Bi-concave meniscus

5

Capillary

Film holder Lower part of the cell

Light source

Beam splitter

Inverted microscope with reflected light

CCD camera

Computer

Fig. 4. Experimental setup for studying thin liquid films.

Fig. 5. An example of the transient digital images of 10−5 M Tetraethylene Glycol Octyl Ether (C8E4) foam films taken at 0.9 s intervals (top left to bottom right) and the corresponding transient thicknesses of the thinning film [26]; Reprinted with the permission from Elsevier, Karakashev S.I. and Ivanova D.S. “Thin liquid film drainage — ionic versus nonionic surfactants”, J. Colloid Interface Sci., 2010, 343, 584–593.

film in line across the latter. In such way one can study the surface waves in the thin films or the formation of black spots (see Fig. 6). Another advanced technique for studying the film thickness inhomogeneity is by making set of pictures of the film interferogram

displaced by equal time interval and processing each one of the picture separately by Image J software (for more information read Ref. [39]), thus obtaining 3D pictures of the thin film (see Fig. 7). 3. Investigations of film thinning The drainage of thin liquid films is accompanied by a number of phenomena, like for example surfactant adsorption onto the film surfaces, surfactant surface diffusion, Marangoni effect, formation of ripples, etc. All of them affect the drainage of the thin films. However, very important for the drainage and the stability of the liquid film is the type of the surfactant used for its stabilization. For this reason, the hydrodynamics of thin film drainage can be split to (i) hydrodynamics with no electrical effects (in the presence of nonionic surfactants); and (ii) hydrodynamics with electrical effects (in the presence of ionic surfactants). We shall follow our observations in this order. We must note here that the first one is trivial and for the case of planar films has been developed in the period 1950−ies–1970−ies [34,40–43], while the second one was developed for the case of plane parallel films recently [29,44,45].

Fig. 6. The film thickness profiles before (filled circles) and after (unfilled circles) the formation of the black spot of foam films stabilized by sodium dodecylbenzene sulfonate (SDBS) [38]; Reprinted with the permission from Elsevier, Karakashev S.I., Nguyen A.V., Manev E.D., and Phan, C.M. “Surface foam film waves studied with high-speed linescan camera”, Colloids Surf. A, 2005, 262, 1–3, 23–32.

3.1. Drainage of planar films stabilized by non-ionic surfactants The conventional theory of thin liquid film (TLF) drainage does not account for the electrical double layer (EDL) in the hydrodynamic differential equations. In this sense the dynamic effects on the EDL and


6


227.5 -- 260.0 195.0 -- 227.5 162.5 -- 195.0 130.0 -- 162.5 97.50 -- 130.0 65.00 -- 97.50 32.50 -- 65.00 0 -- 32.50

200

Y, μm

150

200

100

150 100

200

50

Y, μm

250

0

150

h, nm

0

50

50

100

50 0

150

200

X, μm

100

0

50

100 X, μm

150

200

0

Fig. 7. 3D picture of dimple-like wetting film entrapped between bubble and solid surface.

originating from EDL are not accounted for. Therefore, the following set of differential equations and boundary conditions describes the drainage of such TLF (see Fig. 8):

Thin film bulk mass balance equation (in quasi-steady approximation): 1∂ ∂C ∂2 C r þ 2 ¼ 0; r ∂r ∂r ∂z

ð16Þ

Momentum balance equations (in lubrication approximation): “r” component 2

μ

∂ V r ∂p ¼ ; ∂r ∂z2

ð13Þ

where C is the surfactant concentration in the TLF. Due to the very small Peclet number (Pe b b 1) [7], the convective terms in Eq. (16) are neglected. Thus, the bulk diffusion of the surfactant molecules dominates over their convection during the film drainage. Surface mass balance equation (in quasi-steady approximation): 1∂ 1 ∂ ∂Γ ∂C ; ðrΓU Þ ¼ DS −D r ∂r r ∂r ∂r ∂z z¼h

“z” component

ð17Þ

2

∂p ¼ 0; ∂z

ð14Þ

where p is the isotropic thermodynamic pressure inside the film, Vr is the radial component of the velocity of liquid outflow and μ is the bulk liquid viscosity. Continuity equation:

1∂ ∂V ðrV r Þ þ z ¼ 0; r ∂r ∂z

where U is the velocity of film surfaces during the drainage, Γ is the surfactant adsorption on the film surfaces, C is the surfactant concentration in the film, DS and D are the surface and bulk diffusion coefficients of the surfactant molecules. The film surfaces are moving from the center towards the periphery of the TLF during the drainage thus generating surface convection of the surfactant molecules, opposed by their surface diffusion and accompanied by additional adsorption of surfactant onto the film surfaces.

ð15Þ Tangential stress boundary condition:

where Vz is the axial component of the drainage velocity. Thus the conservation of momentum and mass during TLF drainage is expressed by the upper equations.

μ

∂V r dσ ∂ 1 ∂rU þ μs ¼ dr ∂z z¼h ∂r r ∂r

ð18Þ

2

where σ and μs are the surface tension and surface viscosity. The bulk viscous stress at the very film surfaces is balanced by the surface tension gradient (Marangoni effect) along the film surfaces and surface viscous stress. Additional boundary conditions: V Z jZ¼h ¼ 2

Fig. 8. Sketch of thinning foam film.

∂V r ¼0 ∂z Z¼0

V 2

ð19Þ

ð20Þ



C ðr; zÞ ¼ C ðr; −zÞ

ð21Þ

C ð0; zÞ is finite value

ð22Þ

and the surfactant surface diffusion control the surface velocity U. In such case, the mobility factor f in Eq. (28) is expressed by: f ¼1þ

U ð0Þ is finite value

ð23Þ

C ðz; RÞ ¼ C 0

ð24Þ

where V is the rate of film thinning, R is the film radius, and C0 is the bulk surfactant concentration, h is the film thickness. The governing equation of TLF thinning follows from the upper set of differential equations: ∂h h3 1 ∂ ∂p 1∂ ¼ r − ðrhU Þ r ∂r ∂t 12μ r ∂r ∂r

ð25Þ

Eq. (25) is general and valid for TLF with inhomogeneous film thickness. If we assume planar and stagnant film with immobile surface Eq. (25) can be solved with respect to the pressure p: 2

p ¼ p0 −

3μr dh ; h3 dt

ð26Þ

where p0 is the pressure at the film's rim. Finally we calculate the drag force, which is equal to the force pressing the two surfaces of the film towards each other: 2

ZR

πR Δp ¼

ðp−p0 Þ2πrdr ¼ − 0

3πμR4 dh ; 2h3 dt

ð27Þ

where Δp = pσ − Π. Hence, we arrive at the velocity of thinning of the TLF: 3

V ¼−

dh 2h ¼ ðP −Π Þf ; dt 3μR2 σ

ð28Þ

where Pc = 2σ/Rc and Π are the capillary and disjoining pressures, while f ≥ 1 is the mobility factor, which is related with the film surface velocity U during the drainage. It is affected mainly by the surface tension gradient dσ/dr in terms of Marangoni effect, surfactant surface diffusion and the surface viscosity. With most of the surfactants the Marangoni effect is very powerful in making the film surfaces immobile, starting from their very dilute solutions. In such a case U = 0 corresponding to f = 1. In such a case the theory is reduced to the well-known kinetic equation of Scheludko [46], which is based on the Stefan–Reynolds model for TLF drainage between two rigid plane parallel discs [47]. The application of the Stefan–Reynolds equation to the rate of film thinning is subject to several substantial restrictions, including: 1) Viscosity in the film equal to that of the bulk liquid (meniscus), with which the film is in contact, 2) Negligible rate of thinning through evaporation (much smaller than through drainage), 3) Liquid flow between parallel flat surfaces (planar film), and 4) Radial component of the velocity at the surface equal to zero (tangentially immobile surfaces). It was indicated in Ref. [48] that foam films with radii equal to 50 μm or below, are planar and can be described by the Reynolds model. In a case, at which U N 0(f N 1), but the surface viscosity has negligible effect on TLF the theory can be reduced to the well-known model of Radoev–Dimitrov–Ivanov (RDI) [41], in which the Marangoni effect

7

6D f μ hEG

ð29Þ

where EG = − dσ/d ln Γ is the Gibbs elasticity, which governs the Marangoni stress and, therefore, controls the mobility of the film surfaces. In Eq. (29) Df is an effective film diffusion coefficient and can be expressed by D f ¼ DS þ

Dh dC 2 dΓ

ð30Þ

where dΓ/dC is the first derivative of the adsorption regarding the surfactant concentration, known as the adsorption length. The surface tension isotherm is usually measured to derive the value of the capillary pressure, the Gibbs elasticity and the adsorption length. However, the values of DS and D are usually unknown. Moreover, it is known that the effect of DS on the drainage velocity is significant. There is no experimental way to reliably determine DS. Its value can be calculated by the following approach. The surfactant molecule with the hydrophilic and hydrophobic parts is assumed with cylindrical shape. Its length, L, can be calculated via the HyperChem software for quantum chemistry and MD simulations. In this way, for SDS (sodium dodecyl sulfate) molecule as an example, we obtained LSDS = 1.76 nm for the molecular length and d = 0.137 nm for the distance between two methylene groups. The bulk diffusion coefficient depends on the shape and size of molecules, bulk viscosity and temperature. For cylindrical molecules it can be expressed by [49]: D¼

kB T L ln 3πμL d

ð31Þ

where kB is the Boltzmann constant, T is the temperature, L is the length of the cylinder and d is its diameter. The calculated bulk diffusion coefficient for SDS agrees with the measured value DSDS = 6 × 10−10 m2/s [50,51]. Knowing DSDS for SDS molecules, the bulk diffusion coefficient of other surfactants can be calculated as

D ¼ DSDS

LSDS ln ðL=dÞ : L ln ðLSDS =dSDS Þ

ð32Þ

Regarding the surface diffusion coefficient, DS, it can be calculated from the bulk diffusion coefficient as Ds = 1.5D [52]. The bulk diffusion coefficient of spherical molecules (e.g. TPeAB) can be calculated by the formula D = kBT/6πμRm, where kB is the Boltzmann constant and Rm is the radius of the molecule. The RDI model should be valid for small films, i.e., Rf ≤ 50 μm, because larger films are usually not plane-parallel [7,41,53,54]. If the film surfaces are immobile the RDI model reduces to the Scheludko model. In case at which all the factors (surface viscosity, Marangoni effect and surface diffusion) are accounted for, the mobility factor f in Eq. (28) can be expressed by [42,55]: f ¼

1

∞ X 1−32 k¼1

ð33Þ

2 e 6=λ4k − h=λ k e þ Ma 6 þ Boλ2k h

e h

h=2 1 þ ðN=λk Þ tanh λk e

e ¼ h=R is the dimensionless film thickness, Ma = ER/D μ is where h s the Marangoni number, Bo = μs/μR is the Boussinesq number, N =


8


120

A

Film thickness (nm)

100 80

μ

60 40

0

400

R=50µm

Rf =0.05 mm

0

5

10 Time (s)

15

B

20

200 100

R=50µm

R=0.05 mm 0

ð34Þ

2

4

Time, s

6

8

10

Fig. 9. Example of the level of coincidence between theoretical prediction (Eq. (28)) and experimental data of foam films stabilized by: (A) nonionic surfactant; (B) ionic surfactant [56]; Reprinted with the permission from Elsevier, Karakashev S.I., Ivanova, D. S., Angarska, Z. K., Manev, E. D., Tsekov, R., Radoev, B., Slavchov, R., and Nguyen, A. V., “Comparative validation of the analytical models for the Marangoni effect on foam film drainage”, Colloids Surf. A, 2010, 365, 1–3, 122–136.

(D/Ds)[R/(dΓ/dc)] is the dimensionless diffusion coefficient, and λk is the k-th root of the Bessel function of the first kind and zero order. It ∞

∂p ∂φ εε 0 ∂ ∂φ 2 ; ¼ρ − 2 ∂z ∂z ∂z ∂z

ð35Þ

where ρ is the bulk charge density in the TLF, φ is the electrostatic potential, which is a function of the radial and axial coordinates “r” and “z”, ε is the dielectric permittivity of the medium, while ε0 is the dielectric permittivity of the free space. The continuity equation is identical to Eq. (15). Thin film bulk mass balance equation (in quasi-steady approximation): The simplest case of 1:1 ionic surfactant (e.g. sodium dodecyl sulfate) is taken into account. Therefore, two equations of thin film mass balance equations are needed — one for the cations and one for the anions:

5e-6 M SDS Theory

300

0

∂2 V r ∂p ∂φ εε0 ∂ ∂φ 2 ¼ ; −ρ − 2 ∂r ∂r ∂r ∂r ∂z2 “z” component

20

Film thickness ( nm)

EDL. This causes a correction in the pressure gradient across the TLF [57]. “r” component

1e-6 M C8E4 + 0.02M NaCl Theory

4

is noted that since ∑ 1=λk ¼ 1=32 and tanh(x) = x + O(x3), Eq. (33) k¼1

e simplifies to Eq. (29) in the limit as Bo = 0 and h=λ k bb1. Eq. (33) presents the general drainage equation for foam films which accounts for the Marangoni effect, and the effects of surface shear viscosity and diffusion. The kinetic equations, mentioned above, are valid for planar TLF containing non-ionic surfactants (see Fig. 9) due to the fact that the EDL was not accounted for in the differential equations. The above-mentioned models are based on the hydrodynamic theory in lubrication approximation with excluded electrical effects. It was established [26] that this theory is valid only for thin films stabilized by non-ionic surfactants. In the case of films stabilized by ionic surfactants electrical effects originating from the motion of charged particles emerge. Due to its complexity, this problem has not been solved for decades. 3.2. Drainage of planar films stabilized by ionic surfactants The presence of EDL in thinning TLF makes the problem more complex as compared to the former case. The following set of differential equations is valid for this case. Momentum balance equations (in lubrication approximation): The Maxwell stress tensor is included as a part of the mechanical stress tensor. The physical meaning of the Maxwell stress tensor here is polarization of the liquid of TLF by the electrical charges of

1∂ ∂cþ ∂2 cþ Fcþ þ þ r r ∂r Rg T ∂r ∂z2 − 2 − − 1∂ ∂c ∂ c Fc − þ r 2 r ∂r Rg T ∂r ∂z

! 1∂ ∂φ ∂2 φ r þ 2 ¼ 0; r ∂r ∂r ∂z ! 2 1∂ ∂φ ∂ φ r þ 2 ¼ 0; r ∂r ∂r ∂z

ð36Þ

ð37Þ

where with “+”and “-”are signified cation and anion, F is Faraday number, Rg is gas constant and T is absolute temperature. The potential Surface mass balance equation (in quasi-steady approximation): þ=− 1∂ rΓ U r ∂r " ! þ=− ! þ=− # 1∂ ∂Γ FΓ 1∂ ∂φ ∂cþ=− ¼ DS þ ∓D r r S r ∂r Rg T r ∂r ∂r ∂r ∂z

; z¼h=2

ð38Þ where with “+/−” is signified cationic or anionic surfactant, while with φS is the surface potential, z is the valence of the surface active ion (z = 1 for cationic and z = −1 for anionic surfactant). The convection of surface charges from the center towards the periphery of the film generates radial distribution of the surface potential φS causing local decrease of the EDL overlapping in the central area of the film. Tangential stress boundary condition: μ

∂V r ∂σ ∂ 1 ∂rU ∂φ εε ∂φs 2 ¼ þ μs −ρs s − 0 2 ∂z z¼h=2 ∂r ∂r ∂r ∂r r ∂r

ð39Þ

where σ and μ s are the surface tension and surface viscosity, while ρs is surface charge density. The bulk viscous stress at the very film surfaces is balanced by the surface tension gradient (Marangoni effect) along the film surfaces and surface viscous stress. Poisson equation: ∂2 φ 1 ∂ ∂φ ρ þ : r ¼− εε 0 ∂r ∂z2 r ∂r

ð40Þ



9

The boundary conditions remain identical to the former case (see Eqs. (19)–(25)). However, the involvement of the ELD: ∂φ ¼ 0; ð41Þ ∂z z¼0 ∂φ ρ ¼− s : εε0 ∂z z¼h

ð42Þ

R=42 µm

2

It is known as well that ∂φ/∂z N N ∂φ/∂r. The participation of EDL in TLF drainage generates several basic additional effects: 1. Polarization of the draining liquid, which should diminish the pressure gradient from the center towards the periphery of the film (see Eqs. (34) and (35)). This contributes to diminishing the rate of TLF drainage;

Fig. 11. Level of coincidence between the experimental data of foam film drainage in the presence of ionic surfactant and the theoretical prediction according to the Stefan–Reynolds theory (Eq. (28) with f = 1) and the new theory accounting for the electro-Marangoni effect (Eq. (43)) [29]; Reprinted with the permission from American Chemical Society, Karakashev S.I., and Tsekov R. “Electro-Marangoni Effect in Thin Liquid Films”, Langmuir, 2011, 27, 265–270.

2. The effect of the surfactant surface diffusion on the TLF drainage is increased due to the mutual repulsion of identical charges on the film surfaces (see Eq. (38)). This contributes to increasing the speed of TLF drainage in cases when the Marangoni effect is still weak; 3. The Marangoni effect is weakened by the mutual repulsion of surface charges (see Eq. (39)). This should contribute to increasing the speed of TLF drainage; and 4. The EDL overlapping in the periphery of the film should become stronger than this one in the central area of the film (see Eqs. (38) and (40)). This should weaken the electrostatic repulsion and therefore make the TLF drainage faster.

The problem finally was solved recently in Ref. [29], where the electrostatic and van der Waals contributions into the Navier–Stockes equation were accounted for along with the assumption for non-zero surface velocity. Thus the Marangoni, number was rederived with the account for the electrical effects originating from the motion of charges. The effect of the surface mobility of the ions was captured by the electroMarangoni number (for more details see Ref. [29]). The authors of Ref. [29] arrived at the following kinetic equation:

A number of simplifications can be made [44] in the system of Eqs. (34)–(40). For example, the non-linear terms originating from the Maxwell tensor in Eqs. (34) and (35) can be neglected. It is known that the adsorption of surfactant from the bulk of the film to the film surfaces has weak effect on the TLF drainage, therefore the last term of the right-hand side of Eq. (38) can be neglected. The surface viscosity term in Eq. (39) can be neglected due to the fact that the Marangoni effect (dσ/dr) is substantially stronger than the effects originating from the surface viscosity. Even with these simplifications the equations cannot be solved analytically but only numerically as it was performed in Ref. [44]. The theoretical predictions in Ref. [44] were not validated by any experimental data. In addition, the difference between TLFs containing ionic surfactants and non-ionic surfactants was not shown in the literature due to the absence of consistent theory, which is able to describe such data.

where VRe is the velocity of drainage according to Reynolds equation (Eq. (28) with f = 1), Ma ¼ kB TΓ=μDs κ is the electro-Marangoni number, kB is the Boltzmann constant, T is the absolute temperature, κ is the Debye constant and Γ is the average surfactant adsorption. It must be noted here that the development of outflow during the drainage of the film causes the emergence of streaming potential between the center of the film and its periphery. Hence, the outflow is directed outwards in the middle of the film and inwards close to the film surface (see Fig. 10). This can make the film surfaces flow inwards depending on the value of the electro-Marangoni number. Moreover, one can see that the new theory predicts drainage with significantly better coincidence to the experimental data than the conventional Stefan–Reynolds theory (see Fig. 11). The above-mentioned theory does not operate with ion-specific effects, although it is well-known the type the counter-ion of the added salt affects the state of the adsorption layer (see Fig. 12) and consequently the drainage and the stability of the foam films.

Dimenslionless normal coordinate, x

0.5

0.3

ð43Þ

Ma* =2

0.1

-0.1

V 1 þ Ma 6 ¼ 1− V Re 2 þ Ma κh

Ma* =100

-10

-5

0

Ma* =0

5

10

15

-0.3

-0.5 Dimensionless radial velocity, V r Fig. 10. The dimensionless radial velocity Vr of liquid outflow along the dimensionless normal coordinate x of the film at κh = 10 [29]; Reprinted with the permission from American Chemical Society, Karakashev S.I., and Tsekov R. “Electro-Marangoni Effect in Thin Liquid Films”, Langmuir, 2011, 27, 265-270.

Fig. 12. Surface tension σ as a function of a2/3 of 5 × 10−4 M sodiumpdodecyl sulfate ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi with variation in the concentration of the added salt — LiCl, NaCl, KCl;a ¼ as ðas þ asalt Þ, where as is the activity of the ionic surfactant, while asalt is the activity of the added salt.


10


2

Pumping out

2Rc

2Rc

Double concave drop

Dimpled

2

Non-planar

2Rc

2R

Equilibrium 2R

Fig. 13. Evolution of a foam film in time (top left to right down) [56]. Reprinted with the permission from Elsevier, Karakashev S.I., Ivanova, D. S., Angarska, Z. K., Manev, E. D., Tsekov, R., Radoev, B., Slavchov, R., and Nguyen, A. V., “Comparative validation of the analytical models for the Marangoni effect on foam film drainage”, Colloids Surf. A, 2010, 365, 1–3, 122–136.

However, all the above mentioned theories are valid for plane parallel films. In fact, the formation of planar film occurs rare. Dimple and many ripples can be seen in most of the films during their drainage (see Fig. 13). 3.3. Effect of film thickness non-homogeneity The dimpling is very often met phenomenon during the first stages of thin film drainage especially in the case of the larger films. The evolution of the dimple was first described by Frankel and Mysels [58] who offered separate formulae for the maximum and minimum thickness, the evolution of these two quantities with time and their dependence on the film size (radius) (see Figs. 14 and 15). The problem of the dimple evolution was discussed before Frankel and Mysels by Dejaguin and Kusakov [60], as well as in some later

theoretical works ([61]; Buyevich and Lipkina [62]; Jain and Ivanov [63], etc.). It must be pointed here that according to the Frankel and Mysels theory the velocity of evolution of the two extreme thickness values is different and the dimple does not disappear with thinning; moreover, it must augment upon time. This result, rather unexpected at the time, will be discussed further. Still more pronounced is the dimpling effect in emulsion films. There, most probably due to the lower interface tension on the liquid/liquid boundary, the range of planarity is narrowed down to even smaller radii, as compared to foam films. At larger radii the symmetry of outflow is also broken in foam as well as emulsion films. The dimple formation and evolution in liquid films on solid substrates were studied experimentally by Platikanov [64]. The results are in accord with the theory and are used in the further theoretical developments of the phenomenon [63,65–69]. A mechanism explaining the



11

Fig. 14. Minimum (1) and maximum (2) film thickness as a function of the thinning time of an aqueous (1.0 mM SDS + 0.1 M NaCl) foam film of radius r = 0.5 mm [5,59]; Reprinted with the permission from Elsevier, Manev E.D., and Nguyen A.V. “Critical thickness of microscopic thin liquid films”, Adv. Colloid Interface Sci., 2005, 114–115, 133–146.

asymmetrical drainage (and accelerated thinning) of the films, based on adsorption as a rate-determining process in the mass transport of surfactant, is recently proposed by Joye et al. [70]. The forth (no-slip) boundary condition of a flow between tangentially immobile surfaces, strictly speaking, should never be realized in foam and emulsion films stabilized with soluble surfactants. Deviations from Eq. (28) have been observed experimentally [41], and described theoretically [7,41], always in direction to accelerated thinning. That is, the radial component of the velocity at the surface may differ from zero. In our first experimental investigation [41] only effect of the bulk surfactant diffusion was examined because the theory developed to describe the phenomenon did not contain the surface diffusion. The object of the study with aniline plus dodecanol foam films was adequate to the task and certain quantitative data about the rate of diffusion as well as the value of the ‘dodecanol in aniline’ bulk diffusion coefficient (D = 1.3 × 10− 9 m2/s) were obtained. The latter result was confirmed through direct determination of the diffusion coefficient. One can see in Fig. 13 domains and internal channels (see Fig. 16) within the film at some stage of its evolution. These surface inhomogeneities affect substantially the rate of film drainage as well.

Fig. 15. Amplitude of film thickness heterogeneity (hmax–hmin) versus film radius [5,59]; Reprinted with the permission from Elsevier, Manev E.D., and Nguyen A.V. “Critical thickness of microscopic thin liquid films”, Adv. Colloid Interface Sci., 2005, 114–115, 133–146.

Fig. 16. Evolution of the film profile during the last stages of thinning (time, indicated from 0 to 100 s is counted from an arbitrary “initial” h). The amplitude of thickness heterogeneity is ca. 25 nm. Film of radius r = 1.0 mm from 1.0 mmol/l sodium dodecyl sulfate + 0.1 mol/l KCl aqueous solutions (Manev et al., 1997 [71]).

It was established in Ref. [71] that the dependence of the velocity of drainage of non-planar films is inversely proportional to the radius of the film on power 0.8: V = −dh/dt ~ R−0.8 instead of power 2 (V = −dh/dt ~ R−2) as requires Eq. (28). The latter means that the ripples makes the film thin faster. Hence, different models trying to explain this fact emerged in the literature [56,71–75]. 3.4. The model of Ruckenstein and Sharma (RSh) [72] This drainage model was developed based on the experimental findings of Manev [48] which show that foam films usually thin faster than the Stefan–Reynolds prediction. The model assumes tangentially immobile film surfaces, wrinkled by capillary waves which move from the film center towards its periphery, thus peristaltically pumping out the liquid from the film center faster. Ruckenstein and Sharma [72] introduced the van der Waals disjoining pressure in the tangential and normal components of the Navier–Stokes equations. No electrostatic disjoining pressure was considered by the authors. The film thickness heterogeneity was introduced by the substantial time derivative of the film thickness as dh/dt = ∂h/∂t + vr∂h/∂r, where vr is the radial velocity of the liquid outflow. This stipulates a weak variation of the local film thickness over the film surfaces. In addition, a balance between the local capillary pressure and the pressure in the film at the film surface, p = − (σ/2)∂2h/∂r2, was made. Due to the fact that the capillary waves are naturally periodical, the pressure in the film should be periodically decaying function of the radial coordinate. It is assumed that the capillary waves oscillate around the average film thickness, which leads to two governing differential equations: the first one for the rate of thinning of a foam film with the average thickness (same as the Reynolds model) and the second one for the oscillation of the local film thickness around its average value. The van der Waals disjoining and the local capillary pressures play significant roles in this model equation. The solution of this second differential equation illustrated the existence of decaying and growing surface waves, moving from the center towards the periphery of the film. The authors introduced an empirical correction for the solution of this equation [38], using a characteristic wavelength of the surface waves. The wavelength is equal to the maximum radius (Rf = 50μm), at which the film still can be plane-parallel


12


[48]. An empirical equation for the amplitude, εt, of the inhomogeneity as a function of the film radius was introduced to exploit the experimental data obtained by Manev [48] as εt ¼

0:25 797R f −209

ð44Þ

where the film radius Rf is given in centimeter and the amplitude εt is in Å. In this model, the wavelength (λ = 50μm) and the amplitude of the surface waves do not vary during the film drainage, while the frequency decays as follows: 3π ω¼ λ

rffiffiffi 3 R f dh : 2 h dt

ð45Þ

However, the experimental findings by Karakashev et al. [38] who studied foam film surface waves with high speed linescan camera microscopy indicated that both the wavelength and frequency change during the film drainage. In addition, the film wavelengths measured are in the range of 50–120 μm [38] and to some extent contradict to the stipulations of the RSh model. This does not mean that the model is wrong. After estimating the contributions of the different terms in the general differential equation, the following equation for the drainage velocity was obtained: V ¼−

R f εt dh 2h3 ¼ ðP σ −Π Þ 1 þ 7:35 2 dt 3μR f λ h

ð46Þ

where λ = 5 × 10−5 m is the characteristic wavelength of the thickness inhomogeneity. Finally, it is noted that the surface waves in this model are generated by the liquid outflow during drainage and are suppressed by the hydrodynamic and other repulsions between the film surfaces. This model should be valid for foam films with rigid film surfaces and was compared with experimental data [72,76]. It was shown that the RSh model agrees well with the experimental data when the radii of the films are in the range from 40 to 450 μm. 3.5. The model of Manev, Tsekov and Radoev (MTR) [71] At the first glance this model appears equivalent to the RSh model as far as it stipulates immobile and deformable film surfaces. However, the MTR model is different. The model derivation starts with the general differential equation giving the speed of film evolution with deformable surfaces [7,71]. The normal force balance for a nearly flat thin liquid film ((∂H/∂r)2 b b 1) is expressed as p = − (σ/2)∂2H/∂r2 − Π(H). The local film thickness, H, is assumed to be a homogeneous function of the average film thickness, h, expressed as H¼h

∂H : ∂h

ð47Þ

In the MTR model, the surface corrugations are static rather than running waves as considered in the RSh model. The solution of the governing differential equation for the wave number of the capillary waves is given as vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u u 0 02 2π tΠ Π 24μV : ¼ þ þ k¼ λ σ σ2 σh4

ð48Þ

The equation expresses that there is a steady-state capillary wave, which changes slowly during the film thinning. The drainage rates close to and far from equilibrium present two important limits. At equilibrium the velocity of film drainage is zero and thus 2π λ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2Π 0 =σ

ð49Þ

which corresponds to the Scheludko critical wave [46]. When the film is far from equilibrium and is relatively thick, Eq. (48) reduces to λ ¼ 2π

σh4 24μV

!1=4 :

ð50Þ

Eq. (50) shows that the wavelength decreases with the film drainage and the theoretical wavelength is of the same order of magnitude as the experimental data [38]. With the existence of variable wavelength, this model differs from the RSh model, which stipulates a single (constant) wavelength. When substituting VRe for V into Eq. (50) and following the mathematical procedure given in Ref. [71], one obtains the critical radius of the foam film (about 40μm), below which the film becomes planar and follows the Stefan–Reynolds model for drainage. The latter was confirmed by the experimental data [48]. Integrating the general driving differential equation yields the final equation of the film drainage as [71] dh 1 V ¼− ¼ dt 6μ

h12 ðP σ −Π Þ8 4σ 3 R4f

!1=5 :

ð51Þ

Similar to the RSh model, the MTR model is valid for foam films with any radii and immobile film surfaces and was compared against experimental data [76–78]. The RSh model gives good agreement with the theory with both small and large foam films. In the case of small foam films (Rf ≤ 50 μm) the RSh model agrees with the experimental data even for foam films with mobile film surfaces. At larger film radii the RSh model agrees well with the experiments when Eg ≥ 10 mN/m, which is in accord with the theoretical stipulations of the model (for rigid film surfaces). It should be noted that the film surfaces become practically immobile, starting from Eg = 1 mN/m. The requirement for higher value of the Gibbs elasticity with the RSh model most probably concerns the mechanical strength of the film surfaces. Again Eg ≥ 10 mN/m is fulfilled with most of the cases, which makes RSh model very applicable for variety of different cases. This model does not fit to the experiment in some special cases, as for example, the surfactant concentration is at least 300 times larger than the CMC (see Ref. [56]) or when surface aggregates on the film surfaces are formed (see Table 3 in Ref. [56]). It should be noted here that this model was inducted by the experiments of Manev [48]. In addition, empirical correction originating from Ref. [48] was introduced in the RSh model thus becoming semi-empirical. In this term it is not surprising that this model agrees with the similar experiments in most of the cases. The MTR model was developed for inhomogeneous foam films with immobile surfaces. In most of the cases in the present work, these requirements are fulfilled yet the model predicts significantly faster drainage. The evaluation showed as well that the MTR model agrees fairly (Table 1; see Table 3 in Ref. [56]) with the experiment in some nontrivial cases, when the film radius is below 100 μm and in the presence of the weak surfactant. In addition, surprisingly the model gave good agreement with experimental data with foam films containing surface aggregates (see Table 3 in Ref. [56]). It should be noted that according to Ref. [78] the MTR model agrees better with experiments on films with larger radii (e.g. 200–500 μm) than the other models observed in the present work.

Table 1 Dynamic fractal dimension α obtained by fitting via Eq. (52) of experimental data for the kinetics of TLF thinning. C12E6

R = 50 μm

R = 100 μm

R = 150 μm

1 × 10−5 M 3 × 10−5 M 1 × 10−4 M

2.00 2.00 2.00

0.75 0.88 1.00

0.73 0.66 0.75



100

It was found that the film corrugations emerge when the film radius is larger than certain paffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi transition radius given by the expression [71] RRe ¼ 4 σh=Δp , where σ is the surface tension, h is the film thickness and Δp is the driving pressure. It is evident that the latter varies during the film drainage. It was established [71] that if film radius is below the transition one (R b RRe) the films are planar and they drain in accord with the Reynolds equation. Such films are small practically [48] (R b 50 μm). However, bubbles or droplets in contact in most cases form TLF with radii significantly larger than RRe. Such films exhibit significant dynamic corrugations making the film drain faster than according to the Reynolds equation [71]. These corrugations are surface non-homogeneities as dimples [78], wimples [79] or channels [80] depending on the film radius, thickness and driving pressure. They can be premised as smaller sub-films or domains [78], evolving in one common TLF. It was established [78,81] that in some cases their number depends on both film radius and the transition radius by the expression n = (R/RRe)4/5. As mentioned above, it was also established [48,71,78] that the rate of drainage of some non-planar films is inversely proportional to film radius on power of 4/5 (V ∼ 1/R4/5) instead of power of 2 valid for planar films (VRe ∼ 1/R2). A detailed analysis on the nature of the surface corrugations [48,73, 78,81] shows that spatial correlations between the domains exist. The strength of these correlations affects the functional dependence of the rate of film drainage on the film radius R by the expression V ∼ 1/Rβ, where the power coefficient β is the unknown parameter having values between two and zero. Hence the planar films drain according to the Reynolds equation (β = 2) if the film surfaces are immobile. However non-planar films can drain according to different kinetic equations depending on the power β. A new kinetic theory, accounting for the behavior of the film domains during TLF drainage, was recently created [73]. The spatial correlation between the film surface domains was expressed by the fractal dimension number α having values between two and zero. Hence the rate of TLF drainage is given by the formula: ð52Þ

The strongest spatial correlation between the surface domains corresponds to α = 2. In such a case the film is planar and its drainage obeys the Reynolds equation:

-5

3x10 M C12E6, R=150 m

80 Film thickness, nm

3.6. Fractal model [73,74]

!2−α dh 2h3 Δp R2 Δp 2þα V ¼− ¼ : dt 3μR2 16σh

13

α=0.66

60 40 20 0

0

10

20

30 Time, s

40

50

60

Fig. 17. An illustration about the best fit with Eq. (52) of experimental data of drainage of a foam film with 1 × 10−5 M C12E6. The film ruptures after about 50 s.

Fig. 17). A program was written using the VBA (Visual Basic for Application) programming language available in Microsoft Excel. In such case, the concentration of the background electrolytes was so high that the electrical double layer interaction was fully compressed. Therefore, the surface potential was not essential. A fitting with only one matching parameter α of the experimental data points was conducted using Solver option of Microsoft Excel. Thus the value of α was obtained for TLF with different radii, which were prepared from aqueous solution of hexaethylene glycol dodecyl ether (C12E6) with three different concentrations. Thus, a correlation between the parameter α and the film radius was obtained. All the α values obtained for foam films with different radii are summarized in Table 1 and Fig. 18. An empirical equation α = a/Rb was found reliable to be applied to the experimental data points α vs. R for fitting procedure with two matching parameters — a and b. The fitting procedure resulted in a = 100 and b = 1 (R units are in micrometers, see Eq. (55)). This way for describing the behavior of the parameter α is rather empirical than theoretical, but anyway is helpful for the proper understanding of the nature of parameter α. Eq. (55) shows that the parameter α is inversely proportional to the film radius and it does not depend on the surfactant concentration. The results from Table 1 are presented in Fig. 17, where the line represents the best fit α ¼ 100=R ½μm:

ð55Þ

3

V Re ¼

2h Δp : 3μR2

ð53Þ

In a case when α = 1/2, the film drains according to the equation of Manev, Tsekov and Radoev [71,78]:

V MTR

1 ¼ 6μ

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 12 5 h Δp8 : 4σ 3 R4

ð54Þ

There is not a theory about the fractal dimension number α, hence it is an entity for studies. The parameter α should depend on the film thickness generally. For example, it is well known that despite the pre-history of TLF evolution, the film becomes planar when approaching equilibrium and drains in accord with Reynolds equation (α = 2). The fractal coefficient α was investigated in Ref. [74]. In this study the foam films did not reach equilibrium due to the totally suppressed electrostatic disjoining pressure (ionic strength = 0.024 M). Instead, they drained until rupturing at certain film thickness. Eq. (52) was numerically integrated in Ref. [74] to obtain the transient film thickness using the 4th order Runge–Kutta algorithm (see

Fig. 18. The experimental dependence of α on R.


14


We must acknowledge that the above described theory is the most advanced one and describes very well the drainage of foam films in the presence of non-ionic surfactants in large scope of radii (significantly beyond 100 μm). 4. Perspectives of thin liquid drainage theory Although the topic is quite old, there are some still unresolved problems. For example, the problem for rippled films stabilized by ionic surfactants is not yet solved. Obviously, the ripples will make the film drain faster, while the ionic surfactant will make the film drain slower. Which one of these two opposite factors will dominate depends on the size of the film. Another unresolved problem is the wetting film (bubble against solid) on moving solid surface. Recent publication on this topic [39] indicates the existence of lift force, making the film thicker, however we are still far from proper theoretical description especially in the case of surfactant stabilized film. Another unresolved problem is the dynamic stabilization of foam films in the presence of highly concentrated solutions of “active” salts [82]. In the presence of “non-active” salts the films live shortly. The drainage and stability of foam films from ultrapure water are still a mystery, as far as the film drains until about 30 nm at saturated vapors and ruptures. However, at unsaturated vapors the latter ruptures while being still double concaved drop. In our opinion after solving the above mentioned problem the topic of thin liquid film drainage will be closed. However, their solution is quite difficult and will require a lot of effort and additional investigations. 5. Conclusion The philosophy of this work has been to serve the reader with the basic development of the physical concepts and experimental results on foam film drainage in the recent past. Most of the results belong to outstanding scientists and researchers from the Bulgarian school of physical chemistry. The reader will become familiar with the basic development of the hydrodynamic theory and experimental results on thin liquid film drainage in the recent past. Acknowledgments This work was supported by FP7 project BeyondEverest. References [1] Toshev BV, Ivanov IB. Thermodynamics of thin liquid films. I. Basic relations and conditions of equilibrium. Colloid Polym Sci 1975;253:558–65. [2] Sheludko A. Thin liquid films. Adv Colloid Interface Sci 1967;1:391–464. [3] Levich VG. Physicochemical hydrodynamics. Englewood Cliffs, NJ: Prentice-Hall; 1962. [4] Sheludko A. Certain peculiarities of foam lamellas. I. Formation, thinning, and complementary pressure. Koninkl Ned Akad Wetenschap Proc Ser B 1962;65:76–85. [5] Exerowa D, Kruglyakov PM. Foam and foam films: theory, experiment, application. New York, NY: Marcel Dekker; 1997. [6] Kitchener JA. Progress in colloid chemistry. J Oil Colour Chem Assoc 1954;37:355–76 [discussion, 76-7]. [7] Ivanov IBE. Thin liquid films. New York, N. Y.: Marcel Dekker; 1988 [8] Gibbs JW. The scientific papers of J. Willard Gibbs. New York: Dover; 1961. [9] Plateau JAF. Statique Expérimentale et Théorique des liquides soumis aux seules Forces Moléculaires. Paris: Gauthier-Villars; 1873. [10] Boussinesq JM. The application of the formula for surface viscosity to the surface of a slowly falling droplet in the midst of a large unlimited amount of fluid which is at rest and possesses a smaller specific gravity. Ann Chim Phys (Paris) 1913;29: 357–62. [11] Karakashev SI, Nguyen AV. Effect of sodium dodecyl sulphate and dodecanol mixtures on foam film drainage: examining influence of surface rheology and intermolecular forces. Colloids Surf A 2007;293. [12] Derjaguin B, Obuchov E. Kolloidn Zh 1935;1:385. [13] Deryaguin BV. Theory of the stability of colloids and thin films; 1989. [14] Derjaguin B, Landau L. Theory of the stability of strongly charged lyophobic sols an of the adhesion of strongly charged particles in solutions of electrolytes. Acta PhysChim 1941;14:633–62. [15] Verwey EJW, Overbeek JTG. Theory of the stability of lyophobic colloids. Amsterdam: Elsevier; 1948.

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