On the forced drainage of foam

Colloids and Surfaces A: Physicochem. Eng. Aspects 305 (2007) 1–9

On the forced drainage of foam Paul Stevenson Centre for Advanced Particle Processing, University of Newcastle, Callaghan, NSW 2308, Australia Received 20 September 2006; received in revised form 16 April 2007; accepted 16 April 2007 Available online 21 April 2007

Abstract The method of so-called ‘forced drainage’ has been extensively used to characterise the steady one-dimensional drainage of gas–liquid foams. In this paper, it is shown that the liquid drainage rate is approximately independent of surface dilational viscosity and surface tension (insofar as it creates capillary suction). However, liquid drainage rate is dependent on surface shear viscosity and Marangoni stresses. For transient foam drainage, an independence upon surface dilational viscosity is not demonstrated. Previously reported forced drainage data have been described in a dimensionally consistent form so that different studies can be directly compared with one another, and this has revealed two curious anomalies in the data, some of which may be due to previously reported, but generally overlooked, multi-dimensional effects in forced drainage experiments. © 2007 Elsevier B.V. All rights reserved. Keywords: Foam; Forced drainage; Surfaces viscosity; Marangoni stresses

1. Introduction Leonard and Lemlich [1] presented a solution for the drainage of interstitial liquid from foams that suggested that drainage rate is dependent upon surface shear viscosity and that viscous losses occur only in Plateau borders (or ‘channels’) in the foam. This approach is known as ‘channel-dominated’ foam drainage. Since then there has been a large body of literature that has attempted to relax certain assumptions about foam drainage. Particular highlights include: 1. The development of the so-called ‘forced drainage’ experiment to measure certain drainage characteristics within foam [2]. 2. The development of an alternative hypothesis to ‘channeldominated foam drainage’ that assumed that viscous losses occurred within the nodes (or vertices) at which four Plateau borders meet. This is known as the ‘node-dominated foam drainage equation’ [3]. 3. The demonstration of the dependency of liquid velocity within a single Plateau border upon the surface shear viscosity [4].

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4. The postulation of drainage equations to take into account other surface properties such as surface dilational viscosity and Gibbs elasticity [5]. 5. The direct experimental observation of mobility at the Plateau border walls using confocal microscopy [6]. The method of forced drainage has been extensively used in the literature to investigate foam drainage as it is a convenient method of assessing the drainage characteristics of a foam. In this paper, results gained by using this method will be reviewed and it will be shown that, in the forced drainage experiment, as in systems employing rising columns of pneumatic foam, the only surface property that controls drainage rate is surface shear viscosity and surface tension (insofar as gradients of which cause Marangoni stresses) whereas, in the free drainage of foam, drainage rate may have a dependency upon other surface properties. In addition, apparent dichotomies within the literature will be discussed. The channel-dominated foam drainage equation has found favour with workers who attempt to model froth flotation [7,8]. However, it will be shown the channel-dominated drainage model is not supported by any published data that the author is aware of. Please note that it is not the intention of this work to present any new data or new theory whatsoever. Rather, previously existing data are plotted in a consistent manner so that they can be directly compared, and this will enable

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P. Stevenson / Colloids and Surfaces A: Physicochem. Eng. Aspects 305 (2007) 1–9

the superficial drainage velocity in the foam is Nomenclature A Bo C Fs g jd K1/2 m n Q r rb Re Sk u v0 Vf

column cross-sectional area (m2 ) Boussinesq number ( μs /μrb ) a geometrical factor used in Eq. (12) (≈0.402) surface excess body force (Pa) acceleration due to gravity (m s−2 ) superficial liquid drainage rate (m s−1 ) a dimensionless parameter used in [37] dimensionless number used in Eq. (11) dimensionless index used in Eq. (11) volumetric liquid rate (m3 s−1 ) the radius of curvature of Plateau border walls (m) bubble radius (m) Reynolds number (≡ρjd rb /μ) Stokes number (≡jd μ/rb2 gρ) average velocity of fluid within a rigid vertical Plateau border (m s−1 ) surface velocity (m s−1 ) velocity of the wet front (m s−1 )

Greek letters α dimensionless constant used in Eq. (1) ε volumetric liquid fraction in the foam κs surface dilational viscosity (Pa m s) μ interstitial liquid dynamic viscosity (Pa s) μs surface shear viscosity (Pa m s) ρ interstitial liquid density (kg m−3 )

a quantitative appraisal of the channel-dominated drainage model.

jd =

Q A

(2)

where A is the cross-sectional area of the column. This superficial velocity occurs in a foam of volumetric liquid fraction ε, which can be determined by using the relationship: ε=

jd Vf

(3)

It is therefore apparent that, by measuring Vf as a function of jd for a particular foam, a relationship for the superficial drainage velocity as a function of liquid fraction in a spatially invariant foam can be obtained. Thus we can write: jd ∝ εn

(4)

where n=

1 1−α

(5)

It has recently been shown that, if the relationship between jd and ε is known, the hydrodynamics of pneumatic columns of foam, such as those found in foam fractionation and froth flotation operations, can be modelled [9]. Therefore, forced drainage data expressed in the form of proportionality (4) has arguably greater practical utility than proportionality (1). Note that it is apparent that it is the hydrodynamic condition of the foam upstream of the wet front, rather than the rate at which Plateau borders at the front can expand at the interface between the two zones that controls the front velocity; if it were otherwise the liquid fraction upstream of the front would not exhibit the characteristics of spatial and temporal invariancy.

2. The method of forced drainage 3. Steady versus transient drainage In the forced drainage experiment, a foam is typically prepared within a column by either mechanical agitation or gas sparging. The foam is then allowed to drain under gravity so that it becomes very dry; in these experiments the volumetric liquid fraction typically becomes less than 0.01% [3]. Liquid is then added to the top of the column at a volumetric flowrate Q so that a wet front travels down the column and there are two distinct zones in the column: (1) a dry zone downstream of the wet front, and (2) a wet zone upstream of the front. The liquid fraction in the wet zone is invariant in height and time. The velocity of the wet front, Vf , is measured either optically (with or without the addition of dyes to the liquid applied to the top surface) or by measuring the change in electrical conductivity of the foam at stages down the column. It has been demonstrated by a number of workers [2,3 inter alia] that the velocity of the wet front is proportional to a power of the volumetric liquid rate applied to the surface, i.e. V f ∝ Qα

(1)

where α is a dimensionless constant. Now, it is convenient, for present purposes, to work in terms of superficial velocities so

When a standing foam drains under gravity the liquid drainage rate diminishes over time as the liquid fraction decreases. Both liquid fraction and liquid drainage rates in socalled ‘free drainage’ have been recently measured directly and non-invasively using NMRI methods [10]. Such foams have a spatially and temporally variant liquid fraction and the drainage in such a foam is therefore referred to as ‘transient drainage’ herein. Consider the stress state at the gas–liquid surface. Assuming that the surface exhibits Newtonian rheology, the x-component of the general normal stress boundary condition in cartesian coordinates (Table 4.2-1 of [11]) that is dependent upon surface shear viscosity, μs , surface dilational viscosity, κs and gradients of surface tension (i.e. Marangoni stresses) is   0 ∂v0y ∂σ ∂ ∂v x s + (κs + μs ) + ||P¯ zx || = Fx + ∂x ∂x ∂x ∂y   ∂v0y ∂ ∂v0x + μs (6) − ∂x ∂y ∂y

P. Stevenson / Colloids and Surfaces A: Physicochem. Eng. Aspects 305 (2007) 1–9

where P¯ zx is the z–x element of the pressure tensor, Fxs the xcomponent of the surface excess body force vector, σ the surface tension and v0x and v0y are the x- and y-components of the surface velocity vector, respectively. Due to the dependency of the boundary condition on μs , κs and Marangoni stresses (∂σ/∂x), the flowfield of liquid draining through the foam shares this dependency. Because, in transient foam drainage, the interfacial deformation rate is non-zero, Durand and Langevin [5] developed foam drainage equations based on these normal stress boundary conditions and Stone et al. [12] have shown that a non-dimensionalised permeability of a foam may be expressed as some function of six dimensionless groups, some of which embody these surface properties. The forced drainage experiment is a special case of foam drainage in that the liquid fraction, and therefore structure, of the foam upstream of the wet front, in the absence of Ostwald ripening and bubble coalescence, is temporally invariant. In addition, above the wet front, the liquid fraction does not vary with height (i.e. it is one-dimensionally spatially invariant). Consider the flow within a long and slender Plateau border. (This is the stylised representation of foam structure implied by the channel-dominated foam drainage model in which the Plateau borders have spatially invariant cross-sectional shapes and areas, and is not strictly accurate for foams of finite wetness.) Since there is only an axial component of velocity which does not vary in time, and this velocity is spatially invariant, it is seen that the boundary condition at the surface, and therefore the drainage of liquid through the foam, is independent of κs . In addition, because the structure of the foam is temporally invariant, the surface deformation rate is zero and therefore the drainage rate is not dependent upon any non-Newtonian surface stresses which may exist for a more general case than that considered in the development of Eq. (6). Now consider the foam on a macro-scale. Because the foam is spatially invariant there is no global capillary suction so drainage rate is independent of the absolute value of surface tension. However, we can say nothing about Marangoni stresses that can occur in forced foam drainage because of gradients of the degree of surface adsorption at the interface of a single bubble (i.e. gradients of surface tension). Note that the assertion that drainage rate is independent of surface dilational viscosity in a spatially and temporally invariant foam is strictly valid only for foams in the limit of zero liquid fraction, in which Plateau borders are long and slender and whose cross-sectional area is axially invariant. For practical wetter foams, where the boundary between node and Plateau borders is indistinct and the cross-sectional area of the Plateau border changes (therefore causing changes in the axial component of velocity) it is not possible to assert that liquid drainage rate is independent of surface dilational viscosity, but the above observation might suggest that surface dilational viscosity is of secondary importance in determining liquid drainage rate. In addition, even in very dry foams, there is interfacial expansion and contraction close to the nodes. Thus, the drainage rate from a foam undergoing forced drainage is apparently approximately independent of surface dilational viscosity and surface tension insofar as it governs capillary suction. However, drainage rate will depend upon

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Marangoni stresses, but these will be approximately temporally invariant. The drainage rate is dependent upon μs because this term still appears in the boundary condition. The same argument can be used for the surface dependency of the hydrodynamic condition of columns of rising foam since these are approximately spatially and temporally invariant. Note that Stoyanov et al. [13] experimentally found that drainage rate in forced drainage was independent of surface tension for foam stabilised with SDS (sodium dodecyl sulphate) and CTAB (cetyltrimethylammonium bromide). Thus, if Marangoni stresses are discounted for the time being, for a steady foam we can write that: jd = f (g, rb , μ, μs , ρ, ε)

(7)

where g is the acceleration due to gravity, rb a representative bubble radius, and μ and ρ are the viscosity and density of the interstitial liquid. It has been shown [14], by employing Buckingham’s ␲-theorem, that one can express jd , non-dimensionalised as a Stokes-type number, Sk, as a function of the liquid fraction, a Boussinesq-type number (based on the surface shear viscosity), Bo, and a Reynolds number, Re, thus:   μjd μs ρjd rb Sk = (8) = f (ε, Bo, Re) = f ε, , μrb μ ρgrb2 Making the creeping flow assumption that the flowfield is independent of inertial effects, one can state that Sk is independent of Re to give: Sk = f (ε, Bo)

(9)

In order to quantify Bo, the surface shear viscosity must be known. However, it has been demonstrated that the measurement of μs is most problematic for surfaces stabilised by soluble surfactant. The reported values of μs for surfaces stabilised by SDS vary by a factor of approximately 100-fold (for moderate SDS concentrations) [15]. Pitois et al. [16] have artificially created an over-sized Plateau border and measured drainage rate of liquid through this which allows μs to be inferred. However, the method has not yet been shown to facilitate a priori predictions of the drainage of practical foams. Because of this, μs can only be considered as an adjustable constant and therefore Bo is effectively an adjustable constant that is dependent upon the surfactant type and concentration. This is not to say that Bo is not important but, since it is unquantifiable, we must write that: Sk = f (ε)

(10)

with the proviso that any expression obtained will be specific to a certain type of surfactant at a certain concentration. Note that, if the dependency on surface shear viscosity is described through an adjustable dimensionless constant, this can also act as a proxy to describe the influence of Marangoni stresses on foam drainage, which are also presently unquantifiable in practical systems. Note also that it is assumed that the surface shear viscosity is assumed to exhibit Newtonian rheology; if it does not, in fact, exhibit such rheology, the statement of Eq. (10) is not strictly true even for a specific surfactant system.

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P. Stevenson / Colloids and Surfaces A: Physicochem. Eng. Aspects 305 (2007) 1–9

Thus, following the form of Eq. (4) we write: Sk = mεn

(11)

where m is a dimensionless adjustable constant. With the proviso that one set of m and n values is only appropriate for a specific surfactant at a specific concentration. Note that Eq. (11) employs two dimensionless constants which are the minimum required since: 1. The viscous losses in the nodes are not known (hence the adjustable constant I in [3] which represents a dimensionless measure of viscous dissipation at the nodes). 2. Quantitative values of the surface shear viscosity and Marangoni stresses are not known. To re-iterate, in forced drainage, the drainage rate is approximately independent of surface dilational viscosity and capillary suction. However, drainage rate is dependent upon surface shear viscosity and Marangoni stresses but these are difficult to quantify; instead the drainage rate is expressed in a dimensionless form with adjustable constants that are specific to a certain type and concentration of surfactant. 4. Forced drainage data expressed as Sk = mεn It has been shown [14] that previous data [17] for foams stabilised by SDS and TTAB (tetradecyltrimethylammonium bromide), of various bubble sizes, could be expressed in the form of Eq. (11). Other sets of data for forced drainage experiments are now shown in the same form in Table 1. Note that the forced drainage data [13,18] could not be included since the cross-sectional area of their columns was not reported.

Fig. 1. Sk vs. ε for the experiments of Durand et al. [20]. The lighter solid lines indicates Sk = 0.021ε1.64 , the dashed line indicates Sk = 0.063ε2.10 , and the bold solid line indicates the channel-dominated drainage model represented by Sk = 0.018ε21.92 .

The calculation of m is sensitive to the bubble diameter (and therefore rb ). In some of the reported experiments [20] only approximate values of bubble diameter have been given and, for Koehler’s experiments, rb , has been inferred from reported values of Plateau border edge length using an expression [17] that relates edge length to bubble size and liquid fraction by making the simplifying assumption that the liquid fraction tends to zero. Most other works that report a bubble radius or diameter do not give further definition (i.e. whether an equivalent spherical volume radius is inferred). In addition, the bubble sizes are probably measured at the forced drainage experiment making subsequent analysis subject to errors due to foam coarsening through Ostwald ripening (although some investigators took care to minimise coarsening by using sparingly soluble gases). Fig. 1 shows the data of Durand et al. [20] plotted as Sk versus ε by way of example, with the power-law fits to the data superimposed to demonstrate the efficacy of the approach.

Table 1 Drainage equations in the form of Eq. (11) fitted to previous data sets Source

Solution LiquidTM ?)

Bubble diameter (mm)

Drainage equation

0.77

Sk = 0.046ε2.13

[2]

“Detergent” (Fairy

[17]

SDS (conc. unspecified) TTAB (conc. unspecified)

1.6 and 2.5 1.6 and 2.5

Sk = 0.012ε1.74 Sk = 0.013ε1.78

[10]

2.92 g/l SDS in 0% and 40% glycerol

0.58 and 0.80

Sk = 0.016ε2.00

[3]

0.25 wt% Dawn SoapTM

5.56 est. 2.36 est. 1.42 est.

Sk = 0.0011ε1.56 Sk = 0.0022ε1.56 Sk = 0.0027ε1.55

[20]

0.25 wt% Dawn SoapTM 3.46 g/l SDS, 1.73 mg/l dodecanol 3.46 g/l SDS, 0.86 mg/l dodecanol

1.00 approx. 1.00 approx. 1.00 approx.

Sk = 0.044ε1.63 Sk = 0.063ε2.10 Sk = 0.021ε1.64

[19]

SDS (conc. unspecified) with dodecanol (at conc. 250 times less than SDS)

0.80 0.18

Sk = 0.060ε2.12 Sk = 0.011ε1.45

[21]

10 g/l SDS 1.5 g/l Casein

0.18 and 8.0 0.18 and 8.0

Sk = 0.019ε2.00 Sk = 0.018ε2.10

Source

Models

Drainage equation

[1] [3]

Channel-dominated drainage (rigid walls) employing Eq. (14) Node-dominated drainage

Sk = 0.018ε.1.92 Sk = (0.49/I)ε.1.55

P. Stevenson / Colloids and Surfaces A: Physicochem. Eng. Aspects 305 (2007) 1–9

Fig. 2. Forced drainage data [21] for foams stabilised with 1.5 g/l Casein with the fit Sk = 0.018ε2.10 superimposed. The legend indicates the reported bubble diameter. The bold line represents channel-dominated foam drainage theory (Eq. (15)).

It is seen clearly seen that a lower concentration of dodecanol promotes enhanced drainage due to increased surface mobility [22]. Also shown in Table 1 is the drainage equation obtained using an NMRI method to measure liquid drainage rate in a transient foam as a function of time a position [10]. It was found that the form of Eq. (11) well-described data for foams with interstitial viscosity of both 1 and 4 cP using m = 0.016 and n = 2. In addition, the channel-dominated foam drainage equation with infinite surface shear viscosity [1] and the node-dominated foam drainage equation [3] have been approximately written in the form of Eq. (11) by Stevenson [14] and these expressions are shown in Table 1. (Note that Koehler et al. [3] did give the node-dominated foam drainage equation with liquid fraction as the primary variable.) It is seen that n = 1.55 is consistent with node-dominated foam drainage and n = 1.92 is consistent with channel-dominated drainage. The indices on ε for the experimental programmes of Table 1 have the values of 1.45–2.13. There is little change in n for the three experiments of Koehler et al. [3]. Safouane et al. [18] found that their values of α varied between 0.45 and 0.53 (i.e. values of n are 1.82–2.13) for TTAB solutions whose interstitial kinematic viscosity was enhanced to about 5.3 × 10−6 m2 s−1 . However, Saint-Jalmes and Langevin [19] achieved a large reduction in the index n from 2.12 to 1.45 by using smaller bubbles; this was attributed to a transition from the channel-dominated to node-dominated regimes. Saint-Jalmes et al. [21] have clearly demonstrated that a regime transition between the drainage rate from foams of small bubbles and foams of large bubbles; this regime change is attributed to changes in surface mobility. Therefore, it may, to some, appear curious why the Boussinesq number is absent from Eq. (11) for the liquid drainage from a foam stabilised by a specific surfactant at a specific concentration. However, if the surfactant system is fixed, thereby fixing the value of surface shear viscosity, the dimensional analysis indicates an independence of Stokes number upon Boussinesq number, as can be demonstrated by consideration of the data of Saint-Jalmes et al. who provided forced drainage data for foams stabilised by 1.5 g/l Casein that employed a variety of different bubble sizes, but only the extreme sizes (0.18 and 8.0 mm in diameter) are explicitly given. The data for Casein are plotted as Sk versus ε in Fig. 2. It is seen that the expression Sk = 0.018ε2.10 has efficacy

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for both extreme bubble sizes (i.e. over three decades of Sk), giving support to the bubble size dependency predicted by Eq. (11) and the independence of Sk upon Bo for a foam stabilised by a specific type and concentration of surfactant. Many analyses of foam drainage are explicit in the radius of curvature of the Plateau borders, r, rather than mean bubble radius, rb , so it may appear curious as to why the latter appears in Eq. (7) rather than the former. However, by dimensional arguments, it does not matter whether r or rb is used since they are related via the liquid fraction as shown in any of Eq. (14), (17) or (19). Because ε appears with rb in Eq. (7), r could replace rb in this equation with no loss of generality; rb can act as proxy for r and vice versa (the values of m and n would be modified if r were used). r must not appear with ε and rb however, because the three quantities are not mutually independent of one another. To do so would introduce redundancy into the problem. From a practical point of view it is preferable to use rb since the mean of this quantity is easier to determine experimentally. Most forced drainage studies have focussed upon the value of α inferred from their experiments. However, by expressing predicted drainage rate quantitatively in a dimensionally consistent form we can directly compare the predictions of different workers and this has uncovered two anomalies in the data. 4.1. Anomaly 1: the forced drainage of Dawn SoapTM foam Koehler et al. [3] performed forced drainage experiments on foam stabilised by 0.25 wt% Dawn SoapTM for three different bubble sizes (estimated diameters 1.42, 2.35, 5.56 mm). Durand et al. [20] repeated the same experiment with bubble diameter approximately 1 mm. Drainage expressions of the form of Eq. (11) have been fitted to all four data sets and plotted in Fig. 3. It is clear that Koehler’s data for three bubble sizes very approximately collapses to one curve on a plot of Sk versus ε. The discrepancies that exist can perhaps be explained by the method used to determine the bubble size. However, the expression fitted to Durand’s data predicts a Stokes number that is 10–17 times greater than the expression for Koehler’s smallest bubbles; this discrepancy certainly cannot be explained by errors in bubble size measurement.

Fig. 3. Stokes number vs. liquid fraction for the Dawn SoapTM experiments [3,20] and Sk = 0.044ε1.63 (light solid line), Sk = 0.0022ε1.56 (dashed line). The channel-dominated forced drainage equation (Eq. (15)) is represented by the bold line.

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P. Stevenson / Colloids and Surfaces A: Physicochem. Eng. Aspects 305 (2007) 1–9

The average velocity of fluid flowing in a vertical Plateau border with rigid walls is approximately [25]: u=

(Cr)2 ρg 50μ

(12)

where C is a geometrical factor that takes the value of 0.402 and r is the radius of curvature of the Plateau border walls. To convert this to a liquid superficial drainage velocity we must divide by 3 (to take into account the random orientation of the network of Plateau borders) and multiply by the liquid fraction ε thus: Fig. 4. Stokes number vs. liquid fraction for Fairy LiquidTM experiments [2], the fit of Sk = 0.046ε2.13 superimposed, with three variants of the rigid wall channel-dominated drainage equation (Eq. (15), bold solid line; Eq. (18), light solid line; Eq. (20), dashed line).

jd =

(Cr)2 ρg ε 150μ

(13)

The Plateau border radius is not conveniently measured so we seek a relationship that relates r to rb and ε. The following expression [26] has been used in this work:

4.2. Anomaly 2: the forced drainage of Fairy LiquidTM foam

r = 1.28ε0.46 rb

Koehler et al. [3] presented their node-dominated drainage model as an alternative hypothesis to channel-dominated drainage [1,23]. Note that the channel-dominated model of Verbist et al. [23] is identical to the more general drainage model of Leonard and Lemlich [1] in the limit of infinitely rigid Plateau border walls. Weaire [24] reconciles the discrepancy between the experimental results of Koehler et al. [3] and his own by noting that Koehler used Dawn SoapTM solutions whereas Weaire et al. [2] apparently used a solution of Fairy LiquidTM at an undisclosed concentration [24], although the type of detergent was not disclosed in the original article. Thus Weaire conjectured that the choice of dishwashing liquid brand affected drainage characteristics. Weaire [24] remarked “Both groups, both experiments, both theories would appear to be at least roughly correct.” This statement is curious for two reasons:

which is an explicit approximation of an implicit expression [17] that takes into account the volume of the Plateau borders and nodes. Thus substituting Eq. (14) into Eq. (13) we obtain: Sk = 0.0018ε1.92

In Table 1 the expression Sk = 0.0018ε1.92 is given as an approximation to the channel-dominated drainage model for spatially and temporally invariant foam. It is pertinent to present a full derivation herein.

(15)

However another expression has been proposed to relate r to rb and ε for wet foams [27]: √ r 1.734 ε(1 − ε)1/3 = (16) rb 1 + 0.765ε0.409 to which the expression: r = 0.91ε0.36 rb

(17)

is a good approximation. If Eq. (17) is adopted the channeldominated drainage equation becomes: Sk = 0.00089ε1.72

1. The node-dominated drainage theory of Koehler et al. [3] employs the adjustable constant I to take into account the viscous losses at the nodes and does not claim to have a priori predictive capability. However, estimates of I based on the calculations of slow flow through a packed bed of rigid spheres have previously been presented (see Appendix A). 2. The channel-dominated theory of Verbist et al. [23] appears to claim a priori predictive capability, but it is demonstrated below in Fig. 4. That the model predictions greatly underestimate their own forced drainage data [2]; the drainage rates measured by Weaire et al. [2] are approximately 20 times greater than those predicted by the theory of Verbist et al. [23] represented by Sk = 0.0018ε1.92 (Eq. (15)).

(14)

(18)

In addition the expression: r = 1.91ε0.5 rb

(19)

has been proposed for very dry foam [28]; if this is adopted the channel-dominated drainage equation becomes: Sk = 0.0039ε2

(20)

There also exist expressions that relate r, rb and ε for foams with poly-dispersive bubble size distributions [29]. The three forms of the channel-dominated drainage equation (Eqs. (15), (18) and (20)) are plotted in Fig. 4. It is seen that all three variants of the channel-dominated drainage expression under-predict Weaire’s observed drainage rate. It is therefore apparently difficult to describe channel-dominated drainage theory [23] as “roughly correct” [24] based on a comparison with the data [2]. Eq. (15) is also plotted upon Fig. 1. Again its inadequacy at giving quantitative prediction of liquid drainage rate is evident.

P. Stevenson / Colloids and Surfaces A: Physicochem. Eng. Aspects 305 (2007) 1–9

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5. Multi-dimensional effects in forced drainage Thus far, the forced drainage experiment has been considered as one-dimensional. However, in practise, there are twoand three-dimensional effects that can affect the outcome of forced drainage experiments. Because such multi-dimensional effects have largely been overlooked by those presenting forced drainage data, a brief survey is presented herein. It is supposed that such effects may help explain some of the dichotomies discussed above. 5.1. Wall effects In the forced drainage experiments described above, a number of column cross-sectional shapes and areas have been used. For example, Weaire et al. [2] used a circular section tube of diameter 15 mm, whereas Durand et al. [20] used 40 mm square section column. Brannigan and de Alcantara Bonfim [30] performed forced drainage experiments on foams of undisclosed bubble size made of 0.25 wt% Dawn SoapTM in circular section columns of diameter 12.5, 18, 25 and 37.5 mm to assess whether wall effects were present in the forced drainage experiment. Note that their work is not included in Table 1 because bubble size is not reported. They plotted their data in the form of Eq. (1) and found that α decreased monotonically with increasing diameter. For the largest diameter they found α = 0.33 (which corresponds to n = 1.5, which is consistent with node-dominated foam drainage theory) and α = 0.43 for the smallest diameter (which corresponds to n = 1.75, which tends towards channeldominated foam drainage theory). They stated that wall effects appear to be insignificant for tubes greater than 37.5 mm in diameter. The standard forced drainage experiment is a onedimensional method that does not account for wall effects. Clearly more work is required to investigate wall effects in forced drainage, but the work of Brannigan and de Alcantra Bonfim appears to suggest that experiments on foams in small diameter columns might be compromised due to such effects. 5.2. Bubble size As has already been stated, the liquid drainage rate is sensitive to the value of rb . However, there are sources of error when photographic methods of measuring rb are employed. Cheng and Lemlich [31] identified several sources of error when determining bubble size distribution by photographic methods. Firstly, there is a statistical planar sampling bias where the plane discriminates against the sampling of small bubbles. This has been corrected theoretically thus [32]:  ∞ ∗  f (d) dd −1 f ∗ (d) f (d) = (21) d d 0 where f(d) is the bubble size distribution within the foam and f*(d) is the apparent (uncorrected) size distribution at the column wall. The corollary of this result is that the corrected arithmetic

Fig. 5. The bubble size data [33] re-plotted with a normalised ordinate axis. The dashed and solid lines are best log-normal fits to the sorted and unsorted data, respectively.

mean of the distribution, d1,0 , is equivalent to the uncorrected ∗ harmonic mean, d0,−1 . However, it has been shown that smaller bubbles are able to ‘wedge’ larger bubbles away from the wall [31]. They thought that this effect was able to cancel, or even override, the statistical planar sampling bias. With no way of quantifying this segregation, the effects of statistical planar sampling bias and the ‘wedging’ of larger bubbles away from the wall can only be assumed to be self-cancelling. Cheng and Lemlich [31] also considered that error in determining the bubble size distribution due to bubble distortion is negligible, but that temporal variation due to Ostwald ripening may not be the same at the column wall as it is in the bulk froth [35]. Of course, the coarsening process due to Ostwald ripening causes the harmonic mean to exhibit temporal variation and this should be accounted for in drainage simulations. At this juncture it is pertinent to consider the bubble size data of Hutzler et al. [33] who investigated bubble sorting under forced drainage. They performed an experiment where a foam that had been created by mechanical agitation was introduced into a tube and then experienced forced drainage with liquid being introduced at the top of the column. The added liquid enabled relative bubble displacement, and by extracting foam from three locations showed that smaller bubbles preferentially sorted to the bottom of the column. They also measured the bubble size through the column wall and presented a plot of number of bubbles against bubble size (Fig. 3 of [33]) both before and after the sorting process. The ‘distributions’ looked superficially similar and this was taken as evidence that no coalescence was occurring. However, the sample size after the sorting process was different to that before (see Table 1). In Fig. 5 we re-plot the data of Hutzler et al. [33] with a normalised quantity on the ordinate axis rather than an absolute number. In so doing, we see that the distributions are, in actual fact, quite dissimilar. The Sauter, arithmetic and harmonic means of the two distributions are calculated and presented in Table 2. There are several interesting features to notice about Fig. 5: 1. The poly-dispersivity appears to diminish as a result of the sorting process. This is curious since Ostwald ripening, all other things being equal, causes increased poly-dispersivity.

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P. Stevenson / Colloids and Surfaces A: Physicochem. Eng. Aspects 305 (2007) 1–9

Table 2 Sauter, arithmetic and harmonic mean bubble sizes of the experiment of Hutzler et al. [33]

Unsorted Sorted

Sample size

Sauter mean (mm)

Arithmetic mean (mm)

Harmonic mean (mm)

186 154

2.75 2.05

1.48 1.28

0.97 1.08

2. The arithmetic bubble size decreases. If coalescence occurred, all other things being equal, the arithmetic bubble size would increase. 3. The harmonic mean bubble size increases due to a decrease in poly-dispersivity. The conclusion to be drawn from the above observations is that, whether or not Ostwald ripening and/or coalescence were present, the added liquid caused the smaller bubbles to ‘wedge’ the larger bubbles away form the wall as was noticed by Cheng and Lemlich [31]. In experiments with foams of poly-dispersive bubble size the question arises as to what representative size should be chosen. Stevenson [14] has recently used a residence time argument to show that selection of the harmonic mean bubble size might be appropriate. 5.3. Gross convection Convective instabilities have been noticed by Hutzler et al. [34] where circulation of foam occurs on the scale of the container once ε exceeds about 0.15–0.20 [35]. However, some forced drainage studies have employed foams of wetness greater than 0.20; for example foams of wetness fraction up to about 0.38 were used in the forced drainage experiments of Koehler et al. [3]. Clearly, when gross convection effects occur in forced drainage, the one-dimensional characteristic of the method is compromised. An apparently successful theory for the onset of gross convection has recently been published [36]. 6. Conclusions 1. Forced drainage experiments show that the drainage rate from a foam undergoing forced drainage is proportional to a power of the liquid fraction. 2. Because forced drainage is a steady process, it has been shown that drainage rate is independent of surface dilational viscosity and equilibrium surface tension. 3. The drainage rate is dependent upon surface shear viscosity and Marangoni stresses. However, it is not possible to measure the shear viscosity for surfaces stabilised by soluble surfactants. Therefore the role of surface shear viscosity must be accounted for in a fitting parameter, which can also act as proxy for Marangoni stresses in forced drainage. 4. By employing dimensional analysis, a dimensionless relationship has been suggested for steady foam drainage that employs two adjustable constants which are the minimum

required given uncertainties in surface shear viscosity and quantifying viscous losses at the nodes. 5. Several sets of forced drainage data have been expressed in this dimensionless form so that they can be directly compared. This has revealed two inconsistencies in reported forced drainage data. These may be due to errors in bubble size measurement, gross convection events or wall effects. 6. It is apparent that the rigid wall channel-dominated drainage equation [23] is incapable of quantitative predictions of any of the previously reported forced drainage data. This is a significant observation given its continued use in the modelling of flotation [7], for example. Acknowledgment Prof. Graeme Jameson is thanked for his comments on a draft of this manuscript. Appendix A. Estimating the dissipation in nodes It has been remarked that I is used as a fitting parameter in [3] so that viscous dissipation at the nodes can be estimated. Although a value of this parameter is not given in [3], one can be inferred from the later work of the same researchers [37] who amalgamated both viscous losses at the nodes and the channels into a generalized foam drainage equation. In the latter work, the notation is changed with the introduction of a parameter K1/2 implicitly in place of I but, by comparing Eq. (10) in [3] with Eq. (23) in [37] we can infer by substituting numerical values of relevant constants that I=

0.78 K1/2

(A1)

From experimental data for the forced drainage of foams stabilised by 0.017 M (i.e. 4.9 g/l) SDS [37], the authors fit K1/2 ≈ 0.0023 which corresponds to I ≈ 339. This value is not dissimilar to that estimated for foams of liquid fraction 0.26 using the calculated permeability for viscous flow through a face-centred cubic packing of rigid spheres [38] which gives (in the present notation) I = 284. Substituting I = 339 into the powerlaw approximation of node-dominated theory (Table 1) gives a drainage expression of Sk = 0.0014ε1.55

(A2)

In fact, if one assumes that rb = 1.4L (the relationship utilised in [37] which is valid in the dry limit) rather than the more general but more complicated relationship suggested in [17], the experimental data for forced drainage given in [37] can be expressed as Sk = 0.0018ε1.6

(A3)

which indicates dissipation both in the nodes and the Plateau borders.

P. Stevenson / Colloids and Surfaces A: Physicochem. Eng. Aspects 305 (2007) 1–9

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