Physicochemical properties of foaming slags

Physicochemical properties of foaming slags C. Nexhip*1, Shouyi Sun2 and S. Jahanshahi2 Molten slag foams can be considered very dynamic (non-equilibrium) systems, hence their behaviour in industrial processes can be difficult to predict and control. This paper reviews the role of physicochemical properties of molten slags, and the types of stability mechanisms likely to be operating in slag foams. The emphasis of the review relates to slag foams produced under low superficial gas velocities (such as in an electric arc furnace). Here, the ‘lifetime’ of bubble films in the top layer can be considered the rate limiting step for coalescence and collapse of the foam in a metallurgical reactor. IMR/393

Keywords: Slag foam, Foam drainage, Film thinning, Bubble rupture, Surface tension, Viscosity

List of symbols ai A Af C E g Hf k P9 P0 Pc DP Qg r R R Rf td T V sg VRe z c dc/dC

activity of surface-active solute at equilibrium uniform cross-sectional area of sample container area of bubble lamellae concentration of surface-active solute, mol.-% elasticity coefficient gravitational acceleration constant change in height of foam column reached under constant gas injection rate constant for bubble decay (as defined by Lahiri and Seetharaman) atmospheric pressure within central lamellae lower pressure within curved Plateau border region of films capillary pressure (as defined by Scheludko) pressure differential experienced by lamella [DP5P92P05(2c/r)2P] gas flowrate radius of curvature of Plateau border universal gas constant film radius (as defined by Sharma and Ruckenstein) radius of bubble film (as defined by Reynolds) film drain time, s absolute temperature superficial gas velocity thinning velocity of film as defined by Reynolds distance measured from top of vertical film surface tension of liquid rate of change of surface tension, per unit concentration change of surfactant

1

Formerly of CSIRO Minerals, now Rio Tinto Technical Services, 1 Research Avenue, Bundoora, Victoria 3083, Australia CSIRO Minerals, Box 312, Clayton South, Vic., 3169, Australia

2

*Corresponding author, email [email protected]

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ß 2004 Institute of Materials, Minerals and Mining and ASM International Published by Maney for the Institute and ASM International DOI 1.1179/095066004225021945

Ci d dcr dm dmin e h l m P Pel Pvw r S Q s

surface excess concentration of surfactant i, at liquid surface film thickness of bubble critical thickness of thin bubble film prior to rupture mean film thickness of hydrodynamic surface instabilities minimum film thickness of thermal wave instabilities amplitude of hydrodynamic surface instabilities solid/liquid contact angle critical wavelength of hydrodynamic surface instabilities bulk viscosity of liquid disjoining pressure contribution of electrical double layer interactions, to total disjoining pressure P contribution of van der Waals interactions to total disjoining pressure P density of liquid foaming index void fraction of gas within bubble column (as defined by Pahl and Franke) average void fraction of foam (as defined by Lahiri and Seetharaman)

Introduction Slag foaming phenomena are observed in many ferrous pyrometallurgical processes, including the open hearth furnace, basic oxygen furnace (BOF), blast furnace, hot metal pretreatment practices (such as desiliconisation, dephosphorisation, and desulphurisation of pig iron), vacuum degassing of steels in the ladle, and some other non-ferrous smelting processes (e.g. coppermaking). In recent times, interest in understanding the phenomenon of slag foaming has been reinvigorated, with the development of high intensity metallurgical processes such as new ironmaking, bath smelting, and electric arc furnace (EAF) steelmaking.1–3 New ironmaking is a process whereby coal and partially reduced iron ore are added to a slag or metal phase; the primary chemical

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Physicochemical properties of foaming slags

2 Foam column showing how structure of foam layers differs according to void fraction Q of gas within bubbles:8 bubbles within lower region form assembly of thick-filmed spheres (first termed ‘Kugelschaum’ by Manegold9) and bubbles present in upper foamy layer have well drained and quasipolyhedral films (termed ‘Polyederschaum’)

1 ‘Slopping’ of slag foam from basic oxygen furnace (BOF)

reactions being the cracking of coal and the reduction of the iron ore in the slag by the added carbon, and carbon dissolved in metal4 (FeO)zC(s) ~ ~ > FezCO(g)

(1)

~ > FezCO(g) (FeO)zC~

(2)

The off-gases produced (such as CO and H2) are postcombusted above the bath (reacting with O2 injected from a lance), and the generated heat is transferred to the bath below, to help drive the endothermic reduction reactions occurring in the bath (equations (1) and (2)). Foaming slags are very important for such intense processes, as their large surface areas facilitate the multiphase reactions, leading to improved process kinetics, heat transfer and energy efficiency. Control of slag foaming in these intense metallurgical processes is critical. For example, if the slag foam suddenly becomes unstable in an EAF process, the height of the foam will suddenly decrease, and this can result in increased dust generation, destabilisation of the dc arc, increased anode consumption, and heat loss from the vessel due to radiation. A worst case scenario can occur in the BOF process, where the foamy slag can sometimes deviate from an ideal steady state height, and begin ejecting violently from the furnace; a condition known as ‘slopping’5 (Fig. 1). Gou et al.4 have discussed the intense gas evolution that occurs in new ironmaking processes, and the slag foaming that occurs as a result. Here, the slag layer expands and is ‘lifted’ to the point of overflowing from the vessel (potentially decreasing productivity); the primary forces at work being the drag between phases and the gravitational force acting on the foam. Recently, a review by Guthrie6 has shown that slag foaming can play an important role in determining the overall kinetics of slag based smelting operations. Low temperature simulations of ‘direct steelmaking’ processes were used to demonstrate that superficial gas flowrate plays a significant role in determining overall foam height. It was concluded that BOF, and similar refining systems (where foaming slags are encountered), are governed more by inertial factors, than by viscous effects.

In the current context, it is important to note that the slag foaming studies reviewed have been carried out in the laboratory using low superficial gas velocities, and hence the foaming phenomena described can be considered more relevant to slag foaming in processes such as EAF steelmaking. The review serves to introduce, and define, the types of mechanisms important to the stability and control of slag foams. Naturally, the review also draws heavily on the large body of work relating to aqueous (cold) foaming systems.

Fundamentals Physical nature of foams Foam is a system comprised of a gas, being the dispersed phase, and the liquid slag, being the continuous phase. Foams can be loosely defined as ‘transient’ or ‘metastable’, depending on the extent of drainage that has occurred within the bubble films.7 For example, unstable foams constantly break down as the liquid drains from between the bubbles. Their lifetime depends on the concentration of the surfactants in solution, but at best can be around 20 s for typical aqueous systems. Transient foams form bubbles with relatively thick walls, which subsequently thin and collapse at some definite stage. Metastable foams form lamellae, which thin to a ‘black’ state (too thin to reflect light thus appearing black) and then remain at a more or less constant thickness until they are destroyed by disturbances. In a typical smelting process, the layer of molten slag generally contains gas bubbles, metal droplets and carbonaceous materials. The intensity of gas evolution at the slag/metal interface can be vigorous, facilitating creation of a foamy slag. In practice, direct observation of the internal structure of molten slag foam is not feasible. Drawing analogies to low temperature systems, the layer of slag foam could be considered to be composed of two regions where the structure of the foam differs,8 i.e. a dense foam layer in the lower region of the vessel, and a foamy (less dense) region in the upper part (Fig. 2). Within the lower region the bubbles are likely to be transient, as they jostle together as an assembly of thick-filmed spheres (first termed ‘Kugelschaum’ by Manegold9). The bubbles present in the upper foamy layer are likely to be metastable, i.e. the films are well drained and quasipolyhedral in shape (termed ‘Polyederschaum’), as the thin films separating


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3 Schematic of bubble films (lamellae) and Plateau borders in foams: pressure within planar films is essentially atmospheric, while pressure in border region is lower, creating suction force for liquid drainage

the adjacent bubbles become planar/slightly curved and are of practically uniform thickness.7 The stabilising influence of well drained bubble films can be powerful enough to counterbalance the forces making for collapse, resulting in greater entrainment of the chemically generated gas bubbles in the upper foamy slag layer. Therefore, the extent and rate of drainage of the bubble films present (particularly in the upper layer), will ultimately determine the steady state foam height achieved. Note the term ‘stability’ should be used loosely, as it does not imply complete thermodynamic stability, i.e. a foam is a disperse system having a higher surface area, and hence a higher surface free energy when compared with the segregated gas and liquid. In general, the quantity of chemically generated slag foam will be dependent on: The external energy and gas supplied directly by blowing, e.g. post-combustion of CO with injected O2 (and the physical agitation of the slag layer by the injected gas), or from the intensity of the decarburisation process, which generates myriads of small CO bubbles at the slag/metal interface. The likelihood or propensity for foaming (in the first instance), and the stability of the foam column, i.e. the mean lifetime of the bubbles in the upper layer of the slag foam, before they rupture or coalesce into larger bubbles (and escape from the liquid surface).

N

N

Mechanisms of drainage in foams The surface properties of liquid films can strongly influence the coalescence and rupture time of aqueous foams.10 Drainage of liquid from the bubble lamellae is controlled by two very important mechanisms. The first and more obvious is gravity, because (in every foam) gas is present below liquid, and the centre of gravity of the system descends when liquid flows down and gas rises. The second important drainage mechanism is the Plateau border suction. The intersection where bubble lamellae meet was first described by Plateau,11 after noticing that they were strongly curved when compared to the planar parts of central lamellae. The pressure P0 inside the curved Plateau border region is less than the pressure P9 in the flat region, hence a pressure difference will arise: DP5P92P0 (see Fig. 3). This pressure difference can also be described as DP5(2c/r)2P, where c is the surface tension of the liquid (continuous phase), r is the radius of curvature of the Plateau border, and P is the ‘disjoining pressure’, defined by Derjaguin.12 The disjoining pressure is said to be composed of the long range van der Waals attractive forces, the electric double layer repulsive forces, and the forces due to steric

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interactions of closely packed monolayers. The total pressure difference DP exercises a ‘suction’ on the liquid within the films towards the Plateau border. The border suction has been found to dominate the film drainage of aqueous systems, except in very thin films (of thickness d) where the forces of disjoining pressure become ˚ .12 significant, i.e. d,1000 A When a second layer of bubbles forms under the first and is followed by subsequent layers, the bubbles shift to assume positions dictated by surface tension and capillary pressure.13 To achieve mechanical equilibrium, up to three bubble lamellae only, can radiate at dihedrals of 120u. If more than three films join in a border, the geometry of packing is not stable. Hence, a juncture of four films will instantaneously disproportionate into two junctures by the generation of a new film, until all films again meet at angles of 120u; this is known as Lamarle’s first law.14 Thus, in a reasonably stable 3D (three-dimensional) foam, three and only three lamellae meet in a liquid vein. The angles between veins must be approximately identical, as the (ideal) surface tension forces acting on each Plateau border junction must be equal (Fig. 3). Both gravity and Plateau border suction are thus very important driving forces for liquid drainage in foams. Haas and Johnson15 attempted to delineate the roles played by films and Plateau borders during the drainage of foams in aqueous systems. They showed that gravity played a negligible role within the films themselves, whose drainage was essentially driven by capillary forces. However, they postulated that liquid from the films drains, under capillarity, into the adjoining Plateau borders, which in turn form a network through which the liquid flows downwards under the influence of gravity; hence the two mechanisms operate simultaneously during foam drainage. Although the details of the various models of liquid drainage within the Plateau borders differ,16,17 the above description of foam drainage has essentially been accepted by most of the later investigations. It appears reasonable to assume that the drainage in slag foams proceeds in a similar manner.

Laboratory measurement of slag foam stability The stability of slag foams has mostly been measured using the ‘inert gas injection’ technique, originally developed by Bikerman13 for studies of aqueous systems. Here, Bikerman defined a constant of proportionality termed the ‘foaming index’ S, based on the observation that the volume of foam formed at a steady state was proportional to the gas flowrate Qg. In the past two decades, many have adopted the technique for use on molten slag systems,18–21 where inert gases such as argon or nitrogen were injected into molten slags (Fig. 4). The ‘foamability’ could then be measured by recording the steady state height of the foam column reached under constant gas flowrate. By adopting the parameters described by Bikerman, Ito and Fruehan18 were able to measure the foamability of a slag using a foaming index, as follows Vgs ~Qg =A V sg

(3) 21

where (m s ) is the superficial gas velocity, Qg is the gas flowrate (m3 s21) and A is the uniform crosssectional area of the sample container (m2). The foaming

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4 Schematic of inert gas injection apparatus used to measure slag foaming behaviour21

index S was given by S~Hf =Vgs

(4)

where Hf is the measured change of foam height (m), after reaching a steady state. In practice, S can be obtained from the slope of a plot of foam height versus superficial gas velocity. Physically, S represents the residence time (or average travelling time) of the gas in the foam layer. The inert gas injection technique was used by Ito and Fruehan to quantify the foamability of CaO–SiO2–FeO slags, with the aim of predicting foam heights in metallurgical reactors, based on physical properties of the slags. As these laboratory experiments were carried out under constant gas injection rates, the collapse or rate of decay of the foam column in the absence of gas injection was not quantified. If the slag foam reaches a steady state height during an experiment and the gas injection is subsequently ceased, the rate of decay of the foam column (between two visual marker points for example) as a function of time can be used as a measure of ‘foam stability’. This procedure was also used by Cooper and Kitchener,22 Swisher and McCabe,23 Kozakevitch,24 and Hara and Ogino25 for measuring the stability of foam columns of CaO–SiO2 and CaO– SiO2–FeO slags. Therefore, in the aforementioned laboratory studies, the foamability could be considered a measure of the propensity of the slag to generate foam, while foam stability was a measure of foam decay rate, when gas injection ceased.

Film thinning characteristics and dynamics A great deal of interest lies in understanding the thinning characteristics of single films. Liquid films formed on wire frames (‘free’ films) have been described as the simplest model of a foam unit, and serve as a convenient model for studies of the forces of interactions


in dispersed systems.26 The thinning of liquid films can also be viewed as the rate limiting step of bubble coalescence or film rupture.27 On the basis of their observations, Mysels et al.28 considered three extreme types, corresponding to several (but not all) characteristic classes of aqueous foam films: 1. Rigid (or plastic) films whose surfaces impart a high resistance to any motion within the plane of the film. Such films are very thick when first formed and subsequently drain very slowly. No visible rapid or turbulent motion is observed around the film edges, suggesting the surfactant monolayers that border the film to be quite compacted. 2. Simple mobile films – the most commonly observed film type. They drain much more quickly than rigid films. Rapid turbulent motion is visible along the borders of the film, while horizontal smooth coloured bands (implying equal thickness) form in the film centre. According to Newton,29 these films can thin down to the ‘black’ stage, i.e. they become extremely transparent and thus appear black due to a nearly complete lack of reflected light (when viewed against a black background). 3. Irregular mobile films – formed from concentrated (.4%) surfactant solutions and showing similar motion and colour banding to simple mobile films. However, after the appearance of the black film, the boundary with the highly coloured parts of the film becomes highly irregular, with parts of the film exhibiting the appearance of ‘peacock feathers’. This is due to the formation of streamers of black film extending far down into the coloured bands, each of them terminating around a tiny, highly coloured island of thick film. According to Mysels et al., the thinning behaviour of one rigid, slow draining film is much like that of another, whereas the thinning behaviour among the fast draining films shows considerable differences. This suggests that slow draining films exhibit simple behaviour, while other thinning mechanisms may be operating in varying extents in mobile films. Reynolds,30 in his classical theory of lubrication, derived an expression for the rate of approach of two rigid parallel discs (hence the thinning velocity VRe), under the action of a uniform outside pressure, namely VRe ~{(2d3 =3mR2f )DP

(5)

where d is the intervening film thickness, m is the liquid viscosity, Rf is the radius of the film, and DP is the total pressure causing drainage, which is the sum of the contributions from the border suction 2c/r and the disjoining pressure P DP~2c=r{P

(6)

This expression provided a convenient model for the thinning of tangentially immobile (rigid) foam films, which assumed the interfaces to be non-deformable and plane-parallel. Numerous other models have been developed to describe the thinning behaviour of soap films.31–35 In the initial stages (immediately after bubble formation), the film surfaces are very curved, but as the thinning bubbles approach one another they progressively flatten due to hydrodynamic interactions (see Fig. 3). In most drainage models, the thinning velocity is calculated assuming the film surfaces to be tangentially


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immobile, generally only valid for rigid films. The thinning velocity is not easy to predict for mobile films, as calculations involve the incorporation of surface viscoelastic properties,36 and the assumptions/ requirements of maintaining a plane-parallel interface during thinning remain. The characteristic ‘types’ of bubble films present in slag foam would be dependent not only on the physicochemical properties of the slag (viscosity, surface tension, etc.), but also on the location of the bubbles within the foam column, e.g. thick spherical bubbles of low gas fraction in the lower region, and thinner (longer lifetime) polyhedral bubbles of high gas fraction in the upper region. Bubble film thinning characteristics and dynamics can be viewed as rate limiting steps for coalescence and rupture phenomena.

Influence of physical properties on stability of foams Influence of viscous drainage on stability of foams Quantitative treatment by Mysels et al.28 confirms drainage due to gravity can lead (only) to very slow thinning, rendering it significant only in rigid films. For example, the volume flowrate (per unit width) of liquid in a film Q, of viscosity m, and density r, flowing downward under the influence of gravity between two parallel plates separated by a distance d, is readily obtained by classical hydrodynamics as Q~rgd3 =12m

(7)

and the mean velocity is Q=d~rgd2 =12m

(8)

Mysels et al. showed that, for a supported film with no influx of liquid from the top, the conservation equation dd=dt~{dQ=dz

(9)

along with equation (8), leads to a variation in thickness of the film with time and distance z, from the top. They assumed laminar flow between rigid plane walls, and thus demonstrated that during drainage, the film thickness profile should become parabolic in crosssection (slightly convex), the film thickness d being given by d2 ~(4m=rg)z=td

(10) 21

21

where m is the liquid viscosity (kg m s ), r is the density of the liquid (kg m23), g is the gravitational acceleration (m s22), z is the distance measured from the top of the parabolic portion (m) and td is the drain time (s). Here, liquid drainage is assumed to be impelled by gravity only, the velocity being greater at the film centre and decreasing parabolically to zero at the film surfaces. Mysels et al. concluded that viscous flow plays a significant role in the slow draining of thick/rigid films. To study gravitational drainage of simple mobile films without the complication of Plateau border suction (always present in ‘free’ liquid films), they employed cylindrical films, hanging from a funnel. When comparing their drainage with films formed using rectangular glass wire frames, the cylindrical films drained extremely

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slowly, hence Plateau border suction was determined as the predominant drainage mechanism operating in mobile films. This suggested surface tension was likely to be a stronger influence than viscous flow in the draining of mobile films. A molten slag having a high bulk viscosity would be expected to retard drainage of liquid from the films separating the bubbles in the foam. Typically, CaO–SiO2 slags have higher bulk viscosities with increasing SiO2 content.37 Kozakevitch24 proposed that a slag with a high viscosity would be the most obvious factor stabilising both emulsions and foams, as it is responsible for the very slow elimination of bubbles (or droplets) in the continuous phase. Swisher and McCabe23 found that foam stability increased with decreasing temperature and decreasing slag basicity (CaO/SiO2 ratio) at 1850– 1910 K. They believed that a high bulk viscosity was unlikely to be the primary mechanism stabilising slag foam, although a high bulk viscosity should aid in retarding the drainage of liquid, thus increasing the bubble coalescence time. Cooper and Kitchener22 measured the decay of slag foam columns as a function of time for CaO/SiO250.69, 0.2 mol.-% P2O5 slags at 1770–2020 K. When they plotted the logarithm of foam decay as a function of reciprocal temperature, they observed a straight line relationship, yielding a four-fold increase in slope (activation energy) when compared to a similar plot of the logarithm of slag viscosity versus 1/T. They concluded that foam life (hence stability) had little dependence on bulk viscosity. This statement agreed with that of Swisher and McCabe, who calculated the activation energy for foam collapse of a CaO–SiO2 slag (of basicity 0.64) to be about 2.5 times greater than that for viscous flow in this system. Their observation that viscous CaO–SiO2 slags did not foam appreciably, except in the presence of dilute amounts of P2O5 as a surfactant, agreed with the observations of Cooper and Kitchener. Ito and Fruehan18 studied the temperature dependence of the foaming index S on a 35CaO–35SiO2– 30FeO slag at 1473–1673 K. When the temperature was decreased from 1623 to 1523 K, a two-fold increase in S was observed. They argued that since the temperature coefficient for surface tension is positive (in this system), and that for viscosity is negative, a decrease in temperature would stabilise the slag foam. The temperature dependency of S, in terms of activation energy, was calculated to be about 160 kJ mol21, almost the activation energy required for viscous flow for these slags. However, Ito and Fruehan did acknowledge that measurements previously made by Cooper and Kitchener on CaO–SiO2 slags were likely to differ compared to their18 physical definition of foam life S and, coupled with the different temperature regimes used, may have been the reason for the higher activation energies that Cooper and Kitchener obtained for foam collapse. Ito and Fruehan also stated that the Cooper and Kitchener, and the Swisher and McCabe, measurements were likely to be erroneous due to poor visual observation of the slag foam level at such high temperatures. Ito and Fruehan,18 and Jiang and Fruehan,3 also performed a dimensional analysis, to develop a correlation between S and the physical properties of the slags

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used, such as viscosity m, surface tension c, slag density r and bubble size. An empirical equation was derived for the prediction of S (as a function of the physical properties of the slag) in practical metallurgical processes. From a dimensional analysis of 14 datum points for melts in the CaO–SiO2–FeO system, and 4 datum points in the PbO–SiO2 system, they found that : S~5:7|102 m=(cr)0 5

(11)

for CaO–SiO2–FeO slags, and : S~1:4|102 m=(cr)0 5

(12)

for PbO–SiO2 slags. Although S represented an arbitrarily defined foam life, from these correlations, Fruehan and co-workers concluded bulk viscosity to be the most important physical property determining the foaming index in CaO–SiO2–FeO slags. The presence of a suspension of solids in a slag has also been acknowledged to increase the apparent viscosity of silicate slags, thus retarding the drainage from the foam lamellae; which in turn would increase the stability of the foam.38 Ito and Fruehan18 stated that the presence of fine solid particles (such as Ca2SiO4) in suspension in basic slags was responsible for increasing the foaming index. However, it is interesting to note that the presence of solid particles at the interface of ‘aqueous’ bubble films can in fact increase or decrease foam stability, depending on the solid/liquid contact angle h. For example, Aveyard et al.39 observed rupturing of aqueous films by a ‘bridging–dewetting’ mechanism (where h.90u), resulting in decreased foam stability. Therefore, the presence of suspended solids in a slag may not be a sufficient condition for increasing foam stability; although low solid/liquid contact angle (h,90u) and high apparent slag viscosity would both favour foam stabilisation. After examining the temperature dependence of the foaming index S on the 35CaO–35SiO2–30FeO slag previously discussed by Fruehan and co-workers in equation (11), Gaskell40 subsequently calculated that S~1:08|10{2 exp (7750=T) giving the activation energy for foam decay as about 66 kJ mol21 for this system. This value was less than half the activation energy required for viscous flow previously reported by Ito and Fruehan. Thus, in contrast with the results of Cooper and Kitchener, and Swisher and McCabe, which show the activation energy for foam decay to be much greater than that required for viscous flow, the results of Ito and Fruehan showed activation energy for viscous flow to be much greater than that for foam decay. Gaskell argued that, as the calculated activation energy for foam decay could be higher, or even lower, than that required for viscous flow, the rate of drainage of liquid from a slag foam is not simply controlled by the bulk viscosity of the slag.

Influence of surface related phenomena on drainage and stability of foams A low surface (or interfacial) tension is favourable to both the formation and the stability of foam. Solutes may possess a greater affinity for either the liquid–gas


surface or the bulk phase. Although this effect was initially noted by Gibbs,41 it was further explained by Guggenheim and Adem42 in terms of a ‘surface excess concentration’ of each surface-active component; the adsorption isotherm being Ci ~{1=RT(Lc=L ln ai )

(13)

where ci is the surface excess concentration (moles m2) of a surface-active solute i, ai and c are the activity of the surface-active solute in the bulk solution and the surface tension of the melt (N m21) respectively, R is the universal gas constant (J mol21 K21) and T is absolute temperature (K). The phases are thus considered to be in thermodynamic equilibrium, with temperature, pressure and chemical potential of the surface-active species having a constant value throughout the system. Although Gibbs’ treatment presupposed equilibrium, he demonstrated that the drainage of a bubble film could be uneven, that is, ‘thin spots’ can and do occur. He also demonstrated that a bubble lamella must be able to maintain subtle differences in surface tension, in order to support its own weight. The works of Marangoni43 and Rayleigh44 were the first to adequately describe nonequilibrium surface tension phenomena, and the mechanism of ‘healing’ of local distortions (as an extension to Gibbs’ theory). The ‘Marangoni effect’ is said to become operative when a freshly formed surface has a dynamic surface tension that can be much higher than the equilibrium value.45 For example, as a bubble is formed, the rapid expansion results in a change in the local concentration of surfactant in the film. Thus, a local gradient in surface tension is produced within the film, generating ‘Marangoni flow’. Here the surface will tend to move from a region of lower surface tension to the region of higher surface tension, suppressing further film stretching and rupture. A key feature of transient and metastable foams is the surface property of ‘film elasticity’.46 Gibbs defined the coefficient of (static) surface elasticity as the stress divided by the strain for a unit area. If a bubble lamella of area Af, and thickness d, is stretched by an area dAf, its surface tension rises from c to czdc, and its thickness decreases to d2dd. Surface elasticity, also termed ‘surface dilational modulus’ E, is defined by47,48 E~(2dc=d ln Af )

(14)

where the factor 2 arises from the presence of two interfaces, and the stress is twice the increase of the liquid surface tension. Both the Gibbs and the Marangoni effects should contribute to surface elasticity, but their relative importance is not clearly established. For example, Gibbs’ theory explains surface transport on the basis of an equilibrium value of surface tension, whereas Marangoni’s theory explains it on the basis of an instantaneous (dynamic) value of surface tension, and is only deemed significant in dilute solutions (within a limited concentration range). However the two theories are complementary, and may provide mechanisms of film elasticity (and hence foam stability) under different conditions. Scheludko49 derived an expression for the Gibbs elasticity in terms of the parameters of film thickness, and the surface and bulk concentration of surfactant. Rosen50 simplified Scheludko’s equation for elasticity E,


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approximating it as follows E~4C2 RT=dC

(15)

where d is the thickness of the lamella (containing bulk solution between the surfaces) and C is the concentration of the surface-active solute. According to equation (15), E decreases when d or C increases, with dilute solutions favouring the elasticity mechanism previously described. The Gibbs elasticity mechanism results from an infinitesimal increase in area, while Marangoni elasticity is a dynamic, non-equilibrium property; however both predict a pure liquid will not foam since a surface tension gradient cannot be achieved. Thus, foamability, particularly under dynamic conditions, cannot be correlated simply with surface activity (due to the non-equilibrium coverage likely to be encountered). Experiments by Myers51 and Ross and Haak52 have shown that if the surfactant concentration is too high, then foamability can be impaired if the rate of surface tension lowering is too rapid. For example, if adsorption is rapid, the deficiency of surfactant molecules in the stretched surface may be made up largely by adsorption from the bulk of the film. This will destroy the surface tension gradient, and the amount of solution returned by surface transport will not be sufficient to restore the film.53 Note that a contributory effect of surface elasticity could also be the ‘damping’ of surface ripples/instabilities, which can otherwise cause dangerous local thinning and rupture.54 The role of a high surface or bulk viscosity is generally to dissipate mechanical shocks, by opposing both the disruptive and the restoring forces. Therefore, a high bulk viscosity can invariably be a contributory factor in foam stability (whatever the magnitude of surface viscosity). A high surface viscosity, resulting from a surface excess of surfactant, in conjunction with low bulk viscosity, can lead to quite complex behaviour. For example, Ewers and Sutherland53 suggested that if a stretched film produces a surface tension gradient large enough to produce surface transport, a high viscosity can decrease the rate of surface transport (although adsorption of surfactant from the bulk should proceed unimpeded). As the surface tension gradient can disappear before healing of the film occurs, the presence of a viscous surface layer alone is insufficient for maintaining foam stability; a finding consistent with the experimental results of Brown et al.55 However, according to Malholtra and Wasan,56 who have reviewed most of the evidence, there is a strong case that surface viscosity effects play a significant secondary role in stabilising foams. Obvious effects are the damping of any disturbances such as surface instabilities and the retardation of drainage in the lamellae. Kozakevitch57 found that if P2O5 additions are made to CaO–SiO2 slags (substituting for SiO2 as the adsorbate), the slag surface is likely to become saturated at a concentration of about 1 mol.-% P2O5, with significant lowering of the surface tension of the melt. He also postulated that the surface film of a slag bubble could be composed of a ‘2D surface phase’, distinct from the bulk phase and consisting (when saturated) of more or less closely packed molecules.24,58 In this surface phase, polar groups of molecules may orient themselves towards the bulk phase, with the negative electric charge of silicate rings compensated by cations (such as Ca2z

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or Fe2z), disposed beneath the silicate rings. This could possibly constitute an ionic-double layer, with a very high surface viscosity (the possibility of a mixed surface layer of P2O5 and SiO2 crystals was also postulated). Ogino et al.59 suggested that an increase in foaming was likely to be correlated with the ‘wedge effect’, which is essentially the same phenomenon as electrical double layer repulsion,60 due to the presence of ionic groups 62 32 such as SiO 42 4 , Si2O 7 and PO 4 adsorbed on bubble film surfaces. The suggestions of Kozakevitch, and Ogino et al., tend to agree with the ‘duplex theory’ of Brown et al., where a mixed surface phase was postulated to increase the stability of a bubble film. Using Gibbs’ adsorption isotherm, Swisher and McCabe23 calculated the surface excess concentration Ci, for CaO–SiO2 melts with Cr2O3 added as surfactant in dilute solution (up to 1.5 mol.-%) at 1873 K, to be Ci