15. Microprocesses of Agglomeration, Hetero-coagulation and Particle Deposition of Poorly Wetted Surfaces in the Context of Metal Melt Filtration and Their Scale Up

This Chapter is dedicated to the microprocesses of agglomeration and hetero-coagulation, which both have the potential to support the removal of non-metallic inclusions from metal melts. The removal of inclusions in the size range between 1 and 10 µm is especially problematic, as their movement is not governed by diffusion and their mass is not big enough for the movement to be fully governed by inertia effects. The attachment of particles to each other after collision leads to the formation of bigger particle identities, so called agglomerates. Similarly, the attachment of particles to bubbles is called hetero-coagulation and leads to the formation of bigger identities with reduced density in comparison to the inclusion particles, which supports the cleaning of the melt by flotation of the inclusion particles. For both microprocesses, the likeliness of attachment is influenced by the kinetic energy of the particles and bubbles in the melt, interparticle forces and the surrounding boundary layers. For the hetero-coagulate formation, the deformation of the bubbles also can affect the attachment behavior. Cleaning of the metal melt before the filtration and the subsequent casting process is commonly done by the injection of reactive or inert gases like argon, that are injected by immersion lances or porous plugs. The initial bubble sizes are in the range of 10 mm to 20 mm for porous plugs [1] and in the range of multiple cm for immersion lances and their size increases as the bubbles rise to the surface. The injection of gases does not only serve the removal of inclusions by the flotation and entrainment of inclusion particles but also alters the composition of the melt [1], increases the mixing in the ladle and can decrease the number of gas cavities in the work pieces. Additional to the aforementioned cleaning strategies, the use of active and reactive ceramic foam filters allows an efficient removal of inclusion particles by depth-filtration. The application of reactive foam filters in casting processes leads to the formation of CO-bubbles by a carbothermal reaction between the filter material and the melt. The effect of those bubbles on the filtration efficiency is not fully clear and is neglectable according to numerical simulations of reactive filters in an induction crucible, as the bubbles do not detach from the filter surface [2]. Generally, bubbles inside the melt can partake in the removal of the inclusions by flotation. Additional to the true flotation, which involves particle-bubble-hetero-coagulates, a transfer of inclusions to the slag layer by entrainment has to be expected. The effect of agglomeration of non-metallic-inclusions, mainly alumina, was previously investigated in a number of theoretical and experimental studies in both actual metal melts and a water model system at room temperature. Investigations in the melt are expensive and are limited to relatively rough methods, as the high temperatures and opacity of the melt prevent the precise measurement of forces in the nano- and micro-Newton range, as they occur between micrometer sized inclusions.

The interparticle forces in metal melts are largely unknown and somewhat controversially discussed in the literature. The van der Waals forces cover three types of interactions, the Keesom-, Debye- and London-interactions. The London-interactions are the interactions between two induced dipoles, that are caused by a fluctuation of the charge distributions in the presence of close by molecules and are frequently referred to as dispersion interactions [3]. They are omnipresent in all materials and are the main type of interactions in many systems that have a non-polar character [3]. Many authors have attempted to estimate the van der Waals interactions between inclusion particles in metal melts and two main types of approaches for the estimation of Hamaker constants for metal melt systems are found. The first group of authors makes use of surface energies and utilizes the approaches of Fowkes. The applicability of the Fowkes approach for the determination of the Hamaker-constant for non-metallic inclusion—melt systems remains uncertain because the Fowkes approach only provides reliable results for a selected number of ideal, non-polar liquids [4]. Further it requires the precise measurement of contact angles that later on are used in combination with the Young-Equation, to calculate the surface energies. As the surfaces of inclusions like alumina exhibit high roughness values, it has to be expected that the macroscopic contact angle, obtained from measurements, and the true, microscopic contact angle are deviating. The contact angles of molten metals should be measured in ultra-high vacuum to prevent the formation of an oxide skin. Additionally, chemical reactions between substrate and molten metal are possible. The second group of authors follows along the London theory of dispersion forces to estimate the van der Waals interactions of liquid iron from the available Hamaker-constants for iron at room temperature, which lay in the range 324 to 562 zJ and are well summarized by Gomes de Sousa [5], but ignore any effects of the phase transition. Most notably for this approach are the works of Tanighuchi, who estimated the Hamaker-constant based on a modified Saffman-Turner model, that was applied on agglomeration experiments of alumina, silica and polystyrene latex particles at the IEP [6]. Unfortunately, the London theory of dispersion is formally not valid for interactions in solvents and assumes that the molecules have only 1 ionization potential [3], which is not true for metals. Both approaches have in common, that they require the use of mixing rules with limited applicability for Hamaker-constants and generally a critical discussion of the phase transition, the conducting character of metals, as well as the presence of multiple alloying elements in real metal melts are avoided. A third approach worth mentioning is the use of the Lifshitz and Casimir theories of dispersion in combination with modified oscillator models that have been fitted to optical data of molten and liquid metals. Most commonly Drude-type models are used to describe the dielectric function of metals. The Drude model can be derived from the Lorentz oscillator model for insulators by setting the restoring force equal to zero [7]. The applicability of the Drude model and modified Drude models, as they were suggested by Chen [8] and others to molten metal seems reasonable, as the molten metals lose their electronic band structure when they melt [9]. Daun pointed out that the successful application of the Drude model for molten metallic systems like silver or molten silicon have been reported, but for transition metals like iron and nickel the application still is difficult, due to the d-electron bands, that lead to either unphysical model fits or great deviations from the experimentally determined dielectric functions [9, 10]. The application of such erroneous oscillator models in the Lifshitz- and Casimir-theories would give unpredictable results and only should be attempted with physically sound oscillator models. A possible example of such an application to the liquid metal system is given by Esquivel-Sirvent, who calculated Casimir forces based on dielectric data for gold, mercury and an eutectic indium-gallium alloy, even though extrapolation of the optical data was necessary [11]. The availability of high-quality optical data for molten metal species is very limited. Often the datasets only cover few frequencies and the particularly important part of absorbance spectra is missing. As a consequence of the shortcomings in the theoretical description of the surface forces acting in the melt system, alternative measures of the present surface forces such as contact angle measurements are frequently utilized to characterize the interaction behavior of non-metallic inclusions in melt systems. The Young equation (Eq. 15.1) links the macroscopic contact angle θ to the three energies of the interfaces liquid–gas γ_lg, substrate—liquid γ_sl and substrate—gas γ_sg. The equation underlies the assumption of ideally smooth surfaces and thermodynamic equilibrium between the phases which are rarely present in technical systems.

For molten metal on alumina surfaces, contact angles far above 100° are measured and contact angles as high as 155° are reported [12]. When using water droplets for contact angle measurements, one common convention is the distinction between hydrophilic θ < 90° and hydrophobic θ > 90° materials. In analogy for non-aqueous liquids, materials that exhibit contact angles above 90° are considered poorly wetted. This analogy is used in a water-based model system to investigate the behavior of poorly wetted particles during agglomeration, hetero-coagulation and the deposition of particles in ceramic foam filters. Four main factors have to be considered in a model system, the particle wettability, surface tension, viscosity and density. A common choice is the adjustment of the particle wettability by surface functionalization. Within the CRC920 the particle wettability of oxidic model inclusions is altered by silanization with the fluoroalkyl silane Dynasylan F8261 (1H,1H,2H,2H-perfluorooctyltriethoxysilane) from Evonik (Germany), which reacts with the surface OH-groups of oxidic materials, such as alumina, silica and glass [13]. The silanization procedure leads to contact angles of approximately 105° on smooth naturally oxidized Si-wafers and contact angles up to 134° on rough alumina surfaces.

In aqueous systems such as the water model system relevant interparticle forces include capillary forces, van der Waals forces, electric double layer forces and steric effects, as well as the more controversially discussed hydrophobic forces that are observed in poorly wetted particle systems. Those strongly attractive hydrophobic forces have been reported to act in a range of a couple nm up to several hundreds of nm and have been ascribed to organic contaminations, effects of the electric double layer, nanobubbles and other phenomena, of which many have proven to be unphysical. A physically reasonable approach for the description of hydrophobic forces is the separation of the hydrophobic forces into a short range, true hydrophobic part, that is caused by structuring of water molecules close to surfaces with low surface energy and a long-range part that occurs due to capillary bridging between nanobubbles or gas cavities that are pre-existing on the often rough, poorly wetted surfaces [14]. The wettability behavior of alumina surfaces in the metal melt, the expectation of strongly attractive van der Waals forces between the inclusions and the wettability dependent formation of gas cavities [12] suggest a similar surface force behavior to that observed on hydrophobic surfaces in water-based systems, as both the melt system and the water model system show a low affinity between inclusion and liquid phase. The effects of the capillary force are expected to be more pronounced in the melt system due to the higher surface tension. For molten aluminum at 700 ℃ the surface tension is around \(\gamma_{lg}\) = 865 mN/m [15], but for water at room temperature only \(\gamma_{lg}\)  = 72 mN/m. The presence of gas adsorption layers in the melt system, seems reasonable when considering the theoretical estimates of strongly attractive van der Waals forces and the extremely high contact angles that the molten metals exhibit on ceramic surfaces. Water is a polar molecule and the adsorbed gas layer in the aqueous systems often is considered as a water depletion layer, which is caused by the reorientation of the dipoles parallel to the poorly wetted, hydrophobic surface as it is energetically more favorable [14].

The Atomic Force Microscope (AFM) is one of the most powerful tools for the characterization of surfaces and surface forces. Measurements with the instrument can provide information on the surface topography, as well as the surface forces of solid–solid and solid–fluid systems. The deflection of a cantilever, a thin beam, is measured via the position of an incident laser beam on a position-sensitive photo diode (PSPD). It can be converted to a force by using Hooke’s law [16]. For the determination of the surface topography a soft cantilever with a nanometer thin probe tip is brought into contact with the surface and is moved line wise over it. The measured deflection is used directly to obtain a height-image of the surface from which measures for the characterization of the surface roughness such as the root mean square roughness and the peak to valley roughness can be extracted. The Colloidal-Probe-technique (CP) has frequently been employed in the last decades to investigate particle–particle and particle- bubble interactions by gluing or sintering [17] a commonly spherical three to twenty-five micrometer diameter particle of interest onto a cantilever tip. The CP-cantilevers manufactured by this method are brought into contact with other surfaces to measure the forces acting on the particle.

Alumina surfaces of both, inclusion particles and ceramic foam filters are commonly rough and the roughness has strong implications on the adhesion behavior. Ditscherlein reported root mean square roughness of rms = 0.8 µm for flat alumina samples and rms = 0.08 µm (scan size 3 × 3 µm²) for spherical alumina particles [14]. For rough surfaces in the absence of capillary forces a decrease of the adhesion force is observed, that is caused by an altered contact area [18]. The prominent parts of the surfaces function as spacers that prevent the surrounding areas from contact. For the case of similar surfaces, where the van der Waals forces are attractive the roughness itself leads to a reduction of the adhesion force in comparison to smooth surfaces. Laitinen measured the adhesion force between spherical alumina colloidal probe particles and flat alumina substrates, reporting moderately good agreement of the classical Rabinovic model with AFM experiments in air, but did not quantify the influence of capillary condensation and deviations from the ideal spherical geometry on the measurements [19]. The particle adhesion between rough hydrophobized alumina surfaces in water–ethanol mixtures was investigated by Fritzsche [20], who was able to proof the presence of nanobubbles and the therewith connected different capillary interaction mechanisms using CP-FD-spectroscopy. Examples for the four types of FD-curves are shown in Fig. 15.1, of which three have a capillary character (II-IV) and the remaining type (I) corresponds to van der Waals interactions in the absence of capillary bridging. The FD-curves with capillary character in the most general case show a snap-in (II, IV), at which a sudden increase of the attractive force during the approach phase is measured if sufficiently soft cantilevers are used and which is characteristic for the formation of a capillary bridge. In the subsequent retract phase a snap-off occurs, which marks the loss of contact between the particle and the gas reservoir or surface respectively. Sometimes the contact is not lost at once, but happens in multiple steps with alternating stick and slide phases that are characteristic for the separation of rough particles from gas bubbles and gas reservoirs. The third type of capillary interactions (III) on rough poorly wetted substrates is characterized by a snap-off without the previous occurrence of a snap-in and can be the result of gas nucleation upon perturbation, or of a stable liquid film that does not drain completely during the approach phase, but ruptures upon the retraction of the colloidal probe, which is in line with recent predictions of the Stokes-Reynolds-Young–Laplace model, that predicts the contact formation upon retraction for moderate capillary numbers.

2 line graphs. A. A line graph of force F slash R versus distance h in n m illustrates a decreasing trend. B. A line graph of force F slash R versus distance h in n m for types 1, 2, 3, and 4 illustrates a fluctuating trend.

The overall adhesion characteristics of rough, poorly wetted alumina surfaces are resulting from the superimposition of the four types of interactions. The relative frequency f_R for the occurrence of capillary interactions between silanized alumina surfaces follows f_R = −0.56299 × cos(θ_liquid) + 0.37892, where the wetting angle θ_liquid was altered by variation of the ethanol content between 0 and 20 wt.% [20, 21]. The increased forces that are measured in the presence of nanobubbles are caused by capillary bridges that can form upon particle contact. A numerical model for the calculation of capillary forces caused by gaseous capillary bridges was proposed by Fritzsche [22]. It is based on both, the minimization of the surface energies for the calculation of the three-phase-contact-angle, and on a solution of the discretized Laplace-equation for the calculation of the meniscus outline [22]. Analysis with the model has shown that the total amount of gas molecules in the capillary bridge has only a small influence on the overall adhesion force, which can be explained by the incorporation of the ideal gas law into the set of equations instead of the constant volume constraint that is commonly assumed for liquid capillary bridges [22]. As the pressure increases proportional to the amount of gas molecules, the overall surface area remains almost unaffected which leads to a neglectable influence on the overall adhesion force [22]. Additionally, it was shown that contact line pinning influences the adhesion force when the distance between the particles is altered as shown in Fig. 15.2.

A scatterplot of sealed adhesion force F and absolute capillary pressure in P A versus particle distance in n m for force and pressure. The pressure dots always lie above the force dot plots.

The short-range contribution of the hydrophobic forces that accounts for the water depletion layers can be modeled as adsorbed gas layers on the particle surface with sub nanometer thickness, using a retarded van der Waals layer model [23] described in Chap. 12. The silane coating on the particle surface is modeled as a PTFE-layer, as no optical data for the silane is available and the perfluorinated silane has chemical similarities to PTFE [24]. The attraction between two PTFE coated particles in theory should be weaker, than for pure alumina, but in experiments strong attraction is observed. The vdW-radius of nitrogen is approximately 0.155 nm and serves as a first estimate for the thickness of a monolayer of adsorbed gas on top of the PTFE layer [24].

The modeled curves for the homo-interaction of (non-)silanized particles and the hetero-interaction of a (non-)silanized alumina particle with an air bubble are shown in Fig. 15.3, for the cases of zero to three layers of adsorbed air. For both cases an increase of the Hamaker-function with an increase of the gas layer thickness is observed. For high values of the thickness of the layers, the Hamaker-function of the silanized particles approach the strongly attractive Hamaker-function of the air–water-air configuration [24]. The attraction between two uncoated alumina particles is greater than for the coated particles in the absence of adsorbed gas. At large distances above approximately 1 nm, the van der Waals force for the unfunctionalized particles is stronger than the forces between the functionalized particles with and without gas layer. For separations below 1 nm the particles with gas layers have much higher attractive Hamaker-function values close to 50 zJ, the Hamaker-function of the pure alumina in contrast plateaus at 37 zJ. In practice this means, that for both, the pure alumina and the silanized alumina agglomeration is likely, but due to the higher attraction at contact between the silanized particles their equilibrium agglomerate size will be bigger because the agglomerates can withstand higher shear rates before the contact between the primary particles in the agglomerate breaks.

2 line graphs. A. A line graph of the Hamaker function versus distance h in n m for different alumina surfaces illustrates a decreasing trend. B. A line graph of the Hamaker function versus distance h in n m for different alumina surfaces in the presence of absorbed gas layers illustrates a decreasing trend before increasing.

Similar to the particle–particle interation, the Hamaker-function for a (non-)coated particle interacting with a gas bubble can be calculated by modeling the layers for the first half space that corresponds to the particle and air as the second halfspace to account for the gas bubble. The interaction between the pure alumina and the gas bubble is repulsive at distances above 2 nm, which can easily be seen by the disjoining pressure profiles shown in Fig. 15.4 that correspond to the Hamaker- functions for the particle-bubble interaction shown in Fig. 15.3.

A line graph of disjoining pressure versus distance h in n m for different alumina surfaces in the presence of adsorbed gas layers illustrates an increasing trend before stabilizing.

With increasing gas layer thickness the repulsive contribution of the pure alumina—bubble interaction decreases and strong attraction occurs at separations below 7 nm, when the gas layer exceeds a thickness of 0.31 nm. The modeled gas depletion layers on the silanized alumina particles that are used in the water model system all exhibit a strong exponential decay that concides well in range and slope with previously suggested exponential force law for the short-range hydrophobic forces between a hydrophobic particle and a gas bubble [25], that can be converted to a disjoining pressure \(\Pi_{dis}\) by Eq. 15.2.

The decay length parameter in the exponential force law Eq. 15.3 was previously determined in experiments to be approximately 0.3 nm [25]. The pre-exponential factor in Eq. 15.3 depends on the surface tension \(\gamma_{lg}\) and the macroscopic contact angle \(\theta\) of the particle material with the liquid.

As a first approximation for the attractive force between a gas bubble and poorly wetted alumina in molten metal, the surface tension of the melt and its contact angle are combined with the decay length of 0.3 nm for the particle-bubble interaction, as there is no knowledge about the forces in the melt system. From this, a distance dependent disjoining pressure function for the melt-system is approximated, which is used in combination with the Stokes-Reynolds-Young–Laplace-model for particle-bubble interactions, which currently is the most comprehensive model for bubble-particle interactions under conditions met in AFM experiments. Typically, an approach-retrace-cycle is simulated during which the colloidal probe first is moved towards the bubble with constant velocity, until a certain overlap of the colloidal probe with respect to the undisturbed position of the bubble interface is reached. Subsequently the particle is retracted from the bubble again with constant velocity. The validity of the SRYL model has been shown in numerous experimental and theoretical studies for interactions of deformable drop-drop, bubble–bubble, and drop/bubble-particle interactions [26]. It incorporates the key parameters interfacial tension \(\gamma_{lg}\), fluid viscosity η, contact angle of the bubble θ_b, surface and hydrodynamic forces, as well as the curvatures that determine the collision stability for particle-bubble interactions.

For the description of the film drainage the augmented Young–Laplace-equation (Eq. 15.4) is used in rewritten form for the film thickness in the AFM configuration. It describes the pressure difference over a curved interface in the presence of surface forces and additional hydrodynamic pressure. The latter is obtained by the Stokes-Reynolds-equation (Eq. 15.6) known from the lubrication theory, which describes creeping flows ( Re < 1) in geometries, where one dimension is much smaller than another [26]. The total force between particle and bubble is obtained by integration over the interaction zone via Eq. 15.7 at the apex of the bubble for r \(\in\) [0,r_max] under the assumption of symmetry boundary condition. The boundary condition at the end of this interaction zone given by Eq. 15.8 was derived from Carnie [27] by matching the inner and outer shape of a drop at the end of the interaction zone. It contains contributions for the forces in the interaction zone, the contact angle behavior B(θ₀) and optionally the cantilever spring constant k_c.

The scaled total force over the central film thickness calculated with the SRYL-model and the parameters in Table 15.1, are shown in Fig. 15.5 for the room temperature model system in (a) and the Al-melt in (b) for the interaction between a particle with 10 µm diameter and a sessile bubble of size 200 µm diameter. For the water model system, the disjoining pressure is described with the vdW-layer-model for alumina with a 2 nm PTFE-layer and a gas layer of 0.31 nm thickness. Effects of the electrochemical double layer are not considered. The dynamic viscosity and the surface tension are those of water at 25 ℃, therefore the capillary number (Eq. 15.9) is between Ca = 1.85 × 10^–7 and Ca = 5.56 × 10^–7. At approach velocities below 40 µm/s a rupture of the liquid-film between bubble and the functionalized particle is predicted. With increasing approach velocity, a higher repulsive force acts between particle and bubble, caused by the displacement of the liquid in the film. As a consequence, the bubble interface in the interaction zone deforms, which leads to a slower decrease of the central film thickness. This hydrodynamic influence becomes attractive as the particle is retracted from the surface and the liquid has to flow back into the gap, which can favor the contact upon retraction. If the film thickness is reduced below 4 nm the attractive van der Waals forces take over and particle and the bubble get into contact. In the Al-melt system, shown in Fig. 15.5b, the capillary numbers range from Ca = 1.21*10^–7 to Ca = 6.06*10^–7, as higher approach velocities between 100 µm/s and 500 µm/s have been chosen to identify the transition region between attachment and no attachment. Approach velocities below 350 µm/s lead to an unavoidable contact formation between particle and bubble, as the high interfacial tension of the melt system favors the attachment.

2 line graphs are labeled a and b. A. A line graph of total force F tot slash R c p versus central film height h 0 in n m. A fluctuating trend is plotted. B. A line graph of total force F tot slash R c p versus central film height h 0 in n m of the bubble with R 0 equal to 100, 250, 300, 350, and 500 micrometers per second.

The behavior of particles in the gas–liquid interface can be studied on a sessile drop/bubble with the CP-AFM technique. During an approach-retract-cycle as shown schematically in Fig. 15.6, the colloidal probe is driven towards the interface until a defined push-distance of the piezo is reached and is then retracted again from the interface. Knüpfer et al. investigated the case of a particle interacting with a sessile drop in an AFM experiment [13]. As colloidal probe particles both untreated (hydrophilic) and silanized (poorly wetted) silica particles of high sphericity were used to avoid the difficulties involved with the less regular alumina particles. For simplicity first, the interaction with a sessile drop is analyzed, but the overall force-distance-curves observed in experiments with gas bubbles are similar to the force-distance curves measured on a sessile drop. At first no significant force acts on the particle (1) as it approaches the drop until particle and drop are almost in contact. For small enough distances the interface deforms due to the capillary force, until the interface gets into contact with the particle (2). At this point the wetting of the particle begins (3) and the higher the wettability of the particle is, the faster it gets sucked into the interface of the drop.

2 illustrations. A. An illustration of a particle at a gas liquid interface divided into 7 parts. They represent different stages, such as initial position, equilibrium position, wetting of the particle, and particle after snap off. B. A line graph of force versus distance illustrates a decreasing trend.

If the particle wettability is high enough this leads to the so-called snap-in. Reversely for the interaction of the colloidal probe with a gas bubble a de-wetting of the particle occurs, and the velocity of the de-wetting process increases with a decreased particle wettability. For real inclusion particles that attach to gas bubbles this wettability dependent behavior would translate into an attachment probability that increases with increased three-phase-contact angle. In AFM experiments a strong attractive capillary force is measured at the snap-in. Subsequently the particle is approaching the original position of the interphase line, which leads to a reduction of the attractive force that is exerted on the particle, as the deformation of the interface is reduced. At the point where the force becomes zero, the particle reaches its wettability dependent equilibrium position (4). Any movement of the particle further into the interface results in a repulsive force sensed by the cantilever (5). The distance between the zero crossings of the snap-in and the reaching of the equilibrium position d_snap can be used as a measure for the contact angle upon attachment θ_at of the particle in the interface by using Eq. 15.10 where R_CP is the radius of the colloidal probe particle and d = d_snap. For the drop case, θ_at is the advancing contact angle and for the bubble case θ_at is the receding contact angle. Similarly, the contact angle θ_de during the pull-off can be calculated by Eq. 15.10, using the distance between the first-zero crossing of the retrace curve and the position of the snap-in d = d_re.

The minimum of the force-distance curve marks the detachment or pull-off force, which is defined as the adhesion force F_adh. It depends on the size of the colloidal probe particle, the interfacial tension \(\gamma_{{{\text{lg}}}}\) and the particle wettability, characterized by the contact angle θ_de during the pull-off of the particle from the interface. The corresponding capillary force can be approximated by Eq. 15.11, which underlies the assumption of a constant contact angle and a free movement of the contact line and was suggested by Scheludko and Nikolov [13, 28, 29].

A new model for the calculation of the maximum detachment force, using a blackbox-spring model, was developed by Knüpfer under the assumption of a linear deformation of the interface, which was observed experimentally for the major part of the retraction curve on sessile water droplets with radii in the mm range during the interaction with particles of 30 µm diameter [13]. The model incorporates a free Gibbs energy approach Eq. 15.12 that describes the wetting of the particle but not the deformation of the interface. This leads to the prediction of a linear force trend and the definition of a spring constant with the slope of \(2\pi \gamma_{{{\text{lg}}}}\). The maximum pull-off force can then be approximated by Eq. 15.13 [13].

The scaled force of an approach-retract-cycle for a silanized alumina colloidal probe particle with a radius of R_CP = 5 µm during the interaction with a gas bubble of 120 µm diameter in the water model system is shown in Fig. 15.7 for three approach velocities between 10 and 30 µm/s. The measurements were carried out in an electrolyte solution that contains 1 mM NaCl and 1 mM NaNO₃ for the stabilization of the interface and a hydrophobic Si-wafer with a three-phase-contact angle of 105° was used as a substrate for the bubble. The force-distance curves in Fig. 15.7 have been shifted so that the equilibrium position of the particle in the interface during the approach is located at position zero. For all investigated velocities the particle behaves similar in the interface and no systematic dependency of the wetting behavior on the approach- and retrace-velocities is observed. The snap-in parts of the curves for 10 and 20 µm/s coincide directly with each other, but the maximum scaled force at 20 µm/s is 110 mN/m and at 10 µm/s only 60 mN/m, which is a consequence of the stick- and slide phases that the three-phase-contact-line of the gas bubble on the substrate undergoes during the repeated approach-retract-cycles of the AFM-experiment.

A line graph of F slash R versus position in micrometers for the interaction between a salinized alumina particle and a gas bubble of 120 micrometers in diameter illustrates a decreasing trend before increasing.

At 30 µm/s the observed snap-in distance is a little smaller than for the lower approach velocities, but the depth of the force minimum at the snap in remains unaffected and the difference in the observed snap-in-distances is likely caused by the pinning effects of the gas bubble on the substrate that directly affect the overlap between colloidal probe and initial bubble interface position. The force-distance curves of both, silanized and non-silanized alumina particles have a similar shape and all exhibit a snap-in, that is likely the consequence of gas reservoirs present on the roughness of both, silanized und untreated alumina particles.

The scaled pull-off-force and the scaled force at the snap-in for three colloidal probes of each wettability state, silanized-alumina (CP1-CP3) and pure alumina (CP4-CP6), are shown in Fig. 15.8. The diameters of the particles given in Table 15.2 range from 7.6 to 21.4 µm and the measurements were done on three sessile bubbles with diameters between 85 and 120 µm.

4 histograms for receding contact angle, advancing contact angle, force at snap in F slash R, and adhesive force F slash R versus C P 1, C P 2, C P 3, C P 4, C P 5, and C P 6 are labeled a, b, c, and d, respectively. The highest values are plotted for C P 3 for advancing contact angle.

The difference in particle wettability is evident from all three, the dynamic contact angles, the scaled force at the snap in and the force of adhesion, as the median values of the silanized colloidal probes CP1 to CP3 exceed the median values for the untreated colloidal probes CP4 to CP6. The static macroscopic contact angle for untreated alumina is θ = 90°and θ = 134° for silanized alumina, determined by sessile drop measurements on flat plates. The average of the median receding contact angle of the silanized alumina particles CP1 to CP3 is θ_re = 50.0° and significantly lower than the equilibrium contact angle of θ = 134°. The average advancing contact angle of CP1 to CP3 is θ_adv = 62.8° and also significantly lower than the macroscopic value. Similarly, the average median values θ_re = 31.3° and θ_adv = 43.8° for untreated alumina are lower, which leads to the conclusion that the wetting behavior of particles with slight deviation from an ideal spherical shape and surface roughness is not well described by simple analysis of the force-distance measurements in combination with Eq. 15.10. The median values of the scaled force at the snap-in for silanized alumina lie between 20.5 and 36.8 mN/m and are significantly higher than the median values for the untreated alumina colloidal probes CP4 to CP6 with values of the scaled force between 6.2 mN/m to 13.7 mN/m. The occurrence of the snap-in for pure alumina seems surprising at first glance, but is likely a consequence of the surface roughness and cracks on the particle surface that can host gas reservoirs. Even if initially no gas reservoirs are present on the particle surface, the repeated contacting of the colloidal probe and the gas bubble likely leads to the formation of gas reservoirs that remain in the roughness after the rewetting of the colloidal probe. The scaled adhesion force that has to be overcome to pull the colloidal probe particles out of the bubble interface is less wettability dependent and possibly governed by the particle roughness, as the observed ranges of the pull-off force for both wettability states are overlapping. The median values of the scaled adhesion force for CP1 to CP3 are in the range 116.3 to 179.5 mN/m and for the untreated alumina particles CP4 to CP6 in the range 59.6 to 111.3 mN/m. This result indicates that the likeliness of attachment to a gas bubble is increased with decreased particle wettability and a slight increase of the stability against detachment from the interface for the poorly wetted particle system is observed. Overall the differences in wettability are expected to lead to an increased net-rate of hetero-coagulate formation with decreased particle wettability.

In contrast to the van der Waals and capillary adhesion mechanisms, the water-based model system does not take into account additional effects such as sintering of the inclusion particles on the filter wall due to the extremely high temperatures, especially in the case of molten steel filtration. A general overview of adhesive mechanisms between two surfaces under elevated temperatures is given by Berbner et al. whereby adhesive forces with material bridges (e.g. chemical reaction, sintering, crystallisation) are usually stronger [30].

Adhesive force measurements under elevated temperatures, especially with the atomic force microscope, are comparatively little known, as the high temperatures can very quickly destroy the expensive sensor technology. Specially shielded devices are used in which the samples can be heated, for example, via a filament or directly by direct current supply. The temperature range investigated in the literature is also significantly lower than in the case of the metal melt filtration process; investigations have so far been carried out in the temperature range between 100 and 200 ℃. In an air atmosphere, an increase in adhesive force due to capillary condensation can be observed first, followed by a steady but slight decrease. Even in a nitrogen atmosphere, where no capillary condensation occurs, a slight but steady decrease in adhesive force with increasing temperature can be observed [31‐33]. A stronger atomic oscillation energy is assumed to be the cause. If instead the glass transition temperature of the sample surface is reached, in this case polyester, the adhesive force determined via an apparatus similar to the atomic force microscope first begins to rise moderately, then sharply, and drops slightly at even higher temperatures. It is assumed that the highest energy dissipation rate is achieved by viscoelastic behaviour [34]. Using the centrifuge method, it could be shown that gold particles under mild sintering temperatures at 400 ℃ show an up to 100-fold increase in adhesive strength compared to measurements at 20 ℃, due to plastic deformation and surface diffusion at the contact point [35].

In terms of particle interactions, sintering is the diffusion creep under the influence of capillary forces. Sintering only takes place when the sintering potential, i.e. the Gibbs free energy, is negative; the system thus aims to minimise the Gibbs free energy. Material transport takes place at the interface, where different mechanisms can occur depending on the system conditions, such as temperature, material, particle size or heating regime, which are summarised in Table 15.3.

An important point is that the sintering temperature \(T_{sint}\) can sometimes be significantly below the melting temperature (\(T_{{{\text{sint}}}} \approx 0.50 \ldots 0.95 \cdot T_{{{\text{melt}}}}\)), whereby for individual particles sintering effects also occur in the early range, i.e. at low temperature.

From the work of Kuczynski and Krupp [36, 37], the force needed to separate two sintered surfaces can be determined with the help of a material pair-specific constant \(S\left( {T_{{{\text{sint}}}} } \right)\), the characteristic constants \(m\) und \(n\) (see Table 15.3), the tensile strength \(\sigma_{{{\text{tensile}}}}\), the particle radius \(R\) and, assuming an Arrhenius approach, the force needed to separate two sintered surfaces can be calculated via Eq. 15.14:where \(t_{{{\text{sint}}}}\) corresponds to the sintering time, \(E_{{\text{A}}}\) to the activation energy and \(k_{{\text{B}}}\) the Boltzmann constant. This classical equation, which not only results from the tensile strength of the sinter neck, simultaneously considers the neck growth.

With the aid of the UHV SPM 7500 high-temperature scanning force microscope from RHK Technology, it is possible to realise adhesive force measurements up to 800 ℃, using a developed method for the production of temperature-stable colloidal probe cantilevers, (see Fig. 15.9b) [17]. Results for the alumina system (Al₂O₃ particles and Al₂O₃-coated Si wafer) are shown in Fig. 15.9c).

3 illustrations. A. An illustration of 4 heated samples at 680, 700, 720, and 750 degrees Celsius. B. An illustration of a stable colloidal probe cantilever. C. A scatterplot of F slash R versus temperature in Celsius.

On the one hand, the influence of the atmosphere (ambient air and thus a certain humidity increases the adhesive forces compared to vacuum) and the dwell time of the particle on the sample surface (a longer dwell time also corresponds to a higher adhesive force) must be taken into account. A change in the cantilever resonance frequency after the tests at elevated temperatures was not significant. A peak at about 140 ℃ is noticeable, which can be attributed to burning of residues of the not completely removed dispersant on the particle surface. Up to about 730 ℃, the van der Waals forces decrease slightly, after which they hardly increase. Here, too, as in Lai et al., a stronger fluctuation of the electrons is assumed, which ultimately reduces the attractive interactions. For metal melt filtration, it can therefore be concluded that with pure vdW interactions (i.e. no capillary forces or sintering effects), the adhesion of the inclusion particles to the filter wall is slightly reduced. The opposite is the case with the additional occurrence of sintering effects due to very high temperatures. In order to realise such measurements, a model system had to be used, as temperatures even higher than 800 ℃ are not feasible. Polystyrene particles were therefore chosen as the model inclusion and temperatures up to a maximum of 240 ℃ were set [38]. Another characteristic temperature besides the melting point between 240 and 270 ℃ (depending on the composition) is the glass transition temperature, i.e. the temperature that describes the transition from brittle-elastic to rubber-elastic behaviour; this is about 100 ℃ for polystyrene. The results are shown in Fig. 15.10.

3 illustrations. A. An illustration of the used polystyrene probe cantilever. B. 2 line graphs of F slash R versus distance in m. C. A dashed line graph of F slash R versus temperature in Celsius illustrates an increasing trend.

The SEM image in (a) shows that sintering has already occurred at the selected temperature (here 135 ℃) after contact with the hot surface; the previously almost ideally smooth spherical particle shows several rough elevations. At lower temperatures, a flattening could be observed, which fits well with the behaviour when the glass temperature is exceeded (plastic deformation). Compared to theoretically calculated values (vdW forces), the normalised adhesive force values are somewhat lower, which is due to smallest roughnesses. At about 150 ℃, a force maximum is achieved at 313 mN/m, which corresponds to more than 20 times the value at 85 ℃ and can be explained by stronger sintering effects (Fig. 15.10c)). After that, the adhesive force decreases, which is primarily due to more intensive melting and can be observed, for example, through increased ring formation on the particle. Example curves in Fig. 15.10b) also show the changed shape of the force-distance curves, which further support this assumption.

From the experimentally determined averaged adhesive force values and temperature-dependent surface tension values, a simple equation (Eq. 15.15) can finally be approximated to the measured data via a combination of the above-mentioned sinter model and common capillary force models:

A radius of 13.075 µm can be chosen for the particles. The surface tension is calculated according to Moreira with \(\gamma = f\left( {T in K} \right) = - 0.774T + 66.655\) [39]. This results in an activation energy of 1.2 eV for the model particles with \(C = \frac{{2E_{{\text{A}}} }}{{nk_{{\text{B}}} }}\) assuming volume diffusion, which is within a reasonable size range [35].

The high temperatures in the molten metal can lead to sintering of previously agglomerated inclusion particles which results in an increased agglomerate stability against particle redispersion. Sintering of alumina can be studied by atomistic simulations under the condition that an appropriate interatomic potential is chosen, as the quality and predictability of sintering is strongly depended on the performance of the underlying interatomic potentials. The suitability of the four empirical interatomic potentials, Vashishta potential (Vash) [40], Coulomb-Buckingham (CB) potential of matsui type [41], Born–Mayer-Huggins potential (BMH) [42] and the Charge Transfer Ionic + EAM potential (CTIE) [43] was investigated by Roy [44]. Benchmarking of the potentials showed that most material properties predicted by all four potentials are well within the range of the experimental data available in literature, except for the predictions of elastic constants and surface energies by the BMH and CTIE potentials. An accurate prediction of the experimentally reported melting temperature of Al₂O₃ that lies in the range between 2200 and 2350 K is only predicted with the CB potential where the melting temperature is 2340 K. The melting temperatures obtained with the other potentials are significantly higher and range up to 4200 K for the CTIE potential, therefore the homologous temperature of the corresponding potential was used to facilitate an objective comparison of the sintering process with different potentials.

The simulation setup for the MD simulations of the sintering is shown in Fig. 15.11. The sintering progress is evaluated by six parameters, namely the shrinkage ratio, normalized surface area, mean square displacement, neck curvature, the fraction of ions at the neck and the norm of the stretch tensor.

A flowchart of the simulation setup. It begins with cutting out a sphere from a bulk, followed by equilibrium at the targeted temperature, replication, and the moving of particles close to each other. It ends with sintering test that runs for 1000 p s.

All potentials show an increased amount of sintering with increasing temperature and the sintering is faster for small particles. The Vash potential predicts the material properties after sintering better than the other three potentials, and shows the lowest degree of sintering among the four. The CTIE potential predicts the highest degree of sintering of the potentials considered.

The agglomeration behavior of silanized Al₂O₃ particles under conditions relevant for technical scale processes was investigated in a stirred tank setup by Knüpfer et al., who in addition to the stirred tank experiments provided detailed information on the interplay of flow conditions and surface forces, characterizing the latter by AFM measurements, with special emphasis on the presence of nanobubbles [45]. Nucleation of defined nanobubbles can be triggered by many procedures that include solvent exchange, temperature gradients and perturbation. A solvent exchange procedure is employed in the agglomeration experiments to investigate the influence of nanobubbles on the agglomeration and stability of agglomerates. The presence of nanobubbles can be proved by phase contrast images obtained from AFM as shown in Fig. 15.13b) for nanobubbles on a hydrophobized Al₂O₃ substrate after a solvent exchange with ethanol. The experiments were carried out at the isoelectric point in 0.5 M NaCl solution and with non-spherical Al₂O₃ particles with a maximum particle size of 25 µm. The experimental setup consists of a baffled stirred tank with 4 L volume, a height to diameter ratio of 0.75 and a 6-blade propeller that had half the diameter of the tank. The agglomerate sizes were determined in a bypass by dynamic image analysis, using the QICPIC (Sympatec, Germany). Both, the propeller and the three baffles are required to induce turbulence, which is crucial to enable the collision of inclusions but can also lead to redispersion of agglomerates. An analytical tool for the characterization of the turbulence behavior is the Kolmogorov microscale \(\eta\), given by Eq. 15.16, which describes the length scale of the smallest eddies and depends on the kinematic viscosity ν and the energy dissipation rate ε. It is assumed that the energy dissipates in the smallest eddies and that the flow within those smallest eddies can be described by a laminar shear flow with a Reynolds Number of Re = 1.

The local energy dissipation rate determines the influence of turbulence on the agglomeration, but it is experimentally not accessible and resolving the local turbulence in a stirred tank by CFD is still a formidable task, despite today’s computational capabilities. Therefore, in most experimental investigations the energy dissipation rate which determines the Kolmogorov micro scale is obtained from either power uptake or torque measurements of the stirrer.

The definition of a global root mean shear rate \(\overline{G}\) in Eq. 15.17 as suggested by Camp and Stein [46] and frequently used in other works that are concerned with agglomeration phenomena, can be used to directly relate the equilibrium agglomerate size x to the shear conditions with an allometric fit using Eq. 15.18 [45], as depicted in Fig. 15.13a). The two material specific constants are related to the agglomerate strength and are the agglomerate strength constant C, and the agglomerate strength exponent \(\gamma\) [45].

Agglomerates that are smaller than the Kolmogorov length scale are subject to erosion as a consequence of the viscous shear forces in the smallest eddies and therefore redispersion occurs if the adhesion forces are not sufficiently high [45]. This results in the observation of an equilibrium size distribution after 20 min of agglomeration in the room temperature water model system and is shown in Fig. 15.12 for \(\overline{G}{ } = { }186{ }\,s^{ - 1}\), which correlates well with the measured adhesion forces of the three investigated wettability states of untreated hydrophilic alumina, silanized hydrophobic alumina and silanized hydrophobic alumina after the solvent exchange, which has the highest adhesion force due to the presence of nanobubbles on the surface.

2 line graphs. A. A line graph of q 3 versus x in micrometers for primary particles, ultrasonic treatment, 1, 5, 10, and 20 minutes illustrates an inverted bell curve. B. A line graph of x in micrometers versus t in minutes illustrates an increasing trend.

Two illustrations. A. An illustration of a specific size in micrometers versus the average velocity gradient illustrates a decreasing trend. B. A contrast image highlights the poorly wetted A L 2 O 3 surface.

The agglomeration and dispersion of the agglomerates are fully reversible, as shown by experiments where the agglomeration process is disturbed by an ultrasonic treatment. After the ultrasonic stressing the agglomerates form at the same speed and show the same equilibrium agglomerate sizes, which suggests that the presence of the ultrasound does not affect the stability and surface coverage with nanobubbles, that have heights of 50 to 400 nm and diameters in the range 400 to 2000 nm [45].

The effect of aeration on the agglomerate size distribution of spherical silanized Al₂O₃ is shown in Fig. 15.14. The initial particle size distribution measured after the injection of an ultrasonicated ethanol suspension (t = 0 s) is relatively wide and adjusts to the flow conditions by shifting its modal value to 46 µm at t = 218 s. At this point the number of detected particles is reduced by 40 % relative to the initial number which is a consequence of superimposed effects of agglomeration and deposition of particles in the air–water interface at the top of the tank. After four and a half minutes the stirred tank is aerated for 30 s with microbubbles from a two-phase-nozzle (d_32,bubble = 145 µm), which causes a shift to a multimodal density distribution with two main peaks at 27 µm and 55 µm at t = 330 s. An increase of the detected particle number by 51% compared to the particle number before aeration is measured, as well as the increase of the number of particles that is immobilized inside the gas–liquid interface at the top of the stirred tank, quite similar to the slag layer in the real melt. This indicates both, the flotation of particles to the gas–liquid interface, as well as the destruction of existing agglomerates. The majority of bubbles does not form hetero-coagulates and those that do preferably attach to medium sized agglomerates smaller 50 µm. If the entrainment of the particles in the flow field around the bubbles contributes to the transport of particles to the interface remains unclear, as the redispersion of particles from agglomerates in the bulk is dominant. At t = 495 s an increased number of agglomerates is formed again, accompanied by a decreased number of detected particles. Due to its potential to redisperse particles from existing hetero-coagulates an aeration of the melt by porous plugs or immersion lances should be the last cleaning step of the melt before casting. In this regard the water model system shows great similarities.

2 line graphs. A. A line graph of area equivalent diameter x in micrometers versus t in minutes plots the line for agglomerate size and aeration phase. B. A line graph of q 3 versus area equivalent diameter x in micrometers illustrates an inverted bell curve.