Since it is fairly difficult to accurately measure these times separately, in practice, they have been measured collectively and referred to as the induction time. The induction time plays a critical role in flotation. Under a given hydrodynamic condition, the shorter the induction time is, the higher is the flotation recovery.
Theoretical studies on film thinning, film rupture and three-phase contact line also showed that the induction time measured by moving an air bubble toward and then away from a mineral bed in a liquid is a function of physical properties of the solid particles (e.g. size, density and contact angle), physical properties of the liquid (viscosity, density and surface tension) and the three-phase contact line tension. The key factors controlling the attachment of air bubble-solid particles in a liquid are surface and interfacial properties of the intervening phases, e.g. surface energy of the solid, surface tension of the liquid, interfacial tension of the solid-liquid as well as contact angle of the liquid contacting the solid surface. Only the contact angle and surface tension are directly measurable, all other surface and interfacial properties are derived from these two properties. It is desirable to have an instrument capable of measuring induction time, surface and interfacial properties.
Working principle of the instrument
Figure 1 is a 3D view of the instrument for measuring induction time, surface and interfacial properties, including a motorized vertical main stage holding the main drive (a voice coil motor or a speaker), a gas bubble or liquid drop generation system, an advanced image system including a high speed camera, a regular (or telecentric) zoom lens and a telecentric illuminator, a sample stage with a three-way manual translation, a precision micro (or submicro) displacement sensor, a multifunctional software, Floatimer, that controls all motors, monitors the displacement sensor’s feedback, records and analyzes images to make multifunctional measurements, and a control box enclosing all required electronics. With the gas bubble or liquid drop generation system, an air bubble of controllable size can be generated at the end of a capillary tube using a micro-syringe. The advanced image system is used to assist viewing the precise positioning and contacting process between the air bubble and solids particles, and to provide high quality images for edge detection and digital image processing.
Induction time measurement
The induction time of solid-air bubble can be measured either using an air bubble to attach a solid particle bed in a testing liquid as shown in Fig. 2(a) or using a drop of the testing liquid to attach the solid particles bed exposing in the air directly as shown in Fig. 2(b). The method 2(a) involves the liquid receding process (solid-air interface replacing solid-liquid interface), hence provides a receding induction time, treceding, while the method 2(b) involves the liquid advancing process (solid-liquid interface replacing solid-air interface), hence provides an advancing induction time, tadvancing.
The induction time of solid-air bubble in the testing liquid can also be measured using a solid substrate with a flat surface as shown in Fig. 2(c), attachment between the solid surface and the air bubble is believed to have been established when the air bubble is sticking on the solid surface after the capillary tube retracts to its original place. The induction time of solid-air bubble in the testing liquid can also be measured using a drop of the testing liquid to contact the solid substrate in the air environment as shown in Fig. 2(d), attachment between the solid surface and the testing liquid drop is believed to have been established when the drop of the testing liquid is sticking on the solid surface after the capillary tube retracts to its original place. The method shown in Fig. 2(c) provides a receding induction time, while the method shown in Fig. 2(d) provides an advancing induction time.
Measurements of contact angle, surface and interfacial tension
Surface and interfacial properties can be measured using drop methods (sessile drop method and pendant drop method) and probe methods (Du Noüy ring method, Du Noüy-Padday rodmethod and Wilhelmy plate method).
The sessile drop method is used for characterization of solid surface energies, and in some cases, aspects of liquid surface energies as shown in Fig. 3(a). The shape of a liquid-vapor interface is determined by the Young-Laplace equation, with the contact angle playing the role of a boundary condition via Young’s Equation. The shape profile of the sessile drop is digitized and analyzed using Young-Laplace equation for contact angle calculation.
Figure 3. A schematic illustration of the drop methods: (a) the sessile drop and (b) the pendant drop methods, where θ is the contact angle, and γsg, γlg and γsl represent the solid-gas, liquid-gas, and solid-liquid interfacial energies, respectively, d is the capillary tube diameter, m is the weight of the pendant drop.
γsg = γlg cosθ + γsl, (1)
The pendant drop method is a technique by which a drop of liquid is suspended from a tube (capillary or needle) in a bulk liquid or gaseous phase. The shape of the drop results from the relationship between the surface tension or interfacial tension and gravity. In the pendant drop method, the surface tension or interfacial tension is calculated from the shadow image of a pendant drop using drop shape analysis in accordance with Young-Laplace equation as shown in Fig. 3(b). The force due to surface tension is proportional to the length of the boundary between the liquid and the tube, with the proportionality constant usually denoted, γ. Since the length of this boundary is the circumference of the tube, the force due to surface tension is given by, Fγ = p d γ, where d is the tube diameter, p = 3.14159.
The mass, m, of the drop hanging from the end of the tube can be found by equating the force due to gravity with the component of the surface tension in the vertical direction giving the formula,
mg = p d γ sin θ, (2)
where θ is the contact angle with the tube, and g is the acceleration due to gravity. The limit of this formula, as θ goes to 90°, gives the maximum weight of a pendant drop for a liquid with a given surface tension, γ.
mg = p d γ. (3)
This relationship is the basis of a convenient method of measuring surface tension. More sophisticated methods are available to take account of the developing shape of the pendant as the drop grows.
The du Noüy ring method is one technique by which the surface tension of a liquid or the interfacial tension between two liquids can be measured as shown in Fig. 4(a). The method involves slowly lifting a ring, often made of platinum or platinum-iridium, from the surface of a liquid. The material is also chemically inert and easy to clean. The force, F , required to rise the ring from the liquid’s surface is measured and related to the liquid’s surface tension, γ :
F = 2 p (Ri + Ra) γ, (4)
where Ri is the radius of the inner ring of the liquid film pulled and Ra is the radius of the outer ring of the liquid film.
The Du Noüy-Padday rod method is a minimized version of the Du Noüy ring method replacing the large platinum ring with a thin rod (probe) that is used to measure equilibrium surface tension or dynamic surface tension at an air-liquid interface. The rod is attached to a scale or balance via a thin metal hook. The Padday method uses the maximum pull force method, i.e. the maximum force due to the surface tension is recorded while the probe is first immersed approximately one mm into the liquid and then slowly withdrawn from the interface.
γ = Fmax / (2 p D) (5)
where D is diameter of the probe, g is the surface tension of the liquid. The maximum pull force is obtained when the buoyancy force reaches its minimum. This is observed as a maximum in the force curve, which relates to the surface tension.
Wilhelmy plate method is used to measure equilibrium surface and interfacial tension at an air-liquid or liquid-liquid interface as shown in Fig. 4(c). In this method, the plate is oriented perpendicular to the interface, and the force exerted on it is measured. The Wilhelmy plate is often made from filter paper, glass or platinum which may be roughened to ensure complete wetting. In fact, the results of the experiment do not depend on the material used, as long as the material is wetted by the liquid. The plate is cleaned thoroughly and attached to a balance with a thin metal wire. The force on the plate due to wetting is measured using a microbalance and used to calculate the surface tension, γ, using the Wilhelmy equation:
γ = F / (l cos θ) (6)
where l is the wetted perimeter (2w + 2d, w is the plate width and d is the plate thickness) of the Wilhelmy plate and θ is the contact angle between the liquid phase and the plate.