Chapter 4: Rheology and Viscosity of Carbon Dioxide-in-Water Foams with
4.2 Experimental methods
4.2.4 Microscopy Measurements
The in-situ characterization of bubble sizes of the C/W foam was measured by diverting foam flow after the sand pack or capillary to a high-pressure microscopy cell137 with two six-port injection valves (Valco Instruments, model C6W) and then back to the flow cycle. One valve determined the sampling point for the foam and the other controlled flow through the microscopy cell. The microscopy cell was mounted on a microscope (Nikon Eclipse ME600). The sapphire windows (Swiss Jewel Company, W6.36, 0.635 cm diameter and 0.229 cm thickness) were separated with foil spacers creating a path length of approximately 25 µm.80 Microscopy images were captured when flow through this cell was stopped via a Photometrics CoolSNAP CF CCD camera connected to a computer. Foam was flowed through the microscopy cell for several cell volumes prior to image recording to ensure foam photographed did not age significantly.
The images were analyzed with ImageJ software by setting the scale (microscopy standards were used), adjusting the threshold value of the image, and using the measure particles function. In most cases, the bubble areas with a circularity of 0.60 or greater were obtained, and thus converted to effective spherical radii. Size distribution parameters and average radii were then calculated. The minimum bubble size that could
be measured had a diameter of 0.4 µm at 50x magnification, 0.88 µm at 20x magnification, and 1.8 at 10x magnification; bubbles smaller than these values could not be detected with the microscope and were excluded. Average bubble sizes were obtained by generally analyzing 6-9 microscope images at each condition, which corresponds to at least several 100 bubbles and up to 10,000 bubbles. In Appendix C, typical micrographs of foam are presented.
4.3 THEORY
As foams are a dispersion of two phases, the rheological behavior is more complex than that of either of the phases. The foams often exhibit elastic behavior due to the compressibility of the dispersed phase and reversible deformation of the foam structure.68 The interconnected properties of the continuous phase also influence the rheological properties of the foam, especially when the viscosity of the internal phase is low (as it is for typical air-water foams and C/W foams).180 Thus, a yield stress is commonly found for foams due to the elastic nature of this interconnected network of fluid films.181
The structure of the continuous phase also determines the apparent viscosity of the bulk foam (ηfoam). The foam viscosity is highly dependent on the quality and bubble sizes, which directly determine the number of lamellae in the foam. The shear stresses are concentrated at the lamellae and the thin liquid film that forms on the capillary wall. A diagram magnifying the foam near the capillary wall is presented in Figure 4.2. With the assumption of the no-slip boundary condition, the shear stresses in the liquid phase are translated through the thin film and the lamellae near the capillary wall. Therefore, the
Figure 4.2: Diagram of the magnified view of the thin liquid wetting film and foam lamella at the capillary tube wall.
Plateau Border Lamella
CO2 Bubble
Wetting Film Water
Wall
CO2Bubble
continuous liquid phase is responsible for the majority of the resistance to flow and consequently the high viscosity of the foam. The number of lamellae, the interfacial tension, continuous phase viscosity, and quality are the main factors that determine the foam viscosity.
A proposed equation for the viscosity of a concentrated emulsion (ηHIPE) in 3D derived from a 2D geometry-based model by Princen et al.182 is
( )
⎟⎟1/3 interfacial tension, τo is the yield stress, ηc is the viscosity of the continuous phase, and C(φ) is a numerical factor. Although this equation has been used to describe modulus and yield stress data, viscosity data can be fitted only by changing the exponent to of the last term to ½ from 1/3 and using τo and C as fitting parameters.180,182The apparent viscosities of the bulk foams (ηfoam) investigated here are calculated from the known γ& and measured pressure difference (∆P) across the capillary with a length (L) of 195 cm. The shear stress (τ) and γ& are calculated from ∆PR/2L and the velocity gradient (U/R), respectively. An additional geometric scaling term, λ, is used to calculate the apparent foam viscosity
LU
where R is the capillary tube radius, λ = 0.5 which also absorbs the factor of 2 in the denominator of τ. The average velocity, U, is determined from the total volumetric flow rate of the foam (Qfoam, the sum of the flow rates for the two phases) divided by the cross sectional area of the capillary tube. Unlike Newtonian fluids, plug flow generally occurs away from the wall in foams. The velocity profile of a foam and a Newtonian fluid in a
capillary tube are depicted in Figure 4.3. The addition of a geometric scaling term, λ, takes into account the fact that τ mainly occurs only in the regions near the capillary wall, not throughout the entire radius of the capillary. The scaling factor was set to an arbitrary value of 0.5. Although topological transitions can occur in foams, where foam cells change position and effectively hop to the next metastable disordered state,181,183 plug flow was observed for all the C/W foams investigated here.
The Sauter mean diameter of a given foam, Dsm, and the polydispersity Upoly are calculated as follows
where Di is the diameter of a foam bubble and Dmed is the median bubble diameter of the foam. Number average diameters, Davg, can also be calculated from the Di values.