SALOVEY
4. RÚBRICA EVALUATIVA
Shear rate (γ& ) versus shear stress (τ) plots (calculated from U/R and ∆PR/2L, respectively), known as flow curves, were generated for the bulk foams; typical examples are presented in Figures 4.4-4.7. Additional plots are presented in the Appendix C, in addition to a table of all the ηfoam and Dsm measured for the surfactants investigated.
For a Newtonian fluid, the ratio of τ to γ& gives a constant value equal to the viscosity (η). A C/W foam that fits the Newtonian model is presented in Figure 4.4a for a 5% v/v TMN 6 foam with 80% v/v CO2 at 24 ˚C and 2300 psia at γ& from 140-870 s-1. Figure 4.4b gives the corresponding change in ηfoam withγ& . The flow curve for the Newtonian fluid is a straight line with a slope equal to ηfoam that goes through the origin if no yield stress occurs. This is not the case for most foams as a yield stress is often found.184 In the case of TMN 6 in Figure 4.4, the Newtonian behavior is not expected to occur for all shear rates and thus the slope does not intersect the origin. At low shear rates the elasticity of the foam structure and yield stress are expected to make the foam non-
Table 4.2: The effect of brine (2% NaCl, 0.5% CaCl2, 0.1% MgCl2 w/w in water) on the apparent bulk viscosity and Dsm of C/W foams.
24 ˚C
y = 0.0811x - 7.538
Figure 4.4: Newtonian flow curve (plot A) and ηfoam as a function of shear rate (plot B) for a C/W foam with 5% v/v TMN 6 and 80% v/v CO2 at 2300 psia and 24
˚C.
A
B
y = 0.0128x1.3655 R2 = 0.9991
1 10 100 1000
100 1000
Shear Rate (s-1)
τ (Pa)
1 10 100 1000
100 Shear Rate (s-1) 1000
ηfoam (cP)
1 10 100
100 1000
Shear Rate (s-1)
Dsm
(µm)
Figure 4.5: τ, ηfoam, and Dsm as a function of shear rate for the C/W foam stabilized with 1% v/v 2EH-PO5-EO9 and 90% v/v CO2 at 24 ˚C and 2000 psia (plot A, B, and C, respectively).
A
B
C
y = 0.4895x0.7834 R2 = 0.9821
10 100
100 1000
Shear Rate (s-1)
τ (Pa)
1 10 100 1000
100 1000
Shear Rate (s-1)
ηfoam (cP
1 10 100
100 1000
Shear Rate (s-1) Dsm (µm)
Figure 4.6: τ (plot A), ηfoam (plot B), and Dsm (plot C) as a function of shear rate for the C/W foam stabilized with 1% v/v 2EH-PO12-EO11 with 90% v/v CO2 at 24 ºC and 2000 psia.
C A
B
0 shear rate for the C/W foam stabilized with 1% v/v 2EH-EO5 and 90% v/v CO2 at 24 ˚C and 2000 psia.
C B
A
Newtonian. Different regimes of τ response to a given γ& have been recorded for foams previously.181
Another behavior commonly encountered with foam rheology is that of the power-law model. The power-law fluid model is
kγn
τ = & [6]
where n and k are the two parameters of the model found from experimentally fitting the data. The flow curve of a power law fluid is linear if γ& and τ are plotted on a log scale as seen in Figures 4.5-4.6. The power law model is generally used to represent the behavior of polymer solutions and melts185 and has been used with a yield stress term in some cases to describe foams.181 For example, Hutchins et al. found that shear rate changes near 100 s-1 produced power-law type relationships for C/W foams with additives.178 Figures 4.5 and 4.6 show a power-law relationship between γ& and τ for γ& values in the range 140-870 s-1. Figure 4.5 presents plots of τ, ηfoam, and Dsm as a function of γ&
(respectively) for a C/W foam stabilized with 1% v/v 2EH-PO5-EO9 in the aqueous phase at 24 ˚C and 2000 psia for a quality of 90% v/v CO2. Figure 4.6 displays the change of τ, ηfoam, and Dsm withγ& (respectively) for a foam stabilized with 1% v/v 2EH-PO12-EO11 in the aqueous phase for 90% v/v CO2 at 24 ˚C and 2000 psia.
For many power law fluids, at very low shear rates the materials can become Newtonian as n tends to approach 1.185 When a foam has a combination of Newtonian and power-law properties for the γ& of interest, a polynomial fit of γ& as a function of τ results (as presented in the Appendix C). When n < 1, the power-law fluid is shear-thinning which indicates that the viscosity decreases with increasingγ& as shown in Figure 4.6 for 2EH-PO12-EO11. When n > 1, the fluid is shear-thickening with ηfoam rising on increasing γ& (Figure 4.5). However, the shear-thickening or shear-thinning behavior of the foam does not depend on the γ& -τ relationship.
Due to the specifics of the viscosity equipment, as Qfoam and γ& are increased, the flow rate through the foam generator (the sand pack) is also increased, thus changes in bubble size result. The changes in foam texture, as described by Dsm, need to be investigated in terms of the changes in ηfoam. Figure 4.5 gives τ, ηfoam, and Dsm as a function of γ& for a shear-thickening foam stabilized by 2EH-PO5-EO9 where Dsm
decreases (indicating smaller cell sizes) with an increase in Qfoam (and γ& ). As the bubble size is decreased, more bubbles are formed for a given volume of CO2 and consequently there are more lamellae at the capillary wall. The resulting higher shear stress increases ηfoam (Figure 4.5b). Smaller bubble sizes and a more narrow size distribution have previously been found with higher rates of shear for nitrogen foams when the concentration of surfactant is sufficient to stabilize the newly formed interface.186 Thus, shear-thickening behavior occurs as long as the surfactant can stabilize the smaller bubbles formed with an increase in γ& .59
If the bubble size remains fairly constant with increasing velocity of a foam or HIPE, shear-thinning behavior has been measured.59,174,175 Shear-thinning behavior is seen in Figure 4.6, where a change in γ& from 144 to 384 s-1 decreases Dsm slightly from 35 to 30 µm; however, ηfoam decreases from 170 to 120 cP with the change. An increase in γ& from 384 to 576 s-1, Dsm decreases from 30 to 20 µm and produces a slight increase in ηfoam from 120 to 135 cP as observed for the shear-thickening foams. For the last increase in γ& to 863 s-1, the Dsm remains constant and ηfoam drops again to 110 cP. The shear-thinning behavior at constant cell size is attributed to instability in the disordered foam structure corresponding to rearrangement of the foam cells.187 If the number of lamellae does not increase to cancel out this effect by smaller bubble sizes, ηfoam
decreases.
Another cause of shear-thinning behavior is due to poor stabilization of the foam cells by the surfactant. If the foam lamellae are not adequately stabilized by the surfactant, as γ& is increased coalescence occurs, raising Dsm and correspondingly decreasing ηfoam as seen in Figure 4.7 at the highestγ& . ηfoam and Dsm as a function of γ&
are shown in Figure 4.7 for a C/W foam stabilized with 1% v/v 2EH-EO5 at 24 ˚C and 2000 psia with 90% v/v CO2. Bubbles of all sizes are observed to coalesce in foams stabilized with 2EH-EO5 at all conditions due to the small size of both the head and tail of the surfactant, demonstrating the instability of the foam (which is unusual for the surfactants listed in Table 4.1). There is also a sharp drop in the shear stress response of the foam at the highest γ& as the coalescence disrupts the foam structure (Figure 4.7a).
The combined effects of changing bubble sizes and foam structure can produce complex rheological behavior of foams with changes inγ& , making the measurement of bubble sizes important for understand foam rheology. Interestingly, Harris used a recirculating pipeline viscometer to measure changes in foam texture versus foam viscosity and found that the quality and the continuous phase properties primarily determined the viscosity of the foam, whereas the effect of the texture was smaller for bubbles with an volume mean bubble diameter of 700-1100 µm.186 In our case, for a given quality and the same continuous liquid phase (water and surfactant), the much smaller bubble sizes become important for determining the foam viscosity.