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Mixtures of foam and paste: suspensions of bubbles in yield stress fluids

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Abstract

We study the rheological behavior of mixtures of foams and pastes, which can be described as suspensions of bubbles in yield stress fluids. Model systems are designed by mixing monodisperse aqueous foams and concentrated emulsions. The elastic modulus of the bubble suspensions is found to depend on the elastic capillary number \(\textit{Ca}_{_G}\), defined as the ratio of the paste elastic modulus to the bubble capillary pressure. For values of \(\textit{Ca}_{_G}\) larger than \(\simeq 0.5\), the dimensionless elastic modulus of the aerated material decreases as the bubble volume fraction \(\phi \) increases, suggesting that bubbles behave as soft elastic inclusions. Consistently, this decrease is all the sharper as \(\textit{Ca}_{_G}\) is high, which accounts for the softening of the bubbles as compared to the paste. By contrast, we find that the yield stress of most studied materials is not modified by the presence of bubbles. This suggests that their plastic behavior is governed by the plastic capillary number \(\textit{Ca}_{\tau_y}\), defined as the ratio of the paste yield stress to the bubble capillary pressure. At low \(\textit{Ca}_{\tau_y}\) values, bubbles behave as nondeformable inclusions, and we predict that the suspension dimensionless yield stress should remain close to unity, in agreement with our data up to \(\textit{Ca}_{\tau_y}=0.2\). When preparing systems with a larger target value of \(\textit{Ca}_{\tau_y}\), we observe bubble breakup during mixing, which means that they have been deformed by shear. It then seems that a critical value \(\textit{Ca}_{\tau_y}\simeq 0.2\) is never exceeded in the final material. These observations might imply that, in bubble suspensions prepared by mixing a foam and a paste, the suspension yield stress is always close to that of the paste surrounding the bubbles. Finally, at the highest \(\phi \) investigated, the yield stress is shown to increase abruptly with \(\phi \): this is interpreted as a “foamy yield stress fluid” regime, which takes place when the paste mesoscopic constitutive elements (here, the oil droplets) are strongly confined in the films between the bubbles.

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Notes

  1. Some aspects of the linear and nonlinear behavior of polydisperse suspensions are discussed in Vu et al. (2010).

  2. Note that although flow does not generally have an azimuthal symmetry in a vane-in-cup geometry (Baravian et al. 2002; Ovarlez et al. 2011), azimuthal symmetry seems to be recovered when the shear stress is close to the yield stress (Keentok et al. 1985; Ovarlez et al. 2011).

  3. Since the elastic moduli of the three systems are measured in the linear regime where their behavior does not depend on the strain amplitude, we also note that different \(g(\phi )\)values can be obtained for a same value of \(\textit{Ca}_\tau \) (see Eq. 13); this shows that \(Ca_{\tau }\) is not a relevant parameter in this regime.

  4. We remind that this should be strictly true in the dilute limit only: we do not yet have a prediction for all values of \(\phi \).

  5. If mixing is rapid and if viscous effects are important, for a constitutive behavior of the form \(\tau =\tau_y+f(\dot \gamma )\), the relevant number might rather be \(\frac {\tau_y+f(\dot \gamma )}{2\sigma _t/d}\).

References

  • Ancey C, Jorrot H (2001) Yield stress for particle suspensions within a clay dispersion. J Rheol 45:297–319

    Article  CAS  Google Scholar 

  • Baravian C, Lalante A, Parker A (2002) Vane rheometry with a large, finite gap. Appl Rheol 12:81–87

    CAS  Google Scholar 

  • Blanc F, Peters F, Lemaire E (2011) Local transient rheological behavior of concentrated suspensions. J Rheol 55:835–854

    Article  CAS  Google Scholar 

  • Chateau X, Ovarlez G, Luu Trung K (2008) Homogenization approach to the behavior of suspensions of noncolloidal particles in yield stress fluids. J Rheol 52:489–506

    Article  CAS  Google Scholar 

  • Coussot P (2005) Rheometry of pastes, suspensions and granular materials. Wiley, Hoboken

    Book  Google Scholar 

  • Coussot P, Tabuteau H, Chateau X, Tocquer L, Ovarlez G (2006) Aging and solid or liquid behavior in pastes. J Rheol 50:975–994

    Article  CAS  Google Scholar 

  • Denkov ND, Tcholakova S, Golemanov K, Ananthpadmanabhan KP, Lips A (2009) The role of surfactant type and bubble surface mobility in foam rheology. Soft Matter 5:3389–3408

    Article  CAS  Google Scholar 

  • Dormieux L, Kondo D, Ulm FJ (2006) Microporomechanics. Wiley, Hoboken

    Book  Google Scholar 

  • Dubash N, Frigaard IA (2004) Conditions for static bubbles in viscoplastic fluids. Phys Fluids 16:4319–4330

    Article  CAS  Google Scholar 

  • Dubash N, Frigaard IA (2007) Propagation and stopping of air bubbles in Carbopol solutions. J Non-Newton Fluid Mech 142:123–134

    Article  CAS  Google Scholar 

  • Dutta A, Chengara A, Nikolov AD, Wasan DT, Chen K, Campbell B (2004a) Texture and stability of aerated food emulsions–effects of buoyancy and Ostwald ripening. J Food Eng 62:169–175

    Article  Google Scholar 

  • Dutta A, Chengara A, Nikolov AD, Wasan DT, Chen K, Campbell B (2004b) Destabilization of aerated food products: effects of Ostwald ripening and gas diffusion. J Food Eng 62:177–184

    Article  Google Scholar 

  • Dzuy NQ, Boger DV (1983) Yield stress measurement for concentrated suspensions. J Rheol 27:321

    Article  Google Scholar 

  • Frankel NA, Acrivos A (1970) The constitutive equation for a dilute emulsion. J Fluid Mech 44:65–78

    Article  Google Scholar 

  • Gadala-Maria F, Acrivos A (1980) Shear-induced structure in a concentrated suspension of solid spheres. J Rheol 24:799–814

    Article  CAS  Google Scholar 

  • Gandolfo FG, Rosano HL (1997) Interbubble gas diffusion and the stability of foams. J Colloid Interface Sci 194:31–36

    Article  CAS  Google Scholar 

  • Geiker MA, Brandl M, Thrane L, Nielsen NF (2002a) On the effect of coarse aggregate fraction and shape on the rheological properties of self-compacting concrete. Cem Concr Aggr 24:3–6

    Article  CAS  Google Scholar 

  • Geiker MR, Brandl M, Thrane LN, Bager DH, Wallevik O (2002b) The effect of measuring procedure on the apparent rheological properties of self compacting concrete. Cem Concr Res 32:1791–1795

    Article  CAS  Google Scholar 

  • Gonnermann HM, Manga M (2007) The fluid mechanics inside a volcano. Annu Rev Fluid Mech 39:321–356

    Article  Google Scholar 

  • Goyon J, Bertrand F, Pitois O, Ovarlez G (2010) Shear induced drainage in foamy yield-stress fluids. Phys Rev Lett 128301:104

    Google Scholar 

  • Griffiths RW (2000) The dynamics of lava flows. Annu Rev Fluid Mech 32:477–518

    Article  Google Scholar 

  • Hashin Z, Shtrikman S (1963) A variational approach to the theory of the elastic behaviour of multiphase materials. J Mech Phys Solids 11:127–140

    Article  Google Scholar 

  • Keentok M, Milthorpe JF, O’Donovan E (1985) On the shearing zone around rotating vanes in plastic liquids: theory and experiment. J Non-Newton Fluid Mech 17:23–35

    Article  Google Scholar 

  • Koczo K, Lobo LA, Wasan DT (1992) Effect of oil on foam stability: aqueous foams stabilized by emulsions. J Colloid Interface Sci 150:492–506

    Article  CAS  Google Scholar 

  • Larson RG (1999) The structure and rheology of complex fluids. Oxford University Press, New York

    Google Scholar 

  • Leighton D, Acrivos A (1987) The shear-induced migration of particles in concentrated suspensions. J Fluid Mech 181:415–439

    Article  CAS  Google Scholar 

  • Ley MT, Folliard KJ, Hover KC (2009) Observations of air-bubbles escaped from fresh cement paste. Cem Concr Res 39:409–416

    Article  CAS  Google Scholar 

  • Liddell PV, Boger DV (1996) Yield stress measurements with the vane. J Non-Newton Fluid Mech 63:235–261

    Article  CAS  Google Scholar 

  • Llewellin EW, Mader HM, Wilson SDR (2002) The rheology of a bubbly liquid. P R Soc A 458:987–1016

    Article  CAS  Google Scholar 

  • Mabille C, Schmitt V, Gorria P, Leal Calderon F, Faye V, Deminière B, Bibette J (2000) Rheological and shearing conditions for the preparation of monodisperse emulsions. Langmuir 16:422–429

    Article  CAS  Google Scholar 

  • Mahaut F, Chateau X, Coussot P, Ovarlez G (2008a) Yield stress and elastic modulus of suspensions of noncolloidal particles in yield stress fluids. J Rheol 52:287–313

    Article  CAS  Google Scholar 

  • Mahaut F, Mokéddem S, Chateau X, Roussel N, Ovarlez G (2008b) Effect of coarse particle volume fraction on the yield stress and thixotropy of cementitious materials. Cem Concr Res 38:1276–1285

    Article  CAS  Google Scholar 

  • Mason TG, Bibette J, Weitz DA (1995) Elasticity of compressed emulsions. Phys Rev Lett 75:2051–2054

    Article  CAS  Google Scholar 

  • Mewis J, Wagner NJ (2012) Colloidal suspension rheology. Cambridge University Press, Cambridge

    Google Scholar 

  • Mason TG, Bibette J, Weitz DA (1996) Yielding and flow of monodisperse emulsions. J Colloid Interface Sci 179:439–448

    Article  CAS  Google Scholar 

  • Nguyen QD, Boger DV (1985) Direct yield stress measurement with the vane method. J Rheol 29:335–347

    Article  Google Scholar 

  • Ovarlez G, Bertrand F, Rodts S (2006) Local determination of the constitutive law of a dense suspension of noncolloidal particles through magnetic resonance imaging. J Rheol 50:259–292

    Article  CAS  Google Scholar 

  • Ovarlez G, Rodts S, Ragouilliaux A, Coussot P, Goyon J, Colin A (2008) Wide-gap Couette flows of dense emulsions: local concentration measurements, and comparison between macroscopic and local constitutive law measurements through magnetic resonance imaging. Phys Rev E 036307:78

    Google Scholar 

  • Ovarlez G, Barral Q, Coussot P (2010) Three-dimensional jamming and flows of soft glassy materials. Nat Mater 9:115–119

    Article  CAS  Google Scholar 

  • Ovarlez G, Mahaut F, Bertrand F, Chateau X (2011) Flows and heterogeneities with a vane tool: magnetic resonance imaging measurements. J Rheol 5:197–223

    Article  Google Scholar 

  • Ovarlez G, Bertrand F, Coussot P, Chateau X (2012) Shear-induced sedimentation in yield stress fluids. J Non-Newton Fluid Mech 42:148–157

    Google Scholar 

  • Pal R (2004) Rheological constitutive equation for bubbly suspensions. Ind Eng Chem Res 43:5372–5379

    Article  CAS  Google Scholar 

  • Parsi F, Gadala-Maria F (1987) Fore-and-aft asymmetry in a concentrated suspension of solid spheres. J Rheol 31:725–732

    Article  CAS  Google Scholar 

  • Phillips RJ, Armstrong RC, Brown RA, Graham AL, Abbott JR (1992) A constitutive equation for concentrated suspensions that accounts for shear-induced particle migration. Phys Fluids 4:30–40

    Article  CAS  Google Scholar 

  • Ramamurthy K, Kunhanandan Nambiar EK, Indu Siva Ranjani G (2009) A classification of studies on properties of foam concrete. Cem Concr Compos 31:388–396

    Article  CAS  Google Scholar 

  • Rust AC, Manga M (2002a) Effects of bubble deformation on the viscosity of dilute suspensions. J Non-Newton Fluid Mech 104:53–63

    Article  CAS  Google Scholar 

  • Rust AC, Manga M (2002b) Bubble shapes and orientations in low Re simple shear flow. J Colloid Interface Sci 249:476–480

    Article  CAS  Google Scholar 

  • Salonen A, Lhermerout R, Rio E, Langevin D, Saint-Jalmes A (2012) Dual gas and oil dispersions in water: production and stability of foamulsion. Soft Matter 8:699–706

    Article  CAS  Google Scholar 

  • Sikorski D, Tabuteau H, de Bruyn J (2009) Motion and shape of bubbles rising through a yield-stress fluid. J Non-Newton Fluid Mech 159:10–16

    Article  CAS  Google Scholar 

  • Stickel JJ, Powell RL (2005) Fluid mechanics and rheology of dense suspensions. Annu Rev Fluid Mech 37:129–149

    Article  Google Scholar 

  • Struble LJ, Jiang Q (2004) Effects of air entrainment on rheology. Materials Journal 101:448–456

    Google Scholar 

  • Turner D, Dlugogorski B, Palmer T (1999) Factors affecting the stability of foamed concentrated emulsions. Colloids Surf A 150:171–184

    Article  CAS  Google Scholar 

  • Uhlerr PHT, Guo J, Tiu C, Zhang XM, Zhou JZQ, Fang TN (2005) The shear-induced solid-liquid transition in yield stress materials with chemically different structures. J Non-Newton Fluid Mech 125:101–119

    Article  Google Scholar 

  • van Aken GA (2001) Aeration of emulsions by whipping. Colloids Surf A 190:333–354

    Article  CAS  Google Scholar 

  • Vu TS, Ovarlez G, Chateau X (2010) Macroscopic behavior of bidisperse suspensions of noncolloidal particles in yield stress fluids. J Rheol 54:815–833

    Article  CAS  Google Scholar 

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Acknowledgements

We thank Mohammed Bouricha for help on some of the experiments. We acknowledge funding from Saint-Gobain Recherche.

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Correspondence to Guillaume Ovarlez.

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Special issue devoted to novel trends in rheology.

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Kogan, M., Ducloué, L., Goyon, J. et al. Mixtures of foam and paste: suspensions of bubbles in yield stress fluids. Rheol Acta 52, 237–253 (2013). https://doi.org/10.1007/s00397-013-0677-7

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