PRL 112, 128303 (2014)
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PHYSICAL REVIEW LETTERS
Off-Equilibrium Surface Tension in Colloidal Suspensions 1
Domenico Truzzolillo,1,2,* Serge Mora,1,2,† Christelle Dupas,1,2 and Luca Cipelletti1,2
Université Montpellier 2, Laboratoire Charles Coulomb UMR 5221, F-34095 Montpellier, France 2 CNRS, Laboratoire Charles Coulomb UMR 5221, F-34095 Montpellier, France (Received 12 December 2013; revised manuscript received 16 February 2014; published 24 March 2014) We study the fingering instability of the interface between two miscible fluids, a colloidal suspension and its own solvent. The temporal evolution of the interface in a Hele-Shaw cell is found to be governed by the competition between the nonlinear viscosity of the suspension and an off-equilibrium, effective surface tension Γe . By studying suspensions in a wide range of volume fractions, ΦC , we show that Γe ∼ Φ2C , in agreement with Korteweg’s theory for miscible fluids. The surface tension exhibits an anomalous increase with particle size, which we account for using entropy arguments. DOI: 10.1103/PhysRevLett.112.128303
PACS numbers: 82.70.Dd, 68.05.-n, 83.80.Hj
The surface tension between two fluids quantifies the energetic costs of creating a new interface [1,2]. At equilibrium, surface tension may only exist between immiscible fluids: when two miscible fluids are brought in contact, the initial concentration gradient across the interface rapidly relaxes via diffusion and the system reaches an equilibrated, uniform state. On time scales shorter than that of interface relaxation, however, there are capillary forces at the interface that mimic an effective surface tension, as it was already recognized by Korteweg in 1901 [3]. Similarly to the theory for immiscible fluids, Korteweg’s theory relates the effective surface tension Γe to the gradient of composition across the interface: 2 dφ κ Γe ¼ κ dz ≃ Δφ2 ; δ −∞ dz Z
∞
(1)
where z is the coordinate orthogonal to the interface, κ the Korteweg constant, and φ the concentration of one of the two species. The last approximation holds for a linear concentration profile that increases by Δφ across an interface of thickness δ. Subsequent work by Davis [4] and Joseph [5] has generalized Korteweg’s ideas by suggesting that interfacial stresses may arise whenever gradients of an arbitrary fluid property exist at the interface between miscible fluids, e.g., density or temperature. The existence of a transient effective surface tension has been demonstrated in light scattering experiments probing capillary waves at the interface between miscible fluids [6,7]. Korteweg stresses have also been invoked to explain the shape of drops and bubbles, both under the effect of gravity [5] and in spinning drop measurements [8–10], the onset of a Marangoni-like instability leading to the cellular convective mixing of miscible fluids [11], and the shape of the meniscus between molten silicates of different composition [12]. In spite of the possible relevance of Γe in many situations, including jetting, bubbles and drops formation, coalescence and break-up, plumes and 0031-9007=14=112(12)=128303(4)
convection, precipitation and deposition, experiments that quantitatively probe Korteweg’s theory remain scarce: very few data are available for Γe [6,7,9,10], and large discrepancies between experimental values and those estimated from Eq. (1) have been reported [8]. An increase of Γe with Δφ was reported in spinning drop experiments on water-glycerin [8] and polymer [10] systems, but large deviations with respect to the quadratic scaling of Eq. (1) were observed. The very existence of an off-equilibrium surface tension is debated. Numerical simulations [13] of the fingering instability arising when a less viscous fluid is pushed through a miscible, more viscous one in a HeleShaw cell highlight the role of Γe in stabilizing the interface. By contrast, in earlier works the observed patterns were explained without including the contribution of the surface tension [14], or by explicitly assuming Γe ¼ 0 [15,16]. Colloidal suspensions may be regarded as ideal benchmark systems to investigate surface tension effects because their interfaces may be probed in great detail, down to the particle level [17]. Previous work has focused on the equilibrium interface between phase-separated colloidal fluids [17–20]; however, colloidal suspensions are also excellent candidates for investigating off-equilibrium surface tension. Indeed, diffusion is much slower in colloids as compared to atomic systems, leaving a wider temporal window for probing the transient interface between miscible fluids. Here, we report Hele-Shaw experiments on the fingering instability observed at the interface between two miscible fluids, a colloidal suspension, and its own solvent. We show that the evolution of the interface pattern is governed by both the nonlinear viscosity of the suspension and an effective surface tension, which we measure as a function of the volume fraction of the suspension. Our results confirm Korteweg’s law, Eq. (1). We furthermore show that, for our microgel particles, Γe is governed by the entropy associated with the internal degrees of freedom of the particles, leading to a previously unreported growth of Γe with particle size.
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FIG. 1 (color online). (a) Finger function K exp vs γ_ I in HeleShaw experiments where water is injected in silicon oil. The solid line is the theoretical function K th [right-hand side of Eq. (4)], with no adjustable parameters. Inset: K exp vs K th , as obtained from the analysis proposed in the text (solid circles) or using previous approaches based on the fastest-growing Fourier mode of the instability (open circles). (b)–(e) Water-oil interface (red line), as observed right after the onset of fingering, for various injection shear rates (in s−1 ) as indicated by the labels.
The experiments are performed in a Hele-Shaw cell consisting of two square glass plates of side L ¼ 25 mm, separated by spacers fixing the gap at b ¼ 0.5 mm. The cell is filled with the fluid to be studied, with viscosity η2. A less viscous fluid is injected through a hole of radius r0 ¼ 0.5 mm in the center of the top plate. For all experiments, we use water died with 0.5% w=w of methylene blue as the less viscous fluid, with viscosity η1 ¼ 1.011 mPa s. The _ is controlled via a syringe injected volume per unit time, V, pump. Temperature is fixed at T ¼ 293 0.1 K by means of a Peltier element, with a circular hole of radius 8.5 mm for optical observation. A fast CMOS camera run at 100 to 3000 frames s−1 is used to image the sample during injection. Typical images of the interface between the two fluids are shown in Figs. 1 and 2, where the distinctive instabilities that develop when η1 < η2 are clearly visible. In the framework of linear evolution theory, such an instability is conveniently described by decomposing the interface profile in Fourier modes, the mode of order n being associated with a pattern with n lobes, or fingers. For two Newtonian fluids, the order nf of the mode with the fastest growth rate is given by [21] 0.5 1 4rr0 γ_ I ðη2 − η1 Þ nf ¼ pffiffiffi þ1 ; bΓ 3
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FIG. 2 (color online). Finger function K exp vs γ_ I measured when injecting water in a microgel suspension. Data are labeled by ΦC. The lines are guides for the eye. Small panels: watersuspension interfaces for ΦC ¼ 0.92 and various γ_ I (top row, as indicated by the labels, in s−1 ), or at fixed γ_ I ¼ 570 s−1 and various ΦC (right column). 0.5 _ where r ¼ ½r20 þ Vt=ðπbÞ is the time-dependent radius of the unperturbed interface, Γ the interfacial tension 2 −1 _ between the two fluids and γ_ I ¼ 3Vð2πr the shear 0b Þ rate at the injection hole. This expression is often used to describe the number of fingers experimentally observed at the onset of the instability [22]. However, in experiments the observed number of fingers should be regarded as being related to the mode with maximum amplitude, rather than to the fastest growing one, because in radial Hele-Shaw flow the growth rate is time dependent, as also pointed out very recently [23]. We thus modify the standard linear evolution theory to calculate nA , the order of the experimentally accessible mode with the maximum amplitude, finding [24] 0.5 α 4rr0 γ_ I ðη2 − η1 Þ nA ¼ αnf ¼ pffiffiffi þ1 ; (3) bΓ 3 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where α ¼ −Wð−3e−3 Þ ≃ 0.422, with WðxÞ satisfying x ¼ WðxÞeWðxÞ . Although Eq. (3) is formally derived in the limit nA ≫ 1, we check numerically that it holds to a very good approximation already for nA ≥ 2 [24]. We test the validity of Eq. (3) on two Newtonian fluids for which all the relevant parameters are known: dyed water and silicon oil (η2 ¼ 12.5 Pa s, Γ ¼ 39.8 mN m−1 [34]). Figures 1(b)–1(e) show typical interface patterns observed at various injection rates. We determine nA by counting the number of fingers of the destabilized interface [24]. To compare the experiments to the theory, it is convenient to recast Eq. (3) in the form
(2)
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K exp ðnA ; rÞ ¼ K th ;
(4)
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where the experimental “finger function” K exp is defined by 1 3n2A −1 ; (5) K exp ðnA ; rÞ ¼ r α2 while its theoretical value depends only on the rheological and interfacial properties of the fluids, the cell geometry, and the imposed shear rate: 4r ðη − η1 Þ K th ¼ γ_ I 0 2 : bΓ
(6)
Figure 1(a) shows K exp ðnA ; rÞ vs γ_ I for the water-silicon oil system. The experimental points (symbols) are in excellent agreement with K th obtained from Eq. (6) using the fluids and cell parameters (line). We emphasize that such a quantitative agreement would not hold if K exp was calculated by interpreting the observed number of fingers as the fastest-growing Fourier mode of the destabilized interface, i.e., if nA was replaced by nf ¼ nA =α in Eqs. (4),(5) [inset of Fig. 1(a)]. Having demonstrated that our experiments allow the flow and interfacial parameters to be quantitatively determined, we use the same setup to investigate the offequilibrium interfacial tension between a colloidal suspension and its solvent. We study aqueous suspensions of poly-N-isopropylacrylamide microgel particles [35]. At T ¼ 293 K the particles have hydrodynamic radius Rh ¼ 165 nm, as measured by dynamic light scattering. For the same synthesis, the radius of gyration has been determined to be Rg ≃ 0.5Rh [36]. We perform experiments for several particle concentrations, which we express as the effective volume fraction, ΦC , of the microgels. Experimentally, the polymer mass concentration, c (w=w), is known from the synthesis. For all concentrations, we define the effective volume fraction as ΦC ¼ kc, where k is determined from the viscosity of the suspension in the dilute limit. We find k ¼ 20.1 by matching the c-dependent zero-shear viscosity of the suspension to Einstein’s formula, η ¼ η0 ð1 þ 2.5kcÞ, where η is the viscosity of the suspension and η0 that of the solvent. Viscosity measurements are performed in the range 0 < ΦC ≤ 0.02. Our Hele-Shaw experiments cover volume fractions ranging from ΦC ¼ 0.2, corresponding to diluted, hard spherelike suspensions, up to ΦC ¼ 1.2, where particles are squeezed due to steric constraints and the suspension is fully jammed. For a given ΦC , we perform experiments at various γ_ I , always keeping the injection rate high enough for diffusion-driven mixing between the injected solvent and the suspension to be negligible [37]. Figure 2 shows K exp as a function of the injection shear rate for all the microgel suspensions. A qualitative change is observed when ΦC increases: at low volume fraction K exp ∼ γ_ I , while for jammed suspensions K exp grows sublinearly with γ_ I at high shear rate and tends to a plateau
for γ_ I → 0. This behavior is strongly reminiscent of the shape of the flow curve, σð_γ Þ, in colloidal suspensions, where σ ¼ η_γ is the shear stress when imposing a shear rate γ_ . This suggests that K exp is proportional to the shear stress contrast, i.e., that Eq. (4) may be generalized by K exp ¼
4r0 γ_ I η2 ð_γ r Þ − η1 ; b Γe
(7)
where η2 ð_γ r Þ is the shear-rate dependent viscosity of the suspension, γ_ r ¼ ð4r0 γ_ I =rÞ the shear rate at the position r of the interface (assuming Poiseuille flow), and Γe the (ΦC -dependent) effective surface tension between the suspension and its solvent. In writing Eq. (7) one implicitly assumes that the same kind of patterns at the onset of the instability are observed for our shear-thinning concentrated microgel suspensions as for Newtonian fluids, i.e., that the wavelength of the perturbation at its onset is not drastically changed by the non-Newtonian features of the suspension. Numerical work on the Saffman-Taylor instability in a radial Hele-Shaw geometry supports this scenario [38,39] by showing that the non-Newtonian character of the fluids does not change qualitatively the instability, but just accelerates ( delays) its onset for shear-thinning (shearthickening) fluids. The choice of Eq. (7) is also supported by previous works [40,41] on the Hele-Shaw instability between immiscible non-Newtonian fluids in a rectangular geometry, where the dynamics of the fingers was described by a generalized Darcy law where the Newtonian viscosity was replaced by the shear rate-dependent viscosity. In order to test Eq. (7), we measure the flow curves of the microgel suspensions. Figure 3 shows σð_γ Þ obtained via conventional rheology. The required shear-dependent viscosity is obtained from η2 ¼ σ fit γ_ −1 , where σ fit is a fit to the measured flow curve (lines in Fig. 3). The fits allow the viscosity to be estimated by extrapolation in the whole range of shear rates relevant to the Hele-Shaw experiments. Standard models for Newtonian and non-Newtonian fluids are used for the fits [25,26,36], as detailed in [24]. We find that for all ΦC the reduced finger function, K exp ≡ K exp b½4r0 ðη2 ð_γ r Þ − η1 Þ−1 , is proportional to γ_ I ,
FIG. 3 (color online). Flow curves for the microgel suspensions (same symbols as in Fig. 2). The lines are fits as described in [24].
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polymer mass density, one has φ ¼ 3.6 × 10−2 ΦC . The Korteweg constant has been calculated in [28] for inhomogeneous mixtures of a solvent and ideal-chain polymers. We extend this calculation to cross-linked polymers [24], in the limit φ ≪ 1 relevant to our microgels, finding κ¼
FIG. 4 (color online). Scaled finger function K exp Γe b× ½4r0 ðη2 ð_γ r Þ − η1 Þ−1 for the microgel suspensions whose raw data are shown in Fig. 2. All data collapse on K exp Γe ¼ γ_ I (line), thus confirming Eq. (7).
as predicted by Eq. (7), and we determine the proportionality coefficient Γ−1 by linear fitting. Figure 4 shows e K exp Γe vs the injection shear rate. When using this reduced variable, all the data previously shown in Fig. 2 fall onto a straight line spanning more than three decades, thereby validating Eq. (7). We test Korteweg’s prediction, Eq. (1), in Fig. 5. For our system, the particle volume fraction in the injected phase is zero, so that Eq. (1) reduces to Γe ∼ Φ2C , where we have assumed a linear variation of the concentration profile across an interface of thickness δ. Figure 5 shows that Korteweg’s law holds over a wide range of concentrations, corresponding to a variation of Γe of more than one decade. We go a step further and model our experiments at a microscopic level by calculating κ. To this end, we identify φ in Eq. (1) with the volume fraction of the polymer, rather than that of the particles, since the microgels are highly swollen by the solvent. Using literature values for the
FIG. 5 (color online). Main plot: effective interfacial tension Γe between the microgel suspensions and their solvent, as a function of colloid (polymer) volume fraction [top (bottom), axis]. The lines are quadratic fits to the data for microgels with various Rh , as shown by the legend. The conversion factor between Φc and φ varies slightly with Rh : the scale on the Φc axis is exact only for Rh ¼ 165 nm. Inset: square root of the reduced surface tension Γe =Rh vs φ.
RTR2g ½χ þ 3; 6V w
(8)
with R the gas constant, V w ¼ 18 × 10−6 m3 mol−1 the water molar volume and χ the Flory-Huggins parameter. Reported values of χ for poly-N-isopropylacrylamide microgels in water range from 0.25 to 0.5 [42–44], yielding 5.1 × 10−7 N ≤ κ ≤ 5.49 × 10−7 N. By fitting the experimental Γe vs φ we get κ=δ ¼ 1.40 N m−1 and, hence, 364 nm ≤ δ ≤ 392 nm. The interface thickness thus calculated is in very good agreement with the average distance between particles, which in the range of ΦC studied here varies from 340 to 460 nm. This confirms that in our experiments diffusion at the interface is negligible and validates quantitatively our analysis. To test the robustness of Eqs. (1) and (8), we perform additional experiments on microgels with the same composition but smaller size, Rh ¼ 70 and 100 nm, respectively. From Eqs. (1) and (8) and using δ ∼ Rg ∼ Rh , one expects that data for microgels with different Rh should collapse onto a master curve when normalizing Γe by Rh. The inset of Fig. 5 shows that this is indeed the case. We emphasize that the scaling Γe ∼ Rh is in stark contrast with the usual scaling of the interfacial tension between molecular or colloidal phases, where Γ ∼ a−2 , with a the particle size [2,18]. This highlights the different origin of the surface tension in our experiments where the entropic contribution due to the internal degrees of freedom of the polymeric particles dominates. In conclusion, we have investigated the pattern formation resulting from the injection of the solvent in a colloidal suspension, a model system for investigating the nonequilibrium, effective surface tension between miscible fluids. The interface instability can be rationalized by a remarkably simple expression, depending separately on the rheological properties of the suspension and on the effective, off-equilibrium suspension-solvent surface tension. Our results confirm Korteweg’s law and raise challenging questions on the behavior of κ, and thus Γe , as a function of inter-particle and particle-solvent interactions, as well as particle size and shape. More generally, our findings provide an experimental and theoretical framework for exploring nonequilibrium surface tension effects, a topic relevant in many problems, ranging from material processing to fundamental fluid dynamics. This work has been supported by ANR under Contract No. ANR-2010-BLAN-0402-1. The authors are grateful to E. Bouchaud, O. Dauchot, C. Ligoure, L. Ramos, and V. Trappe for useful discussions.
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domenico.truzzolillo@univ‑montp2.fr Present address: Laboratoire de Mécanique et de Génie Civil—UMR 5508 CNRS et Université de Montpellier 2 Place E.Bataillon, F-34095 Montpellier cedex 5, France. J. E. van der Waals, Z. Phys. Chem. 13, 657 (1894). P.-G. de Gennes, F. Brochard-Wyart, and D. Queré, Capillarity and Wetting Phenomena: Drops, Bubbles, Pearls, Waves (Springer-Verlag, Berlin, 2004). D. Korteweg, Arch. Neerlandaises Sci. Exactes Naturelles 6, 1 (1901). H. T. Davis, Numerical Simulation and Oil Recovery (Springer-Verlag, Berlin, 1988). D. D. Joseph, Eur. J. Mech. B, Fluids 9, 595 (1990). S. E. May and J. V. Maher, Phys. Rev. Lett. 67, 2013 (1991). P. Cicuta, A. Vailati, and M. Giglio, Appl. Opt. 40, 4140 (2001). P. Petitjeans, C.R. Acad. Sci. Paris 322, 673 (1996). J. A. Pojman, C. Whitmore, M. L. Turco Liveri, R. Lombardo, J. Marszalek, R. Parker, and B. Zoltowski, Langmuir 22, 2569 (2006). B. Zoltowski, Y. Chekanov, Masere, J. A. Pojman, and V. Volpert, Langmuir 23, 5522 (2007). P. Garik, J. Hetrick, B. Orr, D. Barkey, and E. Ben-Jacob, Phys. Rev. Lett. 66, 1606 (1991). J. E. Mungall, Phys. Rev. Lett. 73, 288 (1994). C.-Y. Chen, C.-W. Huang, H. Gadelha, and J. A. Miranda, Phys. Rev. E 78, 016306 (2008). L. Paterson, Phys. Fluids 28, 26 (1985). J. Nittman, G. Daccord, and H. Stanley, Nature (London) 314, 141 (1985). J. Nittman and H. Stanley, Nature (London) 321, 663 (1986). D. G. A. L. Aarts, M. Schmidt, and H. N. W. Lekkerkerker, Science 304, 847 (2004). E. H. A. de Hoog and H. N. W. Lekkerkerker, J. Phys. Chem. B 103, 5274 (1999). H. Lekkerkerker, V. Villeneuve, J. Folter, M. Schmidt, Y. Hennequin, D. Bonn, J. Indekeu, and D. Aarts, Eur. Phys. J. B 64, 341 (2008). S. A. Setu, I. Zacharoudiou, G. J. Davies, D. Bartolo, S. Moulinet, A. A. Louis, J. M. Yeomans, and D. G. A. L. Aarts, Soft Matter 9, 10 599 (2013). J. A. Miranda and M. Widom, Physica (Amsterdam) 120D, 315 (1998). E. Alvarez-Lacalle, J. Ortin, and J. Casademunt, Phys. Fluids 16, 908 (2004). E. O. Dias and J. A. Miranda, Phys. Rev. E 88, 013016 (2013).
†
[1] [2]
[3] [4] [5] [6] [7] [8] [9]
[10] [11] [12] [13] [14] [15] [16] [17] [18] [19]
[20]
[21] [22] [23]
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[24] See Supplemental Material at http://link.aps.org/ supplemental/10.1103/PhysRevLett.112.128303, which includes Refs. [25–33] for the explicit calculation of the mode nA , the fits to the flow curves of Fig. 3, and the derivation of the square gradient constant κ. [25] M. Cloitre, R. Borrega, F. Monti, and L. Leibler, Phys. Rev. Lett. 90, 068303 (2003). [26] B. M. Erwin, M. Cloitre, M. Gauthier, and D. Vlassopoulos, Soft Matter 6, 2825 (2010). [27] J. R. Seth, L. Mohan, C. Locatelli-Champagne, M. Cloitre, and R. T. Bonnecaze, Nat. Mater. 10, 838 (2011). [28] N. P. Balsara and E. B. Nauman, J. Polymer Sci. Polymer Phys. Ed. 26, 1077 (1988). [29] K. Ohno, Condens. Matter Phys. 5, 15 (2002). [30] J. W. Cahn and J. E. Hilliard, J. Chem. Phys. 28, 258 (1958). [31] L. P. McMaster, Copolymers, Polyblends and Composites (ACS, Washington, DC, 1975). [32] W. Greiner, L. Neise, and H. Stöcker, Thermodynamics and Statistical Mechanics (Springer-Verlag, Berlin, 1995). [33] M. P. Solf and A. Vilgis, J. Phys. I (France) 6, 1451 (1996). [34] E. Koos, J. Johannsmeier, L. Schwebler, and N. Willenbacher, Soft Matter 8, 6620 (2012). [35] H. Senff and W. Richtering, J. Chem. Phys. 111, 1705 (1999). [36] D. A. Sessoms, I. Bischofberger, L. Cipelletti, and V. Trappe, Phil. Trans. R. Soc. A 367, 5013 (2009). [37] In the worst case, ΦC ¼ 0.2, the longest injection time is 0.3 s, to be compared to tc ¼ 0.6 s, the time it takes a particle to diffuse over its own diameter. [38] J. E. Sader, D. Y. C. Chan, and B. D. Hughes, Phys. Rev. E 49, 420 (1994). [39] L. Kondic, M. J. Shelley, and P. Palffy-Muhoray, Phys. Rev. Lett. 80, 1433 (1998). [40] N. Maleki-Jirsaraei, A. Lindner, S. Rouhani, and D. Bonn, J. Phys. Condens. Matter 17, S1219 (2005). [41] A. Lindner, D. Bonn, and J. Meunier, Phys. Fluids 12, 256 (2000). [42] A. Fernandez-Nieves, H. Wyss, J. Mattsson, and D. A. Weitz, Microgel Suspensions, Fundamentals and Applications (Wiley-VCH Verlag GmbH & Co., KGaA, Weinheim, 2011). [43] T. Hino and J. M. Prausnitz, J. Appl. Polym. Sci. 62, 1635 (1996). [44] J. Wu, G. Huang, and Z. Hu, Macromolecules 36, 440 (2003).
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Off-Equilibrium Surface Tension in Colloidal Suspensions 1
Domenico Truzzolillo,1,2,* Serge Mora,1,2,† Christelle Dupas,1,2 and Luca Cipelletti1,2
Université Montpellier 2, Laboratoire Charles Coulomb UMR 5221, F-34095 Montpellier, France 2 CNRS, Laboratoire Charles Coulomb UMR 5221, F-34095 Montpellier, France (Received 12 December 2013; revised manuscript received 16 February 2014; published 24 March 2014) We study the fingering instability of the interface between two miscible fluids, a colloidal suspension and its own solvent. The temporal evolution of the interface in a Hele-Shaw cell is found to be governed by the competition between the nonlinear viscosity of the suspension and an off-equilibrium, effective surface tension Γe . By studying suspensions in a wide range of volume fractions, ΦC , we show that Γe ∼ Φ2C , in agreement with Korteweg’s theory for miscible fluids. The surface tension exhibits an anomalous increase with particle size, which we account for using entropy arguments. DOI: 10.1103/PhysRevLett.112.128303
PACS numbers: 82.70.Dd, 68.05.-n, 83.80.Hj
The surface tension between two fluids quantifies the energetic costs of creating a new interface [1,2]. At equilibrium, surface tension may only exist between immiscible fluids: when two miscible fluids are brought in contact, the initial concentration gradient across the interface rapidly relaxes via diffusion and the system reaches an equilibrated, uniform state. On time scales shorter than that of interface relaxation, however, there are capillary forces at the interface that mimic an effective surface tension, as it was already recognized by Korteweg in 1901 [3]. Similarly to the theory for immiscible fluids, Korteweg’s theory relates the effective surface tension Γe to the gradient of composition across the interface: 2 dφ κ Γe ¼ κ dz ≃ Δφ2 ; δ −∞ dz Z
∞
(1)
where z is the coordinate orthogonal to the interface, κ the Korteweg constant, and φ the concentration of one of the two species. The last approximation holds for a linear concentration profile that increases by Δφ across an interface of thickness δ. Subsequent work by Davis [4] and Joseph [5] has generalized Korteweg’s ideas by suggesting that interfacial stresses may arise whenever gradients of an arbitrary fluid property exist at the interface between miscible fluids, e.g., density or temperature. The existence of a transient effective surface tension has been demonstrated in light scattering experiments probing capillary waves at the interface between miscible fluids [6,7]. Korteweg stresses have also been invoked to explain the shape of drops and bubbles, both under the effect of gravity [5] and in spinning drop measurements [8–10], the onset of a Marangoni-like instability leading to the cellular convective mixing of miscible fluids [11], and the shape of the meniscus between molten silicates of different composition [12]. In spite of the possible relevance of Γe in many situations, including jetting, bubbles and drops formation, coalescence and break-up, plumes and 0031-9007=14=112(12)=128303(4)
convection, precipitation and deposition, experiments that quantitatively probe Korteweg’s theory remain scarce: very few data are available for Γe [6,7,9,10], and large discrepancies between experimental values and those estimated from Eq. (1) have been reported [8]. An increase of Γe with Δφ was reported in spinning drop experiments on water-glycerin [8] and polymer [10] systems, but large deviations with respect to the quadratic scaling of Eq. (1) were observed. The very existence of an off-equilibrium surface tension is debated. Numerical simulations [13] of the fingering instability arising when a less viscous fluid is pushed through a miscible, more viscous one in a HeleShaw cell highlight the role of Γe in stabilizing the interface. By contrast, in earlier works the observed patterns were explained without including the contribution of the surface tension [14], or by explicitly assuming Γe ¼ 0 [15,16]. Colloidal suspensions may be regarded as ideal benchmark systems to investigate surface tension effects because their interfaces may be probed in great detail, down to the particle level [17]. Previous work has focused on the equilibrium interface between phase-separated colloidal fluids [17–20]; however, colloidal suspensions are also excellent candidates for investigating off-equilibrium surface tension. Indeed, diffusion is much slower in colloids as compared to atomic systems, leaving a wider temporal window for probing the transient interface between miscible fluids. Here, we report Hele-Shaw experiments on the fingering instability observed at the interface between two miscible fluids, a colloidal suspension, and its own solvent. We show that the evolution of the interface pattern is governed by both the nonlinear viscosity of the suspension and an effective surface tension, which we measure as a function of the volume fraction of the suspension. Our results confirm Korteweg’s law, Eq. (1). We furthermore show that, for our microgel particles, Γe is governed by the entropy associated with the internal degrees of freedom of the particles, leading to a previously unreported growth of Γe with particle size.
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FIG. 1 (color online). (a) Finger function K exp vs γ_ I in HeleShaw experiments where water is injected in silicon oil. The solid line is the theoretical function K th [right-hand side of Eq. (4)], with no adjustable parameters. Inset: K exp vs K th , as obtained from the analysis proposed in the text (solid circles) or using previous approaches based on the fastest-growing Fourier mode of the instability (open circles). (b)–(e) Water-oil interface (red line), as observed right after the onset of fingering, for various injection shear rates (in s−1 ) as indicated by the labels.
The experiments are performed in a Hele-Shaw cell consisting of two square glass plates of side L ¼ 25 mm, separated by spacers fixing the gap at b ¼ 0.5 mm. The cell is filled with the fluid to be studied, with viscosity η2. A less viscous fluid is injected through a hole of radius r0 ¼ 0.5 mm in the center of the top plate. For all experiments, we use water died with 0.5% w=w of methylene blue as the less viscous fluid, with viscosity η1 ¼ 1.011 mPa s. The _ is controlled via a syringe injected volume per unit time, V, pump. Temperature is fixed at T ¼ 293 0.1 K by means of a Peltier element, with a circular hole of radius 8.5 mm for optical observation. A fast CMOS camera run at 100 to 3000 frames s−1 is used to image the sample during injection. Typical images of the interface between the two fluids are shown in Figs. 1 and 2, where the distinctive instabilities that develop when η1 < η2 are clearly visible. In the framework of linear evolution theory, such an instability is conveniently described by decomposing the interface profile in Fourier modes, the mode of order n being associated with a pattern with n lobes, or fingers. For two Newtonian fluids, the order nf of the mode with the fastest growth rate is given by [21] 0.5 1 4rr0 γ_ I ðη2 − η1 Þ nf ¼ pffiffiffi þ1 ; bΓ 3
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FIG. 2 (color online). Finger function K exp vs γ_ I measured when injecting water in a microgel suspension. Data are labeled by ΦC. The lines are guides for the eye. Small panels: watersuspension interfaces for ΦC ¼ 0.92 and various γ_ I (top row, as indicated by the labels, in s−1 ), or at fixed γ_ I ¼ 570 s−1 and various ΦC (right column). 0.5 _ where r ¼ ½r20 þ Vt=ðπbÞ is the time-dependent radius of the unperturbed interface, Γ the interfacial tension 2 −1 _ between the two fluids and γ_ I ¼ 3Vð2πr the shear 0b Þ rate at the injection hole. This expression is often used to describe the number of fingers experimentally observed at the onset of the instability [22]. However, in experiments the observed number of fingers should be regarded as being related to the mode with maximum amplitude, rather than to the fastest growing one, because in radial Hele-Shaw flow the growth rate is time dependent, as also pointed out very recently [23]. We thus modify the standard linear evolution theory to calculate nA , the order of the experimentally accessible mode with the maximum amplitude, finding [24] 0.5 α 4rr0 γ_ I ðη2 − η1 Þ nA ¼ αnf ¼ pffiffiffi þ1 ; (3) bΓ 3 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where α ¼ −Wð−3e−3 Þ ≃ 0.422, with WðxÞ satisfying x ¼ WðxÞeWðxÞ . Although Eq. (3) is formally derived in the limit nA ≫ 1, we check numerically that it holds to a very good approximation already for nA ≥ 2 [24]. We test the validity of Eq. (3) on two Newtonian fluids for which all the relevant parameters are known: dyed water and silicon oil (η2 ¼ 12.5 Pa s, Γ ¼ 39.8 mN m−1 [34]). Figures 1(b)–1(e) show typical interface patterns observed at various injection rates. We determine nA by counting the number of fingers of the destabilized interface [24]. To compare the experiments to the theory, it is convenient to recast Eq. (3) in the form
(2)
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K exp ðnA ; rÞ ¼ K th ;
(4)
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where the experimental “finger function” K exp is defined by 1 3n2A −1 ; (5) K exp ðnA ; rÞ ¼ r α2 while its theoretical value depends only on the rheological and interfacial properties of the fluids, the cell geometry, and the imposed shear rate: 4r ðη − η1 Þ K th ¼ γ_ I 0 2 : bΓ
(6)
Figure 1(a) shows K exp ðnA ; rÞ vs γ_ I for the water-silicon oil system. The experimental points (symbols) are in excellent agreement with K th obtained from Eq. (6) using the fluids and cell parameters (line). We emphasize that such a quantitative agreement would not hold if K exp was calculated by interpreting the observed number of fingers as the fastest-growing Fourier mode of the destabilized interface, i.e., if nA was replaced by nf ¼ nA =α in Eqs. (4),(5) [inset of Fig. 1(a)]. Having demonstrated that our experiments allow the flow and interfacial parameters to be quantitatively determined, we use the same setup to investigate the offequilibrium interfacial tension between a colloidal suspension and its solvent. We study aqueous suspensions of poly-N-isopropylacrylamide microgel particles [35]. At T ¼ 293 K the particles have hydrodynamic radius Rh ¼ 165 nm, as measured by dynamic light scattering. For the same synthesis, the radius of gyration has been determined to be Rg ≃ 0.5Rh [36]. We perform experiments for several particle concentrations, which we express as the effective volume fraction, ΦC , of the microgels. Experimentally, the polymer mass concentration, c (w=w), is known from the synthesis. For all concentrations, we define the effective volume fraction as ΦC ¼ kc, where k is determined from the viscosity of the suspension in the dilute limit. We find k ¼ 20.1 by matching the c-dependent zero-shear viscosity of the suspension to Einstein’s formula, η ¼ η0 ð1 þ 2.5kcÞ, where η is the viscosity of the suspension and η0 that of the solvent. Viscosity measurements are performed in the range 0 < ΦC ≤ 0.02. Our Hele-Shaw experiments cover volume fractions ranging from ΦC ¼ 0.2, corresponding to diluted, hard spherelike suspensions, up to ΦC ¼ 1.2, where particles are squeezed due to steric constraints and the suspension is fully jammed. For a given ΦC , we perform experiments at various γ_ I , always keeping the injection rate high enough for diffusion-driven mixing between the injected solvent and the suspension to be negligible [37]. Figure 2 shows K exp as a function of the injection shear rate for all the microgel suspensions. A qualitative change is observed when ΦC increases: at low volume fraction K exp ∼ γ_ I , while for jammed suspensions K exp grows sublinearly with γ_ I at high shear rate and tends to a plateau
for γ_ I → 0. This behavior is strongly reminiscent of the shape of the flow curve, σð_γ Þ, in colloidal suspensions, where σ ¼ η_γ is the shear stress when imposing a shear rate γ_ . This suggests that K exp is proportional to the shear stress contrast, i.e., that Eq. (4) may be generalized by K exp ¼
4r0 γ_ I η2 ð_γ r Þ − η1 ; b Γe
(7)
where η2 ð_γ r Þ is the shear-rate dependent viscosity of the suspension, γ_ r ¼ ð4r0 γ_ I =rÞ the shear rate at the position r of the interface (assuming Poiseuille flow), and Γe the (ΦC -dependent) effective surface tension between the suspension and its solvent. In writing Eq. (7) one implicitly assumes that the same kind of patterns at the onset of the instability are observed for our shear-thinning concentrated microgel suspensions as for Newtonian fluids, i.e., that the wavelength of the perturbation at its onset is not drastically changed by the non-Newtonian features of the suspension. Numerical work on the Saffman-Taylor instability in a radial Hele-Shaw geometry supports this scenario [38,39] by showing that the non-Newtonian character of the fluids does not change qualitatively the instability, but just accelerates ( delays) its onset for shear-thinning (shearthickening) fluids. The choice of Eq. (7) is also supported by previous works [40,41] on the Hele-Shaw instability between immiscible non-Newtonian fluids in a rectangular geometry, where the dynamics of the fingers was described by a generalized Darcy law where the Newtonian viscosity was replaced by the shear rate-dependent viscosity. In order to test Eq. (7), we measure the flow curves of the microgel suspensions. Figure 3 shows σð_γ Þ obtained via conventional rheology. The required shear-dependent viscosity is obtained from η2 ¼ σ fit γ_ −1 , where σ fit is a fit to the measured flow curve (lines in Fig. 3). The fits allow the viscosity to be estimated by extrapolation in the whole range of shear rates relevant to the Hele-Shaw experiments. Standard models for Newtonian and non-Newtonian fluids are used for the fits [25,26,36], as detailed in [24]. We find that for all ΦC the reduced finger function, K exp ≡ K exp b½4r0 ðη2 ð_γ r Þ − η1 Þ−1 , is proportional to γ_ I ,
FIG. 3 (color online). Flow curves for the microgel suspensions (same symbols as in Fig. 2). The lines are fits as described in [24].
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polymer mass density, one has φ ¼ 3.6 × 10−2 ΦC . The Korteweg constant has been calculated in [28] for inhomogeneous mixtures of a solvent and ideal-chain polymers. We extend this calculation to cross-linked polymers [24], in the limit φ ≪ 1 relevant to our microgels, finding κ¼
FIG. 4 (color online). Scaled finger function K exp Γe b× ½4r0 ðη2 ð_γ r Þ − η1 Þ−1 for the microgel suspensions whose raw data are shown in Fig. 2. All data collapse on K exp Γe ¼ γ_ I (line), thus confirming Eq. (7).
as predicted by Eq. (7), and we determine the proportionality coefficient Γ−1 by linear fitting. Figure 4 shows e K exp Γe vs the injection shear rate. When using this reduced variable, all the data previously shown in Fig. 2 fall onto a straight line spanning more than three decades, thereby validating Eq. (7). We test Korteweg’s prediction, Eq. (1), in Fig. 5. For our system, the particle volume fraction in the injected phase is zero, so that Eq. (1) reduces to Γe ∼ Φ2C , where we have assumed a linear variation of the concentration profile across an interface of thickness δ. Figure 5 shows that Korteweg’s law holds over a wide range of concentrations, corresponding to a variation of Γe of more than one decade. We go a step further and model our experiments at a microscopic level by calculating κ. To this end, we identify φ in Eq. (1) with the volume fraction of the polymer, rather than that of the particles, since the microgels are highly swollen by the solvent. Using literature values for the
FIG. 5 (color online). Main plot: effective interfacial tension Γe between the microgel suspensions and their solvent, as a function of colloid (polymer) volume fraction [top (bottom), axis]. The lines are quadratic fits to the data for microgels with various Rh , as shown by the legend. The conversion factor between Φc and φ varies slightly with Rh : the scale on the Φc axis is exact only for Rh ¼ 165 nm. Inset: square root of the reduced surface tension Γe =Rh vs φ.
RTR2g ½χ þ 3; 6V w
(8)
with R the gas constant, V w ¼ 18 × 10−6 m3 mol−1 the water molar volume and χ the Flory-Huggins parameter. Reported values of χ for poly-N-isopropylacrylamide microgels in water range from 0.25 to 0.5 [42–44], yielding 5.1 × 10−7 N ≤ κ ≤ 5.49 × 10−7 N. By fitting the experimental Γe vs φ we get κ=δ ¼ 1.40 N m−1 and, hence, 364 nm ≤ δ ≤ 392 nm. The interface thickness thus calculated is in very good agreement with the average distance between particles, which in the range of ΦC studied here varies from 340 to 460 nm. This confirms that in our experiments diffusion at the interface is negligible and validates quantitatively our analysis. To test the robustness of Eqs. (1) and (8), we perform additional experiments on microgels with the same composition but smaller size, Rh ¼ 70 and 100 nm, respectively. From Eqs. (1) and (8) and using δ ∼ Rg ∼ Rh , one expects that data for microgels with different Rh should collapse onto a master curve when normalizing Γe by Rh. The inset of Fig. 5 shows that this is indeed the case. We emphasize that the scaling Γe ∼ Rh is in stark contrast with the usual scaling of the interfacial tension between molecular or colloidal phases, where Γ ∼ a−2 , with a the particle size [2,18]. This highlights the different origin of the surface tension in our experiments where the entropic contribution due to the internal degrees of freedom of the polymeric particles dominates. In conclusion, we have investigated the pattern formation resulting from the injection of the solvent in a colloidal suspension, a model system for investigating the nonequilibrium, effective surface tension between miscible fluids. The interface instability can be rationalized by a remarkably simple expression, depending separately on the rheological properties of the suspension and on the effective, off-equilibrium suspension-solvent surface tension. Our results confirm Korteweg’s law and raise challenging questions on the behavior of κ, and thus Γe , as a function of inter-particle and particle-solvent interactions, as well as particle size and shape. More generally, our findings provide an experimental and theoretical framework for exploring nonequilibrium surface tension effects, a topic relevant in many problems, ranging from material processing to fundamental fluid dynamics. This work has been supported by ANR under Contract No. ANR-2010-BLAN-0402-1. The authors are grateful to E. Bouchaud, O. Dauchot, C. Ligoure, L. Ramos, and V. Trappe for useful discussions.
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domenico.truzzolillo@univ‑montp2.fr Present address: Laboratoire de Mécanique et de Génie Civil—UMR 5508 CNRS et Université de Montpellier 2 Place E.Bataillon, F-34095 Montpellier cedex 5, France. J. E. van der Waals, Z. Phys. Chem. 13, 657 (1894). P.-G. de Gennes, F. Brochard-Wyart, and D. Queré, Capillarity and Wetting Phenomena: Drops, Bubbles, Pearls, Waves (Springer-Verlag, Berlin, 2004). D. Korteweg, Arch. Neerlandaises Sci. Exactes Naturelles 6, 1 (1901). H. T. Davis, Numerical Simulation and Oil Recovery (Springer-Verlag, Berlin, 1988). D. D. Joseph, Eur. J. Mech. B, Fluids 9, 595 (1990). S. E. May and J. V. Maher, Phys. Rev. Lett. 67, 2013 (1991). P. Cicuta, A. Vailati, and M. Giglio, Appl. Opt. 40, 4140 (2001). P. Petitjeans, C.R. Acad. Sci. Paris 322, 673 (1996). J. A. Pojman, C. Whitmore, M. L. Turco Liveri, R. Lombardo, J. Marszalek, R. Parker, and B. Zoltowski, Langmuir 22, 2569 (2006). B. Zoltowski, Y. Chekanov, Masere, J. A. Pojman, and V. Volpert, Langmuir 23, 5522 (2007). P. Garik, J. Hetrick, B. Orr, D. Barkey, and E. Ben-Jacob, Phys. Rev. Lett. 66, 1606 (1991). J. E. Mungall, Phys. Rev. Lett. 73, 288 (1994). C.-Y. Chen, C.-W. Huang, H. Gadelha, and J. A. Miranda, Phys. Rev. E 78, 016306 (2008). L. Paterson, Phys. Fluids 28, 26 (1985). J. Nittman, G. Daccord, and H. Stanley, Nature (London) 314, 141 (1985). J. Nittman and H. Stanley, Nature (London) 321, 663 (1986). D. G. A. L. Aarts, M. Schmidt, and H. N. W. Lekkerkerker, Science 304, 847 (2004). E. H. A. de Hoog and H. N. W. Lekkerkerker, J. Phys. Chem. B 103, 5274 (1999). H. Lekkerkerker, V. Villeneuve, J. Folter, M. Schmidt, Y. Hennequin, D. Bonn, J. Indekeu, and D. Aarts, Eur. Phys. J. B 64, 341 (2008). S. A. Setu, I. Zacharoudiou, G. J. Davies, D. Bartolo, S. Moulinet, A. A. Louis, J. M. Yeomans, and D. G. A. L. Aarts, Soft Matter 9, 10 599 (2013). J. A. Miranda and M. Widom, Physica (Amsterdam) 120D, 315 (1998). E. Alvarez-Lacalle, J. Ortin, and J. Casademunt, Phys. Fluids 16, 908 (2004). E. O. Dias and J. A. Miranda, Phys. Rev. E 88, 013016 (2013).
†
[1] [2]
[3] [4] [5] [6] [7] [8] [9]
[10] [11] [12] [13] [14] [15] [16] [17] [18] [19]
[20]
[21] [22] [23]
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[24] See Supplemental Material at http://link.aps.org/ supplemental/10.1103/PhysRevLett.112.128303, which includes Refs. [25–33] for the explicit calculation of the mode nA , the fits to the flow curves of Fig. 3, and the derivation of the square gradient constant κ. [25] M. Cloitre, R. Borrega, F. Monti, and L. Leibler, Phys. Rev. Lett. 90, 068303 (2003). [26] B. M. Erwin, M. Cloitre, M. Gauthier, and D. Vlassopoulos, Soft Matter 6, 2825 (2010). [27] J. R. Seth, L. Mohan, C. Locatelli-Champagne, M. Cloitre, and R. T. Bonnecaze, Nat. Mater. 10, 838 (2011). [28] N. P. Balsara and E. B. Nauman, J. Polymer Sci. Polymer Phys. Ed. 26, 1077 (1988). [29] K. Ohno, Condens. Matter Phys. 5, 15 (2002). [30] J. W. Cahn and J. E. Hilliard, J. Chem. Phys. 28, 258 (1958). [31] L. P. McMaster, Copolymers, Polyblends and Composites (ACS, Washington, DC, 1975). [32] W. Greiner, L. Neise, and H. Stöcker, Thermodynamics and Statistical Mechanics (Springer-Verlag, Berlin, 1995). [33] M. P. Solf and A. Vilgis, J. Phys. I (France) 6, 1451 (1996). [34] E. Koos, J. Johannsmeier, L. Schwebler, and N. Willenbacher, Soft Matter 8, 6620 (2012). [35] H. Senff and W. Richtering, J. Chem. Phys. 111, 1705 (1999). [36] D. A. Sessoms, I. Bischofberger, L. Cipelletti, and V. Trappe, Phil. Trans. R. Soc. A 367, 5013 (2009). [37] In the worst case, ΦC ¼ 0.2, the longest injection time is 0.3 s, to be compared to tc ¼ 0.6 s, the time it takes a particle to diffuse over its own diameter. [38] J. E. Sader, D. Y. C. Chan, and B. D. Hughes, Phys. Rev. E 49, 420 (1994). [39] L. Kondic, M. J. Shelley, and P. Palffy-Muhoray, Phys. Rev. Lett. 80, 1433 (1998). [40] N. Maleki-Jirsaraei, A. Lindner, S. Rouhani, and D. Bonn, J. Phys. Condens. Matter 17, S1219 (2005). [41] A. Lindner, D. Bonn, and J. Meunier, Phys. Fluids 12, 256 (2000). [42] A. Fernandez-Nieves, H. Wyss, J. Mattsson, and D. A. Weitz, Microgel Suspensions, Fundamentals and Applications (Wiley-VCH Verlag GmbH & Co., KGaA, Weinheim, 2011). [43] T. Hino and J. M. Prausnitz, J. Appl. Polym. Sci. 62, 1635 (1996). [44] J. Wu, G. Huang, and Z. Hu, Macromolecules 36, 440 (2003).
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