Transition from a simple yield stress fluid to a thixotropi material A. Ragouilliaux 1,2, G. Ovarlez 1, N. Shahidzadeh-Bonn 1, Benjamin Herzhaft 2, T. Palermo 2, P. Coussot 1 1 Université Paris-Est, Laboratoire Navier (UMR CNRS-ENPC-LCPC) 2 Allée Kepler 7742 Champs sur Marne, Frane 2 IFP, 1-4 Av. du Bois Préau, 92852 Rueil Malmaison, Frane Abstrat: From MRI rheometry we show that a pure emulsion an be turned from a simple yield stress fluid to a thixotropi material by adding a small fration of olloidal partiles. The two fluids have the same behavior in the liquid regime but the loaded emulsion exhibits a ritial shear rate below whih no steady flows an be observed. For a stress below the yield stress, the pure emulsion abruptly stops flowing, whereas the visosity of the loaded emulsion ontinuously inreases in time, whih leads to an apparent flow stoppage. This phenomenon an be very well represented by a model assuming a progressive inrease of the number of droplet links via olloidal partiles. PACS: 82.7.-y, 83.6.Pq, 61.43.Hv I. Introdution Various systems suh as foams, emulsions or olloidal suspensions, are made of elements interating via soft interations, i.e. whih vary progressively with the distane between the elements [1-2]. Their jammed harater, namely their ability to undergo a solid-liquid transition when submitted to a suffiient stress, arouses the interest of physiists as it appears typial of a fourth state of matter with some analogy with glass behavior [3]. Their rheologial behavior is usually desribed with the help of simple yield stress models, whih may yield veloity profiles with unsheared regions when the stress distribution is heterogeneous. Reently some works showed that the flow of suh materials exhibits true shear-banding, i.e. two regions with signifiantly different shear rates in geometries in whih the shear stress is homogeneous [4]. Suh a behavior is not predited by usual yield stress models for whih the transition between the solid and the liquid phases is smooth. Two questions thus emerge: (i) how an a material beome a shear-banding material? and (ii) how does the transition towards shear-banding our in time for a given material? The first question was reently addressed by Béu et al. [5] who showed that a simple yielding
emulsion an exhibit shear-banding if it is made adhesive, i.e. the droplets develop some adhesive interations. Here we fous on question (ii), namely the time-dependeny of this shear-banding effet and its origin, as it is likely that the transition from the purely solid (starting from rest) or purely liquid (starting from an intense preshear) regimes towards shearbanding is a omplex phenomenon from whih useful information for our understanding of these materials ould be extrated. Remark that there is ertainly some interplay of these phenomena with the thixotropi harater of these fluids (time variations of the apparent visosity). For example it was shown that a lay-water system an exhibit very different behavior types, and in partiular shear-banding or even fratures, depending on the time it spent at rest before undergoing a stress [6]. Here we ompare the behavior of an emulsion either pure or loaded with olloidal partiles. In order to have a straightforward appreiation of the effetive behavior of these materials we observe their flow harateristis in time with the help of MR (Magneti Resonane) veloimetry. We show that the pure emulsion an be onsidered as a simple yield stress fluid with negligible thixotropy while the loaded emulsion is signifiantly thixotropi. Both fluids have the same behavior in the liquid regime and the thixotropy of the loaded emulsion is due to a progressive aggregation of droplets via olloidal partile links, whih tends to inrease the visosity of the material whih finally stops. As a result the apparent yield stress (below whih no steady flow an be obtained) of the loaded emulsion is larger than the pure emulsion. We finally show that during stoppage the shear rate exhibits a speifi evolution whih is onsistent with the preditions of a model assuming a progressive aggregation of droplets in time. II. Materials and proedures We prepared the pure emulsion by progressively adding the water in an oil-surfatant solution (Sorbitan monooleate, 2%) under high shear. The size distribution of water droplets was almost uniform around a diameter of 1 μm ± 2%. The water droplet onentration was 7%, a onentration larger than the maximum paking fration of uniform spheres, whih means that the droplets were probably slightly deformed. However the droplet deformation due to flow an be negleted during all the flows onsidered here sine, due to the very small droplet size, the interfaial stress is muh larger than the stress due to flow. As a onsequene the average shape of droplets an be onsidered as onstant during all our tests. The loaded
emulsion was prepared by mixing the surfatant and the olloidal partiles (slightly flexible, large aspet ratio lay partiles (Bentone 38, Elementis Speialties ompany) with a mean diameter: 1μm and a thikness.1μm ) at a solid volume fration of 3% in the oil, then progressively adding the water in this suspension under high shear. Remark that the lay-oil suspension (with the same solid volume fration) alone is a simple liquid (whih shows no yield stress) of low visosity (5.1-3 Pa.s) whereas the lay-emulsion mixture is a muh more visous pasty material even at low water frations (e.g. at a volume fration of 1% its apparent visosity is of the order of 5.1-1 Pa.s). As a onsequene it is likely that the lay partiles tend to form links between neighbouring droplets, whih thus form aggregates or even a ontinuous network throughout the material in some ases. The exat nature of these links is not yet understood. We arried out experiments with a wide-gap Couette rheometer (inner ylinder radius: 4m; outer ylinder radius: 6m; height of sheared fluid: 11m) inserted in a MRI set-up. We imposed the rotation veloity of the inner ylinder and measured the torque and the veloity profile ( V (r) in whih r is the distane from the entral axis) inside the sample at suessive times by magneti resonane veloimetry. For a given rotation veloity the torque ( M ) was onstant in time even if for the loaded emulsion the veloity profile signifiantly evolved (see below). The priniple of the magneti resonane veloimetry tehnique used was desribed elsewhere [7]. The typial time needed to get one veloity profile was 15s. The unertainty on the loal veloity measurements was typially 5 5.1 m/s. Slight wall slip effets possibly ourred but did not affet the transient measurements as they remained onstant for a given torque and did not affet our rheologial interpretations based on measurements inside the sample. We also arried out NMR density tests with the help of a new tehnique [8] for separating the oil and the water signals, whih make it possible to detet within 1% some heterogeneity of the water fration resulting from migration effets. No migration was observed in the pure emulsion. With the layey emulsion no suh migration effet was observed during the first 2min. of flow but a signifiant migration developed beyond 3min. of flow. In the following we present only the data orresponding to the first stage (with negligible migration). We also arried out onventional rheometrial tests (dynami tests and reep tests) with a Malvern rheometer equipped with a thin-gap Couette geometry (inner radius: r 1 =17.5mm; outer ylinder radius: r 2 =18.5mm; height: h = 45mm ) with rough surfaes (roughness:.18mm).
III. Experimental results A. Solid regime From reep tests we find that both materials exhibit a solid and a liquid regimes: at low stress the indued deformation tends to saturates (solid regime), whereas at high stress they tend to flow at a onstant rate (liquid regime). This is the typial behavior observed for pasty materials [9]. Let us first ompare the material properties in their solid regime from dynami tests (frequeny: 1Hz, strain amplitude: 1%). For the pure emulsion the elasti modulus ( G ') slightly inreases with time (see inset of Fig.1) (i.e. of 7% after 3s), whih means that, as a first approximate, it negligibly ages in its solid regime. Thus the possible evolution of the droplet onfiguration in time only indues a slight inrease of the network strength. In ontrast for the loaded emulsion G ' signifiantly inreases in time (i.e. of 7% after 3s). Thus, as time goes on, more and more lay links form between droplets, whih strengthens the network of droplet ontats. Shear Stress (Pa) 6 4 2 1.6 G' G' 4 s 1.4 1.2 1 1 2 1 3 t (s) γ 1-2 1-1 1 1 1 Shear rate (1/s) Figure 1: Steady-state shear stress vs shear rate for the pure (irles) and layey (squares) emulsions as dedued from loal measurements within the wide-gap rheometer under n different torque values. The solid line is a Hershel-Bulkley model ( τ = τ e + Kγ ) fitted to.5 data (with τ e = 7.5Pa, K = 6.5Pa.s and n =. 5 ) for the pure emulsion. Inset: Elasti modulus (saled by the value reahed at 4s) of the pure (irles) and layey (squares) emulsions as a funtion of time.
B. Liquid regime Let us now onsider the behavior in the liquid regime. In that aim we preshear the material at a large veloity and leave it at rest few seonds, then we impose a given rotation veloity and observe the veloity profile inside the gap. During a short initial period (say less than few seonds) the rotation veloity varies as a result of inertia and solid-liquid transition effets. Afterwards, for the pure emulsion the veloity profile is onstant. This means that this material exhibits negligible thixotropy in the liquid regime: the speifi (average) droplet onfiguration assoiated with a given flow rate is reahed almost immediately. In ontrast, with the loaded emulsion the veloity profile signifiantly evolves in time: it progressively moves towards the entral axis (f. Fig.2) and finally reahes a steady shape. During this proess the outer region apparently progressively stops flowing but there still might remain some slow flow at veloities smaller than the unertainty on these measurements. In a Couette geometry the shear stress distribution (τ as a funtion of r ) in the material is well known from the momentum equation and we have ( r M hr 2 τ ) = 2π, i.e. at a given torque level the stress dereases with the distane from the axis. Our observations onerning the veloity profiles under a onstant torque imply that in the regions of low stresses (near the outer ylinder) the material tends to stop flowing while in the regions of large stresses (near the inner ylinder) it tends to reah a steady-state flow. This effet is reminisent of the visosity bifuration effet observed with various material types [1]. However here, in ontrast with previous studies, we have a straightforward information onerning the effetive (loal) shear rate in time. Indeed we an ompute the loal shear rate (γ ) for any stress (assoiated with a given distane r ) at any time: we have = r ( V r) r γ. Looking now at the evolutions in time of the shear rate under different stress values (i.e. at different distanes) we effetively observe the visosity bifuration effet (see Fig.3): for τ > τ the shear rate tends to a onstant value, for τ < τ the shear rate progressively tends to zero, with a harateristi time inreasing with τ. As a onsequene there exists a ritial shear rate ( γ ), assoiated with τ, below whih no steady-state flows an be obtained, the material tends to -1 stop. From our data we get γ 3s.
V (m/s) 1-2 1-3 1-4 5 r (m) Figure 2: Veloity profiles at different times in the wide-gap Couette geometry as measured by MR veloimetry for the layey emulsion for a rotation veloity of 15rpm: (from right to left) 3, 6, 9, 12, 15, 18, 21, 24, 27, 33, 39, 45, 54, 66, 78, 9s. The veloity values below the unertainty ( 5.1 5 m.s -1 ) are not represented, whih explains why there is no data beyond some ritial distane for eah profile. Shear rate (1/s) 1 1 1 1-1 1 2 Time (s) 1 3 Figure 3: Effetive shear rate as a funtion of time for different loal stresses as determined from the MR veloity profiles for the layey emulsion for a rotation veloity of 15rpm: (from top to bottom) 34.2, 31.7, 3.1, 29.4, 28.7, 28, 27.3, 26.7, 26.1, 25.5, 24.9, 24.3, 23.8, 22.8, 22.3, 21.8, 21.4, 2.9, 2.5, 2.1, 19.7, 18.9 Pa.
IV Disussion A. Data analysis Let us now ompare the onstitutive equation of these materials in the liquid regime in steady state as it an be dedued from a set of experiments under different torques from MRI data, by omputing the loal shear rate and shear stress for eah value of r as above desribed. The flow urve (steady state τ vs γ ) of the pure emulsion is that of a simple yield stress fluid, i.e. the shear stress tends to a plateau at low shear rates below whih no steady flows are obtained and a Hershel-Bulkley model very well represents the flow urve (see Fig.1). The loaded emulsion exhibits a behavior similar to that of the pure emulsion for shear rates larger than say -1 1 s. This means that the lay links no longer play any role and suggest that there ould be all broken. The striking differene between the two materials onerns the minimum stress level allowing steady state flow: the yield stress of the layey emulsion is signifiantly larger than that for the pure emulsion and just beyond this yield stress the fluid flows at a shear rate larger than γ. Forgetting the short plateau of the flow urve for the layey emulsion we get the following rough piture: both materials have a similar behavior in the liquid regime but the flow urve of the layey emulsion is trunated at a ritial shear stress larger than the yield stress of the pure emulsion, and no stable flows an be obtained at a shear rate below the ritial value assoiated with this yield stress. For suh materials shear-banding will our in shear flows at apparent shear rates smaller than γ. This behavior is reminisent of the trend observed by Beu et al. [5] with an adhesive emulsion, but here we get a preise piture of the different fluid behavior types at a loal sale and we show that the differene finds its origin in some time-dependent effet. With regards to our sheme of the mirostruture it is natural to onsider that this behavior is the result of a dynami effet: a progressive transition from flow to stoppage, whih is likely due to the progressive formation of larger aggregates. Thus as a first approximation we an onsider that all the lay links are broken during a rapid shear, so that the lay fration plays a negligible role (in agreement with our observation of the similar behavior of the two materials in the liquid regime), but below a ritial stress the number of links inreases signifiantly, whih slows down the flow so that the fration of links further inreases and so on until full stoppage.
Previous studies of the thixotropi properties in the liquid regime generally relied on marosopi data under poorly ontrolled onditions. Here, from MR veloimetry we have straightforward information onerning the loal visosity variations in time (see Fig.3). We an remark that the shear rate vs time urves for different stress values below τ look similar in a logarithmi sale: they start with a plateau at short time then rapidly derease after some time, whih dereases as the stress dereases. This similarity definitely appears when plotting the shear rate vs the time saled by appropriate fators, respetively γ ( ) and θ (τ ) (see Fig.4). This is reminisent of an effet reently observed [11]: for various suspensions the elasti modulus vs time urves after the liquid-solid transition under different stress fall along a master urve when the elasti modulus and the time are saled by appropriate funtions of the stress. Here we get the following general result: for a stress imposed below the ritial stress the shear rate evolves as γ γ f ( t θ ) γ τ τ ) ( τ τ ) γ and τ =, with [( ] 1 θ k ( τ τ ). This similarity of the behavior strongly suggests that the strutural state of the material is a funtion of ξ = t θ only, whih thus an be onsidered as desribing the physial age, i.e. the state of struture, of the material during suh a test. It is remarkable that the evolutions of this parameter only depend on the stress. This ontrasts with usual models (e.g. [6]) in that field whih assume that the struture parameter essentially depends on the shear rate. From the above equations we dedue the apparent behavior of the material for all ξ : ( η f ( ξ )) γ ( ) τ = τ (1) + ξ where η τ τ = ( ) γ. In this equation the first term, i.e. τ, is independent of the strutural state and thus an be onsidered as refleting the yield stress of the layey emulsion when there is no droplet-lay links. This yield stress takes its origin in the jammed droplets network with lay possibly affeting the interfaial tension between oil and water, whih explains that it may differ from the yield stress of the pure emulsion. The seond term (the visous term ) is assoiated with both the visous effets and the droplet-lay links in the emulsion. It is worth noting that the behavior desribed by equation (1) does not orrespond to a Bingham model. Indeed the fator of the shear rate ( η f ( ) ), i.e. the so-alled plasti visosity ξ whih is onstant in the Bingham model, in our model depends on the strutural state ξ and thus depends on the shear stress.
B. Model The above analysis suggests that one may see the layey emulsion as a suspension of linked elements in an interstitial homogeneous yield stress fluid. We will make two simplifying assumptions: (i) the stress in this suspension may be expressed as the sum of a yielding term stritly due to the jamming-unjamming proess during flow and a simple visous term with an apparent visosity (η ), resulting from the additional visous dissipation due to flow and funtion of the atual onentration of aggregates; (ii) the presene of aggregates negligibly affets the yielding part. Suh a piture is qualitatively in perfet agreement with the onlusions of our data interpretation (see above). In this ontext the visous term may be found from the visosity of a Newtonian suspension in a liquid of visosity η. We assume that at a given time the droplet-lay links form more or less large (rigid) networks whih trap some volume of the interstitial liquid. Various models exist for the visosity of rigid spherial inlusions [12] but it seems more relevant to use the expression found from energy dissipation onsiderations in the ontext of purely visous suspensions of aggregative partiles [13]: ( ) 2 η η φ φ φ = (1 agg ) 1 ( agg m ). Here φ agg is a parameter whih represents the volume fration of rigid inlusions formed by interstitial liquid trapped in lay-droplet networks. The speifi maximum paking fration φ m used in the Mills model [13] is equal to.57. A different hoie for this value would slightly hange the value of the fitting parameter in the following analysis but not the qualitative results. This visous term an be assoiated with η f ( ) sine both terms desribe the apparent visosity of the suspension in a simple ξ liquid of visosity η, whih provides a orrespondene between φ agg and ξ. For example a kinetis relation of the form 3/ 5 φagg ξ makes it possible to get a good fit of the above equation to the master urve (see Fig.4). Remark that the aggregate size tends towards infinity at stoppage, so that the above theory no longer holds and the orresponding stress ontributes to inrease the yield stress of the material, whih explains that the apparent yield stress of the layey emulsion is larger than pure emulsion.
1 γ γ 1-1 (s -1 ) 12 1 θ 1-2 6 γ 2 25 τ (Pa) 1 2 1 3 t θ Figure 4: Shear rate (data of Figure 3 for τ 27.3 Pa ) saled by a fator γ ( τ ) as a funtion of the time saled by a fator θ (τ ) for the layey emulsion. The ontinuous line is the Mills 3/ 5 model fitted to data (with φ agg =.432ξ ). The inset shows the values of these two parameters as a funtion of the stress. The dotted lines are the empirial models: 1 γ [( τ τ ) ( τ τ )] γ and θ k ( τ τ ) with τ = 17.9Pa, -1 k = 1.75s.Pa, and τ = 27.5Pa. The above equation provides the relation between the effetive advanement of the aggregation (via φ agg ) and the time and the stress applied. It is remarkable that the stress is solely involved in this relationship whereas in general for suh physial phenomena the harateristi time is proportional to 1 γ. One finds a possible beginning of explanation of suh an effet in the usual desription of simple liquids [14], glassy [15] or jammed systems [16] in whih the elements are assumed to be in a potential well with an energy barrier of depth Δ E. We an assume that suh a frame applies for the lay partiles involved in laydroplet links. The probability of esape from this potential well due to thermal agitation is exp ΔE k B T per attempt, in whih k B is the Boltzmann onstant and T the temperature. In this ontext the influene of stress is to derease the energy barrier by a fator τ d in whih d is a harateristi distane of the system, and thus to foster the fluid flow. In order to have the possibility to form a lay bridge two droplets must be neighbours. Over a time of flow Δ t the
number of ourrene of droplet neighbouring is proportional to γ but the time during whih two droplets an be onsidered as neighbours is inversely proportional to γ, so that the rate of flow has no influene on the number of links (this result is valid only beause we negleted possible link breakages). To sum up we find that the rate of variation of the number of links inreases with the stress and is independent of the shear rate, in agreement with our experimental results. However a quantitative predition of the effet of stress on θ requires a more omplete theoretial approah. V. Conlusion We have shown that the transition from a non-thixotropi to a thixotropi emulsion ours by the disappearane of steady flows below a ritial stress assoiated with a ritial shear rate. Here this proess is well explained by the inrease of droplet links via lay partiles. These effets might be general for jammed systems for whih the aging is due to reversible partile aggregation. Referenes [1] P. Coussot, Rheometry of pastes, suspensions and granular materials (Wiley, New York, 25) [2] V. Trappe et al., Nature, 411, 772 (21) [3] A.J. Liu, and S.R. Nagel, Nature, 396, 21 (1998) [4] F. Pignon, A. Magnin, and J.M. Piau, J. Rheol., 4, 573-587 (1996); P. Coussot et al. Phys. Rev. Lett., 88, 21831 (22) [5] L. Béu, S. Manneville, and A. Colin, Phys. Rev. Lett., 96, 13832 (26) [6] P. Coussot et al., Phys. Fluids, 17, 1174 (25) [7] J.S. Raynaud et al., J. Rheol., 46, 79-732 (22) [8] G. Ovarlez, S. Rodts, P. Coussot, J. Goyon, A. Colin, submitted to J. Colloid and Interfae Siene (27) [9] P. Coussot et al., J. Rheol., 5, 975-994 (26) [1] F. Da Cruz et al., Phys. Rev. E, 66, 5135 (22)
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