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Thermophoresis and Thermoelectricity in Surfactant Solutions Daniele Vigolo, Stefano Buzzaccaro, and Roberto Piazza* Dipartimento CMIC, Politecnico di Milano, I-20131 Milano, Italy Received December 4, 2009. Revised Manuscript Received January 14, 2010 In electrolyte solutions, the differential migration of the ionic species induced by the presence of a thermal gradient leads to the buildup of a steady-state electric field. Similarly to what happens for the Seebeck effect in solids, the sample behaves therefore as a thermocell. Here, we provide clear evidence for the presence of thermoelectric fields in liquids by detecting and quantifying their strong effects on colloid thermophoresis. Specifically, by contrasting the effects of the addition of NaCl or NaOH on the Soret effect of micellar solutions of sodium dodecyl sulfate, we show that the presence of highly thermally responsive ions such as OH- may easily lead to the reversal of particle motion. Our experimental results can be quantitatively explained by a simple model that takes into account interparticle interactions and explicitly includes the micellar electrophoretic transport driven by such a thermally generated electric field. The chance of carefully controlling colloid thermophoresis by tuning the solvent electrolyte composition may prove to be very useful in microfluidic applications and field-flow fractionation methods.
Introduction The development of novel and promising methods for external or internal flow control of disperse systems in microfluidic geometry calls for particle manipulation techniques profiting from selective physicochemical properties of a colloidal suspension. Besides electrophoretic and dielectrophoretic methods, which are commonly exploited for tuning particle motion, thermophoresis, namely, particle drift induced by thermal gradients,1 has recently been proposed and tested as a valid alternative.2 When a colloidal suspension is placed in a temperature gradient, the dispersed particles display, on top of Brownian motion, a steady drift velocity given by νT = -DTrT, where the thermophoretic mobility DT is the analog of the thermal diffusion coefficient in simple mixtures.3 Then, depending on the sign of DT, the particles focus either at the cold or the hot side, leading to a steady-state concentration gradient given, for low particle concentration c, by rc = -cSTrT, where ST = DT/D is called the Soret coefficient. With this definition, particles with ST > 0 move to the cold side and are usually dubbed “thermophobic” but colloids with ST < 0 are conversely called “thermophilic”. In the past few years, noticeable experimental effort has been made to investigate thermophoretic effects in a wide class of complex fluids, ranging from model colloidal suspensions to polymers, surfactant solutions, and biological fluids, whereas extensive theoretical investigations have been devoted to unravelling the basic mechanisms underlying thermophoresis. (For a recent review, see ref 1.) The most distinctive feature of these results is the strong dependence of ST on the nature of the investigated system and on solution parameters such as temperature, ionic strength, and pH, rendering in principle thermophoresis much more selective than other particle-driving mechanisms. Yet, many aspects of thermophoresis deserve further investigation, in particular, for what concerns aqueous suspensions of charged particles. A strong sensitivity of the Soret coefficient with *To whom correspondence should be addressed. E-mail: roberto.piazza@ polimi.it. (1) Piazza, R.; Parola, A. J. Phys.: Condens. Matt. 2008, 20, 153102. (2) Geelhoed, P. F.; Lindken, R.; Westerweel, J. Chem. Eng. Res. Des. 2006, 84, 370. (3) Tyrrell, J. V. Diffusion and Heat Flow in Liquids; Buttherworths: London, 1961.
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respec to ionic strength, not only at finite particle concentration but also in the dilute limit, has been singled out ever since seminal studies were performed on charged micellar solutions.4 Although noticeable colligative effects may actually be expected because electrostatic repulsions hinder the buildup of concentration gradients, the single-particle behavior witnesses an intimate relation between thermophoresis and particle/solvent interfacial properties, which is much subtler and more system-specific than for other driven processes such as electrophoresis. In principle, for charged colloids, added electrolytes are themselves subjected to thermal diffusion, which might lead to a nonuniform charge background for particle motion. However, in most cases the Soret coefficient of simple salts is quite small,3 and the latter effects have so far been neglected. Simple acids and bases constitute, however, a noticeable exception because Hþ and OH- are far more responsive to thermal gradients than are most cations and anions.5 Recently, clear indications of ion-specific effects on the Soret coefficient of charged colloids have been presented in a seminal paper by Putnam and Cahill6 and tentatively explained on the basis of a clever model made long ago by Guthrie et al.7 for ternary electrolyte solutions. Considering the latter suggestion, W€urger8,9 later showed that the presence of strongly mobile ionic species indeed yields peculiar thermoelectric effects akin to the Seebeck effect in solids, which can strongly affect colloid thermophoresis. A qualitative explanation of the latter effects can be given by considering, for instance, an aqueous suspension of negatively charged thermophobic particles to which a simple base such as NaOH is added. The stronger tendency of the OH- ions to accumulating toward the cold side, compared to the Naþ cations, generates a steady-state electric field counteracting particle drift.3 It is important to point out that, albeit the latter thermoelectric field remains finite, local charge separation must vanish in the macroscopic limit (the cell behaves basically as a loss capacitor). The former effect can therefore be equivalently (4) (5) (6) (7) (8) (9)
Piazza, R.; Guarino, A. Phys. Rev. Lett. 2002, 88, 208302. Agar, J. N.; Mou, C. Y.; Long Lin, J. J. Phys. Chem. 1989, 93, 2079. Putnam, S. A.; Cahill, D. G. Langmuir 2005, 21, 5317. Guthrie, G.; Wilson, J. N.; Shomaker, V. J. Phys. Chem. 1949, 17, 310. W€urger, A. Phys. Rev. Lett. 2008, 101, 108302. W€urger, A. Langmuir 2009, 25, 6696.
Published on Web 02/10/2010
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envisaged as resulting from the competition between the negative particles and the OH- ions in accumulating toward the cold side, taken together with the requirement of local charge neutrality. The main aim of the present work is therefore to investigate this issue by studying the effects of added NaOH on the thermophoresis of micellar solutions of the anionic surfactant sodium dodecyl sulfate (SDS) and comparing them to the behavior of the same samples in the presence of NaCl. We shall show that thermoelectric effects may be easily detected and can be strong enough to reverse the thermophoretic behavior of SDS micelles fully. Our experimental findings can be reasonably accounted for by a simple model that combines the suggestion made in ref 8 with the already mentioned strong contribution of electrostatic interparticle interactions on the Soret effect. This article is organized as follows. In the next section, we shall briefly review the basic theoretical argument justifying the occurrence of electric fields in ionic solutions subjected to a thermal gradient, and we shall also briefly comment on the concept of charge renormalization for charged colloids, which will be extensively used to analyze our data. After introducing the materials and experimental methods that we have used, we shall first try to give a consistent explanation of the concentration dependence of SDS thermophoresis in the absence of added salt. In the following section, we shall then compare the full concentration dependence of ST for SDS solutions in the presence of NaCl and NaOH, explicitly singling out the strong thermoelectric effects induced by the latter electrolyte. More specifically, we shall show that the difference in the SDS Soret coefficient between the two electrolytes can be envisaged to be a pure electrophoretic contribution whose concentration dependence is directly related to the (equilibrium) osmotic compressibility of the solution. The predictions of the simple model that we have developed is then favorably compared to the results obtained in mixed electrolytes. After a brief discussion of the temperature dependence of SDS thermophoresis in NaCl and NaOH, we shall finally comment on our experimental findings, stressing in particular that the occurrence of thermoelectric effects opens up the opportunity to fine tune thermally induced particle drifts in microfluidic applications.
Theoretical Background Thermoelectricity in Electrolyte Solutions. According to the general consideration made in the former section, in the presence of thermal diffusion the mass flow of a single solute at low concentration c can be written as J ¼ -Drc -cDT rT ¼ -Dðrc -cST rTÞ When considering a solution of a simple electrolyte, however, we are actually dealing with a ternary mixture, so one may wonder whether the response of anions and cations to the thermal gradient can be independently and consistently defined. Experimentally, the only quantity that can be directly probed is actually the Soret coefficient of the whole electrolyte because the requirement of local charge neutrality (at least over spatial scales larger than the Debye-H€uckel screening λDH) implies that the concentration profiles for the two ionic species coincide. Yet, it is physically meaningful to assume that the two ionic species are subjected to distinct responses to rT (envisaged as an effective external field) and local charge neutrality is ensured by an electric field EB¥ that couples the charge motion and reaches a finite value EB¥ at steady state. In other words, similarly to what happens in solids for the Seebeck effect, an electrolyte solution should behave as a thermocell. It is again important to stress that no charge separation is present in the bulk solution so that EB is due only to Langmuir 2010, 26(11), 7792–7801
charge accumulation in a thin layer of thickness d = λDH closed to the cell boundaries. In other words, this is basically a polarization effect. By introducing the reduced coefficients R( = T(ST*)(/2, where ( represents the intrinsic Soret coefficients of cations and (S*) T anions in the absence of charge coupling, the individual ionic fluxes for monovalent electrolytes can therefore be written as rT qi E - c( J ( ¼ -D ( rc ( þ 2c ( R ( T kB T
ð1Þ
where we have used the Einstein expression relating the ionic electric mobility μ( to the diffusion coefficients D( as μ( = (kBT)-1eD( and e is the unit charge. By imposing the local charge neutrality condition at steady state in the form of r(cþ - c-) = 0, eq 1 easily yields for the measured Soret coefficient of both ionic species ST( ¼ -
1 rc ( 1 þ ðST Þþ þ ðST Þ þ R Þ ¼ ðR ¼ c ( rT T 2
ð2Þ
which is just the average of the intrinsic Soret coefficients of the two species, and the steady-state electric field is simply given by E¥ ¼ kB ðRþ- R - ÞrT
ð3Þ
The former description is clearly oversimplified for at least two reasons. First, the coefficients R(, embodying the coupling of a single ion with the thermal field, are assumed to have fixed values that do not depend on the total electrolyte concentration or therefore the ionic strength I. This is surely incorrect when dealing with colloidal macroions and is highly suspect to be so even for small ions. Second, this is essentially a single-ion model where collective effects are totally neglected, which may hold only in the infinite dilute limit. Indeed, at finite concentration, the osmotic pressure associated with the buildup of the electrolyte concentration profile counteracts the driving thermal force, reducing therefore, as we shall see, the Soret coefficient. Experimentally, at variance with what happens for the Seebeck effect in solids, the direct detection of E¥ is moreover quite problematic because it requires the use of reversible electrodes and a careful analysis of electrochemical effects. Notice, however, that all that is required to evaluate E¥ is differences in the intrinsic Soret coefficients, which can be obtained by comparing the Soret coefficients of different salts with a common cation or anion.5 On the contrary, the evaluation of absolute values for R(, which are directly related to the entropy transferred by an ion from a reservoir at temperature T to another at T þ dT,3 requires a detailed microscopic model. The problem that we are considering is analogous to the gravity settling of charged colloids at low ionic strength where, for entropic reasons, the small counterions refuse to accumulate on the bottom of the cell together with the heavy particles. As originally suggested in ref 10 and directly observed by Philipse and co-workers,11 this may induce a polarization field that effectively “lifts” the colloids, yielding a concentration profile that is far more expanded than expected. Recently, van Roij12 has shown that this field can to all extents be envisaged as a consequence of the Donnan effect induced by the gravity segregation of (10) Piazza, R.; Bellini, T.; Degiorgio, V. Phys. Rev. Lett. 1993, 71, 4267. (11) Rasa, M.; Philipse, A. P. Nature 2004, 429, 857. (12) van Roij, R. J. Phys.: Condens. Matter 2003, 15, S3569.
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the particles. Once again, local charge is preserved everywhere except within thin layers close to the bottom of the cell and the top of the sedimentation profile. A recent, more detailed theoretical analysis made using a primitive model (i.e., a multicomponent approach taking into account particles and counterions as separate species) has shown this result to be consistent with a standard effective one-component model, provided that the osmotic equilibrium condition is carefully applied. A consistent model of thermoelectric effects should then proceed along the same line, namely, by solving the PoissonBoltzmann equation for each ionic species in the presence of effective thermal forces fT( = kBT(ST*)(rT. However, the theoretical understanding of thermophoretic forces in charged systems is not complete enough to develop such a rigorous analysis. In what follows, therefore, we shall try to explain our results using a simplified model accounting for changes in the thermophoresis of charged particles by the effect of the electric field associated with the coexistent Soret motion of a background electrolyte. Charge Renormalization in Colloidal Suspensions. We regard it useful to give a brief account of the concept of charge renormalization for highly charged colloids, and we shall profit from what follows. Indeed, although there is by now a clear theoretical understanding of the latter (see, for instance, the very competent review by Belloni13), the way that effective charges are used to fit experimental data is sometimes still rather arbitrary. Moreover, whereas the usefulness of charge renormalization to account, at least partially, for structural properties is unquestionable, the situation for transport properties is more debated. The idea of attributing an effective, or renormalized charge, to a strongly charged particle of radius a stems from the observation that whenever the “bare” particle charge Z satisfies the condition Z . a/l B, where a is the particle radius, l lB = e2/(4πεkBT) is the Bijerrum length, and ε is the solvent dielectric constant so that l lB = 0.7 nm in water, the full Poisson-Boltzmann equation predicts that a large number of counterions will recondense in a thin layer close to the particle surface. More quantitatively, the total charge contained in a spherical shell of radius r > a first shows a very rapid increase followed by a clear inflection, which defines the thickness of the recondensed counterion layer. Such a sharp distinction between condensed counterions bound to the particle surface with an energy much larger than kBT and free, weakly bound counterions holds provided that κa, where κ = 1/λDH is the reciprocal of the Debye-H€uckel screening length, is not very large compared to κa j 5-10. Although this condition is not easy to satisfy experimentally for suspensions of large colloids, unless carefull deionizing protocols are exploited, for the micellar solution we shall examine 0.7 j κa j 2.5. Within this limit, the suspension can therefore be mapped onto an effective system of “dressed” particles with a much lower value of the surface charge, which is on the order of a few times the ratio a/l lB, and therefore of the electric potential φs at the external boundary of the condensed layer. This severe reduction in φs (typically, as we shall see, to a few kBT/e) suggests an alternative definition of the effective charge that, although less physical, is extremely useful in simplifying model interparticle interactions between charged colloidal particles. In this scheme, the renormalized charged is fixed to the value yielding a solution of the linear Debye-H€uckel (DH) equation φDH(r) (which best matches the tail of the corresponding solution for the full (nonlinearized) Poisson-Boltzmann (PB) equation φPB(r)). In other words, we (13) Belloni, L. Colloids Surf., A 1998, 140, 227.
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Figure 1. Effective charges defined according to eqs 4 (Z*) and 5 (Zeff), scaled to the ratio l lB/a (adapted from ref 13). The experimental range that we investigated in this works lies between the two vertical dotted lines. The inset shows the ratio fu/eZeff, where μ is given the numerical values calculated in ref 14.
seek a value Z* such that φPB ðrÞ f φDH ðrÞ ¼ rf¥
eZ/ e -Kðr -aÞ 4πεð1 þ KaÞ r
ð4Þ
Notice that the factor (1 þ κa) in the denominator of φDH, which stems from the boundary condition of the DH equation for a finite-sized macroion, de facto yields a reduction of the particle charge with respect to Z, even for low values of the bare surface charge that do not lead to any counterion condensation (so that Zeff = Z). Therefore, we may also absorb this factor into the definition of the effective charge by writing, more simply, φDH ðrÞ ¼
eZeff e -Kðr -aÞ 4πε r
ð5Þ
Although choosing between these two definition of the effective charge is formally irrelevant (obviously taking into account that in the absence of charge renormalization one should substitute Zeff = Z(1 þ κa)-1), Zeff is much more useful than Z* from a practical point of view. Figure 1, where we plot the numerical values of Z* and Zeff obtained from numerical solutions of the PB equation,13 indeed shows that Zeff is to good approximation constant for κa J 0.8 - 1, with a value Zeff = C (l lB/a), where C = 4 to 5. Conversely, Z* grows linearly in κa, according to its rigorous asymptotic behavior for κa . 1. Because our data are taken within this range, in what follows we shall therefore make use of the effective charge Zeff defined through eq 5. Using this approach, the electrostatic interaction energy between two colloids of effective charge Zeff can tentatively be taken in the simple Yukawa form -2Kax V el ðxÞ 2 lB e ¼ Zeff kB T 2a 1 þ x
ð6Þ
where x = (r - 2a)/2a is the distance between the particle surfaces. Care should be taken, however, because this simple form of Vel(x), although valid over a wide range of κa, is correct only for sufficiently large values of x. In practice, this means that the effective charge obtained by fitting structural data using the effective interparticle potential in eq 6 will be in general slightly larger than the value predicted from Figure 1. Langmuir 2010, 26(11), 7792–7801
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The concept of effective charge is very helpful for getting some general features of the structure factor and phase behavior of charged colloids, although it may not be sufficient to explain more complex features such as the metal-like elasticity of highly charged colloidal crystals15 or the exact location of phase-transition lines in highly charged spherical systems,16 which require more sophisticated approaches such as the modified PB “jellium” model recently proposed by Trizac and Levin.17 To what extent charge renormalization concepts are also useful for colloid transport properties is, however, more controversial. Although accounting for the value of many transport coefficients, such as the suspension conductivity σ or the electrophoretic mobility μ, generally requires the introduction of particle charge values that are much lower than Z, no a priori reason exists for the latter to be related to the effective charge Zeff obtained from structural data. However, the classical concept of the Stern layer in electrophoresis entails the existence of a strongly bound counterion layer whereas the introduction of the ζ potential bears a strong resemblance to the reduction of the effective electrostatic potential due to counterion condensation. Generally, the results of electrophoretic measurements are fitted using standard electrokinetic models based on a full hydrodynamic approach with no approximation of the PB equation, which yields rather complex results of very difficult physical interpretation. Recently, however, several attempts to use charge renormalization concepts explicitly in transport phenomena have been performed. Among the latter, a very interesting theoretical study by Lobaskin et al.14 is worth mentioning, where, by introducing effective charges, experimental18 and numerical data for systems with different particle charges and sizes could be mapped on a single master curve. For the present purposes, it is sufficient to comment on the dependence on κa, in the dilute particle limit, of the values calculated in ref 14 for the electrophoretic mobility. The inset of Figure 1, where we plot the ratio fμ/eZeff, where f = 6πηa is the friction coefficient and η is the solvent viscosity, shows that this quantity varies very little in the range of 0.1 e κa e 2.5 so that the electrophoretic mobility is approximately (within 25%) given by eZeff μ= 6πηa
ð7Þ
notwithstanding that in the same range both μ and Zeff change by almost a factor of 3. This very useful result shows that, to first approximation, μ can be assumed to be approximately constant (κ-independent), simply corresponding to the dilute H€uckel limit (κa , 1) of μ for a particle of charge Zeff.
Experimental Section System. Molecular-biology-grade sodium dodecyl sulfate was obtained from Sigma-Aldrich and used without further purification. The physical properties of SDS, a simple anionic surfactant with a molecular weight of Msds = 288 g 3 mol-1, in aqueous solutions have been extensively investigated in the past. Beyond a critical micellar concentration (cmc) that depends on the ionic strength I (decreasing from 8.3 mM in pure water to 0.9 mM for I = 200 mM), SDS forms globular micelles with a fairly constant hydrodynamic radius of a = 2.5 nm. Their aggregation number (14) Lobaskin, V.; D€unweg, B.; Medebach, M.; Palberg, T.; Holm, C. J. Phys. Chem. 2007, 98, 176105. (15) Reinke, D.; Stark, H.; von Gr€unberg, H.; Schofield, A. B.; Maret, G.; Gasser, U. Phys. Rev. Lett. 2007, 98, 0380301. (16) Wette, P.; Sch€ope, H. J. Prog. Colloid Polym. Sci. 2006, 133, 88. (17) Trizac, E.; Levin, Y. Phys. Rev. E 2004, 69, 031403. (18) Medebach, M.; Shapran, L.; Palberg, T. Colloids Surf. 2007, 56, 210.
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N and therefore the structural charge Z is around 90-100, increasing by no more than 10% up to an ionic strength of I = 0.4 M, beyond which the micellar morphology becomes elongated and N grows consistently. Because the value of Z is much higher than a/l lB = 3.5, SDS micelles should be regarded as macroions in the fully developed counterion condensation regime. Figure 1 indeed yields a theoretical value for the effective charge of Zeff = 16-18 and therefore a degree of dissociation β = Zeff/Z = 0.2, which is in reasonable agreement with both structural19-21 and transport22,23 data. From eq 6, the interaction energy at contact is expected to be on the order of Zeff2l lB/a = 40kBT; therefore, strong intermicellar interactions may be expected even at low surfactant volume fraction. As shown in a seminal study by Corti and Degiorgio,24 interaction effects can be profitably studied by static (SLS) and dynamic (DLS) light scattering. Because the latter data were mostly obtained at higher values of the ionic strength than those used in this work and because they were interpreted without making use of charge renormalization concepts, which were not widespread at that time, we shall discuss interaction effects on the basis of SLS data obtained in our laboratory, which are nonetheless in good agreement, within the common investigation range, both with the results presented in ref 24 and with recent data by Thevenot et al.25 Thermophoretic Measurements. All measurements were performed using a beam-deflection (BD) method that consists of monitoring by a position-sensitive detector the deflection that a laser beam suffers by propagating though a thin cell containing the sample, to which a moderate thermal gradient (on the order of 1 K/mm) is imposed. After a fast initial deflection Δϑth, because of the temperature dependence of the refractive index, the beam undergoes further bending because of the progressive buildup of the Soret-induced concentration gradient, leading to a further steady-state deflection Δϑs, which can be easily distinguished from Δϑth because it takes place on the much longer timescale of particle diffusion. The Soret coefficient can then be simply evaluated from the ratio Δϑs/Δϑth at steady state as ST ¼ -
1Dn=DT ðΔϑÞs c Dn=Dc ðΔϑÞth
ð8Þ
where ∂n/∂T and ∂n/∂c are the temperature and concentration dependence of the refractive index, respectively. In addition, the time dependence of Δϑs(t), which reaches its steady-state value exponentially with a time constant of τ = h2/(π2D), where h is the interplate spacing, allows the evaluation of the particle diffusion coefficient and therefore the thermophoretic mobility DT = STD. BD is therefore an intrinsically differential method that requires only a comparison of the laser deflection for the suspension, with a known refractive index increment, to the deflection because of the T dependence of the refractive index, which is observed for a calibration solvent at a fixed temperature. The sign of the Soret coefficient can be directly obtained by comparing the direction of Δϑth with the fast, purely thermal contribution Δϑth that, in the usual case when the solvent density decreases with T, is directed toward the colder plate.26 Because signal fluctuations are predominantly due to the effect of temperature fluctuations on the density and therefore on the refractive index of the solvent, the BD method is particularly suitable for solvents such as water with a low thermal expansivity. To reduce the duration of BD (19) Sheu, E. Y.; Chen, S. J. Phys. Chem. 1988, 92, 4466. (20) Bucci, S.; Fagotti, C.; Degiorgio, V.; Piazza, R. Langmuir 1991, 7, 824. (21) Bergstr€om, M.; Pedersen, J. S. Phys. Chem. Chem. Phys. 1999, 1, 4437. (22) Stigter, D. In Physical Chemistry; van Olphen, H., Mysels, K. J., Eds.; Theorex: La Jolla, 1975; Chapter 12. (23) Shanks, P. C.; Franses, E. I. Phys. Rev. Lett. 1992, 96, 1794. (24) Corti, M.; Degiorgio, V. J. Phys. Chem. 1981, 85, 711. (25) Thevenot, C.; Grassl, B.; Bastiat, G.; Binana, W. Colloids Surf., A 2005, 252, 105. (26) Piazza, R. Philos. Mag. 2003, 83, 2067.
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cm = 8.3 mM of SDS in monomeric form. Since the first quantitative study of interparticle effects on thermophoresis,4 several experiments have pointed out the close relationship between the collective Soret coefficient of a colloid and the osmotic compressibility κT = (1/c)(∂Π/∂c)-1, where Π is the osmotic pressure of the suspension, which is also found to hold for polymer solutions (e.g., ref 28) and can be qualitatively understood by simply observing that, in the presence of a given external driving force acting on a single particle, the amplitude of the overall concentration gradient that builds up is actually controlled by κT. This basic consideration can be given quantitative grounds by examining the condition of osmotic equilibrium in presence of an effective thermal force fT = kBTST0rT, where ST0 is the singleparticle Soret coefficient in the dilute limit.29 Expressing for convenience the concentration in terms of particle number density F (in our case, micelles per unit volume), the osmotic balance is then given by Figure 2. Soret coefficient of SDS solutions in the absence of added salt as a function of the total surfactant concentration c. The inset shows the concentration dependence of the thermophoretic mobility DT. measurements, we have used a custom-designed microcell with a separation between the hot and cold plates of h = 400 μm over a 20 mm optical path. Beam deflection is measured by a calibrated position-sensitive detector with an accuracy of (5 μm placed along the beam path sufficiently far from the cell. A detailed description of the BD setup can be found in ref 26.
Results and Discussion SDS in the Absence of Added Salt. To discuss thermoelectric effects properly, it is first useful to investigate the thermophoretic behavior of SDS solutions in the absence of any added salt. The concentration dependence of the Soret coefficient for SDS solutions, shown in Figure 2, first shows that ST increases by a factor of 4 within a narrow range around the value csds = 8.3 mM, corresponding to the cmc of the pure surfactant as determined by light scattering.25 Conversely, the value for DT, derived from the transient of the BD signals and shown in the inset, is essentially continuous across the cmc, meaning that the increase in ST just mirrors an equivalent decrease in D due to micellization. This result shows that, in step with the negligible dependence of DT on particle size in colloidal latex suspensions,27 the thermophoretic mobility barely depends on the surfactant aggregation state. In passing, we point out that measurements of the Soret coefficient are therefore a very careful method of detecting the cmc of surfactant solutions. In particular, for charged micelles at low salt concentration, there is a distinct advantage with respect to standard light-scattering methods where, as we shall soon discuss, the strong electrostatic interparticle repulsion yield a very low scattering intensity. Moreover, the BD signals are barely influenced by the presence of dust, which is conversely well known to be a major plaguing effect in SLS and DLS measurements from poorly scattering samples. Beyond the cmc, ST shows a strong decrease with SDS concentration to the point that, for I J 100 mM, ST becomes comparable or even smaller than for single SDS monomers. To inquire about the latter strong concentration dependence, it is useful to consider the contribution to ST due to only micelles, which is simply given by Smic T = STcsds/c, where c = csds - cm is the concentration of associated surfactant, obtained by subtracting from the total SDS concentration csds the fixed amount (27) Vigolo, D.; Brambilla, G.; Piazza, R. Phys. Rev. E 2007, 75, 040401(R).
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rΠðF, TÞ ¼ -FfT ¼ -kB TST0 FrT
ð9Þ
By expanding rΠ ¼
DΠ DΠ rF þ rT DF DT
and assuming rT along z, it is easy to obtain ST ¼ -
-1 1 dF DΠ 1 DΠ ¼ kB TST0 þ F dT DF F DT
ð10Þ
showing that the collective Soret coefficient is indeed expected to be proportional to the osmotic compressibility. Notice that in the dilute limit, using Π = FkBT, one gets ST ¼ ST0 þ
1 T
ð11Þ
The very small, system-independent second term (=3 10-3 K-1) corresponds to the Soret coefficient that has been classically obtained by considering Brownian motion in a temperature gradient without introducing any specific, interfacially driven thermal force fT30 or, more simply, by incorrectly assuming that the osmotic pressure at steady state (which is not equilibrium) is constant along the cell.31 At finite concentration, this additional colligative term can be relevant only to temperature-dependent interparticle forces (such that Π/FkBT strongly depends on T). Because electrostatic forces do not belong to this class, they will not be considered further in what follows. In the situation that we are considering, moreover, ST and ST0 mic should be regarded as contributions Smic T and ST0 to the Soret coefficient of micellized SDS. Keeping this in mind, to keep the notation simpler from now on we shall drop the suffix “mic” and assume therefore that the Soret coefficient of SDS micelles can be written as -1 DΠ ST ¼ kB TST0 ðIÞ DF
ð12Þ
where the single-particle Soret coefficient ST0 depends of course on I. (28) (29) (30) (31)
Enge, W.; K€ohler, W. Phys. Chem. Chem. Phys. 2004, 6, 2373. Parola, A.; Piazza, R. Eur. Phys. J. E 2004, 15, 255. Chapman, S. Proc. R. Soc. London 1928, A119, 34. van’t Hoff, J. H. Z. Phys. Chem 1887, 1, 481.
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Figure 3. (Inset) Comparison of the micellar contribution Smic T to the Soret coefficient of salt-free SDS solutions with the ratio of the scattering intensity to the concentration of SDS micelles, Iscatt/c. (Main panel) The same plot with Smic T rescaled to the square of the Debye-H€ uckel screening length.
In view of eq 12, it is then useful to compare our results with static light scattering data. In the inset of Figure 3, we therefore contrast ST with the ratio Iscatt/c obtained from SLS measurements of salt-free SDS solutions, a quantity that is proportional to (∂Π/∂F)-1. The plot shows that the gross features of the trend for ST can indeed be ascribed to the concentration dependence of the inverse osmotic compressibility. Yet, a moderate difference between the two sets of data can clearly be spotted. The latter can be accounted for by observing that, in the absence of added salt, the ionic strength of the suspension depends on the total surfactant concentration and is given by I = cm þ (β/2)c, where β = 0.2. Former experiments4 and theoretical calculations29 show that ST0 is proportional to the square of the screening length (plus a small constant term of nonelectrostatic origin that can be safely neglected at very low ionic strength) and therefore to I-1. The main body of Figure 3 indeed shows that if ST is rescaled to λDH2 then the agreement with SLS data is satisfactory. Therefore, thermophoresis of SDS in the absence of added salt closely conforms to eq 12, provided that the dependence of ST0 on I is duly taken into account. In view of the former discussion of thermoelectric phenomena, however, we may why, to quantify the concentration behavior of the Soret coefficient of SDS micelles (which are strongly charged), there is no need to invoke any thermoelectric field. Why should macroion and simple electrolyte solutions not be treated on equal footing, introducing a distinct thermal response for micelles and counterions and coupling their transport via a thermoelectric field? The answer stems from a basic result of McMillan-Mayer dilute solution theory, stating that a multicomponent suspension can be exactly mapped onto an equivalent system made of a single component, the colloidal particles, provided that effective (solvent-mediated) interactions are introduced. (For a more modern, comprehensive approach, see ref 32.) In other words, the combination of equilibrium osmotic compressibility effects with a single-particle hydrodynamic model, yielding the ionic strength dependence of ST0, already and correctly embodies the presence of any hypothetical thermoelectric field. Once again, we stress that this result exactly parallels what is found for electric (32) Dijkstra, M.; van Roij, R.; Evans, R. J. Chem. Phys. 2000, 113, 4799.
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Figure 4. (A) Concentration dependence of the Soret coefficient for SDS at 25 C in the presence of 5 (b), 10 (2), 20 (1), 50 (O), and 100 (9) mM NaCl. The Soret contribution to the BD signal for a 1% SDS solution at 50 mM NaCl, plotted with the convention that Δϑs > 0 when ST > 0, is shown in the inset. (B) Corresponding data, using the same symbols, in the presence of NaOH. Inset B clearly emphasizes the thermophilic behavior of SDS micelles brought in by the addition of an equivalent amount of NaOH.
field effects in sedimentation, where the predictions of a primitive (multicomponent) and an effective single-component model coincide,11 at least provided that the approximation of pairwisescreened Coulomb interactions does not break down (which happens only at extremely low ionic strength).33 Conversely, a model where collective effects are exclusively attributed to a concentration-dependent thermoelectric field, at least using the simple model for the thermoelectric effect that we have formerly sketched, would never correctly yield the osmotic compressibility (and would certainly be totally unrelated to light-scattering measurements, where obviously no thermoelectric field is present). SDS Thermophoresis in NaCl and NaOH. We turn now to the main part of this work, where we consider the effect of the addition of electrolytes on SDS thermophoresis, in particular, when the electrolytes are expected to have a large response to thermal gradients. This task will be accomplished by comparing the different effects of the addition of NaOH with respect to NaCl, which has already been investigated in the past.4 We first checked by BD, using the same approach as for salt-free solutions, that the ionic strength dependence of the cmc is exactly the same, regardless of whether I is varied by the addition of NaCl or NaOH. Moreover, DLS and transient BD measurements show that, also for NaOH, R does not significantly change up to a concentration of at least 200 mM. As may be expected, SDS micellization properties in NaCl and NaOH are therefore very similar. The insets in Figure 4, displaying the Soret contribution Δϑs to the BD signal for a 1% SDS solution in the presence of 50 mM (33) Torres, A.; Cuctos, A.; Dijkstra, M.; van Roij, R. Phys. Rev. E 2008, 77, 031402.
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NaCl and NaOH, show that the effects of the two electrolytes on thermophoresis are strikingly different: whereas for NaCl solutions ST > 0, as for salt-free SDS, the addition of NaOH reverses the direction of micellar drift, yielding a thermophilic behavior of comparable magnitude. Moreover, whereas the full concentration dependence of ST for NaCl at different values of I, shown in panel A of Figure 4, qualitatively agrees with the prediction of a milder concentration dependence of ST at higher I, because of the weakening of intermicellar repulsions and therefore increased osmotic compressibility, the trend for sodium hydroxide is very different. Whereas at low NaOH concentration ST(c) seems to show just an overall downshift, for I = 50 and 100 mM the solution displays thermophilic behavior over the whole range and is marginally dependent on SDS concentration. This behavior suggests that a totally different driving mechanism, much less influenced by interparticle interactions, becomes dominant. We shall therefore inquire as to whether the presence of a thermoelectric field may account for this widely different concentration trend. Although there are few accurate experimental data available for the Soret coefficients of the individual ions, at least in the truly dilute limit theoretical calculations of the heat of transport based on a hydrodynamic approach5 allow us to estimate the differences in the R parameters for the two electrolytes as RNa - RCl = þ 0.6 and RNa - ROH = - 2.7. Therefore, we may not only expect thermoelectric effects to be far larger in NaOH than in NaCl but also that in NaOH solutions E¥ strongly counteracts the pure thermophoretic contribution, driving particles to the hot side, in agreement with our experimental findings. By analogy to what has been suggested for the salt-free situation, a rigorous treatment of the concentration dependence of thermophoresis is salt solution would involve solving the Poisson-Boltzmann equation for all of the components in the presence of thermal forces, possibly mapping the system onto an effective one-component model by taking explicitly into account the volume force stemming from the inhomogeneous charge distribution induced by the thermal field. However, we shall limit our analysis to a very simplified model where SDS in considered to be a single species but where interparticle interactions are fully taken into account via the osmotic compressibility whereas the effect of added salt is simply envisaged to be an external thermoelectric field, given by eq 3, that acts on a single SDS micelle treated as a macroion of charge Zeff modifying its thermophoretic motion. Therefore, taking into account eq 7, we shall assume that
DΠ -1 Zeff ðRNa - RCl Þ ðST ÞNaCl ¼ kB T ST0 þ DF T -1 DΠ Zeff ðRNa - ROH Þ ST0 þ ðST ÞNaOH ¼ kB T DF T
ð13Þ
In principle, for moderately concentrated SDS solutions at very low ionic strength, when the counterion density is not negligible compared to the concentration of the added electrolyte, one may expect the released counterions also to be included in the Naþ concentration, contributing to the thermoelectric field (which would then become concentration-dependent). Yet, we point out that concentration effects are already included in the behavior of the osmotic compressibility so that this correction may be regarded as rather suspect. Because of the approximate approach that we are taking, it is actually hard to state how the additional counterion can be properly taken into account. A simple way to dispel any doubt, however, is to trace out any effect of Naþ by just taking differences between the ST values in the two different electrolytes at the same ionic strength and SDS concentration. 7798 DOI: 10.1021/la904588s
We indeed have ΔST def ¼ ðST ÞNaCl - ðST ÞNaOH -1 DΠ ¼ kB Zeff ðROH - RCl Þ DF
ð14Þ
Equation 14 yields two distinctive predictions that can be experimentally tested: (1) No thermoelectric contribution should enter the concentration dependence of ΔST, which is simply related to the osmotic compressibility of the solution and can therefore be profitably compared to the results for κT obtained by equilibrium methods such as SLS measurements. (2) Conversely, the dilute limit of ΔST reflects only the additional thermoelectric effect brought in by the addition of salt, and ST0 is traced out. Therefore, in spite of the widely different behavior of SDS in NaOH compared to that in NaCl, all curves for different values of I should give the same intercept at zero concentration, at least provided that R values and the effective charge are weakly dependent on the ionic strength. Notice also that the value for RNa RCl corresponds to the true dilute limit, which as we already stated is hardly accessible in direct measurements of thermal diffusion in electrolyte solutions. In particular, for sufficiently low SDS concentration, the osmotic pressure can always be written in terms of a virial expansion at second order: Π ¼ FkB Tð1 þ k2 FÞ Introducing the particle volume fraction Φ = VpF, where Vp is the volume of a micelle, we therefore have DΠ ¼ kB Tð1 þ 2B2 ΦÞ DF where B2 = k2/Vp is a dimensionless second virial coefficient of the osmotic pressure. The reciprocal of ΔST is therefore expected to show, for moderately concentrated solutions, linear behavior with respect to the volume fraction of SDS micelles ðΔST Þ -1 ¼
T ð1 þ 2B2 ΦÞ Zeff ðROH -RCl Þ
ð15Þ
with a slope that should coincide with the value that is found from the dependence of the scattering intensity in Φ. The inset of Figure 5 shows that these two basic predictions are basically satisfied. The low-concentration trend of (ΔST)-1 is indeed linear, with intercepts that do not differ by more than (25%, and the values for B2 match quite well with those obtained by SLS for SDS in NaCl solutions, which are in good agreement with former measurements performed in a similar κa range.25 For a more quantitative comparison, let us first consider the values of the virial coefficients. Using eq 6, the electrostatic contribution to the osmotic second virial coefficient is given by Z 1 el ð1 - e -V ðrÞ=kB T Þ d3 r Bel2 ¼ 2Vp Z ¥ el ¼ 12 ð1þxÞ2 ð1 - e -V ðxÞ=kB T Þ dx ð16Þ 0
Although electrostatic effects are dominant in the κa range that we investigated, the additional contributions to B2 due to Langmuir 2010, 26(11), 7792–7801
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Figure 5. (Inset) Volume fraction dependence of (ΔST)-1 for SDS micellar solutions in the presence of 10 (solid dots), 20 (open dots), 50 (solid squares), and 100 (open squares) mM of added electrolyte. (Main panel) Comparison of the virial coefficients obtained from the linear fits (solid dots) to the analogous results obtained by static light scattering (squares). Triangles are the values for B2 obtained in ref 25 reported for the units that we use. The solid and dashed lines are numerical calculations from eq 16 for Zeff = 18 and 16, respectively.
excluded volume (hard sphere) and attractive dispersion forces should not be neglected. Although the former simply yields a contribution of BHS 2 = 4, correctly evaluating the latter would require fixing a value for the Hamaker constant. Yet, a simple way to account for these contributions is based on observing that both of them do not depend on ionic strength (at least provided that salt-specific effects are not present). Because we found, in full agreement with refs 24 and 25 that B2 vanishes for I = 0.45 M, it is then sufficient to subtract from Bel2 a constant nonelectrostatic ne el term Bne 2 , fixed so that B2 = (B2 )I=0.45 M, where the electrostatic value at I = 0.45 M is calculated using eq 16. The main body of Figure 5 shows that the B2 values calculated for Zeff = 16 and 18 are quite close to the experimental data, in good agreement with the theoretical predictions for the effective charge.34 Using Zeff = 18, the absolute values of the intercept, which fall between 14 and 20 K, yield a value of ROH - RCl = 0.8 - 1.2, which is more than a factor of 3 lower than the theoretical value stated by Agar and co-workers.5 By comparing our results for the virial coefficients with those obtained in former studies,4 however, it turns out that the thermoelectric effects in NaCl are probably much smaller than what is expected using RNa - RCl = þ0.6. In other words, as we shall better appreciate in the next section, thermoelectric effects in the presence of NaCl are quite weak. Therefore, the truly dilute limit of RCl might be sensibly higher than the calculated value. Yet, this would only partially account for this discrepancy. It is much more likely that small-ion repulsive interactions, which are totally neglected in our simplified approach, sensibly reduce the thermoelectric field with respect to the value estimated using our simplified model, where the latter interactions are fully neglected. Electrolyte Mixtures. The data in Figure 4 suggest that the thermophoretic behavior of SDS solutions can be carefully tuned from thermophobic to thermophilic by the addition of (34) The nonelectrostatic contributions are respectively given by Bne 2 = 5 and 5.5 and are therefore rather small, as expected, compared to most of the Bel2 values. Note also that this additive decoupling is of course approximately valid provided that the range for dispersion forces is much smaller than for electrostatic forces.
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Figure 6. Soret coefficient of SDS in mixed electrolyte solutions as a function of the molar fraction x of NaOH over the total electrolyte concentration, obtained at four different values of the total ionic strength, indicated in the upper legend. (Inset) Absolute value of the slope of the linear fits as a function of I. The open dots represent an approximate extrapolation to the value I = 0.42 M where SDS micelles behave as an ideal solute (B2 = 0).
electrolytes. This result is better appreciated by examining the SDS behavior at fixed surfactant concentration c = 1% in mixed electrolyte solutions, where the total ionic strength I is kept constant and the relative amount of NaOH is varied. Figure 6 shows that the trend in ST with a NaOH molar fraction of x = cNaOH/I, where cNaOH, the NaOH concentration in moles per liter, is accurately linear for all investigated values of I. The much stronger thermoelectric effect brought in by the hydroxyl ions is easily spotted by comparing the limiting values at x = 0.1. Although the Soret coefficient in NaCl weakly depends on salt concentration, an increase of more than 1 order of magnitude in the absolute value of ST is found by increasing cNaOH from 20 to 200 mM. The former experimental finding can be easily accounted for by considering that, for mixed electrolytes with the common Naþ cation, the thermoelectric field in eq 3 must be written as E¥ ¼ kB ½RNa - ROH x - RCl ð1 - xÞrT ¼ kB ½ðRNa - RCl Þ - ðROH - RCl ÞxrT
ð17Þ
Inserting the latter expression into eq 13 in fact yields the linear behavior of ST versus x, with a slope given by -1 dST DΠ Zeff ðROH - RCl Þ ¼ -kB DF dx
ð18Þ
which is therefore strictly proportional to the osmotic compressibility. Because the latter increases with the total ionic strength, the same is true for the slope. It is interesting to consider the extrapolation of the experimental values for dST/dx under condition where SDS micellar solutions behave as an ideal gas, that is, when B2 = 0, which, as we stated in the former section, takes place for I = 0.45 M. In this limit, we have DΠ dST Zeff ¼ kB T w ¼ ðROH - RCl Þ DF dx T
ð19Þ
From the approximate extrapolated value |dST/dx| = 0.056 0.057 shown in the inset of Figure 6, using Zeff = 16-18 we get an DOI: 10.1021/la904588s
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It is also interesting to point out that rescaling holds provided that A = T*/T0 is constant, which does not depend on the ionic strength or on the nature of the added electrolyte. In other words, our results suggest that, at least for a specific particle system, there are actually only two free fit parameters in eq 20. However, lacking at present any consistent model for the temperature dependence of thermophoresis, we do not have any physical explanation for the dimensionless value A = 16 that is found from a fit according to eq 21.
Conclusions
Figure 7. Temperature dependence of the Soret coefficient of SDS solutions under no-salt conditions and in the presence of variable amounts of NaCl and NaOH, as indicated in the lower-left legend. (Inset) Same curves rescaled and fitted according to eq 21.
independent estimate of ROH - RCl = 1, which lies within the range that we found in the former section. Temperature Effects. Former studies35 have shown that the Soret coefficient of many aqueous systems (including SDS in NaCl solutions) strongly depends on temperature, so much so that the thermophoretic behavior switches from thermophobic to thermophilic with decreasing T. Most experimental data are very well described by the empirical fitting function " ST ðTÞ ¼
ST¥
/ # T -T 1 - exp T0
ð20Þ
where S¥ T represents a high-T thermophobic limit, T* is the temperature where ST switches sign, and the rate T0 of exponential growth embodies the strength of temperature effects. Figure 7 shows that the same trend is shared by SDS solutions in the presence of NaOH. However, both the asymptotic value and the switching temperature strongly depend on the NaOH concentration. In particular, whereas for NaCl solutions T* does not differ very much from the value T* = 5 to 6 C found in the absence of added salt, it consistently increases with cNaOH to the point that, for cNaOH g 50 mM, SDS micelles move to the hot place over the whole investigated temperature range. As shown in the inset, however, all of these different temperature trends nicely superimpose onto a single master curve once the temperature is rescaled to T* and the Soret coefficient is rescaled to ST¥ ~ ST ðTÞ ~ ¼ 1 - exp½Að1 - TÞ ¥ ST
ð21Þ
where T~ = T/T*. Such rescaling clearly implies that the temperature behavior of ST is not influenced by the presence of a thermoelectric contribution or, in other words, that the physical origin of the temperature dependence of thermophoresis is basically unrelated to thermoelectric effects. It is nonetheless interesting that the wide change in T* induced by NaOH allows one to obtain the rescaled curve over a much wider range of T/T* than what would be feasible by adding only NaCl. (35) Iacopini, S.; Rusconi, R.; Piazza, R. Eur. Phys. J. E 2006, 19, 59.
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The experimental results presented and discussed in the former Sections allow to state some general conclusions. (1) The Soret effect of SDS solutions in NaOH, compared to NaCl, shows a strong thermoelectric contribution that can easily lead to sign reversal of the particle motion. (2) A detailed analysis of SDS thermophoresis conversely shows that concentration effects in the absence of added electrolytes can be consistently ascribed to osmotic compressibility effects, with no need to introduce any additional thermoelectric effect stemming from the differential thermal drift of counterion and micelles. In other words, the Soret coefficient of pure SDS depends only on equilibrium colligative effects that can be derived from an effective single-component model. (3) In the presence of electrolytes, a simple approximation based on the model of ref 8 can then be developed, where interacting SDS micelles drift because of a constant electric field induced by the differential migration of cations and anions, in addition to a thermophoretic force. Comparing the values for ST in NaCl and NaOH allows us to study decoupling concentration effects, which coincide with those found by equilibrium SLS measurements and yield an effective micellar charge in good agreement with the predictions of charge renormalization. Conversely, the dilute limit for ΔS entails pure electrokinetic effects, yielding a value for the thermoelectric field that is much lower than predicted. (4) The physical soundness of the former simplified model is further confirmed, in mixed electrolyte solutions, by the strictly linear behavior of ST versus the molar fraction of the two electrolyte components. (5) SDS thermophoresis shows strong temperature effects that cannot be attributed to the thermoelectric effect. Appropriate rescaling suggests that the temperature trend of ST(T) is even more general than what has been previously suggested.35 As final comment, we point out that the opportunity to tune the direction and amplitude of thermophoretic motion by the addition of electrolytes may be quite useful in exploiting thermophoresis as a particle manipulation technique in microfluidic systems, eliminating many drawbacks such as electrode polarization and Joule heating, for electrophoretic methods. Preliminary measurements that we have performed in collaboration with the group of Professor Howard Stone at Harvard on fluorescent polystyrene colloids indeed show that by controlling the amount and nature of the added electrolyte noticeable particle accumulation can be obtained at the cold or Langmuir 2010, 26(11), 7792–7801
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hot wall of a microfluidic channel kept in a transverse temperature gradient.36 We also point out that the methodology that we have discussed allows us to all extents to obtain information on the value of the particle charge without applying any
electric field. Thermoelectric measurements may therefore support standard electrophoretic techniques under conditions of high ionic strength, where the latter are known to perform rather poorly.
(36) Vigolo, D.; Rusconi, R.; Stone, H. A.; Piazza, R. Submitted to Soft Matter.
Acknowledgment. We thank A. W€urger and A. Parola for extensive discussions and useful comments.
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