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Onsager's Reciprocal Relations for Electroacoustic and Sedimentation Onsager’s reciprocal relations for electroacoustic and sedimentation: Application to (concentrated) colloidal suspensions S. Gourdin-Bertin and C. Chassagne Citation: The Journal of Chemical Physics 142, 194706 (2015); doi: 10.1063/1.4921375 View online: http://dx.doi.org/10.1063/1.4921375 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/142/19?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Reciprocal relations in electroacoustics J. Chem. Phys. 141, 044703 (2014); 10.1063/1.4886399 Diffusion, sedimentation, and rheology of concentrated suspensions of core-shell particles J. Chem. Phys. 136, 104902 (2012); 10.1063/1.3689322 Sedimentation equilibrium of a suspension of adhesive colloidal particles in a planar slit: A density functional approach J. Chem. Phys. 116, 384 (2002); 10.1063/1.1421354 Sedimentation potential of a concentrated spherical colloidal suspension J. Chem. Phys. 110, 11643 (1999); 10.1063/1.479103 Suspensions of adhesive colloidal particles in sedimentation equilibrium in a planar pore J. Chem. Phys. 109, 11085 (1998); 10.1063/1.477746 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 131.180.130.187 On: Mon, 15 Jun 2015 08:58:06 THE JOURNAL OF CHEMICAL PHYSICS 142, 194706 (2015) Onsager’s reciprocal relations for electroacoustic and sedimentation: Application to (concentrated) colloidal suspensions S. Gourdin-Bertin1,2 andC. Chassagne 1,3 1Sorbonne Universités, UPMC Université Paris 06, UMR 8234, PHENIX, F-75005 Paris, France 2CNRS, UMR 8234, PHENIX, F-75005 Paris, France 3Department of Environmental Fluid Mechanics, TU Delft, Stevinweg 1, 2628 CN Delft, The Netherlands (Received 31 March 2015; accepted 8 May 2015; published online 20 May 2015) In this article, the relations for electroacoustic phenomena, such as sedimentation potential, sedi- mentation intensity, colloid vibration potential, colloid vibration intensity/current, or electric sonic amplitude, are given, on the basis of irreversible thermodynamics. This formalism allows in particular to discuss the different expressions for concentrated suspensions found by various authors, which are of great practical interest. It was found that some existing expressions have to be corrected. Relations between the electrophoretic mobilities assessed by the different experiments are derived. C 2015 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4921375] I. INTRODUCTION relations were respected in the case of the sedimentation ve- locity and potential theory presented in their paper, for a dilute Electroacoustic phenomena, such as Colloid Vibration suspension of charged spherical colloidal particles. Ohshima8 Potential (CVP), Colloid Vibration Intensity/Current (CVI), later made an analogy between SP and CVP to establish On- or Electric Sonic Amplitude (ESA), allow to assess the prop- sager relationships in this case. Dukhin et al. used the Onsager erties of (concentrated) suspensions and in particular the zeta relation for electroosmosis/streaming potential to derive an potential of individual particles. The commercially available expression for CVI.9,10 devices measuring the CVP and ESA phenomena are routinely Our first goal, in this paper, is to check whether the theo- used by research groups and industry. Since the initial exper- ries found by different authors, using different assumptions, iments of Rutgers on CVP in 1946,1 experimental devices to are in agreement with the Onsager relations. Our second goal measure the CVP response of colloidal suspensions have been is to present the link between the different electrophoretic commercialized by Marlow2 in 1988, by O’Brien3 for ESA the mobilities assessed by different theories/experiments. This last same year, and a few years later for CVP by Dukhin.4 These point has been a matter of debate since 15 years.6,9,10 We hope devices come with different theories to interpret the measured that the present article will clarify this issue. response. Theories for the electroacoustics response of col- In Sec.II, we briefly recall important results regarding loids were developed separately, first by Henry and Booth, the Onsager relations for sedimentation and electroacoustics. Enderby in 1952,5 by O’Brien and coworkers from the 1980s These relations were derived in Refs. 14–16 by writing the onwards,3–6 and by Ohshima and coworkers7,8 and by Dukhin entropy production from which the linear force-flux relations and coworkers9 in the same decades. follow. Using the fluxes-gradients approach then enables us, On the other hand, even though the effect was discovered in Sec. III, to comment on the theories presented by different by Dorn in 1880, Sedimentation Potential (SP) experiments are authors. We will discuss general results regarding the volume- scarce.11,12 Theories for interpreting the sedimentation poten- fraction dependence, which are of high practical importance. tial data have been formulated by Booth in 1954, for low zeta In the conclusion, we present the general relation between the potentials but all particle sizes and double layer thicknesses.13 electrophoretic mobilities found by electrophoretic mobilities In 1984, Ohshima, Healy, White, and O’Brien7 derived the measurements, from theoretical/numerical considerations, and sedimentation velocity of a single charged sphere and the SP(SI)/CVP(CVI)/ESA measurements. sedimentation potential of a dilute suspension, for all zeta potentials and all particle sizes and double layer thicknesses. All the previously cited theories were derived making use II. THEORY of the “electrokinetic set of equations,” which include Pois- son, balance equation for ions, and Navier-Stokes. Another In this section, we give the entropy production derived approach is possible, based on irreversible thermodynamic for sedimentation and electroacoustics. By writing the en- considerations. The corresponding relations between fluxes tropy production, the forces and the fluxes needed to setup the and gradients and the famous “reciprocal relations” were intro- forces-fluxes relations can be correctly defined. In particular, duced by Onsager in 1931. In 1952, de Groot, Mazur, and the forces and fluxes thus defined have the dimensions that Overbeek14 derived the Onsager reciprocal relations for sedi- ensure that the cross coefficients (defined below) have the same mentation. In 2014, Chassagne and Bedeaux15 extended the dimensionality. This also enables to have a coherent definition results of de Groot, Mazur, and Overbeek to the electroacoustic for the signs. The general derivations of the equations given phenomena. Ohshima et al.7 demonstrated that the Onsager in this section can be found in Refs. 14–16. As is discussed 0021-9606/2015/142(19)/194706/11/$30.00 142, 194706-1 © 2015 AIP Publishing LLC This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 131.180.130.187 On: Mon, 15 Jun 2015 08:58:06 194706-2 S. Gourdin-Bertin and C. Chassagne J. Chem. Phys. 142, 194706 (2015) in more detail in Ref. 16, and as was hinted by Ohshima in vp = lPP∆ρφg + lPEE; Ref.8, the relations for sedimentation can be seen as the low- (7) frequency limit of the electroacoustic equations. I = lEP∆ρφg + lEEE: Note that since the system is at mechanical equilibrium, a A. Sedimentation pressure gradient is generated by the gravitational field (rP 16 The total entropy production σ in the case of sedimenta- = ρg). The reciprocal Onsager relation to be verified is lPE tion was found to be = lEP and it follows that the sedimentation intensity (SI) can be defined as I · E g · Jlab σ = + : (1) ( vp ) T T SI = I E=0 = ∆ρφ g: (8) ( ) E g=0 The corresponding force-flux relations derived from the en- l = v =E tropy production were found to be The term PE p g=0 is by definition the electrophoretic mobilityoftheparticlemeasuredatzerototalvolumefluxcondi- tion (in the laboratory frame of reference), i.e., µlab;vol. flux=0, Jlab = m g + m E; E PP PE and can be identified with µ , i.e., the electrophoretic mobility (2) E obtained from standard electrophoretic mobility measurements I = m g + m E; EP EE (seeAppendixA).TheSPofthesuspensionisobtainedfromthe last line of Eq. (7), where Jlab represents the total mass flow in the reference frame lab −µ −µ rP of the laboratory (superscript lab) and J = ρv, where ρ is − r E E lab SP = E I=0 = ∆ρφg = ∆ρφ : (9) the density of the suspension and v = vbar the velocity of the ( ) ( ) K K ρ center of mass of the system in the reference frame of the Note that of course g = g I=0 = g E=0. In setting-up the last laboratory (from now on, the subscript lab and superscript bar equation, we have used the( ) hydrostatic( ) equation: rP = ρg. will be dropped). The electric current is defined by the symbol I, the electric field by E, and the gravitational field by g. The coefficients of proportionality mi j can either be measured or B. CVP/CVI estimated from theories, and Onsager’s relation gives that the The total entropy production σ in the case of electroacous- ffi m = m cross coe cients respect the relation PE EP. tics was found to be For colloids, assuming that the total mass of the colloidal I · E 1 rP particles is much larger than the total mass of the ions, it was σ = + · Jvol: (10) found that in good approximation15 T T ρ The corresponding forces-fluxes relations derived from the ρ = φρp + 1 − φ ρw; entropy production were found to be ( ) (3) vol rP ρv = φρpvp + 1 − φ ρwvw; J = bPP + bPEE; ( ) ρ rP (11) where φ is the volume fraction of the colloidal particles, and I = b + b E; EP ρ EE ρp and ρw are the absolute densities of the colloidal particles and the solvent (water). The bars on the densities are intro- where Jvol represents the total mass flow in the reference frame duced to avoid confusion with the definition of other densities of the total volume flow (superscript vol).
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