We determine the structure of charge-stabilized colloidal suspensions at low
ionic strength over an extended range of particle volume fractions using a
combination of light and small angle neutron scattering experiments. The
variation of the structure factor with concentration is analyzed within a
one-component model of a colloidal suspension. We show that the observed
structural behavior corresponds to a non-monotonic density dependence of the
colloid effective charge and the mean interparticle interaction energy. Our
findings are corroborated by similar observations from primitive model computer
simulations of salt-free colloidal suspensions.Comment: Revised version, accepted to Phys. Rev. Let

A. Stradner

F. Scheffold

L. F. Rojas-Ochoa

P. Schurtenberger

R. Castañeda-Priego

V. Lobaskin

English

arXiv

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Density Dependent Interactions and Structure of Charged Colloidal Dispersions in the Weak Screening Regime

We determine the structure of charge-stabilized colloidal suspensions at low ionic strength over an extended range of particle volume fractions using a combination of light and small angle neutron scattering experiments. The variation of the structure factor with concentration is analyzed within a one-component model of a colloidal suspension. We show that the observed structural behavior corresponds to a nonmonotonic density dependence of the colloid effective charge and the mean interparticle interaction energy. Our findings are corroborated by similar observations from primitive model computer simulations of salt-free colloidal suspensions

Rojas-Ochoa, Luis Fernando

Castañeda-Priego, R.

Lobaskin, V.

Stradner, Anna

Scheffold, Frank

Schurtenberger, Peter

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Density Dependent Interactions and Structure of Charged Colloidal Dispersionsin the Weak Screening RegimeL. F. Rojas-Ochoa,1,2 R. Castan˜eda-Priego,3 V. Lobaskin,4,* A. Stradner,2 F. Scheffold,2 and P. Schurtenberger21Departamento de Fı´sica, Cinvestav-IPN, Av. Instituto Polite´cnico Nacional 2508, 07360 Me´xico D. F., Mexico2Physics Department, University of Fribourg, Chemin de Muse´e 3, 1700 Fribourg, Switzerland3Instituto de Fı´sica, Universidad de Guanajuato, Loma del Bosque 103, 37150 Leo´n, Mexico4Physik-Department, Technische Universita¨t Mu¨nchen, James-Franck-Str., D-85747 Garching, GermanyWe determine the structure of charge-stabilized colloidal suspensions at low ionic strength over anextended range of particle volume fractions using a combination of light and small angle neutronscattering experiments. The variation of the structure factor with concentration is analyzed within aone-component model of a colloidal suspension. We show that the observed structural behaviorcorresponds to a nonmonotonic density dependence of the colloid effective charge and the meaninterparticle interaction energy. Our ﬁndings are corroborated by similar observations from primitivemodel computer simulations of salt-free colloidal suspensions.PACS numbers: 82.70.Dd, 61.05.fg, 61.20.GyDeionized colloidal dispersions have recently attractedmuch experimental [1–3] and theoretical attention [4–7]because of their unique role as simple model systems in theﬁeld of surface charge regulation. In addition to experi-mental challenges posed by these systems, such as the needof control over the ionic strength of the medium, a numberof intriguing observations have been made that challengeclassical theories of electrostatic screening and electro-kinetics (e.g., see [1,8,9] and references therein). Someof the conceptual difﬁculties in the understanding of thesesystems stem from the counterion dominance in the ioniccloud of the colloid, which makes the screening ability ofthe medium dependent on the colloidal concentration. Atthe same time, the counterion condensation, which is con-trolled by the balance of the counterion chemical potentialbetween the bulk suspension and the condensed layer,would depend on the concentration through the state-dependent screening parameters [4,10,11]. Therefore,modeling these systems within the usual one-componentdescription can only be achieved using effective density-dependent interaction parameters, charge, and screeninglength [12]. The dependence of the colloid effective chargeon the volume fraction [4,5,13] or background ionicstrength [14] has been predicted in a few mean-ﬁeldmodels and is expected to be nonmonotonic. However,this nonmonotonic behavior has never been validatedexperimentally.In this Letter, we present a study of the density depen-dence of the colloidal structure and interactions at lowionic strength. Experiments have been carried out over awider range of concentrations than in any previous work.We analyze the particle structure factor obtained from lightand neutron scattering at different particle volume frac-tions and demonstrate that (i) the variation of colloidalstructure in deionized suspensions can be interpreted interms of a one-component model (OCM) that considersYukawa-like pair interactions between colloids with state-dependent parameters, and (ii) the nonmonotonic concen-tration dependence of the interaction energy, as character-ized by the colloid effective charge, is a general feature ofdeionized colloidal dispersions. Primitive model (PM) nu-merical simulations of salt-free suspensions corroborateour observations.Stock suspensions of negatively charged particles (sul-fonate polystyrene latex) were purchased from InterfacialDynamics Corporation (Portland, USA) and master disper-sions were prepared following the protocol described inRef. [15,16]. The samples were prepared by diluting themaster suspensions directly into quartz cells with water-ethanol mixtures, ﬁltered several times (pore size 0:2 m)to remove dust particles. These mixtures were chosen inorder to avoid crystallization by modifying the solventdielectric constant [17], thereby providing a well con-trolled model system of charged dispersions with volumefractions as high as 16%. Accurate values of particleradii and polydispersities, as obtained from DLS andSANS experiments, are given in Table I.The effective interparticle interaction can be quantiﬁedthrough the analysis of the suspension static structure. Thisrequires probing length scales comparable to the size of theparticles. Usually, such information can be obtained fromTABLE I. Polystyrene latex spheres used in the experiments.Bjerrum length, B  e2=40kBT, from the dielectric per-mittivity data in Ref. [17].Sample # Radius Bjerrum Volume Scattering(nm) length (nm) fraction (%) TechniqueS1 54:9 6:4 1.296 0.035–01.15 3D-DLSS2 58:7 8:6 1.383 0.135–01.48 3D-DLSS3 54:7 5:3 1.482 2.500–15.50 SANS1http://doc.rero.chPublished in "Physical Review Letters 100: 178304, 2008"which should be cited to refer to this work.static and dynamic light scattering (SLS/DLS) in the singlescattering regime [18]. In dense suspensions, however,these techniques cannot be applied in most cases due tomultiple scattering of light. An elegant way to overcomethis limitation is the use of modern scattering techniquessuch as 3D dynamic light scattering (3D-DLS) [19] orsmall angle neutron scattering (SANS). While 3D-DLSprovides valuable information at small and intermediatevalues of the scattering vector q, access to large q-vectorsfrom SANS allows to normalize the static structure factorSq and, in that way, unambiguously determine the sus-pension structure. A detailed description of the light scat-tering (LS) experiments is given in Ref. [15].Multiple scattering of neutrons in our SANS experi-ments has been suppressed by partially contrast matchingthe particles using H2O=ethanol–D2O=deuterated ethanolmixtures. The samples were kept in stoppered quartz cells(Hellma, Germany) with a path length of 2 mm and con-taining a mixed-bed of ion-exchanger resins; deionizationwas typically completed within 2 weeks. The measure-ments were performed with a mean neutron wavelengthof 1.27 nm and at a detector distance of 20.3 m, whichcorresponds to a q-range of 0:01–0:1 nm1. For details onthe initial data treatment and data analysis, see Ref. [20]and references therein. The SANS scattering intensitycan be expressed as Iq  APqSANSSqSANS, wherePqSANS is the normalized effective particle form factor,SqSANS is the effective interparticle structure factor, andA is an amplitude, which is proportional to the particlenumber density, n, and to the square of the neutron scat-tering length contrast between the particles and back-ground ﬂuid [21]. The effective SANS static structurefactor can be determined from SqSANS  A0=A	Iq=I0q, where I0q  A0PqSANS is the intensityscattered by a suspension of noninteracting particles withamplitude A0.On the theory side, the static structure factor of colloidaldispersions can be calculated by solving the Ornstein-Zernike (OZ) equation together with the Rogers-Young(RY) closure relation [22] for particles interacting withan effective pair potential ueffr of the Yukawa formueffr  Z2effBexpeffa1 effa2 expeffrr; (1)with an effective charge Zeff , colloid radius a, thermalenergy 1  kBT, screening parameter 2eff 4BZeffn ns, and concentration of salt ions ns  21000NAcs, where cs is the molar concentration [12]. Thisprocedure has been successfully used for calculating thestructure in the strong screening regime at effa  1 [22],but it becomes less accurate in systems with thick doublelayers as the interparticle potential deviates from theYukawa shape [23]. Despite this deﬁciency, an appropriatechoice of Zeff still allows one to predict the structure factorof strongly interacting systems over a wide range of con-ditions as long as the pair interaction energy at the meaninterparticle distance is correctly reproduced [24].Nevertheless, an agreement of the OCM structure and themean interaction energy with the original ones does notguarantee that other thermodynamic properties, such as theosmotic pressure, can be predicted using the effectiveparameters. The difﬁculty with the pressure originatesfrom the loss of the ionic degrees of freedom in thecoarse-grained description. Various solutions to this issuehave been suggested recently. In one, the missing pressurecontributions have been recovered via inclusion of theboundary terms [6]. Alternatively, one can use a modiﬁedOZ-RY approach to get the effective parameters in the ﬁtprocedure [5]. The latter replaces the OCM (Yukawa ﬂuid)pressure by the osmotic pressure known a priori fromprimitive model simulations or mean-ﬁeld models (e.g.,jellium) in calculating the isothermal compressibility. Weuse this route to extract the effective charges and effectivepotentials from the structure data measured in experimentand simulations.Figure 1 displays the normalized scattering intensity fortwo of the samples S3 in Table I: (a) ’  2:5% and(b) ’  7:6%. In (a), the squares are from 3D-DLS andcircles from SANS measurements, thus illustrating thecomplementarity of both experimental techniques. Theexperimental error is estimated from the intensity statisticsin a series of measurements. For both SANS and LS, theFIG. 1 (color online). Normalized scattering intensities ofsamples (a) ’  0:025 and (b) ’  0:076. The squares arefrom 3D-DLS and the circles from SANS. The dashed line isthe calculated Sq and the continuous line its SANS experi-mental convolution, SqSANS. Insets: scattering intensities for(a) a diluted screened suspension and (b) ’  0:076.2http://doc.rero.chexperimental data are in good agreement with theoreticalcalculations (lines) based on integral equations theory us-ing the polydisperse RY closure relation [22] as describedin Ref. [5]. The background salt concentration, cs  2107 M, was determined by measuring the conductivity ofour deionized solvent mixtures as in Ref. [25]. Thus, thefree parameters in our ﬁtting procedure are the volumefraction ’ and the effective charge Zeff . The dashed linesrepresent the calculated Sq and the continuous linescorrespond to the result after its experimental convolution,SqSANS, with the appropriate resolution function [26].The structure factor exhibits a pronounced maximum,whose position qmax varies with the particle volume frac-tion as n1=3 [16]. Additionally, in the insets, we also showthe SANS scattering intensities for: (a) a dilute, ’ 0:005, screened (cs  0:005 M of KCl) suspension, and(b) the sample ’  7:6%. Here, the lines are experimen-tally convoluted ﬁts with polydisperse form, I0q /PqSANS, and structure factor, Iq / PqSANSSqSANS.In Fig. 2(a), we present the ’-dependence of the heightof the main peak of the structure factor, Sqmax, as ob-tained from the LS and SANS analysis. At low volumefractions, the peak height is about 2 and varies weakly withconcentration until ’  0:002; then, it decreases in therange 0:002<’< 0:01 to ﬁnally grow again for ’>0:01. We note that for ’> 0:06, the values of Sqmax>3:0 indicate a glassy state [27]. While the increase ofSqmax with concentration can be readily explained byshortening the distance between neighboring colloids andrising average interparticle repulsion, the decrease at ’ 0:01 is a more complex feature. This depression can beattributed to the maximum of ionic condensation or aminimum of the effective macro-ion charge as discussedbelow. To test whether this behavior is speciﬁc to theexperimental systems considered here, in the inset ofFig. 2(a) we show results of PM simulations (solid sym-bols) of a salt-free charged colloidal suspension withcharge asymmetry 200:1. The simulations were performedfor systems with B=a  0:2 using the same clusterMonte Carlo simulation protocol and settings as inRef. [28] with 80 colloidal particles. The Sqmax fromsimulations shows a similar trend to the experimentsalthough the main features are less pronounced.Nevertheless, one can show that this characteristic trendof the structure factor corresponds to a quite peculiarbehavior of the pair interaction potential. Figure 2(b) de-picts the interaction energy between nearest neighbor col-loids, ueffrm, as extracted from ﬁts to the structure dataof systems S1–S3. ueffrm follows the trend observed forSqmax with more pronounced features: a maximum at’  0:001, a minimum at ’  0:01, and an increase athigher volume fractions. The same holds for the PM results[solid circles in the inset of Fig. 2(b)]. Furthermore, wewere able to self-consistently prove our primitive modelresults by using their associated effective charges andscreening lengths in OCM-NPT simulations [open squaresin the inset of Fig. 2(a)]. Here, we usedN  1000 colloidalparticles and the osmotic pressure as obtained from the PMsimulations.The interactions between colloids are conveniently char-acterized by their effective charge. To compare the resultsbetween different systems, we use the dimensionless ratio~Zeff  ZeffB=a instead of the valency itself. In Fig. 3, weshow the results of ~Zeff as obtained from ﬁtting the scat-FIG. 2 (color online). (a) Sqmax vs ’. Squares and circles arefrom 3D-DLS measurements in S1 and S2, respectively.Triangles are from 3D-DLS (solid) and SANS analysis (open)in S3. Inset: simulations from PM (solid circles) and OCM-NPT(open circles). (b) The colloid interaction energies ueffr at themean interparticle distance rm for experiments and simulations(inset). Lines are guides to the eye.FIG. 3 (color online). ~Zeff vs ’ from experiments (main) andsimulations (inset). The lines are calculations based on renor-malized jellium (dashed line) and PB-cell (dash-dotted line)models, and considering cs  2 107 M (main) and cs  0(inset). Same symbols as in Fig. 2.3http://doc.rero.chtering data with our OZ-RY scheme (same symbols as inFig. 2). Furthermore, in Fig. 3, we compare the data withresults of the PM simulations described above (solid circlesin the inset), where ZbareB=a  40:0 and the ~Zeff areextracted following the same protocol as in the experi-ments. Both sets of data show a pronounced minimum ofthe effective charge at ’  102.The behavior of ~Zeff in the range ’ & 102 stems fromthe variation of the mean counterion concentration. Uponincreasing ’, counterions in average get closer to thecolloidal surface, thus leading to a gradual decrease of~Zeff . At ’> 102; in contrast, the effective charge startsincreasing because the average electrostatic potential in thebulk, or at least at r  rm, becomes comparable to that onthe colloid surface, thus making the counterion condensa-tion less favorable. In Fig. 3, we also present results fromtwo models, which predict a similar density dependencefor ~Zeff : renormalized jellium [4] and Poisson-Boltzmanncell model (PB-cell) [11]. We used cs  2 107 M inthe experiments (main ﬁgure) and cs  0 in the simula-tions (inset). These models correctly reproduce the quali-tative behavior of the effective charge although the valuescorresponding to the saturation regime are too high. Thisquantitative disagreement might be caused by various rea-sons. First, the best ﬁt in our procedure corresponds toslightly lower effective charges than those in the jelliummodel. Moreover, the values are affected by the actualsurface charge density on the colloids in experiment andionic correlation effects in the simulations [28]. The chargedissociation in water-ethanol mixtures is expected to belower than in pure water. Our present data suggest thedissociation on the level of ca. 20% of that in water, whichmeans the actual charges are below the saturation valuesfor these particles at some volume fractions. Finally, weshould note that the nonmonotonic behavior of ~Zeff ischaracteristic for the counterion-dominated screening re-gime. Our numerical estimations show that alreadyamounts of salt comparable to the effective counterionconcentration, ca. 1 M in our systems, might lead todisappearance of the minimum in the effective interactionand effective charge as function of colloidal volumefraction.In summary, we have studied the variation of the colloi-dal structure factor in charged deionized colloidal disper-sions and demonstrated that its density dependencecorresponds to a nonmonotonic variation of the meaninterparticle interaction and particle effective charge,with a minimum at about a volume fraction of 1%. Ourresults cover a wider range of particle concentrations thanin any previous study, thus giving an excellent benchmarkfor models of electrostatic screening in colloidal disper-sions. Furthermore, our ﬁndings are conﬁrmed by com-puter simulations in the counterion-dominated regime ofboth primitive and one-component models.Some of the experiments were performed at the Swissspallation neutron source SINQ, Paul Scherrer Institute,Villigen, Switzerland. We are grateful for the neutronbeam time and help of our local contact personJ. Kohlbrecher. We gratefully acknowledge ﬁnancial sup-port from the Swiss National Science Foundation.L. F. R. O. and R. C. P. thank Conacyt-Mexico for ﬁnancialsupport (Grants Nos. 51669 and 46373).*Present address: School of Physics, University CollegeDublin, Belﬁeld, Dublin 4, Ireland[1] T. Palberg et al., J. Phys. Condens. Matter 16, S4039(2004).[2] L. Shapran et al., J. Chem. Phys. 125, 194714 (2006).[3] C. Haro-Pe´rez et al., J. Phys. Condens. Matter 18, L363(2006).[4] E. Trizac and Y. Levin, Phys. Rev. E 69, 031403 (2004).[5] R. Castan˜eda-Priego et al., Phys. Rev. E 74, 051408(2006).[6] E. Trizac et al., Phys. Rev. E 75, 011401 (2007).[7] V. Lobaskin et al., Phys. Rev. Lett. 98, 176105 (2007).[8] C. P. Royall et al., J. Chem. Phys. 124, 244706 (2006).[9] B. V. R. Tata et al., Solid State Commun. 139, 562 (2006).[10] B. Beresford-Smith et al., J. Colloid Interface Sci. 105,216 (1985).[11] S. Alexander et al., J. Chem. Phys. 80, 5776 (1984).[12] L. Belloni, J. Phys. Condens. Matter 12, R549 (2000); Y.Levin, Rep. Prog. Phys. 65, 1577 (2002).[13] T. Gisler et al., J. Chem. Phys. 101, 9924 (1994).[14] R. R. Netz and H. Orland, Eur. Phys. J. E 11, 301 (2003).[15] L. F. Rojas et al., Europhys. Lett. 60, 802 (2002); L. F.Rojas et al., Faraday Discuss. 123, 385 (2003).[16] L. F. Rojas-Ochoa et al., Phys. Rev. Lett. 93, 073903(2004).[17] D. Bertolini et al., J. Chem. Phys. 78, 365 (1983).[18] P. Schurtenberger and M.E. Newman, in EnvironmentalParticles, edited by J. Bufﬂe and H. P. van Leeuwen(Lewis Publishers, Boca Raton, 1993), pp. 37–115; F.Scheffold and R. Cerbino, Curr. Opin. Colloid InterfaceSci. 12, 50 (2007).[19] 3D-DLS is an extension of traditional SLS/DLS to theregime of weak and moderate multiple scattering based ona cross-correlation scheme [C. Urban and P.Schurtenberger, J. Colloid Interface Sci. 207, 150 (1998)].[20] L. F. Rojas-Ochoa et al., Phys. Rev. E 65, 051403 (2002).[21] C. F. Wu and S.H. Chen, J. Chem. Phys. 87, 6199 (1987).[22] B. D’Aguanno and R. Klein, Phys. Rev. A 46, 7652(1992); G. Na¨gele, Phys. Rep. 272, 215 (1996).[23] M. Brunner et al., Europhys. Lett. 58, 926 (2002); V.Lobaskin et al., J. Phys. Condens. Matter 15, 6693 (2003).[24] V. Lobaskin et al., Phys. Rev. E 63, 020401(R) (2001).[25] M. Evers et al., Phys. Rev. E 57, 6774 (1998); D.Hessinger et al., Phys. Rev. E 61, 5493 (2000).[26] J. G. Barker et al., J. Appl. Crystallogr. 28, 105 (1995).[27] J. P. Hansen and L. Verlet, Phys. Rev. 184, 151 (1969).[28] V. Lobaskin and P. Linse, J. Chem. Phys. 111, 4300(1999).4http://doc.rero.ch

Density dependent interactions and structure of charged colloidal dispersions in the weak screening regime

Density-dependent interactions and structure of charged colloidal
  dispersions in the weak screening regime

Density-dependent interactions and structure of charged colloidal dispersions in the weak screening regime

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