Langmuir 1995,11, 1942-1950
1942
Phase Diagram of Colloidal Dispersions of Anisotropic Charged Particles: Equilibrium Properties, Structure, and Rheology of Laponite Suspensions A. Mourchid,? A. Delville,? J. Lambard,* E. Lecolier,? and P. Levitz*lt CRMD, C N R S , l b Rue de la Fdrollerie, 45071 Orldans Cedex 2, France, and C.E.A., C.E. Saclay, Service de Chimie Moldculaire, 91191 Gif sur Yvette Cedex, France Received December 27, 1994@ We discuss the phase diagram of aqueous dispersions of colloidal platelike charged particles (300 A x 10 A). Particle concentration and ionic strength are the two parameters controlling the system. The suspensions undergo a sougel transition without macroscopic phase separation. Shear rheology is used to monitor this transition and to locate the appearance of the “mechanicalgel”phase. Increasing the ionic strength shifts the sougel transition to lower volume fraction. Direct inspection of this gel phase by cryofracture, TEM and SAXS shows correlated but well-separated particle populations. In order to check the reversibility and the equilibrium properties of this transition, the equation of state was determined by osmotic stress. At fixed ionic strength, the osmoticpressure first increases at low particle concentration, then reaches a “pseudoplateau”,and increases again for higher concentrations. The location of such a singularity in the equation of state of the suspension defines a thermodynamicaltransition coincidingwith the mechanical phase transition. In order to analyze the origin of this gel or “glassy phase”, the role of particle anisotropy, coupled with diffuse layer repulsion, is discussed. isotropic-nematic (I-N) phase separation is known as I. Introduction the Onsager t r a n ~ i t i o n The . ~ biphasic domain extends to Colloidal systems undergo phase transitions such as higher concentrations as the ionic strength increases.1° liq~id-solid,’-~order-disorder, sol-gel: or glass transiHowever, the optimization of the double-layer repulsion t i o n ~ which ,~ are of important technical and scientific is responsible for a relative orientation of the rod-shaped interest. The phase diagrams of the colloidal suspensions particles which counteracts their nematic alignment. For are generally controlled by interparticle interactions. lesser anisotropy, the particles can be immobilized by Although temperature and ionic strength are two intensive strong double-layer repulsion, forming a glassy phase with variables able to modulate these interactions, the morordered regions, well before reaching concentrations where phology of the particles and their anisotropy are more the I-N transition O C C U ~ S . ~ subtle parameters which also play a crucial role in the It is well-known that disklike particles such as Na+ or stability and phase transitions6s7of colloids. Li+homoionic swellingclays11J2form a gel above a specific To illustrate this point, let us consider the behavior of volume fraction. However, the addition of salt does not reference suspensions of spherical charged particles (latex, promote the liquid phase, as observed for spherical or SiOz): they exhibit a transition from a homogeneous liquid rodlike particles, but lowers the particle volume fraction phase to an ordered or disordered soft solid at a volume at which the gel or glassy phase appears.11J2 At first fraction increasing with ionic ~ t r e n g t h . This ~ , ~ transition glance, this result seems to be contrary to what one would is driven by long-range electrostatic repulsion due to the overlap of the diffuse layers surrounding each p a r t i ~ l e . ~ , ~ expect from DLVO theory. Two opposite explanations are proposed in the literature: (i)a microflocculation due The transition line can be predicted from a model of to electrostatic or van der Waals attraction between edges renormalized hard spheres with an effective radius equal and faces of the platelike particles resulting in a linked to the radius of the naked particle plus the thickness of its diffuse l a ~ e r . A ~ rough !~ estimate of this thickness is structure, the so-called “house of cards” configuration,13 given by the Debye length ( K - ~ ) , which is a decreasing and (ii) a gelation mechanism, originally suggested by function of the ionic strength I ( K - ~ I-y2). Norrish14 and supported by others,1°J2J5J6stressing the For nonspherical charged particles such as highly role of long-range electrostatic repulsion between overanisotropic rod^^,^ (tobacco mosaic virus, boehmiteg), an lapping double layers, as revealed by osmotic pressure isotropic-nematic (I-N) phase transition exists, driven measurements, coupled with excluded volume interactions by the loss of rotational entropy of the particle. This induced by the specific anisotropy of the charged disklike particles. * To whom correspondence should be addressed. The choice between attractive and repulsive interparCRMD, CNRS. ticle interactions during gelation of disklike particles is * C.E.A. not easy, and the aim of this work is to provide experiAbstract published in Advance A C S Abstracts, May 15,1995. mental information on the phase diagram of platelike (1) Pieranski, P. Contemp. Phys. 1983,24,25-73. (2)Ottewill, R. H.Ber. Bunsen Ges. Phys. Chem. 1985,89,517-525. charged particles. More specifically, we will focus on
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(3) Okubo, T. Langmuir 1994,10,1695-1702. (4)Buscall, R.; Goodwin, J. W.; Hawkins, M. W.; Ottewill, R. H. J . Chem. SOC.Faraday Trans. 1982,78, 2889-2899. (5) van Megen, W.; Underwood, S. M. Phys. Reu. E 1994,49,42064220. (6)Langmuir, I. J . Chem. Phys. 1938,6, 873-896. (7)Onsager, L. Ann. N.Y. Acad. Sci. 1949,51,627-659. (8)Fraden, S.;Maret, G.; Caspar, D. L. D. Phys. Rev. E 1993,48, 2816-2837. ___ . (9)Buining, P. A.;Philipse, A. P.; Lekkerkerker, H. N. W. Langmuir 1994,10,2106-2114.
(10)Forsyth, P. A.; Marcelja, S.; Mitchell, D. J.;Ninham, B. W.Adu. Colloid Interface Sci. 1978,9,37-60. (11)Sohm, R.; Tadros, Th. F. J. Colloid Interface Sci. 1989,132, 62-71. (12)Rand, B.; Pekenc, E.; Goodwin, J. W.; Smith, R. W. J . Chem. SOC.,Faraday Trans. I1980,76, 225-235. (13)van Olphen, H. A n Introduction to Clay Colloid Chemistry; Wiley: New York, 1977. (14)Norrish, K.Discuss. Faraday SOC.1954,18,120-134.
0743-746319512411-1942$09.00/0 0 1995 American Chemical Society
Phase Diagram of Colloidal Suspensions
A)
suspensions of small (300A x 10 synthetic clay particles (laponite). This choice ensures a correct thermalization of the system and avoids the delicate problem of entanglement of larger sheetlike particles. This system has already been analyzed by different authors and especially by Ramsay15-17and Morvan et aZ.ls In the present work, particle concentration and ionic strength will be the two parameters controlling the system. Shear rheology will be used to monitor the sougel transition and to locate the appearance of the "mechanical gel" phase. Self-organization of the gel phase will be evidenced by cryofracture, TEM and SAXS. Finally, the reversibility and the equation of state of this colloidal system under gelation will be investigated by osmotic stress.
Langmuir, Vol. 11, No. 6, 1995 1943 (USAXS)is described elsewherez2and includes a pair of multiple reflection channel-cut crystals Ge(l,l,l).The full width at halfmaximum of the reflection curve was 80 prad in the horizontal of the direct beam was 200prad, plane and the full width at allowing reliable measurements from 5 x to 10-1 8-1. USAXS are performed by using quasi-linear collimation. The iterative method proposed by Lakezzis used to desmear experimental spectra. All experiments are given in absolute scale (cm-l). For a very dilute suspension of randomly oriented cylinders of thickness W and radius R , the scattering intensity per unit volume may be written asz3
AJQ) = KP(Q)
(2)
where the contrast factor reads
11. Experimental Part 1. Sample Preparation. We use laponite RD, i.e., synthetic hectorite manufactured by Laporte, Ltd., without any purification. The mean chemical composition of this clay is:15J9 66.2%; MgO, 30.2%;NazO, 2.9%; and Li20,0.7%,corresponding to the The most diluted general formula SisMg5.45Li0.4H40~4Na0.~. samples (up to 5%w/w)are directly dispersed in a doubly distilled aqueous solution of fixed ionic strength. The dispersions are stirred at high speed and left at rest for one week prior to any measurement. The pH of the dispersions is adjusted t o 10 by addition of NaOH, in order t o avoid congruent dissolutionlg of the clay particle. Concentrated samples are prepared by osmotic stresszoagainst solutions of Dextran 110 000. The pH of the dextran solution is also adjusted to 10 and all dialysis experiments are performed under Nz to avoid COZ contamination and concomitant decrease of the pH. The membranes used for the dialysis (Visking) have a molecular weight cutoff of 12 000, preventing Dextran contamination of the stressed suspensions, even after a dialysis period of 1month. The absence of dextran contamination is checked by elementary chemical analysis. The clay concentration of the stressed suspensions and the dextran concentration are determined by the loss of weight after desiccation at 140 and 70 "C, respectively. Osmotic pressure calibration of Dextran 110 000 is performed by using the recent determination of Bonnet-Gonnet and Parsegian:zl
log,,
(nddcmz) = 1.826 + 1 . 7 1 5 ~ O . ~ ~ '
where 1OOw is the mass fraction of Dextran in solution. 2. Rheology. Frequency strain response of the dispersions to oscillatory shear stress is performed with an imposed stress rheometer (Carri-Med). The amplitude of the applied stress is chosen in the linear viscoelastic regime. For each sample, the data are recorded after a standing period corresponding to the gelation time subsequent t o the manipulation of the dispersion. The yield stress was recorded under the same stringent conditions. The samples are held at fmed temperature (20 "C) with cone and plate geometry fitted with a guard loop to prevent evaporation. The variations of the viscoelastic moduli G and G with the frequency are measured for different particle concentrations and ionic strength. 3. SAXS. The small-angle X-ray scattering spectra are recorded either with a two-crystal multiple-reflection cameraz2 (CEA, Saclay) or with a conventional SAXS camera allowing a total range of Q vectors between 5 x and 0.8 8-l. Experiments are performed by using Cu Ka radiation. The experimental setup for the ultrasmall-angle X-ray scattering (15)Ramsay, J. D.F. J . Colloid Interface Sci. 1986,109,441-447. (16)Avery, R. G.; Ramsay, J. D. F. J. Colloid Interface Sci. 1986, 109,448-454. (17)Ramsay, J. D.F.; Lindner, P. J . Chem. SOC.Faraday Trans. 1993,89,4207-4214. (18)Morvan, M.; Espinat, D.; Lambard, J.;Zemb, Th. Colloid Surf. A: Physicochem. Eng. Aspects 1994,82,193-203. (19)Thompson, D.W.; Butterworth, J. T. J . Colloid Interface Sci. 1992,151,236-243. (20)Parsegian, V. A.; Rand, R. P.; Rau, D. C. Methods Enzymol. 1986,127,400. (21)Bonnet-Gonnet, C.;Parsegian, V. A. Private communication. (22)Lambard, J.;Lesieur, P.; Zemb, Th. J . Phys. I , Fr. 1992,2,11911213.
(3) be is the diffusion length of the electron and the normalized form factor is given by m2
p'Q' =
s,
sin2(QHcos a)4J;(QR sin a) sin a da (4) (QH cos a)2(QRsin a)'
In eq 3, V, is the volume of the cylindrical paeicle, and N / V is density number. (@solid - @water)is equal to 0.47 A-3, J1is the first order Bessel function, and a is the angle between the normal to the cylinder and the wave vector Q. Below (Q = d R ) , the scattering intensity tends to a plateau and P(Q) converges to 1. For highly asymmetrical particles (p = H / R iz/R), the scattering intensity decreases approximatively as &-z. Departure from this behavior allows the determination of the average thickness (2W, using the approximation16-17
P(Q) = 2 exp(-Q2H213) Q ~ R ~ The average diameter of the cylindrical particles is determined from absolute scaled measurements, using eq 2. The SAXS spectrum of the isolated particle will serve as a reference spectrum to show the organization of concentrated laponite suspensions. 4. Cryofracture. Samples are prepared for cryofracture by addition of 30% of glycerol and hardened by rapid freeze by immersion in liquid propane. Microtomy is performed at -120 atm, and the replica obtained by "C under a pressure of metallization with platinum is observed by TEM. 5. Flow Birefringence. Soft gels (near the sougel transition) flow freely under the action of gravity. The alignment of particles at low Peclet number flow is observed by birefringence. The disappearance of the birefringence pattern after stopping the flow gives information on the relaxation mechanism of the clay suspension. The complete relaxation of soft gels oflaponite requires a long time (above 1 s) compared to the correlation orientation time of a single isolated particle 8). 6. Numerical Simulations. Equilibrium properties of laponite suspensions are modeled by Monte Carlo simulation^.^^ The general purpose of these numerical experiments is the analysis of the clay particles organization induced by excludedvolume and diffuse layer repulsions. In the first case, a large number of particles is treated; as an example, Figure 1shows a snapshot illustrating an equilibrium configuration of48 000 particles. Structural information (radial distribution function, order parameter, SAXS spectrum, random planar section, ...)is directly compared withexperimentalresults. In the second case, two charged laponite particles and their 2000 neutralizing sodium counterions are immersed in a large volume, containing NaCl, in equilibrium with a reservoir at M). These conditions are used to constant ionic strength permit a direct comparison with the results obtained by dialysis. (23)Guinier, A.; Fournet, G. Small Angle Scattering of X-Rays; Wiley: New York, 1955. (24)Metropolis, N.;Rosenbluth, A. W.; Rosenbluth, M. N.; Teller, A. H.; Teller, E.J. Chem. Phys. 1963,21,1087-1092.
Mourchid et al.
1944 Langmuir, Vol. 11, No. 6,1995
n
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10'1
100
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102
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Figure 3. Frequencyresponse of the storage (G') and loss ( G ) M. moduli as a function of particle concentration. I = Figure 1. Snapshot of a typical configuration of 48000 cylindrical particles randomly dispersed in a 4.1-pm3 box (particle concentration 0.011 v/v or 0.03 w/w).
I
0 I
-
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.
L
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c (96 w/w) Figure 4. Variation of the storage modulus (G') as a function of particle concentration and ionic strength of the suspension. All the curves correspondto power laws with the same exponent (2.35) (see text).
o (rad/,)
Figure 2. Frequencyresponseof the storage ( G )and loss (G'") moduli as a function of particle concentration. I = M. For that purpose, we use the grand canonical Monte Carlo (GCMC)samplingintroduced by ad am^.^^ The ionic distribution aroundthe laponite particles and the electrostatic energy clarify the role of the diffuselayer on particle organizationand stability.
III. Results 1. Rheological Behavior. The frequency variation of the storage modulus ( G ) and loss modulus ( G ) are shown in Figures 2 and 3 for clay suspensions between 0.2 and 3% (w/w) at ionic strength M. As already shown by different authors,26-28colloidal suspensions of laponite undergo a gel transition above some solid fraction CO.Below this value, the suspension is slightly viscous and the elastic G(o )and loss modulus G ( o )are weak (around 10-1Pa) (Figure 2). Above CO,the elastic modulus increases markedly and the appearance of a yield stress is observed (Figure 3). G(o)and G ( o )do not vary much with frequency in the dynamical range between and rad/s. The variation of the zero-frequency limiting value of the storage modulus with solid concentration is shown in Figure 4 for.different ionic strengths. At fixed (25) Adams, D. J. Mol. Phys. 1974,28,1241-1252. (26) Neumann, B. S.; Sansom,K. G. Cluy Miner. 1971,9,231-243. (27) Carless, J. E.; Oman, J.J. Phurm. Pharmucol. 1972,24,637644. (28) Perkins, R.; Brace, R.; Matijevic, E. J . Colloid Interface Sci. 1974,48,417-426.
particle concentration, both the elasticity and the yield stress increase with ionic strength. The experimental data nicely fit the power law:
G ( 0 )= A(C - Co)a where A is a constant and C is the solid concentration expressed as mass of solid over mass of liquid. For concentrations less than few percent, C is very close to the mass fraction. Such a power law was already observed for colloidal suspensions of clays.11J5 The exponent a (2.35)and the prefactor A are not sensitive to the ionic strength and eq 6 can be considered as a master curve describing the elastic properties of the laponite gel near the sougel transition. The numerical determination of Co permits an estimation of the concentration at which gelation appears, in good agreement with the appearance of a yield stress. Figure 5 shows the ionic strength dependence of the threshold concentration CO. The transition line in the diagram separates a viscous sol-phase domain from a mechanical viscoelastic gel phase. As ionic strength increases, the clay concentration threshold is shifted toward lower to 2 x values. For an ionic strength ranging from M, rheological properties evolve continuously. Above I=2 x M (for NaCl), flocculation occurs and no measurement is performed on the heterogeous floc. 2. Osmotic Pressure and Equation of State. The equation of state of the suspension involves directly the interparticle interaction. Figure 6 shows the results obtained by osmotic stress of diluted laponite suspensions against Dextran solutions of fixed ionic strength a t pH =
Phase Diagram of Colloidal Suspensions
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Langmuir, Vol. 11, No. 6, 1995 1945
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I . 10. Several remarks can be made. Most notable are: a ' , net repulsion between the laponite particles, even in the 2 / concentrated regime above the sougel transition; the I I -I 'I ,t agreement of the salt effect with the predictions of the c f classical DLVO theory-adding salt reduces the screening ' ( length and thus reduces the swelling p r e s s ~ r e ; the ~~-~~ appearance along the different curves of a break followed , , by a pseudoplateau separating the liquid and the gel phase; - _ the good reversibility of the stressed gels, which can be reswollen to give back the original sol; the lack of macroscopic phase separation along the pseudoplateau. , While the first two observations are characteristic of c( ' # colloidal suspensions of charged particles of any geome* e , try,29-32the last observations address more specifically the sougel transition. As shown in Figure 5, the location Figure 8. Same as Figure 7, for a laponite concentrationof 3% of the first break in the osmotic pressure coincides (gel phase). reasonably well with the sougel transition line. The sol/ gel transition is directly linked with an equilibrium phase performed on laponite suspensions at clay concentrations transition, assigning a thermodynamical statute to this 1%wlw and 3% wlw, respectively, for an ionic strength "mechanical" transition. M (pH stabilized at 10). Under these conditions, the 3% w/w sample is a gel. No aggregation of the clay 3. Microstructure and Particle Organization. particles may be detected. Cryofractures show a homoFigures 7 and 8 show cryofractures and TEM observations geneous dispersion of the particles, without any direct contact. The maximum size of the trace of the laponite (29)Dubois, M.; Zemb, Th.; Belloni, L.; Delville, A.; Levitz, P.; Setton, R. J. Chem. Phys. 1992,96, 2278-2286. particles is around 400 A, in good agreement with former (30) Delville, A. Langmuir 1994,10, 395-402. determinations. Cryofiacture at higher ionic strength and (31)Israelachvili, J.N. Intermolecular and Surface Forces;Academic solid concentration are currently under way and will be Press: New York, 1985. (32)Delville, A,; Laszlo, P. New J. Chem. 1989,13, 481-491. presented elsewhere.
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Mourchid et al.
1946 Langmuir, Vol. 11, No. 6, 1995 loz
the naked particle plus the Debye screening length
, ' " I
'C.,
10'
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a =
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10' 1o
-~ 10'
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Q (A-') Figure 9. Evolution of the SAXS spectra normalized by using the contrast factor K, (see text) of laponite suspensions ( I = 10-2 M), as a function of clay concentration(. * 3.85%and, - -, 12.3%. The simulated spectrum of an isolated particle (-)*is used as a reference (diameter = 300 A and thickness = 10 A). e,
-
SAXS can also be used to quantify the organization of the clay particles, avoiding the alteration of the suspension by the addition of glycerol and by the decrease of temperature. Analysis of the high-Q regime (above 3 x low2A-1) of the SAXS curves, using the theoretical expression of diffusion by a randomly oriented cylinder23 (eqs 2-41, yields an estimation of the average thickness and the diameter of the particle, thus found to be 10 and 300 A, respectively. In Figure 9, the scattering curves are normalized by using the contrast factor Kc (cf. eq 3). Scattering intensities from 3.8% and 12.3% wlw gel M NaC1) are phases a t relatively high ionic strength ( shown in Figure 9. Three regions can be observed in these A-l, all the curves diffusion curves. Above 3 x superimpose, giving information on the shape of the ipdividual particles (see section 11-3). Between 2 x A-l and 3 x the normalized scattering of the gel phases is lower than Ip(Q),the scattering intensity of an isolated particle. In the very low Q region, a positive divergence from I&Q) is observed, going approximately as Q-3. These results, obtained at high ionic strength, are very similar to the observations of Morvan et aZ.18 performed at low ionic strength. However,the comparison between the results displayed in our Figure 9 and Figure 2 of ref 18 shows a difference at low q and for the 3.8% wfw sample, essentially due to a variation ofionic strength.
IV. Discussion 1. The Thermodynamic Statute of the SoVGel Transition. The phase diagram of laponite suspensions, constructed from the rheological data, shows a sol/gel transition coinciding with the phase transition (Figure 5), as deduced from the osmotic measurements. This simultaneity of both transitions assigns a thermodynamical statute to the sougel transition. However, the role of the ionic strength on the soygel transition needs clarification: the sougel and phase transitions are shifted to lower particle concentration by an increase of the ionic strength (Figure 51, while the swelling pressure and thus the interparticle repulsions are reduced (Figure 6). The phase diagram and equation of state of suspensions of spherical charged particles have often been ~ t u d i e d . l - ~ Increasing the ionic strength always reduces the swelling pressure of the colloidal suspensions and shifts the appearance of solJgel transition to higher concentration. This is due to electrostatic interaction between the particles.30 A rationalization was frequently p r ~ p o s e d : ~ the soft repulsion due to the overlap of the counterion diffuse layers is replaced by a renormalized hard-core repulsion with an effective radius equal to the radius of
where e is the electronic charge and I is the ionic strength. Two opposite hypotheses are possible for the interparticle interaction controlling the behavior of suspensions of anisotropic charged particles: either attraction, responsible for particle aggregation, or repulsion between dispersed particles. The results collected on laponite gels do not provide any argument in favor of a local aggregation between the particles forming tactoids or "house of cards" structures almost for ionic strength lower than the flocculation threshold: (i)At pH = 10, all amphoteric acid groups located on the particle border are dissociated and negatively charged. The main mechanism involved in formation of the house of cards is then suppressed. (ii) The pressure is always positive (Figure 6). (iii) Cryofracture shows a complete dispersion of isolated particles (Figures 7 and 8). (iv) SAXS spectra (Figure 9) define an effective particle thickness (10 8)equivalent to the size of the isolated clay p a r t i ~ l e . ~ ~ > ~ * These results and the reversible unswelling of stressed suspensions (even through the line marking the phase transition) are more in favor of a repulsive interparticle interactions. 2. Analysis of the Equation of State. The osmotic pressure may be split in four contributions: (i) the Coulombic long-range interaction between charged particles, their neutralizing counterions, and the dissociated salt; (ii) the entropy associated with the numerous equilibrium configurations available to the ions forming the diffuse layers around each particle; (iii) the ion-ion and ion-particle excluded-volumeeffects; (iv) the particleparticle excluded-volume effects. Whereas the three last contributions are always positive, the energetic contribution to the pressure is negative because the particlecounterion attraction (resulting in counterion condensation) overcomes30the particle-particle and counterioncounterion repulsions. Calculations of the swelling pressure from a solution of the Poisson-Boltzmann (PB) equation29,31,32~35 is at the basis of the DLVO theory. The ion distribution is evaluated in a cell36surrounding each isolated particle, neglecting mutual organization of the particles.37 The derivation of the PB approximation also neglects the ionic ~ o r r e l a t i o n .However, ~~ the PB formalism correctly predicts the equation of state of suspensions oflarge swelling particles (like sodium montmorillonite^^^^^^) neutralized by monovalent counterions. In the presence of divalent counterions, ionic correlation must be taken into ac~ o u n t . The ~ ~numerical ,~~ procedure used to solve the PB equation further imposes a planar symmetry for the computation cell, assuming a parallel alignment of uniformly charged plates with an infinite lateral extension. This approximation is valid for large particles (like m ~ n t m o r i l l o n i t e ~ but ~ , ~ the ~ ) , neglect of border effects40 (33) Grim, R. E. Clay Mineralogy; McGraw-Hill: New York, 1953. (34) Brindley, G. W.; Brown, G. Crystals structures of Clay Minerals and Their X-ray Identification; Miner. Sac.: London, 1980. (35) Delville, A.; Laszlo, P. Langmuir 1990, 6, 1289-1294. (36) Rice, S. A,; Nagasawa, M. Polyelectrolyte Solutions; Academic Press: New York, 1961, p 232. (37) Linse, P.; Jonsson, B. J . Chem. Phys. 1983, 78, 3167-3175. (38) Kjellander, R.; Kkesson, T.;Marcelja, S . J.Chem. Phys. 1992, 97. 1424-1431. (39) Kjellander, R.;Marcelja, S.; Pashley, R. M.;Quirk, J. P. J. Phys. Chem. 1988,92, 6389-6492. (40) Chuang, F.R. C.; Sposito, G. J . Colloidlnterface Sci. 1994,163, 19-27.
Phase Diagram of Colloidal Suspensions
Langmuir, Vol. 11, No. 6, 1995 1947 2.0 10'
..-
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0.1
,
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0.2
0.3
0.4
0.5
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Mass fraction of solid
Figure 11. Equations of state of disklike particles as a function oftheir anisotropy ratio (p= thicknesddiameter]. All the curves are calculated for a particle diameter of 300 A.
may become crucial for small laponite particles. Despite all these approximations, Figure 10 shows qualitative agreement with the experimental data at an ionic strength of M. This simple electrostatic computation ignores the finite size of particles and cannot reproduce the singularity related to the sougel transition also driven by excluded-volume effects. In fact, both electrostatic and excluded-volume contributions must be taken into account since their extents are comparable. For instance, at an ipnic strength of M, the diameter of the particle (300 A), which controls in part the excluded-volumeeffects, is equal to the Debye screening length controlling the range of electrostatic repulsion. Using the strategy discussed above for spherical particles, we also show in Figure 10 the equation of state of t y o suspensions of hard spheres of diameter 300 and 600 A, respectively. The first case corresponds to a model of randomly dispersed naked particles, while the second case roughly integrates the contribution of the diffuse layer. Both equations of state are calculated by using the Carnahan- Starling approximation:
where 7 is the packing fraction (v = 4qR3/3)and e is the density number. The hard-sphere equation of state diverges at an equivalent laponite concentration of 13.3% wfw for the naked particles and 1.66%wfw for the dressed particles. This last concentration is close to the break observed for the solfgel transition at low ionic strength (10-4 MI. This simple computation indicates that laponite particles at the solfgel transition and at low ionic strength M) can interact through their diffuse layers and are not totally free to move separately. One possible way to minimize the total free energy is to align neighboring particles in order to save free volume and reduce their excluded-volume interaction. This is typically the situation encounteredin the Onsager phase transition7of hard disklike particles. As suggested by Forsyth et a1.,I0 a classical but approximate way of taking into account both excluded-volume and ionic condensation effects is a renormalization of the volume of the particles. As for (41)Hansen, J. P.; McDonald, I. R. Theory ofsimple Liquids; Acad. Press: London, 1986. (42) Levesque, D.; Weis, J. J.; Hansen, J. P. In Topics in Current Physics, Monte Carlo Methods in Statistical Physics; Binder, K., Ed.; 1979; Vol. 7, Chapter 2.
spherical colloids, the particle surrounded by its cloud of condensed counterions is treated as a rigid body. The key parameter of this treatment is the effective asymmetry parameter (p = HIR) of the renormalized rigid Barticle, where R is the radius of the naked particle (150 A in our case) and H the global half-thickness of the renormalized particle, i.e;, the sum of the half-thickness of the naked particle (5 A) plus the thickness of its diffuse layer ( K - ~ ) . As it will be shown later, asymetric extension of the double layer (cf. Figure 12) permits justification of the actual definition of the p parameter. By contrast with the rough approximation of the renormalized hard sphere (see above),this approach explicitely includes the shape of the particle in the evaluation ofthe diffuse layer and excludedvolume effects. In Figure 11, the evolution of the osmotic pressure (as given by the Onsager formulation) is shown for different asymmetry parameters ran@ng from the naked particle (high ionic strength, H = 5 A, R = 150 A) to an intermediate situation ( H = 22.5 A, R = 150 A). The comparison with experimental results shown in Figure 6 is interesting. Figure 11shows a first-order phase transition shifting to lower particle concentration when the asymmetry parameter of the renormalized particle is reduced, i.e., when salt is added to the suspension, reducing the thickness of the diffuse layer. This is also observed in Figure 6 , but agreement is only qualitative: the range of concentrations where a phase separation appears on the theoretical curve is 1order of magnitude larger than the experimentalvalues. The computed osmotic pressures are 10 times larger than the experimental values and no real biphasic regime can be observed during the experiment. Furthermore, direct observation of the gel phase at rest hardly shows birefringence, as expected for a nematic phase. Clearly, the former approach suffers from several drawbacks: it is restricted to two-body interactions and its validity is limited to dilute suspensions; it models the soft double-layer repulsion by a hard-core potential; it does not handle the angular dependence of long-range electrostatic interactions and double-layer repulsion which may counteract particle alignment; it considers only extensions of the double layer from the basal surface of the particle. However, these rough computations explain the influence of salinity: increasing ionic strength reveals the intrinsic asymmetry of the solid particles and favors phase transition at lower solid concentrations. To be more quantitative and go a step further, we develop a model to analyze the angular dependence of the Coulombic interaction between two disklike particles surrounded by their atmosphere of counterions. The ionion and clay-ion interactions are described by the primitive model: electrostatic long-range interaction plus short-range hard-core repulsion. The diameter of the
Mourchid et al.
1948 Langmuir, Vol. 11, No. 6, 1995
4
'
3
'
A
I
Y
N
' i ; 2
L AU
- 266
I
E
-
0
-1
A
P 5
60
Figure 14. Average concentration profile (c(z), -) of the counterions condensed on the basal surfaces of the laponite particles. The integration of this profile defines the average charge of the counteGons (Q(z),- - -1, inside a cylinderof height 22 and radius 150 A centered around each laponite particle.
in Figure 13 may be compatible with the repulsion shown by the equation of state (Figure 6). A semiquantitative evaluation of the different contributions to the swelling pressure is obtained by a development of the vinal equation41
\
I'
40
z (A)
Figure 12. Snapshots showingthe equilibrium configurations of all the ions distributed around two laponite particles (see pxt). The distance between the particle centersis fmed at 200 A and two relative orientations (parallel and perpendicular) are presented. A
20
kJ/mol
-2
0
200
400
600
000
D (A)
Figure 13. Variation of the electrostaticenergy as a function of the separation between two parallel laponite particles (see text).
solvated Na+and C1- ions are set to 4.2 A;4Sthis parameter is important since it monitors the minimum approach of the clay counterion. The simulations are performed in a (Np,,us;V;T) canonicdgrand canonical ensemble, where the number of particles (Np)and their corresponding counterions are constant, while the salt is in equilibrium with a reservoir of fixed chemical potential Ole). This ensemble is necessary to describe correctly the experimental conditions of the dialysis. We use the classical Monte Carlo algorithm of Metropolis,24modified by ad am^,^^ to describe the ionic distribution around the particles as a function of their relative orientation. Figure 12 shows projections of two equilibrium configurations of all the ions distributed around two laponite particles isolated in a 8 x lo6 Hi3 box. The clouds of counterions condensed on the basal surfaces of the particles show up. The 10 A thick empty line embedded in each trace of condensed counterions corresponds to the location of a clay particle and proves the absence of noticeable condensation of the counterions on the lateral faces, justifying the approach of Forsyth et a1.lo and the former definition of the /?parameter. Figure 12 shows the results obtained for two relative orientations of the lapoqite particles (parallel and perpendicular), with a 200-A center-to-center separation. The electrostatic energy of the perpendicular configuration is 266 kJ/mol lower than that of the parallel configuration, but there is no difference in energy for an interparticle separation three times larger. An energy difference of 266 kJ/mol may appear excessive, but it corresponds to 133 J/mol of elementary charge, i.e., l/50of the interfacial electrostatic energy previously determined for infinite planar surfaces.32 Figure 13 shows the variation of the electrostaticenergy of two parallel laponite particles as a function of their separation. One can ask how the strong attraction shown (43) Cooker, H.J. Phys. Chem. 1976,80,2084-2091.
where Np and Ni are respectively the total number of particles and ions. Since the external contribution is equal to -3PV, eq 9 is rewritten as41
((wed)
where the indices iland ,u describe the different elements (particle, counterion, or co-ion),@ O i l their average density, NAtheir total number, andgu the distribution function of the (ij)pair of elements (particle or ion). A n estimate of the energetic attractive pressure is given by the slope in Figure 13
dU Penegy = - -1 SdX
with a maximum contribution of -1 x lo6Pa a t a particle separation of 25 A. This attraction is balanced by the entropic repulsion
a t room temperature, with Cop in w/w. Another repulsion is the hard-core interaction between the different entities. As shown previously,3°the main contribution to this hardcore repulsion originates from the collisions between the charged particles and their condensed counterions. This term may be estimated from the average local concentration of counterions in direct contact with the basal surfaces of the particles
Pmnbd % kTeop[ 1 + TR'('H 2n $
+ ai)ci(eontad)]
(13)
>
where ai is the counterion radius. Figure 14 shows the local concentration profile of counterions condensed on the basal surface of the particle. Since q(c0ntact) 4M (see Figure 141, one obtains, a t room temperature:
Langmuir, Vol. 11, No. 6, 1995 1949
Phase Diagram of Colloidal Suspensions .
1.21
-0.2
I
'
-'a
0
200
.
A..
'
I
.
I
o o o
.
I
400
. . .. .. . . -1
I
800
.
1
.
1
2
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800
1000
(A) Figure 16. Radial distribution function (0 and 0) and local order parameter (A and A) calplated from equilibrium configurations of hard disks of 300-A(filled symbols)and 600-A (open symbols) diameter a t the same density (9.9%w/w). r
Figure 15. Random planar section of an equilibrium configuration of 48 000 hard disks distributed in 4.1pm3 (see text).
Assuming a parallel alignment of the laponite particles, the 25-A interparticle separation used to estimate the energetic attraction (eq 11) corresponds to 130% w/w concentration. As discussed previously, the hard-core and entropic repulsions (3 x lo6 Pa) overcome the energetic attraction (-lo6 Pa), leading to the swelling behavior shown in Figure 6. A complete treatment of the orientational electrostatic and entropic contributions will be performed in the near future, to derive an effective interparticle potential which will be used in a global modelization of the dynamical and equilibrium properties of laponite suspensionsin the context of the one-component plasma.42 3. Interparticular Structure of the Suspension. The SAXS spectra of the laponite suspensions shown in Figure 9 contain complementary information on the organization of these gel phasoes. While the high-wavenumber regime (Q> 3 x A-l) is only sensitive to the intrapaGicular correlations, the intermediate regime (2 x A-l < Q < 3 x A-l) depends on interparticle correlations. Monte Carlo simulations of the organization of hard disks (Figure 1)were performed to analyze the structure of laponite suspensions at relatively high ionic strength by a direct comparison between calculated and observed S Y S spectra. The diameter of the simulat2d disks is 300 A, and the thickness of the pa$icle is 13.2 A, i.e., thickness of a dryed clay particle (9 A)3:v34 plus the diameter of a solvated sodium counterion (4.2 A)j3 Figure 15 shows a random planar section of a 4.1-,um3 box containing48 000 particles (cf. Figure l),i.e., at a particle concentration 0.011 v/v ( ~ 0 . 0 3w/w). Figure 15 is very similar to the cryofracture shown in Figure 8. Local order due to particle alignment exists, disappearing a t interparticle separations larger than a few diameters. The local structure is quantified by a computation of the radial distribution function of particle centers, g(r), and the local nematic order parametef14
where the sum is evaluated for the N(r) particles located in a shell of thickness dr at a distance r from the central (44)Eppenga, R.;Frenkel, D.Mol. Phys. 1984,52, 1303-1334.
xu
a \
10"
v
10
I
2
1 0'
1o-2 Q (K')
10'
Figure 17. SAXS spectra calculated from Monte Carlo simulations of 48 000 (W, 3.3%w/w) and 144 000 (0,9.9%w/w) hard disks. The simulated spectrum of anjsolated particle (-1 is used as a reference (diameter = 300 A and thickness for electronic contrast = 10 A).
particle. Evolution of the radial distribution and the order parameter is shown in Figure 16for two different particle diameters (300 and 600 A) a t the same particle density (9.9%w/w). We can observe a local alignment (S(r)G 1) and a strong depletion of particles in the direct vicinity of the central particle. The spatial extension of the particle alignment is limited to the range of the particle and no nematic phase emerges (the order parameter is nearly zero at large distances). As shown in Figure 16,increasing the asymmetry of the particle by a factor 2 corresponds, a t the same volumic fraction, to an apparent extension of the orientational domains. However, these results can be rescaled on the same curve by using the particle size as a normalization parameter. At the limit of infinitely thin platelets, a perfectly nematic phase44forms at lower particle density. To sum up, excluded-volumeinteractions of these platelike particles induce an anisotropic depletion around each particle. In comparison with the situation of fully penetrable disks, many angular configurations are no longer accessible. Anisotropy favors parallel alignment of the particles for distances less than particle size. The undercorrelation observed in Figure 16 for g(r) is mainly responsible for the depletion of the small-angle scattering curves in the intermediate Q range (cf. Figure 9). We show in Figure 17 calculated SAXS spectra obtained from MC simulations of 48 000 ancjl44 000 hard disks (diameter 300 and thickness 13.2 A) in a 4.1-,um3 volume. These two situations correspond to concentrations 3.3% and 9.9% w/w, respectively. The agreement with experimental spectra performed a t high ionic strength is good (see Figure 9). For higher density number, the undercorrelation observed in the
Mourchid et al.
1950 Langmuir, Vol. 11, No. 6, 1995 intermediate Q regime is more pronounced and appears at higher Q values. Clearly, depletion and parallel alignment take place at shorter distances when particle concentration andor ionic strength increases. The low Q regime of the SAXS spectra is not directly accessible, since artifacts due to the finite size of the simulation box are directly apparent in the calculated scattering spectra and mask the diffusive behavior of the infinite system. However, as shown in Figure 16,g(r) is almost constant at large distances. Up to now, M.C. simulations of hard disks do not reproduce experimental density fluctuations at large distances. 4. LongRange Structure of the Suspension. In the low Q regime, SAXS spectra recorded for suspensions of laponite a t high ionic strength (lo+ M) show a Q-3 divergence of the diffused intensity (Figure 9) for a laponite concentration as low as 3.85%wlw. Such a divergence in the low Q regime may be due to large-scale heterogeneitiesls of the suspension. At a lower ionic strength ( MI, the Q-3 divergence is only observed at higher laponite concentration (10% wIw).l8 It is worthwhile to mention that the undercorrelation appearing in the intermediate Q range evolves in parallel with the Q-3 divergence. As already mentioned, this depletion grows and appears at higher Q as the solid concentration andor ionic strength increases. A working hypothesis, compatible with these different observations and able to explain the sollgel transition, considers the appearance of orientational microdomains of platelike particles as already proposed by Ramsay et al.17 at low ionic strength. In order to evaluate some consequences of this hypothesis, let us consider a suspension with specific constant particle concentration and increasing ionic strength. Starting from a homogeneous sol (see Figure 5) of well-dispersed particles, the increase of ionic strength reveals the intrinsic anisotropy and reduces the net electrostatic repulsion between the particles. As in the case discussed by Onsager and Forsyth et al., but a t a more local scale, a particles alignment at shorter distances can be induced. The appearance of microdomains of closer particles will be promoted when ionic strength increases, inducing the formation of large regions having a lower particle density. This long-range density fluctuation can explain the low Q divergence of X-ray scattering from stiff gels. Moreover, the role of ionic strength in shifting the sollgel transition to lower volume fraction can be justified. Particles remaining in the lower density regions will contribute to confine microdomains and impose osmotic pressure. Alignment of these clusters of partially oriented particles during the flow at low Peclet number of a soft gel can justify the observed flow birefringence. Interrelation and/or interconnection of these microdomains will be responsible for the appearance of the gel phase. Finally, this hypothesis implies a hierarchical organization consistent with the large time scale and the nonexponential kinetics of the observed mechanical relaxation of laponite gels.45,46 At this point, several questions still need to be answered: (i) Can the existence of orientational microdomains be proved experimentally (cryofracture and imaging techniques a t different ionic strengths and solid fractions ...)? (ii) What is the orientational correlation length of these microdomains and how can its finite value be explained? (iii) What are the type of interactions inducing the dynamical confinement of particles within these microdomains (van der Waals, ionic correlations, ~~
(45) Mourchid, A.; Levitz, P.; Delville, A,;Van Damme, H. Les Cuh. Rhlol. 1992, 10, 89-97.
(46)Mourchid, A.; Delville, A.; Lgcolier, E.; Levitz, P. Les Cuh. de RhBol. 1994, 13, 130-139.
pressure of the gas of particles left in the low-density regions, ...)? (iv) Is the appearance of the low Q divergence of X-ray scattering continuous or is it strongly related to the soygel transition line and how is this divergencerelated to the long-range organization of the gel?
V. Conclusions The gelation of clay suspensions was previously discussed in terms of mechanical and structural transition. In this work, we have shown that this transition is also a reversible thermodynamical transition. For ionic strength lower than the flocculation threshold (the horizontal line in Figure 5)) available structural, viscoelastic, and thermodynamics properties of the laponite gels evolve continuously with ionic strength and solid fraction. It is worthwhile to recall that (i) specific structural features such as the q-3 divergence and the undercorrelation in the intermediate q range are observed either at low solid fraction and relatively high ionic strength or at high particle and low salt concentrations, (ii) elastic properties of the gel phase are well described by eq 6, where the parameter A and the exponent a are independent of the ionic strength, and (iii) the sougel transition line, location of rapid change in the suspension, is a continuous decreasing curve from I = to M. In this range of ionic strength, it is reasonable to assume the same mechanism of gelation driven by the interplay of some interparticle interactions. Flocculation involving direct contact between particles does not seem appropriate to explain the transition. On the contrary, electrostatic interactions and excluded-volume effects of anisotropic charged particles seem to play a major role in the sollgel transition. In contrast with the predietions of Forsyth et aZ.I0 concerning the Onsager first-order transition,' gels of laponite do not show either birefringence (at rest) or demixing in two well-separated phases (as observed for highly anisotropic bentonite6). In addition to possibly antagonistic electrostatic effects (cf Figure 121, the large extent of particle alignment, necessary to form a macroscopic nematic phase, seems impossible for laponite, because of its reduced anisotropy. The same situation was recently reported for suspensions of another slightly anisotropic particles (boehmiteg). A possible hypothesis explaining the coincidence of the thermodynamical transition with a mechanical sougel transition involves the formation of orientational microdomains of particles able to be connected in space. This proposal raises several questions on the finite size and the long-range organization of these microdomains. Experiments and numerical simulations are currently under way in our laboratory to test this hypothesis.
Acknowledgment. We thank Dr. T. Zemb (C.E.A., Saclay) for his help in performing U.S.AX.S experiments and Mr. Tranchant (Dior-Orlbans) for cryofractures and TEM observations. It is a pleasure to acknowledge Drs. M. Dubois, B. Cabane, C. Bonnet-Gonnet, M. Nabavi, F. Bergaya, R. Setton; and H. Van Damme, for helpful discussions. The Monte Carlo calculations were performed locally a t the CRMD on a workstation (HP 9000/720) for the simulations of hard disks and on a Cray YMP supercomputer (IDRIS, Orsay), for the diffuse layer calculations. We thank Mr. G . Bourhis (IDRIS, Orsay) for his contribution in the drawing of Figure 1. LA9410422
