Colloidal Gas−Liquid Condensation of Polystyrene Latex Particles

pubs.acs.org/Langmuir © 2009 American Chemical Society

Colloidal Gas-Liquid Condensation of Polystyrene Latex Particles with Intermediate Ka Values (5 to 160, a . K-1) Masamichi Ishikawa*,† and Ryota Kitano*,‡ †

Department of Innovative and Engineered Materials, Tokyo Institute of Technology, 4259 Midori-ku, Nagatsuta-cho, Yokohama, Kanagawa 226-8502, Japan, and ‡AGC Techno Glass Co. Ltd., 3583-5 Kawajiri, Yoshida-cho, Haibara-gun, Shizuoka 431-0302, Japan Received August 6, 2009. Revised Manuscript Received September 22, 2009 Polystyrene latex particles showed gas-liquid condensation under the conditions of large particle radius (a . κ-1) and intermediate κa, where κ is the Debye-H€uckel parameter and a is the particle radius. The particles were dissolved in deionized water containing ethanol from 0 to 77 vol %, settled to the bottom of the glass plate within 1 h, and then laterally moved toward the center of a cell over a 20 h period in reaching a state of equilibrium condensation. All of the suspensions that were 1 and 3 μm in diameter and 0.01-0.20 vol % in concentration realized similar gas-liquid condensation with clear gas-liquid boundaries. In 50 vol % ethanol solvent, additional ethanol was added to enhance the sedimentation force so as to restrict the particles in a monoparticle layer thickness. The coexistence of gas-liquid-solid (crystalline solid) was microscopically recognized from the periphery to the center of the condensates. A phase diagram of the gas-liquid condensation was created as a function of KCl concentration at a particle diameter of 3 μm, 0.10 vol % concentration, and 50:50 water/ethanol solvent at room temperature. The miscibility gap was observed in the concentration range from 1 to 250 μM. There was an upper limit of salt concentration where the phase separation disappeared, showing nearly critical behavior of macroscopic density fluctuation from 250 μM to 1 mM. These results add new experimental evidence to the existence of colloidal gas-liquid condensation and specify conditions of like-charge attraction between particles.

Introduction Gas-liquid phase separation analogous to vapor-liquid condensation in atomic systems has been studied in dilute and deionized colloidal suspensions of polystyrene particles.1 After the theoretician’s acceptance of the void formation in the densitymatched colloidal suspension, the role of like charge attraction between colloidal particles in causing colloidal phase separation was debated.2 However, it is often claimed that the range of the miscibility gap in the gas-liquid condensation experiments has not been clearly identified in any kind of phase diagram such that it presents the range of instability. Van Roij et al., Levin et al., and Warren proposed a van der Waals-like instability of charge-stabilized colloidal suspensions based on linear Poisson-Boltzmann theory.3-9 The essence of their argument is the full evaluation of free-energy terms that relate the macroion-small ion, macroion-macroion, and small ion-small ion interactions. Tamashiro and Schiessel raised objections to the validity of their linear theory, in particular, as to whether the gas-liquid phase instabilities are mathematical artifacts of the linearization approximation because the evaluation of nonlinear Poisson-Boltzmann theory showed no miscibility gap.10 *Corresponding author. Tel: þ81-45-924-5561. Fax: þ81-45-924-5562. E-mail: [email protected]. (1) Tata, B. V. R.; Jena, S. S. Solid State Commun. 2006, 139, 562–580. (2) (a) Ito, K.; Yoshida, H.; Ise, N. Science 1994, 263, 66–68. (b) Yoshida, H.; Ise, N.; Hashimoto, T. J. Chem. Phys. 1995, 103, 10146–10151. (c) Dosho, S.; Ise, N.; Ito, K.; Iwai, S.; Kitano, H.; Matsuoka, H.; Nakamura, H.; Okumura, H.; Ono, T.; Sogami, I. S.; Ueno, Y.; Yoshida, H.; Yoshiyama, T. Langmuir 1993, 9, 394–411. (3) van Roij, R.; Hansen, J. P. Phys. Rev. Lett. 1997, 79, 3082–3085. (4) van Roij, R.; Dijkstra, M.; Hansen, J. P. Phys. Rev. E 1999, 59, 2010–2025. (5) Zoetekouw, B.; van Roij, R. Phys. Rev. E 2006, 73, 021403. (6) Levin, Y.; Barbosa, M. C.; Tmashiro, M. N. Europhys. Lett. 1998, 41, 123– 127. (7) Tmashiro, M. N.; Levin, Y.; Barbosa, M. C. Physica 1998, A258, 341–351. (8) Diehl, A.; Barbosa, M. C.; Levin, Y. Europhys. Lett. 2001, 53, 86–92. (9) Warren, P. B. J. Chem. Phys. 2000, 112, 4683–4698. (10) Tmashiro, M. N.; Schiessel, H. J. Chem. Phys. 2003, 119, 1855–1865.

2438 DOI: 10.1021/la9029084

Addressing that point, Zoetekouw and van Roij again reported the positive correspondence of the nonlinear version of their theoretical instability with experimental results, introducing the criteria of ZlB/a g 24, where Z is the charge, a is the radius, and lB is the Bjerrum length.11 These discussions assumed the weak coupling limit of a low volume fraction of colloids, low surface charge density, low counterion valency, and high temperature. Netz further analyzed the opposite limit of strong coupling and showed the existence of the attraction of like-charged macroions by considering the counterion condensation process that directly affects the effective interactions.12,13 Counterion condensation has been known in polyelectrolyte solutions. Manning proposed a model of simple polyelectrolytes.14-16 This condensation results in effective charge saturation due to the screening of the surface charge by counterions and coions, which are treated theoretically as charge renormalization in the Debye-H€uckel theory.17,18 The effective charge changes drastically according to the bare charge of the particle because with increasing bare charge the ionic density in the electrical double layer grows nonlinearly too large. In the field of electrophoresis measurements, it is well known that counterion condensation around charged colloids has an important effect on zeta potential measurements. Fluid flow of the surrounding solution and the external electrical field deform the double layer when its thickness is approximately located at the surface of shear. This phenomenon, called the relaxation effect, has been extensively (11) Zoetekouw, B.; van Roij, R. Phys. Rev. Lett. 2006, 97, 258302. (12) Netz, R. R. Eur. Phys. J. E 2001, 5, 557–574. (13) Naji, A.; Netz, R. R. Eur. Phys. J. E 2004, 13, 43–59. (14) Manning, G. S. J. Chem. Phys. 1969, 51, 924–933. (15) Groot, R. D. J. Chem. Phys. 1991, 95, 9191–9203. (16) Levin, Y. Rep. Prog. Phys. 2002, 65, 1577–1632. (17) Alexander, S.; Chaikin, P. M.; Grant, P.; Morales, G. J.; Pincus, P. J. Chem. Phys. 1984, 80, 5776–5781. (18) (a) Trizac, E.; Bocquet, L.; Aubouy, M. Phys. Rev. Lett. 2002, 89, 248301. (b) Bocquet, L.; Trizac, E.; Aubouy, M. J. Chem. Phys. 2002, 117, 8138–8152.

Published on Web 10/15/2009

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studied to relate the electrophoretic mobility to the ζ potential around a particle.19-21 The effect is prominent in the range of intermediate κa with values ranging from 5 to 150. Latex particles with a diameter exceeding 1 μm show a relaxation effect in the relatively low ionic strength solution. Colloidal gas-liquid phase separation is proposed to be a consequence of the nonlinear many-body interaction between the particles with the surrounding ionic atmosphere as described in detail by Warren.9 When a . κ-1 (κ is the Debye-H€uckel parameter), the nonlinear effects are confined to be in the immediate vicinity of macroions with an extension of κ-1. Therefore, the charged colloids with a large diameter and intermediate κa may behave according to the linear Poisson-Boltzmann theory. This kind of experimental condition has the potential to examine the phase instability because nonlinear interactions activate the attraction between particles and the linear approximation is still valid in the analytical treatment. Unfortunately, there are few gas-liquid condensation experiments using large particles with such intermediate κa. We have found an equilibrated phase separation of the colloidal suspension under the condition of intermediate κa. Although the particles used were relatively large, they still migrated upward against the gravitational force as a result of Brownian agitation. To confine their presence in a monoparticle layer for further quantitative treatment, we enforced the sedimentation of the particles by adjusting the solvent density. This caused monoparticle layer condensation at the bottom of the experimental cell, but the layer still floated above the bottom glass surface because of electrostatic repulsion. As a result, all of the particles initially sedimented to the bottom and then laterally condensed toward the center of the cell. An important parameter to consider is the effective surface charges of the macroions in the analytical evaluation of the phase instabilities under the imposed experimental conditions. We determined the effective surface charges using an electrophoretic mobility measurement, and their validity was checked by comparison with the relationships derived from the linear Poisson-Boltzmann theory.

Experimental Section The colloidal particles of sulfonated polystyrene (PS) microspheres (diameters, 1 and 3 μm) were obtained from Duke Sci. Corp. and treated with a mixed-bed ion-exchange resin, BioRAD-501-X8(D), for approximately 1 month. The experimental cell was made of crown glass with a disk-shaped inner volume with a 10.0 mm diameter and a 0.5 mm thickness as shown in Figure 1. The glass surface was processed with a flatness of 4 nm. The lower and middle glass plates were bonded via a plastic agent. The upper glass was sealed using vacuum grease, and air bubbles were carefully excluded when each sample was filled. The cell was located on a flat horizontal metal plate to diminish temperature gradients and an inclination to gravity. The particle concentration was varied from 0.001 to 0.23 vol %. The sedimentation force was enhanced to realize the monoparticle layer condensation on the bottom plate of the cell by mixing water with ethanol in a volume ratio of 0-77%. Both solvents were fully deionized by the ionexchange resin in advance. The density of the solvent was determined such that nearly 100% of the particles were distributed within the thickness of their diameter, assuming the sedimentation (19) (a) O’Brien, R. W.; Hunter, R. J. Can. J. Chem. 1981, 59, 1878–1887. (b) O'Brien, R. W.; White, L. R. J. Chem. Soc., Faraday Trans. 2 1978, 74, 1607–1626. (20) Hunter, R. J. Zeta Potential in Colloid Science: Principles and Applications; Academic Press: New York, 1981. (21) Hunter, R. J. Foundations of Colloid Science; Oxford University Press: Oxford, U.K., 2001. (22) Russel, W. B.; Saville, D. A.; Schowalter, W. R. Colloidal Dispersions; Cambridge University Press: Cambridge, U.K., 1989. (23) Beckham, R. E.; Bevan, M. A. J. Chem. Phys. 2007, 127, 164708.

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Figure 1. Experimental setup of colloidal phase separation. The optical cell, was made of crown glass with 10 mm diameter and 0.5 mm thickness (A). The Gravitational sedimentation of the polystyrene latex was enhanced by adjusting the solvent density by mixing 50:50 water/ethanol, and the monoparticle layer was maintained during the liquid-phase condensation (B). equilibrium density profile n(z)22,23
 zPe nðzÞ ¼ n0 exp h

ð1Þ

where z is the height from the bottom of the cell, n0 is the particle number at z = 0, h is the height of the cell, Pe = hU0/D, U0 is the Peclet number, U0 is the stationary-state sedimentation velocity, and D is the diffusion coefficient. The monoparticle layer sedimentation was confirmed using confocal laser scanning microscopy (CLSM). Optical microscopy was used for in situ observation of the colloidal condensation at the bottom of the cell. KCl was added as a salt with concentration ranging from 1 to 1000 μM. All experiments were performed at room temperature. The experimental conditions are summarized in Table 1. The measurements of the zeta potential of colloids in various salt solutions were accomplished by a electrophoretic mobility measurement. The particle mobility under an applied electric field was determined by tracing their loci in real time using a ZEECOM zeta potential analyzer (Microtec Co. Ltd. Japan). Using the average mobility data of 100 particles, zeta potentials of the particles in each salt concentration were evaluated using analytical eq 4.20 in the paper by O’Brien and Hunter.19 When the measured sample was in the range of intermediate κa, this equation has a maximum mobility above 40 mV in the absolute value of the zeta potential so that two zeta potentials (low and high values) were found. The zeta potential of crown glass was evaluated using ELSZ-1 (Otsuka Electronics Co. Ltd. Japan), which enabled the measurement of the zeta potential of plate samples as a function of salt concentration. The sample glass with dimensions of 15 mm (W)  35 mm (L)  1 mm (T) was located on the bottom of a measurement cell with an HPC (hydroxyl propyl cellulose)-coated PS particle solution. The mobility was measured by the laser Doppler method. Zeta potentials were evaluated according to the method of Mori and Okamoto.24 The surface charge densities of colloids and (24) Mori, S.; Okamoto, H. Fusen 1980, 27, 117-126, in Japanese.

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Table 1. Experimental Conditions and the Appearance of Phase Transitions at Room Temperature

no.

PS vol %

EtOH vol %

Pe

salt (μM)

G-L trans.

L-S trans.

σ = 1 μm 1 2 3 4 5 6

0.001 0.005 0.01 0.05 0.1 0.2

50 50 50 50 50 50

77 77 77 77 77 77

0 0 0 0 0 0

yes yes yes yes yes yes

no no no no no no

yes yes yes yes yes yes yes yes yes yes yes yes yes yes weak weak very weak yes yes yes yes yes yes yes yes yes yes yes yes

no no no no no no no no no no no no no no no no no no yes yes no no yes yes no no yes yes yes

σ = 3 μm 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35

0.01 0.04 0.06

0.08 0.1

0.12 0.14 0.18 0.23

50 50 50 60 68 77 50 0 50 50 50 50 50 50 50 50 50 60 68 77 50 50 60 68 0 50 60 0 50

1990 1990 1990 2340 2660 3060 1990 1010 1990 1990 1990 1990 1990 1990 1990 1990 1990 2340 2660 3060 1990 1990 2340 2660 1010 1990 2340 1010 1990

0 0 0 0 0 0 0 0 0 1 5 10 50 100 250 500 1000 0 0 0 0 0 0 0 0 0 0 0 0

glass were calculated according to the linear Poisson-Boltzmann theory as described later.

Results and Discussion Gas-liquid condensation was observed for both 1 and 3 μm suspensions over wide ranges of concentration, 0.001-0.20 and 0.01-0.23 vol %, respectively. The detailed experimental conditions and the appearance of phase transitions are summarized in Table 1. Homogeneous samples with varying particle, EtOH, and salt concentrations equally sedimented to the bottom plate of the cell within 1 h and gradually condensed over 20 h. In the aqueous solvent containing more than 50 vol % EtOH, it was confirmed using CLSM that 3 μm suspensions settled to the bottom plate in a monoparticle layer thickness. However, 1 μm suspensions formed thick condensed layers at the bottom. Interestingly, 3 μm suspensions under the conditions of more than 0.10 and 50 vol % EtOH showed a coexistence of the gas, liquid, and solid (ordered) phases located from the periphery of the condensates to the center (sample nos. 25, 26, 29, 30, 33, 34, and 35 in Table 1). We observed the phase separation in detail using 3 μm suspensions at 0.10 vol %, 50% EtOH, and with no added salt. Figure 2A shows an example of such gas-liquid phase separation. Because sedimentation was enhanced by reducing the solvent density, the timescale of the sedimentation became much shorter than that of the condensation, and as a result, a separate 2440 DOI: 10.1021/la9029084

Figure 2. Series of photographs of liquid-phase condensation (white part) after the homogeneous dispersion of the colloidal particle (diameter 3 μm) at 0.10 vol % as a function of time (A). The small white circles in the photographs are air bubbles. The change in the particle condensation area (white part) was determined by the imageprocessing method (B). The dotted line shows the bottom plate area. The magnified gas-liquid phase boundary after 25 h (C).

observation was possible. Quantitative measurements of the particle-existing area (white area) were made by the imageprocessing method. This area decreased with time and reached a constant value (77% of the bottom plate) as shown in Figure 2B. A magnified boundary between the liquid phase and the gas phase is also indicated in Figure 2C. The particle arrangements showed sparseness at the edge of the boundary and more dense condensation inside the boundary, indicating the attractive interaction of the periphery particles with the inside particles. To determine the region of the observed miscibility gap, the salt-colloid phase diagram was investigated. A phase diagram was constructed at a constant particle number (2.77  106, 0.10 vol %) at room temperature. Figure 3 shows the phase diagram in water/ EtOH (50:50) with respect to volume fraction and added salt concentration. The volume fraction φ was calculated according to the relationship φ = (4/3)πa3N0/2aS, where N0 is the number of added particles and S is the area of the bottom plate. The dotted line in Figure 3 is the initial volume fraction calculated assuming that the total particles settled to a monoparticle layer on the bottom plate. The low and high error bars were drawn using the densities at the peripheral and the center of a condensate, respectively. Thus, the difference between the dotted line and the volume fraction of the condensed phase shows a miscibility gap extending from 1 to 1000 μM salt concentration as shown in Figure 3. An interesting feature of the instability is the appearance of a near-critical density fluctuation in the vicinity of the upper bound of the miscibility gap (250-500 μM, KCl). Figure 4A indicates a wavelike inhomogeneity extended over the whole area of the cell. Figure 4B also shows a magnified view of the inhomogeniety, where dense and depleted areas were striated nearly 100 μm in width. Nabutovskii et al. reported the appearance of a charge density wave (CDW) phase near the critical point.25 The wavelike (25) Nabutovskii, V. M.; Nemov, N. A.; Peisakhovich, Y. G. Phys. Lett. A 1980, 79, 98–100.

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Figure 3. Miscibility gap found in the volume fraction-salt concentration expression. Polystyrene particles were sedimented in the monoparticle layer at the bottom of the observation cell at an initial homogeneous particle concentration of 0.10 vol %. The error bars indicate the minimum and maximum volume fractions about the mean values observed over the condensation areas. The symbol \ indicates the appearance of the near-critical phase separation showing the large-scale wavelike density fluctuation as shown in Figure 4A.

Article

theory, the effective charges on the particles are necessary for the accurate estimation of the free-energy term of the interactions. We measured the effective charges by the electrophoresis method. Zeta potentials were calculated from the electrophoretic mobility data using the equation derived by O’Brien and Hunter.19 Because small ions accumulate around highly charged particles, it is not the bare charge that determines the mobility but rather the effective (or renormalized) charge. To derive a formula for the mobility of a large particle, it is necessary to take the distortion of the double layer into account. As pointed out by O’Brien and White,19 there is a maximum in the electrophoretic mobility as a function of the zeta potential in the intermediate κa value (5 to 150), that is, the low and high values of the zeta potential are assigned when the mobility is obtained. The ionic environment surrounding a macroion deforms by both particle movement and the external electric field. The appearance of the mobility maximum is formulated by introducing the relaxation term of the fluid mechanical deformation of the electrical double layer whose periphery coincides with the shear plane as a result of decreasing Debye length. The fluidity of the electrical double layer indicates a many-body interaction between macroions via redistributing surrounding counterions. Table 2 shows the results of zeta potential measurements at both low and high values for each salt concentration. The absolute values increased with increasing salt concentration. According to the numerical solution of the Poisson-Boltzmann equation of spherical colloids, the effective surface charge increases with increasing structural charge of macroions before finally saturating at what is called the effective charge saturation.17,18 Because the saturated effective charge increases with κ, we can test the charge saturation under our experimental conditions using the obtained data. Solving the linearized Poisson-Boltzmann equation with the condition that the electrical potential at the surface matches the surface charge density gives for the electrostatic potential ψ(r)26 ψðrÞ ¼

Figure 4. Photograph of an abnormal density fluctuation (white

part) observed at a salt concentration of 500 μM KCl (A) and magnified photograph of a density fluctuation showing a striated structure (B).

inhomogeneity is thought to appear as the result of a long-range colloidal density fluctuation in the vicinity of the critical point to form the macroscopic dipolar structure of the colloid with rich and depleted phases, where the electrical double layer formed around the rich phase by a charge neutralization requirement. Here, we mention only the appearance of the abnormal density fluctuation in the experiment and its relevance to Warren’s theoretical prospect,9 leaving its detailed analysis for future study. Figure 5 shows another interesting aspect of the gas-liquid phase separation. When the liquid-phase condensation proceeded under enhanced sedimentation, the increase in particle concentration at the center of the cell caused an order-disorder transition as shown in Figure 5B. The inset of the Figure shows a Fourier transform pattern of the ordered structure. The observation clearly shows a mechanism of colloidal crystallization where the attractive force between the particles realized a packing density that triggered the Kirkwood-Alder transition.22 The experimental results showed the attractive nature of particle interactions. Next, we evaluated the origin of the miscibility gap according to the linear Poisson-Boltzmann theory. We assume that the validity of linear theory is based on a large particle radius (a . κ-1)15 and weak coupling with salt ions.12 To evaluate the appearance of the miscibility gap according to linear Langmuir 2010, 26(4), 2438–2444

ZlB kT e -Kðr -aÞ r 1 þ Ka

ð2Þ

where Z is the charge on the particle, lB is the Bjerrum length defined as lB = e2/4πεkT, and ε is the permittivity of the solvent. Note that zeta potential measurements are an approximate method to obtain the effective charge of polystyrene latex.27 The effective surface charge Zeff was evaluated using the following relationship, setting r = a and ψ(a) = ζ in eq 2. ζ¼

1 Zeff 1 4πε a 1 þ Ka

ð3Þ

The obtained values are listed in Table 2 and plotted against κa, as shown in Figure 6. The charges linearly increased with κa for both low and high values of the effective surface charge. As mentioned before, it is useful to test the validity of the values according to whether they are saturated. The values are compared in Figure 6 using the theoretically derived relationship in which the saturated effective charge is a linear function of κa.18 sat Zeff ¼

4a ð1 þ KaÞ lB

ð4Þ

The above function is derived on the basis of the Debye-H€uckel theory in the colloidal limit where the particle size is large (26) McQuarrie, D. A. Statistical Mechanics; University Science Books: Sausalito, CA, 2000. (27) Ito, K.; Ise, N.; Okubo, T. J. Chem. Phys. 1985, 82, 5732–5736.

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Figure 5. Coexistence of the gas-liquid and liquid-solid phase transitions. The left image is a total view of the gas-liquid-solid phase appearance. The intermediate image is a magnified view of a liquid region. A indicates the region where the view was taken. The right image shows ordered structures of particles taken from region B. The insets are Fourier transform images of each particle arrangement. Table 2. Measurements of the Zeta (ζ) Potential and Estimated Effective Surface Charges, Assuming the Electrical Potential of the Debye-H€ uckel Theorya low value KCl (M)

κa

ζ (mV)

-6

high value ζ (mV)

Zeff

Zeff

10 4.9 -39.5 1.95  10 N.D. N.D. 15.6 -52.2 7.26  104 -235 3.27  105 10-5 10-4 45 -76.8 2.96  105 -213 8.21  105 10-3 156 -93.9 1.23  106 -258 3.39  106 a N.D. indicates a divergence of the mobility function by O’Brien and Hunter.19 4

To evaluate the instability conditions, osmotic compressibility under the experimental parameters of observed phase instabilities was calculated following the method of the free-energy calculations for highly asymmetric electrolytes.9 We assume colloidal particles or macroions with a diameter of σ=2a, negative charge |Z| . 1, and number density nm. There are small co-ions and counterions at number density n- and nþ, respectively. The solvent is taken to be a dielectric continuum of permittivity ε= ε0εr, where ε0 and εr are the permittivity of vacuum and the relative permittivity of the solvent, respectively. The Coulomb interaction between a pair of univalent charges in units of kT is lB/r. The densities are related by the electroneutrality condition Znm þ n- =nþ and n- =ns. The Debye screening length is defined as κ2 =4πlB(nþ þ n-)=4πlB(Znm þ 2ns). The free energy of colloidal phase separation is analytically defined by summing five elementary energy terms: ideal gas, macroion-small ion interaction, macroion-macroion interaction, hard core interaction, and small ion-small ion interaction, in order of relative importance. If we restrict our analysis to gas-liquid phase separation, then two free-energy terms are enough to estimate the upper limit of the salt concentration, that is, the ideal gas term Fid of macroions and small ions and the electrostatic term Fms of macroions surrounded by small ions. The ideal gas term of macroions and small ions is Fid ¼ nm log nm þ ns log ns þ ðZnm þ ns ÞlogðZnm þ ns Þ VkT ð5Þ

Figure 6. Surface charges of colloidal particles as a function of

κa (radius of particle a = 1.5 μm, κ-1 = 4.9-156). The effective charges were obtained assuming the electrical potential of the Debye-H€ uckel theory, and the low and high values due to the maximum in mobility function with respect to the zeta potential are plotted by the symbols b and 9, respectively. The data used are listed in Table 2. The theoretical relationship deduced by Bocquet et al.18 is indicated by the dotted line.

compared to both the screening length κ-1 and the Bjerrum length lB. The theoretical estimate is in good agreement with the case of low effective charge. The results show that the linear theory gives a consistent method by which to evaluate the effective charge of particles by the zeta potential measurement and that the charges obtained are correct in using the various relationships obtained from linear theory. In addition, because the bare charge of polystyrene latex is more than 100 times larger than the effective charge and thus leads to a very large potential nonlinearity near the solid surface, the obtained high charges (∼3 times the low charges) are still considered to be within the error of charge estimation. An analysis of both the low and high charges is included in the following discussion. 2442 DOI: 10.1021/la9029084

The electrostatic term for macroions is obtained as eq 6, assuming the pair correlation method and the Debye-H€uckel approximation:9 Fms 2 ¼ - Z 2 lB Knm f ðKaÞ 3" VkT # 3 x2 where f ðxÞ ¼ 3 logð1 þ xÞ -x þ x 2 Osmotic compressibility (F) is defined in eq 7.
 1 DV Dð1=nm Þ ¼ -nm F¼ V DP DP

ð6Þ

ð7Þ

If F is negative, then the colloidal suspension is unstable, indicating that a miscibility gap is present. Then, the osmotic pressure P is defined according to the following relations F ≈ Fid þ Fms

ð8Þ


 F βP ¼ nm μm þ ns μs β V

ð9Þ

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Figure 8. Estimated phase diagram of polystyrene latex (diameter 3 μm) in a 50 vol % ethanol/water mixed solvent at 25 °C. Figure 7. Ratios of the surface charge density between the glass substrate and colloidal particles as a function of salt concentration. σglass is the surface charge of glass, and σparticle is that of colloidal particles. The solid lines marked by 9 and b show the calculated relationships using linear theory with the low and high values of zeta potentials, respectively.

where μi = (∂F/∂Ni)V,T, i = {m, s}, and β = 1/kT. We can evaluate F from the analytical equation of osmotic pressure as a function of nm and ns, which is obtained by combining eqs 5, 6, 8, and 9. The maximum salt concentrations of immiscibility were calculated by evaluating the values changing the sign of F from negative to positive with increasing salt concentration. The parameters that were used were Z = Zeff (1.23  106 and 3.39  106, respectively), nm = 1.18  1010/cm3, and lB = 1.09 nm (relative permittivity 51.43 for 50 vol % ethanol/water at 25 °C, Handbook of Chemistry and Physics, CRC). The 1.31 mM (1.23  106) and 10.1 mM (3.39  106) concentrations were obtained as the maximum salt concentrations. The 1.31 mM concentration was in good agreement with the upper limit of gas-liquid immiscibility as shown in Figure 3, and the 10.1 mM value at high charge was too large to compare with the experimental observation. We also obtained the critical salt concentration by calculating the spinodal curve as a function of nm at a charge of 1.23  106. The maximum pressure of the spinodal curve was obtained at nm = 0.9501010/cm3 and [KCl]=1.31 mM, indicating that the experimental particle concentration was in the vicinity of the critical point and the observed abnormal density fluctuation near the critical concentration showed an aspect of CDW. The results enabled the estimation of the phase diagram of polystyrene particles in Figure 8. The effective charge of 1.23106 was 100 times larger than the highest value used in the theoretician’s work. If we compare the calculated critical concentration of 1.31 mM with, for example, 25 μM in the case of low surface charge particles (radius=326 nm and Z=7300),5 then the experimental immiscibility extended to the higher salt concentration is quite reasonable. The effects of the cell wall have several features, such as the pair interaction between like-charged particles attractive and accumulating suspended colloidal particles near the wall.28-30 To examine the extent to which the wall effects are working in the gas-liquid condensation, we evaluated the effective charges of the bottom wall as a function of salt concentration. As studied by Grier and co-workers,28 the effective pair potential of (28) Polin, M.; Grier, D. G.; Han, Y. Phys. Rev. E 2007, 76, 041406. (29) Muramoto, T.; Ito, K.; Kitano, H. J. Am. Chem. Soc. 1997, 119, 3592–3595. (30) Goulding, D.; Hansen, J.-P Europhys. Lett. 1999, 46, 407–413.

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like-charged colloids becomes negative (attractive) when the charged particles are confined/located near the glass plates and the attractive interaction was rationalized within a density functional calculation.30 The configuration reported was quite similar to the case here, where we evaluated the contribution of the glass wall to the gas-liquid condensation as follows. The zeta potential of crown glass was measured as a function of the salt concentration using the electrophoretic method. The results are shown in Table 3. The absolute value of the zeta potential decreased with increasing salt concentration, indicating no charge saturation of the glass surface, unlike the case for latex particles. For comparison, the intensity of the particle-glass wall interaction with respect to the particle-particle interaction and the surface charge densities of glass σglass and the particle σparticle were evaluated. σparticle was calculated using eq 3, and σparticle = Zeff/4πa2. σglass was calculated using the following relationship derived from the Poisson-Boltzmann equation of planar geometry and the Gauss theorem.31
 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi eψglass σglass ¼ 8ns εkT sinh 2kT

ð10Þ

where ψglass is the surface potential of glass. The ratios of the surface charge density listed in Table 3 are plotted in Figure 7 with respect to salt concentration. The ratio σglass/σparticle ranged from 0.202 to 1.34 for low values of the surface potential and 0.0734 to 0.214 for high values of the surface potential. In both cases, the salt concentration increased with decreasing values of σglass/ σparticle. Below 10 μM salt concentration, σglass/σparticle exceeded unity, indicating a greater contribution from the glass wall than from the surrounding particle; otherwise, particle-particle interactions dominate in the range of high ionic strength. Although the presence of the confining glass wall introduces a significant complication, such as an effective long-range attraction between the colloidal particles close to the wall, we can safely conclude that the major driving force of the gas-liquid condensation is the macroion-small ion interaction and that the wall effect may assist the effective attraction between the particles, especially in the range of low ionic strength. Zoetekouw and van Roij proposed a condition of gas-liquid spinodal instability of ZλB /a g 24 in the colloid-salt phase diagram, including the nonlinear interaction between particles.11 The calculated value of ZlB/a was 450 at the upper limit of salt concentration where the instability was observed, a value that is (31) Ohshima, H., Furusawa, K., Eds.; Electrical Phenomena at Interfaces; Marcel Dekker: New York, 1998.

DOI: 10.1021/la9029084

2443

Article

Ishikawa and Kitano Table 3. Ratios of the Effective Surface Charge Density between Crown Glass and Particles in Various Salt Concentrationsa crown glass

KCl (M)

κa

ζ (mV)

particle (low value) -2

σglass (C m )

-6

10 4.9 -27.49 15.6 -23.71 10-5 45 -15.58 10-4 156 -8.74 10-3 a N.D. is the same as in Table 1.

-4

1.69  10 3.96  10-4 8.06  10-4 1.41  10-3

-2

σparticle (C m ) 1.10  10 4.10  10-4 1.67  10-3 6.95  10-3

quite high as as result of the high effective charge of particles. The validity of the mean-field approximation in our experiments was also tested by calculating the coupling parameter Ξ = 2πq2lBσs, where q is the valency of the counterion and σs is the surface charge density.12 If the value is below unity, then Poisson-Boltzmann theory is expected to be valid because each ion interacts with a diffuse cloud (mean field) of other ions. When Ξ > 1, mean-field theory is expected to break down and attractive many-body interaction dominates. The calculated Ξ ranged from 0.33 to 0.91 and is considered to be at the border between weak and strong coupling. It is considered that the observed gas-liquid phase change lies in the region where the linear PoissonBoltzmann theory is still valid, but the contribution of manybody interactions due to strong coupling may not be small in higher ionic strength solutions. On the basis of the experimental observation and the theoretical evaluation of the critical salt concentration, an estimated phase diagram of polystyrene particle was constructed as shown in Figure 8. The diagram gives wider stability range up to high critical salt concentration than theoretical estimations by van Roij et al.,3,4 Levin et al.,6 Diehl et al.,8 Warren,9 and Denton.32 Although our calculation was at the same level of approximation as used by these authors, the reason that the phase transition appeared in our experiment is due to the extraordinarily high value of the effective charge. The other aspects are the appearance of gas-liquid-crystal coexistence (triple point) at low ionic strength and the abnormally large density fluctuation near the critical point. These phase behaviors are still under active discussion. Further modeling work is undoubtedly necessary.

Conclusions We studied the anomalous gas-liquid condensation of colloidal suspensions and attributed its primary driving force to the electrostatic macroion-small ion interaction proposed by van Roij, Warren, and others. The appearance of the instability was confirmed by calculating the osmotic pressure of the colloidal suspensions as a function of salt and particle concentrations, (32) Denton, A. R. Phys. Rev. E 2006, 73, 041407.

2444 DOI: 10.1021/la9029084

-4

σglass/σparticle 1.34 0.966 0.483 0.202

particle (high value) σparticle (C m-2)

σglass/σparticle

N.D. 1.85  10-3 4.63  10-3 1.92  10-2

N.D. 0.214 0.174 0.0734

where the effective charges of colloids obtained by the zeta potential measurement were used in the application of the Debye-H€uckel theory. The calculation showed a miscibility gap below a maximum KCl concentration of 1.31 mM and explained the experimental limit of the colloidal phase separation. The effect of the glass wall was examined by measuring the ratio of charge density between the glass wall and colloidal particles. Although it is known that a charged glass wall induces an attractive effective potential between particles at low ionic strength, we estimated it to be a secondary factor that assisted the phase separation. The electrical double layer of highly charged particles of intermediate κa is characterized by a highly screened, dense ionic atmosphere so that the nonlinearity of the Poisson-Boltzmann equation of the charged interface is confined within the length of the Debye length (much smaller than the particle radius). This enables one to replace the effective charges by the renormalized surface charges. In this study, the charges experimentally obtained by the zeta potential measurement were confirmed to match the saturated surface charges of large particles well. Such saturation indicates the strong charge renormalization by counterions, which may make it possible to explain gas-liquid condensation within the linear theory of the Poisson-Boltzmann equation. Acknowledgment. We have benefited from discussions with Kaoru Tsujii and Ikuo S. Sogami and are grateful to Tsutomu Kajino for supporting the zeta potential measurement. This research was partially supported by the Ministry of Education, Science, Sports and Culture through a grant-in-aid for 4901, 20200004, 2008. Supporting Information Available: The possible involvement of other simple mechanisms, such as bowing, evaporation, and thermal convection in the cell, were discussed using a series of photographs of colloidal gas-liquid condensation with increasing particle concentration. Photographs showing gas phases inside liquid phases that were similar to void structures are shown. This material is available free of charge via the Internet at http://pubs.acs.org.

Langmuir 2010, 26(4), 2438–2444