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Colloidal Dispersion Stability of CuPc Aqueous Dispersions and Comparisons to Predictions of the DLVO Theory for Spheres and Parallel Face-to-Face Cubes Jiannan Dong, David S. Corti, and Elias I. Franses* School of Chemical Engineering, Purdue University, 480 Stadium Mall Drive, West Lafayette, Indiana 47907-2100
Yan Zhao, Hou T. Ng, and Eric Hanson Commercial Print Engine Lab, HP Laboratories, Hewlett-Packard Co. 1501 Page Mill Road, Palo Alto, California 94304 Received November 6, 2009. Revised Manuscript Received December 12, 2009 The dispersion stability and the ζ potentials of nonspherical crystalline (β-form) copper phthalocyanine (CuPc) particles of hydrodynamic diameter dh ≈ 90 nm were investigated at 25 C in water and in aqueous solutions of NaNO3. The electrolyte concentrations c ranged from 1 to 500 mM and the particle concentrations ranged from 50 to 10000 ppm (0.005 to 1 wt %). In each case, the Fuchs-Smoluchowski stability ratio W was determined from dynamic light scattering (DLS) data and the Rayleigh-Debye-Gans (RDG) scattering theory. The data suggest that electrostatic effects play a major role in the stability of CuPc-based dispersions. The calculated particle charge z per CuPc particle based on the ζ potential data and the area of the particles (assumed to be cubical) suggests that there is preferential adsorption of NO3- ions on the uncharged CuPc surface, and the surface charge increases with increasing electrolyte concentration. Furthermore, two new models of the DLVO theory, for spheres and for parallel face-to-face cubes, were reformulated in dimensionless form. The Hamaker constant of CuPc particles was calculated by the same authors on the basis of theoretical models in J. Chem. Theory Comput. (2010, in press). The key dimensionless group is the ratio N of the electrostatic double layer energy to the Hamaker constant. The two DLVO models were used to predict the value of a dimensionless maximum potential energy Φ h max, the conditions when it may exist, and then the value of W. In water, the DLVO model for spheres overpredicted the stability, while the model for cubes underpredicted the stability. At c = 1 mM, both models overpredicted the stability. At c = 10 mM, the model for spheres underpredicted the stability, whereas the model for cubes overpredicted the stability. Hence, there seems to be some significant shape effects on the electrostatic stabilization of CuPc particle dispersions. At c = 100 and 500 mM, both models underpredicted the stability substantially, suggesting the existence of additional short-range repulsive forces, which may primarily control the stability. Simple sensitivity analysis on the calculations supported these conclusions.
1. Introduction For the past 60 years the Derjaguin, Landau, Verwey, and Overbeek (DLVO) theory has been used as a basis for explaining or interpreting, at least in part, the stability of colloidal dispersions of charged particles in water.1 In the context of this theory, the van der Waals attractive “long-range” forces cause particles to agglomerate, and the electrostatic double layer generates longrange repulsive forces which tend to keep particles from agglomerating. Usually, the stability is better the higher the absolute value of the surface (or ζ) electrical potential or the surface charge density, and the lower the electrolyte concentration. Moreover, in what was known empirically as the Schulze-Hardy (S-H) rule, the stability worsens substantially as the valence of the electrolyte counterions (with opposite charge from that of the surface) increases from 1 to 2 to 3. Despite the initial success in explaining the semiquantitative S-H rule and other observations, the DLVO theory seems to be unable to properly account for the observed colloidal dispersion stability and rates of agglomeration for a broad range of *Corresponding author. E-mail:
[email protected]. Telephone: 765494-4078. Fax: 765-494-0805. (1) Verwey, E. J. W.; Overbeek, J. T. G. Theory of the Stability of Lyophobic Colloids; Elsevier: Amsterdam, 1948.
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dispersions. The DLVO theory is a “macroscopic theory” that is valid for particle separation distances d at which the fluid solution between the particles can be considered as a “continuum.” This means that the d-range, at which this theory may apply, should be much larger than the ion or solvent molecular dimensions, practically above ca. 2 nm.2 Indeed, several direct tests, with the use of a surface force apparatus, have provided important qualitative and quantitative validation of the DLVO theory for large d-values. Nonetheless, for understanding collision and agglomeration events, one needs data at d-values of the order of 0.2-2 nm, at which the continuum assumption is no longer valid. At such short distances, though, additional forces can be associated with the solvent: “hydration” forces, solvent-moleculepacking-type forces, etc. Strong and broad evidence has been generated for such forces,2,3 which can be both repulsive and attractive, and may strongly fluctuate with the separation distance.3 It is unclear whether these forces may cause a net increase or decrease in the rate of agglomeration and the overall dispersion stability. One goal of this article is to test the applicability of the DLVO theory combined with the Fuchs-Smoluchowski theory in (2) Israelachvili, J. N.; McGuiggan, P. M. Science 1988, 241, 795. (3) Israelachvili, J.; Wennerstrom, H. Nature 1996, 379, 219.
Published on Web 01/14/2010
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predicting the stability of certain commercially important copper phthalocyanine (CuPc) cyan blue pigment dispersions. Such pigments show favorable color strength, brightness of shade, fastness properties, and relatively low cost.4 The stability of CuPc dispersions tends to decrease with increasing electrolyte concentration, as expected qualitatively from the DLVO theory and as shown in some results below. In testing this theory, it is essential to determine the Fuchs-Smoluchowski stability ratio W quantitatively. This is done by the use of dynamic light scattering (DLS) data and a method developed in Refs. 5-7. A second goal of this article is to present two new simple dimensionless analytical formulations of the DLVO theory at constant surface potential, for monodisperse spherical and parallel face-to-face cubical particles. The formulation for cubes is needed to examine possible shape effects on the DLVO theory predictions, since the CuPc particles used are crystalline and have flat edges, making them closer in shape to cubes than to spheres. Although the new model formulations are approximate, they make the application and testing of the DLVO theory easier physically and mathematically. For simplicity of calculation, only one orientation of the cubes is considered. This article is organized as follows. In section 2, experimental details and methods, including the theory used for extracting W from DLS data, are presented. Section 3 covers the experimental results. The method for estimating the Hamaker constant of CuPc in water is briefly discussed in section 4, and has been described in more detail in a submitted publication. The new dimensionless formulations of the DLVO theory for spheres and cubes are presented in section 5. The results are given in terms of how the DLVO dimensionless potential energy maximum Φ h max, depends on the Hamaker constant A, the surface potential c0, and the concentration c or the ionic strength I of a 1:1 electrolyte. This value of Φ h max is linked to W for spheres on the basis of the Fuchs-Smoluchowski model.8,9 The same relationship between Φ h max and W is assumed to hold for cubes. In section 6, the data-based W values are compared with the predicted values for the two shapes, and the observed differences are discussed. Section 7 summarizes the conclusions.
2. Experimental Details and Methods Used 2.1. Materials. CuPc pigment particles (Cabot-O-JET250C) were obtained from Cabot Corp. (MA, USA) as a 10 wt % stable dispersion in water, and were used as received. According to the manufacturer specifications, these particles were crystalline, in the β-CuPc polymorphic form. The particles surfaces were stabilized by chemically attached sulfonate groups. The nominal pH was 6.5 at 10 wt %, and the nominal mean diameter of the particles was 110 nm. Some dispersion parameters were also determined independently in this article. Water was first distilled and then passed through a Millipore four-stage cartridge system; it had a resistivity of 18 MΩ 3 cm at the exit port. Sodium Nitrate (99%, p.a., Fluka) was purchased from Sigma-Aldrich (MO, USA).
2.2. Transmission Electron Microscopy (TEM) Imaging. A 400-mesh copper grid with a carbon-coated Formvar resin was glow-discharged prior to use. A 50 ppm of CuPc dispersion was deposited on the grid. After about 3 min, the excess water was drawn off with a filter paper, and the particles were dried. (4) Mather, R. R. J. Porphyrins Phthalocyanines 1999, 3, 643. (5) Vaccaro, A.; Sefcik, J.; Morbidelli, M. Polymer 2005, 46, 1157. (6) Holthoff, H.; Egelhaaf, S. U.; Borkovec, M.; Schurtenberger, P.; Sticher, H. Langmuir 1996, 12, 5541. (7) Berka, M.; Rice, J. A. Langmuir 2004, 20, 6152. (8) Smoluchowski, M., Phys Z 1916, 17, 557-571 and 585-599. (9) Hiemenz, P. C.; Rajagopalan, R. Principles of Colloid and Surface Chemistry; Marcel Dekker, Inc.: New York, 1997.
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Dong et al. Samples were imaged using a Philips CM-10 TEM operated at 80 kV, with a spot condenser aperture of 2.200 μm and an objective aperture of 50 μm. Images were captured on Kodak SO-163 electron image film. For high resolution (HR) TEM imaging, a JEOL 3100 TEM equipped with a LaB6 emission source was used. An accelerating voltage of 300 KV was used for routine HRTEM imaging.
2.3. Dynamic Light Scattering (DLS) Measurements. The hydrodynamic diameter of CuPc particles dh was measured at 25 C, at a wavelength λ0 of 659 nm, and at a scattering angle θ of 90, with a Brookhaven ZetaPALS dynamic light scattering instrument, which has a BI-9000AT digital autocorrelator. Standard square acrylic cells with volume of 4.5 mL were used. Each measurement took about 1 min. Six measurements were taken, and the averages are reported. All measurements were conducted at the weight fraction w = 50 ppm (0.005 wt %) or after diluting higher weight fractions to 50 ppm, since the paticle absorbances due to absorption at w > 50 ppm at this wavelength are so large that they diminish significantly the scattering signal. 2.4. Zeta Potential (ζ) and pH Measurements. Data were obtained at 25 C with the above ZetaPALS instrument, using the software PALS ζ potential analyzer, version 3.29. Each measurement consisted of 20 cycles, and 10 measurements were made with each sample. All measurements were conducted at w = 50 ppm, for the same reason mentioned above. The pH of the CuPc dispersions was probed at 25 C using a pH meter (InPro3250) from Mettler Toledo Corp.
2.5. Determining the Stability Ratio (W) from DLS Data. The aggregation or coagulation in a dispersion of monodisperse colloidal particles can be described by a general set of kinetic population balance equations, which are called the “Smoluchowski coagulation equation.”8 For a dilute dispersion, it is often reasonable or convenient for one to assume that for sufficiently short times only dimerization is significant, and that the presence of all higher aggregates can be neglected. Then the Smoluchowski coagulation equation is reduced to dN 1 ¼ -k 11 N 1 2 ð1Þ dt and dN 2 1 ¼ k 11 N 1 2 2 dt
ð2Þ
where k11 is a coagulation rate constant, t is the time, and N1 and N2 are the number densities of the monomers and the dimers, respectively. For slow coagulation, the rate constant k11 is equal to the slow coagulation rate constant ks, which is given by the FuchsSmoluchowski equation k 11 ¼ k s ¼
8kT 3ηW
ð3Þ
where k is the Boltzmann constant, T is the absolute temperature, η is the viscosity of the solvent, and W is the stability ratio.9 The quantity kf = 8kT/3η is the fast coagulation rate constant, for which W = 1 by definition. For stable dispersions, one should have W . 1. With the initial conditions that N1 = N0 and N2 = 0, the solutions to eqs 1 and 2 are N 1 ðtÞ 1 ¼ N0 1 þ ksN 0t
ð4Þ
N 2 ðtÞ 1 k s N 0 t ¼ N0 2 1 þ ksN 0t
ð5Þ
and
where N0 is the initial number density of the monomers. Langmuir 2010, 26(10), 6995–7006
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To estimate N1(t), N2(t), and ks, one may use DLS measurements, from which the average diffusion coefficient of the particle population is obtained. For polydisperse particle sizes, one determines not the number-average but the intensity-averaged diffusion coefficient:6 P N i I i Di D ¼ Pi ð6Þ i N iI i where Ii and Di are the scattered intensity and diffusivity of an aggregate containing i primary particles, respectively. The Stokes-Einstein equation RH i = kT/6πηDi relates the hydrodynamic radius of the i-fold aggregate to Di. From the average diffusion coefficient, one obtains the average hydrodynamic radius RDLS by assuming that the Stokes-Einstein equation is valid for polydisperse spherical particles, and also for nonspherical particles and aggregates: P kT N iI i RDLS ¼ ¼P i ð7Þ H 6πηD i N i I i =Ri The value of the scattered intensity Ii depends on the scattering “regime.” The two simple regimes, Rayleigh scattering or Rayleigh-Debye-Gans (RDG) scattering, are considered here. For each scattering regime, two cases, coalescence of two spherical particles to a single spherical particle and noncoalescence, will be examined below. For spherical particles, the intensity Ii can be expressed in general for both Rayleigh and RDG scattering as I i V i 2 Pi ðqÞ Ri 6 Pi ðqÞ
ð8Þ
where Vi ¼ ð4π=3ÞRi 3 is the particle volume, Ri is the sphereequivalent radius, and Pi(q) is the overall “form factor” of a particle or a noncoalesced aggregate. Pi(q) is a measure of intraparticle interference. For Rayleigh scattering, Pi(q) = 1, and hence Ii is independent of the particle or aggregate shape. For RDG scattering, Pi(q) can be approximated as the product of the form factor of the primary spherical particle, P1(q), times the “structure factor”, S(q),10 Pi ðqÞ ¼ P1 ðqÞS i ðqÞ
ð9Þ
where ( P1 ðqÞ ¼
3
ðqR1 Þ
)2 ½sinðqR1 Þ -ðqR1 Þ cosðqR1 Þ 3
ð10Þ
and " # 1 X X sinðrlm qÞ : S i ðqÞ ¼ 2 ðrlm qÞ i m l
ð11Þ
The parameter q = (4πn0/λ0) sin(θ/2) is the magnitude of the scattering vector, where n0 is the refractive index of the suspending medium, λ0 is the wavelength in vacuum, θ is the scattering angle, and rlm is the center-to-center distance between the particles l and m in an ifold aggregate. For higher aggregates than dimers, this form factor is an average of all configurations of such aggregates. When only monomers and dimers are present in a dispersion, as in the early stages of agglomeration of initially momodisperse particles, RDLS can be related to ks by substituting eqs 4 and 5 (10) Kerker, M. The Scattering of Light and Other Electromagnetic Radiation; Academic Press: New York, 1969.
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into eq 7,
RDLS R1
I2 2 þ ksN 0t I1 ! ¼ I2 R1 2 þ ksN 0t I1 RH 2
ð12Þ
For low Reynolds numbers, the hydrodynamic radius RH i is different from the sphere-equivalent radius Ri for noncoalesced aggregates. RH 2 /R1 is found to be 1.38 for two touching particles with no free rotation,11 whereas the ratio p offfiffiffithe sphere-equivalent radii of a dimer vs a monomer R2/R1, is 3 2 or 1.26. The equations that result from evaluating eq 12 for these various cases are shown in Table 1. Their possible applicability to the CuPc dispersions studied here is now evaluated. In this study, CuPc particles have R1 ≈ 45 nm. The conditions for Rayleigh scattering are R1 [πdsn0/λ0] , 1 and R2 [πds|np|/ λ0] , 1, where ds is the longest linear dimension through the scattering particle, λ0 is the wavelength in vacuum (λ0 = 659 nm here), n0 is the refractive index of water (n0 = 1.33), and np is the complex refractive index of CuPc particles at 659 nm (estimated to be 1.5-0.1i.12,13) These conditions are not satisfied since R1 = 0.57 and R2 = 0.64 for the monomers and 1.14 and 1.28 for the dimers. The conditions for the RDG scattering, which are |m - 1| , 1 and 2R1|m - 1| , 1 (where m np/n0) are better satisfied, since |m - 1| = 0.15 and 2R1|m - 1| = 0.13 for the monomers and 0.26 for the dimers.12 Since the CuPc particles are rigid and crystalline, their aggregates are unlikely to coalesce. Hence, eq 16 (see Table 1) is used to interpret the data. Nevertheless, using any of the other three equations in Table 1 would have a small impact on the conclusions. Finally, the value of ksN0 can be determined directly from a linear fit of the data of [RDLS/ R1 - 1] vs time. By using the value of N0, one can then find the stability ratio W from eq 3.
3. Experimental Results 3.1. TEM Images and Particle Shapes. The TEM images showed discrete microcrystalline flat-edged particles, some of which probably aggregated during the sample preparation (Figure 1). Their dimensions ranged from 30 to 50 nm. The single-crystalline character of particles was clear in a high-resolution TEM image (Figure 2). Distinct lattice fringes, extending up to the edges of the pigment particles, were observed, with a spacing of about 1.87 nm, which corresponded to the lattice constant a of the β-CuPc crystal.14 Individual particles looked globular and nonspherical. Their shape resembled that of short right cylinders or parallelepipeds. For this reason, and for mathematical convenience, the particle shape was modeled below either as spheres or as parallel face-to-face cubes (Figure 3). Among various orientations for two short right cylinders, we only considered two simple cases. Configuration 1 for short cylinders was approximated as two spheres. Configuration 2 for short cylinders was approximated as two parallel face-to-face cubes. Among various orientations for two parallelepipeds, configuration 3, in which the length of either parallelepiped mattered little (because the attractive forces diminish with distance and the repulsive forces do not depend on the length), can also be approximated as two parallel face-to-face cubes. If there were strong repulsive forces among cylinders or parallelepipeds, the model for cubes may provide an upper bound of the stability ratio, and the model for spheres may yield a lower bound, as detailed below (11) Happel, J.; Brenner, H. Low Reynolds Number Hydrodynamics; PrenticeHall: Englewood Cliffs, NJ, 1965. (12) van de Hulst, H. C. Light Scattering by Small Particles; Dover Publications: New York, 1981. (13) Peiponen, K. E.; Kontturi, V.; Niskanen, I.; Juuti, M.; Raty, J.; Koivula, H.; Toivakka, M. Meas. Sci. Technol. 2008, 19. (14) Lozzi, L.; Santucci, S.; La Rosa, S.; Delley, B.; Picozzi, S. J. Chem. Phys. 2004, 121, 1883.
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Dong et al. Table 1. Dependence of RDLS(t)/R1 on Scattering Regime and Coalescence vs Noncoalescence Type Aggregation value of RDLS(t)/R1
case
description
general equation pffiffiffi 2 þ k s N 0 tð 3 2Þ6 pffiffiffi 2 þ k s N 0 tð 3 2Þ5 pffiffiffi 2 þ k s N 0 tð 3 2Þ6 ffiffi ffi p 2 þ k s N 0 tð 3 2Þ6 =1:38
CuPc case
asymptotic forma
1 þ 2:00k s N 0 t 1 þ 1:59k s N 0 t
1 þ 0:41k s N 0 t ð13Þ
1 þ 2:00k s N 0 t 1 þ 1:45k s N 0 t
1 þ 0:55k s N 0 t ð14Þ
1
Rayleigh coalescence
2
Rayleigh noncoalescence
3
RDG coalescence
PðqR1 Þ þ 2PðqR2 Þk s N 0 t PðqR1 Þ þ 1:59PðqR2 Þk s N 0 t
1 þ 1:85k s N 0 t 1 þ 1:47k s N 0 t
1 þ 0:38k s N 0 t ð15Þ
4
RDG noncoalescence
1 þ 2S 2 ðqÞk s N 0 t 1 þ 1:45S 2 ðqÞk s N 0 t
1 þ 1:62k s N 0 t 1 þ 1:17k s N 0 t
1 þ 0:45k s N 0 t ð16Þ
a
The asymptotic forms are valid for ksN0t , 1.
Figure 1. TEM images of CuPc pigment particles dried from a 50 ppm dispersion in water, at magnifications of (a) 28500 and (b) 52000.
Figure 2. HRTEM images of Cabot-250 CuPC particles. (section 6). Nonface-to-face or nonparallel orientations are, of course, possible, but they lead to numerical calculations. In addition, they may produce weaker attractive and repulsive interactions. 3.2. Particle Density. The ideal, or theoretical, particle density for β-CuPc was calculated as F1 =1.62 g/cm3, based on the following crystal lattice parameters:14 a = 19.407 A˚, b = 4.79 A˚, c = 14.628 A˚, β = 120. The weight fraction of the particles was confirmed to be 10.1 wt %. The experimentally determined density was F1 = 1.56 ( 0.03 g/cm3 (average of n = 3 measurements). The reasons for the small, about 4%, difference could be due to (i) the presence of crystal imperfections or voids in the particles, (ii) not taking account of the surface sulfonate groups and associated counterions, or (iii) other experimental errors. The impact of this discrepancy on the calculated value of the Hamaker constant A11 was small. This experimental value was used in the calculation of A11.
3.3. DLS data and Determination of the Stability Ratios (W). The average hydrodynamic diameter of the CuPc particles
as received was initially 90 ( 3 nm, with a range of about 60 to 120 nm and no evidence of a bimodal distribution from DLS 6998 DOI: 10.1021/la904224g
measurements (Figure 4). This value was slightly smaller than the nominal diameter, and larger than the range of 30-50 nm estimated from the TEM images. The difference cannot be explained even if the hydrodynamic diameter of a cube of side c were equal to that of the sphere which encloses the cube. It was possible that the “primary particle” determined from the DLS measurements was on average an aggregate of a few smaller particles, such as those seen in the TEM images. To test whether any such possible aggregate can be broken by intense sonication, the dispersion was sonicated for 7 h. No difference in the size was detected (Figure 4). Hence, the 90 nm was considered to be the initial size of the (primary) particles. In addition the following test was done, to examine the redispersibility of the particles. A dispersion was completely dried in an oven at 50 C for 3 days, and then the particles were redispersed by gentle shaking by hand. Even though this dispersion looked the same to the naked eye as the one before drying, the size of the redispersed particles (dh ≈ 178 nm) was almost double that of the initial ones. Vigorous stirring could not reduce this size. Nonetheless, intense sonication for increasing times produced progressively Langmuir 2010, 26(10), 6995–7006
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Figure 3. CuPc particle geometries and orientations and models considered for three possible configurations.
Figure 5. Effects of NaNO3 concentration (c = 10, 100, and 500 mM) on the hydrodynamic sizes of CuPc dispersion: A, w = 50 ppm; B, w = 1000 ppm.
Figure 4. The hydrodynamic diameter of CuPc particles in a 1000 ppm dispersion: 1, original reference, not redispersed; 2, original reference, sonicated for 7 h; 3, redispersed, not sonicated; 4, redispersed, stirred; 5, redispersed, sonicated for 1 h; 6, redispersed, sonicated for 3 h; 7, redispersed, sonicated for 5 h; 8, redispersed, sonicated for 6 h; 9, redispersed, sonicated for 7 h. smaller sizes. After 7 h of sonication, the dh value almost returned to 90 nm. We inferred that the aggregates formed during drying were broken by this sonication procedure. The CuPc dispersions in water were quite stable. The particle sizes did not change over a couple of months. Dispersions at various NaNO3 electrolyte concentrations c, or ionic strength I, were prepared, by first mixing a bulk CuPc dispersion in water and a bulk NaNO3 solution, and then adding water to reach the desirable electrolyte concentrations. The average hydrodynamic diameter was followed with time (Figure 5). For c = 1, 10, or 100 mM, the dispersions were quite stable, changing little even after several days, for both w = 50 and 1000 ppm. For c = 500 mM and w = 50 or 1000 ppm, the dispersions were quite unstable, with aggregation and settling seen after only about 20 and 10 min, respectively. The fast aggregation at 500 mM NaNO3 made the experimental determination of RDLS more uncertain. The results of the strong effect of c on the dispersion stability were qualitatively consistent with the predictions of the DLVO theory. Langmuir 2010, 26(10), 6995–7006
Plotting the data as [RDLS/R1 - 1] versus time, and fitting them to eq 16 provided the value of ksN0 (Figure 6), and N0 was estimated indirectly from w and the particle density. The values of ks and W are listed in Table 3 in section 6. Comparisons with the predictions from the DLVO models will be discussed in section 6. In certain cases of low c, the dispersions were so stable that only an upper estimate of the curve slope could be obtained, if the measurements were limited to 1 week or less. For c = 500 mM, for which the dispersions were quite unstable, only the initial slope was obtained. The value of Wexp was 1.4 108 in water. At c = 1 mM, Wexp was greater than 3.2 106. At c = 10 mM, the Wexp values were quite high (see Table 3) at w = 50 and 1000 ppm. At w = 10000 ppm (1 wt %), the rate of increase in RDLS was sufficiently fast to allow a fairly accurate determination of Wexp, calculated to be 7.9 108. At c = 100 mM and w = 1000 ppm, Wexp was about 2.6 107, which indicates a significant effect of c in decreasing the value of Wexp. At c = 500 mM, the effect of c was more dramatic, and Wexp ≈ 1.0 103 or 9.0 103 for w = 50 and 1000 ppm, respectively. This W value implies that the dispersion was more stable than at the “critical flocculation concentration” of the electrolyte.9 3.4. Zeta Potential (ζ) and pH Measurements. The ζ potential of a CuPc dispersion in water was found to be negative as expected, ζ = -42 ( 3 mV. At c = 10 mM and w = 50 ppm, ζ=-37 ( 5 mV. At w = 1000 ppm, which was diluted to 50 ppm for measurements, ζ = -32 ( 3 mV, which is within the measurement error. At w = 10000 ppm, ζ = -37 ( 2 mV, which is consistent with the above values. At c = 500 mM, the two values of ζ at w = 50 and 1000 ppm are different. The reasons for this DOI: 10.1021/la904224g
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Dong et al. Table 2. ζ Potentials of CuPc Dispersions at Various NaNO3 Concentrations and Some Calculated Properties Based on ζ Potentials c(NaNO3), mM 1 10
100 500
w, ppm 50 50 1000a 10000a 50 1000a 50 1000a
ψ0 ≈ ζ, mV
σ, C/m2
-38 ( 4 -3.0 10-3 -37 ( 5 -9.2 10-3 -32 ( 3 -7.8 10-3 -37 ( 2 -9.2 10-3 -31 ( 4 -2.4 10-2 -31 ( 4 -2.4 10-2 (-17 ( 3)b -2.8 10-2 (-7 ( 2)b -1.2 10-2
a, nm2/ ion
z
di , nm
53 17 21 17 6.8 6.7 5.7 14
-891 -2793 -2365 -2855 -6875 -6900 -8591 -3492
7.3 4.2 4.5 4.2 2.6 2.6 2.4 3.7
a Measurements were conducted after diluting the dispersions to 50 ppm. b Not reliable because of fast aggregation during measurements.
Figure 6. Plots of [RDLS/R1 - 1] vs time for a 1000 ppm of CuPc dispersion at different concentrations of NaNO3: A, c = 10 mM; B, c = 100 mM; C, c = 500 mM. (The dashed lines represent the upper bound of the slope in part A, the best fitting line in part B, and the initial slope in part C.) Plots for other data are not shown. discrepancy are unclear, but may be related to the pronounced instability of the dispersion. On the basis of the assumption that the surface Stern potentials are equal to the ζ potentials, we calculated σ, a, z, and di to obtain more information about the surface chemistry of the CuPc particles, where σ (C/m2) is the surface charge density calculated from ζ using eq 23, a (nm2/ion) is the average area per monovalent ion on the surface (a = 0.16/|σ|), z is the particle charge calculated from σ and the surface area of the particles (assumed to be cubical), and di (nm) is the average distance between two neighboring negative ions on the CuPc surface. For reference, we have done the following calculation. If σ were -0.15 C/m2, corresponding to one sulfonate group per CuPc molecule (with an area 1.1 nm2 if it lies flat on the surface),15 the ζ potentials would be -224, -165, -107, and -68 mV for a 50 ppm of CuPc dispersion at c = 1, 10, 100, and 500 mM, respectively. The (15) Brown, C. J. J. Chem. Soc. A: Inorg. Phys. Theor. 1968, 2488.
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differences between the measured values (Table 2) and the predicted values of ζ may be due to the smaller values of the sulfonate surface density, or to only partial dissociation of the sulfonic acid groups, or to the adsorption of Naþ ions on the sulfonate groups. As the concentration of NaNO3 increases, |ζ| and a decrease. di also decreased, but its value is still quite high, ranging from 7.3 to 2.6 nm. The drop in |ζ| does not result in a decrease in |σ|. In fact, σ and z became more negative; |z| was about 9 times larger at c = 500 mM than that at c = 1 mM. If only Naþ ions had adsorbed, then z should become less negative. Hence, some anions may have adsorbed on that portion of the CuPc surface that is not covered by the sulfonate groups. A qualitative schematic diagram is shown in Figure 7. We postulate that a substantial concentration of NO3- ions adsorbed on the uncharged polar CuPc surface. We cannot exclude the possibility that some Naþ ions may also have adsorbed on the uncharged polar surface, in addition to those adsorbed on the sulfonate groups. However, the fact that z became more negative suggested that the number of adsorbed positive and negative ions were not equal, and that there was an excess number of adsorbed NO3- ions. Reddy and Fogler have also reported that dispersions of uncharged hydrocarbon particles in water can be negatively charged by preferential adsorption of OH- ions over H3Oþ ions from water.16 Elimelech and Omelia have shown that the increase of the mobility with increasing electrolyte concentration is due to the approach of some co-ions close to the surface of the negatively charged particles.17 The pH of CuPc dispersions in water decreased from 6.9 at 50 ppm (0.005 wt %) to 6.2 at 10 wt %. This minor difference may indicate that as the weight fraction of CuPc dispersion increased, some hydrogen ions were released into the aqueous solution or some hydroxyl anions were adsorbed onto the particles surfaces.
4. Estimation of Hamaker Constant of CuPc in Water Although no direct measurements were available in the literature for the Hamaker constant A11 of CuPc, two experimentally derived values for A11 have been reported. A value of 0.2 10-20 J was estimated from certain mechanical strength properties of a CuPc powder, where the distance between presumed flat-faced particles was estimated to be uniformly 0.4 nm.18 Such a method seemed potentially problematic. This value was quite small, and smaller than the range of 4 to 7 10-20 J reported for most hydrocarbons.9 Another value of 3.7 10-20 J for “CuPc green,” or chlorinated CuPc, was reported recently.19 Since CuPc contains copper and benzene rings, its density is higher than those of hydrocarbons, and hence the value for CuPc should be higher than 7 10-20 J. Therefore, the accuracy of these two values was questionable. (16) Reddy, S. R.; Fogler, H. S. J. Phys. Chem. 1980, 84, 1570. (17) Elimelech, M.; Omelia, C. R. Colloids Surf. 1990, 44, 165. (18) Li, Q.; Feke, D. L.; Manas-Zloczower, I. Powder Technol. 1997, 92, 17. (19) Hui, D.; Nawaz, M.; Morris, D. P.; Edwards, M. R.; Saunders, B. R. J. Colloid Interface Sci. 2008, 324, 110.
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Article Table 3. W Values Calculated from the Experiment and W Values Predicted from the DLVO Theory for Two Models DLS Data
c(NaNO3), mM w, ppm 0 1 10
100 500
50 50 50 1000 10 000 50 1000 50 1000
3
ks, m /s
DLVO model for cubesb
DLVO model for spheres Wexp
Wsphere DLVO
9.1 10-26 1.4 108 5.2 1030 e3.8 10-24 g3.2 106 1.4 1014 2.5 106 6.5 105 3.8 107 2.1 102 1.6 10-26 7.9 108 7.8 105 2.1 106 1a 4.7 10-25 2.6 107 1a 1.2 10-20 1.0 103 1a 1.4 10-21 9.0 103 1a
Φsphere max , 73 38 19 11 19 n/a n/a n/a n/a
kT d1, nm 19 2.3 1.5 1.8 1.5 n/a n/a n/a n/a
conclusion
Wcube DLVO
overpredict overpredict (underpredict)c
3.1 102 1.9 1065 5.0 10106 8.3 1061 1.5 10109 1a 1a 1a 1a
underpredict underpredict
Φcube max , kT d2, nm 3.9 159 256 153 262 n/a n/a n/a n/a
89 5.1 2.7 3.1 2.7 n/a n/a n/a n/a
conclusion underpredict overpredict overpredict
(underpredict)c underpredict
a When no positive Φmax was predicted, fast coagulation limit (W = 1) is assumed. b The equation of calculating W for the sphere-model was used here c since the equation for predicting W from Φ h cube max was not available. This conclusion is not fully supported by the sensitivity/error analysis. (See Appendix).
The parameter a (=0.6815) was used to account for the many body nonadditive interactions and forces, which were neglected in the derivation of the Hamaker equation. Using the experimental value of F1 = 1.56 g/cm3, A11 was found to be 13.3 10-20 J. This value differed by 7.0% from the value of 14.3 10-20 J calculated using the theoretical density of F1 = 1.62 g/cm3. The first value will be used in the DLVO models below. An experimental value of the nonretarded Hamaker constant for water, A22, was found in the literature to be 3.7 10-20 J, which was close to a theoretical prediction of 3.8 10-20 J.23 Using these values of A11 and A22 and the well-known approximate relation9 pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi A121 ≈ ð A11 - A22 Þ2
ð18Þ
The effective Hamaker constant of the CuPc particles in water, A121, or A for simplicity, was found to be 2.9 10-20 J. The same value of A is used for CuPc in all NaNO3 solutions studied. In the sensitivity analysis of the calculations of W, values of 1.9 10-20 J and 3.9 10-20 J were also used (see Appendix).
5. New Dimensionless Formulations of the DLVO Theory
Figure 7. Schematic of surface charges on CuPc particles in contact with the aqueous solution.
To obtain a reliable value of A11, the following method was used. Details were described in a recently accepted paper by the same authors.20 The method used is time-dependent density functional theory (TDDFT) to predict the nonretarded London dispersion constant C11,21,22 and an empirical modification of the Hamaker constant equation with an empirical constant a to arrive at the relation A11 ¼ aπ2 C 11 F1 2
ð17Þ
The effects of the surface sulfonate groups were ignored, because the surface layer atoms comprised no more than 1% of the total. (20) Zhao, Y.; Ng, H.; Hanson, E.; Dong, J.; Corti, D.; Franses, E. J. Chem. Theory Comput. 2010, in press. (21) Marques, M. A. L.; Castro, A.; Malloci, G.; Mulas, G.; Botti, S. J. Chem. Phys. 2007, 127. (22) Botti, S.; Castro, A.; Andrade, X.; Rubio, A.; Marques, M. A. L. Phys. Rev. B 2008, 78.
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5.1. Review of Dimensional Equations for Spheres and Parallel Face-to-Face Cubes. The approximate expression for between two identical spherical particles the total energy Φsphere T of radius RS at a distance d, between the surfaces along the line that connects their centers, is given by eq 19, which is valid for small distance such that d , RS,9 Φsphere ðA, RS , Y 0 , n, d, TÞ ¼ T
ARS þ 64πkTRS nY 0 2 K -2 expð -KdÞ 12d ð19Þ
where the dimensionless surface potential y0 and the Debye length κ-1 are Fψ0 ψ0 ¼ RT ðRT=FÞ
ð20Þ
jy j exp 0 -1 2 Y0 jy0 j þ1 exp 2
ð21Þ
y0
(23) Hough, D. B.; White, L. R. Adv. Colloid Interface Sci. 1980, 14, 3.
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K -1
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi RTε0 εr 2F 2 I
A kT
ð28Þ
RS
RS ¼ KRS K -1
ð29Þ
d
d ¼ Kd K -1
ð30Þ
A
ð22Þ
and the surface charge density σ is jy j 1=2 σ ¼ ð8ε0 εr nkTÞ sinh 0 2
ð23Þ
Here n is the molecular concentration n = cNA (molecules/m3) for a 1:1 electrolyte and NA is the Avogadro number, F is Faraday’s constant, R is the gas constant, ε0 is the permittivity of vacuum, and εr is the dielectric constant of the medium, taken here to be equal to 78.5, that of water. Equation 23 is used to estimate σ from c0 ≈ ζ in Table 2. Equation 19 was derived based on the assumption that the effective surface Stern potential c0 was equal to the measured potential ζ.24 The Derjaguin approximation was used to estimate the repulsive double layer energy from the long-range analytical expression for parallel plates. The constant-c0 case, in which the surface potential remained the same as the particles approach, was considered. Moreover, it was assumed that there were no specific ion effects. For simplicity, the strong short-range repulsive forces, which prevented the particles from overlapping,9 were not considered. The expression for the total potential energy of interaction between parallel face-to-face cubical particles with identical sizes can be obtained by modifying the expression for interacting between two interacting plates. The attractive energy Φplate A parallel plates is expressed, in J/m2, as follows9 "
Φplate A ðA, d, cÞ ¼ -
A 1 1 2 þ 12π d 2 ðd þ 2cÞ2 ðd þ cÞ2
# ð24Þ
where d is the distance between the plates surface, and c is the thickness of the plates. For c . d, eq 24 reduces to a simpler form, which together with the repulsive energy yields the following (in J/m2) expression for Φplate A Φplate ¼ T
A þ 64kTnK -1 Y 0 2 expð -KdÞ 2πd 2
A c2 þ 64kTnc2 K -1 Y 0 2 expð -KdÞ 12πd 2
sphere
Φsphere T kT
ð27Þ
(24) Hunter, R. J. Zeta Potential in Colloid Science; Academic Press: San Diego, CA, 1981.
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EDL ¼ 12 64πnY 0 2 kT
ð32Þ
The last term is a measure of the repulsive double layer potential energy. If we also define another dimensionless number N, as the ratio of E DL to A, N
EDL EDL ¼ A A
ð33Þ
then eq 19 reduces to the simpler equation. sphere
Φ hT
ðA, RS , N, dÞ ¼ þ
ARS 1 - þ N expð -dÞ 12 d
ð34Þ
The dimensionless number N, which is the ratio of the repulsive double layer energy to the attractive energy, or the Hamaker constant, arises naturally in the DLVO theory, and could be called the “DLVO number,” or the “Verwey-Overbeek number.” This number has been identified as a key dimensionless group affecting stability by Russel et al.25 for small surface potentials, as εε0 ψs 2 RS A
to have a maximum (and a minimum), the first In order for Φ h sphere T with respect to d has to be zero. Equation 34 derivative of Φ h sphere T yields the following condition: 2
ð26Þ
This equation also can be used to approximately evaluate the interaction energy of two rectangular blocks with identical parallel oriented surface, but different lengths (e.g., c/2 or 2c), since the outer parts of the blocks contribute little to the total attractive interaction energy. 5.2. Dimensionless Formulations for Spheres and Parallel Face-to-Face Cubes. For monodisperse spherical particles, various dimensionless groups are defined with an overbar, as follows Φ hT
EDL
Nr
2 ¼ Φplate Φcube T T c
ð31Þ
and
ð25Þ
In order to have a finite-size particle, we consider the total interaction energy for two cubical particles in a parallel, face-toface orientation with an interaction area of c2. Equation 25 yields
¼ -
n nK -3
d expð -dÞ -
1 ¼0 N
ð35Þ
For convenience in assessing the solution to the above equation, we now define the function f1(d) to be equal to 2
f 1 ðdÞ d expð -dÞ
ð36Þ
Moreover, in order for Φ h sphere max to be positive, the following condition should be satisfied: 1 d expð -dÞ - > 0 N
ð37Þ
(25) Russel, W. B.; Saville, D. A.; Schowalter, W. R. Colloidal Dispersions; Cambridge Univ. Press: Cambridge, U.K., 1989.
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W ¼
pffiffiffi π sphere exp½ Φ h max 4RS p
ð41Þ
where vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 1ffi u u uARS @ 1 2 A - 2þ 3 p ¼t 24 d1 d1
Figure 8. Plots of functions f1(d), f2(d), and f3(d) vs d for the DLVO models. The intersection d 1 of curve f1(d) with 1/N represents the distance at the maximum Φ h sphere for the sphere model. max The intersection d 2 of curve f3(d) with 1/N represents the distance cube at the maximum Φ h max for the cube model.
A large positive value of Φ h sphere yields a substantial value max sphere of W. Then Φ h max is interpreted as an “activation energy” for coagulation. Here we neglect hydrodynamic interactions, which could decrease the coagulation rate by a factor of only 2 to 3.28 For monodisperse cubical particles, we define the same dimensionless variables A, d, and EDL as those for the spherical particles, and introduce two more dimensionless quantities, cube
Φ hT
ð38Þ
Plots of f1(d) and f2(d) are shown in Figure 8. The function f1(d) has a maximum of 0.5413 at d = 2. This implies that if N < 1/ 0.5413 = 1.8473, then eq 35 has no solution, and the function has no maximum. The function f2(d) has a maximum of Φ h sphere T h sphere can be positive only if N > 2.72, for 0.368 at d = 1. Hence, Φ T which there is a maximum. For N > 2.72, there is a positive Φ h sphere max , h sphere varies with one can solve eq 35 to find how the distance d 1 at Φ max N. Using eq 35 in eq 34 we find the values of the positive maximum Φ h sphere max . 2 3 ARS 4 1 15 sphere Φ h max ¼ ð39Þ - þ 2 > 0, where d 1 < 1:0 12 d1 d1
ch
c ¼ Kc K -1
ð43Þ
where r = d þ 2Rs.26 For strong repulsive interactions, when W g 103, there is a useful approximation to W,27 (26) Evans, D.; Wennerstr€om, H. The Colloidal Domain: Where Physics, Chemistry, Biology, and Technology Meet, 2nd ed.; Wiley-VCH Inc.: New York, 1999. (27) Overbeek, J. T. G. Adv. Colloid Interface Sci. 1982, 16, 17.
ð44Þ
Equation 26 can then be reduced to the simpler form 0
cube Φ h T ðA,
1" # 2 A c h 1 A - þ N expð -dÞ c, h N, dÞ ¼ þ @ 2 12π d
ð45Þ
to have a positive maximum, eq 45 yields In order for Φ h sphere T the following two conditions: 3
d expð -dÞ 1 - ¼0 2 N
ð46Þ
1 >0 N
ð47Þ
and 2
The negative minimum Φ h sphere min , which may be important in flocculation, can also be obtained, but it is not considered here further. When N is very large, Φ h sphere max has an asymptotic solution of sphere Φ h max ≈ ARs N=12 ¼ Rs EDL =12 ¼ 64πRs nY 0 2 . In this limit, 2 Φ h sphere max is independent of A, and proportional to Y0 , and to the dimensionless concentration n. From eqs 29 and 31, we find that √ the term RSn = κRSnκ-3 = RSnκ-2. Since κ is proportional to n, 2 the value of Φ h sphere max in this limit is proportional to RS and Y0 , and it is independent of electrolyte concentration. The stability ratio W has been predicted by Fuchs and Smoluchowski to depend on the following integral of Φ h sphere T Z ¥ dr sphere W ¼ 2Rs exp½ Φ h T ðr -2Rs Þ 2 ð40Þ r 2Rs
Langmuir 2010, 26(10), 6995–7006
2 Φplate T c kT
and
The function f2(d) is defined also for convenience as f 2 ðdÞ d expð -dÞ
ð42Þ
d expð -dÞ -
and similarly the function f3(d) is defined for convenience. 3
d expð -dÞ f 3 ðdÞ 2
ð48Þ
The function f3(d) has a maximum of 0.672 at d = 3 (Figure 8). This implies that eq 46 has a solution only if N > 1/0.672 = 1.488. The function f4(d) d 2 exp(-d) = f1(d) has a maximum of h cube can be positive only if N > 1.8473. 0.5413 at d = 2. Hence, Φ T For such values, one can solve eq 46 to find how the dish cube tance d 2 at the maximum Φ max varies with N. Similarly as for the sphere-model, the positive maximum Φ h cube max for the cube-model is 2 3 2 A c h 1 2 cube 4 - 5 > 0, where d 2 < 2:0 ð49Þ Φ h max ¼ 12π d 2 d 3 2
2
The above dimensional and dimensionless models for the cubes seem to be novel. For very large values of N, Φ h cube max also has an (28) Honig, E. P.; Roeberse, G.; Wiersema, P. H. J. Colloid Interface Sci. 1971, 36, 97.
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2
2
asymptotic solution Φ h max ≈ A ch N=12π ¼ ch EDL =12π ¼ 64 2
2 2 2 2 -3 ch nY 0 2 From eqs 31 and 44, the term √ c n c n = κ c nκcube = 2 -1 Φ h max is c nκ . Since κ is proportional to n, the value of √ proportional to c2, Y02, and now also proportional to n. The relationship between Φ h cube max and W is assumed to be the same as that for spheres, eqs 40-42.
6. Comparisons of Experimental W Values with Those Predicted by the DLVO Theory In water, the model for spheres predicted Φsphere max =73 kT, and W = 5.2 1030. Since d1 ≈ 19 nm, the DLVO theory was expected to be accurate, and indeed predicted high stability. The 10 wt % dispersion was stable for months, and its stability was surely consistent with the DLVO prediction. The model for spheres overpredicted the stability (Table 3). The model for cubes underpredicted the stability, however, for this extremely low ionic strength. For a ζ potential of -42 mV, the total particle charge was z = -10. For such small charge the assumption of uniform surface charge density became poor, and the application of the DLVO theory was uncertain. The stability ratio W generally decreases with increasing ionic strength I at fixed c0, at which σ increases based on eq 23. In the CuPc particles tested, |c0| decreases and |σ| increases as I increases from 1 to 500 mM (Table 2). Generally, the DLVO model predicts a less stable dispersion at higher ionic strength (Table 3). At c = 1 mM, both models overpredicted the stability. At c = 10 mM, the sphere-model underpredicted the stability ratios by 2-5 orders of magnitude. In contrast, the cube-model overpredicted the stability. Hence, the shape of the particles seemed to have some effects on the stability. If the CuPc particles were spherical, the model predicted that they would be less stable than they were. It would indeed be interesting to test this prediction if spherical CuPc particles were available. At c = 100 or 500 mM, both models predicted no electrostatic stabilization at all. No Φmax is predicted, because the values of N were below the thresholds of N = 2.72 for spheres and N = 1.85 for cubes. Even if there were a predicted value of Φmax, the values of d1 or d2 would be so small (0.2 to 1 nm) that the application of the DLVO theory would be invalid. Since the dispersions were stable at c = 100 mM, and somewhat stable at c = 500 mM, we inferred that their stability was caused not by the DLVO-type electrostatic forces but by other short-range repulsive forces. For c = 100 mM, such forces can produce W-values of 106-107. For c = 500 mM, the reduction of effective DLVO repulsive forces due to the charge screening by the electrolyte (small Debye length) allowed the particles to approach quite close and led to fast coagulation. Nevertheless, there was still some repulsive barrier, the nature of which was unknown. The values of the key parameters Rs (or c), A, and ζ for cases of 50 ppm of CuPc dispersion at various concentrations of NaNO3, were varied first one at a time, and then three at a time, to examine their effects on the predicted stability ratios (See Appendix). The variations were (5 nm (or (10 nm), ( (1 10-20) J, and (5 mV, respectively. For the sphere-model, ζ was found to be the most sensitive parameter. At c = 0 mM (no salt) or at c = 1 mM, the model overpredicted the stability in all cases, validating the conclusions. At c = 10 mM, the differences between the experimental W values and the predicted W values did not seem to be statistically significant. At c = 100 and 500 mM, no positive Φsphere max values were predicted for all values tested, validating the conclusion. 7004 DOI: 10.1021/la904224g
For the cube-model, at c = 0 mM, all model cases underpredict the stability. For c = 1 or 10 mM, all cases overpredict the stability. For c = 100 mM, the predictions are quite sensitive to the values of A and ζ, but not c. Hence, the inference that the model underpredicts the stability is not certain. For c = 500 mM, all predictions are much smaller than the data. If the polydispersity of the primary particles were substantial, the time evolution of the concentration of the monomers, dimers and other aggregates at the early stages of aggregation would deviate from the simple models we considered.29 Consequently, the W values would be quite different from those for monodisperse particles. The predicted stability ratio could increase by 2- to 3-fold as the size polydispersity increases. The effect will be more pronounced at lower ionic strength.30
7. Conclusions The dispersion stability and the ζ potentials at 25 C of nonspherical CuPc particles (dh ≈ 90 nm) in water, and in various NaNO3 solutions (1, 10, 100, and 500 mM) were investigated. The initial stability ratios W were determined quantitatively from DLS data for the RDG scattering regime. As the NaNO3 concentration increased from 1 to 500 mM, the W values decreased from 3.2 106 to 1.0 103. This indicates that the stability of CuPc dispersion depends on the electrolyte concentration and that electrostatic effects play an important role. The absolute particle charge |z| per cubic CuPc particle increased from 891 to 8591, which implies that some NO3- ions adsorbed preferentially on the uncharged portion of the CuPc surface. Two new models of the DLVO theory, for spheres and for parallel face-to-face cubes, at constant potential, were reformulated in dimensionless form. The conditions for the existence of a positive dimensionless potential energy maximum Φ h max were found. The value of N, which was defined as the ratio of the double layer repulsive energy to the attractive energy, or the Hamaker constant, was the key dimensionless group. When N > 2.72 for spheres or N > 1.85 for cubes, Φ h max existed and dispersion stabilization by electrostatic forces was possible. When N . 2.72 and the dimensionless size RS or ch, were large enough for Φ h max to be greater than about 15, then W can exceed 106 and the dispersions were stabilized substantially. The value of Φ h max when it existed, was obtained for two models, and was used to predict W for these two shapes. The model for cubes, for the systems tested at 1 and 10 mM NaNO3 concentrations, predicted higher stability than the model for spheres. In water only, the model for spheres overpredicted the stability, while the model for cubes highly underpredicted the stability. By contrast, at c = 1 mM, both models overpredicted the stability. At c = 10 mM, the model for spheres underpredicted the stability whereas the model for cubes overpredicted the stability. This suggests some significant shape effects on the stability. At c = 100 and 500 mM, both models underpredicted the stability substantially. Shape and orientation effects and the effects of other shortrange forces need to be investigated further, by molecular-scale theories or simulations. In particular, at higher ionic strengths, some additional short-range forces barrier may play a major role in the dispersion stability. Acknowledgment. This work was support by the HewlettPackard Company Innovation Research Program. We are grateful to Prof. Won at Purdue for allowing the use of the Brookhaven (29) Holthoff, H.; Borkovec, M.; Schurtenberger, P. Phys. Rev. E 1997, 56, 6945. (30) Yato, A.; Papadopoulos, K. D. Colloids Surf. 1985, 16, 55.
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ZetaPALS instrument. We thank Mrs. Debbie Sherman for help with the electron microscopy in Figure 1.
Appendix: Sensitivity Analysis of Predictions of Stability Ratios Such an analysis was done for both spheres and cubes, to examine effects of possible errors in Rs, A, or ζ on the predicted
Φ h max and the stability ratios W. For spheres (Table 4), the ζ potential had the most impact for 0, 1, and 10 mM NaNO3. For 100 or 500 mM, the predicted values of N were below the limit of 2.72, in all cases, and no stabilization was predicted. For cubes, the results were sensitive to the errors considered for c = 1 and 10 mM, but not for c = 0, 100, and 500 mM. (Table 5).
Table 4. Sensitivity Analysis in Predictions of Energy Maximum and Stability Ratios for the Sphere Model c(NaNO3), mM
Rs, nm
A, 10-20 J
ζ, mV
d1, nm
Φsphere max , kT
45 2.9 -42 19 73 50 2.9 -42 19 81 40 2.9 -42 19 65 45 3.9 -42 22 72 45 1.9 -42 15 73 45 2.9 -47 17 89 45 2.9 -37 21 57 40 3.9 -37 24 51 50 1.9 -47 14 99 1 45 2.9 -38 2.3 38 50 2.9 -38 2.3 42 40 2.9 -38 2.3 33 45 3.9 -38 2.7 34 45 1.9 -38 1.8 42 45 2.9 -43 2.0 50 45 2.9 -33 2.6 26 40 3.9 -33 3.1 21 50 1.9 -43 1.6 61 10 45 2.9 -37 1.5 19 50 2.9 -37 1.5 21 40 2.9 -37 1.5 17 45 3.9 -37 1.8 13 45 1.9 -37 1.1 26 45 2.9 -42 1.3 29 45 2.9 -32 1.8 11 40 3.9 -32 2.2 5.3 50 1.9 -42 1.0 41 100 39 to 49 1.9 to 3.9 -36 to -26 n/a n/a 500 40 to 50 1.9 to 3.9 -12 to -22 n/a n/a a No positive Φsphere max were predicted and the fast coagulation limit (W = 1) is assumed. 0
Wsphere DLVO 5.2 1030 1.5 1034 2.1 1027 3.3 1030 7.3 1030 5.1 1037 1.5 1024 2.2 1021 1.4 1042 1.4 1014 8.3 1015 2.5 1012 4.1 1012 1.1 1016 3.1 1019 2.8 109 1.2 107 1.7 1024 6.5 105 4.5 106 9.4 104 3.0 103 5.8 108 1.0 1010 2.1 102 1.7 1.4 1015 1a 1a
Wexp 1.4 108
g3.2 106
>2.5 106
>2.1 106 1.0 103
sensitivity base case small small very small very small large large large large base case small small small small large large very large very large base case small small small small large small very large very large very small very small
Table 5. Sensitivity Analysis in Predictions of Energy Maximum and Stability Ratios for the Cube Model c(NaNO3), mM 0
1
10
l, nm 90 100 80 90 90 90 90 80 100 89 99 79 89 89 89 89 79 99 90 100 80 90 90 90 90 80 100
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A, 10-20J 2.9 2.9 2.9 3.9 1.9 2.9 2.9 3.9 1.9 2.9 2.9 2.9 3.9 1.9 2.9 2.9 3.9 1.9 2.9 2.9 2.9 3.9 1.9 2.9 2.9 3.9 1.9
ζ, mV -42 -42 -42 -42 -42 -47 -37 -37 -47 -38 -38 -38 -38 -38 -43 -33 -33 -43 -37 -37 -37 -37 -37 -42 -32 -32 -42
d2, nm
Φcube max , kT
Wcube DLVO
89 89 89 99 77 83 97 107 72 5.1 5.1 5.1 5.8 4.3 4.7 5.7 6.4 3.9 2.7 2.7 2.7 3.1 2.2 2.4 3.1 3.6 2.0
3.9 4.8 3.1 3.8 4.0 4.8 3.1 2.4 6.1 159 196 125 141 182 214 110 76 299 256 317 203 195 342 382 153 84 602
3.1 102 5.7 102 2.0 102 3.1 102 3.1 102 6.8 102 1.6 102 1.2 102 1.6 103 1.9 1065 3.1 1081 6.5 1050 5.5 1057 2.1 1075 1.6 1089 2.6 1044 4.5 1029 8.4 10125 5.0 10106 4.8 10132 3.0 1083 1.2 1080 9.0 10143 1.8 10161 8.3 1061 1.5 1032 3.6 10256
Wexp 1.4 108
g3.2 106
>2.5 106
sensitivity base case very small very small very small very small very small very small very small small base case very large very large very large very large very large very large very large very large base case very large very large very large very large very large very large very large very large
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Dong et al. Table 5. Continued
c(NaNO3), mM
l, nm
A, 10-20J
ζ, mV
d2, nm
Φcube max , kT
88 2.9 -31 n/a n/a 98 2.9 -31 n/a n/a 78 2.9 -31 n/a n/a 88 3.9 -31 n/a n/a 88 1.9 -31 1.6 89 88 2.9 -36 1.7 47 88 2.9 -26 n/a n/a 78 3.9 -26 n/a n/a 98 1.9 -36 1.3 335 500 80 to 100 1.9 to 3.9 -12 to -22 n/a n/a a No positive Φsphere max were predicted and the fast coagulation limit (W = 1) is assumed. 100
7006 DOI: 10.1021/la904224g
Wcube DLVO a
1 1a 1a 1a 1.7 1033 8.2 1014 1a 1a 3.3 10139 1a
sensitivity
Wexp >2.1 10
6
1.0 103
base case very small very small very small very large very large very small very small very large very small
Langmuir 2010, 26(10), 6995–7006
