Paradox of Stability of Nanoparticles at Very Low Ionic Strength

Jun 28, 2012 - Department of Civil and Environmental Engineering, Center for the Environmental Implications of NanoTechnology (CEINT),. Duke Universit...
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Paradox of Stability of Nanoparticles at Very Low Ionic Strength Shihong Lin and Mark R. Wiesner* Department of Civil and Environmental Engineering, Center for the Environmental Implications of NanoTechnology (CEINT), Duke University, P.O. Box 90287, Durham, North Carolina 27708, United States ABSTRACT: The scaling of electrical double layer interaction energy from a plate−plate system to a sphere−plate system was reexamined, and it was found that accurate scaling without resorting to the Derjaguin approximation theoretically predicts the destabilization of nanoparticles in water depleted of added electrolyte and, consequentially, a maximum stability at a moderate ionic strength. This theoretical feature re-emphasizes the dual-role nature of added electrolyte that was supported by experimental results of direct surface force measurement but not by those of colloidal stability of nanoparticle deposition/aggregation. Inconsistences between the theoretical prediction and the experimental observation and between experimental observations in different systems were discussed. Possible reasons leading to the inconsistences were explored, including the effect of curvature, the contribution from counterions, the mode of interaction, and the applicability of an equilibrium model to describe the colloidal interaction of a nanoparticle suspension.

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double layer of an increased ionic concentration.17 In addition, the Columbic contribution (or the Maxwell stress) between two similarly charged surfaces is, perhaps contrary to intuition, attractiveif the attractive Columbic force between the counterions and the two charged surfaces is also taken into account.17,18 For two surfaces of similar charges, the excess osmotic pressure is usually significantly stronger than the attractive Maxwell stress, and thus, the overall EDL interaction between such two surfaces is repulsive. According to such an understanding, the EDL interaction will become weak if the electrolyte concentration in the solution is very low, because the excess osmotic pressure (i.e., the difference between the pressure in the gap and that outside the gap) should be roughly proportional to the electrolyte concentration according to the Morse equation. 19 Following such an argument, it is hypothesized that nanoparticles might become relatively unstable at very low ionic strength due to the absence of sufficient ions to provide a strong enough osmotic repulsion to counter the vdW attraction. This may be a different mechanism, if proven to be existent, to destabilize nanoparticles, as compared to that of adding excessive indifferent electrolytes for the screening of the EDL interaction. It is the purpose of this study to examine this hypothesis from both theoretical and experimental perspectives.

INTRODUCTION The Derjaguin−Landau−Verwey−Overbeek (DLVO) theory1−3 has been widely used in quantitatively explaining a plethora of phenomena in intersurface or interparticle interactions in electrolyte solutions. In brief, the DLVO theory states that the interaction between two charged lyophobic surfaces in an electrolyte solution is simply the sum of the van der Waals (vdW) interaction and the electrical double layer (EDL) interaction. Applying the DLVO theory to the interaction between two identical or similar surfaces (or particles), the vdW force is always attractive and the EDL interaction is usually repulsive if the two interacting surfaces are of similar charges.4 Significant effort has been spent on identifying conditions with which the classic DLVO theory alone is insufficientas exemplified by the situations when steric force,5−7 hydration force,8,9 oscillation force,10,11 hydrophobic interaction,12,13 or ion-correlation force4,6 are present. Except for some forces, such as the ion-correlation force that may become significant at high ionic strength (IS), many of the forces mentioned do not actually invalidate the classic DLVO theory, as they can be added as independent forces to the original DLVO theory to formulate the corresponding extended-DLVO (xDVLO) theories that can be successfully applied to analyze measured force curves or stability curves.16 In short, DLVO theory has been proven to be valid provided the conditions used to formulate the theory are satisfied. It is well-understood that the repulsive force between two surfaces of similar charges in an electrolyte solution arises primarily from the increased osmotic pressure in the gap between the surfaces, as the surrounding solvent has the tendency to migrate into the gap to dilute the overlapping © 2012 American Chemical Society

Received: August 11, 2011 Revised: June 25, 2012 Published: June 28, 2012 11032

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Figure 1. (A) EDL interaction energy per area between two parallel flat plates based on LSA (eq 1). The surface potentials for both surfaces are both −25.7 mV (kBT/e). (B) The total interaction energy (EDL + vdW) between the same two plates in part A, assuming a Hamaker constant of 5 × 10−21 J.

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THEORY One interesting and important feature that DLVO theory predicts for the interaction between two parallel plates was that, with increasing ionic strength, the EDL interaction is stronger at very close separation (H→0) but it also decays faster with increasing separation H. This feature is obviously demonstrated by the presence of κ in both the prefactor and exponential term in the expression (eq 1) for such an interaction20

Coupled with the nonretarded vdW interaction between two flat surfaces17 as given by eq 2, it is demonstrated in Figure 1B that the energy barrier increases with increasing ionic strength but the repulsion also decays more slowly. vPP,vdW (H ) = −

(2)

where vPP,vdW(H) is the per-area vdW interaction energy between two parallel plates, and Ah is the Hamaker constant. Such a pattern of interaction energy curves has in fact been observed by experiments using surface force apparatus (SFA) with which the force between the surfaces of two crossed cylinders is measured in the presence of electrolytes.10 The results of these experiments were reported as F(H)/2πR (the measured force at a given separation distance F(H) over the radius of the crossed cylinders R), which is equivalent to the potential energy per area between two flat plates.21 The reported results of F(H)/2πR had a pattern essentially similar to that shown in Figure 1B. These observations appeared to validate the predicted dependence of surface−surface interaction profile as a function of the ionic strength described by the classic DLVO theory. It is natural to ask then whether a similar pattern also applies to the dependence of colloidal stability on the concentration of indifferent electrolyte (or κ). The answer given by Verwey and Overbeek was “no” for a sphere−sphere interaction.22 They concluded, on the basis of the calculated potential energy of EDL interaction between two spheres as obtained by applying Derjaguin approximation (DA), that when the Debye constant κ approaches zero (low IS), the potential energy of EDL interaction at any separation, and consequently the barrier of the total interaction energy, could only increase. They attempted to explain such a theoretical discrepancy between sphere−sphere interaction and plate−plate interaction in their response to the change of IS by the “divergence of lines of force of the electric field in the case of spherical particles”,22 thereby postulating that curvature would lead to a monotonic response of colloidal stability to the change of ionic strength. Mathematically, it is clear that when the plate−plate interaction as described by eq 1 is scaled to a sphere−sphere

⎛ zeψ ⎞ ⎛ zeψ ⎞ ⎛k T ⎞ vPP,EDL − LSA(H ) = 32εκ ⎜ B ⎟ tanh⎜ P1 ⎟tanh⎜ P2 ⎟ ⎝ ze ⎠ ⎝ 4kBT ⎠ ⎝ 4kBT ⎠ 2

e−κH

Ah 12πH2

(1)

where vPP,EDL−LSA(H) is the per-area EDL interaction energy between two parallel plates as evaluated using linear superposition approximation (LSA), ε is the absolute dielectric constant of water, κ is the Debye constant, and is equal to (IS)1/2/0.304 numerically, kBT is the thermal energy, z is the valence of the symmetric electrolyte, e is the charge of an electron, ψP1 and ψP2 are the surface potential of plate 1 and plate 2, respectively, and H is the separation distance. This expression, and the expressions scaled from this in the following discussion, is applicable for the interaction between asymmetric double layers.20 Plotting vPP,EDL−LSA(H) at different ionic strengths again clearly illustrates such a feature as shown in Figure 1A. The pattern of change of vPP,EDL−LSA(H) in response to the change of κ can be explained mathematically on the basis of eq 1, and the indifferent electrolytes play two roles in determining the strength EDL interaction: (1) The osmotic pressure between the approaching surface arises primarily from the presence of the indifferent ions, and thus the concentration of these ions regulates the overall strength of the osmotic repulsion, as reflected by the prefactor κ. (2) The electric field stemming from the charged surface is screened by the indifferent ions and thus increased ionic concentration decreases the range of interaction, as reflected by the decay constant κ in the exponential decay term e−κH.This physical interpretation of such phenomenon emphasizes the entropic contribution to the EDL interaction and the dual-role nature of added electrolyte as both the constituting and screening factors of the EDL. 11033

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Figure 2. (A and C) EDL interaction between a sphere of a = 10 nm and a plate, both with a surface potential of −25.7 mV (1 kBT/e), as evaluated by eq 4 (A) and eq 5 (C), respectively. (B and D) Total interaction energy (EDL+vdW) for the same sphere−plate system as evaluated by eqs 4 and 6 (B) and eqs 5 and 6 (D), respectively. The Hamaker constant is 5 × 10−21 J.

interaction using DA, the Debye constant κ only remains in the exponential decay term but not as a prefactor, as shown in eq 3.

It should be noted that although eqs 3 and 4 are obtained using LSA, expressions based on the assumption of constant potential interaction essential yield the same feature.20 It should also be noted that while scaling the plate−plate interaction to sphere−sphere or sphere−plate interaction, DA was employed and thus eqs 3 and 4 are both only applicable when κa ≫ 1, i.e., only when the particle is large compared to the thickness (characteristic decay length) of the double layer. Recently, more accurate expressions were proposed for calculating the energy of EDL in a sphere−plate system based on the LSA24,25 (and also on constant potential and constant charge interactions based on linearized Poisson−Boltzmann equation). The new expression was obtained via applying the surface element integration (SEI) method26 that captures the essense of DA but avoids the geometric simplification that leads to the large κa requirement and is thus applicable for the whole range of κa. This expression (eq 5) is accurate to the extent that it yields the same result as that given by evaluating the sum of osmotic pressure and Maxwell stress over an arbitrary surface enclosing the particle using the potential distribution obtained from LSA, which is considered as the benchmark method for evaluation of interaction energy.

VSS,EDL − LSA,DA(H ) = 64πε

2 ⎛ zeψ ⎞ ⎛ zeψ ⎞ a1a 2 ⎛ kBT ⎞ ⎜ ⎟ tanh⎜ S1 ⎟tanh⎜ S2 ⎟e−κH a1 + a 2 ⎝ ze ⎠ ⎝ 4kBT ⎠ ⎝ 4kBT ⎠

(3)

where VSS,EDL−LSA,DA(H) is the EDL interaction energy between a sphere of radius a1 and a sphere of radius a2 based on LSA with the Derjaguin approximation. Therefore, with eq 3, κ only regulates the decay rate but not the overall strength of the EDL interaction, which means increasing IS can only weaken the EDL interaction by an electrostatic screening effect.20 A similar conclusion naturally extends to the a sphere−plate interaction described by eq 4 that was obtained by having one of the radii in eq 3 become infinity:23 VSP,EDL − LSA,DA(H ) ⎛ zeψS ⎞ ⎛ zeψP ⎞ −κH ⎛ k T ⎞2 = 64πεa⎜ B ⎟ tanh⎜ ⎟tanh⎜ ⎟e ⎝ ze ⎠ ⎝ 4kBT ⎠ ⎝ 4kBT ⎠

(4) 11034

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VSP,EDL − LSA(H ) =

2 ⎛ zeψS ⎞ ⎛ zeψP ⎞ 64πε ⎛ kBT ⎞ ⎜ ⎟ tanh⎜ ⎟tanh⎜ ⎟ κ ⎝ ze ⎠ ⎝ 4kBT ⎠ ⎝ 4kBT ⎠ ⎡(κa − 1)e−κH ⎤ ⎢ ⎥ ⎢+(κa + 1)e−κ(H + 2a)⎥ ⎣ ⎦ (5)

VSP,Total(H) also determines the affinity between a particle and a flat surface, which controls the kinetics of the particle deposition. For nanoparticle aggregation, the relation between aggregation kinetics and VSS,Total(H) can be obtained by solving the diffusion problem under a force field from the mutual interaction between particles as described by VSS,Total(H).3 For nanoparticle deposition in a porous media, the kinetics of particle deposition can be evaluated by solving the convectivediffusion problem under the influence of the force field that is determined by VSP,Total(H).29,30 The predictive model using this approach is named the interaction force boundary layer (IFBL) model. Alternatively, it is also a practical and simpler approximation to use the Maxwell−Boltzmann energy distribution to assess the probability for a nanoparticle to overcome the energy barrier Vmax in VSP,Total(H).31,32 The resulting expression for particle attachment efficiency is

This simple expression based on LSA, like the more complicated charge regulation model, yields an intermediate interaction energy between that of a constant charge (CC) interaction and a constant potential (CP) interaction and was considered a more appropriate approximation (than CC or CP) to the realistic interaction.20,27 It is chosen here for discussion also because it has the simplest mathematical form such that the impact of κ on the interaction energy is comparatively obvious. One very important feature of the new expression VSP,EDL−LSA(H) is that it again reveals a characteristic dependence of interaction energy on κ similar to that given by eq 1 although the κ appears in multiple places in the prefactors and the exponentials, the overall effect is that κ regulates the strength of EDL interaction as well as its decay rate in a very similar way to that in a plate−plate interaction (Figure 2C). It is worth noting that an expression based on the linearized Poisson−Boltzmann (L-PB) equation was also derived for constant-potential (CP) interaction between a sphere and a plate using the SEI method.25 Such an expression, yielding the same result as that given by evaluating the sum of osmotic pressure and Maxwell stress, reveals the same feature of the dependence of VSP,EDL−CP(H) on κ. To evaluate the total interaction energy, the approximate expression for retarded vdW interaction for the sphere−plate system as proposed by Gregory28 was used VSP,vdW(H ) =

⎞ Ah a ⎛ 1 ⎜ ⎟ 6H ⎝ 1 + 14H /λ ⎠

α = 1 − erf( Vmax ) +

2 −Vmax e Vmax π

(7)

A higher Vmax leads to a stronger stability or a lower affinity between nanoparticles or between a nanoparticle and a surface. Therefore, if eq 5 appropriately models the EDL interaction between a particle and a flat surface, in the case of a small particle that allows small κa over a range of realistic ionic strengths (note that minimum ionic strength in any aqueous solution is 10−7 M due to H2O self-dissociation), the affinity between the particle and the surface will be higher at a very low ionic strength as compared to that at a moderately low ionic strength. In other words, a maximum stability should occur at a moderately low ionic strength provided that κ regulates the EDL interaction between a particle and a plate in a way similar to plate−plate interaction. To test this hypothesis, we conducted column filtration experiments in which gold nanoparticles (AuNPs) were forced to pass through a porous medium composed of silica beads at different ionic strengths. The affinity between the particle and the surface can be quantified by the attachment efficiency (α), which is defined as the probability of successful attachment per collision. On the basis of the filtration theory, the relative effluent concentration (Ceff/C0) is related to the attachment efficiency (α) by the following equation33

(6)

where VSP,vdW(H) is the retarded vdW interaction between a sphere and a plate, λ is the characteristic wavelength of interaction, and all other parameters have been defined in the above equations. With eqs 5 and 6, we obtain total interaction energy curves at different ionic strengths (0.1−100 mM) for a nanoparticle of radius a = 10 nm with the surface potentials of both the particle and the plate being equal to −25.7 mV (Figure 2D). As comparison, the EDL interaction energy given by VSP,EDL−LSA,DA is also presented in Figure 2A for different ionic strengths, with the understanding that VSP,EDL−LSA,DA is not accurate for small κa due to DA. The corresponding interaction energy curves taking into account the vdW interaction are given in Figure 2B. The major difference between parts A and C of Figure 2 is the interaction energy at separation H = 0: VSP,EDL−LSA,DA(0) is independent of κ while VSP,EDL−LSA(0) increases with increasing κ. Correspondingly, with VSP,EDL-LSA,DA Figure 2B shows a monotonically decreasing energy barrier with increasing ionic strength, whereas with VSP,EDL-LSA the energy barrier reaches a maximum at a certain ionic strength, as illustrated by Figure 2D. If the predictions given by Figure 2C,D are indeed correct, theory would predict that the characteristic dependence of EDL interaction energy and the total interaction energy on ionic strength for a plate−plate interaction remains valid with the sphere−plate scaling, provided a more accurate evaluation of the interaction energy is used. Colloidal stability (against homoaggregation) is controlled by the profile of interaction energy VSS,Total(H).3 Similarly,

α=−

⎛C ⎞ 4aC ln⎜ eff ⎟ (1 − f )η0L ⎝ C0 ⎠

(8)

where aC is the radius of the collectors (silica beads), L is the length of the column, and ηo is called the single collector efficiency and can be calculated using the Tufenkji−Elimelech (T−E) correlation.34 Therefore, by experimentally measuring the percentage of nanoparticles passing through the column under specific conditions, we can evaluate the affinity between the nanoparticle and the surface under those conditions.

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MATERIALS AND METHODS

Particle Synthesis and Characterization. The AuNPs used in this study were synthesized using the borohydride reduction method.35 In brief, 0.7 mL of freshly prepared solution of 0.1 M sodium borohydride (NaBH4) was added dropwise into a rigorously stirring solution of 50 mL of 0.01% (w/v) HAuCl4. Using a goniometer (ALV/CGS-3), the hydrodynamic radius of the AuNPs synthesized was measured by dynamic light scattering (DLS) to be 7.7 ± 0.1 nm with a polydispersity (PDI) of 0.27 ± 0.01. Nanoparticles of very small size (a) are preferred for two reasons: (1) if a is large, κa will be large for any realistic ionic strength, in which case VSP,EDL−LSA approaches 11035

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Figure 3. (A) Measured attachment efficiencies from column filtration experiment (squares) and attachment efficiency predicted from Maxwell− Boltzmann model using energy curves calculated by summing the VSP,vdW and VSP,EDL that is described by either eq 5 (blue dotted curve) or eq 4 (green dashed curve). a = 7.7 nm, ΨS = −56.6 mV, ΨP = −45.3 mV. (B) Attachment efficiencies based on similar column experiments collected from the literature. The Darcy velocity (or apparent velocity) of the mixed flow in the column was 0.02 cm/s. More than 10 pore volumes (PV) of the background electrolyte solution was allowed to pass through the column before the actual deposition experiment to equilibrate the silica collector surface. Solution Chemistry. Since some of the transport experiments were conducted at very low ionic strengths and the AuNPs had to be used without purification, the impact of the residual ions from the AuNPs synthesis on the ionic strength has to be assessed. An exact calculation of the IS from the reaction stoichiometry was difficult due to the challenges in evaluating the degree of reaction completion and the unknown equilibrium of BO3− adsorption on the formed AuNPs. Therefore, a pragmatic approach was adopted to estimate the contribution of residual ions to the IS by measuring the electric conductivity of the AuNPs stock and mapping that to the corresponding NaCl concentrations. The IS of the AuNPs was estimated to be 1.46 mM which, by taking into account of the dilution by the electrolyte suspension, contributed an IS of 0.12 mM to the mixture in the column experiment (This might be still be an overestimation, as the charged AuNPs also contributed to the conductivity). The concentrations of NaCl in the electrolyte solution were 0, 0.1, 0.5, 1, 5, 10, 20 and 50 mM, respectively. These ionic strengths were added to that contributed by the AuNPs suspension (~0.12 mM as estimated) to yield to estimated ionic strengths of the mixture in Figure 3A. The pH of the mixture was measured to be 6.51 ± 0.11.

the limit of VSP,EDL−LSA,DA and the maximum in attachment efficiency would not be predicted, even theoretically, to occur; and (2) if a is large, entrapment of nanoparticles in the secondary minima of the energy curve becomes more likely, which will complicate the interpretation of the result of the deposition experiments.31,36 The AuNPs were used as synthesized without any further purification because attempts to remove the excess ion by either dialysis or centrifuge-redispersion would inevitably result in particle aggregation due to the simultaneous removal of the potential-determining ions. However, the ionic strength contributed by the excess ion is assessed to be insignificant after dilution. Using a ZetaSizer (Nano ZS, Malvern, Worcestershire, UK) the electrophoretic mobility (EPM) of the AuNPs was measured to be −2.95 ± 0.22 × 10−8 m2 V−1 s−1, which is equivalent to a ζ-potential of −56.6 ± 4.4 mV according to the Huckel equation.37 Also measured was the EPM of colloids dislodged from the silica collector surface by sonication. To obtain a colloidal suspension of the silica, 10 g of silica beads (Potters Industries Inc. Berwyn, PA) that were used in the column experiments were soaked in 20 mL of DI water. The mixture was sonicated in a bath sonicator for 20 min and the supernatant was collected. The average hydrodynamic radius of the silica colloidal particles was measured to be 187.08 ± 24.74 nm (PDI = 0.35 ± 0.07). The EPM of the demobilized silica colloid was measured to be −3.54 ± 0.15 × 10−8 m2 V−1 s−1, which is equivalent to a ζpotential of −45.3 ± 3.5 mV on the basis of the Smolokowski equation.37 Such a method was found to yield the same results as that given by streaming potential measurements conducted directly on the beads.38 The UV−vis spectrum of the AuNPs shows a maximum absorption at the wavelength λmax ≈ 511 nm, at which there existed a linear relationship between the AuNPs concentration and the absorbance. Nanoparticle Transport in Porous Media. A chromatography column of 1.6 cm diameter (C10/10, GE healthcare, Piscataway, NJ) was packed with the aforementioned silica beads of average diameter of about 360 μm. Thorough cleaning of the silica beads was conducted following a procedure reported previously in literature.39 The packing length was 6 cm and the porosity of the packed column was 0.38, as determined by the gravimetric method. The background electrolyte solution of NaCl concentration ranging from 0 to 5 × 10−2 M was delivered using a syringe pump (Harvard Apparatus, Holliston, MA) at a flow rate of 0.9 mL/min. The nanoparticle stock suspension was also delivered by a syringe pump at a flow rate of 0.08 mL/min. The AuNPs stock solution and the background electrolyte solution mixed in a Y-connector (BioChem Fludics, UK) of a volume of microliters before entering the column; thus, aggregatin of AuNPs was negligible.

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RESULT AND DISCUSSION Figure 3A shows the attachment efficiencies (α) at different ionic strengths as calculated from the measured Ceff/C0 using the T−E correlation. It also shows the predicted attachment efficiency calculated from interaction energy curves VSP,Total constructed using eq 6 for vdW interaction and either VSP,EDL−LSA (eq 5, dotted line) or VSP,EDL−LSA,DA (eq 4, dashed line) for EDL interaction, based on the Maxwell−Boltzmann model given by eq 7. It is not our purpose to provide a good fit to the data using the DLVO theory, as it has been wellunderstood that there are numerous sources of error and nonidealness that would affect the validity of the energy curve calculation. These sources of error include, but are not limited to, the usage of the ζ-potential as surrogate to surface potential, heterogeneous distribution of particle size, shape and surface 11036

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potentials, hydration force, etc.23 There are also sources of error in evaluating the α from the relative effluent concentration in the column experiments using the T−E correlation due to the polydispersity of the particle size and nonidealness of the porous media. However, taking these limitations into account, two pieces of useful information are still extractable from Figure 3A. First, the application of DA-based expression for EDL inteaction would usually yield an unrealistically low attachment efficiency (e.g., ∼5 × 10−10 at 10−3 M in this case), regardless of whether a Maxwell−Boltzmann model or an IFBL model is applied.23,31 The discrepancy may have been caused by secondary minimum deposition for larger colloidal particles. However, for small particles with which secondary minimum deposition is unlikely, the discrepancy is most likely due to the application of the DAbased expression for EDL interaction VSP,EDL−LSA,DA that is known to overestimate the EDL repulsion.26 Using a more accurate expression, VSP,EDL−LSA, seems to yield a much better estimation at least for the magnitude of the α. The second important observation, which is most pertinent to the current study, is the significant discrepancy between the measured α and the predicted α at very low ionic strength. On the basis of eq 5, with a low concentration (e.g., 0.1 mM) of added electrolyte, VSP,EDL−LSA as well as the energy barrier Vmax in VSP,Total should almost vanish (Figure 2D). As a consequence, the theoretically predicted α should be close to unity. Even by assuming surface potentials of −100 mV for the surfaces of both the silica and AuNPs in the calculation, the predicted α should still be around 0.4, which is over an order of magnitude higher than the experimentally determined value. In conclusion, destabilization of AuNP at very low ionic strength, as predicted by the DLVO theory with more accurate evaluation, was not observed in the current study. Similarly, previous studies have not shown destabilization of nanoparticles at very low ionic strength due to the depletion of indifferent ions (Figure 3B). For example, a recent study by Quevedo and Tufenkji revealed a monotonic decrease of attachment efficiency with decreasing ionic strength down to 0.1 mM for different particles of sizes ranging from 10 to 24 nm;40 results from studies on the carbon nanotubes (particles of a very large local curvature) in porous media also showed a similar monotonic dependence of α on ionic strength.41,42 Similar results but for deposition of larger particles are also available in the literature.43−46 It maybe argued that with large particles such as the ferryhydrite NP used in ref 42, the Derjaguin approximation, and correspondingly eq 4, can be applied, and thus, a monotonic increase of affinity with respect to the increase of ionic strength was reasonable. However, experiments with smaller nanoparticles such as CdTe-QD of a diameter of 10 nm, which corresponded to a κa = 0.16 with an IS = 0.1 mM, and CdSe-QD of a diameter of 14 nm, which corresponded to a κa = 0.23 with an IS = 0.1 mM, still did not reveal an affinity minimum predicted by eq 5. Neither was such an affinity minimum observed with the single-wall carbon nanotubes (SWCNT) that were of the largest local curvature (small a) of all nanoparticles, even though with the radius of a SWCNT cross section the calculated κa could be very small. Finally, although most aggregation experiments in the literature usually do not report data on ionic strength lower than 1 mM,47 similar conclusion can be drawn for particle aggregation aggregation was not observed using DLS in practice in which electrostatically stabilized nanoparticles were dispersed using DI water.

The reason for not observing the existence of a maximum stability at moderately low ionic strength is yet to be clarified. Although LSA is an approximation used in evaluating EDL interaction energy, more rigorous expressions or numerical results based on exact solution of linearized PB equation for energy of EDL interaction with constant surface potential also supported the existence of an ionic strength that results in a maximum stability25,26 (Although the cited papers only presented comparisons between VEDL at different ionic strengths or κa, maximum energy barrier can be observed by simply coupling VEDL and VvdW to obtain the total interaction energy VTotal). Therefore, such a prediction is by no means an artifact of an approximate expression, especially considering the fact that its physical interpretation is rather straightforward: the added electrolytes not only screen the EDL interaction but they also contribute to the osmotic pressure. In addition, such a theoretically predicted phenomenon has in fact been observed using surface force measurement,10 which confirms the validity of the theory. Although we observe a high stability of nanoparticles at low concentrations of added electrolyte with which the nanoparticles should be theoretically unstable, several reasons that are usually considered to lead to increased stability apparently fail to resolve the discrepancy. For example, the argument of existence of hydrodynamic interaction, i.e., the slow drainage of water between the approaching surfaces that would cause an impediment to the mutual contact between two surfaces,48,49 cannot serve as a satisfactory explanation. First of all, the effect of such a hydrodynamic interaction on particle deposition kinetics was found to be rather small.50 In addition, such an effect, if ever significant, would only reduce the attachment efficiency in a manner that is independent of ionic strength and thus would not invalidate the existence of a maximum stability. Another possibility that is usually employed in explaining increased stability is the presence of hydration force. But according to the results of Pashley,8 the hydration force only becomes significant at high ionic strength. Thus, for the range of ionic strength being considered, the hydration force is unlikely a key factor. In the following discussion, we attempt to explore several other possible explanations for the apparent paradox presented by theory and observation. Effect of Curvature. An important difference between a sphere−sphere or a sphere−plate system and plate−plate system is that, with a spherical surface, the electric field propagates along the radial direction and diverges, whereas with a plate−plate system the intersurface electric field is conserved in the gap between the parallel plates. This is also the case for the distribution of counterions: for parallel plates, the counterions are physically confined in the gap, whereas the counterions of a charged spherical surface can diffuse in a semiinfinite space the geometric boundary of which is defined by the other interacting particle or surface. However, were this argument valid, dependence of stability on κ should be reflected in the EDL interaction energy calculated via the more stringent force method (osmotic pressure plus Maxwell stress) based on the electric potential distribution obtained by using either LSA or by numerically solving the PB, which is considered as an benchmark approach independent of DA or any DA-like scaling method. But the results for such an evaluation based on either LSA or constant potential interaction still evidently indicated the dual role of added electrolyte on both composing and screening the electric double layer.24 To conclude, the dual-role feature of added electrolyte on EDL interaction should not 11037

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low (note that 0.01 C/m2 corresponds to about one charge per 60 nm2). For a rough comparison, the pressure between these charged surfaces in water with no added electrolyte is comparable to pressure between two such flat surfaces of very high surface potentials in median to high concentration of added electrolyte, as approximated using LSA method, and is by no means insignificant. It should be noted, however, that if the results from two successive work by Pashley are compared,8,52 the force curve measured in water with no added electrolyte has a lower energy barrier and decays more slowly52still consistent with the concept of dual-role nature of the indifferent electrolytes. It seems that a more elaborate theoretical consideration needs to be provided for better describing the EDL interaction in solution of low concentration of added electrolyte. The description of local charge density in deriving the PB equation may have to be modified by taking into account the counterion concentrations.53 However, for dilute suspension of nanoparticles, such a correction seems to be negligible due to the small volume fraction of the particles. Constant Potential vs Constant Charge, and Nonequilibrium Interaction. In addition to the curvature effect, another important difference between the experiments in force measurment and those in colloidal aggregation/deposition is that colloidal particles are always under Brownian motion. This is especially true for nanoparticles, as the diffusivity of the particle is inversely proportional to particle size according the Stokes−Einstein equation. The fast kinetic movement of nanoparticles may have strong impact on their colloidal interaction. For example, because the surface discharging process in a constant potential interaction takes finite time to complete, if two surfaces approach each other quickly, as happens in nanoparticle aggregation and deposition, the collision time may be much shorter than the time needed to complete the surface discharge, in which case constant charge interaction may be a more appropriate model to describe the interaction, even if the interaction is indeed a constant potential when equilibrium is allowed to be established54,55 (as more likely in force measurement experiments). Because the excess osmotic pressure P is roughly proportional to ρ∞ψm2(ρ∞) for a constant potential interaction (ρ∞ is the concentration of the added electrolyte and ψm is the midplane potential),17 ψm is limited by the upper bound of surface potential ψ0 due to surface discharging (Figure 5B); therefore, the ionic strength may have greater influence on the osmotic pressure via affecting ρ∞ than on affecting ψm in the range of low ionic concentration. With constant charge interaction, however, the counterions are conserved in the narrowing gap and the midplane potential can become significantly higher than ψ0 (Figure 5C), in which case the role of ionic strength on double layer screening may become more important than its role on constituting the double layers when the strength of double layer interaction is to be determined. In fact, on the basis of the expressions for EDL interacion between two flat plates under constant charge by Usui56 and for the exact scaling of such an expression to a sphere−plate system,25 the VPP,EDL−CC and VSP,EDL−CC both decrease monotonically with increasing κ for any given H. However, these expressions suffer from problem of using a linearized PB equation to describe the potential distribution, yet in a constant charge interaction the potential between the gap can easily become so high (Figure 5B) that the linearization of PB equation is simply unacceptable.20 Gregory suggested an approximate expression, based on what was called the

vanish due to the scaling from plate−plate interaction to sphere−sphere or sphere−plate interaction. Contribution of Counterions to the Double Layer. A more compelling explanation may be that the conventional theory developed for EDL interaction in electrolyte solution can no longer be applied to model the electrostatic interaction when the added electrolyte concentration is too low. This has actually been pointed out by Beresford-smith et al., who suggested that the contribution of the counterions resulting from the surface charges has to be taken into account,51 although the model developed in Beresford-smith et al.’s work focused primarily on the effect of volume fraction on colloidal interaction in a concentrated colloid suspension. An interesting extreme scenario was considered by Israelachvili, who evaluated the osmotic pressure between two charged surfaces in water with no added electrolyte.17 Such an osmotic pressure stems solely from the overlapping of EDL formed by the counterions dissociating from the interacting surfaces, but it can be surprisingly strong. By solving eqs 9 and 10, the excess osmotic pressure between two parallel plates of similar charges (σ) can

Figure 4. Excess osmotic pressure (in atm) between two parallel plates at different charge densities in water with no added electrolyte (curves with labels) and excess osmotic pressure between two plates of surface potentials of −150 mV at different ionic strengths based on LSA (eq 5).

be numerically evaluated (Figure 4) by the following expressions ⎛ k T ⎞2 P(H ) = 2ε⎜ B ⎟ K (H )2 ⎝ ze ⎠ −

⎛ K ( H )H ⎞ 2kBT σ = ⎟ tan⎜ ⎝ 2 ⎠ ε zeK (H )

(9)

(10)

where P(H) is the pressure between two charged plates with a separation distance H, σ is the charge density of the plates, and K(H) is equivalent to the Debye length, but it is dependent on both the charge density and separation distance. According to such a calculation, the osmotic pressure at a close separation can be well above the attractive pressure of vdW interaction as long as the surface charge density is not too 11038

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still cannot rectify the discrepancy, as the interaction energy given by such a charge-regulation model usually falls between that given by a constant potential interaction and that given by a constant charge interaction,27 both of which show a significantly reduced energy barrier with a very low ionic strength. Therefore, regardless of interaction modes, the prediction of a low colloidal stability in a solution depleted of ions seems to be theoretically robust in the mean-field theory framework as describe by the Poisson−Boltzmann equation. In addition to the possible incomplete surface charging, there may be another aspect of nonequilibrium interaction especially for very small particlesthe electrical double layer relaxation.57−59 It was found, in an experiment using atomic force microscopy (AFM), that there is an increase in repulsive double layer force with increasing approaching velocity (although the effect of hydrodynamic hindrance cannot be ruled out, especially for the flat AFM tip to which the colloidal probe attached); this double layer relaxation effect may contribute to the high stability in very low ionic strength, which cannot be fully accounted for by the equilibrium double layer interaction model employed in classic DLVO theory. Finally, it is worth mentioning that describing the nanoparticles interaction as a screened Coulomb interaction does predict a monotonic decrease of repulsive interaction energy with increasing ionic strength. For interaction between nanoparticles, such as in aggregation, the screen Coulomb interaction given by eq 11 is often employed to describe the electrostatic interaction between two particles when κa is small53,60

Figure 5. Electric potential distribution in a parallel plate system when (A) the plates are infinitely apart and no EDL overlapping occurs, (B) EDLs overlap between the plates at constant surface potential, and (C) EDLs overlap between the plates at constant charge.

compression approach, for EDL interaction between two plates of constant charge,20 which is of remarkable accuracy when compared to that evaluated on the basis of the numerical results of the full PB equation. A comparison between VPP,EDL−LSA and VPP,EDL−CC evaluted using the compression approach is given in Figure 6A. The total interaction energy curve for constant charge interaction is also given in Figure 6B. Unlike the results by Usui that showed a monotonic decrease of EDL interaction energy with respect to increasing ionic strength for any given separation, the more accurate evaluation for VPP,EDL−CC again reflects evidently the dual-role of added electrolyte, and the interaction energy curves (VPP,Total−CC) demonstrate increasing energy barriers with increasing ionic strength. Therefore, the observed experimental results cannot be reconciled with the theory simply by assuming a constant charge interaction. Similarly, a model of interaction with charge-regulating surface

VSS,SC(H ) =

Q 2e 2 1 e−κH 2 4πε (1 + κa) H + 2a

(11)

where Q is the total effective surface charge of the particles of a radius a. Using a similar approach, the energy of screened Coulomb interaction between a sphere of total charge QS and a plate of charge density σp can be expressed as VSP,SC − CC(H ) =

Q SσP εκ

e−κH

(12)

It seems, at least from the dependence of interaction on κ, that the screened Coulomb interaction is more appropriate in

Figure 6. (A) EDL interaction energy per area between two infinite flat plates evaluated using LSA (red curves) and the “compression approach” for constant charge interaction (blue curves). (B) Total interaction energy for constant charge interaction (Ah = 5 × 10−21 J). For all curves it is assumed that the unperturbed surface potentials are all −25.7 mV. 11039

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(3) Verwey, E. J. W.; Overbeek, J. T. G.; van Nes, K. Theory of the Stability of Lyophobic Colloids: The Interaction of Sol Particles Having an Electric Double Layer; Elsevier: New York, 1948. (4) Hunter, R. J. Foundations of Colloid Science; Oxford University Press: Oxford, 2001. (5) Napper, D. Steric stabilization. J. Colloid Interface Sci. 1977, 58, 390−407. (6) Zhulina, E. Theory of steric stabilization of colloid dispersions by grafted polymers. J. Colloid Interface Sci. 1990, 137, 495−511. (7) de Gennes, P. G. Polymers at an interface; a simplified view. Adv. Colloid Interface Sci. 1987, 27, 189−209. (8) Pashley, R. M. DLVO and hydration forces between mica surfaces in Li+, Na+, K+, and Cs+ electrolyte solutions: A correlation of double-layer and hydration forces with surface cation exchange properties. J. Colloid Interface Sci. 1981, 83, 531−546. (9) Israelachvili, J.; Wennerström, H. Role of hydration and water structure in biological and colloidal interactions. Nature 1996, 379, 219−225. (10) Horn, R.; Israelachvili, J. Direct measurement of forces due to solvent structure. Chem. Phys. Lett. 1980, 72, 192−194. (11) Israelachvili, J. N.; Pashley, R. M. Molecular layering of water at surfaces and origin of repulsive hydration forces. Nature 1983, 306, 249−250. (12) Israelachvili, J.; Pashley, R. The hydrophobic interaction is long range, decaying exponentially with distance. Nature 1982, 300, 341− 342. (13) Tsao, Y.; Evans, D.; Wennerstrom, H. Long-range attractive force between hydrophobic surfaces observed by atomic force microscopy. Science 1993, 262, 547−550. (14) Ninham, B. W.; Yaminsky, V. Ion binding and ion specificity: The Hofmeister effect and Onsager and Lifshitz theories. Langmuir 1997, 13, 2097−2108. (15) Guldbrand, L.; Jönsson, B.; Wennerström, H.; Linse, P. Electrical double layer forces. A Monte Carlo study. J. Chem. Phys. 1984, 80, 2221−2228. (16) Ninham, B. W. On progress in forces since the DLVO theory. Adv. Colloid Interface Sci. 1999, 83, 1−17. (17) Israelachvili, J. N. Intermolecular and Surface Forces; Academic Press: New York, 2010. (18) Ohshima, H. Biophysical Chemistry of Biointerfaces; Wiley: New York, 2010. (19) Lewis, G. N. J. Am. Chem. Soc. 1908, 30, 668−683. (20) Gregory, J. Interaction of unequal double layers at constant charge. J. Colloid Interface Sci. 1975, 51, 44−51. (21) Israelachvili, J. Measurement of forces between two mica surfaces in aqueous electrolyte solutions in the range 0−100 nm. J. Chem. Soc., Faraday Trans. 1 1978, 74, 975−1001. (22) Verwey, E. J. W.; Overbeek, J. T. G.; van Nes, K. Theory of the Stability of Lyophobic Colloids: The Interaction of Sol Particles Having an Electric Double Layer. Elsevier: New York, 1948; pp 162−163. (23) Elimelech, M.; O’Melia, C. R. Effect of particle size on collision efficiency in the deposition of Brownian particles with electrostatic energy barriers. Langmuir 1990, 6, 1153−1163. (24) Zypman, F. R. Exact expressions for colloidal plane−particle interaction forces and energies with applications to atomic force microscopy. J. Phys.: Condens. Matter 2006, 18, 2795−2803. (25) Lin, S.; Wiesner, M. R. Exact analytical expressions for the potential of electrical double layer interactions for a sphere−plate system. Langmuir 2010, 26, 16638−16641. (26) Bhattacharjee, S.; Elimelech, M. Surface element integration: A novel technique for evaluation of DLVO interaction between a particle and a flat plate. J. Colloid Interface Sci. 1997, 193, 273−285. (27) Pericet-Camara, R.; Papastavrou, G.; Behrens, S. H.; Borkovec, M. Interaction between charged surfaces on the Poisson−Boltzmann level: The constant regulation approximation. J. Phys. Chem. B 2004, 108, 19467−19475. (28) Gregory, J. Approximate expressions for retarded van der Waals interaction. J. Colloid Interface Sci. 1981, 83, 138−145.

describing the repulsive electrostatic interaction for describing the stability of nanoparticles (against aggregation and deposition), whereas equilibrium double layer interaction (either constant charge or constant potential) offers a more reasonable description for the interaction between two macroscopic bodies of slow approaching velocity. However, more detailed elucidation is yet to be provided in future research to fully resolve such a paradox. But it is now clear that there are fundamental and physical reasons rather than the inapplicability of Derjaguin approximation alone that render the classic DLVO theory inappropriate in the case of small κa.

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CONCLUSION This study shows that the reduced energy barrier between two charged surfaces at very low ionic strength, as well-explained by DLVO theory and supported by results from surface force measurements, is not observed as a reduced stability of nanoparticles against aggregation or deposition at very low ionic strengths. Possibilities that may lead to this apparent paradox were explored, including the effect of curvature, the contribution of counterions to the ionic strength, and the effect of interaction modes. It seems that, theoretically, the effect of curvature or the geometric scaling from plate−plate interaction to sphere−plate intraction should not invalidate the dual-role nature of added electrolyte that is evident in plate−plate interaction. The excellent stability of charged nanoparticles in water with no (or low concentration of) added electrolyte is widely observed, but the reason is not fully clear. It is postulated that such a phenomenon results from the fact that the equilibrium condition assumed in formulating the Poisson− Boltzmann equation in the DLVO theory may not be satisfied with fast moving nanoparticles. Although the conclusion of the current study is, paradoxically, rather inconclusive, the scenario considered in this study is of theoretical interest, as it stimulates the reexamination of the classic DLVO theory and its limitation when applied to very small particles in solution of low ionic strength (κa→0).

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Phone: 919-660-5292. Fax: 919660-5219. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This material is based upon work supported by the National Science Foundation (NSF) and the Environmental Protection Agency (EPA) under NSF Cooperative Agreement EF0830093, Center for the Environmental Implications of NanoTechnology (CEINT). Any opinions, findings, conclusions, or recommendations expressed in here are those of the authors and do not necessarily reflect the views of the NSF or the EPA. This work has not been subjected to EPA review and no official endorsement should be inferred.

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