Effects of Brownian motion and structured water on aggregation of charged particles

Colloids and Surfaces A: Physicochemical and Engineering Aspects 177 (2001) 111 – 122 www.elsevier.nl/locate/colsurfa

Effects of Brownian motion and structured water on aggregation of charged particles Janet L. Baldwin a, Brian A. Dempsey b,* b

a Roger Williams Uni6ersity, USA Department of Ci6il and En6ironmental Engineering, The Pennsyl6ania State Uni6ersity, 212 Sackett Building, Uni6ersity Park, PA 16802, USA

Abstract Coagulation processes are integral to water treatment for removal of colloids and colloid-associated pollutants and therefore must be well understood. However, current research shows a disparity between experimental results and theoretical predictions. In some conditions, colloids that are predicted to be stable exhibit significantly less stability in experiments. To date, reasons offered for this disparity have not completely resolved the problem. In this paper, three issues that have previously been ignored are proposed as contributors to the disparity. The first is a model for estimating Brownian jump length which is anisotropic, with which it is shown that particles take longer to move away than once thought, and therefore, have a greater opportunity for attachment. The second issue develops a method for estimating the impact on the calculation of electrostatic repulsion due to variation in dielectric constant of water near surfaces. It is shown that including the low dielectric at the surface will greatly decrease the predicted electrostatic repulsion, and therefore, will decrease stability. The third issue concerns the impact of surface-induced water structure on the effective kinetic energy of particles near another surface. A model is developed for estimating the impact of kinetic energy on Brownian jump length and with the model it is shown that the impact can be great. Particles will have more energy to jump toward the interacting surface, than away, thus increasing the probability of aggregation. © 2001 Elsevier Science B.V. All rights reserved. Keywords: Brownian motion; Structured water; DLVO theory

1. Introduction Colloidal particles are ubiquitous in aqueous environments and can be associated with various types of contamination. Colloids are small particles (B 1 mm) with high specific surface area and can either be adsorbents for pollutants, or may be pollutants themselves. Common colloid associated * Corresponding author.

pollutants include trace metals, synthetic organic pollutants, pathogenic microorganisms, asbestos fibers, and humic precursors of trihalomethanes [1]. This has serious implications for both natural systems, where distribution and transport of contaminants are of concern, and for engineered systems such as water treatment, where removal is of primary concern. Removal of colloids in water treatment is accomplished through coagulant-induced aggregation and filtration, and therefore,

0927-7757/01/$ - see front matter © 2001 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 7 - 7 7 5 7 ( 0 0 ) 0 0 6 6 4 - 6

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aggregation processes are important to understand. Current theories of aggregation of charged particles do not compare well with results of aggregation experiments. The commonly used DLVO theory predicts a sharp drop-off of collision efficiency (a) with decreasing ionic strength, but experimental results shows a more gradual decline [2]. In addition, some experiments show a weak, or non-existent, dependence of aggregation on particle size e.g. [2,3], while theory predicts a strong dependence. Although many researchers have proposed reasons for this e.g. [2,4], none have completely explained the differences. This paper begins with a brief discussion of DLVO theory and a summary of research undertaken to modify the theory. This leads into a discussion and analysis of three molecular-level phenomena that can influence aggregation and disaggregation. The first phenomenon deals with anisotropic Brownian jump length close to the surfaces due to particle–particle interactions. The second and third phenomena are variation of dielectric constant close to the surface and anisotropic variations in Brownian impulses due to structured water.

2. DLVO theory and modifications In the 1940s, two groups of scientists concurrently and independently developed a theory of colloidal stability now known as the DLVO theory [5,6]. For particles of like charge, an electrostatic repulsion occurs when the diffuse layers

Fig. 1. DLVO energy vs. separation distance.

overlap while attraction is due to van der Waals energy. In the DLVO theory, it is assumed that these long-ranged forces (van der Waals and electrostatic) control the stability of colloids. However, the DLVO theory does not always provide accurate predictions of colloid stability, and this is discussed below. Fig. 1 shows a generic plot of DLVO interaction energy shown as a function surface separation distance. The primary energy well is due to strong attraction at short separation distances. For colloids with a surface charge, the electrostatic repulsion energy gives rise to an energy barrier at intermediate separations, and that energy barrier has been used to quantitatively predict colloid stability. When the electrolyte concentration is lower the energy barrier is higher, which may bring about colloid stability. The DLVO theory allows for the calculation of the critical coagulation concentration (CCC) which is the theoretical electrolyte concentration where fast aggregation should occur. Refinements to the DLVO theory include solvation forces e.g. [7], hydrodynamic retardation e.g. [8], heterogeneity of surface charge e.g. [9], steric interactions e.g. [10], and surface roughness e.g. [11,12]. At larger separation distances, a secondary minimum may exist and aggregation in the secondary energy minimum has also been described e.g. [13].

3. Conclusions of previous experimental and theoretical work Even when incorporating recent modifications, the DLVO theory does not predict experimental results well. Generally, experimental results show that suspensions have lower stability (W) than is predicted. Also, theory predicts a strong dependence of stability on particle size, while experiment shows a weak, or in some cases, lack of dependence. Researchers often invoke surface roughness and heterogeneity of surface charge to explain discrepancies between theory and results. However, research in which these characteristics were con-

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sidered did not completely resolve the problems, and these phenomena are not well understood e.g. [11]. Kihira et al. [9] modeled coagulation by including a factor for surface charge segregation. They were able to resolve differences between predicted coagulation rates and measured by invoking charge heterogeneity but the necessary heterogeneity was large and was a function of particle size. Variation of heterogeneity with particle size is questionable. Interfacial water has also been cited as a factor in colloid stability. For example, repulsive solvation forces operate at very short distances due to the structuring of liquid molecules when confined between two surfaces [7]. Israelachvili [20] experimentally measured repulsive solvation forces for hydrophilic surfaces, and attractive solvation forces for hydrophobic surfaces. There are both oscillatory and monotonic components to these forces. Compression of two surfaces forces the solvent molecules to leave quasi-discrete layers, which explains the oscillatory nature. In addition, the structuring of water brought about by surface–solvent interaction gives rise to a monotonic force, which decays exponentially with surface separation. For Hamaker constants greater than 10 − 21 J, the monotonic solvation force is significantly greater than the oscillatory force. Solvation forces have been measured experimentally, and an exponential relationship based on empirical constants and separation distance was determined. Repulsive hydration forces between mica and silica surfaces in water had a range 3 – 5 nm, which is about twice the amplitude of the oscillatory solvation force for water. This solvation force is repulsive for hydrophilic surfaces, and would, therefore, increase colloid stability. While other modifications have been suggested, none can completely resolve the discrepancy between theory and experiment. Modifications such as inclusion of hydrodynamic retardation and solvation forces decrease the collision efficiency, which is the opposite of the effect that is often observed. In addition, hydrodynamic analysis by Honig et al. [14] showed that the effect of hydrodynamic interaction on the slope of the stability curve was negligible. Inclusion of steric interactions has been useful for systems that contain

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polymers, but not for others. Adding the possibility of aggregation in the secondary minimum results in a decrease in stability, but the problem of particle size dependence has not been resolved. Electrostatic interaction has also been modified by including a Stern layer or with other changes, but these models have not succeeded in predicting the observed aggregation behavior for ‘stable’ particles. The influence of Brownian motion and water structure on aggregation has not been adequately addressed in previous research. A molecular-level understanding of the motion of particles might provide a better understanding of aggregation in systems that are incorrectly predicted to be stable.

4. Effect of Brownian jump length on aggregation Brownian motion dominates the mass transfer of particles that are much less than 1 mm in diameter. Elimelech et al. [15] compared collision rate constants as a function of particle diameter for three transport mechanisms, perikinetic (Brownian); orthokinetic; and differential settling. Brownian transport dominated for particle diameter less than approximately 500 nm. Primary particles in most coagulation processes are less than 10 nm [16,17]. It is typically assumed that a fraction of colliding particles (described by the Boltzmann distribution) have sufficient kinetic energy to overcome the electrostatic barrier. However, DLVO models describe Brownian motion through application of a macroscopic diffusion analogy that does not account for the extremely short mean path length of diffusion. It is argued in this paper that this fundamental concept is wrong when interfacial forces are important and especially incorrect when the interfacial forces are anisotropic. Brownian impulse is due to the impact of water molecules and the kinetic energy of a particle is determined by the kinetic energy of the colliding water molecules. The kinetic energy of each particle changes with each collision, and so changes with time. The motion of Brownian particles can be described by a momentum-balance equation, the solution of which shows that the time scale for

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Brownian jumps is very short, on the order of 10 − 9 s for a 50 nm particle [15]. The length of Brownian jumps is also very small (B10 − 10 m) and depends on particle size. Brownian jump length is typically computed using only inertial and viscous forces, but at close separations the particle interaction forces can also be significant. The combination of very short jump lengths and variable jump lengths can have a significant impact on the tendency of particles to aggregate and to disaggregate, and this is discussed below.

5. Dynamic description of Brownian motion Brownian motion can be modeled as a series of random walks, where each walk is independent of previous walks. The particle collides with fluid molecules and the collisions produce the initial velocity. Langevin-type equations have been used to describe the balance of Brownian impulse with particle inertia and the viscous drag forces, e.g. m

dx d2x +j = fB(t) 2 dt dt

(1)

where m is particle mass, x(t) the position of the center of the particle at time, t, j the Stokes friction coefficient (equal to 6pmr for single, hard spheres) and fB(t) is the fluctuating Brownian force. After the initial impulse, fB is 0 and the subsequent motion of the particle can be described by Newton’s second law of motion [15]. The velocity at any time during the walk is

 

u(t)=u0 exp −

t tB

(2)

where tB equals m/j and can be considered the typical timescale for a single random walk. The mean initial velocity (u0) can be estimated from kinetic theory: 1 1 m(u 20)= kT 2 2

(3)

The distance traveled in t and in tB are then described by Eqs. (4) and (5), respectively.



 n

x(t)=tBu0 1− exp − lB :

t tB

(4)

(mkT)1/2 j

(5)

This model applies to particles with no charge or at large separation distances so that interfacial interaction forces are negligible. If inter-particle forces become significant compared with Newtonian forces, then the jump length will be a function of the direction of the jump relative to the interface. To determine the jump length for interacting particles, the interaction forces were included in the motion equation as follows: m

dx d2x = − j +Fb elec + Fb VDW 2 dt dt

(6)

where Fb elec and Fb VDW are the force vectors representing the electrostatic repulsion and van der Waals attractive force vectors, respectively. Both interactive forces are functions of separation distance, but because the jump length is so small, the assumption was made that the inter-particle forces would remain constant for the jump. Solving this equation yields:



x(t)= tB u0 − tB

Fb T m



 n

1− exp −

t tB

+ ttB

Fb T m (7)

The Brownian jump length can be computed using Eq. (7) by solving for x when t= tB. lB : 0.63tBu0 + 0.37t 2B

Fb T m

(8)

where Fb T, is the total interaction force and tB. is the Brownian relaxation time. Expressions for electrostatic energy and van der Waals energy can be found in many texts e.g. [18–20]. Force is the first derivative of energy with respect to distance. The electrostatic energy equation used here is based on the Derjaguin approximation for equal size particles with equal potentials. The force expression was derived from the energy expression found in Israelachvili [20]: Fe(H)=

32nokTg 2pD exp(− kH) k

(9)

J.L. Baldwin, B.A. Dempsey / Colloids and Surfaces A: Physicochem. Eng. Aspects 177 (2001) 111–122

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Fig. 2. Brownian jump length vs. particle separation distance.

where no is the ion concentration in the bulk solution, k the Boltzmann constant, D the particle diameter, T the absolute temperature, k the reciprocal of the Debye length, and g is the surface charge constant, relating the surface potential, co, and charge per indifferent electrolyte ion, z, and is calculated as follows:

 

g= Tanh

zco 4kT

(10)

The Debye length 1/k is considered the characteristic length of the diffuse electric double layer, and is calculated as follows: 0.304 1

ooorkT = 1/2 = (I) k e(% ni z 2i )1/2

(11)

where oo and or, are the permittivities in vacuum and water respectively, e the elementary charge, and I is the ionic strength. The right hand side of the equation is the calculated value for 25°C in a monovalent electrolyte. van der Waals force was derived from the expanded Hamaker expression [18]: FVDW(H)= −





A D 2R D2 2D 2 + − 6 (R 2 −D 2)2 R 3 R 3 −D 2R (12)

where A is the Hamaker constant (3E-20 J was used for this study), R the distance between cen-

ters (D+ H), and D is the particle diameter. The van der Waals attractive energy can be retarded at all but very small separation distances because the finite time of propagation causes a reduced correlation between oscillations [15]. However, Gregory [21] has shown that the expanded and unretarded Hamaker expression accurately predicts van der Waals energy for small spheres (B 100 nm). The Brownian jump length was calculated with Eq. (8) for jumps normal to the interface (positive-x is separation and negative-x is approach) and using the average kinetic energy (1/2) kT to compute u0. Jump lengths as a function of separation distance are shown in Fig. 2 for two 100 nm diameter spherical particles with surface potential of − 50 mV in a 0.05 M solution. At the peak of the energy barrier, the separation jump was more than twice the jump length predicted with no interaction. The approach jump length was close to 0 at the same location. The jump length without interaction was approximately 0.1 nm for 10 nm particles, and, therefore, many jumps in the same direction would be required to traverse the energy barrier. The model predicted that particles would be detained even in very shallow secondary minima due to the application of realistically short jump lengths. Conversely, aggregation in the primary minimum was inhibited (compared with the mod-

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eled use of long diffusion times relative to a Brownian jump time) when the electrostatic repulsion exceeded approximately kT. A similar conclusion with regard to primary and secondary aggregation was reached by Hahn and O’Melia [4], using statistical representations of jump energy. Experimentally, Prieve et al. [22] measured apparent diffusion coefficients in a secondary minimum that was due to gravitational and electrostatic forces, and discovered values that were only 2–3% of diffusion coefficients in the bulk solution. However, it must be recalled that diffusion coefficients are statistically lumped parameters, reflecting translocation after a multitude of individual Brownian steps.

6. Effect of vicinal water structure on dielectric constant The dielectric constant of water changes with proximity to a surface, due to surface-induced structure. Debye – Hu¨ckel, DLVO, and van der Waals models all ignore the influence of water structure [23] although the effects of hydrodynamically interacting particles has been described [15] and the Stern modification of DLVO acknowledges the effect of hydrated ions adjacent to the particle surface. Decreasing dielectric constant of water close to the interface should produce higher predicted collision efficiencies, because c would decrease more rapidly with distance from the surfaces. Water molecules become structured at interfaces because they are polar and because nonbonding electrons can bond with the surface. Bulk water is organized into ‘icebergs’ with loose hexagonal structures of water molecules [24]. The hexagonal structure is apparent in ice, but when melted, the heat of melting is only 13% of the sublimation energy of the solid, suggesting that only about 13% of hydrogen bonds are ruptured upon melting [25]. The lowest energy configuration of bulk water is anisotropic. Water molecules in the vicinity of a surface have a more isotropic structure. This structured water is called vicinal, or interfacial water, and appears to exist near most (if not all) solid surfaces, regardless of the

specific chemical nature of the surface [23]. Vicinal water can not be directly observed, and so the exact nature of bound water is not known. Estimates of the extent of structuring are varied from 2 nm from the surface [26] up to 50 nm [23] with most researchers estimating 5 nm e.g. [27]. It is believed that the degree of structuring is highest at the surface and decreases exponentially with distance from the surface [23]. The structuring of the water molecules at the surface alters the physical properties of water, including density, dielectric constant, viscosity, and heat capacity. Because of the isotropic structure at the surface, the density is approximately 3% lower than in the bulk and the effects are apparent to about 3–5 nm, decreasing in an exponential-like manner [27]. Viscosity also follows an exponential function, with the maximum viscosity occurring at the surface and decreasing with increasing distance [28]. The heat capacity of vicinal water is 25% greater than the heat capacity of bulk water, and seems to be independent of the solid material [29]. Zettlemoyer and McCafferty [30] reported that the decreased dielectric constant close to the surface was due to structuring of the layers of solvation water, the structure does not allow the molecules to freely rotate to orient themselves with an applied field. The surface layer of water molecules can also be strongly sorbed to the surfaces of oxides, further limiting rotational freedom and decreasing the dielectric constant. The dielectric constant rises sharply in the second layer because the molecules are more mobile, and rises less sharply for outer layers. A model to describe the variation of dielectric constant was derived from published experimental data. Zettlemoyer and McCafferty [30] measured the dielectric constant of water vapor at 25°C on hydroxylated a-Fe2O3 and determined that the monolayer (at the surface) of water molecules has a very low dielectric constant, approximately equal to 2, while the dielectric constant of the second layer is approximately 40. Using the average distance between oxygen atoms in bulk water of 0.3 nm [25], and the dielectric constant of bulk water equal to 78, these data were fit with an exponential function yielding the following model:

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o(x)=78−76 exp(1 − 1.35x)

(13)

where x is the distance from surface in nm and o is the dielectric constant (dimensionless) as a function of distance. The computed variation of dielectric constant using Eq. (13) is plotted in Fig. 3 versus distance from the surface. The model predicts that the surface-induced structural effect extends approximately 3 nm from the surface. Use of an exponential function is substantiated by several researchers e.g. [29], and the predicted characteristic decay length of 0.74 nm is more conservative than the 1 nm decay length predicted by previous researchers [23]. A step function was used to simplify the computation of electric potential (see Fig. 3). Each step length was the thickness of a water layer (3 A, ). Two cases were examined, with and without a Stern layer. For the case with the first layer of water in the Stern layer, the surface potential for the diffuse layer was the potential after the Stern layer. The electrical potential field and charge density are co-dependent on the dielectric constant, and this was modeled using the Poisson equation. The effect of the variable dielectric was calculated by estimating the electrical potential field, and then by calculation of electrostatic energy. For a symmetrical electrolyte, the solution to the Poisson– Boltzmann equation for a planar interface is as follows Eq. (18):

c=

2kT 1+ g exp(− kx) ln ze 1− g exp(− kx)



where k=

2noz 2e 2 oookT



117

(14)

1/2

(15)

and g= tanh

zeco 4kT

(16)

The electrical potential was computed for three conditions (see Fig. 4). Initially, a constant dielectric constant was used (the bulk dielectric). Then the two cases with variable dielectric were calculated, one with o in the first layer of 6 and the ‘Stern’ case with o of 40 in the second layer. The potential drop for each step (water layer) was computed separately, starting with the surface. Both intuition and Fig. 4 indicate that the lower the dielectric constant for the surface water layer, the steeper the decrease in potential with distance from the surface. The calculated potential was used to predict electrostatic repulsion between two similarly charged spheres. The parameters k and g were optimized for each case in order to obtain the desired surface potential (−0.03 V for Fig. 5). In order to simplify these calculations, the dielectric was held constant over the critical separation distances, but with different values for each condi-

Fig. 3. Variation of dielectric constant with distance from surface. Solid line is the computed exponential function, dashed line is the computed step function.

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Fig. 4. Electrical potential vs. distance from surface. Potential computed using step function for dielectric constant for no Stern layer case (1), Stern-layer (2), and using constant dielectric constant equal to bulk value. Initial diffuse-layer potential is 0.03 V and ionic strength is 0.0 1 M.

Fig. 5. Total interaction energy computed using varying dielectric. The electrostatic energy was computed three ways, non-varying dielectric constant (1); varying dielectric with Stern layer (2); and varying dielectric with no Stern layer (3). Particle diameter of 50 nm, ionic strength of 0.01 M, surface potential = − 0.03 V.

tion. Electrostatic interactions were predicted using Eqs. (9) and (12). The results are shown in Fig. 5 and show that variation of dielectric constant has a large impact on the primary energy barrier. The effect is dependent on the system conditions, specifically ionic strength, surface potential, and particle diameter. For ionic strength

of 0.01 M, particle diameter of 50 nm, and surface potential of 0.03 V, the impact was large. The primary barrier was lowered from approximately 5 to 1.25 kT for the ‘Stern’ case, and there was no barrier for the ‘Gouy–Chapman’ case. These calculations demonstrate the importance of structured water on coagulation. A lower

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dielectric constant at the surface will lower the primary energy barrier, thus increasing collision efficiency for slow coagulation conditions. For the conditions shown in Fig. 5, the collision efficiencies were raised from approximately 0.1 when using the bulk dielectric to 1.0 when using either of the reduced dielectric models.

calculated using an exponential decay model with distance from the surface. The bond strength for vicinal water was estimated at 1E-19 J from the cited literature and an exponential decay constant was presumed to be 1 nm − 1, based on decays in the previously discussed physical properties of water at interfaces. Thus B.E.(H)= 1E− 19 exp(− H)

7. Effect of structured water on Brownian motion Structured interfacial water can also influence the Brownian motion of particles. Structured water molecules are less mobile, as indicated by the increased viscosity and the enthalpy of surface hydration. It is reasonable, therefore, that these molecules will be less capable of exerting a Brownian impulse, compared with bulk water. The result is a diffusion coefficient that is dependent on both position and jump direction (towards or away from the interface). This phenomenon could inhibit migration of particles away from the interface, resulting in higher apparent collision efficiencies than predicted by diffusion-based analogies for Brownian motion at interfaces. These effects are inherently different from hydrodynamic effects e.g. [15] which have been used to explain changes in approach vectors due to the extrapolation of bulk viscosity effects to the approaching surfaces. For an interfacial water molecule to produce a Brownian impulse, it must first break away from the surface. The bond strength of surface water molecules has been estimated to be between 3.2E20 and 1E-19 J based on several types of measurements. According to Horne [31] a water molecule at a surface gains 1E-19 J due to orientation. This value is close to the estimation of 7E-20 J which was derived from the strength of hydrogen bonds of 3.5E-20 J [25] assuming that molecules at the surface participate in two hydrogen bonds [32]. Measurements of heat of immersion also support these estimations, heat of immersion for iron oxide was approximately 350 erg cm − 2 [30], which is 3.2E-20 J per water molecule (assuming 0.3 nm diameter with hexagonal packing). The probability of a water molecule having kinetic energy greater than the bond energy was

119

(17)

The predicted bond energy decays to about 0 at 5 nm from the surface, which is consistent with previous research e.g. [27]. The probability of a water molecule having sufficient kinetic energy to break away from the structured water layer was then computed using the Boltzmann equation: P(KE\E)= e − (E/kT)

(18)

Inserting the bond energy expression Eq. (17) into Eq. (18) for E resulted in the plot shown in Fig. 6. The predicted impact of water structure was substantial. The 50% probability of a water molecule having kinetic energy great enough to break away occurred at just over 3.5 nm from the surface. Water molecules closer to the surface are less likely to break away and thus are less likely to provide a Brownian impulse, which results in a lower transfer of kinetic energy to a particle from the interfacial side compared with the bulk water side. The effect of this interfacial anisotropy in Brownian impulses is illustrated using the description of the Brownian jump model that was derived above. The mean initial velocity was estimated from kinetic theory (Eq. (5)). Since there is a lower effective kinetic energy for a jump in the direction away from the interacting particle, the average separation velocity is lower than the average approach velocity. Quantification of this effect was based on the probability of water molecules having sufficient kinetic energy to break away from the interfacial bonds. Using Eq. (18), the velocity for jump in the direction away from the interacting surface was estimated by u0(H)=

'



kT B.E. exp − m kT



(19)

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Fig. 6. Probability of kinetic energy of water molecules greater than bond energy.

where B.E. is the bond energy, as described in Eq. (17). Because the structured water exists on both approaching surfaces, H was taken to be half of the separation distance. Although the examination of the effect of the structured water on kinetic energy is conducted in 1-dimensional terms, the influence of the curved shape of a sphere must be accounted for. Water molecules separating two surfaces will have the most structure, and therefore the greatest decrease in effective kinetic energy, at the point of closest separation distance (H). When the surfaces are spheres, the greatest impact will be on the line of centers, and will decrease on either side because the separation distance increases. Therefore, an average separation distance, x, was estimated as the separation distance at the half-point between the center and the edge. x=

H+ (0.134 × Dia) 2

(20)

Results are shown in Fig. 7 for two particle sizes, 10 and 100 nm. The jump lengths were strongly anisotropic for 10 nm particles at separation distances less than 6 nm. Although there

was almost no effect of structured water on the attraction jumps, the separation jumps were 0 for distances less than 3 nm. The effects of structured water on jump lengths were not as great for 100 nm particles.

8. Summary Three molecular-level phenomena that affect aggregation of charged particles were evaluated in this paper. Consideration of these phenomena could help to resolve the disparity between experimental results and theoretical predictions. Brownian jump lengths were calculated, considering interfacial forces. Jump lengths were anisotropic. This phenomenon favored aggregation in a secondary minimum and inhibited coagulation in a primary minimum. Low dielectric at surfaces greatly decreased the predicted electrostatic repulsion, and would greatly increase aggregation in a primary minimum. Structured water at interfaces reduced the Brownian impulse for disaggregation, and the effect was particularly large for very small particles.

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Fig. 7. Effective kinetic energy effect on jump length profile. (a) Particle diameter =10 nm, (b) particle diamter = 100 nm. Jump length profile is shown as a function of jump direction, (1) jump towards interacting particle (negative x direction); (2) jump away from interacting particle; and (3) jump out of range of particle interactions. Surface potential = − 30 mV, I= 0.05 M, and A = 3E-20 J.

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