Again, P2(pix) is not an absolute, but rather a probability distribution function . In this case, the probability of a reaction occurring is based on the relative reaction rate . The solution for the distribution therefore can be simulated using a second uniform random variable (r2) and solving for p in the relation whereby:

X^*» ) < r^X^D') ) r2 e Uniform {0 < r2 < 1} (6 . 21)

The order of the summation is irrelevant. Therefore, this relation can be solved mathematically by successive summations until the following condition is met:

M n r^X^o, )"X(h'kD' )< 0 n e {1,2,3,..., M} (6.22)

When this is satisfied, the value of p is therefore that corresponding to the prior value of i (pi-1) .

Although theoretically complex, this approach allows for the prediction of the rate of aggregate formation regardless of the number of separate types of reactions or the number of intermediates involved . It also foregoes the need to solve a generalized master equation by considering all potential interactions simultaneously It is very powerful; however, it is also very computationally intensive .

An example for the application of Gillespie's model to predict the collision kinetics for an agglomeration reaction is illustrated in Figure 6 .5. As expected, the lower the probability of product formation, the longer the process of chain reaction agglomeration It is interesting that the uncertainty also increases This uncertainty is not the result of prediction (experimental) error, but rather represents differential reaction pathways and is a true measure of the variance expected if such a reaction were repeated an infinite number of times This again is the result of the large number of potential intermediates possible in the aggregation between the slowest linear aggregation pathway (X + X, XX + X, XXX + X, ...) and the fastest geometric

FIGURE 6.5 Examples of projected reaction probabilities based on stochastic kinetics: (a) representation of variability in product formation for the agglomeration of a 10-nm particle with P(r) = 0 . 1; (b) example of projected probability of agglomeration at differing particle size at an assumed P(r) of 0 . 5

FIGURE 6.5 Examples of projected reaction probabilities based on stochastic kinetics: (a) representation of variability in product formation for the agglomeration of a 10-nm particle with P(r) = 0 . 1; (b) example of projected probability of agglomeration at differing particle size at an assumed P(r) of 0 . 5

aggregation pathway (X + X, XX + XX, XXX + XXX, ...) . Using this approach, not only is the range recognized, but also the relative probability, which is a function of the relative collision kinetics of the intermediates, is retained

6.3.2 Predicting Temporal Reaction Rates: Estimating Particle Affinities

In addition to a method to determine the time course of the collision/diffusion kinetics, prediction of the fate of nanomaterials requires derivation of the probability of a reaction resulting in the formation of a product per collision event, P(r) . Experimentally, this is reasonably easy to determine within the confidence of the collision kinetics as the ratio of observed product formation given the determined rate of collision:

kDx where k'Dx is the rate of collision based on diffusion and k'x- is the rate of product formation . Deriving P(r) from thermodynamic principles is difficult because of the number of competing forces and from the limited knowledge regarding near-body interactions in solution Hence, the methods described below should only be considered a means of estimation

It is generally true that the more thermodynamically advantageous a reaction, the more likely it is to occur, and therefore the faster the rate of product formation With respect to the agglomeration of nanoparticles, product formation occurs when the forces of attraction outweigh the forces of repulsion . This summation, however, is not straightforward because the molecular force fields around each nanoparticle vary with distance from the particle The energy required to overcome these force fields depends on the kinetic energy of the particles, which is neither constant nor uniform Derivation of predictive values for the free energy of solvation — and its inverse, the free energy of precipitation — takes into account the affinity of the solvent (in this case, air or water) for the solute relative to the affinity of the solute particle for other solute particles These affinities are chemical specific However, it is possible to generalize the interactions of a nanoparticle with its solvent medium

Consider an example of a nanoparticle introduced to an aqueous medium:

• If the nanoparticle's surface affinity for like nanoparticles is low relative to the affinity for the water molecules, then the material will disperse

• If the nanoparticle has a low affinity for like nanoparticles but its affinity for polar water molecules is insufficient to overcome the water-water affinity, then the material will be hydrophobic and will not disperse in water but will disperse in nonpolar environments at the solvent interface

• If the nanomaterial has a high affinity for like nanoparticles, the material will not disperse in either aqueous or nonaqueous environments

These situations are never absolute In general, the stronger the affinity of the nanoparticle for water, the higher the equilibrium concentration — and vice versa (Recall that if the free energy of solvation is less than zero, then a material will disperse spontaneously in water )

Dispersion in air (aerosol) differs from hydrosol formation principally because (1) the fluid medium has a lower density and higher particle velocities; (2) the medium has a low dipole moment; and (3) the medium has a low dielectric constant . Therefore, the primary factors in air dispersion are particle size and inter-particle affinities that are related to inducible net zeta potential in air

In both cases — dispersion in water and dispersion in air — the fate of the nanoparticle results from the interplay of competing interactions at the nanoparticle interface To predict the probability of agglomeration and thereby the stability of the nanomaterial, the force fields at this interface must be described in thermodynamic terms that then can be converted to a probability density function

6.3.3 Nanoparticle Affinity and Inter-Particle Force Fields

Interactions between nanoparticles and environmental constituents such as fluid media are expected to result predominantly from Coulomb (electrostatic) forces and van der Waals interactions That is not to say that nanomaterials will not undergo covalent reactions within the environment. An example of such a reaction is the application of zero-valent iron in groundwater remediation where the iron nanoparticles undergo direct redox reactions with groundwater contaminants [8] However, this is the exception and specific to the type of nanoparticles involved Coulomb forces will occur in any situation where the particle/medium system permits the formation of a charge imbalance van der Waals interactions are universal to nanoparticles and will differ among type only with regard to their magnitude

In agglomeration reactions, the Coulomb force is almost always repulsive This occurs because it is most common that like particles in the same medium will acquire the same type of charge, although the charge density may vary with the particle size Charges can arise as the result of charge separation producing a dipole situation, but unlike molecular dipoles, this is usually aligned between the outside surface of the particle and its interior As such, steric hindrance inhibits differential charge interactions . The potential energy (E(C)xx-) arising from the Coulomb forces between the two particles, X and X', can be defined as follows:

4 n • z -eo • es where q is the net particle charge on X or X', e0 is the electric constant (8 . 85 x 10-12 C2 N-1m-2), es is the dielectric constant of the medium, and z is the particle separation [9] This can be optimized for the interaction between two spheres as follows [10]:

where k is the inverse Debye screening length (» 1 .43 nm) .

A positive energy is repulsive; a negative energy is attractive . 6.3.3.2 van der Waals Energy

The net van der Waals force is a balance of weak attractive and repulsive interactions between either nanoparticle surfaces or the nanoparticle surface and other medium constituents . Over molecular distances the net force is always attractive and exothermic, with the change in free energy being the result of the enthalpy of adsorption CH) . (Over atomic distances, the force is always repulsive.) This balance can be approximated by the Lennard-Jones (12-6) relation [11], where intermolecular potential energy (E(w)) is given by:

where z is the distance between two particles, and z0 is the most thermodynamically favorable distance at which E(w) is equal to ~Ha. The derivation of the Lennard-Jones relation comes from the differences between the attractive forces that vary with the 6th power of the inverse distance, and the repulsive force that varies with the 12th power. Note that the parameters represent the summation of paired potentials across the interacting surface . Therefore, the values for -Ha and z0 will not be the same in an agglomerate such as a nanoparticle, as they would for the individual molecular or atomic constituents

The relationship changes when dealing with a molecular/nanoparticle interaction This is because the potential is based on the summation of paired interactions of one body acting on multiple single points . As a result, the relation changes from a (12-6) to a (9-3) [12] as follows:

f \9 |
1 |
/ | ||

Zo | ||||

v 2 J |
6 |
V 2 |
where n is the number of binding sites upon the nanoparticle . Examples of the differential relations are provided in Figure 6 . 6 for C60 fullerene-fullerene [A] and C60 fullerene and water [B] . Determinations of the van der Waals energy are difficult, particularly for opaque materials . However, the energy can be predicted for a binary system of two like particles (x) in a solvent (s) based on the Hamaker constant (A) . The Hamaker constant can be estimated within a given system based on the reference dielectric constant in a vacuum (e0,n) using the Tabor Winterton approximation [13] as follows: where: 12nz" |

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