Environmental Fate and Transport - Nanotechnology and the Environment - page 134

with.time.are.based.on.the.kinetics.of.important.competing.reactions.that.occur.as.
the.system.moves.from.a.state.of.disequilibrium,.usually.at.the.point.of.introduction.
to.the.environment,.to.equilibrium..A.quantitative.approach.to.dynamic.prediction.
in.risk.assessment.is.discussed.in.the.next.section.

6.3.1 p rediCting t empOral r eaCtiOn r ates : C hain i nteraCtiOns
Chemical.reaction.kinetics.is.a.quantitative.generalization.between.the.rate.of.a.reac-
tion.going.iguratively.forward,.and.the.rate.of.the.reaction.going.backward..Take,.
for.example,.the.agglomeration.of.two.nanoparticles. X :

k

X + X   XX

xx

k

XX   → X + X

− xx

. . (6.9)

X + X ↔ XX

1
2

d X
d

[ ]

2

Rate = - = k [ X ] − − k [ XX ] xx xx

t

The.accumulation.rate.of.the.agglomerate. XX is.the.difference.between.the.rate.
of.agglomeration.( k xx [ X ] 2 ).and.the.stability.of.the.agglomerate.( k -xx [ XX ])..Many.of.the.
engineered.nanoparticles.currently.in.use,.particularly.the.carbonaceous.nanomateri-
als,.form.stable.aggregates.because.the.combined.electrostatic.repulsion.and.energy.
of.solvation.cannot.overcome.the.van.der.Waals.forces.under.typical.ambient.condi-
tions.(i.e.,. k xx .>>. k -xx )..This.allows.the.following.simpliication:.the.rate.at.which. X .
agglomerates.to. XX .is.merely.the.product.of.the.rate.of.interaction.between. X s.and.the.
probability.that.a.given.interaction.will.result.in.the.formation.of.the.product. XX .
The.rate.of.interaction.between. X s,.or.the.collision.kinetics,.is.governed.by.the.
particle. size. of. X . and. the. balance. between. the. system's. energy. (temperature). and.
resistance.to.movement.(viscosity)..With.an.estimate.of.the.rate.of.collision,.the.rate.
of.product.formation.can.be.quantiied.based.on.the.rate.of.reaction.per.collision.as.
follows:

2
3

kT 2

k = P r ( ) ⋅ ⋅ 2 r

XX X

η r

X

. . (6.10)

8
3

kT

= P r ( )

η

where.η.is.the.viscosity.of.the.solvent.and. P ( r ).is.the.probability.of.a.reaction.result-
ing.in.product.formation.on.a.per-collision.basis.
Because.each.productive.interaction.in.an.agglomeration.reaction.will.increase.
the.particle.size.by.the.sum.of.the.two.particles,.the.agglomeration.reaction.becomes.
asymmetric.very.quickly..It.must.be.described.as.an.interaction.between.unlike.par-
ticles. X .and. X ´,.where. X ´.is.the.product.of.a.deined.number.of.agglomeration.steps.
with.a.rate.constant.of.( k XX ´ ):