Environmental Engineering Reference
In-Depth Information
( )
2
3
kT
r
r
r
r
X
X
′
.
k
=
P r
⋅
2
+
+
.
(6.11)
XX
′
η
X
′
X
Because.of.the.relatively.large.size.of.nanoparticles.(compared.to.typical.mol-
ecules),. the. asymmetry. between. the. initial. particle. radius.
r
X
. and. the. radius. of. the.
agglomerated.particle.
r
X
´,
.grows.very.large.as.the.result.of.a.relatively.small.number.
of.agglomeration.reactions..Hence,.even.if.there.is.no.change.in.the.probability.of.
agglomeration.
P
(
r
),.the.reaction.rate.will.change.signiicantly.with.time.and.inde-
pendent.of.relative.concentrations..This.is.further.compounded.by.the.large.number.
of.coupled.agglomeration.reactions.involved.(
X
+.
X
,.
X
.+.
XX
,.
XX
.+.
XX
,.
X
.+.
XXX
,.
XX
.+.
XXX
,.…).in.the.evolution.of.suspended.nanoparticles.into.large.particles.that.
cannot.remain.in.suspension.
Fortunately,.the.need.to.estimate.overall.reaction.rates.with.time-variable.reac-
tion.constants.is.not.unique.to.nanomaterials..It.was.a.problem.irst.encountered.in.
nuclear.physics.in.solving.multi-stage.chain.reactions..Nuclear.physicists.overcame.
this. problem. using. multiple. stochastic. reaction. simulations. with. randomized.
iterations,. also. referred. to. as. Monte. Carlo. simulation.. Gillespie. [7]. proposed. one.
approach,.originally.developed.to.predict.water.droplet.aggregation.in.clouds,.that.
is.particularly.applicable.to.the.agglomeration.of.nanomaterials.in.suspension..It.is.
a.sequential.stochastic.simulation.that.predicts.the.concentration.of.various.deined.
products/.reactants.by.determining.the.probability.of.the.most.likely.reaction.(
P
(µ)).
to.occur.between.time.t.and.time.t+τ.based.on.the.competitive.values.for.the.respec-
tive.reaction.rates.(
k
´).speciic.for.time.
t
.(
P
(τ,µ)).
The.stochastic.probability.model.divides.the.reaction.probability.into.two.prob-
abilities:.(1).the.independent.probability.of.any.reaction.occurring.in.the.duration.of.
τ.(
P
1
(τ)),.and.(2).the.dependent.probability.of.a.speciic.reaction.(µ).occurring.given.
a.speciic.value.for.τ.(
P
2
(µ|τ)):
.
P
(τ,µ).=.
P
1
(τ).·.
P
2
(µ|τ).
(6.12)
The.ininitesimal.of.the.probability,.
P
(τ,µ).
d
τ,.represents.the.probability.at.time.
t
.
that.the.next.reaction.will.occur.in.the.differential.time.interval.of.
t
+τ.to.
t
+τ.
d
τ..For.
any.speciic.reaction,.µ,.the.probability.of.co-occurrence.within.
d
τ.if.the.product.of.
the.rate.of.diffusive.interaction.(
k
D
µ
).and.the.number.of.distinct.reactant.combina-
tions.found.present.at.time.
t
(
h
μ
).is.as.follows:
.
P
(µ)
d
τ.=.
h
µ
.·.
k
D
µ
.
d
τ.
(6.13)
The.value.of.
h
µ
.can.be.determined.by.the.nature.of.the.reaction.as.to.how.the.respec-
tive. reactant. concentrations. change. with. production. of. the. product. (
Y
). with. each.
reaction.event.μ.using.the.following.relations:
(
)
[
X
]
⋅
[
X
]
−
1
X
+
X
→
Y
h
=
m
2
.
.
(6.14)
X
+
X
′ →
Y
h
=
[
X
] [
⋅
X
′
]
m
