Environmental Engineering Reference
In-Depth Information
table .
empirical Coeficients for M in the
tabor winterton approximation
geometry
M
Molecular.point-to-point
6
Two-plane.parallel.bodies
2
Two.spherical.particles
1
Note:
.Table.data.taken.from.French.[1].
6.3.4 p
rediCtiOn
OF
p
rObability
OF
p
rOduCt
F
OrmatiOn
Predicting. the. kinetics. of. nanoparticles'. mobility. in. the. environment. requires. the.
quantiication.of.the.probability.of.product.formation.relative.to.the.collision.kinet-
ics..The.enthalpy.of.adsorption.is.the.critical.factor.in.predicting.the.probability.of.
product.formation..For.simplicity,.the.derivation.begins.with.an.assumption.of.uni-
form.dispersion.within.an.aqueous.medium..While.this.assumption.is.not.necessary.
to.validate.the.solution,.it.removes.considerations.of.steric.hindrances.while.present-
ing.a.more.intuitive.model.
From.this.initial.dispersed.situation,.the.free.energy.of.a.single.agglomeration.
reaction.has.three.components,.which.sum.as.follows:
( )
1 2
X H O
·
↔
2
X
+
2
H O
2
2
( )
2
H O
+
H O
↔
H O H O
·
2
2
2
2
.
.
(6.29)
( )
3
X
+
X
↔
X
·
X
( )
4 2
X H O
·
↔
X X
·
+
H O H O
·
2
2
2
For.simplicity,.it.can.be.assumed.that.the.free.energy.of.
X
·
H
2
O
.is.independent.of.the.
number.of.water.molecules.present,.and.the.free.energy.of.
X
·
X
.is.independent.of.the.
number.of.nanoparticles.previously.combined.
Every. interaction. will. result. in. either. the. formation. of. a. product. (
X
·
X
). or. the.
elastic.rebound.of.the.reactants.(
X
·
H
2
O
)..Hence,.the.expression.for.the.probability.of.
outcome.per.collision.can.be.described.as.follows:
.
P
(
X
·
X
).+.
P
(2
X
·
H
2
O
).=.1.
(6.30)
Because.van.der.Waals.energy.is.always.negative,.the.probability.of.(
X
·
X
).will.be.
0.5,.provided.that.no.steric.hindrance.or.electrostatic.potential.inhibits.the.agglom-
eration..Considering.this.is.a.simple.particle.agglomeration,.it.can.be.assumed.that.
steric.hindrance.is.not.an.issue..Therefore,.as.the.electrostatic.repulsion.increases,.
the. probability. favors. the.
P
(
X
·
H
2
O
). over. the.
P
(
X
·
X
),. and. vice. versa.. Using. DLVO.
kinetics,.the.aggregation.eficiency.(a),.sometimes.referred.to.as.the.inverse.stability.
(1/W),.can.be.expressed.as.follows:
