Environmental Engineering Reference
In-Depth Information
slow.settling.kinetics..As.a.result,.nanoparticles.can.be.said.to.possess.an.apparent.
solubility. (
k
as
). that. can. be. described. in. a. manner. similar. to. that. for. a. solution. as.
follows:
.
[
X
X
]
f
.
(6.3)
k
=
as
[
]
s
where.[
X
]
f
.represents.the.concentration.of.nanoparticle.
X
.in.sol.and.[
X
]
s
.represents.
the.concentration.in.the.solid,.non-sol.form..If.it.is.assumed.that.the.material.is.ini-
tially.introduced.into.the.luid.medium.in.the.nanoparticulate.form,.the.settling.rates.
are.within.a.range.of.thermal.kinetics,.and.hence.absolute.temperature.(
T
).becomes.
a.factor.in.determining.the.equilibrium.concentration.of.the.particles.in.the.sol..An.
expression.for.
k
as
.can.be.derived.using.the.Boltzmann.equation.as.follows:
ln
[
X
X
]
∫
V g
(
ρ
−
ρ
)
.
f
x
x
f
.
(6.4)
ln
k
=
= −
⋅
hdh
as
[
]
kT
S
where.
k
. is. the. Boltzmann. constant,.
T
. is. absolute. temperature,. and.
h
. is. the. linear.
measure. of. particle. separation.. At. saturation,. the. amount. in. non-suspension. (i.e.,.
[
X
]
s
).will.have.no.real.effect.on.the.amount.in.suspension,.Hence. the.equilibrium.
equation.can.be.expressed.solely.based.on.the.aqueous.concentration.of.the.nanopar-
ticle.as.follows:
∫
V g
(
ρ
−
ρ
)
.
.
(6.5)
x
X
f
ln
k
= −
⋅
hdh
as
kT
The.integration.of.the.Boltzmann.equation.allows.a.irst.approximation.of.the.total.
suspended.nanoparticulate.concentration.at.equilibrium.as.follows:
x
=
0 01
.
m
∫
V g
(
ρ
−
ρ
)
(
)
=
x
x
aq
ln
k
−
⋅
hdh
as
kT
x
=
0
m
V g
(
ρ
−
ρ
)
x
X
aq
0 01
2
= −
⋅
.
.
.
(6.6)
2
kT
Therefore:
V g
(
ρ
−
ρ
aq
)
x
x
0 01
2
−
⋅
.
[
X
]
=
e
2
kT
aq
This. derivation. shows. that. the. particulate. concentration. and. temporal. stability. of.
heterogeneous.sols.depend.on.the.size.of.the.particles..If.the.nanoparticles'.size.is.
stable,. then. the. suspension. will. be. stable. (excluding. disruption. by. outside. forces)..
Thus,. nanoparticles. can. form. metastable. suspensions.. However,. if. the. particles.
agglomerate.with.like.particles.or.other.constituents.in.air.or.water,.then.the.suspen-
sion.will.not.be.stable..This.phenomenon.is.discussed.further.in.Section.6.2.2.
This. method. provides. a. means. to. predict. the. concentration. of. nanomaterials.
in. a. hydrosol. or. aerosol. based. on. the. physical. properties. of. the. materials. and. the.
interplay.of.particle.size.and.density.(Figure.6.2)..For.materials.with.a.density.less.
