Environmental Engineering Reference
In-Depth Information
To determine the γ
sl
(
D
) function, we consider a compressible spherical particle, or a cube
with cube side taken as
D
, immersed in the corresponding bulk liquid. According to Laplace-
Young equation [66], we have
P
= 2
fA
/(3
V
) = 4
f
/
D
(2.15)
where
A
is the surface area of the particle,
P
is the difference in pressure inside and outside of
the particle and
f
is interface stress. Using the definition of compressibility κ = -Δ
V
/(
VP
), ε =
Δ
D
/
D
= Δ
A
/(2
A
) = Δ
V
/(3
V
) under small strain and
A
/
V
= 6/
D
where Δ denotes the difference,
ε
= -4κ
f
/(3
D
).
(2.16)
In terms of a scalar definition of
f
, there exists
[5,75-76]
ƒ
= ∂
G
/∂
A
= ∂(γ
sl
A
)/∂
A
= γ
sl
+
A
∂γ
sl
/∂
A
≈ γ
sl
+
A
Δγ
sl
/Δ
A
(2.17)
where
G
= γ
sl
A
states the total excess Gibbs surface free energy, or Δγ
sl
= (Δ
A
/
A
)(
f-
γ
sl
).
To find mathematical solutions of
f
and γ
sl
or γ
sl
(
D
), two boundary conditions of γ
sl
(
D
) are
needed. An understandable asymptotic limit is that as
D
→ ∞, γ
sl
(
D
) → γ
sl0
. As
D
→ ∞, let
Δγ
sl
= γ
sl
(
D
)-γ
sl0
.
(2.18)
Substituting Eq. (2.18) into Eq. (2.17) and taking in mind that
V
/
A
=
D
/6 and Δ
A
/
A
= 2ε =
-8κ
f
/(3
D
) in terms of Eq. (2.16), it reads [77],
γ
sl
(
D
)/γ
sl0
= [1-8κ
f
2
/(3γ
sl0
D
)]/[1-8κ
f
/(3
D
)].
(2.19)
Eq. (2.19) is consistent with general calculations of thermodynamics [15,78-79] and
quantum chemistry [80] for particles.
To determine
f
, we assume that when almost all atoms of a low-dimensional crystal
immersed in fluid is located on its surface with a diameter of
D
′
0
, the crystal is
indistinguishable from the surrounding fluid where the solid-liquid interface is at all diffuse.
Note that the crystal is now similar to a cluster produced by an energetic fluctuation of the
fluid. This assumption leads to a limit case: As
D
→
D
′
0
, γ
sl
(
D
) → 0 where
D
′
0
depends on
the existence of curvature [81]. Note that when a crystal has plane surface, such as a film, the
possible smallest value of
D
is 2
h
. For any crystals with curved surfaces, such as a particle or
a wire, the possible smallest
D
is 3
h
[77]. According to the above definition of
D
′
0
,
hA
/
V
= 1-
V
i
/
V
= 1-[(
D
′
0
-2
h
)/
D
′
0
]
3-
d
= 1 where
V
i
is the interior volume of the crystal. The solution of the
above equation is
D
′
0
= 2
h
. For the particle and the wire usually having a curved surface, the
value of
D
cannot be smaller than 3
h
. In this case, there is no exact solution of the equation.
As a first order approximation,
D
′
0
= 3
h
is taken. Note that this approximation does not lead
to big error. For instance, for a spherical particle,
hA
/
V
= 26/27 ≈ 1. In summary,
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