Chemistry Reference
In-Depth Information
The second important aspect of the model [31] in reference to nano-
filler particles aggregation simulation is a finite nonzero initial particles
concentration
c
or ϕ
n
effect, which takes place in any real systems. This
effect is realized at the condition x
≈
R
ag
, that occurs at the critical value
R
ag
(
R
c
), determined according to the relationship [31]:
cR
−
.
(19)
ag
f
dd
c
~
The Eq. (19) right side represents cluster (particles aggregate) mean
density. This equation establishes that fractal growth continues only, un-
til cluster density reduces up to medium density, in which it grows. The
calculated according to the Eq. (19) values
R
c
for the considered nanopar-
ticles are adduced in Table 6.1, from which follows, that they give rea-
sonable correspondence with this parameter experimental values
R
ag
(the
average discrepancy of
R
c
and
R
ag
makes up 24%).
Since the treatment [31] was obtained within the frameworks of a more
general model of diffusion-limited aggregation, then its correspondence to
the experimental data indicated unequivocally, that aggregation processes
in these systems were controlled by diffusion. Therefore, let us consider
briefly nanofiller particles diffusion. Statistical walkers diffusion constant
z can be determined with the aid of the relationship [31]:
(
)
1/ 2
,
(20)
xz
≈
t
where
t
is walk duration.
The Eq. (20) supposes (at
t
=const) z increase in a number technical
carbon-nanoshungite-microshungite as 196-1069-3434 relative units,
that is, diffusion intensification at diffusible particles size growth. At the
same time diffusivity
D
for these particles can be described by the well-
known Einstein's relationship [33]:
kT
,
(21)
D
=
6
πh α
r
n
where
k
is Boltzmann constant,
T
is temperature, h is medium viscosity, α
is numerical coefficient, which further is accepted equal to 1.
In its turn, the value h can be estimated according to the equation [34]:
2.5
φ
h
h
n
n
,
(22)
=+
−
1
1
φ
0
