Chemistry Reference
In-Depth Information
A few remarks should be made. First, the constant value of deformation amplitude
ε
0
corresponds to the same strain amplitude for any frequency, because we consider
only deforming layer of the matrix. Strain amplitude depends on depth and it is higher
at the surface (graph not shown), so that the maximum strain equals to
ε
0
. Also note
h
depends on frequency, thus the characteristic time of reaching
x
2
that
h
by the
=
diffusing solute front will also depend on frequency.
Now, as we have determined the length of the gel to solve for, it is possible
to determine the working range of other parameters. The value of the dissocia-
tion rate
k
r
can be of the order of magnitude about
10
−2
10
−5
1
s
. The simpli¿ cation
÷
h
2
D
4
R
g
f
k
r
⋅
H
A
k
⋅
4
10
4
. The value of the association rate
k
f
can be of the order of magnitude about
10
−2
yields
k
2
=
10
−5
÷
k
2
=
=
k
r
=
k
r
D
⋅
f
1
s
. To evaluate
k
1
further as-
÷
1
μ
M
⋅
sumption is needed, that will limit the value of
c
0
. If
N
′
=
1
is assumed, then, using
0.01
1
the data from Table 1,
k
f
⋅
c
0
=
10
÷
s
. Hence,
4
R
g
f
10
−2
10
7
k
1
=
k
f
⋅
c
0
=
÷
(27)
N
,
′
Or, for any
10
−2
′
10
7
′
(28)
k
1
=
÷
N
N
Note that the ratio
k
k
1
< 1
, and it depends on
c
0
, but not on
f
and
R
g
.
Similarly the dimensionless time can be represented as:
f
4
R
g
t
′
=
t
(29)
In this non-dimensional consideration the dimensionless frequency will be set to
4
(see Equation (25)). Hence, the time dependence of gain will be evaluated with respect
to the following parameters:
R
g
,
k
1
,
k
2
k
1
′
f
=
,
N
,
ε
0
. Note, that the frequency dependence
′
is included in
k
1
, not in
f
. In other words, for any frequency we consider respective
thickness of the gel, in which fluid flow is significant. Such consideration helps to
keep the same precision of calculations for different values of parameters and to save
computing time.
First let us set
′
1
,
k
2
100
,
0.05
,
R
g
=
ε
0
=
k
1
=
10
−3
′
N
=
Figure 3 shows the gain of free, bound, and total solute mass uptake by the gel (see
Equation (26)) as a function of time for
k
1
=
10
3
. The oscillations of the gain function
repeat the same behavior of the solute concentration at loading condition. Hence, its
frequency equals to
R
g
⋅
f
. On Figure 4 the average of gain function for free solute
′
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