Chemistry Reference
In-Depth Information
t
)
(16)
ε
⊥
= ε
0
⋅
sin(2
π ⋅
f
⋅
where
0
is a deformation amplitude (relative to the width of the gel), which will be
not more than 0.05,
f
is the loading frequency. The argument could be made against
this choice of applied strain expression, that the real loading is often only compression
with strain changing from zero to maximum negative value and back. But after the
relaxation time this load can be described as a superposition of a prestressed state (in
which strain equals half maximum value) and compression-tension sinusoidal load.
By defining applied deformation by Equation (16) we make the following simplifica-
tions: skip the transition state during relaxation time and do not account for a prestress
value.
As the solute concentration changes only in
x
-direction, and the problem con-
sidered is one-dimensional, the boundary conditions for the displacement
u
(
x
,
t
)
are:
ε
⎧
⎨
⎩
u
x
(0,
t
)
=
0
u
x
(
h
,
t
)
=
0
and yield the boundary conditions for
u
(
x
,
t
)
=
u
x
(
x
,
t
)
+ ε
⊥
(
t
)
⋅
x
:
u
(0,
t
)
⎧
⎨
⎩
=
0
u
(
h
,
t
)
(17)
=
h
⋅ ε
0
⋅
sin(2
π
ft
)
The boundary conditions for the solute concentration are derived from the assumption
that the bath solution is well mixed and therefore the concentration of free solute on
the surface of the gel is constant. Impermeable wall at
x
0
yields zero flux condition:
=
⎧
⎨
⎪
⎩
⎪
c
F
(
h
,
t
)
=
c
0
∂
c
F
∂
(18)
x
x
= 0
=
0
7.3.5 Initial Conditions
The process of solute accumulation will be analyzed by the present model that is why
the initial conditions reflect zero concentration of the solute in the gel and no displace-
ment:
⎧
⎨
⎩
c
F
(
x
,0)
=
0
(19)
c
B
(
x
,0)
=
0
u
(
x
,0)
=
0
(20)
7.3.6 Nondimensionalization of Governing Equations
In order to find general solution and to determine governing parameters of the model
system, all the variables for the set of equations (Equation (8), (12), (15) with conditions
Search WWH ::
Custom Search