Biomedical Engineering Reference
In-Depth Information
mixture framework, variables that satisfy such continuity requirements are the solid
displacement
u
and the mechanoelectrochemical potentials
μ
α
of the solvent and
solutes. In general, neither fluid pressure
p
nor solute concentrations
c
α
are contin-
uous across boundaries (Lai et al.,
1991
). However, since
˜
μ
α
's are less practical to
use as nodal variables, it is possible to define related nodal variables that represent
effective measures of fluid pressure,
p
=
p
−
RθΦ
α
˜
c
α
,
(17.12)
=
s,w
and solute concentration,
c
α
κ
α
,α
=
s,w.
c
α
=
(17.13)
These equivalent nodal variables are obtained by making use of standard constitutive
relations from physical chemistry, for the chemical potential
μ
α
of the solvent and
solute in dilute solutions (Sun et al.,
1999
; Yao and Gu,
2007
; Ateshian et al.,
2011
).
In these expressions,
Φ
is the
osmotic coefficient
, a non-dimensional property that
describes the deviation of the osmotic pressure from the ideal behavior known as
van't Hoff's law (McNaught and Wilkinson,
1997
);
κ
α
is the
partition coefficient
of solute
α
relative to an ideal solution. This partition coefficient may be further
described by
γ
α
exp
,
κ
α
z
α
F
c
ψ
Rθ
κ
α
˜
=
−
(17.14)
where the non-dimensional property
κ
α
is the
solubility
of solute
α
in the mixture,
representing the fraction of the interstitial pore volume which is accessible to the
solute (Mauck et al.,
2003
); and
γ
α
is the
activity coefficient
of solute
α
, a non-
dimensional property that describes the deviation of the solute chemical potential
from the ideal behavior of a very dilute solution (Tinoco et al.,
1995
). The ratio
κ
α
≡
κ
α
/γ
α
may be interpreted as the
effective solubility
of solute
α
(Ateshian
et al.,
2011
). Constitutive relations must be provided for
Φ
and
κ
α
. For a neutral
ˆ
solute (
z
α
0), the partition coefficient reduces to the effective solubility. For
ideal
mixtures
in the context of physical chemistry,
Φ
=
κ
α
1.
Physically, since
RθΦ
α
=
s,w
c
α
is the osmotic (chemical) contribution to the
fluid pressure,
=
1 and
ˆ
=
p
may be interpreted as that part of the total (mechanochemical)
fluid pressure which does not result from osmotic effects; thus, it is the mechanical
contribution to
p
. Similarly, the effective solute concentration
c
α
represents the true
contribution of the molar solute content to its electrochemical potential. When using
˜
˜
c
α
in lieu of mechanoelectrochemical potentials, the mass fluxes given in
Eqs. (
17.9
), (
17.10
) may be represented by the equivalent fluid volume flux,
p
and
˜
grad
c
β
,
Rθ
β
κ
β
d
0
=−
k
d
β
w
·
p
˜
+
·
grad
˜
(17.15)
=
s,w
