Biomedical Engineering Reference
In-Depth Information
Figure 2.10
ComparisonbetweenNewtonian(NWT)andnon-Newtonian,shearthinningbehavior:
(a)shearstressversusshearrate(one-dimensionalcase),and(b)viscosityagainstshearrate.Viscosity
remainsconstantuntilthecriticalshearrate;abovethisshearrate,viscositydecreasesquickly.
where
η
is the dynamic viscosity of the solution (units Pa.s or kg/m/s) and
η
s
is that
of the solvent (carrier fluid). The specific viscosity has no unit; at a small concentra-
tion, the specific viscosity goes to zero. One also defines the intrinsic viscosity by
æ
ö
η η
-
1
[ ]
s
η
=
lim
(2.18)
ç
÷
c
®
0
η
c
è
ø
s
It is shown that the very general Martin relation [9] usually applies for solutions
like alginates solutions
[ ]
[ ]
k
¢
η
c
(
)
(2.19)
where
k
¢ is the Huggins coefficient. A simplified expression for (2.19) stems from a
Taylor expansion
η
=
c
η
e
sp
2
k
¢
2
3
[ ]
(
[ ]
)
(
[ ]
)
η
=
c
η
+
k c
¢
η
+
c
η
+
...
(2.20)
sp
2!
Truncature at the rank 2 yields the Huggins law
)
2
[ ]
(
[ ]
= +
(2.21)
Because the Taylor expansion has been limited to the second order, Huggins
law applies for very dilute solutions only. For semidilute solutions, more terms in
the expansion (2.20) should be kept. However, it has been shown that the specific
viscosity can generally be approached by the power law
η
c
η
k c
¢
η
sp
[ ]
)
n
(
η
=
a c
η
sp
(2.22)
where
a
is a coefficient. Taking alginate solutions as an example, it can be shown
that relation (2.22) fits well the experimental results with
a
= 0.1, [
η
] of the order
of 3-7 L/g and n of the order of 3-4 depending on the type of alginate [9, 10]. For
