90 FLUIDITY AND PLASTICITY

at a distance <pi/(<pz — <PI) to the left of the origin to which
Eq. (27) is referred.

3. We conclude therefore that the curve obtained by plotting
viscosities against volume concentrations is not normally linear
but a part of an equilateral hyperbola.

From Eq. (30)-we find the curvature for any mixture to be

, __ 2Q2 - <ll)2 [<Pl + fa —

" {[^i + (92 - *i)a]4 + fa
By differentiating this curvature in respect to the concentration
a and equating to zero, we find the concentration where the
curvature is a maximum to be

a = V?, -?!-»' (32) i

<P2 — <P1 I'

Substituting this value in Eq. (31), the amount of the curvature
where the curvature is a maximum, is found to be

* = v ™~ <*• (33)
4. The curvature of the viscosity-volume concentration curves
is greatest when the difference between the fluidities, i.e., <p2 — PI,
is large, and becomes .zero when ^2 — <PI = 0.
5. The curvature must continually decrease as the concentra- \
tion increases unless the square root of ^2 — (pi is greater than <pi, \
in which case the point of greatest curvature will be found at \
some positive concentration (cf. Eq. (32)). }
6. Mathematically considered, the curvature is dependent only '
upon the difference in the fluidities of the components, i.e.,
<pz — <pi and not upon (pi, but since we can only realize positive
values of a\, it follows that for a given value of 92 ~ <PI the
curvature at any concentration will be greatest when <p\ is very
small.
LIQUID MIXTUKES
The first conclusion is confirmed by the repeated observation
that viscosity-concentration curves of homogeneous mixtures are
normally sagged. The fourth conclusion offers an explanation
of the fact that they are sometimes very nearly linear. In
particular, this conclusion is confirmed by Thorpe and Rodger,
who, in commenting on the data of Linebarger, make the interest-