0Introduction
III Theoretical Physics of Soft Condensed Matter
0.1 The physics
Unsurprisingly, in this course, we are going to study models of soft-condensed
matter. Soft condensed matter of various types are ubiquitous in daily life:
Type Examples
emulsions mayonnaise pharmaceuticals
suspensions toothpaste paints and ceramics
liquid crystals wet soap displays
polymers gum plastics
The key property that makes them “soft” is that they are easy to change in
shape but not volume (except foams). To be precise,
–
They have a shear modulus
G
of
∼
10
2
–10
7
Pascals (compare with steel,
which has a shear modulus of 10
10
Pascals).
–
The bulk modulus
K
remains large, with order of magnitude
K ∼
10
10
Pascal. As K/G ∼ ∞, this is the same as the object is incompressible.
Soft condensed matter exhibit viscoelasticity, i.e. they have slow response to
a changing condition. Suppose we suddenly apply a force on the material. We
can graph the force and the response in a single graph:
t
σ
0
σ
0
/G
τ
η
−1
here the blue, solid line is the force applied and the red, dashed line is the
response. The slope displayed is
η
−1
, and
η ≈ G
0
τ
is the viscosity. Note that
the time scale for the change is of the order of a few seconds! The reason for
this is large internal length scales.
Thing Length scale
Polymer 100 nm
Colloids ∼ 1 µm
Liquid crystal domains ∼ 1 µm
These are all much much larger than the length scale of atoms.
Being mathematicians, we want to be able to model such systems. First of
all, observe that quantum fluctuations are negligible. Indeed, the time scale
τ
Q
of quantum fluctuations is given by
~ω
Q
=
~
τ
Q
' k
B
T.
At room temperature, this gives that
τ
Q
∼ 10
−13
s
, which is much much smaller
than soft matter time scales, which are of the order of seconds and minutes. So
we might as well set ~ = 0.
The course would be short if there were no fluctuations at all. The counterpart
is that thermal fluctuations do matter.
To give an example, suppose we have some hard, spherical colloids suspended
in water, each of radius
a ' 1 µm
. An important quantity that determines the
behaviour of the colloid is the volume fraction
Φ =
4
3
πa
3
N
V
,
where N is the number of colloid particles.
Experimentally, we observe that when Φ
<
0
.
49, then this behaves like fluid,
and the colloids are free to move around.
In this regime, the colloid particles undergo Brownian motion. The time scale of
the motion is determined by the diffusivity constant, which turns out to be
D =
k
B
T
6πη
s
a
,
where
η
s
is the solvent viscosity. Thus, the time
τ
it takes for the particle to
move through a distance of its own radius
a
is given by
a
2
=
Dτ
, which we can
solve to give
τ ∼
a
3
η
s
k
B
T
.
In general, this is much longer than the time scale
τ
Q
of quantum fluctuations,
since a
3
η
S
~.
When Φ > 0.55, then the colloids fall into a crystal structure:
Here the colloids don’t necessarily touch, but there is still resistance to change
in shape due to the entropy changes associated. We can find that the elasticity
is given by
G ' k
B
T
N
V
.
In both cases, we see that the elasticity and time scales are given in terms of
k
B
T
. If we ignore thermal fluctuations, then we have
G
= 0 and
τ
=
∞
, which
is extremely boring, and more importantly, is not how the real world behaves!