Definition
FourPoint Susceptibility of Colloids
Here I will only discuss how to compute the four point
susceptibility using the known trajectory of many particles. In our
work we focus on tracking the diffusive motion of individual
colloids in the supercooled or glassy state. So this definition will
apply to such situations. This definition will also apply to
molecular dynamics or Monte Carlo simulations where the
trajectories of the molecules are known. Other methods for
measuring the fourpoint susceptibility include using light scattering and
electric susceptibility data.
Let's start by assuming that we have N particles of known trajectory
r(t), where r is the position of the particle at
time t. The simplest definition of the fourpoint
susceptibility χ_{4} is the fluctuations in time of the
percentage of particles that are mobile. To clarify this definition
we need to define which particles are mobile and which are not. We
define a particle to be mobile if its displacement over a time interval
Δt is larger than the some threshold displacement
ΔL.
So how do you choose Δt and
ΔL? If there are any particular length scales and time
scales for which you want to know the dynamic heterogeneity you can pick
those values. A more common approach is to vary the two numbers until
you find which pair of values
maximizes χ_{4} because this is the length scale and
time scale where the dynamics are most heterogeneous.
Ok, back to the math. Once Δt is
chosen, we can define a displacement
Δr_{i}(t) =
r_{i}(t + Δt) 
r_{i}(t),
where i indexes the ith particle at time
t. Using the displacements and the chosen value of
ΔL we can define which particles at time t are
mobile. After labeling each particle mobile or immobile the percentage
Q(t) of mobile particles can be computed. Note
that for each time t there is a value of Q
associated with that time. For instance, if there are 100 different
moments in time where you can compute a displacement, then you
should have 100 different values of Q.
Now comes the simple part. The fourpoint susceptibility is the
product of N and the variance in Q(t)
with time, where we write this as
χ_{4} =
N(⟨Q(t)^{2}⟩  ⟨Q(t)⟩^{2}).
The reason we introduce the factor of N is because the
variance in Q is inversely proportional to
N. Therefore if we have 10 times as many particles the
variance in Q decreases by a factor of 10, but the
χ_{4} remains constant.
There are other slight mathematical ways to calculate
χ_{4} by changing the definition of Q. We can
write our previous definition of Q mathematically as
Q = ⟨Θ(Δr  ΔL)⟩_{i},
where the average is carried out over all the particles at time
t and Θ is a
heaviside function. The way we think of Q is that it's
an average of weights where we use the displacements to compute the
weights. In the previous example we used the heaviside function to
weight the displacements. However there are other ways we can weight the
displacements. To give one example we could use an exponential such
that
Q = ⟨exp(Δr/ΔL)⟩_{i}.
Besides the two examples given so far, there are other ways the
displacements can be weighted.
Finally, what does χ_{4} tell us? In the first definition
I provide, χ_{4} is a measure of the average number of
particles whose dynamics are correlated. This is important because in both molecular and
colloidal glasses correlated dynamics are an essential feature that
gives rise to the anomalous behavior of glassy
materials. Quantifying how the correlated dynamics change with
temperature for molecular glasses or with volume fraction for colloidal
glasses provide insight on how the correlated dynamics change as the glass
transition is approached. The fourpoint susceptibility is just one
tool out of many for quantifying the correlated dynamics.
To see how we use χ_{4} to study correlated dynamics
in supercooled colloidal glasses read the article below. This article
also discusses other methods we used to quantify the correlated
dynamics.
 "Spatial and temporal dynamical heterogenities approaching the binary colloidal glass transition"
T Narumi, SV Franklin, KW Desmond, M Tokuyama, & ER Weeks, to be published in Soft Matter
Abstract /
PDF
