HA DANG: Today, I am going to use Mathematica
to visualize a nanoparticle polymer network
and to investigate how the polymer length affects
of probability that polymer would cross the nanoparticles
and how the polymer length affects
the nanoparticle polymer network formation.
Polymers and macromolecules composed of hundred to thousand
subunits.
They can adopt many different configuration,
depending on the identity of their subunits.
Common everyday examples of polymers
are PTC, polystyrene, Teflon, and so on.
When particles and polymers of are
favorable interaction are mixed, it is not always apparent
whether they will form a cross-linked network.
If the polymer configuration makes the end
to end distance of the polymer shorter
than the distance between two particles,
no polymer-particle network will form.
This project will focus on visualizing the interaction
between particles and polymers given polymer's length,
or the number of subunits.
In stage 1, we make polymer by use the random walk method.
We consider a particle as many lattice sites.
Polymer with start a random lattice site
This initiation part just ensure that the polymer will
truly random walk at a certain distance
away from the surface of the particle.
In the random walk simulaton, the step size
is the length of the subunit of the polymer,
which is about 10 times smaller than the size of the particles.
The step angle tell you the direction
that the poem will walk.
The step tell you how much the polymer will walk
in the x and the y direction.
The next position of the polymer will be the current position
of the polymer plus step.
Each time the new polymer position is computed,
it is stored inside on list which we call path.
Once we have the polymer path, we
can visualize it inside the particle matrix using Graphic.
In stage 2, we want to generate many polymers
and visualize how they distribute and interact
with particles inside a lattice of many particles.
And in order to do so, we combine all the steps
in state 1 into one function, polymerRandomwalk,
which takes in the polymer length
and the lattice of particles and returns a polymer path.
We will generate 100 random random walk polymers of 1,000
subunit by using Table.
We visualize the polymers and particle matrix using Graphic.
In stage 3, we want to calculate the probability of crossover
between nanoparticle in polymers by counting
how many polymers that wrap around more than two particles.
In a nutshell, we arrive a function and crosslinkSuccess,
which test whether each subunit of the polymer
would touch a neighboring particle by using RegionMember.
If there is a least a subunit that successfully cross
a particle, the polymer connects two or more particles
and the function return 1.
If the polymer fails to cross a particle,
the function will return 0.
So we run this function over many polymers
by using the PR function.
And this is a resource that we got.
And we really want to know the ratio of the cross-link polymer
to a total number of polymers.
And to do that, we just do a simple math
by dividing the number of successfully
cross-linked polymer to the total number of polymers.
Below is a function that calculate
the ratio of polymer that link two or more particles together.
Next, we calculate the ratio of cross linking for polymer
of various lengths--
100, 200, 500, 800, 1,000 and 1,500.
Since the operation take a while,
I'll show just the results.
The most interesting thing that we can obtain for this data
is actually plotting the ratio of cross-link
versus the polymer length.
What we find out is that as the polymer length
increases, the probability to cross-link increases.
This makes sense because as a polymer gets longer,
it can walk a larger distance and touch a particle.
What is even more interesting is that the graph
has a shape of a lock function.
We go ahead and use Fit to generate
least square fit equation.
And then we plot it with the data.
As you can see, the square fit function
give you a pretty good fit of the data.
What this fit tell you is that first,
there's a minimum number of subunits
that cross-link will occur.
In our case, the minimum polymer length
will have to be between 100 and 200 subunits
to be able to provide you some type of cross-linking
between polymer and nanoparticles.
Second, the graph plateau at a certain polymer length.
Any polymers that are longer than those
would not increase the probability of cross-link.
Although the analysis of probability for polymer
to cross-link it's useful, it does not
tell any information about the network of polymers
and particle as a whole.
So in stage 4, I will use cluster analysis
to analyze network formation given polymer length
and particle distribution.
In a nutshell, the function singlePolymerCrossLinkTest will
return all the particles' lattice site which the input
polymer can connect.
Then we use a built-in function, UndirectedEdge and Graph
to visualize this connection.
We want to do this simulation for all the polymers
and particle to visualize the connectedness with the polymer
nanoparticle network then use a cluster analysis to analyze
the network formed by mixing polymers
and nanoparticles together.
In here, we write a little function
that basically run the single polymer
cross-link test over many polymer of the same length.
What I want to show you is this GraphCommunityPlot
and FindGraphCommunities.
In this graph, you see that there's
many clusters which has a very dense cross-linking
between polymer and nanoparticles.
However, most of the clusters do not connect it together.
This suggests that a network of polymer and nanoparticle
may not be formed, even though you
have many cluster of densely cross-linking
between nanoparticles and polymers.
What I want to show next is a comparison
between a cluster network of 100 subunit polymer
and a cluster network of 1,000 subunit polymers.
As you can see, as a polymer length increase,
there is a higher chance that cluster can connect together.
So definitely the polymer length can
affect the connectedness of the system
and give a higher chance that polymer
can connect nanoparticle and form a cohesive network.
In a realistic system of polymers and nanoparticles
which have favorable interaction, one
a polymer hits a nanoparticle, it
will stick to that nanoparticle.
In the system that we used above,
polymers are allowed to walk in such a way that does not
take into account the interaction
between nanoparticles and polymers.
In state 5, we'll attempt to take
into account that interaction by using this algorithm step.
First, we make a lattice of nanoparticle sites.
One lattice is randomly selected.
And a random walker will spawn from the site and can walk.
If polymer walker hits a charge site,
including the original one, then the polymer stick to that side.
The modifiedPolymerRandomWalk function
is very similar to the random walk polymer function above,
except for its use RegionMember to test whether or not
a subunit touched a particle.
If the subunit touched a particle,
the function will stop and return the current path.
Here I generate hundreds polymer of 1,000 subunit
by using this modified polymer random walk function
and using Graphic to visualize it.
As you can look into this picture,
the probability that polymers will cross-link nanoparticles
decreases because you restricted it to stopping once the polymer
hitting a nanoparticle.
Although this approach takes into account the interaction
between nanoparticles and polymers,
it is still not a realistic representation
of a real system.
In this approach, we let a polymer subunit
spawn on the particle surface and grow into a polymer.
In a real system, polymers are thrown into a disperse solution
of particles.
The confirmation of polymers in a dynamic process in which
the entire polymer twist, turns so
that the two ends of a polymer find nanoparticle surfaces.
This project has not taken into account
the dynamic confirmation of polymers in a dilute solution.
For future research, I would like
to improve this project by taking into account polymer
confirmation when doing polymer random walk
and finding how polymer concentration
and nanoparticle concentration affect the nanoparticle polymer
network by doing cluster analysis.
I would like to thank Dr. Keane for your guidance
on this project.
I would like to thank Professor Carter for teaching 3016, which
I found very helpful for the other project
I'm working on my UROP right now.
And I would like to tell you all the TA's in 3.016
say for helping me during office hour.