Self Assembly – a viable manufacturing technique??

Self-assembly is scientifically interesting as it occurs inside the most basic form of life – the cell. The cell contains an astonishing range of complex structures such as lipid membranes, folded proteins, structured nucleic acids, protein aggregates, molecular machines, and many others that form by self-assembly. At the same time, self-assembly is technologically important as it provides routes to a range of materials with regular structures: molecular crystals, liquid crystals, and semicrystalline and phase-separated polymers are examples.Self-assembly is a process in which components, either separate or linked, spontaneously form ordered aggregates. Self-assembly can occur with building blocks from molecular to the macroscopic scales, provided that appropriate conditions are met. For customized materials on macroscopic length scales to be made, self-assembly is essential because the number of building blocks that would have to be precisely placed in the absence of a self-directed method are too large and the process is prohibitively time consuming. A Fermi approximation of the time required for building a 1cm cube using 10 nm building blocks with the help of optical tweezers would be in the order of a hundred million days.Self-Assembly is an alternative technique by which materials can be formed. The basic idea behind using self-assembly as a manufacturing technique is to develop building blocks that have sufficient information content inherent in them, such that a desired structure can form spontaneously from a collection of building blocks in a thermal bath.Amongst interesting building blocks that have been used to self-assemble custom materials are colloids. These mixtures of immiscible materials form fine suspensions that have potential photonic applications because the particles that make up the stabilized phase can be tuned to have diameters on the same scale as visible light (500nm) and can form periodic structures. Polystyrene colloidal crystals, consisting of charged monodisperse polystyrene microspheres suspended in water, organize into a face-centered-cubic (fcc) lattices with lattice spacings comparable to the wavelength of light (Clark, N.A. et al, Nature 1979, 281, 57-60). Such colloidal structures may also find applications in nanoscale electronics, miniature diagnostic systems and hierarchically structured catalysts amongst others. However, for the assembly of more complex structures, it has been recognized that anisotropic shape and specific interactions (or “patchiness”) are useful (Glotzer S.C., & Solomon M.J. Nature materials 2007, 6(8), 557-62).

A recent work by Chen et al., Nature (2011), 469 (7330), 381-384, has shown that triblock Janus particles can self-assemble into a Kagome lattice (see Figure 1). A triblock Janus particle has two hydrophobic poles of tunable area and an electrically charged middle band. There are two competing forces that drive self-assembly for suspensions of these particles: the hydrophobic poles agglomerate in aqueous solvents, while the charged middle bands repel. The balance between these forces can be tuned by screening the charged repulsion via the addition of an ionic compound like NaCl, and the particles can stabilize in a kagome lattice. Furthermore, sheets of kagome lattices are reported to stack in parallel layers. An interesting video of this self assembly process can be found at: http://www.nature.com/nature/journal/v469/n7330/extref/nature09713-s3.mov Thus self-assembly seems to offer a viable strategy for generating nanostructures. Recent advances in nanoparticle synthesis have led to the availability of various building blocks including spheres,cubes, rods, and tetrapods at both the nano and micro scales. Specific interactions between these building blocks can also be obtained by modifying chemical composition, changing shape, and by adding functional groups. With such a vast parameter space available to experimentalists, there is much opportunity for simulation to aid in the understanding of how such complicated building blocks assemble.

- N. Khalid Ahmed

PhD Pre-Candidate,

Glotzer Group,

Department of Chemical Engineering,

University of Michigan, Ann Arbor – 48109

nkahm@umich.edu

Solving First-order simultaneous ODEs

Solving a system of ordinary differential equations (ODEs) has always been a cakewalk. Given a pen and paper, it takes almost no time to solve a simple system such as this:

$ \frac{dx}{dt} + ax + by = 0$

$ \frac{dy}{dt} + cx + dy = 0$

One can introduce the differential operator $D$ and solve as if it were a set of algebraic equations. But when the system of equations gets bigger, things become messy. They can be solved only by an in-built solver routine in a high-level language like MATLAB. However, feeding such a huge system can still be troublesome. The purpose of this post is to expose a cute little trick in mathematics and exploit it to our advantage.

To explain the trick, we consider the above set of differential equations. We rewrite them in matrix form as:

$
\begin{bmatrix}
x'\\
y'
\end{bmatrix}
+
\begin{bmatrix}
a &b\\
c &d
\end{bmatrix}
\begin{bmatrix}
x\\
y
\end{bmatrix}
=
\begin{bmatrix}
0\\
0
\end{bmatrix}
$

or simply,

$ X' + AX = 0$

where, $X$ is the unknown vector whose elements are $x$ and $y$.
Now, here is the trick:

Split the matrix $A$ into 3 matrices as $P^{-1} \Lambda P$, where $\Lambda$ is a diagonal matrix, with the eigenvalues of $A$ as the diagonal elements. $P$ is a matrix, which has the eigenvectors of $A$ as its columns.

This is called as Eigenvalue decomposition. This is actually an one-liner in MATLAB. Now, the matrix equation becomes:

$ X' + P^{-1}\Lambda P X = 0$

Pre-multiply by $P$ to get

$ PX' + \Lambda PX = 0$

Now, take $PX$ to be $U$, and owing to the fact that $\Lambda$ is diagonal, we get a system of the form:

$ \frac{du_1}{dt} + \lambda_1 u_1 = 0$

$\frac{du_2}{dt} + \lambda_2 u_2 = 0$

Now that we have separated the dependent variables, solving this should be a no-brainer. Finally, apply the reverse transformation $P^{-1}U$ to get the solution.

This works even when the right-hand side is non-zero. We just have to pre-multiply it by $P^{-1}$. Also, remember to transform the boundary conditions as well.

A variation to this problem may be something like this:

$ AX' + BX = c$

where $A$ and $B$ are square matrices. In such a case, first pre-multiply by $A^{-1}$ and convert it to the 'standard form' discussed above. The rest should be easy.

Always remember: The trick behind the trick is knowing when to use the trick. Else there is no point in knowing the trick.