טכניון מכון טכנולוגי לישראל
הטכניון מכון טכנולוגי לישראל - בית הספר ללימודי מוסמכים  
M.Sc Thesis
M.Sc StudentTepper Tal
SubjectThe Kinetics of Linear Polymerization of a Fixed Number
of Identical Monomers
DepartmentDepartment of Nanoscience and Nanotechnology
Supervisor Professor Uri Sivan
Full Thesis textFull thesis text - English Version


Abstract

The concept of self-assembly is encountered in diverse branches of science from supramolecular chemistry to biophysics, and in recent years, nanotechnology. Inspired by nature, self-assembly is investigated as an alternative route to the production of mesoscopic scale structures, based on a set of carefully designed interactions between elementary building blocks. In statistical mechanics terms, self-assembly amounts to the flow of the assembling system from its initial state, to the equilibrium assembled state. Significant efforts in diverse applications have been invested in a careful design of the equilibrium state. Less is known on the assembly kinetics, namely, the time taken by the flow from separated blocks to the assembled equilibrium state.  Motivated by the quest for the general principles that govern the sought kinetics we study here a simple one-dimensional polymerization model and discover that the kinetics is governed by entropy barriers. Remarkably, we find that these entropy barriers prevail also in more permissive one-dimensional polymerization models. The kinetics of one-dimensional self-assembly in its later stages is thus shown to be generic.

The simple model studied here is termed the Edge Polymerization Model (EPM). In this model, polymer elongation and disassembly is restricted to sequential addition or removal of a single monomer at the polymer's end. Being interested in the assembly of predominantly single long polymers, we focus our attention on the low temperature limit. We find three distinct assembly regimes. In the first, instantaneous regime, monomers readily assemble to give a factorially narrow distribution of short polymers. At the end of this phase, the monomers are consumed. Now, the assembly kinetics is governed by the rate of dimer disassembly and the attachment of the freed monomers to longer polymers. This intermediate phase concludes with the consumption of short polymers and convergence of the polymer distribution function to a universal curve. The consumption of short polymers also marks the beginning of the third phase where assembly decelerates due to the growing entropy barriers. This phase dominates the overall assembly time, which is calculated to be proportional to the number of monomers squared.

The kinetics in the three phases is analyzed by solution of the Reaction Rate Equations (RRE) where they hold, and the Chemical Master Equations (CME) in the later stages of self-assembly where the RRE fail. The CME are solved exactly for a small number of monomers and numerically using a stochastic approach for a large number of monomers.