Polymer melts

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Polymer melts - the deGennes model

The Nobel prize in Physics, 1991:

"For the latter cases, de Gennes has also established a number of predictions regarding how polymer chains and their individual parts can move, i.e. what physicists call polymer dynamics. These predictions often have the character of "scaling laws": they say that conditions shall be similar for certain combinations of the starting variables (such as polymer concentration in a solution, and temperature). These are properties that can sometimes be controlled experimentally, and many works on polymer dynamics have been performed using neutron-scattering techniques. In experiments of this kind, it is possible to distinguish how individual parts of a polymer chain move by noting how an oscillation of a selected wavelength initiated by the neutron collision is damped during a certain measurable time. Such measurements have helped to confirm de Gennes' models for polymer-chain motion. One of these models, theblob model, states that a certain typical segment of a chain can move as if it were free, even in more concentrated solutions. Another is the reptation model, which describes the serpentine motion of a polymer chain within a "tangle" of surrounding polymer chains.



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Viscosity of polymer melts

Polymer melts are polymers above their glass transition temperatures.

Experimentally it is found that high molecular weights the zero shear viscosity scales as the 3.4 power of molecular weight. For low molecular masses, the viscosity must be corrected for the relative mass dependance on the glass transition temperature.

<math>\eta _{0}\sim N^{3.4}</math>
Jones, Fig. 5-8

This relation is found for many linear polymers. Why?

The model proposed by de Gennes and others is called the reptation model:

  • Because the interactions between chains is independent of MW, the change in viscosity must be a variation in molecular-weight dependent relaxation time.
  • <math>\tau \sim N^{3.4}</math>
  • Polymers are linear and cannot pass through one another.
  • Polymers entangle, but only transiently.
  • Suppose they are trapped in “tubes” from which they must diffuse.

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Reptation model of polymer flow in melts

Jones, Fig. 5.9

In present, one may describe the polymer melt viscosity dependence on the polymerisation degree (the number of chain monomers per one macromolecule) in terms of the reptation model suggested by P.G. de Gennes in 1971. The reptation theory explains why the power law for the melt viscosity relation to the molecular weight of polymer can be observed. The tube model: crosses (top) represent chains coming out of the plane of the screen. The test chain cannot cross these chains, and so its is confined to move in an effective tube (bottom)

Polymer motion along the tube is unhindered, i.e. purely viscous. We can try the following: <math>D_{tube}=kT\mu _{tube}=\frac{kT\mu _{seg}}{N}</math>
If L is the length of the tube, the time to escape is: <math>\tau _{T}=\frac{L^{2}}{D_{tube}}</math>
The length of the tube is proportional to its molecular weight: <math>L\sim N</math>
We find something reasonable: <math>\tau _{T}\sim N^{3}</math>
Since experimental is <math>\tau \sim N^{3.4}</math>

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Extensions to reptation model

Constraint release: surrounding chains move to reorganize the tube (Rouse motion).


Reptation leads to simultaneous tube dilation:


Some polymer branching impedes motion:

Jones, Fig. 5-11

Extreme branching!!!:



DNA Sequencing using Reptation

In the race to the $1,000 genome, many different methods for high throughput DNA sequencing are being researched. One of the most promising is electrophoretic DNA sequencing which partly relies on DNA reptation. Dr. Daniel Branton from Harvard University says that "When it comes to sequencing, nanopores hold great promise, and have held great promise for such a long time that I think a lot of people are getting impatient." The original idea showed up about five years ago and scientists are still working to successfully sequence by measuring electric signal of nucleotides translocating through a carbon nanotube. In fact, Dr. Branton was recently awarded a grant to research the development of nanopores that can recognize individual DNA bases by their electrical signals.


Figure 1. A biased nanopore in an insulating membrane that separates two ionic solution-filled compartments translocates DNA molecules in sequencial nucleotide order between probes that serve as emitter and collector of a tunneling “microscope.” In response to a voltage bias (labeled “ - ” and “+”) across the membrane, ssDNA molecules (yellow) in the “-” compartment are driven, one at a time, into and through the nanopore. Elevated temperatures and denaturants maintain the DNA in an unstructured, single-stranded form. (Note: figure and explanation taken from Dr. Branton's research page)

References http://www.nature.com/nmeth/journal/v5/n3/full/nmeth0308-267.html http://www.mcb.harvard.edu/branton/index.htm http://euler.phys.cmu.edu/widom/research/rep/repton.html http://www.aesociety.org/meetings/2006/PosterAbstracts2006.pdf

The team is among several grant winners who are developing nanopores (holes about two nanometers in diameter) that may be able to recognize individual DNA bases by their electrical or ionic signals to achieve high-accuracy sequencing of individual DNA molecules. The goal of the Harvard scientists is to design and optimize nanopore technology using novel electronic control and sensing methods to create a nanopore detector chip capable of sequencing a mammalian genome within a day on a single instrument.

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