Department of Applied Mathematics FACULTY OF MATHS AND PHYSICAL SCIENCES Tubes: mean paths, constraint release and bimodal blends Daniel Read, Alexei Likhtman, Kamakshi Jagannathan, and others

The “solution” (tubes, mean path, bending energy) The “problem” (convective constraint release models in non-linear flow) The solution helps! Thinking further – bimodal blends Overview

What do we mean by the “tube”? It represents the localisation of chains due to “entanglements” with other chains. What makes us think we can define a tube length, tube tangent, etc? It is a “coarse-grained” random walk. Is it defined at all on lengthscales smaller than tube diameter? R(s) s but we draw this: which suggests a smooth path in space and we claim the chain moves along this path! monomers fluctuate about this path by a distance of order the tube diameter NB – monomer motion is not 1D. Is the tube some “average position” of the monomers??

A model where I can do the maths: the Warner-Edwards “tube”. Look for “mean path” about which monomers fluctuate….. BENDING ENERGY!!! fluctuationsmean path

sR(s) A “tube” model for non-linear flow Graham, Likhtman, McLeish, Milner (J. Rheol. 47, 1171, 2003) Equation of motion for tube-tangent correlation function (stretch and orientation) Reptation +CLF flow CR retraction (stretch relaxation)

Focus on the constraint release term - noise - Essentially a “Rouse” model with renormalisations for chain stretch Rouse relaxation term noise Noise correlation has finite width to give a smooth tube...

FDT: noise and relaxation terms are intimately linked. If the noise has a finite width, then so does the relaxation term. It should have the form Model: detailed balance for finite width hops of tube “textbook” 1D free energy for chain in a tube There are some problems with this….

“Rouse” CCR dynamics with bending energy and finite width noise 4 th derivative comes from bending energy Appropriate noise function obtained by looking for the change in the mean path obtained by moving one of the springs. NB – no 1/ seems necessary: indeed, it seems to be unnatural within the above picture.

“Rouse” CCR dynamics with bending energy and finite width noise Points: previous model of Graham et al (which fits experiments quite well) Lines: current model with bending energy (but no 1/ ). So: we retain same ability to fit data – to the extent that the previous equations did – but on a sounder footing that can be taken further. Z=6Z=20Z=15

“Thin tube” is the tube representing entanglements with ALL other chains “Fat tube” is the tube representing entanglements with LONG CHAINS ONLY Bimodal blends tube picturesprings picture rare slow springs, or weak slow springs, give “fat tube”

Bimodal blends – scaling in limit of large number of springs hence, to get “fat” tube diameterwe need to have i.e. weak slow springs are a factor of  2 weaker, or rare slow springs are a factor of  2 rarer (c.f. limit of small number of spings per tube diameter, or point-like constraints, which give factor of  ).

Bimodal blends – a happy scaling accident? r SG (Ratio of reptation time in thin tube to CR- Rouse time of thin tube) Number of entanglements along “fat tube” CR on thin tube vs CR on fat tube Reptation in thin tube vs reptation in fat tube Reptation in thin tube vs constraint release of thin tube Long chains self- entangled? “Viovy-Colby-Rubinstein” diagramthis line (predicted gradient –2) comes from competition between fat tube and thin tube constraint release. Line (+ correct gradient) emerges naturally from rare slow spring picture (recall springs are factor  2 rarer). by contrast, assigning fast/slow dynamics to point- like constraints (slow ones a factor  rarer) does not give a line within the sub-space of this diagram!

Some things that I like about a mean path with a bending energy It is the path about which monomers fluctuate. It is a smooth, “coarse grained” random walk. It has a free energy – you can define an equilibrium distribution of configurations. If the mean path is in this equilibrium distribution, then chains are in equilibrium gaussian distribution!! Tangent vector is defined. Dynamics that satisfy detailed balance naturally relax towards the equilibrium. sounds like a tube???

A very suggestive picture from a simulation….. ….courtesy of Dr Alexei Likhtman sliplinks + localising potentials suggests you can get somewhere with the “average monomer position” idea?