Polymer Dynamic

The de Gennes Reptation Theory While the Rouse–Bueche theory was highly successful in establishing the idea that chain motion was responsible for creep, relaxation, and viscosity, quantitative agreement with experiment was generally unsatisfactory. De Gennes model consisted of a single polymeric chain, P, trapped inside a three-dimensional network,G, such as a polymeric gel. The gel itself may be reduced to a set of fixed obstacles—O1, O2, . . . , On. . . . The chain P is not allowed to cross any of the obstacles; however, it may move in a snakelike fashion among them. P. G. de Gennes, J. Chem. Phys., 55, 572 (1971).

A model for reptation. The chain P moves among the fixed obstacles, O, but cannot cross any of them.

t<τ0 τ0<t<τe τe<t<τRept τRept<t

t<τ0 τ0<t<τe τe<t<τRept τRept<t Time Scale Tube effect Time Scale Length Scale t<τ0 τ0<t<τe τe<t<τRept τRept<t

L= b[NNe(-1/2) in experiment ττ

Stress Relaxation Modulus According to Rouse model

Stress Relaxation Modulus At τe Independent of Time According to Doi and Eward The main contribution1 comes from the first mode p = l and the function is almost a single exponential:

Time-Temperature superposition Temperature related parameters The temperature dependence of all relaxation times is controlled by the ratio of friction coefficient and absolute temperature: The temperature dependence of the modulus at any relaxation time τ is proportional to the product of the polymer mass density p and absolute temperature T: Therefore, the viscosity:

Time-Temperature superposition Temperature related parameters The kinematic viscosity: ρ Has little temperature dependence therefore dependence of viscosity to temperature is as high as ζ.

Time-Temperature superposition Master Curves The relaxation times of all modes have the same temperature dependence. The relaxation time ratio at different temperatures depends on viscosity ratio at the same temperatures. Therefore, it should be possible to superimpose linear viscoelastic data taken at different temperatures. This is commonly known as the time-temperature superposition principle.

The WLF Equation Time shift factor:

The basic Time-Temperature Equation: bT, modulus shift factor; In oscillatory experiments:

Master Curve Construction

The End

When the defects move, the chain progresses When the defects move, the chain progresses. There are tubes made up of the urrounding chains. The velocity of the nth mer is related to the defect current Jn by Reptation as a motion of defects. (a) The stored length b moves from A toward C along the chain. (b) When the defect crosses mer B, it is displaced by an amount b

Using scaling concepts, de Gennes found that the self-diffusion coefficient, D, of a chain in the gel depends on the molecular weight M as: polyethylene of 1 x 104 g/mol at 176°C , D= 1 x 10-8 cm2/s. Polystyrene of 1 x 105 g/mol , D= 1 x 10-12 cm2/s at 175°C. The chain is considered within a tube. (a) Initial position. (b) The chain has moved to the right by reptation. (c) The chain has moved to the left, the extremity choosing another path, I2J2. A certain fraction of the chain, I1J2, remains trapped within the tube at this stage . The tubes are made up of the surrounding chains.

Fickian and Non-Fickian Diffusion The three-dimensional self-diffusion coefficient, D, of a polymer chain in a melt is given by: where X is the center-of-mass distance traversed in three dimensions, and t represents the time. For one dimensional diffusion, say in the vertical direction away from an interface, the six in equation is replaced by a two. According to the scaling laws for interdiffusion at a polymer–polymer interface, the initial diffusion rate as the chain leaves the tube goes as t1/4, representing a case of non-Fickian diffusion. An important example involves the interdiffusion of the chains in latex particles to form a film;