A Deterministic PDE Model describing

POLYMER CRYSTALLIZATION PROCESSES

In 2000, Vincenzo Capasso (Milan, Italy) derived a deterministic model describing polymer crystallization

processes from a stochastic model which was under study in his group at Università degli Study di Milano.

The problem was given to me during my post-doc in Milan in 2002, and is still interesting, 10 years after.

The model

A cooling object is applied to the side (single cooling) or both sides (double cooling) of a polymer melt. If the applied temperature is

lower than the critical freezing threshold, small crystals begin to appear and start to grow. A wide crystallization front –a band– then

emerges where all the crystallization process takes place. Under constant applied temperatures, the crystallization is nonhomogeneous,

in the sense that spatial intervals of the same length take different times to crystallize. Moreover, the temperature field describes slight

oscillations which makes the crystallization band to advance by*bumps*.

- Original formulation:

The model consits of two PDEs for the degree of crystallinity*y(x,t)*and the temperature*T(x,t)*, with the special feature

that the dependence of the nucleation and growth rates upon the crystallinity and the temperature has been decoupled.

as it appeared the first time in [*], as the expression (3.57) in p. 62 and the definition of*b*,_{b}(T)*b*,_{g}(T)*β(y)*and*κ(y)*in p. 63.

[*] "*Mathematical models for polymer crystallization processes*",

V. Capasso,*in*V. Capasso, H. Engl, J. Periaux*(Eds.)*, Computational

Mathematics Driven by Industrial Problems (Springer Berlin, 2000) pp 39–67.

- Present formulation:

The truncation to zero of the rate functions*b*and_{b}(T)*b*when the temperature_{g}(T)*T(x,t)*is greather

than the freezing threshold*T*(and then no nucleation nor growth can take place) was later introduced._{f}

- Numerical solution:
Single (left) cooling with a constant applied temperature of
*u(t)*= 40°C

- Original formulation:
Numerical simulations ( right-clic on the picture to see a larger image )

- Three different cooling strategies:

A) and B): single (left) cooling at different constant temperatures

C): double cooling: left side, constant temperature, right side, exponentially decreasing temperature

A) *u(t)*= 40°C

B) *u(t)*= 0°C

C) *u*= 40°C,_{L}(t)*u*= exp._{R}(t)

Final configuration: (note the spatial inhomogeneity)

A) *u(t)*= 40°C

B) *u(t)*= 0°C

C) *u*= 40°C,_{L}(t)*u*= exp._{R}(t)

- Three different cooling strategies:
The Stefan problem formulation

The crystallization band can be identified to a free boundary whose time-evolution obeys a Stefan equation.

The associated Stefan problem can be solved analytically for constant and exponential applied temperature;

for arbitrary applied temperature profiles, the*pseudo-steady state approximation*can be used.

- Analytical or PSS solution of the Stefan problem
For the single (left) cooling with
*u(t)*= 40°C

Temperature field for (A) the full crystallization model, and (B) the Stefan approximation

- Numerical solution of the full model together with the Stefan problem approximation

The vertical yellow line denotes the position of the free boundary, and the blue line is the approximation

of the temperature profile provided by the PSS approximation to the solution of the stefan problem.

A) *u(t)*= 0°C

B) *u(t)*= arbitrary

- Analytical or PSS solution of the Stefan problem
For the single (left) cooling with