1) DEFINITION OF TERMS :

Crystallized phase : orderded solid phase according to a network composed by several identic shapes at long distances.
Amorpheous phase : desordered solid phase. Some order may occur but only at short distances.

2) CRYSTALLIZATION IN POLYMERS :

2.1) Morphological observations :

The polymers are more or less crystalline. It depends on the temperature and on their chemical structure. The crystallization will not give massive crystals such as in case of metals or molecular solids. The low mobility of the long macromolecular polymer chains leads to a partial crystallization. The lamellar structures starting from central germs can be visualized in scanning electronic microscopy or in polarization microscopy. Radial lamellar structures are called spherulites. Between spherulites and also crystalline lamellae, an amorphous phase is founds. Spherulites diameters are ranged from 1 to 10 Ám. Lamellae thicknesses are ranged from 10 to 100 nm.


Cristalline lamellae grow in only one direction from the central germ and have given thickness and broadness. The dimensions of the cristalline lamella depends on polymer chemical structure and thermodynamic conditions in particular during cooling.

The crystallization rate is defined as the molar fraction of the crystalline phase in the whole polymer. It is commonly lower than 1. It depends also on the chemical structure and on the thermomechanical story of the polymer.

2.2) Mechanical properties of crystallized polymers :

Crystallized polymers are more mechanically resistent than non crystallized polymers even they are in a their rubbery state. For exemple, polethylen is then a good material for current mechanical applications at ambiancy temperature even it is not charged with carbon particles like in rubbers, because it is crystallized.
Crystallized parts are more dense then amorpheous parts and then the total volume of the polymer diminishes during crystallisation. This may lead to a method for following evolution of the crystallinity rate by measuring the variation of the molar volume of the polymer.

2.2) Thermodynamic considerations :

Crystallization temperature is an intrinsic parameter which is independant from the glass transition temperature but it is usually higher. So when a melted polymer is frozen, it usually first crystallise and then becomes vitreous. Crystallization involves heat releasing with a decrease of polymer heat capacity as it may be observed by differential scan calorimmetry analysis.

2.3) Kinetic considerations :

Polymer crystallization kinetics depends on the temperature. The increase of the crystallization rate in a frozen polymer may be related either to the apparition of new spherulites in melted phase, either to the spherulites growth.

2.3.1) Apparition of the spherulites :
a)If Apparition of spherulites is only induiced by spontaneous germination during time : this case is called homogeneous kinetics.
b)If germs preexist in the melted polymer (more often they are due to some impurities) and no germs appear later : this case is called heterogeneous kinetics.

Sometimes the kinetics evoluates between these to extreme cases and then its study may be difficult.

2.3.2)Spherulite growth :
Spherulite growth is related to the crystalline lamellae growth inside the spherulites. The growth speed depends on the temperature. Its activation energy is also not constant and may depend on the temperature too : This is explained by the crystallization mechanism at the extremity of a crystalline lamella : When a chain segment crystallise, a new crystalline layer may born or a crystalline layer already born may be completed. The speed of birth and filling may have different variations. This may lead to different growth regimes with a global activation energy which depend on the temperature.
It is also noticed that the density of crystalline lamellae in the spherulite is homogeneous and that in fact new lamellae can germ against the other lamellae when the diameter of the spherulite increases. This is called the secondary germination. The structure of the whole spherulite is then partially ramified.

Thus, some different degrees of modelization can be involved : global one through Avrami's model, or detailed ones taking in acount more parameters.

3) CRYSTALLIZATION MODELS :

3.1) Global approach : Avrami's Model

Avrami's model is based on the observation of the percolation of growing spherical structures from germs. One regards the total volume V(t) occupied by the spheres at a time t.

Avrami showed that : V(t) = 1 - exp(-K t^n)

where :
K is a kinetic constant.
n is the Avrami's exponent

All the spheres/disks grow at the same speed even they may born at different times.

In a polymer, the rate of crystallinity Tx(t) may replace V(t) and the relation becomes :

Tx(t) = Txoo * (1 - exp(-K t^n)

where : Txoo is the maximum of the crystallization rate. Usually, One draw the linearized expression :

Ln(-Ln (1 - Tx(t) /Txoo)) = f(Ln(t))

If the obtained curve is linear and has got an entire value for its slope n, the kinetics of crystallization is assumed to follow Avrami's model. The ordinate at the origin Yo allows to calculate the kinetic constant K :

K = exp(Yo)

In a three demensional space : for an homogeneous kinetics n = 4, for an heterogeneous kinetics n = 3.
In a two dimensional space : for an homogeneous kinetics n = 3, for an heterogeneous kinetics n = 2.

The kinetic constant K is global. It depends on the temperature. it can be also related to number of initial germs, speed of germination, and speed growth of the spherulites.

3.2) Detailed approach : secondary germination and crystalline lamella growth

Growth of the spherulites can be described through secondary germination and growth of the cristalline lamellae :

a) Secondary germination occurs when a new crystalline lamella is borned against another cristalline lamella. Some ramification then occurs.

b) Growth of crystalline lamellae may be also related to the local crystallization mechanism through the following parameters :
- The speed of birth of a new crystalline layer on the extremity of the lamella.
- The speed of filling a crystalline layer.

The relative variations of the speed values of birth and filling crystalline layers lead to different growth regimes.

4) MONTE-CARLO SIMULATIONS :

4.1) Simulation of the global model :

This simulation is a two dimensional one and represents the spherulites as colored disks which born and grow at parametrized speeds. Localization of born spherulites is randomly chosen. Germs are represented by single pixels on the visalization screen.

The input parameters taken in acount are :
- The number of initial germs,
- The speed of germination (birth) of new spherulites.
- The speed of growth of the sperulites (speed growth of their radii),
the maximum of the crystallization rate Txoo is assumed to be equal to 1.

Germination speed and growth speed are controlled by random numbers compared to threshold values.

No representation of the crystalline lamellae is done and no consideration is made about the secondary germination speed and growth regimes.

4.2) Simulation of the detailed approach by a reptation model :

In this simulation, the chain segments which crystallize are represented by small two pixels segments which displace themselves by reptation and may be vertically or horizontally positioned. Thus when they crystallize, the chain segments may be immobilized horizontally or vertically. Crystalline lamellae are then formed by several adjacent parallel segments and can be thus horizontal (and formed by several vertically positioned segments) or vertical (and formed by several horizontally positioned segments). Secondary germination is considered to occur when a chain segment come along a lamella and is perpendicularly positioned against the crystallized chain segments of this lamella. Germs may appear when two mobile segments are parallel and then crystallize.

The input parameters taken in acount are :
- The number of initial germs,
- The germination speed (birth) of new spherulites,
- The secondary germination speed,
- The growth speed of the crystalline lamellae.
the maximum of the crystallization rate Txoo is assumed to be equal to 1.

The different speeds are controlled by random numbers which are compared to threshold values.

No consideration is done about growth regimes.

 
 
   
  Administrator SAFE FOOD PACKAGING PORTAL
Introduction proposed Dr Jean-Yves Dolveck
Jan 2008