**3.3 Thermokinetics as a means to control macrometric length scale molecular organizations through molten to solid transitions under mechanical stress**

A newly developed phenomenological model for pattern formation and growth kinetics of polymers uses thermodynamic parameters, as thermo-mechanical constraints and thermal gradient. It is a system of physically-based morphological laws-taking into account the kinetics of structure formation and similarities between polymer physics and metallurgy within the framework of Avrami's assumptions.

Polymer crystallization is a coupled phenomenon. It results from the appearance (nucleation in a more or less sporadic manner) and the development (growth) of semi-crystalline entities (*e.g.*, spherulites) (Gadomski & Luczka, 2000; Panine et al., 2008). The entities grow in all available directions until they impinge on one another. The crystallization kinetics is described in an overall manner by the fraction *(t)* (surface fraction in two dimensions (2D) or volume fraction in three dimensions (3D)) transformed into morphological entities (disks in 2D or spheres in 3D) at each time *t.*

The introduction of an overall kinetics law for crystallization into models for polymer processing is usually based on the Avrami-Evans's (AE) theory (Avrami, 1939, 1940, 1941; Evans, 1945). To treat non-isothermal crystallization, simplifying additional assumptions have often been used, leading to analytical expressions and allowing an easy determination of the physical parameters, *e.g.*, Ozawa (1971) and Nakamura et al. (1972) approaches. To avoid such assumptions, a trend is to consider the general AE equation, either in its initial form as introduced by Zheng & Kennedy (2004), or after mathematical transformations as presented by Haudin & Chenot (2004) and recalled here after.

## **General equations for quiescent crystallization**

The macroscopic mechanism for the nucleation event proposed by Avrami remains the most widely used, partly because of its firm theoretical basis leading to analytical mathematical equations. In the molten state, there exist zones, the potential nuclei, from which the crystalline phase is likely to appear. They are uniformly distributed throughout the melt, with an initial number per unit volume (or surface) *N0*. *N0* is implicitly considered as constant. The potential nuclei can only disappear during the transformation according to activation or absorption ("swallowing") processes. An activated nucleus becomes a growing entity, without time lag. Conversely, a nucleus which has been absorbed cannot be activated any longer. In the case of a complex temperature history *T(t)*, the assumption of a constant number of nuclei *N0* is no more valid, because *N0 = N0(T) = N0(T(t))* may be different at each temperature. Consequently, additional potential nuclei can be created in the nontransformed volume during a cooling stage. All these processes are governed by a set of differential equations (Haudin & Chenot, 2004), differential equations seeming to be most suitable for a numerical simulation (Schneider et al., 1988).

### **Avrami's Equation**

654 Thermodynamics – Interaction Studies – Solids, Liquids and Gases

The local affinities of AuNPs with PEO/SCCO2 stabilize the thermodynamically unstable SCCO2-plasticized network and keep it stable with time, which cannot be observed without the insertion of gold nano-particles mainly because of diffusion effect of the solvent (Boyer et al., 2006a)**.** The mean height of AuNPs layer is about 3 nm, which is 20 times smaller than PEO cylinders with a 60 nm in length. Thus PEO channels could be considered as nano-dots receptors, schematically as a "compact core–shell model" consisting of a spherical or isotropic AuNP "core" embedded into a PEO channel "shell", consequently leading to isotropic two- and three-dimensional materials. Nicely, AuNPs clusters on PEO channel heads can be numerically expressed. The presence of, 4, 5 and 8 single Au nano-clusters for *m* = 114, 272 and 454 is identified, respectively. It represents a linear function between the number of AuNPs on swollen PEO *versus* SCCO2-swollen diameter with half of ligands of

From this understanding, a fine thermodynamic-mechanical control over extended *T* and *P* ranges would provide a precious way to produce artificial and reliable nanostructured materials. SCCO2-based technology guides a differential diffusion of hydrophilic AuNPs to cluster selectively along the hydrophilic PEO scaffold. As a result, a highly organized hybrid metal-polymer composite is produced. Such understanding would be the origin of a 2D

A newly developed phenomenological model for pattern formation and growth kinetics of polymers uses thermodynamic parameters, as thermo-mechanical constraints and thermal gradient. It is a system of physically-based morphological laws-taking into account the kinetics of structure formation and similarities between polymer physics and metallurgy

Polymer crystallization is a coupled phenomenon. It results from the appearance (nucleation in a more or less sporadic manner) and the development (growth) of semi-crystalline entities (*e.g.*, spherulites) (Gadomski & Luczka, 2000; Panine et al., 2008). The entities grow in all available directions until they impinge on one another. The crystallization kinetics is

or volume fraction in three dimensions (3D)) transformed into morphological entities (disks

The introduction of an overall kinetics law for crystallization into models for polymer processing is usually based on the Avrami-Evans's (AE) theory (Avrami, 1939, 1940, 1941; Evans, 1945). To treat non-isothermal crystallization, simplifying additional assumptions have often been used, leading to analytical expressions and allowing an easy determination of the physical parameters, *e.g.*, Ozawa (1971) and Nakamura et al. (1972) approaches. To avoid such assumptions, a trend is to consider the general AE equation, either in its initial form as introduced by Zheng & Kennedy (2004), or after mathematical transformations as

The macroscopic mechanism for the nucleation event proposed by Avrami remains the most widely used, partly because of its firm theoretical basis leading to analytical mathematical equations. In the molten state, there exist zones, the potential nuclei, from which the crystalline phase is likely to appear. They are uniformly distributed throughout the melt,

*(t)* (surface fraction in two dimensions (2D)

**3.3 Thermokinetics as a means to control macrometric length scale molecular organizations through molten to solid transitions under mechanical stress** 

AuNPs linked with PEO polymer chain.

within the framework of Avrami's assumptions.

described in an overall manner by the fraction

presented by Haudin & Chenot (2004) and recalled here after.

**General equations for quiescent crystallization** 

in 2D or spheres in 3D) at each time *t.*

nanocrystal growth.

Avrami's theory (Avrami, 1939, 1940, 1941) expresses the transformed volume fraction ( )*t* by the general differential equation **eq. (8)**:

$$\frac{d\alpha(t)}{dt} = (1 - \alpha(t)) \frac{d\tilde{\alpha}(t)}{dt} \tag{8}$$

( )*t* is the "extended" transformed fraction, which, for spheres growing at a radial growth rate *G(t)*, is given by **eq. (9)**:

$$\tilde{\alpha}(t) = \frac{4\pi}{3} \frac{t}{0} \left[ \int\_{\tau}^{t} G(u) du \right]^3 \frac{d\tilde{N}\_a(\tau)}{d\tau} d\tau \tag{9}$$

*dN t dt <sup>a</sup>*( )/ is the "extended" nucleation rate, 3 <sup>4</sup> ( ) <sup>3</sup> *t G u du* is the volume at time *<sup>τ</sup>* of a

sphere appearing at time *t* , and ( ) *<sup>a</sup> dN* are spheres created per unit volume between *τ* and *τ* + *dτ.* 

### **Assumptions on Nucleation**

The number of potential nuclei decreases by activation or absorption, and increases by creation in the non-transformed volume during cooling. All these processes are governed by the following equations:

$$\frac{dN(t)}{dt} = -\frac{dN\_a(t)}{dt} - \frac{dN\_c(t)}{dt} + \frac{dN\_g(t)}{dt} \tag{10a}$$

$$\frac{dN\_a(t)}{dt} = q(t)N(t) \tag{10b}$$

$$\frac{dN\_c(t)}{dt} = \frac{N(t)}{1 - \alpha(t)} \frac{d\alpha(t)}{dt} \tag{10c}$$

$$\frac{dN\_{\mathcal{S}}(t)}{dt} = (1 - \alpha(t)) \frac{dN\_0(T)}{dT} \frac{dT}{dt} \tag{10d}$$

Thermodynamics and Thermokinetics to Model Phase Transitions of Polymers

**General equations for shear-induced crystallization** 

volume fraction is written as (Haudin et al., 2008):

spherulites and into shish-kebabs, respectively.

Modification of **eqs. (8)** and **(10a)** gives:

functions, *e.g.*:

*t* and

*t* and

*N t* 

*A A*<sup>1</sup> , with

shear rate

**Spherulitic Morphology** 

' is becoming:

the shear rate.

over Extended Temperature and Pressure Ranges Under Various Hydrostatic Fluids 657

the frequency of activation *q* of these nuclei and the growth rate *G* . In isothermal conditions, they are constant. In non-isothermal conditions, they are defined as temperature

Crystallization can occur in the form of spherulites, shish-kebabs, or both. The transformed

*dt dt dt dt dt dt*

( ) ( ) (1 ( )) *dt dt <sup>t</sup> dt dt*

( ) () () ( ) ( ) *<sup>g</sup> a c dN t dN t dN t dN t dN t dt dt dt dt dt*

is the number of nuclei per unit volume generated by shear. Two situations are

if ( ) 0 *aAN*

if ( ) 0 *aAN*

*a* and *A*1 are material parameters, eventually thermo-dependent. As a first approximation,

If crystallization proceeds during shear, only the liquid fraction is exposed to shear and the

1/3

  possible, *i.e.*, crystallization occurs after shear or crystallization occurs during shear. If crystallization during shear remains negligible, the number of shear-generated nuclei is:

( )

*aAN*

0

 

*dN*

*dt* 

> *dN dt*

 

 

*t* are the thermo-dependent volume fractions transformed *versus* time into

 

*t* are the actual and extended volume fractions of spherulites, respectively.

*N N N TT* 0 00 exp ( ) 01 0 (20a)

*q q qT T* 0 10 exp ( ) (20b)

*G G GT T* 0 10 exp ( ) (20c)

(21)

(22)

(24a)

(24b)

' /(1 ) (25)

(23)

( ), ( ), ( ), ( ) *Nt N t N t N t acg* are the number of potential, activated, absorbed and generated (by cooling) nuclei per unit volume (or surface) at time *t*, respectively. *q(t)* is the activation frequency of the nuclei at time *t*. The "extended" quantities , *N Na* are related to the actual ones by:

$$N = (1 - \alpha)\bar{N}\tag{11a}$$

$$\frac{dN\_a}{dt} = qN = (1 - \alpha)\frac{d\tilde{N}\_a}{dt} \tag{11b}$$

#### **The System of Differential Equations**

The crystallization process equations are written into a non-linear system of six, **eqs. (12, 13a, 14-17)**, or seven, **eqs. (12, 13b, 14-18)**, differential equations in 2D or 3D conditions, respectively (Haudin & Chenot, 2004):

$$\frac{dN}{dt} = -N\left(q + \frac{1}{1-a}\frac{da}{dt}\right) + (1-a)\frac{dN\_0(T)}{dT}\frac{dT}{dt} \tag{12}$$

$$\frac{d\alpha}{dt} = 2\pi (1 - \alpha) \mathbf{G} (F\tilde{\mathbf{N}}\_a - P) \tag{13a}$$

$$\frac{da}{dt} = 4\pi (1 - a) \mathbf{G} (F^2 \tilde{N}\_a - 2FP + \mathbf{Q})\tag{13b}$$

$$\frac{dN\_a}{dt} = qN\tag{14}$$

$$\frac{d\tilde{N}\_a}{dt} = \frac{qN}{1 - \alpha} \tag{15}$$

$$\frac{dF}{dt} = G \tag{16}$$

$$\frac{dP}{dt} = F \frac{d\tilde{\mathbf{N}}\_a}{dt} = F \frac{qN}{1 - a} \tag{17}$$

$$\frac{d\mathbf{Q}}{dt} = \mathbf{F}^2 \frac{d\tilde{\mathbf{N}}\_a}{dt} = \mathbf{F}^2 \frac{qN}{1 - a} \tag{18}$$

The initial conditions at time *t* = 0 are:

$$N(0) = N\_0$$

$$\alpha(0) = N\_a(0) = \tilde{N}\_a(0) = F(0) = P(0) = Q(0) = 0\tag{19}$$

*F*, *P* and *Q* are three auxiliary functions added to get a first-order ordinary differential system. The model needs three physical parameters, the initial density of potential nuclei *N0*, the frequency of activation *q* of these nuclei and the growth rate *G* . In isothermal conditions, they are constant. In non-isothermal conditions, they are defined as temperature functions, *e.g.*:

$$N\_0 = N\_{00} \exp\left(-N\_{01}(T - T\_0)\right) \tag{20a}$$

$$q = q\_0 \exp\left(-q\_1(T - T\_0)\right) \tag{20b}$$

$$G = G\_0 \exp\left(-G\_1(T - T\_0)\right) \tag{20c}$$

### **General equations for shear-induced crystallization**

Crystallization can occur in the form of spherulites, shish-kebabs, or both. The transformed volume fraction is written as (Haudin et al., 2008):

$$\frac{d\alpha\left(t\right)}{dt} = \frac{d\beta\left(t\right)}{dt} + \frac{d\kappa\left(t\right)}{dt} \tag{21}$$

 *t* and *t* are the thermo-dependent volume fractions transformed *versus* time into spherulites and into shish-kebabs, respectively.

### **Spherulitic Morphology**

656 Thermodynamics – Interaction Studies – Solids, Liquids and Gases

( ), ( ), ( ), ( ) *Nt N t N t N t acg* are the number of potential, activated, absorbed and generated (by cooling) nuclei per unit volume (or surface) at time *t*, respectively. *q(t)* is the activation frequency of the nuclei at time *t*. The "extended" quantities , *N Na* are related to the actual

> *N N* (1 )

(1 ) *a a dN dN qN dt dt*

The crystallization process equations are written into a non-linear system of six, **eqs. (12, 13a, 14-17)**, or seven, **eqs. (12, 13b, 14-18)**, differential equations in 2D or 3D conditions,

> *dN <sup>d</sup> dN T dT N q dt dt dT dt*

*dt* 

> 

<sup>0</sup> <sup>1</sup> ( ) (1 ) <sup>1</sup>

2 (1 ) ( ) *<sup>a</sup> <sup>d</sup> G FN P*

<sup>2</sup> 4 (1 ) ( 2 ) *<sup>a</sup> <sup>d</sup> G F N FP Q dt*

1 *<sup>a</sup> dN qN dt*

*dF <sup>G</sup>*

*<sup>a</sup> dP dN qN F F*

2 2

<sup>0</sup> *N N* (0)

(0) (0) (0) (0) (0) (0) 0

*F*, *P* and *Q* are three auxiliary functions added to get a first-order ordinary differential system. The model needs three physical parameters, the initial density of potential nuclei *N0*,

*<sup>a</sup> dQ dN qN F F*

*dt dt*

*dt dt*

1

1

*N N FPQ a a* (19)

(12)

(13a)

(13b)

*<sup>a</sup> dN qN dt* (14)

(15)

*dt* (16)

(17)

(18)

(11a)

(11b)

ones by:

**The System of Differential Equations** 

respectively (Haudin & Chenot, 2004):

The initial conditions at time *t* = 0 are:

Modification of **eqs. (8)** and **(10a)** gives:

$$\frac{d\beta(t)}{dt} = (1 - \alpha(t)) \frac{d\bar{\beta}(t)}{dt} \tag{22}$$

$$\frac{dN(t)}{dt} = -\frac{dN\_a(t)}{dt} - \frac{dN\_c(t)}{dt} + \frac{dN\_g(t)}{dt} + \frac{dN\_\gamma(t)}{dt} \tag{23}$$

 *t* and *t* are the actual and extended volume fractions of spherulites, respectively. *N t* is the number of nuclei per unit volume generated by shear. Two situations are possible, *i.e.*, crystallization occurs after shear or crystallization occurs during shear. If crystallization during shear remains negligible, the number of shear-generated nuclei is:

$$\frac{dN\_{\gamma}}{dt} = a\dot{\gamma}(A - N\_{\gamma}) \quad \text{if} \quad a\dot{\gamma}(A - N\_{\gamma}) \ge 0 \tag{24a}$$

$$\frac{dN\_\gamma}{dt} = 0 \text{ if } \ a\dot{\gamma}(A - N\_\gamma) \le 0 \tag{24b}$$

*a* and *A*1 are material parameters, eventually thermo-dependent. As a first approximation, *A A*<sup>1</sup> , with the shear rate.

If crystallization proceeds during shear, only the liquid fraction is exposed to shear and the shear rate ' is becoming:

$$
\dot{\gamma}' = \dot{\gamma} \left/ \left( 1 - \alpha \right)^{1/3} \right. \tag{25}
$$

Thermodynamics and Thermokinetics to Model Phase Transitions of Polymers

*dN*

*dt* 

1/3

nuclei for the oriented structure is given as:

activation frequency of the nuclei, *b* and *B1* the material parameters:

*dt* 

1

1/3

*dM <sup>d</sup> M w dt dt*

The initial conditions at time *t* = 0 are:

**Shish-Kebab Morphology** 

 and 

*Mt* , *Ma t* , *Mc t* , *M*

shish-kebab,

over Extended Temperature and Pressure Ranges Under Various Hydrostatic Fluids 659

<sup>1</sup> ( )

 

1 1

*N N FPQ a a* (38)

(40)

*dt*

(42)

*dt* (43)

(44)

(41)

*t* are the numbers of potential, activated, absorbed and

(37a)

(39)

 

(37b)

*aA N*

 

<sup>1</sup> 1/3 1

<sup>0</sup> *N N* (0) (0) (0) (0) (0) (0) (0) 0

> *N* (0) 0

Firstly are introduced the notions of real and extended transformed volume fractions of

 (1 ) *dt dt dt dt*

 *a c dM t dM t dM t dM t dt dt dt dt*

generated (by shear) nuclei per unit volume, respectively. In the same way as for the spherulitic morphology, a set of differential equations can be defined where *w* is the

> 1 1

> >

 

2 (1 ) ( ) *<sup>a</sup> <sup>d</sup> H RM S*

1 1

1 1/3

*<sup>M</sup> M d b B*

*<sup>a</sup> dM wM*

1 *<sup>a</sup> dM wM dt*

( )*t* is the total transformed volume fraction for both spherulitic and oriented phases. Shish-kebabs are modelled as cylinders with an infinite length. The growth rate *H* is deduced from the radius evolution of the cylinder. The general balance of the number of

 

, respectively. Both are related by **eq. (39)**:

*dN N N <sup>d</sup> a A dt dt*

 

 

By defining *N* as the extended number of nuclei per unit volume generated by shear in the total volume, then:

$$\frac{d\tilde{N}\_{\gamma}}{dt} = a\dot{\gamma}'(A\_1\dot{\gamma}' - \tilde{N}\_{\gamma})\tag{26}$$

The number *N*of nuclei generated by shear in the liquid fraction is:

$$N\_{\gamma} = (1 - \alpha)\bar{N}\_{\gamma} \tag{27}$$

Under shear, the activation frequency of the nuclei increases. If the total frequency is the sum of a static component, *st q* , function of temperature, and of a dynamic one, *flow q* , then:

$$
\eta = \eta\_{st} + \eta\_{flow} \tag{28}
$$

*flow q* is given by **eq. (29)** where as a first approximation 2 02 *q q* and 3 *q* is constant.

$$q\_{flow} = q\_2(1 - \exp(-q\_3 \dot{\gamma})) \tag{29}$$

The system of differential equations **(12, 13b, 14-18)** is finally replaced by a system taking the influence of shear into account through the additional unknown *N* and through the dynamic component of the activation frequency *flow q* . Two cases are considered, *i.e.*, crystallization occurs after shear **(37a)** or crystallization occurs under **(37b)** shear.

$$\frac{dN}{dt} = -N\left(q + \frac{1}{1-\alpha}\frac{d\alpha}{dt}\right) + (1-\alpha)\frac{dN\_0(T)}{dT}\frac{dT}{dt} + \frac{dN\_\gamma}{dt} \tag{30}$$

$$\frac{d\mathcal{J}}{dt} = 4\pi (1 - \alpha) \mathbf{G} (\mathbf{F}^2 \tilde{\mathcal{N}}\_a - 2FP + \mathcal{Q}) \tag{31}$$

$$\frac{dN\_a}{dt} = qN\tag{32}$$

$$\frac{d\tilde{N}\_a}{dt} = \frac{qN}{1-a} \tag{33}$$

$$\frac{dF}{dt} = G \tag{34}$$

$$\frac{dP}{dt} = F \frac{d\tilde{N}\_a}{dt} = F \frac{qN}{1 - \alpha} \tag{35}$$

$$\frac{dQ}{dt} = \mathbf{F}^2 \frac{d\tilde{\mathbf{N}}\_a}{dt} = \mathbf{F}^2 \frac{qN}{1 - \alpha} \tag{36}$$

$$\frac{dN\_{\mathcal{V}}}{dt} = a\dot{\boldsymbol{\chi}}(A\_1\dot{\boldsymbol{\chi}} - N\_{\mathcal{\boldsymbol{V}}}) \tag{37a}$$

$$\frac{dN\_\gamma}{dt} = a\dot{\gamma} \left( \left( 1 - \alpha \right)^{1/3} A\_1 \dot{\gamma} - \frac{N\_\gamma}{\left( 1 - \alpha \right)^{1/3}} \right) - \frac{N\_\gamma}{1 - \alpha} \frac{d\alpha}{dt} \tag{37b}$$

The initial conditions at time *t* = 0 are:

$$\begin{array}{c} N(0) = N\_0\\ \alpha(0) = N\_a(0) = \tilde{N}\_a(0) = F(0) = P(0) = Q(0) = 0\\ N\_\gamma(0) = 0 \end{array} \tag{38}$$

#### **Shish-Kebab Morphology**

658 Thermodynamics – Interaction Studies – Solids, Liquids and Gases

*dN*

*dt* 

*flow q* is given by **eq. (29)** where as a first approximation 2 02 *q q*

the influence of shear into account through the additional unknown *N*

crystallization occurs after shear **(37a)** or crystallization occurs under **(37b)** shear.

of nuclei generated by shear in the liquid fraction is:

*N N* (1 ) 

Under shear, the activation frequency of the nuclei increases. If the total frequency is the sum of a static component, *st q* , function of temperature, and of a dynamic one, *flow q* , then:

2 3 (1 exp( )) *flow qq q*

The system of differential equations **(12, 13b, 14-18)** is finally replaced by a system taking

dynamic component of the activation frequency *flow q* . Two cases are considered, *i.e.*,

<sup>0</sup> <sup>1</sup> ( ) (1 ) <sup>1</sup> *dN d dN T dT dN N q dt dt dT dt dt* 

> <sup>2</sup> 4 (1 ) ( 2 ) *<sup>a</sup> <sup>d</sup> G F N FP Q dt*

> > 1 *<sup>a</sup> dN qN dt*

> > > *dF <sup>G</sup>*

*<sup>a</sup> dP dN qN F F*

2 2

*<sup>a</sup> dQ dN qN F F*

*dt dt*

*dt dt*

1

1

 

as the extended number of nuclei per unit volume generated by shear in the

<sup>1</sup> ( )

(26)

*st flow qq q* (28)

(27)

and 3 *q* is constant.

(29)

(31)

*<sup>a</sup> dN qN dt* (32)

*dt* (34)

(35)

(36)

(30)

and through the

(33)

*aA N*

 

By defining *N*

The number *N*

total volume, then:

Firstly are introduced the notions of real and extended transformed volume fractions of shish-kebab, and , respectively. Both are related by **eq. (39)**:

$$\frac{d\kappa\left(t\right)}{dt} = (1 - \alpha)\frac{d\tilde{\kappa}\left(t\right)}{dt} \tag{39}$$

( )*t* is the total transformed volume fraction for both spherulitic and oriented phases. Shish-kebabs are modelled as cylinders with an infinite length. The growth rate *H* is deduced from the radius evolution of the cylinder. The general balance of the number of nuclei for the oriented structure is given as:

$$\frac{dM\left(t\right)}{dt} = -\frac{dM\_a\left(t\right)}{dt} - \frac{dM\_c\left(t\right)}{dt} + \frac{dM\_\gamma\left(t\right)}{dt} \tag{40}$$

*Mt* , *Ma t* , *Mc t* , *M t* are the numbers of potential, activated, absorbed and generated (by shear) nuclei per unit volume, respectively. In the same way as for the spherulitic morphology, a set of differential equations can be defined where *w* is the activation frequency of the nuclei, *b* and *B1* the material parameters:

$$\begin{split} \frac{dM}{dt} &= -M \left( w + \frac{1}{1 - \alpha} \frac{da}{dt} \right) \\ &+ b\dot{\gamma} \left( \left( 1 - \alpha \right)^{1/3} B\_1 \dot{\gamma} - \frac{M}{\left( 1 - \alpha \right)^{1/3}} \right) - \frac{M}{1 - \alpha} \frac{da}{dt} \end{split} \tag{41}$$

$$\frac{d\kappa}{dt} = 2\pi (1 - a) H (R\tilde{M}\_a - S) \tag{42}$$

$$\frac{dM\_a}{dt} = wM\tag{43}$$

$$\frac{d\tilde{M}\_a}{dt} = \frac{wM}{1 - \alpha} \tag{44}$$

$$\frac{d\mathcal{R}}{dt} = H\tag{45}$$

Thermodynamics and Thermokinetics to Model Phase Transitions of Polymers

over Extended Temperature and Pressure Ranges Under Various Hydrostatic Fluids 661

conditions **(eqs. 20a-c)**. The exponential temperature evolution of the three key parameters *N*0, *q*, *G* is possibly calculated from the values of the physical parameters obtained in three different ways: firstly, an approximate physical analysis with direct determination from the experiments *(APA)*, secondly, the use of the Genetic Algorithm method for an optimization based on several experiments (at least 5) done with the same specimen, thirdly, an optimization based on several experiments (at least 8) involving different polymer samples for which an important dispersion of the number of nuclei is observed (Haudin et al., 2008, Boyer et al., 2009). These sets of optimized temperature functions made it possible to validate the mathematical model in the 2D version, as illustrated in **Fig. 5.a-b-inserts**. The selected polymer is a polypropylene that is considered as a 'model material' because of its

aptitude to crystallize with well-defined spherulitic entities in quiescent conditions.

kinetics under low shear with enough accuracy, when the entities are spherulitic.

d *N*/dt ( *N*

structure have to be optimized.

not exceed 19%.

Shear-induced crystallization, with a spherulitic morphology, gives access to the function

time and to the shear dependence of the activation frequency for different relatively low shear rates (up to 20 s-1). A set of seven optimized parameters are identifiable: *N*00, *q*0, *G*<sup>0</sup> from quiescent isothermal crystallization, and ( 02 3 1 *q qAa* ,, , ) from isothermal shear-induced crystallization. The agreement between experiment and theory is better for higher shear rates associated with a shorter total time of crystallization. The mean square error does not exceed 12 %, the average mean square error for 5 s-1 is equal to 6.7 %. The agreement between experiment and theory is less satisfactory for the number of spherulites, the mean square error reaches 25 %. Then, the new model is able to predict the overall crystallization

Shear-induced crystallization, with both a spherulitic and an oriented morphology, is a different task. High shear rates (from 75 s-1) enhance all the kinetics (nucleation, growth, overall kinetics) and lead to the formation of micron-size fibrillar (thread-like) structures immediately after shear, followed by the appearance of unoriented spherulitic structures at the later stages **(Fig. 6insert)**. The determination of the parameters for this double crystallization becomes a complicated task for a twofold reason: the quantitative data for both oriented and spherulitic structures are not available at high shear rate, and the double crystallization kinetics model requires to additionally determine the four parameters ( <sup>1</sup> *wHB b* ,,, ). So, optimization is based only on the evolution of the total transformed volume fraction **(eq. 21)**. Parameters characterizing quiescent crystallization ( 00 0 0 *N qG* , , ) and shearinduced crystallization with the spherulitic morphology ( 02 3 1 *q qAa* ,, , ) are taken from the previous 'smooth' analysis, so that four parameters ( <sup>1</sup> *wHB b* ,,, ) characterizing the oriented

**Fig. 6.** gathers the experimental and theoretical variations of the total transformed volume fraction for different shear rates. At the beginning, the experimental overall kinetics is faster than the calculated one most probably because the influence of shear rate on the activation frequency of the oriented structure is not taken into account. Since with higher shear rate thinner samples (~30 µm at 150 s-1) are used, and since numerically the growth of entities is considered as three dimensional, the condition of 3D experiment seems not perfectly respected and the experiments give a slower evolution at the end. The mean square errors between numerical and experimental evolutions of the total transformed volume fraction do

is the number of nuclei per unit volume generated by shear **(eq. 23)**) *versus*

$$\frac{dS}{dt} = R\frac{d\tilde{M}\_a}{dt} = R\frac{wM}{1-a} \tag{46}$$

*F*, *P, Q*, *R* and *S* are five auxiliary functions giving a first-order ordinary differential system. The initial conditions at time *t* = 0 are:

$$M(0) = M\_0$$

$$\kappa(0) = M\_a(0) = \tilde{M}\_a(0) = R(0) = S(0) = 0\tag{47}$$

### **Inverse resolution method for a system of differential equations**

The crystallization, and especially the nucleation stage, is by nature a statistical phenomenon with large discrepancies between the sets of experimental data. The analytical extraction of the relevant crystallization parameters must be then considered as a multicriteria optimization problem. As such the Genetic Algorithm Inverse Method is considered. The Genetic Algorithm Inverse Method is a stochastic optimization method inspired from the Darwin theory of nature survival (Paszkowicz, 2009). In the present work, the Genetic Algorithm developed by Carroll (Carroll, "FORTRAN Genetic Algorithm Front-End Driver Code"*,* site: *http://cuaerospace.com/ga*) is used (Smirnova et al., 2007; Haudin et al., 2008). The vector of solutions is represented by a parameter *Z*. In quiescent crystallization **(eqs. 20a-c)**, 00 01 0 1 0 1 *Z N N qqGG* [ , ,,, , ] with *N00*, *N01*, *q0*, *q1*, *G0*, *G1* the parameters of non-isothermal crystallization for a spherulitic morphology. In shear-induced crystallization, 00 01 0 1 02 3 0 1 0 1 1 *Z N N q q q q G G M wHA aB b* [ , , , , , , , , , , , ,, ,] with ( 02 3 1 *q qAa* ,, , ) the parameters of shear-induced crystallization for a spherulitic morphology **(eqs. 26,29)** and ( 0 1 *M* ,, , , *wHB b* ) the parameters of shear-induced crystallization for an oriented, like shish-kebab, morphology **(eqs. 41,43,45,47)**.

The optimization is applied to the experimental evolution of the overall kinetics coupled with one kinetic parameter at a lower scale, the number of entities (density of nucleation *Na(t)*). The system of differential equations is solved separately for each experimental set and gives the evolutions of *(t)* and of the nuclei density defining a corresponding data file. The optimization function *Qtotal* is expressed as the sum of the mean square errors of the transformed volume fraction *Q<sup>α</sup>* and of the number of entities *QNa* .

#### **Model-experiment-optimization confrontation**

The structure development parameters are identifiable by using the optical properties of the crystallizing entities. The experimental investigations and their analysis are done thanks to crossed-polarized optical microscopy (POM) (Magill, 1962, 1962, 2001) coupled with optically transparent hot stages, a home-made sliding plate shearing device and a rotating parallel plate shearing device (*e.g.*, Linkam). Data accessible directly are: *i)* the evolution of the transformed fraction *(t)*, and the number of activated nuclei *Na(t), ii)* the approximate values of the initial number of potential nuclei *N*0*(T),* activation frequency *q(T),* and growth rate *G(T)* for isothermal conditions and their functions of temperature for non-isothermal

*dR <sup>H</sup>*

*<sup>a</sup> dS dM wM R R*

*F*, *P, Q*, *R* and *S* are five auxiliary functions giving a first-order ordinary differential system.

<sup>0</sup> *M*(0) *M*

The crystallization, and especially the nucleation stage, is by nature a statistical phenomenon with large discrepancies between the sets of experimental data. The analytical extraction of the relevant crystallization parameters must be then considered as a multicriteria optimization problem. As such the Genetic Algorithm Inverse Method is considered. The Genetic Algorithm Inverse Method is a stochastic optimization method inspired from the Darwin theory of nature survival (Paszkowicz, 2009). In the present work, the Genetic Algorithm developed by Carroll (Carroll, "FORTRAN Genetic Algorithm Front-End Driver Code"*,* site: *http://cuaerospace.com/ga*) is used (Smirnova et al., 2007; Haudin et al., 2008). The vector of solutions is represented by a parameter *Z*. In quiescent crystallization **(eqs. 20a-c)**, 00 01 0 1 0 1 *Z N N qqGG* [ , ,,, , ] with *N00*, *N01*, *q0*, *q1*, *G0*, *G1* the parameters of non-isothermal crystallization for a spherulitic morphology. In shear-induced crystallization, 00 01 0 1 02 3 0 1 0 1 1 *Z N N q q q q G G M wHA aB b* [ , , , , , , , , , , , ,, ,] with ( 02 3 1 *q qAa* ,, , ) the parameters of shear-induced crystallization for a spherulitic morphology **(eqs. 26,29)** and ( 0 1 *M* ,, , , *wHB b* ) the parameters of shear-induced crystallization for an oriented, like shish-kebab,

The optimization is applied to the experimental evolution of the overall kinetics coupled with one kinetic parameter at a lower scale, the number of entities (density of nucleation *Na(t)*). The system of differential equations is solved separately for each experimental set

The optimization function *Qtotal* is expressed as the sum of the mean square errors of the

The structure development parameters are identifiable by using the optical properties of the crystallizing entities. The experimental investigations and their analysis are done thanks to crossed-polarized optical microscopy (POM) (Magill, 1962, 1962, 2001) coupled with optically transparent hot stages, a home-made sliding plate shearing device and a rotating parallel plate shearing device (*e.g.*, Linkam). Data accessible directly are: *i)* the evolution of the transformed fraction *(t)*, and the number of activated nuclei *Na(t), ii)* the approximate values of the initial number of potential nuclei *N*0*(T),* activation frequency *q(T),* and growth rate *G(T)* for isothermal conditions and their functions of temperature for non-isothermal

*dt dt*

 (0) (0) (0) (0) (0) 0 

**Inverse resolution method for a system of differential equations** 

The initial conditions at time *t* = 0 are:

morphology **(eqs. 41,43,45,47)**.

and gives the evolutions of

**Model-experiment-optimization confrontation** 

transformed volume fraction *Q<sup>α</sup>* and of the number of entities *QNa* .

1

*M M RS a a* (47)

*(t)* and of the nuclei density defining a corresponding data file.

*dt* (45)

(46)

conditions **(eqs. 20a-c)**. The exponential temperature evolution of the three key parameters *N*0, *q*, *G* is possibly calculated from the values of the physical parameters obtained in three different ways: firstly, an approximate physical analysis with direct determination from the experiments *(APA)*, secondly, the use of the Genetic Algorithm method for an optimization based on several experiments (at least 5) done with the same specimen, thirdly, an optimization based on several experiments (at least 8) involving different polymer samples for which an important dispersion of the number of nuclei is observed (Haudin et al., 2008, Boyer et al., 2009). These sets of optimized temperature functions made it possible to validate the mathematical model in the 2D version, as illustrated in **Fig. 5.a-b-inserts**. The selected polymer is a polypropylene that is considered as a 'model material' because of its aptitude to crystallize with well-defined spherulitic entities in quiescent conditions.

Shear-induced crystallization, with a spherulitic morphology, gives access to the function d *N* /dt ( *N* is the number of nuclei per unit volume generated by shear **(eq. 23)**) *versus* time and to the shear dependence of the activation frequency for different relatively low shear rates (up to 20 s-1). A set of seven optimized parameters are identifiable: *N*00, *q*0, *G*<sup>0</sup> from quiescent isothermal crystallization, and ( 02 3 1 *q qAa* ,, , ) from isothermal shear-induced crystallization. The agreement between experiment and theory is better for higher shear rates associated with a shorter total time of crystallization. The mean square error does not exceed 12 %, the average mean square error for 5 s-1 is equal to 6.7 %. The agreement between experiment and theory is less satisfactory for the number of spherulites, the mean square error reaches 25 %. Then, the new model is able to predict the overall crystallization kinetics under low shear with enough accuracy, when the entities are spherulitic.

Shear-induced crystallization, with both a spherulitic and an oriented morphology, is a different task. High shear rates (from 75 s-1) enhance all the kinetics (nucleation, growth, overall kinetics) and lead to the formation of micron-size fibrillar (thread-like) structures immediately after shear, followed by the appearance of unoriented spherulitic structures at the later stages **(Fig. 6insert)**. The determination of the parameters for this double crystallization becomes a complicated task for a twofold reason: the quantitative data for both oriented and spherulitic structures are not available at high shear rate, and the double crystallization kinetics model requires to additionally determine the four parameters ( <sup>1</sup> *wHB b* ,,, ). So, optimization is based only on the evolution of the total transformed volume fraction **(eq. 21)**. Parameters characterizing quiescent crystallization ( 00 0 0 *N qG* , , ) and shearinduced crystallization with the spherulitic morphology ( 02 3 1 *q qAa* ,, , ) are taken from the previous 'smooth' analysis, so that four parameters ( <sup>1</sup> *wHB b* ,,, ) characterizing the oriented structure have to be optimized.

**Fig. 6.** gathers the experimental and theoretical variations of the total transformed volume fraction for different shear rates. At the beginning, the experimental overall kinetics is faster than the calculated one most probably because the influence of shear rate on the activation frequency of the oriented structure is not taken into account. Since with higher shear rate thinner samples (~30 µm at 150 s-1) are used, and since numerically the growth of entities is considered as three dimensional, the condition of 3D experiment seems not perfectly respected and the experiments give a slower evolution at the end. The mean square errors between numerical and experimental evolutions of the total transformed volume fraction do not exceed 19%.

Thermodynamics and Thermokinetics to Model Phase Transitions of Polymers

optimization of experimental and calculation procedures.

support in the development of «CRISTAPRESS» project.

10.1016/0022.5096(93)900 13-6

between polymer and metal transformations.

physico-chemical nature.

**5. Acknowledgments** 

**6. References** 

18.1141

respect.

over Extended Temperature and Pressure Ranges Under Various Hydrostatic Fluids 663

thermokinetics explicitly applied over extended temperature and pressure ranges, particularly under hydrostatic stress generated by pressure transmitting fluids of different

Clearly, such an approach rests not only on the conjunction of pertinent coupled experimental techniques and of robust theoretical models, but also on the consistency and

Illustration is made with selected examples like molten and solid polymers in interaction with various light molecular weight solvents, essentially gases. Data obtained allow evaluating specific thermal, chemical, mechanical behaviours coupled with sorption effect during solid to melt as well as crystallization transitions, creating smart and noble hybrid metal-polymer composites and re-visiting kinetic models taking into account similarities

This work generates a solid platform for polymer science, addressing formulation, processing, long-term utilization of end-products with specific performances controlled via a clear conception of greatly different size scales, altogether with an environmental aware

The principal author, Séverine A.E. Boyer, wishes to address her grateful acknowledgments for financial supports from Centre National de la Recherche Scientifique CNRS (France) ; Institut Français du Pétrole IFP (France) with Mrs. Marie-Hélène Klopffer and Mr. Joseph Martin ; Core Research for Evolutional Science and Technology - Japan Science and Technology Agency CREST-JST (Japan) with Prof. Tomokazu Iyoda (Tokyo Institute of Technology TIT, Japan) ; ARMINES-CARNOT-MINES ParisTech (France) ; Conseil Régional de Provence-Alpes-Côte d'Azur and Conseil Général des Alpes-Maritimes (France) for

Séverine A.E. Boyer wishes to expresses her acknowledgements to Intech for selectionning the current research that has been recognized as valuable and relevant to the given theme.

Ahzi, S.; Parks, D.M.; Argon, A.S. (1991). Modeling of deformation textures evolution in

Arruda, E.M.; Boyce, M.C. (1993). A three-dimensional constitutive model for the large

Asta, M.; Beckermann, C.; Karma, A.; Kurz, W.; Napolitano, R.; Plapp, M.; Purdy, G.;

semi-crystalline polymers. *Textures and Microstructures,* Vol.14-18, No1, (January 1991), pp. 1141-1146, ISSN 1687-5397(print) 1687-5400(web); doi: 10.1155/TSM.14-

stretch behaviour of rubber elastic materials. *Journal of the Mechanics and Physics of Solids,* Vol.41, No2, (February 1993), pp. 389-412, ISSN 0022-5096; doi:

Rappaz, M.; Trivedi, R. (2009). Solidification microstructures and solid-state parallels: Recent developments, future directions. *Acta Materialia*, Vol.57, No4, (February 2009), pp. 941-971; ISSN 1359-6454; doi: 10.1016/j.actamat.2008.10.020

Fig. 5. Experimental (symbols) and numerically predicted (lines) of **(a)** the overall kinetics and **(b)** the number of activated nuclei *vs.* temperature at constant cooling-rate. The inserts illustrate the events at 10, 3 and 1 °C.min-1. Sample: iPP in 2D (5 μm-thick layer).

Fig. 6. Experimental (dashed-line curves) and numerically predicted (solid curves) total overall kinetics, *i.e.*, spherulitic and oriented structures, *vs.* time in constant shear, *T* = 132 °C. The insert illustrates the event at 150 s–1. Sample: iPP in 2-3D (~30 μm-thick layer).

The present differential system, based on the nucleation and growth phenomena of polymer crystallization, is adopted to describe the crystalline morphology evolution *versus* thermomechanical constraints. It has been implemented into a 3D injection-moulding software. The implementation allows us to estimate its feasibility in complex forming conditions, *i.e.*, anisothermal flow-induced crystallization, and to test the sensitivity to the accuracy of the values of the parameters determined by the Genetic Algorithm Inverse Method.

### **4. Conclusion**

Fundamental understanding of the inherent links between multiscale polymer pattern and polymer behaviour/performance is firmly anchored on rigorous thermodynamics and thermokinetics explicitly applied over extended temperature and pressure ranges, particularly under hydrostatic stress generated by pressure transmitting fluids of different physico-chemical nature.

Clearly, such an approach rests not only on the conjunction of pertinent coupled experimental techniques and of robust theoretical models, but also on the consistency and optimization of experimental and calculation procedures.

Illustration is made with selected examples like molten and solid polymers in interaction with various light molecular weight solvents, essentially gases. Data obtained allow evaluating specific thermal, chemical, mechanical behaviours coupled with sorption effect during solid to melt as well as crystallization transitions, creating smart and noble hybrid metal-polymer composites and re-visiting kinetic models taking into account similarities between polymer and metal transformations.

This work generates a solid platform for polymer science, addressing formulation, processing, long-term utilization of end-products with specific performances controlled via a clear conception of greatly different size scales, altogether with an environmental aware respect.
