Simulating polymer crystallization in thin films ...

European Polymer Journal 73 (2015) 1–16

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European Polymer Journal journal homepage: www.elsevier.com/locate/europolj

Simulating polymer crystallization in thin films: Numerical and analytical methods Audrey Durin a, Jean-Loup Chenot b, Jean-Marc Haudin c, Nicolas Boyard a,⇑, Jean-Luc Bailleul a a

Université de Nantes, Laboratoire de Thermocinétique de Nantes (LTN), UMR CNRS 6607, Nantes, France TRANSVALOR S.A., Sophia Antipolis, France c MINES ParisTech, PSL – Research University, Centre de Mise en Forme des Matériaux (CEMEF), UMR CNRS 7635, Sophia Antipolis, France b

a r t i c l e

i n f o

Article history: Received 27 July 2015 Received in revised form 18 September 2015 Accepted 1 October 2015 Available online 3 October 2015 Keywords: Crystallization Simulation Thin film Transcrystallization

a b s t r a c t In this paper, a general numerical method to simulate polymer crystallization under various conditions is proposed. This method is first validated comparing its predictions with well-validated analytical models in infinite volumes. Then, it is compared to Billon et al. validated model for thin films, without or in presence of transcrystallinity on the films surfaces. It is also compared with Chenot et al. model for thin films, proposed in a conference in 2005 and never yet compared with other methods. Finally, it is also compared with an extension of this model for the transcrystalline case. These models are valid for general nucleation cases (not only sporadic or instantaneous), and can be used for any thermal conditions. All the numerical and analytical results are consistent, except in a case which is shown to be out of the validity domain of the transcrystalline case extension of Chenot et al. model. Ó 2015 Elsevier Ltd. All rights reserved.

1. Introduction Many analytical and numerical modeling methods have been proposed to simulate the crystallization of polymers under different conditions and configurations, as in thin films or for non-isothermal conditions. Such simulations are very useful to understand DSC measurements, to predict the final microstructure and the physical properties of polymer parts or to compute temperature evolution in process simulations [1–6]. 1.1. Global kinetic theories Polymer crystallization can be described by overall crystallization kinetic theories. These models are based on a general description of the transformation of the polymers from the liquid to the solid state. From the molten state, many polymers develop a semi-crystalline spherulitic morphology. In this kind of microstructure, the crystallization starts at particular points of the volume: the nuclei. These are the starting points of mono-crystalline lamellae growing and multiplying in all directions of space, including between them some amorphous polymer. The combination of the crystalline lamellae and the amorphous polymer defines, in three dimensions, a spherical semi-crystalline entity: the spherulite. Spherulites tend to occupy the whole volume without covering each over. This description is the basic principle of the overall kinetic theories, neglecting the possibility of a secondary crystallization of the amorphous part. The studied quantity is generally the trans⇑ Corresponding author. E-mail address: [email protected] (N. Boyard). http://dx.doi.org/10.1016/j.eurpolymj.2015.10.001 0014-3057/Ó 2015 Elsevier Ltd. All rights reserved.

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A. Durin et al. / European Polymer Journal 73 (2015) 1–16

formed fraction a, defined as the ratio of the volume occupied by spherulites to the total volume. This quantity is 0 at the beginning and reaches 1 at the end of the transformation. Thus, it has to be differentiated from the crystalline fraction, as this one will never be equal to 1. One of the most used among the overall kinetics theories is Avrami’s theory [7–9]. It is based on a number of hypotheses concerning nucleation and growth. There are pre-existing sites in the molten liquid, called potential nuclei, with an initial density N 0 (number per unit volume) function of temperature [10,11]. Each potential nucleus in a non-transformed area can be activated, i:e. generate an entity (e.g., a spherulite). On the other hand, a potential nucleus that has been absorbed by an entity before getting activated can no longer be activated. The activation frequency q is defined as the probability per unit time for any nucleus to be activated. The new entity generated by this activation immediately starts to grow. The spherulite growth rate G is the lamella growth rate. The impingement of two growing spherulites stops their growth at the contact point. G and q are only functions of temperature, as the crystalline growth is governed by secondary nucleation phenomena, and not by diffusion phenomena [12]. Therefore, their time dependence is only through the time variation of temperature. In the limiting case of instantaneous nucleation, all the potential nuclei are instantaneously activated and the number of spherulites per unit volume remains constant and equal to N 0 . In the other limiting case of isothermal spoa radic in time nucleation, the number of activated nuclei per unit time and unit volume is constant: dN ¼ qN ¼ cte (with N the dt number of remaining potential nuclei per unit volume). That implies that q is small and N 0 is high. Sometimes, the presence of surfaces or inclusions will generate a fast and massive local nucleation. In this case, the numerous spherulites limit the growth of their neighbors and then oblige them to grow along one direction perpendicular to the surface on which they were generated. Then, they form a crystallization front propagating at the spherulite growth rate G. This phenomenon is called transcrystallinity and can lead to very different crystallization kinetics from those obtained in bulk infinite volume. Especially, it can play an important role in DSC measurement results, and thus has to be taken into account in their interpretations [1]. Sample geometry, especially volume restriction, as in thin films, can also lead to changes in crystallization kinetics, and models have to be adapted to the sample geometry to perform accurate predictions [13]. Furthermore, it has been shown that transcrystallinity and volume restriction effects are tightly connected [2,14]. If Avrami’s theory [7–9] is very popular, there are other well-known theories [15–18], all of them based on statistical approaches allowing the calculation of the average transformed volume fraction a. For all of these models, there is only one kind of nuclei, uniformly distributed in an infinite volume. All the entities are growing at the same rate, and are spheres in 3D, disks in 2D and rods in 1D. A critical review of overall crystallization kinetics theories has been done by Piorkowska et al. [19]. It has been demonstrated that Evans’ and Avrami’s approaches are equivalent [20], whereas Tobin’s one is generally recognized as incorrect [19]. Based on the hypothesis of an instantaneous or a sporadic in time nucleation, simplified formulations are frequently used either in isothermal (Avrami–Evans’ simplified model) or in non-isothermal (Nakamura’s and Ozawa’s models) conditions. Their interest is to replace the three parameters (N 0 ; q; GÞ by two more easily measurable ones [7–9,21–24].

1.2. Extensions to non-isothermal and thin films cases In their general form, the overall crystallization kinetics theories can be used or extended to treat non-isothermal crystallization, to consider a non-uniform distribution or several populations of nuclei [25] or to include the influence of parameters other than temperature (shear rate, for example) on the kinetic parameters (N 0 ; G; qÞ. An important extension has been the modelling of the crystallization in thin films [13], with the possibility to consider transcrystallinity on the film surfaces [26] and non-isothermal cases [2,20,25]. In these models, transcrystallinity is introduced through a second set of parameters (number of potential nuclei per surface unit N s (m2) and activation frequency s, equivalent to N 0 (m3) and q in the volume, G remaining the same), describing the behavior of nuclei generated on surfaces. These parameters are chosen so that the nucleation on the surfaces is faster than in the volume, in agreement with experimental observations. Some experimental methods, based on these simulations, have been proposed to find the overall kinetics parameters [11,27]. Some keys for the interpretation of DSC experiments have been highlighted using a thin film simulation method [1,14]. Piorkowska has also proposed another approach of analytical simulation, based on the spherulitic pattern, which can be characterized by the distance between the center of a spherulite and its boundary at the interface with another spherulite. This approach has been proposed for instantaneous and sporadic in time activations, for both isothermal and non-isothermal conditions, in infinite and finite volumes, and allows focusing on the structure weaknesses that are the spherulites boundaries [4–6,28,29]. In the present study, the focus is put on the model proposed by Chenot et al. in 2005 [30], which is an extension for thin films of the model proposed by Haudin and Chenot [10], and which till now, has been only presented in an international conference. Haudin–Chenot’s model allows computing the evolution of the transformed fraction, the number of activated nuclei and the distribution of spherulite sizes during the crystallization, whatever the temperature conditions and for any type of nucleation i:e. not only for instantaneous or sporadic in time nucleation. In the case of non-isothermal conditions, most of the classical models neglect the possibility for new potential nuclei to appear during cooling. Monasse et al. [11], who have experimentally validated the model, proved that these appearances have an influence on the crystallization kinetics, justifying the need to take them into account. This model can also consider the influence of shear rate on the kinetic parameters [31]. In 2005, Chenot et al. [30] proposed a generalization of this model to thin film configuration. However, this


3

model has never been confronted to other simulation methods. An extension of this last model for thin films with transcrystallinity on the surfaces has recently been developed and will be presented in this paper. Many of the previous studies used numerical methods to validate their models [32–34], conjointly with experiments. In this study, a pixel coloring method is proposed [35]. This kind of method has been previously used by Ruan et al. for nonisothermal polymer crystallization in 3D, and for short-fiber reinforced polymer crystallization in 2D, in isothermal and nonisothermal conditions [36–38]. The method will be first validated by comparing its results to well-established models (Evans’ generalized (Billon) and Haudin–Chenot models) for isothermal and non-isothermal conditions in an infinite volume [10,25]. Then, the Chenot et al. models (without or in presence of transcrystallinity) and this method will be confronted to Billon et al. analytical model for thin films [20]. 2. Analytical models for the validation of the numerical method In this section, classical models will be presented, both for infinite bulk volume and thin film cases. These models will later be used to validate our numerical method and Chenot et al. analytical models for thin films, without or in presence of transcrystallinity. 2.1. Evans’ generalized model [25] A generalized version of Evans’ model, for any type of thermal conditions, has been proposed by Billon and Haudin [25]. This model is based on the fact that the untransformed volume fraction is the same in the actual case of crystallization and in a fictitious case where the spherulites appear and grow unconstrained by spherulites around them. It is equivalent to Avrami’s model. On an average, at time t, any point in the volume has been reached by EðtÞ spherulites, with E the mathematical expectation of the Poisson probability law P n ðtÞ for any point to has been reached by n spherulites at time t. If the nuclei are homogeneously distributed, and under the isokinetic assumption i:e. if G/q is constant in the studied range of temperature, this finally leads to:

8 aðtÞ ¼ 1 exp ½EðtÞ > > > h i3 < EðtÞ ¼ 8pN0 Gq f ðgðtÞÞ > > > 2 3 : f ðgðtÞÞ ¼ exp ½gðtÞ 1 þ gðtÞ g 2ðtÞ þ g 6ðtÞ

ð1Þ

where gðtÞ is defined as:

gðtÞ ¼

Z

t

qðuÞdu

ð2Þ

0

2.2. Billon et al. model [20] Billon et al. also proposed an extension of Evans’ model to thin film crystallization, without or in presence of transcrystallinity on the surfaces [20,25,26]. The film thickness is h. The location in the film is measured using the distance X a from the half distance between surfaces 2h 6 X a 6 2h . In the case where there is symmetric transcrystallinity on both surfaces, these nuclei are considered as a new family of nuclei, with parameters N s (m2) and s instead of N 0 (m3) and q, and:

EðX a ; t Þ ¼ Ev ðX a ; t Þ þ Es ðX a ; t Þ

ð3Þ

with Ev and Es the mathematical expectations in the volume and on surfaces respectively. The transformed fraction is locally equal to

al ðX a ; tÞ ¼ 1 exp ðEðX a ; tÞÞ

ð4Þ

and the global transformed fraction is

aðtÞ ¼

2 h

Z

h 2

al ðX a ; tÞdX a

ð5Þ

0

The isokinetic hypothesis is done and the two types of nuclei are supposed homogeneously distributed in the film and on the surfaces, respectively. The values of Ev depending on the location and on the value of the maximal radius of the spherulites at time t, defined as

q~ max ðtÞ ¼

Z

t

GðuÞdu; 0

ð6Þ

4

A. Durin et al. / European Polymer Journal 73 (2015) 1–16 Table 1 Ev depending on the location and on the value of the maximal radius of the spherulites at time t. Ev ðX a ; t Þ=8p

q~ max ðtÞ 6 h2 h ~ 2 6 qmax ðtÞ 6 h ~ max ðtÞ h6q

cðX a Þ ¼ Gq h2 jX a j gx ðX a ; t Þ ¼ gðtÞ cðX a Þ

h i3 G q

N0

f ðgðtÞÞ (Eqs. (1) and (2)) Xa Þ 3 gx ðX a ; tÞ f ðgðtÞÞ þ cðX4 a Þ 12 f ðgx ðX a ; t ÞÞ cð24 0 0 ðX Þ Xa Þ 3 a gx ðX a ; tÞ þ c ð4X a Þ 12 f g0x ðX a ; t Þ c 24 g03 f ðgðtÞÞ þ cðX4 a Þ 12 f ðgx ðX a ; t ÞÞ cð24 x ðX a ; t Þ c0 ðX a Þ ¼ Gq 2h þ jX a j g0x ðX a ; t Þ ¼ gðtÞ c0 ðX a Þ

are given in Table 1. If there is no transcristallinity on the surfaces, Es ¼ 0. Else, its value depending on the location and on the value of the maximal radius of the spherulites at time t is given in Table 2. 2.3. Haudin–Chenot differential model [10] Haudin and Chenot proposed another approach, generalizing Avrami’s model and writing it in a differential form [7–10]. The basic assumptions and the general framework of the theory are kept, but the mathematical derivation is quite different. The following notations are used: a is the transformed fraction, N is the number of remaining potential nuclei per unit volume, N a is the number of activated nuclei per unit volume. Avrami considers the fictitious case where impingement is disregarded and absorbed nuclei can be still activated [7–9]. e N e a which are related to the actual ones by the following equations: ~ ; N; This introduces the ‘‘extended” quantities a

~ da da ¼ ð1 aÞ dt dt

ð7Þ

e N ¼ ð1 aÞ N

ð8Þ

ea dNa dN ¼ ð1 aÞ dt dt

ð9Þ

The fictitious transformed fraction and its time derivative can be calculated using:

a~ ¼

4p 3

Z

~ da ¼ 4p dt

t

q~ 3 ðt; sÞ

0

Z

t

ea dN ds ds

q~ 2 ðt; sÞGðtÞ

0

ð10Þ

ea dN ds ds

ð11Þ

~ ðt; sÞ is the fictitious radius at time t of an spherical entity created at time s: where q

q~ ðt; sÞ ¼

Z

s

t

ð12Þ

GðuÞdu

In the general case, when the physical parameters q; G and N 0 depend on the temperature T and the process is not isothermal, the equations must be numerically integrated. To facilitate the numerical treatment, a set of 7 differential equations is established:

Table 2 Es depending on the location and on the value of the maximal radius of the spherulites at time t. Es ðX a ; t Þ ~ max ðtÞ 6 2h q

~ max ðtÞ q ~ max ðtÞ jX a j 2 q jX a j
2h q ~ max ðtÞ 2h jX a j < q

~ max ðtÞ h2 jX a j P q cs ðX a Þ ¼ Gs h2 jX a j gsx ðX a ; tÞ ¼ gs ðtÞ cs ðX a Þ R gs ðtÞ ¼ 0t sðuÞdu

0

i 2 h 2 2p Gs N s ðcs ðX a Þ 1Þg gsx ðX a ; tÞ þ 12 cs ðX a Þgsx ðX a ; tÞ i 2 h 2 2 2p Gs N s ðcs ðX a Þ 1Þg gsx ðX a ; tÞ þ 12 csðX a Þ gsx ðX a ; t Þ þ c0s ðX a Þ 1 g g0sx ðX a ; tÞ þ 12 c0s ðX a Þg0sx ðX a ; tÞ i 2 h 2 2p Gs N s ðcs ðX a Þ 1Þg gsx ðX a ; tÞ þ 12 cs ðX a Þgsx ðX a ; tÞ c0s ðX a Þ ¼ Gs 2h þ jX a j g0sx ðX a ; t Þ ¼ gs ðtÞ c0s ðX a Þ g ðgÞ ¼ exp ðgÞ 1 þ g g2

2


5

dN 1 da dN0 ðTÞ dT ¼ N q þ þ ð1 aÞ dt 1 a dt dT dt

ð13Þ

da e a 2FP þ Q Þ ¼ 4pð1 aÞGðF 2 N dt

ð14Þ

dNa ¼ qN dt

ð15Þ

ea dN qN ¼ 1a dt

ð16Þ

dF ¼G dt

ð17Þ

ea dP dN ¼F dt dt

ð18Þ

ea dQ dN ¼ F2 dt dt

ð19Þ

The initial conditions at time t = 0, with temperature T 0 , are:

Nð0Þ ¼ N 0 ðT 0 Þ;

e a ð0Þ ¼ Fð0Þ ¼ Pð0Þ ¼ Q ð0Þ ¼ 0 að0Þ ¼ Na ð0Þ ¼ N

ð20Þ

e a and the three auxiliary functions F; P and Q are added to get a first-order ordinary differThe main variables are N; a; N a ; N ential system. These non-linear differential equations can be numerically integrated using a classical integration scheme (Runge–Kutta, for example). It is worth noticing that, from the number of spherulites per unit volume N a and the transformed fraction a, one can compute an approximation of the mean spherulite radius value through the expression: qffiffiffiffiffiffiffiffiffiffiffiffi q ðtÞ ¼ 3 4p3aNðtÞ . a ðtÞ 3. Chenot et al. differential models From Haudin and Chenot approach, Chenot et al. proposed an extension to films crystallization [30] and then to films with transcrystallinity. 3.1. Finite thickness When the crystallization takes place between two parallel plates separated by a small thickness h, corrections must be introduced. Escleine et al. and Billon et al. have analyzed this situation, from Avrami’s model in the isothermal case and from Evans’ model in non-isothermal case with or without transcrystallinity [2,13,20,25,26]. Chenot et al. treatment [30] is more general than Escleine et al. one, but it is initially restricted to the case where the maximum extended radius R ~ max ðtÞ ¼ FðtÞ 6 h. q~ max ðtÞ ¼ 0t GðuÞdu of the entities verifies: q At this point, the surfaces are not supposed to enhance nucleation (no transcrystallinity), thus the spherulite distribution is equiprobable along the z-axis perpendicular to the surfaces. For a spherulite of radiusR located at z < R, the deleted volume V s ðzÞ due to the presence of a surface can be calculated with the expression (Fig. 1):

V s ðzÞ ¼ p

Z

R

z

2 1 R2 u2 du ¼ p R3 zR2 þ z3 3 3

ð21Þ

If h > R, the average suppressed volume by spherulite due to the two surfaces is estimated by:

VM ¼

2 h

Z

R

V s ðzÞdz ¼ 0

p R4 2 h

ð22Þ

By analogy, the expression of the ‘‘fictitious” average removed volume is:

~4 eM ¼ p q V 2 h

ð23Þ

The total ‘‘fictitious” deleted volume is then:

~4 eS ¼ N eM ¼ N eaV ea p q V 2 h

ð24Þ

6


Fig. 1. Intersection between a sphere and a plane.

Eqs. (10) and (11) are rewritten taking into account this correction, leading to the new expressions:

a~ ¼

4p 3

Z t

q~ 3 ðt; sÞ

0

~ da ¼ 4p dt

Z t

e ~ 4 ðt; sÞ d N 3q a ðsÞds 8 h dt

q~ 2 ðt; sÞ

0

ea ~ 3 ðt; sÞ 1q dN GðtÞ ðsÞds 2 h dt

ð25Þ

ð26Þ

The authors develop the different terms of (Eq. (26)), multiplying it by ð1 aÞ (see (Eq. (7))). Then, they replace the corresponding expressions by F; P and Q (Eqs. (17)–(19)) and introducing:

ea dU dN ¼ F3 dt dt they finally obtain:

da ¼ 4pð1 aÞG dt

ð27Þ 1 3 e 3 2 3 1 F N a 2F F Pþ 1 F Qþ U F2 2h 2h 2h 2h

ð28Þ

This equation replaces the equation (Eq. (14)) in a system of eight equations. The equation (Eq. (27)) is the eighth one and the six remaining stay the same as previously (Eqs. (13), (15)–(19)). The initial conditions remain the same (Eq. (20)) and Uð0Þ ¼ 0. 3.2. Transcrystallinity Then, one considers that there are two transcrystalline fronts, parallel to the surfaces, propagating along the perpendicular direction with a growth rate equal to G (Fig. 2). These fronts are supposed to appear instantaneously. The solidified thickness on each surfaces is, according to (Eq. (17)):

Z e¼

t

GðuÞdu ¼ FðtÞ

ð29Þ

0

Fig. 2. Propagation of transcrystalline fronts in a thin polymer film of thickness h (half of the figure is represented). The gray lines represent the transcrystalline morphology. During a small progression de of the front with a growth rate G, an existing spherulite will become completely occluded, while another one begins to be entrapped.


7

Two types of morphology are envisaged: transcrystalline and spherulitic. Let us call aTr the volume fraction transformed into transcrystalline phase. Spherulites appear in the still liquid phase (volume fraction aL Þ, in the region of thickness h2e ¼ 2h e in the half-thickness of the polymer film, considered as symmetrical (Fig. 2). Some of them remain immersed in the liquid phase (volume fraction aS Þ, whereas others are captured by the growing surface layer (Fig. 2). Therefore, this solidified layer contains both spherulites and true transcrystallinity. The transcystalline growth is expressed by:

daTr G ¼ 2ð1 aSd Þ h dt where aSd ¼

ð30Þ

aS aL þ aS

ð31Þ

is a new spherulite fraction, equal to the spherulite volume fraction in the zone of thickness he . It is worth noticing that the liquid fraction is related to the total transformed fraction a (spherulites and transcrystallinity) by:

aL ¼ 1 a

ð32Þ

The volume fraction of the two surface layers of thickness e is:

e h

2 h

aF ¼ 2 ¼ F

ð33Þ

Then, the volume represented by liquid and spherulites immersed in the liquid phase is related to this ‘‘front” volume by:

2 h

aL þ aS ¼ 1 aF ¼ 1 F

ð34Þ

Using (Eqs. (32) and (34)), the spherulite fraction in the ‘‘liquid phase” is:

2 h

as ¼ a F

ð35Þ

According to (Eqs. (31), (34) and (35)), we obtain:

aSd ¼

a 2h F

ð36Þ

1 2h F

The generation of spherulites in the ‘‘liquid phase” of thickness he is similar to the crystallization between two surfaces, so, according to (Eq. (28)), the total conversion rate of the melt into spherulites is given by:

da0S he ð1 aSd ÞG ¼ 4p dt h

1 3 e 3 2 3 1 F2 F N a 2F F Pþ 1 F Qþ U 2he 2he 2he 2he

ð37Þ

R ~ max ðtÞ ¼ 0t GðuÞdu. It is worth noticing that this equation will no longer be valid at t if he ðtÞ 6 q The total transformed fraction is given by the addition of the contributions of the transcrystalline phase and of the spherulites:

Thus:

da daTr da0S ¼ þ dt dt dt

ð38Þ

da G he 1 3 e 3 2 3 1 F N a 2F F Pþ 1 F Qþ U ¼ 2ð1 aSd Þ þ 4p ð1 aSd ÞG F 2 h 2he 2he 2he 2he dt h

ð39Þ

The other equations remain the same, except that 1 a is replaced by 1 aSd , as all as spherulites appear in a he thick film. In (Eq. (13)), the expression ddta is kept because the potential nuclei in the liquid zone are captured by both transcrystalline phase and spherulites growth. So there is a system of eight equations, with (Eq. (28)) replaced by (Eqs. (39) and (13)) replaced by:

dN 1 da dN0 ðT Þ dT þ ð1 aSd Þ ¼ N q þ dt 1 aSd dt dT dt

ð40Þ

and (Eqs. (16), (18), (19), and (27)) replaced by:

ea dN qN ¼ 1 aSd dt

ð41Þ

8


ea dP dN qN ¼F ¼F dt 1 aSd dt

ð42Þ

ea dQ dN qN ¼ F2 ¼ F2 dt 1 aSd dt

ð43Þ

ea dU dN qN ¼ F3 ¼ F3 dt 1 aSd dt

ð44Þ

The initial conditions and (Eqs. (15) and (17)) remain the same as previously. 4. Numerical method All the computations are done using MatlabÒ, the differential equations systems are solved using ode45 MatlabÒ function (Runge–Kutta 4, 5 [39]). 4.1. Method principle: the pixel coloring method The pixel coloring method is a ‘‘natural” method, as it does not make more assumptions than overall kinetic theories: it considers the nucleation and growth of morphological entities, such as spherulites, and calculates the volume fraction occupied by these entities. Therefore, it does not take into account the internal microstructure of the entities, i.e., the semicrystalline character of polymer spherulites. A virtual parallelepiped sample is divided into pixels (Figs. 3 and 4). The pixels size is chosen to avoid two nuclei being in the same pixel (as a function of N 0 and N s Þ. The minimal number of nuclei along each direction to obtain representative results (i.e. no need to perform several computations) has been determined around 10, comparing the simulation results for different sample sizes with Evans’ model in an infinite bulk volume. Thus, the numerical sample size is set according to N 0 . Then, some potential nuclei are randomly set on this grid according to the potential nuclei density N 0 . At each time step, each potential nucleus has a probability to be activated, according to q. It is worth noticing that the time step much be set so that this probability remains small, in order not to skip activations. The time step is also set in order to keep a growing step smaller than 1 pixel (according to GÞ. If the nucleus is activated, a random number is allowed to it, which can be translated into a color for visualization. The activation time s and the location of this spherulite origin are also recorded. Then, at each time step, for each ‘‘liquid” (uncolored) pixel, the neighboring pixels are checked. If there is an already colored pixel in contact with this ‘‘liquid” pixel, and that the center (original nucleus location) of this spherulite Rt is less distant than s GðuÞdu, the ‘‘liquid” pixel becomes solid and takes the same color. The transformed fraction a is computed by dividing the number of colored pixels by the total number of pixels. In the cases where there is no ‘‘physical” borders (infinite volume), the sample boundary conditions are periodic, i:e:; the colored pixels located on a boundary are taken into account for the computation of the evolution of a ‘‘liquid” pixel located on the opposite boundary. In the case where there are ‘‘physical” boundaries, as film surfaces, the periodic conditions are not used for these surfaces. If transcrystallinity is induced by surfaces, the potential nuclei density and the activation frequency for the pixels located on these surfaces are different from those in the volume (N s (m2) and s instead of N 0 (m3) and qÞ. In the case of a film with transcrystallinity and with few nuclei in the volume (low N 0 Þ, the length of the virtual film is chosen in order to be representative of the transcrystalline nucleation instead of the volume nucleation (N s instead of N 0 Þ, because, if there is only few or even no nuclei in the volume, the sample cannot be defined infinitely long in order to obtain

Fig. 3. Parallelepiped sample divided into pixels.


9

Fig. 4. 2D slice of a crystallized 3D virtual sample.

ten nuclei in the two directions of the film plan. Furthermore, the pixel size has to be small in transcrystalline cases in order to correctly describe the front growth without apparent numerical ‘‘steps”, and extending the sample size would lead to a too high number of pixels to compute. Thus, in an intermediate case with some nuclei in the volume, if the sample size is representative of the transcrystalline phase, it may be too small to correctly describe the nuclei in the volume and that could lead to less good numerical results. A simple but time expensive solution to this problem is to perform several computations and to compute an average value of the transformed fraction. In the present study, only one computation is done for each case, so the error can be tested in the worst configuration. 4.2. Validation for an infinite volume In order to validate the numerical method, first comparisons are performed for an infinite volume, in both isothermal and non-isothermal conditions. The numerical sample is chosen sufficiently large, i:e., with enough potential nuclei along each dimension (according to N 0 Þ to minimize the influence of frontiers and the conditions on sample surfaces are periodic. As an additional precaution, a margin is taken from sample boundaries for the transformed fraction calculation. The results of the numerical simulations are compared to Evans’ (generalized) model (see [25] and Section 2.1 above) and Haudin–Chenot’s model (see [10] and Section 2.3 above), using the transformed fraction a as the comparison value. It is worth noticing that Evans’ model neglects the apparition of new potential nuclei during cooling. Therefore, it is not computed in the case where N 01 – 0 (see (Eq. (45)) below). The evolution of crystallization parameters evolution is defined by equations similar to those determined by Monasse et al. for a polypropylene [11]:

N0 ðT Þ ¼ N00 exp N 01 T T ref

ð45Þ

qðT Þ ¼ q0 exp q1 T T ref

ð46Þ

GðT Þ ¼ G0 exp G1 T T ref

ð47Þ

with T ref = 90 °C. One can notice that (Eq. (56)) is an approximation of Hoffman-Lauritzen equation [40]. The equation parameters are arbitrarily set in the order of magnitude of polymer crystallization. The default values of the parameters are: N 00 = 1015 m3, N 01 = 0 °C1, q0 = 0.1 s1, q1 = 0.1 °C1, G0 = 106 m s1, G1 = 0.1 °C1. The modified parameters are specified in the figure captions. For isothermal cases, T is set to 100 °C. For non-isothermal cases, T goes from T 0 = 110 °C to T f = 100 °C with a cooling rate of c = 5 °C/min. The results obtained in the different conditions (displayed in the figure heading) are presented in (Fig. 5). One can see that the maximal difference between models and simulation for the transformed fraction is for the isothermal case and is lower than 0.02 (or 2% of the final fraction) (Fig. 5(a)). Thus, one can consider that the numerical method gives results equivalent to those of Evans and Haudin–Chenot analytical models. Considering that these models are well-validated, one can consider that the numerical method is validated for infinite volume cases.

10


Fig. 5. Comparison between the transformed fractions obtained with the numerical method and with the analytical ones (Evans, Haudin–Chenot) in an infinite volume: (a) in an isothermal case (T = 100 °C), (b) in an non-isothermal case (T 0 = 110 °C, c = 5 °C/min) and (c) in an non-isothermal case with potential nuclei evolution (T 0 = 110 °C, c = 5 °C/min, N 01 = 0.1 °C1, Evans’ model not computed).


11

5. Validation for thin films Chenot et al. models and our numerical method are now compared with a validated analytical model (Billon et al. [25]) in the case of a thin film, generating transcrystallinity on its surfaces or not, for isothermal and non-isothermal conditions. The transformed fraction will be the parameter used for comparison purpose between the models. In the cases where the number of potential nuclei evolves with temperature (N 01 – 0), only the numerical method will be computed as Billon et al. model does not take into account the temperature evolution of the potential nuclei. Thus, the numerical method and Chenot et al. models will not be strictly validated in these cases but a good consistence of the results would be an encouraging sign.

5.1. Thin film without transcrystallinity The values of the kinetic parameters are the same as those presented in Section 4.2, but one of the three dimensions of the sample is set to 50 lm with no possibility for the program to change it (Fig. 6). The surfaces perpendicular to this direction are considered as ‘‘solid”, thus there are no more periodic boundary conditions along this direction. The results obtained in different conditions (displayed in the figure heading) are presented in (Fig. 7). First of all, one can notice the good agreement between the three simulations in the first two cases (isothermal and non-isothermal) (Fig. 7(a) and (b)). The maximal error is observed in the non-isothermal case between our numerical method and Billon et al. model but remains low since it is only slightly above 0.02 (2% of the final fraction). One can consider that these models provide similar results in these cases. In the case of non-isothermal conditions with potential nuclei evolution, Billon et al. model cannot be applied, so the comparison is possible only with the numerical model. The maximal error value is again slightly above 0.02 (2% of the final fraction) (Fig. 7(c)). Therefore, one can consider that the Chenot et al. model and the numerical method give equivalent results in this case too.

5.2. Thin film with transcrystallinity New parameters describing the kinetics of crystallization on the film surfaces (transcrystallinity) are introduced: N s = 1013 m2, s = 100 s1 and attributed to pixels neighboring the film surfaces. They are equivalent to N 0 (m3) and q in the volume but they do not depend on temperature. These values are chosen to induce a massive instantaneous nucleation on the film surfaces, according to usual observations of transcrystallinity phenomenon [14] (Fig. 8). The other parameters keep the same values than those presented in Sections 4.2 and 5.1. First of all, the three simulations are in quite good agreement for the two cases considered here, with a maximal error value of 0.095 (9.5% of the final fraction) between the numerical method and Chenot et al. model (Fig. 9(a) and (b)). The results of the analytical models (Billon and Chenot) are quasi-identical, so the error probably comes from the numerical method, because of a too small virtual sample, as explained in Section 4.1. Even if a maximal error of 10% remains acceptable, some improvements in the method of the sample size choice can probably be done. In the isothermal case and in the non-isothermal case with N 01 ¼ 0, N 0 does not depend on temperature and is always equal to N 00 (Eq. (45)). In the non-isothermal case with N 01 ¼ 0:1 °C1, N 0 is initially (at 110 °C) smaller than N 00 as T ref ¼ 90 °C < 110 °C (Eq. (45)). Thus, the crystallization is slower in this case. One can notice that the value calculated by the model has reached a plateau that is below 1 (Fig. 10). In fact, this case is out of the validity domain of the model, as ~ max has reached the he decreasing value of space between transcrystalline fronts before the maximal extended radius q the crystallization was completed (Fig. 11). Thus the computation of the spherulite growth in a thin film limited between the transcrystalline fronts is no longer accurate. This model is thus valid for sufficiently high nucleation rate in the volume (sufficiently small spherulites) i:e. for high values of N 0 and q compared to G. In the cases in which there is transcrystallinity, this one has to be instantaneous to be well depicted by the model. In the other cases, Billon et al. model could be used instead, but it does not allow taking into account the temperature dependence of the number of potential nuclei. For the cases in which the transcrystallinity is almost the only crystallization process, a simpler model could be used, as ddta ¼ 2 Gh for example. A possible improvement could be to extend Chenot et al. model of crystallization between two parallel planes to the case q~ max ðtÞ ¼ FðtÞ P h. It has been done analytically (see Appendix A), but not yet implemented in numerical calculations, nor applied to transcrystallinity, which could be source of difficulty.

Fig. 6. 2D slice of a crystallized 3D thin film virtual sample.

12


Fig. 7. Comparison between transformed fraction obtained by the numerical method and by analytical ones (Billon et al., Chenot et al.) in a thin film: (a) in an isothermal case (T = 100 °C), (b) in an non-isothermal case (T 0 = 110 °C, c = 5 °C/min) and (c) in an non-isothermal case with potential nuclei evolution (T 0 = 110 °C, c = 5 °C/min, N 01 = 0.1 °C1, Billon et al. model not computed).


13

Fig. 8. 2D slice section of a crystallized 3D thin film virtual sample with transcrystallinity on its surfaces.

Fig. 9. Comparison between transformed fraction obtained with the numerical method and with the analytical ones (Billon et al., Chenot et al.) in a film generating transcrystallinity on its surfaces: (a) in an isothermal case (T = 100 °C), (b) in an non-isothermal case (T 0 = 110 °C, c = 5 °C/min).

14


Fig. 10. Transformed fraction obtained with Chenot et al. method in a film generating transcrystallinity on its surfaces in an non-isothermal case with potential nuclei evolution (T 0 = 110 °C, c = 5 °C/min, N 01 = 0.1 °C1).

~ max ðdashed lineÞ. Fig. 11. The theoretical limit of the model is reached when he ðsolid lineÞ ¼ q

6. Conclusion A new numerical method has been proposed to simulate crystallization of polymers in different conditions. This methodology, combining theoretical approach and numerical simulation, has a general character, since it is based on the general framework of overall kinetic theories, initially developed for metals. Nevertheless, the application done here is specific of polymers, since we consider that growth rate is only function of temperature, which means that growth is governed by an interfacial mechanism and not diffusion-controlled. Of course, our approach could be extended to materials obeying the same type of growth kinetics. This model has been first validated for infinite volumes comparing it to well validated analytical models. These comparisons show very good agreement in all cases. This new method, along with an already validated analytical model (Billon et al.), has then be used to validate Chenot et al. analytical models for crystallization in thin films simulation (without or in presence of transcrystallinity). The numerical and analytical models are in good agreement but, as expected, Chenot et al. model for transcrystalline cases reaches its limit in the cases in which the spherulites are too large compared to the film thickness or compared to the space between transcrystalline fronts. In these cases, another method should be used instead or Chenot’s model should be extended. A following step in crystallization modeling could be the study of crystallization in polymers with fiber reinforcement simulation. As there is no general analytical model for fiber reinforcement yet, this numerical method could be very useful to predict crystallization kinetic in such kind of complex configuration. Acknowledgements This work was carried out during the project STIICPA. We thank the ADEME (French agency for environment and energy control) for its financial support.


15

Appendix A. Crystallization in a thin film for FðtÞ >h The treatment is a generalization of the model of Escleine et al. [13]. Let us define the time s2, function of t, by:

q~ ðt; s2 Þ ¼

Z

t

s2

GðuÞdu ¼ h

ðA1Þ

The extended volume fraction becomes:

a~ ðtÞ ¼ since

p 3

Z s2 h

~ 2 ðt; sÞ 3q

0

2

h 2

!

e Z e a ðtÞ ~ ðt; sÞ d N dN 4p t 3 3q a ðtÞ ds þ q~ ðt; sÞ 1 ds ds 8 h ds 3 s2

ðA2Þ

s 6 s2 ) q~ ðt; sÞ h

s s2 ) q~ ðt; sÞ 6 h By derivation with respect to time, one obtains:

! ! 2 ~ ðtÞ da F 3 ðtÞ e e a ðs2 Þ Pðs2 Þ þ 4pGðtÞ e a ðs2 Þ 2FðtÞ 3F ðtÞ ðPðtÞ Pðs2 ÞÞ N a ðtÞ N F 2 ðtÞ ¼ 2ph GðtÞ FðtÞ N dt 2h 2h ! 3FðtÞ UðtÞ U ðs2 Þ ðA3Þ ðQ ðtÞ Q ðs2 ÞÞ þ þ 1 2h 2h

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