Modelling multi-dimensional crystallization of polymers in interaction with heat transfer

Nonlinear Analysis: Real World Applications 3 (2002) 139–160www.elsevier.com/locate/na

Modelling multi-dimensional crystallization ofpolymers in interaction with heat transfer

Martin Burgera, Vincenzo Capassob;c; ∗, Claudia Salanib;caIndustrial Mathematics Institute, Johannes Kepler Universit�at Linz,

Altenbergerstr. 69, 4040 Linz, AustriabMIRIAM, Milan Research Centre for Industrial and Applied Mathematics, Universit%a di Milano,

Via C. Saldini 50, 20133 Milano, ItalycDipartimento di Matematica, Universit%a di Milano, Via C. Saldini 50, 20133 Milano, Italy

Received 29 April 1999; accepted 29 November 2000

Keywords: Averaging; Crystallization of polymers; Di0usion; Growth; Nonlinear evolution equa-tions; Nucleation

1. Introduction

A crystallization process consists, in general, of the superposition of two features,namely nucleation and growth of crystals. While the growth process may be considereddeterministic (with speed G = G(x; t), the growth rate), nucleation occurs randomlyin space and time. We will consider a crystallization process in a bounded domainE ⊂ Rd (d = 1; 2; 3) and assume that nucleation takes place only in the interior ofE with some rate � = �(x; t) (nucleation rate). In practice, one should also expectboundary nucleation (cf. [11]), which is neglected almost completely in literature sofar. Nevertheless, we note that the approach presented here might also be adapted forthe case of boundary nucleation, which will be an important future task.

In general, we will denote by �t the crystalline phase at time t and by �t(x0; t0) acrystal born at point x0 at time t0 and freely grown up to time t. In a crystallizationprocess with nucleation events {(Xj; Tj)|0 6 T1 6 T2 6 : : :}, the crystalline phase isgiven by

�t =⋃Tj¡t

�Tj ;

∗ Corresponding author. Dipartimento di Matematica, Universit;a di Milano, Via C. Saldini 50, 20133Milano, Italy. Fax: +39-02-706-30346.

E-mail address: [email protected] (V. Capasso).

1468-1218/01/$ - see front matter ? 2001 Elsevier Science Ltd. All rights reserved.PII: S1468 -1218(01)00019 -0

140 M. Burger et al. / Nonlinear Analysis: Real World Applications 3 (2002) 139–160

where �tj denotes the crystal born at time Tj at Xj (again freely grown up to time t).

It has turned out (cf. [4,10,12]) that on a macroscopic scale the quantity

�(x; t) :=E[I�t (x)] = P(x ∈ �t); x ∈ E; t ∈ R+

is suitable for the description of the crystallization process.In the simple case of spatially homogeneous growth and nucleation rate, equations

for � can be derived based upon the approaches by Avrami, Kolmogorov and Evans(cf. [1–3,13,15]). These models yield a good description of isothermal processes and havebeen investigated frequently in bounded and unbounded domains (cf. [6,7,11,12,16]).

We develop a model for heterogeneous growth based upon experimentally veriHedprinciples in Section 2. By using this growth model we can derive the crystallizationkinetic equations for arbitrary growth and nucleation rates, which are given functions oftime and space. This derivation uses a generalization of the classical theory by Avrami,Kolmogorov and Evans, the main tool being the causal cone. This theory cannot beextended easily to the case of interaction with temperature, since the causal cone cannotbe deHned in a deterministic way if the temperature Held is random. In Section 5.3we provide a formal derivation of the deterministic system by an approximation atdi0erent scales. A rigorous derivation will be left to subsequent work (cf. [9]).

2. Growth of a nucleus in an heterogeneous �eld

The classical Kolmogorov–Avrami–Evans theory for isothermal crystallization heav-ily used the fact that crystals are of a spherical shape if the growth rate is constant; thesame is true if the growth rate depends upon time only. In the case of heterogeneousgrowth, i.e., if the growth rate G depends on space and time, the shape of a polymericcrystal (before impingement) is no longer a ball centered in the origin of growth. Inthe case of a growth rate with constant gradient it has been veriHed experimentally thatthe growing nucleus is the union of lines (the growth lines) which lead to growth inminimal time (cf. [19]). This principle can be adapted for the case of arbitrary growthrates (cf. [20]).

Assumption 2.1 (Minimal-time principle). A nucleus grows from its origin to any otherpoint such that the needed time is minimal.

The minimal-time principle is obviously satisHed for homogeneous growth, sincethere the growth lines are just straight lines, whose union forms the spherical shape.In general, the growth of a nucleus in R2 between its origin (x0; y0) and another point(x1; y1) due to Assumption 2.1 may be formulated as follows:

t1 = min(x;y;�)

s.t.

x(t) = G(x(t); y(t); t) cos�(t); t ∈ (t0; t1);

y(t) = G(x(t); y(t); t) sin �(t); t ∈ (t0; t1);

M. Burger et al. / Nonlinear Analysis: Real World Applications 3 (2002) 139–160 141

x(t0) = x0; y(t0) = y0;

x(t1) = x1; y(t1) = y1:

The necessary Hrst order conditions for this control problem (cf. e.g. [14]) lead tothe following equation for the control variable �:

� = 〈∇G(x; y; t); (− sin �; cos�)T〉:For the growth of a nucleus in R3 we obtain the control problem

t1 = min(x;y;�;�)

s.t.

x(t) = G(x(t); y(t); z(t); t) cos�(t) cos �(t);

y(t) = G(x(t); y(t); z(t); t) sin �(t) cos �(t);

z(t) = G(x(t); y(t); z(t); t) sin �(t);

x(t0) = x0; y(t0) = y0; z(t0) = z0;

x(t1) = x1; y(t1) = y1; z(t1) = z1

which leads to the necessary conditions:

�(t) = 〈∇G(x(t); y(t); z(t); t); (− sin �(t); cos�(t); 0)T〉;

�(t) = 〈∇G(x(t); y(t); z(t); t); (− cos�(t) sin �(t);− sin �(t) sin �(t); cos �(t))T〉:By eliminating the angles we may deduce a second order ODE for the growth lines

given by

ddt

(x

G(x; t)

)= −∇G(x; t) +

⟨∇G(x; t);

xG(x; t)

⟩x

G(x; t); (1)

where x denotes the vector (x; y)T in R2 and (x; y; z)T in R3, respectively.The crystal at time t is now given as the union of all growth lines, i.e.

�t0 = {x(�)|x solves (1); x(t0) = x0; � ∈ (t0; t)}:

Each growth line is determined uniquely by its derivative at initial time, which maybe written as

x(t0) = G(x0; t0)n0;

where n0 is an arbitrary vector in Rd with ‖n0‖ = 1. Thus, we may introduce aparametrization for the crystal based on the initial direction, namely, in R2

�t0 = {x(�; �)|� ∈ (t0; t); � ∈ [0; 2�)}; (2)


where x(�; �) denotes the solution of (1) with initial values

x(t0) = x0;

x(t0) = G(x0; t0)(cos �; sin �)T

and in R3

�t0 =

{x(�; �1; �2)|� ∈ (t0; t); �1 ∈ [0; 2�); �2 ∈

[−�

2;�2

]}; (3)

where x(�; �1; �2) denotes the solution of (1) with initial values

x(t0) = x0;

x(t0) = G(x0; t0)(cos �1 cos �2; sin �1 cos �2; sin �2)T:

Eq. (1) yields a description of crystal growth based on growth lines, which arecomputed independently. The parametrizations introduced in (2) or (3) also provideanother view upon the growing nucleus. For the sake of simplicity we concentrate onthe case of a crystal in R2, but similar reasoning is possible in higher dimensions. ByHxing the time t we obtain the set

@�t0 = {x(t; �)|� ∈ [0; 2�)};

which is called the growth front. In the following, we will deduce equations for thecrystal growth using information about the surface @�t

0 and its normal vector, whichwill be denoted by n(t; �).

Let x′(t; �) denote the derivative of x with respect to �. Then ‖x‖2 =G(x; t)2 implies⟨xG

;@@tx′⟩

= 〈∇G; x′〉:

Multiplying (1) with x′ yields⟨@@t

xG

; x′⟩

= −〈∇G; x′〉 +⟨∇G;

xG

⟩⟨xG

; x′⟩

and by combining these equations we obtain the initial-value problem@@t

⟨xG

; x′⟩

= p(t)⟨xG

; x′⟩

; (4)⟨xG

; x′⟩t=0

= 0; (5)

where p(t) := 〈∇G; x=G〉. Since we may suppose that x and x are known as the solu-tions of (1) we may interpret (4), (5) as an equation for the function 〈x=G; x′〉, where� is Hxed now. If p(t) is suLciently smooth we may conclude that the homogeneousinitial-value problem (4), (5) has only the trivial solution, i.e.,⟨

xG

; x′⟩

(t; �) = 0; ∀t ¿ t0; ∀� ∈ [0; 2�):

Since x=G is a unit vector this immediately implies that x=G is just the normal vectorof the growth front.


Consequently, we rewrite Eq. (1) as the Hrst order system

x(t; �) = G(x(t; �); t)n(t; �); (6)

n(t; �) = −∇G(x(t; �); t) + 〈∇G(x(t; �); t); n(t; �)〉n(t; �); (7)

which clearly shows that the growth is determined by the actual normal direction ofthe growth front as well as by the growth rate and its gradient. The initial values aregiven by

x(t0; �) = x0; (8)

n(t0; �) = (cos �; sin �)T:

In R3 Eq. (6) may be deduced in an analogous way by di0erentiating with respect toboth angles separately, hence system (6), (7) still remains valid. The initial conditionsare given by (8) and by

n(t0; �) = (cos �1 cos �2; sin �1 cos �2; sin �2)T:

As opposed to the original minimal-time principle, the derivated system (6), (7)needs only information about the shape of the crystal at the actual time, but not aboutthe history of growth. Hence, this description seems to be suitable not only for thecase of growth in a given Held, but also for growth in interaction with an evolvingHeld.

2.1. Examples of growing crystals

For a Hrst simulation of crystal growth in an heterogeneous Held we used a typicalparabolic temperature proHle (i.e., the solution of the heat equation without sourceterm) and data for the growth rate obtained by measurements of i-PP (cf. [17]). Theresults are presented in Fig. 1, which shows the growth front in the Hrst time steps.The deviation from the spherical shape obviously increases with time, nevertheless, thecrystals still remain convex and do not produce exotic shapes.

3. The causal cone approach

A classical way of deducing model equations on a macroscopic scale is based on theinvestigation of the causal cone A(x; t), i.e., the set of all points y and times s, suchthat nucleation at y at time s would lead to coverage of x at time t. The probability�(x; t) that the point x is covered at time t (which is the same as the expected valueE[I�t (x)]) is substituted by the probability that no nucleation occurs in the causal cone.

This approach was successfully applied in the case of crystallization leading to theAvrami–Kolmogorov–Evans equation (cf. [1–3,13,15]). Recently, this approach hasbeen generalized to heterogeneous crystallization (cf. [4,10]). The growth rate G andthe nucleation rate � are usually treated as a given Held, their kind of temperature


Fig. 1. Crystal shapes in a typical temperature Held.

dependence being introduced a posteriori. In the case of non-isothermal crystallization,this is a very delicate point, since growing crystals inNuence temperature and thus thegrowth rate. This problem will be discussed in detail in the following section.

3.1. The structure of the causal cone

It has been shown (cf. [4,10]) that the degree of crystallinity � satisHes

�(x; t) = 1 − e−w(x; t);

w(x; t) =∫ t

0

(∫E(x; t; s)

�(y; s) dy)

ds

with E(x; t; s) deHned by

E(x; t; s) = {y ∈ E|(y; s) ∈ A(x; t)} = {y ∈ E|x ∈ �t(y; s)}if the condition

P(x �∈ �t) = P(no nucleation occurs in A(x; t)) (9)

is satisHed. In order to verify (9), we have to show that any nucleation in the causalcone leads to coverage independent of the crystalline phase at that time. More precisely,we have to show that even if (x1; t1) ∈ A(x; t) is already covered by another crystal, thepoint x will be covered at time t. This can be done by comparing two crystals—the one


Fig. 2. Two hitting crystals in the situation of Lemma 3.1.

already existing before (born at (x0; t0) and denoted by �t0) and the (virtual) new-born

crystal �t1 (see Fig. 2). We show that the later born will always stay ‘inside’ the other

one. DeHning the cone of in;uence by

I(x0; t0) := {(x; t)|x ∈ �t0};

we can express this statement mathematically in the following lemma:

Lemma 3.1. If (x1; t1) ∈ I(x0; t0); then I(x1; t1) ⊂ I(x0; t0).

Proof. Suppose I(x1; t1) �⊂ I(x0; t0). Then there exist t ¿ t1 and t ¿ t such that

�t1 ⊂ �t

0;

�s1 �⊂ �s

0;∀s ∈ (t; t):

For all points x ∈ @�t0 ∩ @�t

1, the normal with respect to both crystals must havethe same direction, because otherwise there would exist a point x in �t

1 − �t0. By

investigating the growth line y0 from (x0; t0) to (x; t) and the growth line y1 from(x1; t1) to (x; t), the equality of the normal vector implies

y0(t) = x = y1(t);

y 0(t) = G(x; t)n = y 1(t):

Since y0 and y1 satisfy the second order ODE (1), the uniqueness of the Cauchy-problem(for suLciently smooth G) implies y0(s) =y1(s) for all s¿ t. Hence, the crystal born


at (x1; t1) can never grow beyond the boundary of the crystal born at (x0; t0), i.e.,

�s1 ⊂ �s

0; ∀s¿ t1;

a contradiction.

Lemma 3.1 can be applied to any situation, where the growth rate G(x; t) is a givenHeld independent of the actual crystalline phase, e.g. in the case of a non-homogenousmaterial, where G = G(x). In the case of a really non-isothermal situation, this isnot true anymore, since G depends upon temperature and the temperature dependsupon the crystalline phase. Especially, two growing nuclei cannot be compared as inLemma 3.1, since both inNuence temperature and consequently, their growth is drivenby di0erent growth rates. In a situation like that, even the causal cone cannot bedeHned independent of the process anymore, because of the reasons described above italways depends on the pattern of the crystalline phase via the temperature. Nevertheless,the results obtained by this generalized Avrami–Kolmogorov–Evans theory yield goodapproximations under typical crystallization conditions as we will see in the followingsection.

3.2. Model equations in the multi-dimensional case

As we have seen in the previous section, the degree of crystallinity is given by

�(x; t) = 1 − e−w(x; t);

w(x; t) =∫ t

0

∫E(x; t; s)

�(y; s) dy ds

=∫ t

0

∫E�(x; t; y; s)�(y; s) dy ds

where � is the indicator function of the s-section of the causal cone of (x; t), moreprecisely,

�(x; t; y; s) = IE(x; t; s):

In a preceding paper [2] we have deduced the hyperbolic initial–boundary valueproblem(

1G

wt

)t= (Gwx)x + 2� in E × R+;

wt + Gwn = 0 on @E × R+;

w = wt = 0 in E × {0};

(10)

in the case of E ⊂ R1 by applying a method of characteristics upon �. In the following,we will deduce similar model equations in the multidimensional case, making use ofEqs. (6) and (7).


Since the solution x(t; �) of (6) and (7) represents the surface on which the value 1of the indicator function is propagated, � satisHes

�t + G〈∇�; �〉 = 0;

〈∇�; � ⊥〉 = 0;

〈�; �〉 = 1

for t ¿ s, with initial values

�(x; t; y; t) = 0 for x �= y;

�(x; t; x; s) = 1 for t ¿ s:

In the above equations, � = �(x; t; y; s) represents the outer normal of the causal conewith respect to x.

The same technique as applied to the growth of one nucleus yields

�t = G�1;

�1t = ∇x:(G�2) + d;

�2t = ∇x(G�1)

in Rd, where �1 and �2 may again be interpreted as densities of the boundary of thecausal cone and d(x; t) is the functional

〈 2(x; t); �〉 := 2�∫ t

0�(x; s) ds

for d = 2, respectively,

〈 3(x; t); �〉 := 4�∫ t

0G(x; s)

∫ s

0�(x; �) d� ds

for d = 3.Now, we may easily verify that w is a solution of the system

wt = Gu; (11)

ut = ∇:(Gv) + 〈 d; �〉; (12)

vt = ∇(Gu) (13)

with homogenous initial values for u, v and w.In a similar way, we may also derive the boundary condition

u(x; t) + 〈v(x; t); n(x; t)〉 = 0 for x ∈ @E

(assuming that no boundary nucleation occurs), where @=@n denotes the normal deriva-tive with respect to @E.

It turns out that system (11)–(13) can be transformed to the form of (10) if G¿ 0.We note that also the higher-dimensional model contains the well-known system ofrate equations derived by Schneider et al. [18] for the space-homogeneous case.


4. Interaction with heat transfer

In an experimental situation, where heterogeneities are caused only by the heat trans-fer in the material, we may model growth and nucleation rates as certain temperature-dependent material functions (cf. [11])

G(x; t) = G(T (x; t));

�(x; t) =@@t

N (T (x; t)):

Vice versa, the growing crystalline phase inNuences the heat conduction process inthe material as follows:

(%cT )t = ∇ · ('∇T ) + (hI�t )t in E × R+;

Tn = )(T − Tout); on @E × R+;

where again the derivative of the indicator function I�t has to be understood in a weaksense. Here % denotes the density, c the heat capacity, ' the heat conductivity, ) theheat transfer coeLcient and h the latent heat released at the moment of phase change.

The parameters in the heat equation may depend upon the phase, i.e., if %1; c1; '1

and )1 denote the parameters of the crystallized material and %2; c2; '2 and )2 the onesof the non-crystallized, we may write

% = I�t%1 + (1 − I�t )%2;

c = I�t c1 + (1 − I�t )c2;

' = I�t'1 + (1 − I�t )'2;

) = I�t)1 + (1 − I�t ))2:

(14)

This heat transfer model is a random di0erential equation, since all parameters dependupon the random variable I�t . A direct consequence is the stochasticity of temperature,whose evolution depends upon the geometry of the crystalline phase.

The typical scale of the heat transfer problem is given by

xT =√

'0t0%0c0

; (15)

where t0 is the length of the considered time interval, '0, %0 and c0 are typical scalesfor ', % and c. The typical scale for the growth of a nucleus is given by

xG = G0t0; (16)

where G0 is a typical value for the growth rate G. In practical applications it turnsout that xT�xG, which is due to the fact that crystal growth is very slow, whereas theheat conduction is rather fast. This means that there exist two signiHcant scales in theproblem, i.e.

• Microscale xG for growth.• Macroscale xT for heat conduction.


Fig. 3. Schematic representation of the scales in polymer crystallization.

It is a direct consequence that if one is interested only in local microscopic e0ects,temperature variations can be neglected, whereas for a pure macroscopic descriptionthe growth e0ects are not important (see Fig. 3). The scale of real interest in polymerprocessing is a mesoscale between xT and xG, which is Hne enough for morphologicaldetails without paying attention to each microscopic crystal. In the subsequent sectionswe will derive a hybrid model on such a mesoscale.

5. A stochastic model and its average

A di0erent approach to the computation of � by ‘looking back’ via the causal coneis to investigate the crystalline phase directly and take expected values later, using theso-called method of characteristics (cf. [20]). The aim is to write a di0erential equationfor the indicator function of the nucleus, which grows at its boundary with growth rateG. Since the non-continuous indicator function cannot satisfy such an equation in aclassical sense, characteristics are used.

5.1. The evolution of a single crystal

We Hrst consider the crystal �tj born at time t = Tj and point x = Xj, its indicator

function will be denoted by fj. The evolution of fj is determined by the Hrst orderPDE (cf. [4])

fjt + G‖∇fj‖ = 0 (17)

for x ∈ E, t ¿Tj with additional conditions

fj(x; Tj) = 0 for x �= Xj;

fj(Xj; t) = 1 for t ¿Tj:


We may also consider the same equation for t ¡Tj with the additional conditionfj(Xj; t) = 0.

Eq. (17) may be written equivalently as

fjt + G∇fj · n = 0;

∇fj · n⊥ = 0;

n · n = 1; (18)

where n is the normal vector of the growth front. As the normal vector multiplied bythe growth rate is just x, we have

dfj

dt(x(t; �); t) = fj

t + ∇fj · x = fjt + G∇fj · n = 0;

i.e., fj is constant along the growth lines.For any smooth function � = �(x) we have

@@t

∫Efj(x; t)�(x) dx =

@@t

∫�t

j

�(x) dx

= limSt→0

1St

[∫�t+St

j

�(x) dx −∫�t

j

�(x) dx

]

= limSt→0

1St

∫�t+St

j −�tj

�(x) dx

= limSt→0

1St

∫@�t

j

�(x) d+(x)G(x; t)St

=∫@�t

j

G(x; t)�(x) d+(x)

If we consider the crystal �tj as an open set, i.e. 1 − fj equals 1 on @�t

j, this yields

@@t

∫Efj(x; t)�(x) dx =

∫@�t

j

(1 − fj(x; t))G(x; t)�(x) d+(x):

Let us denote by u j the density of the crystal surface, that is

u j(x; t) dx = ,d−1(@�tj ∩ dx);

where ,d−1 is the Lebesgue measure on Rd−1. We have that

〈u j; �〉 =∫Rd

u j(x; t)�(x) dx

=∫@�t

j

�(x) d+(x)


for any suLciently smooth function �. We may write, in a weak sense,

fjt = (1 − fj)Guj: (19)

Let the function v be the density of the surface direction, i.e.

v j(x; t) dx = −n(x; t),d−1(@�tj ∩ dx);

so that

〈v j; �〉 = −∫@�t

j

�(x)n(x; t) d+(x):

Note that u j and v j satisfy the compatibility condition

v j · n = −u j:

From Eqs. (18) and (19), we obtain

G∇fj · n = −fjt = (1 − fj)Guj = (1 − fj)Gvj · n

and we may conclude

∇fj = (1 − fj)v j:

With the above deHnitions for u j and v j, we may write

fjt = (1 − fj)Guj; (20)

∇fj = (1 − fj)v j: (21)

By deriving (20) with respect to the space variable and (21) with respect to time,we have

∇(fjt ) = −∇fjGuj + (1 − fj)∇(Guj)

= −(1 − fj)v jGu j + (1 − fj)∇(Guj)

= (1 − fj)(∇(Guj) − v jGu j);

(∇fj)t = −fjt v

j + (1 − fj)vjt

= −(1 − fj)Gujv j + (1 − fj)vjt

= (1 − fj)(vjt − Gujv j):

For the compatibily condition, u j and v j must satisfy

vjt = ∇(Guj): (22)

The time derivative of u j can be calculated as

@@t〈u j; �〉 = lim

St→0

[∫@�t+St

j

�(x) d+(x) −∫@�t

j

�(x) d+(x)

]

= limSt→0

∫@�t

j

[�(x + Gn dt) − �(x)] d+(x)


= limSt→0

∫@�t

j

[�(x) + G dtn · ∇�(x) − �(x)] d+(x)

=∫@�t

j

G(x; t)n(x; t)∇�(x) d+(x)

= −〈Gvj;∇�〉= 〈∇ · (Gvj); �〉

for t ¿Tj which implies

ujt = ∇ · (Gvj) (23)

in a distributional sense.In order to extend the equations to the whole time domain t ¿ 0 we examine the

time derivatives at t = Tj using

@@t〈u j; �〉|t=Tj =

@@t

∫@�t

j

�(x; t) d+(x)|t=Tj ; k = 0; 1; 2: (24)

Since at time t = Tj, the grain �tj reduces to the point Xj, we may expand the

function �(x; t) in a neighborhood of (Xj; Tj), thus obtaining

�(x; t) = �(Xj; Tj) + �t(Xj; Tj)(t − Tj) + ∇�(Xj; Tj) · (x − Xj) + o(t − Tj)

= �(Xj; Tj) + �t(Xj; Tj)(t − Tj)

+∇�(Xj; Tj) · G(Xj; Tj)n(x; t)(t − Tj) + o(t − Tj):

By substituting in (24), we have

@@t〈u j; �〉|t−Tj =

@@t

(�(Xj; Tj)|@�tj|+�t(Xj; Tj)(t−Tj)|@�t

j|+o((t−Tj)d))|t=Tj ;

because |@�tj| = O((t − Tj)d−1) and, for the isotropy of n around the nucleus,

limt→Tj

∫@�t

j

n(x; t) d+(x) = 0:

In R1 we have |@�tj| = 2 and, consequently,

@@t〈u j; �〉|t=Tj = 2�t(Xj; Tj):

In R2, |@�tj| = 2�r(Xj; Tj; t) + o(t − Tj), where

r(Xj; Tj; t) =∫ t

Tj

G(Xj; s) ds

and thus

@@t〈u j; �〉|t=Tj = 2�G(Xj; Tj)�(Xj; Tj):


In R3 we have |@�tj| = 4�r(Xj; Tj; t)2 + o((t − Tj)2), hence

@@t〈u j; �〉 =

@@t

(�(Xj; Tj)4�r(Xj; Tj; t)2 + o((t − Tj)2))

= 4��(Xj; Tj)r(Xj; Tj; t)@@t

∫ t

Tj

G(Xj; s) ds + o(t − Tj)

= 8��(Xj; Tj)r(Xj; Tj; t)G(Xj; Tj) + o(t − Tj)

and, since r(Xj; Tj;Tj) = 0,@@t〈u j; �〉|t=Tj = 0:

With the notation@@t〈u j; �〉 = 〈Gu j; �〉;

where

〈u; �〉 = 8��(Xj; Tj)r(Xj; Tj; t)

and thus@@t〈u j ; �〉|t=Tj = 8�G(Xj; Tj)�(Xj; Tj);

we obtain

@2

@t2〈u j; �〉|t=Tj = G(Xj; Tj)

@@t〈u j; �〉|t=Tj = 8�G(Xj; Tj)2�(Xj; Tj):

Similar reasoning yields that (22) is satisHed at t = Tj, without any point source,which is due to the isotropy of the additional factor n(x; t) at the point of birth of thenucleus.

Obviously (22) and (23) hold for 0¡t¡Tj with homogeneous initial values att = 0, since both distributions are equal to zero in this region.

Hence, we have shown that, in a distributional sense, in Rd; d = 1; 2,

ujt = ∇ · (Gvj);

v jt = ∇(Guj);

t ¿Tj;

ujt = Sd

j ;

v j = 0;t = Tj;

u j = 0;

v j = 0;t 6 Tj; (25)

where Sdj is the generalized function deHned by

〈Sdj ; �〉 =

@@t〈u j; �〉|t=Tj :


In R3, (25) has to be replaced by

u jtt = ∇ · (Gvj)t ;

v jt = ∇(Guj);

t ¿Tj;

ujtt = GS3

j ;

u jt = 0;

v j = 0;

t = Tj;

u j = 0;

v j = 0;t 6 Tj; (26)

where S3j is the generalized function deHned by

〈S3j ; �〉 =

@@t〈u j ; �〉|t=Tj :

5.2. The evolution of the union of the crystals

Based on the equations for fj we may deduce a system for the indicator function fof the whole crystalline phase in order to investigate the crystallization process directlyand take expected values later.

Let f(x; t) := I�t (x); fj(x; t) := I�tTj

(x), then

1 − f(x; t) =∏Tj¡t

(1 − fj(x; t));

since f equals 0 if and only if all fj are 0.DeHning the sum of densities

u =∑Tj¡t

u j;

v =∑Tj¡t

v j;

we may write

ft =∑Tj¡t

fjt

∏k �=j

(1 − fk)

=∑Tj¡t

(1 − fj)Guj∏k �=j

(1 − fk)

= G∑Tj¡t

u j∏Tk¡t

(1 − fk) = G(1 − f)u


and, similarly,

∇f = (1 − f)v:

As an immediate consequence of the fact that u and v are linear superpositions ofthe functions u j and v j, respectively, the quantities u and v can be computed directlyby solving, for E ⊂ Rd; d = 1; 2,

ut = ∇ · (Gv) +∑Tj=t

Sdj in E × R+;

vt = ∇(Gu)

u + 〈v; n〉 = 0 on @E × R+;

u = 0

v = 0in E × {0}: (27)

For E ⊂ R3, Eqs. (27) become

utt = ∇ · (Gtv) + ∇ · (Gvt) +∑Tj=t

GSdj

vt = ∇(Gu)

in E × R+: (28)

The distributions u and v are of special interest by themselves, since they provideadditional information about the morphology of the material, e.g. the total surface ofnuclei in a domain B in the absence of impingement is given by∫

Bu dx := 〈u; IB〉 =

∑Tj¡t

∫@�t

j

IB(x) d+(x) =∑Tj¡t

,d−1(@�tj ∩ B):

The expected value of u is just the mean free metric density, which has been introducedin [8] as one of the most important measures for the morphology of the Hnal material.

5.3. Averaging on macroscopic scales

By coupling with temperature we obtain, in R1 and R2 the system

ft = G(T )(1 − f)u

vt = ∇(G(T )u)

ut = ∇ · (G(T )v) +∑Tj=t

Sdj

(%cT )t = ∇ · ('∇T ) + (hf)t

in E × R+

u + 〈v; n〉 = 0

Tn = )(T − Tout)on @E × R+;


f = 0

u = 0

v = 0

T = T 0

in E × {0}; (29)

whose stochasticity is caused by the sources Sdj only (we obtain an analogous system

in R3, coupling (28) with the temperature).At this point, by the remarks made in Section 4, we may consider a multiple scale

approach, in order to obtain a deterministic system from (29).At the level of the mesoscale, it makes sense to consider a region B small enough

that the space variation of temperature inside B may be denied and such that there isa suLciently high number of “small” crystals in B, so that a law of large numbersallows the approximation

∑Xj∈B

∑Tj6t

Sdj ≈ E

∑

Xj∈B

∑Tj≤t

Sdj

; (30)

i.e. we may approximate the source term by its expected value, which is the mean rateof surface production.

In Eq. (29), at each time t, there is the contribution of the nuclei born exactly atthat time. But at every time t, the probability of birth of a nucleus is zero, so that fora rigorous interpretation of (29), we need to use an integral version of it and introducethe approximation in (30):

∫B�(x; t)u(x; t) dx =

∫B�(x; 0)u(x; 0) dx

+∫ t

0

∫B∇ · (G(T )v(x; s))�(x; s) dx ds +

∑Tj6t

∑Xj∈B

〈Sdj ; �〉

≈∫B�(x; 0)u(x; 0) dx +

∫ t

0

∫B∇ · (G(T )v(x; s))�(x; s) dx ds

+E

∑

Tj6t

∑Xj∈B

〈Sdj ; �〉

≈∫B�(x; 0)u(x; 0) dx +

∫ t

0

∫B∇ · (G(T )v(x; s))�(x; s) dx ds

+∫ t

0

∫BFd[G; �; T (x; s)] dx ds: (31)


More precisely, for a test function �(x; t) smooth enough and s.t. �(x; t) = 0 on @B,

∑Xj∈B

∑Tj6t

〈S1j ; �〉 ≈ E

∑

Xj∈B

∑Tj6t

2�t(Xj; Tj)

=∫ t

0

∫B

2�(x; s)�t(x; s) dx ds

= −2∫ t

0

∫B�t(x; s)�(x; s) dx ds;

∑Xj∈B

∑Tj6t

〈S2j ; �〉 ≈ E

∑

Xj∈B

∑Tj6t

2�G(Xj; Tj)�(Xj; Tj)

= 2�∫ t

0

∫B�(x; s)G(x; s)�(x; s) dx dt;

∑Xj∈B

∑Tj6t

G(Xj; Tj)〈S3j ; �〉 ≈ E

∑

Xj∈B

∑Tj6t

8�G2(Xj; Tj)�(Xj; Tj)

= 8�∫ t

0

∫B�(x; s)G2(x; s)�(x; s) dx ds:

So we have that the source terms in (31) are deHned by

F1[G; �; T ](x; t) := −2�(T (x; t))t ;

F2[G; �; T ](x; t) := 2�G(T (x; t))�(T (x; t));

F3[G; �; T ](x; t) := 8�G2(T (x; t))�(T (x; t)):

In (31), all randomness is eliminated and we obtain a nonlinear initial–boundaryvalue problem for a system of one parabolic and one hyperbolic equation:

@�@t

= G(T )(1 − �)u; (32)

@u@t

= ∇ · (G(T )v) + Fd[G; �; T ]; (33)

@v@t

= ∇(G(T )u); (34)

@@t

(c%T ) = ∇ · (k∇T ) +@@t

(h�); (35)

in E × R+; E ∈ Rd; d = 1; 2, supplemented by the boundary conditions

u + 〈v; n〉 = 0; (36)


@T@n

= )(T − Tout); (37)

on @E × R+ and initial values given by

� = 0; (38)

u = 0; (39)

v = 0; (40)

T = T 0 (41)

in E × {0}, usually with T 0(x) ¿ Tm for all x ∈ E. In order to express the essentialdi0erence between the (generalized) functions f; u; v and T and their equivalents inthe averaged equations we now write �; u; v and T .

In R3, introducing the variable w = @u=@t, the system becomes@�@t

= G(T )(1 − �)u;

@u@t

= w;

@w@t

= ∇ ·(

@@t

G(T )v)

+ ∇ ·(G(T )

@@t

v)

+ F3[G; �; T ];

@v@t

= ∇(G(T )u);

@@t

(c%T ) = ∇ · (k∇T ) +@@t

(h�):

The parameters c; %; k; h are consequently averaged by substituting � to I�t in (14).For a rigorous derivation of (33) as a limit of a suitable stochastic counterpart, the

reader is referred to [9].

6. Conclusions and future work

We have seen that the Avrami–Kolmogorov–Evans theory is applicable to any crys-tallization process, where the growth and nucleation rates are arbitrary functions oftime and temperature, but independent of the actual crystalline phase. If the rates de-pend upon the actual pattern of the crystalline phase by some reason, e.g. because ofinteraction with temperature, the theory fails, since the causal cone cannot be deHnedanymore as a deterministic region. Nevertheless, the results obtained that way yielda good approximation under typical conditions, which is conHrmed by the alternativedirect approach presented in Section 5, which clearly shows the averaging included inthe model using the di0erent scales of the process. In addition, the direct approachseems to be more Nexible for other situations, like age-dependent growth, too.


We note that the averaging procedure presented in this paper was only qualitative, aquantitative estimation of the error made by using the averaged model seems to be animportant task for future work. Together with this estimate, one might also examineother possible simpliHcations of the model, since perturbations of the same order as themodel error can obviously be disregarded, too. Especially, the terms including spatialderivatives with the ‘small’ factor G might possibly be disregarded.

Another important task for future investigations will be the identiHcation of modelparameters from temperature measurements and phase observations in (32)–(41). In thepast, this problem had been treated only in the case of one-dimensional crystallization(cf. [5]).

Acknowledgements

This work has been carried out in the framework of the ECMI Special Interest Group‘Polymers’ with partial support by the EU under the TMR research network Di=eren-tial Equations in Industry and Commerce and the Austrian Fonds zur FWorderung derwissenschaftlichen Forschung, project P 13478-INF. Further support by ASI, contractARS-96-121, MURST 40% ‘Stochastic Calculus’ and GNFM-CNR is also acknow-ledged.

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Modelling multi-dimensional crystallization of polymers in interaction with heat transfer

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