JEC COMPOSITES MAGAZINE - Issue #114 - July 2017 - 55

A novel physics node for Nakamura crystallization kinetics

(1)
where t is time, n is the Avrami index and K(T) is the
Nakamura kinetics crystallization function, which is related
to the Avrami isothermal kinetics function, KAvrami(T), by
K(T)=Kavrami^(1/n). In the following, the G(α) function is
called the Nakamura function. It is represented in Figure 2.
Nakamura crystallization kinetics is widely used to predict the
advancement of the crystallization transformation in polymer
forming processes [1, 4, 5, 6, 7].
Modified Nakamura kinetics
The Nakamura function, G, presents a singular derivative in the vicinity of α=0 and α=1. Therefore, a modification of the Nakamura
function is proposed by extrapolating it below αmin=10-6 and
above αmax =1-10-6. These modifications, discussed in a recent
work [8], are illustrated in Figure 2. They (i) ensure a robust numerical integration by forcing real values of G when α is not in the
[0, 1] interval, and (ii) force an artificial germination when α=0.

Fig. 2: The original Nakamura function G (Equation 1) is singular when α=0
and α=1 (vertical tangent, represented by the red arrows). In the proposed
modified extrapolation, a thresholding is performed below αmin, and a
linear extrapolation above αmax

Heat transfer coupling
In real industrial processes, as illustrated in Figure 1, the coupling
of crystallization kinetics and heat transfer is of primary interest.
Crystallization kinetics and its domain-dependent variable, α, is
strongly coupled to heat transfer and its domain-dependent variable: the temperature, T. On the one hand, crystallization is an
exothermic reaction such that a source term appears in the heat
transfer equilibrium equation:

(2)
where ρm is the material density (matrix density in the case of a
composite material), φm the matrix volume content and H the
enthalpy of crystallization. Moreover, the thermal properties
(conductivity, density and specific heat) depend on the degree of
crystallization. These dependencies are imposed by the material
behaviour.
On the other hand, temperature appears to drive crystallization
kinetics. It appears explicitly with the Nakamura kinetics crystallization function, K(T), in Equation 1. This function is also
temperature-dependent. A typical example for PEEK is given in
Figure 3.

Implementation of a dedicated physics in COMSOL
In a physical domain, the degree of crystallization, α, has to be
solved for each location in space. The ruling ODE (Equation 1)
could be implemented in any COMSOL model using the domain ODE node. After having done so for several years, with
a multitude of models, the team decided to implement this
Nakamura kinetics in a novel physics node, using the Physics

Fig. 3: Avrami kinetics crystallization function, KAvrami, and Avrami index,
n, versus temperature for a PEEK polymer, as characterized by Tardiff et
al [9]

Builder in COMSOL. First, this makes it possible to implement
the Nakamura function modification once and for all (see Figure
2). Then, this simplifies the possible coupling with heat transfer
(exotherm computation, for instance). And finally, this facilitates
the creation of new models by providing readily available physical
material properties (see Figure 3). In addition, all the variables
then have physical names and consistent units. This prevents mistakes while developing new models.
Domain and dependent variable
The Nakamura kinetics equation (1) has to be solved at each material location to obtain the field of degree of crystallization, α.
Thus, α is defined as the dependent scalar variable over a domain
whose dimension can be either 0, 1, 2 or 3D. A crystallisation domain feature node is added in the Physics Builder as the default
node.
Material properties and user input
Equation (1) calls for two material properties, the Avrami index,
N°114 July 2017

/ jec composites magazine

55



Table of Contents for the Digital Edition of JEC COMPOSITES MAGAZINE - Issue #114 - July 2017

Cover
Edito
Opinion: Composite music
Contents
NEWS
In brief
Raw materials
Agenda
BUSINESS
Thermoset resin - Part B
Glass fibre
Construction
MANUFACTURING
Control
Out-of-autoclave
Recycling
Feature Sports & Leisure - Composites: the key to performance?
Software
CFRP
Simulation
Golf
Biocomposites
Natural fibre
SOLUTIONS
Infrastructure
Roofing and siding
In the world
TECHNOLOGY
Software
A novel physics node for Nakamura crystallization kinetics
Index
Advertisers
JEC COMPOSITES MAGAZINE - Issue #114 - July 2017 - 60
JEC COMPOSITES MAGAZINE - Issue #114 - July 2017 - Cover
JEC COMPOSITES MAGAZINE - Issue #114 - July 2017 - 2
JEC COMPOSITES MAGAZINE - Issue #114 - July 2017 - Edito
JEC COMPOSITES MAGAZINE - Issue #114 - July 2017 - Opinion: Composite music
JEC COMPOSITES MAGAZINE - Issue #114 - July 2017 - 5
JEC COMPOSITES MAGAZINE - Issue #114 - July 2017 - Contents
JEC COMPOSITES MAGAZINE - Issue #114 - July 2017 - 7
JEC COMPOSITES MAGAZINE - Issue #114 - July 2017 - In brief
JEC COMPOSITES MAGAZINE - Issue #114 - July 2017 - 9
JEC COMPOSITES MAGAZINE - Issue #114 - July 2017 - 10
JEC COMPOSITES MAGAZINE - Issue #114 - July 2017 - Raw materials
JEC COMPOSITES MAGAZINE - Issue #114 - July 2017 - AD1
JEC COMPOSITES MAGAZINE - Issue #114 - July 2017 - AD2
JEC COMPOSITES MAGAZINE - Issue #114 - July 2017 - 12
JEC COMPOSITES MAGAZINE - Issue #114 - July 2017 - AD3
JEC COMPOSITES MAGAZINE - Issue #114 - July 2017 - AD4
JEC COMPOSITES MAGAZINE - Issue #114 - July 2017 - 13
JEC COMPOSITES MAGAZINE - Issue #114 - July 2017 - Agenda
JEC COMPOSITES MAGAZINE - Issue #114 - July 2017 - Thermoset resin - Part B
JEC COMPOSITES MAGAZINE - Issue #114 - July 2017 - 16
JEC COMPOSITES MAGAZINE - Issue #114 - July 2017 - 17
JEC COMPOSITES MAGAZINE - Issue #114 - July 2017 - 18
JEC COMPOSITES MAGAZINE - Issue #114 - July 2017 - Glass fibre
JEC COMPOSITES MAGAZINE - Issue #114 - July 2017 - 20
JEC COMPOSITES MAGAZINE - Issue #114 - July 2017 - Construction
JEC COMPOSITES MAGAZINE - Issue #114 - July 2017 - 22
JEC COMPOSITES MAGAZINE - Issue #114 - July 2017 - 23
JEC COMPOSITES MAGAZINE - Issue #114 - July 2017 - Control
JEC COMPOSITES MAGAZINE - Issue #114 - July 2017 - 25
JEC COMPOSITES MAGAZINE - Issue #114 - July 2017 - 26
JEC COMPOSITES MAGAZINE - Issue #114 - July 2017 - 27
JEC COMPOSITES MAGAZINE - Issue #114 - July 2017 - Out-of-autoclave
JEC COMPOSITES MAGAZINE - Issue #114 - July 2017 - 29
JEC COMPOSITES MAGAZINE - Issue #114 - July 2017 - Recycling
JEC COMPOSITES MAGAZINE - Issue #114 - July 2017 - 31
JEC COMPOSITES MAGAZINE - Issue #114 - July 2017 - 32
JEC COMPOSITES MAGAZINE - Issue #114 - July 2017 - Software
JEC COMPOSITES MAGAZINE - Issue #114 - July 2017 - 34
JEC COMPOSITES MAGAZINE - Issue #114 - July 2017 - CFRP
JEC COMPOSITES MAGAZINE - Issue #114 - July 2017 - 36
JEC COMPOSITES MAGAZINE - Issue #114 - July 2017 - Simulation
JEC COMPOSITES MAGAZINE - Issue #114 - July 2017 - Golf
JEC COMPOSITES MAGAZINE - Issue #114 - July 2017 - 39
JEC COMPOSITES MAGAZINE - Issue #114 - July 2017 - Biocomposites
JEC COMPOSITES MAGAZINE - Issue #114 - July 2017 - 41
JEC COMPOSITES MAGAZINE - Issue #114 - July 2017 - Natural fibre
JEC COMPOSITES MAGAZINE - Issue #114 - July 2017 - 43
JEC COMPOSITES MAGAZINE - Issue #114 - July 2017 - 44
JEC COMPOSITES MAGAZINE - Issue #114 - July 2017 - AD5
JEC COMPOSITES MAGAZINE - Issue #114 - July 2017 - AD6
JEC COMPOSITES MAGAZINE - Issue #114 - July 2017 - SOLUTIONS
JEC COMPOSITES MAGAZINE - Issue #114 - July 2017 - 46
JEC COMPOSITES MAGAZINE - Issue #114 - July 2017 - 47
JEC COMPOSITES MAGAZINE - Issue #114 - July 2017 - Infrastructure
JEC COMPOSITES MAGAZINE - Issue #114 - July 2017 - Roofing and siding
JEC COMPOSITES MAGAZINE - Issue #114 - July 2017 - In the world
JEC COMPOSITES MAGAZINE - Issue #114 - July 2017 - Software
JEC COMPOSITES MAGAZINE - Issue #114 - July 2017 - 52
JEC COMPOSITES MAGAZINE - Issue #114 - July 2017 - 53
JEC COMPOSITES MAGAZINE - Issue #114 - July 2017 - A novel physics node for Nakamura crystallization kinetics
JEC COMPOSITES MAGAZINE - Issue #114 - July 2017 - 55
JEC COMPOSITES MAGAZINE - Issue #114 - July 2017 - 56
JEC COMPOSITES MAGAZINE - Issue #114 - July 2017 - 57
JEC COMPOSITES MAGAZINE - Issue #114 - July 2017 - Advertisers
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