Journal of Modern Physics, 2013, 4, 28-37
http://dx.doi.org/10.4236/jmp.2013.47A2005 Published Online July 2013 (http://www.scirp.org/journal/jmp)
Thermophysical Characterization and Crystallization
Kinetics of Semi-Crystalline Polymers
Matthieu Zinet, Zakariaa Refaa, M’hamed Boutaous*, Shihe Xin, Patrick Bourgin
Université de Lyon, CNRS, INSA-Lyon, CETHIL, UMR 5008, F-69621, Villeurbanne, France
Received April 15, 2012; revised May 18, 2013; accepted June 23, 2013
Copyright © 2013 Matthieu Zinet et al. This is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
ABSTRACT
Final properties and behavior of polymer parts are known to be directly linked to the thermomechanical history experi-
enced during their processing. Their quality depends on their structure, which is the result of the interactions between
the process and the polymers in terms of thermomechanical kinetics. To study the actual behavior of a polymer during
its transformation, it is necessary to take into account all the thermal dependencies of their thermophysical properties. In
this paper, a complete experimental thermal characterization of a semi-crystalline polymer is performed. Thermal con-
ductivity is measured using the hot wire method. The PVT diagram is obtained by means of an isobaric piston type di-
latometer. Heat capacity is characterized versus temperature by differential scanning calorimetry (DSC). A modification
of the Schneider rate crystallization equations is proposed, allowing to identify in a simple way all the crystallization
kinetics parameters, using only DSC measurements. Finally, a multiphysical coupled model is built in order to numeri-
cally simulate the cooling of a polypropylene plate, as in the cooling stage of the injection molding process. Calculated
evolutions of temperature, crystallinity, pressure and specific volume across the plate thickness are presented and com-
mented.
Keywords: Thermophysical Characterization; Heat Transfer; Crystallization Kinetics; Polymer; Multiphysical
Modeling
1. Introduction
Polymers materials play an important role in the industry,
especially polymer based composite materials, which are
increasingly used in many industrial domains such as au-
tomotive, aeronautics, building and civil engineering. They
represent solutions for weight and cost reduction, energy
saving, increasing safety and possibilities of innovation
in design.
Due to a lack of experimental data (characterization
experiments are costly and difficult) and to the complex-
ity of the multiphysical coupling involved in the numeri-
cal simulation of the behavior of polymers in processing
condition, only few complete parametric studies are avai-
lable in the literature. The thermal and mechanical rates
experienced by the polymer during processing determine,
to a great extent, the final material structure which in turn
governs its final thermomechanical behavior. In semi-
crystalline polymers, the crystallization phenomenon plays
a crucial role in the quality of industrial parts. It strongly
affects rheological properties and influences mechanical
and barrier properties of obtained objects [1].
Many efforts have been devoted to the characterization
of these materials and to the understanding of the funda-
mentals concerning their physical structure, their ther-
momechanical behavior and the link between the struc-
ture, the process and the final properties [2-5]. Neverthe-
less, there are still many phenomena to investigate in
order to master the polymer final behavior, to explain the
mechanisms involved in polymer transformation and/or
crystallization in real industrial cases and to propose in-
novative industrial solutions. For instance, in the injec-
tion molding process (one of the most used in polymer
processing industry), crystallization kinetics directly in-
fluences the part behavior regarding both its dimensional
stability and its mechanical and thermal performances.
Indeed, the different crystalline morphologies and sizes
are responsible for the residual stresses within the parts,
leading to shrinkage and deformation of the material. In
other words, the final physical structure is responsible for
the thermal and mechanical characteristics of the obtain-
ed parts.
For the modeling of such material transformation,
*Corresponding author.
C
opyright © 2013 SciRes. JMP

M. ZINET ET AL. 29
crystallization kinetics must be accurately described and
characterized. The first global theory was developed by
Avrami in 1939 [6] and generally serves as the basis for
all the subsequent global theories of crystallization from
the melt.
In processing conditions, the crystallization kinetics
depends on the thermodynamical state of each point of
the material: temperature, temperature rate, pressure, spe-
cific volume (V) and intrinsic thermophysical properties
such as thermal conductivity (λ) and specific heat (Cp).
Furthermore, all these characteristics change depending
on whether the polymer is in a solid or liquid state. A
simple way to model the crystallization rate is to mimic
the metallic materials behavior. This approach considers
that the solidification process of the polymer in the mold
is similar to the quench of a slab, due to sudden contact
between the mold cold surfaces and the hot polymer.
This quenching process assumes that the crystallization
front moves from the interface to the core zone as in me-
tallic materials, i.e. proportionally to the square root of
time. According to this theory, there is no delay in crystal-
lization when the melting point is reached during cooling
[7].
It is now established that this theory is not consistent.
At least, the temperatures at which crystallization occurs
depend drastically on the cooling rate [8]. In addition, as
it was developed in our previous work [3,9], the pressure
and the flow enhance the crystallization rate during pro-
cessing by orienting the polymer chains. This creates a
structural gradient in the part thickness, which leads in
turn to other consequences on the part behavior. The spe-
cific volume evolution is another influencing parameter
which should be taken into account when modeling the
cooling of semi-crystalline polymers. Although it can be
assumed that the cooling rate does not affect the specific
volume of amorphous polymers, the specific volume of
semi-crystalline polymers is strongly related to the de-
gree of crystallinity, which itself depends on the tempe-
rature evolution and pressure [10]. Hence, modeling of
polymer cooling is a multidisciplinary and multiphysical
problem that requires careful parameters identification
and numerical couplings.
In this paper, we propose an experimental analysis
protocol in order to estimate the intrinsic thermophysical
properties p of a semi-crystalline polymer, in-
cluding their thermal dependency from the melt state to
the solid state. The calorimetric measurements needed to
identify all the required parameters to model the crystal-
lization kinetics using a modified Schneider approach [11]
are also presented. The methodology and the measure-
ments constitute a very helpful complete set of data for
the literature. The modified Schneider model gives ac-
cess to an easy way of identifying the crystallization pa-
rameters, using only calorimetric (DSC) measurements.
Moreover, this model based on the spherulitic nucleation
and growth could provide a mean to predict the size dis-
tribution of the crystallites and thus to quantify the mi-
crostructure gradient within the thickness of the part. An
example case of numerical simulation of the cooling of a
polypropylene (PPH) plate in quiescent condition is then
described and analyzed. In this multiphysical coupled
model, all the thermophysical parameters are temperature
dependant. The PVT behavior is also introduced. The ef-
fect of pressure on the crystallization kinetics parame-
ters is measured or identified in both isothermal and non
isothermal conditions.
2. Experimental Measurements
The investigated material is an injection molding grade
homopolymer polypropylene (PPH) supplied by Atofina.
Several experimental techniques have been used to esti-
mate or identify the various parameters: calorimetric mea-
surement by means of a Dynamic Scanning Calorimeter
(DSC) apparatus for the crystallization kinetics, transi-
tion temperatures and heat capacity of the material from
the melt to the solid state; dilatometry measurements for
the specific volume; and hot wire apparatus to estimate
the thermal conductivity.
2.1. Thermal Conductivity
The thermal conductivity was measured using the hot
wire technique from “K-system II” apparatus [12] during
slow cooling from liquid state to solid state for several
temperatures. The advantage of this apparatus is its large
temperature measurement range. The thermal conductiv-
ity values obtained for the PPH from the melt to the solid
state are represented in Figure 1.
The measured thermal conductivity was then fitted to a
linear relation versus temperature for the amorphous liq-
uid phase, Equation (1), and the semi-crystalline solid
phase, Equation (2):

,,lC V
Figure 1. Measured thermal conductivity evolution versus
temperature of PPH.

M. ZINET ET AL. 30

1.071 10
aT
 
40.1455T 

40.1858T 
11
K

 
(1)
1.377 10
sc T
 
Wm
(2)
with T in ˚C and λ in .
During solidification, the thermal conductivity also
varies also with the relative crystallinity (α). Hence, a
simple mixing rule was used to describe its evolution as a
function of temperature and relative crystallinity, Equa-
tion (3):


,1
a
TT T   

V

 
1
ca
VX V 

 

sc (3)
2.2. PVT Measurements
The PVT diagram (Figure 2) represents the specific vo-
lume variations versus temperature and pressure. Two
measurement techniques are available: the piston type
and the immersion type dilatometers.
For this study, the PVT experimental results were ob-
tained by using a piston-type dilatometer (PVT100) com-
merciallized by SWO HAAKE Polymertechnik GmbH.
Experiments were performed in isobaric cooling mode
from 260˚C to 30˚C, for different pressures ranging from
20 MPa to 160 MPa, and for a 5˚C/min cooling rate. The
specific volume is modeled by a mixture law between the
fitted amorphous specific volume a and the crystal-
line specific volume . Following the work of Ful-
chiron [13], the specific volume can be written as:

c
V
,X  VT (4)
The amorphous specific volume was described by the
empirical Tait equation from [14]:
 
0
,1
a
VT
P VTCln1
P
BT



 




(5)
Figure 2. PVT diagram of PPH, obtained in isobaric mode
with a cooling rate of 5˚C/min.
where C is a universal constant: C = 0.0894. The pa-
rameters of Tait equation were obtained by fitting the
liquid part of the PVT measurements:




4
01.2003exp 9.104110VT T

 (6)




3
81.29exp4.81 10BT T



(7)
with T in ˚C, P in MPa and Vc in cm3/g. The specific
volume of the crystalline phase was described by a linear
variation, where parameters were adjusted together with
the absolute crystallinity of the polymer
X
, in order
to obtain the best fit of the experimental results:


44
,1.0723.481 103.861 10
c
VTPT P


0.477X
(8)
where T is in ˚C, P is in MPa, Vc is in cm3/g and

, from the literature.
2.3. DSC Measurements
A dynamic scanning calorimeter (DSC) measures the
temperatures and heat flux associated with transitions in
the polymer as a function of time and temperature. These
measurements provide qualitative and quantitative in-
formation on physico-chemical transformations such as
crystallization, melting or glass transition. This technique
is also suitable to measure changes in the heat capacity of
the polymer as a function of temperature.
A DSC (Perkin Elmer PYRIS Diamond™ DSC) was
used to measure enthalpy changes in a sample, during se-
veral thermal cycles. The apparatus was first calibrated in
temperature and heat flow.
The following thermal cycles were used to measure
the heat capacity: first the sample was heated from 20˚C
to 240˚C and maintained at this temperature to erase the
thermomechanical history, and then the sample was cool-
ed at 5˚C/min, 20˚C/min and 50˚C/min.
Using the obtained heat flux curves, the heat capacity
was calculated using a DSC calibration as follows:



dd 60
dd
p
r
Ht EH
CE
mTt Vm


 (9)
where E is a calibration constant, H is the heat flux (mW),
Vr is the cooling rate (˚C/min) and m is the mass of the
sample (mg).
Heat capacity (Figure 3) is fitted to a linear relation
versus temperature for the liquid state Equation (10) and
the solid state Equation (11).


3.4656 1774.2
pa
CT T
(10)


8.3619 1329.2
psc
CT T (11)
with T in ˚C and Cp in J·kg−1·K−1
Just as thermal conductivity, heat capacity varies with
the relative crystallinity. The same mixing rule was used

M. ZINET ET AL. 31
to describe its evolution as a function of temperature and
relative crystallinity, Equation (12).
 


1
pa
CT  
0
T

cf
TfT
,
ppsc
CT CT   (12)
In order to study the crystallization kinetics, the same
DSC apparatus was used to perform isothermal and non-
isothermal crystallization measurements, for several tem-
peratures and cooling rates.
From the obtained heat flow curves Figures 4 and 5,
the relative crystallinity was calculated from the partial
area method (the area under the crystallization peak at in-
stant t over the total area of the peak), see further in Fig-
ures 6 and 7.
The DSC measurements also give an estimation of the
thermodynamic equilibrium melting temperature m of
the polymer, using the Hoffman-Weeks method [5] and
the analysis of isothermal measurements.
In this approach, the equilibrium melting temperature
is given by the intersection between the linear curve of
the evolution of the crystallization temperature versus the
melting temperature of the polymer and the
curve of Tm = Tc.
Figure 3. Heat capacity of PPH versus temperature.
Figure 4. Measured heat flux for isothermal crystallizations.
Figure 5. Measured heat flux for constant cooling rate crys-
tallizations.
Figure 6. Measured and simulated isothermal crystalliza-
tion rates.
Figure 7. Measured and simulated non isothermal crystal-
lization rates.
After summarizing all the DSC measurements, m T
was found to be 200˚C as it is shown in Figure 8. The
glass transition temperature Tg was found to be −10˚C.
0

M. ZINET ET AL. 32
Figure 8. Determination of the equilibrium melting tem-
perature.
3. Crystallization Parameters
3.1. Crystallization Kinetics Model
The evolution of the relative crystallinity


t 
 
1exptt 
 

Nt


Gt
as a
function of time and temperature history is described by
means of the quiescent crystallization kinetics theories.
They generally belong to two types of approaches: the
geometrical approach [6] expresses the volume occupied
by the semi-crystalline part of the polymer, whereas the
probabilistic approach expresses the probability that an
element of volume crystallizes [15,16]. The mathemati-
cal developments inferred by these approaches differ;
nevertheless, they lead to identical results.
In the geometrical approach, the free growth of sphere-
ical crystalline entities (representing the spherulites) is
modeled, then a correction is introduced to take into ac-
count the fact that the growth is actually not free but lim-
ited by the contact between adjacent entities (the socal-
led impingement). This results in the following expres-
sion:


 (13)
where  ’ is the extended crystalline fraction correspond-
ing to the fictitious free growth of the crystallites. The set
of equations developed by Schneider et al. [11] describes
the evolution of the extended crystalline fraction on the
basis of the spherulitic nucleation and growth process.
According to these concepts, is the nuclei activa-
tion rate (in s−1·m−3) and is the linear growth rate
of the spherulites (m·s−1). Ri being the spherulite radius,
the extended crystalline fraction (i.e. the crystalline vo-
lume per material volume unit if impingement is disre-
garded) is expressed as:
 
33
4
π
33
ii
ii
Rt 

0
4
πt tRt 




 (14)
Successive time derivations of this expression lead to:
 
2
0
1
dd
4π
dd
i
i
R
RGtt
tt





 (15)



 
12
2
dd
4π
dd
i
i
tRtGt t
tt





 (16)

 
2
3
dd
8π
dd
i
i
tRt Gtt
tt





 (17)


3
d8π
d
tNt
t

 (18)
with the following intermediate variables:


3
14πitot
i
tRtSt  

(19)
representing the total surface area of the spherulites Stot,




28π8π itot
i
tRtRt  

(20)
2

is proportional to the sum of the spherulites radii Rtot,
and:


38πtNt
  (21)
3

is proportional to the number of spherulites per unit
volume.
The final set of these so-called rate equations reads:



 
 
3
23
12
01
8πtNt
tGt t
tGt t
tGt t
















(22)
and the relative crystallinity is ultimately given by:

 
0
1exptt  

(23)
The main benefit of the Schneider modeling approach
is that the simulated geometrical characteristics of the
crystalline phase (number of spherulites, radii, surface
area) can be retrieved thanks to the intermediate vari-
ables of the system.
3.2. Modified Rate Equations
For polypropylene, nucleation is known to be heteroge-
neous: for a given supercooling, the activated nuclei ap-
pear simultaneously; their number per unit volume is re-
lated to temperature according to the following relation-
ship [17]:

 


0
exp m NTtaT Ttb



 (24)


0
T where m is the equilibrium melting temperature, a and
b are two parameters to be experimentally determined.
The linear growth rate is temperature dependant and

M. ZINET ET AL. 33
given by the Hoffman-Lauritzen model [18]:
 

0exp exp
U
GT GRT T








0
g
m
K
TT T






U
30 CTT
0
T
(25)
where is the activation energy, R is the gas constant,
gis the temperature below which molecu-
lar motion becomes impossible, Tg is the glass transition
temperature, m is the equilibrium melting temperature,
and G0 and Kg are two polymer parameter to be deter-
mined. Thus, the time dependant growth rate appearing
in Equation (22) can be written:


 
0
GGTt

Gt  (26)
Insertion of Equations (24) and (25) in the set of rate
equations, Equation (22) leads to:
 



  


 




 




0
3
20
02
2
10 2
2
01
3
00 1
3
00
d
8πexp exp
d
8πexp exp
8πexp
8πexp
8πexp
8πexp
8πexp
m
taTT
t
tGbGTt
Gbt
tGbGTtt
Gbt
tGbGTt
Gbt











































0
m
tb
aTTt
t
 


i
(27)
A modified set of rate equations with new intermediate
variables




0
m
aTTt
is thus obtained:
 

   
 


2
12
01
exptGTt
tGTt t
tGTt t




 
 







000
8πtCt 


3
00
expCG b

G
0
T
0
T
0
m
T









(28)
and the link between this modified set of rate equation
and the previously defined extended crystalline fraction
is:

(29)
where C0 is a polymer parameter defined as:
(30)
Note that the parameter C0 is a combination of a
growth related parameter 0 and a nucleation related
parameter (b). On account of this, it has no strict physical
meaning, but it allows the experimental characterization
of the crystallization kinetics by means of differential
scanning calorimetry only. Indeed, the nucleation rate
effect and the growth rate effect on crystallinity evolu-
tion measurements obtained from DSC experiments can-
not be completely discriminated: G0 and exp b cannot be
individually identified as they always appear together as
a product in the rate equations.
Thanks to this modified Schneider formalism, the cry-
stallization kinetics of the polymer under quiescent con-
ditions is completely described when the parameters m,
Tg, U*, a, Kg and C0 are known. The equilibrium melting
temperature m is determined by the Hoffman-Weeks
method as shown in Section 2. The glass transition tem-
perature Tg is known to be 0˚C for polypropylene. The
activation energy U* can be assigned a universal value of
6270 J·mol−1. The parameters a, Kg and C0 are identified
simultaneously from isothermal crystallization and con-
stant cooling rate crystallization DSC experiments, as de-
scribed in the next section.
3.3. Parameter Identification
In isothermal crystallization experiments, the polymer is
melted above the equilibrium melting temperature, then
the temperature of the polymer is lowered as fast as pos-
sible and maintained at the constant crystallization tem-
perature T (lower than ). Consequently, the nuclea-


0
exp aTT
m

tion term  

and the growth term G* re-
main at constant values during crystallization. Hence,
integrating the set of rate equations, Equation (28) leads
to:


3
30
0exp 6
m
t
GTaT T 





 (31)


0t

where t is the current time, corresponding to the
instant at which the polymer melt reaches the crystalliza-
tion temperature T. Using Equations (23) and (29), the
evolution of the relative crystallinity is explicitly given
as a function of time by:


303
0
4π
1exp exp
3m
tCGTaTTt 





 





(32)
In constant cooling rate crystallization experiments, the
polymer is melted above the equilibrium melting tem-
perature, and then the temperature of the polymer is de-
creased at a controlled constant rate whereas crystalliza-
tion occurs. Contrary to the isothermal case, the set of
rate equations cannot be explicitly integrated to obtain
the crystallinity evolution expression. Hence, the itera-
tive identification procedure must be run in conjunction
with a numerical time integration algorithm of the rate
equations.
The parameter identification procedure consists of
finding a single set of values for parameters a, Kg and C0
that lead to the overall smaller deviation between the ex-
perimental crystallinity evolutions and the calculated cry-

M. ZINET ET AL. 34
stallinity evolutions given either by Equation (32) (iso-
thermal cases) or by the numerical integration procedure
(constant cooling rate case). The optimization algorithm
used here is the Levenberg-Marquardt method [19,20].
Designed as an enhancement of the Gauss method, it is
based on the minimization of a quadratic variance crite-
rion (least square method).
In the present application, the objective function is de-
fined as the sum of the squared differences between the
experimentally determined relative crystallinity values and
the calculated ones. Since isothermal experiments at sev-
eral temperatures and non-isothermal experiments at se-
veral cooling rates are considered jointly in the identifi-
cation procedure (care being taken that each single ex-
periment has an equivalent weight in the objective func-
tion), the resulting set of parameters is likely to yield the
best overall description of the polymer crystallization ki-
netics over the possible experimental range allowed by
the DSC technique. After application of this procedure,
the values obtained for the parameters are:
15 1
11
52
015 10s
13 10K
4.83610K





02.
1.1
g
C
a
K







Figures 6 and 7 show comparative curves of the mea-
sured and modeled relative crystallinity evolution, under
isothermal and constant cooling rate conditions. The mo-
deled curves have been obtained using the identified set
of parameters.
The agreement between the modeled evolutions and
the experimental ones is fairly good. The reduction of the
crystallization half time as temperature decreases is well
predicted by the model. Thanks to the simultaneous op-
timization of the model against isothermal and constant
cooling rate experiments, the validity range (in terms of
crystallization temperature) of the identified parameters
can be estimated to at least [90˚C - 140˚C].
4. Application to the Cooling of a PPH Plate
In this section, an example of simulation using the char-
acterization data obtained in the previous section is pre-
sented. The analyzed case is drawn from the injection
molding process, and especially the cooling stage. Start-
ing from a thermally homogeneous mold and polymer
melt under high pressure, the isochoric cooling phase (i.e.
constant material volume), which ends as soon as atmos-
pheric pressure is reached and polymer shrinkage begins,
is modeled and simulated. The model introduces a coupling
between heat transfer, crystallization, compressibility and
the temperature evolution of all the intrinsic thermophy-
sical parameters.
First, the equations describing pressure evolution, heat
transfer and crystallization in the cavity are introduced.
Then these equations are coupled together in order to si-
mulate the isochoric cooling process. Note that in this
work, the pressure history is not an input data, but a com-
putational result. Indeed, pressure evolution is a direct
consequence of the thermodynamical behavior of the po-
lymer. The assumption of hydrostatic (i.e. homogeneous)
pressure in the cavity is made. This is quite realistic as
long as the polymer is in the melt state, but the reality is
far more complex for the solidified state as physical pro-
perties variations rapidly occur in the crystallizing mate-
rial.
Figure 9 shows the simulated geometrical configura-
tion and the thermal boundary conditions. The cavity
(thickness = 2ep = 3 mm) is filled with polypropylene
(PPH) melt and surrounded by two steel blocks (thick-
ness = e
m). For symmetry reasons, only one half of the
system is modeled. In addition, the lateral insulation boun-
dary conditions account for an infinite length in the x di-
rection. Although the model geometry is bidimensional,
for the sake of simplicity, the boundary conditions are
chosen such that the problem reduces to a 1D case.
In the isochoric cooling phase, both total cavity vol-
ume and total polymer mass present in the cavity are
conserved. Consequently, there exists a coupling between
pressure and specific volume, which is taken into account
by considering mass and momentum conservation equa-
tions:


,, ,, 0 PT PT
t
  

  

u

(33)

T
,,
2
11
3
PT t
P












 



uuu
uu u
(34)

Figure 9. Geometry and boundary conditions of the thermal
model.

M. ZINET ET AL. 35
where the density  is the inverse of the specific volume
V, is a velocity vector, and  is the dynamic viscosity.
Note that the motion occurring in the polymer is only due
to the local contraction and expansion caused by the
evolution of the density gradients in the material. Con-
sequently, the viscosity value (taken here as 106 Pa) has
no effect on the calculation results.
u

It is assumed that the only heat transfer mode is con-
duction, because no significant convective motion can
occur in the packed polymer and the material is opaque
to radiation. Furthermore, the thermal contact between
the polymer and the mold is assumed to be perfect. Heat
transfer in the polymer is modeled by the energy conser-
vation equation in a compressible and phase-changing
medium [21]:

 


,
,,
,,
p
CT T
VPT t
VPT t









,TT
HP
TT





(35)
where the two right-hand source terms are respectively
the latent heat of crystallization release and the compres-
sibility heat release. In Equations (33) to (35), the ther-
mophysical properties Cp, V,  were specified in section
3. By analogy with polymer composites [22], the coeffi-
cient of thermal expansion is approximated by:

11
a
1
c
P
P a
V
X
VT






11
W mKl


c
V
X
VT
 



 


 (36)
The crystallization kinetics of the polymer is modeled
by the Schneider modified rate equations, Equation (28)
and the relative crystallinity is calculated according to
Equations (23) and (29). Heat transfer in the metal is de-
scribed by the heat conduction equation with constant
thermophysical properties and no source term, since no
phase change occurs and the material is assumed to be
incompressible:
11
475 JkgK,45
p
C


33
0.12710m /kgV 
and
. In the present analysis, the ac-
tual injection molding cooling stage have been somewhat
simplified. At initial time t0, the temperature T0 is as-
sumed to be homogeneous in the whole system, i.e. me-
tallic block and polymer plate, and an initial pressure P0
is fixed. At temperature T0, the polymer is supposed to be
fully amorphous. The thermal boundary conditions are
presented on Figure 9. A convection boundary condition
is applied outside the mold; the thermal exchange coeffi-
cient h is set at 5000 W·m−²·K−1 and the fluid tempera-
ture Tc = 20˚C (typical values for steel-water heat transfer
in a mold cooling circuit). The whole set of equations is
then solved over the considered geometry using the finite
element method. At each time step, the thermal boundary
condition and the heat source terms influence the tem-
perature field. Crystallization progression is computed
and thermophysical properties are updated (specific vo-
lume in particular), such that the mass and momentum
equations are satisfied and yield an updated pressure va-
lue. This coupling is a strong one: the equations are not
solved sequentially but together iteratively. When the pres-
sure reaches the atmospheric value, convergence is no
longer obtained as the mass conservation equation cannot
be satisfied, and calculations are stopped. This is the end
of the isochoric cooling stage.
Since the following stage of the process does not take
place at constant volume but at constant pressure, the pre-
sent mathematical model is no longer relevant. In the
actual process, if the temperature continues to decrease,
an air gap between the polymer and the mold wall will
appear because of the polymer shrinkage.
A simulation with initial temperature: T0 = 220˚C and
initial pressure: P0 = 200 MPa yields the temperature,
relative crystallinity, pressure and specific volume evolu-
tions. Figure 10 shows temperature evolutions across the
polymer thickness


p
YYe
. Due to the thermal in-
ertia of the mold, temperature starts to decrease at the
wall/polymer interface after 20 s. At the core, the quasi-
plateau of crystallization appears at 80 s, which corre-
sponds to 158˚C. Figure 11 clearly shows that crystalli-
zation near the mold wall (skin layer) is faster than at the
center (core). After 140 s, the polymer is completely so-
lidified.
Pressure evolution is shown on Figure 12. The curve
exhibits 4 distinct parts. In part one, the pressure remains
constant: cooling has not yet started in the polymer due
to the mold thermal inertia. In the second part, the poly-
mer is in the amorphous state, its temperature decreases
and pressure decreases in an almost linear way due to the
isochoric cavity assumption. When crystallization occurs
(third part), the specific volume strongly decreases and
pressure falls faster. Finally, as crystallization is com-
Figure 10. Temperature evolution at several locations across
polymer thickness (T0 = 220˚C, P0 = 200 MPa).

M. ZINET ET AL. 36
Figure 11. Evolution of relative crystallinity at several loca-
tions across polymer thic kness (T0 = 220˚C, P0 = 200 MPa).
Figure 12. Pressure evolution (T0 = 220˚C, P0 = 200 MPa).
pleted, the polymer is entirely solidified, and temperature
continues to drop while becoming more homogeneous
across the thickness. Pressure still decreases until it
reaches Patm, corresponding to the unsticking of the po-
lymer from the mold. The evolution of the local specific
volume (Figure 13) results from the interaction of two
phenomena within the material: a contraction due to the
temperature decrease, and an expansion due to the pres-
sure drop. At the beginning of cooling, the skin tempera-
ture decreases faster than at the core, so does the specific
volume, which also implies a pressure drop in the cavity.
For core locations, this relaxation first induces an in-
crease in specific volume as pressure drops there faster
than temperature, contrary to what occurs close to the
metal/polymer interface. The phenomenon is even more
noticeable during crystallization: the skin layers, whose
temperatures are lower, are the first to crystallize and to
contract. Pressure falls quickly, allowing polymer at the
core (still in liquid phase) to dilate. When the core fi-
Figure 13. Specific volume evolution at several locations
across polymer thickne ss (T0 = 220˚C, P0 = 200 MPa).
nally crystallizes, its specific volume decreases; there-
fore, to compensate this, the volume of the cavity being
constant, the crystallized skin layer dilates. After the end
of the crystallization, the polymer is in semi-crystalline
phase, pressure is stabilized, and the specific volume
variations are mainly controlled by the temperature evo-
lution.
It is clear that under lower pressures, crystallization
cannot be completed before the end of the isochoric
phase: the lower the initial pressure, the sooner shrinkage
begins. This is precisely the case in the actual injection
molding process, for which the order of magnitude of
pressure at the end of the packing phase is about 50 MPa.
By the way, the end of the isochoric simulation is able to
yield an initial state, in terms of temperature and density
distributions, for a further simulation of the isobaric
cooling stage and shrinkage development.
5. Conclusion
After highlighting the importance of an accurate knowl-
edge of the thermophysical characteristics of semi-crys-
talline polymers and their role in the process-material in-
teraction, a protocol for the determination of thermal
conductivity, heat capacity, PVT diagram and crystalli-
zation kinetics parameters is proposed. A modified Sch-
neider model is developed in order to identify all the ne-
cessary parameters for the crystallization kinetics, using
only DSC measurements. Finally, a numerical simulation
of the cooling of a polymer plate is carried out to illus-
trate the interdependency of the thermophysical parame-
ters and the importance of taking into account their evo-
lution versus temperature and crystallinity. This work con-
stitutes a complete dataset and a clear protocol for char-
acterizing and modeling heat transfer in polymers sub-
jected to phase change as usually encountered in Indus-
trial processes.

M. ZINET ET AL. 37
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