Solidification criteria and

rheology during solidification

Giuseppe TitomanlioUniversity of Salerno Italy

PIAM Winter School, January 15-19, 2007 Aussois, France

gtitomanlio@unisa.it

What do we mean by solidification criteria?

Non flow Non flow temperature, Ttemperature, Tnf nf :: Solidification Solidification temperature, Ttemperature, Ts s ::

The two temperatures in principle can be very different

ViscosityViscosity is is one or two one or two orders of magnitude orders of magnitude largerlarger than every where than every where else in the same cross else in the same cross sectonsecton

Relaxation timeRelaxation time is much is much largerlarger than the cooling than the cooling timetime

They are often regarded as a single temperature

and . . . . they can also be very close

About non flow and solidification criterion

Amorphous polymers Amorphous polymers

Tg: WLF Tg: WLF Semicrystalline polymersSemicrystalline polymers

Tg changes with

Pressure and cooling rate

Both non flow and Both non flow and solidification conditions are solidification conditions are determined by crystallinity, determined by crystallinity, XXnfnf and and XXss

and thus by and thus by crystallizationcrystallization temperature and temperature and kineticskinetics

CTT

TTA

TTAD g

; exp2

110

Amorphous polymers, aPS

reported by Zoetelief. Solidification temperature was chosen reported by Zoetelief. Solidification temperature was chosen as indicated by Zoetelief asas indicated by Zoetelief asTs(P)=100°C+0.051K/barTs(P)=100°C+0.051K/bar 8.68.6

As mentioned above, a solidification temperature should As mentioned above, a solidification temperature should depend also on cooling rate. If indeed this approach is depend also on cooling rate. If indeed this approach is followed, the following expression is obtained for Tsol:followed, the following expression is obtained for Tsol:

8.78.7where both the reference temperature and the constant where both the reference temperature and the constant describing the effect of describing the effect of

Tsol 81C 2logqsol

1Cs 1

0.051

Cbar

P

•Dependence upon cooling rate q: 2log (q/1°C/s)

•Dependence upon pressure P: P *0.05 °C/bar

Often non flow and solidification temperatures are

regarded as a single temperature,

for amorphous polymers

T, [ºC]

85

90

95

100

105

110

115

120

125

130

135

0 200 400 600 800 1000

1 ºC/s100 ºC/s10000 ºC/s

P, bar

Tg (P, T’)Tg (P, T’)

• DOW PS 678E

Outline

2.2. Observations and modelling of rheology Observations and modelling of rheology evolution during crystallization evolution during crystallization

3.3. Role of non flow criterion in the simulation of Role of non flow criterion in the simulation of injection moulding and identification of the injection moulding and identification of the proper crystallization kinetic modelsproper crystallization kinetic models

4.4. Solidification Criterion and its relevance on Solidification Criterion and its relevance on internal stresses and warpageinternal stresses and warpage

1.1. Non flow and solidification temperaturesNon flow and solidification temperatures

Solidification ProcessSolidification Process amorphous vs Crystalline behaviouramorphous vs Crystalline behaviour

Amorphous Polymer Viscosity

Model Extrapolation

Measurements

SolidificationTemperature

Melt Temperature

MoldTemperature

Depend on the Solidification Conditions

Vis

cosi

ty

Semicrystalline Polymer Viscosity

Viscosity increase with crystallinity is always sharp

iPP T30GiPP T30G

An example of quiescentcrystallization

Sferulites are seen when they are already big

RheologyRheology vs crystallinity , suspension vs crystallinity , suspension viewview

1. Small molecules: solid particles suspension

Rheology changes with time because particles grow, with very small interactions.

Interactions became relevant only at the end

NUCLEI ACT AS PHYSICAL CROSSLINKS NUCLEI ACT AS PHYSICAL CROSSLINKS

3. A different, melt structure-based view

The small crystalline nuclei ACT AS physical crosslinks which produce an apparent molecular

weigth increase with a parallel fast viscosity change.

A nucleus

Crystallization determines a network?Crystallization determines a network?

physical

Gel Point Gel Point [Winter et al.1986][Winter et al.1986]

Suspension vs Crosslinks based views? Suspension vs Crosslinks based views?

Suspension-like microstructure for low melt connectivity:

Low molecular weightLow nuclei density

Crosslinks for high melt connectivity:

High molecular weightHigh nuclei density

Eterogeneous nucleation tTTEtTE m exp

Nuclei density depends upon temperature

Nuclei density changes with temperature and cooling rate

E(T)decreasing crystallization temperature or increasing cooling increasing cooling raterate produces an increase of the number of nuclei and a decrease of particle dimensions

, s

iPP T30G

123°C121°C

tTTEtTE m exp0

Crystallization takes place at the temperature where crystallization time equals cooling time

Eterogeneous nucleation: density =

thus, connettivity depends also upon cooling rate

SEMSEM

0.02 K/s

50 K/s90 K/s

.

T

.

T

2 K/s

Morphology vs cooling rates, Morphology vs cooling rates, iPP T30GiPP T30G

AVERAGE DIAMETER OF AVERAGE DIAMETER OF SPHERULITESSPHERULITES iPP iPP T30GT30G

0.1

1

10

100

1000

0.01 0.1 1 10 100- (dT/dt) at 343K [K/s]

Dia

mete

r of

sp

heru

lite

s [

m]

Experimental - SEM

1 phase model - (Optical and Calorimetric Measurements)

2 phases model - (Full Data Set)

Diametro(T) Diametro(T)

34

3

aNR

calorimetry

iPP T30G

Quenching esperiments

time (s)

0 2000 4000

(P

a s

)

0

4e+6

8e+6

T (

°C)

80

100

120

140

160

180

annealing at 160°C to erase any crystalline memory

time (s)

0 2000 4000

(P

a s

)

0

4e+6

8e+6

T (

°C)

80

100

120

140

160

180

rapid cooling to 98°C

time (s)

0 2000 4000

(P

a s

)

0

4e+6

8e+6

T (

°C)

80

100

120

140

160

180

constant stress is applied, polymer viscosity is monitored

time (s)

0 2000 4000

(P

a s

)

0

4e+6

8e+6

T (

°C)

80

100

120

140

160

180

crystallization determines a

viscosity upturn

PB200

RHEOLOGICAL RHEOLOGICAL EVIDENCEEVIDENCE of crystallization of crystallization

T

Effect of flowEffect of flowPB200

T=105°C

time [s]

0 2000 4000 6000 8000

0

2

4

6

8

10

4500 Pa10000 Pa18000 Pa

Flow enhances

crystallization rate

Polipropilene T30GPolipropilene T30G

Viscosity upturn during crystallization

Crystallinity increases during calorimetric measurements

Crossing both informations at the same temperature and time, the evolution of viscosity with crystallization is obtained

o

Only total crystallinity?

ViscosityViscosity Models and relationshipsModels and relationshipsEquation Author derivation parameters

/0=1+a0 a Katayama 85 Suspensions a=99

/0= (1- /a0)-2 Metzner 85 also Tanner 2002 Suspensions a=0.68 for smooth spheres

/0=1+(/a1)a2/(1-/a1)

a2 Tanner 2002 Empical, based on suspensions

a1=0.44 for compact

a1=0.68 for spherical

crystallites

/0= exp(a1 a2) Shimizu 85; also Zuidema 2001, and Hieber 2002

Empirical

/0=1/(-c)a0 Ziabicki 88 Empirical c=0.1

/0=1+a1 exp(-a2/ a3) Titomanlio 97; also Guo 2001, and Hieber 2002

Empirical

/0= exp(a1 + a2 2) Han , 97 Empirical

/0= 1+a1 +a22 Tanner 2003 Empirical a1=0.54 , a2=4, <0.4

Only total crystallinity is considered ! !

Shapes reproduced by equations of the Shapes reproduced by equations of the ModelsModels

1

10

100

1000

0.0% 2.5% 5.0% 7.5% 10.0% 12.5% 15.0%

Relative Crystallinty

Vis

co

sit

y [

Eta

(Xc

)/E

ta(0

)]

Eta(Xc)/Eta(0) Katayama

Eta(Xc)/Eta(0) Ziabicki

Eta(Xc)/Eta(0) Titomanlio

Eta(Xc)/Eta(0) Shimizu

All equations were adopted with a factor of about 20 at crystallinity sligtly above 5%

1,E+02

1,E+03

1,E+04

1,E+05

1,E+06

1,E+07

0,001 0,01 0,1 1 10 100 1000 10000

Shear rate [s-1]

[P

a*s]

Exp. 431K Exp. 453K Exp. 473KExp. 503K Model 431K Model 453KModel 473K Model 503K 500 bar, 431K

Effect of pressure and temperature on viscosityEffect of pressure and temperature on viscosity

iPP T30G

Viscosity and relaxation time increase with pressure

When the viscosity increases the curve shifts also on the left

1.E+02

1.E+03

1.E+04

1.E+05

1.E+06

1.E+07

0.1 1 10 100

Xc= 0% Xc= 0.1%Xc= 0.3% Xc= 0.4%Xc= 0.6% Xc= 1.1%Model Xc= 0% Model Xc= 0.1%Model Xc= 0.3% Model Xc= 0.4%Model Xc= 0.6% Model Xc= 1.1%

', [s-1]

, [Pa s]

T30G: effect of cristallinity on viscosity

m

ffh

21 exp 1 )(

22

32110 exp

DTA

PDDTAD

n

xPTC

xPTxPT

1.

0

0.

,,1

,,,,,

).(h

The viscosity curve becomes higher and shifts on the left

Non flow and solidification conditions

For crystalline polymers:

are determined by crystallinity, Xnf Xs

For amorphous polymers:Tg(P, cooling rate)

for both conditions, most commercial codes adopt a single constant temperature :

Ts=Tnf=const.

Outline

1.1. Non flow and solidification temperatures Non flow and solidification temperatures

2.2. Observations and modelling of rheology Observations and modelling of rheology evolution during crystallization evolution during crystallization . . . 24: suspensions or . . . . physical crosslinks

3.3. Role of Role of non flownon flow criterion in the simulation of criterion in the simulation of injection moulding and identification of the injection moulding and identification of the proper crystallization Kinetic modelsproper crystallization Kinetic models

4.4. Solidification Criterion and its relevance on Solidification Criterion and its relevance on internal stresses an warpageinternal stresses an warpage

Pressure evolution during injection moulding, BA238G

GATE:thickness: 1.5mm or 0.5mm

P2

P3

P4

P1

SPRUE:initial diam: 4.7mmfinal diam: 7mmlength: 80mm

8mm

9mm

68mm

6mm

120mm

15mm

60mm

105mm

30mm

CAVITYthickness: 2mm

0

50

100

150

200

250

300

350

400

450

0 5 10 15 20

P0

P1

P2

P3

P4

t, s

P, bar

Thick gate

experimental

0

50

100

150

200

250

300

350

400

450

0 5 10 15 20

P0

P1

P2

P3

P4

t, s

P, bar experimental

Thin gate

1,E-05

1,E-04

1,E-03

1,E-02

1,E-01

1,E+00

1,E+01

1,E+02

0 2 4 6 8t, s

Flo

w R

ate

, cc

/s

P4

P2

Thermomechanical history changes with position

Termomechanical history (dT/dt, pressure, flowdT/dt, pressure, flow) is a strong function of position, in injection moulding

P3P2 P4

z

y

1.E+00

1.E+01

1.E+02

1.E+03

1.E+04

0 0.2 0.4 0.6 0.8 1

0

50

100

150

200

250

-T'@100ºC

P@100ºC

y, mm

-T', [ºC/s] P, [bar]

Cooling rate and pressure at 100°C; Simulated

Morphology changes mainly with the distance from mould wall and also along the flow path

“Standard” sample

P2 P3 P4

Micrographs taken in a polarized optical microscope of “Standard” sample along flow direction.

 

Morphology distribution in inj. moulded samplesMorphology distribution in inj. moulded samples

P3P2 P4

x

y

iPP T30G

Morphology changes with the distance from the skin and slowly along the flow direction

Sferulite dimensions increase with the distance from the skin

DIAMETER OF SPHERULITESDIAMETER OF SPHERULITES

0

5

10

15

20

25

30

35

40

0 0.2 0.4 0.6 0.8 1Distance from the skin [mm]

Dia

mete

r of

sph

eru

lite

s [ m

]

SEM

Model prediction

End of dark zone

FastFast

0

5

10

15

20

25

30

35

40

0 0.2 0.4 0.6 0.8 1

Distance from sample skin [mm]

Dia

mete

r of

spheru

lite

s [

m] SEM

Model prediction

End of dark zone

Both non flow and solidification conditions

For crystalline polymers:

are determined by crystallinity For amorphous polymers:Tg(P, cooling rate)

In order to calculate Xnf and Xs, the crystallization kinetics has

to be defined and implemented in the codes

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

10 100 1000 10000t [sec]

410K

413K

416K

418K

0102030405060708090

0 500 1000 1500 2000 2500 3000t [sec]

410K

413K

416K

418K

410K

413K

416K

418K

Calorimetric isotherms, BA238G :

Calorimetric and PVT cooling scans, BA238G

1.00E-03

1.05E-03

1.10E-03

1.15E-03

1.20E-03

1.25E-03

1.30E-03

1.35E-03

1.40E-03

273 323 373 423 473T [K]

iPP; C-Mold data baseHigh temperature fit: amorphous phaseCrystal phaseLow temperature fit

Room pressure

1.00E-03

1.05E-03

1.10E-03

1.15E-03

1.20E-03

1.25E-03

1.30E-03

1.35E-03

1.40E-03

273 323 373 423 473T [K]

iPP; C-Mold data baseHigh temperature fit: amorphous phaseCrystal phaseLow temperature fit

P=50MPa

0

5

10

15

20

340360380400420T [K]

1.33K/sec

1.00K/sec

0.83K/sec

0.33K/sec

0.16K/sec

0.10K/sec

0.02K/sec

cooling rate, q

0%

10%20%

30%

40%50%

60%

70%

80%90%

100%

340360380400420T [K]

cooling rate, q

BA230g-Comparison between expermental and simulated pressure curves with Xnf = 5%

0

50

100

150

200

250

300

350

400

450

0 5 10 15 20

P0

P1

P2

P3

P4

t, s

P, bar

Thick gate

experimentalThermomechanical model with Kinetics calibrated by calorimetric and PVT experiments

0

50

100

150

200

250

300

350

400

450

0 5 10 15 20 25

Trsd 1

Trsd 2

Trsd 4

Trsd 5

Trsd 6

t, s

P, barsimulated

Poorcomparison !

WHAY ?

BA238g; non-flow temperatures and final crystallinities obtained by simulation with Xnf 5%

Crystallization kinetics was identifies by calorimetric tests (low cooling rate)

Kinetics needs to account of behaviour at high cooling rates

20

30

40

50

60

70

80

90

100

0 0.2 0 .4 0 .6 0 .8 1

0%

5%

10%

15%

20%

25%

30%

35%

40%

45%

T so lXc

Xc, [ - ]T , [ º C ]

y, [m m ]

Thick gate

Non

-flo

w te

mpe

ratu

re

Distance from the skin

Characterised Quenching experiments

1.100E-03

1.105E-03

1.110E-03

1.115E-03

1.120E-03

1.125E-03

1.130E-03

1.135E-03

1.140E-03

0.01 0.1 1 10 100 1000 10000

cooling rate at 343K [K/sec]

0.01

0.1

1

10

100

1000

10000

250 300 350 400 450 500T [K]

47

61 68

19

2DSC1.7

DSC0.1

DSC0.51

C1000

250

300

350

400

450

500

0.1 1 10 100 1000 10000time [sec]

47

6168

19

2

DSC1.7

DSC0.5DSC0.1

1C1000

nozz le

spray

sample

Coolant outlet

thermocoup le

Copper sample holder

Sliding rod

Electical heater

Oven

Bath

Final crystallinity in quenched samples: comparison

1.110E-03

1.120E-03

1.130E-03

1.140E-03

1.150E-03

1.160E-03

1.170E-03

1.180E-03

0.01 0.1 1 10 100 1000 10000

cooling rate at 343K [K/sec]

Meas. at298K

Pred.: Fulldata set

Pred: onlyDSC data

0%

10%

20%

30%

40%

50%

60%

70%

0.01 0.1 1 10 100 1000 10000

cooling rate at 343K [K/sec]

Meas. at 298KMeas. at 318KMeas. At 298KPred: full data setPred: only DSC data

….. model 1: Calorimetry

___ model 2: full data set

models obtained by calorimetry usually give poor results at high cooling rates

and should not be adopted for injection moulding

Calorimetric and PVT results: comparison

1.00E-031.05E-03

1.10E-031.15E-031.20E-031.25E-03

1.30E-031.35E-031.40E-03

300 350 400 450 500T [K]

Equilibrium

1E-3 K/sec

1K/sec

500 K/sec

Room Pressure

1.00E-031.05E-03

1.10E-031.15E-031.20E-031.25E-03

1.30E-031.35E-031.40E-03

300 350 400 450 500T [K]

Equilibrium

1E-3 K/sec

1K/sec

500 K/sec

P=50MPa

330340350360370380390400410420430

0.01 0.1 1 10Cooling rate [K/sec]

peak from DSC cooling rampsPredictions: Full data setPredictions: only DSC data

Model 1: Calorimetric

Model 2: full data set

Results of Simulation with full data crystalliztion kinetics

0

20

40

60

80

100

120

140

160

0 0.2 0.4 0.6 0.8 1

0

50

100

150

200

250

300

350

T(t sol)

P(t sol)

y, mm

T, [ºC] P, [bar]

Sol

idifi

catio

n t

empe

ratu

re

Distance from the sample skin

Sol

idifi

catio

n p

ress

ure

BA238G, solidification temperatures and pressures

0

1

2

3

4

5

6

7

0 0.2 0 .4 0 .6 0 .8 1

0

50

100

150

200

250

300

350

t so lP( t so l)

y, m m

t, [ s ] P, [ bar]

Distance from the sample skin

So

lidifi

catio

n t

ime

accounting of the full data setfull data set in the

crystallization Kinetics was required In

order to acheive non-flow temperature

on the wholeon the whole cross section, in the

simulation0

20

40

60

80

100

120

140

160

0 0.2 0.4 0.6 0.8 1

Dsc

Dsc&Quenches

y, mm

T, [ºC]

Non-flow temperatures

Xnf=5%

BA230G - Comparison between expermental and simulated pressure curves with Xnf = 5%

Kinetics from full data set (quenches included)

0

20

40

60

80

100

120

140

160

0 0.2 0.4 0.6 0.8 1

Dsc

Dsc&Quenches

y, mm

T, [ºC]

0

50

100

150

200

250

300

350

400

450

0 5 10 15 20 25

Trsd 1

Trsd 2

Trsd 4

Trsd 5

Trsd 6

t, s

P, bar

Kinetics: full data set

0

50

100

150

200

250

300

350

400

450

0 5 10 15 20 25

Trsd 1

Trsd 2

Trsd 4

Trsd 5

Trsd 6

t, s

P, bar

Kinetics: DSC

0

50

100

150

200

250

300

350

400

450

0 5 10 15 20

P0

P1

P2

P3

P4

t, s

P, bar

Thick gate

experimental

Crystallization kinetics calibrated by calorimetric & PVT experiments usually is not adequate to describy injection moulding

Comparison for final crystallinity in position P3, BA238g

0%

10%

20%

30%

40%

50%

60%

70%

0 0.2 0.4 0.6 0.8 1

DSC&QuenchDSC

y, mm

Xc, [-]

Cystallinity distribution is essentially constant on the cross section consistently with experimental risults

50%

52%

54%

56%

58%

60%

62%

0 0.2 0.4 0.6 0.8 1

DSC&QuenchExperimental

y, mm

Xc, [-]

Dettailed comparison

BA230G -Comparison between expermental and simulated pressure curves with Xnf = 5%

0

50

100

150

200

250

300

350

400

450

0 5 10 15 20

P0

P1

P2

P3

P4

t, s

P, bar

Thin gate

Experimental

0

50

100

150

200

250

300

350

400

450

0 5 10 15 20 25

Trsd 1

Trsd 2

Trsd 3

Trsd 4

Trsd 5

P, bar

DSC Kinetics

0

50

100

150

200

250

300

350

400

450

0 5 10 15 20 25

Trsd 1

Trsd 2

Trsd 3

Trsd 4

Trsd 5

t, s

P, bar

Kinetics: full data set(Quenches)

Crystallization kinetics calibrated by calorimetric & PVT experiments usually is not adequate to describy injection moulding

Outline

1.1. Non flow and solidification temperatures Non flow and solidification temperatures

2.2. Observations and modelling of rheology Observations and modelling of rheology evolution during crystallization evolution during crystallization . . . 24: suspensions or . . . . physical crosslinks

3.3. Role of non flow criterion in the simulation of Role of non flow criterion in the simulation of injection moulding and identification of the injection moulding and identification of the proper crystallization Kinetic modelsproper crystallization Kinetic models

4.4. Solidification Criterion and its relevance on Solidification Criterion and its relevance on internal stresses an warpageinternal stresses an warpage

The solidification cristallinityThe solidification cristallinity1. consider a layer which goes under stress during the

cooling of the object

2. as long as relaxation time is small with rspect to cooling time, stresses relaxe and the solid will have the new geometry as reference configuration

This is a simplification, which replaces a dettailed knowledge of the evolution of rheology with crystallization

3. If, viceversa, relaxation time is long with respect to cooling time, relaxation will be negligible and the final solid will keep its initial reference configuration (under stress)

4. at crystallinities higher than that which gives rise to condition 2 the material behaves as a solid, this identies Xs

5. a simplified model for cooling stresses build up would consider the polymer as a melt at crystallinities lower than Xs and as a solid at higher crystallinities

Solidification criterion: How big is Xs?

0

10

20

30

40

50

60

70

80

90

100

1000000

10000000

100000000

1000000000

10000000000

0 50 100 150 200

T [°C]

X/Xeq

E''

E'

A slight melting may reduce the moduli by orders of magnitude

Xs is probably close to Xeq

BA230G

Schematic of cooling stesses build up and thus of warpage

distributions of solidification temperature and Xs are relevant to cooling stresses distribution and to warpage

If they are constant over the whole section trey contribute only on shrinkage

.Pressure free configuration

Also contraction due to cooling and crystallization

Small points to remember

1. Viscosity has been related only to total crystallinity

2. Non-flow condition is different from solidification condition, which is determined by the value of the relaxation time compared to cooling time

3. A low value of Xnf (5%) was often found adequate to describe experimental viscosity increase, Xs is larger than Xnf

4. Crystallization kinetics calibrated by calorimetric & PVT experiments usually is not adequate to describe injection moulding (crystallization, flow, pressure evolution, orientation, morphology)

5. Experiments performed at high cooling rates (100-1000k/s) need to be considered

6. Solidification pressure, temperature and crystallinity are relevent to shrinkage, treir distribution are relevance to internal stresses and warpage

I would be happy to discuss any comment

Thank You