Polymer Properties

Exercise 4

Viscoelasticity and rheology

Effect of molecular weight on viscosity

• Above the critical molecular weight the zero viscosity of polymer can be calculated using equation:

• Below the critical molecular weight the constant is 1.0, above the critical molecular weight = 3.4.

• Zero viscosity is determined by rheology measurement: dynamic viscosity at zero frequency.

Effect of temperature on viscosity• For amorphous polymers above

the glass transition, WLF equation can be applied:

wMk0 s

s

s TTC

TTC

2

1lg

Reference temperature Ts

C1 = 8.86 and C2 = 101.6

Reference temperature Ts = Tg

C1 = 17.44 and C2 = 51.6

1. Viscosity

• Viscosity of an amorphous PVC was measured to be 3.9105 Pas at temperature 122 oC. For processing, the viscosity should be below 2104 Pas, but at least 5000 Pas.

• At what temperature should the processing be done?

1)

• Amorphous PVC follows WLF equation in the temperature range T = Ts 50 °C.

• Solve for the temperature at which the viscosity is at most 2104 Pas :

s

s

s TT

TT

6.101

86.8lg

s

s

s TT

lg86,8

lg6,101

CT

KK

sPa

sPasPa

sPa

T

o139

3.412395

109.3

102lg86.8

109.3

102lg6.101

5

4

5

4

1)

• When the lowest acceptable viscosity is 5000 Pas, the temperature is:

• Processing should be done within the temperature range 139 - 149 oC.

CKK

sPa

sPasPa

sPa

T o1496.422395

109.3

5000lg86.8

109.3

5000lg6.101

5

5

2. Viscosity and Mw

Zero viscosity of a linear polyethylene was determined to be 676000 Pas at 190°C. For polyethylene the constants for comparison of Mw and zero viscosity are k = 3.410-15 Pas and = 3.5. The temperature dependence of the viscosity of PE in melt can be estimated with Arrhenius-type equation.

a) What is the molecular weight Mw of PE?

b) How much should the temperature be altered in order to reduce the viscosity by half?

2a)

• The weight average molecular weight for the polymer can be calculated from equation:

wMk0

mol

g

mol

g

Pas

Pas

kM w 630000

10*4.3

6760005.3

150

2b)

• Temperature dependence for the viscosity by Arrhenius:

• When viscosity is reduced by half by altering temperature:

RT

Ek exp

21

2

1

2

1 11exp

exp

exp

1

2

TTR

E

RT

Ek

RT

Ek

2b)• Activation energy for HDPE is 27 kJ/mol, the temperature can be

solved:

=1/0.001946

T2=513.8K = 241°C

• The temperature should be increased by 51 °C in order to reduce the viscosity by half.

2ln27000

314.8

463

112ln

11

212 molJmolKJ

KTR

E

TT

3. Stress-strain (Creep)

• Tensile stress shear stress • Strain

• Shear rate

• Viscosity

• Creep compliance

0l

lt

dt

dy

t

tJ

A

F

A

F

3. Creep

• Polypropylene PP rod attached to the ceiling (length 200 mm, width 25.0 mm, thickness 3.0 mm) is loaded with 30 kg´s. How much will the polymer creep in two minutes when the creep compliance J(t) follows the equation (t is time in minutes)?

J(t) = 1.5 - exp(-t/6min) GPa-1

3)

• Stress imposed on the cross section of the polymer rod is:

• Creep at the moment t is obtained from the Strain:

• Where

26

2

109.30250.00030.0

81.930

m

N

mms

mkg

A

F

tJt

11 783.0min6

min2exp5.1

GPaGPatJ

3)

• Creep at two minutes:

• PP rod has strained during the two minutes time:

0031.0109.31

10783.0 69 PaPa

t

mmmml 62.00031.0200

4. Viscosity and chain length

• When the polymer chain are long enough to form stable entanglements, longer than the critical chain length Zw > Zc,w, the polymer viscosity and chain length Zw can be connected by:

where K is a constant

4.30 wKZ

4)

• The usual processing temperature of polystyrene cups is 160 oC and the melt viscosity is then 1.5102 Pa s, provided that the mainchain length of PS is Zw = 800. The quality of the polymer however varies and one day the Zw = 950. Processing is tuned for a particular viscosity range.

• How should the processing temperature be altered so that the melt viscosity would still be 1.5102 Pa s?

Glass transition temperature of PS is 100oC.

4)

• Viscosity is increased when the molecular weight increases. By increasing the temperature the viscosity can be kept lower.

• Solving the constant K first:

• Viscosity of the novel polymer grade at 160oC:

sPasPa

ZKKZ

ww

8

4.3

2

4.31,

1,04.30 1002.2

800

105.1

sPasPaKZw 24.384.32,2,0 1069.29501002.2

4)

• The viscosity of this polymer at the glass transition temperature can be obtained using WLF equation:

• The new processing temperature T2 can be solved from WLF equation:

sPa

sPa

TT

TT

KK

KK

TT

TTgg

g

gg

g

12

3734336.51

)373433(44.17

2

6.51

44.17 1040.6

10

1069.2

106.51

44.17lg

g

g

g TT

TT

6.51

44.17lg

CK

sPa

sPa

KKsPa

sPa

T

TT

T

TTTT

TT

TT

o

g

ggg

ggg

g

g

g

g

6.1638.436

44.171040.6

105.1lg

37344.176.513731040.6

105.1lg

44.17lg

44.176.51lg

lg6.5144.17lg

6.51

44.17lg

12

2

12

2

2

2

22

2

2

4)

The processing temperature should be about 4oC higher so that the viscosity would remain the same.