Polymer Physics Chapter 6

Chapter 6. Molten State

6.1 Introduction

Kinetic energies of molecules ≈ potential energies of interaction : liquid " ≫ " : gas " ≪ " : solid (crystal)

Molten state of polymers depends on MW

Flexible-chain polymers - random conformation in molten state chain entanglements (important)

Liquid-crystalline polymers - orientational order

Polymer Physics Chapter 6

6.2 Fundamental concepts in rheology

"Rheology" : the study which deals with the flow and deformation of fluids.(or ) relationships : (rheological) constitutive eq'ns

Balance equation : Law of mass conservation (Continuity equation) Law of momentum conservation (Momentum equation) Law of energy conservation (Energy or Heat equation)

* Incompressible, steady flow,

(mass) (momentum) (energy)

γσ − γσ &−

STkTC

p

p&+∇+∇=∇⋅

+⋅∇+−∇=∇⋅=⋅∇

vσv

gσvvv

:

0

2ρ

ρρ

Polymer Physics Chapter 6

• Steady simple shear flow

0,0,

/

322

1

21

===

=

vvdxdv

AF

γ

σ

&

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

33

2221

1211

0000

σσσσσ

σstresses normal:,,

),symmetric(

332211

2112

σσσσσ =

Polymer Physics Chapter 6

* Material parameters

tcoefficien stress normal secondary)(or second:

tcoefficien stress normal primary)(or first :

viscosity:

23322

2

22211

1

21

γσσ

γσσ

γση

&

&

&

−=Ψ

−=Ψ

=

Polymer Physics Chapter 6

• Elongational flow

* Uniaxial elongational flow

for an incompressible fluid

: elongational (or extensional) strain rate

: elongational viscosity (fiber spinning)

1

2

3

332211 2,

2, xvxvxv εεε &&& −=−==

ε&

εσση&

2211 −=

Polymer Physics Chapter 6

* Biaxial elongational flow

(film blowing)

1

2

332211 2,, xvxvxv

3

εεε &&& −===

εσσ

εσση

&&33223311 −=

−=B

Polymer Physics Chapter 6

* Planar extensional flow

(sheet casting)

0,, 32211

2

=−== vxvxv εε &&

εσση&

2211 −=EP

1

Polymer Physics Chapter 6

• Dynamic flow

: phase angle

・Complex shear modulus

・Complex viscosity

))(exp(

"')sin(cos)exp(

0*

00*

δωσσ

γγωωγωγγ

+=

+=+==

ti

ititti

δ

modulus) phaseof(out modulus loss:"modulus) phaseinor modulus, (elastic modulus storage:'

"'sincos0

0

0

0*

**

--G-G

iGGiG +=+== δγσδ

γσ

γσ

storage)(energy viscosity complex ofpart elastic:"n)dissipatio(energy viscosity dynamic:'

"''"**

*

ηη

ηηωωγ

ση iGiG −=−==&

Polymer Physics Chapter 6

- Rheological behaviors of simple shear

most polymer melts (pseudoplastic) wet beach sand (dilatant) oils, cement slurries, margarine (Bingham)

γ&

σ

Polymer Physics Chapter 6

(Newtonian)

(power-law) Ostwald-de Waele equation

K : consistency factor, n : power-law index

- Time dependence of a non-Newtonian liquid

γησ &=21

γγσ && 121−= nK

Polymer Physics Chapter 6

6.3 Measurement of rheological properties:

Couette viscometer, capillary viscometer, parallel-plate viscometer,

cone-and-plate viscometer

(first normal stress difference) =

(second normal stress difference) =

(first normal stress coefficient) =

(second normal stress coefficient) =

21 &, NNη

1N

2N

1Ψ

2Ψ

2211 σσ −

3322 σσ −

21

γ&N

22

γ&N

Polymer Physics Chapter 6

ωφ

θ

cθ

r) :3 , :2 , :(1scoordinate Spherical θφ

2221112RFN

πσσ =−=

33221121 22lnσσσσθθ −+=+=

∂∂

=∂∂ NN

rpr

r

cRT

θωγ

πγησ === && ,

23)( 312

Polymer Physics Chapter 6

6.4 Flexible-chain polymers

Flow properties ~ function of MW & chain branching

cMMM ∝ for4.3

0η

nt)entanglemechain ofpoint (startingmassmolar critical:cM

일반적으로온도

압력

↓→↑ η

↑→↑↑ η( ) ( )⎟⎟⎠

⎞⎜⎜⎝

⎛−

∆=⎟⎟

⎠

⎞⎜⎜⎝

⎛−

=00

0 expexp TTREA

TTBA

αη

Polymer Physics Chapter 6

• Rouse model

flexible repeating units in viscous surroundings

long enough to obey Gauss distribution

Forces acting on each repeating unit

i) drag force w.r.t. surrounding medium

relative velocity of the repeating unit

ii) force from the adjacent repeating units

iii) force from Brownian motion

Not applicable to polymer melts of MW > Mc

∴ Model for dilute solution

∝

Polymer Physics Chapter 6

• Reptation model

coiled chain trapped in a network

du Gennes : theory of reptation of polymer chains

Doi & Edwards : the relationship of the dynamics of repeating chains to mechanical properties

(rubbery plateau shear modulus)

(zero-shear rate viscosity)

experimental data :

∴Model for entangled polymer chains in the absence of a permanent network

30

0

M

MGp

∝

∝

η

4.30 M∝η