Cross-linked Polymers and Rubber Elasticity

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Definition

• An elastomer is defined as a cross-linked amorphous polymer above its glass transition temperature.

1. Capability for instantaneous and extremely high extensibility

2. Elastic reversibility, i.e., the capability to recover the initial length under low mechanical stresses.when the deforming force is removed.

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Crosslinking effect

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Defects in crosslinks

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For the purpose of the theoretical treatments presented here, the elastomer network is assumed to be structurally ideal, i.e., all network chains start and end at a cross-link of the network.

Force and Elongation

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Hookian

Rubber elasticity Stress

induced crystallinity

Rubber Elasticity and Force

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The origin of the forceAt constant V

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Under isothermal conditions

Entropy origin

Eneregy origin

Entropy change or internal energy change is important?

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Since F is a function of state:

The change in internal energy in effect of l change

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Experimental data

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Experimental data

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Thermodynamic Verificationat constant p

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According to the first and second laws of thermodynamics, the internal energy change (dE) in a uniaxially stressed system exchanging heat (dQ) and deformation and pressure volume work (dW) reversibly is given by:

The Gibbs free energy (G) is defined as:

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The partial derivatives of G with respect to L and T are:

The partial derivative of G with respect to L at constant p and constant T

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The derivative of H with respect to L at constant p and constant T

Experiments show that the volume is approximately constant during deformation, (V /L)p,T= 0 . Hence,

Statistical Approach to the Elasticity

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Elasticity of a Polymer Chain

relates the entropy to the number of conformations of the chain Ω

Entropy of the chain

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the probability per unit volume, p(x, y, z)

<r2>o represents the mean square end-to-end distance of the chain

The entropy decreases as the end-to-end distance increases

The work required for change in length

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It can be concluded that (1)is proportional to the temperature, so that as T increases the force needed to keep the chain with a certain value of r increases, and (2)the force is linearly elastic, i.e., proportional to r.

Elasticity of a Netwrok

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Assumptions

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l. The network is made up of N chains per unit volume. 2. The network has no defects, that is, all the chains are joined by both

ends to different cross-links. 3. The network is considered to be made up of freely jointed chains,

which obey Gaussian statistics. 4. In the deformed and undeformed states, each cross-link is located

at a fixed mean position. 5. The components of the end-to-end distance vector of each chain

change in the same ratio as the corresponding dimensions of the

bulk network. This means that the network undergoes an affine deformation.

Model of deformation

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And the chain

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The entropy change

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For N chain

And

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the work done in the deformation process or elastically stored free energy per unit volume of the network.

The total work;

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True and Nominal stress

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The Phantom Model

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When the elastomer is deformed, the fluctuation occurs in an asymmetrical manner. The fluctuations of a chain of the network are independent of the presence of neighbor in chains.

Other quantities:Young Modulus

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RTr

rnE

LLE

i

VT

20

2

,

3

RTr

rnE

LLE

i

VT

20

2

,

3

RTr

rnG

EG

i2

0

2

5.0),1(2/

RTr

rnG

EG

i2

0

2

5.0),1(2/

?

Statistical Approach to the Elasticity

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a) For a detached single chain

VTVT r

TrkT

r

Ff

TrkTconsF

TSUF

,,

),(ln

),(ln.

VTVT r

TrkT

r

Ff

TrkTconsF

TSUF

,,

),(ln

),(ln.

A Spherical Shell and the End of the Chain in it

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The probability for finding the chain end in the spherical shell between r and r+r

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2

0

4

),(

),()( r

drTr

drTrdrrW

2

0

4

),(

),()( r

drTr

drTrdrrW

Recall=>

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)2(3

4)(

20

2

2

3

21

22

r

rerW r

)2(3

4)(

20

2

2

3

21

22

r

rerW r

20

3

r

kTrf 2

0

3

r

kTrf Retractive force for a

single chain

Gaussian distribution

Recall again =>

b) For a Macroscopic Network

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3

1izyx rr 22222

3

1izyx rr

The Stress-Strain Relationship

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2

12

212

)(

)(20

3 r

rel

el

i

rdrr

nRTF

FW

2

12

212

)(

)(20

3 r

rel

el

i

rdrr

nRTF

FW

20

20

20

2222

0

2

2 zyxzyxi

el r

rnRTF 2

02

020

2222

0

2

2 zyxzyxi

el r

rnRTF

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22

2

,

1

o

i

VT r

rnRT

F

22

2

,

1

o

i

VT r

rnRT

F

21

20

20

20

1,

1

1

zyx

zyx

zxyx

21

20

20

20

1,

1

1

zyx

zyx

zxyx

3

2

22

2

2

o

iel r

rnRTFW

3

2

22

2

2

o

iel r

rnRTFW

We have:

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22

2

,

1

o

i

VT r

rnRT

F

22

2

,

1

o

i

VT r

rnRT

F

And the stress-strain eq. for an elastomer

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Eq.Hookean -non a 1

2

G Eq.Hookean -non a

12

G

Equibiaxial tension

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such as in a spherical rubber balloon, assuming ri2/r 2

0 = 1, and the volume changes of the elastomer on biaxial extension are nil.

The Carnot Cycle for an Elastomer

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2

0

0 L

L

L

LnRT

2

0

0 L

L

L

LnRT

Work and Efficiency

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dLe dLe

PdVg PdVg

II

IIIg q

qq

II

IIIg q

qq

II

III

IIe Q

QQ

Q

dL

II

III

IIe Q

QQ

Q

dL

A Typical Rubber Network

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Vulcanization with sulfur

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Radiation Cross-linking

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Using Multifunctional Monomers

Comparison between Theory and Experiment

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Thermodynamic Verification

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At small strains, typically less than = L/ L0 < 1.1 (L and L0are the lengths of the stressed and unstressed specimen, respectively), the stress at constant strain decreases with increasing temperature, whereas at λ values greater than 1.1, the stress increases with increasing temperature. This change from a negative to a positive temperature coefficient is referred to as thermoelastic inversion. Joule observed this effect much earlier (1859). The reason for the negative coefficient atsmall strains is the positive thermal expansion and that the curves are obtained at constant length. An increase in temperature causes thermal expansion (increase in L0 and also a corresponding length extension in the perpendicular directions) and consequently a decrease in the true λ at constant L. The effect would not appear if L0 was measured at each temperature and if the curves were taken at constant λ (relating to L0 at the actual temperature). The positive temperature coefficient is typical of entropy-driven elasticity as will be explained in this section.

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Stress at constant length as a function of temperature for natural rubber.

Thermodynamic Verification

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The reversible temperature increase that occurs when a rubber band is deformed can be sensed with your lips, for instance. It is simply due to the fact that the internal energy remains relatively unchanged on deformation, i.e. dQ=-dW (when dE=0). Ifwork is performed on the system, then heat is produced leading to an increase in temperature. The temperature increase under adiabatic conditions can be substantial. Natural rubber stretched to λ=5 reaches a temperature, which is 2-5 K higher than thatprior to deformation. When the external force is removed and the specimen returns to its original, unstrained state, an equivalent temperature decrease occurs.

At constant V and T

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TSUF TSUF

VLVT

VTVTVT

T

f

L

S

L

ST

L

U

L

Ff

,,

,,,

VLVT

VTVTVT

T

f

L

S

L

ST

L

U

L

Ff

,,

,,,

Wall’s differential mechanical mathematical relationship

VLVT L

fT

L

Uf

,,

VLVT L

fT

L

Uf

,,

Thermodynamic eq. of state for rubber elasticity

A Similar Equation

Analysis of Thermodynamic Eq.

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Stress-Temperature Experiments

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se fff se fff

End of Chapter 9

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