Physical Properties of Polymers

Muhammad Zafar Iqbal

Lect-701-04-2008

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Today

• The Elasticity of polymers• The Kinetic theory of rubber like elasticity• Configuration of a single chain• Gaussian distributions of existance

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The Elasticity of Polymers• Two types of elastic deformations: DOE and DHE

• These are caused by some definite molecular mechanisms• The main boundaries between these types are arbitrary and

depend totally upon the time scale of experiment• DOE is assumed to be instantaneous

• DHE is very much time dependent

• DOE becomes very small as time interval becomes infitesimal.

• At high enough frequency of stresses, DOE has a minimum critical value which is then independent of all the variables.

• At this state,GOE = 102 – 104 Kg/mm2 (?) for almost all hard plastics

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• DOE in amorphous and crystalline polymers and in metals

• Small work has been done on amorphous polymers• In polymers, mainly DHE occurs bcoz of uncoiling of chains

• DHE > DOE in fully developed conditions

GHE =~ 0.2 Kg/ mm2 why?

• As a general rule, overall modulus will be:

• Since DHE is time dependent, so equilibrium value is also time dependent and it is found that:

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The Kinetic Theory of High Elasticity• High elasticity is often called Rubber Like elasticity• Common in Amorphous polymers

Consider an linear chain of C-atoms where no side groups occur.The adjacent members are tetrahedrally bonded (angle=109.50).

If C1 and C2 are fixed, thenC3 can have three possiblePositions on the circle andC4 has 32.Similarly if only C1 is fixed thenC2 will have three possible Positions and C3 will have 32 andC4 will 33

Each separate position of the chain can be explained by a set of Cartesian coordinates or by polar coordinates with “r” as the characteric distance from the centre. 5

Configuration of a single Chain• The best approach is the Random walk model approach.• Let us develop a random model for a chain in one axis and

then extend it to three axis:

Consider a man who takes “n” steps, each of length “l” . Each step is independent of the previous one. What is the probability that after these total steps he has covered a distance “x”?

We develop the model for uni-direction

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Let, A= no. of forward steps, n is very large number B = no. of backward steps A+B = n = total steps

Let x is +ive, then A(l) + B (-l) = xA = ½ (n+x/l)B = ½ (n-x/l)

We can write the probability of reaching “x” as:

Applying the stirling’s approximation, we getLn W= n ln2 – ½ (n+x/l) ln(1+x/nl) – ½ (n-x/l) ln(1-x/nl)

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Since for polymer, characteristic length < contour length of chainSo x << nl, and from algebra ln (1+_x/nl) ~ +_(x/nl)Then

ln W= n ln2 – x2/nl2

and p(x) = constant. exp (-x2/nl2)Since p(x) = 1 max. value, so by taking (nl infinity )

And after integration,where = 1/ nl2

For random flight in 3-axis,= 3/2 nl2

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C-chain and Random Flight Model• C-chain is a special case of random flight• As c-atoms are tetrahedrally bonded, so angle = 109.5 0,• This valence bond angle induces the steric hindrance to the

chain and the beta factor is changed as:

• This relationship holds even of is small enough• As n increases, the distribution rapidly approaches that of a

random flight. ?• The basic chain can not be identified with the monomeric

units. If a large unit is present then there would be a fewer effective links in the chain but rc is constant. So when the no. of compound links is large, random models correctly applicable. ?

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Graphical Representation

If “r” is the distance between the ends and a single direction in space then p(x,y,z) is the probability of a length “r “in a single direction i.e. p(r). We need only r-values not the p(r) but P(r) the probability in any direction.

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Why P(r) not p(r)?Let p(r) be the population distribution in lahore at various

Muhallas each say between 5 - 5.2 miles from Kalma Chowk, while P(r) gives the population distribution in a zone drawn between two circles of radii 5 and 5.2 miles.

Concept: P(r) is independent of any direction Calculation of P(r):

Where r2= x2 + y2 + z2

dx.dy.dz = 4.pi. R2. dr

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• As “r” increases, (4.pi.r2) increases from zero (left hand side) and exp (- 2 r2) falls to zero (R.H.S). It means there must be some max. value of P(r) which would be compromise between the above two. This max. value will show the most probable state of the system.

• From maximization theorem, at d P(r)/dt =0 rm = 1/

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