Lecture No.02 Polymer engineering

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MM – 538 Dr. Kausar Ali SyedPolymer Engineering Lecture No 2

July 7, 2010

Structure of Polymers

Various polymer characteristics are affected by the magnitude of the molecular weight

or degree of polymerization. Melting point, softening point is increased with increasingmolecular weight. Melting temperature also rises with an increase in intermolecular

forces or bonds which are generally Van der Waals and/or hydrogen.

The properties of polymers depend not only on molecular weight but are also strongly influencedby shape and details of the structure of the molecular chains. These details include differentskeletal structures, the overall chemical composition and the sequence of monomer units in thecase of copolymers, the stereochemistry or tacticity of the chain, and the geometric isomerizationin the case of diene-type polymers. Modern polymer synthesis techniques permit considerablecontrol over various structural possibilities.

Chain Dimension

The subject of chain dimension is concerned with relating the sizes and shapes of

individual polymer molecules to their chemical structure, chain length, and molecular

environment. Polymer chains in solution are free to rotate around individual bonds,

and almost a limitless number of conformations or chain orientations in three-

dimensional space are possible for long, flexible macromolecules.

Molecular shape (Conformational Disorder)

We can calculate that the average length of a PE molecule with n = 500 [i.e., (—C2H4—)500] is 154nm, because each of the 1000 C-C bonds is 0.154 nm long. However, this calculation needs acorrection, because polymer chains are not strictly straight.

Consider the following figure. Third carbon atom may be at any point on the cone of

revolution and still subtend about a 109° angle with the bond between other two

atoms.

Fig. 1

A straight chain segment results when successive chain atoms are positioned as below

Fig. 2



Supposing the bond angles across the carbon atoms as approximately 120˚, the maximum limit for the end-to-end "sawtooth" length of the polyethylene chain (Fig. 2 & 3) with n = 500 would be 154nm x sin 60˚ = 135 nm. This is only one of the conformation possible. Single chain bonds are

capable of rotation and bending in three dimensions. This rotational freedom is available athigher temperatures in amorphous (noncrystalline) polymers and in polymers dissolved in a liquidsolvent.

Fig. 5

Fig. 6

Chain bending or twisting takes place when there is a rotation of the chain atoms into

other positions as illustrated below:

Fig. 3

Thus, a single chain molecule compound of many chain atoms might assume a shape

similar to that represented schematically below, having a multitudes of bends, twists,

and kinks.

Fig. 4

With only three C-C bonds in butane (Fig. 5), the a-d distance is 0.4 nm,whereas the a-d' distance is less than 0.3 nm. The length varies randomlybetween these limits as the thermal agitation rotates the bond angles.



The maximum limit for the end-to-end length of the polyethylene as calculated above is 135 nm;the minimum end-to-end vector length r would be a fraction of a nanometer i.e., much smaller

than the total chain length (in the unlikely event that the molecule were twisted or kinked tobring the two ends into contact).

This end to end chain vector is calculated using statistical methods.

The average, or root-mean square length, √ ˂ r 2

˃ ), in noncrystalline polymers calculated on astatistical basis as

√ ˂ r 2 ˃ ) = l √xwhere

√ ˂ r 2 ˃ = root mean square end to end distance of the chain (also written as r rms )

I = the individual bond length and x = the number of bonds in the chain.

Thus, the doubling of the degree of polymerization of polyvinyl increases the mean end-to-endlength by 40 percent. Some polymers consist of large numbers of molecular chains, each

of which may bend, coil, and kink in the manner shown above (Fig. 4). This leads to

extensive intertwining and entanglement of neighbouring chain molecules, a situation

similar to that of a fishing line that has experienced backlash from a fishing reel or likespaghetti thrown into a bowl.

The twisting and coiling arising from bond rotation is called conformational entropy, or disorder.This rotation is limited to single C-C bonds; however, it is important to properties. For example,above the glass-transition temperature, Tg , many polymers may be stretched from their kinkedconformation to give high strains without changing the interatomic distances. Rubbers possess thischaracteristic, so they develop high strains at relatively low stresses. Furthermore, the moleculesrecoil to their kinked conformation when the stress is removed. Rubber is visibly elastic because ithas a very low elastic modulus.As for all materials, the mechanical strength of polymers depends on structure. The strength of polymers can vary greatly, and the origin of this variation is the structure of the molecule. Thereare a great number of covalent bonds in a polymeric molecule. There is a very small possibility thata polymeric chain will be one straight line. Rather each individual chain will be jumbled , and thecombination of chains will be randomly entangled like spaghetti thrown into a bowl.

When the polymer is subjected to a mechanical load, the chains shown in fig. disentangle. If thisdisentanglement is difficult, the polymer is strong. If not, the polymer is weak. In order for disentanglement to occur, the chains must slide past each other. Thus, polymers in which chainsliding is easy are weak, and those in which chain sliding is difficult are strong. These random

coils and molecular entanglements are also responsible for many important

characteristics of the polymers e.g. large elastic extensions by rubber.

All polymers have some strength because of the large number of intermolecular forces

(secondary bonds) between the polymer chains.



Because polymers are such large molecules, the large number of secondary bonds

inhibit chain sliding and give the polymer strength.

There are four features which inhibit chain sliding:

• Connecting the chains through cross-linking is the most effective method of

inhibiting chain sliding. Therefore, cross-linked polymers are among the

strongest.

• Hydrogen bonding is the strongest of the intermolecular forces (secondary

bonds). If the polymer has a chemical structure in which hydrogen bonding can

occur between the polymer chains, chain sliding will be inhibited and the

polymer will be strong. This is the case for nylons.

• If the polymer has benzene ring (aromatic) groups in the chain, chain sliding will

be inhibited.

• If the polymer has large side groups, chain sliding will be inhibited, as the bulky

side groups will interfere chain sliding.

Some mechanical and thermal properties of polymers are a function of the ability of

chain segments to rotate in response to applied stresses or thermal vibrations.

Rotational flexibility is dependent on mer structure and chemistry. C ═ C is rotationally

rigid. Also bulky or large group like benzene ring in polystyrene restricts rotational

movement.

Skeletal Structure

Different skeletal structures like linear, branched, cross-linked, and network are possibledepending on the manner in which structural units are joined together.

Linear structures are those in which mers are joined together end to end in single chains. Thesechains are flexible and even though we say “linear’, the chains are actually not in the form of straight lines but may be thought of as a mass of spaghetti. These are only loosely bondedtogether by secondary bonds. Ex. PE, PVC, PS, PMMA, nylons, etc.

Branched structures are those in which there are primary polymer chains and secondaryoffshoots of smaller chains that stem from these main chains. The chain packing efficiency isreduced with the formation of side branches, resulting in lowering of the polymer density.Branching of linear polymers thus weakens secondary bonds between the chains and lowers thetensile strength of the bulk polymeric material.



Crosslinked structure are those in which adjacent linear chains are joined to one another atvarious points by covalent bonds. This is achieved either during synthesis or by a non reversiblechemical reaction usually at elevated temperature. Cross-linking is the bonding of linear chainstogether by low molecular weight compounds. In rubber it is called vulcanization when it is throughadditive atoms or molecules like S.

Network structure: Trifunctional mer units, having three covalent bonds, form threedimensional networks instead of linear chain as assumed by bifunctional mers.

Variations in skeletal structure give rise to major differences in properties. For example, linear polyethylene has a melting point about 20°C higher than that of a branched polyethylene. Unlikelinear and branched polymers, network polymers do not melt upon heating and will not dissolve,though they may swell considerably in compatible solvents. The importance of crosslink densitycan be seen in the vulcanization (i.e. sulphur-crosslinking) of natural rubber. With low crosslinkdensities (i.e. low levels of sulphur) the product is a flexible elastomer, whereas it is a rigid material



when the crosslink density is high. Similarly 3D-network structures of epoxies and phenol-formaldehydes have distinctive mechanical and thermal properties.

Copolymers

In addition to these polymer structures, we can also modify the structures by combining more thanone monomer in the same chain, thus obtaining different properties. Polymers with two different

repeating units (two different monomers) in their chains are called copolymers (e.g., styrene-acrylonitrile) and one that contains three monomers a terpolymer (e.g., ABS, or acrylonitrile-butadiene-styrene). Commercially, the most important copolymers are derived from vinylmonomers such as styrene, ethylene, acrylonitrile, vinyl chloride, and vinyl acetate.

These combinations can be in different sequences depending upon the relative reactivities of eachmonomer during copolymerization process.

Although the monomers in most copolymers are randomly arranged, four distinct types of copolymers have been identified: random, alternating, block , and graft.

Random Copolymers:

Different monomers are randomly arranged within the polymer chains. If A & B are different

monomers, then an arrangement might be.

-AAABBBABBABABBAAA-

(r 1 = r 2 = 1) ideal Ex. Copolymerization of styrene and 4-chlorostyrene.

Alternating Copolymers

Different monomers show a definite ordered alternation, as

-ABABABABAB-

(r 1 = r 2 = 0) Ex. Copolymerization of styrene and maleic anhydride.

If both reactivity ratios are small but not exactly zero, the comonomer sequence will not be

completely alternating but will have an alternating tendency (i.e., segments of alternating

sequences). Ex. styrene and acrylonitrile.

Block Copolymers:



Different monomers in the chain are arranged in relatively long blocks of each monomer.

-AAAAAA-BBBBBB----

( both r 1 and r 2 are very much larger than unity - block copolymers or a mixture of twohomopolymers will form)

Graft Copolymers:

Appendages of one type of monomer are grafted to long chain of another.

Chain reaction polymerization can take place between two or more different monomers if they canenter the growing chains at relatively the same energy level and rates. e.g. 85% PVC and 15%PVA are used to form a copolymer to be used as a basic material for vinyl records.

m

H H

C C

H Cl

n

H H

C C

H O C

O

CH3

+

H H

C C

H Clx

C

H

H

H

C

H C

O

CH3y

PVC – PVA Copolymer.

An ABS copolymer consists of 30 wt % polyacrylonitrile, 40 wt% polybutadiene, and 30 wt

% polystryrene. Calculate the mole fraction of each component in this material.

Let there be 100g of terpolymer.Therefore we have 30g of PAN, 40g of PB, and 30g of PS. first determine the no. of moles of eachcomponent and then we calculate the mole fraction of each.

• Moles of PAN

BBB

BAAAAAAAAAAA

AAAAB BB BB BB B

B



H

C

H

H

C

C N

Mol.wt. of PAN mer = 3x12+14+3x1 = 53 g/molNo. of moles of PAN in 100g of terpolymer =

• Moles of PB

H H H H

C C C C

H H

Mol.wt.of PB mer = 4x12+6x1 =54 g/mlNo. of moles of PB in 100g of terpolymer =

• Moles of PS

H

C

H

H

C

COH5

Mol.wt. of PS mer 12x8+1x8=104g/ml

No. of moles of PS in / wg of terpolymer =

Mole fraction of PAN = = 0.355

Mole fraction of PB = = 0.465

Mole fraction of PS = = 0.180

Determine the mole fraction of PVC and PVA in a copolymer having a molecular weight of 10,000 g/ml and degree of polymerization of 140.

H H

C C

H Cl

Mol.wt. of the VC mer = 2 X 12 + 1 X 3 + 1 X 35.5 = 62.5 g/ml



H

C

H

H

C

O C

O

CH3

Mol.wt. of the VA mer = 4 X 12 + 6 X 1 + 2 X 16 = 86 g/ml

Since the sum of the mole fractions of PVC + PVA = 1

Average mol.wt. of the copolymer mer is

Average Mol.wt. of the Copolymer mer =

g/mol.mer

=

or

Molecular Configuration

In addition to the effects of skeletal structure and of the chemical composition of the repeat units,type, number, and sequential arrangements of monomers along the chain, the properties of apolymer are strongly influenced by its molecular microstructure, i.e., the spatial arrangement of substituent groups.

Different molecular microstructures arise from there being several possible modes of propagation.For polymers having more than one side atom or group of atoms bonded to the main chain, theregularity and symmetry of the side group arrangements can significantly influence the properties.

Consider the mer unit

H H│ │

─ C ─ C─ │ │

H ®



® is a group other than H e.g. Cl, CH3

One arrangement is possible when R side group of successive mer unit is bonded to alternate Catom e.g.

H H H H

│ │ │ │ ─ C ─ C ─ C ─ C ─ │ │ │ │

H ® H ®This is known as head to tail configuration.

Other is head to head when ® is bonded to adjacent chain atom

H H H H│ │ │ │ ─ C ─ C ─ C ─ C ─ │ │ │ │

H ® ® H

Head to tail configuration predominates in most polymers because of both steric and energeticreasons. There is often polar repulsion between (R) groups in head to head configuration.

Isomerism is also found in polymer molecules, wherein different atomic configuration are possible

for the same composition. Two isomeric subclasses are stereoisomerism and geometrical isomerism

Tacticity

Atoms are linked together in the same order (head to tail) but differ in their spatial arrangement.For one stereoisomerism, all the (R) groups are situated on the same side of the plane formed bythe extended-chain backbone.

H H H H H H H│ │ │ │ │ │ │

─ C ─ C ─ C ─ C ─ C ─ C ─ C ─

│ │ │ │ │ │ │H ® H ® H ® H



This is called isotactic configuration.

If the substituent groups regularly alternate from one side of the plane to the other, as shownbelow

H H H ® H H H ®│ │ │ │ │ │ │ │

─ C ─ C ─ C ─ C ─ C ─ C ─ C ─C│ │ │ │ │ │ │ │

H ® H H H ® H H

it is syndiotactic configuration.

And when groups are randomly positioned as

H H H H H H H| | | | | | | |

—C — C — C — C — C — C — C — C—| | | | | | | |

H H H H H

It is called atactic configuration.

Conversion from one stereoisomer to another (e.g., isotactic to syndiotactic) is not possible by asimple rotation about single chain bonds; these bonds must first be severed, and then, after theappropriate rotation, they are reformed.In reality, a specific polymer does not exhibit just one of these configurations; the predominantform depends on the method of synthesis.

R

RRR

R



H3

C

H

CH3 H

Geometrical Isomerism:

Another important chain configuration are possible within mer units having a double bondbetween chain Carbon atoms. Bonded to each of the carbon atom of the double bond is a singleside-bonded atom or radical which may be situated on one side of the chain or its apposite.Consider the isoprene mer.

C ═ C

—CH2 CH2—in which CH3 group and H atom are positioned on the same side of the chain. This is termed as Cisstructure. The resulting polymer is Cis-isoprene or natural rubber.Alternatively CH3 & H reside on opposite chain sides. It is trans structure. Trans isoprene, or guttapercha has distinctively different properties from natural rubber as a result of this configuration.Conversion of trans to Cis or vice versa is not possible by simple chain bond rotation because the

chain double bond is extremely rigid.

C ═ C

—H2C CH2—

Polymer molecule may be characterized in terms of their size, shape, and structure.

Molecular Characteristics

Chemistry Size Shape Structuremer Composition (Mol. wt) (Chain twisting entanglements etc)

Linear Branched Crosslinked Network

Isomeric States



Stereoisomers Geometrical Isomers

Isotactic Syndiotactic Atactic Cis Trans

Classification scheme for the characteristics of Polymer Molecules.

Chain Conformation

If three single bonds in an all-carbon backbone lie in a plane and are arranged as shown in figs 1(a) and(b), the middle bond is said to be a trans bond. This is usually the conformation of lowest energy. If three single bonds in an all-carbon backbone are arranged as shown in figs 1 (c) and (d), the middlebond is said to be a gauche bond.

Fig. 1 Trans/gauche isomerism: (a) a trans bond viewed normally to the plane of the three bonds required todefine it; (b) a view of (a) from the righth and side, looking almost end-on to the trans bond; (c) and (d)nearly end-on views of left- and right handed gauche bonds, respectively.

Figure 2 shows how the energy of the n-butane molecule CH3—CH2—CH2—CH3 varies with the torsionalangle around the central C—C bond.

This angular variation is due partly to the inherent lowering of the energy of the bonds when the shape,or conformation, of the molecule corresponds to any of the three angles 60°, 180° and 300°in fig. 2 and partly due to the repulsive interaction of the groups of atoms attached to the two inner Catoms.

The energy in the trans conformation is lower than that in the gauche conformations, which are theconformations of next lowest energy, because in this conformation the two large CH 3 groups are as faraway from each other as possible.



Fig. 2 The variation of the energy of the n-butane molecule CH3—CH2—CH2—CH3 as a function of the torsion angle

of the central C—C bond measured from the eclipsed conformation.

The two gauche conformations can be imagined to be obtained by rotation of one end of the moleculewith respect to the other by ±120° around a central bond that is originally trans. The highest-energyconformation is the eclipsed arrangement which brings the CH3 groups most closely together and isobtained by a rotation of ±180° around a central bond that is originally trans. Note that, although theword gauche means ‘left’ in French, it also means ‘awkward’ in both English and French; gauche bondscan be either right- or left-handed. Butane molecules with right- and left-handed gauche bonds aremirror images of each other.

As has already been implied, similar energy considerations apply to each of the single C—C bonds in apolymer backbone, but the lowest-energy states (and hence the conformation of the backbone) may bemodified by the factors considered below.



Fig. 5

Conformations and chain statistics

In the previous section the idea of the conformation of a small molecule, n-butane, was discussed, aswas the idea of the local conformation of a polymer chain. It was tacitly assumed that such states werein fact different from each other, i.e. that they each had a certain degree of permanence. The plot in fig.2 shows that the difference in energy between the trans and gauche states is about 3 kJ mol —1, which isquite close to thermal energies even at room temperature. This means that, even though the transstate has lower energy, there will be a considerable fraction of molecules in the gauche state. Thevarious conformations of a molecule are often called rotational isomers and the model of a polymerchain introduced by Flory in which the chain is imagined to take up only discrete conformational statesis called the rotational isomeric-state approximation. Polymers typically have a very large number of single bonds around which various conformational states can exist, and a polymer molecule as a whole

therefore has a very large number of conformational states. Even for a molecule with ten C-C bonds inthe backbone the number is 38 = 6561, assuming that all possible trans and gauche states could bereached, and the corresponding number for a typical polyethylene chain with about 20 000 C—C bondsis about 109540. It is clear from this that even though the number of conformational states that couldpotentially be reached is reduced somewhat by steric hindrance or overlapping of the chain with itself,only statistical methods can be used in discussing the conformations of whole polymer chains. In thisdiscussion it is usual to start with a simplified model, that of the freely jointed chain.

The single freely jointed chain

Rotation around a single bond in a molecular chain generally gives a cone of positions of one part of thechain with respect to the other. If the rotation is not totally free, as for example when trans or gauchebonds have lower energy than other conformations for an all-carbon backbone, certain positions on thiscone are favoured. Nevertheless, in order to develop the simplest form of the statistical theory of polymer-chain conformations the following simplified model, called the freely jointed random linkmodel, is used.

(a) The real chain is replaced by a set of points joined by n equal one dimensional links of length l. The contour or fully extended length of the chain is then nl.

(b) It is assumed that there is no restriction on the angles between the links; the angles are notrestricted to lie on cones.

(c) It is assumed that no energy is required to change the angles.



An important quantity in the theory is the root-mean-square (RMS) length of an unperturbed randomlycoiled chain, which can be calculated as follows (see fig. 6):Let r be the end-to-end vector of the chain. Then

r x = ∑ l cosθi and r 2 x = l2 ( ∑ 2θi + ∑ cosθi cosθ j )

i i≠jIf rx

2 is averaged over a large number of chains, the last term averages to zero, because there is norestriction on the angles between the links. Thus

˂ r 2 x ˃ = nl2 ˂ cos2θi ˃ = 1/3 nl2

where the angle brackets ˂ ˃ denote the average over all chains. However,˂ r 2 ˃ = ˂ r 2 x ˃ + ˂ r 2 y ˃ + ˂ r 2 z ˃ and ˂ r 2 x ˃ = ˂ r 2 y ˃ = ˂ r 2 z ˃

Thus the RMS length r rms is given byr rms = √ ˂ r 2 ˃ = n1/2l

The fully extended length of the chain is equal to nl so that the maximum extensibility of a randomcoil with r = r rms is from n1/2l to nl, i.e. a factor of n1/2.

Fig.6. Calculation of the RMS chain length. The thick lines represent five actual linkssomewhere in the chain, the rest of which is represented by the curved lines.The angle θi is the angle between the ith link and the Ox axis.

The value of rrms also gives a measure of the spatial extent of a chain. A second useful measure of this is

the radius of gyration, rg, which is the RMS distance of the atoms of the chain from the centre of gravityof the chain. Debye showed that, provided that n is very large, rg = rrms / √6.

In order to simplify the theory further it is usual to assume that, for all chains that need to beconsidered, the actual end-to-end distance is very much less than the fully extended length, i.e. r ˂˂ nl,which becomes true for all chains when n is sufficiently large. A chain for which the assumption is validis called a Gaussian chain. Consider such a chain with one end fixed at the origin and let the other end,P, be free to move (see fig.10). With OP ˂˂ nl it can then be shown that the probability p(x, y, z) that Plies in the small element of volume dx dy dz at (x, y, z) is

p (x, y, z,) dx dy dz = ( b3 / π3/2 ) exp (─b2 r2 ) dx dy dz

where b2 = 3 / ( 2nl2 ) and r2 = x2 + y2 + z2 = 0

The function p (x, y, z ) is the Gaussian error function or normal distribution.The maximum valueof p (x, y, z ) is at the origin, corresponding to r2 = x2 + y2 + z2 = 0. The most probable value of r isnot, however, zero. The value of rrms also gives a measure of the spatial extent of a chain. A second useful measure of this isthe radius of gyration, rg, which is the RMS distance of the atoms of the chain from the centre of gravityof the chain. Debye showed that, provided that n is very large, rg = rrms / √6.



Fig. 10 The Gaussian chain with one endcoincident with the origin.

In order to simplify the theory further it is usual to assume that, for all chains that need to beconsidered, the actual end-to-end distance is very much less than the fully extended length, i.e. r ˂˂ nl,which becomes true for all chains when n is sufficiently large. A chain for which the assumption is validis called a Gaussian chain. Consider such a chain with one end fixed at the origin and let the other end,

P, be free to move (see fig.10). With OP ˂˂ nl it can then be shown that the probability p(x, y, z) that Plies in the small element of volume dx dy dz at (x, y, z) is

p (x, y, z,) dx dy dz = ( b3 / π3/2 ) exp (─b2 r2 ) dx dy dz

where b2 = 3 / ( 2nl2 ) and r2 = x2 + y2 + z2 = 0

The function p (x, y, z ) is the Gaussian error function or normal distribution.The maximum valueof p (x, y, z ) is at the origin, corresponding to r2 = x2 + y2 + z2 = 0. The most probable value of r isnot, however, zero. The probability P(r) dr that the end P lies somewhere in the spherical shell of radius r and thickness dr

is given by

P ( r ) dr = 4πr 2 (p) dr = 4πr 2 ( b3 / π 3/2 ) exp (─b2 r 2 ) dr

For r = 0 this probability is zero, and the probability peaks at r = 1 / b = √2n / 3l = √2 / 3(r rms ).

By rearrangement, above equation can be written

P(br) d(br) = (4π 1/2 ) (br)2 exp[ (─br 2 )] d (br)

where br is dimensionless. P (br) is plotted against br in fig. 7.

Fig. 7



The probability for a chain of length r in the Gaussian approximation.See the text for discussion.

As discussed further in the following section, it can be shown that the statistical distribution of end-to-end distances for any real chain reduces to the Gaussian form if the number of rotatable links issufficiently large. By suitably choosing n and l for the freely jointed random-link model, both r rms and thefully extended length can be made equal to the corresponding values for the real chain. These valuesdefine the equivalent freely jointed random chain. For example, if it is assumed that in a realpolyethylene chain (i) the bonds are fixed at the tetrahedral angle and (ii) there is free rotation around

the bonds, it can be shown that one random link is equivalent to three C—C bonds in the real chain.Evidence based on stress–optical measurements suggests that, for real polyethylene, the number of C—C bonds in the equivalent random link is much greater than this, as might be expected, because therotation around the bonds is not free.

Setting r2 = rrms gives the ratio as x2 exp [─3(x2 ─ 1) / 2], where x = r1 / rrms.

Substituting x = 2 or 3 gives the ratios 0.044 and 5.5 10-5.

Example 3.2Calculate the ratio of the probability of the end-to-end separation of a Gaussian chain being within asmall range dr near (a) 2rrms and (b) 3rrms to that for it being within a small range of the same size nearrrms.

SolutionFor any two values r1 and r2 of the end-to-end separation the ratio of the probabilities, from equation(3.8), is

( r1 / r2 )2 exp [─b2(r21 ─ r2

2)] = ( r1 / r2 )2 exp[─3( r21 ─ r2

2) / (2r2rms)].

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Lecture No.02 Polymer engineering