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Dr. Z. Maghsoud

Physical Chemistry of Polymers(5)

THE AMORPHOUS STATE

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The glass transition temperature (Tg) is defined as the temperature at which the polymer softens because of the onset of long-range coordinated molecular motion.

On the simplest level, the structure of bulk amorphous polymers has been likened to a pot of spaghetti, where the spaghetti strands weave randomly in and out among each other.

The model would be better if the strands of spaghetti were much longer, because by ratio of length to diameter, spaghetti more resembles wax chain lengths than it does high polymers.

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Short-Range Interactions in Amorphous Polymers

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One of the most powerful experimental methods of determining short-range order in polymers utilizes birefringence.

Birefringence measures orientation in the axial direction.

The birefringence of a sample is defined by

axial and radial order

Short-Range Interactions

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n1 and n2 are the refractive indexes for light polarized in two directions 90° apart.

If a polymer sample is stretched, n1 and n2 are taken as the refractive indexes for light polarized parallel and perpendicular to the stretching direction.

For stretching at 45° to the polarization directions, the fraction of light transmitted is given by

d represents the thickness, λ0 represents the wavelength of light in vacuum, and π’= 3.14

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Short-Range Interactions

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The stress–optical coefficient (SOC) is a measure of the change in birefringence on stretching a sample under a stress σ.

The change in birefringence that occurs when an amorphous polymer is deformed yields important information concerning the state of order in the amorphous solid.

Experiments show no appreciable changes, leading to the conclusion that the order within a chain (axial correlation) does not change beyond a range of 5 to 10 Å.

Long-Range Interactions in Amorphous Polymers

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Small-Angle Neutron Scattering (SANS) The basic theory of SANS follows the development of light-scattering. For small-angle neutron scattering, the weight-average molecular

weight,Mw, and the average radius of gyration, Rg, may be determined

where R() is the scattering intensity known as the “Rayleigh ratio”.

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Small-Angle Neutron Scattering (SANS)

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In SANS experiments a deuterated polymer is dissolved in an ordinary hydrogen-bearing polymer of the same type (or vice versa).

A Zimm plot with extrapolations to both zero angle and zero concentration.

Note that the second virialcoefficient is zero, because polystyrene is essentially dissolved in polystyrene .

SANS of polyprotostyrenedissolved in polydeuterostyrene.

Small-Angle Neutron Scattering (SANS)

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The quantity (R2g/M)1/2 is a constant characteristic of each polymer.

The values in θ-solvents and in the bulk state are identical within experimental error; the polymer chain theoretically is unable to distinguish between a solvent molecule and a polymer segment with which it may be in contact.

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Models of Polymer Chains in the Bulk Amorphous State

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The most important reason why some polymer scientists are suggesting nonrandom chain conformations in the bulk state includes the high amorphous/crystalline density ratio.

MACROMOLECULAR DYNAMICS

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The basic notions of chain motion in the bulk state are required to understand much of physical polymer science, such as chain crystallization, the onset of motions in the glass transition region and the extension and relaxation of elastomers.

Small molecules move primarily by translation. A simple case is of a gas molecule moving in space, following a straight line until hitting another molecule or a wall.

In the liquid state, small molecules also move primarily by translation, although the path length is usually only of the order of molecular dimensions.

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MACROMOLECULAR DYNAMICS

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Polymer motion can take two forms: (a) the chain can change its overall conformation, as in relaxation after strain, or (b) it can move relative to its neighbors. Both motions can be considered in terms of self-diffusion.

Polymer chains find it almost impossible to move “sideways” by simple translation, for such motion is exceedingly slow for long, entangled chains.

Sideways diffusion can only occur by many cooperative motions.

Thus polymer chain diffusion demands separate theoretical treatment.

The Rouse–Bueche Theory

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The first molecular theories concerned with polymer chain motion were developed by Rouse and Bueche.

This theory begins with the notion that a polymer chain may be considered as a succession of equal submolecules.

These submolecules are replaced by a series of beads of mass M connected by springs with the proper Hooke’s force constant.

Rouse–Bueche bead and spring model of a polymer chain.

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The Rouse–Bueche Theory

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the restoring force, f, on a chain or chain portion is given by

where Δx is the displacement and r is the end-to-end distance of the chain.

The segments move through a viscous medium (other polymer chains and segments) in which they are immersed. This viscous medium exerts a drag force on the system, damping out the motions.

The viscous force on the ith bead is given by fi, where ρ is the friction factor

The Rouse–Bueche Theory

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Zimm advanced the theory by replacing the friction factor by the macroscopic viscosity of the medium. This leads to relaxation times of:

where η0 is the bulk-melt viscosity, p is a running index, and c is the polymer concentration.

The Rouse–Bueche theory is useful especially below 1% concentration. However, only poor agreement is obtained on studies of the bulk melt.

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Reptation and Chain Motion

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The de Gennes Reptation Theory

More recently, de Gennes introduced his theory of reptation of polymer chains. His model consisted of a single polymeric chain, P, trapped inside a three-dimensional network, G, such as a polymeric gel.

The chain P is not allowed to cross any of the obstacles; however, it may move in a snakelike fashion among them.

The de Gennes Reptation Theory

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The snakelike motion is called reptation. The chain is assumed to have certain

“defects,” each with stored length, b. These defects migrate along the chain in a

type of defect current. When the defects move, the chain

progresses. The tubes are made up of the surrounding chains.

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The de Gennes Reptation Theory

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The reptation motion yields forward motion when a defect leaves the chain at the extremity. The end of the chain may assume various new orientations.

In zoological terms, the head of the snake must decide which direction it will go through the bushes. De Gennes assumes that this choice is at random.

Physical entanglements define a tube of some 50 Å diameter.

The de Gennes Reptation Theory

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Using scaling concepts, de Gennes found that the self-diffusion coefficient, D, of a chain depends on the molecular weight M as

Numerical values of the diffusion coefficient in bulk systems range from 10-12 to 10-6 cm2/s.

The de Gennes theory of reptation as a mechanism for diffusion has also seen applications in the dissolution of polymers, termination by combination in free radical polymerizations, and polymer–polymer welding.

3M ~τ

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The de Gennes Reptation Theory

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Nonlinear Chains For nonlinear chains, sideways motion of the chain is effectively

restricted. So how do branched, star, and cyclic polymers diffuse?

Two possibilities exist for translational motion in branched polymers. First, one end may move forward, pulling the other end and the branch into the same tube.

This process is strongly resisted by the chains as it requires a considerable decrease in entropy to cause a substantial portion of a branch to lie parallel to the main chain in an adjacent tube

Nonlinear Chains

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Instead, it is energetically cheaper for an entangled branched-chain polymer to renew its conformation by retracting a branch so that it retraces its path along the confining tube to the position of the center mer.

Then it may extend outward again, adopting a new conformation at random

The basic requirement is that the branch not loop around another chain in the process, or it must drag it along also.

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Nonlinear Chains

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The result is that diffusion in branched-chain polymers is much slower than in linear chains.

Since many commercial polymers are branched or star-shaped, the self-diffusion of the polymer is correspondingly decreased, and the melt viscosity increased.

The Glass-rubber Transition

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Qualitatively, the glass transition region can be interpreted as the onset of long-range, coordinated molecular motion.

While only 1 to 4 chain atoms are involved in motions below the glass transition temperature, some 10 to 50 chain atoms attain sufficient thermal energy to move in a coordinated manner in the glass transition region

When Tg became relatively independent of Mc in a plot of Tg versus Mc (the molecular weight between cross-links) the number of chain atoms was counted

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Measurement of Tg

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Dilatometry The polymer is confined by a liquid and the change

in volume is recorded as the temperature is raised.

The usual confining liquid is mercury, since it does not swell organic polymers and has no transition of its own through most of the temperature range of interest.

The results may be plotted as specific volume versus temperature

The straight lines above and below Tg, yield the volumetric coefficient of expansion

Measurement of Tg

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Dilatometry

Dilatometric studies on branched poly(vinyl acetate)

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Measurement of Tg

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Differential scanning calorimetry (DSC) The DSC method uses a supply of energy at a varying rate to the

sample and the reference, so that the temperatures of the two stay equal.

The areas under the peaks can be directly related to the enthalpicchanges quantitatively.

The hysteresis peaks are most often associated with some physical relaxation, such as residual orientation. On repeated measurement, the hysteresis peak usually disappears.

Measurement of Tg

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Mechanical Methods Measurements by DMA refer to any one of several methods where the

sample undergoes repeated small-amplitude strains in a cyclic manner.

Molecules perturbed in this way store a portion of the imparted energy elastically and dissipate a portion in the form of heat.

E’, Young’s storage modulus: a measure of the energy stored elastically

E”, Young’s loss modulus: is a measure of the energy lost as heat

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The effect of frequency (time) on Tg

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The “Deborah number,” is defined as

τ is the characteristic material time and t is the timescale of the deformation.

Response will appear elastic at high Deborah numbers (De>>1) and viscous at low Deborah numbers (De 0).

In fact the observed glass transition temperature depends very much on the time allotted to the experiment, becoming higher as the experiment is carried out faster.

So the heating or cooling rate is important in determining exactly where the transition will be observed

The effect of frequency (time) on Tg

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Tg of polystyrene increases steadily in temperature as the frequency of measurement is increased. Since Tg is usually reported at 10 s (or 1 × 10-1 Hz), a glass transition temperature of 100°C may be deduced from Figure, which is, in fact, the usually reported Tg.

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The effect of frequency (time) on Tg

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The straight line for the α relaxation process, as drawn, corresponds to an apparent activation energy of 84 kcal/mol.

The β relaxation possesses a corresponding apparent energy of activation of 35 kcal/mol.

As a first approximation for many polymers, Tβ= 0.75Tg at low frequencies.

The WLF equation says that Tg will change 6 to 7°C per decade of frequency

THEORIES OF THE GLASS TRANSITION

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The Free-Volume Theory As first developed by Eyring and others, molecular motion in the bulk

state depends on the presence of holes, or places where there are vacancies or voids.

For polymer chains, the main difference being that more than one “hole” may be required to be in the same locality, as cooperative motions are required.

The important point is that molecular motion cannot take place without the presence of holes. These holes, collectively, are called free volume.

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The Free-Volume Theory

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Simha and Boyer then postulated that the free volume at T = Tgshould be defined as

The Free-Volume Theory

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Simha and Boyer concluded that

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The WLF Equation

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The WLF (Williams–Landel–Ferry) equation is derived here because the free volume at Tg arises as a fundamental constant.

AT is called the reduced variables shift factor .

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The making of a master curve, illustrated with polyisobutylene data. The classical Tgof this polymer is -70°C.

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Engineering Materials (Polymer Part), P.C. Painter

٣۶Engineering Materials (Polymer Part), P.C. Painter

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٣٧Engineering Materials (Polymer Part), P.C. Painter

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Engineering Materials (Polymer Part), P.C. Painter

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٣٩Engineering Materials (Polymer Part), P.C. Painter

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Engineering Materials (Polymer Part), P.C. Painter

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۴١ Engineering Materials (Polymer Part), P.C. Painter

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Engineering Materials (Polymer Part), P.C. Painter

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Engineering Materials (Polymer Part), P.C. Painter

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Engineering Materials (Polymer Part), P.C. Painter

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The Tg of one-phase systems can be estimated from the Tg values of the components, Tg1 and Tg2, with the Fox equation:

The w’s are weight fractions of the components.

Effect of copolymerization and blending on Tg

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Experimentally, two general cases may be distinguished: where one phase is retained and where two or more phases result.

Effect of copolymerization and blending on Tg

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Most polymer blends, as well as their related graft and block copolymers and interpenetrating polymer networks, are phase-separated.

In this case each phase will exhibit its own Tg.

If appreciable mixing between the component polymers occurs, the inward shift in the Tg of the two phases

polystyrene–block–polybutadiene–block–polystyrene

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Effect of crystallinity on Tg

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Semicrystalline polymers exhibit glass transitions, though only in the amorphous portions of these polymers.

Sometimes Tg appears to be disappeared, especially for highly crystalline polymers.

number of times a given value ofTg for linear polyethylene has been reported in the literature by various standard methods indicated

Effect of pressure on Tg

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Since an increased pressure causes a decrease in the total volume, an increase in Tg is expected based on the prediction of decreased free volume.