physical chemistry of polymers - zahra...

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Physical Chemistry of Polymers

Dr. Z. Maghsoud

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THE CRYSTALLINE STATE

The crystalline state is defined as one that diffracts X‐rays and exhibits the first‐order transition known as melting.

A first‐order transition normally has a discontinuity in the volume–temperature dependence.

The most important second‐order transition is the glass transition, in which the volume–temperature dependence undergoes a change in

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dependence undergoes a change in slope.

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THE CRYSTALLINE STATE

Polymers crystallized in the bulk, however, are never totally crystalline, a consequence of their long‐chain nature and subsequent entanglements.

The melting (fusion) temperature of the polymer, Tm, is always higher than the glass transition temperature, Tg. Thus the polymer may be either hard and rigid or flexible.

The development of crystallinity in polymers depends on the

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regularity of structure in the polymer .

Thus isotactic and syndiotactic polymers usually crystallize, whereas atactic polymers, with a few exceptions (where the side groups are small or highly polar), do not.

Melting Phenomena

There are several possible complications in the melting of polymers.

First of all, simple liquid behavior may not be immediately apparent , p q y y ppbecause of the polymer’s high viscosity. If the polymer is cross‐linked, it may not flow at all.

It must also be noted that amorphous polymers soften at their glass transition temperature, Tg, which is emphatically not a melting temperature but may resemble one.

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If the sample does not contain colorants, it is usually hazy in the crystalline state because of the difference in refractive index between the amorphous and crystalline portions.

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METHODS OF DETERMINING CRYSTAL STRUCTURE

There are four basic methods in wide use for the study of polymer crystallinity:

X‐ray diffraction, electron diffraction, infrared absorption, and Raman spectra.

By considering crystals as reflection gratings for X‐rays, Bragg derived his now famous equation for the distance d between successive identical planes of atoms in the crystal:

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where l is the X‐ray wavelength, q is the angle between the X‐ray beam and these atomic planes, and n represents the order of diffraction, a whole number.

THE UNIT CELL OF CRYSTALLINE POLYMERS

When polymers are crystallized in the bulk state, the individual crystallites are microscopic or even submicroscopic in size.

h l f h l d d b l d They are an integral part of the solids and cannot be isolated. Hence studies on crystalline polymers in the bulk were limited to powder diagrams of the Debye‐Scherrer type, or fiber diagrams of oriented materials.

Of course a major difference between polymers and low‐molecular‐weight compounds relates to the very existence of the

l l ’ l h i

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macromolecule’s long chains.

These long chains traverse many unit cells. Their initial entangled nature impedes their motion, however, and leaves regions that are amorphous. Even the crystalline portions may be less than perfectly ordered.

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One of the most important polymers to be studied is polyethylene.

The PE unit cell is orthorhombic, with cell dimensions of a = 7.40, b = , ,4.93, and c = 2.534 Å. The unit cell contains two mers.

The unit cell dimensions are substantially the same as those found for the normal paraffins of molecular weights in the range 300 to 600 g/mol.

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the unit cell of polyethylene


The chains are in the extended zigzag form, the carbon–carbon bonds are trans rather than gauche.

The polymer chain passes through many unit cells. Because of the known high molecular weight, the polymer chain was calculated to be even longer than the crystallites.

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A single crystal of polyethylene,precipitated from xylene

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STRUCTURE OF CRYSTALLINE POLYMERS

The fringed micelle model

According to the fringed micelle model, the crystallites are about 100 Å long. The disordered regions separating the crystallites are amorphous.

The chains wander from the amorphous region through a crystallite, and back out into the amorphous region. The chains are long enough to pass through several crystallites, binding them together.

The fringe micelle model was used with great

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The fringe micelle model was used with great success to explain a wide range of behavior in semicrystalline plastics, and also in fibers.


The Folded‐Chain Model

In 1957 Keller succeeded in preparing single crystals of polyethylene by precipitation from extremely dilute solutions of hot xylene. These crystals tended to be diamond‐shaped and of the order of 100 to 200 Å thick.

Electron diffraction analysis showed that the polymer chain axes in the crystal body were essentially perpendicular to the large, flat faces of the crystal.

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Since the chains were known to have contour lengths of about 2000 Å and the thickness of the single crystals was in the vicinity of 110 to 140 Å, Keller concluded that the polymer molecules in the crystals had to be folded upon themselves

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For many polymers, the single crystals are not simple flat structures.

The specific surface energy of the fold surface of polyethylene is about seven times greater than the specific surface energy of the lateral surfaces.

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Polymer single crystals in flat lamellae

Single crystal of polyamide 6 precipitated from a glycerol solution

CRYSTALLIZATION FROM THE MELT

Spherulitic Morphology

When polymer samples are crystallized from the melt, the most obvious of the observed structures are the spherulites, which are sphere‐shaped crystalline structuressphere‐shaped crystalline structures.

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Percent Crystallinity in Polymers

most crystallizing polymers are semicrystalline; a certain fraction of the material is amorphous, while the remainder is crystalline.

The reason why polymers fail to attain 100% crystallinity is kinetic, resulting from the inability of the polymer chains to completely disentangle and line up properly in a finite period of cooling or annealing.

There are several methods for determining the percent crystallinity in

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such polymers. The first involves the determination of the heat of fusion of the whole sample by calorimetric methods such as DSC

Percent Crystallinity in Polymers

Another method involves the determination of the density of the crystalline portion via X‐ray analysis of the crystal structure, and determining the theoretical density of a 100% crystalline material.

The density of the amorphous material can be determined from an extrapolation of the density from the melt to the temperature of interest.

The percent crystallinity is given by

١۴

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KINETICS OF CRYSTALLIZATION

The melting temperature of poly(ethylene oxide) is 66°C, where the rate of crystallization is zero. The rate of crystallization increases as the temperature is decreased.

١۵the isothermal crystallization of poly(ethylene oxide) as determined dilatometrically.


Crystallization rates may also be observed microscopically, by measuring the growth of the spherulites as a function of time.

This may be done by optical microscopy or by transmission electron microscopy of thin sections.

The low‐molecular‐weight atactic polypropylene acts as “impurity” .

Spherulite radius as a function of time, grown isothermally (at 125°C) in a blend

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g y ( )of 20% isotactic and 80% atactic (M = 2600) polypropylene

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The isothermal radial growth of the spherulites is usually observed to be linear.

The more impurity, the slower is the growth rate. However, linearity of growth rate is maintained.

In such a steady state, the radial diffusion of rejected impurities is outstripped by the more rapidly growing lamellae so that impurities diffuse aside and are trapped in interlamellar channels.

Blends of unextracted Isotactic polypropylene with atactic polypropylene

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When the radial growth rate is plotted as a function of crystallization temperature, a maximum is observed between Tg and Tm.

Several experimental findings form the basis for three theories of polymer crystallization kinetics.

Keller postulated that the chains had to be folded back and forth.

The rate of radial growth of the spherulites

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g pwas linear .

The rate of growth goes through a maximum as the temperature of crystallization is lowered.

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Theories of Crystallization Kinetics

The Avrami Equation

In the Avrami theory, the concept of raindrops falling in a puddle is taken into account.

These drops produce expanding circles of waves that intersect and cover the whole surface. The drops may fall occasionally or all at once.

The expanding circles of waves, of course, are the growth fronts of the spherulites, and the points of

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impact are the crystallite nuclei.

The probability px that a point P is crossed by x fronts of growing spherulites is given by an equation originally derived by Poisson:

The Avrami Theory

Where E represents the average number of fronts of all such points in the system.

The probability that P will not have been crossed by any of the fronts, and is still amorphous, is given by

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Since E0 and 0! are both unity. Of course, p0 is equal to 1 ‐ Xt, where Xt is the volume fraction of crystalline material, known widely as the degree of crystallinity.

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The Avrami Theory

In athermal nucleation all nuclei are formed and start to grow at time t

= 0. The spherical crystals grow at a constant rate r..

All nuclei within the radius r.t from point P have formed spherical waves which pass the arbitrary point P during time t.

The average number of crystal fronts (E) passing under these conditions is given by:

N)tr(4

)t(E 3

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where N is the volume concentration of nuclei.

N)tr(3

)t(E

)Ntr3

4exp(x1 33

t

The Avrami Theory

A slightly more complex case involves thermal nucleation.

The nuclei are here formed at a constant rate both in space and time, similar to 'normal’ rain.

The number of waves (dE) which pass the arbitrary point (P) for nuclei within the spherical shell confined between the radii rand r + dr is given by:

drL)r

rt(r4dE *2

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where L* is the nucleation density (number of nuclei per cubic metreper second). The total number of passing waves (E) is obtained by

integration of dE between 0 and r.t :

43*

tr

0

*2 t3

rLdr)

r

rt(Lr4E

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The Avrami Theory

Which gives

)trL

exp(x1 43*

t

Crystallization based on different nucleation and growth mechanisms can be described by the same general formula, the general Avramiequation:

)t3

exp(x1 t

)Ktexp(x1 n

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The above derivation suggests that the quantity n should be either 3 or 4.

)Ktexp(x1 t

The Avrami Theory

The constants in the Avrami equation are obtained by taking the double logarithm of the last equation:

tlnnKln)x1ln(ln t

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Keith–Padden Kinetics of Spherulitic Crystallization

Although the Avrami equation provides useful data on the overall kinetics of crystallization, it provides little insight as to the molecular organization of the crystalline regions, structure of the spherulites, dand so on.

The first theory to address the kinetics of spherulitic growth in crystallizing polymers directly was developed by Keith and Padden.

According to Keith and Padden, a parameter of major significance is

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the quantity


D is the diffusion coefficient for impurity in the melt and G represents the radial growth rate of a spherulite.

The quantity δ, whose dimension is that of length, is a measure of the internal structure of the spherulite, or its coarseness.

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By logarithmic differentiation of δ equation

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The derivative dD/dT always has a positive value. However, dG/dTmay be positive or negative.

The coarseness of the spherulites depends on which of the two terms on the right of equation is the larger.

If the quantity on the right‐hand side of the equation is positive, an increase in coarseness is expected as the temperature is increased.

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Hoffman’s Nucleation Theory

The major shortcoming of the Keith–Padden theory resides in its qualitative nature.

The free energy of formation of a single chain‐folded crystal may be set down in the manner of Gibbs as

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l: the thin dimension of the crystal,X: the large dimension,σe:the fold surface interfacial free energy, σ: the lateral surface interfacial free energy,

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The quantity Δf represents the bulk free energy of fusion, which can be q y p gy ,approximated from the entropy of fusion, ΔSf, by assuming that the heat of fusion, Δhf, is independent of the temperature.

At the melting temperature of the crystal, the free energy of formation is zero.

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For x >>l ,


Hoffman defined three regimes of crystallization kinetics from the melt, which differ according to the rate that the chains are deposited on the crystal surface.

Regime I. One surface nucleus causes the completion of the entire substrate of length L ; that is, one chain is crystallizing at a time.

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Regime II. In this regime multiple surface nuclei occur on the same crystallizing surface, because the rate of nucleation is larger than the crystallization rate of each molecule.

This results from the larger undercooling necessary to reach regime II. As in regime I each molecule is assumed to fold back and forth to give adjacent reentry.

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Regime III. This regime becomes important when the niche separation characteristic of the substrate in regime II approaches the width of a stem.

Regime III is very important industially, where rapid cooling is employed. In this regime, the crystallization rate is very rapid.

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As the temperature is lowered through regimes I, II, and III, substrate completion rates per chain decrease. However, more chains are crystallizing simultaneously.

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THERMODYNAMICS OF FUSION

Melting is a first‐order transition, ordinarily accompanied by discontinuities of such functions as the volume and the enthalpy

Owing to the range of crystallite sizes and degrees of perfection in the real case, a range of melting temperatures is usually encountered.

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Dilatometric behavior of polymer melting

Theory of Melting Point Depression

The melting point depression in crystalline substances from the pure state Tºf is given by the general equation

ΔHf is the heat of fusion per mole of crystalline mers.

For small values of XB,

٣۴

There are three important cases in which the melting temperature may be depressed.

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1. If XB represents the mole fraction of noncrystallizable comonomerincorporated in the chain,

2. The mer unit at the end of the chain must always have a different chemical structure from those of the mers along the chain.

If M0 is the molecular weight of the end mer (and assuming that both

٣۵

If M0 is the molecular weight of the end mer (and assuming that both ends are identical), the mole fraction of the chain ends is given approximately by 2M0/Mn.


3. If a solvent or plasticizer is added, the case is slightly more complicated.

)(VR11 2u

V1 is the molar volume of the solvent, and Vu is the molar volume of the polymer repeat unit. χ1 the polymer‐solvent interaction. v 1 is the solvent volume fraction.

)(VHTT

2111

1

u

f0ff

٣۶

The quantity Tºf may be determined either directly on the pure polymer or by a plot of 1/Tf versus v1.

χ1 and ΔHfof the polymer are calculated from the slope and intercept of a plot of (1/Tf) ‐ (1/ Tºf) versus (1/Tf).

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EFFECT OF CHEMICAL STRUCTURE ON THE MELTING TEMPERATURE

The actual values of the enthalpy and entropy of fusion are, of course, controlled by the chemical structure of the polymer.

In general, high melting points are associated with highly regular structures, rigid molecules, close packing capability, strong interchainattraction, or several of these factors combined.

The effect of structural irregularities can be illustrated by a study of polyesters:

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EFFECT OF CHEMICAL STRUCTURE ON THE MELTING TEMPERATURE

The effect of bond flexibility may also be examined utilizing the polyester structure . In this case substitutions in h i id i dthe rigid aromatic group are made:

Interchain forces can be illustrated by the following substitutions of increasingly

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polar groups:

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