Fundamentals of Polymer Engineering Part 9 doc

A variety of methods are available for molecular-weight determination andthey are applicable in different ranges of molecular weight.. Thus, end-group analysis andcolligative property me

in both the crystalline and amorphous phases This variation can be brought abouteither by changing material variables or process conditions The former includethe chemical structure, the molecular weight and its distribution, the extent ofchain branching, and the bulkiness of the side groups The latter include thetemperature and the deformation rate It is the interplay within this multitude ofvariables that leads to the physical structure visible in the finished product Thisstructure, in turn, determines the properties of the solid polymer In this chapter,

we examine the methods of measuring the polymer’s molecular weight and itsdistribution These quantities were defined inChapter 1, and knowledge thereofcan be helpful to the process engineer in optimizing desired polymer properties.These properties include mechanical properties such as impact strength, flowproperties such as viscosity, thermal properties such as the glass transitiontemperature, and optical properties such as clarity

There are several other reasons why we might want to measure themolecular weight The molecular weight and its distribution determine theviscous and elastic properties of the molten polymer This affects the processi-

bility of the melt and also the behavior of the resulting solid material (see alsoChapter 12) To cite a specific example [1], a resin suitable for extrusion musthave a high viscosity at low shear rates so that the extrudate maintains itsintegrity To be suitable for injection molding, however, the same resin must have

a low viscosity at high shear rates so that the injection pressure not be excessive.Both of these requirements can be satisfied by a proper adjustment of themolecular-weight distribution More often, though, different grades of the samepolymer are marketed for different products that are fabricated via differentpolymer processing operations; the resin used for making polycarbonate waterbottles, for example, differs significantly in molecular weight from the poly-carbonate that goes into compact disks Differences in molecular-weight distribu-tion also influence the extent of polymer chain entanglement and the amount ofmelt elasticity, as measured by phenomena such as extrudate swell The effect ofswell shows up during processing, wherein flow results in different amounts ofchain extension and orientation, which remain frozen within the solidified part

As a consequence, two chemically similar polymers, processed identically, thathave the same molecular weight but different molecular-weight distributions mayresult in products that show significantly different shrinkages, tensile properties,and failure properties [2] For this very important reason, it is advantageous toknow the molecular weight and molecular-weight distribution of the polymersused Furthermore, because polymers can mechanically degrade during proces-sing and during use (polymers such as nylon can also increase in molecularweight), a second measurement of the molecular weight can reveal the extent ofchain scission or postcondensation These measurements are also useful inverifying that the various kinetic schemes postulated for polymer synthesis inChapters 3– do, indeed, produce the molecular-weight distributions predictedtheoretically Other situations where the molecular weight and its distributiondirectly influence results include phase equilibrium and crystallization kinetics

A variety of methods are available for molecular-weight determination andthey are applicable in different ranges of molecular weight Also, they providedifferent amounts and kinds of information Thus, end-group analysis andcolligative property measurements yield the number-average molecular weight.Light scattering, on the other hand, furnishes the weight-average molecularweight and the size of the polymer in solution Intrinsic viscosity supplies neithernumber-average molecular weight (
Mn) nor weight-average molecular weight(
Mw); it gives a viscosity-average molecular weight The entire distribution can beobtained using either ultracentrifugation or size-exclusion chromatography.However, the former technique is an absolute one, whereas the latter is indirectand requires calibration All of these methods mandate that the polymer be insolution Other, less commonly encountered methods are described elsewhere [3]

8.2 END-GROUP ANALYSIS

The simplest conceptual method of measuring polymer molecular weight is tocount the number of molecules in a given polymer sample The product of thesample weight and Avogadro’s number when divided by the total number ofmolecules gives the number-average molecular weight This technique works bestwith linear molecules having two reactive end groups that each can be titrated insolution Consequently, linear condensation polymers made by step-growthpolymerization and possessing carboxyl, hydroxyl, or amine chain ends arelogical candidates for end-group analysis

Nylon 66, a polyamide and one of the earliest polymers to be synthesized,contains amine and carboxyl end groups The number of amine groups in asample can be determined by dissolving the polymer in a phenol–water solvent[4] Typically, ethanol and water are added to this solution and the mixture istitrated to a conductometric end point with hydrochloric acid in ethanol Becausethe number of amine end groups may not equal the number of carboxyl endgroups, the acid groups are counted separately by dissolving the nylon in hotbenzyl alcohol, and titration is carried out with potassium hydroxide in benzylalcohol to a phenolphthalein end point Finally, assuming that the reaction goes tocompletion and that each nylon molecule has two titratable ends, it is possible tocalculate

In addition to polyamides, end-group analysis has been used with esters, polyurethanes, and polyethers Besides titration, counting methods thathave been employed include spectroscopic analyses and radioactive labeling.Because the number of chain ends for a given mass of sample reduces withincreasing molecular weight, the method becomes less and less sensitive as thesize of the polymer molecules increases The molecular weight of most conden-sation polymers, however, is less than 50,000, and in this range, end-groupanalysis works fine [6] Note also that the amount of polymer needed for end-group analysis is relatively small

poly-Example 8.1: In order to determine the number of carboxyl end groups in asample of polyethylene terephthalate, Pohl dissolved 0.15 g of the polymer in hotbenzyl alcohol, to which some chloroform was subsequently added [7] Thissolution, when titrated with 0.105 N sodium hydroxide, required 35mL of the

alkali If a blank solution of the benzyl alcohol plus chloroform required 5mL ofthe base, how many carboxyl end groups were contained in the polymer sample?Solution: Because 30mL of 0.105 gram equivalent per liter of the base reactedwith the polymer, the concentration of gram equivalents of end groups was

Phase equilibrium is the basic principle used to obtain expressions for themagnitude of the different colligative properties It is known from thermo-dynamics that when two phases are in equilibrium, the fugacity, ^ff, of a givencomponent is the same in each phase Thus, if, as shown inFigure 8.1, pure vapor

A is in equilibrium at temperature T and pressure P with a liquid mixture of A and

B, where B is a nonvolatile solute,

where the superscriptsv and L denote vapor phase and liquid phase, respectively,and xA is the mole fraction of A in the liquid phase Also, a ‘‘hat’’ (^) on fA

signifies a component in solution as opposed to a pure component

If the mixture of A and B is sufficiently dilute, it will behave as an idealsolution, for which the following holds [8]:

Integrating Eq (8.3.6) from temperature Tb to temperature T at constantpressure and noting that Tb T and, therefore, TTb T2

1

Tb

1T

¼h0
h

RT2 b

FIGURE8.1 Pure solvent vapor in equilibrium with a polymer solution

Applying Eq (8.3.7) to pure A in the vapor phase and then to pure A in the liquidphase and introducing the results in Eq (8.3.5) gives

h0
hv

RT2 b

DTb¼
h0
hL

RT2 b

whereDTb equals T
Tb, the elevation in boiling point

Because ln xA equals lnð1
xBÞ, which for small xBis the same as
xB, wesee the following:

DTb¼ðRTb2xBÞ

in whichDhvequals hv
hL, the molar latent heat of vaporization of pure solvent

A From the definition of the mole fraction,

If we consider the situation depicted inFigure 8.2instead of that shown inFigure 8.1, then an analysis similar to the one carried out earlier leads to anexpression for the depression in freezing point, which is identical to Eq (8.3.11)except thatDT is now Tf
T, where Tf and T are the freezing points of the puresolvent and the solution, respectively Also,Dh becomes the molar latent heat offusion of the pure solvent, and Tbis replaced by Tf This measurement is known

as cryoscopy

For an ideal solution, the vapor pressure pA of the solvent in solution isgiven by Raoult’s law as follows [8]:

where PAis the vapor pressure of the pure solvent at temperature T This lowering

in vapor pressure is utilized for the measurement of molecular weight in thetechnique known as vapor-pressure osmometry

Figure 8.3 shows a schematic diagram of a vapor-pressure osmometer Twothermistor probes are positioned inside a temperature-controlled cell that issaturated with solvent vapor If syringes are used to introduce drops of puresolvent on the thermistor probes, then at thermal equilibrium, the temperature ofthe two probes is the same and an equal amount of solvent vaporizes andcondenses at each probe If, however, one of the solvent drops is replaced by adrop of solution, there is an initial imbalance in the amount of solvent condensingand vaporizing at that probe Because of the lowering in vapor pressure, lesssolvent vaporizes than condenses, which leads to a rise in temperature due tothe additional heat of vaporization When equilibrium is reestablished, thetemperature T at this probe is higher than the temperature TS at the other probewhich is in contact with the drop of pure solvent Under these conditions, thevapor pressure of pure solvent at temperature TSequals the vapor pressure of thesolvent in solution at temperature T , and the situation is analogous to ebullio-metry Therefore, we can use Eq (8.3.11) again if we defineDT as T
TS Thetemperature difference itself is measured as a difference in electrical resistance by

FIGURE8.2 Pure solid solvent in equilibrium with a polymer solution

FIGURE8.3 Schematic drawing of a vapor-pressure osmometer

making the two thermistors be the two arms of a Wheatstone bridge Commercialinstruments can, at best, measureDT down to about 5  10
5C Because of heatlosses and solution nonidealities, the measured DT does not equal the valuecalculated based on Eq (8.3.11) It is necessary to calibrate the instrument using

a material of known molecular weight The range of commercial vapor pressureosmometers is from 40 to 50,000 g=mol, with the lower limit being set by solutevolatility [6]

For polymer molecular weights of 100,000 and greater, the temperaturedifferences predicted by Eq (8.3.11) for a dilute polymer solution in a typicalorganic solvent are about 10
5C (see Table 8.1) Such small changes intemperature are very difficult to measure with any degree of precision Conse-quently, when working with high-molecular-weight polymers, we turn to othertechniques of molecular-weight determination, especially osmotic pressure.When a polymer solution is separated from the pure solvent by asemipermeable membrane that allows passage of the solvent but not the solute,then (as shown in Fig 8.4) the tendency to equalize concentrations results in flux

of the solvent across the membrane and into the solution As mass transferproceeds, a pressure head builds up on the solution side, tending to slow downand ultimately stop the flow of solvent through the membrane At equilibrium, theliquid levels in the two compartments differ by h units; the difference in pressure

p is known as the osmotic pressure of the solution Note that if additionalpressure is applied to the solution, solvent can be made to flow back to the solventside from the solution side; this is known as reverse osmosis As the followinganalysis demonstrates, osmotic pressure can be employed to measure the number-average molecular weight of a polymeric solute

If we designate solvent properties by subscript 1 and solute properties bysubscript 2, then the following relations hold at thermodynamic equilibrium,using the condition of phase equilibrium:

100,000 3:9  10
4 4:25  10
4 2.9

500,000 7:8  10
5 8:5  10
5 0.58

5,000,000 7:8  10
6 8:5  10
6 0.058

where the second equality follows from the Lewis and Randall rule quently,

where c is the mass concentration of the solute Again, we typically extrapolate

p=c to c ¼ 0 to ensure that ideal solution behavior is obtained and Eq (8.3.18)

FIGURE8.4 Osmosis through a semipermeable membrane

holds Expected values of the osmotic pressure for dilute solutions of polystyrene

in toluene are listed in Table 8.1

A typical plot of experimental data for aqueous solutions of polyethyleneoxide at 20C is shown in Figure 8.5 [9]; these data are extremely easy to obtaineven though 2 days are required for equilibrium to be reached It is seen that theplot has a nonzero slope, and significant error can occur if extrapolation to zeroconcentration is not carried out This nonzero slope can be theoretically predicted

if real solution theory is used instead of assuming ideal solution behavior Forinstance, if we employ the Flory–Huggins theory (considered in detail in Chapter

9) and equate the fugacities (or, equivalently, the chemical potentials) of thesolvent on both sides of the membrane, the use of Eq (9.3.30) along with theknown dependence of the chemical potential on temperature leads to thefollowing result (see Chapter 9):

where f1and f2are the volume fractions of the two components, respectively, m

is the ratio of the molar volume of the solute to the molar volume of the solvent,and w1 is the interaction parameter

FIGURE8.5 Osmotic pressure of aqueous polyethylene oxide solutions at 20C (FromRef 9.)

On expanding ln f1 in a Taylor series about f2¼ 0 and noting that themass concentration c of the polymer equals
Mnf2=mv1, Eq (8.3.19) becomesp

and it is seen that Eq (8.3.18) is obtained by letting c tend to zero in Eq (8.3.21)

As discussed in Chapter 9, the latter equation can be used to estimate theinteraction parameter if the polymer number-average molecular weight is known.Because the Flory–Huggins theory is not strictly valid at low polymerconcentrations, it is common practice to rewrite Eq (8.3.21) in the form of a virialequation (as is done in thermodynamics):

106 can be measured at temperatures exceeding 100C The lower limit on themolecular weight is set by solute permeability, whereas the upper limit isgoverned by the sensitivity and accuracy of the pressure measuring system.Additional details may be found in the literature [6,10]

A beam of light is a transverse wave made up of sinusoidally varying electric andmagnetic field vectors that are perpendicular to each other and also to thedirection of propagation of the wave Such a wave contains energy that ismeasured in terms of the wave intensity I, defined as the power transmitted perunit area perpendicular to the direction in which the wave is traveling Using theprinciples of physics [11], it is easy to show that the average intensity or thepower averaged over one cycle is proportional to the square of the waveamplitude When such a beam travels through a polymer solution, it can go

through unaltered, but, more commonly, it is either absorbed or scattered.Absorption occurs only if the wave frequency is such that the energy of radiationexactly equals the energy gap between, say, the electronic or vibrational energylevels of the molecules making up the liquid medium; this phenomenon is thebasis of methods such as infrared and nuclear magnetic resonance spectroscopy.Scattering, on the other hand, involves attenuation of the incident beam withsimultaneous emission of radiation in all directions by the scattering moleculesdue to the presence of induced instantaneous dipole moments In this case, theenergy of the incident beam equals the sum of the energies of the transmittedbeam and all of the scattered beams Here, scattered light has the same frequency

as the incident light, and the process is called elastic light scattering Sometimes,though, scattered light has a different frequency, which is called inelastic orRaman scattering From an observation of the time-averaged intensity ofelastically scattered light (called static light scattering), we can get informationabout the weight-average molecular weight, the second virial coefficient, and thesize or radius of gyration (root-mean-square distance of chain elements from thecenter of gravity of the molecule) of macromolecules Instantaneous scatteringintensity or dynamic light scattering reveals, in addition, the translationaldiffusion coefficient [12] Note that, in recent times, dynamic light scatteringhas also been applied to, among other things, studies of bulk polymers, micelles,microemulsions, and polymer gels [13] Here, however, we consider the applica-tion of static light scattering to the determination of polymer molecular weightunder conditions where absorption effects are not important

The theory of light scattering was developed many years ago; a review [14],excellent books [15,16], and an elementary treatment [17] on the subject areavailable The essential features of elastic light scattering can be understood withreference to Figure 8.6, which shows an unpolarized beam of light of intensity I0and wavelength l passing through a cylindrical sample cell of unit volume Theintensity Iyof the scattered beam is measured at a distance r from the cell and at

an angle y with the direction of the transmitted beam If the cell contains Nnoninteracting, identical particles of an ideal gas (polymer solutions are consid-

FIGURE8.6 Schematic diagram of static light scattering

ered later) and if the size of the scattering particles is small compared to thewavelength of the incident light, then we have the following [15]:

c equals NM=NA Thus, if there were a mixture of two kinds of particles, with onekind being much larger than the other, the contribution of the larger particles tothe scattered light intensity would be the dominant one This fact is used to greatadvantage in determining the molecular weight of polymeric solutes in solution

In this situation, for light scattering from an ideal polymer solution, Eq (8.4.1) ismodified to read as follows [15]:

Equation (8.4.2) is valid only at infinite dilution For finite concentrations,the use of a virial expansion of the type introduced in Eq (8.3.22) leads to

For most polymer molecules, the limitation that the particle size be muchsmaller than the wavelength of light, which, in practice, means that all moleculardimensions should be less than l=20, is too restrictive When the particle sizebecomes comparable to the wavelength of the incident beam, scattering occursfrom different parts of the same molecule, resulting in interference due to phasedifferences This tends to progressively reduce Iyas y increases The result can beseen inTable 8.2, which lists data for polystyrene-in-toluene solutions However,because Eq (8.4.3) still holds when y is zero, we can either make measurements

at different y values and extrapolate data to y equal to zero, or make ments at very small values of y The latter situation is practical because the use oflasers as light sources allows us to conduct experiments at y values as small as 4.Note that light scattering at nonzero y values depends on the geometric shape ofthe scattering particle; it is from the deviation of the data from the predictions of

measure-Eq (8.4.3) that we estimate the radius of gyration of the polymer molecule

It is common practice to define a quantity Ry(called the Rayleigh ratio) asfollows:

lim

which yields the polymer molecular weight Also, the slope of the plot allows us

to compute the second virial coefficient If measurements are made at severalconstant temperatures, the temperature value at which A2 equals zero is the ytemperature

TABLE8.2 Light-Scattering Data, Iy, in Arbitrary Units for Solutions of Polystyrene inToluene at 20C

Concentration (g=cm3)Scattering angle, y (deg) 0.0002 0.0004 0.001 0.002

25.8 3.49 5.82 7.86 8.6536.9 2.98 4.88 7.38 8.0253.0 2.19 3.82 6.37 7.4166.4 1.74 3.12 5.58 6.8890.0 1.22 2.25 4.42 5.95113.6 0.952 1.80 3.73 5.35143.1 0.763 1.48 3.15 4.79

Source: Ref 18.

If the polymer sample is polydisperse, then Rycan be written as a sumP

Riover all of the molecular-weight fractions; so, Eq (8.4.7) becomes

Ry¼ K P

Mici¼ K P

Miwi

because the mass concentration of each molecular-weight fraction equals the ratio

of the respective mass divided by the solution volume The total mass tion c, however, is

In order to understand the theory of the ultracentrifuge, let us first consider ananalogous situation, that of a single sphere settling under gravity in a long tubefilled with a Newtonian liquid, as shown inFigure 8.7 If the sphere of mass mand volume V is dropped from a state of rest, it initially accelerates, but soonreaches a constant velocity, known as the terminal velocity, at which point thevector sum of all the forces acting on the sphere is exactly zero As long asthe tube radius is large compared to the sphere radius, the forces that act on the

sphere are gravity, buoyancy, and the viscous drag of the liquid Fdtending to slowdown the sphere At equilibrium, therefore,

where R is the universal gas constant It can be seen that a measurement of theterminal velocity makes it possible to compute the molecular weight if the otherquantities in Eq (8.5.4) are known.

If, instead of a single particle, a large number of particles are dropped intothe tube, then, in the absence of particle–particle interactions, the mass flux ofspheres at any cross section is given by

where c is the local mass concentration of spheres

As time proceeds, spheres build up at the bottom of the tube, and thetendency to equalize concentrations causes a diffusive flux of spheres upward inthe tube The magnitude of the flux is given by Fick’s law (see Chapter 13) asfollows:

Flux ¼ D dc

where D is the same diffusion coefficient appearing in Eq (8.5.2) and z is thedistance measured along the tube axis For a steady state to be reached in thesphere concentration, the fluxes given by Eqs (8.5.5) and (8.5.6) have to be equal

in magnitude Equating these two quantities and replacing the terminal velocity

by an expression obtained with the help of Eq (8.5.4) gives

of Eq (8.5.4) is that the value of the diffusion coefficient is not needed

If we try to apply the foregoing theoretical treatment to the determination ofpolymer molecular weight from the sedimentation of a dilute polymer solution,

we discover that, in practice, polymer molecules do not settle This is the casebecause the equivalent sphere radii are so small that colloidal forces [notaccounted for in Eq (8.5.1)] predominate over gravitational forces and keepthe polymer molecules from settling However, the situation is not irredeemable

If the polymer solution is placed in a horizontal, pie-shaped cell and the cell isrotated at a large angular velocity o about a vertical axis as shown inFigure 8.8,the centrifugal force that develops can exceed the force of gravity by a factor of afew hundred thousand Indeed, the centrifugal force can and does causesedimentation of polymer molecules in the direction of increasing r Because