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Vol. 10 MOLECULAR WEIGHT DETERMINATION 445

MOLECULAR WEIGHTDETERMINATION

Introduction

This article discusses the theory and methods for the determination of molecularweight averages and molecular weight distributions of oligomeric and polymericmaterials. The more important methods are described in greater detail in separatearticles (see CHROMATOGRAPHY, SIZE EXCLUSION; FRACTIONATION; CHARACTERIZATION

OF POLYMERS).

Molecular Weight Averages and Distributions

Most polymeric materials are comprised of mixtures of molecules of various sizes.This distribution of molecular weights is a result of the statistical nature of thepolymerization proces (1). A complete description of the molecular weight dis-tribution of a homopolymer is necessary to understand its physical, rheological,and mechanical properties. Structural variations, such as chain branching, andcopolymer or higher mer polymers with various degrees of structural randomnessfurther complicate a complete description of the molecular ensemble. Copolymersand higher mer polymers also need to be characterized for molecular composi-tion as a function of molecular weight. Block copolymers and polymer mixturesalso offer significant challenges to evaluate their molecular weight distribution.Failure mechanisms and practical lifetimes of polymeric materials may also beestablished through molecular weight measurements.

Most classical techniques for molecular weight determination are only ca-pable of yielding one of the molecular weight averages of the distribution. Thesetraditional methods are used nowadays only on a limited basis, and only a limitednumber of commercial instrumentation is available.

These averages are defined in terms of the molecular weight Mi and thenumber of moles ni or the weight wi of the component molecules. The molecularweight averages are defined by equations 1-4.

Number-average molecular weight:

Mn =∑

niMi∑ni

=∑

wi∑wi/Mi

(1)

Weight-average molecular weight:

Mw =∑

niM 2i∑

niMi=

∑wiMi∑

wi(2)

z-Average molecular weight:

Mz =∑

niM 3i∑

niM 2i

=∑

wiM 2i∑

wiMi(3)

Encyclopedia of Polymer Science and Technology. Copyright John Wiley & Sons, Inc. All rights reserved.

446 MOLECULAR WEIGHT DETERMINATION Vol. 10

(z + 1)-Average molecular weight:

Mz + 1 =∑

niM 4i∑

niM 3i

=∑

wiM 3i∑

wiM 2i

(4)

Viscosity-average molecular weight:

Mv =[∑

niM1 + ai∑

niMi

]1/a

=[∑

wiMai∑

wi

]1/a

(5)

Equation 5 is an important practical molecular weight average derived fromviscometry. To calculate this average, the exponent a and the constant K of theMark–Houwink relationship relating intrinsic viscosity [η] to molecular weightmust be known.

[η] = KMa (6)

The value of a lies between 0.5 and 1.0 for random coils, depending on thesolvent employed to determine the relationship of the intrinsic viscosity to molec-ular weight. For rigid rod molecules, the value of a is expected to be ∼1.8. Withthese limits for a, it may be seen that Mv is always larger than Mn but can equalMw when the upper limit of a is reached for random coils.

If molecular weight is considered a continuous variable, the molecular weightdistribution may be described by a set of moments µr given by the integrals

µr =∫ x

0Mrf (M) dM

where r = 0, 1, 2, 3, etc; f (M) is the number-density distribution; and f (M)dMis the number of moles of molecules with molecular weight between M and(M + dM).

The average molecular weights are defined as

Mn = µ1

µ0

Mw = µ2

µ1

Mz = µ3

µ2

Mz + 1 = µ4

µ3

Molecular weight averages can be determined from these moments (2).


Molecular Weight Distribution Functions

Various mathematical functions have been employed to describe the distributionof molecular weights. Some of the more common functions are shown in Table 1in terms of the mole fraction X. Further details are available (3).

Width of Molecular Weight Distributions. The width of the Gaussiandistribution function may be expressed in terms of the standard deviation of themole fraction MWD function σ n or the mass fraction MWD function σw:

σn = (MwMn − Mn)0.5

σw = (MzMw − Mw)0.5

The standard deviation is an absolute measure of the width for the Gaussianfunction only. Widths of molecular weight distribution for other functions have tobe calculated for each case from the distribution function itself.

The relationships between molecular weight averages and parameters in themolecular weight distribution functions are as follows:which leads to

exp(σw)2 = Mw

Mn= Mz

Mw

Thus the ratios of two adjacent averages are constant.

Poisson distribution function.The ratio of the weight-average to number average molecular weight is givenby

Mw

Mn= 1 +

(1

Mn

)−

(1

Mn

)2

Table 1. Molecular Weight Distribution Functions

Name Function Comments

Gaussian X(M) = 1

σn(2π )1/2 exp

[(M − Mm)2

2σ 2n

]Mm = median value equal to Mn

Log-normal X(M) = 1

σn(2π )1/2 exp

[(lnM − lnMm)2

2σ 2n

]Mm = geometric mean

Poissona X(M) = νM − 1

�(M)exp( − ν) ν = Mn − 1

Flory–Schulzb X(M) = βk + 1Mk − 1 Mn

�(k + 1)exp( − βM) k = degree of coupling

β = k

Mn

a�(M) = γ -function.b�(k + 1) = γ -function.


Gaussian distribution functionmole-fraction distribution function median value = Mn

mass-fraction distribution function median value = Mw

Log-normal distribution functionmass-fraction distribution function median value = Mm

Mn = Mm exp[(σw)2/2]

Mw = Mm exp[3(σw)2/2]

Mz = Mm exp[5(σw)2/2]

and therefore depends only on Mn.

Flory–Schulz distribution functionThe molecular weight averages this function are related by

Mn

k= Mw

k + 1= Mz

k + 2

where k is the coupling constant, defined as the number of independently growingchains required to form one dead chain.

PolydispersityTraditionally, polydispersity Q has been defined as

Q = Mw/Mn = U + 1

where U, the molecular inhomogeneity, has a numerical value of one less than Q.The width of molecular weight distributions increases with increasing Q and U.Except for the Gaussian distribution function, the standard deviation is only arelative measure of distribution width (4,5).

Determination of Molecular Weight

Several books have been published since this article was last written (6–9).Number-Average Molecular Weight.End-Group Analysis. In end-group analysis the concentration of an end

group is measured by a suitable technique, and then, from the known structureof the polymer, the value for Mn may be calculated (10–15). In condensation poly-mers one end of the polymer should have a specific group that can be titrated byappropriate means. Polyesters (13), polyamides (13), and polyurethanes (16) havebeen subjected to this analysis. Polyethers have been analyzed by hydroxyl-grouptitrations similar to those used for polyesters or by reaction with excess phenylisocyanate followed by reaction with excess di-n-butylamine; the latter was back-titrated with perchloric acid (17). Nonaqueous titrations (18) have been used to


characterize polysulfones with Mn values of up to 25,000. Potentiometric titrationin aqueous and nonaqueous media have been used to determine molecular weightsof acrylamide and acrylonitrile polymers (19). Spectrophotometric analyses havealso been widely utilized (14), notably uv–visible, infrared techniques, and NMR(20). Examples include the determination of phenolic end groups in polymers byuv spectroscopy (21) and potentiometric titration (18). Infrared spectroscopy hasbeen used (22) to estimate carboxyl groups in the presence of carbonyl groups byreaction of the former with SF4; the resulting thionyl halides were quantitativelymeasured. NMR spectroscopy (1H) has been used to characterize Mn of hydroxylend-capped polystyrenes (A. D. Edwards and M. J. R. Loadman, Malaysian RubberProducers Research Association, unpublished data, 1976). Reaction of the singleproton on the hydroxyl group yields a trimethylsilyl group containing nine pro-tons. The enhanced sensitivity allows determination of Mn values of up to 80,000.A similar method has been reported for hydroxy-terminated polybutadienes (23),polyester polyurethanes (24), epoxides (25), and other polymers (26–28). Acid endgroups in polyethylene terephthalate have been characterized (29) by reactionwith hexafluoroisopropanol and 19F NMR.

In certain cases extremely high sensitivity may be attained with radioac-tive labeling (30). Using radiolabeled (35S) bisulfite initiator, the number-averagemolecular weight of Teflon samples was determined in the range of 389,000–8,900,000 (31). This is a particularly useful technique for insoluble polymers.

Measurement of Colligative Properties. Colligative properties are thosethat depend on the number of species present rather than on their kind (32,33).From thermodynamic arguments it may be shown that for very dilute idealsolutions

ln a1 = − X2

where a1 is the activity of the solvent and X2 is the mole fraction of solute. Fromthis relationship the solute molecular weight may be calculated if the weightfraction w2 is known.

x2 = n2

n1= w2M1

M2w1

This equation demonstrates that the activity, measurable by several meth-ods, is proportional to the number of solute molecules. Thus in the case of polydis-perse solutes, the number-average molecular weight is the average determined.At higher concentrations and molecular weights, the solutions become nonideal,and higher powers of the solute concentration are introduced into the equations.Measurements of several solute concentrations are required, and appropriate ex-trapolation techniques must be used.

Lowering of Vapor Pressure. The partial vapor pressure p1 of solvent 1 overa solution is lower than the vapor pressure over the pure solvent p1

0. This isexpressed by Raoult’s law:

P1 = X1p 01


where X1 is the mole fraction of the solvent.For a binary solution containing a mole fraction X2 of solute then,

X2 = p 01 − p1

p 01

= �p1

p 01

(7)

For a dilute solution,

X2 = n2

n1= w2 M1

M2w1(8)

Combining equations 7 and 8 yields

M2 = w2M1

w1

p01

�p1

The unknown molecular weight of solute M2 may be calculated from thelowering of the vapor pressure caused by the addition of w2 grams of solute toform a binary solution; Mn values of up to 1000 may be determined (34–37).

A variation of this technique, the isopiestic method (38–40), was devised toavoid the difficulty of accurately determining the small difference in vapor pres-sure caused by the addition of solute. In this method (41) two limbs of a containerjoined by a common vapor space are filled, one with a reference solution of knownweight concentration ws and molecular weight Ms and the other with the unknownsolution of known weight concentration w2 and unknown molar concentration. Theapparatus is immersed in a constant-temperature bath controlled to ±0.001◦C; theabsolute temperature is not important. Distillation of the solvent continues untilthe vapor pressures above the solutions in each limb are equal, and thus the so-lutions contain equal mole fractions of solute. At equilibrium, which may requireseveral weeks, the final volumes Vs, and V2 of the known and unknown solutionsare read from the calibrations on each limb.

Mn = w2MsVs

V2ws

Mnvalues of up to 20,000 have been determined by this method.Ebulliometry. Ebulliometry (34,42–47) is another technique for determin-

ing the depression of the solvent activity by the solute. In this case the elevationof the boiling point is determined. The boiling-point elevation �Tb is measuredwith sensitive thermocouples or matched thermistors in a Wheatstone bridge. Themolecular weight Mn is calculated from

Mn = Kbc�Tb

where c is the concentration of solute in g/1000 g of solvent and

Kb = RT 2b M

1000 �Hv(9)


D

A

N

T

P

M

S

V

C

R

B

Q Boiler

Insu

latio

n

Insu

latio

n

(a) (b)

Boiler

Fig. 1. (a) Ebulliometer design for molecular weight determination of up to Mn = 100,000.A denotes ebulliometer cooler; B, boiler; C, Cottrell pump; D, adiabatic jacket cooler; M,thermopile; N, upper thermopile weldings; P, thermopile terminals connected with detector;Q, adiabatic jacket boiler; R, platinum resistance; T, lower thermopile weldings; and V,vacuum jacket circulated by solvent vapor. (b) Complete ebulliometer with adiabatic jacketsand insulating apparatus (50). Courtesy of Huethig and Wepf Verlag.

is the molal ebullioscopic constant. M is the molecular weight of the solvent andTb its boiling point; �Hv is the molar latent heat of vaporization of the solvent.

The ebullioscopic constant may be evaluated directly from equation 9 andthus provides an absolute method. However, the value for Kb is often determinedby using a high purity solute of known molecular weight.

Currently, a sensitivity of 1 × 10− 5◦C or 2.4 × 10− 6◦C is attainable us-ing a 160-junction thermocouple (48) or thermistors (49), respectively. The upperlimit of molecular weight, which may be determined by this method, also dependson the solvent and solution nonideality. With careful measurements and goodebulliometer design, Mn values of up to 100,000 may be determined on isotacticpolypropylene (50). The apparatus employed is shown in Figure 1. This techniquehas been applied to aqueous solutions (51). An ebulliometer of simple design hav-ing a temperature sensitivity of 50 × 10− 6◦C has recently been reported (52). Arotating ebulliometer attempts to overcome the superheating problem (53). Er-rors in the methods for treating ebullioscopic data to obtain molecular weighthave been discussed (54).

Cryoscopy. The freezing point of a solution is depressed below that of thepure solvent by an amount proportional to the mole fraction of solute. The value


for Mn is obtained from

Mn = Kfc�Tb

where c is the concentration of solute in g/1000 g of solvent and

Kf = RT 2m M

1000 �Hfus

is the molal cryoscopic constant; M is the molecular weight of the solvent and Tmits melting point; �Hfus is the molar latent heat of fusion of the solvent.

Some materials, such as camphor, have very large cryoscopic constants andmay be used to increase the sensitivity of the method. The method is used todetermine Mn of polymers (55–59), such as polyesters (60) and ethylene-vinylacetate copolymers (61).

A novel variation of this technique (62) involves depression of thefirst-order, nematic–isotropic melting transition of N-(p-ethoxybenzylidene)-p-n-butylaniline. Polystyrene and poly(ethylene oxide) are soluble in both phases, andMn values of up to 106 have been studied.

Vapor-Pressure Osmometry. When a drop of solution is exposed to pure sol-vent vapor, the solvent vapor condenses onto the droplet, because the solute low-ers the vapor pressure of the drop (63–68). The heat of condensation causes thetemperature of the drop to rise, until theoretically the solution vapor pressure isincreased to equal that of the solvent. In practice this does not occur, but a state ofequilibrium is reached, where the heat losses from the droplet are matched by theheat of condensation (GB/T 6579-86). The temperature increase is proportionalto the number of moles of solute present. A standardized method is available(H. Knauer, Berlin, unpublished data).

The design of a commercial instrument is given in Figure 2 (H. Knauer,Berlin, unpublished data). The temperature stability needed is 0.001◦C. In an-other design, by Wescan Instruments, Inc., small platinum mesh screens ensurethat a constant amount of solvent or solution is present on the thermistor. Theresistance value �R, which is proportional to the temperature difference, is mea-sured with a bridge circuit in all instruments. A calibration curve of �R vs molalitym is determined by employing a solute of known molecular weight. After severalminutes �R reaches a constant value; the calibration curve is often highly linear,but may not go through the zero point (66).

�R = a + bm

The number-average molecular weight of the unknown sample may then becalculated from equation 10:

Mn = bc�R − a

(10)


Probe

Aluminum block forthermostating syringes

Gasket

Wick

Thermistors

Window

Glass vesselfor solventAluminummeasuring cell

Fig. 2. Vapor-pressure osmometer apparatus (H. Knauer, Berlin, unpublished data,1987). Courtesy of Dr. -Ing. Herbert Knauer GmbH.

where c is the concentration of solute in g/1000 g of solvent. In older osmometersthe calibration curve depended on the material used to perform the calibration.For the Wescan instrument and another research vapor-pressure osmometer (69),both based on the same design (70), the calibration constant was independentof the material. These instruments have determined polystyrene Mn values of400,000 (71), 100,000 (72), and 100,000 (69); polyolefins have been characterized at140◦C (73) and styrene/methylmethacrylate copolymers (32). For higher molecularweight solutes, the virial coefficients become important. In this case �R/c mustbe plotted against c in order to extrapolate the �R/c value to infinite dilution.When the plot is not linear, it is necessary to plot (�R/c)1/2 vs c to obtain a linearextrapolation to infinite dilution.

Membrane Osmometry. Membrane osmometry is a well-established tech-nique (68,74–81). Commercial instrumentation is available (Gonotec GmbH,Knauer). A standardized method is available (GB/T 6596-86). The main problemis membrane permeation by lower molecular weight species (82). Improvementsin membranes with well-defined, low molecular weight cutoffs can be expected asa consequence of the renewed interest in membrane-separation techniques (83,84)(see MEMBRANE TECHNOLOGY). Membrane osmometry has been applied to a widevariety of polymers, including operation at high temperatures for polymers dif-ficult to dissolve. Comb-like polymers have been characterized and the resultscompared with those from gel permeation chromatography (GPC) (85). Prelimi-nary results of an online osmometer for GPC detection have been described (86,87).The upper limit for molecular weight determination is ca 1 × 106. The lower limit


depends on the molecular weight distribution, ie, the details of the composition ofthe low molecular weight tail. In most cases the concentration dependence of os-motic pressure � must be determined and �/c extrapolated to zero concentrationby plotting �/c vs c, or (�/c)1/2 vs c.

Viscosity-Average Molecular Weight. The viscosity of dilute polymersolutions may be related to the molecular weight of the polymer by the appropriatecalibration (see Viscometry). The polymer is usually separated into narrow molec-ular weight distribution fractions, which are characterized by absolute molecularweight methods. The molecular weight is related to the intrinsic viscosity [η] bythe Mark–Houwink relationship (eq. 6).

The intrinsic viscosity is obtained using viscometer types of viscometer tubesand plotting the reduced or inherent viscosity of a series of polymer solutions withvarious concentrations against the solution concentration. Extrapolation to zeroconcentration yields the intrinsic viscosity. The technique has found widespreadapplication (88–92), and compilations of Mark–Houwink constants are available(93). A novel method to determine intrinsic viscosity using a flow piezoelectricquartz crystal has been developed (94). Static and continuous measurements of in-trinsic viscosity using a differential viscometer (Viscotek) have been reviewed (95).If the Mark–Houwink constants are determined using narrow molecular weightdistribution materials, the average molecular weight obtained from equation 6 isthe viscosity-average molecular weight Mv. The range of molecular weights thatmay be characterized is very large. Special precautions are required for extremelyhigh molecular weight samples to avoid degradation during the solution processor the measurement (96). Dilute solution viscometry for measurement of molec-ular weight, branching, and other physical parameters have been reviewed (97).A method to calculate the viscosity average molecular weight from GPC for apolymer with unknown Mark–Houwink constants has been reported (98).

Weight-Average Molecular Weight.Light Scattering. Laser Light Scattering (qv) is a technique (99–106) widely

used to characterize polymeric materials. Standardized methods are available(ASTD D4001 and GB/T 6598-86). Commercial instrumentation is available tomeasure the light-scattering properties of polymer solutions and the refractiveindex increment (107). Older instrumentation includes the Brice Phoenix Pho-tometer, the Sofica Light Scattering Photometer, as well as the Chromatix prod-uct. Wyatt Technology and Malvern Instruments currently produce instruments.Lasers are used as light sources to measure scattering at small angles to the beam.This is important since the data obtained must be extrapolated to zero concentra-tion and zero scattering angle using Zimm plots (108). The weight-average molec-ular weight is determined by this technique, and the molecular weights accessiblerange from a few hundred to several million. The solutions employed must be freeof dust and are usually subjected to filtration or centrifugation. Solutions of highmolecular weight degradable polymers are the most difficult to clarify. The solventis selected to give a reasonable difference in refractive index with the solute. Thetechnique has been applied to determine the molecular weight of difficultly solu-ble polymers such as polyethylene (109,110) and of polytetrafluoroethylene and itscopolymers with ethylene (111–113). Copolymers with homogeneous distributionof monomer units in the polymer chain may be analyzed by the same methods ashomopolymers. If the composition varies as a function of molecular weight, theanalysis is more complicated (114). Terpolymers have also been considered (115).


True molecular weights and heterogeneity of styrene-butyl acrylate copolymershave been determined using static low angle laser light scattering (LALLS) andGPC/LALLS (116) and styrene–polydimethylsiloxane block copolymers (117) us-ing GPC/LALLS. Multi-angle laser light scattering (MALLS) has been used as aGPC detector to characterize molecular weight, molecular size, and branching oforganic and water soluble polymers (118).

Dynamic light scattering (119) has been used to determine the number av-erage molecular weight (120) and molecular weight distribution of polymers insolution (121–123). There are several mathematical approaches to determine themolecular weight distribution from PCS data. Comparisons with GPC and numer-ical simulation have been made (124). The resolution of bimodal distributions hasbeen demonstrated (125).

Pulse-induced critical scattering (126,127), although not utilized as a methodfor molecular weight characterization, may be a sensitive procedure for detectingsmall differences in molecular weight distribution.

X-ray (128) and neutron scattering (qv) (129,130) may also be used to de-termine weight-average molecular weights. The main drawback is the greaterexpense and the lack of suitable facilities.

Ultracentrifuge. The ultracentrifuge is used to determine molecularweights (131–133). Sedimentation velocity employs a sufficiently high centrifu-gal field so that the sedimentation rate may be measured. The sedimentationcoefficient S is empirically corrected for concentration and pressure effects to giveS0, the sedimentation coefficient at zero concentration. The method also requiresthe measurement of the diffusion coefficient at infinite dilution D0. The molecularweight is calculated from the Svedberg equation

S0

D0= M(1 − νρ)

RT

where ν is the partial specific volume of the solute and ρ the density of the so-lution. For a polydisperse solute the correct averages for S and D are combinedin the Svedberg equation to yield a well-defined average molecular weight. If Dis obtained from intensity-fluctuation spectroscopy (134) and combined with theweight average S value, the weight-average molecular weight is obtained. Theapplication of the sedimentation velocity method to calculate molecular weightdistribution is extremely complex, but has been successful in special studies(135).

In the sedimentation equilibrium method, a lower centrifugal field is main-tained for a period of time in such a way that sedimentation is balanced by diffusionand an equilibrium distribution of polymer is established in the cell. Although Mwand Mz are easily determined, the length of time of the experiment is a disadvan-tage. In contrast to light scattering, this method is not affected by dust particles,and no calibration is needed. The molecular weight distribution may be obtainedfrom the sedimentation velocity data, but not without mathematical difficulties(136) or requiring additional data at other rotor speeds (137,138). An improveddetection system for the analytical ultracentrifuge has been reported (139,140). Anew 8-cell rotor that allows increased throughput of samples has been described(141).


Improved data analysis has led to the simultaneous determination of thesedimentation coefficient and diffusion coefficient, and a calculation of the molec-ular weight from the Svedberg equation (142). The Svedberg method is capableof determining molecular weights of up to 40 × 106 (96) on an absolute basis.Sedimentation equilibrium in a density gradient has been used to determine thecompositional analysis of copolymers, eg, butadiene-styrene (143).

Munk (144) has recently reviewed the use of sedimentation velocity and sed-imentation equilibrium to characterize molecular weight. A new absolute methodfor the determination of molecular weight distribution based on band sedimenta-tion and LALLS has been applied to water-soluble polymers (145).

Higher Molecular Weight Averages. Classical techniques often cannotbe used for higher molecular weight averages (Mz, Mz + 1, etc). These are usu-ally measured by determining the molecular weight distribution by fractionation(qv) or affinity chromatography (qv) and back-calculating the averages from thedistribution.

Determination of Molecular Weight Distribution

Fractionation. Polymers may be separated on the basis of molecularweight by precipitation or phase-separation techniques (146–149). The polymerthat is originally in solution separates on the basis of molecular weight when thethermodynamic quality of the solvents is changed, ie, by changing the solventcomposition or temperature. The principal techniques are fractional precipitationand coacervate extraction, both solution-based (Fig. 3). The former is based on theremoval of material from the polymer-rich precipitated phase and the latter onthe removal of fractions from the polymer-lean solution phase. Computer simu-lations and experimental results have shown the coacervate extraction technique(150) to yield narrower molecular weight distribution fractions than the fractionalprecipitation method (151–153).

1

2

3

1

2

3

spf ssf

Fig. 3. Successive precipitation (SPF) and solution fractionation (SSF) (coacervate extrac-tion): , mother solution; , polymer rich phase; and , polymer-lean phase. Numbersdenote fraction numbers (148). Courtesy of Society of Chemical Industry.


To obtain the molecular weight distribution, the fractions of known massmust be characterized by a molecular weight technique. The integral and differ-ential molecular weight distribution may be evaluated using a method (154) thatassumes no particular molecular weight distribution function for the fraction ortreatments that assume a particular model for the molecular weight distributionof the fractions (155). Fractionation may also be achieved by crystallization fromdilute solution (156).

In column fractionation (157,158) the polymer is precipitated onto an inertsupport, which is placed at the top of a packed column (159). A solvent mixtureof increasing solvent power is pumped through the column; a temperature gra-dient is often maintained. This is known as Baker–Williams fractionation (160).This technique is applicable to all amorphous homopolymers and crystalline ho-mopolymers above the melting point. For copolymers and more complex composi-tions, the same technique may be employed, but the analysis is considerably moredifficult.

Temperature rising elution fractionation (TREF) (161,162) is a useful frac-tionation technique which gives information on the compositional distribution.

Chromatography. The determination of molecular weight distributionand chemical composition by chromatographic techniques has been discussed(163).

Gel Permeation (Size-Exclusion) Chromatography. GPC has been in usesince the 1960s and has been a most important development in molecular weightdetermination (164–169). For this technique the polymer must be soluble, and cal-ibration with suitable standards is required. The method is extremely efficient; itrequires 0.5–2 h per sample and only a few milligrams of material. Standardizedmethods are available (GB/T 6599-86, ASTM D5296-97, and A83 ASTM D6747-99). Commercial instrumentation capable of operating up to 150◦C is availablefrom Waters and Polymer Laboratories. Detection is normally accomplished bydifferential refractometry, although uv and ir (170) absorption spectrometry havealso been employed. The chromatographic columns are packed with spherical-or irregular-shaped porous beads. The packing may be a crosslinked polymer,a porous glass, or a silica material. The pore size distribution determines theseparation range of the columns. Calibration is normally performed using well-characterized narrow MWD standards. Calibration can be performed using broadMWD materials (171). Typical calibration curves (172) are shown in Figure 4.By combining columns in series, a separation range may be achieved coveringthe molecular weight distribution of the sample. Mathematical treatments of thechromatogram and calibration curve to calculate the molecular weight distribu-tion are given (173). Broadening by the chromatographic process must be cali-brated for and corrections applied to produce true molecular weight distributions.Chemimetrics has been applied to GPC calibration equations (174) and helped toestablish the best equations. Data reduction in GPC (175) has been described. Anew solvent system which allows polyesters and polyamides to be characterizedby GPC at room temperature has been developed (176). High speed GPC of lowmolecular weight polymers has been described (177).

The design and use of a continuous, viscometric detector (178,179) has fur-ther advanced the application of the GPC technique, and a commercial viscometricdetector is on the market from Viscotek Corp (180,181).


20102

103

104

105

106

Wei

ght-

aver

age

mol

ecul

ar w

eigh

t of p

olys

tyre

ne s

tand

ard

30 40

Elution volume Ve, mL

50

Fig. 4. GPC calibration curve for Corning porous glasses with various pore sizes: �, CPG10–240; �, CPG 10–370; ×, CPG 10–1250; and �, CPG 10–2000 (172). Courtesy of JohnWiley & Sons, Inc.

Direct determination of the polymer molecular weight in the detector cell af-ter separation by GPC is possible by LALLS. The technique (182–184) is applicableto the dilute polymer solutions emerging from the chromatographic column; com-mercial equipment is available, older models (Chromatix KMX6 and LSD100 fromLDC/Milton Roy) and newer instruments (Wyatt and Polymer Labs). A review ofprior development and a description of a methodology for correcting for dispersionin the concentration detector and the light scattering detector has been published(185). A review of GPC and the application of molecular weight detectors andrelated separation techniques has been published (186). The characterization ofcomplex polymers by GPC and high performance liquid chromatography (HPLC)has been discussed (187). Quantitative high temperature GPC of polyolefins usingrefractive index and differential viscometer detection (188) and differential vis-cometer and light scattering detectors (189) has been reported. This method hasbeen used for on-line molecular weight measurement and branching characteriza-tion of water-soluble polymers (185,190). Shear degradation during the chromato-graphic process may be a problem (191), but can be detected by viscometry andLALLS. A triple detector employing a concentration detector, a viscometer, and alight scattering detector is available (Viscotek). Three detectors, an evaporativelight scattering detector to measure concentration, a refractometer, and LALLS,have been used to determine MWD of copolymers (EPM and EPDM) (192). LALLShas proven useful in detecting and eliminating aggregates in dilute polymersolutions which would affect the GPC results (193). Nuclear magnetic resonance,


NMR, has been employed as an on-line detector for GPC (194). Goldwasser (195)developed an analysis which allows the calculation of absolute values for Mn fromthe combined results from GPC and differential viscometry. Mass spectrometryhas been coupled with GPC to characterize a variety of polymers (196–198). Acomparison of the most probably peak value of polymer distributions has beenevaluated between MALDI MS and GPC (199).

Analyses by GPC of block and statistical copolymers (200,201) and copoly-mers (202) and polymer mixtures (203) have been reported. Full copolymer char-acterization by combining GPC-NMR with GPC-MALDI has been demonstrated(204).

Other detectors that are capable of determining the polymer concentrationin the effluent are based on dielectric constant (205) and density (206–213). Masscan be determined directly with a piezoelectric quartz sensor having a sensitivityof 10− 10 g (214). A universal mass detector, evaporative light scattering Detector(ELSD) based on the formation of droplets in a nebulizer gas and detection by lightscattering using a laser, has been developed and applied to GPC (215). CommercialELSD units are available from Alltech, Varex Corp., and Polymer Laboratories.

A direct molecular weight method (216) employs the LALLS detector, butuses sedimentation velocity instead of GPC to separate the polymers.

Round robin tests for reproducibility of GPC results have been reported (217).Twenty-four laboratories used calibration standards from two different sourcesand three test polymers. The results for the three samples varied from 3 to 14%depending on which calibration was used. Similar results for number-averagemolecular weights have been documented (218). Round robin characterization ofpolyethylene by high temperature GPC in 15 laboratories has been documented(219).

High Performance Liquid Chromatography and Supercritical Fluid Chro-matography. For lower molecular weights, HPLC and supercritical fluid chro-matography (SFC) (220) offer higher resolution than GPC. Higher molecularweights may produce multiple peaks from monodisperse standards (221). Theoligomers are resolved into individual peaks, and molecular weight averagesare calculated (222). HPLC is more common and has been applied to phenol-formaldehyde resins (223), polyurethanes (224), polyesters (225), and polycar-bonates (226). Since solvent gradients are often used in HPLC the refractometerdetector cannot be used, and ultraviolet/visible spectrometers and/or ELSDs mustbe used. The ELSD may also be used in SFC. SFC can resolve higher oligomersthan HPLC. Polystyrene oligomers of up to n = 50 may be resolved by SFC;the process requires 18 h (227). Several commercial SFC units are availableand their availability should rapidly increase the application of this techniqueto characterize oligomeric species. A review of supercritical fractionation of poly-mers and copolymers has been published (228). Mass spectrometry has been usedin conjunction with SFC and capillary GPC to characterize polypropylene glycol(229).

Liquid adsorption chromatography (LAV) and GPC have been used todetermine chemical composition distribution (CCD) and MWD of styrene–methyacrylate copolymers (230).

Thin-Layer Chromatography. Thin-layer chromatography (TLC) is widelyapplicable to polymer characterization and has been used to characterize


molecular weight distribution and compositional distribution (231–233). Cellu-lose nitrate has been fractionated by this method on the basis of molecular weightor nitrogen content (234).

Mass Spectrometry. Advances in soft ionization techniques and develop-ments in high resolution mass spectrometers has led to the possibility of deter-mining intact molecular ions up to 1 × 106 Da and the measurement of the entireMWD (235,236). The application of mass spectrometry techniques to character-ize polymer molecular weight distribution has evaluated (237–243). The MWD ofpolystyrene SRM 1487 and a narrow MND polymethylmethacrylate (MW 6300)determined by MALDI TOF mass spectrometry have been compared with resultsfrom the ultracentrifuge and GPC (244).

Phase-Distribution Chromatography. A chromatographic fractionation isachieved in this method (245,246), by partitioning a solute between a solvent flow-ing through a column, packed with a support coated with a thin layer of high molec-ular weight, non-cross-linked polymer. The temperature of operation is below the�-temperature of the solute, and separation efficiency increases sharply with de-creasing temperature. A fully automated chromatograph has been designed (247),and applications have been reported (248–250).

Field–Flow Fractionation. Field-flow fractionation (FFF) employs a one-phase chromatographic system (251,252). Commercial instrumentation is avail-able from Postnova Analytics and Tecan. Separation occurs in a thin channelcontaining a single moving fluid. The field applied across the channel may be se-lected on the basis of the solute. Possible fields include sedimentation, cross-flow,concentration, dielectric, thermal, and magnetic. A book (253) and a review (254)of this technique and its comparison with GPC for the characterization of polymermolecular weights have been published.

Thermal fields have been used for the separation of polymers in organic sol-vents (255). Methods for calculation of MWD from thermal FFF have been pub-lished (256). For aqueous systems a cross-flow of solvent has been used to separatepolyelectrolytes (257) and high molecular weight polyacrylamides (258). Water-soluble polymers have also been characterized by this technique (259,260), in con-junction with LALLS detector (261) and MALLS (262). Extremely high molecularweights of up to 1012 can be determined (263); because no abrupt changes occurin the forces imposed on the polymer molecules, no degradation takes place. Sed-imentation fields were investigated (264), and small samples were quickly frac-tionated. Compositional heterogeneity has been determined by sequential GPCand FFF (265).

Electrophoresis. Electrophoresis is widely practiced on biopolymers(266–268) for the determination of molecular weights of proteins and their sub-units. Typically, a polyacrylamide bed is used to separate or characterize thesesolutes under the influence of an electric field, on the basis of charge and molec-ular size. The proteins are frequently denatured with sodium dodecyl sulfonate(SDS), and these subunit-SDS complexes are separated on the basis of molecu-lar size. A densitometer employing a soft laser scanning device is commerciallyavailable to automate the data-reduction process (269).

Rheological Measurements. Some polymers are difficult to dissolve. Me-chanical spectroscopy or rheological measurements (qv) offer a way to determinethe molecular weight distribution of these materials. Mechanical spectroscopy has


been applied to characterize the MWD of polyolefins (270) Rheological data hasbeen used to determine the MWD of polybutadiene and polypropylene (271). Mea-surements of ethylene–tetrafluorethylene copolymers using dynamic mechanicalanalysis of the melts showed broader distributions than those determined by hightemperature laser light scattering (272).

Other Techniques. Atomization of polymer solutions to form singlemolecules deposited in a solvent-free state on a surface has been demonstratedfor a wide variety of polymers (273). The electron microscope is used at 20,000–30,000 × after platinum shadowing to determine size and size distribution (274).The scanning transmission electron microscope is employed for biological macro-molecules (275). For extremely high molecular weight polymers hydrodynamicchromatography (HDC) may be a viable technique. Columns are packed withsolid spheres and the diameter determines the separation range. Solid particlesbetween 5 and 300 nm may be separated. Columns are available from PolymerLaboratories, and using a high temperature infrared detector cell the compositionof polyolefin copolymers can be characterized. A mathematical model for HDC andGPC for porous column packings has been published (276). Methods to character-ize insoluble polymers have been discussed (277,278).

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ANTHONY R. COOPER

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MOLECULAR WEIGHT DETERMINATION - Freenguyen.hong.hai.free.fr/EBOOKS/SCIENCE AND ENGINEERING... · 2006-11-09 · is the number of moles of molecules with molecular weight between