ASTM D4001-20

Standard Test Method for Determination of Weight-Average Molecular Weight of Polymers by Light Scattering

SIGNIFICANCE AND USE
4.1 The weight-average molecular weight is a fundamental structure parameter of polymers, which is related to many physical properties of the bulk material, such as its rheological behavior. In addition, knowledge of the weight-average molecular weight, together with knowledge of the number-average molecular weight from osmometry, provides a useful measure of the breadth of the molecular-weight distribution.  
4.2 Other important uses of information on the weight-average molecular weight are correlation with dilute-solution or melt-viscosity measurements and calibration of molecular-weight standards for use in liquid-exclusion (gel-permeation) chromatography.  
4.3 To the extent that the light-scattering photometer is appropriately calibrated, light scattering is an absolute method and is therefore be applied to nonionic homopolymers that have not previously been synthesized or studied.
SCOPE
1.1 This test method describes the test procedures for determining the weight-average molecular weight Mw  of polymers by light scattering. It is applicable to all nonionic homopolymers (linear or branched) that dissolve completely without reaction or degradation to form stable solutions. Copolymers and polyelectrolytes are not within its scope. The procedure also allows the determination of the second virial coefficient, A2, which is a measure of polymer-solvent interactions, and the root-mean-square radius of gyration (s2)1/2, which is a measure of the dimensions of the polymer chain.  
1.2 The molecular-weight range for light scattering is, to some extent, determined by the size of the dissolved polymer molecules and the refractive indices of solvent and polymer. A range frequently stated is 10,000 to 10,000,000, is often extended in either direction with suitable systems and by the use of special techniques.  
1.2.1 The lower limit to molecular weight results from low levels of excess solution scattering over that of the solvent. The greater the specific refractive increment dn/dc (difference in refractive indices of solution and solvent per unit concentration), the greater the level of solution scattering and the lower the molecular weight that shall be determined with a given precision.  
1.2.2 The upper limit to molecular weight results from the angular dependence of the solution scattering, which is determined by the molecular size. For sufficiently large molecules, measurements must be made at small scattering angles, which are ultimately outside the range of the photometer used.  
1.3 The values stated in SI units are to be regarded as standard.  
1.4 This standard does not purport to address all of the safety concerns, if any, associated with its use. It is the responsibility of the user of this standard to establish appropriate safety, health, and environmental practices and determine the applicability of regulatory limitations prior to use.
Note 1: There is no known ISO equivalent to this standard.  
1.5 This international standard was developed in accordance with internationally recognized principles on standardization established in the Decision on Principles for the Development of International Standards, Guides and Recommendations issued by the World Trade Organization Technical Barriers to Trade (TBT) Committee.

General Information

• Status: Published
• Publication Date: 30-Jun-2020
• ICS: 83.080.01 - Plastics in general
• Technical Committee: D20 - Plastics
• Drafting Committee: D20.70 - Analytical Methods

Overview

ASTM D4001-20 is a standard test method developed by ASTM International for the "Determination of Weight-Average Molecular Weight of Polymers by Light Scattering." This standardized protocol provides detailed procedures and guidelines to accurately measure the weight-average molecular weight (Mw) of nonionic homopolymers-both linear and branched-using light scattering techniques. Understanding the Mw of polymers is fundamental, as this parameter is intrinsically linked to the material’s bulk properties, such as rheological behavior and overall performance in various applications. The test method also supports the determination of other key polymer characteristics, including the second virial coefficient (A2) and the root-mean-square radius of gyration, offering a comprehensive profile of polymer-solvent interactions and chain dimensions.

Key Topics

• Test Scope and Applicability

  • Applicable to nonionic homopolymers that dissolve completely without degradation or reaction.
  • Not intended for copolymers or polyelectrolytes.
  • Suitable for polymers with molecular weights from approximately 10,000 to 10,000,000, a range determined by molecular size and refractive indices.

• Significance and Use

  • Provides a measurement of Mw, which is critical for correlating polymer structure with physical properties like viscosity and processability.
  • Mw data facilitates the assessment of molecular-weight distribution when combined with number-average molecular weight values from osmometry.
  • Enables calibration of molecular-weight standards for gel-permeation chromatography and supports research in polymer solution or melt rheology.

• Light Scattering as an Analytical Method

  • Recognized as an absolute method, provided proper calibration of the light-scattering photometer.
  • Especially valuable for novel nonionic homopolymers not yet extensively studied.

• Measurement Parameters

  • In addition to Mw, the test also yields the second virial coefficient (A2), indicative of polymer-solvent interactions, and the mean square radius of gyration, reflecting the dimensions of polymer chains.

Applications

• Material Characterization: Manufacturers and researchers use this standard to quantify molecular weight, which is a critical determinant of polymer performance in end-use applications, such as plastics, films, and fibers.
• Quality Control: Provides a reliable and reproducible method to ensure batch-to-batch consistency in polymer production.
• Product Development: Facilitates comparative studies of new polymer formulations or synthesis methods, ensuring desired molecular characteristics are achieved.
• Viscosity Correlation: The standard allows for robust comparison and calibration with other measurement techniques, like dilute-solution or melt viscosity, enhancing the precision of empirical models.
• Regulatory Compliance: Adhering to standardized test methods like ASTM D4001-20 supports compliance with international and customer-specific quality requirements.

Related Standards

• IEEE/ASTM SI-10: Governs the use of International System of Units (SI), ensuring consistency in measurements and reporting.
• ASTM D5296: Standard Test Method for Molecular Weight Averages, Distribution, and Molecular Size Distribution of Polymers by Size-Exclusion Chromatography.
• ASTM D3536: Standard Test Method for Molecular Weight Averages and Molecular Weight Distribution of Polymers by Capillary Viscometry.

Note: There is no known ISO equivalent for ASTM D4001-20.

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Adopting ASTM D4001-20 supports accurate, reproducible measurement of polymer weight-average molecular weight by light scattering, driving innovation and quality in polymer research, manufacturing, and quality assurance. Use this standard to gain detailed insights into polymer characteristics and ensure industry best practices are met.

Standards Content (Sample)

This international standard was developed in accordance with internationally recognized principles on standardization established in the Decision on Principles for the
Development of International Standards, Guides and Recommendations issued by the World Trade Organization Technical Barriers to Trade (TBT) Committee.
Designation: D4001 − 20
Standard Test Method for
Determination of Weight-Average Molecular Weight of
Polymers by Light Scattering
This standard is issued under the fixed designation D4001; the number immediately following the designation indicates the year of
original adoption or, in the case of revision, the year of last revision.Anumber in parentheses indicates the year of last reapproval.A
superscript epsilon (´) indicates an editorial change since the last revision or reapproval.
1. Scope* priate safety, health, and environmental practices and deter-
mine the applicability of regulatory limitations prior to use.
1.1 This test method describes the test procedures for
determining the weight-average molecular weight M of poly-
NOTE 1—There is no known ISO equivalent to this standard.
w
mers by light scattering. It is applicable to all nonionic
1.5 This international standard was developed in accor-
homopolymers (linear or branched) that dissolve completely
dance with internationally recognized principles on standard-
without reaction or degradation to form stable solutions.
ization established in the Decision on Principles for the
Copolymers and polyelectrolytes are not within its scope. The
Development of International Standards, Guides and Recom-
procedure also allows the determination of the second virial
mendations issued by the World Trade Organization Technical
coefficient, A , which is a measure of polymer-solvent
Barriers to Trade (TBT) Committee.
interactions, and the root-mean-square radius of gyration
2 1/2
(s ) , which is a measure of the dimensions of the polymer
2. Referenced Documents
chain.
2.1 ASTM Standards:
1.2 The molecular-weight range for light scattering is, to
IEEE/ASTMSI-10American National Standard for Use of
some extent, determined by the size of the dissolved polymer
theInternationalSystemofUnits(SI):TheModernMetric
molecules and the refractive indices of solvent and polymer.A
System
range frequently stated is 10,000 to 10,000,000, is often
extended in either direction with suitable systems and by the
3. Terminology
use of special techniques.
3.1 Definitions—Units, symbols, and abbreviations are in
1.2.1 The lower limit to molecular weight results from low
accordance with IEEE/ASTMSI-10.
levelsofexcesssolutionscatteringoverthatofthesolvent.The
greater the specific refractive increment dn/dc (difference in
4. Significance and Use
refractive indices of solution and solvent per unit
concentration), the greater the level of solution scattering and 4.1 The weight-average molecular weight is a fundamental
structure parameter of polymers, which is related to many
the lower the molecular weight that shall be determined with a
given precision. physical properties of the bulk material, such as its rheological
behavior. In addition, knowledge of the weight-average mo-
1.2.2 The upper limit to molecular weight results from the
angular dependence of the solution scattering, which is deter- lecular weight, together with knowledge of the number-
average molecular weight from osmometry, provides a useful
mined by the molecular size. For sufficiently large molecules,
measurements must be made at small scattering angles, which measure of the breadth of the molecular-weight distribution.
are ultimately outside the range of the photometer used.
4.2 Other important uses of information on the weight-
average molecular weight are correlation with dilute-solution
1.3 The values stated in SI units are to be regarded as
or melt-viscosity measurements and calibration of molecular-
standard.
weight standards for use in liquid-exclusion (gel-permeation)
1.4 This standard does not purport to address all of the
chromatography.
safety concerns, if any, associated with its use. It is the
responsibility of the user of this standard to establish appro- 4.3 To the extent that the light-scattering photometer is
appropriately calibrated, light scattering is an absolute method
This test method is under the jurisdiction ofASTM Committee D20 on Plastics
and is the direct responsibility of Subcommittee D20.70 on Analytical Meth-
ods.70.05). For referenced ASTM standards, visit the ASTM website, www.astm.org, or
CurrenteditionapprovedJuly1,2020.PublishedJuly2020.Originallyapproved contact ASTM Customer Service at [e-mail]. For Annual Book of ASTM
in 1981. Last previous edition approved in 2013 as D4001-13. DOI: 10.1520/ Standards volume information, refer to the standard’s Document Summary page on
D4001-20. the ASTM website.
*A Summary of Changes section appears at the end of this standard
Copyright © ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959. United States
D4001 − 20
and is therefore be applied to nonionic homopolymers that 7. Sample
have not previously been synthesized or studied.
7.1 The sample must be homogeneous, and must be thor-
oughly free of all foreign impurities. If at all possible, samples
5. Apparatus
to be used for light-scattering measurements must be specially
treated from synthesis on to minimize exposure to or contami-
5.1 Volumetric Flasks, 100-mL, or other convenient size.
nation with particulate impurities. Gels that consist of very
5.2 Transfer Pipets.
high-molecular-weight particles, are sometimes formed during
synthesis and will interfere with the analysis.All such particu-
5.3 Photometer, whose major components, described in
late matter must be removed, sometimes with considerable
Appendix X1, are a light source, a projection optical system, a
difficulty. It should be understood that when this is done, the
sample-cell area, a receiver optical system, a detector system,
remainingsampleisnolongertrulyrepresentativeoftheentire
and a recording system.Typical photometers are described and
polymer. The extent of the difference from the original sample
summarized (1) in the literature.
will depend on the removal techniques employed.
5.4 Differential Refractometer, with sensitivity of approxi-
–6
NOTE 4—Reduction of sample particle size in a clean Spex or Wiley
mately 3×10 refractive-index units, capable of measuring
millspeedssolutionand,withslow-dissolvingmaterials,isessentialifthe
the specific refractive increment dn/dc at the wavelength and
measurements are to be made in a reasonable time. Overheating with
temperature of the scattering measurements (2).
consequent sample degradation must be avoided during the milling
process.Hard,toughsamplesorthosewithlowmeltingpointsarehandled
NOTE 2—Specific refractive increments are tabulated (2,3) for many
by mixing with clean dry ice, milling the mixture, and then allowing the
polymer-solvent systems.
dry ice to sublime. Clean dry ice may be obtained by opening a tank of
5.5 Refractometer, Abbé type or equivalent, capable of carbon dioxide to the atmosphere. Commercial dry ice has often been
shown to be contaminated.
measuringtherefractiveindicesofsolventsandsolutionsatthe
wavelength and temperature of the scattering measurements.
8. Preparation of Dust-Free Cell and Contents
5.6 Spectrophotometer, capable of measuring the absor-
8.1 Clean all glassware, including the scattering cell, with a
banceofsolutionsatthewavelengthofthescatteringmeasure-
suitable detergent to remove grease and other contaminants.
ments.
Use of an ultrasonic cleaning bath is recommended. Rinse
glassware at least four times with distilled water to remove all
5.7 Laminar-Flow Clean-Air Station, to provide a dust-free
traces of detergent, and dry in a clean, dust-free drying oven.
area for preparing and cleaning solutions and filling the
scattering cell. NOTE 5—A laminar-flow clean-air station is recommended for provid-
ing a dust-free area for solution preparation and filtration. If a clean-air
5.8 Filters and Filter Holders, for cleaning solvents and
station is not used, a closed area in a location free of drafts and of
solutions. Membrane filters with pore sizes from 0.10 to 0.45 sufficient size to hold the filter unit, scattering cell, and other glassware
shall be used.
µm, used in glass or plastic filter holders, are recommended.
8.2 Filter solvent directly into the scattering cell. First rinse
5.8.1 For water and aqueous solutions, and for organic
the cell several times with 5 to 10 mLof filtered solvent each,
solvents that do not attack the material, the use of polycarbon-
to remove dust particles. Upper surfaces of the interior of the
ate (Nucleopore) filters is recommended.These filters have the
advantages of high flow rate without the use of gas pressure, cell shall be well washed down. Close the cell with a cap
similarly rinsed with filtered solvent.After rinsing, fill the cell
minimal retention of solute on the filter, and efficient cleaning
action. For other solvents, the use of cellulosic filters (Milli- with the minimum amount of solvent required to bring the
liquid level above the point where the light beam in the
pore or equivalent) is recommended.
photometer passes through the cell.
NOTE 3—Sintered-glass filters is sometimes used, but these are rela-
NOTE 6—Use of a small filter holder fitting between a hypodermic
tively expensive and difficult to clean between uses. Centrifugation is
syringeandneedleisconvenientwhereonlysmallquantitiesofliquidsare
sometimes used, but this step requires special care and techniques, or
filtered. A cell cap, with a hole just large enough to insert the needle, is
special scattering cell design, to be satisfactory.
used.
6. Reagents and Materials 8.3 Place the scattering cell in the photometer, or in an
equivalent strong light beam, and examine it in the dark,
6.1 Solvents, as required. Since dn/dc is a function of
viewing at small scattering angles. Bright specks of dust must
composition, solvents shall be of high purity. Significant errors
notbevisible;iftheyare,thecellwasnotrinsedcompletelyor
in molecular weight, which depends on the square of d n/dc,
the filtration procedure is inadequate.
will be incurred if literature values of dn/dc are employed and
8.4 Subsequent use of the clean cell for adding increments
the actual value of this quantity is different because of
of filtered solution or for replacing solvent with solution
impurities in the solvent.
requires no further rinsing, except to ensure that residual
6.2 12-Tungstosilicic Acid, as standard for calibration of
solvent remaining, after the cell is emptied, is removed and
photometer.
replaced with solution.
9. Procedure
9.1 Calibrate the light-scattering photometer. This calibra-
The boldface numbers in parentheses refer to the list of references at the end of
this test method. tion is required to convert measurements of scattered light
D4001 − 20
intensity from arbitrary to absolute values, an essential step in do not readjust these variables, but change amplifier gain by
thecalculationofmolecularweight.Thecalibrationprocedure, known factors or insert neutral filters of known transmittance
which is lengthy and requires great care to obtain accurate as required to maintain readings on scale.
results, is given in Appendix X2. The calibration constant of 9.4.4 Solvent Measurement—After the cell has remained
most photometers remains stable for long periods of time, undisturbed in the photometer for 10 to 15 min to allow
however, so making the calibration procedure infrequent. residual dust to settle out, read and record the scattered
intensity at angles of 30°, 90°, 150°, and at least three pairs
9.2 Prepare a stock solution of polymer, noting the precau-
between,symmetricallyplacedwithrespectto90°,asavailable
tions of Sections 7 and 8, at a concentration estimated as
on the photometer used.
follows: For a polymer of M =100,000 in a solvent
w
9.4.5 Reference—Turn the phototube to the specified refer-
such that dn/d c ≈ 0.2 mL/g (for example, polystyrene in
ence angle, adjust amplifier gain or insert neutral filters as
2-butanone),thestocksolutionshallbeintherangefrom10to
required, insert the reference standard, and read and record the
20 g/L. Since scattered intensity is proportional to M and to
w
indicated reference intensity.
the square of dn/dc, estimates of the stock-solution concentra-
9.4.6 SolutionMeasurement—Prepareandfilterintothecell
tion required for other samples and systems is made. Prepare
the solutions required in 9.3. Mix thoroughly, allow a few
no more stock solution than is required by the following
minutes for residual dust to settle out, and measure each
procedure.
solution as in 9.4.4.
9.3 Select one of the following measurement schemes:
9.5 Determine solution concentrations. Since filtration
9.3.1 Where the volume of liquid required for measurement
through membrane filters has been known to result in retention
in the photometer is varied by at least a factor of two, it is
of some polymer on the filter, it is necessary to determine the
recommendedthatthescatteringfromtheminimumvolumeof
solution concentrations after filtration.
solvent be measured first, followed by measurement of solu-
9.5.1 If successive concentrations are generated in the cell
tions prepared in the cell by the addition of weighed or
from a stock solution filtered under constant conditions, only
volumetrically measured aliquots of filtered stock solution.
the concentration of the filtered stock solution need be deter-
From four to six such solutions shall be measured, the most
mined; otherwise, the concentration of each solution measured
concentrated consisting of approximately equal volumes of
must be determined.
solvent and stock solution if its concentration is selected in
9.5.2 Determinetheconcentrationsofsolutions,asrequired,
accordance with 9.2, and the least concentrated being about
by one of the following methods. Use standard analytical
one fourth this concentration. A specific example is given in
techniques where applicable.
Appendix X3.
9.5.2.1 Evaporate a portion of the solution to constant
9.3.2 If the volume of liquid in the scattering cell cannot be
weight. It is necessary to do this at high temperatures, namely,
varied as in 9.3.1, it is necessary to prepare and filter into the
above the glass transition temperature and under vacuum, to
cell from four to six separate solutions covering the range
remove tightly bound solvent. Because solvent is sometimes
suggested in 9.3.1.
very difficult to remove, such a procedure for determining
9.3.3 A further alternative is to measure the most concen-
concentration must be verified by other techniques before
trated solution first (for this purpose, the stock solution
being adopted.
concentrationestimatedin9.3.1shallbereducedbyafactorof
9.5.2.2 Determine the ultraviolet absorbance of the solution
two), followed by successive dilutions with solvent. The
at a suitable wavelength.
scatteringfromthepuresolventmustbemeasuredinaseparate
9.5.2.3 Determinethedifferenceinrefractiveindexbetween
step. If necessary, start dilution sequences at two or more
solution and solvent, using a differential refractometer, for
concentration levels to obtain the range specified in 9.3.1.
cases where the specific refractive increment is known.
9.5.3 Forcaseswhereaseriesofsolutionsisproducedinthe
9.4 Measure the scattering of the pure solvent, filtered into
cell, calculate the actual solution concentrations from that of
the cell as described in Section 8, and of each of the series of
the stock solution by standard volumetric or gravimetric
filtered solutions described in 9.3, following the instructions
analytical methods.
providedwiththephotometerorintheliterature (4),beingsure
that the following steps are included. (This procedure is based 9.6 If the specific refractive increment dn/dc is not known,
on the scheme of 9.3.1.) determineitusingsolutionsofknownconcentrations;thesame
solutions used for light scattering measurements shall be
9.4.1 Instrument Check—See that the photometer is pre-
utilized. The specific refractive increment is the slope of the
paredformeasurement,withthelamplit,highvoltagesupplied
straight line relating solution-solvent refractive-index
to the photomultiplier detector, and all components fully
difference, ∆n,tosolutionconcentration,c.Sincetherelationis
warmed up and stabilized.
linear, determination of ∆n for one value of c suffices, but
9.4.2 Solvent Preparation—Fill the cleaned scattering cell
multiple determinations are recommended to reduce the uncer-
with filtered solvent as described in Section 8, insert it in the
tainty of the value of dn/dc. For use and calibration of the
instrument, and align it as required.
differential refractometer, follow the instructions supplied with
9.4.3 Intensity Level—Select the wavelength-isolating filter
the instrument.
to be used. Turn the detector to the specified angle and set the
level of high voltage, or adjust the slit openings, as called for 9.7 Iftherefractiveindexofthesolventisnotknownforthe
to provide an appropriate solvent reading. In subsequent steps, wavelength and temperature of the measurements, determine it
D4001 − 20
using a conventional refractometer. If the refractive indices of 10.3.6 Byuseofthecalibrationconstantdeterminedin10.2,
the polymer solutions used differ significantly from that of the convert the data of 10.3.5 (or 10.3.4 if no additional factors
solvent, determine them also. were applied in 10.3.5) to values of the Rayleigh ratio ∆R .
θ
Typical data are shown in Table X3.1, Section E.
9.8 Ifthepolymerabsorbslight,orissuspectedofabsorbing
light, at the wavelength of the scattering measurement, an 10.4 Prepare data for graphical treatment by the Zimm-plot
absorption correction (Appendix X4) must be applied.
method.
9.9 If the polymer or solvent fluoresces, or is suspected of
NOTE 7—An alternative method, known as the dissymmetry method, is
doing so, the possibility of fluorescence must be eliminated. used when the angular dependence of the Rayleigh ratio is small (for
example ∆R <2 ∆R ).The method is less general in application than
45° 135°
9.9.1 Fluorescence is detected by placing in the detector
the Zimm-plot method, however, and is not recommended. Further details
optical system a sharp-cutting short-wavelength-cutoff filter
of the dissymmetry method is found in the literature (5).
thatabsorbscompletelyatthewavelengthoftheincidentlight.
10.4.1 Divide each polymer concentration by the corre-
The scattered-light reading will drop to zero if there is no
sponding values of ∆R from step 10.3.5 to obtain the
θ
fluorescence, but will remain finite if fluorescence is present.
quantities c/∆R . Typical data are given in Table X3.1, Section
θ
9.9.2 If fluorescence is present, place a narrow-bandpass
F.
interferencefiltertransmittingatthewavelengthoftheincident
10.4.2 Select an appropriate value of the quantity k in the
light in the detector optical system.Alternatively, but with less
expression sin (θ/2)+kc, such that for the highest value of c
certainty of success, place in the detector optical system an
utilized,kcisintherangefrom0.2to0.4.Tabulatethequantity
absorbing filter that absorbs at wavelengths longer than that of
sin (θ/2)+kc. Typical data are shown in Table X3.1, Section
the incident light. (Such filters are not usually sharp-cutting,
G.
and hence are less efficient than the use of an interference
filter.)
10.5 Plot the data of steps 10.4.1 and 10.4.2 to yield the
Zimm plot. A typical Zimm plot is shown in Fig. X3.1.
10. Calculation
10.5.1 Plot, on graph paper with scales suitably selected,
corresponding values of c/∆R and sin (θ/2)+kc.
10.1 Calculate Correction Factors—Using the methods of θ
10.5.2 For each angle θ, connect points at various values of
Appendix X4, calculate the following correction factors, as
ctoform(ifpossible)astraightline.Extrapolatethislinetothe
required.
point corresponding to c=0, that is, to the value of sin (θ/2)
10.1.1 Factors that must be considered for each data point:
for that angle.
amplification, filter, and reflection factors.
10.5.3 For each concentration c, connect points at various
10.1.2 Factors to be applied at each concentration: absorp-
values of θ to form (if possible) a straight line. Extrapolate this
tion and depolarization factors.
line to the point corresponding to θ=0, that is, to the value of
10.1.3 Factors to be applied at each angle: polarization and
kc for that concentration.
volume factors.
10.5.4 Connectpointsatc=0toform(ifpossible)astraight
10.1.4 Factor to be applied to calibration constant: refrac-
line. Connect points at θ=0 to form (if possible) another
tion correction.
straightline.Extrapolatetheselinestotheordinateaxis(where
10.2 Calculate the Calibration Constant— Following the
c=0 and θ=0), where they must meet at a single point,
procedure of Appendix X2, calculate the calibration constant
denoted the intercept.
for converting light-scattering intensities into Rayleigh ratios.
NOTE 8—It is possible to produce the plots of 10.5.2, 10.5.3, and the
10.3 Calculate Rayleigh Ratios—Apply the necessary cor- 2
twolinesof10.5.4onseparategraphs,usingthevariablessin (θ/2)andkc
rection factors and otherwise treat the data of 9.4 and 9.5 as
separately as required instead of together, but when the principles of the
Zimm plot have been mastered, plot the combined data on a single graph
follows. Typical data are shown in Appendix X3.
as indicated.
10.3.1 Obtain original data from recorder chart,
galvanometer, or other readout device, at each concentration
10.6 Assess the quality of the Zimm plot.
and angle utilized. Correct for zero-signal level as required.
10.6.1 If the families of lines in 10.5.2 and 10.5.3 are all
Apply amplification and filter factors, if any, to obtain a
straight and reasonably parallel, and if the lines in 10.5.4 are
self-consistentsetofdata.TabulateasinTableX3.1,SectionA.
straight and meet at a point on the ordinate axis, the quality of
10.3.2 Apply the volume correction factor determined from
the Zimm plot is satisfactory and 10.7 shall be carried out.
Appendix X4 to the data of 10.3.1. Typical data are shown in
10.6.2 If the data of 10.5.1 cannot be made to fit straight
Table X3.1, Section B.
lines, due to scatter or systematic deviations, the quality of the
10.3.3 Correct the data of 10.3.2 for the reference by
Zimm plot is not satisfactory, and conclusions must be drawn
dividing by the reference intensity. Typical data are shown in
from the experiment only with extreme caution. The recom-
Table X3.1, Section C.
mendedactionistorepeattheexperiment,payingmorecareful
10.3.4 Subtract the solvent readings from 10.3.3 from the
attention to sample preparation and solution filtration, since
corresponding data for each concentration of polymer. Typical
dustorotherunwantedscatteringmaterialisthemostprobable
data are shown in Table X3.1, Section D.
cause of the difficulty. Reference to the literature (6) is
10.3.5 As required, apply absorption, depolarization, recommended. In some cases, a microgel component, which is
polarization, and reflection factors to the data of 10.3.4. really part of the distribution of polymer, has been known to
D4001 − 20
cause severely distorted Zimm plots (7), and few valid conclu-
n = therefractiveindex.“Slope”referstotheinitialslopeof
sions are drawn from the data.
the line of c=0 points in the Zimm plot (10.5.2) and
10.6.3 If the data of 10.5.2 fall on straight, reasonably
“intercept” is the value of (c/∆R ) determined
θ c=0, θ=0
parallel lines, and the data of 10.5.3 fall on gently curved but
from 10.5.4.
stillparallellines,thedistortionresultsfromlargemoleculesor
NOTE11—Thederivationsoftheequationsin10.7,andtypicaldata,are
given in Appendix X3.
a broad distribution of molecular sizes, and conclusions drawn
have been shown valid if data at small enough angles are
11. Report
included in the analysis (8).
11.1 Report the following information:
10.7 Calculate the molecular parameters.
11.1.1 Identification of the sample.
10.7.1 Calculate the Debye constant, K, as follows:
11.1.2 Conditioning of the sample, if any.
2 2 2 4
K 5 2π n ~dn/dc! /N λ
o
11.1.3 Solvent, temperature, and instrument used.
11.1.4 Filtration technique.
where:
11.1.5 Basic data, including wavelength, d n/dc, n, verti-
n = the refractive index from 9.8 or tables (2,3),
cally polarized or unpolarized light, nature of reference, and
dn/dc = the specific refractive increment from 9.6 or tables
calibration constant.
(2,3),
11.1.6 Correction factors and any basic data (absorbance,
N = Avogadro’s number, and
o
depolarization) used in deriving them.
λ = the wavelength of the incident light as measured in
¯
11.1.7 Results, including K, M , and (optionally) A and
air.
w 2
1 ¯
(s¯ ) ⁄2 . Unless otherwise agreed, M shall be reported to three
NOTE 9—This treatment assumes that the refractive index n is essen-
w
tially the same (within 0.01) for the solvent and all polymer solutions. If significant figures, and A and (s ) ⁄2 to two significant figures.
not, separate values of K must be calculated for each polymer
11.1.8 If agreed, a table of data similar to Table X3.1, and
concentration, and applied to the corresponding data in 10.4.1 rather than
the Zimm plot.
at this stage.
¯
10.7.2 Calculate the weight-average molecular weight M 12. Precision and Bias
w
as follows:
12.1 For most polymer-solvent systems where the solutions
¯ 21
are reasonably free of dust or other extraneous scattering
M 5 K C/∆R
~ !
w @ θ #
c50, θ50
material, the weight-average molecular weight will be deter-
where:
mined with a standard deviation for reproducibility of about
K is the calibration constant of 10.7.1 and (c/∆R ) is
θ c=0, θ=0
5% of its value. A typical statistical analysis is given in the
the intercept of the Zimm plot from 10.5.4.
literature (9). Bias in the strict sense are seldom estimated
10.7.3 Calculate the second virial coefficient A as follows:
because of the lack of absolute standards in the molecular
weight ranges involved. No round-robin data have been ob-
A 51/2 K c/∆R 2 c/∆R / c 2 c
@~ ! ~ ! # ~ !
2 θ c2 θ c1 2 1
tained.
where the quantities have the same meanings as before and
12.2 In accordance with 12.1, “absolute standards” means
c and c are two concentrations, at the high and low ends of
1 2
¯
polymers for which M is accurately determined by methods
the range encompassed, respectively. w
other than light-scattering. There are few such polymers
NOTE 10—Since A has been known to vary slightly with the angle θ,
capable of being used interchangeably with the synthetic
calculate it from the data extrapolated to θ=0.
polymers to which this practice is directed. Relative standards
10.7.4 Calculate the radius of gyration (s ) ⁄2 as the square
doexist,intheformofwell-characterizedpolystyrenesamples
root of the mean square radius of gyration obtained from the
(10,11), and it is recommended that one of these be measured
following equation:
at periodic intervals to ensure that the results obtained are
2 2 2 2
consistent with the body of experience elsewhere (12,13).
s¯ 5 ~3λ /16π n ! 3 ~slope/intercept!
where:
13. Keywords
s¯ = the mean square radius of gyration,
13.1 light scattering; polymers; weight-average molecular
λ = the wavelength of incident light measured in air, and
weight
D4001 − 20
APPENDIXES
(Nonmandatory Information)
X1. PHOTOMETERS
X1.1 Typical Photometers drical cross section. For use with the Zimm-plot method in
conventional photometers, use of a cylindrical cell with flat
X1.1.1 Typical photometers have as major components, a
entrance and exit windows is recommended.
light source, a projection optical system, a sample cell area, an
optical receiver system, a detector system, and a recording
X1.4.2 Reflection or scattering from the air-glass interfaces
system. They are surveyed (1) and described (14-18) in the
of the sample cell is one source of stray light. Care must be
literature.
takentokeepsuchsurfacesclean,andproperdesignofthecell
and the detector optical system is important to minimize the
X1.2 Light-Source System
problem. In some photometers, the cell is immersed in a
X1.2.1 Apowerful and stable source is required because of
containerofliquidhavingthesamerefractiveindexastheglass
the low level of scattering from typical polymer solutions.
of the cell, with the container walls out of the field of view of
Mercuryarclampshavebeenthemostcommonlyusedsource.
the detector system.
Solid State lasers are the most commonly used source as they
X1.4.3 The sample-cell compartment is usually placed in a
provide stable, powerful, monochromatic, fully polarized
light-tight box to shield the detector system from extraneous
sources.
light. Such a box should be amply large. The cell should be
X1.2.2 Lasers with various wavelengths can be used.
reproducibly and rigidly mountable with its center on the optic
Shorter wavelengths result in stronger scattering (from both
axis of rotation of the detector system and its entrance face
solvent and polymer). Shorter wavelengths also reduce the
precisely perpendicular to the optic axis of the incident beam,
lower and upper limits for accurate determination of RMS
but easily removable for cleaning and filling.
radius (and also the upper limit for molar mass determination),
X1.4.4 Provision must be made for replacing the scattering
and increase the possibility of absorbance and fluorescence.
cell with a stable, reproducible scatterer or diffuser, which can
For accurate molar mass, dn/dc must be known at the wave-
serve as a constant arbitrary reference for periodic measure-
length used for the measurement. If not available in the
ment. Scattered intensities from solutions and solvent are
literature, it can be measured using a differential refractometer
adjusted to correspond to constant intensity from the reference
with a light
...

This document is not an ASTM standard and is intended only to provide the user of an ASTM standard an indication of what changes have been made to the previous version. Because
it may not be technically possible to adequately depict all changes accurately, ASTM recommends that users consult prior editions as appropriate. In all cases only the current version
of the standard as published by ASTM is to be considered the official document.
Designation: D4001 − 13 D4001 − 20
Standard Test Method for
Determination of Weight-Average Molecular Weight of
Polymers Byby Light Scattering
This standard is issued under the fixed designation D4001; the number immediately following the designation indicates the year of
original adoption or, in the case of revision, the year of last revision. A number in parentheses indicates the year of last reapproval. A
superscript epsilon (´) indicates an editorial change since the last revision or reapproval.
1. Scope*
1.1 This test method describes the test procedures for determining the weight-average molecular weight M of polymers by light
w
scattering. It is applicable to all nonionic homopolymers (linear or branched) that dissolve completely without reaction or
degradation to form stable solutions. Copolymers and polyelectrolytes are not within its scope. The procedure also allows the
determination of the second virial coefficient, A , which is a measure of polymer-solvent interactions, and the root-mean-square
2 1/2
radius of gyration (s ) , which is a measure of the dimensions of the polymer chain.
1.2 The molecular-weight range for light scattering is, to some extent, determined by the size of the dissolved polymer
molecules and the refractive indices of solvent and polymer. A range frequently stated is 10,000 to 10,000,000, is often extended
in either direction with suitable systems and by the use of special techniques.
1.2.1 The lower limit to molecular weight results from low levels of excess solution scattering over that of the solvent. The
greater the specific refractive increment dn/dc (difference in refractive indices of solution and solvent per unit concentration), the
greater the level of solution scattering and the lower the molecular weight that shall be determined with a given precision.
1.2.2 The upper limit to molecular weight results from the angular dependence of the solution scattering, which is determined
by the molecular size. For sufficiently large molecules, measurements must be made at small scattering angles, which are ultimately
outside the range of the photometer used.
1.3 The values stated in SI units are to be regarded as standard.
1.4 This standard does not purport to address all of the safety concerns, if any, associated with its use. It is the responsibility
of the user of this standard to establish appropriate safety safety, health, and healthenvironmental practices and determine the
applicability of regulatory limitations prior to use.
NOTE 1—There is no known ISO equivalent to this standard.
1.5 This international standard was developed in accordance with internationally recognized principles on standardization
established in the Decision on Principles for the Development of International Standards, Guides and Recommendations issued
by the World Trade Organization Technical Barriers to Trade (TBT) Committee.
2. Referenced Documents
2.1 ASTM Standards:
IEEE/ASTM SI-10 American National Standard for Use of the International System of Units (SI): The Modern Metric System
3. Terminology
3.1 Definitions—Units, symbols, and abbreviations are in accordance with IEEE/ASTM SI-10.
4. Significance and Use
4.1 The weight-average molecular weight is a fundamental structure parameter of polymers, which is related to many physical
properties of the bulk material, such as its rheological behavior. In addition, knowledge of the weight-average molecular weight,
together with knowledge of the number-average molecular weight from osmometry, provides a useful measure of the breadth of
the molecular-weight distribution.
This test method is under the jurisdiction of ASTM Committee D20 on Plastics and is the direct responsibility of Subcommittee D20.70 on Analytical Methods.70.05).
Current edition approved Nov. 1, 2013July 1, 2020. Published November 2013July 2020. Originally approved in 1981. Last previous edition approved in 2006 as
D4001-93 (2006).-13. DOI: 10.1520/D4001-13.10.1520/D4001-20.
For referenced ASTM standards, visit the ASTM website, www.astm.org, or contact ASTM Customer Service at [e-mail]. For Annual Book of ASTM Standards
volume information, refer to the standard’s Document Summary page on the ASTM website.
*A Summary of Changes section appears at the end of this standard
Copyright © ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959. United States
D4001 − 20
4.2 Other important uses of information on the weight-average molecular weight are correlation with dilute-solution or
melt-viscosity measurements and calibration of molecular-weight standards for use in liquid-exclusion (gel-permeation)
chromatography.
4.3 To the extent that the light-scattering photometer is appropriately calibrated, light scattering is an absolute method and is
therefore be applied to nonionic homopolymers that have not previously been synthesized or studied.
5. Apparatus
5.1 Volumetric Flasks, 100-mL, or other convenient size.
5.2 Transfer Pipets.
5.3 Photometer, whose major components, described in Appendix X1, are a light source, a projection optical system, a
sample-cell area, a receiver optical system, a detector system, and a recording system. Typical photometers are described and
summarized (1) in the literature.
–6
5.4 Differential Refractometer, with sensitivity of approximately 3 × 10 refractive-index units, capable of measuring the
specific refractive increment dn/dc at the wavelength and temperature of the scattering measurements (2).
NOTE 2—Specific refractive increments are tabulated (2,3) for many polymer-solvent systems.
5.5 Refractometer, Abbé type or equivalent, capable of measuring the refractive indices of solvents and solutions at the
wavelength and temperature of the scattering measurements.
5.6 Spectrophotometer, capable of measuring the absorbance of solutions at the wavelength of the scattering measurements.
5.7 Laminar-Flow Clean-Air Station, to provide a dust-free area for preparing and cleaning solutions and filling the scattering
cell.
5.8 Filters and Filter Holders, for cleaning solvents and solutions. Membrane filters with pore sizes from 0.10 to 0.45 μm, used
in glass or plastic filter holders, are recommended.
5.8.1 For water and aqueous solutions, and for organic solvents that do not attack the material, the use of polycarbonate
(Nucleopore) filters is recommended. These filters have the advantages of high flow rate without the use of gas pressure, minimal
retention of solute on the filter, and efficient cleaning action. For other solvents, the use of cellulosic filters (Millipore or equivalent)
is recommended.
NOTE 3—Sintered-glass filters is sometimes used, but these are relatively expensive and difficult to clean between uses. Centrifugation is sometimes
used, but this step requires special care and techniques, or special scattering cell design, to be satisfactory.
6. Reagents and Materials
6.1 Solvents, as required. Since dn/dc is a function of composition, solvents shall be of high purity. Significant errors in
molecular weight, which depends on the square of d n/dc, will be incurred if literature values of dn/dc are employed and the actual
value of this quantity is different because of impurities in the solvent.
6.2 12-Tungstosilicic Acid, as standard for calibration of photometer.
7. Sample
7.1 The sample must be homogeneous, and must be thoroughly free of all foreign impurities. If at all possible, samples to be
used for light-scattering measurements must be specially treated from synthesis on to minimize exposure to or contamination with
particulate impurities. Gels that consist of very high-molecular-weight particles, are sometimes formed during synthesis and will
interfere with the analysis. All such particulate matter must be removed, sometimes with considerable difficulty. It should be
understood that when this is done, the remaining sample is no longer truly representative of the entire polymer. The extent of the
difference from the original sample will depend on the removal techniques employed.
NOTE 4—Reduction of sample particle size in a clean Spex or Wiley mill speeds solution and, with slow-dissolving materials, is essential if the
measurements are to be made in a reasonable time. Overheating with consequent sample degradation must be avoided during the milling process. Hard,
tough samples or those with low melting points are handled by mixing with clean dry ice, milling the mixture, and then allowing the dry ice to sublime.
Clean dry ice may be obtained by opening a tank of carbon dioxide to the atmosphere. Commercial dry ice has often been shown to be contaminated.
8. Preparation of Dust-Free Cell and Contents
8.1 Clean all glassware, including the scattering cell, with a suitable detergent to remove grease and other contaminants. Use
of an ultrasonic cleaning bath is recommended. Rinse glassware at least four times with distilled water to remove all traces of
detergent, and dry in a clean, dust-free drying oven.
NOTE 5—A laminar-flow clean-air station is recommended for providing a dust-free area for solution preparation and filtration. If a clean-air station
The boldface numbers in parentheses refer to the list of references at the end of this test method.
D4001 − 20
is not used, a closed area in a location free of drafts and of sufficient size to hold the filter unit, scattering cell, and other glassware shall be used.
8.2 Filter solvent directly into the scattering cell. First rinse the cell several times with 5 to 10 mL of filtered solvent each, to
remove dust particles. Upper surfaces of the interior of the cell shall be well washed down. Close the cell with a cap similarly rinsed
with filtered solvent. After rinsing, fill the cell with the minimum amount of solvent required to bring the liquid level above the
point where the light beam in the photometer passes through the cell.
NOTE 6—Use of a small filter holder fitting between a hypodermic syringe and needle is convenient where only small quantities of liquids are filtered.
A cell cap, with a hole just large enough to insert the needle, is used.
8.3 Place the scattering cell in the photometer, or in an equivalent strong light beam, and examine it in the dark, viewing at small
scattering angles. Bright specks of dust must not be visible; if they are, the cell was not rinsed completely or the filtration procedure
is inadequate.
8.4 Subsequent use of the clean cell for adding increments of filtered solution or for replacing solvent with solution requires
no further rinsing, except to ensure that residual solvent remaining, after the cell is emptied, is removed and replaced with solution.
9. Procedure
9.1 Calibrate the light-scattering photometer. This calibration is required to convert measurements of scattered light intensity
from arbitrary to absolute values, an essential step in the calculation of molecular weight. The calibration procedure, which is
lengthy and requires great care to obtain accurate results, is given in Appendix X2. The calibration constant of most photometers
remains stable for long periods of time, however, so making the calibration procedure infrequent.
9.2 Prepare a stock solution of polymer, noting the precautions of Sections 7 and 8, at a concentration estimated as follows: For
a polymer of M = 100,000 in a solvent
w
such that dn/d c ≈ 0.2 mL/g (for example, polystyrene in 2-butanone), the stock solution shall be in the range from 10 to 20 g/L.
Since scattered intensity is proportional to M and to the square of dn/dc, estimates of the stock-solution concentration required
w
for other samples and systems is made. Prepare no more stock solution than is required by the following procedure.
9.3 Select one of the following measurement schemes:
9.3.1 Where the volume of liquid required for measurement in the photometer is varied by at least a factor of two, it is
recommended that the scattering from the minimum volume of solvent be measured first, followed by measurement of solutions
prepared in the cell by the addition of weighed or volumetrically measured aliquots of filtered stock solution. From four to six such
solutions shall be measured, the most concentrated consisting of approximately equal volumes of solvent and stock solution if its
concentration is selected in accordance with 9.2, and the least concentrated being about one fourth this concentration. A specific
example is given in Appendix X3.
9.3.2 If the volume of liquid in the scattering cell cannot be varied as in 9.3.1, it is necessary to prepare and filter into the cell
from four to six separate solutions covering the range suggested in 9.3.1.
9.3.3 A further alternative is to measure the most concentrated solution first (for this purpose, the stock solution concentration
estimated in 9.3.1 shall be reduced by a factor of two), followed by successive dilutions with solvent. The scattering from the pure
solvent must be measured in a separate step. If necessary, start dilution sequences at two or more concentration levels to obtain
the range specified in 9.3.1.
9.4 Measure the scattering of the pure solvent, filtered into the cell as described in Section 8, and of each of the series of filtered
solutions described in 9.3, following the instructions provided with the photometer or in the literature (4), being sure that the
following steps are included. (This procedure is based on the scheme of 9.3.1.)
9.4.1 Instrument Check—See that the photometer is prepared for measurement, with the lamp lit, high voltage supplied to the
photomultiplier detector, and all components fully warmed up and stabilized.
9.4.2 Solvent Preparation—Fill the cleaned scattering cell with filtered solvent as described in Section 8, insert it in the
instrument, and align it as required.
9.4.3 Intensity Level—Select the wavelength-isolating filter to be used. Turn the detector to the specified angle and set the level
of high voltage, or adjust the slit openings, as called for to provide an appropriate solvent reading. In subsequent steps, do not
readjust these variables, but change amplifier gain by known factors or insert neutral filters of known transmittance as required to
maintain readings on scale.
9.4.4 Solvent Measurement—After the cell has remained undisturbed in the photometer for 10 to 15 min to allow residual dust
to settle out, read and record the scattered intensity at angles of 30°, 90°, 150°, and at least three pairs between, symmetrically
placed with respect to 90°, as available on the photometer used.
9.4.5 Reference—Turn the phototube to the specified reference angle, adjust amplifier gain or insert neutral filters as required,
insert the reference standard, and read and record the indicated reference intensity.
9.4.6 Solution Measurement—Prepare and filter into the cell the solutions required in 9.3. Mix thoroughly, allow a few minutes
for residual dust to settle out, and measure each solution as in 9.4.4.
9.5 Determine solution concentrations. Since filtration through membrane filters has been known to result in retention of some
polymer on the filter, it is necessary to determine the solution concentrations after filtration.
D4001 − 20
9.5.1 If successive concentrations are generated in the cell from a stock solution filtered under constant conditions, only the
concentration of the filtered stock solution need be determined; otherwise, the concentration of each solution measured must be
determined.
9.5.2 Determine the concentrations of solutions, as required, by one of the following methods. Use standard analytical
techniques where applicable.
9.5.2.1 Evaporate a portion of the solution to constant weight. It is necessary to do this at high temperatures, namely, above the
glass transition temperature and under vacuum, to remove tightly bound solvent. Because solvent is sometimes very difficult to
remove, such a procedure for determining concentration must be verified by other techniques before being adopted.
9.5.2.2 Determine the ultraviolet absorbance of the solution at a suitable wavelength.
9.5.2.3 Determine the difference in refractive index between solution and solvent, using a differential refractometer, for cases
where the specific refractive increment is known.
9.5.3 For cases where a series of solutions is produced in the cell, calculate the actual solution concentrations from that of the
stock solution by standard volumetric or gravimetric analytical methods.
9.6 If the specific refractive increment dn/dc is not known, determine it using solutions of known concentrations; the same
solutions used for light scattering measurements shall be utilized. The specific refractive increment is the slope of the straight line
relating solution-solvent refractive-index difference, Δn, to solution concentration, c. Since the relation is linear, determination of
Δn for one value of c suffices, but multiple determinations are recommended to reduce the uncertainty of the value of dn/dc. For
use and calibration of the differential refractometer, follow the instructions supplied with the instrument.
9.7 If the refractive index of the solvent is not known for the wavelength and temperature of the measurements, determine it
using a conventional refractometer. If the refractive indices of the polymer solutions used differ significantly from that of the
solvent, determine them also.
9.8 If the polymer absorbs light, or is suspected of absorbing light, at the wavelength of the scattering measurement, an
absorption correction (Appendix X4) must be applied.
9.9 If the polymer or solvent fluoresces, or is suspected of doing so, the possibility of fluorescence must be eliminated.
9.9.1 Fluorescence is detected by placing in the detector optical system a sharp-cutting short-wavelength-cutoff filter that
absorbs completely at the wavelength of the incident light. The scattered-light reading will drop to zero if there is no fluorescence,
but will remain finite if fluorescence is present.
9.9.2 If fluorescence is present, place a narrow-bandpass interference filter transmitting at the wavelength of the incident light
in the detector optical system. Alternatively, but with less certainty of success, place in the detector optical system an absorbing
filter that absorbs at wavelengths longer than that of the incident light. (Such filters are not usually sharp-cutting, and hence are
less efficient than the use of an interference filter.)
10. Calculation
10.1 Calculate Correction Factors—Using the methods of Appendix X4, calculate the following correction factors, as required.
10.1.1 Factors that must be considered for each data point: amplification, filter, and reflection factors.
10.1.2 Factors to be applied at each concentration: absorption and depolarization factors.
10.1.3 Factors to be applied at each angle: polarization and volume factors.
10.1.4 Factor to be applied to calibration constant: refraction correction.
10.2 Calculate the Calibration Constant— Following the procedure of Appendix X2, calculate the calibration constant for
converting light-scattering intensities into Rayleigh ratios.
10.3 Calculate Rayleigh Ratios—Apply the necessary correction factors and otherwise treat the data of 9.4 and 9.5 as follows.
Typical data are shown in Appendix X3.
10.3.1 Obtain original data from recorder chart, galvanometer, or other readout device, at each concentration and angle utilized.
Correct for zero-signal level as required. Apply amplification and filter factors, if any, to obtain a self-consistent set of data.
Tabulate as in Table X3.1, Section A.
10.3.2 Apply the volume correction factor determined from Appendix X4 to the data of 10.3.1. Typical data are shown in Table
X3.1, Section B.
10.3.3 Correct the data of 10.3.2 for the reference by dividing by the reference intensity. Typical data are shown in Table X3.1,
Section C.
10.3.4 Subtract the solvent readings from 10.3.3 from the corresponding data for each concentration of polymer. Typical data
are shown in Table X3.1, Section D.
10.3.5 As required, apply absorption, depolarization, polarization, and reflection factors to the data of 10.3.4.
10.3.6 By use of the calibration constant determined in 10.2, convert the data of 10.3.5 (or 10.3.4 if no additional factors were
applied in 10.3.5) to values of the Rayleigh ratio ΔR . Typical data are shown in Table X3.1, Section E.
θ
10.4 Prepare data for graphical treatment by the Zimm-plot method.
NOTE 7—An alternative method, known as the dissymmetry method, is used when the angular dependence of the Rayleigh ratio is small (for example
D4001 − 20
ΔR <2 ΔR ). The method is less general in application than the Zimm-plot method, however, and is not recommended. Further details of the
45° 135°
dissymmetry method is found in the literature (5).
10.4.1 Divide each polymer concentration by the corresponding values of ΔR from step 10.3.5 to obtain the quantities c/ΔR .
θ θ
Typical data are given in Table X3.1, Section F.
10.4.2 Select an appropriate value of the quantity k in the expression sin (θ/2) + kc, such that for the highest value of c utilized,
kc is in the range from 0.2 to 0.4. Tabulate the quantity sin (θ/2) + kc. Typical data are shown in Table X3.1, Section G.
10.5 Plot the data of steps 10.4.1 and 10.4.2 to yield the Zimm plot. A typical Zimm plot is shown in Fig. X3.1.
10.5.1 Plot, on graph paper with scales suitably selected, corresponding values of c/ΔR and sin (θ/2) + kc.
θ
10.5.2 For each angle θ, connect points at various values of c to form (if possible) a straight line. Extrapolate this line to the
point corresponding to c = 0, that is, to the value of sin (θ/2) for that angle.
10.5.3 For each concentration c, connect points at various values of θ to form (if possible) a straight line. Extrapolate this line
to the point corresponding to θ = 0, that is, to the value of kc for that concentration.
10.5.4 Connect points at c = 0 to form (if possible) a straight line. Connect points at θ = 0 to form (if possible) another straight
line. Extrapolate these lines to the ordinate axis (where c = 0 and θ = 0), where they must meet at a single point, denoted the
intercept.
NOTE 8—It is possible to produce the plots of 10.5.2, 10.5.3, and the two lines of 10.5.4 on separate graphs, using the variables sin (θ/2) and kc
separately as required instead of together, but when the principles of the Zimm plot have been mastered, plot the combined data on a single graph as
indicated.
10.6 Assess the quality of the Zimm plot.
10.6.1 If the families of lines in 10.5.2 and 10.5.3 are all straight and reasonably parallel, and if the lines in 10.5.4 are straight
and meet at a point on the ordinate axis, the quality of the Zimm plot is satisfactory and 10.7 shall be carried out.
10.6.2 If the data of 10.5.1 cannot be made to fit straight lines, due to scatter or systematic deviations, the quality of the Zimm
plot is not satisfactory, and conclusions must be drawn from the experiment only with extreme caution. The recommended action
is to repeat the experiment, paying more careful attention to sample preparation and solution filtration, since dust or other unwanted
scattering material is the most probable cause of the difficulty. Reference to the literature (6) is recommended. In some cases, a
microgel component, which is really part of the distribution of polymer, has been known to cause severely distorted Zimm plots
(7), and few valid conclusions are drawn from the data.
10.6.3 If the data of 10.5.2 fall on straight, reasonably parallel lines, and the data of 10.5.3 fall on gently curved but still parallel
lines, the distortion results from large molecules or a broad distribution of molecular sizes, and conclusions drawn have been shown
valid if data at small enough angles are included in the analysis (8).
10.7 Calculate the molecular parameters.
10.7.1 Calculate the Debye constant, K, as follows:
2 2 2 4
K 5 2π n dn/dc /N λ
~ !
o
where:
n = the refractive index from 9.8 or tables (2,3),
dn/dc = the specific refractive increment from 9.6 or tables (2,3),
N = Avogadro’s number, and
o
λ = the wavelength of the incident light as measured in air.
NOTE 9—This treatment assumes that the refractive index n is essentially the same (within 0.01) for the solvent and all polymer solutions. If not,
separate values of K must be calculated for each polymer concentration, and applied to the corresponding data in 10.4.1 rather than at this stage.
10.7.2 Calculate the weight-average molecular weight M¯ as follows:
w
¯ 21
M 5 K~C/ΔR !
w @ θ #
c50, θ50
where:
K is the calibration constant of 10.7.1 and (c/ΔR ) is the intercept of the Zimm plot from 10.5.4.
θ c = 0, θ = 0
10.7.3 Calculate the second virial coefficient A as follows:
A 5 1/2 K@~c/ΔR ! 2 ~c/ΔR ! #/~c 2 c !
2 θ θ 2 1
c2 c1
where the quantities have the same meanings as before and c and c are two concentrations, at the high and low ends of the
1 2
range encompassed, respectively.
NOTE 10—Since A has been known to vary slightly with the angle θ, calculate it from the data extrapolated to θ = 0.
10.7.4 Calculate the radius of gyration (s ) ⁄2 as the square root of the mean square radius of gyration obtained from the
following equation:
2 2 2 2
s¯ 5 3λ /16π n 3 slope/intercept
~ ! ~ !
D4001 − 20
where:
s¯ = the mean square radius of gyration,
λ = the wavelength of incident light measured in air, and
n = the refractive index. “Slope” refers to the initial slope of the line of c = 0 points in the Zimm plot (10.5.2) and “intercept”
is the value of (c/ΔR ) determined from 10.5.4.
θ c = 0, θ = 0
NOTE 11—The derivations of the equations in 10.7, and typical data, are given in Appendix X3.
11. Report
11.1 Report the following information:
11.1.1 Identification of the sample.
11.1.2 Conditioning of the sample, if any.
11.1.3 Solvent, temperature, and instrument used.
11.1.4 Filtration technique.
11.1.5 Basic data, including wavelength, d n/dc,n, vertically polarized or unpolarized light, nature of reference, and calibration
constant.
11.1.6 Correction factors and any basic data (absorbance, depolarization) used in deriving them.
11.1.7 Results, including K,M¯ , and (optionally) A and (s¯ ) ⁄2. Unless otherwise agreed, M¯ shall be reported to three
w 2 w
significant figures, and A and (s ) ⁄2 to two significant figures.
11.1.8 If agreed, a table of data similar to Table X3.1, and the Zimm plot.
12. Precision and Bias
12.1 For most polymer-solvent systems where the solutions are reasonably free of dust or other extraneous scattering material,
the weight-average molecular weight will be determined with a standard deviation for reproducibility of about 5 % of its value.
A typical statistical analysis is given in the literature (9). Bias in the strict sense are seldom estimated because of the lack of
absolute standards in the molecular weight ranges involved. No round-robin data have been obtained.
12.2 In accordance with 12.1, “absolute standards” means polymers for which M¯ is accurately determined by methods other
w
than light-scattering. There are few such polymers capable of being used interchangeably with the synthetic polymers to which this
practice is directed. Relative standards do exist, in the form of well-characterized polystyrene samples (10,11), and it is
recommended that one of these be measured at periodic intervals to ensure that the results obtained are consistent with the body
of experience elsewhere (12,13).
13. Keywords
13.1 light scattering; polymers; weight-average molecular weight
APPENDIXES
(Nonmandatory Information)
X1. PHOTOMETERS
X1.1 Typical Photometers
X1.1.1 Typical photometers have as major components, a light source, a projection optical system, a sample cell area, an optical
receiver system, a detector system, and a recording system. They are surveyed (1) and described (14-18) in the literature.
X1.2 Light-Source System
X1.2.1 A powerful and stable source is required because of the low level of scattering from typical polymer solutions. Mercury
arc lamps have been the most commonly used source. The mercury lines at 435 nm (blue) and 546 nm (green) are the most
commonly used.Solid State lasers are the most commonly used source as they provide stable, powerful, monochromatic, fully
polarized sources.
X1.2.2 Absorbing filter combinations have often been used to isolate the mercury line used. Such filters typically have halfband
widths of 2550 nm and often transmit no more than 20 % of their peak wavelength. Interference filters, which have halfband widths
as low as 0.51 nm and typically transmit over 80 % at their peak wavelength, are recommended to replace the older filters.
D4001 − 20
X1.2.2 TEM-00 lasers, such as He-Ne (632.8 nm) or He-Cd (441.6 nm), are optionally used as sources. They provide stable,
powerful, monochromatic, fully polarized sources, but because of their different wavelengths, the large body of values ofLasers
with various wavelengths can be used. Shorter wavelengths result in stronger scattering (from both solvent and polymer). Shorter
wavelengths also reduce the lower and upper limits for accurate determination of RMS radius (and also the upper limit for molar
mass determination), and increase the possibility of absorbance and nfluorescence. and For accurate molar mass, dn/dc in the
literature cannot be directly used. Argon-ion lasers can be obtained with a tunable dye attachment and adjusted to operate at the
same wavelengths as the mercury lines.must be known at the wavelength used for the measurement. If not available in the
literature, it can be measured using a differential refractometer with a light source which matches the wavelength of the light
scattering detector.
X1.2.3 Absorbance by a polymer at the laser wavelength will result in a lower measured molar mass, but this can be corrected
by measuring the decrease in transmitted light intensity when the sample is in the sample cell. Fluorescence causes an increase
in the measured molar mass and can be avoided by selecting a longer wavelength and installing interference filters between the
cell and the detectors.
X1.2.4 Neutral-density filters are normally required to aid in keeping the intensity of scattered light in the range of the detector
system. Typically, four filters transmitting approximately 50, 25, 12, and 6 % are used.Because the scattering intensity at zero
degrees is necessary for the calculation of molar mass, intensity must be measured at a minimum of three angles in order to reliably
extrapolate the zero-degree insensity with an estimate of uncertainty and rejection of noise. For polymers with a higher RMS radii,
a larger number of angles is required due to the curvature in the relationship of intensity with scattering angle.
X1.3 Projection Optical System
X1.3.1 The function of the projection optical system is to provide, by the use of lenses and slits, a sharply defined uniform beam
passing through the cell compartment with a minimum o
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