A Polymer Viscosity Experiment with No Right Answer Lois C. Rosenthal Santa Clara University, Santa Clara, CA 95053 Polymer viscosity measurements provide an interesting experiment for physical chemistry lab. The measurements themselves are easy and seem to offer a straightforward probe of polymer molecular weight and conformation. Unfortunately, the results are often characterized by high uncertainties and a disappointing lack of reproducibility. Whatever the causes-perhaps the simplifying assumptions built into the data analvsis schemes are too unrealistic. or perhaps the measurements are extraordinarily sensitive to s techniaue-students eenerallv find i t frussmall ~ o i n t of trating to end up with highly impre2se resuits from such easily Derformed measurements. ~ f v e nsuch difficulties, it is constructive to take another point of view and have students make a study of the data analysis schemes themselves. Then the data becomes primary, and the theories are tested against it. This is closer to real science than the usual student ex~eriment.and it reminds students that science is fundamentally em&rical. Furthermore, since the theories are to be evaluated in lieht of the data,the right answer does not even exist until tLe experiments are done. The students must draw conclusions obiectively, without knowing what the answer "is supposed to be". The answers are correct only to the degree that they are consistent with the data a t h&d. T h e experiment entails a class effort to prepare a calibration curve relating molecular weight and intrinsic viscosity, and use of the calibration curve to determine an unknown molecukr weieht. I n both the determination of intrinsic viscosity and the construction of the molecular weight calibration curve, commonly used equations are tested and evaluated. Some published guidelines and applications of the equations are also judced against the data. The unknown extra mot&aGon for use of good scientific judgement in these evaluations. Descrlptlon of Experiment A convenient s.ystem for this experiment is poly(ethyleneglycol)' (PEG)dissolved in waterat 25T.Other systemsareappropriatp,as long as the polymer forms a flexihle wil in solution. Five or six molecular weight standards are distributed among the members of the class; it is advantageous to assign two or mare independent measurements for each sample, so that erroneous results are easier to spot and do not unduly affect the calibration curve. Two threehour lab periods are sufficient for each student to do two or three standards and one unknown. While narrow molecular weight distribution standardszare usually recommended for calihration curves, all of the major features of the analysis are observable with inexpensive, polydisperse samples3. Themolecular weights should cover the range from 1000 to 20,000gJ mol or higher, with three standards above 10,OW if possible. A simpler experiment results if only molecular weights above 10,000 are used. as the lower molecular weiehts give Less reoroducihle results. ~imuleOstwald viscometers (ASTM size 100)are admuate. . - . ~ ~~ hut l'bhelbhde v i r r u m e t e r * size ~ ~ 11 ~ are ~ ~much appreciated by thestudentsand givc htterresults. A dilution v~scomaer'prwides the ultimate in convenienceand will probably improve the dataeven further. The basic experiment wnsista of viscometer flow time measurements on a series of dilutions of stock solutions for each oolvmer . . standard. For the system described here,studrnts prepare their own stock solutions at runcenlration 0.015 gtml.. Three successive I : ] dilutionn plus the stock provrde a series of four concentrations. The ~~
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Journal of Chemical Education
~
,
~~
flow time of pure water is also needed for viscometer calibration. Viacometer sizes were chosen so that flow times were about 100 s, to minimize kinetic enerm effects while maintaining some convenience. All measureme& must be done in a const&t-temperature water bath. Density measurementswere found to be unnecessary; all of the solution densitiee are dose to that of water.. and the variation of density with concentration affects the results by only a few perrent. Standard prucedurer are followed for the measurements 1131. ~
~
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Theories The basic equations for this study are outlined briefly below. Derivations and further details are given in standard references (14). T h e various terms for polymer viscosity measurements used here are as follows: the viscosity of the solution the viscosity of the solvent nap, the specifieviscosity, = (q - qo)/qo q,, the relative viscosity, = q/qo [q], the intrinsic viscosity e, the concentration of the polymer solution, in g/mL q,
70,
The Poisseuille equation was used for computing viscosity from capillary flow time, t: q = Bpt
where B is the viscometer c o n s w t , measured by calibration with water, and p is the solution density, assumed to he equal to t h e ~ o l v e ndensity. t Neither the correction term for kinetic energy nor effects of non-Newtonian flow were considered. The two equations used for finding intrinsic viscosities from viscosity vs. concentration data were the Huggins equation and the Kraemereauation ( I ) . Both are based on a linear plot, with the intrinsk viscosity appearing as the y intercept. q,Jc = [nl + kH[qI2c
(lk)In?,
3
+
[q) k,[nl2c
Huggins
(1)
Kraemer5
(2)
The k values in these equations are said to be constant for a particular polymer-solvent system over a wide range of molecular weights (3).The two k's are mathematically related5
I n theory, the k values depend on conformation (1,5):k~
-
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= 0.73-0.77 for a rigid rod. 0.60 for a coil. and 2.0-2.26 for a ~~
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sphere. Empirically, however, for common polymer systems, ku ..is described (6)as eenerallv hetween 0.3 and 0.5. Additional eiuations for finding [ q ] have been described (3,6)and could be tested in this type of an experiment.
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' Synonym: poly(oxyethylene).
Ressure Chemical Co.. 3419 Smallman St.. Pittsburgh. PA 15201. (412)682-5882. Aldrich Chemical Co. 'Thomas Scientific19723159. 'Some references use a negative sign before the second tern on the right-hand side of eg 2. In mat case, eq 3 would read: k kK = \,.
+
Two calibration curves relating [TI] to molecular weight were tested. The first is the Mark-Houwink-Sakurada (MHS) equation,
Id = K.W
Hvggino and ~raemerConstants
PEG In Water. 25 OC
(4)
usually plotted in logarithmic form. This equation is recommended for molecular weights above 10,000 glmol (2, 4). Both K and a are specific to the polymer-solven~temperature system. Theoretically, the exponent a depends on the polymer conformation (2,6): for glohular particles, a = 0; for random coils, a = 0.5-1.0 depending on solvent, with a = 0.5 at theta conditions; for rigid rods, a = 2. The constant K contains molecular parameters such as effective segment length, radius of gyration, and degree of interaction with solvent (1). A second equation, linear in form, has been suggested for polymers with molecular weight "up to about 6000-8000" (2):
The two constants are again specificto the polymer-solvent system.
SIudent Task The students' first task is to test the Kraemer and Huggins equations with their data, looking for linearity, agreement in [ q ] , and uncertainty in [?].The k values are also studied with resoect to orecision, constancy with molecular weight, and usefhness for conformation determination. The validity of eq 3 is tested. Each student derives a "hest" [7] for each sample, consistent with findings on the merits of eqs 1and 2. In the lab report, the approach is to he justified and backed up with data. Findings on the k values are also discussed. The class results on [q] vs. molecular weight are then compiled, and the two proposed calibration curves are tested. Qualities of linearity and range of validity are considered, as is the numerical result for a. Finally, each student determinesa molecular weieht result for his or her unknown, based on a study of the relative merits of the two calibration curves. Aeain. the aooroach is to be justified in the discussion section of the l i b report. Most students find the lab report for this experiment a challenge, and a series of questions to guide the students' thinking and writing may be helpfd6 Sample Results
Some student results are shown here for illustration. These results are from an experiment on inexpensive, polydisperse samples done by one class of students. All of the student data are shown or otherwise accounted for below. Huggins-Kraemer Analysis The Huggins-Kraemer plots closely resembled textbook illustrations (6). In all cases the two equations gave consistent results for [?I, often with agreement to the first decimal place. The actual uncertainty in [ q ] ,taken as one standard deviation in they intercept, was often 10X higher than such agreement, and reproducibility was poor (see Fig. 3). With the high molecular weight samples (>10,000) the described (3, 4) trends of a positive slope for the Huggins plot and a near-zero slope for the Kraemer plot were ohserved. The lower molecular weight samples often did not follow this trend. The collected k values are shown in Figure 1. (The Kraemer constant is reported such that values below zero correspond to a negative slope.) Uncertainties are not shown: thev were aenerallv vew hieh, sometimes reaching 100%.one &dent;esult was omitted as grossly in error, and five others, all on molecular weights 8000 and below, were beyond the scale of Figure I , with k H from 4.5 to 7.1. Two of
Figure 1. Huggins (circles) and Kraemer (squares) constants vs. molecular weight tw PEG in watw at 25 ' C . The larger symbols denote uncertainties greater than 100%. Open symbols: polydlsperse samples. Solid symbols: nearly manodisperse standsrds run by an experienced student. Six data sets are onscale and not shown (see ted).
these five cases came from measurements on monodisperse standards by an experienced student. Some generalizations may be made about the k values. Above molecular weights of 10,000, the Huggins constants tend to cluster between 0.3 and 0.8, with the Kraemer constants close to zero. The lower molecular weight measurements fluctuate much more widely: below 8000 glmol, k~ values vary from +1to -1, lacking even sign consistency. In all data sets, k~ is less than k". The wide variation found in the data seems to be intrinsic to the measurements, rather than aresult of sample polydispersity or student inexpertise. Just as much variation was found on runs repeated by an experienced student on nearly monodisperse standards (Fig. 1, solid symbols plus two points off scale). It appears that with ordinary equipment, techniaue. and samole handline. k values are measurable only t d within an eider of magLitude. Published k values also suffer from irreproducibility and lack of precision (5); even so, some texts (2,4) quote k~ as "about 0.35". It is interesting to note that the quantity (k" - k ~shows ) less fluctuation ihan its factors over the entire molecular weight range. The data validate eq 3 within about 25%. The poor reproducihility and high uncertainty of the k values make it impossible to use them to distinguish conformations. And. statements such as "ku..is constant over a wide range of molecular weights" are not reliable or even easily tested. Aeain...oublished data corroborate the class findines: k has been reported to increase, decrease, and remain constant with molecular weieht (5). These results provides concrete example of the limitations of both theory and experiment, and it is important that students he encouraged to&ticulate such limititions as valid conclusions of the experiment. Molecular Weight Callbration Plots The two molecular weight calibration plots, shown in Figures 2 and 3, are more straightforward to interpret. The MHS plot appears linear (lower left point omitted) over the whole molecular weight range. Apparently, the recommended guideline underestimates its usefulness. The "linear" plot, on the other hand, has a break in it and appears to be
Instructions to the student gladly s e w upon request. Volume 67 Number 1 January 1990
79
10,
5,
15,
~olecularweight (lo3 @moll
Flgure2 Mark-Howmk-Sakurads molecuhr we~ghtcal~bratloncwvefor PEG m water at25 'C. wlth least-squaresltne From me slope, a = 0 9 i 0 2
meaningful for molecular weights of 8000or below, a s recommended. Student Performance Students performed hetter on finding their unknown molecular weights than on evaluating the theories. Ahout half of t h e s t u i e n t s came within %of the correct molecular weight, much hetter than a formal uncertainty analysis woild imply. Both high and low molecular weights were successful. T h e quality of laboratory technique was clearly t h e determining factor here. On the other hand, almost all t h e students h A trouble with t h e more qualitative aspects of t h e exneriment. D e s ~ i t et h e leading questions supplied t o guide h e i r discussions, many s t u d i t s h a d prohlems making generalizations about the data. Few saw a break in t h e "linear" calihration curve; most just drew a least-squares line through all the data. As is common with college students (7),there was much difficulty with t h e actual writing of a discussion calling for generalizations hacked u p with evi-
Figure 3. [ q ] vs. molecular welght for PEG In water at 25 OC dence. And, there was extreme reluctance t o state or even recognize that the high uncertainty i n k valuescould makeit impossible to use them for conformation information. T h e timidity with which students approached this experiment underscores the need to brine t o the curriculum more work requiring independent t h o u i h t so t h a t students can make t h e transition from "givine the teacher the right answer" t o making ohjective observations about realit; Literature Cited 1. McCaffwy, E. M. Lobrotary PrDporotion lor Mmromolarulor Chamistry:McGrsvHill: New York. 1970: pp 25fT 2. Salrberg, H.W.; Morrov,J.I.;Cahen,S.R.;Grccn,M. E. Phyaieolchemisfry~borofory: MacMillan: New York, 1978:pp 113-123. 3. Johnson,J. F.:Marfin,J. R.;Porter. R.S. In Techniqu~~o1Chemialry:A. Weisbberger, Ed.: Wile)-New York, 1977; Val. I, Part VI. pp 64-121. 4. Allcock. H. R.: Larnpe, F. W. ConfampornryPo1ymerChemistry;~rentice- all: ~ n g l e wood Cliff% NJ, 1981. 5. Sutferlin, N. In The Polymer Handbook, 2nd ed.: Brandrup, J.; Immenut, E. H., Eds.; Wiloy: New York, 1975:pp IV-135 If,
6. Biltingham. N.C. Molor Moss Measurements in Poiymsr Science: Wiles New York, 1977:oo 172-185. 7. ~ o s e n t h a iL. , C. J. Chem. Edue.
1987.64,996.
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11th BCCE-Program Description Plan now toattend the 11th Biennial Conferenceon Chemical Education r u be heldat GeorgiaTech inAtlanta.August 5 9 , 1990. In addition to poster and genrral-paper sessions. the program will include: Plenary Lecturers:Mark Wrighton, the Brasted Lecturer, Marjorie Gardner, Walter McCrane, and Bassam Shakhaahiri. Symposia:Education for the New Technology;Forensic Science; Paper Chemistry; Chemical Demonstrations;Modern IR
Spectroscopy; the Environment; Computers: Interfacing, Hypercard and Hypertext, State-of-the-Art Hardware, Graphin, Artificial Intelligence,Spreadsheets, Software,and Videodisc Applications; UndergraduateFaeulty Enhaneement Programs; Improving the Public Image of Chemistry; Anti-AIDS Drugs; Plastin; Teaching Physical Chemistry; Low Cast Instructional Materials; General Chemistry Laboratories; Nuclear Chemistry: Nuclear Medicine, Nuclear Power, and Radioactivity in Industry and Research; Microscale; Basic and Advanced Inorganic Chemistry; Polymers; Chemical Microscopy; World of Chemistry; General Chemistry Textbook Authors and Editors; Reactivity Network; AP Chemistry; Organic Chemistry;Laboratory Assessment Builds Success; Geochemistry; ChemSource; Graduate Study in Chemical Education; CEPUP; Science Education Research; Chemistry in the Toy Store; Caamie Chemistry; ChemCom; ChemicalResourcesin Museums; Travelling Chemistry Demonstration Presentation; History of the Chemistry Set; PreHigh-SchoolChemistry; Reform in Science Edueation; Chemical Competitions;and the FIPSE Follow-Up Symposium. Workshops: Minoscale; Computer Data Acquisition and Analysis; Project SERAPHIM: Game-port Interfacing, Appleworks, One-Computer Classroom, and KC? Discoverer and the Periodic Table Videodisc; Laboratory Safety; Fires and Explosions; ChemCom; Institute of Paper Science and Technology Site Visits; Applied Chemistry for Inner-City High School Students; Polymer Chemistry Demonstrations and Experiments; Environmental Chemistry; Modem IR Spectroscopy;Fun with Polymers; Demonstrations, Novel and Easy; World of Chemistry; Superconductors, Semiconductors, and Metals; Chemical Microscopy; Chemistry in the Toy Store; Chemistry Can Be Fun (ICE Workshop); Travelling Chemistry Presentation; and the FIPSE Follow-Up Symposium. For more information about the program, registration, and local arrangements, rontart: Dr. Toby F. Block. General Chair, School of Chemistry. Georgia Institute of Technology. Atlanta, GA 303324400.
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