Polymer Analysis. Chapter 2. Error Analysis/Statistical Descriptions of Data.

Transcription

1 Polymer Analyss Chapter. Error Analyss/Statstcal Descrptons of Data. All Polymer Propertes are Dsperse: Polymerc materals are subject to dsperson n all analytc propertes. For example, the meltng pont n a low molecular weght organc or norganc s a fxed value constant whch mght dsplay some varablty over or degrees. The meltng pont for hgh-densty polyethylene (HDPE), for example can vary from about 0 C to about 60 C dependng on processng and generally dsplays a broad dsperson over about 0 to 0 C. Spectroscopc analyss of polymers usng technques such as nfra red adsorpton (R) and nuclear magnetc resonance (MR) rely mostly on local chemcal groups, so can dsplay farly sharp absorpton bands. n these spectroscopc technques, dsperson s shown by the exstence of peak splttngs or the presence of a number of chemcal speces n small amounts. Addtonally, many absorpton bands n polymers are dffcult to descrbe analytcally and pertan to varous acoustc modes assocated wth the conformaton of long chan structures. These are typcally broad bands due to wde dsperson n these long chan conformatons. X-ray dffracton from polymer crystalltes generally dsplays broad dffracton peaks assocated wth small crystals and a hgh degree of dsorder wthn crystalltes. Polymers also dsplay dsperson n structural and chemcal orentaton whch should be vewed wth statstcs. Dspersons n mechancal propertes are always seen n polymers. Statstcs are also used to descrbe the dsperson of chan sze, molecular weght, and topologcal arrangement of tactcty. All descrptons of polymer chan sze are based on statstcs. t s crtcal to realzed at the onset of any analyss of a polymerc system that every physcal property wll be descrbed by a dstrbuton. Error Analyss n Analytc Methods: n the physcal scences each analytc measurement must be assocated wth an assessment of the confdence whch should be assocated wth the analytc descrpton. A value for some property of a materal s of no use unless some estmate of the expected error and dstrbuton s provded. For example, for a commercal sample of HDPE a techncan mght report the meltng pont of a blown flm as 35 C. You mght be nvolved n usng ths materal n a packagng applcaton where the materal wll be subjected to shppng at a maxmum temperature of 95 C. From the reported meltng pont and operatng temperature would you feel confdent that ths materal would meet the specs of the packagng company? The correct answer s that the measured value s of no use wthout a descrpton of the statstcal dstrbuton n the samples as well as a statstcal descrpton of the dstrbuton of meltng ponts,.e. onset, peak and maxmum meltng pont. The techncan should be requred to measure at least 0 samples (preferably more) and calculate a standard devaton for the reported value. Addtonally, smlar determnatons of the onset of meltng and the maxmum meltng pont would be needed to determne f ths materal meets the specfcatons of the packager. There are a number of useful texts whch descrbe the correct handlng of expermental data. The most commonly cted reference s "Data Reducton and Error Analyss for the Physcal Scences" by P. R. Bevngton, 969 McGraw Hll. Ths secton of the course wll summarze some of the major ponts of Bevngton wth an emphass on applcatons n polymerc systems. Types of Error: Statstcans categorze three types of error. Ths framework may be useful n consderng a specfc experment such as determnaton of the d-spacng from an XRD peak. llegtmate Error: These errors nvolve an operator error, e. g. you have placed the wrong sample n the dffractometer and are determnng the d-spacng for the wrong sample. There s no

2 statstcal descrpton for these errors unless you want to consder socology. Your man protecton aganst llegtmate errors s to always consder, when faced wth extremely unexpected results, that the results nvolve a human error. Always look carefully at completely unexpected results. Systemc Error: There errors are also not subject to a statstcal analyss. They result from faulty calbraton of the nstrument or other problems whch result n a constant shft of the data. n XRD a systematc error mght nvolve confuson of θ wth θ leadng to roughly a doublng of the d- spacngs. Systemc errors can sometmes be corrected after the fact f one s careful. t s mportant to keep a good record of the analyss that was performed partly for the purpose of correcton of systemc errors of ths type. Random Error: Random errors are the man area whch statstcs can deal wth (.e. standard devaton, mean). Random errors have two sources. ) Random varablty n the samples themselves. ) Lmted precson of the analytc equpment. These two sources can be dstngushed f more precse equpment s avalable or a standard sample s avalable. n descrbng the error nvolved n an analytc measurement all types of error should be consdered. Accuracy of a measurement pertans to how close a measurement s to the actual value. Precson refers to the reproducblty of the measurement whether or not t s close to the actual value. f systematc error s present a measurement mght be extremely precse but completely naccurate! Also, f the equpment s well calbrated a measurement could be extremely accurate but not partcularly precse. Mostly the mportance of these terms s n communcaton of your confdence n a partcular value and what the possble sources of error are. Statstcal Analyss of Measurements: Generally a gven measurement wll be conducted several tmes n order to determne the statstcal dstrbuton for the measurement. The values most commonly determned are the mean, µ, and the standard devaton, : µ = x = ( x µ ) s the number of measurements made, the value from each measurement s x for measurement. The square of the standard devaton s called the varance. Dstrbuton Functons (see P. C. Hemenz, Polymer Chemstry, 984, Marcel Dekker, pp. 34): f a measurement s not sngle valued then t s common to ft the number dstrbuton of values to a dstrbuton functon. The smplest dstrbuton functons wll nvolve two parameters, the mean, µ and the standard devaton,. More complcated contnuous dstrbuton functons wll nvolve hgher moments of the dstrbuton. The k'th moment of a dstrbuton s gven by: k' th moment = f x x s k

3 where f s the fracton of all measurements whch have the value x,.e. /, x s s bass for the moment and k s the order of the moment. For example, the mean, µ, s the frst moment (k = ) about the orgn (x s = 0). The varance,, s the second moment (k = ) about the mean (x s = µ). The molecular weght dstrbutons commonly used n polymer scence can be descrbed n terms of the more broadly used statstcal descrpton of moments. The number average molecular weght, M n s the same as the frst moment about the orgn or the mean. The weght average molecular weght (mass average), M w, s gven by: M w M = M = fm fm or the rato of the second to the frst moment about the orgn. The polydspersty ndex, M w /M n, s the rato of the second moment to the square of the frst moment about the orgn. Gven M w and M n the standard devaton of a dstrbuton of molecular weghts can be determned: = M M n M w n Several specfc dstrbuton functons are mentoned below. A dstrbuton s usually consdered unmodal f one of these contnuous dstrbuton functons descrbes the dstrbuton of measured values. f the data s best descrbed by several of these functons t s termed bmodal, trmodal etc. A bmodal dstrbuton n lamellar thcknesses n polyethylene mght be generated f crystallzaton occurred n two dstnct steps such as prmary crystallzaton and secondary eptaxal spherultc decoraton for nstance. Ths dstrbuton mght be descrbed by two Gaussan functons, so a total of 4 parameters, means and standard devatons. Bnomal Dstrbuton: Consder a tactc polymer such as polypropylene. As dscussed n class, on passng along the man chan of the polymer there s a choce n handedness for the substtutent methyl group whch could be consdered smlar to a con toss,.e. f the substtutent occurs wth the same handedness as the prevous mer unt ths would be smlar to a con tossed heads (R for tactcty) and f t occurs wth the opposte handedness (S for tactcty) ths would be smlar to a con tossed tals. f the tactcty of the polymer s determned at random (atactc) then there s equal probablty for R and S handedness. The bnomal dstrbuton descrbes such a stuaton where the "probablty of success" s gven by p, (consder R a success, here p = 0.5 for atactc polymers). The probablty of observng "n" R handedness mer unts n a chan of mer unts when the probablty of seeng an R handness mer unt s p s gven by: P Bnomal ( np,, ) =! n n p!! p n n 3

4 the mean, µ, and standard devaton,, for the bnomal dstrbuton are gven by: µ = p = p p P Bnomal could be used to plot a dstrbuton functon of tactctes for example. ote that there s a fnte probablty for a completely R polymer (0! =). Posson Dstrbuton: When s large and the mean, µ, s constant wth sample sze, the bnomal dstrbuton smplfes to the Posson Dstrbuton: P Posson ( n, µ ) = n µ µ n! e where the standard devaton s gven by: = µ The Posson Dstrbuton s used to descrbe small samples of large populatons, so s approprate n analytc technques whch nvolve countng of events, such as XRD and lght counters, and mass spectrometers whch measure events. The standard devaton n such a countng measurement (countng of events) s the square root of the number of counts. For example, an ntegrated dffracton peak mght generate 0,000 counts on a proportonal detector. The error n ths value s ± 00. f an analytc nstrument does not report "counts" or "events" then the Posson dstrbuton value for the standard devaton can not be used drectly. Gaussan Dstrbuton: For very large samples,, wth a fnte probablty of success, p, a smooth dstrbuton s usually observed. Such a dstrbuton can be descrbed by a Gaussan functon: PGaussan n, µ, exp π = n µ The Gaussan dstrbuton s the bass of the "bell-shaped curve". t s also used to descrbe random walks n dffuson as well as the path of a polymer col under theta-condtons. ntegraton of the Gaussan dstrbuton as a weghtng factor for r, the square of the chans end-to-end dstance yelds the mean square end-to-end dstance for a "Gaussan" chan, l, where l s the step sze and s the number of randomly arranged steps n the non-nteractng chan. For the Gaussan chan µ = 0 and = nl. Other Dstrbutons: There are many other dstrbuton functons commonly used n polymer scence such as the Lorentzan Dstrbuton (see hand out) and the Maxwellan Dstrbuton (see Polymer Materals Scence by J. Schultz for nstance). The defnton of these dstrbutons wll rely on an understandng of the moments of a contnuous dstrbuton descrbed above. 4

5 Covarance: n many analytc experments several parameters are determned from a sngle measurement through the use of a ft to expermental data. For nstance, a lght scatterng curve can be used to determne the molecular weght (frst moment about the orgn) and second vral coeffcent, A, through the Zmm plot (fgure.8 n our text). Often the two parameters whch are measured are not completely ndependent n terms of the ft to the data. Under such condtons t s necessary to determne the covarance of the two parameters. The covarance reflects the degree to whch two parameters effect each other. f more than two parameters are unknown, then a covarance matrx can be constructed to determne the extent two whch any two parameters are assocated. For two parameters the covarance s defned by: ( ) uv = u u v v where <> ndcates a mean. Covarance s ncluded for completeness here. Determnaton of the covarance s generally rare n the lterature. Propagaton of Errors: n many analytc technques a value s measured, an error s determned, and ths value s used to calculate a parameter of nterest. For example, n the determnaton of the modulus of a sample the extenson of the sample s measured wth some error and s normalzed aganst the orgnal length to determne the engneerng stran. The force appled to the sample s measured wth an expermental error and s normalzed by the cross sectonal area to determne the stress. These two parameters are plotted (stress versus stran) and a curve ft s used to determne the modulus at low stran. n order to determne the standard devaton n the modulus the expermentally determned errors n length, force and area must be propagated to the stress and stran and the error n these parameters must be propagated to a lnear ft of the data ponts. Propagaton of errors s a rudmentary tool necessary to perform polymer analyss. Consder a measurement of the absorpton, A, and absorpton coeffcent, α, usng a sngle wavelength of lght whch passes through a sample and a photomultpler tube whch reports counts. The relevant equaton s the Beer-Lambert Law for lnear absorpton (pp. 40 and 54 n Cambell): A = ln cl = ln T = α 0 where c s the concentraton of absorbng speces and l s the sample thckness. T s called the transmsson. For a sold sample c =. Two measurements are necessary, 0 wth no sample and wth a sample of thckness l. f the two measurements are 0 = 00,000 counts, and = 0,000 counts, the respectve standard devaton s gven by a Posson Dstrbuton functon as the square root of the number of events, 0 = 00,000 ± 30 counts; = 0,000 ± 40 counts. To propagate the error n 0 and to /T the general error propagaton rule n dfferental form can be used: 5

6 x u v uv x x = + + u v x x u v Here t s safe to assume that and 0 are uncorrelated (unrelated) so the covarance, uv = 0. The two standard devatons are gven above. For x = /T, δx/δ 0 = /, and δx/δ = - 0 /. The above equaton yelds: or / T 0 = / T 0 = + ( / T) 0 ths follows the general rule for ratos gven by Bevngton (see handout). The propagated value and error for /T s, /T = 5 ± The absorpton, A, s the natural log of /T and the propagated error s gven by: A = / T A / T = ( / T) ( / T ) So, A =.6 ± 0.0. The sample thckness, l, s 0. ± 0.0 cm as calculated from a seres of 0 measurements usng the mean and number of samples equaton gven above. The absorpton coeffcent, α, s calculated from α = A/cl, where c s for a sold sample. Replacng the varables n the equaton above for the error n /T, α = 6 ± 3. otce the reducton n the number of sgnfcant fgures assocated wth the large error. t should also be noted that often the largest source of error s related to factors whch are of mnor sgnfcance to the measurement, e.g. here error n the sample thckness domnates the error n the absorpton coeffcent. Purpose of Error Analyss: Although t s assumed that you can determne the error value for any quantty you measure n scence, the most mportant part of error analyss nvolves nterpretaton of the sgnfcance of the analyss n vew of the error and n vew of logc and the reasonableness of the results. Error analyss s a crtcal factor n both demonstratng the scentfc reasonableness of a result, as well as formng a bass of a scentfc crtque of work performed for you by a techncan or outsde lab. Error analyss s always at the heart of a scentfc argument. A measured value has no meanng wthout an analyss of the assocated error. Such an analyss can be quanttatve, as above, or qualtatve, based on your scentfc judgment. The value n ether case s based on the reasonableness and logc of the approach. n the remander of ths course specal emphass wll be gven to qualtatve and quanttatve assessments of the error n the analytc technques covered. 6

7 Least-Squares Fts: n order to determne analytc parameters t s often necessary to perform a curve ft to a raw data set. For example the determnaton of modulus from a stress-stran measurement requres a lnear ft of the type = Eε where s s the stress, = F/A, and e s the stran, ε = l/l, and E s the Youngs modulus. The error n the stress measurement could be propagated from the estmated error n the Force and area measurements. These errors can be propagated to the modulus n a lnear ft through a least squares mnmzaton of χ. χ s a measure of the dfference between the actual data ponts and the projected ponts assocated wth the ft parameter. t bears resemblance to the varance, χ = y f x ( ) The propagated uncertanty n the coeffcents for the least squares ft can be obtaned n a computer program by calculaton of the second dervatve of χ wth respect to varaton n the parameter: aj = χ a j A full dscusson of least-squares fttng routnes and propagaton of error s gven n Bevngton. For lnear functons a relatvely smple algorthm for propagaton of error s gven n the handout. 7