I~
`POLYMER
`CHEMISTRY
`
`G.S. MISIM
`
`Former Professor & Head
`Department of Chemistry, Universities of Jabal pur and Janunu
`and
`Ex-Director, Indian Lac Research Institute
`Namkum, Ranchi
`
`JOHN WILEY & SONS
`
`NEW YORK CHICHESTER BRISBANE TORONTO SINGAPORE
`
`RBP_TEVA05017958
`
`TEVA EXHIBIT 1038
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`
`

`
`First Published in 1993 by
`WILEY EASTERN LIMITED
`4835/24 Ansari Road Daryaganj
`New Delhi 110002, India
`
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`
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`
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`
`Copyright© 1993 WILEY EASTERN LIMITED
`New Delhi, India
`
`Library of Congress Cataloging-in-Publication Data
`
`ISBN 0-470-21720-0 John Wiley & Sons Inc.
`ISBN 81-224-0471-5 Wiley Eastern Limited
`
`Typeset by Printek India, C4F/271B, Janakpuri, New Dehli-110058
`Printed in India at Taj Press, Mayapuri, New Delhi 110 064
`
`RBP_TEVA05017959
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`TEVA EXHIBIT 1038
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`
`

`
`MOLECULAR WEIGHT DETERMINATION
`
`CHAPTER 6
`
`6.1 INTRODUCTION
`
`One of the most important measurements to be carried out on a polymer is the
`determination of its molecular weight. Most of the useful mechanical properties of
`polymers depend on their molecular weig_hts. Nearly all the properties of a polymer
`change with its degree of polymerization, liP. The control of the molecular weight of
`a polymer is, therefore, essential for the preparation of products with different end
`uses.
`
`6.2 MOLECULAR WEIGHT AVERAGES
`In contrast to the low molecular weight-compounds, a polymer is .usually a complex
`mixture of .molecules of different molecular weights. The polymers are thus
`polydisperse and heterogeneous in composition. Therefore, the molecular wei~ht of a
`polymer is actually an average of the molecular weights .of constituent molecules.
`Different averages are obtained depending on the method' of ·measurement of the
`molecular weight.
`The number average molecular weight Mn, is obtained by the measurement of
`the colligative properties of a polymer by QSmometry or end group analysis and is
`defined as,
`
`M =LN;M1
`"
`l:N;
`
`(6.1)
`
`\
`
`where N1 is the number of molecules of molecular weight M1.
`The weight average (M w) molecular weight is obtained from light scattering
`measurements and is defined as,
`
`-
`l:N;Ml
`Mw =LN;M;
`
`(6.2)
`
`To explain these molecular weight averages, one can take an example. Suppose
`there are 50 polymer molecules of molecular weight 102, 200 polymer molecules of
`molecular weight 1()3, and 100 molecules of molecular weight 104, then,
`M =50x102 +200xl03 +100x104
`"
`50 +200+ 100
`= 3443 "" 3440 app.·ox.
`M _ 50x 104 -t 200x 106 + lOOx 1011
`w -50 X 16'+ 200 X 103 + 100 X 104
`
`= 8465 "" 8470 approx.
`
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`

`
`Molecular Weight Determination
`
`99
`
`It is evident from this example that the number average and weight aver~e ~olecular
`weights are not the same except in monodisperse systems where M~ = M..;, M~ is very
`sensitive to the molecules of lo:wer molecular weight wnereas Kt is sensitive to the
`presence of molecules of higher molecular weight.
`The viscosity average moleeuiar weight, -M .. is obtained by viscosity measure-
`ments and is defined as,
`·
`
`_ [~NM!t+a)]tta
`M = .L,
`I
`, .
`~N;M;
`
`v
`
`.
`
`(6.3)
`
`where a is the exponent in the Mark-Howink-Sakurada equation: [11] = KM<X (see
`Section 6.5 and Eq. 6.12).
`a can vary between 0.5 and 1.0. The following inequality occurs:
`
`The z-average molecular weight is defined as,
`
`-
`~NM-3
`M =--'-'
`' ~N;Ml
`
`(6.3a)
`
`Q.6.1.
`
`(i) If two polymers of molecular weights 10,000 and 100,000 are mixed
`together in equal parts by weight, deteimine the number average and weight
`average molecular weights. (ii) If the above p~mers are mixed so that
`equal number of molecules are added, determine M" and !J' ...
`
`Q.6.2.
`
`If you mix 1000 g of a polymer of molecular weight 1000 g/mole and·lOOO g
`o{ a polymer of molecular weight 1Q6 g!mole, what is the ratio Mwlfi:{.?
`
`6.3. MOLECULAR WEIGHT DISTRIBUTION (ADDITION POLYMERS)
`
`When sufficiently sharp fractions have been isolated from a polymer and the
`molecular weights of the fractions have been determined, the distribution of the
`fractions of the various molecular weights may be represented in a number of ways.
`This is illustrated by taking a sample of poly(styrene). R. Signer and H. Gross took a
`100 g sample of poly(styrene) and separated the various fractions with the aid of
`ultra-centrifuge and determined their molecular weights. The original molecular
`weight of the sample of poly(styrene) was 80,000. The molecular weights of different
`fractions are listed in Table 6.1.
`
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`

`
`100
`
`Polymer Chemistry
`
`Table 6.1. Molecular Weight Distribution Data on Poly(styrene)
`
`Fraction
`· number
`
`Wt.of
`po:~r
`
`Percent of
`polymer used
`on total
`
`Cumull<tive
`percent of
`polymer
`
`1
`2
`3
`4
`5
`6
`7
`8
`9
`10
`11
`12
`13
`
`0.2
`1.7
`3.6
`8.4
`20.0
`23.8
`20.2
`10.4
`6.0
`3.3
`1.7
`0.5
`0.2
`
`0.2
`1.7
`3.6
`8.4
`20.0
`23.8
`20.2
`10.4
`6.0
`3.3
`1.7
`0.5
`0.2
`
`0.2
`1.9
`5.5
`13.9
`33.9
`57.7
`77.9
`88.3
`94.3
`97.f.
`99.3
`99.8
`100.0
`
`Mol
`wt. x J(f-3
`
`25- 35
`35- 45
`45- 55
`55- 65
`65- 75
`75- 85
`85- 95
`95-105
`105-115
`115-125
`125- 135
`135- 145
`145-155
`
`MOLECULAR WEIGHT
`
`Fig. 6.1. Population curve of a poly(styrene) sample.
`
`17""
`1\
`\
`
`I
`
`~ 2.5
`N
`0
`E 2. o
`1-JsTEP
`CURY~_,. I
`~
`1/
`\!)
`i:ij 1 . 5
`~
`...J
`._
`g 1. 0
`z
`L&J a::
`~ o. s
`....--
`0
`0
`0 ..n
`
`1&.
`a
`
`hL
`b.l.
`f;Z
`
`C>
`0
`0
`11'1.
`-t
`
`0
`0
`0
`11'1'
`\1)
`
`N
`
`[.!: SMOOTH Cl R\ I"'
`
`~
`~
`~1::::.
`0
`0
`0
`0
`0
`0
`11'1'
`11'1
`~
`~
`MOLECULAR WEIGHT
`
`0
`0
`0
`Ul'
`<D
`
`0
`0
`0
`u:l
`0
`
`0
`0
`0
`11'1
`
`"'
`
`Fig. 6.2. Differential weight distribution curve of a poly( styrene) sample.
`
`The data in Table 6.1 is represented in Fig. 6.1 as a population curve, in Fig. 6.2 as a
`differential weight distribution curve, in Fig. 6.3 as differential number distribution
`
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`

`
`Molecular Weight Determination
`
`101
`
`curve and in Fig. 6.4 as integral weight distribution curvt:. The data of Table 6.1 has
`been plotted as population curve in Fig. 6.1 by plotting the percentage weights of the
`fractions against their molecular weights. This is not of much practical value. The
`integral distribution curve (Fig. 6.4) is obtained by plotting the cumulative weight
`percentages Of the polymer fractions against their molecular weights. Such a plot
`results in a step curve through which a continuous curve is obtained. The differential
`weight distribution curve shown in Fig. 6.2 is obtained by graphic differentiation of
`the continuous integral weight distribution curve (Fig. 6.4) by dividing the change in
`the weight fraction by AMW, the change in molecular weight.
`
`J
`
`a::
`Lll
`m
`:::E
`=>2 z
`...I
`<l
`.....
`~1
`a::
`Lll
`1&.
`1&.
`00 1/
`
`0
`0
`0
`U'l~
`N
`
`C)
`0
`0
`U'l~
`
`...,
`
`0
`0
`0
`0
`0
`0
`U'l~ U'l
`10
`
`..,
`
`0
`0
`0
`U'l~
`0
`
`0
`0
`0
`0
`0
`0
`c.n' U'l
`
`"' ...,
`
`MOLECULAR WEIGHT
`
`Fig. 6.3. J:>ifferential number distribution function of a poly(styrene) sample.
`
`,_., ::::--
`
`/
`
`·-·-
`
`HE PCURV 7
`I SMOO H Cl RVE
`
`~ 100
`~ 90
`~ 80
`(.) 70
`a::
`Lll
`liO
`ll..
`LiJ so
`>
`..... 4 0
`<l
`...I JO
`:;)
`20
`:::E
`:;) 1 0
`u
`0
`0
`0
`0
`~~
`N.
`
`I
`I
`
`~
`
`1-:-L
`...... ~ J.-:-.
`
`0
`0
`0
`
`0
`0
`q
`U'l
`li)
`
`0
`0
`0
`
`<P
`
`0
`0
`q
`""
`"''
`"'
`:}
`....
`0
`"'
`MOLECULAR WEIGHT
`
`0
`0
`
`~
`
`C)
`0
`0
`.,;-
`\0
`
`0
`0
`0
`
`"'
`-.;.
`
`Fig. 6.4. Integral weight distribution curve of a poly(styrene) sample.
`
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`

`
`102
`
`Polymer Chemistry
`
`Q.6.3: From the following data, construct the integral and differential distribution
`•curves for 2-vinyl pyridine.
`
`Sample No.
`
`Wt. ofpolymer
`(g)
`
`1.
`2.
`3.
`4 . .
`5.
`6.
`7.
`8.
`9.
`10.
`11.
`12.
`
`3.57
`1.17
`3.43
`3.74
`4.93
`8.88
`2.82
`4.72
`1.89
`2.78
`4.80
`4.64
`
`DP
`
`466
`840
`1000
`1130
`1270
`1480
`1480
`1640
`1650
`1810
`2420
`2880
`
`Q.6.4.
`
`If a polymerization reaction is carried out so that initiation takes place at a
`constant rate, the chain grows at a constant rate and there is no chain
`termination, what will be the ratio MwiM.?
`
`6.4. FRACTION OF POLYMERS
`
`Separation of a polymer sample into fractions which are more homogeneous is known
`as polymer fractionation. In suitable solvents the solubility of a homogeneous
`polymer fraction decreases with increasing molecular weight. Thus, the addition of a
`non-solvent to a solution of heterogeneous polymer precipitates out fractions in the
`order of decreasing molecular weights. Fractionation procedures are of two types:
`(i) precipitation in which actual fractions are isolated for further study, and
`(ii) analytical in which the distribution curves are obtained without isolating
`fra~tions.
`
`6.4.1. Precipitation Methods
`Following are the main precipitation methods:
`(A) fractional precipitation by non-solvent addition,
`(B) triangular precipitation method,
`(C) fractional precipitation by evaporation of solvent, and
`(D) fractional precipitation by cooling.
`These methods are self-explanatory. The method (B) is the most elaborate and
`can lead to more or less pure fractions. It will be discussed here in some detail.
`
`6.4.1 .1 Fractional Precipitation by Non-solvent Addition
`In this procedure, usually a solution of 0.5% or lesser concentration of the high
`polymer to be fractionated is prepared in a suitable solvent. Precipitation of the
`polymer molecules in solution is then carried out by the slow addition of a precipitant
`until an initial turbidity is reached at a controlled temperature.
`The mixture is then warmed slightly after the initial turbidity is observed to aid
`the attainment of thermal equilibrium conditions. It is then cooled slowly to the
`original precipitation temperature. The precipitate is then recovered usually by
`
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`
`Molecular Weight Determination
`
`103
`
`centrifuging to effect a clearer separation. The average molecular weight as well as
`the weight percent is determined for the fraction.
`After separating the fltSt fraction from the polymer molecules still remaining in
`the solution, a further · increment of precipitant is now added as before to the
`remaining solution, and the process repeated to obtain the second fraction, and so
`forth, for as many fractions as are desired or required.
`The solubility of a chain polymer in suitable solvents or a mixture of solvents
`falls off with.increasing molecular weight Hence, the longer the chain length of the
`polymer, the smaller is the volume of the precipitant required to cause precipitation.
`
`6.4.1.2 Triangular Precipitation Method
`In this method sufficient.·precipitant is added to polymer solution (P) at the first
`addition to bring about the precipitation of the polymer to get two phaseS: gel (a)
`(precipitated phase) and solution (b) (supernatant layer). After separation, the gel (a) .
`is brought into solution to get solution (a). The two solutions (a) and (b) are treated
`similarly as the original solution (P..) was treated to get four approximately equal
`fractions. The solution (a) gives gel'(c) and the solution (d), and the solution (b) gives
`gel (e) ad solution (f). Solution (d) and gel (e) are combined to get the solution die,
`and gel (c) is again dissolved. Each of the three solutions thus obtained are subjected
`to further fractionation in a way shown in Fig. 6.S. This procedure is continued until a
`number of fractions are obtained. The larger the number of fractions, the more
`accurate is the fractionation.
`
`Fig. 6.5. The triangular fraction scheme.
`
`6.4.1.3 Fractional Precipitation by Evaporation of the Solvent
`An alternative method for precipitation of a polymer ·fraction from a solution is the
`preferential removal of solvent by evaporation. Certain advantages over the non(cid:173)
`solvent addition procedure are:
`(i) Local complete precipitation is avoided.
`(ii) The volume of the system decreases throughout the fractionation.
`(iii) Fraction size can be controlled from 'the turbidity developed.
`
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`
`104 Polymer Chemistry
`
`6.4.1.4 Fractional Precipitation by Cooling
`
`The solubility of a high polymer in a solvent is temperature-dependent. As the
`temperature is lowered, for example, the thermodynamic properties of the solvent
`change, making it a poorer solvent. Eventually, a temperature will be reached at'
`which precipitation of the least soluble components will occur because at that stage
`the solvent and polymer are no longer miscible in all proportions. A stepwise
`precipitation and · recovery of respectively precipitated fractions may then be
`performed and the data handled as in fractional precipitation procedure.
`
`6.4.2. Analytical Methods
`
`These include methods like turbidimetric titrationslultra centrifugation, gradient
`elution analysis and gel permeation chromatography. The last mentioned method will
`be described in detail.
`
`6.4.2.1 Gel Permeation Chromatography
`
`This is one of the latest techniques employed and is accurate. It is used with
`advantage for fractionating pcilymer samples and also for determining molecular
`weight distribution without collecting the different fractions. The fractionation is
`done iri a chromatographic column packed with rigid 'gel' .beads. These gel beads are
`made of porous highly cross-linked poly(styrene). These gels and porous glass beads
`form the packing material. A dilute solution of the polymer is allowed to flow down
`the column. Fig. 6.6 shows·how a simple GPC column works.
`
`SOLVENTl
`FLOW
`
`SAMPLE- QX•X ,X~
`MIXTURE
`
`• X
`
`SEPARATION- ~f"::l~@
`~!);\ X
`BEGINS
`
`• d
`00
`G)~@x
`·aQ·. 66
`xxQQx galR
`.... ': . X- SMALL MOLECULES
`IX\Q'
`v
`
`s~~~~~~~~-
`
`• _LARGE MOI .ECULES
`
`Fig. 6.6. The working of a simple GPC column.
`
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`

`
`Molecular Weight Determination
`
`105
`
`6.4.2.1.1 Principle of the Method
`
`When a dilute solution of the polymer to be fractionated is allowed to stream down
`the GPC column, followed by pure solvent, the desired number of fractions can be
`collected and characterized. In this process one obtains the fraction of the polymer
`with highest molecular weight first and that with the lowest molecular weight at the
`last This is because the 'gel' beads used in the column are full of capillaries. Th~sl.l
`capillaries or pores are of different diameters. Of these capillaries, some are accessi(cid:173)
`ble only to very small polymer .molecules present in the solution, some others are
`accessible to polymer molecules of medium size or molecular weight. A few are
`accessible to macromolecules of very high molecular weights. if V<;;. 100> is the total
`volume of the pores which are just accessible only to polymer molecules of degree of
`polymeri~ation (P = 100) and V(fi= 10> is the total volume of pores accessible_to mole(cid:173)
`cules of P = 10, then it becomes evident that the larger the V;; the smaller is P, that is,
`
`>
`
`Vp =too
`
`>
`
`Vp= woo
`
`>
`
`Vp= too.ooo
`
`The highly cross-linked poly(styrene) gel of the above micrograph (Fig. 6.7) was
`prepared by polymerizing 30% styrene and 10% divinylbenzene in the presence of a
`60% diluent (mixture of 20% diethylbenzene and 80% of isoamyl alcohol).
`
`( \ 0 0 . ~ ~ [f]@J
`
`dark polystyrene
`a M=6•10~
`b M =6•106
`c M =5• 105
`d M =5•105
`
`Light pores
`
`Fig. 6.7. Electron micrograph of a highly cross-linked poly(styrene) gel.
`
`Thus, macromolecules with degree of polymerization 10 can enter all the
`capillaries in the gel, whereas polymers with P = 10,000 can enter into very few
`capillaries (those with diameter large enough to permit entering of macromolecules
`with P = 10,000 or more). It, therefore, follows that when a dilute solution of a
`polymer sample is allowed to flow through a column, the probability of a molecule
`entering the capillaries (available volume) is the larger and, hence, the residence time
`
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`

`
`106
`
`Polymer Chemistry
`
`is the greater for macromolecules with lower degree of pelymerization than for those
`with higher molecular weight (greater P). Hence, macromolecules with higher P are
`eluted first and those oflowest P at the last.
`
`This method can be used for a wide variety of polymers and solvents. When
`poly(styrene) gels are used, nonpolar polymers can be analysed, using solvents
`such as tetrahydrofuran or toluene. The results of carefully performed GPC
`fractionations . for molecular weight distribution agree closely with those obtained
`by other methods such as solvent gradient elution method. The accuracy of the
`method is further established by the fit obtained between the experimentally obtained
`data and the theoretically calculated distribution curve on the basis of polymerization
`kinetics.
`
`6.4.2.1.2. Determination of Molecular Weight Distribution
`
`In many cases, fractionation is carried out to determine the molecular weight distribu(cid:173)
`tion. This is conveniently done by collecting fractions, drying them, weighing to
`determine the amount of polymer and measuring its molecular weight by some
`independent method. A major advantage of GPC is that these steps can be eliminated
`by use of a detector that determines the concentration of the polymer in the effluent
`since neither a solvent nor temperature gradient is present. A differential refracto(cid:173)
`meter is usually used because a change in refractive index is directly proportional to
`concentration in dilute solutions of polymer. Moreover, it is usually independent of
`the molecular weight in the range of molecular weights above a few hundreds. Fig.
`6.8 shows an example of the chart record produced by a differential -refractometer.
`The displacement from the base line is proportional to the difference in refractive
`index and, hence, the concentration.
`
`cr-----------~~----------------~
`.Q
`ti
`.!!!
`Q;
`
`"0 ... Q)
`'E
`0
`~~------------------------------~
`Elution volume
`
`Fig. 6.8. GPC curve for poly( styrene) dissolved in tetrahydrofuran.
`
`The vertical lines measure the elution volume. A siphon at the end of the column
`is dumped when 5 cc of the effluent has been collected. This dumping action sends an
`impulse to the recorder to automatically produce the spike on the curve. Thus, if the
`concentration is recorded directly, then the molecular weight remains to be
`determined. This may be done by reference to a calibration curve (Fig. 6.9). If the
`calibration is done in terms of molecular size parameter, for example [11] M, it can be
`applied to a wide variety of both linear and branched polymers.
`
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`

`
`Molecular Weight Determination
`
`101
`
`~
`·;;
`-to'
`
`• Lino&~r polyslyron~
`o Urunche<J poly!;lyruno
`(comb typo)
`+ Branched polystyrene
`(star type)
`• Branched block copolymer
`PS/PMMA.
`• Poly (methyl mclhncrylnlc)
`o.Poly (vinyl chloride)
`• Gran copolymer PS/PMMA
`• Poly (phenyl siloxanc)
`• POlybuladlcno
`
`Fig. 6.9. Calibration cuJVe for gel penneation chromatography b:o.sed on hydrodynamic
`volume expressed by the product [T\] M.
`
`Fig. 6.10. Apparatus for gel penneation chromatography.
`
`A suitable apparatus for gel permeation chromatography is shown in Fig. 6.10.
`
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`

`
`108
`
`Yolymer Chemistry
`
`Separations have been achieved at temperatures from 20° to 150°C; higher
`temperatures lower the viscosity of the solvent, which can lead to more rapid analysis
`and higher resolution in the column. Temperature control of the columns is usually
`achieved in the thermostatted, stirred air baths.
`
`Q.6.5. How would you fmd the detailed molecular weight distribution of a polymer
`without fractionating it?
`
`6.5. DETERMINATION OF MOLECULAR WEIGHTS
`
`6.5.1. Determination of Number Average Molecular Weights
`
`A colligative property which depends on the number of molecules is measured to get
`number average molecular weight of a polymer. Some of the common methods used
`for determining number average molecular weight are desc1ibed below.
`
`6.5.1.1. Osmometry
`The number average molecular weight of many polymers can be measured
`conveniently by osmometry. The use of this colligative ·property is based on the fact
`that certain semi-permeable membranes may be constructed which permit penetration
`by solvent molecules but which prevent the transport of macromolecules. The Gibb's
`free energy of the solvent is known to be lowered by the presence of solute. Pure
`solvent, thus, passes. through the membrane in order to lower the free energy of the
`system. lf a thermostatted cell is constructed and pure solvent is placed on one side of
`the membrane while a solution of the polymer in the same solvent is placed on the
`other side of the membrane, a pressure gradient will develop. The process . will
`continue until an equilibrium is reached in which the free energy change due to the
`pressure rise just equals the free energy change due to dilution of the solution. The
`equilibrium pressure developed is called the osmotic pressure (1t). The principle of an
`osmometer is illustrated in Fig. 6.11.
`
`OSMOTIC
`
`..... f~-~IssuRE
`'··· .· .. · ..
`·.·
`.·
`.... 1::-
`:;.;
`·.: .·.-·. •. :· : .. ~ ....... -. ,"': .......... _-':-_---
`- """----
`-------
`- ----
`---- --·
`SOLVENT
`------
`------
`------
`------
`
`~SEMIPERMEABLE
`MEMBRANE
`
`Fig. 6.11. The basic principl~ of an osmometer.
`
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`
`Application of elementary thermodynamics to the osmotic equilibrium gives the
`following relationship.
`
`Molecular Weight Determination
`
`109
`
`( I
`n
`-=RT =-+A2c +A3c + .. .
`2
`)
`M.
`c
`
`(6.4)
`
`where c is the concentration of the solution in g /cc, R is the gas constant and Tis the
`absolute temperature. It is assumed in this derivation that solutions are dilute and
`temperature is constant. It is noted that Eq. (6.4) is a power series ~ the
`concentration. In order to obtain the number average molecular weight (M,J an
`extrapolation procedure yielding nlc at the limit of infmite dilution is required. As c
`approaches zero, all intermolecular interactions vanish and the theory becomes exact.
`Eq. (6.4) shows that a graphical treatment cf the osmometric data produces a line
`having zero concentration intercepts inversely !)roportional to M •. Often (n/c) 112
`versus c plot is more nearly linear and provides exactly the same extrapolated point.
`
`6.5.1.1.1. Osmometers
`
`The membranes usual!:,· used in a membrane osmometer are collodion (ll-13.5%)
`nitrogen containing nitrocellulose), regenerated cellulose, ~ell cellophane, poly(vil}yl
`alcohol), poly(urethane) etc. The choice of a membrane is critical for the accurate
`determination of molecular weight by this method. The presence of low molecular
`weight polymers in the unfractionated polymer vitiates the results. These have to be
`removed by fractionation or exu·action. A simple calculation will show that 0.1%
`(Wt.) of an impurity having molecular weight uf 100 g/mole greatly distorts the
`osmotic result of a polymer having M.W. = J05 g!mole (an M. of slightly greater than
`50,000 g!mole is the result).
`In the static "equilibrium procedure, the osmotic pressure is determined by
`measuring the difference in heights of the two capillaries (connected to the cells) due
`to the diffusion of the solvent across the semi-permeable membrane. The main
`disadvantage of these osmometers is the length of time required to attain equilibrium.
`In the dynamic method, the counter pressure required to prevent the diffusion of
`the solvent through the membrane is measured. In high speed membrane osmometers,
`an optical system in the solvent chamber detects flow through the membrane and
`automatically adjusts pressure by means of ar electro-mechanical system to prevent
`any net flow. Since practically no flow of the s0lvent is needed to establish osmotic
`pressure, the whole process takes only minutes and causes no dilution of the solution
`on the other side of the membrane. This not only shortens the overall time per
`determination but also gives better results for materials which can diffuse through the
`membrane.
`
`6.5.1.2 End Group Analysis
`
`The number average molecular weight of linear polymers can be determined by
`estimating the number of end groups by chemical analysis. For the successful
`application of the end group method, the number and nature of end groups per
`polymer molecule should be reliably known. This method is apl}ljcable to linear
`condensation polymers and also to addition polymers. For the meth!!'l to be reliable,
`the molecular weight of the polymer should not exceed 25,000.
`
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`
`110
`
`Polymer Chemistry
`
`6.5.1.2.1. Condensation Polymers
`Condensation polymers usually contain functional groups which can lend themselves
`to chemical analysis. Thus, poly(esters) and poly(amides) contain carboxy groups,
`which can be estimated by titration with a base in alcoholic or phenolic solutions. The
`amino groups in poly(amides) can be estimated by titration against an acid under
`similar conditions. For the determination of hydroxyl groups in a polymer, it is
`reacted with a titrable reagent.
`A recent inethod involves derivatization and use of NMR spectroscopy. For
`example, a poly(carbonate) derived from Bisphenol-A was capped at the ends by a
`trimethyl silyl ether. By observing the ratio of the peak intensity of methyl groups
`bonded to silicon to that of the aromatic protons of Bisphenol-A, it was possible to
`get a very good estimate of M., the number average molecular weight.
`
`6.5.1.2.2. Addition Polymers
`As already stated the end-group method can be applied to addition polymers also
`when the polymerization is initiated by initiators containing identifiable groups or
`radioactive labelled groups. The initiator fragments get attached to one or both the
`ends of the chain depending on the mode of termination. If the mechanism of
`termination involves disproportionation, only one fragment gets attached .. On the
`other hand, if it is by coupling, two such fragments usually get attached.
`
`6.5.1.3. Vapour Phase Osmometry
`
`This method has been found to be useful in determining the number average
`molecular weight of those polymers whose molecular weight is too low for
`measurement in a membrane osmometer. In an instrument called 'vapour pressure
`osmometer', use is made of the small temperature difference~resulting from ditferent
`rates of solvent evaporation from and condensation onto droplets of pure solvent and
`polymer solution maintained in an atmosphere of solvent vapour. The prinCiple of the
`method is that the temperature difference is proportional to the vapour pressure
`lowering of the polymer solution at equilibrium point and, thus, to the number
`average molecular weight. As there are heat losses, the measurement is conducted at
`.several concentrations and results extrapolated to zero concentration. To get accurate
`results, the instrument is calibrated with low molecular weight standards. Further,
`such variables as drop size and line of measurement have to be carefully standardised.
`The method is useful in measuring M. upto 40,000.
`In a suitable instrument there is a vapour phase chamber. Droplets of solvent and
`polymer solution are placed with the aid of hypodermic syringes on the beads of two
`thermistors used as
`temperature sensing elements. These are maintained in
`equilibrium with an atmosphere of solvent vapour. The lower activity of the solvent
`in the solution droplet leads to an excess of soivent condensation over evaporation
`· there, compared to the droplet of pure solvent The excess heat of evaporization thus
`leads to a rise in temperature.
`Ebulliometry and cryoscopy have also been used to determine number average
`molecular weights by using accurate temperature sensing devices.
`
`Q. 6.6. Explain why colligative property measurements give number average
`molecular weight.
`
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`
`Molecular Weight Determination
`
`111
`
`6.5.2. Determination of the Weight Average Molecular Weight
`
`6.5.2.1. Light Scattering Measurements
`If a bP.am of light is passed through a colloidal solution, it is possible to see the light
`beam from the sides. 11iis is the well known Tyndall effect which results from the
`scattering of a part of the beam of light by the colloidal particles in all directions.
`Light scattering measurements can be used · to determine the · weight average
`molecular weights of polymers.
`
`6.5.2.1.1. Light Scattering from Particles with Diameter less than 'A/20
`In the case of macromolecules, the colloidal particles are highly solvated and the gel
`particles consist of about 90 to 99% of the solvent. The refractive ind~x of such
`colloidal particles is not very different from that of the solvent.
`According to P. Deby,e, the following power function applies in such cases.
`
`RT c (n dn)
`
`3
`32 1t
`11't = - -. - - - - (drrJdc)
`31..4 N0
`de
`
`(6.5)
`
`where 1t is the turbidity, l11t represents the excess turbidity of the solution over that of
`the pure ·solvent, A. is the wave length and n is the refractive index. In the absence of
`absorption, 1t is related to the primary beam intensity before and after it passes
`through the scattering medium. If the incident intensity 10 is reduced to 1 in a length I
`of the sample,
`
`_,,
`1
`-=e
`lo
`
`(6.6)
`
`The turbidity, which is the total scattering integral over all angles, is often
`referred by the Rayleigh rado R (6) [R (8) = R (6) - R (8)solveoJ which relates the
`·scattered intensity at angle e to the incident beam intensity. For particles small
`compared to A., the Debye equation is obtained as,
`
`(6.7)
`
`where N0 is the Avogadro number.
`Eq. (6.7) forms the basis of determination of polymer molecular weight by light
`scattering. Beyond the measurement of 't or R6, only the refractive index (n) and the
`specific refractive index increment (dnldc) require experimental determination. The
`latter quantity is constant for a given polymer, solvent and temperallire and is
`measured with an interferometer or differential refractometer.
`Eq. (6.7) is correct only for vertically polarized incident light and for optically
`isotropic particles (substances in which the velocity of light and, hence, the refractive
`index is the same in all directions are said to be isotropic, e.g., water or glass). The
`use of unpolarized light requires that 't be multiplied by (I + cos2 8).
`I
`
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`
`112
`
`Polymer Chemistry
`
`For systems with particles size < IJ20. any observation angle may be selected
`since the scattering function is spherically symmetrical.* However, for experimental
`reasons (to avoid interference of the primary beam), one usually selects values of e
`not very different from 90°.
`
`/Solution
`.. e~--------~~~0
`h~------------~~~
`
`I DETECTOR I
`
`Fig. 6.12. Essential parts of a light scattering instrument.
`
`In this simple way, through the measurement of scattered light intensity at a
`single angle e, one obtains the molecular weight, but only for particle of diameters
`smaller than /J20. This is true only for glycogen, a number of proteins and linear
`polymers of relatively low molecular weights in poor solvents.
`
`6.5.2.1.2. Light Scattering from Particle of Diameters Larger than A/20
`For polymer molecules whose coil diame