POLYMER
`MOLECULAR WEIGHTS
`
`(in two parts)
`
`PART I
`
`Philip E. Slade, Jr.
`MONSANTO TEXTILES COMPANY
`PENSACOLA, FLORIDA
`
`MARCEL DEKKER, INC.
`
`New York
`
`RBP_TEVA05017947
`
`TEVA EXHIBIT 1037
`TEVA PHARMACEUTICALS USA, INC. V. MONOSOL RX, LLC
`
`

`
`Copyright© 1975 by Marcel Dekker, Inc.
`
`ALL RIGHTS RESERVED
`
`Neither this book nor any part may be reproduced or transmitted
`in any form or by any means, electronic or mechanical, including
`photocopying, microfilming, and recording, or by any information
`storage and retrieval system, without permission in writing from
`the publisher.
`
`MARCEL DEKKER, INC.
`270 Madison Avenue, New York, New York 10016
`
`LIBRARY OF CONGRESS CATALOG CARD NUMBER: 74-80625
`ISBN: 0-8247-6227-4
`
`Current printing (last digit):
`10 9 8 7 6 5 4 3 2 1
`
`PRINTED IN THE UNITED STATES OF AMERICA
`
`RBP_TEVA05017948
`
`TEVA EXHIBIT 1037
`TEVA PHARMACEUTICALS USA, INC. V. MONOSOL RX, LLC
`
`

`
`Chapter 1
`INTRODUCTION
`
`Philip E. Slade, Jr.
`Monsanto Textiles Company
`Pensacola, Florida
`
`I. MOLECULAR WEIGHT AVERAGES
`A. Number Average Molecular Weight
`B. Weight Average Molecular Weight
`C. Z and Z + 1 Average Molecular Weights
`D. Viscosity Average Molecular Weight
`II. MOLECULAR WEIGHT DISTRIBUTION
`References
`
`2
`3
`4
`5
`5
`
`7
`
`7
`
`Macromolecules have many of the characteristies of simple
`molecules; that is, most will dissolve in common solvents, they
`exhibit colligative properties, have chemically reactive functional
`groups, and absorb energy at specific wavelengths. One problem
`
`unique to polymers, ~' i~~-that_'t:!t~~II_I()~!~El~:;~IIIB:L~:--~-~~~~=ed
`
`1
`
`RBP_TEVA05017949
`
`TEVA EXHIBIT 1037
`TEVA PHARMACEUTICALS USA, INC. V. MONOSOL RX, LLC
`
`

`
`2
`
`PHILIP E. SLADE, JR.
`
`~-va~;y:!_ng_ amounts of sma!l~:r: !l!()!l_()m_e_~ units. These changes in the
`degree of polymerization result in molecules with differing molec(cid:173)
`ular weights and, indeed, in differing distributions of those mo(cid:173)
`lecular weights. Polymeric materials with molecular weights
`ranging from a few hundred to several million are not uncommon,
`even within the same population group made in the same reaction
`vessel at the same time. This infamous phenomenon has initiated
`many hours of research and is the reason that this book exists.
`The determination of polymer molecular weights can become a
`difficult task for the novice in polymer physical chemistry. The
`actual laboratory experimentation is not complex in most cases,
`but the treatment of the data can be very involved. Fortunately,
`the use of electronic computers has resolved this problem to some
`extent. However, many laboratories are not blessed with this aid
`nor with competent programers to insert the proper numbers in the
`correct sequence;
`In order to make the best use of the data that
`we obtain from the laboratory by any of the techniques discussed
`in later chapters, we must understand the basic concepts of poly(cid:173)
`mer molecular weight and how to calculate the numbers we use.
`It
`is hoped that this book will help clarify unanswered questions and
`resolve some of the difficulties in measuring this macromolecular
`parameter.
`
`I. MOLECULAR WEIGHT AVERAGES
`
`Extreme care must be exercised when molecular weight termi(cid:173)
`nology is used since· s.everal systems for expressing this infor(cid:173)
`mation exist. Different types of experimental procedures will
`yield data that vary widely, since they will measure completely
`'
`independent properties. Accordingly, molecular weight averages
`must be defined to accomodate the particular data considered. The
`most commonly used terms describing the experimental data are~
`per av~_age. ... m()lecular weight, M ; viscosity average molecular
`..... n. .. -
`r-
`- . ------------- ···--·-··
`
`RBP_TEVA05017950
`
`TEVA EXHIBIT 1037
`TEVA PHARMACEUTICALS USA, INC. V. MONOSOL RX, LLC
`
`

`
`Introduction
`
`3
`
`weight, Mv; weight averagSLl!!.QLe_c.ular_w_ei-gh.t.,-Mw; the "Z-average,"
`Mz and "Z + 1 average," Mz + 1 . Each of these will be considered
`for their meaning and interelationships.
`
`A. Number Average Molecular Weight
`
`'------·~-~·-----
`
`If a technique for molecular weight determination is selected
`that is de;eendent ~e_nwnb_e:LQ_f _ _mo.l.e_cules __ pr.es_~pj:, that is, col-
`ligative properties, the function obtained is the number average
`molecular weight. Examples of these techniques are membrane os(cid:173)
`mometry, boiling point elevation, and end group analyses. All
`colligative property measurements depend on the mole fraction of
`solute present, N2, assuming that the concentration of the solute
`is small enough that .. Ji_e!lry' s law is obeyed. Polymers, however,
`have a number of different species of solute dissolved in this
`dilute solution, since we have a distribution in the degree of
`polymerization, and each of these species will be present at its
`own concentration.
`It is thus necessary to add all of these mole
`fractions together to determine the total solute mole fraction.
`To accomplish this, we first total the number of moles E N of
`i
`i
`solute present. Here N. is the number of moles of solute of spe-
`1
`cies i. Likewise, the total weight of the sample, W, is
`W = E W.
`
`1
`
`1
`
`(1-1)
`
`where ~ is the actual weight in gr@ls .. _of speci~. Since moles
`times molecular weight is equal to weight in grams, we may also
`write
`
`W = E N.M.
`
`1
`
`1 1
`
`(1-2)
`
`The number average molecular weight, Mn' is now defined as the
`total weight of all solute species, divided by the total number
`of moles presbnt. Thus
`
`(1-3)
`
`RBP_TEVA05017951
`
`TEVA EXHIBIT 1037
`TEVA PHARMACEUTICALS USA, INC. V. MONOSOL RX, LLC
`
`

`
`4
`
`PHILIP E. SLADE, JR.
`
`or by combining (1-2) and (1-3)
`
`M = E N.M./E N.
`n
`1
`1 1 1
`1
`
`(1-4)
`
`If we take a simple hypothetical example of a polymer with 100
`moles of a species with a molecular weight of 103, 100 moles with
`a molecular weight of 104 , and 100 moles with a molecular weight
`of 105, we find
`
`M
`n
`
`(100) (103) + (100) (104) + (100) (105)
`300
`
`3.7 X 104
`
`B. Weight Average Molecular Weight
`
`Let us now consider the situation where a property, such as
`light scattering, based on the weight of solute present is measured.
`These techniques will give the weight average molecular weight, M •
`w
`In an expression similar to Eq,. (1-4), Mw is defined as
`
`M = E M.W./E W.
`w
`1
`1 1 1
`1
`
`(1-5)
`
`with Wi being the weight of the "i" species and Mi its molecular
`weight.
`It is much more convenient to use the same terms to ex(cid:173)
`~2ress each molecular weight function; thus a simple conversion can
`be made. Since
`
`W. = N.M.
`1
`1 1
`
`it follows that
`
`M = E N.M. 2/E N.M.
`w
`1
`1 1
`1
`1 1
`
`• (1-6)
`
`(1-7)
`
`It may be noted that, like Eq. (1-4), this expression contains
`one more power of molecular weight in the numerator than in the de(cid:173)
`nominator, thus providing the "dimensions" of molecular weight in
`each case. Clearly, the expressions could be generalized to in(cid:173)
`clude other molecular weight averages, as we shall see.
`This equation also shows that with the squared molecular
`weight ~w-_ji.!J_Lbe-la-r,g.er,_th,_,an~,_,Mn-'-~lQ our polymer example,
`we find
`
`RBP_TEVA05017952
`
`TEVA EXHIBIT 1037
`TEVA PHARMACEUTICALS USA, INC. V. MONOSOL RX, LLC
`
`

`
`Introduction
`
`5
`
`M
`w
`
`(100) (103) 2 + (100) (104) 2 + (100) (105) 2
`(100) (103) + (100) (104) + (100) (105)
`
`9.1 X 104
`
`C. Z and Z + 1 Average Molecular Weights
`
`Much more infrequently used are the Z and Z + 1 molecular
`weight averages that are higher than M and are obtained from
`w
`sedimentation experiments. The equations of definition are
`M = E N.M.3/E N.M.2
`z
`1
`1 1
`1
`1 1
`
`(1-8)
`
`and
`
`(1-9),
`
`It is possible to calculate even higher averages by increasing
`the exponents of Mi in Eq. (1-9).
`For our polymer system described above, M is 9.9 x 104 and
`z
`Mz + 1 near 1 x 10 5.
`
`D. Viscosity Average Molecular Weight
`
`The Mark-Houwink relationship of viscosity to molecular
`weight states that
`
`(1-10)
`
`where~Jis the limiting viscosity number (or intrinsic viscosity),
`'M' is the molecular weight, and 'K' and 'a' are constants which
`depend on the polymer-solvent system in use. With heterogeneous
`systems, the intrinsic viscosity is weighted by the amount of 'i'
`species present; thus
`rnl. W. = K M.a w.
`L:IJ 1
`1
`1
`1
`
`(1-11)
`
`Summing over all species present, we find that
`a
`[nQ= K EM. W./"£ W.
`1
`1
`1
`1
`since Eq. (1-2) states that t Wi
`
`"£ N.M.
`1 1
`
`(1-12)
`
`RBP_TEVA05017953
`
`TEVA EXHIBIT 1037
`TEVA PHARMACEUTICALS USA, INC. V. MONOSOL RX, LLC
`
`

`
`6
`
`and
`
`[n] = K L N.M.M. a/L N.M.
`1 1 1
`1
`1 1
`1
`
`[nJ = K L N.M.a+ 1/L N.M.
`
`1
`
`1 1
`
`1 1
`
`PHILIP E. SLADE, JR.
`
`(1-13)
`
`(1-14)
`
`(1-15)
`
`(1-16)
`
`If we now let the molecular weight in Eq. (1-10) be Mv
`a
`a+1
`K M = K L N.M.
`/L N.M.
`v
`1
`1 1
`1
`1 1
`
`Mv is now defined as
`[L N.M. a+ 1/L N.M.] 1ia
`
`1
`
`1 1
`
`1
`
`1 1
`
`M v
`
`It should be noted that Mv isj10~_a true average number but
`
`gc.tually a_ range depen4e!!Le>_~J:J:t~-Y-l!_!~~-?.-~-~-~ which in turn will
`change with polymer-solvent interaction paramet~rs.
`The principle difficulty in obtaining a value for 'a', are(cid:173)
`lationship for [n] vs M when M is unknown, may be circumvented by
`v
`v
`first making a log-log plot of~]vs Mw for samples with the same
`molecular weight distribution; Mw is selected as the absolute
`average for the graph since it is nearest Mv in value.
`It is then
`necessary to assume some distribution function to obtain a rela(cid:173)
`tionship between Mv and Mw. One such distribution that is of use
`is the generalized Schultz-Zimm (1) for calculating Q, the ratio
`of Mw/Mv.
`In this system
`[r (Z + 1) I r: (Z + 1 + a)J1/a
`Z = (Z + 1)
`where Z is defined as z- 1 =
`(M /M ) - 1 and r(f) the gamma function.
`w n
`The
`'a' in Eq. (1-9) is then the slope from the[n]- Mw plot. Mv
`can then be calculated for each data point, another graph of
`~]- Mv can be drawn, and 'a' can be obtained from
`a
`[nJ = K Mv
`
`(1-17)
`
`(1-18)
`
`If a = 1, Mv = Mw.
`the equation of the line.
`If we assume a value of 'a' of 0.75 for our hypothetical
`polymer, Mv is then
`
`RBP_TEVA05017954
`
`TEVA EXHIBIT 1037
`TEVA PHARMACEUTICALS USA, INC. V. MONOSOL RX, LLC
`
`

`
`\
`
`Introduction
`
`7
`
`Mv
`
`[CIOO) (10 3) 1. 75 + (100) (10 4) 1. 75 + (100) (lOs) 1. 751
`
`(100)(103)+(100)(104)+(100)(105~ l/0· 75 = 8.9 X 104
`
`II. MOLECULAR WEIGHT DISTRIBUTION
`
`From the above relationships, it is obvious that
`
`M<M<M<M<M
`n v- w
`z
`z+l
`
`and, by summarizing molecular weight averages calculated for the
`hypothetical polymer, M = 3.7 x 104, M = 8.9 x 104, M = 9.1 x
`w
`n
`v
`10 4 M = 9.9 x 104 and M
`~ 1 x 10s.
`'
`z
`'
`z + 1
`Se_ve.r_~t_indexes. for _)!lole~_e!_~-~-~- distribution have been
`prqQosed with the M /M rat~n~only used. Others include
`w n
`------.:__...._ __
`.---
`U(2), the Uneinheitlichkeit or inhomogeneity factor; ~(3), the
`square root of U; and g(4), a polydispersity index for the higher
`molecular weight ends. These may be defined mathematically as
`
`u M /M
`w n
`
`- 1
`
`~
`
`(M/Mn
`
`!,.:
`1) 2
`
`g = (M /M
`z w
`
`!,.:
`- 1) 2
`
`(1-19)
`
`(1-20)
`
`(1-21)
`
`It is hoped that this brief discussion will help to summarize
`some of the terminology used in describing molecular weights and
`will introduce the reader to the discussion of measuring techniques
`following in subsequent chapters.
`
`References
`
`1. B. H. Zimm, J. Chern. Phys. 3 ~' 1099 (1948).
`2. G. v. Schultz, z. Physik. Chem.3 B47, 489 (1963).
`3. G. G. Lowry, J. Polymer Sai. 3 ~' 489 (1963).
`4. R. Hosemann and W. Schramek, J. Polymer Sai. 3 ~' 29 (1962).
`
`RBP_TEVA05017955
`
`TEVA EXHIBIT 1037
`TEVA PHARMACEUTICALS USA, INC. V. MONOSOL RX, LLC
`
`

`
`8
`
`PHILIP E. SLADE, JR.
`
`General References
`
`P. W. Allen, ed., Techniques of Polymer Characterization,
`Butterworths, London, 1959.
`G. L. Beyer, "Molecular Weight Determinations" in F. D. Snell and
`C. L. Hilton, ed., Encyclopedia of Industrial Chemical Analyses,
`Vol. 2, p. 611, Interscience, New York, 1966.
`R. v. Bonnar, M. Dimbat, and F. H. Stross, Number Average
`Molecular Weights, Interscience, New York 1958.
`M. J. R. Cantow and J. F. Johnson, "Molecular Weight Determination,"
`in H. F. Mark and N. Gaylord, ed., Encyclopedia of Polymer Science
`and Technology, Vol. 9, p. 182, Interscience, New York, 1968.
`
`M. J. R. Cantow, ed., Polymer Fractionation, Academic Press,
`New York, 1967.
`P. J. Flory, Principles of Polymer Chemistry, Cornell University
`Press, Ithaca, 1953.
`F. W. Billmeyer, Jr., Textbook of Polymer Science, Interscience,
`New York, 1965.
`H. Morawetz, Macromolecules in Solution, Vol. XXI of High Polymers,
`Interscience, New York, 1965.
`S. G. Weissberg, S. Rothman, and M. Wales, "Molecular Weights and
`Sizes," in G. M. Kline, ed., Analytical Chemistry of Polymers,
`Part II, Vol. XXI, p. 1, of High Polymers, Interscience,
`New York, 1962.
`C. Tanford, Physical Chemistry of Macromolecules, John Wiley & Sons,
`New York, 1961.
`
`RBP_TEVA05017956
`
`TEVA EXHIBIT 1037
`TEVA PHARMACEUTICALS USA, INC. V. MONOSOL RX, LLC