Smectroscopy of Polimers

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Preface to the second edition

How time flies! How rapidly things change! It is hard to believe that the first edition of
not only out of date but also out of print. Therefore, I took the

opportunity to rewrite major sections of the book in an effort to bring in the large number
of recent advances in the spectroscopy of polymers. The challenges of characterizing the
molecular structure of synthetic polymers continue to grow as the polymer chemists for a
broad range of new materials applications are synthesizing exciting new structures.
The first chapter on the theory of polymer characterization is essentially unchanged
with only minor editions and tightening of some of the text. The second chapter has been
substantially modified to include both Raman and Infrared in the description of the molecular
basis of vibrational spectroscopy. This makes for a more consistent approach and will better
guide the student to the more subtle differences in the applications of the two complementary
methods of vibrational spectroscopy. The chapter on infrared instrumentation and sampling
techniques has been modified to include innovations in instrumentation such as step-scan and
focal array detector imaging interferometers. The sections in this chapter on data processing
and quantitative analysis have been updated to reflect the major impact in this field of fast
computers with large memories. The chapter on applications of infrared has been updated
by including some of the new developments in the continuing astronomical growth in the
applications of infrared to polymers. It is possible to include only a small number of the
numerous new examples that are available and those selected were based on the utility as
pedagogical examples for the student.
Raman spectroscopy has become a most important tool for characterization of polymers as
low frequency lasers have improved so Raman spectroscopy excited in the infrared frequency
range can be used to minimize the effect of fluorescence. Advances in Raman instrumentation
have been dramatic and the polymer spectroscopists must now seriously consider the use of
Raman as complementary to infrared. In many cases Raman spectroscopy will be the preferred
method as the many new applications in this revised chapter illustrate.
Advances in solution NMR arise from the greater availability of high field instruments
and the fast computers with large memories. The higher applied magnetic field results in
greater dispersion so the resolution is higher and the information content is extended as longer
stereo and comonomer sequences are observed. The improved computers allow the complex
computational problems associated with two-dimensional techniques to be overcome so the
multidimensional NMR techniques are available to the ordinary user. The spectral editing

chapter of the first edition has been incorporated in this chapter. The applications of solution
NMR chapter has been expanded to reflect the advances made using the higher applied
magnetic field. A large number of new chemical structures are resolved including endgroups,
branches, crosslinks and stereosequences.
The utility of solid state NMR continues to grow. Many new applications, particularly those
based on motional differences of the phases, are developing, as experiments are easier to do
with better spinners and decoupling methods.

Spectroscopy of Polymers is


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viii

Preface to the second edition

The final chapter is an entirely new one describing mass spectrometry and its increasing
importance in polymer characterization. Vaporization and ionization techniques are being
developed that make macromolecules fly with little fragmentation. Therefore, the advantages
of short measurement and analysis times make mass spectrometry an important tool in polymer
characterization. I wish to thank Robert Latimer of B.E Goodrich for introducing me to the
potential of the technique and for kindly making a number of valuable suggestions on the
manuscript.
I want to thank my professional colleagues and those many students from around the world
that have made comments on the first edition of the Spectroscopy of Polymers. I have made an
effort to incorporate many of the constructive suggestions in the second edition.
I want to thank Barbara Leach and June Ilhan for their efforts in turning rough text and
figures into a readable manuscript and pursing all of the innumerable details associated with
production of a second edition.

Again, I want to thank my wife, Jeanus, for her giving up time that we would ordinarily
share together for me to work on the book. Her constant support is appreciated.
Finally, it is unusual to have a second edition of a book published by a different publisher.
The first edition was published by ACS publications. However, ACS decided not to publish
monographs any longer and kindly released the copyright, of the first edition to me. Elsevier
has agreed to publish the second edition. Jonathan Glover was instrumental in making the
selection of Elsevier as the publisher and I thank him for his efforts.


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Chapter 1

Theory of polymer characterization

The primary motivation for determining the molecular structure of a polymer chain is to relate the structures to the performance properties of the
polymer in end use. If the polymer chains are completely characterized and
the structural basis of its properties are known, the polymerization reaction
can be optimized and controlled to produce the optimum properties from the
particular chemical system.

Elements of polymer structure
The following basic terms are used for defining a polymer structure.
9 The composition (or constitution) of a molecule defines the nature of the atoms
and the type of bonding irrespective of the spatial arrangement.
9 The configuration of chemical groups characterizes the chemical state of a polymer. Different configurations constitute different chemical entities and cannot be
interconverted into one another without rupture of chemical bonds.
9 The conformation of chemical groups characterizes the geometrical state of a
polymer. Different conformations of a polymer can be produced by rotation about
single bonds without rupture of chemical bonds. Changes in conformation arise

from physical considerations such as temperature, pressure, or stress and strain.
Polymer chains are made up of sequences of chemical repeating units that may be
arranged regularly or irregularly on the backbone.
9 The chemical microstructure is defined as the internal arrangement of the different
chemical structures or sequences on the polymer chain.
Polymers can exhibit phase transitions and show a number of fundamental spatial
distributed macroconformations that define the crystalline and amorphous phases.
9 The polymer morphology defines the intermolecular packing of the polymer
molecules as crystals or spherulites in the bulk.
9 Polymer chains can exhibit different chain topologies where topology describes
the molecular packing of the chains. Chain alignments, orientation and entanglements are topologic features.


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2

Theory of polymer characterization

The mechanical properties such as modulus, tenacity, and yield are properties
which depend on the dynamics of the polymer chain.
9 The molecular motion depends on the intramolecular and intermolecular constraints imposed on the structure of the chain and the non-bonded neighbors in the
vicinity.
From a structural point of view polymers are chainlike molecules.

X-A-A-A-A-A-A-A-A-

A-A-A-A-A-A-A-

A-A-Y

The structural elements of an ideal polymer molecule with a single structural
repeating unit can be represented by the molecular formula, X(A)nY, where A represents the repeating unit of the polymer molecule and n is the number of repeating
units of A, X and Y are the end-group units which are chemically different from
A. The number of connected repeating units, n, can range from 2 to greater than
100,000. The chemical nature of the repeating unit A determines the chemical properties of the polymer. The chemical structure of the repeating unit can be very simple
(e.g., CH2 for polyethylene) or very complicated.
The end-group units X and Y can be substantially different in chemical structure
from A or very similar depending on the nature of the polymerization process.
Structural variations within the chain can be represented by the letters B, C, etc., to
indicate the differences in the chemical, configurational, or conformational structure.
The polymerization reaction converts the initial bifunctional monomers into a
chain of chemically connected repeating units. The process of polymerization can be
written in the following form:
nM --+ ( - A - A - A - A

........

A),,

(1.1)

where M is the monomer. We have neglected for the moment the fact that the end
groups X and Y are different structures from A.
However, this representation of the ideal polymerization reaction is incomplete,
as the polymerization reaction is statistical in nature and does not generate a single
molecule of a specified length n. Rather millions of polymerization reactions are occurring simultaneously, generating millions of molecules of various lengths ranging
from 1 to a very large number (e.g., 100,000) depending on how many reactions
have occurred between the individual molecules during the polymerization. So more
precisely, the polymerization reaction must be written

t/~OO

Z
n~-2

nM--+ Z

(-A-A-A-A

........

A)n

(1.2)

n=2

Thus, in contrast to chemistry of simple molecules, which produce a single molecular


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Approach to polymer structure determination using probability considerations

3

species, the polymerization batch contains a mixture of chain molecules ranging in
length from very short to very long. For this simple ideal polymer system, the only
structural variables are the lengths n of the chains and the number of molecules of
these various lengths N (n). In other words, to determine the structure of the polymer

we need to know the number fractions of molecules having different specific lengths.

Approach to polymer structure determination using probability
considerations
Let us start with the simplest structural example: the degree of polymerization
(DP). With the polymerization model just described, and assuming an equal likelihood for the selection of any polymer molecule from the mixture, it is possible to
calculate the probability distribution function for the chain length n. The probability
distribution function for n is the probability of finding a molecule with a given chain
length n in a polymer sample. For experimental purposes, the probability function is
the fraction of all polymer molecules that possess the stated chain length, n.
You can visualize the probability approach as one of reaching into the reaction mixture and pulling out a single polymer molecule. You must then
calculate the probability that the molecule selected has a specified length.

Let P be the probability that a propagation polymerization reaction has occurred
and Q be the probability of termination; that is, the molecule has not undergone
propagation and is terminated from further polymerization. This is a simple description of an ideal simple chain/addition polymerization (a chain reaction in which the
growth of a polymer chain proceeds exclusively by reactions between monomers and
reactive sites on the polymer chain with regeneration of the reactive sites at the end
of each growth step). This is illustrated in the following diagram:
Polymerization
Propagation
Termination

Event
A n + A - - + An+ 1
A n ~ An

Probability
P
Q

The instantaneous probability of propagation, P, in this case, is given by
pz

Number of propagation e v e n t s

Rate of propagation

Total number of events

Rate of propagation + Rate of termination

For termination

Q

Number of termination events

Rate of termination

Total number of events

Rate of propagation + Rate of termination


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4

Theory of polymer characterization

We will make the Flory assumption that the probabilities of propagation and
termination are independent of chain length. We will also neglect the contribution of
the initiation reaction, in other words, we will assume a high molecular weight chain
and the contribution of the initiation can be neglected.
There are only two events occurring, propagation and termination, so for completeness (something must happen) it is necessary that
P+Q=I
SO

Q=l-P
Let P(n) be the probability that a molecule of length n has been formed. At
any given instant of the polymerization process, the probability that a chain will
propagate and terminate to give a length n is the product of the probability of the
individual propagation steps and the probability of termination at that chain length.
If the propagation polymerization probabilities are independent (i.e., occur randomly
and do not depend on chain length), the probability of forming a molecule of length
n = 2 is the probability P that one propagation reaction coupling two monomer units
has occurred times the probability Q (= 1 - P) that termination or no further reaction
has occurred. Hence,
P
P(2) = P ( 1 -

P)

(1 - P)

Product

A* --+ A - A * --+ A - A

A2

(1.3)

A chain molecule of n = 3 is formed by two propagations and a termination; the
probability of the first propagation, P, times the probability of the second propagation, P, forming the trimer times the probability of termination, (1 - P), or that no
reaction has occurred. Hence,
P
P(3) = p 2 ( 1 -

P)

P

(1 - P)

A* --+ A - A * --+ A - A - A *

Product

--+ AAA

A3

(1.4)

A chain molecule of n - 4 is formed by three propagations, P3, and a termination.
So
P
P(4) = p 3 ( 1 -

P)

P

P

A* -+ A - A * --~ A - A - A *

(l-P)

-+ AAA* -+ AAAA

Product
A4

(1.5)

A general trend is emerging. The number of propagations is always one less than
the length of the chain and one termination step is always required, so the general
formulation for any chain length n can be written
p

P(n) = P ~ - I ( 1 - P)

p

p

p

p,,-1

(l_p)

A* -+ -+ -+ -+ . . . . . . A * - + An

Product
An

(1.6)


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Structure calculations using the probability distribution function

5

Equation (1.6) is the probability distribution function for the structural parameter of interest, which in this case, is the chain length n of the polymer.
From the probability distribution function for n, any desired information
or property arising from the chain-length distribution of the system can be
obtained by appropriate mathematical calculations. In this case, we can
calculate the molecular weight averages and distributions. Inversely, and
equally important from our perspective, experimental determinations of the
molecular weights and molecular-weight distributions allow a determination
of the probability of propagation, P, which is the characteristic quantity
controlling the ideal polymerization process we are using as a model in this
case. More complicated polymerization models lead to different probability
distribution functions as we will see.

These chain-length probabilities must sum to 1 (i.e., the complete condition, meaning
that something must happen):
0(3

P (n) -- 1

(1.7)

n=l

because n must have some value between 1 and infinity. This relationship indicates
that the individual P (n) are simply the number fraction of individual molecules in
the mixture:

P(n) --

N.

(1.8)
N
where Nn is the number of molecules of length n, and N is the total number of
polymer molecules in the polymerization batch.

Structure calculations using the probability distribution function

A simple example: degree of polymerization [1]
The structural variable in this case is the length of the polymer chain or
the degree of polymerization, DP, defined as the number of similar structural units linked together to form the polymer molecule. This number is
converted to molecular weight by multiplying by the molecular weight of a
single structural or repeating unit. Measurement of the molecular weight of

the polymer system by any physical means yields numbers representing the
weighted averages of the DPs of all the molecules present.
A colligative property measures the number of molecules in solution. A colligative
property measurement, for example, osmotic pressure, freezing-point depression,


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6

Theory of polymer characterization

boiling-point elevation, or vapor pressure lowering, of a polymer solution yields a
'number-average' DP or molecular weight (simply by multiplying by the molecular
weight of a single repeating unit). The effective DP measured is the sum of the DPs
of all the molecules divided by the number of molecules present.
1N1 + 2N2 + 3N3 + 4N4 + - - .

DP. =

DP

-

NI -+- N2 -+- N3 + N4 -k . . .
Z

nNn

(1.9)

(1.10)

where Nn is the number of molecules present whose DP is n. The N. - N is the total
number of molecules in the system. The number of molecules having a specific length
Nn corresponds to the probability of finding molecules of this length multiplied by
the total number of molecules
Nn -- N P ( n ) -- Np~-I(1 - P)

(1.11)

Substituting Eq. (1.11) in Eq. (1.10) yields
nNP(n)

(1.12)

DP = - - -

N
Substituting for P(n)
DP-- Znpn-l(1

- P) -( 1 - P)

(1.13)

Equation (1.13) demonstrates that a measurement of the average degree of polymerization, DP, allows a determination of the probability of propagation for this model
of chain polymerization. The DP can also be written
DP =

No
1
=
N
(l-P)

(1.14)

where No is the number of monomer units at the start of the polymerization, so
N-

No(1 - P)

(1.15)

Therefore, the number distribution function can be written
Nn -- No(1 - p)2pn-1

(1.16)

The number distribution function is plotted in Fig. 1.1.
With knowledge of P, the derived probability distribution function can be used
to calculate the other parameters of the polymerization, including the various types
of average molecular weights such as the weight and z average and the moments
of the molecular-weight distribution. For this simple ideal polymerization model, a
determination of P is all that is required for a complete structural evaluation because
the only structural variable is the length of the chains.


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Structure calculations using the probability distribution function

7

N

Chain

Length,

n

Fig. 1.1 The number distribution function as a function of chain length.

Number-average molecular weights
For a monodisperse system, the molecular weight is given by
W
M~ = ~
N

(1.17)

where W is the weight of the sample in grams and N is the number of molecules. For
a polydispersed system,

W -- ~

N~M~

(1.18)

N~

(1.19)

and
N -- Z

so the number-average molecular weight is given by
oo

~NnMn
Mn = n=0o ~

(1.20)

n=O

The number-average molecular weight is an average based on the number of
molecules, Am, of a particular size, Mn, taken over the total number of molecules.
This definition of the number-average molecular weight in grams contains Avogadro's number of molecules and on this basis is entirely consistent with the definition of molecular weight for a monodisperse molecular species.
Now, for a single repeating unit, it is obvious that

M,, -- nMo

(1.21)

where M0 is the molecular weight of the repeating unit. The number distribution
function can be used in these equations to calculate M~ in terms of the probability of

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8

Theory of polymer characterization

condensation, P,

M0
m~ = 1 - P

(1.22)

Note that the number-average molecular weight is the molecular weight of the repeating unit divided by the probability of termination, (1 - P). So a determination of
M~ yields an evaluation of P, the probability of propagation for this simple model.

Weight-average molecular weight measurements
Averaging on the basis of weight fractions, w,, of molecules of a given mass,
M~, gives a description of the weight-average molecular weight. For example, with
light-scattering measurements of polymer solutions, the effect is proportional to the
molecular weight of the molecules in solution, so a weight-average molecular weight
is measured. The weight-average molecular weight is given by
OO

OO

Mw = n=0

= n=0

O0

(2O

n =0

n =0

(1.23)

In Fig. 1.2 is shown the wt% of molecules of n length versus molecular weight.
M w is shown as the maximum on the curve.
It is apparent then that
OO

Z

wn -- W

wn -- n M o N o

n=0

i

Mw

Wt%

Molecular Weight
Fig. 1.2. The weight-fraction molecular weight distribution as a function of molecular weight.

(1.24)


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Structure calculations using the probability distribution function

9

(1.25)

W - MoNo

Substitution and evaluation reveals
M w -- M 0 ( 1 -

P)

(1.26)

1-p
With this equation, from a measurement of the weight-average molecular weight, the
probability of propagation, P, can be determined.

Distributions of molecular weight
The distributions of molecular weight that can be measured with gel permeation

chromatography and sedimentation can also be calculated for the simple model given,
and in fact correspond to the molecular distribution functions previously described.
Moment analysis is commonly used to characterize the distributions, where in our
case the first moment is the number-average molecular weight, M~; the second moment is the weight-average molecular weight, M~; and the third moment is the M z ,
the z-average molecular weight. The ratio of the weight-average to number-average
molecular weights is termed the dispersity.
It can be recalled from Eq. (1.6) that P ( n ) - (1 - p ) p , - 1 ; then one can write
d[P(n)]

= In P
(1.27)
dn
This indicates that a plot of In chain length distribution [ln P(n)] vs. n should give a
straight line with slope In P [12].

Chemical heterogeneity in the polymer chain
Consider a copolymer chain with only two different structural elements, A and B;
the number of different chain species possible is 2n, where n is the polymerization
index or the number of units in the chain, whichever is greater. The value of n is generally a large number, > 10,000. Consequently, the number of different possible chemical structures is enormous. There is no point in trying to completely define the spatial
coordinates of the atoms with such a large number of possible structures. To make
matters even worse, most real synthetic polymers have more than two different structural elements in the same chain, so the possible number of structures is even larger.
Some of the possible structural variables found in synthetic polymers are as follows:
9 chain length: molar mass of polymer chain reflecting variations in number of
repeating units in the chain;
9 c h e m i c a l defects: impurities in feed, monomer isomerization, and side reactions;
9 e n c h a i n m e n t defects: positional, stereospecific, branches, and cyclic isomers;
9 chain conformations: stiff ordered chains and flexible amorphous chains;
9 m o r p h o l o g i c a l effects: crystal phases, interfacial regions, and entanglements.

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lO

Theory of polymer characterization

This list of structural possibilities is long, and some polymers exhibit a number of
these structural variables simultaneously. Hence the total number of possible structures for a single chain is very large, indeed.
The problem is further complicated by the fact that synthetic polymers exhibit
chemical heterogeneity along the chain. Chemical heterogeneity reflects the distribution of the structural entities in the chain. The nature of the chemical heterogeneity
influences the ultimate properties of the polymer as it produces heterogeneity in the
molecular mobility of the chains.
Let us consider the case of a copolymer of two monomer units A and B. A
simple copolymerization model is represented by the following reactions and their
corresponding rates in terms of kinetic rate constants and concentrations of reacting
species:
Terminal group

Added group

Rate

Final group

"~A*
~'B*
~A*

[A]
[A]

[B]

kAA[A*][A] AA*
kBA[B*][A] BA*
kAB[A*][B] AB*

~B*

[B]

kBB[B*][B] BB*

where A* and B* are the propagating terminal species. The amount and type of
chemical heterogeneity of the copolymer chain is determined by the relative chemical
reactivity of the monomers A and B with the propagating terminal groups ~-,A* and
B* (more about this later in the chapter).
The systematic types of distributions of structural elements that can occur are as
follows:
Type

Nature of polymerization process

Random
Skewed toward alternation
Skewed toward blockiness
Block (A)N-- (B)M

No chemical correlation
A - B chemical correlation
A - A or B - B chemical correlation

A - A followed by B - B chemical correlation
by change of monomers
Exclusive A - B chemical correlation
Fluctuations in process variables

Alternating [ ( A ) l - ( B ) I ] ,
Irregular

These structural distributions are determined by the nature of the polymerization
process; that is, the random distributions arise from a copolymerization which shows
no differences in chemical reactivity between the chemical monomers A and B and
the growing chain ends A* and B*. The skewed toward alternation sequence distributions are nonrandom systematic structures determined by differences in chemical
reactivity between the chemical monomers A and B and the growing chain ends
A* and B*. The block structure is produced by chemical design (monomer feed)


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Characterization of polymer microstructure

11

of the polymerization process in which monomer A polymerizes to depletion and
monomer B is added. The alternating ordered distribution can only be obtained when
monomer A does not react with A* and B with B*. The irregular distribution reflects
the nonstationarity of the polymerization process arising from fluctuations in the
polymerization process variables such as temperature, pressure, flow, monomer feed,
etc.
The chemical sequence heterogeneity influences the characterization of the polymer in two ways:
(1) The chain structure is highly variable because the polymerization process is

a statistical process determined by probability considerations. Thus, polymer
samples are always multicomponent complex structural mixtures as well as made
up of different chain lengths.
(2) Detailed molecular pictures of the structure are not possible because our experimental measurements are going to provide only some weighted average structure
which depends on the nature of the response of the measurement technique.

Characterization of polymer microstructure [3]
Structural model of the polymer chain

For a microstructural model of the polymer chain, we use a model made up of
connected repeating units of similar or different structures. Letters A and B designate
the different structural types of the repeating units.
The microstructure of the chains are structurally presented as sequences of similar
or different units, that is A, AA, AAA, B, BB, BBB, AB, AAB, BBB, etc. Additional
structural components can be indicated by the use of additional letters, C, D, E, and
so forth. The application of this model to copolymers A and B and terpolymers A,
B, and C is obvious. Positional, conformation, and configurational isomerism as well
as branching and crosslinking are considered as copolymer analogs although they are
not generated by copolymerization.
In polymer characterization, the goal is to generate the structural sequence distribution function of the polymer chain in order to calculate the structurally significant
information, which represents the chain and can be used to correlate with the performance properties of the system.
Our aims are:
9 to relate these average sequence structures to the performance properties
of the polymer under consideration, and
9 to seek the polymerization mechanism and parameters that generate the
sequence structure in order to optimize and control the polymer structure.


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Theory of polymer characterization

12

For simplicity, we begin with the experimental measurements that are possible on
the microstructure of a polymer chain made up of only two structures, A and B. The
following is a portion of such a chain:

Sample copolymer chain
-,-.-,-,-.-,-.-,-.-,1

2

3

4

5

6

7

8

9

10

which is a chain of 10 repeating units made of A and B. We assume that this portion

of the chain is representative of the complete chain.

In mathematical terms, this assumption is called the stationary condition,
that is, the distribution of the two structural elements of the chain does not
change as the polymerization proceeds. We could have selected any length
of segment that we desired, but the counting process becomes tedious if the
segment is too long, and we cannot demonstrate the various points if the
chain is too short.
Again for simplicity, we also assume that the molecular weights are sufficiently
large so that the end groups need not be taken into consideration.

Measurement of polymer structure using composition
If spectroscopically we have infinite structural resolution or the ability to detect
the differences between the two structural elements A and B, we can measure the
individual number of A and B structural elements in the polymer. That is, we can
count the relative number of A and B elements taken one at a time, which we will
term mono-ads or 1-ads and write N1 (A) as the number of mono-ads of A. Using the
perfect resolution of the human eye, in this case for the model copolymer segment
shown previously, the number of the two types of mono-ads are
N1 (A) = 4

N~ (B) = 6

(1.28)

Spectroscopically, the number of mono-ads cannot actually be counted, but the
fraction of units in the chain can be measured. This result can be expressed in terms
of the experimental number fraction for mono-ads (A) and (B) by dividing by the
total number of segments (N = 10). So
(A) =

NA

(B) =

NB

(1.29)

N
N
and for our model copolymer system
(A)--

NA
4
N = 10 = 0 " 4

(B)--

NB
6
N = 1--0=0"6

(1.30)


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Characterization of polymer microstructure

13

This result, (A) or (B), is the fractional composition in terms of A and B units
of the polymer when we assume that the 10-segment portion of the polymer is
representative of the total chain. Also,
(A) + (B) = 1

(1.31)

as is required because for our model copolymer only A and B repeating units are
allowed (end groups are ignored as the molecular weight is assumed to be large).

Measurement of polymer structure using dyad units
Chain
Dyads

A - B - B - A - B - B - B - A - B - A - I - A - next sequence
AB-BB-BA-AB-BB-BB-BA-AB-BA-AA-

On a slightly more sophisticated level, we can analyze this 10-unit segment by
counting sequences two at a time, that is, counting the number of dyads (2-ads) in the
chain. The final AA dyad arises because the adjacent segment that is identical to this
segment starts with A. There are four possible types of dyads, AA, BB, AB, and BA,
and for this chain
N2(AA)
N2(BB)
N2(AB)
N2(BA)

=
=
---

1
3
3
3

Can we truly experimentally distinguish between the heterodyads AB and BA? In
this particular case they are differentiated with our eyes by the direction of the counting (left to right), but what if the chain segment were reversed? The AB dyads would
become BA dyads and vice versa. Which direction of the chain is the proper one?
Clearly, both directions are equally likely, and we will never know from examination of the final polymer chain in which direction it grew during the polymerization.
Therefore, AB and BA are equally likely outcomes and so [4]
N2 (AB) -- N2 (BA)

(1.32)

The equality saves the day for spectroscopic measurements, because no
known method of spectroscopic measurement can distinguish between AB
and BA, and so from a spectroscopic point of view they are indistinguishable.
Consequently any practical experimental measurement of the AB dyad units measures the sum Nz(AB) + Nz(BA), which we will designate as:
N(AB) = N2(AB) + N2(BA)

(1.33)


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Theory of polymer characterization

14

The experimental number fraction of each dyad (AA), (BB) and (AB) can be
determined by using N2 (total) = 10.
(AA) =

(BB) --

(AB) --

N2 (AA)

(1.34)

N
Nz(BB)

(1.35)

N
N: (AB)
N

+

N: (BA)

(1.36)

N

For our 10-segment polymer chain
(AA) -

(BB) --

(AB) --

Nz(AA)
N
Nz(BB)
N
N2 (AB)
N

= 0.1

(1.37)

= 0.3

(1.38)

4-

N: (BA)
N

= 0.6

(1.39)

The experimental completeness condition is

(1.40)

(AA) + ( A B ) + ( B A ) + (BB) -- 1
is satisfied. Or, in terms that can be measured spectroscopically

(1.41)

(AA) + ( A B ) + (BB) -- 1

Measurement of polymer structure using triad segments
We can now proceed to dissect the polymer structure in terms of higher n-mers
in spite of the fact that we might introduce a certain amount of boredom with the
process. Let us determine the structure of the polymer chain in terms of triads or
3-ads, that is, counting the units as threes. For triads, there are a total of eight
possible structures.

Chain
Triads

A- B- B- A- B- B- B- A- B- A-I-

A - B - next sequence

- AB B - B B A - B AB - AB B - B B B - B BA - B AB - AB A - B A A - A A B -

Once again, the final two triads are recognized by the adjoining segment that corresponds to the first two units in the beginning of this segment. Well, how many do