Predicting and Measuring the Sequence Distribution of Addition Polymers - Mathematical Approaches to Polymer Sequence Analysis and Related Problems - page 200

where s is the size of the sequence, MMD .s/ is the distribution of molar masses and

I XXXX is the normalized probability. MMD .s/ has the same dimensions of W XXXX ,

whereas I XXXX is dimensionless. In AB copolymers, X can take two values X D A

and X D B. In ABC copolymers, X can take three values X D A X D BandX D C.

A sequence made of three repeat units is called triad. Tetrads, pentads, hexads and
heptads are sequences made of four, five, six and seven repeat units, respectively.
W XXXX and

;

I XXXX are square matrices for dyads in binary copolymers, for triads in
ABC copolymers, for tetrads in ABCD copolymers, for L-ads when the length of
the sequence is L. In the other cases,

I XXXX are rectangular matrices.
The MMD can be measured by chromatography, by MS or by other means
[ 1 - 3 ]. MMD averages are very important. The number-average molar mass,

W XXXX and

Q n

,

is given by:

X m i N i . X N i

N Y

n D

(10.2)

Q w , is given by:

In a similar manner, the weight-average molar mass,

X . X m i N i

N Y w D

m i / 2 N i . ; (10.3)

where all summations span over all masses (from one to infinity). The above equa-
tions can be used to measure the MMD of a polymer sample by simply measuring
the abundance N i of each macromolecular chain with mass m i . The most impor-
tant MMD is the Schulz-Flory MMD, which is a decreasing exponential. The ratio
Q w = Q n

takes the value 2. The Schulz-Zimm MMD function is given by the product
of a power-law and a decreasing exponential:

.s/ D a nofa .s/ ˛ exp

MMD .s=/; (10.4)

where a nofa is a suitable normalization

factor. The MMD averages, computed using the definition, turn out to be:
N Y
n

˛ and are two adjustable parameters and

D .˛ C 1/= (10.5)

N Y w D .˛ C 2/=;

(10.6)

where is the mass of the repeat unit. In the case of poly(styrene), poly(methyl

methacrylate) and poly(ethyleneoxide) (see figure),

is 108, 100 and 44 g mol 1 ,

respectively

10.3 Free-Radical Copolymerization

Among chain-growth polymerization reactions (also referred to as addition poly-
merizations), free-radical copolymerization is the most common [ 1 - 3 ]. Figures 10.2
and 10.3 report in a concise manner the chemical reactions that occur in free-radical