Visualizing Molecular Weight Averages and Distributions Ferdinand Rodriguez School of Chemical Engineering, Olin Hall. Cornell University, Ithaca. NY 14853 With rare exceptions, all real polymers are polydisperse, that is, they consist of a mixture of molecules with the same chemical composition but with various sizes. Some natural polymers are essentially monodisperse. Collagen is one example. Svnthetic ~ o l v m e r scan be made bv methods that yieid theF'oisson disiribution, which is almost distinguishable from a monodis~ersesvstem. When a distribution is made up of a popula$on with a mole fraction N(x) of each species of size x, various averaae values of size can be defined. The two most commonly i s e d are x, and I,. If x is the number of repeat units in a homopolymer, i t is called the degree of polymerization. The molecular weight (or mass), MW, is the product of x and the molecular weight - of a r e ~ e a t unit. The number-average degree of polymerization is defined as total moles of monomer in system x, = I E ~ N ( N E N ( x )=l per mole of polymer (1) The weight-average degree of polym&ation
is defined as
X, = IXX~N(X)IIIZXN(X)I (2) The polydispersity index, I,, is the ratio x,/x.. A highmolecular-weight, synthetic polymer with a measured I, of 1.04 would be regarded as possessing about as narrow a distribution as is readily attainable. However, even a population consisting of equal molar quantities of two species differing in size by a factor of 1.5 would have the same polydispersity. That is,
+ 0.5(1.5y) = 1 . 2 5 ~
x, = 0 . 5 ~
Dlstrlbutlons A distribution of molecular sizes is a complete description of all the molecules making up a population encompassing all values of x from 1to infinity. Two distributions encountered in real situations arise' from the situations where a polymerization is conducted in which 1. All polymer chainsare started at the same time.Theadditionof a
bifunctional monomer to each "seed" is equally probable, but the addition takes place only on a seed or a chain emanating from a seed that was present at zero time. This results in the Poisson distribution. The seeded polymerization of anionic polymer chains made from styrene starting from butyllithium "seeds" is a practical example. 2. All bifunctional monomer and polymer chains resulting from additions of monomer are equally reactive. A reaction between two monomers, two polymers, or a monomer with a polymer of any size all are equally probable. This results in the "most probable" distribution. The ~olvcondensationof 3-hvdroxwrooan, -. ~. oic acid rs a n example ~Hchdimer,trimer, and higher multiple still wntalns one reactive hydrmyl group and one reactive carboxd group no matter how many ester groups haw heen formed. ~
Paper presented at ACS Meeting, Miami Beach, 1985. Flory, P. J. Principles of Polymer Chemistry; Cornell University: Ithaca. NY, 1953; Chapter 8.
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In either case, the distribution can be described in terms of the mole fraction of x-mer, N(x), which is the same as the probability that a randomly selected molecule from that population will have a degree of polymerization equal to x. There is only one additional parameter in the two theoretical distributions being considered here. The parameter can be expressed as ihr number-average degree of polymeri7ation, x,. For the Poisson d~stribution,the theoretical modrl is
Thus the mole fraction of monomer, x = 1, when a population of x. = 10 is considered, will be e-9 (1.23 X lo-'), whereas the mole fraction of decamer, x = 10, is much larger, 0.132. For the "most probable" distribution, the model is N(z) = (xn - I ) ~ ' ~ ' ~ I ( . T ~ Y
(4)
In this distribution, the mole fraction of monomer is always the greatest. For comparison with the Poisson, consider again the case where x. = 10. Now, when x = 1, N(1) = 0.100, and when x = 10, N(10) = 0.0387. Slmulatlon of the Polsson Dlstrlbutlon In this simulation, we use 12 "seeds". Each seed represents a molecule with one monomer unit (x = 1). Now we wish to add a total of 24 more monomer units. There now will be a total of 36 monomer units in the population, but still only 12 molecules. I t makes no difference whether each seed now has three units or whether 11seeds are still monomeric and only one seed has a total of 25 units. The x, will he 3.0 in either case. However, x, will not be the same. T o add the monomers in a random fashion, two dies are rolled for each unit. The seed is selected by letting the value of die A represent seeds 1through 6 when die B shows an even number and seeds 7 through 12 when die B shows an odd number. The first few tosses of the dies might yield: Die A
Die 8
Seed selected
3
4 2
10
5 6
3
8 3
An actual example is difficult to compare directly with the theoretical model because of the limited number of seeds involved (Table 1, Fig. 1). A cumulative mole-fraction plot is less harsh a test (Fig. 2). Also shown are the "experimental" results for an additional 24 monomers (for which x. = 5). The polydispersity index I, is not far from the theoretical even for these small samples (Fig. 1). The theoretical I, is
I t can he seen that I, rapidly approaches unity as x, increases. When x, = 10,Ip = 1.083. Slrnulatlon of the "Most Probable" Dlstrlbutlon One way of illustrating the generation of a most probable distribution, MPD, is to use any item that is bifunctional and can be attached only to another unit (not to itself). Alligator clips and pop-it heads are two such items. These can correspond to a hydroxy acid which "attaches" by formation of an ester unit,
Table 1.
x
Simulating the P o i s o n Dlstrlbutlon: Ordered Frequency for the Distrlbutlon Generated (Fig. 1 )
Table 2. Slmulatlng the "Most Probable" Dlstrlbutlon: Ordered Frequency for the Dlstrlbutlon Generated (Fig. 3)
Calc'd from Eq 3b (with x. = 3) N(x) EN(*
Distribution of Figure 1. NO.01 molecules N(x) XN(x)
x
1 2 3 4
37 14 5 2 0
5
6 7
0.617 0.233 0.083 0.033 0.000 0.033 0.000
2
0
0.617 0.850 0.933 0.966 0.966 0.999 0.999
12
Total Calculatedla generated dlstrlbutlon. x, = 3.57 and b = 1.19. 'Caiculated bomeq 5 wnh x. = 3, x, = 3.66, and b = 1.22.
Cab'd from Eq 4b (with x. = 5/31 N(x1 m d
Disbibution of Figure 3. NO.of molecules N(x) xN(d
~Calcuiatedfor generated distribution: x. = 2.42 and b = 1.45. ~Calculstedlromeqs 2and 4wlm & = 513. x, = 2.33, and b = 1.40
mole Fract.
o
0.51
( 1
o'---d-= Seed Number Figure I. The Poissondistributionis simulated by random addition of 24 units to 12 starting molecules (seeds) to give x. = 3.
2HO-R-CO-OH monomer, x = 1
-
Xn= Polydispersity Index, I p :
Xn
Simulation
Equation
119
1.22
1.14
1.16
5 1
2
3
4
5
6
7
8
Degree of Polymerization,
9 x
1
0
Figure 2. Simulation of "experimental" Poisson distribution (symbols) compared to mathematical model (lines) for population of 12 molecules. Also shown is the result of adding 24 more units to give & = 5
+ HOH
HO-R-CO-0-R-CO-OH dimer, x
A
=2
(6)
Since the dimer, too, is a hydroxy acid, i t can react in the same way as a monomer. If a beaker with 100 pop-it beads is "reacted" truly at random, each joining corresponds to one ester group being formed. A maximum of 100 reactions can take place. The fraction of the total possible reactions that have taken place is p, which, in turn, is related to x, by Also, I t is hard to be completely objective in this kind of simulation since the demonstrator can distinguish rather easily hetween monomer, dimer, etc. A table of random numbers gives a less subjective generation of the MPD. A list of numbers from l to 100 is made up, each representing a reactive bond. They can be pictured as monomer units arranged in a circle (Fig. 3). The first two digitsof eachentry in the table of random numbers is used to choose the bonds that react. Startine anv olace in the table and proceeding in some methodical Fashion (down columns, across row. or skiwoinelines). .. . . eachentrv" gives the location of a reaction. If the first number encountered is 62, it signifies that bond number 62 in the list is reacted. ioinina- toeether the two monomer units on either side of 62:~s an example, after 40 reactions have taken d a c e (discarding numbers that reappear after once being &acted), the number of x-mers can be evaluated (Table 2). In Figure 3;an isolated circle is a
-
Figure3. The most probable distributionis simulated by selectinga succession of monomer units numbered from 1 to 100 by their occurrence in a table of random, two-digit n ~ m b e r s . ~
dimer. two circles in succession is a trimer. and so on. In this example, since 40 reactions out of a possible 100 have taken ulnce. o = 0.40andx. = 51.1. Thecumulntivedistribution can be compared with tdkory (Fig. 4). Another way of using the table of random numbers is to Abramowitz, M.; Stegun, I. A., Eds. Handbook Functions: Dover: New York. 1965; p 991. Volume 64
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June 1987
489
Table 3.
Vlsuatlzatlon
of a Typlcal Dlstrlbutlon (Flg. 5)
A typical "rounded-otV distribution simllar N(x) values from to a "most-probable" distribution' eq 4 wimp = 213 NO.Of x molecules N(4 xN(x) .?~(x) N(x)
1 2 3 4 5 6 7 8 9
10 7 4 3
Totals
1 1 1 1
0.333 0.233 0.133 0.100 0.067 0.033 0.033 0.033 0.033
0.333 0.467 0.400 0.400 0.333 0.200 0.233 0.267 0.300
0.333 0.933 1.200 1.600 0.667 1.200 1.633 2.133 2.700
0.333 0.222 0.148 0.099 0.066 0.044 0.029 0.020 0.013
30
0.998
2.933
13.399
0.974'
2
aCalculated far typical distribution: h = 2.93,x. = 1339912.933 = 4.57.and b = l 66
a PIUSiarser r term!
1.0 Cum.
mole Fract. Figure 5. Average molecular size of population in Table 3 visualized as the balance paint of a weightless beam. (a) with number of molecules at each x, and (b) with mass of moleculesateach x. An actual model u*s3/,in. steel nuts for monomer units and a foamed polystyrene beam.
0.5
EN(x)
0 Degree
of Polymerization,
x
Figure 4. Simulation of "experimental" most probable distribution (symbols) compared to mathematical model (line) for population of 100 monomer units and X. = 513.
cause random scission. In either case, the process leads to a "most probable" distribution no matter how narrow or how broad the initial distribution might have been. For example, start with aloop of 250 randomintegersfrom a table or a computer generator. Each integer represents one repeat unit in a single polymer molecule. 1. Assume that scission takes place after each integer k, say k = 9. The original loop now is broken up into about 25 or so individual chains with an average of ahout 10 units each.
choose any combination of numbers that represent reactions or "bits". For instance. if all the sinele dieits in the first column of such a table kxcept 0 and f a r e regarded as hits, then onlv two 0's. two 9's. or a 0 and a 9 in succession represent unreacted monomer. This game has the advantage of beina carried through a greater number of units since the numbe; of monomersat tKe start is not specified (like the 100 in the earlier example). A very rapid generation of an entire distribution can be accomplished when marbles are used. If 60 white and 40 black marbles are mixed together and then poured out into a group, the situation is just like the random number table. Let black marbles represent reacted groups. Now isolated black marbles are dimers, two in succession constitute a trimer, etc. This method of eeneratine the MPD is verv convenient for a lecture demon&ation, whereas the random number table is suitable as a homework assignment. The same random number table game can be played t o simulate random chain scission. Radiation of some nolvmers . . with gamma rays or energetic electrons leads almost exclusively to random scission. Polv(methvlmethac~late). . .DOly(isohurylenej, and pdy(butyient. suifonej are &an,ples:' Hydrolysisof a polyester suchas poly(glycolir m d J also may
2. Repeat the scissioning with integers k plus j where j is, say, I . Now there should he about 50 chains with an average of about five units in each. Usuallv a sinale simulation will illustrate the principle. Iteration and e&nhin&on of results from a number of such experiments will give a very close simulation to the most probahle distribution.
Schnabel. W. In Aspects of begradation and Stabilization of Polymers;Jellinek. H. H. G..Ed.: Elsevier: New York. 1978: Chapter 4.
Equation 10 can be derived from eq 9 by simple rearrangement and substitution in the following way:
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Journal of Chemical Education
Mechanical Analogue lor Obtalnlng Averages A histogram for adistribution (Table 3, Fig. 5a) is simply a plot of the number of molecules of various sizes versus size itself. The number-average molecular size can be visualized as the balance point when the distribution is placed on a beam which has no mass of itself (Fig. 5a). Almost every student will recognize the principle that one mass a t distance of 10 units exerts a torque equal to 10 masses a t a distance of one unit on the other side of the balance point. Mathematically, i t follows that the balance point (x.) can be found by summing up the products x.N and dividing by the sum of N . I .
= IXx.MI(XM
(9)
From the viewpoint of the balance beam, i t may be easier to see that
and
Once the balance point is determined, the student can visualize it more clearly by summing up masses, x.N times distance from the balance point, x - x,, in analogy wit,h the previous example.
0 = (Xx.Ml(XM - x. =
- Xx,.MI(XM
= (X[x- x.l.Ml(XM
(12)
While the number average is represented hy the balance point when the number of molecules is arranged on the beam, the weight average becomes apparent when the molecules are replaced by total mass for each x (Fig. 5b). One molecule of x = 6 is the equivalent mass of six molecules with x = 1 at the same distance from the balance point. The balance point for the entire distribution, x, can he determined by summing up masses and distances in the following way:
Unfortunately, this simple mechanical analogue is not easily extended to higher averages. Of course, there are many mathematical uses of higher moments in mechanics, but the student who is already familiar with mechanics, especially beyond freshman physics, seldom has difficulty with molecular averages.
Volume 64
Number 6
June 1967
491
