CHAPTER
2.9 Typical Molecular Weight Distributions
As mentioned, different polymerization techniques yield different molecular weight distributions. There exist three typical molecular weight distributions and they are the Poisson distribution (anionic polymerization described in Chapter 12), exponen- tial (condensation polymerization described in Chapter 8) distribution, and log- normal distribution. Since the mathematical descriptions of these distributions are known, one can calculate the corresponding molecular weight averages.
In the Poisson distribution, the number fraction of chains withirepeating units is given by
ni
n 5e2aai
i! (2-46)
where n is the total number of chains and a is a constant. And i50, 1, 2,. . .. Substituting Eq. (2-46)into the original definitions of Mn andMw (Eqs. 2-8 and 2-14) and after some derivation,Mn5M1a, whereM1is the monomer molecular weight and Mw5M1(a11). The polydispersity index is 11M1=Mn: Obviously, the polydispersity index approaches 1 asMnincreases.
In the exponential distribution, the number fraction of chains with i repeating units is given by
ni
n 5e2bbi
i! (2-47)
where b,1 and i51, 2,. . .. The corresponding Mn and Mw are M1
ð12bÞ and M1ð11bÞ
ð12bÞ , respectively. And the polydispersity index is 22M1
Mn
. Here, the polydis- persity index approaches 2 asMn increases. It is obvious that the exponential distri- bution has a broader distribution than the Poisson distribution whenMnis low.
The log-normal distribution resembles the one shown in Fig 2.4;Mn,Mw, and the polydispersity index are given by the following equations.
Mn5Með2σ2=2Þ (2-48)
Mw5Meðσ2=2Þ (2-49)
Polydispersity index5eσ2 (2-50) whereMis the peak molecular weight andσ2is the variance.
Appendix 2A
Molecular Weight Averages of Blends of Broad Distribution Polymers
When broad distribution polymers are blended,Mn;Mw;Mz;etc., of the blend are given by the corresponding expressions listed in Table 2.2 but the Mi’s in this case are the appropriate average molecular weights of the broad distribution com- ponents of the mixture. Thus, for such a mixture,
ðMnÞmixture51 X wi
ðMnÞi (2-11a)
ðMwÞmixture5X
wiðMwÞi (2-13a)
and so on.
81 Appendix 2A
As a “proof,” consider a mixture formed ofa grams of a monodisperse poly- mer A (with molecular weight MA) and b grams of a monodisperse polymer B (with molecular weight MB). This blend, which we call mixture 1, contains weight fraction (wA)1of polymer A and weight fraction (wB)1of polymer B.
ðwAÞ15a=ða1bÞandðwBÞ15b=ða1bÞ The number average molecular weightðMnÞ1of this mixture is
ðMnÞ15 a1b a=MA1b=MB
(2-11b)
ðMnÞ15 a
ða1bÞMA1 b
ða1bÞMB
21
(2A-1) If mixture 2 is produced by blending c grams of A and d grams of B, then similarly
ðwAÞ25c=ðc1dÞandðwBÞ25d=ðc1dÞ while
ðMnÞ25 c
ðc1dÞMA1 d
ðc1dÞMB
21
(2A-2)
Now we blend e grams of mixture 1 withf grams of mixture 2. The weight fraction w1 of mixture 1 is e/(e1f) and that of mixture 2 is w25f/(e1f). The weight fraction of polymer A in the final mixture is w1(wA)11w2(wA)25wAand that of polymer B isw1(wB)11w2(wB)15wB.
w1ðwAÞ11w2ðwAÞ25 e e1f
a a1b 1 f
e1f c c1d
5wA (2A-3)
w1ðwBÞ11w2ðwBÞ25 e e1f
b a1b
1 f e1f
d c1d
5wB (2A-4)
The number of average molecular weight of the final blend is
ðMnÞblend5 wA
MA
1wB
M 21
by definition (2-11c)
Substituting
ðMnÞblend5 e e1f 0
@ 1 A a
a1b 0
@ 1 A
MA1 f e1f 0
@ 1 A c
c1d 0
@ 1 A
MA
2 4
1 e e1f 0
@ 1 A b
a1b 0
@ 1 A
MB1 f e1f 0
@ 1 A d
c1d 0
@ 1 A
MB
21
5 w1 a a1b 0
@ 1 A
MA1w2 c c1d 0
@ 1 A
MA
2 4
1w1
b a1b 0
@ 1 A
MB1w2
d c1d 0
@ 1 A
MB
21
5 w1
MAða1bÞ a 2 4
3 5
21
1 MBða1bÞ b 2 4
3 5
21
0
@
1 A 2
4 1w2
MAðc1dÞ c 2 4
3 5
21
1 MBðc1dÞ d 2 4
3 5
21
0
@
1 A21
(2A-5)
ðMnÞblend5 w1
ðMnÞ1
1 w2
ðMnÞ2
21
(2A-6) This is equivalent toEq. (2-11)withðMnÞI substituted forMi.
Similar expressions can be developed in a straightforward manner forMw;Mz and so on.
PROBLEMS
2-1 If equal weights of polymer A and polymer B are mixed, calculate Mw
andMnof the mixture:
Polymer A:Mn535;000; Mw590;000 Polymer B:Mn5150;000; Mw5300;000
2-2 Calcium stearate (Ca(OOC(CH2)16CH3)2) is sometimes used as a lubricant in the processing of poly(vinyl chloride). A sample of PVC compound containing 2 wt% calcium stearate was found to have Mn525000: What isMnof the balance of the PVC compound?
2-3 If equal weights of “monodisperse” polymers with molecular weights of 5000 and 50,000 are mixed, what isMzof the mixture?
83 Problems
2-4 CalculateMnandMwfor a sample of polystyrene with the following com- position (i5degree of polymerization; % is by weight). Calculate the vari- ance of the number distribution of molecular weights.
i 20 25 30 35 40 45 50 60 80 .80
wt% 30 20 15 11 8 6 4 3 3 0
2-5 What value wouldMz have for a polymer sample for whichMv5Mn? 2-6 Given that homopolymers formed by the following monomers (a e)
have the following molecular mass distribution:
wi Mi
0.05 10,000 0.25 50,000 0.20 80,000 0.20 100,000 0.15 150,000 0.10 200,000 0.05 500,000 (a) CH25CH(CH3)
(b) CH25CHCl (c) CF25CF2 (d) CH25CH(OH)
(e) CH25CH2
Calculate the number average molecular weightMn[10 marks] and the corresponding degree of polymerization of each polymer.
2-7 If 200 g of polymer A, 300 g of polymer B, 500 g polymer C, and 100 g of polymer D are mixed, calculate both Mnand Mwof the blend.
Polymer A Mn545,000; Mw565,000 Polymer B Mn5100,000; Mw5200,000 Polymer C Mn580,000; Mw585,000 Polymer D Mn5300,000; Mw5900,000
What are the polydispersity index and the standard deviation of the number distribution of molecular weight of the mixture?
2-8 The measured diameters of a series of spherical particles are shown as follows:
Number of particles Diameter (mm)
1000 2.5
1800 3.0
1700 3.2
1500 3.5
(a) Calculate the number average diameterðDnÞ:
(b) Calculate the volume average diameterðDvÞ:
(c) Calculate the weight average diameter ðDwÞ: [weight of a sphere α volumeα(diameter)3].
2-9 The measured diameters of a series of spheres follow:
Number of spheres Diameter (cm)
2 1
3 2
4 3
2 4
(a) Calculate the number average diameterðDnÞ:
(b) Calculate the weight average diameter ðDwÞ:[Weight of a sphere
~ volume ~ (diameter)3.]
2-10 A chemist dissolved a 50-g sample of a polymer in a solvent. He added nonsolvent gradually and precipitated out successive polymer-rich phases, which he separated and freed of solvent. Each such specimen (which is called afraction) was weighted, and its number average molecular weight Mn was determined by suitable methods. His results follow:
Fraction no. Weight (g) Mn
1 1.5 2,000
2 5.5 50,000
3 22.0 100,000
4 12.0 200,000
5 4.5 500,000
6 1.5 1,000,000
Assume that each fraction is monodisperse and calculateMn;Mw;and a measure of the breadth of the number distribution for the recovered poly- mer. (Note: This is not a recommended procedure for measuring molecular weight distributions. The fractions obtained by the method described will not be monodisperse and the molecular weight distributions of successive fractions will overlap. The assignment of a single average molecular weight to each fraction is an approximation that may or may not be useful in particular cases.)
2-11 The degree of polymerization of a certain oligomer sample is described by the distribution function
wi5Kði32i211Þ
85 Problems
wherewiis the weight fraction of polymer with degree of polymerizationi andican take any value between 1 and 10, inclusively.
(a) Calculate the number average degree of polymerization.
(b) What is the standard deviation of the weight distribution?
(c) Calculate thezaverage degree of polymerization.
(d) If the formula weight of the repeating unit in this oligomer is 100 g mol21, what isMzof the polymer?
2-12 Molecular weight distributions of polymers synthesized using the tech- niques of living polymerization and condensation polymerization can be described by the Poisson and exponential distributions, respectively, as shown in the following equations:
ni
n 5e2aai
i! ði50;1;2;. . .Þ ðPoisson distributionÞ
ni
n 5ð12bÞbi21 ði51;2;. . .Þ ðExponential distributionÞ where ni is the number of chains having a degree of polymerization ofi andn is the total number of chains. Here, a andb are constant. Note that Mi5iM1where M1 is the monomer molecular weight. In the case of the Poisson distribution,MnandMwcan be shown as follows:
Mn5M1a Mw5M1ð11aÞ
while the corresponding expressions for the exponential distribution are:
Mn5 M1
ð12bÞ Mw5M1ð11bÞ ð12bÞ
(a) Show that the polydispersity indices of the polymers prepared by the aforementioned polymerization techniques approach different limiting values asMnincreases.
(b) Using the standard deviation of the number distribution of molecular weight, show that polymers synthesized by the living polymerization technique exhibit a considerably narrower molecular weight distribu- tion than those by the condensation polymerization whenMnis large.
(c) Given that two samples of oligomers synthesized, respectively, by liv- ing and condensation polymerization techniques have the same Mn
and Mw values of 500 and 900, calculate the molecular weights of their monomers.
2-13 Given that the number distribution of the molecular weight of a polymer (fN(M)) is given by the following expression:
fNðMÞ5k1e2k2M
wherek1andk2are constants.
(a) Sketch qualitatively the distribution function.
(b) Sketch qualitatively the corresponding normalized integral number distribution curve (i.e., cumulative mole fraction against molecular weight).
(c) How would you calculateMnandMwof the polymer sample using the given distribution (show the relevant equations but do not do the calculations)?
(d) Assuming that you do not know how to do the calculations in part (c), can you determine the variance of the distribution if the polydispersity index of the sample is given? Why or why not?
(e) Given that the variance of the weight distribution (i.e.,σ2w) is given by P
iwiðMi2MwÞ2, show that σ2w=M2w5ðMz=MwÞ21 (note that P
iwiMi25MzMw).
Reference
[1] G. Herdan, Small Particle Statistics: An Account of Statistical Methods for the Investigation of Finely Divided Materials, second ed., Academic Press, London, 1960 (p. 281).
87 Reference
3
Practical Aspects of Molecular Weight Measurements
Now what I want is, Facts. Facts alone are wanted in life.
—Charles Dickens,Hard Times
In this chapter, analytical methods that are commonly used for the measurements of molecular weight averages (Mn and Mw) and molecular weight distribution will be described. It is interesting to note that it is possible to determine one of the average molecular weights of a polymer sample without knowing the molecu- lar weight distribution. This is accomplished by measuring a chosen property of a solution of the sample.