4.7. Molecular Weight
The atoms in polymer chains, as in metals and all other materials, consist of electrons orbiting a nucleus containing protons and neutrons. The atomic mass (weight) of an element is the sum of the masses of the protons and neutrons in its nucleus, since the mass of the electrons is several orders of magnitude smaller and therefore negligible. Note that although atomic mass is the more appropriate (and ISO standard) term, by common usage atomic weight is most often found in polymer literature. The number of protons defines the element, but for some elements several isotopes are possible, all having the same number of protons but different numbers of neutrons. The atomic mass of such atoms is given in the periodic table as the weighted average (according to abundance in nature) of the atomic masses of the naturally occurring isotopes. A proton and a neutron have the same mass to three significant digits and the atomic mass unit (amu) is defined on the basis of Carbon-12, the most common isotope of Carbon
containing 6 protons and 6 neutrons, with an atomic mass of exactly 12.000 amu. The atomic mass shown in the periodic table for Carbon is slightly higher (12.011) as it accounts for small amounts of the isotope 13C.
Since one does not typically work with single atoms or molecules, quan¬tities of chemical substances are given in moles. A mole of an element is defined as 6.02214 x 1023 (Avogadro’s number) atoms; a mole of a given type of molecule is 6.02214 x 1023 molecules. Avogadro’s number is de¬fined to provide a simple conversion to grams: 6.022 x 1023 atoms (or molecules) have the mass in grams of the atomic mass of a single atom (molecule). For example, 1 mole (6.022 x 1023 atoms) of 12C has a mass of exactly 12.0g. The conversion is therefore
6.02214 x 1023 amu = 1 gram
or 1 amu = 1.66054 x 10-24 g
As an example, consider a mole of water molecules (H2O) which contains 6.022 x 1023 atoms of oxygen and 2 x (6.022 x 1023) atoms of hydrogen. The atomic masses of oxygen and hydrogen are 15.9994 amu and 1.0079 amu respectively. Therefore a mole of water has a mass of 2x(1.0079g) + 15.9994g = 18.015 grams. This example also emphasizes that moles are the necessary units to use for chemical reactions as the proper number of atoms must be tracked: e.g. one mole of oxygen and two moles of hydro¬gen can be combined to form 1 mole of water; 1 gram of oxygen and 2 grams of hydrogen are not in the proper ratio to form a gram of water ow¬ing to the differing masses of the elements.
Historically, the terms gram-atom and gram-molecule were originally coined to refer to the mass in grams of Avogadro’s number of atoms or molecules, respectively; with the introduction of the term mole, these early terms are less used but can still be found in the literature. The term “mo¬lecular weight” is by far the most common expression used to refer to the mass of a molecule. A 1992 ISO standard dictates that the term “relative molecular mass” should replace “molecular weight” in all publications, but in practice adoption of the terminology has been slow. The word “relative” is used in the expression to convey that the mass is given relative to 1/12 the mass of an atom of Carbon-12. Since the mass of 12C is exactly 12.00amu, the relative molecular mass provides the mass of a molecule in amu although technically the quantity is unitless. Another term sometimes seen is “molar mass” or “relative molar mass”. Both of these latter terms refer to the mass per mole of a substance and are expressed in grams/mole.
To give an example for a polymer, a single polyethylene chain with a degree of polymerization of 104 (or 104 mer units) has a relative molecular mass (molecular weight) of
Mass of 1 PE chain: 104 (2 x 12 + 4 x1) = 280,000 amu
A mole of polyethylene chains, where each chain is 104 mer units long, has a molar mass of
Mass of 1 mole of PE chains: 280,000 grams (or grams/mole)
neglecting chain end effects. Note that the molecular mass of a chain end (or at a branch point) is not the same as the molecular mass of a mer unit but the difference is neglected because the effect is small in terms of the total molecular mass of a chain.
While “relative molecular mass” is the official and more correct termi¬nology for polymers (as used in McCrum, 1997), in the following the term molecular weight will be most often used as is common in many polymer texts.
A useful term to describe the extent of polymerization in polymers is the “degree of polymerization” (DP) which is defined as the number of mer units per chain or,
M
n ?
Mr ??DP (4.6)
where M is the molar mass (weight) of a chain and Mr is the molar mass (weight) of a mer or repeat unit. (Number average and weight average de¬grees of polymerization are also used as will be evident directly.)
The degree of polymerization or the length of a polymer chain is an in¬dicator of the nature and mechanical characteristics of a polymer com¬posed of similar length chains. The following table illustrates the relation¬ship between chain length and the character of a polymer at 25 ?C and a pressure of one atmosphere.
134 Polymer Engineering Science and Viscoelasticity: An Introduction
Table 4.6 Degree of polymerization – phase relationship (data from Clegg and
Collyer (1993), p. 11)
Number of –CH2-CH2 Molar Mass Softening Character at
Repeat units per chain kg mol-1 Temperature 25°C and 1
(Degree of Polymerization °C at.
1 28 -169 Gas
6 170 -12 Liquid
35 1000 37 Grease
140 4000 93 Wax
430 12000 104 Resin
1350 38000 112 Hard Resin
It is now clear how to calculate the molecular weight of a single chain or of a mole of polymer chains of identical lengths. Unfortunately, however, the lengths of chains in a polymer vary greatly and depend to a large de¬gree on the circumstances and the manner in which the polymerization re¬action proceeds. That is, a wide distribution of chain lengths (DP’s or chain molecular weights) exist in a typical polymer as shown in Fig. 4.38. The distribution is seldom symmetrical and the breath of distribution var¬ies with the type of reaction. For example, the distribution is often quite broad for polyethylene while the distribution for polystyrene may be quite narrow (Fried, 1995).
Because of the distributed nature of the lengths of chains in a polymer it is necessary to define the molecular weight using an averaging process. The most common averaging processes used are the number average, the weight average and the z-average. Only the number and weight average methods will be described here. Both discrete and continuous distributions are possible. For
example the continuous distribution in Fig. 4.38(b) is obtained by drawing a smooth curve through the discrete distribution shown in Fig. 4.38(a).
For a discrete distribution the number average molecular weight is de¬fined as,
where Ni is the number of chains within an interval, Mi is the median (middle) molecular weight in an interval, k is the total number of intervals and N is the total number of chains. If a continuous curve is fit to the dis¬crete data such that N is given as a function of M, i.e., N=N(M), the sum-mations can be replaced by an integration to obtain (Kumar and Gupta, (1998)),
The product of the number of chains in an interval, Ni, and the molecular weight of an interval, Mi, equals the total weight of an interval. Exchang¬ing Ni in Eq. 4.10 by Ni Mi defines the weight average molecular weight and can be written as,
where M is the total molar mass of the sample. Some have likened the number average and weight average molecular weights to the first and sec¬ond moments of masses (or areas) in elementary mechanics courses. Such an analogy is appropriate if the number of chains, Ni, is replaced by a lever arm di with units of length. One text incorrectly relates the weight average molecular weight to a radius of gyration.
Consider the example where,
i Mi Ni
Interval No. g/mole of chains
in interval No. of chains in interval
1 5,000 2
2 15,000 4
3 30,000 5
4 50,000 1
The number average and the weight average molecular weight from Eqs. 4.10 and 4.11 will be,
(4.10a)
M n ??2(5,000)???4(15,000) ??5(30,000) ?1(50,000)
12 ??22,500 g/mole (4.10b)
Some experimental approaches separate the chains in a polymer into dis¬crete number or weight fractions. The number fraction, xi, is defined as the ratio of the number of chains in an interval to the total number of chains in the sample,
Ni (4.11)
N
and the weight fraction, wi, is the ratio of the total weight of the chains in an interval to the total weight of the sample,
Mi (4.12)
M
An example illustrating this approach is shown in the hypothetical distri¬bution given below,
Fig. 4.39 Size distributions of a hypothetical polymer
Using this definition, the number average molecular weight or weight av¬erage molecular weight can be written as,
k
Mn ??xi
??Mi (4.13)
i?1
k
Mw ??wiMi
? (4.14)
i?1
where all quantities are as previously defined and (4.13) and (4.14) yield identical results to (4.10) and (4.12) respectively.
The number average emphasizes the importance of the smaller molecu¬lar weight chains while the weight average emphasizes the higher molecu¬lar weight chains. This is demonstrated in Fig. 4.40.
Fig. 4.40 Distributions of molecular weight in a typical polymer.
The ratio of the weight average molecular weight to the number average molecular weight is defined as the polydispersity index,
PDI??M w (4.15)
M n
which is often used as a measure of the breadth of the molecular weight distribution. Typical ranges in the PDI for polymers are shown in Table 4.7.
Table 4.7 Typical ranges of M w M n in synthetic polymers. (Data from Bill-meyer (1984), p. 18)
Polymer Range
Hypothetical monodisperse polymer 1.0
Actual monodisperse living polymers 1.01 – 1.05
Addition polymer, termination by coupling 1.5
Addition polymer termination by disproportionation 2.0
High conversion vinyl polymers 2-5
Polymers made with autoacceleration 5-10
Addition polymers made by coordination polymerization 8-30
Branched polymers 20-50
When M n is high and PDI is low there are more chance for entanglements which in turn increases strength and rigidity because the strain is lower for a given stress. When M w or PDI is high, chains are likely longer and the temperature resistance is increased.
Molecular weight is an important indicator of mechanical properties. For example the variation of tensile strength of a lightly crosslinked rubber is shown in Fig. 4.41 and the variation of the elastic modulus above the glass transition temperature is shown in Fig. 4.42. As may be observed, above the Tg the modulus becomes very small when the molecular weight is low but increases to a plateau when the molecular weight is very high. This plateau extends to relatively high temperatures until sufficient energy is input to begin to degrade cross-links and the backbone chain. This is of¬ten indicated by a change in color of the polymer due to charring. The rea¬son for the different behavior as a function of molecular weight is due to increased entanglements for higher molecular weights (Clegg and Collyer (1993)).
Tg Temp.
Fig. 4.41 The effect of molecular weight on the elastic modulus of an amorphous thermoplastic polymer above the Tg.
Fig. 4.42 Approximate tensile strength of a lightly crosslinked rubber as a function of number average molecular weight. (Data from Clegg and Collyer (1993))
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