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Wi Ni M i .
• Molecular weights that are important in determining polymer properties are the number-average, Mn (=1), the weightaverage, Mw (=2), and the z-average, Mz (=3), molecular weights.
Introduction to Polymer Physics
Prof. Dr. Yiwang Chen
School of Materials Science and Engineering, Nanchang University, Nanchang 330047
Chapter 4 Molecular weight
0
2 2
N ( M ) MdM
W (M ) dM M
1
Mw
N ( M ) M dM N (M )MdM
0 0
0
N ( M ) M dM W ( M ) MdM
0
Mz
0
W ( M ) M 2 dM W ( M ) MdM
0
M 0 W ( M ) M dM
i i i i
2 i
i
Wi M i
i
Mz
Viscosity-average molecular weight
zi M i
i
z
i
2 w M i i
i
ቤተ መጻሕፍቲ ባይዱ M
i i
1
i
3 n M i i
i
n M
i i
i
2 i
M Wi M i i
Mn
n M n
be as low as 1.06.
M M M M N (M )dM
2 2 n n 0 n
0
M
i
2
2M M n M 2 N (M )dM M 2 2M n M n M 2 M 2 n M n
i i
2
2
n
2
分子量分布宽度
and Pearson distributions, and standard probability functions, such as the Poisson and logarithmic-normal distributions.
Mn
0
N ( M ) MdM 0 0 N (M )dM
1
N(M) and W(M) are the total number and weight of molecular-
weight species in the distribution，if I(M) is the weightintegral function of molecular-weight distribution, then
i i i i
i
w
i
i
wi i Mi
W
i
i
Wi i Mi
1 Wi i Mi
M Wi M i i
1
When =-1, When =1,
M
1 Mn Wi i Mi
M Wi M i M w
i
KM
In general, is in the range of 0.5 to 1, therefore,
Number-average molecular weight
ni M i w Mn i Ni M i n ni i
i
Weight-average molecular weight
Mw
z-average molecular weight
w M w
i i i i
i
n M n M
i
1
W 1
i
wi ni M i
• For a discrete distribution of molecular weights, an average molecular weight, M, may be defined as NM
M N M
i i i i i i
4.1 Statistics of Molecular Weight of Polymers
Molecular-Weight Averages
高分子试样中若干种分子量不等的分子，其重量和摩尔数等各物理量之间的
关系为：
ni n w w
i i i
ni Ni n wi Wi w
N
i i
1
• Where Ni indicates the number of moles of molecules with a molecular weight of Mi and the parameter is a weighting factor that defines a particular average of the molecular-weight distribution. • The weight, Wi, of molecules with molecular weight Mi is then
estimated.
• Such mathematical forms include theoretical distribution functions derived on the basis of a statistical consideration of
an idealized polymerization, such as the Flory, Schultz, Tung,
M n M M w
• Since the molecular-weight distribution of commercial polymers is normally a continuous function, molecularweight averages can be determined by integration if the proper mathematical form of the molecular-weight distribution (i.e., N as a function of M) is known or can be
dI ( M ) W (M ) dM
I (M ) W (M )dM
0
M
I () W (M )dM 1
0
Polydispersity index
• A measure of the breadth of the molecular-weight distribution is given by the ratios of molecular-weight averages • For this purpose, the most commonly used ratio is Mw/Mn, which is called the polydispersity index or PDI. • The PDIs of commercial polymers vary widely. For example, commercial grades of polystyrene with a Mn of over 100,000 have polydispersities indices between 2 and 5, while
Mn
i 1
n
i 1
3 i 1 3 i
3
i 1
i
w M
i i 1
3
i
i
1 2 2 31250 1 2 2 10000 50000 100000
Mw
n M
i 1 i
2 i
n M
i
w M
i 1 i
3
w
i 1
3
10000 2(50000 ) 2(100000 ) 62000 5
polyethylene synthesized in the presence of a stereospecific
catalyst may have a PDI as high as 30. In contrast, the PDI of some vinyl polymers prepared by “living” polymerization can
重均分布宽度指数
多分散系数
2 w M Mw
M
2 2 w
w M w M w M z M w 1
2
2
d d
Mw Mn Mz Mw
分子量均一的试样 分子量非均一的试样
M z M w M M n
M z M w M M n
Example Problem • A polydisperse sample of polystyrene is prepared by mixing three monodisperse samples in the following proportions: • 1g 10,000 molecular weight • 2g 50,000 molecular weight • 3g 100,000 molecular weight • Using this information, calculate the number-average molecular weight, weight–average molecular weight, and PDI of the mixture. 3 3 • Solution ni M i wi
