1 a i

1/ a
Example:
ni
nM n
i i
10
10 20
10 10
Mi(×10-4) 30
Mn
i
10 30 10 20 10 10 104 20 104 10 10 10
10 302 10 202 10 102 Mw 104 23.3 104 ni M i 10 30 10 20 10 10
i
n M
i i
i
i
Wi M i
i
Z-average molecular weight
Mz
3 N M i i
N M
i i
i
2 i

2 w M i i
w M
i i
i
i
Viscosity-average molecular weight
Ni M Mv i Ni M i i
Method
Natural Proteins Anionic Polymerization Free Radical Polymerization
Step Polymerization Ziegler-Natta Polymerization Cationic Polymerization
2.0-4
2 n M i i
10 303 10 203 10 103 4 4 Mz 10 25.7 10 2 2 2 2 n M 10 30 10 20 10 10 i i
3 n M i i
Schematic of a simple molecular weight distribution, showing the various averages
• 分子量分布是聚合物最基本的结构参数之一：
• ①高分子材料加工条件的控制 如：熔体强度与弹性与样品中高分子量部分有较大关系
• ②高分子材料使用性质 如：拉伸强度和冲击强度与样品中低分子量部分有较大关系
• ③溶液性质 如：溶液粘度与样品中高分子量部分有）高分子材料才具有机械强度 极性聚合物 非极性聚合物 临界聚合度40 临界聚合度80
T f
1 M n
Latent heats of fusion per gram of solvent
Vapor-phase osmometry (VPO)
气相渗透压法
Vapor pressure lowering
Vapor pressure of the solution
M1
n1 m1
M2
n2 m2
…
… …
Mi
ni mi
mi ni M i
Mn
Number-average molecular weight Weight-average molecular weight
n M n
i i i i
i
Ni M i
i
Mw
mi M i
i
m
i

2 n M i i
Tb RT 2 lim c 0 c H v
1 K M M
K M Tb c c0
证明：沸点升高法测定的是数均分子量
Tb c0
ci Ni ci Kc i Mi i K Kc Kc i Mi ci Ni M i M n
其它方法 Other method
5.2 Determination of the number-average molecular weight
数均分子量的测定
End-group analyses
端基分析
• Many types of syntheses leave a special group on one or both ends of the molecule, such as hydroxyl and carboxyl. These can be titrated or analyzed instrumentally by such methods as NMR.
Colligative methods
依数性方法
• The colligative properties include:
– Boiling point elevation – Melting point depression – Vapor pressure lowering – Osmotic pressure
i
Mn
• For molecular weights above about 25,000 g/mol, the method becomes insensitive because the end groups are present in too low a concentration
DP PEG 3( I f Ii ) / 4I g
=n+1
DP PCL 2DP PEG [ I e /( I f I i )]
=m
•
注意点：
1. 要求聚合物结构明确(每条聚合物链含几个末端 基，端基结构是什么)。
2. 结合其它方法，可以测定聚合物支链数目。 3. 采用的方法 化学滴定法: 缩聚产物, 如聚酯, 聚酰胺等 放射化学法: 末端具有放射性同位素 波谱法: 末端具有特定吸收的基团
若部分聚合物分子透过半透膜，所测分子量偏高还是偏低？
Comparison of colligative methods
依数性方法的比较
A comparison of the colligative solution properties of a 1% polymer solution with M=20,000 g/mol
化学或波谱方法 Chemical method 端基分析法 End group analysis, or end group measurement
热力学方法 Thermodynamics method
光学方法 Optical method 动力学方法 Dynamic method
沸点升高，冰点降低，蒸汽压下降， 渗透压法 Osmotic method
K M T c c0
T solvent
Mn
Membrane osmometry
膜渗透压法
Osmotic pressure
RT lim c 0 c Mn

Schematic illustration of an osmometer
Schematic illustration of the dependence of osmotic pressure on concentration
Ebulliometry
沸点升高法
Boiling point elevation
Boiling point elevation Gas constant Temperature
Tb RT 1 A2 c ... c H v M
2
Solute concentration in grams per cubic centimeter Solvent density Latent heats of vaporization per gram of solvent
2-40 Broad
None
High None
• 离散型分子量分布
只含有限个级分，可粗略地描述各级分的含量和分子 量的关系。
• 分子量分布的连续函数表示 微分分布曲线：
W (M ) 或 W (M ) N (M ) 或 N (M )
M
M
M1 M 2
M1 M 2
重量微分分布曲线
数量微分分布曲线
聚合物分子量的测定方法
Mz > Mw > Mv > Mn
Molecular weight distribution
分子量分布
Number-average distribution width
M M n
2 n

2

n
Mw M M 1 n
2 n
Weight-average distribution width
1 2 RT A2 c A3c ... c M
若A30
c
1/ 2
2 c 1 2 c c 0
1/ 2
2=MnA2
膜的选择
• 使聚合物分子不能透过 • 溶剂分子透过速率较大 • 不与聚合物和溶剂发生化学作用 (不适合未分级的含有大量低分子的聚合物)
H2N(CH2)5CO[NH(CH2)5CO]nNH(CH2)5COOH
• Molecular weight determined by end-group analyses is number- average molecular weight
W M N
W N M N N
i i i i i i i i
i i
Cryoscopic method
冰点降低法
Melting point depression
Freezing point depression
Gas constant
Temperature
RT 2 lim c 0 c H f
Solute concentration in grams per cubic centimeter Solvent density
P10 P1 X2 0 P1
Vapor pressure of the pure solvent Mole fraction of the solute