1. Introduction to Polymer Physics
Prof. Dr. Yiwang Chen
School of Materials Science and Engineering, Nanchang University, Nanchang

2. Chapter 4 Molecular weight
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：

3. For a discrete distribution of molecular weights, an average molecular weight, M, may be defined as
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
Molecular weights that are important in determining polymer properties are the number-average, Mn (=1), the weight-average, Mw (=2), and the z-average, Mz (=3), molecular weights.

4. Number-average molecular weight
Weight-average molecular weight
z-average molecular weight
Viscosity-average molecular weight

5. When =-1,
When =1,
In general,  is in the range of 0.5 to 1, therefore,

6. Since the molecular-weight distribution of commercial polymers is normally a continuous function, molecular-weight 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 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, and Pearson distributions, and standard probability functions, such as the Poisson and logarithmic-normal distributions.

7. N(M) and W(M) are the total number and weight of molecular-weight species in the distribution，if I(M) is the weight-integral function of molecular-weight distribution, then

8. 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 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 be as low as 1.06.

9. 分子量分布宽度
分布宽度指数：各个分子量与平均分子量之间的差值的平方平均值。
数均分布宽度指数
重均分布宽度指数
多分散系数
分子量均一的试样
分子量非均一的试样

10. Example Problem
A polydisperse sample of polystyrene is prepared by mixing three monodisperse samples in the following proportions:
1 g 10,000 molecular weight
2 g 50,000 molecular weight
3 g 100,000 molecular weight
Using this information, calculate the number-average molecular weight, weight–average molecular weight, and PDI of the mixture.
Solution

11. 4.2 Measurement of Molecular Weight
There are three important molecular-weight averages-number-average (Mn), weight-average (Mw), and z-average (Mz). Absolute values of Mn, Mw, and Mz can be obtained by the primary characterization methods of osmometry, scattering, and sedimentation, respectively.
In addition to these accurate but time-consuming techniques, there are a number of secondary methods by which average molecular-weights can be determined provided that polymer samples with narrow molecular-weight distributions are available for reference and calibration.
The most important of these secondary methods is gel-permeation chromatography (GPC), sometimes called size-exclusion chromatography (SEC).
Another widely used secondary method is the determination of intrinsic viscosity from which the viscosity-average molecular weight can be determined.

12. Terminal determination
如果高聚物的化学结构已知，高分子链末端带有可以用化学方法定量分析的基团，则从化学分析的结果，可以计算分子量。
试样的分子量越大，单位重量聚合物所含端基数就越少，测定的精度就越差，所以端基分析法只适用于分子量小于2X104的高聚物。该分子量为数均分子量。
适用于明确的端基官能团
可用于测定支链数目

13. Boiling point increase and Freezing point lower
在溶液中加入不挥发性的溶质后，溶液的蒸汽压会下降，使得溶液的沸点升高及冰点下降。
高分子溶液在无限稀释的情况下符合理想溶液的规律，可以在各种条件下测定溶液的沸点升高 Tb 和冰点降低 Tf 的数值，然后以 T/C 对 C 作图并外推，来计算分子量。由于温差测量精度的限制，使分子量测量的上限为1X104。 所测得为数均分子量。
注意：溶液的浓度C以千克溶剂中所含溶质的克数，如果浓度C用克/毫升，则Kb和Kf值需要乘1000/, 为溶剂的密度。

14. Membrane Osmometry
The osmotic pressure, , of a polymer solution may be obtained from the chemical potential, 1, or equivalently from the activity, 1, of the solvent through the basic relationship
Where V1 is the molar volume of the solvent. Substitution of the Flory-Huggins expression for solvent activity and subsequent arrangement gives

15. Simplification of this relation can be achieved by expansion of the logarithmic term in a Taylor series and the substitution of polymer concentration, c, for volume fraction, 2, through the relationship
Where  is the specific volume of the polymer. Substitution and rearrangement give the expression
The classical van’t Hoff equation for the osmotic pressure of an ideal, dilution solution
may be seen as a special or limiting case when 12=1/2 and second- and higher-order terms in c can be neglected (i.e., for dilute solution).

16. For high-molecular-weight, polydisperse polymers, the appropriate molecular weight to use is the number-average molecular weight, Mn.
Where A2 and A3 are the second and third virial coefficients, respectively. Comparison of two equations reveals the following relations for the virial coefficients:
In the limit of dilute solution (typically less than 1 g/dL), terms containing second- and higher-order powers of c can be neglected, and therefore a plot /RTc versus c yields a straight line with an intercept, 1/Mn, and slope, A2.

17. As shown by the relation between A2 and 12, the second virial coefficient is a convenient measure of the quality of polymer-solvent interactions. In good solvents in which the polymer chains are expanded (i.e., >1), A2 is large and, therefore, 12 is small (e.g., <0.5). At  conditions (i.e., =1), A2=0 and 12=0.5.

18. 如果  的单位是 g/cm2, C的单位是 g/cm3, 则计算 M 时所用的 R 应为8.478104 g cm/C mol
由于半透膜的存在，使两池存在化学位的差别。
稀溶液中渗透压与分子量的关系为：
van’t Hoff equation
分子量的测定范围：1X104 ~ 1.5 X104

19. Experimental procedures to determine osmotic pressure are relatively simple although often very time consuming. In operation, pure solvent and a dilute solution of the polymer in the same solvent are placed on opposite sides of a semipermeable membrane, typically prepared from cellulose or a cellulose derivative. Regenerated cellulose is an especially good membrane polymer because it is insoluble in most organic solvents.
Normally, the membrane is first preconditioned in the solvent used in the measurements. An ideal membrane will allow the solvent to pass through the membrane but will retain the polymer molecules in solution.

20. The resulting difference in chemical potential between solvent and the polymer solution causes solvent to pass through the membrane and raise the liquid head of the solution reservoir.
The osmotic pressure is calculated from the height, h, of the equilibrium head representing the difference between the height of solvent in the solvent capillary and the height of solution in the opposite capillary at equilibrium as
Where  is the solvent density.

21. One intrinsic problem with membrane osmometry is the performance of the membrane. No membrane is completely impervious to the passage of small molecules, and any migration of smaller polymer molecules across the membrane during measurement will not contribute to the osmotic pressure and, therefore, an artificially high value of Mn will be obtained.
For this reason, membrane osmometry is accurate only for polymer samples with molecular weights above about 20,000. The upper limit for molecular weight is approximately 500,000 due to inaccuracy in measuring small osmotic pressures.
For the characterization of low-molecular-weight (i.e.,<20,000) oligomers and polymers, an alternative technique called vapor-pressure osmometry (VPO) is preferred, particularly when molecular weight is less than about 10,000.

22. Vapor Pressure Osmometry
When a polymer is added to a solvent, the vapor pressure of the solvent will be lowered due to the increase in solvent activity. The relation between the difference in vapor pressure between solvent and solution, p=p1-p10, and the number-average molecular weight, Mn, of the polymer is given as
Where p10 and V10 are, respectively, the vapor pressure and molar volume of the pure solvent.

23. Due to the inverse dependence of p on Mn, the effect of even a low-molecular-weight polymer on the lowering of vapor pressure will be very small and, therefore, direct measurement of the vapor pressure is a very imprecise method of molecular-weight determination. For this reason, an indirect approach, based upon thermoelectric measurements, is used in commercial instrument as described below.
A commercial vapor pressure osmometer uses two matched thermistors that are placed in closed, constant-temperature (0.001C) chamber containing saturated solvent vapor. A drop of solvent is placed by syringe on one thermistor and a drop of dilute polymer solution on the other.

24. As a result of condensation of solvent vapor onto the solution, the temperature of the solution thermistor increases until the vapor pressure of the solution equals that of the solvent.
The difference in temperature between the two thermistors is recorded in terms of a difference in resistance (R), which is calibrated by use of a standard low-molecular-weight sample.
Extrapolation of R/c over range of dilute-solution concentrations to zero concentration yields Mn through
Where KVPO is the calibration constant obtained by measuring R for a low-molecular-weight standard whose molecular weight is precisely known.

25. 气相渗透压法是利用间接测定溶液的蒸汽压降低，来测定溶质分子量的方法。
如图，溶液滴和溶剂滴之间的温度差和溶液中溶质的摩尔分数成正比。测量范围：40 ~ 3X104。

26. Light-Scattering Methods
The weight-average molecular weight can be obtained directly only by scattering experiments. The most commonly used technique is light scattering from dilute polymer solution. It is also possible to determine Mw by small-angle neutron scattering of specially prepared solid samples. Although this technique has great current importance in polymer research, it is not routinely used for molecular-weight determination because of the difficulty and expense of sample preparation and the specialized facilities required.

27. The Fundamental relationship for light scattering is given as:
In this equation, K is a function of the refractive index, n0, of pure solvent, the specific refractive increment, dn/dc, of the dilute polymer solution, and the wavelength, , of the incident light according to the relationship
Where NA is Avogadro’s number (6.0231023 molecules mol-1).

28. The specific refractive increment is the change in refractive index, n, of the dilute polymer solutions with increasing polymer concentration. The term R() is called the Rayleigh ratio, which is defined as
In this equation, I0 is the intensity of the incident light beam and I() is the intensity of the scattered light measured at a distance of r from the scattering volume, V, and at an angle  with respect to the incident beam.
The parameter P() is called the particle scattering function, which incorporates the effect of chain size and conformation on the angular dependence of scattered light intensity.

29. Spherical particles smaller than the wavelength of light act as independent scattering centers generating a symmetrical envelope of scattered light intensity. In this case of small particles, P() is unity, but in the case of polymer chains whose dimensions are >/20, scattering may occur from different points along the same chain and P()<1. For this reason, diminution of scattered light intensity can occur due to interference, and the scattering envelope is no longer spherically symmetrical. In this case, the angular dependence of scattered light intensity is given by the particle scattering function, which, for a monodisperse system of randomly-coiling molecules in dilute solution, is given by the expression.

30. s2 is called the mean-square radius of gyration
s2 is called the mean-square radius of gyration. For linear-chain polymer, s2 is related to the mean-square end-to-end distance as
To determine Mw, it is necessary to know the value of P() at each angle for which R() has been measured.

31. Zimm Method
The most rigorous approach to determine Mw from light-scattering data is a Zimm plot. This procedure has the advantage that chain conformation need not be known in advance. However, Zimm plots require tedious measurements of scattered light-intensity at many more angles than needed by the dissymmetry technique.
A double extrapolation to both zero concentration and zero angle is used to obtain information concerning molecular-weight, second-virial coefficient, and chain dimensions.

32. In the limit of small angles where P() approaches unity, it can be shown by means of a series expansion of 1/P():
Data is plotted in the form of Kc/R() versus sin2(/2)+kc for different angles and concentrations (where k is an arbitrary constant added to provide spacing between curves).
A double extrapolation to =0 and c=0, for which the second and third terms on the right equation become zero, yields Mw as the reciprocal of the intercept.
A2 is then obtained as one-half of the slope of the extrapolated line at =0.
s2 is obtained from the initial slope of the extrapolated line at c=0.

33. One obvious difficulty with the Zimm method is that a large number of time-consuming measurements is required; however, the method provides a great deal of information-Mw, A2, and s2.

34. Low-Angle Laser Light-Scattering (LALLS)
In recent years, helium-neon (He-Ne) lasers (=632.8 nm) have replaced conventional light sources in some commercial light-scattering instruments. The high intensity of these light sources permits scattering measurements at much smaller angles (2 to 10) than possible with conventional light sources and for smaller samples at lower concentrations.
Since at low angles the particle for scattering-function, P(), approaches unity, the equation reduces to the classical Debye equation for scattering by small spherical particles as
Therefore, a plot of Kc/R() versus c at a single angle gives Mw as the inverse of the intercept and A2 as one-half of the slope.

35. Intrinsic Viscosity Measurements
A method widely used for routine molecular-weight determination is based upon the determination of the intrinsic viscosity, [], of a polymer in solution through measurements of solution viscosity. Molecular weight is related to [] by the Mark-Houwink-Sakurada equation give as
Where M is the viscosity-average molecular weight defined for a discrete distribution of molecular weights as

36. Both K and  are empirical (Mark-Houwink) constants that are specific for a given polymer, solvent, and temperature. The exponent  normally lies between the values of 0.5 for a  solvent and 1.0 for a thermodynamically good solvent.
The values of M normally lies between the values of Mn and Mw obtained by osmometry and light-scattering measurements, respectively.

37. Relative viscosity-increment Reduced viscosity Inherent viscosity
粘度法是测定分子量的常用方法。溶液的粘度一方面与聚合物的分子量有关，同时也决定于聚合物分子的结构、形态和在溶剂中的扩张程度。因此，粘度法测分子量只是一种相对方法。
当液体流动时，剪切应力与剪切速率成正比，比例常数为粘度，其数值相当于流速梯度为1 秒–1、面积为1厘米2 时两层液体的内摩擦力。
Relative viscosity
Relative viscosity-increment
Reduced viscosity
Inherent viscosity
Intrinsic Viscosity

38. In actual practice, reduced viscosity is obtained at different concentrations not by direct measurement of solution and solvent viscosities but by measurement of the time required for a dilute solution (t) and pure solvent (t0) to fall from one fiducial mark to another in a small glass capillary.
If these efflux times are sufficiently long (e.g.,>100s), the relative viscosity increment can be obtained as

39. In addition to determination of molecular weight, measurement of intrinsic viscosity can also be used to estimate chain dimensions in solution.
The mean-square end-to-end distance is related to intrinsic viscosity through the relationship
Where  is considered to be a universal constant (2.10.21021 dL mol-1 cm-3) known as the Flory constant.

40. 端基分析法：（数均分子量，分子量在 2 X104 以下的高聚物）
膜渗透压法：（数均分子量，分子量的测定范围为 1 X104~ 1.5 X104）
气相渗透压法：（数均分子量，分子量的测定范围为40 ~ 3 X 104 ）
光散射法：（重均分子量，分子量的测定范围为10 4~ 107 ）
超速离心沉降法：（数均分子量和重均分子量） Ultracentrifugation and sedimentation
粘度法：（粘均分子量）

41. 思考题
已知聚苯乙烯-环己烷体系(I)的温度为34C，聚苯乙烯-甲苯体系(II)的温度低于34C，假定于40C在此两种溶剂中分别测定同一聚苯乙烯试样的渗透压与粘度，问两种体系的 ，A2, 和 ， ， ， 的大小顺序如何？问两种体系、两种方法所得试样的相对分子质量之间有什么关系？
作业：
P330: (4) (5) (6)