Introduction to Polymer Physics

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

2 Chapter 3 Properties of Polymer Solutions
Solvents are frequently used during the polymerization process, during fabrication (e.g., film casting, fiber formation, and coatings), and for determining molecular weight and molecular-weight distributions. Interactions between a polymer and solvent influence chain dimensions (i.e., conformations) and, more importantly, determine solvent activities. The measurement of osmotic pressure and scattered-light intensity from dilute polymer solutions-techniques based upon the principles of polymer-solution thermodynamics-are the primary methods used to determine number-average and weight-average molecular weights, respectively.

3 3.1 Solubility of Polymers
Polymer solution: 高聚物以分子状态分散在溶剂中形成的均匀混合物 不同类型的高聚物的溶解方式不同： 非晶态高聚物：溶剂分子容易渗入高聚物内部使之溶胀和溶解。 晶态高聚物：溶剂分子渗入高聚物内部非常困难，其溶胀和溶解困难。 非极性晶态高聚物：室温时难溶，升温至熔点附近，使晶态高聚物转变为非晶态高聚物后，再溶解。 极性晶态高聚物：室温时采用极性溶剂溶解。

4 Thermodynamics for Dissolution of Polymers
Changes of Gibbs free energy during dissolution: 该过程的S > 0，所以 F 的正负，取决于 H 的正负及大小。 溶解过程可分为几种情况： 极性高聚物溶解在极性溶剂中： 过程的 H< 0 ， 则 F < 0。 非极性高聚物： 过程的 H > 0 ，要使 F < 0，则必须使 | H | < T |S | 。

5 Calculation for enthalpy, H, of solution mixture:
Assume that no volume change occurs in mixture of two solutions (V = 0). Hildebrand equation Solubility parameter, , is the root of cohesive energy density Therefore, Hildebrand equation: As the form of equation indicates, the solubility parameter approach can be used to estimate the heat of mixing when Hm0 but not when Hm<0 (i.e., for exothermic heat of mixing). Namely, only applicable to mixing of nonpolar solvents and nonpolar polymers.

6 Predications of Solubilities
The solubility parameter is related to the cohesive energy-density, or the molar energy of vaporization of a pure liquid, E, as =(E/V)1/2, where E is defined as the energy change upon isothermal vaporization of the saturated liquid to the ideal gas state at infinite dilution and V is the molar volume of the liquid. Units of  are (cal cm-3)1/2 or (MPa)1/2. Since it is not reasonable to talk about an energy of vaporization for solid polymers, the solubility parameter of a polymer has to be determined indirectly or calculated by group-contribution methods.

7 Experimentally, the solubility parameter of a polymer can be estimated by comparing the swelling of a crosslinked polymer sample immersed in different solvents. The solubility parameter of the polymer is taken to be that of the solvent resulting in maximum swelling. Alternatively, the solubility parameter of a polymer can be estimated by use of one of several group-contribution methods, such as those given by Small and by Hoy. An extensive presentation of group-contribution methods for estimating polymer properties, including those for solubility parameters, is given by van Krevelen.

8 Calculation of  by a group-contribution method requires the value of a molar attraction constant, Fi, for each chemical group in the polymer repeating-unit. The solubility parameter of a polymer is then calculated from these molar attraction constants and the molar volume of the polymer, V (units of cm3 mol-1), as: Where the summation is taken over all groups in the repeating unit.

9 Example Problem: Estimate the solubility parameters, in units of (MPa)1/2, for poly(methyl methacrylate) (PMMA) by the method of Small. The density of PMMA is reported to be g cm-3 at 25C. Solution The structure of PMMA is From the available chemical groups listed in literature, the molar-attraction constant for repeating unit of PMMA can be obtained as follows:

10 Group F Number of Groups Fi -CH3 438 2 867 -CH2- 272 1 >C< -190 -COO- 634 1592 The formula weight of a PMMA repeating unit is calculated from atomic weights as The molar volume, V, is then /1.188=84.28 cm3 mol-1. The solubility parameter is then calculated as =Fi/V=1592/84.28=18.9 MPa1/2.

11 Another approach that can be used to calculate  is based upon knowledge of the equation of state, V(p,T), for the polymer: where  is the (isobaric) thermal-expansion coefficient, and  is the (isothermal) compressibility coefficient, Equations of state are now available for most commercial polymers.

12 溶度参数 估算： 利用物理常数估算： 实验法：常用稀溶液的粘度法或测定交联网溶胀度的方法来估算  。 由重复单元中各基团的摩尔引力常数直接计算得到：

13 Selection of Solvents for Polymers
选择溶剂时主要应该遵守两个原则： 极性相近原则：当溶质和溶剂极性相近时，两者容易互溶。 溶度参数相近原则：一般，当 | 1-2 | > 1.7 ~ 2.0 时，高聚物不溶解。 各种类型的聚合物对溶剂选择的要求： 非晶态非极性高聚物：选择溶度参数相近的溶剂。 非晶态极性高聚物：既要选择溶度参数相近，也要选择极性相近的溶剂。 晶态非极性高聚物：选择溶度参数相近溶剂的同时，又要提高溶解的温度。 晶态极性高聚物：选择溶度参数相近和极性相近的溶剂。

14 Calculation for Mixing Solvents
The matching of polymer and solvent solubility parameters to minimize Hm is a useful approach for solvent selection in many cases but often fails when specific interactions such as hydrogen bonding occur. To improve the prediction, two- and three-dimensional solubility parameters, which give individual contributions for dispersive (i.e., van der Waals), polar, and hydrogen bonding interactions, are sometimes used. In the case of the three-dimensional model proposed by Hansen, the overall solubility parameter can be obtained as =(d2+p2+h2)1/2, where d, p, and h are the dispersive, polar, and hydrogen-bonding solubility parameters, respectively. Calculation for Mixing Solvents

15 3.2 Thermodynamics of Polymer Solutions
It was recognized in the 1940s that the thermodynamics of polymeric systems need to be treated in a special way. In 1942, Gee and Treloar reported that even dilute polymer solutions deviated strongly from ideal-solution behavior. In these early experiments, a high-molecular-weight rubber was equilibrated with benzene vapor in a closed system and the partial pressure of the benzene (the solvent), p1, was measured. The solvent activity, 1, was calculated as the ratio p1 to the saturated vapor pressure of pure benzene, p10, at the system temperature as:

16 Experimental benzene activity is plotted as a function of the volume fraction of rubber, 2.
These data are compared with predictions of Raoult’s law for an ideal solution given as p1=x1p10, where x1 is the mole fraction of the solvent. For an ideal solution, 1=x1. Ideal solution: there is equal interactions between solute and solute, solvent and solvent, and solute and solvent in solution, there are no volume change and enthalpy change in mixing. Therefore: Raoult’s law Mixing entropy of ideal solution: Mixing free energy of ideal solution:

17 The Flory-Huggins Theory
In the early 1940s, Paul Flory and Maurice Huggins, working independently, developed a theory based upon a simple lattice model that could be used to understand the nonideal nature of polymer solutions. In the Flory-Huggins model, the lattice sites, or holes, are chosen to be the size of the solvent molecule. As the simplest example, consider the mixing of a low-molecular-weight solvent (compound 1) with a low-molecular-weight solute (compound 2). The solute molecule is assumed to have the same size as a solvent molecule, and therefore only one solute or one solvent molecule can occupy a single lattice site at a given time.

18 The increase in entropy due to mixing of a solvent and solute, Sm, may be obtained from the Boltzmann relation Sm=kln, where k is Boltzmann’s constant (1.3810-23 J K-1) and  gives the total number of ways of arranging N1 indistinguishable solvent molecules and indistinguishable N2 solute molecules, where N=N1+N2 is the total number of lattice sites. The probability function is given as =N!/(N1!N2!). Use the Stirling approximation lnn!=nlnn-n leads to the expression for the entropy of mixing per molecule as

19 Alternatively, the molar entropy of mixing can be written as
where R is the ideal gas constant and x1 is the molar fraction of the solvent given as Equation is the well-known relation for the entropy change due to mixing of an ideal mixture.

20 The entropy of mixing a low-molecular-weight solvent with a high-molecular-weight polymer is smaller than given by above equation for a low-molecular-weight mixture. This is due to the loss in conformational entropy resulting from the linkage of individual repeating units along a polymer chain compared to the less ordered case of unassociated low-molecular-weight solute-molecules dispersed in a low-molecular-weight solvent. In the development of an expression for Sm for a high-molecular-weight polymer in a solvent, lattice is established by dividing the polymer chain into r segments, each the size of a solvent molecule, where r is the ratio of polymer volume to solvent volume (i.e., a lattice site).

21 For N2 polymer molecules, the total number of lattice sites is then N=N1+r N2.
The expression for the entropy change due to mixing obtained by Flory and Huggins is given as: where 1 and 2 are the lattice volume fractions of solvent (compound 1) or polymer (compound 2), respectively. These are given as

22 For a polydisperse polymer, equation can be modified as
when the summation is over all polymer chains (N) in the molecular-weight distribution. The Gibbs free energy of mixing, Gm, of a polymer solution is given as: Once an expression for the enthalpy of mixing, Hm, is known, expressions for the chemical potential and activity of solvent can be obtained as:

23 Where Gm is the partial-molar Gibbs free energy of mixing and the activity is related to the chemical potential as: For an ideal solution, Hm=0. Solutions for which Hm0 but for which Sm are termed regular solutions and are the subject of most thermodynamic models for polymer mixtures.

24 The expression that Flory and Huggins gave for the enthalpy of mixing is:
Where z is the lattice coordination number or number of cells that are first neighbors to a given cell, and 12 is the change in internal energy for formation of an unlike molecular pair (solvent-polymer or 1-2) contacts given by the mean-field expression as Where ij is the energy of i-j contacts. It is clear from above equation that an ideal solution (Hm=0) is one for which the energies of 1-1, 1-2, and 2-2 molecular interactions are equal.

25 Since z and 12 have the character of empirical parameters, it is useful to define a single energy parameter called the Flory interaction parameter, 12, given as The interaction parameter is a dimensionless quantity that characterizes the interaction energy per solvent molecule divided by kT. As the equation indicates, 12 is inversely related to temperature but is independent of concentration. 12称为Huggins 参数，它反映高分子与溶剂混合时相互作用能的变化。12kT的物理意义表示当一个溶剂分子放到高聚物中去时所引起的能量变化。

26 The expression for the enthalpy of mixing may then be written as:
Combining the expression for the entropy and enthalpy of mixing gives the well-known Flory-Huggins expression for the Gibbs free energy of mixing: From this relationship, the activity of the solvent can be obtained as: In the case of high-molecular-weight polymers for which the number of solvent-equivalent segment, r, is large, the 1/r term within parentheses on the right-hand side of equation can be neglected to give

27 与实验结果的比较： 根据溶剂的化学位 1的关系式，得： 从高分子溶液蒸汽压p1 和纯溶剂的蒸汽压p1o 的测量，可计算高分子—溶剂相互作用参数 12。 12应与高分子溶液的浓度无关，但这一结果与实验事实不符。实验表明，只有天然橡胶-苯溶液的12 值与浓度 2无关

28 Smx>Sm>Smi The entropy of mixing for polymer solution:
The entropy of mixing for ideal solution: If the polymer is divided into x structure units which have same lattice site as solvent, the entropy of mixing for this solution is given as Smx, therefore: Smx>Sm>Smi

29 The Flory-Huggins equation is still widely used and has been largely successful in describing the thermodynamics of polymer solutions; however, there are a number of important limitations of the original expression that should be emphasized. The most important are the following: Applicability only to solutions that are sufficiently concentrated that they have uniform segment density. There is no volume change of mixing (whereas favorable interactions between polymer and solvent molecules should result in a negative volume change). There are no energetically preferred arrangements of polymer segment and solvent molecules in the lattice. The interaction parameter, 12, is independent of composition.

30 似晶格模型理论的推导过程存在不合理的地方：
没有考虑到由于高分子的链段之间，溶剂分子之间以及链段与溶剂之间的相互作用不同会破坏混合过程的随机性，从而引起溶液熵值的减少，使S 偏高。 高分子在解取向态中，分子之间互相牵连，有许多构象不能实现，而在溶液中原来不能实现的构象有可能表现出来，因此，推导时高估了S高聚物 ，使 S 偏低。 高分子链段分布均匀的假定只在浓溶液中才合理。因此 S 的结果只适用于浓溶液。

31 There have been a number of subsequent developments to extend the applicability of the original Flory-Huggins theory and to improve agreement between theoretical and experimental results. For example, Flory and Krigbaum have developed a thermodynamic theory for dilute polymer solutions. From values of the solubility parameters for the solvent and polymer, the heat of mixing can be estimated by the Scatchard-Hildebrand equation as: Where V is the volume of the mixture. The interaction parameter can be estimated from the value of Hm as:

32 Flory-Krigbaum Theory
Flory and Krigbaum have provided a model to describe the thermodynamics of dilute polymer solution in which individual polymer chains are isolated and surrounded by regions of solvent molecules. In contrast to the case of a semidilute solution addressed by the Flory-Huggins theory, segmental density can no longer be considered to be uniform. In their development, Flory and Krigbaum viewed the dilute solution as a dispersion of clouds consisting of polymer segments surrounded by regions of pure solvent.

33 From above equation can be written as
把N2个刚性球分布在体积V的溶液中，排列方法数为 For a dilute solution, the expression for solvent activity was given as: Where K1 and 1 are heat and entropy parameter, respectively. They defined an “ideal” or theta () temperature as From above equation can be written as

34 Where 1E is the excess chemical potential of solvent and  is the Flory temperature (theta temperature)

35 When T=, 1E =0, 12=1/2, ideal solution
When T>, 1E <0, 12<1/2, good solvent for polymer When T<, 1E >0, 12>1/2, poor solvent for polymer At the  temperature, the dimensions of a polymer chain collapse to unperturbed dimensions (i.e., in the absence of excluded-volume effects). If the state using given solvent and temperature has 1E =0, the state is called  state, the solvent is called  solvent, and the temperature called  temperature.

36 According to Flory-Krigbaum theory
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 rearrangement gives

37 Equation can be then rearranged to give the widely used relation:
Where A2 and A3 are the second and third Virial coefficients, respectively. Comparison equations reveals the following relations for the Virial coefficients:

38 When T=, 12=1/2, A2=0, ideal solution,  solvent
When T>, 12<1/2, A2>0, good solvent for polymer, polymer becomes loose coils When T<, 12>1/2, A2<0, poor solvent for polymer, polymer coils become shrinkage and precipitation 当 T >  时，A2 > 0 ，高分子处于良溶剂之中，这时，由于溶剂化作用，卷曲的高分子链能充分伸展，且温度愈高，溶剂化作用愈强，链也愈伸展。 高分子链的均方末端距和均方旋转半径，与 状态下相比较，都会有相应的扩大。可以用一个参数  来表示高分子链扩张的程度。 称为扩张因子或溶胀因子，其值与温度、溶液性质、高分子的分子量、溶液的浓度等有关。

39 3.3 Subconcentrated Solution of Polymers
For very dilute solutions, polymer coils are widely separated and do not overlap. At a critical concentration, c**, marking the transition from the extremely dilute to dilute regions, the hydrodynamic volumes of individual coils start to touch. As concentration is further increased (c>c*), coils begin to overlap and finally entanglements are formed that increase viscosity.

40 The critical concentration, c
The critical concentration, c*, may be marked by an abrupt increase in the relative viscosity increment. Below c*, viscosity is proportional to concentration, But, above c*, viscosity is approximately proportional to the fifth power of concentration.

41 C<<C* 高分子溶液热力学性质，u, 12, A2只与高分子分子量、溶质和溶剂性质以及温度有关，与浓度无关
在浓度达到临界值时，高分子在溶液中已处于密堆积程度，所以溶液的浓度可按每个高分子在溶液中的体积（S3）中该分子的重量M计算 在良性溶剂的稀溶液中 说明分子量越大C*越小

42 亚浓溶液的渗透亚 在良溶剂中，Flory-Krigbaum推导出稀溶液(C<C*)的渗透亚 C>>C*，溶液中的高分子的链段分布均一，由许多分子量为M的链组成的溶液和由一个分子量为无穷大的单链充满整个容器所组成的溶液相比，只要两者浓度相等，其热力学性质应该没有差别，即热力学性质只与浓度有关，而与分子量无关。 这是de Gennes 用标度理论得出的结果

43 3.4 The concentrated Solution of Polymers
Typically, commercial plastics are mixtures of one or more polymers and a variety of additives such as plasticizers, flame retardants, processing lubricants, stabilizers, and fillers. The exact formulation will depend upon the specific application or processing requirement. For example, poly(vinyl chloride) (PVC) is a thermally unstable polymer having high modulus, or stiffness, typical of other glassy polymers at room temperature. In order to obtain a flexible-grade resin of PVC for use as packaging film or for wire insulation, the polymer must be blended with a plasticizer to reduce its Tg and with a small amount of an additive to improve its thermal stability at processing temperatures.

44 In many cases, certain properties of a polymer can be enhanced by blending it with another polymer.
For example, the engineering thermoplastic, poly(2,6-dimethyl-1,4-phenylene oxide) (PPO), is high-Tg polymer that is difficult to process because of its susceptibility to thermal oxidation at high temperatures. Commercial resins of this polymer (Noryl) are blends of PPO and high-impact polystyrene (HIPS) and may also include different additives, such as lubricants, thermal stabilizers, flame retardants, or fillers. Since the Tg of PS (ca. 100C) is lower than that of PPO (ca. 214C), the overall Tg of the PPO/HIPS blend is lower than that of PPO alone and, therefore, the resin can be melt-processed at reduced temperatures where the oxidative instability of PPO is not a problem.

45 Polymeric composites are physical mixtures of a polymer (the matrix) and a reinforcing filler (the dispersed phase) that serves to improve some mechanical property such as modulus or abrasion resistance. Fillers may be inorganic (e.g., calcium carbonate) or organic (graphite fiber or an aromatic polyamide such as Kevlar). Virtually any material can be used as the composite matrix, including ceramic, carbon, and polymeric materials. Typically, matrices for polymeric composites are thermosets such as epoxy or (unsaturated) polyester resin; however, some engineering thermoplastics with high Tg and good impact strength, such as thermoplastic polysulfones, have been used for composites.

46 Plasticizers Plasticizers, particularly for PVC, constitute one of the largest segments of the additives market. The principal function of a plasticizer is to reduce the modulus of a polymer at the use temperature by lowering its Tg. Increasing the concentration of the plasticizer causes the transition from high-modulus (glassy) plateau region to the low-modulus (rubbery) plateau region to occur at progressively lower temperatures. Typically, plasticizers are low-molecular-weight organic compounds having a Tg in the vicinity of -50C. In some cases, a miscible high-molecular-weight polymer having a low Tg (e.g., polycaprolactone or copolymers of ethylene and vinyl acetate) can be used as a plasticizer.

47 高聚物因加入高沸点、低挥发性并能与高聚物混溶的小分子物质而改变其力学性质的行为，称为增塑，所用的小分子物质称为增塑剂。增塑剂不仅限于小分子，某些柔性的高分子物质也可起到增塑作用。
从化学结构上可以把增塑剂分成几类： ⑴. 邻苯二甲酸酯类 ⑵. 磷酸酯类 ⑶. 乙二醇和甘油类 ⑷. 己二酸和葵二酸类 ⑸. 脂肪酸酯类 ⑹. 环氧类 ⑺. 聚酯类 ⑻. 其它如氯化石蜡、氯化联苯、丙稀晴—丁二稀共聚物

48 不同类型的增塑剂对不同类型的聚合物的增塑作用不同：
非极性增塑剂溶于非极性聚合物中，使高分子链之间的距离增大，减弱了高分子链之间的相互作用力和链段间相互运动的摩擦力，从而影响到玻璃化温度等性质。非极性增塑剂使非极性高聚物玻璃化温度降低的数值T =  与增塑剂的体积分数成正比。 极性增塑剂溶于极性聚合物中，增塑剂分子进入大分子链之间，其本身的极性基团与高分子的极性基团相互作用，破坏了高分子间的物理交联点，使链段运动得以实现，使高聚物的玻璃化温度降低值与增塑剂的摩尔数成正比，T = n，与体积无关。

49 选择增塑剂时要考虑的因素： 互溶性 (miscibility)：要求增塑剂与高聚物之间的互溶性好。 互溶性与温度有关，高温互溶性好，低温互溶性差。 雾点 (Fog point) ：刚刚产生相分离时的温度，称雾点。 雾点愈低，制品的耐低温性能愈好。 2. 有效性 (efficiency)：对增塑剂有效性的衡量，应综合考虑其积极效果和消极效果。 3. 耐久性 (durativity)：要求增塑剂能稳定的保存在制品之中。 这就要求增塑剂要具备：沸点高、水溶性小、迁移性小及具有一定的抗氧性及对热和光的稳定性。

50 凝胶和冻胶 冻胶是由范德华力交联形成的，加热破坏范德华力可使冻胶溶解。冻胶有两种： 分子内部交联的冻胶，其形成分子内的范德华交联，高分子链为球状结构，溶液的粘度小。 分子间交联的冻胶，其形成分子间的范德华交联，高分子链为伸展链结构，溶液的粘度大。 用加热的方法可以使分子内交联的冻胶变成分子间交联的冻胶。 凝胶是高分子链之间以化学键形成的交联结构的溶胀体，加热不能使凝胶溶解也不能熔融。 交联结构的高聚物不能被溶剂溶解，但能吸收一定量的溶剂而溶胀，形成凝胶。

51 交联聚合物在溶剂中不能溶解，但是能发生一定程度的溶胀，溶胀取决于聚合物的交联度。
当交联聚合物与溶剂接触时，由于交联点之间的分子链段仍然较长，具有相当的柔性，溶剂分子容易深入聚合物内，引起三维分子网的伸展，使其体积膨胀； 但是交联点之间分子链的伸展却引起了它的构象熵的降低，进而分子网将同时产生弹性收缩力，使分子网收缩，因而将阻止溶剂分子进入分子网。 当这两种相反的作用相互抵消时，体系就达到了溶胀平衡状态，溶胀体的体积不再变化。

52 在溶胀过程中，溶胀体内的自由能变化应为：
其中，GM为高分子-溶剂的混合自由能；Gel为分子网的弹性自由能。当达到溶胀平衡时。 根据高分子溶液的似晶格模型理论，高分子溶液的稀释自由能可以表示为： 式中，1、2分别为溶剂和聚合物在溶胀体中所占的体积分数；1为高分子－溶剂分子相互作用参数；T为温度；R为理想气体常数；x为聚合物的聚合度。对于交联聚合物，x，因此上式化简为：

53 交联聚合物的溶胀过程类似于橡皮的性变过程，因此可直接引用交联橡胶的储能函数公式，即：
式中，N为单位体积内交联链的数目；k为波尔兹曼常数；为聚合物的密度；Mc为两交联点之间分子链的平均分子量；1、2、3分别为聚合物溶胀后在三个方向上的尺寸（设试样溶胀前是一个单位立方体）。假定该过程是各向同性的自由溶胀，则设： 因此偏微摩尔弹性自由能为： 当达到溶胀平衡时：

54 设橡皮试样溶胀后与溶胀前的体积比，即橡胶的溶胀度为Q，显然，
当聚合物交联度不高，即较大时，在良溶剂中，Q值可超过10， 此时2很小。因此可将ln1=ln(1-2)近似展开并略去高次项， 所以，在已知、1和V1的条件下，只要测出样品的溶胀度Q，利用上式就可以求得交联聚合物在两交联点之间的平均分子量Mc。

55 Determination of the Interaction Parameter
Experimentally, 12, in the Flory theory can be determined by a variety of techniques, including several scattering methods and inverse gas chromatography. Swelling technique: 溶胀过程伴随两种现象发生，一方面溶剂力图进入高聚物内使其体积膨胀，另一方面，由于交联高聚物体积膨胀导致网状分子链向三维空间伸展，使分子网受到应力而产生弹性收缩能，力图使分子链收缩。 当两种相反的倾向互相抵消时，达到了溶胀平衡。交联高聚物在溶胀平衡时的体积与溶胀前体积之比，称为溶胀比 (Swelling ratio) ，用 Q 表示。溶胀比与温度、压力、高聚物的交联度及溶质、溶剂的性质有关。

56 3.5 The Solution of Polyelectrolyte
在侧链中有许多可电离的离子性基团的高分子称为聚电解质。当聚电解质溶于介电常数很大的溶剂中时，就会产生电离，结果生成高分子离子和许多低分子离子。低分子离子称为抗衡离子。

57 Polycation Polyanion Polyzwitterion

58 聚电解质的溶液性质与所用溶剂关系很大。 采用非离子化溶剂，其溶液性质与普通高分子相似。 在离子化溶剂中，其溶液性质与普通高分子不同，且表现出在低分子电解质中也看不到的特殊行为。 溶液中的聚电解质和中性聚合物一样，呈无规线团壮，离解作用所产生的抗衡离子分布在高分子离子的周围。但随着溶液浓度与抗衡离子浓度的不同，高分子离子的尺寸会发生变化。 (a). 稀溶液时，高分子链更为舒展，尺寸较大。 (b). 浓度增加，高分子链发生卷曲，尺寸缩小。 (c). 加入强电解质时，高分子链更卷曲，尺寸更小。

59 3.6 Phase Equilibria Whether or not a polymer and solvent are mutually soluble, or miscible, is governed by the sign of the Gibbs free energy, Gm, which is related to the enthalpy and entropy of mixing. Three different dependencies of Gm on solution composition (i.e., volume fraction of polymer) at constant temperature are illustrated below.

60 If Gm is positive over the entire composite range, as illustrated by curve I, the polymer and solvent are totally immiscible over the complete composition range and will coexist at equilibrium as two distinct phases. Two other possibilities are those of partial and total miscibility, as illustrated by curves II and III, respectively. For total miscibility, it is necessary that both Gm<0 and that the second derivative of Gm with respect to the volume fraction of solvent (component 1) or polymer (component 2) be greater than zero over the entire composition range as formally expressed by

61 Both conditions for miscibility are satisfied by curve III but not curve II, which exhibits two minima in Gm, and therefore the derivative criterion expressed by above equation is not satisfied at all points along the Gm-composition curve. A solution that exhibits such minima will phase-separate at equilibrium into two phases containing different compositions of both components. The compositions of two phases are given by the points of common tangent, where the composition of the solvent-rich phase is identified as 2A and that of the polymer-rich phase as 2B.

62 Phase equilibrium is strongly affected by solution temperature
Phase equilibrium is strongly affected by solution temperature. In fact, any of three types of phase behavior may result by a change in the temperature (or pressure) of the system. Our usual experience with solutions of low-molecular-weight compounds is that solubility increases with an increase in temperature. In this example, the solution is homogeneous (i.e., the two components are totally miscible) at temperatures above the point identified as UCST, which stands for the upper critical solution temperature.

63 At lower temperatures (i. e
At lower temperatures (i.e., below the UCST), phase separation may occur depending upon the overall composition of the mixture. At a given temperature below the UCST (e.g., T1), compositions lying outside the curves are those constituting a homogeneous phase, while those lying inside the curves are thermodynamically unstable and therefore the solution will phase-separate at equilibrium. 临界共溶温度Tc与分子量有关，溶质的分子量越大，溶液的临界共溶温度愈高

64 Although the UCST behavior of dilute polymer solutions had been observed for many years, it was not until 1961 that phase separation of polymer solutions was first reported to occur with an increase in temperature. In this case, the binodal and spinodal curves coincide at a temperature and composition called the lower critical solution temperature or LCST. One serious limitation of the Flory-Huggins theory is that it fails to predict LCST behavior.

65 12值比较小时，1随2单调下降 12值比较大时，1随2有极大值和极小值 当两个极值点重合成为拐点，即临界点
M不太大时，12可以超过1/2 M无穷大时， 12 1/2，体系处于状态，也就是M趋于无穷大时的温度就是临界温度Tc 相分离的分子量依赖性，可以用逐步降低温度法把聚合物按分子量大小分离开来

66 3.7 Thermodynamics of Polymer Blends
Whether a particular polymer blend will be homogeneous or phase-separated will depend upon many facts, such as the kinetics of the mixing process, the processing temperature, and the presence of solvent or other additives; however, the primary consideration for determining miscible of two polymers is a thermodynamic issue that is governed by the same (Gibbs) free-energy considerations that were discussed before for polymer-solvent mixtures.

67 当 1 很小或为零时: 曲线在  =1/2 处有一极小值。说明两种聚合物可以以任何比例互溶。
当 1 等于某一临界值 1c 时：曲线在   =1/2 处有极大值。在两侧对称位置上出现两个极小值与拐点。 当 1 值很高时：曲线有极大值。此时两种聚合物在任何组成下都不互溶

68 The relationship between the change in Gibbs free energy due to mixing (Gm) and the enthalpy (Hm) and entropy (Sm) of mixing for a reversible system was given as Gm= Hm-TSm. If Gm is positive over the entire composition range at a given temperature, the two polymers in the blend will separate into phases that are pure in either component, providing that a state of thermodynamic equilibrium has been reached. For complete miscibility, two conditions are necessary: Gm must be negative and the second derivative of Gm with respect to the volume fraction of component 2 (2) must be greater than zero over the entire composition range.

69 If Gm<0, but above equation is not satisfied, the blend will separate at equilibrium into two mixed-composition phases. This means that each phase (i.e., the dispersed and continuous phase) will contain some of each polymer. At a given temperature, equilibrium concentrations are given as a pair of isothermal points along the binodal (iA=iB). For polymer blends in the solid state, recent studies have shown that LCST behavior is quite common and needs to be considered during melt processing when elevated temperatures can cause phase separation and result in deleterious changes in the properties of the blend.

70 An example of LCST phase behavior is a blend of polystyrene with the polycarbonate of tetramethylbisphenol-A. In this case, a phase diagram was obtained by determining the temperature at which a particular blend composition first scatters light due to the incipient phase separation. Data obtained over the entire composition range are used to form a cloud-point curve. If the blend is heated above the LCST and rapidly cooled, the two-phase morphology will be retained in the solid state. As will be discussed shortly, thermal and mechanical properties will be very different depending on whether the blend forms a homogeneous mixture or a two-phase structure.

71 Compared to LCST phase behavior, UCST behavior is much more difficult to observe since an UCST may fall below the blend Tg, at which temperature all long-range (cooperative) segmental motions cease. Such chain mobility is required to achieve phase separation. For this reason, UCST behavior may be observed only in a solution of the blend in a low-molecular-weight solvent or under other special circumstances.

72 The bulk of experimental evidence indicates that most polymer pairs are immiscible.
Immiscibility is a consequence of the very small combinatorial entropy change that results when two high-molecular-weight polymers are mixed. Scott has extended the Flory-Huggins (F-H) lattice model to obtain an expression for Sm for a polymer blend as: Where R is the ideal gas constant, V is the volume of the blend, Vr is a reference volume (often taken as the molar volume of the smallest polymer repeating unit), i is the volume fraction of polymer 1 or 2, and ri is the degree of polymerization of polymer 1 or 2 relative to the reference volume.

73 The corresponding equation for the Gibbs free energy of mixing is then
Assume: Polymer 1 与 2  链段的摩尔体积相等，均为 Vr 。则： The corresponding equation for the Gibbs free energy of mixing is then Where 12 is the (Flory) interaction parameter of the blend of polymer 1 and 2. If r1=r2=r, and =1=(1-2), then

74 极小值与极大值出现条件是函数的一阶导数为零
拐点出现的条件是函数二阶导数为零 临界条件应该是三个极值与两个拐点同时出现，判别是三阶导数为零 12<12c时，两种聚合物才有在某些组成范围内形成均相容混物 12>12c时，任何组成下都不能形成均相混合物 对于高分子量的聚合物共混，12值常常大于12c，因而大多数的共混高聚物不相容

75 Since the degree of polymerization (r1 and r2 appearing in the denominators on the RHS) of the high-molecular-weight polymers is large, Sm is small. In order for Gm to be negative as a necessary condition of miscible, the enthalpic contribution to mixing, Hm, must be either negative or zero, or have a small positive value. This implies that blend miscibility requires favorable interactions between the two blend polymers. Examples of favorable interactions include dispersive and dipole-dipole interactions, hydrogen bonding, or charge-transfer complexation.

76 3.8 Hydrodynamics of Polymer Solutions
处于溶液中的高分子，其迁移时会携带溶剂分子一起迁移，随高分子一起迁移的溶剂分子由两部分组成，一部分是与高分子有溶剂化作用的，另一部分是单纯携带的。 Diffusion of Polymers in Solution 平移扩散 (Shifting diffusion)：高分子在溶液中由于局部浓度或温度的不同，引起高分子向某一方向的迁移，这种现象称平移扩散。 旋转扩散 (Rotatory diffusion)：高分子的构象是不对称的，呈棒状或椭球状，在溶液中高分子会绕其自身的轴而转动，称旋转扩散。

77 平移扩散，各种因素与扩散系数的关系：Fick 第一扩散定律和 Fick 第二扩散定律
Stockes Law 旋转扩散与旋转摩擦系数的关系：

78 Flow of Polymers in Solution
小分子的流动为层流，剪切应力与剪切速率成正比，比例常数为粘度。(Viscosity) 高分子溶液的粘度较大，常用一个相对值—极限粘数来表示高分子对溶液粘度的贡献。(Limiting viscosity number or Intrinsic viscosity) 极限粘数与分子量的关系为：Mark-Houwink 方程式

79 溶剂中： Flory-Fox黏度公式 Flory普适常数

80 思考题 聚甲基丙烯酸甲酯在丙酮中，20℃，其极限粘数与分子量的关系式为 ，试计算其分子的无扰尺寸 ，极限特征比C∞,空间位置参数，链段长度le和链的持续长度a. 作业： P330: (6), (7), (8)