Molecular weight control in linear polymerization 2.6 Molecular weight control and molecular weight distribution in linear polymerization Molecular weight control in linear polymerization Molecular weight distribution in linear polymerization

Molecular weight control in linear polymerization Molecular weight control in linear polymerization is the main content of the study of kinetics There are many influence factors on the molecular weight of polycondensations.

1. The effects of the balanced characteristic Closed system： = + 1 由式(2-23)得 Unclossed system： = 由式(2-24)得 Nw是体系中残留水的量

2. The effects of the Catalyzed system Self-Catalyzed system： = 2k3 t + 1 External Catalysis system： = k2C0t + 1

3. The effects of the degree of the reaction 基团是等摩尔比的条件下： The groups is the same mole ratio： =

2.6.1 Molecular weight control in linear polymerization Making the functional groups not react is a effective method of controlling molecular weight. The method of stabilizing molecular weight is used to control the molecular weight.

Two kind of the methods: 非化学计量的 The concentrations of the two monomers or two groups are slightly non-stoichiometric. One of the reactants is present in slight excess. Another reactant is completely used up and all the chain ends posses the same functional group. 相同官能团 Another method is adding a small amount of a monofunctional monomer. 单官能团

（1）The concentrations of the groups of monomer are non-stoichiometric The method fits for the systems of a-R-a + b-R‘-b, but not for the systems of a-R-b The numbers of A and B functional groups are given by Na and Nb, respectively. And b-groups is present in excess. The stoichiometric imbalance r of the two functional groups is given by r=Na/Nb ( r≤1).

Non-stoichiometric a-R-a b-R'-b t=0 The number of monomer The number of group A is Na The number of group B is Nb The number of monomer molecules (a-R-a) is given by Na/2 The number of monomer mole-cules (b-R-b) is given by Nb/2 The total number of monomer molecules(namely the total number of construction unit)is given by （Na＋Nb）/ 2

The total number of polymer molecules t = t the degree of reaction of group A is given by Pa Reacted a ： NaPa b : NaPa Unreacted a ：Na(1－ Pa) b : Nb－ NaPa The total number of unreacted a and b is given by: Na＋Nb－2NaPa The total number of polymer molecules is : (Na＋Nb－2NaPa ) /2

The moles of the two groups are equal（r=1） Thus Substitute r=Na/Nb The group A is used up ( Pa=1) The moles of the two groups are equal（r=1） If r =1

If the equal mole ratio of the two monomers or two functional groups can always be kept，and if the reaction tends to the end, the average degree of the condensation polymer will reach the maximum.

Excessive mole-percent q r=Na/Nb ( r≤1) Excessive mole-percent q Besides r，the Excessive percentage q of b-R‘-b molecule is often used. The q is defined as: namely

（2） Adding a small amount of a monofunctional monomer(R"-b) for locking the end group R”-b can reacts with group A in the polymer. Then this method can lock the end group and stabilize molecular weight. The molecular weight of polymer can be adjusted by the number of R”-b.

This method not only fits the system of bifunctional monomer a-R-a and b-R’-b where mole values of a-R-a and b-R’-b are equal，but also fits the system of a-R-b.

① The systems of a-R-a + b-R‘-b with the same mole ratio The number of the R"-b is NC The coefficient 2 in front of Nc is required since one B molecule has the same quantitative effect as one excess B-B molecule on limiting the growth of a polymer chain.

② The system of a-R-b The number of the R"-b is NC

If the numbers of r and P are substituted for the equation （2-32） , the average degree of polymerization will be obtained.

Example： 1 mol aRa + 1mol bR’b + 0.1%R”b

The high purity of monomer Exact measurement Proper temperature 惰性气体 Protecting by inert gas, using catalytic agent and evacuating the system 排出系统

2.6.2 Molecular Weight Distributions in linear condensation polymers Molecular Weight Distributions is： The ratio of molecules of different to the total (the number of molecules and the weight) of polymers. Three kinds of expression methods: Tabulation, Distribution Curve and Distribution index 分子量分布指数 作表格 分布曲线

MWD is determined by the experiment. MWD can be derived by the theory of probability statistics.

Example As an example the amino acids whose construction is H2NRCOOH forms the polyamide. This is a kind of the system of homogeneous polycondensation whose functional group is always in the same mole. The construction unit is only －NHRCO－in the chains of polymer. 均缩聚

The results derived from this system suit the system of a-R-a～b-R-b , too.

This is a problem of probability. Consider the probability of finding a polymer molecule containing x structural units. X个结构单元 In other words, if the end group is amino group(NH2)，it has a probability of making up x structural units.

Expanding the meaning of degree of the reaction(P) The P is the percentage of the end groups which have reacted at time t. the percentage of the end groups unreacted is 1－P.

The probability that the groups have reacted by time t is defined as the extent of reaction, P. The probability of the groups being unreacted is 1 - P.

The course of forming polyamide molecular formula 分子式 The number of structural units The number of the reacted groups COOH 1 2 3 4 x x-1 The course of forming polyamide

The reaction begins from the terminal group(－NH2 ) The reaction begins from the terminal group(－NH2 ). （x－1）carboxyl groups(－ COOH) have reacted and one carboxyl group (－ COOH) has not reacted as the end group. Thus：The probability that (x-1) carboxyl groups have reacted is Px-1 The probability of a carboxyl group unreacted is 1－P

The probability that the molecule has exactly x units, P(x), is : P(x)=Px-1(1－P) P(x) is synonymous with the mole or number fraction of molecules in the polymer mixture which are X-mers, then Nx/N =Px-1 (1-P) where N is the total number of polymer mole- cules and Nx is the number of X-mers. Nx =NPx-1 (1-P) The total number of structural units present initially is N0 , then N＝N0（1－P） Nx=N0Px-1 (1-P)2

Nx=N0P x-1 (1-P)2 The equation gives the number-distribution function for linear step polymerizations at the extent of reaction P. It is seen that on a number when X=1 basis there are more monomer molecules than in any polymer species regardless of the extent of the reaction. 不论反应程度如何，单体分子比任何X-聚体大分子都要多，这是分子量数量分布的特征。

Nx=N0P x-1 (1-P)2 The Molecular Wight Distribution becomes broad and the average molecular weight increases as the extent of reaction goes.

Number fraction distribution curve for linear polymerization

The weight of the end-group( H and OH) can be neglected. –如果忽略端基的质量 Thus: Mx=xM0 Where: (M0 is the molecular weight of a structural unit) The number average molecular weight:

Where: The result:

The weight fraction Wx of X-mers The Wx is the weight of all X-mers in the weight of all units Because

The weight average molecular weight since

Polydispersity Index The breadth of the molecular weight distri- bution (also called Polydispersity Index) is then given by:

The value of D increases with increase of he extent of As P→1，D approaches two. The value of D increases with increase of he extent of reaction. The breadth of the molecular weight distribution is broader，the molecular weight is not uniformer.