Kinetic Considerations

The rate of polymerization can be expressed as the rate of disappearance of one of the functional groups. In reactions of polyesterification, this can be the rate of disappearance of carboxyl groups, d[CO₂H]/dt.

In the above equation [CO₂H], [OH], and (C(OH)₂) represent carboxyl, hydroxy, and protonated carboxyl groups, respectively. Also, it is possible to write an equilibrium expression for the proton ation reaction of the acid as follows:

This equation can be combined with the above rate expression:

If there is no catalyst present and the dicarboxylic acid acts as its own catalyst, HA is replaced by [COOH] and the expression becomes:

In the above expression k₁, k₂, k₃, and the concentration of the ½A ions have been replaced by an experimentally determined rate constant, k. In most step-growth polymerization reactions, the concentrations of the two functional groups are very close to stoichiometric. This allows writing the above rate equation as follows:

In this equation, M represents the concentration of each of the reacting species. They can be hydroxy and carboxylic acid groups in a polyesterification reaction, or amino and carboxylic acid groups in polyamidation reaction, and so on. The above equation can also be written as follows:

k dt = d(M)/(M)³

after integrating of the above, we get: 2kt = 1/(M)² + constant The constant in the above equation equals 1/[Mo]2, where [Mo] represents the initial concentration of the reactants (of hydroxyl or carboxyl groups in a polyesterification) at time t ¼ 0. At the start of the polymerization, there are [Mo] molecules present. After some progress of the reaction, there are [M] molecules left; [Mo] [M] is then the number of molecules that participated in the formation of polymeric chains. The conversion, p, can be written, according to Carothers [6], as

or, the concentration of [M] at any given time t is (M)=(M_o) (1-p) and the degree of polymerization,

It is important to realize from the above equation that in order to a achieve 98% conversion (p must equal 0.98). The value of DPofonly 50, it is necessary to DP, at any given time, t is equal to the ratio of monomer molecules that were present at the start of the reaction divided by the number of molecules that are still present at that particular time:

By combining the above expression with Caruthers equation and solving for [M], one obtains

For a second-order rate expression, the above equation can be written as

and by replacing 1/(- p) with DP one obtains DP =(M_o)kt+1 Using the above equation, it is possible to calculate from the rate constant (if it is known) and the concentration of monomers the time required to reach a desired number average molecular weight. When there is no catalyst present and the carboxylic acid assumes the role of a catalyst itself, then a third-order rate expression (shown above) must be employed:

By integrating the third-order rate expression, one obtains:

and, by substituting for [M] the Carothers equation and then rearranging the resultant equation, one obtains: this can also be written as:

The above equation shows that without a catalyst the molecular weight increases more gradually. It can be deduced from the above discussion that a high stoichiometric balance is essential for attaining high molecular weight. This means that any presence of a monofunctional impurity has a strong limiting effect on the molecular weight of the product. The impurity blocks one end of the chain by reacting with it. This is useful, however, when it is required to limit the DP of the product. For instance, small quantities of acetic acid are sometimes added to preparations of some polyamides to limit their molecular weight. In polymerizations of monomers with the same functional groups on each molecule, like A — A and B—B(i.e.,adiamineandadiacid),the number of functional groups present can be designated as No A for A type and No B for B type. These numbers No A and No B represent the number of functional groups present at the start of the reaction. They are twice the number of A — A and B — B molecules that are present. If the number No B is slightly larger than No A, then we have a stoichiometric imbalance in the reaction mixture. This imbalance is designated as r.

(It is common to define the ratio r as less than or equal to unity, so, in the above, B groups are present in excess.) The total number of monomers at the start of the reaction are (N^o_A + N^o_B)/2or N^o_A(1 +(1/r))/ 2.

The extent of the reaction, p, can be defined as the portion of the functional groups A that reacted at any given time. The portion of the functional groups B that reacted at the same time can be designated by rp. The unreacted portions of A and B groups can then be designated as 1 p and 1 rp, respectively. The total number of unreacted A groups in the reaction mixture would then be N^oA(1-p). This reaction mixture also contains N^oB(1-rp) unreacted B groups. The total number of chain ends on the polymer molecules is the sum of the unreacted A and B groups. Because each polymer molecule has two chain ends, the total number of chain ends is then N^o _A (1-p) +N^o_B (1-rp))/2. The number average degree of polymerization is equal to the total number of A—A and B—B molecules present at the start of the reaction divided by the number of polymer molecules at the end. This can be represented as follows:

the expression can be reduced (since r = N^o_A/ N^o_B) to

The molecular weight of the product can be controlled by precise stoichiometry of the polymeri zation reaction. This can be done by simply quenching the reaction mixture at a specified time when the desired molecular weight is achieved. Flory derived statistical methods for relating the molecular weight distribution to the degree of conversion [1, 3]. In these polymerizations, each reaction step links two monomer molecules together. This means that the number of mers in the polymer backbone is always larger by one than the number of each kind of functional groups, A or B. If there are x monomers in a chain, then the number of functional groups that have reacted is x-1. The functional groups that are unreacted remain at the ends of the chains. If we designate p as the extent of the reaction or the degree of conversion, as above, then the probability that x-1 of A or B has reacted is p^x-1, where p = (No – N)/No and the probability of finding an unreacted functional group is p 1. The probability of finding a polymer molecule that contains x monomer units and an unreacted functional group A or B is p^x-1(1-p). At a given time t, the number of molecules present in the reaction mixture is N. The fraction that contains x units can be designated as Nx and can be defined as: Nx = Np ^x-1 (1-p) The Carothers equation defines N/No = 1-p. The above expression for Nx can, therefore, be written as:

where No is, of course, the number of monomer units that are present at the start of the reaction. To determine the molecular weight distribution of the polymeric species formed at any given degree of conversion, it is desirable to express the weight average and number average molecular weights by terms, like p.BydefiningMo as the mass of the repeating unit, the number average molecular weight is:

and the weight average molecular weight is:

where w_x is the weight fraction of molecules containing x monomer units. It is equal to xNx/No and that can be written, based on the above equation for Nx,aswx ¼ xð1 pÞ2px 1. The weight average molecular weight can now be expressed as:

It can be shown that summation,

pÞ3.Based on that, the weight average molecular weight

and, the molecular weight distribution is:

It is interesting that this equation tells us that at high conversion, when p approaches 1, the molecular weight distribution approaches 2. There is experimental confirmation of this. Until now this discussion was concerned with formation of linear polymers. The presence, however, of monomers with more that two functional groups results in formation of branched structures. An example is a preparation of a polyester from a dicarboxylic acid and a glycol, where the reaction mixture also contains some glycerol. Chain growth in such a polymerization is not restricted to two directions and the products are much more complex. This can be illustrated further on a trifunctional molecule condensing with a difunctional one

Further growth, of course, is possible at every unreacted functional group and can lead to gelation. The onset of gelation can be predicted from a modified form of the Carothers equation [1]. This equation includes an average functionality factor that averages out the functionality of all the functional groups involved. An example is a reaction mixture of difunctional monomers with some trifunctional ones added for branching or cross-linking. The average functionality, fave, may be (2 + 2 + 2 + 3)/4 = 2.25. The Carothers equation, discussed above, states that p =(No-N)/No where No and N represent the quantities of monomer molecules present initially and at a conversion point p. The number of functional groups that have reacted at that point is 2(No-N). In the modified equation, the number of molecules that were present initially is Nofave. The equation now becomes:

No/N can be replaced by DP and the above expression becomes:

It is generally accepted that gelation occurs when the average degree of polymerization becomes infinite. At that point, the second term in the above equation becomes zero. When that occurs, the conversion term becomes pc. It is the critical reaction conversion point:

Gelation, however, is less likely to be a major concern in polymerization reactions where only small quantities of tri- or multifunctional monomers are present. In the preparation of alkyds, for instance (described further in this chapter), some glycerin, which is trifunctional, is usually present.

If the amount of glycerin is small, then the product is only branched. In addition, there might be only one branch per molecule:

Statistical methods were developed for prediction of gelation [7]. These actually predict gelation at a lower level than does the Carothers equation shown above. As an example, we can use a reaction of three monomers, A, B, and C. We further assume that the functionality of two monomers, fA and fB, is equal to two, while that of f_C is greater than two. The critical reaction conversion can then be written as:

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خدمات صحية متكاملة قدَّمها قسم الشؤون الطبية لزائري ليلة النصف من شعبان في مجمَّع الشيخ الكليني

خدمات صحية متكاملة قدَّمها قسم الشؤون الطبية لزائري ليلة النصف من شعبان في مجمَّع الشيخ الكليني

في ذكرى ولادة الإمام المهدي (عجّل الله فرجه).. قسم الشؤون الفكرية يصدر عددًا خاصًّا من مجلة العقيدة

للاطّلاع على خدماته المقدمة للزائرين مركز السيد جعفر الحلي يستقبل المدير العام لمديرية العمليات والخدمات الطبية في وزارة الصحة

المزيد

الآخبار الصحية

دراسة تكشف "سر الوسادة".. ودورها في حماية البصر

بالأشعة تحت الحمراء.. "ثورة علمية" في علاج اللاعبين

مشروب أحمر يخفض ضغط الدم "بشكل ملحوظ"

الصحة العالمية تكشف حقيقة انتشار فيروس نيباه خارج الهند

المزيد

الآخبار التكنلوجية

دراسة تدفعك لشراء ساعة ذكية "فورا".. هذا ما تقوله عن قلبك

علماء حاصلون على "نوبل": الذكاء الاصطناعي لن يستبدل الإنسان

وانغ هو: القمة العالمية للعلماء تجسد تقدير الإمارات للعلم

رئيس رابطة كبار العلماء يعلن تأسيس مقر في الإمارات

المزيد

مواضيع ذات صلة

Reactions of Functional Groups

Poly (vinyl alcohol) and Poly (vinyl acetal) s

Poly (vinyl acetate)

Poly (vinylidine chloride)

Copolymers of Vinyl Chloride

Poly (vinyl chloride)

Miscellaneous Fluorine Containing Chain-Growth Polymers

Copolymers of Fluoroolefins

Poly chloro tri fluoro ethylene

Poly tetra fluoro ethylene

Polyacrylamide, Poly(acrylic acid), and Poly(methacrylic acid)

Acrylonitrile and Methacrylonitrile Polymers