Thermodynamics of Polymerization

1. PSC 480/740 Kinetics 1 Thermodynamics of Polymerization Thermodynamics of polymerization determines the position of the equilibrium between polymer and monomer(s). Thus, it impacts both polymerization, depolymerization, and degradation. The thermodynamics of polymerization of most olefins is favorable due to the exothermic nature of converting ? bonds into ? bonds. For cyclic compounds, the driving force for polymerization can vary over a much wider range, and one observes a variety of behaviors ranging from completely unreactive to spontaneously polymerizable under all conditions. The well known thermodynamic expression: ?G = ?H - T?S yields the basis for understanding polymerization/depolymerization behavior. For polymerization to occur (i.e., to be thermodynamically feasible), the Gibbs free energy of polymerization ?Gp < O. If ?Gp > O, then depolymerization will be favored. Thus: Any factor that lowers the enthalpy, H (i.e., makes ?Hp more negative), or raises the entropy, S (i.e., makes ?S more positive), will shift the equilibrium towards polymerization. Standard enthalpy and entropy changes, ?Hop and ?Sop are reported for reactants and products in their appropriate standard states. Generally: Temperature = 25oC = 298K Monomer – pure, bulk monomer or 1 M solution Polymer – solid amorphous or slightly crystalline Polymerization is an association reaction such that many monomers associate to form the polymer Regardless of mechanism, there is a large loss in the total number of rotational and translation degrees of freedom in the total system as the monomers associate. This occurrence thus yields a major loss in entropy upon polymerization. Thus: ?Sp < O for nearly all polymerization processes. Since depolymerization is almost always entropically favored, the ?Hp must then be sufficiently negative to composite for the unfavorable entropic term. Only then will polymerization be thermodynamically favored by the resulting negative ?Gp. In practice: Polymerization is favored at low temperatures: T?Sp is small Depolymerization is favored at high temperatures: T?Sp is large Therefore, thermal instability of polymers results when T?Sp overrides ?Hp and thus ?Gp > O; this causes the system to spontaneously depolymerize (if kinetic pathway exists).

2. PSC 480/740 Kinetics 2 Thermodynamics of Polymerization (continued) Since most polymerizations are characterized by an exothermic propagation reaction and an endothermic depropagation reaction, the activation energy for the depropagation reaction is higher, and its rate increases more with increasing temperature compared to the propagation reaction. As shown below, this results in a ceiling temperature, defined as the temperature at which the propagation and depropagation reaction rates are exactly equal at a given monomer concentration. At long chain lengths, the chain propagation reaction is characterized by the following equilibrium expression: The standard-state enthalpy and entropy of polymerization are related to the standard-state monomer concentration, [M]o (usually neat liquid or 1 M solution) as follows:

3. PSC 480/740 Kinetics 3 Thermodynamics of Polymerization (continued) At equilibrium, ?G = 0, and T = Tc (assuming that ?Hpo and ?Spo are independent of temperature). Or: Or: At [M]c = [M]o, Tc = ?Hpo/?Spo Notice the large variation in the -?H values. ethylene > isobutylene - attributed to steric hinderance along the polymer chain, which decreases the exothermicity of the polymerization reaction. ethylene > styrene > ?-metylstyrene - also due to increasing steric hinderance along the polymer chain. Note, however, that 2,4,6-trimethylstyrene has the same -?H value as styrene. Clearly, the major effect occurs for substituents directly attached to the polymer backbone.

4. PSC 480/740 Kinetics 4 Other possible effects on ?Hp loss of resonance stabilization upon polymerization changes in bond hybridization changes in H-bonding between M and P states Notice the small changes in the ?Sp values. This small variation is attributed to the loss of translational entropy which is about constant from system to system. For the systems in the table above, the equilibrium at 25 oC (i.e., at the standard state condition) favors the formation of polymer. This may be verified using the equation we examined previously. ?Go = -RT lnKeq As the temperature increases, the equilibrium constant decreases (characteristic of an exothermic reaction). When Tc is exceeded, Keq becomes less than 1, and thus, depolymerization becomes the dominant process. It is very important to note that the Tc concept applies only to closed systems at equilibrium. For open systems, monomer may volatilize away, and thus, depolymerization may occur well below the predicted Tc. In fact, few polymers actually match their thermal stability as predicted from the Tc approach. Thermodynamics of Polymerization(continued)

5. PSC 480/740 Kinetics 5 Experimental Determination of ?Hop and ?Sop ?Hop - by direct calorimetric measurement of amount of heat evolved when known amount of the monomer is converted to a known amount of polymer. or by heats of combustion of M and P which yields ?Hof (enthalpy of formation) of M and P. The ?Hop is thus obtained by the relationship: ?Sop - from the absolute entropies of M and P, such that: The absolute entropies may be obtained from calorimetric measurements of heat capacities of M and P over a wide T range, as given by:

6. PSC 480/740 Kinetics 6 Floor Temperature Behavior Although the vast majority of all polymerizations possess negative ?H and ?S, and hence display ceiling temperature behavior, four distinct possibilities exist as outlined in the table: As stated earlier, -?S for polymerization is almost universal. Therefore, for olefins and small cyclics, polymerization is possible at low temperatures. However, many compounds are never spontaneous toward polymer due to +?H (e.g. cyclohexane, tetrasubstituted olefins) +?S for polymerization is rare, but known examples exist (see below). This rare behavior leads to floor temperature behavior or entropy-driven polymerizations. Floor temperature monomers are invariably large cyclics containing large atoms from the third row and below of the periodic table, that yield polymers with highly flexible chains. Examples of monomers possessing a floor temperature:

7. PSC 480/740 Kinetics 7 The Reactivity of Large Molecules In general, when considering growing polymer chains (i.e., regardless of the type of polymerization mechanism), the reactivity of the chain ends will be the primary focus in studying the kinetics of the polymerization reaction. Thus, investigations of the kinetics of polymerization may be simplified by assuming that the rate constant of the chain growth reaction is independent of the size of the molecule to which the reactive functional group is attached. The validity of the assumption that the rate of polymerization is independent of changes in molecular size of the reactants may be rationalized by observing the behavior of several small molecule reactions. For reactions involving homologous series of reactants, the rate constant levels off and becomes independent of molecular size when n > 2. Note that this behavior is quite analogous to step-wise polymerization. Further physical rationalizations for the underlying assumption include: 1. The larger and heavier the molecule, the greater the distance between the center of mass of the molecule and the reactive chain end. Thus, the mobility of the reactive end group in solution is much greater than the mobility of the molecular center of mass (i.e., the average mobility of the total chain). This enhanced mobility of the reactive sites yields an "encounter rate" which is much greater than that predicted by the total molecular mass and is approximately independent of the molecular size. 2. In most polymerization reactions, the diffusion rate of reactants (i.e., the reactive chain ends and monomers) is much more rapid than the chemical reaction.

8. PSC 480/740 Kinetics 8 Dependence of kp on Molecular Size

9. PSC 480/740 Kinetics 9 The Reactivity of Large Molecules(continued) Consider the following kinetic scheme: where A is the reactive site, M is a monomer, (A+M) represents the pair of reactants trapped in the "liquid cage", and P is the product polymer. The rate constants k1 and k-1 represent diffusion rate constants into and out of the liquid cage, while k2 is the rate constant for the chemical reaction. Assuming a steady-state concentration of the trapped reactants, the rate of polymer formation is given by: If the diffusion is much more rapid than the chemical reaction, such that k-1>>k2, then: Since diffusion into the cage is affected by molecular size in the same way as diffusion out of the cage, the effect of molecular size cancels out of the rate expression.

10. PSC 480/740 Kinetics 10 Kinetics of Condensation (Step-Growth) Polymerization Step-Growth polymerization occurs by consecutive reactions in which the degree of polymerization and average molecular weight of the polymer increase as the reaction proceeds. Usually (although not always), the reactions involve the elimination of a small molecule (e.g., water). Condensation polymerization may be represented by the following reactions: Monomer + Monomer Dimer + H2O Monomer + Dimer Trimer + H2O Monomer + Trimer Tetramer + H2O Dimer + Dimer Tetramer + H2O Dimer + Trimer Pentamer + H2O Trimer + Trimer Hexamer + H2O Generally, the reactions are reversible, thus the eliminated water must be removed if a high molecular weight polymer is to be formed. Based on the assumption that the polymerization kinetics are independent of molecular size, the condensation reactions may all be simplified to: ~~~~COOH + HO~~~~ ? ~~~~COO~~~~ + H2O Note that there are many types of step-growth polymerization reactions which yield a wide variety of polymers including proteins, nylons, and polyesters. Although similar treatments apply to all step-growth polymerizations, this section will focus on the kinetics of polyesterification.

11. PSC 480/740 Kinetics 11 Kinetics of Condensation (Step-Growth) Polymerization Polyesterification reactions are catalyzed by acid and the mechanism is given by: Step 1: Fast Equilibrium Step 2: Nucleophilic attack – slow, rate determining step Step 3: Loss of water Step 4: Regeneration of catalyst In this mechanism, step 1 is a fast equilibrium and step 2 is the slow, rate-determining step, which follows the rate law: By applying the fast equilibrium assumption, the rate law becomes:

12. PSC 480/740 Kinetics 12 Polyesterification Without Acidic Catalyst In this case, the carboxylic acid groups must themselves function as the catalyst such that [H+] ? [COOH] and thus, where kexp includes k2, Keq1, and other constants of the acid-base equilibrium of the carboxylic acid. For a stoichiometric feed ratio of the reactants at the beginning of the reaction (t = 0), such that [COOH] = [OH] at all times, and the rate equation becomes which upon integration yields:

13. PSC 480/740 Kinetics 13 Polyesterification Without Acidic Catalyst (continued) Consider the fractional conversion of the polymerization reaction, P, expressed in terms of the fraction of COOH groups (or OH groups) that have reacted: Substitution into the integrated rate expression yields: Note that experimental data for esterification reactions show that plots of 1/(1-p)2 vs. time are linear only after ca. 80% conversion. This behavior has been attributed to the reaction medium changing from one of pure reactants to one in which the ester product is the solvent. Thus, the true rate constants for condensation polymerizations should only be obtained from the linear portions of the plots (i.e., the latter stages of polymerization). For example, the kinetic plots for the polymerization of adipic acid and 1,10-decamethylene glycol show that at 202oC, the third-order rate constant for the uncatalyzed polyesterification is k = 1.75 x 10-2 (kg/equiv)2 min-1.

14. PSC 480/740 Kinetics 14 Uncatalyzed Polyesterification

15. PSC 480/740 Kinetics 15 Acid-Catalyzed Polyesterification Recall that the rate law from the acid catalyzed polyesterification is given by: If acid is added to the system, then by definition of a catalyst, the acid concentraion remains constant. Furthermore, at the stoichiometric feed, [RCOOH] = [OH], the rate expression becomes: and in terms of their fractional conversion of the reactive groups, Thus a second-order plot of 1/(1-p) vs. time yields a linear relationship. Note that experimental data are usually linear only beyond ca. 80% conversion. The polyesterification of adipic acid catalyzed by p-toluene sulfonic acid shows the the rate constant for reaction with 1,10-decamethylene glycol at 161 oC and 0.4% p-toluene sulfonic acid is k = 9.7 x 10-2 (kg/equiv) min-1. Note that this rate constant is significantly larger than the noncatalyzed rate constant.

16. PSC 480/740 Kinetics 16 Catalyzed Polyesterification

17. PSC 480/740 Kinetics 17 Time Dependence of the Degree of Polymerization Consider a polyesterification of bifunctional monomers, at a stoichiometric feed ratio. In general, a polymer of (AB)n may be formed in the reaction: HO-(C=O)-R-(C=O)-OH + HO-R'-OH ? HO-(C=O)-R-(C=O)-O-R'-OH + H2O or HO-A-OH + H-B-H ? HO-A-B-H + H2O where A and B are the structural units -(C=O)-R-(C=O)- and -O-R-O-, respectively. If water is efficiently removed during the reaction (which must be done to obtain high polymer), then the number of COOH groups present is equal to the number of molecules present, at all times. where N is the total number of molecules in the system and V is the volume. Since the structural units A and B are never removed during the reaction, the total number of structural units present at all times is constant and equal to the number of initial molecules.

18. PSC 480/740 Kinetics 18 The Number Average Molecular Weight in Polycondensation By defining the average degree of polymerization of the system, Xn, as the average number of structural units per molecule, the relationship becomes: This relationship is a special case of the Carother's Equation. Note that for condensation polymers prepared from two reactants, the average number of repeating units per molecule is one-half the average degree of polymerization. If Mo is the average molecular weight of the structural units, then the number average molecular weight, Mn may be defined as: where Nx is the moles of x-mer of mass Mx, and 18 is added to account for the unreacted (HOH) groups at the ends of each polyester chain. The following figure demonstrates the dependence of the number average molecular weight on the fractional conversion. Clearly, very high conversions are required in order to obtain useful polymers of molecular weights greater than 10,000.

19. PSC 480/740 Kinetics 19 Mn as a Function of Conversion

20. PSC 480/740 Kinetics 20 Using the kinetic relationships derived earlier, a dependence of the molecular weight on reaction time may be given by: For large reactions times (i.e., for conversions greater than 80%) the following approximations are reasonable. The Number Average Molecular Weight (continued)

21. PSC 480/740 Kinetics 21 Molecular Weight Distributions of Linear Condensation Polymers While the average degree of polymerization may be determined at any time t using the above relationships, it is equally important to know the distribution of molecular weights and the dependence of this distribution on reaction time. Given a reacting system composed of an A-B type monomer, we wish to define the number fraction of molecules, at a given conversion, p, which contain exactly x structural units. A key question becomes: What is the probability that a molecule selected randomly from the polymerization mixture will contain exactly x structural units? p = conversion = fraction of COOH groups that have reacted at time = t, and (1-p) = fraction of COOH groups remaining at time = t Thus, the probability of obtaining the molecule shown above is given by Prob(x) = px-1(1-p) The chance that a randomly selected molecule contains exactly x structural units is equal to the fraction of molecules composed of x-mers, such that

22. PSC 480/740 Kinetics 22 where Nx is the number of x-mers in a system of N molecules. Thus, the relationship becomes: Therefore, we can see that Prob(x) is the mole fraction of molecules containing x structural units If the evolved water is completely removed during the polymerization, then NCOOH = N = No(1-p) where No is the initial number of molecules. Combining eqs. (3) and (4) yields: Nx = No (1-p)2 px-1 As shown in the following Figure, for any given conversion, p, low molecular weight polymers (i.e., the low values of x) have the highest probability of being formed in the total distribution. However, the distribution becomes broader and the average molecular weight increases as the conversion increases. Molecular Weight Distributions of (continued)

23. PSC 480/740 Kinetics 23 Effect of Conversion on the Number Distribution of Structural Units

24. PSC 480/740 Kinetics 24 Molecular Weight Distributions of (continued) The number average molecular weight is obtained from Prob(x) and the definition of an average. Neglecting the weight of water on the terminal groups of the condensation polymer, the molecular weight of an x-mer is given by: Mx-mer = xMo where Mo is the average molecular weight of the structural units. Thus, we have: Now, it can be shown that for p = 1, Combining eqs. (8) and (9) yields:

25. PSC 480/740 Kinetics 25 Molecular Weight Distributions of (continued) The weight fraction of x-mers, Wx, may be defined as the total weight of molecules containing exactly x structural units divided by the total weight of polymer: The following is true for p = 1: Combination of eqs. (11) and (12) yields the simplification: Again, the following Figure shows that this distribution of Wx favors low molecular weight polymer at low conversions. In addition, the weight average molecular weight, Mw , may be defined as: In view of eq. (11) we have:

26. PSC 480/740 Kinetics 26 Effect of Conversion on the Weight Distribution of Structural Units

27. PSC 480/740 Kinetics 27 Combination of eqs. (10) and (15) shows that the polydispersity is given by: Molecular Weight Distributions of (continued)

28. PSC 480/740 Kinetics 28 Effect of Non-Stoichiometric Reactant Ratios The highest possible molecular weight is achieved in polycondensation reactions using equal concentrations of reacting groups. However, it is often desirable to produce a specific molecular weight in polymerization. This is accomplished by designing the system so that unreacted or unreactive end groups are incorporated into the polymer. Since molecular weight is inversely proportional to the number of end groups, this offers a means for molecular weight control. We will consider three types of systems. Type 1 – A system of A-A and B-B monomers in which the total number of A functional groups, NA, is less than (or equal to) the total number of B functional groups, NB. We define a stoichiometric imbalance parameter, r, where, In this situation, reaction proceeds until the A groups are completely consumed and all the chain ends possess unreacted B groups. It is obvious that the greater the stoichiometric imbalance, the more leftover B groups there will be, and the lower the molecular weight. Type 2 – A system of A-A and B-B monomers in which molecular weight control is achieved by the addition of small amounts of a monofunctional monomer containing either a single A or B group. Type 3 – A system of A-B monomers in which molecular weight control is achieved by addition of small amounts of mono- and/or polyfunctional monomers containing only A or only B groups.

29. PSC 480/740 Kinetics 29 Effect of Non-Stoichiometric Reactant Ratios (continued) For all types of systems, the polymerization can be designed to yield the desired through the use of the Carother’s Equation. The key to this method is the concept of number average functionality, favg. To compute favg, one must first identify which is the minority or deficient type of group, A or B. Thus for the case in which the A groups are deficient in number, where, N(A)’s = the number of moles of each type of monomer carrying an A group f(A)’s= functionality of each type of monomer carrying an A group N’s= the number of moles of each type of monomer present (A and B) The Carother’s Equation is: or where, p = the fractional conversion of the deficient groups = the number average degree of polymerization

30. PSC 480/740 Kinetics 30 For Type 1 systems, the total number of molecules at any time is given by: With the degree of polymerization defined as the average number of structural units per molecule, the average degree of polymerization in terms of conversion and feed ratio is now given by: Note that at a stoichiometric ratio r = 1 the above relationship reduces to the previous form: In addition, the maximum average degree of polymerization possible corresponds to a complete conversion of the A groups (i.e., p = 1), such that: Effect of Non-Stoichiometric Reactant Ratios (continued)

31. PSC 480/740 Kinetics 31 Effect of Non-Stoichiometric Reactant Ratios (continued) For Type 2 systems (with NA = NB + NB’) where, NB’ = number of B groups contributed by a monofunctional monomer, and For Type 3 systems (with NA = NB) where, NBf = number of B groups contributed by a polyfunctional monomer, and f = functionality of polyfunctional monomer, and, All other cases should be treated using favg and the general Carother’s Equation.

32. PSC 480/740 Kinetics 32 Branched and Cross-Linked Condensation Polymers Mono and bifunctional monomers yield linear polymers; however, if one of the reactants is a tri- or multifunctional monomer, then a branched or crosslinked polymer will result. The general form of the Carothers equation allows the possibility of calculating the conditions needed to avoid or ensure the reaching of the gel point (i.e., the point of extensive crosslinking). Since gelation is presumed to occur when the average degree of polymerization becomes infinitely large, the Carothers equation reduces to: where pc is the critical conversion. In practice, it is important to note that this approach often overestimates the reaction point at which gelation occurs. This overestimation is attributed to the broad molecular weight distribution in which the high molecular weight molecules reach the gelation point before those which have the average value of the molecular weight.