CHAPTER 1: Kinetics and Thermodynamics of Radical Polymerization

Radical polymerization processes are of great scientific and economic importance. Knowledge about their kinetic principles is a prerequisite for the efficient synthesis of a wide range of polymeric products. Since the dawn of macromolecular chemistry in the 1920’s, the study of these principles has been a central topic of academic research. Although a radical polymerization process is basically constituted by just four types of reactions, which are initiation, propagation, transfer and termination, the coupled nature of these reactions leads to a complexity that makes it difficult to determine the individual rate constants and to evaluate their effects on the properties of the final polymer, like its molecular weight distribution. Scheme 1.1 shows a set of fundamental reaction equations for a radical polymerization process.

There is a kinetic rate law expression for each of these reactions. Determination of the corresponding rate coefficients is the main task of all kinetic experiments in this field. The employed experimental techniques can roughly be separated into two classes: one approach focuses on accurate measurements of the overall polymerization rate or time-resolved species concentration, while the other one is based on the analysis of the resulting molecular weight distributions. Provided that all relevant rate coefficients for a certain polymerizing system are known, it is possible to make precise predictions about the kinetics of the entire process, and therefore also about the molecular weight distribution. Today, computer simulations are an important tool in polymer research, allowing for precise numerical simulations of even very complex polymerizing systems and thus contributing to a deeper understanding of radical polymerization kinetics.

The intention of the following chapter is to give a general overview of our knowledge about the kinetics of conventional radical polymerization and its implications for the process and the formed product. This basic knowledge is also mandatory for the understanding and optimization of controlled polymerization processes.

For a radical polymerization to occur, the first thing needed are free radicals. These are initially provided by some agent, the initiator, during the reaction step called initiation. The initiation step is commonly characterized by two coefficients, the initiation rate coefficient k_i and the initiator efficiency f. Knowledge of these parameters is of crucial importance for both industrial applications and theoretical studies of radical polymerizations.

The vast majority of initiators belong to one of two groups, thermal initiators or photoinitiators. Thermal initiators form radical species upon heating, while photoinitiators decompose when exposed to visible or UV light. While in commercial processes mainly thermal initiators are used, kinetic studies are preferentially performed using photoinitiators. This is because the irradiation can precisely be timed, defining a sharp starting point for the polymerization reaction. A general scheme for the decomposition reaction, regardless of the type of initiator, is given in Scheme 1.2.

The initiator (I) decay, be it caused by light absorption or heating, follows a first order rate law:

For the polymerization kinetics, the initiator concentration is not the important quantity. More important is the concentration of primary radicals formed by the initiation process. The rate R_d of formation of radicals that can start chain growth can be expressed by the following first order rate law:

where k_d is the rate coefficient for the initiator decomposition reaction, and f is the initiator efficiency. Integration of eqn (1.2) leads to the following expression, showing the exponential decrease of initiator concentration with time:

Initiator decay alone is not sufficient to start a new polymer chain. The formed radical has to react with a monomer unit. Right after decay, the (usually two) freshly formed radicals I₁˙ and I₂˙ are still in close proximity of each other and surrounded by solvent molecules. The primary radicals’ ability to leave the solvent cage unreacted and then react with a monomer is expressed by the initiator efficiency f, with values ranging from zero (no initiation) to unity. In a real system, not every primary radical will actually start a polymer chain. Radicals can recombine before leaving the solvent cage or undergo a side reaction before they encounter a monomer molecule. Typically, f has a value between 0.5 and 0.8 and depends on the viscosity of the system, indicating that the diffusion-controlled escape from the solvent cage is the crucial factor.

If the initiator molecule is asymmetric, i.e. I₁˙≠I₂˙, the formed radical species generally do not show identical reactivity towards the monomer. Thus, the initiation process will take on the form shown in Scheme 1.3, where M indicates a monomer molecule, R₁˙ refers to a macroradical of chain length 1 and k_i⁽¹⁾ and k_i⁽²⁾ refer to the rate coefficients of initiation for the respective initiator fragments. The overall rate of initiation, R_i, can be calculated according to eqn (1.4):

The rate coefficient of initiation k_i can be expressed as the arithmetic mean of the coefficients for the individual fragments, since

There are mainly two types of thermally activated initiators: azo-type, and peroxy-type. Their general structures are shown in Scheme 1.4.

Thermal initiators decompose in a first order reaction upon heating, displaying a characteristic half life at a certain temperature. It is correlated to the decomposition rate coefficient by eqn (1.6). The half-life, t_1/2, is the amount of time it takes for half of a sample of initiator to decompose.

A large variety of initiators has been described in the literature, and many of them are available commercially, so that the initiator can be chosen according to the desired decomposition rate. For practical use, initiators are often characterized by the temperature where they show a half life of 10 h. Typical values for these temperatures lie in the range from 20 to 120 °C. A large collection of data on initiator properties has been published in the Polymer Handbook.¹  However, the decomposition rate is not the only property to consider when selecting an initiator. Possible side reactions with monomer or solvent might also play a role, or the ability of the initiator to act as a transfer agent (compare Section 4.2).

Photoinitiators are seldomly used in commercial applications, because large reaction volumes cannot be irradiated easily in a uniform way. Still, there are some applications in the area of UV hardening lacquers and printing inks. In research, on the other hand, photoinitiators are used frequently, for they allow to precisely define the beginning and end of the initiation process by flash photolysis of the initiator. Also, most photoinitiators show almost no temperature dependence of the decomposition rate, but a strong dependence on the (UV) light intensity.

A good photoinitiator for a given polymerization should have the following properties:

1. An irradiation wavelength should be available were the initiator shows strong absorption, but the monomer and the solvent show almost no absorption.

2. The initiator should show a high efficiency.

3. At best it should only generate a single radical species.

There are two groups of photoinitiators that differ in the mechanism of radical formation. Type I photoinitiators generate radicals via unimolecular bond cleavage after irradiation, similar to thermal initiators. Type II initiators show no bond cleavage directly after irradiation. Instead, they enter a bound excited state which reacts with a so-called co-initiator molecule to generate free radicals, mostly via H-abstraction. Most visible light active photoinitiators belong to type II.

Type I photoinitiators show considerable structural variety, one of the most common motives being the acetophenone group. The general structure of this kind of photoinitiators is shown in Scheme 1.5.

Upon irradiation, the initiator molecules will absorb a certain amount of energy changing from the electronic ground state to an excited state. The accessible electronic states of a molecule are usually shown in a Jablonski diagram. An example of such a diagram is shown in Figure 1.1.

In most cases, absorption will cause the initiator molecule to enter the first singlet excited state, commonly denoted as S₁. In general, more than one deactivation channel will be active for the excited species, and not all of them do necessarily lead to free radical generation. The fraction of the excited molecules that are actually converted into primary radicals is expressed by the overall quatum yield Φ, which is the product of the quantum yields of three successive elementary processes: intersystem crossing from the lowest excited singlet to the lowest triplet state (1.8), bond scission in the triplet state (1.9), and reaction of the formed radical with a monomer molecule (1.10).

where

For ketones, the intersystem crossing step usually has a rather high quantum yield,²  so setting the value of to one is a good approximation in most cases. The overall quantum yield is then determined by the values of and . may be reduced due to alternative deactivation pathways from the T₁ triplet state. The dominating deactivation reactions in free radical polymerizations are the reaction with molecular oxygen and the deactivation by monomer molecules (compare ref. 3). The influence of the former may be reduced by thorough degassing of the polymerization mixture prior to initiation.

A long life time of the triplet state tends to lead to a reduced overall quantum yield, because chances are higher that alternative deactivation routes may successfully compete with radical formation. 2,2-Dimethoxy-2-phenylacetophenone (DMPA) is an example for an initiator with a rather short-lived first triplet state (τ<0.1 ns),⁴  causing a high quantum yield . The quantum yield for the formation of macroradicals from the initiator fragment radicals is also known as the initiator efficiency, f, which is defined analogously to the efficiency of thermal initiators.

Most acetophenone-type initiators decompose into more than one kind of radical upon irradiation. Typically, two different radical species and are formed, showing very different reactivities towards the monomer. While the carbonyl radical is efficiently forming macroradicals, the additional radical does not add much to the initiator efficiency. On the contrary, for DMPA the methoxy radical is thought to be involved exclusively in termination steps.⁵  In such a case, it is common to say that the carbonyl radical is the “effective” and the methoxy radical the “ineffective” primary radical. Their initial concentrations are equal and denoted as ρ, while the overall initial radical concentration is 2ρ. Polymerization kinetics can strongly be influenced by the formation of two sorts of primary radicals with markedly different reactivities.

Azoinitiators may also be used as photoinitiators. They decompose upon irradiation by a mechanism that is markedly different from the acetophenone type. By example of 2,2-azobisisobutyronitrile (AIBN), this mechanism is shown in Scheme 1.6.

UV irradiation leads to a cis-trans isomerization reaction. Since the rate of this isomerization reaction is finite, it is observed in experiments that the time of incidence of the laser pulse is not identical with the time of primary radical formation, but there is a certain delay, usually in the order of microseconds.

The concentration of effective primary free radicals that is generated by irradiation may be calculated by:

with Φ the primary quantum yield, n_abs the number of absorbed photons and V the volume considered.

The number of absorbed photons is given by the Beer–Lambert Law:

• E_p: energy absorbed by the sample

• E_λ: molar energy of photons at the irradiation wavelength λ

• ε: molar absorption coefficient of the initiator molecule at wavelength λ

• c: photoinitiator concentration

• d: optical path length

Detailed information about different photoinitiators and their decomposition behaviour has been gathered by Gruber.⁶ 

It is not strictly necessary for a radical polymerization to be started by an initiator. It might also be initiated by impurities, by peroxy compounds that are formed in the presence of molecular oxygen, or even by the monomer itself. A prominent example for the latter is the self-initiation of styrene, which proceeds via Diels–Alder reaction of two monomers, as depicted in Scheme 1.7.⁷  Such self-initiated polymerization processes are typically limited to elevated temperatures, and can often be prevented under very pure conditions. Few monomers are capable of self-initiation even under very pure conditions, one of which is styrene, reacting via a self Diels–Alder cycloaddition mechanism.^8–10  The self-initiated bulk polymerization of styrene has a substantial activation energy: a 50% monomer conversion needs 400 days at 29 °C, but only 4 h at 127 °C. However, the produced polystyrene is very pure due to the absence of initiators and other additives.

Redox-initiation is most frequently used in polymerizations in aqueous systems but may be used in organic solvents as well. A redox-initiator consists of an oxidizing and a reducing agent. In most redox initiators, the redox reaction leads to the formation of only one radical, avoiding cage termination processes and thus enhancing the initiator efficiency. As an example, Scheme 1.8 shows the radical forming reaction in an iron-(ii)-peroxide system.

For more information on redox-initiation, the reader is referred to more specialized literature, for example the review article of Sarac¹¹  and the sources cited therein.

The propagation step, that is the addition of another monomer unit to a macroradical, can be described by a rate law expression as shown in eqn (1.13).

where k_p is the rate coefficient of propagation of a macroradical. It is well known that up to high monomer conversions, the propagation step is controlled chemically. Therefore, the rate coefficient is independent of monomer conversion as long as it stays lower than about 80%. The chemical control of the propagation reaction becomes obvious when comparing the frequency of propagation reactions, typically on the order of 10³ s⁻¹, to the average collision frequency in a liquid at room temperature, which is about 10¹² s⁻¹, meaning that only one in 10⁹ collisions is reactive.

While the rate coefficient of propagation does not depend on conversion (except in the high viscosity regime), it is doubtlessly dependent on chain length. The first few addition steps are particularly fast and may reach rate coefficients many times the long chain limit, as in the case of methyl methacrylates, where the first propagation step at 60 °C exceeds the long chain limit by a factor of about 16.^12–14  It has also been found that the product of monomer concentration and propagation rate coefficient is strongly dependent on chain length up to several hundred monomer units.^15–18  Since currently no method is known to determine the propagation rate coefficient independently of the concentration of monomer, the apparent chain length dependence of k_p might actually reflect a structuring of monomer concentration in the surrounding of the propagating chain end, rather than a chemical effect.

The absolute value of the propagation rate coefficient is determined by the reactivity of the propagating radical as well as by the properties of the monomer unit. The relationship between monomer reactivity and reactivity of the propagating radical is roughly reciprocal, i.e. a very reactive monomer corresponds to a rather unreactive radical and vice versa.

An important condition for chain growth is that the macroradical is sufficiently stable, which means it has to survive a large number of ineffective collisions with monomer or solvent in order to finally react with a monomer molecule. Any possible decomposition or side reaction has to be sufficiently slow so that it will not compete with propagation.

When going from ethene as the simplest possible monomer for radical polymerization to monomers with higher reactivity, one or more hydrogen atoms are formally substituted by groups activating the double bond and stabilizing the macroradical. Of course, steric hindrance is added at the same time, so a complete separation of entropic and enthalpic effects on the propagation rate seems impossible.

Nevertheless, a certain degree of separation is achievable by comparing Arrhenius parameters of different monomers, as shown in Scheme 1.9. It is found that electronic changes are reflected mainly by the activation energy E_A, while steric effects tend to change the pre-exponential factor A. In comparison with methyl methacrylates (MMA), dimethyl itaconate (DMI)^19,20  shows a considerably lower pre-exponential factor, while the activation energy is almost identical.

This effect is attributed to the greater sterical hindrance in DMI. If MMA is compared to ethyl α-hydroxy methacrylates (EHMA) instead, the activation energy is reduced due to the electronic effect of the additional β-oxygen atom, while the pre-exponential factor remains unchanged. In a monomer that combines both effects in one molecule, like ethyl α-acetoxymethyl acrylate, a decrease in both Arrhenius parameters is observed (compare Scheme 1.9).

Structurally similar monomers often display similar values for the propagation rate coefficient. For example, acrylate monomers with linearly increasing ester groups, as in methyl, ethyl, propyl acrylate, and so on, have virtually identical k_p values. The same is true for a series of methacrylate systems. Data on rate coefficients and Arrhenius parameters for a large range of common monomers can be found in the literature.²¹  Selected values are presented in Table 1.1. These values cover data up to 2000 and were obtained via pulsed-laser polymerization size-exclusion chromatography (PLP-SEC),²²  which is the IUPAC-recommended and most accurate kinetic method for determining k_p values. Since then, many other k_p values have been obtained and published, e.g., for bulky acrylates,²³  butyl acrylate,²⁴  for methacrylic acid,²⁵  for acrylonitrile,²⁶  and for radical polymerization from solid surfaces,²⁷  to name but a few.

It is important to note here that there is sometimes more than one type of propagating radicals in one monomer system. These radicals may exhibit significantly different reactivities. This is true, for instance, for acrylates, vinyl acetate or ethylene, where a backbiting reaction (intramolecular transfer for polymer, see below) induces the transformation of secondary propagating radicals (SPR) to tertiary mid-chain radicals (MCR). These two radicals may have very different k_p values, differing by orders of magnitude.²⁴  The ratio of these two types of radicals depends on the reaction temperature^35,36  and will impact the overall rate of propagation, which is lower than expected for pure terminal radicals. Assuming that backbiting is the only cause of MCR formation, an effective propagation rate k_p^eff may be calculated when the backbiting rate k_bb (see below) and monomer addition rate of SPR and MCR are known:^24,37 

For most monomers, a strong pressure dependency of the propagation rate coefficient is observed, with k_p increasing at higher pressures. The termination rate coefficients, on the contrary, typically show a decrease with increasing pressure, leading to an increased overall polymerization rate at elevated system pressures (see below).

The solvent usually has no large influence on the propagation rate, since the reaction is under chemical control up to high conversions. Studies of the solvent dependence of propagation rates generally found only minor effects.^38–40  These are rationalized by assuming a difference in monomer concentration between the solution and the surrounding of the growing chain end, depending on the difference of molar volumes of monomer and solvent.⁴¹ 

In contrast to these relatively weak influences of many solvents, water has a pronounced impact on aqueous radical polymerization. It is frequently found that the polymerization rates of water-soluble monomers in aqueous solutions are higher than in organic solvents.⁴²  PLP-SEC studies into different systems (methacrylic acid (MAA)^43–45  acrylic acid (AA),^46–48  N-isopropyl acrylamide,⁴⁹  acrylamide,⁵⁰  and N-vinyl pyrrolidone⁵¹  revealed a huge solvent effect on k_p. Figure 1.2 demonstrates this effect on the example of N-vinyl pyrrolidone.

Nonionized methacrylic acid (MAA), for instance, has been studied within the entire concentration range from bulk to dilute aqueous solution.⁴⁴  Hydrogen-bonded interactions between the propagating MAA macroradical and an environment that, depending on the particular MAA concentration, consists of different relative amounts of MAA and water molecules do not significantly affect the activation energy for propagation reaction, E_A(k_p), but largely influence the pre-exponential factor, A(k_p) by about a factor of 10. This is assigned to the increased friction that the relevant degrees of rotational motion of the transition state structure experience⁵²  upon replacing H₂O by MAA molecules. This finding for MAA appears to be a general kinetic effect for hydrogen-bonded monomers in aqueous solution. It is, however, not consistent with the data that the variation in k_p is due to a deviation in local monomer concentration at the site of the propagating radical. For protic monomers, like MAA and acrylic acid (AA), polymerizations at different degrees of ionic dissociation revealed that ionization also strongly affects k_p, which also can be explained by the above-mentioned entropic effect, which primarily acts on the Arrhenius pre-exponential factor.⁵³  For both AA and MAA, a decrease of k_p with increasing degree of dissociation is observed. A solution of 5 wt% of monomer in water, at a temperature of 6 °C, shows a decrease of k_p by about an order of magnitude in the range α=0–1.⁵⁴ 

In the presence of ionic liquids, radical polymerization rates and polymer molecular weights have also been reported to be significantly enhanced as compared to reaction in conventional organic solvents or in bulk.^55–60  This is partly due to an increased k_p value when using this specialty solvents. By applying the PLP-SEC method it was found that k_p for methyl methacrylate (MMA) in mixtures containing 50 vol% 1-butyl-3-methylimidazolium hexafluorophosphate ([bmim]PF₆) is about a factor of 2.5 above the bulk polymerization value.^61,62  The k_p in ionic liquid solution containing only 20 vol % MMA was found to be enhanced even by a factor of 4 in comparison to the bulk value.⁶³  The analysis of the k_p data from PLP-SEC experiments on two methacrylate monomers dissolved in four ionic liquids suggests that the observed, rather pronounced solvent effects are primarily due to a lowering of the activation energy for propagation upon gradually replacing monomer molecules by ionic liquid species (Figure 1.3).

Models that include only initiation, propagation, and termination reactions generally predict higher average molecular weights than experimentally observed. The main reason for this discrepancy is stopped chain growth due to transfer reactions. A transfer reaction is a reaction that transfers the radical function from a chain end to another molecule, as shown in Scheme 1.10.

A transfer agent T contains a weakly bonded atom X that can be abstracted by a propagating macroradical of chain length i, thus generating a dead polymer molecule and a new radical T˙. The newly formed radical might initiate new chain growth or undergo a termination reaction with some other radical. The transfer agent might be a purposefully added molecule, but transfer to monomer, solvent or initiator are well-known and mostly undesirable side reactions. The rate law for a general transfer reaction is given by the following equation

where k_tr is the rate coefficient of the transfer reaction and c_T and are the concentrations of the transfer agent and the macroradical, respectively. It is common to express the influence of transfer reactions as the transfer constant C, which is the ratio of the transfer rate coefficient to the propagation rate coefficient:

A large collection of data on initiator properties has been published in the Polymer Handbook.¹ In special cases, transfer has a measurable impact on the overall reaction kinetics, i.e., the polymerization rate R_p depends on these two rate coefficients as well as the coefficient k_re-in for re-initiation, meaning the initiation of a new polymerizing chain by the transfer agent radical. Four different scenarios may occur, depending on the relative sizes of the rate constants:

• k_p≫k_tr and k_re-in ≈ k_pThis is the most common case. The average molecular weight is reduced, while the transfer reaction has no significant influence on the polymerization rate.

• k_p≫k_tr and k_re-in < k_pIn this case, the transfer reaction displays an inhibitory effect, slowing the polymerization reaction and decreasing the average molecular weight.

• k_p≪k_tr and k_re-in ≈ k_pTelomerization is observed, the molecular weight is severely reduced while the overall reaction rate remains unaltered.

• k_p≪k_tr and k_re-in < k_pDegradative chain transfer occurs: both the average molecular weight and the overall reaction rate are decreased.

(These scenarios will also be addressed within the context of inhibition and retardation, see below.) Although transfer can be an inconvenient side reaction, it can also be used as a tool to deliberately influence a polymer's molecular weight distribution, for example in a case (i) scenario, where the overall reaction kinetics is not disturbed, because the number of free radicals remains unaltered, but the molecular weight distribution is shifted to a lower average weight. Case (i) is also assumed in the justification of the Mayo equation, which is commonly employed for determination of transfer rate coefficients (see below).

Only one type of transfer reaction is unavoidable, and that is transfer to monomer. In absence of other transfer reactions, transfer to monomer sets an upper limit to achievable chain lengths. Typical rate coefficients for this process are in the range of 3×10⁻⁵ and 20×10⁻⁵ L mol⁻¹ s⁻¹, which thankfully is slow enough to allow for appreciable polymer growth. It is not possible to reduce the influence of transfer to monomer by lowering the monomer concentration, because both the transfer rate and the propagation rate are proportional to monomer concentration, so it cancels out in the ratio. Nevertheless, the activation energies of propagation and transfer to monomer are generally different, typically being greater for the transfer reaction, so a decrease in temperature typically results in a reduced monomer transfer constant C_M. Table 1.2 lists C_M values for a selection of usual monomers.

For monomers containing aliphatic hydrogen atoms, the transfer reaction usually occurs via abstraction of one such hydrogen atom. Ethene is an example for a monomer not containing aliphatic hydrogens. Nevertheless, transfer occurs also by abstraction of a hydrogen atom from a monomer molecule by a propagating radical, as shown in Scheme 1.11.

In the case of styrene, direct hydrogen abstraction is unlikely, since the chain transfer coefficient of ethylbenzene is significantly lower than that of styrene. So the hydrogen is rather abstracted from a Diels–Alder adduct of two styrene molecules, where the carbon–hydrogen bonds are comparatively weak.

In the case of vinyl chloride, transfer proceeds via chlorine abstraction: a monomer molecule adds head-to-head to a propagating radical. The resulting radical is highly unstable and reacts with another monomer molecule in a chlorine abstraction step.⁷²  This special reaction pathway is the reason for C_M being much greater for vinyl chloride than for most other industrially important monomers.

Allylic compounds (CH₂CHCH₂X) are usually reluctant to homopolymerization, due to the activation of the allylic hydrogen atom towards abstraction. The generated allylic radical is highly stabilized by both the substituent X and by delocalization of the free electrons into the double bond. These radicals add to monomer very slowly and perform side reactions that in turn lead to retardation.

Except in the rare case of self-initiated polymerization, transfer to initiator is also unavoidable, but as long as initiator concentration is kept low, its impact on overall reaction kinetics is small. For some common initiators, transfer constants C_I are given in Table 1.3.

Since the possibilities of influence on transfer to monomer and transfer to initiator are very limited, interest is often focused on the transfer to solvent reaction. In typical industrial polymerization processes, the solvent is the major component of the reaction mixture, so even comparatively slow reactions gain importance, due to the high concentration. For many solvents, transfer rate coefficients are similar to those of common monomers given above, but there are exceptions, like tetrachloromethane, showing considerably higher transfer constants.

Solvents displaying very high transfer constants cannot be used as solvent for a polymerization reaction mixture, but can be useful as a purposefully added transfer agent. For a transfer constant greater than unity, even small amounts of the respective substance can significantly influence molecular weights, making it a useful tool to control viscosity and thus enhance heat flow in an industrial process.

Another means to control the average molecular weight is to increase the initiator concentration. But this has the drawback of a considerable increase in polymerization rate, which might lead to uncontrollable reaction conditions. Examples for transfer agents displaying very high transfer constants are thiols and compounds with high halogen contents, like tetrabromomethane. If large quantities of such efficient transfer agents are applied, this results in telomerization, that is, the formation of extremely short-chained products, mainly consisting of di- and trimers. Table 1.4 gives an overview of some important transfer agents.

Thiols may be used as transfer agents in a wide variety of free radical polymerization processes.⁸⁵  Scheme 1.12 shows the general reaction mechanism for this class of transfer agents. Nucleophilic radicals react more readily with thiols than electrophilic radicals, so transfer coefficients are higher for vinyl esters and styrene than for acrylates and methacrylates. Aromatic thiols react more readily than aliphatic ones, i.e., the chain transfer constant is higher, but they also show a stronger retardation effect as the resulting S-centered radicals are less prone for monomer addition due to their increased stability. The product of the transfer reaction is a thiyl radical, which is electrophilic and will react preferably with the more electron rich monomer in copolymerizations.

It is feasible to use functionalized thiols as transfer agents to prepare end-functional polymers. Such materials have successfully been used in the synthesis of block and graft copolymers.

A broad spectrum of organic disulfides has been employed as transfer agents. Among these are dialkyl, diaryl and diaroyl disulfides as well as xanthogens. For monomers such as styrene or methacrylates, transfer coefficients of disulfides are very low, but in the polymerization of vinyl esters they may come close to unity. For xanthogens and thiurams, transfer coefficients are usually higher than for disulfides, which is commonly explained by their iniferter capabilities.^86,87 

Monosulfides are less effective transfer agents than disulfides, due to steric reasons and because of the greater strength of the C–S bond compared to the S–S bond.

Some metal complexes show the ability to catalyze transfer to monomer reactions, so they lead to very high transfer coefficients without being consumed in the transfer reaction or built into the polymer chain.^88–90  The transfer catalysts that are most commonly employed are based on low-spin cobalt macrocycles.⁹¹  Scheme 1.13 shows some typical structures of such complexes.

While catalytic chain transfer is very efficient for tertiary radicals, as for example in methyl methacrylate polymerization, its efficiency is much lower for secondary radicals. This is explained by cobalt–carbon bonding as a side reaction, which significantly reduces the concentration of the catalytically active species. Adducts containing cobalt–carbon bonds have been detected via matrix-assisted laser desorption ionization (MALDI) mass spectrometry.⁹² 

The mechanism is thought to operate as shown in Scheme 1.14 for the example of methyl methacrylate. A β-hydrogen atom is abstracted by the cobalt complex, thus forming a terminally unsaturated polymer chain and a cobalt hydride species that re-initiates a monomer unit by hydrogen atom transfer.

Catalytic chain transfer has become a useful tool in commercial synthesis, allowing for the production of low molecular mass polymers with terminal double bonds. Such materials may be used as chain transfer agents for block structure synthesis or as macromonomers in the synthesis of comb- or star-shaped polymers.

Alkyl halides may also be used as transfer agents in radical polymerization processes. Their reactivity depends not only on the nature of the halogen atom and the alkyl unit, but also on the character of the propagating radical. For nucleophilic radicals like styrene, the reactivity is much higher than for electrophilic radicals like acrylates. The haloalkyl radicals that are formed in the transfer reaction are, in turn, electrophilic in nature.

The transfer coefficients increase with increasing period of the halogen, Cl<Br<I, and also increases with the number of halogen atoms. For example, in the polymerization of styrene, transfer coefficients for iodoacetic acid, bromoacetic acid and chloroacetic acid are C_I=0.8, C_Br=0.043 and C_Cl=0.020, respectively, whereas the values for dibromo- and tribromoacetic acid are C_Br2=0.27 and C_Br3>10. Halogenated organic acids and their esters and halomethanes are the most commonly employed groups of alkyl halides.

The application of high concentrations of effective transfer agents leads to the exclusive formation of oligomeric species, also known as telomerization. Both halomethanes and halogenated esters have found widespread application in the synthesis of telomers, due to their low cost and high transfer efficiencies. The kinetics of 35elomerisation reactions are rather difficult, since the reactivities of the various involved radical species scatter over a broad range. The tetrahalomethanes react according to the mechanism shown in Scheme 1.15, leading to a trihaloalkyl radical that may re-initiate polymerization. Of course, the polymer chain started by the trihalogenomethyl radical may again take part in a transfer reaction, but the reactivity decreases rapidly with the number of halogen atoms, as mentioned above.

In the case of hydrohalomethanes, two alternative routes are possible for the transfer reaction, either halogen- or hydrogen-atom abstraction. The hydrogen transfer is favoured for chloroform, because the C–H bond is weaker than the C–Cl bonds.

Transition metal halides can also act as transfer agents. For example, copper(ii) chloride or iron(iii) chloride may be applied. Transfer coefficients for these two halides have been determined in DMF at 60 °C. For copper(ii) chloride, the transfer coefficients are C=10⁴, C=10³ and C=10² for styrene, MMA and acrylonitrile, respectively.⁹³  Iron(iii) chloride is less efficient and gives values of C=626, C=306, C=86, C=4 and C=2 for vinyl acetate, styrene, vinyl chloride, MMA and acrylonitrile, respectively.⁹⁴  The transfer process forms alkyl halides and metal species in lower oxidation states, the latter occasionally being capable of activating the former. Such a mechanism of reversible activation and deactivation is utilized in atom transfer radical polymerization (ATRP).

Transfer to polymer occurs via intra- or intermolecular abstraction of a hydrogen from the polymer backbone by the chain-end macroradical. By this transfer reaction, the number of growing chains is not increased. Hence, the average molecular mass of the obtained polymer is not intrinsically lowered by transfer to polymer. A lowering of molecular mass, as may be expected from the ideal kinetic chain-length (ratio between propagation and termination rate) in conjunction with chain-end radical k_p, is generally observed due to the decreased propagation reactivity of radical species (MCR) produced by transfer to polymer (see above). Further, short- and long-chain branches are introduced in the chains. Transfer to polymer is thermodynamically favored as the position of the radical at the chain-end is less stabilized as compared to a radical position along the polymer backbone. In ethene polymerization, the driving force comes from the transformation of a primary chain-end radical into a secondary mid-chain radical.^95,96  Thus, the transfer to polymer rate constants are considerably higher (approximately by a factor of 10) than those observed for the corresponding monomer.

Transfer coefficients to polymers are not as readily determined as other transfer coefficients because the process does not necessarily lead to a reduction of the molecular weight. Polymerization in the presence of polymer yields a mixture of the polymer initially present and the new polymer formed, and thus the decrease in the molecular weight cannot be accurately evaluated. However, chain transfer coefficients to polymers are accessible via the structural investigations of the generated branched macromolecules, e.g. via NMR.^97,98  An alternative approach utilizes the reversible addition-fragmentation chain transfer (RAFT) mechanism via modelling of Z-RAFT star polymerizations, in which long chain branching can be detected selectively via RAFT-group absorbance of a star-star couple containing two living cores (see Scheme 1.16)⁹⁹  By this approach, the rate coefficient of intermolecular transfer to polymer at 60 °C was estimated to be k_tr^P,inter= 0.33 L mol⁻¹ s⁻¹ in butyl acrylate polymerization and k_tr^P,inter=7.1 L mol⁻¹ s⁻¹ in dodecyl acrylate polymerization.

As mentioned above, it is known that intramolecular chain transfer, in particular, 1,5-hydrogen shift, does also occur during the polymerization of monomers that yield very reactive macroradicals, such as acrylates^100–103  and acrylic acid.¹⁰⁴  This so-called backbiting reaction, by which a secondary radical (SPR) is transformed into a more stabilized tertiary (MCR) one, proceeds via a six-membered cyclic transition state with rate coefficient k_bb (see Scheme 1.17). In principle, intramolecular chain transfer to a remote chain position and intermolecular chain transfer to another polymer molecule may also take place.¹⁰⁵  These latter processes are, however, found to be not significant in butyl acrylate polymerization at low and moderate degrees of monomer conversion and temperature.^106,107 

The overall kinetics is rather complex: tertiary mid-chain radicals, MCRs, are produced by backbiting reactions of secondary propagating chain-end radicals, SPRs. On the other hand, monomer addition to an MCR with rate coefficient k_p^t, again produces an SPR under simultaneous formation of a short-chain branch. SPRs rapidly propagate, undergoing backbiting or termination with another SPR or an MCR with rate coefficients, k_t^ss and k_t^st, respectively. Propagation from an MCR is approximately two orders of magnitude slower than the one from an SPR. Also MCR homo-termination with rate coefficient k_t^tt is slower than SPR homo-termination.

For acrylate and acrylic acid monomers, the polymerization kinetics and polymer properties are affected decisively by the formation of less reactive MCR species. The polymerization rate is significantly lower than would be expected from the ideal process with propagation of SPRs only, as already mentioned above. The fraction of MCRs may be estimated by assuming that dynamic equilibrium has been reached and making a quasi-steady-state assumption on dc_MCR/dt.¹⁰² 

Derivation of eqn (1.15) neglect other side-reactions such as transfer to monomer and β-scission (see below). By implementing the so-called long-chain hypothesis (k_p^tc_M≫2k_t^ttc_MCR+2k_t^stc_SPR) into eqn (1.16) yields a simplified expression for the fraction of MCRs:

Eqn (1.16) is particularly suitable for describing x_MCR at high monomer concentrations and higher temperatures, whereas eqn (1.15) is usually a more reliable description for x_MCR at low temperatures where high concentrations of SPRs are present. Experimental methods for determination of reliable rate coefficients for k_bb are (i) ¹³C NMR,^108–110  (ii) frequency-tuned (ft)-PLP-SEC¹¹¹  and (iii) single-pulse pulsed-laser polymerization coupled with online time-resolved electron-spin resonance spectroscopy (SP-PLP-EPR).¹¹²  The propagation rate coefficient for MCRs may be obtained via ft-PLP-SEC¹¹¹  and SP-PLP-EPR.¹¹²  Termination rate coefficients k_t^s,s, k_t^s,t and k_t^t,t are only accessible from SP-PLP-EPR,¹¹²  in which different types of radicals can simultaneously be traced as a function of time. Remaining kinetic coefficients can then be obtained via computer modeling. Table 1.5 collates kinetic coefficients for butyl acrylate polymerization as an example.

Bonds in the β-position to radical functionalities are relatively labile, particularly at higher temperatures. Due to the associated high activation barrier, β-scission reactions may often be neglected below 80 °C. The mechanism of β-scission is illustrated in Scheme 1.18.

By β-scission, a macroradical is cleaved into a chain-end radical and a double-bond-terminated molecule. In the special case that m=0 and Z=H (see Scheme 1.18), a chain-end radical with degree of polymerization n is cleaved into a similar chain-end radical with degree of polymerization (n−1) and a monomer molecule. This reaction is often referred to as depropagation, since it is the back reaction of a propagation step. For m≥1, β-scission of a midchain radical (typically m≥2 for an MCR produced via backbiting) produces an SPR of chain length n and a so-called macromonomer, MM, of chain length m. It needs to be noted that MMs will subsequently add to SPRs during radical polymerizations. Further details on β-scission and macromonomer synthesis are provided in ref. 113, 114 and 115.

The termination reaction in free radical polymerization is arguably the most complex reaction step of the polymerization process. The corresponding termination rate coefficient, k_t, is influenced by a variety of different factors, i.e., (i) the system viscosity, (ii) the chain length of the terminating free macroradicals, (iii) the temperature, (iv) the pressure and the (v) monomer conversion. These parameters are very difficult to separate and experimental access to k_t is thus far from being trivial. In addition, many of these parameters vary during the course of a radical polymerization reaction. The enormous scatter of reported termination rate coefficients over several orders of magnitude, as e.g. compiled in the Polymer Handbook,¹  for the same monomer at the same reaction temperature is a direct manifestation of the influence of these various parameters on the termination event. More recent data of k_t is addressing these influences with greater care and also employ more precise k_p values obtained by PLP-SEC,²²  which are often needed for the determination of k_t. It is for this reason that the quality of k_t data has been improved enormously during the last years.

There are two modes of termination: one is the direct coupling (combination) of two free macroradicals yielding a dead polymer chain of chain length i+j. The associated rate coefficient is k_t,comb. The second mode is disproportionation, where a hydrogen atom is transferred from one of the radical chain end to the other radical, giving two dead polymer chains of which one carries a double bond. This reaction is associated with the rate coefficient k_t,dis. These processes are illustrated in Scheme 1.19 on the example of poly(methyl methacrylate) macroradicals.

The reactions between two carbon-centered radicals generally give a mixture of disproportionation and combination. Which termination mode dominates depends largely on the structure of the monomer unit, but also – however to a lesser extent – on the reaction temperature, pressure and solvent.¹¹⁶  Disproportionation is (slightly) favoured at higher reaction temperatures. The reasons for this behaviour have yet to be clarified, but there is some evidence pointing towards a different temperature dependence of the corresponding pre-exponential factors in the Arrhenius expression for k_t,comb and k_t,dis. Combination and disproportionation are clearly two different reactions with two distinct transition states, which is supported by theoretical studies.^117–120  However, the measured effects are rather small and associated with a large experimental scatter, as indicated in Figure 1.4 for the example of the temperature dependence of the contribution of disproportionation to the overall termination process, λ, for a MMA polymerization at ambient pressure.

This impressively indicates that λ is a very difficult parameter to measure accurately. Recently, the situation was somewhat improved by applying high-resolution mass spectrometry for determining λ, which allows a very precise measurement of end-groups originating from the two termination processes.¹²¹  This method arrived at a value for MMA at 85 °C of λ=0.63, which is very close to the average value λ of all data obtained before, which is an indication of this method's accuracy.

For a given series of radicals, λ increases with the number of β-hydrogen atoms. However, there is no direct correlation and other factors are involved.^116,122  In addition, it is generally observed that the extent of disproportionation increases with rising substitution at the radical centre. Steric effects play a very important role as can be demonstrated on the self-reaction of cumyl radicals (Scheme 1.20(a)) and the tert-butyl substituted equivalent (Scheme 1.20(b)). The termination reaction of radical (a) shows predominantly combination, whereas radical (b) gives predominantly disproportionation, although there are less β-hydrogen atoms. This finding may imply that the combination reaction is more suppressed by the steric hindrance than the disproportionation reaction. The statistical effect that favours disproportionation when more β-hydrogen atoms are available is hence to be considered less pronounced than the steric effect.¹²³  The steric crowding can lead in extreme cases to persistent radicals (e.g. di-tert-butyl methyl radical¹²⁴  and triisopropylmethyl radical¹²⁵  that are relatively reluctant to perform radical–radical reactions.

The mode of termination has no direct impact on the rate of the free radical polymerization. However, the generated molecular weight distributions are influenced by the termination mode. Since some methods for the determination of the termination rate coefficient rely on the analysis of the full molecular weight distributions, it is mandatory to have reliable data on λ. The termination mode also determines whether one (disproportionation) or two (combination) of the end groups are initiator-derived. λ also determines whether dead polymer produced during a controlled radical polymerization occurs at the same chain length as the “living” polymer (and remains invisible) or occurs as hump at the doubled chain length. Table 1.6 presents selective data on termination modes for various monomers at ambient pressure and various temperatures.

A more detailed review of the literature known data for various monomers and model systems can be found in ref. 140. However, unambiguous numbers for λ are scarce and there is in most cases only a qualitative agreement between different literature values. Despite these divergences, the following statements generally hold:

1. Polymerizations of vinyl monomers predominantly terminate via combination.

2. Polymerizations of α-methyl-vinyl monomers always show a contribution of disproportionation.

3. The hydrogen atoms of the α-methyl-group are more prone to abstraction during the disproportionation reaction than the methylene hydrogen atoms.

4. Within a series of vinyl or α-methyl-vinyl monomers, λ apparently decreases according to the radical stabilization ability of the substituent.

The rate law expression for termination in its simplest form is

with the radical concentration and the termination rate coefficient, k_t. There sometimes is confusion whether to incorporate the factor 2 of the rate law expression into the termination rate coefficient.¹⁴¹  The factor 2 is needed if the rate law describes the rate of macroradical loss, but unnecessary if termination events are considered. Nevertheless, the IUPAC ruling¹⁴²  on this is clear: termination rate coefficients are to be reported without the factor two. All termination rate coefficients given in this chapter are in accordance with this IUPAC guideline.

It is, however, very difficult to tabulate single values for k_t, because of the above-mentioned chain-length and monomer conversion dependence of the termination rate. It is however possible to give chain-length averaged k_t value for a specific monomer conversion, <k_t>, which may be sufficient for rough estimations of the polymerization process. Many simulations of technical processes indeed rely on this simplified concept, as the average chain-length of the radical population is not changing drastically in conventional polymerization. This simplified approach, however, cannot be used in controlled radical polymerization, where chain lengths systematically increase during polymerization. The impact of monomer conversion, on the other hand, has to be considered with both types of radical polymerizations in case it changes drastically during the reaction. In conclusion, k_t is best given as functional dependence both of chain-length and monomer conversion. This function, however, is to date not available for more than one or two systems (see below) and simpler approximations are still in use.

For low and intermediate monomer conversion, the termination reaction itself can be broken down into three consecutive stages (see Scheme 1.21):^143–146 

1. Translational centre-of-mass diffusion (TD) of both species towards each other travelling through the reaction medium.

2. Segmental diffusion (SD) of the radical chain ends towards each other. This occurs when a coil pair is penetrating each other in an entangled state. Segmental diffusion brings the chain ends into a position that enables them to react.

3. The chemical reaction (CR) of the two radical sites yielding the polymeric product(s).

The diffusion-controlled termination rate coefficient for low and intermediate monomer conversion, k_t,D, is expressed via eqn (1.19) were k_TD, k_SD and k_CR denote the rate coefficients associated with the reaction steps in Scheme 1.21.

At low and moderate conversions, overall k_t is adequately represented by k_t,D, i.e. k_t=k_t,D. Since the chemical reaction between two (macro)radical functionalities is extremely fast (associated with a high value of k_CR), termination is diffusion controlled from the initial phase of polymerization on. Termination usually depends on the rate-determining diffusion step, denoted by SD- or TD-controlled k_t.

At high degrees of conversion, termination may in parallel occur to a significant extent via so-called reaction diffusion (RD). Termination via RD, with rate coefficient k_t,RD, occurs by chain-end encounter after successive addition of monomer units. This mechanism plays a major role when macroradicals are immobilized (trapped) in a polymer network. The termination rate coefficient k_t is thus given by eqn (1.20).

The conversion dependence of termination was experimentally investigated in detail via the single-pulse pulsed-laser polymerization coupled with online time-resolved near infrared spectroscopy (SP-PLP-NIR) technique.^142,147  This method provides access to chain-length-averaged termination rate coefficients, 〈k_t〉. The variation of 〈k_t〉 towards increasing monomer-to-polymer conversion, X, is exemplified in Figure 1.5 for bulk polymerization of MMA.

The plot of 〈k_t〉 vs. X in Figure 1.5 reveals distinct regimes of rather different conversion dependencies, which can be assigned to 〈k_t〉 being controlled by specific termination mechanisms, denoted as SD, TD or RD control.

The SD control of 〈k_t〉 in the initial stage of a polymerization is often called plateau-regime, since 〈k_t〉 remains more or less constant with increasing conversion. The plateau level depends on (mostly the viscosity of) monomer and solvent. SD control is characterized by fast centre-of-mass diffusion of macroradicals through the environment of mostly monomer and solvent and subsequent segmental re-orientation, which also occurs against the friction of monomer and solvent environment. The formation of polymer induces an increase in bulk viscosity, even though this does not to a major extent influence the mobility of the terminating radicals, since the large mesh-size of the polymer chains allows for macroradical diffusion essentially controlled by monomer and solvent fluidity. Center-of-mass diffusion in the SD regimes is not correlated with bulk viscosity, but rather determined by a so-called “microviscosity” of the monomer–solvent mixture. The past section is of particular relevance in view of chain-length dependent termination.

The pronounced decrease of 〈k_t〉 in the TD regime is associated with the occurrence of the so-called gel-effect.¹⁴⁸  Also known as the ‘Trommsdorff’, ‘Norrish–Smith’ or ‘Norrish–Trommsdorff’ effect, this effect can cause problems within both an industrial and scientific context ranging from a product mixture to reactor explosion, due to its exothermic nature.^148,149  Increasing polymer content induces overlap of polymer chains and decreases the mesh-size in between the polymer chains beyond a critical limit. As a consequence, TD may become the rate-determining step in Scheme 1.21 for the majority of macroradicals, thus 〈k_t〉 decreases by orders of magnitude in some cases. It is important not to confuse the gel effect with the auto-acceleration that is observed when a polymerization is carried out under non-isothermal conditions, so that the reaction temperature increases with increasing monomer conversion, due to the exothermic nature of the polymerization reaction. The gel effect is observed under isothermal reaction conditions. The cause of the gel effect has been discussed extensively and various theories have emerged which can explain all or part of the experimental data (excellent reviews on the topic can be found in ref. 150 and 151).

The RD regime is indicated by a less pronounced dependence of 〈k_t〉 on conversion. As termination depends on propagation rate, k_t,RD is directly proportional to (constant) k_p and to monomer concentration. Decreasing monomer concentration with ongoing conversion essentially explains the decay of 〈k_t〉 in the RD region. The stronger decrease of termination beyond conversions of ca. 90% is explained by strong deceleration of propagation rate. Buback has developed a model for the dependence of the termination rate on the monomer conversion, which considers segmental, translational and reaction diffusion processes.¹⁵²  This model has been successful in describing a large set of data up to high monomer conversions.^153–155 

Caused by the diffusion-controlled termination steps in the full course of polymerization, k_t depends on the chain lengths i and j of the associated terminating radicals. During radical polymerization carried out under continuous initiation, termination generally occurs between macroradicals of different chain lengths, thus, termination rate coefficients k_t(i,j) need to be considered. During the course of a radical polymerization, the average chain-lengths of terminating macroradicals may be altered, even under continuous initiation. This is especially true for controlled radical polymerizations, in which the radical population continuously grows in chain length. Consideration of CLD-T may thus significantly improve kinetic models used for modeling and simulation.

Three averaging models are commonly used to describe k_t(i,j) as a function of the individual chain lengths i and j, of k_t(1,1) associated with termination of two monomeric radicals and of the exponent value α, in which the strength of CLD-T is expressed. Values for k_t(1,1) and α can be obtained in a reliable way by experimental techniques (see below).

The individual models: harmonic-mean (hm), diffusion-mean (dm) and geometric-mean (gm) include different weighting of the contribution of shorter and longer chains. For example, the dm-model is directly based on the Smoluchowski equation, eqn (1.27), i.e. the extent of contribution to k_t(i,j) of the individual macroradical refers to the size of diffusion coefficients associated with i and j. Simulation of polymerization processes by implementation of one of these models is however extremely complex and time-consuming, since the chain-length distribution of macroradicals present at any stage during the radical polymerization has to be implemented into the model in addition to an adequate function for k_t(i,j). Averaging-model-free determination of k_t(i,j) is in principle possible but also rather difficult from an experimental point of view.

Chain-length averaged termination rate coefficients 〈k_t〉 may be estimated from eqn (1.21)–(1.23) via eqn (1.24), provided that data for the concentration of macroradicals as a function of chain-length, , is available.

Enormous progress has been made in the past decade in determining chain-length-dependent termination rate coefficients for two macroradicals of almost identical chain length, k_t(i,i), which is somewhat easier to handle. Barner-Kowollik and Gregory have recently presented a comprehensive review on this.¹⁵⁶  Termination between radicals of identical chain-length plays an important role in controlled radical polymerization, since the chain-length distributions of active chains are narrowly distributed and increase constantly with conversion. Hence, rational modeling of controlled radical polymerization intrinsically relies on the availability of chain-length dependent k_t data. Practical approaches to implement experimental data for k_t(i,i) into the kinetic schemes used for simulations of technical relevant processes have also been made e.g. by eqn (1.25).

The chain length, i, in eqn (1.25) refers to the number average degree of polymerization of macroradicals. The correction factor a is found to be much smaller than unity which empirically expresses the impact of short-long termination.

Specially designed techniques for determination of k_t(i,i) are based on controlling the radical chain length either by laser single-pulse initiation^157–159  or by RAFT polymerization.^160,161  All being well, these techniques induce a narrow size distribution of radicals with degree of polymerization increasing linearly with time and with monomer conversion. Thus, the obtained termination rate coefficients, k_t(i,i), vary with time and refer to the length, i, of radicals present at each instant.

It is found that the so-called composite model for termination, eqn (1.26),¹⁶²  seems to be obeyed by all monomers:

This model postulates that there are two distinct regimes of chain-length dependence. For short radicals, k_t(i,i) strongly decreases with i, and the exponent α_s is found to be between 0.50 and 0.65 for styrene, methacrylates and some other monomers. This is consistent with termination being controlled by centre-of-mass diffusion. These values of α_s are consistent with the power-law exponents found in measurements of diffusion coefficients, D_i, as a function of chain length i of deliberately synthesized oligomers.^163,164  For pure translational diffusion, theory predicts the exponent α_s to be 0.5 or 0.6 for the diffusion of coiled chains in theta and athermic solvents, respectively, and to be 1.0 for stiff, rod-like chains.^165–167  Figure 1.6 shows the function of k_t vs. chain-length for two acrylates, which was obtained via pulsed-laser RAFT polymerization.¹⁶⁸  It can easily be seen that k_t is indeed changing systematically within two distinct regimes, which supports the concept of the composite model. This model, however, is somewhat semi-empirical, and not all aspects of the differences in chain-length dependencies of k_t in different systems are fully understood yet. Table 1.7 collates chain-length dependent k_t data for some important monomers for the segmental diffusion regime, i.e., for low monomer conversion roughly up to 20 to 30 %.

For radicals of size above a certain crossover chain length i_c of around 50, the dependency becomes much weaker, with observed values of α_l mostly falling in the range 0.15–0.30.¹⁵⁶  Such values are in accord with theoretical predictions of α_I=0.16 for control of (long-chain) termination by segmental diffusion in a good solvent.¹⁶⁹ 

When measuring k_t(i,i) during the course of a controlled radical polymerization – which e.g. may be realized by online rate measurement via DSC during a RAFT polymerization (RAFT-CLD-T method)¹⁶⁰  – one obtains k_t data that is also inherently linked to a distinct monomer conversion as it is impossible to determine k_t(i,i) at fixed conversion. This difficulty can be turned into a virtue by using different RAFT agent concentrations during a set of experiments on the same monomer system. This approach was introduced by Barner-Kowollik and co-workers and leads to different paths of k_t through the (i,X)-space. By combining all such data and fitting it, one may map out k_t(i,X), as illustrated in Figure 1.7 on the example of methyl acrylate polymerization.¹⁸⁶ 

This method was applied to MA,¹⁸⁶  vinyl acetate,¹⁸⁷  and to MMA in order to study the gel regime.^188,189  Such 3D-plots are naturally difficult to tabulate here and the interested reader is referred to the original literature. It is, however, without doubt the wish of all scientists working in this field that such generalized data become available for many other monomers and systems at various reaction parameters in the future and that computer-based models for k_t become available to apply the correct values of k_t for every i,X-point that is passed during a (controlled) radical polymerization.

Because monomeric radicals are so small, their termination k_t(1,1) must be via centre-of-mass diffusion. This situation can adequately be described by the Smoluchowski equation (1.27)

where N_A is the Avogadro number, D₁ is the self-diffusion coefficient of the monomer, i.e. radical of chain length unity, R_c is the capture radius for termination, and P_spin is the probability of encounter involving a singlet pair: on straight statistical grounds this value will be 0.25.^190,191  The most important quantity in eqn (1.27) is D₁. Its behavior should be captured by the well-known Stokes–Einstein equation:

Here k_B is the Boltzmann constant, T is (absolute) temperature, r₁ is the hydrodynamic radius of monomer, and η is the viscosity of the reaction mixture. For polymerization systems, η should be understood as the microviscosity (or solvent viscosity), because it is well known that termination rate coefficients do not vary according to bulk viscosity (see above). From the above considerations one expects that k_t(1,1)∼(r₁η)⁻¹. The additional expectation is that E_A(k_t(1,1))≈E_A(η⁻¹), where E_A denotes activation energy. Table 1.8 gives activation energies and activation volumes () for selected acrylate and methacrylate termination rate coefficients.

Inspection of Table 1.8 shows that the activation energies are rather low which is consistent with the diffusion (either segmental or translational) controlled nature of the termination reaction, i.e. the fact that these activation energies correspond to temperature dependence of the inverse system viscosity. Currently, there is no data on the activation energy of the actual termination reaction itself. However, it is very likely that the termination reaction itself has a close to zero activation energy. This conclusion may be deduced from the activation parameters observed for small radical termination.^195,190  Importantly, the activation volume of the termination reaction is positive, i.e. the value of k_t decreases with increasing pressure. This observation can also be connected with the pressure dependency of the viscosity. The reaction medium tends to be more viscous at higher reaction pressures, thus slowing the rate of termination, i.e. the activation volume of the termination rate coefficient is very close to the corresponding activation volume that characterizes the pressure dependence of the inverse of the monomer viscosity.¹⁹⁶  It is important to note that the pressure dependencies of the termination and propagation rate coefficients display opposite behaviour, i.e. allowing for increased rates of polymerization at elevated pressures.

In general, solvent effects on k_t are rather small. There are, like with k_p, two important exceptions to this rule, namely ionic liquids and water. When using ionic liquids as solvent, it is observed that polymerization rate and polymer molecular weight are enhanced compared to polymerizations in conventional organic solvents or in bulk.^197–200  One reason for this is the enhancement of the propagation rate coefficient, k_p (see above).^201,202  In addition to an increase in k_p, k_t decreases in the highly viscous solution containing ionic liquid.²⁰¹  Both rate coefficients, k_p and k_t, thus contribute to an enhancement of polymerization rates in the presence of ionic liquids. Barth and Buback²⁰³  studied this effect by time-resolved EPR spectroscopy after single laser pulse initiation in detail and found a decrease in k_t by around one order of magnitude (see Table 1.9).

k_t in both ionic liquids are well below the value measured for MMA bulk polymerization. The direction of change agrees with the inverse of the bulk viscosities at 10 °C: 0.67 cP for MMA,²⁰⁴  55.9 cP for dry [emim] NTf₂ and 171 cP for dry [bmim] BF₄.²⁰⁵  This indicates that the increased microviscosity of the solution phase is the cause of the reduced termination rate. Chain length dependency does not change with solvent.

The situation is somewhat more complex when using water as the solvent. Data on k_t in aqueous solution, however, is scarce²⁰⁶  and not all effects are fully understood. Studies into k_t in polymerization of 1-vinylpyrrolidin-2-one in a solution of water, for instance, has revealed that the full mechanism of termination is shifted with increasing content of water (see Figure 1.8)²⁰⁷  Termination is being enhanced with increasing concentration of water and the transition between segmental and translational diffusion control is shifting simultaneously.

A conventional radical polymerization is in nearly all cases a steady-state polymerization system, which is characterized by a constant free radical concentration over time as given by eqn (1.28)

A simple but rather general expression for the rate of polymerization, R_p, can be derived when assuming the following approximations:

• –All reactions are irreversible.

• –Monomer species are consumed only by propagation of radical species.

• –All macroradicals are of identical reactivity, regardless of their chain length and the degree of monomer-to-polymer conversion.

• –Termination of macroradicals takes place either by bimolecular combination or disproportionation.

Under steady state conditions, the rate of initiation, R_i, is equal to the rate of termination eqn (1.29), which is an assumption necessary for the establishment of a constant free radical concentration

Rearrangement of eqn 1.29 and insertion into the simplified form of eqn 1.30

yields the final expression for the rate of polymerization, R_p:

Eqn 1.31 indicates a reaction order of one of monomer concentration on the rate of polymerization and a reaction order of 0.5 of the initiator concentration. These dependencies have been confirmed experimentally on the example of many polymerizing systems, although it is important to note that deviations from ideality, such as chain length dependent rate coefficients, back-biting, and primary radical termination, lead to a change in the exponents associated with the initiator and monomer concentrations.^208,209  In such cases, the rate of polymerization will scale with a weaker than square root dependence on c_I and a stronger than linear dependence on c_M. Very low monomer concentrations can also alter the exponents of monomer and initiator concentration. For many conventional polymerization systems, however, this equation holds and can be integrated to yield an expression which correlates the monomer conversion with an observed overall kinetic rate coefficient, k_obs.

X is the fractional monomer conversion. This equation is the famous “pseudo first-order rate law” of radical polymerization, which is very often used to characterize the kinetics of a radical polymerization process. The finding that a plot according to eqn (1.32) is linear indicates that the assumptions made above about steady-state polymerization are fulfilled. A linearity of a first order rate plot primarily indicates that the concentration of active sites – radicals in the case of radical polymerization – is constant over time. Deviations from linearity are mainly due to depletion of the initiator. It must be stressed here, that every ideal conventional radical polymerization gives a linear pseudo first-order rate plot. Very often, a linear pseudo first-order rate plot is taken as proof for a successful controlled radical polymerization, which is absolutely not correct. In contrast, controlled radical polymerizations based on the persistent radical effect, such as NMP and ATRP, do in principle not give a linear pseudo first-order rate plot. RAFT polymerization – as it rests on degenerative chain transfer – does show linear pseudo first-order rate plots, on the other hand. It is thus not possible to evaluate the quality of a controlled radical polymerization by inspecting pseudo first-order rate plots. This misconception originates from living anionic polymerization, where the linearity of the pseudo first-order rate plots is indication of the fact that active species do not terminate. This concept cannot be transferred to controlled radical polymerization. Unfortunately, this misconception can still be found very often in literature.

The temperature dependence of the overall polymerization rate is given by the temperature dependencies of the individual rate coefficients. Each rate coefficient follows its own Arrhenius law, k=Aexp(–E_A/RT), where A is the pre-exponential factor and E_A denotes the activation energy. The overall activation energy of the rate of polymerization, , p, equals the sum of the weighted activation energies of the elementary reactions, propagation (), initiation () and termination ().

Activation energies for commonly used thermally decomposing initiators, , are in the order of 120 to 150 kJ mol⁻¹. The values for most common monomers lie within the range of 20 to 40 kJ mol⁻¹, and is generally in the range of 4 to 10 kJ mol⁻¹. Hence, typical values for overall activation energies for the rate of polymerization initiated by a thermally decomposing initiator are close to 80 kJ mol⁻¹. This corresponds to a two or threefold rate increase in rate for a 10 °C temperature increase. Photochemical polymerization rates have a much lower activation energy of about 20 kJ mol⁻¹, according to close to zero activation energy of the photoinitiation process.

A deviation from ideality in conventional radical polymerization, which is often observed and which also occurs frequently in controlled radical polymerization, is the effect of inhibition and retardation. These effects are due to reactions of macroradicals with transferring or terminating species. In order to derive the kinetics of retardation and inhibition, these effects are described in terms of chain transfer, and termination is considered as transfer to species with negligible re-initiation ability. The chain transfer process stops the chain growth and the transfer agent itself becomes a radical. If the re-initiation ability of the generated radical is in the order of that of the macroradical and if the time necessary for the chain transfer process is within the range of one propagation step, ‘normal’ chain transfer occurs (see above). This normal or conventional chain transfer does not lead to any change in the polymerization rate. If the generated radical is less reactive than the propagating radical, retardation (degradative chain transfer) takes place, which slows down the rate of polymerization. This effect is e.g. observed when using phenol as retarder in MMA polymerization. If the retardation is very effective, the polymerization process is completely suppressed and this is referred to as inhibition. This can for instance be observed with oxygen in MA polymerization. Inhibition leads to an induction period where no polymerization takes place at all, until the inhibitor is consumed. The reaction then starts with the conventional rate of polymerization (see Figure 1.9).

The kinetics of the inhibition effect for a steady-state polymerization, i.e. the scenario that the generated radical does not reinitiate, can be analyzed by adding an additional reaction step to the basic scheme of polymerization.^210,211 

where Q is the retarder or inhibitor and k_Q is the associated rate coefficient of the inhibition reaction. The generated radical, Q˙, is assumed to not reinitiate, but remains persistent. This scenario may e.g. be realized by addition of typical “stabilizers” such as substituted phenols (e.g. p-methoxyphenol, 2,6-di-tert-butyl-4-methyl phenol) used as stabilizer for styrene²¹²  or as benzoquinone, which gives a phenoxy radical as a result of a rapid addition of the radicals present in the system to the CO double bond.^213–215  The steady state assumption, which is only a very rough approximation until all inhibitor is consumed,²¹⁶  can then be written as

which yields in combination with the expression (1.30), assuming that the rate of monomer loss, –dc_M/dt, equals the rate of polymerization, R_p

The ratio of the rate coefficients for retardation, k_Q, and propagation, k_p, is often referred to as the inhibition constant, z=k_Q/k_p, which reflects the ability of a molecule to cause inhibition. If the inhibition constant is large (z≫1), the second term of the l.h.s. of eqn 1.36 will become much smaller than the third one. In this case, the rate of inhibition is much larger than the rate of termination. Eqn (1.36) then reads

Eqn 1.37 shows that the polymerization rate is inversely proportional to the inhibitor concentration. It should be kept in mind that the inhibitor concentration will decrease during the reaction, since each radical generated by the initiation process will consume one inhibitor molecule. Propagation can become competitive with the inhibition reaction, when the inhibitor concentration becomes low.

Dividing the rate law for the loss of inhibitor, , by the rate law of propagation, , leads to

and subsequent integration with c_Q⁰_˙ and c_M⁰ being the concentration of inhibitor and monomer at the beginning of the reaction.

It is apparent from eqn 1.39 that, if z is large, the monomer conversion remains nearly zero until the inhibitor is consumed.

The rate coefficients of the individual reaction steps of radical polymerization are determining the rate of polymerization, R_p. The same kinetic parameters may be employed to calculate the lengths of macroradicals and the final polymer. The chain length distribution of a polymer is defined as the fraction of molecules x_P that contains P basic monomer units. The degree of polymerization P is equivalent to the chain length i. The propagating macroradicals by which the dead polymer is generated through any chain-stopping event exhibit a chain length distribution, too. Both distributions are closely related to each other and the chain length distribution of the dead polymer can be calculated via the derivative of the distribution of the living macroradicals.

Like any other distribution function, the chain length distribution may be described by its statistical moments, which are defined as

The combination of such moments define mean values for the degree of polymerization, . In practice, there are two mean values calculated by the first three statistical moments, which are extensively used. The number average degree of polymerization, , and the weight average degree of polymerization, .

To calculate the number average degree of polymerization, , of a polymer produced by a steady-state polymerization, it is mandatory to know how many propagation steps occur before the chain mechanism is stopped. It has to be distinguished between the term ‘chain’ used in a molecular sense and ‘chain’ used as a kinetic concept. The kinetic chain length, ν, is defined as

With the assumption of chain length independent rate coefficients, eqn (1.42) can be rewritten as

Elimination of c_R by means of eqn (1.29) leads to an expression for the kinetic chain length, ν, which shows the dependence on the various kinetic parameters.

This equation illustrates one important characteristic of the free radical polymerization: the sizes of the macromolecules are inversely proportional to the square root of the initiator concentration. Increasing the initiator concentration leads to shorter chains. Disregarding any transfer effect, as a first approximation, correlates the kinetic chain length with the number average degree of polymerization, . In the case of termination by disproportionation one polymer molecule is produced per every kinetic chain

Termination by combination leads to one polymer molecule per two kinetic chains, reflecting the combination mechanism.

Any mixture of these both mechanisms can be described by using the value λ (see Section 5.1), the contribution of disproportionation to the overall termination process.

The transfer process does not change the radical concentration, but the chain length of the polymer. Without changing the free radical concentration, chain transfer processes remain hidden in any experiment measuring the rate of polymerization alone. The kinetic chain length is also unaffected by transfer, because the growing free radical centre stays active after the transfer, although more than one polymer chains are produced. For this reason eqn (1.46) does not hold true if chain transfer occurs. Taking chain transfer into account, the number average degree of polymerization, , can be described as

“End group” here means any group at the end of a polymer chain that is not propagating. The various reactions within the polymerization process generate different amounts of end groups per initiation step:

• initiation 1 end group

• propagation 0 end groups

• transfer 2 end groups

• termination by disproportionation 1 end group

• termination by combination 0 end groups

At steady-state, the number of polymerized monomer units can be substituted in eqn (1.47) with the rate of polymerization and the numbers of end groups by the rate of their formation.

Insertion of the simplified rate laws of the different processes

and subsequent inversion leads to

is the concentration of any molecule that is capable of taking part in a chain transfer reaction, including solvent S, initiator I, polymer P and added chain transfer agent T. It is usual to define chain transfer constants for the different molecules

Thus, eqn (1.53) becomes

This equation gives the fundamental correlation of the number average degree of polymerization with the rate of polymerization and the various chain transfer constants. Performing a polymerization experiment with only low conversion of monomer to polymer, the concentration of polymer is often too low to show significant chain transfer. The same holds true for the initiator, which is mainly used in the range of low concentrations. Without addition of solvent and additional chain transfer agent, eqn (1.55) reads after introduction of eqn (1.17)

Hence, a plot of the inverse number average degree of polymerization, against the rate of polymerization R_p, – the rate of polymerization can be easily varied by the concentration of the initiator – yields the monomer chain transfer constant C_M as intercept of a linear plot. The value of C_M⁻¹ constitutes a limit for the maximum number average degree of polymerization, , as transfer to monomer cannot be avoided. Methyl methacrylate, for instance, has a monomer chain transfer constant of about C_M=5 × 10⁻⁵ at 60 °C, leading to a maximal chain length of about 20 000, whereas in a radical polymerization of vinyl acetate with C_M=2 × 10⁻⁴ at 60 °C, the limit is already reached at 5000.

So far, only the average degree of polymerization has been considered. To calculate the distribution function itself for a steady state polymerization it is convenient to choose a statistical approach based on kinetic parameters. A probability factor α of propagation is defined as the probability that a radical will propagate rather than terminate. The factor α is the ratio of the rate of propagation over the sum of the rates of all possible reactions the macroradical can undergo.

First, we assume that termination occurs only by disproportionation and that the propagation probability factor is equal for each chain length. The probability for the occurrence of a polymer chain – hence its distribution function – with the length P is given by the probability of P – 1 propagation steps and the probability of one chain stopping event (termination or transfer).

The molecular weight averages can be evaluated by calculating the moments of this distribution function by insertion of eqn (1.58) into eqn (1.59–1.61),

and subsequent insertion into eqn (1.62).

The ratio of the weight average and the number average degree of polymerization, , equals the dispersity index, Đ, of a chain length distribution and can be expressed by

It should be noted that the propagation step must be highly favoured over chain stopping events to produce polymer with a significant chain length and the value of α must be near to one. Hence, eqn (1.63) shows that for a chain-length distribution of a polymer produced in a steady-state experiment, where chain stopping events are termination by disproportionation or transfer, the dispersity becomes nearly 2.

Expressions may also be derived for the chain length distribution produced, when the termination process is by combination. The expression for the probability of the occurrence of a chain with the chain length P is now given by the contributions of two chains with the chain length n and m, which form the desired molecule by combination. Hence, the auxiliary condition n+m=P must hold true.

Evaluating the moments of this distribution function by insertion of eqn (1.64) into eqn (1.65–1.67) as above

leads to

The dispersity can then be given by

Keeping in mind that α has a value close to one, 1.69 leads to a dispersity of 1.5 for a polymer produced in a polymerization process where termination is by combination. The corresponding chain length distribution is somewhat narrower than that generated by disproportionation, because of the statistical coupling of two chains with different sizes.

Almost every polymerization system shows both disproportionation and combination modes. In order to combine the two modes the general expression for the dispersity of any given termination controlled chain length distributions reads

Because the value of α is close to one, the expression ln(α)≈α−1 holds, leading to α≈exp[–(1 – α)]. With this in mind, the combination of eqn (1.58) with the l.h.s of eqn (1.62) gives

with the factor (P – 1) substituted by P, since P ≫ 1. eqn (1.71) demonstrates that the chain length distribution of the polymer formed by disproportionation or chain transfer follows an exponential function in the limit of infinite chain length.

The same calculation procedure, starting with eqn (1.58), also leads to an exponential expression for the chain length distribution for termination by combination.

However, eqn (1.72) exhibits the chain length P additionally in the pre-exponential factor.

Evaluation of eqn (1.56) immediately leads to

All derived distribution functions and average degrees of polymerization may now be expressed via the kinetic coefficients.

For further reading on the topic of molecular weight distributions of polymers the reader is referred to more specialized literature, e.g. the works of Peebles,²¹⁷  or Bamford et al.²¹⁸ 

The propagation step in radical polymerization cannot be seen as being irreversible at higher temperatures, where it is reversible leading to a thermodynamic equilibrium. This equilibrium can be described by the free energy difference, ΔG_p, between polymer and monomer. The polymerization process is thermodynamically favoured if ΔG_p is negative. The value of the free energy difference is given by the fundamental equation,

The enthalpy and entropy changes in the propagation reaction are effectively those of the overall polymerization reaction for long polymer chains.^219,220  The polymerization enthalpy, ΔH_p, of radical polymerizations are mostly negative with typical values of –30 to –100 kJ mol⁻¹. The values for the standard polymerization entropies are negative, too, because the monomer is losing degrees of freedom when becoming part of the polymer chain. Typical values for the polymerization entropies are –100 to –120 J K⁻¹ mol⁻¹. The two terms on the r.h.s. of eqn (1.74) are thus antagonistic. The exothermicity of the reaction exceeds the entropic term at normal temperatures and ΔG_p becomes negative. At elevated temperatures, however, the entropic term becomes significantly larger and finally equals the enthalpic term at the so-called ceiling temperature, T_c. At the ceiling temperature, the free energy difference becomes zero and polymerization does not occur any longer to a significant extent.²²¹  Only a few systems are known in which both the enthalpy and entropy change of the polymerization are positive. The polymerization of sulfur of the eight-membered ring conformation is one example:^222–224  the entropy increases during polymerization as the S₈-ring is rigid and the ring opening reaction makes additional conformations available. It follows from eqn (1.74) that in such cases there exists a floor temperature, T_f, with polymerization being feasible only above a certain temperature.

Polymerization reactions can also be described in view of kinetics with propagation being reversible:^225,226 

with the rate coefficient of depropagation, k_dp. Since depropagation is a β-scission reaction (see above), its activation energy, E_A,dp, is much higher than that of the propagation reaction, E_A,p. The difference between these two activation energies is equal to the enthalpy change of the polymerization reaction (see eqn 1.77):

At standard conditions, the standard free energy difference of polymerization can be related to the equilibrium constant, K, of polymerization according to eqn (1.78)

where the kinetic expression for K is the ratio of the rate coefficients of the forward reaction to the rate coefficient of the backward reaction. This can be set equal to the thermodynamic definition of K. If the degree of polymerization is very large, the concentrations of growing chains i and i+1 can be considered as being nearly identical, which leads to

where [M]_e is the equilibrium monomer concentration (e.g. 10⁻⁶ mol⁻¹ L⁻¹ for styrene at 25 °C). With the standard reaction enthalpy, , and the standard reaction entropy change for [M]=1 mol L⁻¹, it follows that

With the assumption that the reaction enthalpy is independent of temperature, the reaction enthalpy equals the standard reaction enthalpy, . Inserting the fact that at T_c then leads to

Standard polymerization enthalpies, ΔH_p⁰, and standard polymerization entropies, ΔS_p⁰, of various monomers are collated in Table 1.10. Eqn (1.81) indicates that the ceiling temperature is a function of the equilibrium monomer concentration. This implies that there exists a specific ceiling temperature for every given monomer concentration. The maximum ceiling temperature is reached for the bulk polymerization system. It is determined by thermodynamic parameters and is independent of the polymerization mechanism. The applicability of the above relations depends on depropagation being the exact reverse of propagation. Hence, the observation of a ceiling temperature requires the presence of active centres. In their absence, polymers can exist at temperatures above the ceiling temperatures. The introduction of active centres, e.g. via UV-radiation and/or high temperatures, may however open up the pathway for depropagation, which is one pathway for the degradation of polymers.

Depropagation hardly occurs in most of the typical radical polymerization systems. In some 1,1-disubstituted monomer systems, however, the effects of the reverse reaction cannot be ignored at some conditions. α-Methyl styrene, for instance, has a low ceiling temperature of around 60 °C in bulk polymerization, which originates from its relatively low heat of polymerization.²²⁷  Methacrylate and styrene monomers also exhibit depropagation, although at much higher temperatures (220 °C and 310 °C, respectively, for bulk polymerizations). The depropagation process lowers the rate of polymerization according to

The effective rate coefficient of propagation can therefore be defined as

It can be seen that the impact of depropagation^228–230  (right term in eqn (1.83)) is inversely proportional to the monomer concentration, which is part of the thermodynamic equilibrium. The effective propagation rate coefficient can be directly measured by PLP-SEC. The deviation of k_p^eff from the linear slope of an Arrhenius plot at higher temperatures is impressively demonstrated in Figure 1.10.²³¹