Polymers and Network Characteristics

Coatings contain thermoplastic polymers for the larger volume of applications (e.g. in particular waterborne wall paints). However, in more advanced applications, one uses almost universally polymers that are crosslinked, that is, thermosets. In order to be somewhat self‐contained, in this chapter we discuss first briefly the nature of polymers. Thereafter, we discuss the basics of network formation for various combinations of starting materials, leading to some rules for their usage in practice. In the next chapter we apply these rules to the most frequently used polymer system for coatings.

Polymers consist of long molecular chains of covalently bonded atoms. Typically a *polymer* is constructed from a large number of *repeating units*, which contains one or two species, the *monomers*, leading to a high molecular mass molecule. With a small number of repeating units, a low molecular mass molecule is obtained, referred to as an *oligomer*. In solid noncrosslinked polymers, intermolecular bonding occurs via secondary interactions, such as van der Waals interactions and hydrogen bonds, and via entanglements between the chains. The *van der Waals interaction* is mainly the attraction resulting from charge distribution fluctuations in different molecules mutually influencing each other. The energy of this interaction typically ranges from 10 to 40 kJ mol^{−1} for polymers,^{1} dependent on molecular mass. In a number of cases also hydrogen bonding is present with similar binding energy. *Hydrogen bonds*arise from the bonding of a hydrogen atom to two other atoms, either symmetrically A-H⋯A or asymmetrically A-H⋯B. These bonding types result in soft materials with low melting temperature. By joining the chains at points along their length with a covalent bond, a *crosslinked structure* (Figure 2.1) arises, leading to (somewhat) harder but elastic materials showing no flow when heated.

Generally two classes of polymers can be distinguished: *chain‐grown* and *step‐grown polymers*, or chain and step polymers for short, in the past often erroneously addressed as *addition* and *condensation* polymers, respectively. Polymers of the first class are typically made by activating a monomer by a catalyst to provide it with an initiating end site via opening of a double bond and then growing the chain by addition of monomers until growth is terminated, either by exhaustion of the monomer supply or via a side reaction. At any time there are essentially only monomers, *dead* or nongrowing, and *living* or growing polymer chains present. In order to have control over the polymer final properties, the number of the latter should always be low. Possibly the simplest example is polyethylene (PE), which consists of long chains of a -[CH_{2}-CH_{2}]- repeating unit. The monomer is ethylene, CH_{2}=CH_{2}. If the monomer is modified to CH_{2}=CXY, where X and Y represent a certain chemical group, the polymers are called *vinyl polymers*. If X is a methyl, phenyl, or chloride group and Y is a hydrogen atom, the resulting polymer is indicated by polypropylene (PP), polystyrene (PS), and polyvinyl chloride (PVC), respectively. If X is a COOH group and Y is a methyl group, we have poly(methyl methacrylate) (PMMA). PE and PP form about 50% of all polymers produced, while about 60% of all polymers is produced via chain growth [1]. Linear members of the second class (step growth) are made by reacting predominantly bifunctional molecules with the elimination of a low molar mass condensation product, for example, water. At any moment the mixture contains growing chains and small molecules split off (*condensate*), such as water or an alcohol. The number of reactive groups decreases with increasing chain length. Examples are polyamides, polyesters, and polycarbonates.

The chemical structure of the chains is influenced by isomerism. A simple but infrequently occurring example of chemical isomerism is provided by the vinyl polymers for which one may have *head‐to‐head* (-CH_{2}-CHX-CHX-CH_{2}-) or *head‐to‐tail* (-CH_{2}-CHX-CH_{2}-CHX-) addition. A somewhat more complex case involves steric isomerism. Consider again the case of vinyl polymers in which a side group is added to every alternate carbon atom. If the groups are all added in an identical stereochemical way, we obtain an *isotactic* polymer (Figure 2.1). If on the other hand there is a stereochemical inversion for each monomer unit, we obtain a *syndiotactic* polymer. Finally, an irregular addition sequence leads to an *atactic* polymer.

Each sample of polymer will consist of molecules with chains of varying length and consequently of varying molecular (or molar) mass. One can distinguish between the *number average M*_{n} and *mass average M*_{w}, defined by

2.1

and

2.2

Here *N*_{i} is the number of molecules (or moles) with molecular (or molar) mass *M*_{i}, the summation is over all masses, and the *number distribution P*(*M*) d*M* provides the number of masses between *M* and *M* + d*M*. Since the second moment of the number distribution function is given by

we have

2.3

and the coefficient *D* describes the relative width of the distribution. Since the mass average is always larger than the number average, *D* is always positive. In practice, instead of *D*, the ratio *Đ* = *M*_{w}/*M*_{n}, labeled as the *dispersity*, is often used as an indicator for the width.

The average molar mass and the molar mass distribution are important for many (if not all) properties of polymers and can in fact be used to optimize the material behavior. For example, the viscosity of a molten polymer (see Section 9.3.2) or a polymer in solution (see Section 10.1.2), the glass transition temperature of a solid polymer (see Section 9.3.1), and the surface tension of a polymer (see Section 7.3.2) all increase with increasing molecular mass. For further discussions, see [2].

After having indicated the molecular mass distribution, we turn to the molecular conformations for which the *gauche* and *trans* conformations are worth explaining. To elaborate a bit, consider first the central bond between two C atoms in ethane, C_{2}H_{6}. Figure 2.2 shows the two extremes in conformation, namely, the *cis* and *trans*, in a view along the C-C bond axis. In ethane, three equivalent minimum energy or *trans* conformations are present. To rotate the two CH_{3} groups with respect to each other, a force has to be applied to overcome the energy barrier between the two *trans* states. Substituting on each C atom one H atom by a CH_{3} group, so that we get butane, C_{4}H_{10}, the equivalence between the *trans* states is lost, and we obtain one *trans* (t) conformation and two equivalent (g^{+}, g^{−}) *gauche* conformations with dihedral angles *φ* = 0° and *φ* = +120° and *φ* = −120°, respectively, for the minimum energy conformations (Figure 2.2). Continuing with substitution of end H atoms with CH_{3} groups results in PE. Although the details for each C-C bond for this molecule may slightly differ, one *trans* and two *gauche* conformations are present for each C-C bond. They all have to be specified for a complete description of the molecule. For PE the lowest energy conformation is the all‐*trans* conformation with a zigzag structure of the C-C bonds.

This is no longer true for other polymers where the H atoms have been replaced by other atoms or groups. Consider for concreteness polytetrafluoroethylene (PTFE) where all H atoms have been replaced by F atoms. Since the F atoms are larger than the H atoms, the nonbonded repulsive interactions between CF_{2} groups of second nearest carbon atoms become much more important, and the repulsive energy can be lowered by rotating a bit along the C-C axis of each bond. Of course, this increases the bond rotation energy, and in this way equilibrium is reached. For the case of PTFE, an optimum dihedral angle of *φ* ≅ 16.5° is obtained. Consecutive bonds rotate in the same direction to minimize repulsion. The result of all this is that the molecule forms a helix along its axis in which the positions of the side groups (the F atoms in the case of the PTFE) rotate along the molecular axis. After *n* screws along the axis, the position of the *m*th monomer regains the position of the first monomer apart from a shift along the axis. Described in this way we refer to them as *m*/*n helices*. For example, PE has a 2/1 helix (the zigzag structure), and PTFE has a 13/6 helix below 19 °C and a 15/7 helix above 19 °C. The description is not as exact as it appears since the *periodicity* along the chain may vary slightly [2]. Although single bonds rotate relatively easily along the bond axis, this rotation is somewhat hindered. For a unit like -CHY=CHX-, the preferred orientations are *cis*, with the groups X and Y on the same side of the double bond, and *trans*, with the groups X and Y on different sides of the double bond. In this case the rotational barrier is considerably higher.

The formation of helices is not only important for crystallization but occurs in solution as well, both for biopolymers (e.g. proteins) and for synthetic polymers (e.g. polyethers). Moreover, for less regular polymers the availability of many configurations leads to an overall configuration described as a coil, the details of which depend on their chemical constitution and length. Considering the arrangement of the chains with respect to each other, there are three relevant topics – entanglements, crystallinity, and amorphous polymers – which are discussed in the next sections.

To describe entanglement, we note that the individual chains, having a relatively large internal flexibility, form *coils*, the size of which depends on temperature and on whether the chain is in a melt or in solution. In the latter case the size also depends on the solvent used. This will be briefly discussed in Section 2.1.4. Polymers in the molten state are amorphous, and the individual chains are entangled, that is, the chains get mixed up and are difficult to unravel since at various positions a kind of nonpermanent or transient knots, *entanglements*, sometimes called *physical crosslinks*, are formed. A crude analogy is that of a bowl of wriggling spaghetti with length to diameter ratio of 10^{4} or more. The number of entanglements per molecule increases with increasing molecular mass.

Many polymers, in particular atactic polymers, random copolymers, and highly branched polymers, when cooled down from the molten state, remain in the disordered or *amorphous* state (Figure 2.3), but others, in particular those with a more regular chain structure, when cooled down sufficiently slowly from the melt or from solution, will grow *crystals*. The crystallized state will be briefly discussed in Section 2.1.3, while the amorphous state is dealt with in Section 2.1.4. Finally, we mention *crosslinking*, that is, creating covalent bonds between the individual polymer chains so that a polymer network is created and can be used to obtain *rubbers* (or *elastomers*). In this state the material can be extended many times its own length and will return upon unloading rapidly to its original shape. The prototype material here is natural rubber, consisting of *cis*‐isoprene, which crystallizes but with difficulty. Crosslinking of rubbers, or as the jargon reads *vulcanization*, was originally done by sulfur but nowadays usually with peroxides. The most well‐known synthetic rubber is a random copolymer of styrene and butadiene (SBR), often reinforced with particles such as carbon black and used, for example, in vehicle tires. Network formation as a result of crosslinking is dealt with in Section 2.3.

For coatings crystallization generally is of limited significance, but for certain coatings, for example, PE coil coatings as used in the food industry, as well as in laboratory practice, crystallization of polymers is often encountered. In particular polymers with a more regular chain structure can crystallize, when cooled down sufficiently slowly from the melt or from solution [3]. Since the cross section of such molecules is more or less rectangular, they tend to crystallize in an orthorhombic crystal structure. A feature closely related to the helix structure is *polymorphism*, that is, more than one crystal structure can be observed. As an example we take polyoxymethylene (POM) ([-CH_{2}-O-]_{n}). For this molecule the *gauche* conformation is the most stable. The energy difference between *gauche* and *trans* is about 8 kJ mol^{−1}. The all‐*gauche* conformation (…g^{+}g^{+}g^{+}… or …g^{−}g^{−}g^{−}…) with a torsion angle of 60° generates a 2/1 helix with the aforementioned rectangular cross section, leading to an orthorhombic unit cell. However, a small change in the torsion angle to about 77° leaves the chain essentially in an all‐*gauche* conformation but leads to a 9/5 helix with a more or less circular cross section. This leads to hexagonal packing (Figure 2.4).

So, we see that polymers may crystallize given sufficient regularity along the chain, which usually implies linear, isotactic, or syndiotactic polymers. These crystallized polymers are macroscopically isotropic but microscopically nonhomogeneous. Generally, crystallization is incomplete though, that is, amorphous regions exist between the crystallites (Figure 2.3). The origin of this effect can be found in the chain‐like nature of polymers, which generally precludes full orientation of all the molecules. In fact, the amorphous regions typically contain the nonregular parts, for example, the chain ends, the defective parts of the chains, and the crossovers to other crystals. The density is, correspondingly, between the theoretical density of the crystals and that of the fully amorphous polymer, and on X‐ray diffraction patterns, apart from relatively sharp diffraction rings, also diffuse halos appear (see Section 8.1.6). Originally this semicrystalline behavior was described by the *fringed micelle model* (Figure 2.5). In this model the molecular chains alternate between regions of order (the crystallites) and disorder (the amorphous regions). The lateral dimensions of the crystals so formed can be several tens of micrometers, while the thickness is about 10–20 nm. In view of this shape, these crystallites are often referred to as *lamellae*. Adjacent chains not only align, but also an individual chain participates in several lamellae. Later findings, in particular using solution‐grown crystals, suggest that lamellae are formed according to the *regular fold model* (Figure 2.5), where the large surfaces of the lamellae contain the folds. The answer to the question whether the folds at the surface of lamellae are sharp and regular or that there is some deviation from regular reentrance is complex. From small‐angle scattering, the end‐to‐end distance in solution‐grown crystals has been determined to be much smaller than in the liquid, leading to a high *regular fold* fraction. From infrared measurements an estimate of 75% regular folds in linear PE was made, essentially in agreement with scattering data. However, the end‐to‐end distance upon melt crystallization is not dramatically changed, and therefore it seems logical to conclude that the entangled structure of the melt is largely preserved in the semicrystalline state. Nevertheless, also in this case a high fraction of regular folds is present as, for example, can already be assessed from density measurements. Likely, in general there is a (varying) mixture between the *pure* micellar and *pure* lamellar structure, dependent on crystallization conditions and type of polymer.

Typically in a melt‐crystallized polymer, the lamellae are organized further, and the description of this organization, addressed as *morphology* (although the noun morphology originally was a synonym for shape), contains generally three levels [4]:

- The lamellae of folded chains.
- Stacks of nearly parallel lamellae separated by amorphous material.
- Superstructures, the most important one of which is the spherulite.

A *spherulite* (Figure 2.6) is a part of the material in which all the lamellar stacks have grown radially, leading to a spherical shape. This feature requires a mechanism for branching and splaying of the lamellae, which is different for different polymers. While for linear PE a screw dislocation is suggested as the initiating factor, for PS a sheaflike central initiating part is identified.

After this excursion to crystallinity of polymers, we turn to the more general case of the amorphous state of polymers. Any polymer that consists of more than three monomers in substantial amounts has such an irregular sequence distribution that crystallization is virtually absent. Examples are unsaturated polyester resins, polyurethanes, polycarbonates, and polyacrylates. A special case is PMMA where only one monomer is employed. In the latter case we can have isotactic or syndiotactic molecules although in many syntheses atactic molecules result. These molecules, like the isotactic and syndiotactic molecules, have their own preferred conformation, depending on the sequence of preferred local conformations along the chain. This irregularity of the sequence leads to a coil‐like conformation, different for each molecule since the sequence is different for each molecule, and prevents regular packing in a lattice. Atactic polymers are thus generally also *amorphous*. Obviously complete random organization is impossible in view of the covalent bonds between the atoms in the chain. Generally this slight orientational preference is nondetectable by X‐ray diffraction. We deal first with amorphous solids obtained from polymer melts and from solutions. After that we deal with various aspects of long‐chain coils, ending with the equivalent chain.

A polymer molecule, whether in solution or in the melt, can further be characterized by the *end‐to‐end distance r*. In a good solvent the polymer–solvent attractions prevail, the coil expands, and *r* increases. The effective monomer–monomer interaction is always repulsive. In a poor solvent the polymer–polymer attractions, irrespective of whether they are due to parts from the same or from different chains, prevail. The coil shrinks and *r* decreases until the effective monomer–monomer repulsion due to excluded volume forces sets in. Under certain conditions the intramolecular interactions are similar in magnitude to the intermolecular interactions. In other words the enthalpy and entropy contributions from solvent–monomer and monomer–monomer interactions to the Helmholtz energy of the assembly of molecules under consideration compensate, and one part of the molecule does not seem to notice other parts of the molecule. The molecules behave like phantoms and are sometimes referred to as phantom chains. The temperature for which this happens is the *Flory temperature**θ*, and one speaks of *theta conditions*. Under theta conditions the coil neither shrinks nor expands with respect to the molten state and has unperturbed dimensions. If 〈*r*^{2}〉 denotes the mean of *r*^{2} (second moment), the influence of the solvent can be described by

2.4

where the subscript denotes the theta conditions and *α* a parameter dependent on solvent, temperature, and molecular weight. The parameter *α* can be estimated from , where *A* combines constants and thermodynamic parameters [5]. At the Flory temperature *T* = *θ*, theta conditions hold and *α* = 1. For *T* > *θ*, *α* > 1 and depends at most on *M* as *M*^{1/10}, and this effect contributes to the intrinsic viscosity of polymer solutions (see Chapter 10). Important for solids is now the *Flory theorem*: In a dense polymeric system, theta conditions prevail. Describing theta conditions as the configuration where intra‐ and intermolecular interactions compensate and since the solvent is the polymer melt itself, the theorem is highly plausible. Rephrasing, on the one hand, the monomers of a certain reference chain are subjected to a repulsive potential due to the excluded volume effect of its own monomers, and this leads to an expansion of the coil. On the other hand, neighboring chains, interpenetrating the reference chain, generate a counteracting attractive potential acting inward on the reference chain, and under theta conditions the two effects cancel, leading to (pseudo)unperturbed chains. Small‐angle neutron scattering experiments support the theorem. Since we deal mainly with solids, we omit the subscript *θ* in 〈*r*^{2}〉_{θ} from now on.

Focusing on the chains themselves, a first estimate of the end‐to‐end distance^{2} is made via the *freely jointed chain* model: *n* bonds, each of length *l*, connected without any restriction. The probability distribution of the end‐to‐end vectors for long‐chain molecules is described by the random walk model, resulting in

2.5

For such a model chain, one obtains, in the limit of a large number of atoms,

where 〈*r*^{2}〉 = 〈*x*^{2}〉 + 〈*y*^{2}〉 + 〈*z*^{2}〉 is the mean square end‐to‐end distance of the chains. The end‐to‐end distance *r* = 〈*r*^{2}〉^{1/2} is thus proportional to *n*^{1/2}.

However, we know that the bonds are not freely connected but have a certain bond angle *τ*. Leaving the bonds otherwise unrestricted, we obtain the *freely rotating chain* model for which it holds in the limit of a large number of bonds that

2.6

As expected the square root dependence on *n* is preserved, but the proportionality factor is changed. For sp^{3}‐hybridized carbon atoms, for example, in a PE chain, with a bond angle of *τ* = 109.5°, we have approximately 〈*r*^{2}〉 = 2.0*nl*^{2}.

A further improvement is obtained by introducing the *independent hindered rotation model*, that is, a rotating chain but with a preferential orientation for the dihedral (bond rotation) angle *φ* around the central bond between four consecutive atoms in the chain. In this model one obtains

2.7

Again the square root dependence on *n* is preserved, and the proportionality factor changes. For the PE chain we have one *trans* (t) configuration with a dihedral angle *φ* = 0° and two equivalent *gauche* (g^{+}, g^{−}) configurations with a dihedral angle *φ* = 120° and *φ* = −120°, respectively (compare Figure 2.2). The latter have a higher energy by an amount *E*_{gau}. Denoting the Boltzmann factor by *σ* = exp(−*E*_{gau}/*RT*), we obtain as an estimate for the average dihedral angle

2.8

For the end‐to‐end distance, we thus have

2.9

For PE at 140 °C, using *E*_{gau} = 2.1 kJ mol^{−1}, we find *σ* = 0.54, leading to 〈*r*^{2}〉 ≅ 3.4*nl*^{2}.

Finally we recognize that the hindered rotation around a bond is correlated, and this is taken into account in the *correlated hindered rotation model*. The final expression for the mean square end‐to‐end distance becomes

2.10

where the *characteristic ratio C* is a function of the correlation of the rotations along the chain and therefore a measure of the stiffness of the chain. Taking into account the correlation up to two bonds away, Flory calculated for PE *C* = 6.7 ± 0.2, in good agreement with experiment. Values of *C* for other polymers are given in Table 2.1. Equivalently, one uses the *radius of gyration R*_{g} defined as the root mean square average of the distance of atoms from the center of mass of the chain. For a freely jointed chain, *R*_{g}^{2} = *R*_{0}^{2}/6.

Table 2.1 Values for characteristic ratio *C* for various polymers.

Material |
C |
Material |
C |
Material |
C |

PEO | 4.0/4.1 | i‐PP |
5.8 | PVC | 13 |

PE | 6.7/6.8 | s‐PP |
5.9 | a‐PVAc |
8.9/9.4 |

a‐PS |
10.0 | a‐PMMA |
8.4 | PDMS | 6.2 |

i‐PS |
10.7 | i‐PMMA |
10 | a‐PiB |
6.6 |

a‐PP |
5.5 | s‐PMMA |
7 | PC | 2.4 |

PE, polyethylene; PEO, polyoxyethylene; PS, polystyrene; PP, polypropylene; PMMA, poly(methyl methacrylate); PVC, poly(vinyl chloride); PVAc, poly(vinyl acetate); PDMS, poly(dimethylsiloxane); PiB, poly(isobutylene); PC, poly(carbonate); *a*, atactic; *i*, isotactic; *s*, syndiotactic.

The description given with respect to dimension/size is common for many polymers. This has led to the introduction of the *equivalent chain*, in which a real chain, containing *n* correlated and rotation hindered bonds of (average) length *l*, is described as a freely jointed chain of *n*′ segments of length *l*′. Each of the *segments* thus represents a number of real bonds, but since the correlation along the chain is limited to a few bonds, these segments can be considered as freely jointed. Hence, for this description we match 〈*r*^{2}〉 with *n*′ *l*′^{2} and the maximum projected length of the chain *r*_{max} with *n*′ *l*′. This can be done in a unique way, leading to *l*′ = 〈*r*^{2}〉/*r*_{max} and *n*′ = *r*_{max}^{2}/〈*r*^{2}〉. Let us take again PE as an example. The maximum projected length of the PE chain *r*_{max} is *r*_{max} = *nl* sin(*τ*/2) ≅ 0.83 *nl*, and using 〈*r*^{2}〉 = 6.7*nl*^{2} leads to *l*′ ≅ 8*l* and *n*′ ≅ 0.1*n*. The segment thus contains about 10 (real) bonds, and its length, often addressed as *Kuhn length**,* reflects the stiffness of the molecular chain. In discussing the properties of polymers, frequent use is made of the equivalent chain model. For other polymers than PE, other equivalent lengths are of course obtained.

Depending on the degree of crystallinity, polymers may or may not show a clear melting point, but, depending on the temperature, a polymer can exhibit elastic/brittle behavior, viscoelastic behavior, or viscous behavior, in a similar way as inorganic glasses. Consequently, amorphous polymers do show a *glass transition temperature T*_{g} rather than a melting point. The *T*_{g} is a characteristic temperature for viscoelastic materials below which the polymer behaves largely elastically and is called a *glass*, while above this temperature the polymer behaves largely viscous and is called a *rubber*. The *T*_{g} depends on chemical structure, molecular mass, branching, and degree of crosslinking. More chain flexibility leads to a lower *T*_{g}, more bulky, and/or polar side groups to a higher *T*_{g}. Since viscous flow depends on the rate of disentanglement, a lower cooling rate leads to (slight) decrease in *T*_{g}. Moreover, confinement decreases *T*_{g}. For example, when the thickness of a PS film applied on silicon decreases below 50 nm, *T*_{g} drops from the bulk value of 375 °C to about 350 °C at 15 nm [6].

Since there is generally more free volume associated with the chain ends than with the chain middle, the glass transition temperature increases with increasing molecular mass *M*_{n}, up to *M*_{n} ≅ 2 × 10^{4} g mol^{−1}, whereafter the effect is limited. One equation to describe the behavior approximately is the *Fox*–*Flory equation* [7]:

2.11

where *T*_{g}^{∞} indicates the *T*_{g} for very high *M*_{n} and *k*_{1} is an empirical constant with typically a value between 1 × 10^{4} and 5 × 10^{4} K mol g^{−1}. Similarly, a small amount of branching reduces *T*_{g}, while a large amount restricts mobility and therefore increases *T*_{g}. Crosslinking increases the density, implying that the free volume decreases. Hence crosslinking increases *T*_{g}, described approximately by the *Fox–Loshaek equation*:

2.12

where *M*_{sub} is the molecular mass of the subchains between the crosslinks and *k*_{2} is another empirical constant. While a noncrystalline polymer above *T*_{g} behaves like a viscous liquid because the chains can slide along each other, crosslinking suppresses such liquidity. The presence of a low molar weight additive increases the free volume and thus lowers the *T*_{g}, as approximately described by the *Fox equation*:

2.13

where *w*_{1} and *w*_{2} are weight fractions of components 1 and 2, respectively.

As indicated in Section 2.1, most polymers grow by either the step‐growth or the chain‐growth mechanism. For step growth usually a low polymerization rate and a low molecular weight^{3} (500–50 000) result. All polymer chains grow equally fast, and one obtains homogeneous systems. The functionality is usually *telechelic*, that is, the polymer is an end‐functional polymer where all ends possess the same functionality, typically with *f*_{n} = 2–3. With chain growth one usually obtains a high molecular weight (10^{4}–10^{6}) and (very) high polymerization rates. The process is kinetically determined, and *dead chains* and monomers coexist with *fast‐growing chains*, leading likely to heterogeneous systems. The functionality is usually *pending*, that is, has a functionality distribution with *f*_{n} ≫ 2. Exceptions are *living chain‐growth* and ring‐opening reactions. The resulting low molecular mass polymers (*oligomers*), typically with a molecular weight of a few hundred to a few thousand and still having at least two reactive groups, are often called *resins*.

For later reference the definition of functionality requires some fine‐tuning. A functional group is a chemical moiety, reactive under given conditions, able to bond *once* to another polymer or monomer molecule. Typical examples of *mono*‐functional groups are OH, COOH, and NH groups, while NH_{2} (in reactions with epoxies) and C=C (in acrylates) provide examples of *bis‐*functional groups. Finally, we mention a *chain stopper*. This is a mono‐functional reactive monomer that ends propagation, that is, stops chain growth. A chain stopper may or may not contain another functionality that is unreactive at the given conditions, for example, an unsaturated fatty acid may be present in an alkyd resin. This functionality acts as a chain stopper in polycondensation, while in the drying process of an alkyd paint, it participates as a multifunctional group in the autoxidation.

In this section we discuss the basics of both the step‐ and chain‐growth mechanisms. To do so we need two concepts: first, the *number average molar weight M*_{n}, defined by the ratio of sample weight over number of moles of polymer formed, and, second, the *number average degree of polymerization* (DP)*X*_{n}, defined by the ratio of the number of monomers incorporated over the number of polymer chains. For chain‐growth polymerization for both definitions, the number of unreacted monomers is not to be included.

Let us now consider step growth and suppose we have a mixture of monomers A_{2} and B_{2} that can react with each other, for example, a diol and a diacid or a diamine and a diacid. Let us denote the initial concentration by *C*_{A}^{0} and *C*_{B}^{0} and their actual concentrations at any moment by *C*_{A} and *C*_{B}, respectively. In this case we have *X*_{n} = ½(*C*_{A}^{0} + *C*_{B}^{0})/½(*C*_{A} + *C*_{B}). Defining *r* = *C*_{B}^{0}/*C*_{A}^{0} and using *C*_{A}^{0} − *C*_{A} = *C*_{B}^{0} − *C*_{B}, one easily obtains

where in the last step excess B_{2} is assumed for the limit when *C*_{A} = 0. For a large value of the DP, we can neglect end‐group weight differences and approximate *M*_{n} = *X*_{n}(*m*_{A} + *m*_{B})/2, where *m*_{X} is the molecular weight of X_{2}.

More generally, we have the (number) distribution *P*_{n} describing the number of moles of polymer *n*_{j} with DP = *j*, that is, *P*_{j} = *n*_{j}/Σ_{j}*n*_{j}. In terms of *P*_{j}, *X*_{n} and *M*_{n} are given by

2.15

where *M*_{j} is the molecular weight for DP = *j*. The probability that any monomer is selected at random at a chain end is *C*_{A}/*C*_{A}^{0}. Hence, the probability that a monomer is in the interior of the chain, the *extent of reaction*, is *p* = 1 − (*C*_{A}/*C*_{A}^{0}). The chance that a selected chain contains *j* monomers is thus

2.16

where in the last but one step Eq. 2.14 is used, so that

2.17

In the last step *p*^{j−1} is approximated by *p*^{j−1} ≅ [exp(−*X*_{n}^{−1})]^{j−1}, which, in combination with *j* taken as continuous since *j* ≫ 1 anyway, simplifies a number of calculations. The function *p*_{j} is a monotonically decreasing function of *j* with as most probable value *j* = 1 and as average value *j* = *X*_{n}.

Using *w*_{j} = *M*_{j}*n*_{j} the *weight distribution* becomes

2.18

having a maximum close to *X*_{n}. For the *weight average molecular weight*, one obtains

2.19

Furthermore the *weight average DP* becomes

Approximately therefore, the weight average molecular weight is twice the number average molecular weight.

One can show that for chain termination by disproportionation, labeled *Z* = 0, and by combination, labeled *Z* = 1 (see Section 2.2.4), one obtains *P*_{j} = *Cj*^{Z}exp(−*j*/*y*) with *y* = *X*_{n}/(*Z* + 1). Using this expression as an empirical function for any *Z* value, it is frequently addressed as the *Schulz–Zimm distribution*. It has the property *M*_{w}/*M*_{n} = (*Z* + 2)/(*Z* + 1) so that *Z* acts as a dispersity parameter. Many practical considerations on step growth are given in [8].

The kinetics of step‐growth polymerization is in principle relatively simple. For a typical reaction such as of an alcohol B with an acid A, the reaction is catalyzed by acids. This may be either the reacting acid or another deliberately added acid. Denoting the concentration of the *catalyst* C with *C*_{C}, the change in concentration is given by

2.21

If the reacting acid A itself acts as catalyst, this expression becomes

which, upon integration, leads for *C*_{B} = *C*_{A} to

2.23

Generally this behavior is also observed experimentally. Deviations at low and high DP occur due to, respectively, solvent effects and reaching a limiting value, as given by Eq. 2.14. However, frequently a catalyzing acid is deliberately added because the noncatalyzed reactions are sluggish. In this case, assuming *C*_{C} is constant, integration of Eq. 2.20 for *C*_{B} = *C*_{A} leads to

2.24

In the absence of ring formation and depolymerization, also this relation is experimentally observed reasonably well. Generally, molecular weight buildup is much faster than for the self‐catalyzed case, because *X*_{n} ∼ *t* (instead of *X*_{n} ∼ *t*^{1/2}) and the effective rate constant *kC*_{C} is comparable with *k* in Eq. 2.22, even at a lower temperature.

So far we discussed only linear chains, the *functionality f* of the monomers being restricted to *f* = 2. Generally however, *f* ≥ 2, and *branching* during polymerization may occur (Figure 2.1). In this case at low extent of polymerization, already large molecules appear, which are no longer molecularly dissolvable. The limit to branching is *gelation*, where a solvent‐swollen, viscous (incipient) *gel* appears in the remaining lower molecular weight polymers in solution, called the *sol*. The point where a steep increase in viscosity is observed is denoted as the *gel point*. Polymers of industrial relevance should have an upscalable production process. The reactions involved therefore should proceed in a finite time, while the risk for gelation should be excluded or the reaction should lead to intrinsically nongelling compounds. Moreover, gelation is undesirable as it may lead to defects during application of a coating as well. Hence, one would like to be able to predict gelation on the basis of the monomer composition (and reaction conditions). Here the statistical theory of branching comes to the rescue. Interpreting the gel point as the point with the first appearance of infinite molecules in the system as a result of the ongoing polymerization occurs, the gel point (as function of conversion) can be calculated.

The phenomenology of the gel point is different for the step‐growth and chain‐growth mechanisms (Figure 2.7). In step growth, gradual polymerization occurs in which the number of monomers is always larger than that of the oligomers, which on its turn is larger than that of the polymers. A broad molar mass distribution is present near the gel point, and the critical extent of reaction *p** has usually a relatively high value. In chain growth, *sudden* boosts in polymerization occur for which the number of monomers is larger than that of the *living* polymers, but with many *dead* polymers present. When gelation does occur, it occurs early in the process, so that *p** is usually low. This situation is best treated with kinetic models and/or simulations (see Section 2.2.4).

The (classical) *statistical theory of branching* is based on a number of crucial assumptions:

- Functional groups of certain type have equal reactivity, regardless of their position in the chain or the fate of the other groups on the same molecule.
- No intramolecular reactions occur (e.g. cycle formation) in
*finite*molecules. They are excluded for mathematical*simplicity*. Obviously this introduces a (fortunately usually small) systematic error, since in reality cycles do occur. Due to this*omission*the gelation point is usually underestimated. - No side reactions occur. Near the gelation point side reactions can be important. Side reactions that lead to chain extension are influencing the gel point, while those that lead to branching are of lesser importance with respect to gelation.

In fact, the classical Flory–Stockmayer theory [9], the Gordon–Dusek cascade theory [10], the Miller–Macosko recursive approach [11], and the Durand–Bruneau propagation‐expectation theory [12] all use the same set of assumptions. In the Flory–Stockmayer approach,*M*_{w} is calculated from the distribution functions, resulting in tedious or impossible calculations for complex systems, while the Gordon–Dusek approach uses the probability generating functions, also leading to complex equations to solve. The Miller–Macosko approach is simpler but requires involved substitutions to solve the relevant equations. In contrast, the Durand–Bruneau approach uses a concept called the propagation expectation, resulting directly in *M*_{w} and an effective functionality. Finally, a completely different approach is based on percolation theory [13]. In this section we discuss the basics of the statistical theory and in the next section application of the effective functionality concept.

To discuss gelation we introduce the *branching index**α*, representing the probability that from a given branch point, a selected chain continues to another branch point rather than terminating in loose end. To estimate whether an infinitely long path in a branched polymer can be found, we consider that there are *f* − 1 ways to continue a path at each branch point. Given *α* as the probability to successfully proceed to a next branching point, *α*(*f* − 1) is the probability to successfully continue to any of the *f* − 1 branch points possible. For a connected series of *j* branch points, the total probability is [*α*(*f* − 1)]^{j}, so that, if this probability must remain finite, we should have.

2.25

Given the functionality *f*, we thus should calculate the branching index *α*.

Consider now as example a mixture of difunctional monomers A_{2} and *f*‐functional monomers A_{f} (*f* > 2) reacting with difunctional monomers B_{2}, but where no reaction occurs between molecules of component A or molecules of component B. They have the initial concentrations (1 − *ρ*)*C*_{A}^{0}, *ρC*_{A}^{0} (where *ρ* is the fraction *f*‐functional monomer), and *C*_{B}^{0}, respectively. Denoting the probability that an X group has reacted with *p*_{X} = 1 − (*C*_{X}/*C*_{X}^{0}), the probability to find *j* − 1 A_{2} and *j* B_{2} monomers providing a connection between a selected branch point and another branch point is [(1 − *ρ*)*p*_{A}*p*_{B}]^{j−1}*ρp*_{A}*p*_{B}. The probability that such a sequence connects a selected branch point with another, regardless of length, is thus

Summing using Σ_{j}*x*^{j} = (1 − *x*)^{−1} and *r* = *C*_{B}^{0}/*C*_{A}^{0}, so that *rp*_{B} = *p*_{A}, one obtains

Using this expression one can calculate the branching index *α*(*r*, *ρ*). Solving *α* = (*f* − 1)^{−1} leads to the critical extent of the reaction

One can similarly calculate the number average DP. For the above situation the total number of units *N*_{0} is given by *N*_{0} = *ρN*_{A}^{0}/*f* + (1 − *ρ*)*N*_{A}/2 + *N*_{B}^{0}/2, where *N*_{A}^{0} and *N*_{B}^{0} are the total number of A and B groups. The total number of chains is the total number of units minus the number of bonds formed during polymerization. Hence,

and

2.30

As example, take the stoichiometric mixture A_{3} + B_{2} so that *f* = 3, *ρ* = 1, and *r* = 1. In this case *α* = *p*_{A}^{2}, and from *α*(*f* − 1) = 1, we see that gelation starts when *α** = 1/2 or when *p*_{A}* = 2^{−1/2} ≅ 0.71. Calculating *X*_{n} we obtain *X*_{n} = (1 − 6*p*_{A}/5)^{−1}, and consequently at the gel point, *X*_{n} ≅ 6.6, which is a rather low DP. We can also deal with the self‐condensation of A_{f} by taking *p*_{B} = 1 in Eq. 2.26. With *f* = 3, *ρ* = 1, and *r* = 1, the result is *α* = *p*_{A} and thus *p*_{A}* = ½. For calculating *X*_{n} we have to replace the term *r*/2 in Eq. 2.29 by 1/*f* (because B = A), and *X*_{n} becomes *X*_{n} = (1 − *p*_{A}*f*/2)^{−1}. For *f* = 3 at the gel point, therefore *X*_{n} = 4, again a rather low value.

To calculate the sol and the gel fraction, consider the probability *Q* that a randomly selected branch point is not connected to the gel. This probability equals the probability 1 − *α* that two consecutive branch points are not connected plus the probability that they are connected (probability *α*) but that that this reaction did not lead to a connection to the gel (probability *Q*^{f−1}), that is, *α Q*^{f−1}. Hence, the *recurrence relation*

2.31

is obtained. The sol fraction *p*_{sol} equals the probability that a randomly selected site is not connected to the gel along any of its *f* bonds, that is, *p*_{sol} = *Q*^{f} or *Q* = *p*_{sol}^{1/f}. Substitution in the recurrence relation leads to *p*_{sol}^{1/f} = 1 − *α* + *α Q*^{(f−1)/f}. For *α* < *α**, the solution is *p*_{sol} = 1, but for *α* > *α** another solution exists if *f* ≥ 3. The solution is easy for *f* = 3, leading to *p*_{sol} = [(1 − *α*)/*α*]^{3} and therefore to the gel fraction *p*_{gel} = 1 − [(1 − *α*)/*α*]^{3}. One can show that the number average DP *X*_{n}(*α*) = [1 − (*αf*/2)]^{−1} and is thus finite at *α* = *α** = (*f* − 1)^{−1}, while the weight average DP *X*_{w}(*α*) = (1 + *α*)/[1 − (*f* − 1)*α*] and thus becomes infinite at *α* = *α**.

If one is not interested in the details but only wants to estimate whether gelation occurs or not, a somewhat simpler approach can be followed. Recall that gelation occurs when *α*(*f* − 1) = 1. If we thus can estimate an effective functionality *f*_{eff}, *α* can be estimated as well. Now we distinguish between *number average functionality f*_{n} (compare *M*_{n}) and *end‐group average functionality f*_{end} (compare *M*_{w}). The definitions are, respectively,

2.32

where *ν*_{j} denotes the number of monomers with functionality *f*_{j}, and identify *f*_{end} as the effective functionality *f*_{eff}. However, typically we have species A_{f} and B_{g} (i.e. for a polyester, a diacid + diol) with functionalities *f* and *g*, respectively. In a random sequence along a chain, they will occur with equal probability, and therefore we take heuristically

2.33

Finally, we have to take into account off‐stoichiometry (which is the usual situation, with either excess of diol or of diacid) via the ratio of monomers *r* = *C*_{B}/*C*_{A} (or vice versa, so that here *r* < 1 always) with *C*_{X} denoting the number of monomers X. As soon as the deficient component, that is, the least present component, is exhausted, the reaction stops. This is accomplished by adding the factor *r*, so that we have in total

and where *f*_{eff} has to be calculated at time *t* = 0. In fact, Eq. 2.34 has been derived by Durand and Bruneau using solid statistical arguments [12].

Let us now consider a few examples and take as the first one an A_{f} system for which molecules of component A_{f} with functionality *f* react with each other. Practically speaking such systems are rare, but we can consider *living* vinyl addition polymerization^{4} as an example [14]. Here *f* = *g* = 2 (each carbon atom of the C=C bond reacts with another monomer). For the critical branching index *α**, we have, using *f*_{end} = 2 at time *t* = 0, so that

As expected, we predict that an A_{2} system forms linear chains. Suppose we slightly modify the system to A_{2} + A_{4} (e.g. by a small addition of diacrylate) in a ratio of, say, 99 acrylate monomers to 1 diacrylate monomer. For *f*_{end} at *t* = 0 we calculate *f*_{end} = (99 × 2^{2} + 1 × 4^{2})/(99 × 2 + 1 × 4) = 2.04, so that *α** = (*f*_{end} − 1)^{−1} = 0.96. Hence gelation occurs. This appears to be generally true: Any A_{f} system with *f* > 2 causes gelation, as could also have been seen from Eq. 2.27.

As a second example, consider an A_{f} + B_{g} system as occurs for the polycondensation of polyesters. Here we have species A_{2} and B_{2} (i.e. for a polyester, a diacid + diol) with functionalities *f* = *g* = 2. At stoichiometry, linear chains are formed because *r* = 1 and *f*_{end} = *g*_{end} = 2, so that *f*_{eff} = 2. For an off‐stoichiometry situation (which is the usual situation, with either excess of diol or of diacid), we need to take into account the ratio of monomers *r* = *C*_{B}/*C*_{A}.

Suppose we modify the system, for example, by adding a triol B_{3} to A_{2} instead of the diol B_{2}. In this case we have *f* = 2 and *g* = 3 and obtain for two (arbitrary) off‐stoichiometric cases

- 3A
_{2}+ 2B_{3} *r*= 2B_{3}/3A_{2}= 1,*f*_{end}= 2,*g*_{end}= 3,*α**= 0.71 < 1, hence gelation- 6A
_{2}+ 2B_{3} *r*= 2B_{3}/6A_{2}= 0.5,*f*_{end}= 2,*g*_{end}= 3,*α**= 1, hence*no*gelation.

This appears to be generally true: For an A_{f} + B_{g} system gelation can be circumvented by having excess of one of the components.

Another option is the system A_{2} + A_{3} + B_{2} (e.g. a diol, a triol, and a diacid). For example, for (slightly) branched OH‐functional polyesters,

- 5A
_{2}+ 6A_{3}+ 10B_{2} *r*= 20/28,*f*_{end}= 74/28,*g*_{end}= 2,*α**= 0.92 < 1, hence gelation- 10A
_{2}+ A_{3}+ 10B_{2} *r*= 20/23,*f*_{end}= 49/23,*g*_{end}= 2,*α**= 1.11 > 1, hence*no*gelation.

Also the addition of chain stoppers, that is, an A_{1} component, can prevent gelation. For example, for the system A_{1} + A_{3} + B_{2} (i.e. a monoalcohol, a triol, and a diacid), we obtain

- 10A
_{1}+ 6A_{3}+ 10B_{2} *r*= 20/28,*f*_{end}= 64/28,*g*_{end}= 2,*α**= 1.34 > 1, hence*no*gelation.

Obviously, this approach can be extended to, say, A_{2} + A_{f} + B_{g} systems. The most important examples in this class are the polyesters and polyurethanes (Figure 2.8). Another example is the alkyds in which glycerol (A_{3}), penta (A_{4}), fatty acids (B_{1}), or an anhydride (B2) reacts (Figure 2.9). In summary, for A_{f} + B_{g} systems, various options exist to prevent gelation during the fabrication of the polymers.

Chain‐growth polymerization can be classified into radical, ionic, and ring‐opening polymerization. We limit the discussion to radical polymerization though, which occurs with three distinct steps: initiation, propagation, and termination. To discuss these steps, we use as generic labels I_{2} for the initiator, M for the monomer, and *k* for the rate constants, while concentrations are indicated with [X]. Generally we have

The activation energies for these three steps are in the order of approximately 35, 5, and 3 kcal mol^{−1}, respectively. In the following we provide only the basic equations and refer for further details again to the literature.

Since initiation is a two‐step process, the rate of initiation *R*_{ini} will be a combination of the dissociation of the initiator and the initial addition of the initiating radical. If these rates are equal, we have

2.40

with *f* the efficiency factor for the radicals to initiate polymerization. This factor may be less than 1 if radicals are wasted by so‐called cage reactions. *Cage reactions* comprise primary recombination within about 10^{−11} s after dissociation and secondary recombination within after 10^{−9} s after dissociation. If recombination leads to the original initiator, *f* = 1, but many initiators dissociate (with short half‐lives) under simultaneous elimination of a small molecules. In such a case, *f* < 1. The most frequently encountered initiators are peroxides, but compounds with -O-N- bonds, -S-S- bonds, and -N-N- bonds (azo compounds) are also used. Initiators can dissociate thermally or photochemically. Examples of the thermal type are dibenzoyl peroxide (BPO) and dicumene peroxide (Dicup), dissociating according to

and

respectively. An example of the photochemical type is azobisisobutyronitrile (AIBN) dissociating via

upon illumination with 350 nm radiation.

The rate of propagation *R*_{pro} is given by

2.41

For the termination we have

2.42

where *k*_{ter} represents either *k*_{com} or *k*_{dispro}, or their sum if both processes occur simultaneously, and [R^{•}] = Σ_{j}[R_{j}^{•}]. These steps lead to the rate equations

2.43

2.45

2.46

Since the concentration of radicals and the number of chains that are actually undergoing propagation are small, the rate of production of polymer is close to the consumption rate of monomer. Neglecting the monomers used in the initiation step, the consumption rate of monomers is given by

2.47

while summing Eqs. 2.35, 2.36, 2.37 yields

2.48

In the steady‐state approximation we take d[R^{•}]/d*t* ≅ 0, leading to

2.49

Similarly from Eq. 2.39, we obtain

2.50

Hence,

2.51

The rate constants *k*_{pro} and *k*_{ter} are determined by the choice of the monomers, the temperature, and possibly the solvent, while the type of initiator and its concentration [I_{2}] can be chosen at will.

To calculate *X*_{n} we consider first termination by disproportionation in which for any two chains formed, two radicals are terminated by the transfer of an H atom. Therefore the rate is *k*_{dispro}[R^{•}]^{2}. For termination by combination, two radicals terminate one chain, and thus the rate is ½*k*_{comb}[R^{•}]^{2}. Combining we have

2.52

with *δ* = 1 for disproportionation and *δ* = ½ for combination. The 1 is added since the rate of polymerization in the numerator excludes the formation of R_{1}^{•} from I^{•} and M. Thus, *X*_{n} will be larger, the larger *k*_{pro}/*k*_{ter}, the larger [M], and the smaller [I_{2}]. Note that the rate of polymerization d[M]/d*t* is oppositely influenced by [I_{2}].

Applying the steady‐state approximation to Eqs. 2.35, 2.36, 2.37, 2.38 individually leads to

2.53

The quantity *γ* is often referred to as the *kinetic chain length* and, given by the ratio of the rate of polymerization over the rate of termination, represents the ratio of the number of monomers added to the chain during the lifetime of the initiating radical. Using Eq. 2.44 we have *X*_{n} = *γ* + 1 for termination by disproportionation and *X*_{n} = 2*γ* + 1 for termination by combination. For disproportionation with rate constant *k*_{dispro}, we have d[*M*_{j}]/d*t* = *k*_{dispro}[R_{j}^{•}][R^{•}], leading to

Since *X*_{n} = *γ* + 1, Eq. 2.54 equals *n*_{j} = *X*_{n}^{−1}(1 − *X*_{n}^{−1})^{j−1}, as also obtained for the step‐growth mechanism. Therefore the limiting ratio is also *M*_{w}/*M*_{n} ≅ 2. For combination with rate constant *k*_{comb}, we have d[*M*_{j}]/d*t* = *k*_{recomb}∑_{p+q=j}[R_{p}^{•}][R_{q}^{•}], leading to

2.55

often denoted as the *Schulz distribution*. In this case the limiting ratio is *M*_{w}/*M*_{n} = 3/2.

Finally, we mention *inhibitors*, molecules that react with radicals to saturated molecules or stable radicals that do not react further. Inhibitors are important not only to terminate reactions, in which case they are often referred to as chain stoppers, but more importantly to prolong *shelf life*. As inhibitors typically nitro compounds and quinones are used. Further practical details on radical polymerization can be found in [15, 16], while [17] provides a concise discussion of the basics.

In Sections 2.2.1 and 2.2.4, we discussed linear polymers, while in Section 2.2.2 we dealt with chain branching, where a secondary chain initiates from a point on the main chain, leading to a more entangled structure as compared with linear polymers, and therefore somewhat more difficult to deform than linear polymers. Lightly branched and linear polymers together are generally called *thermoplastics*, or *thermoplasts* for short. These materials are relatively easily deformed at elevated temperature since thermal motion in combination with mechanical load can change the entanglement structure relatively easily.

Now we have to say a few words about *crosslinking* or the *network formation* of polymers, that is, the formation of chemical bonds at certain points along a particular molecular chain to neighboring chains. Crosslinking can be random or controlled implying the formation of bonds at random points or at well‐controlled points along the chain. Such a bond is often referred to as a *junction*, and the part of the original chain between junctions is called a *subchain* (or *network chain*).

Heavily crosslinked polymers (*thermosets*) are relatively difficult to deform, even at elevated temperature. While with increasing temperature the mobility for a thermoplast increases, for a thermoset at a certain temperature the mobility does not increase any further significantly (Figure 2.10a). Such a thermoset is usually built up from *crosslinker* molecules and *resin* chains, leading to a unique structure, denoted as *network*. Both types of molecules have reactive functional groups, such hydroxyl, epoxy, or amine groups, that can react with other so that a network is formed. This network restricts the mobility, but a glass transition temperature *T*_{g} still may exist. The network formed is largely independent of polymer entanglements, while the crosslink density and the wetting of fillers or pigments can be controlled by a proper choice of chains and crosslinkers. Moreover, in this way the starting materials are easy to process as they are small molecules applied in the form of a liquid with low viscosity *η*. The final properties, such as elasticity and heat resistance, are obtained only after the *chemical reaction* between the resin and crosslinkers has proceeded, a process usually addressed as *crosslinking* or *curing*.

Actually, network formation can be treated similarly as branching with as difference that positioning the branch point is controlled by the crosslinking chemistry rather than by the polymerization. Using the same arguments as for gelation, this means that for a path in a network, the probability to continue on another chain from an arbitrary one must exceed one. With a fraction crosslinked chains *ρ*_{X}, the probability in a chain of *j* units to continue the path is *ρ*_{X}(*j* − 1). Hence, if *ρ*_{X}(*j* − 1) > 1, an infinite network is possible. Denoting the probability that the selected crosslink occurs in a chain with *j* units by *p*_{j}, we calculate

2.56

The probability *p*_{j} will be proportional to the number of units *j* in the chain and the number of chains *n*_{j} containing *j* units. Therefore

2.57

with *X*_{w}^{0} the weight average DP before crosslinking. For *ε* = 1, the critical value *ρ*_{X}* = 1/*X*_{w}^{0}, and thus gelation occurs if there is one crosslinker for every weight average chain. With *ν* crosslinked units, we have *ρ*_{X} = *ν*/∑_{j}*jn*_{j}, so that the *crosslinking index**γ*, defined as the number of crosslinked units per initial chain, becomes.

2.58

For the critical value of *γ*, we have *γ** ≤ 1, because *X*_{n}^{0} ≤ *X*_{w}^{0} always.

For the typical situation where linear (or slightly branched) polymers A react at their end with multifunctional crosslinkers, the argument runs as follows. With *N*_{end}^{0} the initial number of reactive end groups, *N*_{end} the number of end groups at any moment in time, *N*_{A}^{0} the initial number of reactive groups A, and *N*_{A} the number of remaining groups A, we define.

2.59

The probability of two crosslinks being connected being equal to *p*_{end} *p*_{A}, we obtain from the branching argument the criterion for an infinity long chain as

2.60

This expression also follows directly from Eq. 2.28 using *ρ* = 1. As an example, consider the conditions *N*_{A}^{0} = *N*_{end}^{0} and *f* = 3. In this case, gelation starts when the extent of reaction of the end groups is (1/2)^{1/2} ≅ 0.71.

If the crosslinkers are connected with the polymer chains, we have thus a network (Figure 2.10b). Such a network is (partially) characterized with the molecular mass *M*_{sub} of the molecular chains between the crosslinks, for which normally the number average *M*_{n} is used. This average is often determined using simple rubber theory that predicts that the storage (shear) modulus *G*′ = *ρRT*/*M*_{sub}, where *ρ* is the mass density, *R* the gas constant, and *T* the temperature. Networks are usually not ideal, however, and in a *nonideal network* defects are present. A chain connected to a junction at only end is a *dangling chain*, and the one that is attached to the same junction at both ends is called a *loop* (Figure 2.10b). A dangling chain does not contribute to the elasticity of the network, neither does a loop that is not penetrated by another chain that itself is elastically active. A network with no dangling bonds or loops and no junctions with functionality less than 3 is a *perfect* (or *ideal*) *network*. *Entanglements* can also be considered as a kind of defect acting similarly as a crosslink and are quite important in the deformation of both semicrystalline and amorphous polymers. Crosslinks and entanglements are sometimes collectively addressed as *elastically active knot**s* (EAKs).

Various models for rubber elasticity exist, such the affine network and ghost network model. They all lead to the same functional expression for *G*′ but with a different numerical prefactor. Both the presence of defects and the existence of various models render the comparison of *M*_{sub} as given in various papers not as straightforward as might be expected. Differences in using different models and/or corrections yield numerical values that may easily lead to a difference of a factor 2–3. The classic reference for elastomers is [18], while [19, 20] provide a more compact and extensive treatment, respectively.

To describe networks somewhat further, we denote the total number of subchains (network chains) and junctions in a network by *ν* and *μ*, respectively. A network can be thought as being formed in two steps. In the first step all subchains are joined at the junctions to macromolecule in the form of a *tree*. There are *ν* + 1 ≅ *ν* junctions in such a tree. To some of the junctions, chains are connected that can react with one another in the second step to form a network. In this process the number of junctions is reduced to *ν* + 1 − *ξ* ≅ *ν* − *ξ*, where we introduced the number of independent paths *ξ*, generally addressed as *cycle rank*. It is the number of bonds that has to be cut to change the network to a tree. For an ideal network of the five parameters that characterize the network (*ν*, *μ*, *M*_{sub}, *φ*, and *ξ*), only two are independent. It can be shown that [21]

where *V*_{0} is the volume of the network in the formation stage, *ρ* is the corresponding density, and *N*_{A} is Avogadro's number. For imperfect networks we have to identify the *active* (or *effective*) chains and junctions. Flory [5] defined an active chain as the one that contributes to the elasticity of the network and related their number *ν*_{eff} to the cycle rank *ξ* by *ν*_{eff} = 2*ξ*. It appears that general expressions relating *ν*_{eff} to other network parameters are not available at present. However, for an imperfect tetrafunctional network, the number of effective chains is approximately *ν*_{eff} = *ν*(1 − 2*M*_{sub}/*M*_{n}) with *M*_{n} the molecular weight of the primary molecules (initial polymers).

To conclude this part, we note that for an elastomer a typical subchain contains between 100 and 1000 skeletal bonds. Below 100 bonds the material is likely to be a thermoset, while above 1000 bonds long times are required to reach equilibrium under mechanical load. In a typical elastomer a subchain containing 500 bonds has a root mean square end‐to‐end distance 〈*r*^{2}〉^{1/2} of about 7–8 nm. Such a domain contains about 40 crosslinks and the associated chains. Thus, a subchain shares its available space with many other subchains, resulting in entanglements permanently trapped in the network.

Application of thermosets occurs via two routes. The first is the 1K or *one‐component* route.^{5} In this case we have all the ingredients mixed in one storage container that occurs at the paint manufacturer plant. A high shelf life is desired requiring latent reactivity. The curing reaction then proceeds after thermal activation or by using an external trigger, such as radiation, moisture, or oxygen. The second one is the 2K or *two‐component* route, where ingredients must be stored separately. In principle, this 2K route has an *infinite* shelf life, just because the reactants are separated. Mixing occurs at the paint user's location just before application and results in a limited *time‐to‐application window* as a result of sufficient reactivity for the application conditions.

The thermoset kinetics determine whether a 1K or 2K formulation should be used. Obviously this depends on the balance between stability (storage) and reactivity (cure). For a chemical reaction like the curing reaction A + B **→** A − B, the Arrhenius equation provides the *rate r* via (Figure 2.11)

2.61

in which the concentrations are again indicated by [X] and the rate constant *k* reads

2.62

where *k*_{0} is the frequency factor, *R* the gas constant, and *E*_{act} the activation energy.^{6} For an ideal 1K system, one requires simultaneously a high reactivity upon curing, that is, a high *k*_{c} at *T*_{c}, and a low reactivity when stored, that is, a low *k*_{s} at *T*_{s}. So, one requires a reaction with a high *k*_{0} and a high *E*_{act}. The precise data depend clearly on what degree of conversion *a* one allows during shelf life. Let us assume that we allow *a*_{s} < 0.05 at 30 °C after 6 months and we require that *a*_{c} > 0.90 at *T °*C after 10 min. Table 2.2 provides the required rate constants and activation energies at various temperatures to fulfill these conditions. Most practical chemical reactions have *k*_{0} < 10^{16} s^{−1} and *E*_{act} < 150 kJ mol^{−1} so that fulfilling these requirements may not be easy below 125 °C curing temperature.

Table 2.2 Kinetic window for *a*_{s} < 0.05 and *a*_{c} > 0.90 after 10 min.

T (°C) |
k_{0} (s^{−1}) |
E_{act} (kJ mol^{−1}) |

175 | 10^{10} |
109 |

150 | 10^{12} |
121 |

125 | 10^{17} |
146 |

100 | 10^{24} |
188 |

In this chapter we provided an overview on polymers and the resulting networks. Many of the aspects indicated will return with some frequency at various places in subsequent discussions. For each of these topics, a great deal more can be said. For that, we refer to the literature.

- 1 PlasticEurope Market Research Group (PEMRG), Consultic Marketing Industrieberatung GmbH, data August 2016.
- 2 Boyd and Phillips (1993).
- 3 For a review of early work, see Geil, P.H. (1963). Polymer Single Crystals. New York: Wiley.
- 4 Gedde (1995).
- 5 Flory (1953).
- 6 Howard, R.N. and Young, R.J. (1997). The Physics of Glassy Polymers. London: Chapman and Hall.
- 7 Fox, T.G. (1956). Bull. Am. Phys. Soc. 1: 123.
- 8 Stoye and Freitag (1998). Chapter 6.
- 9 (a) Flory, P.J. (1953). Principles of Polymer Chemistry. Ithaca, NY: Cornell University Press.(b) Stockmayer, W.H. (1943). J. Chem. Phys.11: 45.(c) Stockmayer, W.H. (1943). J. Chem. Phys. 12: 125.(d) Stockmayer, W.H. (1952). J. Polym. Sci. 9: 69.
- 10 (a) Gordon, M. (1962). Proc. Roy. Soc. A268: 240.(b) Dusek, K., Scholtens, B.J.R. and Tiemersma‐Thoone, G.P.J.M. (1987). Polym. Bull. 17: 239.
- 11 Miller, D.C. and Macosko, C.W. (1976). Macromolecules 9: 199; (1978), ibid.
**11**, 656. - 12 (a) Durand, D. and Bruneau, C.M. (1981). Brit. Polym. J. 13: 33.(b) Durand, D. and Bruneau, C.M. (1982). Polymer 23: 69.
- 13 Stauffer, D., Coniglio, A. and Adam, M. (1982). Adv. Polym. Sci. 44: 104.
- 14 Braunecker, W.A. and Matyjaszewski, K. (2007). Prog. Pol. Sci. 32: 93.
- 15 Stoye and Freitag (1998). Chapter 8.
- 16 Paul (1995). Chapter 1.
- 17 Boyd and Phillips (1993). Chapter 4.
- 18 Treloar, L.R.G. (1975). The Physics of Rubber Elasticity, 3e. Oxford: Clarendon.
- 19 Boyd and Phillips (1993), Chapter 8.
- 20 Erman, B. and Mark, J.E. (1997). Structures and Properties of Rubberlike Networks. Oxford: Oxford University Press.
- 21 Mark, J.E. and Erman, B. (1988). Rubber Elasticity. A Molecular Primer. New York: Wiley.

- Boyd, R.H. and Phillips, P.J. (1993). The Science of Polymer Molecules. Cambridge: Cambridge University Press.
- Flory, P.J. (1953). Principles of Polymer Chemistry. Ithaca, NY: Cornell University Press.
- Gedde, U.W. (1995). Polymer Physics. London: Chapman and Hall.
- Hiemenz, P.C. (1984). Polymer Chemistry. New York: Marcel Dekker.
- Odian, G. (2004). Principles of Polymerization, 4e. Hoboken, NJ: Wiley.
- Paul, S. (1995). Surface Coatings: Science and Technology, 2e. Chichester: Wiley.
- Rubinstein, M. and Colby, R.H. (2003). Polymer Physics. Oxford: Oxford University Press.
- Sperling, L.H. (2006). Introduction to Physical Polymer Science, 4e. Hoboken, NJ: Wiley.
- Stoye, D. and Freitag, W. (1998). Paints, Coatings and Solvents. Weinheim: Wiley‐VCH.
- Young, R.J. and Lovell, P.A. (1991). Introduction to Polymers, 2e. Chapman and Hall.

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