Polymer Networks from Preformed Precursors Having Molecular

Article pubs.acs.org/Macromolecules

Polymer Networks from Preformed Precursors Having Molecular Weight and Group Reactivity Distributions. Theory and Application Karel Dušek,†,* Miroslava Dušková-Smrčková,† Jos Huybrechts,‡ and Andrea Ď uračková†,§ †

Institute of Macromolecular Chemistry, Academy of Sciences of the Czech Republic, 162 06 Prague 6, Czech Republic Du Pont Belgium, S.A., Mechelen, Belgium

‡

S Supporting Information *

ABSTRACT: High-performance cross-linked polymeric materials are now prepared from preformed precursors typical by distributions of molecular weights, number and reactivities of functional groups, and specific architectures. This makes theoretical treatment of networks evolution and their final structure difficult. This paper describes kinetically controlled cross-linking of a precursor formed from polyfunctional cores by arm extension by which molecular weight distribution develops and new groups of different reactivity are formed. These precursors are then cross-linked with a polyfunctional cross-linker. If the precursor groups react independently, a random (binomial) distribution of reactive groups results and the gel point conversion and other network parameters are independent of the differences in reactivity the groups with the cross-linker. If the condition of random (binomial) distribution is not met (fixed numbers of groups or substitution effect in the precursor molecules), this independence does not exist. Relations for molecular weight averages prior to gelation and gel fraction and concentration of elastically active network chains in the postgel state are derived. This general treatment applies to precursors obtained by a wide variety of chain extension chemistries and any of the family of cross−linking reactions of A + B type. In the second part, the general form of the theory was adapted to describe polyether precursors prepared by addition of an epoxy ester (glycidyl pivalate) to multifunctional polyols and their curing with a tri-isocyanate. Some of the primary OH groups of the core are chain−extended to form a polyether chain terminated by a secondary OH group. The distributions are altered by additional reactions − transesterification and alcoholysis. The branching theory was modified and the results compared with experiments. Gel point conversions were affected by these additional reactions, but the concentration of elastically active network chains (EANCs) (calculated from equilibrium elastic modulus) did not change much. The fraction of formed bonds wasted in cycles amounted to 12−22%.

1. INTRODUCTION The cross-linked state is a necessary condition for a wide range of applications of polymeric materials. Rather than simple organic compounds, the current precursors of polymer networks are mostly predesigned and preprepared polyfunctional materialsoften mixtures characteristed by molecular weight, composition, functionality, and group reactivity distributions.1,2 Hyperbranched polymers, highly branched off-stoichiometric copolyadducts, functional stars with variable arm lengths and group reactivities, functional copolymers of various architectures and prereacted systems involving polyisocyanates and polyepoxides, reaction products of phenols, urea, or melamine with formaldehyde, etc. (cf., e.g., refs 1−10) often developed in multistage processes (cf., e.g., refs 11 and 12) are examples of precursors of this category. It is not at all obvious how the conditions of precursor formation, i.e., the conditions of the stages prior to network formation, affect the network structure. Theoretical analysis can help understanding the role of various distributions in cross-linking and to optimize them for the given purpose. There are several practical reasons for variation of precursor structure and © 2013 American Chemical Society

composition such as control of reactivity and viscosity (viscosity profile when approaching the gel point), emission of volatile organic compounds, introduction of specific properties into the resulting network, processing characteristics, or price. Existence of distributions of various properties in precursors (molecular weight, composition, functionality, and group reactivity) obtained by prereactions determines the choice of strategy in modeling of network formation. Here, we will consider kinetically controlled structure build-up which is the most common way of synthesis. A full overlap of growing molecules (well over c* concentration) is guaranteed by a relatively high concentration of reactive groups. For dynamic simulations in 3D space, the multicomponent systems with distributions are not well manageable. Application of the kinetic theory based on (infinite) sets of differential equations is possible13 but not easy and often Monte Carlo methods are Received: November 22, 2012 Revised: February 19, 2013 Published: March 19, 2013 2767

dx.doi.org/10.1021/ma302396u | Macromolecules 2013, 46, 2767−2784

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used for the solution.14 Inapplicability of the kinetic method to the description of the gel components is its main disadvantage. The statistical method working with building units in different reaction states is very suited for treatment of chemically complex reactions. The lack of rigor of the statistical methods applied to the kinetically controlled cross-linking comes from erasing of information about the composition of sequences of bonds as they develop in time. In many cases, the statistical solution is identical with the kinetic one. In some other cases, the difference does not exceed experimental uncertainty, but in some stage of the process the difference can be important. The “combined method” was designed in which some stages are described by kinetic method and the formed structures (superspecies) are linked by the statistical method.15−23 However, which of the processes are statistically fully independent must be carefully analyzed.24,25 The statistical build-up is essentially a Markov process and it is imperative that the transition probabilities, identical with linking probabilities, be determined by chemical kinetics.25 Cyclization is important especially in high-functional systems. Despite of the progress reached in the past decades (cf., e. g., refs 26−30), none of the theoretical approaches is sufficiently predictive. This refers mainly to systems of more complicated architectures as well as to the postgel state, where formation of closed circuits occurs by reactions among functional groups of the gel component. Only some network chains can be classified as elastically inactive which are not stretched in equilibrium as a result of macroscopic strain. The ring-free case is the basic reference state to which cyclization “correction” can be related. The ring-free critical conversion at the gel point can be calculated and also reached by extrapolation of the experimental dependence of the gel point conversion on dilution. In this way, the extent of cyclization at the gel point can be determined. This value can serve for estimation of fraction of elastically inactive cycles in the gel.31,32 Such strategy is also applied in this work. This contribution is composed of two parts. The first, theoretical one, deals with a class of polyfunctional precursors (A) obtained by chain extending reactions of a multifunctional core by which arms of different length are formed and the functional groups of the core are partly transformed into other functional groups (A1 groups transformed into A2 groups). This distribution of precursor molecules is cross-linked in the second step with a polyfunctional cross-linker (B). Extensions of hydroxy-functional cores with epoxide compounds to polyethers, cyclic anhydride-epoxide combinations to polyesters, or chain extensions with lactones, or lactams can serve as examples. The cores can have other functional groups than OH such as carboxyl, epoxide, or amine (numerous other examples can be found in refs 3 and 33−35). Thus, the resulting precursors are nonuniform in molecular weights and numbers of reactive groups A1 and A2. The molecular weight and functionality distributions are important for the cross-linking step, because the groups A1 and A2 have a different reactivity toward groups of the cross-linker. Our attention is concentrated on how the variations in precursor structure affect the structure evolution during cross-linking and properties of the network. The distribution functions are obtained in closed forms and used as input information for the cross-linking stage. In the second part, an experimental system of star-like polyols chain extended to form polyether chains having potential for high-performance automotive coatings is studied with the aim to understand relations of the composition to

properties. It was found that the chain extension is complicated by inherent transesterification and alcoholysis and the branching theory was modified accordingly. The gel point conversions and equilibrium elastic moduli are compared with predictions made by the modified branching theory. The whole study using combination of kinetic and statistical theoretical approaches (combined method) and structure-sensitive experimental methods can serve as a paradigm for helping understand and develop complicated but important systems.

2. MODELING OF PRECURSOR STRUCTURE AND ITS CROSS-LINKING 2.1. Development of Precursor Structure. An A f A molecule having fA groups A1 reacts with group E of chainextending molecule R−E by which group A2 is formed; the A2 group can further react with another E group by which A2 group is reformed. In the cross-linking step, A1 and A2 groups react with B groups of molecules Bf B. As an example, formation of distribution of hexafunctional oligomers by successive addition of the E molecules is shown in Figure 1.

Figure 1. Polyaddition of RE molecules to A1 groups of the A6 molecule.

The resulting oligomers have always fA functional groups A but the number of A1 and A2 groups of a molecule and the molecular weight averages depend not only on the initial molar ratio [E]:[A], but also on the relative reactivities of A1 and A2 groups with E groups. We will establish relations between the initial composition, relative reactivities of A1 and A2 groups, and conversion of E groups (input) and number of A1 and A2 groups per molecule and molecular weight averages of the oligomers (output). The basic assumptions used are as follows: 1. The reactivities of all A1 groups of AfA are the same and do not depend on whether neighboring A1 groups have reacted or not. This condition can be relaxed if necessary, and substitution effect can be considered 2. The reactivities of the A2 groups formed by the reaction of E are the same and do not depend on the length of the branch. Modification can be made in that the reactivity of the first A2 is different compared to A2 groups more distant from the AfA-core. We will first examine the degree-of-polymerization distribution of A2-functional chains formed by reaction of RE molecules with an A1 group controlled by reaction kinetics. In the next step, the multifunctional (AfA) case will be considered when several chains grow simultaneously from 2768


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⎛ ⎞ 1 Pn = 1 − κ2⎜1 + ln(1 − ξA1E)⎟ ξA1E ⎝ ⎠

one molecule. The treatment of the initiated chain growth is somewhat related to our previous treatment of side reactions accompanying amine-epoxide curing16−19 and also to ref 36). 2.2. Distribution of Reaction States of the Precursor. 2.2.1. Polymerization of RE Molecules Released by an A1 Group. These reactions are kinetically considered as bimolecular reactions

The second-moment (weight-) average degree of polymerization of E units, Pw, is derived from the second derivative of g(Z) ⎛ ∂ 2g (Z) ⎞ ⎡ 1 ⎤ ⎟ ⎜ ln 2(1 − ξA1E)⎥ ≡ g ″(1) = κ22⎢ 2 ⎣ ξA1E ⎦ ⎝ ∂Z ⎠Z = 1

d[A1] = −k1[A1][E] dt

⎤ ⎡ 1 ln(1 − ξA1E)⎥ − 2κ2(1 − κ2)⎢1 + ξA1E ⎦ ⎣

d[A1E1] = k1[A1][E] − k 2[A1E1][E] dt ⋮ d[A1Ek] = k 2[A1Ek − 1][E] − k 2[A1Ek][E] dt ⋮

Pw = (1)

d[E] = −k1[A1][E] − k 2([A1]0 − [A1])[E] dt i.e.

i

([A1]0 − [A1]) d[E] = 1 + κ2 d[A1] [A1]

where Z is the auxiliary variable of the gf. The transformation is done by multiplication of both sides of each of the equations of the set (1) by the respective power of the variable Z, Zi. Summation of the left-hand sides and right-hand sides gives the equation

∂t

= k 2[E](Z − 1)gc (Z) + k1[E][A1]Z

∂[A1]

= −κ2

1 (Z − 1)gc (Z) − Z [A1]

[E]0 − [E] = (1 − κ2)([A1]0 − [A1]) ⎛ [A1]0 ⎞ + κ2[A1]0 ln⎜ ⎟ ⎝ [A1] ⎠

(2)

(3)

ξE =

⎛ ⎛ [A1]0 ⎞ × ⎜⎜[A1]0 ⎜ ⎟ ⎝ [A1] ⎠ ⎝

ξA1E =

[A1]0 − [A1] ; [A1]0

[A1]0 [E]0

ξErE = (1 − κ2)ξA1E − κ2 ln(1 − ξA1E)

(10)

Figure 2 shows that for low values of κ2, possible polymerization of E groups occurs only after all A1 groups have reacted (i.e., for [E]0/[A1]0 > 1). At κ2 = 1, about 62% of the A1 groups have reacted and, consequently, have been transformed into A2 groups and 38% remained as A1; for κ2 = 10, 70% of A1 groups remain unreacted (see Figure 1). The average degrees of polymerization of sequences of E units reacted with an A1 group were calculated using eqs 6−8 and 10. With increasing ξE, Pn, Pw, and Pw/Pn increase (Figure 3) When the number of E groups exceeds the number of A1 groups (Figure 4), a maximum develops at low conversions of E groups. In this section, we describe the kinetically controlled growth of a collection of single E chains initiated by reaction of A1 group. Each E chain is terminated by A2 group while some A1 groups

(4)

where Ng is the normalizer; Ng = ([A1]0 − [A1])−1, such that g(Z = 1) = 1. The degree-of-polymerization averages are obtained by differentiation of g(Z): ⎡ ⎛ ∂g (Z) ⎞ Pn = ⎜ ≡ g ′(1) = Ng ⎢(1 − κ2)([A1]0 − [A1]) ⎟ ⎝ ∂Z ⎠Z = 1 ⎣ ⎛ [A1]0 ⎞⎤ + [A1]0 κ2 ln⎜ ⎟⎥ ⎝ [A1] ⎠⎦

[E]0 − [E] ; [E]0

rA =

⎡ Z g (Z) ≡ NE(Z) = Ng gc (Z) = Ng ⎢ ⎢ κ2(Z − 1) + 1 ⎣ ⎞⎤ − [A1]⎟⎟⎥ ⎥ ⎠⎦

(9)

Equation 9 can be expressed as conversions of E groups and A1 groups

where κ2 = k2/k1. The solution for gc(Z), normalized so that g(Z) is a probability generating function (pgf), reads

κ2(Z − 1)

(8)

The solution reads

since d[A1]/dt = −k1[E][A1] ∂gc (Z)

(7)

d[A1] = −k1[A1][E] dt

i=1

∂gc (Z)

g ″(1) +1 g ′(1)

2.2.1.1. Relation between Concentrations of A1 and A2 Groups. The change in concentration of A1 and E groups is given by the equations

∞

∑ [AEi]Z

(6)

Thus

where [AEi] is the concentration of oligomer composed of i molecules of E and [E] is the concentration of the chain extender and k1 and k2 are rate constants. This infinite set of equations can be expressed by a differential equation for the generating function (gf) gc(Z) defined as gc (Z) ≡

(5a)

(5) 2769


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Figure 2. Dependence of conversion of A1 groups with E groups, ξA1E, on conversion of E groups, ξE, for different values of the reactivity ratio κ2 indicated; [E]0/[A1]0 = 1, eq 10.

Figure 4. Nonunifomity of E unit sequences for different molar ratios [E]0:[A1]0 (indicated) in dependence on conversion of E groups; reactivity ratio κ2, = 0.3.

remain unreacted. The distribution is generated by irreversible second-order reaction kinetics (sets of eqs 1 and 7). 2.2.2. Branches in Polyfunctional Precursors. Initially, polyfunctional molecules carry fA groups A1. When they react with E groups, the AfA molecules are transformed by a series of parallel and consecutive reactions. The fA functional compounds are symmetrical, so that the intrinsic reactivity of all A groups is the same, but there may exist a substitution effect. The transformation scheme for an A4 molecule is shown in Figure 5. First, we will be considering the substitution effect to be absenta case often encountered for reactions of small polyols.37 In that case, the rate constants k40 = k31 = k22 = k13 = k1 and the reactions are described by infinite sets kinetic differential equations (eqs 11 and 12).

Figure 5. Reaction scheme for transformation of A40 molecule, having initially 4 A1 groups. Coding of molecules: Axy(k), x number of A1 groups, y number of A2 groups, t reaction time, and k number of E units per A4 molecule. Substitution effect in Axy unit, equal reactivity of all A2 groups.

d[A22(2)] = 3k1[A31(1)][E] − (2k1 + 2k 2)[A22(2)][E] dt

d[A40] = −4k1[A40][E] dt

x>2 d[A22(x)] = 3k1[A31(x − 1)][E] + 2k 2[A22(x − 1)] dt − (2k1 + 2k 2)[A22(x)][E]

d[A31(1)] = 4k1[A40][E] − (3k1 + k 2)[A31(1)][E] dt

x>1

(11)

d[A31(x)] = k 2[A31(x − 1)][E] dt − (3k1 + k 2)[A31(x)][E]

d[A13(3)] = 2k1[A22(2)][E] − (k1 + 3k 2)[A13(3)][E] dt

Figure 3. Dependences of the weight-average degrees of polymerization of E unit sequences, Pw, and the nonuniformity, Pw/Pn, on conversion of E groups, ξE, for different values of the reactivity ratios of A2 to A1 groups, κ2, indicated (upper curve, κ2 = 10; lowest curve, κ2 = 0.02). Initial ratio [E]0 = [A1]0. 2770


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x>3

generated by solution of the kinetic differential equations using sets of equations of the type of eq 12 (cf. also Figures 5 and 6).

d[A13(x)] = 2k1[A22(x − 1)][E] dt + 3k 2[A13(x − 1)][E] − (k1 + 3k 2)[A13(x)][E]

d[A04(4)] = k1[A13(3)][E] − 4k 2[A04(4)][E] dt x>4 d[A04(x)] = k1[A13(x − 1)][E] + 4k 2[A04(x − 1)][E] dt − 4k 2[A04(x)][E] (12)

An example of the concentration profiles of A22(k) as a function of ξE(t) illustrates the evolution of the distributions with increasing conversion of E groups. 2.2.3. Analysis of Kinetic Simulation and Its Significance for the Cross-Linking Stage. The analysis of the results of numerical solution of systems of differential eqs 11 and 12) was performed for the range of values of reactivity ratio κ2 = 0.1− 1.0. It gave the following conclusions: (1) The distribution of reaction states given by number fractions, n(X,( fA-X)), of A units having X A1 and ( fA − X) A2 groups irrespective of the number of attached E units is identical with the binomial coefficients ⎛f ⎞ fA − X n(X , fA − X ) = ⎜ A ⎟(1 − ξA1E)X ξA1E ⎝X⎠

Figure 6. Distribution of concentrations of fragments A22(k) as a function of conversion ξE(t) where k is the number of E units per unit A22, Initial concentration of groups [A1]0 = [E]0 = 1, and κ2 = 0.3; dashed curve corresponds to ΣkkcA22(k).

The results were compared with those obtained by eq 14. For κ = 0.3, both methods gave the same results with an accuracy of ±0.1% given by the precision of numerical solution (tabular results are available in the Supporting Information). The existence of convolution can be proved analytically (Supporting Information, also cf. ref 38). Thus, for the random case the functionality and molecular weight distributions can be described by a fA-fold convolution of distributions for a single branch of the AF precursor (eq 16). However, the convolution theorem is not valid if there is a substitution effect in the AfA component. The case with substitution effect has been analyzed separately (Supporting Information) for the bifunctional core as an example. The distribution of the number of E units in two branches of A02 depends on the fraction of E monomer consumed in the buildup of single branch of A11, i.e., on what is left for the formation of the second branch of A02. The case of substitution effect can be solved by the method of transformation of sets of kinetic differential equations into partial differential equations for the generating function (cf. Supporting Information). 2.2.4. Distributions of States of the Polyfunctional Precursor and Molecular Weight Averages. The degrees of polymerization, molecular weights and number and type of functional groups are essential for processing of network buildup. Following the fully kinetic model of schemes of Figure 6 with the absence of substitution effect and eqs 11 and 12, the distributions can be described by the following number-fraction generating function

(13)

where ξA1E is the conversion of A1 groups in transforming them into A2 groups. (2) The number-average degree of polymerization of E sequences per A4 unit increases linearly with the number of reacted A1 groups. This means that the numberaverage number of E units per one E sequence is the same as for a single E sequence. This implies that the degrees of polymerization of E chains are also independent of each other. This is, however, not a sufficient proof; one has to analyze higher averages of the polymerization degree. This question is answered in entry 3. (3) The distribution of E units linked to A units having k E branches (and, therefore, have k reacted groups A1), C(Z), can be described by a distribution function which is obtained by k-fold convolution of distribution function, N(Z) for reaction of a single A1 group, i.e.

fA

C(Z) = (N (Z))k

NAf (Z) = ZAXA ∑ nA(x , f

38

It has been shown elsewhere that the distribution C(Z) must be narrower than N(Z). Equation 9 of ref 38 offers the relation between the nonuniformity for the degree of polymerization of all E units linked to an A unit having k E branches, D = Pw/Pn, as a function of the nonuniformity D0 = P0w/P0n of the primary distribution N(Z) reads D +k−1 D = 0 D0 kD0

x=0

A

x (fA − x) NE, x(Z EXE) − x)ZA1ZA 2

(15)

The variables of NAf(Z), ZA,ZA1,ZA2, refer to units A, functional groups A1, and functional groups A2, respectively; the variable ZE is associated with E units. X A and XE characterize certain properties of units A and E, respectively; XA = 1, XE = 1 gives the degree of polymerization distribution, XA = MA, XE = ME the molecular weight distribution (MA and ME are molecular weights of A and E units, respectively). The E function NE,x(ZM E )is a pgf describing the number-fraction distribution of molecular weights of E branches in a molecule having x E branches. The fraction nA(x,fA−x) is the normalized

(14)

In the next step, the distributions for the number of E units bound to various states of A4 units, i.e., to A22, A13, A04, was 2771


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Figure 7. Weight-average molecular weight, Mw, and nonuniformity, Mw/Mn, of A4-E adducts as a function of conversion of E groups for initial molar ratio [A1]:[E] = 1; Mn is independent of κ2; Mw - for κ2 = 10, 3, 1, 0.3, 0.1, 0.03 from top to the bottom; MA = 136, ME = 158 g/mol.

Figure 8. Dependence of the nonuniformity of A4−E adducts for (a) several initial molar ratios of A1 to E groups, rE indicated, and (b) several functionalities of the precursor; κ2 = 0.3. Other parameters are the same as for Figure 7.

concentration of A units having x A1 groups in the unreacted state and fA − x groups in the reacted state. The precursor has, therefore, fA − x groups A2. The pgf NAf(Z) (eq 13) can be formulated as a fA-fold convolution NAf (Z) = ZAMA((1 − ξA1E)ZA1 + ξA1EZA2NE(Z EME)) fA

′ (1) (MAf )w = W Af 0 = [MA 2 + 2fA MA MEξA1EPEn 0 + fA (fA − 1)(MEPEn ξA1E)2 0 0 0 + fA ME 2ξA1EPEn PEw ]/[MA + fA ξA1EMEPEn ]

(16)

(19)

The molecular weight averages are obtained from NAf(Z)

P0Ew

where = Pw of eq 7 Figures 7 and 8 illustrate the dependences of Mw, and nonuniformity Mw/ Mn on the reactivity ratio, κ2, molar ratio of E to A groups, rE, and functionality of the precursor, fA. The number-average molecular weight is independent of κ2 and increases linearly with ξE. The nonuniformity, Mw/Mn, passes through a maximum. This is because the A1 groups react gradually and there exist certain occupancy of A by E at which the nonuniformity is maximal (Figure 8). With decreasing κ2, the nonuniformity decreases because the A1-E reactions are preferred compared to A2-E reactions. Very interesting is the finding (Figure 8) that at higher conversions of E groups the nonuniformity decreases with increasing precursor functionality, fA. This is in agreement with the conclusion already made above the n-fold convolution of any degree-of-polymerization distribution: it is narrower that the original distribution. This is opposite to the effect of increasing functionality on random branching. 2.3. Cross-Linking of the fA-Functional Precursor A with f B-Functional Component B. Next, the fA-functional precursor is cross-linked with an f B-functional component B (cross-linker). Let us consider the same and independent

⎛ ∂N′ (Z) ⎞ ⎛ ∂N′ (Z) ⎞ + ⎜ Af (MAf )n = ⎜ Af ⎟ ⎟ ⎝ ∂ZA ⎠Z = 1 ⎝ ∂Z E ⎠Z = 1 0 = MA + fA ξA1EMEPEn

(17)

P0En

where is the number-average degree of polymerization of an E unit sequence per reacted A1 group (eqs 5 and 5a, P0En = Pn). The weight-average molecular weight is obtained from the weight-fraction generating function, WAf(Z), ⎛ ∂N (Z) ⎞ ⎛ ∂N (Z) ⎞ ≡ ZN ′(Z)/N ′(1) W (Z ) = Z ⎜ ⎟/⎜ ⎟ ⎝ ∂Z ⎠ ⎝ ∂Z ⎠Z = 1

WAf (Z) =

MA ZAMAY fA + fA ZAMAZ EMEMENE′(Z EME)ξA1EY fA − 1 MA + fA ξA1EMENE′(1) (18) M ξA1ENE(ZE E))

where Y = (1 − ξA1E + The weight-average molecular weight is obtained by differentiation of WAf(Z). 2772


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equality of the determinant D is zero (this equality also determines the divergence of Mw)

reactivity of all B groups of component B. The distribution of precursor units with respect to their molecular weights and numbers of issuing bonds is described by the probability generating function (pgf) F0n(z,Z), where the auxiliary variables Z refer to contribution by units to the molecular weight and the variables z to bonds to other units. Thus F0n(z , Z) = nA FA0n(zA , ZA) + nBFB0n(z B, Z B)

D=

(20)

where nA and nB are, respectively, are molar fractions of all A units and B units (nA + nB = 1). The vectors of auxiliary variables have the following composition z = (zA , z B), ZA = ZA , Z E ;

BA1 −FA2B

BA2 −FBA1

BA2 1 − FBA2

BA2 −FA1B

BA2 −FA2B

A1B −FBA1

A1B −FBA2

A1B 1 − FA1B

A1B −FA2B

A2B −FBA1

A2B −FBA2

A2B −FA1B

A2B 1 − FA2B

=0

⎛ ∂F (z , Z = 1) ⎞ XY FAB = ⎜ AB ⎟ ∂z XY ⎠z = 1 ⎝

z B = (z BA1 , z BA2)

ZB = ZB

Because several members of the determinant (23) are either zero or equal to one another, the solution of D is relatively simple BA1 A2B A1B D = 1 − FA2B (FBA1 + FBA1 )=0

(24)

and the gel point condition reads (fA − 1)(fB − 1)αB[(1 − ξA1E)αA1 + ξA1EαA2] = 1

(25)

The sum [(1 − ξA1E)αA1 + ξA1EαA2] is equal to the average number of bonds A→B which must be equal to the number of B→A bonds, so that the gel point eq 25 simplifies to

ZAMA((1

FB0n(z , Z B) =

BA1 −FA1B

The meaning of the values of derivatives is as follows

The subscripts at z mean the type and direction of bond, e.g., zA1B is associated with the bond extending from unit A via reacted group A1 to reacted group B of unit B. This statistical treatment is based on the first-order Markovian statistics; it is related to the concentration of dyads A1B and A2B. More elaborated treatment, needed for the unequal reactivity/ substitution effects case would be based on triads: A1B1A1, A1B2A1, A1B1A2, A1B2A2, A2B1A2, A2B2A2, etc. Thus FA0n(z , Z) =

BA1 −FBA2

(23)

Z = (ZA , Z B)

zA = (zA1B , zA2B),

BA1 1 − FBA1

− ξA1E)(1 − αA1 + αA1zA1B) f + ξA1E(1 − αA2 + αA2zA2B)NE(Z EME)) A

(fA − 1)(fB − 1)αB 2rB = 1,

Z BMB[1

− αB + αB(ψBA1z BA1 + (1 − ψBA1)z BA2)] fB

rB = [B]0 /[A]0

(26)

which is the same as that for the A + B system composed of A units having A groups of the same reactivity. This surprising result is a consequence of the random (binomial) distribution of the fractions of precursor molecules having specified numbers of groups A1 and A2. Unlike the critical conversions, the critical reaction times depend on differences of reactivities A1 and A2 groups with the B groups. When the binomial distribution for the numbers of groups A1 and A2 in the precursor A does not hold, the gel point and other network parameters do depend on the reactivity ratio τ2 (eq 27 and fractions of groups nA1 and nA2 =1 − nA1. Fixed numbers of A1 and A2 groups fA1 and fA2, and fraction of groups nA1 and nA2 =1 − nA1 can serve as an example (cf. Appendix). For expressing the gel point in terms of conversions of A1 or A2 groups, one uses the interrelation obtained from reaction kinetics

ψBA1 = ξA1EαA1/(ξA1EαA1 + (1 − ξA1E)αA2) (21)

where αA1 and αA2 are, respectively, conversions of groups A1 and A2 with the cross-linker B, and αB is the conversion of E groups B with A1 and A2 groups. The distribution NE(ZM E ) is ME equal to the distribution function (4) substituting ZE for Z. The simple form of the transition probability ψBA1 for the kinetically determined bond formation arises from the fact that ξA1E is a constant for a given cross-linking system and independent (not equal!) reactivity is assigned to the precursor and cross-linker groups. The distribution of additional number of bonds issuing from a unit already connected by one bond, F, is obtained by differentiation of F0n(z,Z)with respect to zA1B, zA2B, zBA1, zBA2, and normalization:

1 − αA2 = (1 − αA1)τ2 ; kA2B/kA1B = τ2

(27)

Examples of the effect of the reactivity ratios τ2 and κ2 are shown in Figure 9. 2.3.2. Pregel Region. The structure development in the pregel region is characterized by increasing molecular weight averages. The number-average molecular weight, Mn, is equal to the ratio of number-average weight of components per molecule. The number of molecules is given by the number of components minus number of bonds connecting them. For alternating reaction, the number of bonds is equal to

FBA1(z , Z) = ZAMA((1 − ξA1E)(1 − αA1 + αA1zA1B) f −1 + ξA1E(1 − αA2 + αA2zA2B)NE(Z EME)) A FBA2(z , Z) = FBA1(z , Z) FA1B(z , Z) = Z BMB[1 − αB + αB(ψBA1z BA1 + (1 − ψBA1) z BA2)] fB − 1 FA2B(z , Z) = FA1B(z , Z)

nA fA [(1 − ξA1E)αA1 + ξA1EαA2] = nBfB αB

(22)

2.3.1. Gel Point. The distribution of the number of bonds of the given types determines the gel point conversion irrespective of molecular weights of the components. For the gel point, it is necessary that the average number of additional bonds extending to other units reach 1. This condition is fulfilled by

Thus Mn = 2773

0 nA (MA + fA ξA1EMEPEn ) + nBMB

1 − nBfB αB

(28)


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The superscripts of WX, X = A, E, B, mean, respectively, the variable ZX with respect to which the differentiation has been performed. The derivatives of u are obtained by differentiating eqs 31 and solving the system of linear equations. Explicit results are given in the Supporting Information. 2.3.3. Postgel Stage. For the postgel state, the extinction probabilities, v = vBA1, vBA1, vA1B, vA2B, are the key quantities. They are the conditional probabilities that, given a bond exist, it has a finite continuation. Finite continuation means that the substructure extending from the bond is only finite. The extinction probabilities are obtained from pgfs F vBA1 = ((1 − ξA1E)(1 − αA1B + αA1BvA1B) + ξA1E(1 − αA2B + αA2BvA2B)) fA − 1 vBA1 = vBA1

Figure 9. Critical gel point conversions of functional groups A1 and A2 when reacting with cross-linker B: A1 groups, (αA1)crit (full curves), A2 groups (αA2)crit (dashed curves), and B groups, (αB)crit, (dotted line) on the ratio of rate constants for the reaction of A1 and A2 with B groups, τ2; the reactivity ratio κ2 indicated, initial ratio [A]0/[E]0 = 1, ξE = 0.99; fA = 4, f B = 3.

vA1B = [1 − αB + αB(ψBA1vBA1 + (1 − ψBA1)vBA2)] fB − 1 vA2B = vA1B (33)

by numerical solution. Simplification is possible because The weight-average molecular weight of the branching system is obtained from the weight-fraction generating function W(Z) W (Z) = wAWA(Z) + wBWB(Z),

1/(f − 1)

vBA1 =

vA1B B

− (1 − αB) αB

The extinction probabilities make possible a probabilistic characterization of sol and substructures of the gel: for instance, dangling chains, elastically active network chains, and elastically active junctions.1,2 The sol fraction, wsol, is composed of units participating in bonds exclusively with finite continuation. It is obtained from W(u(Z)), eqs 29 and 30 by substituting u = v while Z = 1:

Z = (ZA , Z E , Z B) (29)

where wA and wB are weight fractions of component A and B, respectively. The variables Z count the mass contribution of units in the molecules. The contributions by adjoining units are secured by the recursive variable u(Z) defined below. Thus, WA(Z) → WA(u(Z))WB(Z) → WB(u(Z))

wAsol = ((1 − ξA1E)(1 − αA1 + αA1vA1B) + ξA1E(1 − αA2 + αA2vA2B)) fA

and the explicit form of the components of W(Z) reads WA(u(Z)) = ZAMA((1 − ξAPE)(1 − αA1 + αAPuA1B) f + ξAPE(1 − αA2 + αA2uA2B)WE(Z EME)) A

wBsol = [1 − αB + αB(ψBA1vBA1 + (1 − ψBA1)vBA2)] fB = [1 − αB + αBvBA1] fB

WB(u(Z)) = Z BMB[1 − αB + αB(ψBA1uBA1 + (1 − ψBA1)uBA2)] fB

wsol = wAwAsol + wBwBsol (34) (30)

The elastically active network chains (EANC) are contributed by elastically active junctions (having 3 or more bonds with infinite continuation). Each such bond contributes by 1/2 to the number of EANCs. Thus, defining a pgf T(zinf), for the number of bonds with infinite continuation (corresponding variable zinf), one can count the contribution by components A and B (not distinguishing the bond types) as

WE(ZEME) describes the weight-fraction distribution of E sequences extending from reacted A1 groups. It is given by E eqs 5−7 when ZE is replaced by ZM E . The variables u of the recursive relations are obtained from the pgfs F (eq 22) uBA1 = ZAMA((1 − ξA1E)(1 − αA1B + αA1BuA1B) f −1 + ξA1E(1 − αA2B + αA2BuA2B)WE(Z EME)) A uBA2 = uBA1

fA orfB

T (z inf ) ≡

∑

i tiz inf = F0n(Z = 1 , z = v + (1 − v)z inf )

i=0

uA1B = Z BMB[1 − αB + αB(ψBA1uBA1 + (1 − ψBA1)uBA2)] fB − 1

Explicitly, it gives TA(z inf ) = [(1 − ξA1E)(1 − αA1 + αA1 (vA1B + (1 − vA1B)z inf ) + ξA1E (1 − αA2 + αA2(vA2B + (1 − vA2B)z inf ))] fA

uA2B = uA1B (31)

The weight-average molecular weight is obtained by differentiation of the pgf W(Z)

TB(z inf ) = [1 − αB + αB(vBA1 + (1 − vBA1)z inf )] fB

M w = W ′(Z = 1) ≡ W ′(1) = wAW A′ (1) + wBW B′ (1)

(35) (32)

and

where WA′ (1) = WAA(1) + WEA(1) + WBA′(1); WB′ (1) = WAB(1) + WEB(1) + WBB(1).

T (z inf ) = nA TA(z inf ) + nA TA(z inf ) 2774

(36)


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Figure 10. System A4 + B3: (a) number- (Mn) and weight- (Mw) average molecular weights and (b) concentration of EANCs as a function of conversion of B groups, αB; full curves Mw, dashed curve Mn, precursor prepared at molar ratio [E]0:[A1]0 = rE = 1, reactivity ratio for reactions of E groups with A2 relative to A1 groups, κ2 = 0.1, 0.3, and 10, respectively. Molecular weight of the components: MA = 136; ME = 158; MB = 524 g/mol.

Figure 11. Systems A4 + B3.A, with increasing number of E units characterized by rE = [E]0/[A1]0: (a) nonuniformity Mw /Mn; (b) concentration of EANCs as a function of conversion of B groups for systems of Figure 10; dashed curve for rE = 0.5.

precursor, fA, and the reactivity ratio for the reaction of A1 and A2 groups with the cross-linker groups B, τ2. High conversion of E groups, ξE = 0.99 and equimolar ratios of A to B groups are considered. Most interesting and important is the finding that all important network formation parameters - number- and weight-average molecular weights, critical gel point conversion, sol fraction, and concentration of elastically active network chains (EANC) as functions of the conversion αB do not depend on the difference in reactivity of A1 and A2 groups with B groups (parameter τ2) for any value of ξA1E. This is, however, valid if and only if the distribution of A units having various numbers of A1 (i) and A2 ( fA − i)) groups is random and given by the binomial expansion of ((1 − ξA1E)ZA1 + ξA1EZA2)fA (cf. eqs 13, 16, and 21). If the A unit has a certain fixed number of A1 and A2 groups, the dependences of structural parameters on conversion do depend on the reactivity ratio τ2! This is shown in the Appendix. In any case, the dependences on reaction time are functions of τ2. The reactivity ratio κ2 determines the numbers of A1 and A2 groups per precursor molecule and its molecular weight averages, and in this way it also affects the structure of reaction products with B (Figure 10). The number-average molecular weight is independent of reactivity ratio κ 2 . The dependences of M w and the concentration of EANCs on κ2 are not strong. The dependences for the weight fraction of gel (not displayed) are not identical but they differ only little (90−95%) of the signals in the examined region of molecular masses can be assigned to oligomers X−PIi−GLj (with i = j, as well as i ≠ j). This means that no other significant side reactions occur. (2) In addition to “stoichiometric” compounds i = j, compounds richer in i as well as j are detected of intensities not negligible compared to the signals of the diagonal “stoichiometric” ones. This is due to transesterification. (3) In oligomers, there is a slight excess of glycidyl (GL) over pivalate (PI) fragments. Alcoholysis by small amount of OH-containing impurities in GLPI is the reason. Visually, serious discrepancies between experimental and calculated abundances are sometimes observed and it is difficult to fine-tune the values of xTE and xH. However, one should realize that abundance 10, corresponds to the content of substance less than 1 mol %. The discrepancies are given by (a) the generally strong deviations from proportionality between abundances and molar fractions already discussed in section 3.1.2 and (b) the approximate form of incorporation of transesterification and alcoholysis effects used in eq 39. Despite of that it is possible, by generating distributions for various combinations of xTE and xH (more than those shown as examples in Table 1), to set by trial and error a guaranteed range of values of xTE and xH as follows: TME MPE DPE xTE 0.10 ± 0.05 0.15 ± 0.05 0.03 ± 0.01 xH 0.08 ± 0.03 0.10 ± 0.05 0.03 ± 0.01

(fA − 1)(fB − 1)αB[(1 − ξA1E)αA1 + ξA1EαA2] = 1

derived from FA0n(z,ZA) and FB0n(z,ZB) (eq 21). The letter B refers to the polyisocyanate component. To include transesterification and alcoholysis, one starts from distribution of numbers of bonds issuing from building units (eq 39) with the exception that now bonds issuing to polyisocyanate units are counted. Since we are interested in gelation, the variables counting mass contributions are dropped (ZGL = ZPI = ZB = 1): FA0n(z) = ((1 − ξA1E)βA1 + ξA1EβA2NETH(z)) fA FB0n(z) = [1 − αB + αB(ζBA1z BA1 + (1 − ζBA1)z BA2)] fB − 1 (40)

with βA1 = (1 − x TH)A1 + x TH ; A1 = 1 − αA1 + αA1zA1B;

βA2 = (1 − x TH)A 2 + x TH A 2 = 1 − αA2 + αA2zA2B;

X = (1 − x H)x TE + x H(1 − x TH) ∞

NETH(z) =

∑ ni[(1 − xH)(1 − xTE + xTEA1) i=1

+ x H((1 − x TH)A1 + x TH)]i ζBA1 =

((1 − ξA1E)(1 − x TH) + XξA1EPn)αA1 ((1 − ξA1E)(1 − x TH) + XξA1EPn)αA1 + ξA1E(1 − x TH)αA2

In these equations, the bond-related variables zA1B, zA2B, zBA1, zBA2 denote bonds A1 → B, A2 → B, B → A1, and B → A2, respectively, and A1 and A2 denote, respectively, primary and secondary OH groups. Using the same procedure as described before (eqs 22−24), one arrives at the gel point condition A1B A2B A2B A1B (fB − 1)(αB)g [ζ(FBA2 + FBA2 ) + (1 − ζ )(FBA1 + FBA1 )]

=1

In the selection process, the highest weight was put on comparison of sums of abundances of diagonal (i = j) and offdiagonal compounds (i > j) and (i < j). Without any doubt, for DPE, xTE and xH are smaller than for TMP and MPE.

(41)

containing values of derivatives of the generating functions FBA2,FBA1 obtained by differentiation of the basic functionFA0n(z). 2779


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Table 2. Comparison of Extrapolated Values of Critical Conversion of NCO Groups (Figure 1) with Values Calculated Using Equation A-1 for Various Precursorsa NPG (αB)g experimental extrapolated (αB)g calculated (eq 41) xTE xH

TME

0.530 0.660 0 0

0.520 0.10 0.10

0.379 0.529 0.15 0.05

0.504 0.15 0.10

0.466 0 0

0.390 0.10 0.08

MPE (αB)g experimental extrapolated (αB)g calculated (eq 41) xTE xH

0.335 0.10 0.10

0.379 0.13 0.10

0.276 0.05 0.03

0.262 0.10 0.05

DPE

0.320 0.381 0 0

0.385 0.10 0.10

0.276 0.320 0.15 0.06

0.310 0.15 0.10

0.295 0 0

0.278 0.03 0.03

a

The experimental system consisted of polyolGLPI precursor (initial molar ratio [OH]0/[E]0 = 1, conversion of [E] > 0.96), cross-linked with trimer of HDI, and linear extrapolation shown in Figure 2. For calculation, the following values were used: ratio of rate constants for reaction of secondary and primary OH groups with GLPI, κ2 = 0.3 (giving ξA1E = 0.78, Pn = 1.25, Pw = 1.45), ratio of rate constants for DBTD catalyzed reaction of aliphatic NCO groups (trimer with HDI) with secondary and primary OH group, and τ2 = 0.25.45,46 Various extents of transesterification and alcoholysis were used for calculation of (αB)g: boldface, values corresponding to experimental gel points; boldface italics, optimum values from MS (see section 3.1.3).

Figure 14. (a−d) Dependence of the gel-point conversions of isocyanate groups (component B) for cross-linking of various precursors on reciprocal concentration of functional groups (NCO+OH) (1/c0 [g/mol]). Dilution with diglyme, 25 °C. A1B = αA1(fA − 1)((1 − ξA1E)(1 − x TH) + ξA1EXPn) FBA1

+

A2B = (fA − 1)αA2ξA1E(1 − x TH) FBA1

αA1ξA1EX2Pn(Pw − 1) (1 − ξA1E)(1 − x TH) + ξA1EXPn

+

2780

ξA1EX(1 − x TH)αA2Pn ((1 − ξA1E)(1 − x TH) + ξA1EXPn) dx.doi.org/10.1021/ma302396u | Macromolecules 2013, 46, 2767−2784

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A1B FBA2 = (fA − 1)αA1((1 − ξA1E)(1 − x TH) + ξA1EXPn)

+ αA1XPn A2B FBA2 = (fA − 1)αA2ξA1E(1 − x TH)

A more detailed derivation of eq 41 is available in the Supporting Information. The gel point conversion also depends on cyclization which lowers the branching efficiency and shifts the gel point to higher conversions. Cyclization is not accounted for in the branching theory used here. Modeling of cyclization is difficult and unreliable as discussed in ref 32. The ring-free value of critical conversion can be accessed through extrapolation of the dependence of the critical conversion on the reciprocal concentration of NCO + OH groups (c0−1) to c0−1 = 0. The concentration dependences of the gel point conversions for the present systems and their extrapolation are shown in Figure 16. In Table 2, the extrapolated values to 1/c0 → 0 (ring-free) are compared with values obtained by simulation using eq 41 for several extents of transesterification and alcoholysis corresponding to ranges obtained from analysis of MS. This table also contains extrapolated (αB)g data calculated for the case when transesterification and alcoholysis are absent. Except of DPE, their effect is noticeable (20−30 rel.%). The values of xTE and xH calculated from extrapolated(αB)g (boldface) should be compared with values of xTE and xH estimated from MS (section 3.1.3) (boldface italics). The difference is well within experimental error of determination of xTE and xH from MS and from (αB)g. As shown in section 2.2.3, the gel point conversions of B (NCO) groups and other structural parameters were independent of the difference of the reactivity of A (primary and secondary OH) groups (τ2) when the distribution for the precursor was based on convolution of distribution for one branch. Now, when xTE ≠ 0 and xH ≠ 0, this is no longer true. The dependence is the stronger, the lower is τ2. 3.2.2. Extent of Cyclization at the Gel Point. The dependences in Figure 14 allow us to characterize the extent of cyclization at the gel point and to compare it with other systems cross-linked with the same polyisocyanate. Usually the extent of cyclization is characterized by the slopes of the dependences of critical conversion on 1/c0 or by fraction of formed bonds wasted in closing cycles, s, s=

Figure 15. Fraction of bonds wasted in cycles at the gel point for NPG, TME, MPE, DPE based precursors cross-linked with triisocyanate Desmodur N 3600 at 80% solids (calculated from dependences in Figure 14) compared with previously published32 data for cross-linking of polycaprolactone diol (PCLD), polycaprolactone triol (PCLT) and 4-functional polyester star (4f-star) with the same cross-linker recalculated to the same concentration of solids. The numbers at PCLT and PCLD are their nominal molecular weights.

The fraction of bonds engaged in cycles increases with increasing functionality of the precursor and it is higher than in the previous series. This is generally due to longer arms of precursors of the former series,32 because the probability of ring-closing decreases with the −3/2 power of the number of statistical units in the connecting paths of bonds.45,47 3.3. Concentration of Elastically Active Network Chains (EANC). The concentration of EANCs, νe, for the ring-free case (no elastically inactive cycle) can be calculated as a function of conversion as shown in section 2.2.3 with the difference that the extinction probabilities vA1B, vA2B, vBA1, vBA2 (eq 33 are the pgf T(z) (eqs 35, 36) are now derived from the pgfs F0An and F0Bn (eq 40). Here, we offer neither explicit formulas for these quantities nor examples of dependences of νe on conversion. However, we compare experimental equilibrium moduli of final samples for which the conversion of NCO groups has reached 100%. For such ideal networks prepared with stoichiometric amount of NCO and OH groups and not containing any elastically inactive cycles, the extinction probabilities approach zero and the respective pgfs get simpler as shown in the Supporting Information. The values of νe calculated in this way for xTE = xH = 0 and for the optimum values found from gelation experiments (boldface type data in Table 2) are displayed in Table 3. Application of MS optimum values (boldface italics in Table 2) changes the figures only little (not more than by 2%). Unlike the gel point conversions, the concentrations of EANCs are only little affected by the side reactions, except for the low functionality system (NPG), where the bifunctional precursor unit can be made higher-functional by alcoholysis and partly by transesterification. The general conclusion is that transesterification and alcoholysis affect the concentration of EANCs of final samples much less than they affect the gel point conversion. The experimental values of νe were calculated from the equilibrium small-strain tensile modulus of samples swollen in diglyme (Table 3) using the Flory−Erman junction-fluctuation rubber elasticity theory48 in the suppressed fluctuation limit, A

αg(c0) − αg,extrapol αg(c0)

(42)

Because the oligomeric precursors contain a certain amount of diluent, the reference concentration of groups c0 was set to 0.00445 mol/g which roughly corresponds to 80% solids (20% diluent). The experimental gel point conversions depend markedly on dilution which means that cyclization can be rated as moderate (10−25% bonds lost in cycles, s in Figure 15) ranking between weak of low functionality/long star arms systems, and high cha racteristic of f ree-radical cross-linking (c o)polymerization.2,45 The linear dependence of the 1/c0 plot is fairly well obeyed. The results displayed in Figure 15 are amended by data obtained earlier using larger polyols and the same polyisocyanate.32 The data of ref 32 have been recalculated for the same concentration of solids (80%). 2781


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Table 3. Calculated Concentrations of Elastically Active Network Chains (EANC), νe, For a Network Not Containing Any Elastically Inactive Cycles and 100% Conversion of Functional Groups in the Absence and Presence of Transesterification and Alcoholysis, And Comparison with Values Calculated Using Experimental Value of E and Eq 44 νe mol/cm3

a

code

fA

xTE = 0, xH = 0

xTE, xH gelation optimala

experimental ϕ02 = 0.8

νe/(νe)max

αEANC,inter

ϕ2 diglyme 25 °C

DPE MPE TME NPG

6 4 3 2

0.00326 0.00333 0.00327 0.00159

0.00338 0.00334 0.00302 0.00178

0.00280 0.00243 0.00113 0.00079

0.84 0.73 0.38 0.44

0.93 0.88 0.81 0.91

0.70 0.64 0.49 0.42

Boldface type data in Table 2

initiated by the reaction of E with A1 groups and propagation by reaction of E with the formed A2 groups was simulated kinetically. Depending on the ratio of reactivities of groups A2 and A1 toward E, κ2, the degree-of-polymerization distribution gets wider (starting from Poisson distribution at κ2 → 0) when κ2 increases over 1. A fully kinetic simulation for a fA-functional monomer was found to give the same distribution as the fA-fold convolution of distribution for one branch. For such random distribution of groups A1 and A2 in the precursor, the gel point conversion of B groups and other structural parameters (Mn, Mw, wg, νe) are independent of the difference in reactivities of groups A1 and A2 with the B group. To understand this result fully: although generally the gel point conversion and other parameters do depend on the numbers and reactivity differences of A2 groups compared to A1 groups, they get independent of this reactivity ratio, if the groups A1 and A2 are distributed in the precursor molecule randomly. We call this “randomization effect” imposed on the reactivity distribution. This theoretical treatment of a real systema set of star-like polyols chain extended with an epoxyester and then crosslinked with a triisocyanatewas performed to help understanding the relation between initial composition and network formation and properties. However, the analysis of the precursor composition has revealed that it is more complicated by occurring secondary reactionstransesterification and alcoholysis. Mass spectrometry and gelation experiments helped to characterize the intensity of these secondary reactions. The branching theory was modified in an approximate way to include the effect of these secondary reactions on network build-up. While the effect of the secondary reactions on gel point conversion is not small, the secondary reactions affect the cross-link density only little. Cyclization is also operative and the fractions of bonds wasted in cycles at the gel point amount to 12−22% and fit well the series of other cross-linking systems studied so far.31,32 The combined theoretical and experimental treatment presented here can serve as a paradigm for modeling of a real-life thermosetting systems where the chemistry is often more complicated than initially assumed and intended.

= 1. According to this theory, the tensile stress is related to macroscopic deformation relative to the isotropic state as σsw = f /Ssw = RTAνeϕ21/3(ϕ20)2/3 (Λ − Λ−2) = 3RTAνeϕ21/3(ϕ20)2/3 ε + ....

(43)

where f is the tensile force, Sw, is the cross-section area of swollen sample, Λ is the deformation ratio relative to the isotropic swollen state and ε = Λ − 1, ϕ02 is the volume fraction of polymer forming material during network formation. Thus, the small-strain Young modulus E reads E ≅ E′ = 3RTAνeϕ21/3(ϕ20)2/3

(44)

The calculated values of νe are displayed in Table 3. The small difference between concentrations of EANCs for tri-, tetra-, and hexafunctional precursors is due to the fact that the weight equivalents per OH group are very similar. The drop of νe for NPG is caused by its bifunctionality. Experimentally by FTIR, the final conversion of NCO groups was >97%. The values calculated for near to 100% conversion of functional groups, (νe)max, can be considered as the maximum attainable values for system of the given functionality if formation of elastically inactive cycles could be prevented. The experimental concentration of EANCs and the conversion of groups into bonds closing elastically active circuits αEANC,inter calculated from it are lower due to the formation of elastically inactive loops. One could conclude that the relative decrease in modulus and νe gets larger with decreasing functionality of the precursor due to larger fraction of bonds wasted in elastically inactive cycles. However, such conclusion would be incorrect. The narrow range of conversions available for the network build-up in systems of low functionality is the reason: the same amount of defects gives larger relative decrease of νe.

4. CONCLUSIONS Branching, gelation, and network evolution are theoretically described for a system obtained by one of the important ways of formation of polymer networks from preformed precursors. The precursors are formed by chain extension of core molecules AfA and are characteristic of distributions of molecular weights and numbers of groups differing in reactivity. The chain extending molecule E is a crypto-bifunctional molecule which upon reacting with core groups A1 generates linear oligomeric branches terminated by a new reactive group A2 capable of reaction with E. This results in a distribution in numbers of reactive groups and molecular weights. The control by chemical kinetics is operative. One of the important objectives of this study was to find when and how the statistical way of generation of structures based on first order Markovian statistics could be used. The growth of E chains

■

APPENDIX

Cross-Linking of Precursor Molecules Having Exactly fA1 Groups A1 and fA2 Groups A2: fA1 + fA2 = fA

Assumption: All A1 groups and A2 groups, respectively, are of the same and independent reactivity, all B groups are of the same reactivity; ring-free case. The reactivity difference between A1 and A2 groups with B groups is given by the reactivity ratio τ2 (eq 27). The distribution of reaction states now reads (cf. eqs 21 and 22) as follows: 2782


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Figure 16. (a) Molar conversion of functional groups A1, A2, and B at the gel point for a precursor having fA1 = 2 A1 groups and fA2 = 2 A2 groups, and f B = 3 B groups, in dependence on the reactivity ratio τ2. (b) Detail of variation of critical conversion of B groups for the same case, compared with the constant value 0.408 (horizontal line) for random (binomial) distribution of groups A1 and A2

■

FA0n(z , ZA) = ZAMA(1 − αA1 + αA1zA1B) fA1 × (1 − αA2 + αA2zA2B) fA2 FB0n(z , Z B) =

FBA1(z , Z) =

§

Polymer Institute, Slovak Academy of Sciences, Dúbravska cesta, Bratislava, Slovakia.

Z BMB[1

− αB + αB(ψBA1z BA1 + (1 − ψBA1)z BA2)] fB

ψBA1 = fA1 αA1/(fA1 αA1 + fA2 αA2)

ZAMA(1

Notes

The authors declare no competing financial interest.

■

(A-1)

ACKNOWLEDGMENTS K.D. and M.D.-S. acknowledge participation in European Commission FP7 PITN-GA-2009-238700 NANOPOLY project.

fA1 − 1

− αA1 + αA1zA1B) × (1 − αA2 + αA2zA2B) fA2

■

FBA2(z , Z) = ZAMA(1 − αA1 + αA1zA1B) fA1 × (1 − αA2 + αA2zA2B) fA2 − 1

LIST OF MOST FREQUENTLY USED SYMBOLS AND ABBREVIATIONS c concentration C(Z) generating function of variable Z D Pw/Pn E Young modulus [E] concentration of chain extender [Ei] concentration of extender chain composed of iE units fX functionality of component X F0(Z,z) probability generating function for the properties of units (Z) and the number of issuing bonds (z) FXY(z) probability generating function for the properties of units (Z) and the number of additional bonds (z), the unit is already bonded by one of its bonds G shear modulus GL glycidyl fragment GLPI glycidyl pivalate k1, k2 rate constants for reaction of group A1 and A2, respectively, with E group kA1B, kA2B rate constants for reaction of group A1 and A2, respectively, with B group mX mass (weight) fraction of component X Mn, Mw number- and weight-average molecular weight nX molar fraction of component X N(Z) number fraction generating function Ne number of elastically active network chains (EANC) Pn, Pw number- and weight-average degree of polymerization PI pivalate fragment

FA1B(z , Z) = Z BMB[1 − αB + αB(ψBA1z BA1 + (1 − ψBA1)z BA2)] fB − 1 FA2B(z , Z) = FA1B(z , Z)

(A-2)

Proceeding as before, i.e., finding the values of derivatives of components of the vector F (eq A-2) and finding the value of the determinant D formulated analogously to eq 23, we get the gel-point condition in the following form (fB − 1)αB[fA1 αA1 + fA2 αA2 − ψBA1αA1 − ψBA2αA2] = 1 (A-3)

which, for the equal reactivity case (τ2 = 1, αA2 = αA1 = αA), transforms into (fB − 1)αB(fA1 + fA2 − 1)αA = 1

Figure 16 shows the dependence of the critical value of the conversion of B groups, (αB)crit on τ2. It is seen that (αB)crit depends now on the reactivity ratio τ2 . The limiting value of (αB)crit for τ2 = 0 is 1/2 and the dependence passes through a shallow minimum at τ2 = 1 (equal reactivity system), at which it is equal to the value for the random (binomial) distribution case of eq 21 independent of the value of τ2 (cf., also ref 35).

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AUTHOR INFORMATION

Present Address

ASSOCIATED CONTENT

S Supporting Information *

Additional content for sections 2.2.3, 2.3.2, 2.3.3, 3, 3.1.1, 3.1.2, 3.2.1, 3.3). This information is available free of charge via the Internet at http://pubs.acs.org/. 2783


Macromolecules rA, rE t T(zinf) xTE, xTH, xH z, Z αX ϕ2 ϕ02 κ2 Λ νe ρ τ2 ξE ξEA1

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Article

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initial ratio of functional groups, [A]0/[B]0, [E]0/ [A1]0 time probability generating function for bonds with infinite continuation extents of transesterification and alcoholysis auxiliary variables of the generating functions conversion of groups X volume fraction of polymer in swollen network volume fraction of diluents at network formation ratio of rate constants k2/ k1 deformation ratio in tension concentration of elastically active network chains (EANC) specific gravity ratio of rate constants kA2B/ kA1B conversion of groups E conversion of groups A1 with groups E

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Polymer Networks from Preformed Precursors Having Molecular

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