4-Dimensional Modeling Strategy for an Improved Understanding of

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4‑Dimensional Modeling Strategy for an Improved Understanding of Miniemulsion NMP of Acrylates Initiated by SG1-Macroinitiator Paul H. M. Van Steenberge,†,‡ Dagmar R. D’hooge,*,† Marie-Françoise Reyniers,† Guy B. Marin,† and Michael F. Cunningham*,‡ †

Laboratory for Chemical Technology (LCT) Ghent University, Technologiepark 914, B-9052 Gent, Belgium Department of Chemical Engineering, Queen’s University, Kingston, Ontario K7L 3N6, Canada

‡

S Supporting Information *

ABSTRACT: For the first time, a kinetic model considering fourdimensional Smith−Ewart equations is presented to simultaneously calculate the time evolution of the conversion, number-average chain length, dispersity, end-group functionality (EGF), and short chain branching (SCB) content for the miniemulsion NMP of n-butyl acrylate (nBuA), initiated by poly(nBuA)-(N-tert-butyl-N-(1-diethylphosphono2,2-dimethylpropyl) at 393 K ([nBuA]0:[poly(nBuA)-SG1]0 = 300). On the basis of literature kinetic and diffusion parameters, model analysis reveals that backbiting cannot be neglected for an accurate description of the NMP characteristics, despite the low number of SCBs formed per chain (ca. 2) and that the small loss of EGF at low conversions is mainly caused by chain transfer to monomer. SG1 partitioning (partitioning coefficient Γ = 50) between the organic and aqueous phase increases the dispersity and polymerization rate at low particle diameters (dp < ca. 50 nm) with a limited effect on the EGF profile. However, the extent of these increases is very sensitive to the Γ value, highlighting the relevance of its accurate experimental determination in future studies.

1. INTRODUCTION Controlled radical polymerization (CRP) has emerged as a promising technique for the synthesis of next-generation polymers1−5 due to excellent control over chain length, topology, composition and end-group functionality (EGF).6−10 Envisaged high-tech CRP architectures are well-defined block and star copolymers, which can be used for the improved production of for instance coatings, adhesives and dispersants.11 In contrast to conventional free radical polymerization (FRP), in CRP, a mediating agent temporarily deactivates macroradicals into dormant macrospecies, simultaneously incorporating EGF. For high deactivation rates, the polymer product mainly consists of these dormant macrospecies and, hence, unwanted dead macrospecies, formed by termination as in FRP, are of minor importance. In addition, for fast CRP initiation, the dormant macrospecies possess almost uniform chain length at least at sufficiently high monomer conversions, allowing the synthesis of low dispersity functionalized polymers. In the simplest form, as illustrated in Figure 1, CRP deactivation occurs via a nitroxide X, which is initially formed at elevated temperature upon activation of an alkoxyamine initiator R0X. The resulting polymerization is typically denoted as nitroxide-mediated polymerization (NMP)12 and allows in particular the synthesis of well-tailored (co)polymers, consisting of styrene and/or (meth)acrylate monomer units. Despite the well-known potential of NMP, application at industrial scale remains a key issue.13 One of the challenges prohibiting industrial realization is identifying robust experimental protocols © XXXX American Chemical Society

Figure 1. Principle of nitroxide-mediated polymerization (NMP); initially R0X = initiator (alkoxyamine) and M = monomer are present; P = dead polymer molecule; X = nitroxide (mediating agent); ka,da,p,t = rate coefficient for activation, deactivation, propagation and termination; Keq = activation/deactivation equilibrium coefficient (ratio of ka to kda) is very small, favoring the formation of dormant macrospecies (left part).

for the synthesis of well-defined, high average chain length (co)polymers with a high yield.14 Typically, a too slow NMP results and diffusional limitations disturb the regular CRP growth pattern at sufficiently high monomer conversion,14−17 as the composition of the polymer environment strongly varies along the polymerization. In addition, side reactions can interfere, leading to a loss of control over the polymer properties. For example, with styrene NMP, the control is diminished by thermal Received: August 25, 2014 Revised: September 30, 2014

A

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raising the polymerization temperature to 125 °C. Alternatively, these authors proposed that a NMP macroinitiator can be used instead of a low molar mass alkoxyamine, as such initiator minimizes the exit of NMP initiator radicals. In this work, a detailed deterministic kinetic modeling study is presented for the isothermal miniemulsion NMP of n-butyl acrylate (nBuA) with SG1 as nitroxide, using 4-dimensional Smith−Ewart equations to account for the potential compartmentalization of all radical species, i.e., NMP initiator and secondary, tertiary, and nitroxide radicals. Hence, the kinetic model allows explicit determination of whether the rate of radical reactions is influenced by the particle diameter. To limit the influence of exit/entry of oligomeric species, the NMP is initiated by poly(nBuA)-SG1. The kinetic model allows, for the first time, the simultaneous calculation of the evolution of the conversion, number-average chain length (xn), dispersity, EGF and short chain branching amount (SCB) with polymerization time. Literature data are used to account for nitroxide partitioning and to calculate the intrinsic reactivities while differentiating between secondary and tertiary macrospecies. It is shown that backbiting reactions and nitroxide partitioning cannot be neglected for an accurate simulation of the NMP characteristics. Moreover, simulations indicate that accurate nitroxide partitioning coefficients are crucial for understanding and improving the NMP minemulsion kinetics.

initiation and chain transfer reactions, at least at polymerization temperatures above 100 °C,14−19 whereas for the controlled NMP of methacrylates conditions have to be selected to avoid disproportionation reactions involving nitroxide moieties.20−23 Furthermore, in the NMP of acrylates, both secondary and tertiary macroradicals are present due to backbiting reactions.24,25 Short chain branches (SCBs) can thus be formed upon tertiary propagation (or recombination), changing the topology of the polymer chains from linear to branched. Importantly, the degree of branching can vary depending on the individual secondary and tertiary activation/deactivation rate coefficients. Hence, in general, a detailed reaction scheme is required to understand and improve the NMP of acrylates. A second challenge prohibiting industrial realization is the limited knowledge of the NMP kinetics in aqueous dispersed media, which are favored because of environmental constraints, a higher mixing efficiency and a better heat transfer capacity.26−28 Specifically for aqueous emulsion polymerization systems, in which the reaction loci are dispersed “nanoreactors”, it has been indicated that an improved control over the NMP process is possible. In contrast to bulk polymerization, for a sufficiently low particle diameter (dp < ca. 100 nm), radical segregation becomes kinetically relevant, allowing a significant suppression of termination reactions and thus tuning of the polymer properties.29−32 In the limit, a zero-one system could be obtained, implying that the number of each radical type is either zero or one. Unfortunately, for low dp values (dp < ca. 50 nm), the miniemulsion kinetics are complicated by exit/entry phenomena of low molar mass species,33,34 such as oligomeric radicals and nitroxides, and the so-called confined space or single molecule concentration effect,35−40 which typically relates to the higher rates of bimolecular reactions, such as deactivation, in smaller particles. For instance, for miniemulsion NMP of styrene generating the surfactant potassium oleate in situ (dp = 20 nm), Zetterlund39 studied nitroxide partitioning for three nitroxides of vastly different water solubility, i.e. 4-stearoyl-TEMPO (TEMPO: 2,2,6,6-tetramethylpiperidinyl-1-oxy), TEMPO, and 4-hydroxyTEMPO, at 130 °C and highlighted that partitioning can only be ignored for very hydrophobic nitroxides. In particular, for TEMPO, a partitioning coefficient Γ between the styrene and water phase of 98.8 was reported,33,41 implying a ca. 1% leakage of the mediating species from the polymer particles under equilibrium conditions. This partitioning was shown to be important for low dp values (dp < 40 nm) by means of simulations limited to low conversions (ca. 0.20).33 However, in general, limited data have been reported on Γ values for CRP mediating agents, complicating the complete understanding of CRP in miniemulsion. The relevance of the dp value to understand the competition between deactivation and nitroxide exit was also highlighted by Delaittre and Charleux42 for the synthesis of poly(acrylic acid)-bpolystyrene amphiphilic block copolymers. These authors indicated that TEMPO exit for active particles is only much faster than NMP deactivation if dp is higher than 20 nm, whereas for N-tert-butyl-N-(1-diethylphosphono-2,2-dimethylpropyl (SG1), which is more water-soluble, exit dominates by far even for active polymer particles with a dp of 10 nm. In addition, Farcet et al.43 focused on identifying reaction conditions achieving a high initiator efficiency for acrylate NMP. For the miniemulsion NMP of n-butyl acrylate (nBuA) using as NMP initiator N-tert-butyl-N-(1-diethylphosphono-2,2-dimethyl)propyl-O-1methoxycarbonylethyl hydroxylamine (MONAMS), this efficiency displays a value below one at 112 °C, which can be increased by

2. 4D-MODELING METHODOLOGY 2.1. Reaction Scheme and Rate Coefficients. The miniemulsion NMP of nBuA mediated by SG1 and initiated by poly(nBuA)-SG1 at 393 K is modeled using the reactions in Table 1. In this table, typical radical polymerization reactions, such as propagation, chain transfer to monomer and termination, and NMP specific activation/deactivation reactions are listed. Side reactions, such as chain transfer to polymer and β-scission, have been neglected based on respectively a maximum conversion of 0.744 and a relatively limited polymerization temperature of 393 K.45,46 For simplicity, thermal self-initiation47 has been neglected as well. Since secondary acrylate macroradicals may backbite to form tertiary macroradicals possessing a lower chemical reactivity, a distinction is made between secondary and tertiary macroradicals in Table 1. At the considered polymerization temperature, the tertiary propagation reactivity is ∼600 times lower than the secondary one and the tertiary equilibrium coefficient for NMP activation/deactivation is 5 orders of magnitude higher than the one for secondary species. Previous kinetic studies on acrylate CRP25,48 demonstrated that the tertiary activation/deactivation intrinsic kinetic parameters strongly influence the branching degree. High SCB amounts, as found in FRP, can only be obtained under CRP conditions when the tertiary activation rate is high and/or the tertiary deactivation rate is low, the latter corresponding with an increased lifetime of the tertiary radicals. It is further assumed that a macroinitiator molecule generates solely a secondary radical upon NMP activation. Also, its chain length is not explicitly taken into account for the calculation of the chain length characteristics, implying that the chain length relates only to the newly incorporated monomer units upon polymerization and that the simulated dispersity values must be seen as maximum values in particular at low conversions, as longer chains imply a higher uniformity in chain length. Furthermore, in Table 1, the rate coefficients for the FRP reactions at 393 K are taken from literature.49,50 For termination, a relatively low value of ca. 4 × 107 L mol−1 s−1 is used, which is B

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reactions as a function of the particle diameter, ultimately allowing the identification of the optimal particle diameter for given product specifications. In the kinetic model, the aforementioned rate coefficients can be used directly, unless the viscosity effect cannot be ignored, which is the case for termination reactions. Diffusional limitations on termination are accounted for via an average apparent termination rate coefficient:18

Table 1. Reactions and Rate Coefficients to Model the SG1-Mediated NMP of nBuA at 393 K in Miniemulsionc reaction NMP activation

equation

3.0 × 10−4

2

ka , s

3.0 × 10−4

2

ka , t

2.8 × 10−1

2

kda ,0, s

8.0 × 108

2

kda , s

8.0 × 108

2

kda , t

2.6 × 106

2

k p ,0, s

8.8 × 104

49

k trM ,0, s

1.3 × 101

49

k trM , s

1.3 × 101

49

k trM , t

1.5 × 10−3

49, 50

kp,t

8.8 × 104

49

kp,t

1.5 × 102

49

4.7 × 103

49

k tc ,00, ss

4.5 × 107

49

k tc ,0, ss

4.5 × 107

49

k tc , ss

4.5 × 107

49

k tc ,0, st

4.1 × 107

49

k tc , st

4.1 × 107

49

k tc , tt

3.8 × 107

49

R 0, sX ⎯⎯⎯⎯→ R 0, s + X

R i , tX ⎯→ ⎯ R i ,t + X R 0, s + X ⎯⎯⎯⎯⎯→ R 0, sX R i , s + X ⎯⎯⎯→ R i , sX

R i , t + X ⎯⎯⎯⎯→ R i , tX chain initiation chain transfer to monomera

R 0, s + M ⎯⎯⎯⎯→ R1, s R 0, s + M ⎯⎯⎯⎯⎯⎯⎯→ P0 + R 0, s R i , s + M ⎯⎯⎯⎯⎯→ Pi + R 0, s R i , t + M ⎯⎯⎯⎯⎯→ Pi + R 0, s

propagation

R i , s + M ⎯⎯→ R i + 1, s R i , t + M ⎯⎯→ R i + 1, s

backbiting termination by recombinationb

ref

ka ,0, s

R i , sX ⎯→ ⎯ R i ,s + X

NMP deactivation

k ((L mol1) s‑1)

kbb

R i , s ⎯→ ⎯ R i ,t R 0, s + R 0, s ⎯⎯⎯⎯⎯⎯⎯→ P0

R i , s + R 0, s ⎯⎯⎯⎯⎯→ Pi R i , s + R j , s ⎯⎯⎯⎯→ Pi + j R i , t + R 0, s ⎯⎯⎯⎯⎯⎯→ Pi + j R i , t + R j , s ⎯⎯⎯⎯→ Pi + j R i , t + R j , t ⎯⎯⎯⎯→ Pi + j

kt = kt 0 exp( −0.4404wpol − 6.362wpol 2 − 0.1704wpol 3) (1)

in which wpol is the polymer mass fraction. This correlation was originally developed for styrene radical polymerizations but can be used to a first approximation in the current work, after correcting the value at zero conversion, as explained above. Since the simulated conversion is at most 0.7, diffusional limitations on the NMP activation/deactivation process can be omitted in the simulations, at least to a first approximation. Literature data15,17,53 indicate that diffusional limitations on the activation/ deactivation process are only operative under very viscous conditions for relatively bulky mediating agents and high intrinsic deactivation reactivities (>107 L mol−1 s−1), justifying the previous assumption. Similarly, diffusional limitations on the other reaction steps can be neglected to a first approximation. 2.2. Exit and Entry Coefficients. It has been indicated in the literature that oligomeric and nitroxide species can partition over the organic and water phase during miniemulsion NMP,32,33,41,42 complicating the kinetic description because exit and entry rates have to be accounted for too. However, in the current work, a NMP macroinitiator is used, implying that the number of oligomeric chains is limited and thus only nitroxide entry and exit rate coefficients have to be considered, at least to a first approximation. In agreement with Bentein et al.,32 a linear and inverse quadratic relationship with respect to the polymer particle diameter are considered for respectively the entry and exit rate coefficient (Table 2). To a first approximation the SG1 diffusion coefficients can be assumed the same in both phases, taking into account the maximum conversion of 0.7 and the semiquantitative scope of this work. It should be stressed that the exit and entry rate coefficients cannot be varied independently, as they are interrelated by the partitioning coefficient Γ (Table 2), which is a thermodynamic quantity for a solute in a two-phase system. As indicated above, data on Γ are scarce and mostly relate to TEMPO derivatives. For example, for TEMPO a value of ca. 100 was reported with styrene as organic phase at 135 °C. In the current work, the latter value is initially lowered by a factor 2 (Γ = 50) to describe the higher water solubility of SG1 compared to TEMPO, as indicated by Zetterlund31 and ignoring a potential solubility effect in the organic phase by switching from styrene to nBuA. In an additional step, the sensitivity of the modeling results with respect to a variation in Γ is studied. 2.3. Smith−Ewart Equations for Polymer Particles and Moment Equations. As mentioned before, compartmentalization of nonabundant species, i.e., radicals, can alter the miniemulsion kinetics significantly compared to the bulk case. Hence, polymer particles must be distinguished at least based on the number of radicals they contain. On the other hand, for the more abundant species, uniformized concentrations are used in agreement with previous modeling studies.32,50,55 In the current work, initially no free nitroxide species are present in either the organic or water phase. For simplicity, the particle diameter dp is considered monodisperse with a constant particle volume vp. The corresponding constant total number of polymer particles

a

For simplicity, the radicals formed upon chain transfer to monomer reactions are grouped with the macroinitiator radicals. The value for tertiary species (ktrM,t) is calculated based on the value for secondary species (ktrM,s; ref 49) corrected with a factor taken from ref 50. bThe k values for termination reactions are corrected as a function of the polymer mass fraction/conversion to account for diffusional limitations (see Supporting Information). The subscript “0” pertains to a chain length of 0, excluding the monomer units of the macroinitiator. For termination involving a secondary and tertiary radical the geometric average is taken of the corresponding “homotermination” rate coefficients in ref 49. cR0,s denotes the poly(nBuA)-SG1 macroinitiator of which the chain length is not explicitly tracked; Ri,s/t = secondary/tertiary macroradical with chain length i; Pi = dead polymer chain with chain length i; X = SG1; M = monomer; reactions occur only in the organic phase. M: monomer.

more consistent with the high molar mass NMP initiator considered. As indicated in literature51,52 such approach is suited at least to a first approximation. The rate coefficients for the NMP (de)activation reactions in Table 1 are taken from the extensive review of Nicolas et al.,2 considering small molecules as model compounds. Such approach is justified as the current kinetic modeling study aims only at a qualitative description of the miniemulsion NMP of nBuA toward an improved understanding of the relevance and interplay of the polymerization C

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and dead macrospecies (τ0, ω0, μ0), related to these population balances are obtained by simultaneously integrating the corresponding continuity equations, as explained in Supporting Information. Additionally, this integration yields the average number of initiator radicals, secondary macroradicals, tertiary macroradicals and nitroxide radicals per polymer particle via:

(Np) is calculated based on the monomer density only. It should be stressed that the kinetic model can however be extended to allow for the average evolution of the particle size. For the calculation of the conversion, the fraction of SCBs with respect to the number of monomer units or polymer chains, and EGF as a function of polymerization time t, 4-dimensional Smith−Ewart equations are integrated, accounting for the potential compartmentalization of all radical types, i.e. a distinction is made between the potential compartmentalization of initiator radicals (R0), nitroxide radicals (X), secondary radicals (Rs), and tertiary radicals (Rt). This 4-dimensional modeling strategy can be seen as an extension of the model of Bentein et al.32 for miniemulsion NMP of styrene in which a 3-dimensional description was sufficient, due to the absence of tertiary radicals. Denoting the Avogadro constant as NA, the corresponding 4-dimensional Smith−Ewart equations, which allow description of the conservation of polymer particles Nk,l,m,n having k secondary macroradicals, l nitroxide radicals, m initiator radicals, and n tertiary macroradicals, are specified by the following set of population balances (k, l, m, n ≥ 0): dNk , l , m , n dt

n ̅ (R 0) =

m

k ,l ,m,n

n ̅ (R s ) =

∑

k

k ,l ,m,n

n ̅ (R t ) =

∑

n

k ,l ,m,n

n ̅ (X ) =

∑ k ,l ,m,n

l

Nk , l , m , n Np

Nk , l , m , n Np Nk , l , m , n Np

Nk , l , m , n Np

(3)

where Np = ∑k,l,m,nNk,l,m,n is the total number of polymer particles. For the calculation of the number-average chain length (xn) and dispersity as a function of polymerization time, higher order moment equations are simultaneously integrated with the Smith−Ewart equations and the continuity equations for the noncompartmentalized species, in agreement with earlier work of Bentein et al. for the miniemulsion NMP of styrene. For completeness, these moment equations have been included in Supporting Information. All differential equations are integrated numerically using the LSODA solver, i.e., Livermore solver for ordinary differential equations (ODE) with automatic switching for stiff and nonstiff ODEs. Importantly, in the present study, the radicals are not assumed to be in quasi-steady state a priori, highlighting the generic nature of the proposed 4-dimensional modeling strategy.

= (ka0, sNAvp)[R 0X ](Nk , l − 1, m − 1, n − Nk , l , m , n)

+ (ka , sNAvp)τ0(Nk − 1, l − 1, m , n − Nk , l , m , n) + (ka , tNAvp)ω0(Nk , l − 1, m , n − 1Nk , l , m , n) ⎛k ⎞ da0, s ⎟⎟((m + 1)(l + 1)Nk , l + 1, m + 1, n − mlNk , l , m , n) + ⎜⎜ ⎝ NAvp ⎠ ⎛k ⎞ da , s ⎟⎟((k + 1)(l + 1)Nk + 1, l + 1, m , n − klNk , l , m , n) + ⎜⎜ N ⎝ Avp ⎠ ⎛k ⎞ da , t ⎟⎟((n + 1)(l + 1)Nk , l + 1, m , n + 1 − nlNk , l , m , n) + ⎜⎜ N ⎝ Avp ⎠ ⎛ ktc 00, ss , app ⎞⎛ (m + 1)(m + 2) ⎞ ⎟⎟⎜ Nk , l , m + 2, n − (m − 1)mNk , l , m , n⎟ + ⎜⎜ ⎝ ⎠ N v 2 ⎝ A p ⎠

3. RESULTS AND DISCUSSION In this section, 4-dimensional Smith−Ewart equations are first applied to study the importance of nitroxide partitioning between the polymer particles and aqueous phase and the impact of backbiting reactions for the NMP of nBuA initiated by poly(nBuA)-SG1 at a typical particle diameter of 50 nm at which compartmentalization effects are relevant. Next, the effect of the particle diameter is included, aiming at the identification of an optimal particle diameter with respect to control over polymer properties and polymerization time. In all simulations, the initial molar ratio of NMP macroinitiator to monomer ([nBuA]0: [poly(nBuA)-SG1]0) is 300 and the polymerization temperature is 393 K. 3.1. Importance of Nitroxide Partitioning. For a particle diameter of 50 nm, Figure 2 shows a comparison between the simulation results obtained with the model parameters listed in Table 1 and 2 (green full lines; Γ = 50; reference case) and those in case nitroxide partitioning would be neglected (red dashed lines, Γ = ∞ instead 50). For the reference case, a conversion of 0.7 is reached in less than 4 h (Figure 1a) with a moderate control over chain length, as evidenced by the relatively high numberaverage chain length (xn) values throughout the NMP and the dispersity of ca. 1.6 at a conversion of 0.7 (Figure 2c). As will be illustrated further, the latter value is also influenced by the selected Γ value, highlighting the qualitative nature of the simulation results. In addition, it should be remembered that the dispersity values have not been corrected for the number of

⎛ ktc 0, ss , app ⎞ ⎟⎟((k + 1)(m + 1)Nk + 1, l , m + 1, n − kmNk , l , m , n) + ⎜⎜ ⎝ NAvp ⎠ ⎛ ktc , ss , app ⎞ ⎟⎟((k + 1)(k + 2)Nk + 2, l , m , n − (k − 1)kNk , l , m , n) + ⎜⎜ ⎝ NAvp ⎠ ⎛ ktc , tt , app ⎞ ⎟⎟((n + 1)(n + 2)Nk , l , m , n + 2 − (n − 1)nNk , l , m , n) + ⎜⎜ ⎝ NAvp ⎠ ⎛ ktc , st , app ⎞ ⎟⎟((k + 1)(n + 1)Nk + 1, l , m , n + 1 − knNk , l , m , n) + ⎜⎜ ⎝ NAvp ⎠ ⎛ ktc 0, st , app ⎞ ⎟⎟((m + 1)(n + 1)Nk , l , m + 1, n + 1 − mnNk , l , m , n) + ⎜⎜ ⎝ NAvp ⎠ + (ktrm , s)[M ]((k + 1)Nk + 1, l , m − 1, n − kNk , l , m , n) + (ktrm , t )[M ]((n + 1)Nk , l , m − 1, n + 1 − nNk , l , m , n) + (kp0, s)[M ]((m + 1)Nk − 1, l , m + 1, n − mNk , l , m , n) + (kp , t )[M ]((n + 1)Nk − 1, l , m , n + 1 − nNk , l , m , n) + (kentry , X )[X ]aq (Nk , l − 1, m , n − Nk , l , m , n) + (kexit , X )((l + 1)Nk , l + 1, m , n − lNk , l , m , n)

∑

(2)

The concentrations for the “noncompartmentalized” species, i.e., M, R0X, Xaq, and the secondary dormant, tertiary dormant D

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Table 2. Model Parameters to Calculate the Exit and Entry Rate Coefficient for X = SG1 from/into a Polymer Particle with Diameter dp in the NMP of nBuA at 393 K in Miniemulsion Initiated by Poly(nBuA)-SG1 Macroinitiator at 393 Kd coefficients diffusion nitroxide entry nitroxide exit

equation

ref

2.0 × 10−7 dm2 s−1 2.0 × 10−7 dm2 s−1 3.0 × 10−6 1.8 × 105 L mol−1 s−1

14 − 32 −

Cexit

1.1 × 10−7 8.8 × 100 s−1

− −

5.0 × 101

this work

kexit , X = CexitDX , org dp partitioningc

Γ≡ a

value at dp = 50 nm

DX,org DX,aqa Centryb kentry , X = CentryNADX , aqdp

[Xorg ] [Xaq]

=

−2

kentry , X kexit , XvpNA

≈

6Centry πCexit

b

For simplicity, taken equal as the one for the organic phase. Typical value as reported by Bentein et al.32 cFor TEMPO partitioning over styrene and water at 135 °C a value of 98.8 has been reported in refs 33 and 41. Since SG1 has a higher solubility in water compared to TEMPO, this value has been lowered with a factor 2, hence, a leakage of 2% to the aqueous phase is assumed under equilibrium conditions; this value thus fixes Cexit. d [Xorg/aq] = SG1 concentration in the organic/aqueous phase (mol L−1); Cexit/entry = proportionality constant for SG1 exit/entry from/into a polymer particle (−); DX,org/aq = SG1 diffusion coefficient in the organic/aqueous phase (dm2 s−1); NA = Avogadro constant (mol−1); vp = polymer particle volume (L).

Figure 2. Effect of nitroxide partitioning (green full lines: reference case with Γ = 50; red dashed lines: Γ = ∞) on (a) the conversion profile and (b−f) the main polymer properties (xn, dispersity, EGF, SCB) as a function of conversion for the NMP of nBuA in miniemulsion initiated by poly(nBuA)-SG1 macroinitiator ([nBuA)]0:[poly(nBuA)-SG1]0 = 300) at 393 K and for a dp of 50 nm. xn = number-average chain length; EGF = end-group functionality; SCB = short chain branch (e, per monomer unit; f, per chain); chain length characteristics calculated ignoring the monomer units in the macroinitiator part. Model parameters as given in Tables 1 and 2 (Γ excepted).

Despite the moderate control over chain length in both simulated cases, it follows from Figure 1d that a highly living polymer is synthesized (EGF > 0.95), allowing chain extension and indicating a limited importance of termination and chain transfer reactions, which lead to the formation of dead polymer chains. This low contribution of termination reactions implies a weak persistent radical effect, which is in agreement with an earlier statement of Nicolas et al.56 As illustrated in Supporting Information (Figures S1 and S2), most termination reactions occur between tertiary species because the lower termination reactivity of tertiary macroradicals (Table 1) is overcompensated by the dominant presence of these tertiary radicals. Indeed, as shown in Figure 3f, from low conversions onward the average fraction of tertiary macroradicals is already above 0.80, while at high conversions even a value close to 0.95 is observed. Further inspection of Figure 2d reveals that the EGF profile under reference conditions (green full line) displays a steep initial decrease of almost 2%, after which a steady decrease is observed. As shown in Supporting Information, this decrease is caused by the increased importance of chain transfer to monomer (Figure S3; Γ = 50 vs Γ = ∞) and to a minor degree of termination reactions

monomer units in the macroinitiator part, and, hence, in reality lower values can be expected, in particular at lower conversions. Since the NMP is relatively fast (Figure 2a), the NMP initiator consumption is slow on a conversion basis (Figure 3e), leading even to incomplete NMP initiation at a conversion of 0.65 and thus a strongly nonuniform incorporation of the monomer units into the dormant species, explaining the relatively low control over chain length in Figure 2b−c. Note that in case nitroxides would not be able to exit (Γ = ∞), the relative importance of the deactivation rate increases leading to a slower NMP (red dashed line in Figure 2a) and thus to a faster NMP initiation on a conversion basis (red dashed line in Figure 3e). Consequently, an increase of the control over chain length is obtained for Γ = ∞. In particular, at a conversion of 0.70 a lower dispersity of 1.3 results (red dashed line in Figure 2c). For completeness, it is mentioned here that at higher conversions it can be expected that the difference in polymerization rate between both cases becomes less pronounced, as in reality the nitroxide exit rate is influenced by the viscosity increase in the polymer particles,32 an effect which is ignored in the current kinetic modeling study, as mentioned above. E

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Figure 3. Effect of nitroxide partitioning (green line: reference case with Γ = 50; red line: Γ = ∞) on the on the average number of (a) initiator radicals, (b) secondary macroradicals, (c) tertiary macroradicals, and (d) nitroxide radicals per polymer particle (eq 2) as a function of conversion for the NMP of nBuA in miniemulsion initiated by poly(nBuA-SG1) macroinitiator ([nBuA]0:[poly(nBuA)-SG1]0 = 300) at 393 K and for a dp of 50 nm; (e) NMP initiator concentration; (f) fraction of macroradicals being tertiary as a function of conversion. Model parameters as given in Table 1 and 2 (Γ excepted).

(Figure S3; Γ = 50 vs Γ = ∞). It should however be stressed that this observation is a direct consequence of the absolute values of the rate coefficient of the NMP system under investigation and cannot by directly generalized. When Γ = 50, nitroxide radicals can exit the polymer particles and thus deactivation becomes less important compared to the case where Γ = ∞. The previous observation is corroborated by the initial steep increase of the number-average chain length (green full line in Figure 2c) and the higher simulated initial EGF values in the case where nitroxide partitioning is neglected (dashed red line in Figure 2d). Moreover, as illustrated in Supporting Information (Figure S4 and S5), the dominant contribution of chain transfer to monomer for dead polymer formation would be obtained even in the case when a ten time times higher termination reactivity is assumed. Interestingly, the number of SCBs per monomer unit (Figure 2e) agrees well with values of ca. 1% reported in literature for homogeneous systems,56 highlighting the physical relevance of the simulation results. On average, only two SCBs per polymer chain are formed at a conversion of 0.7 (Figure 1f), which implies the presence of one quaternary C atom per ca. 100 monomer units, taking into account that the number-average chain length is 210 (Figure 1b). Hence, the simulation results show that, despite the high tertiary activation/deactivation equilibrium coefficient (Table 1), the amount of SCBs is still low, supporting the statement that under well-chosen CRP conditions SCB formation is suppressed.24 Parts a−d of Figure 3 show the corresponding average number of radical types (eq 2) per polymer particle. Again a distinction is made between the case in which nitroxide partitioning is accounted for (full green lines; Γ = 50) and neglected (dashed red lines: Γ = ∞). Clearly, all these average numbers are below one with the average number of initiator radicals (Figure 3a) being the lowest. For Γ = 50, the latter number steadily decreases in line with the earlier-mentioned near-complete initiation at a conversion of 0.65. Similarly, for Γ = ∞, a decrease is observed but only until a lower conversion of 0.35, in agreement with the NMP initiator concentration profile in Figure 3e (dashed red lines). Note that about 1 out of 106 polymer particles contain an −6 NMP initiator radical as n(R corresponds to ̅ 0) ≈ 10 , which 12 ca. 10 polymer particles per liter (Np ≈ 1018 L−1), highlighting that such low fraction of 10−6 is still relevant on an overall basis.

Figure 3b shows that the average number of secondary radicals also decreases in both cases, while the average number of tertiary radicals keeps accumulating (Figure 3c), due to their lower tertiary propagation reactivity (see Table 1) and the increased importance of unimolecular backbiting reactions at lower monomer concentrations. Furthermore, parts c and d of Figure 3 confirm the lower number of macroradicals in the case where nitroxide exit is neglected. On the other hand, irrespective of the Γ value, the fraction of tertiary macrospecies (Figure 2f) remains around 0.80, in agreement with the theoretical work of Ahmad et al.24 in which a balance between the tertiary propagation and backbiting rate under controlled CRP conditions was derived (see Figure S2 in Supporting Information). It can be seen in Figure 3d that the average number of nitroxides is the highest among all the radical types and increases steadily as a function of conversion. As a consequence of termination reactions at very low conversions, an excess of nitroxide is created but part of those species diffuse to the aqueous phase, keeping the nitroxide concentration of most polymer particles particles near zero. As shown in Figure 4a, the ratio of the average nitroxide concentration per polymer particle to the nitroxide

Figure 4. (a) Ratio of the average concentration of nitroxide in the polymer particles to the concentration of nitroxides in the aqueous phase and (b) the marginal nitroxide distribution for the NMP of nBuA in miniemulsion initiated by poly(nBuA-SG1) macroinitiator ([nBuA]0:[poly(nBuA)-SG1]0 = 300) at 393 K and for a dp of 50 nm. Model parameters as given in Tables 1 and 2 (Γ = 50). F

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Figure 5. Effect of backbiting rate coefficient kbb (25% (dotted blue lines), 50% (dashed red lines), 100% (full green lines) of the value specified in Table 2 on the main polymer properties as a function of conversion for the NMP of nBuA in miniemulsion initiated by poly(nBuA)-SG1 macroinitiator ([nBuA]0:[poly(nBuA)-SG1]0 = 300) at 393 K and for a dp of 50 nm. Model parameters as given in Table 1 and 2 (kbb excepted); xn = number-average chain length; EGF = end-group functionality; SCB = short chain branch; average chain length characteristics calculated ignoring the monomer units in the macroinitiator part.

acrylate radical polymerization,24,25 less backbiting implies a lower importance of tertiary macroradicals (Figure 6, parts d and f) and thus less retardation by tertiary propagation, explaining the faster polymerization (Figure 5a). However, a faster NMP implies a slower NMP initiation on a conversion basis (Figure 6e), which as explained above, leads to higher dispersity values, as can be derived from Figure 5c. Moreover, Figure 5b shows that the number-average chain length values are well above the target values (300 multiplied by the conversion), since for low backbiting rate coefficients NMP initiation is not completed even at this high conversion (Figure 6e). The lower EGF values for a lower backbiting rate coefficient (Figure 5d) are a direct consequence of the increased importance of secondary macroradicals (Figure 6, parts b and f) and the large difference in the activation/deactivation reactivity of secondary and tertiary macrospecies (see Table 1). Despite the lower total average number of macroradicals (sum of average amounts of secondary and tertiary species in Figure 6, parts b and c), which is consistent with the lower average number of nitroxide species (Figure 6d) for a lower kbb value, secondary chain transfer and termination are promoted (Figure S6 in Supporting Information). The latter reactions are characterized by a higher reactivity, compensating the decreased average amount of radicals and thus explaining the increased formation of dead polymer molecules, as witnessed in Figure 6d. Hence, based on the differences between the simulated lines in Figure 5 and 6, it can be concluded that backbiting cannot be ignored for an accurate simulation of the CRP miniemulsion kinetics, despite the low number of SCBs formed. 3.3. Importance of the Particle Diameter and Partitioning Coefficient. As discussed above for a single dp of 50 nm, nitroxide partitioning greatly affects the NMP characteristics. Figure 7 shows that the previous observation can be generalized to experimentally accessible particle diameters ranging between 20 and 80 nm,40 focusing on a conversion of 0.7 and considering a broader range of Γ values (Γ = 50, 100, 200, 500, and 5000). For the reference case, as discussed above, a Γ value of 50 was selected based on an arbitrary factor of 2 compared to the Γ value of ca. 100 for the TEMPO nitroxide with styrene as organic phase. Hence, it is worthwhile to investigate the sensitivity of

concentration in the water phase indeed decreases at very low conversions until equilibrium is obtained, as evidenced by the constant value of 50, being the value for Γ (Table 2). However, the asymptotic value in Figure 4a at sufficiently high conversions does not imply equilibrium for nitroxide partitioning per polymer particle, as almost no polymer particles can possess a sufficiently high number of nitroxide species with respect to the Γ value, taking into account that the NMP is started in the absence of free nitroxide. As shown in Figure 4b (green circles) at a conversion 0.7 a distribution exists of polymer particles with a different number of nitroxide species, ranging from 0 to 4. Most polymer particles contain no mediating agent, which can be expected since an acrylate monomer is selected. On the other hand, a substantial amount of particles possess a small amount of nitroxide species, highlighting the non-“zero-one” nature of CRP miniemulsion systems. A correct simulation of polymer particles with for example four nitroxides is only possible if the simulations account for the formation of more than one radical per polymer particle. In other words, it can be expected that termination reactions cannot be ignored as these reactions lead to a buildup of nitroxide species. Indeed, as shown in Figure S5 in Supporting Information, incorrect simulation results are obtained in case termination is ignored. On the basis of Figures 2−4, it can thus be concluded that at the selected particle diameter of 50 nm nitroxide partitioning cannot be ignored assuming a leakage of 2% of SG1 to the water phase (Γ = 50), as it influences both the polymerization rate and control over polymer properties. 3.2. Importance of Backbiting. For a particle diameter of 50 nm, the effect of the backbiting rate coefficient kbb on the main NMP miniemulsion characteristics is presented in Figure 5 (Γ = 50). The green full lines are the same as before and are thus obtained using the kbb value specified in Table 1 (kbb = 4.7 × 103 s−1), whereas the red dashed and blue dotted lines represent the simulation outcomes respectively in case a kbb value twice and four times as low is used. It can be seen that a weaker backbiting tendency leads to a significantly faster NMP (Figure 5a) and concomitant loss of control, as higher dispersity values (Figure 5c) and lower EGF values (Figure 5d) result. In agreement with literature reports on G

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Figure 6. Effect of backbiting rate coefficient kbb (25% (dotted blue lines), 50% (dashed red lines), 100% (full green lines) of the value specified in Table 2) on the average number of (a) initiator radicals, (b) secondary macroradicals, (c) tertiary macroradicals, and (d) nitroxide radicals per polymer particle (eq 2) as a function of conversion for the NMP of nBuA in miniemulsion initiated by poly(nBuA-SG1) macroinitiator ([nBuA]0:[poly(nBuA)-SG1]0 = 300) at 393 K and for a dp of 50 nm; (e) NMP initiator concentration; (f) fraction of macroradicals being tertiary as a function of conversion. Model parameters as given in Table 1 and 2 (kbb excepted).

Figure 7. Effect of the nitroxide partition coefficient Γ on the main polymer properties as a function of the particle diameter dp for the NMP of nBuA in miniemulsion initiated by poly(nBuA)-SG1 macroinitiator ([nBuA)]0:[poly(nBuA)-SG1]0 = 300) at 393 K. Model parameters as given in Table 1 and 2 (Γ excepted); xn = number-average chain length; EGF = end-group functionality; SCB = short chain branch; conversion = 0.7; average chain length characteristics calculated ignoring the monomer units in the macroinitiator part.

the key NMP characteristics to this partitioning coefficient. A minimal value of 50 is selected for Γ as otherwise highly uncontrolled NMPs are obtained at typical polymer particle diameters. The maximal value of 5000 represents negligible nitroxide solubility in water. From Figure 7 it can be seen that at high particle diameters (dp > 70 nm) the effect of nitroxide partitioning is limited, in agreement with the work of Bentein et al.32 on the NMP of styrene. On the other hand, large differences are obtained for low particle diameters, with the highest impact on the polymerization time and control over chain length. However, depending on the Γ value, a different NMP behavior is obtained. For a negligible nitroxide solubility in water (Γ = 5000) at a dp of 20 nm it takes more than 50 h to reach a conversion of 0.7 albeit with an excellent control over chain length (dispersity =1.2) and livingness (EGF = 0.97). On the other hand, for a low Γ of 50, ca. 2 h are sufficient to reach the same conversion but a dispersity above

1.5 results, while a similar livingness is still obtained. Close inspection of Figure 7 also reveals that for low Γ values (Γ ≤ 200) a minimum for the polymerization time and a maximum for the dispersity are observed at a dp of 30 nm, indicating the difficulty to improve both the polymerization rate and control over chain length in CRP processes. The strong dependence of the key NMP characteristics in Figure 7 on the particle diameter and Γ value can be understood on the basis of Figure 8 showing the evolution of the ratio of the extensive total deactivation rate to the extensive total propagation rate at a conversion of 0.5. It can be seen that only for a sufficiently low water solubility of the nitroxide (Γ ≥ 500) deactivation is strongly favored over propagation for small particles, leading to an increased control over chain length and a slower NMP. Note that for a Γ of 500 this beneficial effect on control is even accompanied by a limited increase of the polymerization time (Figure 7a), highlighting the potential of the H

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not obtained. Furthermore, model analysis shows that the limited loss of end-group functionality (EGF) at low conversions can be attributed predominantly to chain transfer to monomer reactions and to a lesser extent to termination reactions, involving mainly tertiary macrospecies. The current theoretical study also underlines the necessity of accurate values for the nitroxide partitioning coefficient toward the identification of an optimal particle diameter with respect to a minimal polymerization time for a given polymer yield or maximal control over polymer properties. Only for sufficiently high Γ values, the use of low particle diameters, which are also encountered during the nucleation stage in emulsion NMPs, is beneficial. Under such conditions, an excellent control over chain length and EGF can be obtained within a reasonable polymerization time.

Figure 8. Effect of the partition coefficient Γ on (a) the ratio of the extensive total deactivation rate to the total extensive propagation rate as a function of the particle diameter dp for the NMP of nBuA in miniemulsion initiated by poly(nBuA)-SG1 macroinitiator ([nBuA)]0:[poly(nBuA)-SG1]0 = 300) at 393 K and a conversion of 0.5. Model parameters as given in Table 1 and 2 (Γ excepted).

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ASSOCIATED CONTENT

S Supporting Information *

NMP miniemulsion technique for improved control within industrially relevant polymerization time spans. On the other hand, for a too high water solubility of the nitroxide (Γ ≤ 100), the relative importance of the deactivation rate is reduced for low polymer particle diameters (Figure 8; dp < 70 nm) as nitroxide partitioning becomes relevant, leading to reduction of the control over chain length albeit with an increased polymerization rate, in agreement with the work of Charleux on styrene NMP.57 However, as explained in Supporting Information (Figure S7), at very low particle diameters (dp < 30 nm) deactivation gains again in importance, since the NMP initiation is slowed down, explaining the observed minimum in Figure 7a and the observed maximum in Figure 7c. It can thus be concluded on the basis of Figures 7 and 8 that the particle diameter strongly influences the miniemulsion NMP characteristics and that availability of accurate Γ data is crucial to fully assess the potential of low particle diameters for the optimization of NMP miniemulsion of nBuA. Moreover, it can be expected that the previous observations are also valid for emulsion NMPs, as small particles determine the NMP behavior, at least during the nucleation stage. Hence, the reported insights are important for both small particle miniemulsion NMP and emulsion NMP.

(i) Additional continuity and moment equations not specified in the main text, (ii) the correlation used to account for diffusional limitations on termination, (iii) importance of chain transfer to monomer rate for Figures 1 and 5 in the main text and relevance of the non-“zero-one” nature of the studied miniemulsion NMP, (iv) and a detailed explanation of extrema in Figure 7 in the main text. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Authors

*(M.F.C.) E-mail: [email protected]. *(D.R.D.) E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS P.H.M.V.S., D.R.D., M.-F.R., and G.B.M. acknowledge financial support from the Long Term Structural Methusalem Funding by the Flemish Government, the Interuniversity Attraction Poles ProgrammeBelgian StateBelgian Science Policy, and the Fund for Scientific Research Flanders (FWO; G.0065.13N). D.R.D. acknowledges the FWO through a postdoctoral fellowship. M.F.C. gratefully acknowledges support from the Ontario Research Chairs program.

4. CONCLUSIONS For the miniemulsion NMP of nBuA initiated by poly(nBuA)SG1 at 393 K, a kinetic model has been developed which allows to evaluate the possible compartmentalization effect of four radicals types, i.e., the NMP initiator radicals, the secondary macroradicals, the tertiary macroradicals, and the nitroxide radicals, while accounting for nitroxide partitioning and diffusional limitations on termination. Clearly, the NMP cannot be treated as a “zero-one” system, as termination reactions cannot be ignored despite the low extensive termination rates, implying the necessity of tracking polymer particles possessing more than one nitroxide and macroradical. The performed simulations reveal that backbiting reactions cannot be neglected, despite the limited formation of short chain branches (SCBs), and that backbiting allows a faster NMP initiation on a conversion basis, achieving lower dispersity values. Nitroxide partitioning mainly leads to a rate acceleration and a lower control over chain length, in particular at lower particle diameters. On an average basis, the nitroxide species are partitioned between the organic and aqueous phase under equilibrium conditions but per polymer particle equilibrium is

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