Prediction of Chain Propagation Rate Constants of Polymerization

Article pubs.acs.org/JPCB

Prediction of Chain Propagation Rate Constants of Polymerization Reactions in Aqueous NIPAM/BIS and VCL/BIS Systems Leif C. Kröger, Wassja A. Kopp, and Kai Leonhard* Chair of Technical Thermodynamics, RWTH Aachen University, D-52062 Aachen, Germany S Supporting Information *

ABSTRACT: Microgels have a wide range of possible applications and are therefore studied with increasing interest. Nonetheless, the microgel synthesis process and some of the resulting properties of the microgels, such as the cross-linker distribution within the microgels, are not yet fully understood. An in-depth understanding of the synthesis process is crucial for designing tailored microgels with desired properties. In this work, rate constants and reaction enthalpies of chain propagation reactions in aqueous N-isopropylacrylamide/ N,N′-methylenebisacrylamide and aqueous N-vinylcaprolactam/N,N′-methylenebisacrylamide systems are calculated to identify the possible sources of an inhomogeneous cross-linker distribution in the resulting microgels. Gas-phase reaction rate constants are calculated from B2PLYPD3/aug-cc-pVTZ energies and B3LYPD3/tzvp geometries and frequencies. Then, solvation effects based on COSMO-RS are incorporated into the rate constants to obtain the desired liquid-phase reaction rate constants. The rate constants agree with experiments within a factor of 2−10, and the reaction enthalpies deviate less than 5 kJ/ mol. Further, the effect of rate constants on the microgel growth process is analyzed, and it is shown that differences in the magnitude of the reaction rate constants are a source of an inhomogeneous cross-linker distribution within the resulting microgel.

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INTRODUCTION Microgels are soft polymer particles with a wide range of possible applications, such as drug delivery systems1 or as catalyst matrix.2 Because of their promising applications, microgels are investigated with increasing interest.3 Microgels based on N-isopropylacrylamide (NIPAM) and N-vinylcaprolactam (VCL) are thermoresponsive and have a volume phase transition temperature near physiological temperature.4 As a consequence, these microgels swell at low temperatures and collapse at higher temperatures. This unique characteristic makes them attractive for biotechnological applications.4 Therefore, NIPAM and VCL serve as building blocks of a large amount of microgels reported in the literature.4 To tailor microgels with specific properties for applications, knowledge of the microgel synthesis process is essential. Nevertheless, the formation and the growth of microgels in particular are not yet fully understood and are the objectives of this study. One notyet-fully understood property of microgels is the inhomogeneous cross-linker distribution within the microgel, which studies have reported for NIPAM-5 and VCL6-based microgels. In addition to experimental research approaches, modeling of the polymerization and particle growth processes has become a popular technique in investigating and understanding the macroscopic properties of polymers and microgels.6−16 These kinds of models allow identifying variables that can be used to design the properties of microgels. A recent example includes a mathematical model for particle formation and evolution of © XXXX American Chemical Society

particle size distribution in the emulsion polymerization of vinyl chloride monomer.9 Other examples are kinetic Monte Carlo approaches to predict the characteristics of the polymer network formation during the pre- and postgelation regimes of free-radical cross-linking copolymerization8 and to simulate the miniemulsion copolymerization of n-butyl acrylate with a water-soluble monomer.10 The modeling of free-radical copolymerization kinetics with cross-linking of vinyl/divinyl monomers in supercritical carbon dioxide12 and the modeling of precipitation polymerization by employing the method of finite molecular weight moments13 have also been studied. To provide realistic insights into the formation and the characteristics of microgels, all models need both reliable reaction kinetics and reliable thermochemistry data. Moreover, the knowledge of global reaction rates is not sufficient for sophisticated models because several elementary reaction steps are lumped into global reactions, and thereby, intermediates are neglected. Instead, the rates of elementary reactions, such as individual chain propagation or chain termination reactions, are of interest in sophisticated models. Presumably, this is especially valid for those polymerization systems in which previous studies identified heterogeneous growth processes and heterogeneous microgel structures.5,6 Additionally, enthalpies Received: September 9, 2016 Revised: March 8, 2017 Published: March 13, 2017 A

DOI: 10.1021/acs.jpcb.6b09147 J. Phys. Chem. B XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry B Table 1. Schemes of the Reactions under Study

of reactions are of interest to model reactors, to allow a comparison to calorimeter experiments, or to evaluate the heat release.17 The experimental determination of such elementary reaction rates and reaction enthalpies is possible for the polymerization of a pure monomer.18 However, it is extremely challenging to identify the elementary reaction rates and the reaction enthalpies experimentally in case of cross-linked systems and copolymerization because of the large number of possible reactions. Compared with the variety of reactions, the number of available independent experiments is usually smaller, making the deduction of elementary reaction rates mathematically underdetermined. As a consequence, the resulting reaction rates may contain uncertainties of several orders of magnitude.19,20 An additional challenge is the changing environment during the synthesis. Usually, the polymer content changes, especially if precipitation takes place, a reactant

vanishes, or a product accumulates. These different reaction environments influence the solubility of the reactants and products and thus influence the elementary reaction rates. This makes it even more difficult to determine the desired reaction rates experimentally. Therefore, the direct calculation of these elementary reaction rates by quantum mechanical (QM) methods is a reasonable alternative. This is a common and widespread approach in combustion chemistry and astrochemistry21 and was already successfully applied to polymerization reactions.22−24 NIPAM7 and VCL6 often serve as basis components for microgels. Although NIPAM is in principle able to form microgels via self-cross-linking,25,26 a cross-linker is usually added to both, NIPAM and VCL, for producing microgels. A frequent choice is N,N′-methylenebisacrylamide (BIS)15,27−29 as cross-linker along with one of either monomers, NIPAM5,7,15,27,30 and VCL,6,28,29 respectively. The formation B


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The Journal of Physical Chemistry B

Computational Details. Liquid-phase reaction rates can be determined by employing continuum models, such as COSMO38 or CPCM,39,40 to treat solvation effects and by updating gas-phase reaction rate constants. This is computationally more efficient but less accurate than the direct incorporation of several solvent molecules in the QM calculations. Combinations of both approaches are used as well to increase the accuracy (at the cost of increasing computational effort). This is done especially for ionic species41 because continuum models are often not parametrized to ionic species. In this work, the approach of Deglmann et al.22 is followed, in which molecular structures are obtained in gas phase first and COSMO-RS42 is used to incorporate solvation effects later, thus providing liquid-phase molecular structures. To acquire the desired reaction rate constants, the conventional transitionstate theory (TST) is employed as RT ̃ RT qTS −ΔẼ/ RT RT −ΔG̃ / RT k(T ) = K= · e = e h h qA qB h (1)

of microgels from aqueous NIPAM/BIS and aqueous VCL/BIS solutions comes with most problems described in previous paragraphs. Not only have previous studies reported inhomogeneities regarding the BIS distribution inside the resulting NIPAM-5 and VCL6-based microgels, but also the number of possible reactions is too large for an accurate experimental determination of the related reaction rate constants. Furthermore, the reaction environment changes during the growth of a microgel particle. Although the polymer mass fraction is low at the beginning, it increases during the growth process. The reported water content of the final microgel particles varies between 70 and 35%27,31−34 for the collapsed state, which corresponds to a polymer content of 30− 65%. Consequently, the rate constants and reaction enthalpies of the most relevant chain propagation reactions in aqueous NIPAM/BIS systems and the rate constants of the most relevant chain propagation reactions in aqueous VCL/BIS systems are calculated in this study. The influences of temperature and polymer content on the reaction rate constants are consequently evaluated with a focus on microgel formation.

where K̃ is the quasi-equilibrium constant between reactants and the TS, whereas the consumption rate of the TS is the product of the universal gas constant (R) and the temperature (T) divided by Planck’s constant h. Parameters qTS, qA, and qB denote the partition functions of the TS, reactant A, and reactant B, respectively. The difference between the electronic energies of the TS and the reactants is represented by ΔẼ , and the difference between their free energies is ΔG̃ . Variational effects are expected to be of minor importance because of the rather high free-energy barriers and imaginary frequencies. Because the changes in the free-energy barrier height must be viewed as the dominant solvent effect on the TST rate constant,43 the difference in free energy (ΔG̃ ) can be split into a gas-phase contribution and a contribution from solvation. Accordingly, eq 1 can be rewritten as

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METHODOLOGY Reaction Details. The polymer chain elongation in aqueous NIPAM/BIS and VCL/BIS microgel synthesis systems takes place via radical polymerization. The major chain propagation reactions in the NIPAM/BIS system are the addition of a NIPAM or BIS monomer to the radical site of an existing polymer chain. All reactions studied here are shown in Table 1. While the monomers are built from just about 20 atoms, the polymer chains contain several hundreds or thousands of atoms and are therefore too large for an accurate QM treatment. Consequently, in this study, the chains are represented by their last component, which is then a NIPAM radical or a BIS radical. This is known as a terminal copolymerization model,35 in which the propagation kinetics of each system can be described by four propagation rate coefficients: two of homopropagation reactions and two of cross-propagation reactions. Obviously, representing polymer chains with a single molecule introduces an error of unknown magnitude. In general, the reaction rate constants are dependent on chain length. Nevertheless, in a polymer, this chain-length dependence usually decays rapidly so that the reaction rate constants can be assumed to be independent of chain length after the first couple of monomers,23 and the expected error due to neglecting the chain-length effect rapidly decays to less than 1 order of magnitude. Therefore, the presented approach is commonly used in the literature.22,24,36,37 It was pointed out by one of the article’s reviewers that the carbon atom bearing the radical is positioned next to a primary carbon because of the rather rough cutting of the radical chain. Consequently, the reviewer suggested to add an additional CH3 group to the monomeric radical to analyze the effect. This suggestion was tested on two transition states (TS). It was shown that the gas-phase free-energy barrier heights differed by a maximum of 2 kJ/mol among the two radical models. Thus, the addition of a CH3 group to the radical does not change the reaction kinetics significantly. Because of the small effect and for computational efficiency, the QM calculations in this study are performed with the smaller radical.

k(T ) =

RT −ΔGgas ̃ / RT −ΔGsolv ̃ / RT ̃ e e = kgas(T ) e−ΔGsolv / RT h (2)

Now, the temperature-dependent reaction rate constants in the gas phase, kgas(T), and ΔG̃ solv can be determined independently and combined to determine the liquid-phase reaction rate constant, k(T). In this work, kgas(T) was obtained from TST calculations as implemented in the TAMKin software package,44 along with the RRHO approximation and Eckart tunneling. In some of the reactions examined, TSs were found containing a negative potential-energy barrier. In these cases, there are van der Waals complexes of the reactants that are energetically more favored than the reactants themselves. The reaction barrier here is defined as the barrier between reactants and the TS, and the free-energy barrier is still large so that TST can be applied. Eckart tunneling, however, is employed between the van der Waals complexes and the product, which increases the obtained rate constants by 10−15%. The geometric structure and frequencies of the molecules are obtained at the B3LYPD3/TZVP level of theory with an ultrafine grid, whereas energies are determined at the B2PLYPD3/aug-cc-pVTZ level of theory. Geometries and barrier heights (Table S1) can be found in the Supporting Information. We checked using IRC scans whether the TSs connect the desired reactants and products. Dihedral scans were performed at the B3LYPD3/TZVP level with 30° C


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The Journal of Physical Chemistry B increments around the reaction coordinate. The dihedral scans for the NIPAM polymerization unveiled a large number of different TS conformations with deep minima and different properties orthogonal to the reaction coordinate. Therefore, reaction rate constants are determined for each TS separately without one-dimensional (1D) hindered rotor treatment and are subsequently summed up to obtain the total reaction rate. Obviously, the pathway via the TS with the lowest barrier will be predominant and will have the largest rate constant. Nevertheless, all other pathways (TSs) are also possible. Therefore, all rates of parallel pathways have to be added up to obtain the overall rate for the elementary reaction.45 Gaussian 0946 is used for geometry, frequency, and dihedral scans, whereas Turbomole 7.0.147,48 is employed for single-point energy calculations. COSMO-RS calculations used to obtain ΔG̃ solv were performed with COSMOtherm,49 whereas the necessary files were created at the BP-TZVP-COSMO level with COSMOconf in case of reactants and products and turbomole48 for the TSs. The exact composition of the liquid phase can be specified in COSMO-RS. This allows for obtaining a concentrationdependent ΔG̃ solv and in turn yields empirical rate constants that depend on the liquid-phase composition in contrast to the thermodynamic ones. In this context, it is irrelevant whether the liquid-phase components are reactants or not. In most modeling approaches, the utilization of these liquid-phase composition-dependent rate constants is much more convenient. Similarly, the enthalpies of reaction (ΔhR) are calculated in the gas phase and then updated by including the solvation contribution for the liquid phase. The gas-phase reaction enthalpy is obtained from the ab initio geometries, frequencies, and single-point energies using the python package TAMKin44 directly, whereas the solvation contribution can be calculated using the Gibbs−Helmholtz equation ΔHR = ΔHR,gas + ΔHR,solv

The two major error contributions add up to a total error (ΔEErr) of 3.49 kJ/mol ΔE Err =

j

(5)

Considering that the calculation of the rate constants involves three molecular structures (two reactants and one TS), the error factor of the reaction rate constants (ΔkErr) can be estimated as ⎛ 3 ΔE Err ⎞ ΔkErr = exp⎜ ⎟ ⎝ RT ⎠

(6)

where R is the universal gas constant and T is the temperature. For the temperature range used in this study (283−463 K), ΔkErr lies between 5 and 13. We would like to mention that this is an upper estimate of the error in the rate constant computations. Calculation of the NIPAM Reaction Rate Constants. As stated before, several rotational conformations of the TS are used individually for each reaction. In case of NIPAM, for instance, eight different TSs are considered. Treating them individually revealed that the TSs are stabilized very differently in the liquid phase. In Figure 1, two TSs of the NIPAM

Figure 1. Two different TSs of the reaction of a NIPAM radical with a NIPAM monomer. The green dashed line represents the new bond, which will be formed during the reaction, and the white dashed line indicates a hydrogen bond. Carbon atoms, hydrogen, oxygen, and nitrogen are depicted in dark gray, white, red, and blue, respectively.

(3)

⎛ ∂ ⎛ ΔG R,solv ⎞⎞ =ΔHR,gas − T 2⎜ ⎜ ⎟⎟ ⎝ ∂T ⎝ T ⎠⎠ p ,{n }

3.262 + 1.252 kJ/mol = 3.49 kJ/mol

polymerization reaction are shown exemplarily. TS1 has the lowest gas-phase free-energy barrier of all TSs found, whereas the free-energy barrier at 298 K associated with TS2 is more than 15 kJ/mol higher. As a consequence, TS1 accounts for more than 50% of the total gas-phase rate, whereas TS2 accounts for roughly 1%. However, this difference changes completely in the liquid phase. Here, because of different stabilization and hence different changes of the free energy due to solvation, TS2 accounts for almost 60% of the total liquidphase reaction rate constant, whereas TS1 accounts for only 5%. This observation itself is not surprising, and it is well known that conformers can be stabilized very differently. In the present case, the different solubility can be attributed to the internal hydrogen bond, which is clearly visible in TS1 (cf. Figure 1) and which is broken in TS2 (cf. Figure 1). This hydrogen bond stabilizes TS1 in the gas phase but prevents hydrogen bonds to solvent molecules, thus leading to a weaker stabilization in the liquid phase. These results emphasize the importance of treating several TSs individually, at least when applying a continuum model for updating gas-phase results to the liquid phase. When the gas-phase reaction rate constants are determined solely with the global minimum TS of the gas phase, the update to the liquid phase would not consider the

(4)

ΔG

R,solv The derivative of with respect to T is computed T numerically as a central difference quotient with 1 K increments. The overall error of the used approach regarding reaction rate constants can be estimated by adding the errors of the individual methods using the Gaussian error propagation law.50,51 Zheng et al.52 determined the mean unsigned error of the B2PLYP method with MG3S basis set to be 3.26 kJ/mol. The MG3S basis set used in ref 52is significantly smaller than the aug-cc-pvtz basis set, which is used in this work. In addition, the empirical dispersion correction, D3, is employed in this work. Both the larger basis set and the dispersion correction will significantly decrease the error given by Zheng et al.52 The used B3LYP geometries should not have a significant influence on the accuracy of the energy calculation because B3LYP geometries are known to be in close agreement with B2PLYP geometries. In addition, small changes of the geometries at the stationary point do not affect the energy significantly. Also, B3LYP frequencies are known to be very accurate. The error by COSMO-RS in calculating the solvation free energy is given at 1.25 kJ/mol for neutral compounds.53

D


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Given that the assumption of Kalugin et al.54 regarding the termination rate is valid, the deviation amounts to a factor of 2.5 in the case of VCL and is of the same order as the NIPAM case. The deviation from the literature values is slightly smaller than the previously derived expected error of the used method but is of the same order of magnitude. This slightly smaller error can be attributed to the larger basis set and the dispersion correction.

global minimum TS in the liquid phase and thus give inaccurate reaction rate constants. This is valid even if 1D hindered rotor treatment is applied when calculating the gas-phase reaction rate constants. In principle, one could think of updating the hindered rotor scan itself to the liquid phase or performing the scan directly in the liquid phase. However, this is computationally and technically very expensive and hence the treatment of several individual TSs for each reaction is a more reasonable and practicable approach. Validation of the Computational Results. Only few data on NIPAM or VCL reaction rate constants are available in the literature. Often, the overall reaction rates or the ratio of propagation to termination rate constants, such as kp/k0.5 t , are experimentally determined. As a consequence, these rates usually incorporate a variety of elementary reactions and thus cannot be used for a direct validation of the chain propagation rate constants. Nevertheless, Ganachaud et al.18 performed pulsed laser polymerization experiments of NIPAM in water to obtain propagation rate coefficients. As stated by the authors, these rate constants depend on the monomer and initiator concentrations as well as temperature.18 Figure 2 shows a

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RESULTS AND DISCUSSION Reaction Rate Constants. In the NIPAM system (Figure 3), the reaction of a NIPAM radical and a BIS monomer has

Figure 3. Polymerization rate constants of chain propagation reactions in an ideally diluted aqueous solution for the NIPAM/BIS system over a temperature range of 283−463 K. The solid black line refers to the rate constant of the reaction of a NIPAM radical with a NIPAM monomer, the dotted green line to a NIPAM radical with BIS monomer, the dashed blue line to a BIS radical with a BIS monomer, and the dot-dashed red line to the rate constant of the reaction of a BIS radical with a NIPAM monomer.

the largest rate constant, although the reaction of a NIPAM radical and a NIPAM monomer is just slightly slower. The rate constants of the reactions of a BIS radical with a NIPAM monomer and a BIS monomer are almost identical and at the same time roughly 1−2 orders of magnitude smaller than those of the other two reactions. Therefore, one can assume that the magnitude of the reaction rate constants is mainly influenced by the radical that is involved in the reaction. The rate constants of reactions involving a NIPAM radical are larger than those of the reactions involving a BIS radical. In the context of NIPAM-based microgel synthesis, this indicates that the consumption of available BIS and NIPAM monomers should be very similar and linked to the concentrations of the monomers. Nevertheless, Figure 3 implies that the percental consumption of BIS is slightly higher than that of NIPAM. A similar observation was made by Wu et al.,5 who measured the conversion versus time curves for polyNIPAM microgel latexes and found that BIS was consumed faster than NIPAM. Further, their results showed that the presence of BIS increases the total monomer conversion compared with the polymerization of NIPAM in the absence of BIS.5 The first observation can be explained with the presented reaction rates. The second observation of Wu et al.5 can be explained with the slightly higher rate constants of the reaction

Figure 2. Comparison of calculated and measured rate constants for polymerization of NIPAM over various monomer concentrations. The solid black line refers to the rate constant of a NIPAM radical with a NIPAM monomer calculated in the present work, and the orange crosses indicate the corresponding values obtained by Ganachaud et al.18

comparison of the rate constants by Ganachaud et al.18 and of rates obtained by the presented method of this work for NIPAM polymerization. Although the decay of the rate coefficients with increasing monomer concentration is not as distinctive in the present work compared to that in the experiments, the calculated rate coefficients agree within a factor of 2−10 with the experiments. In case of VCL, Kalugin et al.54 performed calorimetric studies of the radical polymerization of VCL in benzene. Although they did not determine the reaction rate constants directly, they concluded from their experiments that the polymerization rate constants of VCL must be around 1 m3/ mol s, by assuming that the rate constant of the bimolecular termination is of the same order as for most vinyl monomers.54 The presented methodology yields a rate constant of 0.411 m3/ mol s under the same conditions as the study of Kalugin et al.54 E


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is, two amides (NIPAM and BIS), and stronger when the reactants are of different chemical groups, that is, one amid and one lactam (BIS and VCL). In principle, initiation and termination could also have an impact on the inhomogeneous cross-linker distribution. However, this seems not likely: Each subchain is initiated by an initiator radical and will contain on average tens or hundreds of monomers. Therefore, the majority of reactions in the systems are described by the reactions calculated in this work; thus, knowledge of the chain propagation reactions allows for estimations of the consumption of BIS, VCL, and NIPAM. The reaction conditions are similar to those of the termination reactions. The vast majority of termination reactions will be radical combination reactions. In general, these reactions are barrierless and hence the reaction rate constant will be large. In other words, whenever two radicals “meet”, they will recombine. As a consequence, the termination reactions will be limited mostly by diffusion. Diffusion on the other hand depends mainly on the size of the molecules. The sizes of the NIPAM, BIS, and VCL are very similar. Therefore, the size of the radical chains will depend on their length but not on their composition. Hence, the diffusion will be of the same order for all radicals. Thus, the effective rate constants for termination will be nearly the same for all radical pairs in the two considered systems. That is why, termination reactions do not favor any of the monomers. Consequently, an effect that affects all monomers equally cannot be the origin of an inhomogeneous cross-linker distribution. Furthermore, the results show that the rate constant of the reaction of a BIS radical with a BIS monomer is up to 1 order of magnitude smaller than that of the reaction of a BIS radical with a VCL monomer. This alone makes reactions of a BIS radical with a BIS monomer very unlikely. Together with the fact that the BIS concentration is usually a 100 times smaller than the VCL concentration in most microgel synthesis solutions, one could think of neglecting this reaction completely in a modeling approach. Figures 5 and 6 show the reaction rate constants of the NIPAM and VCL systems over polymer mass fraction at a temperature (T) of 343 K, which is commonly used in microgel synthesis.6,55,56 The reaction rate constants depend on the polymer mass fraction according to eq 3. The corresponding values at a polymer mass fraction of 0 wt % correspond to the results shown in Figures 3 and 4. In Figures 5 and 6, one can see that both systems show the same trend: The reaction rate constants decrease with increasing polymer content down to minimal values at 60−80 wt %. Then, the rate constants increase again. In fact, all systems show similar rate constants at polymer contents of 100 and 0 wt %. In each system, the characteristics of the trend are similar for all reactions except for the reaction of a BIS radical with a BIS monomer, for which the increase in reaction rates at high polymer mass fractions is less pronounced. This observation may be due to the fact that the polymer content in studied systems is modeled via trimers that do not contain BIS. However, this particular reaction was found to be of minor importance in general and occurs only rarely. Therefore, one can conclude that a change of polymer mass fraction during a polymerization process, for example, due to precipitation, does not affect the ratio of the elementary reaction rate constants. Enthalpy of Reaction. To set up sophisticated microgel synthesis models capable of predicting the heat flow in the synthesis process and being compared with calorimetric

of a NIPAM radical with a BIS monomer and the small amount of BIS used in the study conducted by Wu et al.5 (123.7 mmol/ L NIPAM vs 9.08 mmol/L BIS). The lower rate constant of a BIS radical reaction with a BIS monomer should be of minor importance at low BIS concentrations. In general, it can be concluded from a reaction kinetics point of view that the NIPAM and BIS monomers do have a similar behavior in the NIPAM/BIS system. In case of the VCL system, the reaction rate constants show a different behavior (Figure 4). Here, the rate constants of a

Figure 4. Polymerization rate constants of chain propagation reactions in an ideally diluted aqueous solution for the VCL/BIS system over a temperature range of 283−463 K. The solid black line refers to the rate constant of the reaction of a VCL radical with a VCL monomer, the dotted green line to a VCL radical with a BIS monomer, the dashed blue line to a BIS radical with a BIS monomer, and the dotdashed red line refers to the rate constant of the reaction of a BIS radical with a VCL monomer.

reaction of a VCL radical with a VCL monomer are the smallest of all rate constants. The rate constant of a VCL radical reacting with a BIS monomer is almost 2 orders of magnitude larger. Additionally, there is a significant difference in the rate constants of the reactions involving a BIS radical. Hence, the reaction rate constants in the VCL system depend on both the involved radical and the involved monomer. The large difference in the rate constants of a VCL radical reacting with a BIS monomer and a VCL monomer indicates that the BIS monomers are consumed significantly faster than the VCL monomers. In the context of microgel synthesis, the results suggest that BIS is consumed mainly at the beginning of the polymerization process. Consequently, the cross-linking density should be higher in the core of the resulting VCL-based microgels and lower in the shell of the microgels. Balaceanu et al.6 proved the existence of a heterogeneous morphology in VCL-based microgel particles. Further, through a combined experimental and modeling approach, they determined the cross-linker amount to be higher in the core of the microgel.6 The presented results suggest that differences in reaction rate constants are at least one source of these inhomogeneities. A difference of several orders of magnitude between the rate constants of several competing reactions must lead to inhomogeneities unless another noninvestigated phenomenon counteracts on them. The cause of such inhomogeneities seems to be weaker when two chemically similar substances react, that F


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In this study, the enthalpies of reaction are calculated under the same condition, and the comparison is shown in Table 2, which reveals that the calculated enthalpies of reaction agree within 2−5 kJ/mol with the experimental values. Table 2. Comparison of Calculated Enthalpies of Reaction to the Literature under Various Conditions

Figure 5. Polymerization rate constants of chain propagation reactions in polymer/water solutions at T = 343 K for the NIPAM/BIS system. The polymer is represented by NIPAM trimers. The solid black line refers to the rate constant of the reaction of a NIPAM radical with a NIPAM monomer, the dotted green line to a NIPAM radical with a BIS monomer, the dashed blue line to a BIS radical with a BIS monomer, and the dot-dashed red line refers to the rate constant of the reaction of a BIS radical with a NIPAM monomer.

ΔHLit r − calc ΔHr

ref

conditions

−86.7

2.2

57

−76.0 ± 0.9

−80.8

4.8

54

−82.0 ± 2.9

−77.8

−4.2

58

water, 333 K benzene, 333 K molten BIS, var. temp.

reaction

ΔHLit r (kJ/mol)

ΔHcalc r (kJ/mol)

NIPAM/NIPAM

−84.5 ± 0.3

VCL/VCL BIS/BIS

Now, the enthalpies of reactions that have not been reported in the literature and that are experimentally difficult to obtain can be calculated with the same accuracy. An overview of the reaction enthalpies of the chain propagation reaction of this study is presented in Table 3. Here, the reaction enthalpies are Table 3. Calculated Enthalpies for Different Reactions at T = 343 K in Infinitely Diluted Monomer reaction

ΔHcalc (kJ/mol) r

BIS radical/BIS BIS radical/NIPAM BIS radical/VCL NIPAM radical/BIS NIPAM radical/NIPAM VCL radical/BIS VCL radical/VCL

−77.8 −81.7 −74.8 −89.3 −87.5 −87.4 −83.2

obtained at 343 K in an infinitely diluted aqueous solution. In general, all calculated reaction enthalpies are in a similar range, which was expected due to the similarity of the reactions. The smallest heat of polymerization was calculated for the reaction of a BIS radical with a VCL monomer, whereas the largest reaction enthalpy was identified for the reaction of a NIPAM radical with a BIS monomer. Nevertheless, all presented values are in the area of other reported values for vinylic polymerization reactions.59 Again, differences in the VCL/BIS system are slightly more distinct than those in the NIPAM/BIS system. The reaction enthalpies differ by about 12.6 kJ/mol in the VCL system and 7.6 kJ/mol in the NIPAM system. The reaction of a BIS radical with a BIS monomer is not taken into account. Further, reactions involving a BIS radical have lower reaction enthalpies.

Figure 6. Polymerization rate constants of chain propagation reactions in polymer/water solutions at T = 343 K for the VCL/BIS system. The polymer is represented by VCL trimers. The solid black line refers to the rate constant of the reaction of a VCL radical with a VCL monomer, the dashed blue line to a BIS radical with a BIS monomer, the dot-dashed red line to a BIS radical with a VCL monomer, and the dotted green line refers to the reaction of a VCL radical with a BIS monomer.

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experiments, reaction enthalpies are required along with the reaction rate constants. Again, the experimental determination of cross-polymerization reaction enthalpies is challenging because of the large number of possible reactions. Some have been evaluated in the literature when only a single type of monomers is involved: Virtanen et al.57 determined the heat of polymerization of NIPAM in water at 333 K to be −84.5 kJ/ mol by integrating calorigrams for various NIPAM concentrations. Similarly, Kalugin et al.54 obtained the heat of polymerization of VCL in benzene at 333 K calorimetrically to be −76.0 kJ/mol. The heat of polymerization of BIS was studied by Verneker et al.58 and determined to be −82.0 kJ/ mol.

CONCLUSIONS In this study, the polymerization rate constants of the major chain propagation reactions in the widely studied NIPAM and VCL systems were calculated quantum-mechanically. For most of these elementary reactions, no data has been available before, neither experimental nor computational. The calculated rate constants deviate less than 1 order of magnitude from the literature values in the few cases where a comparison is possible. The results show that the reaction rate constants in the NIPAM system mainly depend on the radical that is involved in the reaction. The presented reaction rate constants reveal that G



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BIS and NIPAM have similar reactivities. The aqueous VCL/ BIS system behaves fundamentally different because the reactivities of VCL and BIS differ significantly. The rate constant of the reaction of a BIS monomer reacting with a VCL radical is 2−3 orders of magnitude larger than that of a VCL radical and a VCL monomer, suggesting that the consumption of BIS is faster than the consumption of VCL in the microgel synthesis. The rate constants of the two systems decrease with increasing polymer content up to a polymer mass fraction of 60−80%, after which the rate constants increase again. Nevertheless, the ratio of rate constants remains constant. Thus, a change in the polymer concentration does not change the composition of the products but changes the overall speed of polymerization. Also, it was pointed out that BIS is consumed mainly at the beginning of a microgel synthesis process. Unless this effect is not coincidentally canceled by contributions, for example, diffusion effects or hindrance due to folding of the polymer, this early BIS consumption leads to an inhomogeneous cross-linker distribution within microgels. Thus, differences in the rate constants of elementary reactions influence the cross-linker density within VCL- and NIPAM-based microgels and result in a higher cross-linking density in the core of microgels and a less cross-linked microgel shell. On the basis of the presented results, this effect is more pronounced in VCL-based microgels. Further, we present the first prediction of the heat of polymerization for cross-propagation reaction in aqueous NIPAM/BIS and VCL/BIS systems. The deviation to the three experimental values yields an error of less than 5 kJ/mol. The presented results deepen the understanding of the synthesis process of NIPAM- and VCL-based microgels and allow for more sophisticated, physically based modeling of the same.

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REFERENCES

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpcb.6b09147. Geometries of the transition states, reactants, and products (ZIP) Plots of the reactivity ratios, Mayo−Lewis plots, reaction schemes, potential-energy and free-energy barriers, and rate constants (PDF)

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Article

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Phone: +49 241 80 98174. Fax: +49 241 80 92255. ORCID

Leif C. Kröger: 0000-0003-0463-1726 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors thank the Deutsche Forschungsgemeinschaft (DFG) for support within SFB 985 “Functional microgels and microgel systems”. Simulations were performed with computing resources granted by RWTH Aachen University under project rwth0083. The authors also thank Dr. Annett Schwarz and Malte Döntgen, M.Sc., for fruitful discussions. H


Article

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Prediction of Chain Propagation Rate Constants of Polymerization

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