Anionic Ring-Opening Polymerization Simulations for Hyperbranched

Oct 16, 2013 - We have investigated the anionic ring-opening multibranching polymerization for hyperbranched polyglycerol using slow monomer addition ...
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Anionic Ring-Opening Polymerization Simulations for Hyperbranched Polyglycerols with Defined Molecular Weights Florian Paulus,†,⊥ Maximilian E. R. Weiss,‡,⊥ Dirk Steinhilber,† Anatoly N. Nikitin,‡,§ Christof Schütte,‡,∥ and Rainer Haag*,† †

Institut für Organische Chemie und Biochemie, Freie Universität Berlin, Takustraße 3, 14195 Berlin, Germany Institut für Mathematik und Informatik, Fachbereich Mathematik, Freie Universität Berlin, Arnimallee 6, 14195 Berlin, Germany § Institute on Laser and Information Technologies, Svyatoozerskaya 1, Shatura, Moscow Region 140700, Russia ∥ Zuse-Institut Berlin (ZIB), Takustraße 7, 14195 Berlin, Germany ‡

S Supporting Information *

ABSTRACT: We have investigated the anionic ring-opening multibranching polymerization for hyperbranched polyglycerol using slow monomer addition at 120 °C. Different molecular masses were targeted, and the reaction mixture was probed at regular intervals for the experimental data. The resulting polymers were characterized by gel permeation chromatography, mass spectrometry, and NMR spectroscopy. A computational PREDICI model of the polymerization was developed to describe the experimental parameter dependencies. The rate coefficients were determined for the thermal and base-catalyzed, intra- and intermolecular reactions by fitting simulated number- and weight-average molecular weights to the experimental values. Although the main reaction was expected to be base-mediated, thermal propagation proved to play a crucial role in the dynamics of the investigated system. Both thermal and base-catalyzed self-initiation significantly increased the dispersity for targeting molecular masses exceeding 10 kDa, whereby the size of the growing polymer species was affected by polymerization kinetics.

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INTRODUCTION In the past decade, hyperbranched polyglycerol (hPG) was the subject of various studies on polymer therapeutics and diagnostics.1,2 hPG is highly biocompatible because of its flexible polyether backbone and large number of peripheral hydroxyl groups.3−5 Its main advantage over perfectly branched polyglycerol dendrimers is that it can be easily synthesized on a large scale.6,7 Our main research focus was therefore to synthesize various molecular weights with low dispersity.8,9 Other studies have focused on the dimensions of hPG architecture,10 which directly influence cellular uptake mechanisms, pharmacological activity in vitro and in vivo, and transfection capability.11−14 After Flory’s15 theoretical calculations on branched polymer architectures and Sandler and Berg’s16 work from 1966 using glycidol as a monomer, Vandenberg17 discovered the reaction conditions for branched structures in 1985. Tokar et al.18 and Dworak et al.19 reported a cationic polymerization of glycidol that indicated the coexistence of an active chain end (ACE) and active monomer (AM) mediated mechanism. Sunder et al.20 finally introduced the anionic ring-opening multibranching (ROMB) polymerization mechanism in solution using glycidol as a latent AB2 monomer that was attached to a partially © 2013 American Chemical Society

deprotonated multifunctional Bf core (with f = number of functional groups). The process was based on the slow monomer addition (SMA) method, where the monomer was diluted with nonpolar solvent to keep the local monomer concentration in the reaction mixture low. Hence, the number of thermally opened epoxides could be significantly reduced and the control of the reaction increased. The elevated dispersity ranging between 1.25 and 1.5 was explained by the formation of cyclic species. Recently, Brooks et al.5,21 published a different approach for synthesizing high molecular weight hPG which is based on an emulsion polymerization strategy using 1,4-dioxane, tetrahydropyran, ethylene glycol diethyl ether, and decane as emulsifying agents. They were able to synthesize hPG with molecular masses from Mn = 10 to 540 kDa with dispersity from 1.05 to 2.1, depending on the solvent/ glycidol ratio and the resulting homogeneity of the system. These experimental findings were explained by viscosity effects and emulsifier polarities influencing the solvent−ion complex stability and therefore the proton transfer/counterion exchange Received: August 15, 2013 Revised: September 30, 2013 Published: October 16, 2013 8458

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for the MWD and dispersity. These simulations can be employed to develop strategies to improve polymer properties obtained via the ROMB polymerization of glycidol. The executed simulations reflect the high potential of the developed PREDICI model for optimizing the ROMB polymerization of glycidol by carrying out numerical experiments under diverse conditions.31

equilibrium. Comparing this mechanism to the ROMB polymerization in solution is rather difficult as the kinetics and thermodynamics in heterogeneous and homogeneous systems are supposed to be different. Simulations became the preferred method to explore the complex ROMB polymerization parameters and to study the underlying mechanisms during the ROMB polymerization was developed and finally introduced. Radke et al.22 calculated the effect of core functionality on the molecular weight distribution (MWD) and the degree of branching (DB) for hyperbranched polymers in a self-condensing vinyl polymerization (SCVP) of AB* monomers. They showed that a semibatch process yielded a lower dispersity compared to a batch reaction and that the core functionality was crucial for controlling both dispersity and DB. Hö lter et al. 23 and Hanselmann et al. 24 studied simultaneously a similar polymerization process of an ABm monomer that was slowly added to a diluted Bf core (“coredilution/slow addition technique”) by performing computer simulations. Although reasonable results were obtained, the accessible molecular weight was still limited to approximately 10 kDa due to cyclization reactions of autopolymerized species that reduced the molecular weight.25 On the basis of the preliminary publications, Wilms et al.8 investigated the polymerization of hPG by applying a two-step process using hPG macroinitiators of 500 and 1000 Da. They were able to synthesis hPG with molecular masses up to 24 kDa and dispersity ranging from 1.3 to 1.8. With the same initial degree of deprotonation (DD), the greater range of accessible sizes was explained by an elevated DD throughout the process due to a larger number of initial hydroxyl groups of the macroinitiator. Hence, fewer thermally induced side reactions occurred, and the reaction control was increased. We recently investigated the spontaneous polymerization of glycidol at elevated temperatures.26 In this earlier study, experiments were carried out without an initiator to identify the influence of thermal side reactions on the kinetics of the polymerization. At higher reaction temperatures or energy input, we obtained self-initiated hPG with molecular masses up to 10 kDa and dispersity up to 8. For a description of the experimental dependencies, we used the software package PREDICI, which was developed to perform modeling and dynamic simulations of macromolecular processes, e.g., emulsion, suspensions, living, radical polymerizations, and polycondensations.27−30 With the developed PREDICI model of the glycidol polymerization, the kinetic parameters of the side reactions were estimated by fitting simulated number- and weight-average molecular weights to the experimental values measured at different reaction times during the polymerization. The PREDICI simulation confirmed that thermal side reactions lead to a high dispersity of the final product and are highly sensitive to the reactor operating temperature. The obtained results serve as a basis for further investigation of the basecatalyzed polymerization in the presence of an initiator. In this paper, the ROMB polymerization of glycidol at 120 °C is described. The polymerization was conducted at conditions that resulted in different defined number-average molecular weights. By probing the reaction mixture in regular intervals, we obtained time profiles of important parameters (molecular masses, degree of branching, etc.) characterizing the polymerization and the complete MWD as well. The PREDICI model for the ROMB polymerization of glycidol was developed and used to fit the experimental results in computer simulations, in particular to determine limitations

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EXPERIMENTAL SECTION

Materials. All chemicals were purchased from Acros Organics and used as received if not otherwise indicated. Glycidol was dried over CaH2 and distilled under reduced pressure. The purified monomer was stored at 4 °C under an inert atmosphere and only used up to 2 months. All solvents were purchased from Sigma-Aldrich and used without further purification. Dry tetrahydrofuran (THF) was obtained from a solvent purification system. Dry N-methyl-2-pyrrolidone (NMP) and potassium tert-butoxide (KOtBu) were used and stored under AcroSeal conditions and used as received. Polymerization. The polymerization was carried out under a similar procedure reported elsewhere.32,33 A batch reactor (V = 1 L) was predried at 140 °C and with reduced pressure overnight. The initiator 1,1,1-tris(hydroxymethyl)propane (TMP) was melted at 65 °C under reduced pressure. Applying inert conditions, KOtBu (1 M in THF) was added to deprotonate 30% of all hydroxyl groups. The precipitated initiator salt was directly dissolved in 10 mL of dry NMP. The stirring was adjusted to 250 rpm, and the temperature was increased to 120 °C to distill off THF and the byproduct tert-butanol (tBuOH). The monomer glycidol (n = 1.35 mol, m = 100 g) was diluted with dry THF (v/v = 1/2.5) and slowly added to the reactor over 18 h using a precision dosing pump (ProMinent g/5a; model no. 400150). In regular intervals of 2 h, the reaction mixture was probed by using a pipet to extract approximately 0.3 mL. The samples were diluted with methanol (MeOH), and the solvent was evaporated under reduced pressure. The samples were characterized by gel permeation chromatography (GPC), matrix-assisted laser desorption/ionization time-of-flight mass spectrometry (MALDI-TOF MS), and NMR. Three polymerizations at monomer-to-initiator ratios ([M]/[I] = degree of polymerization (DPn) ∼ MWtarget) targeting 2.5 kDa (DPn = 33.8, n(TMP) = 39.9 mmol, m = 5.35 g), 5.0 kDa (DPn = 67.5, n(TMP) = 20.0 mmol, m = 2.68 g), and 20 kDa (DPn = 270, n(TMP) = 5.0 mmol, m = 67.1 mg) were carried out. Additionally, the final MWD from a polymerization targeting 10 kDa (DPn = 135, n(TMP) = 10.0 mmol, m = 1.34 g) without probing was used to evaluate the established model. Hyperbranched Polyglycerol. 1H NMR (700 MHz, CD3OD, δ): 4.00−3.20 (m, hPG backbone), 1.4 (q, methylene group, TMP core), 0.91 (t, ethyl group, TMP core). 13C NMR (175 MHz, CD3OD, δ): 82.0−81.0 (hPG backbone, linear 1−3 units), 80.5−79.5 (hPG backbone, dendritic units), 74.5−73.5 (hPG backbone, linear 1−4 units), 73.5−72.0 (hPG backbone, linear/dendritic units) 72.0−70.5 (hPG backbone, linear 1−3/1−4 units), 65.0−64.0 (hPG backbone, terminal units), 63.5−62.0 (hPG backbone, linear 1−3 units), 66.8, 44.6, 24.2, 8.0 (TMP core). GPC. Molecular weight distributions were determined by means of GPC coupled to a refractive index detector (RI) obtaining the complete distribution (Mn, Mp, Mw, dispersity). Measurements were carried out under highly diluted conditions (10 mg/mL, injected volume 20 μL) from a GPC consisting of an Agilent 1100 solvent delivery system with pump, manual injector, and an Agilent differential refractometer. Three 30 cm columns (PPS: Polymer Standards Service GmbH, Germany; Suprema 100 Å, 1000 Å, 3000 Å with 5 and 10 μm particle size) were used to separate aqueous polymer samples using water with 0.05% NaN3 as the mobile phase at a flow rate of 1 mL/ min. The columns were operated at room temperature with the RI detector at 50 °C. The calibration was performed by using certified standards pullulan (linear) and dextran (branched) from PSS. WinGPC Unity from PSS was used for data acquirement and interpretation. 8459

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MALDI-TOF-MS. Polymer masses and starter incorporation were analyzed by MALDI-TOF MS using an Ultraflex-II TOF/TOF instrument (Bruker Daltronics, Bremen, Germany) equipped with a 200 Hz solid-state smart beam laser. The mass spectrometer was operated in a positive linear mode. Spectra were acquired over a m/z range of 500−20 000. α-Cyano-4-hydroxycinnamic acid (CHCA) was used as the matrix, and samples were spotted using the dried droplet technique. NMR. NMR spectra were recorded on a Delta Joel Eclipse 700 MHz spectrometer. Proton and carbon NMR were recorded in ppm and referenced to indicated deuterated solvents. NMR data are reported as follows: chemical shift, multiplicity (s = singlet, d = doublet, t = triplet, q = quartet), integral. Multiplets (m) are reported over the range in which they appear at the indicated field strength.

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Scheme 1. Thermally and Anionically Ring-Opening of an Epoxide and Oligoformation Is Induced by Either an Alcohol (a) or Alkoxide (b)a

MODELING AND SIMULATIONS

Basic Model Components. For the simulation of the TMP-initiated and base-catalyzed polymerization process of glycidol in solution, all components of the system except for THF have to be considered. This includes TMP, NMP, the number of active sites (alkoxides), glycidol, and the volume change of the system regarding the added volume of glycidol with its consumption and the density change due to polymerization. The presence of THF as a solvent to dilute the monomer can be neglected as it is immediately distilled as soon as it enters the reactor at the applied temperature. The coefficients for the simulation of thermally induced ringopening reactions of glycidol are based on our previously published data.26 This strategy significantly reduces the number of variables for the simulation and consequently the complexity of the model. Ring-Opening Reactions. Ring-opening reactions of epoxides can be thermally induced in the presence of an alcohol.34 This requires a certain energy input (e.g., temperature) to open the strained, three-membered epoxide ring. Furthermore, ring-opening reactions can be induced via a cationic or an anionic pathway.16,18 As the polymerization discussed in this paper proceeds via a thermal or anionic pathway, the cationic pathway will not be further considered. In general, the opening or consumption of an epoxide ring with an alcohol (from an initiating molecule or a monomer) does not change the total number of hydroxyl groups in the system but only reduces the number of available epoxides. Possible ring-opening reactions and first oligo-formations are depicted in Scheme 1. For this study, the nature of the hydroxyl group (primary or secondary) is neglected by assuming the same reactivity. This assumption significantly reduces the complexity of the model. The theoretical simplification is based on the delocalization of the potassium counterion. This solvent shared ion pair is not only located on the hydroxyl group but also rapidly transfers between all the hydroxyl groups in the investigated system. This is based on the so-called proton transfer equilibrium that is well-known for hydroxyl group containing systems20,35,36 in solution, which will be discussed later. It was further determined with systematic studies that rate coefficients and activation energies are most dependent on the polarity of the solvent rather than the nature of the ion pair.37 Although primary hydroxyl groups are supposed to be preferably deprotonated, the ratio of primary to secondary hydroxyl groups remains constant during the polymerization process as will also be discussed later. Cyclization Reaction. Cyclization reactions can occur in any ABm type molecule that contains the right number of atoms to form a thermodynamically favorable ring state. In general,

a

R represents the trifunctional initiator TMP, an epoxide, or any growing polymer species. The active sites are marked in red (proton, thermal propagation) and blue (potassium, base-mediated propagation).

ABm type molecules can react as follows: (a) an A group reacts intermolecularly with another B group, (b) a B group reacts intermolecularly with another A group, and (c) a B group reacts intramolecularly with an A group. Whereas the first two reactions are the most popular polymerization/propagation reactions for forming the desired polymer species, the last reaction (c) is not so favored because it leads to lower molecular masses and higher dispersity.25 Possible cyclization reactions for the investigated ROMB polymerization system are shown in Scheme 2. Scheme 2. Cyclization Reaction of the Smallest Possible Dimer:26 Thermal (a) and Base-Mediated (b) Cyclization Reactions Possibly Occurring during the ROMB Polymerization Processa

a

The cyclization reaction for the smallest thermodynamically stable ring size is presented.

To form a cyclic species, a hydroxyl and an epoxide group have to be in close proximity to each other, i.e., the formed backbone of a dimer or an oligomer has to bend accordingly. Under the applied conditions the monomers cannot undergo cyclization because of the low number of carbon atoms and a resulting lack of flexibility. The number of possible conformations in a molecule grows with the molecular mass, i.e., growing number of repeating units. Consequently, the number of conformations that result in a cyclization also increases proportionally to the molecular mass. It is not trivial to find an appropriate expression for the probability of a cyclization reaction to occur. There is still no clear answer in the literature for hyperbranched macromolecules. In this publication, we therefore follow the same argumentation as in our previous 8460

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study that bigger polymers are more likely to undergo cyclization.26 Self-Initiation of Glycidol. For the SMA strategy, both base-mediated and thermally induced self-initiation reactions (Scheme 1, R representing an epoxide) lower the targeted molecular weight and increase the dispersity due to possible cyclization. However, if a cyclic species does not form, the chain can grow similarly to a starter initiated species, i.e., with TMP as the starting moiety, which is irrelevant for the growing polymer chain. In consequence, the focal epoxide ring can combine with any alcohol or alkoxide group at any time during the reaction. These functional groups can originate from another monomer thus forming secondary polymers, i.e., another focal epoxide or a growing polymer chain (primarily, by incorporating TMP). If a TMP-initiated species recombines with a focal epoxide species, it is still possible to obtain a polymer with the target molecular mass. However, it is more likely that the focal epoxide will be consumed by a cyclization reaction before it can recombine, as seen in our previous study.26 Ion Transfer Reactions. As mentioned above, the whole ROMB polymerization mechanism is based on a fast proton exchange. In other words, the active alkoxide with its potassium counterion is delocalized over the whole system. Assuming this proton transfer equilibrium is independent from the nature of the hydroxyl group, the probability for each hydroxyl group in the system to deprotonate is equal. This is regardless of the size of the growing species (monomer, oligomer, polymer). From this assumption, the active site is not only able to transfer intramolecularly between hydroxyl groups on the one-chain or “tree-like” hPG architectures but also intermolecularly to the hydroxyl groups from other species (see Scheme 3).

Scheme 4. General ROMB Polymerization Mechanism for the Synthesis of hPGa

a

The scheme includes both base-catalyzed and thermal pathways: (i) deprotonation/initiation, (ii) propagation, (iii) intramolecular proton transfer, and (iv) intermolecular proton transfer to primary or secondary polymer or monomer.

The underlying mechanism is depicted in Scheme 4. For the self-initiation step, the core−OH moiety resembles an epoxide ring. Considering cyclization reactions, the core resembles a macrocyclic species (according to Scheme 2).

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COMPUTER SIMULATIONS In order to calculate the reaction rates correctly, the different molecular species are defined by the number of functional groups they contain. In the following we will denote an epoxide derivative as an A group and any hydroxyl group as a B group. Therefore, glycidol can be considered as AB molecule which can undergo a ring-opening reaction with any molecule that has at least one B group. Because of the fact that any ring-opening reaction consumes one A and one B group and results in the formation of a new B group, one can conclude that any such reaction will lower the total number of A groups by one and the total number of B groups will remain constant. In consequence, the systems can only contain molecules that have either one or zero A groups. As a result, the system can be described by two different polymer species: (a) ABs and (b) Bs (s: number of OH groups), of which the monomers AB (s = 1) and the trifunctional initiators B3 (s = 3) present special cases. Here the index s does not denote DPn unlike does the literature. However, DPn of a polymer is closely related to the number of B groups because the addition of a monomeric subunit will also increase the number of B groups by one. For clarification, we differentiate polymers by their functional groups and not their DPn because the key dependencies of the reaction rate equation will become more easily eluded this way. The reaction rates for the anionic ring-opening polymerization also depend on the distribution of the counterions which cause the presence of partially deprotonated hydroxyl groups, i.e., the number of active sites of a molecule. Because the number of active sites m is strictly related to the number of hydroxyl groups of a molecule, this quantity is modeled as superscript, and all possible polymers can be described by two 2-dimensional polymer species Bms and ABms . Knowing the number of epoxide derivatives, hydroxyl groups, and counterions of any molecule in the system, the reaction rates can be approximated very accurately. Assuming that every A group can react with any B group, all propagation, combination, and cyclization reactions can be modeled with the following set of reaction equations:

Scheme 3. Intra- and Intermolecular Proton Transfer Equilibrium of the Delocalized Potassium Counteriona

a

Three representative polyglycerol oligomers are presented with R and R′: initiator, epoxide, and polymer of different species.

Ion transfer reactions are essential for a regular, spherical growth. However, a transfer to monomer reaction (Scheme 4) can introduce a second species to the system. These additional polymers react very similarly to the TMP-initiated species and thus compete for free monomer. This leads to a reduced average molecular weight and higher dispersity. ROMB Polymerization Mechanism. In summary, the ROMB polymerization mechanism includes the following main and side reactions: thermal or base-catalyzed initiation, thermal or catalyzed propagation, intra- and intermolecular ion transfer to polymer or monomer, and thermal or catalyzed cyclization. 8461

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Macromolecules k ion,1m

Bms + ABnr ⎯⎯⎯⎯⎯→ Bms ++rn k thr,1(s − m)

Bms + ABnr ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯→ Bsm++r n k ion,1(m + n)

ABms + ABnr ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯→ ABms ++rn k thr,1(s − m + r − n)

ABms + ABnr ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯→ ABms ++rn k ion,2m

ABms ⎯⎯⎯⎯⎯→ Bms k thr,2(s − m)

ABms ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯→ Bms

Article

trajectories of the concentrations of every molecular species in the system.42 The described system is especially challenging because it incorporates a two-dimensional polymer species which could only be simulated in very simple cases.43 For the purpose of performing efficient and reliable simulations that allow the use of sophisticated parameter estimation methods, we chose the simulation software PREDICI. Although the treatment of two-dimensional polymer species is currently being studied and there is a realistic hope that such problems can be solved in the near future,44 the current version of the software is not capable of simulating such a complex system. The software package only provides possible ways to circumvent the former issue by estimating the number of active sites as a function of the number of hydroxyl groups that also depends on the overall ratio of counterions to hydroxyl groups. This ratio, which will from now on be referred to as p, is easily calculated because (a) the number of counterions is defined solely by the amount of anionic initiators and remains constant throughout the polymerization process and (b) the number of hydroxyl groups can be calculated in dependence of the total amounts of monomers and initiators being added to the system. Using this approximation technique, the molecules can be modeled with only two one-dimensional polymer species, whereas the index denotes the total number of hydroxyl groups. The system of reaction equations then reduces to the following:

(1a) (1b) (2a) (2b) (3a) (3b)

Although all second-order reactions were depicted by polymer combinations, the initiation, normal propagation, and spontaneous monomer combination are also included as special cases. The elemental initiation, propagation, and combination mechanisms are equivalent to those in the cyclization reactions, but the reaction rates are calculated differently for first- and second-order reactions. Hence, the rate coefficients of the intramolecular cyclization reactions (eqs 3a and 3b) are denoted by a different subscript (2 instead of 1). For reactions with a reaction rate of zero (m = 0 in eqs 1a and 3a, if m = s in eqs 1b and 3b, if m + n = 0 in eq 2a or if m + n = s + r in eq 2b), the system remains unchanged. However, they were still included to illustrate the complete system. Although the reaction equations describe all possible ringopening reactions, the ion transfer reactions have to be incorporated in order to estimate the distribution of counterions over the existing hydroxyl groups. As only a small amount of the hydroxyl groups in the system is deprotonated, the postulated fast proton exchange mechanism (inter- and intramolecular) is crucial for a uniform growth of the spherical architectures in all three dimensions.38 To the best of our knowledge, there are no publications presenting empirical data on the proton exchange equilibrium of a ROMB polymerization or any comparable polymerization system. Hence, we refrained from modeling the intramolecular transfer reactions and assured that the potassium ions were equally delocalized over all the hydroxyl groups of a molecule and, as a result, only considered intermolecular ion transfer reactions. The corresponding reaction rates were proportional to the product of active sites on the donor and the number of “inactive sites” on the acceptor, as can be seen in the following set of reaction equations: k tm(r − n)

Bms + Bnr ⎯⎯⎯⎯⎯⎯⎯⎯→ Bms − 1 + Bnr + 1 k tm(r − n)

Bms + ABnr ⎯⎯⎯⎯⎯⎯⎯⎯→ Bms − 1 + ABnr + 1 k tm(r − n)

ABms + Bnr ⎯⎯⎯⎯⎯⎯⎯⎯→ ABms − 1 + Bnr + 1 k tm(r − n)

ABms + ABnr ⎯⎯⎯⎯⎯⎯⎯⎯→ ABms − 1 + ABnr + 1

k feed(t )

∗ ⎯⎯⎯⎯⎯⎯→ AB1

(I)

1 k ion (s + r )p

ABs + ABr ⎯⎯⎯⎯⎯⎯⎯⎯⎯→ ABs + r 1 k thr (s + r )(1 − p)

ABs + ABr ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯→ ABs + r 1 k ion sp

Bs + ABr ⎯⎯⎯⎯→ Bs + r 1 k thr s(1 − p)

Bs + ABr ⎯⎯⎯⎯⎯⎯⎯⎯⎯→ Bs + r 2 k ion sp

ABs ⎯⎯⎯⎯→ Bs 2 k thr s(1 − p)

(IIa) (IIb) (IIIa) (IIIb) (IVa)

ABs ⎯⎯⎯⎯⎯⎯⎯⎯⎯→ Bs

(IVb)

The illustrated system can easily be implemented in the simulation software, thus allowing the estimation of the four remaining reaction coefficients. By implicitly accounting for the ion transfer reactions, the corresponding reaction coefficients no longer have to be considered. For the parameter estimation procedure, we use the built-in tools that the simulation software provides. This includes a parameter variation method, which is a Gauss−Newton method for local and simulated annealing for a global optimization strategy.45 In order to assess the quality of a simulation run, we calculated the total residual sum of squared differences between the simulated and the experimental data. This quantity then served as the objective function that was minimized by the parameter estimation method.

(4a) (4b) (4c) (4d)

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The time-dependent changes in the molecular species can now be simulated using eqs 1a−4d for the kinetic reaction rate coefficients kion,1, kthr,1, kion,2, kthr,2, and kt. The set of reaction equations can easily be transformed into ordinary, previously discussed differential equations (ODEs).30,39,40 The ODEs, which are often referred to as population balance equations (PBE),41 can be used to solve an initial value problem (IVP) using numerical methods to simulate time

RESULTS AND DISCUSSION Experimental Results. The MALDI-TOF-MS characterization yields a rough and incomplete picture of the synthesized polymer (see Figure S1). Because of its hyperbranched nature and an elevated dispersity, it is not possible to capture all the molecules. Up to a certain threshold (around 5 kDa with low 8462

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0.41. As the reaction proceeded, however, this ratio only fluctuated around a constant value of 0.38 ± 0.021, as depicted in Figure 1. Hence, the overall reaction rates were estimated

dispersity), one can track the growth of the spherically and branched polymer chains. However, beyond this threshold it seems that the number of flying and detected molecules decreases. As the measured number of molecules might only represent a certain molecular weight cut out, MALDI-TOF-MS characterization is not included into the simulation process. Although it is possible to distinguish between TMP-initiated and self-initiated species, it cannot be determined whether the propagation occurred anionically or thermally. Therefore, both pathways have to be equally considered. The NMR measurement gives a picture of the growing polymer and the development of the DB (see Figures S2 and S3). To our knowledge, the number of secondary hydroxyl groups in the system is about 2.5−3 times higher than the number of primary hydroxyl groups due to their lower reactivity.8,26 The nature of the hydroxyl group can be determined by 13C NMR spectroscopy based on the signal of the branching (linear, dendritic, terminal) and hence the DB (see Figure S2).38 Referring to the type of branching to the hydroxyl groups, linear 1−3 connected monomers (L13) represent a primary alcohol (pOH) and linear 1−4 connected monomers (L14) a secondary alcohol (sOH). Table 1 summarizes the DB−time profile of a hPG polymerization with a target molecular mass of 5 kDa.

Figure 1. Representative time profile for the ratio of primary (pOH) to secondary (sOH) hydroxyl groups in the ROMB polymerization with target molecular mass 5 kDa.

with the four mentioned coefficients kion,1, kion,2, kthr,1, and kthr,2 for base-catalyzed and thermal propagation, assuming that there was a rapid proton transfer equilibrium. Keeping in mind that there is no comparable standard for hyperbranched, water-soluble polymers, the remaining characterization method GPC still delivers the best data for the simulation process. Pullulan has a linear structure, whereas dextran is a partially branched polymer varying from 0 to 40% branching. Nevertheless, there were no obvious deviations in the analyzed molecular weight range after characterizing the synthesized polymer with both standards. Comparing both accessible standards, it is more reasonable to refer to the linear rather than the branched standard, as the dispersity range in the investigated region was around 2.50. In general, using a linear polymeric standard leads to overestimated molecular masses, whereas branched standards lead to underestimation. Unfortunately, determination by multiangle linear laser light scattering (MALLS) is not possible as the investigated polymers possessed only a small hydrodynamic volume below the measuring threshold. To obtain the complete MWD, all samples were characterized by GPC, and the number averages (Mn) as well as the weight-average (Mw) molecular weights were processed by parameter simulation. A complete overview of the time profiles for the three target molecular weights is presented in Table 2. Additionally, Figure 2 depicts the time profiles with a diagram comparing the measured MWD to the calculated overall molecular mass. To reduce any viscosity effects induced by altering amounts of glycidol, the amount of TMP was varied to obtain different molecular masses. The molecular weight for the small hPGs (Figure 2a,b) linearly grew with the theoretical molecular mass. As the monomer was linearly added and apparently immediately consumed, the reaction proceeded as expected. For the polyglycerol targeting 2.5 kDa the Mn increases slower and the Mw faster than the theoretical molecular weight. The MWD development may be caused by complex viscosity effects. From the experimental setup to target different molecular weights, the initial concentration of TMP is changed. Consequently, the relative viscosity change throughout the polymerization is different for each molecular weight. This may influence the growth behavior of the spherical polymer, the monomer distribution, the general diffusion of all species, or the heat transfer within the system.

Table 1. Experimental Time Profiles for a hPG Polymerization (MWtarget = 5 kDa) Obtained from Inverse Gated 13C-NMR Characterizationa time [h] 2 4 6 8 10 12 14 16 18

L13 (%) 9.14 10.6 11.5 10.3 10.1 10.9 10.8 10.5 10.8

± ± ± ± ± ± ± ± ±

1 1 1 1 1 1 1 1 1

L14 (%) 25.7 28.0 27.8 27.9 27.8 27.7 27.1 28.2 28.8

± ± ± ± ± ± ± ± ±

1 1 1 1 1 1 1 1 1

D (%) 12.9 19.1 21.5 23.5 24.3 24.5 25.2 25.5 25.4

± ± ± ± ± ± ± ± ±

1 1 1 1 1 1 1 1 1

T (%) 52.3 42.2 39.2 38.4 37.9 37.0 37.0 35.8 35.8

± ± ± ± ± ± ± ± ±

DB (%) 1 1 1 1 1 1 1 1 1

59.9 59.4 62.0 61.9 64.2 62.0 61.6 60.0 63.5

± ± ± ± ± ± ± ± ±

1 1 1 1 1 1 1 1 1

a

L13: content of linear 1−3 connected monomers; L14: content of linear 1−4 connected monomers; D: ratio of dendritic units; T: ratio of terminal units; DB: degree of branching was calculated according to an equation describing general AB2 polymerization systems (see Supporting Information).8

It is obvious that after 2 h reaction time the ratio of dendritic units was comparably low. Considering the Mn (MWtarget = 5 kDa) the number of repeating units is