Polydispersity and Molecular Weight Distribution of Hyperbranched

Jul 29, 2013 - polymer topologies with defined molecular weight, degree of branching. (DB), and ... In addition, the complete molecular weight distrib...
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Polydispersity and Molecular Weight Distribution of Hyperbranched Graft Copolymers via “Hypergrafting” of ABm Monomers from Polydisperse Macroinitiator Cores: Theory Meets Synthesis Christoph Schüll,†,§ Hauke Rabbel,‡ Friederike Schmid,*,‡ and Holger Frey*,† †

Institute of Organic Chemistry, Johannes Gutenberg-University, Duesbergweg 10-14, 55128 Mainz, Germany Institute of Physics, Johannes Gutenberg-University, Staudinger Weg 9, 55128 Mainz, Germany § Graduate School Materials Science in Mainz, Staudinger Weg 9, 55128 Mainz, Germany ‡

S Supporting Information *

ABSTRACT: The hypergrafting strategy designates the synthesis of hyperbranched graft copolymers (HGCs) in a grafting-from approach, using ABm monomers, from multifunctional, polydisperse macroinitiator cores by slow monomer addition. Hypergrafting leads to complex polymer topologies with defined molecular weight, degree of branching (DB), and polydispersity (PD). By a generating function formalism, a generally applicable equation for the PD of HGCs (PD = PDf + (m − 1)/ f ̅) is derived, where PDf is the polydispersity of the core and f ̅ its average functionality. In addition, the complete molecular weight distribution function has been calculated for varied m and f ̅ as well as for a given distribution of initiator functionalities f. For comparison of the theoretical predictions with experimental results, a series of novel linear polyglycerol-graf t-hyperbranched polyglycerol (linPG-g-hbPG) HGCs (Mn = 1000−4000 g mol−1) were synthesized and characterized as a model system. An increase in polydispersity occurred as a consequence of the hypergrafting process, confirming the theoretical predictions of the novel equation. Moreover, the model system allows for the determination of the DB of hbPG prepared by hypergrafting from linear polyglycerol macroinitatiors (DB = 0.59−0.61). The theoretical results presented are key to achieve control over the branch-on-branch topology of hyperbranched blocks in nonconventional polymer architectures, such as linear−hyperbranched block copolymers.

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INTRODUCTION

If hyperbranched macroinitiators are employed for hypergrafting, hyperbranched-graf t-hyperbranched copolymers (HHGCs) can be generated, which are potentially interesting as core−shell structures for biomedical transport applications.6,7 In a work by our group, low molecular weight hbPG was used as a macroinitiator core for the polymerization of glycidol to enlarge the available molecular weights of hbPG up to 24 kg mol−1.8 Moreover, the hypergrafting of ABm monomers was applied in several works to synthesize linear−hyperbranched block copolymers (LHBCs).9,10 By using linear−linear block copolymer macroinitiators with one block containing multiple initiating groups for the polymerization of ABm monomers, the hyperbranched block can by synthesized by hypergrafting from the linear “macroinitiator block”. Using this approach, polystyrene-block-hyperbranched polyglycerol (PS-b-hbPG)11 or poly(ethylene glycol)-block-hyperbranched polyglycerol (PEG-b-hbPG)12 has been synthesized by hypergrafting of glycidol. PEG-b-hbPG copolymers are interesting for various applications in bioconjugation13 or the preparation of liposomes for drug delivery14 due to the excellent biocompatibility

One of the current key challenges in polymer science is the controlled synthesis of multifunctional, complex polymer architectures.1 Hyperbranched polymers are characterized by a branch-on-branch topology and can usually be synthesized in one reaction step.2,3 Moreover, they are interesting building blocks for nonconventional copolymer topologies such as block and graft copolymers, which all exhibit properties that differ entirely from their linear counterparts. In this context, the hypergrafting strategy (grafting-f rom), i.e., the synthesis of a hyperbranched block copolymer by using a multifunctional macroinitiator core for the polymerization of ABm monomers, is a useful synthetic method to attain control over the key parameters molecular weight, polydispersity (PD), and degree of branching (DB).4a Scheme 1 shows different hyperbranched graft copolymer architectures with linear or hyperbranched macroinitiators used for the hypergrafting reaction, exemplified for an AB2 monomer. Glycidol is an important AB2 monomer, which can be polymerized in a controlled manner by ring-opening multibranching polymerization (ROMBP) to give hyperbranched polyglycerol (hbPG),4b,5 which has been used as a building block for several hyperbranched graft copolymers (HGCs). © 2013 American Chemical Society

Received: May 29, 2013 Revised: July 12, 2013 Published: July 29, 2013 5823

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(SCVP)31 of AB2 inimers in the presence of multifunctional cores Bf (f: number of core functionalities) was developed (eq 1).32

Scheme 1. Hypergrafting of AB2 Monomers from Linear or Hyperbranched Macroinitiator Cores for the Synthesis of Hyperbranched Graft Copolymer (HGC) Topologies

PD = 1 +

1 f

(1)

According to expression 1, with increasing core functionality f the polydispersity of the resulting hyperbranched polymer can be reduced significantly. In another work based on simulation studies, Hanselmann et al. proposed a more general expression to describe the polydispersity for the SMA of ABm monomers (eq 2).33 PD = 1 +

m−1 f

(2)

In this work and others,34 only monodisperse cores in the range of f = 2−12 were investigated. In addition, eq 2 was not derived systematically, but only conjectured, based on simulation results for the cases m = 2 and m = 3. Accordingly, a general mathematical derivation of eq 2, which can also be used to calculate other statistical quantities of interest and which can be extended to include polydisperse cores, is highly desirable. This work represents a combined theoretical and experimental study for HGCs prepared by hypergrafting, using the slow monomer addition (SMA) technique. First, the key structural parameters of HGCs are defined, which are applicable for any reaction system. Second, two equations for the prediction for the PD for any monodisperse and polydisperse macroinitiator core B f and arbitrary AB m monomers are derived. Using the same approach, the molecular weight distribution function is calculated. Third, the synthesis of a novel HGC by hypergrafting, consisting of a linear polyglycerol core and hyperbranched polyglycerol side chains, is presented. This synthetic model system is then used for the experimental evaluation of the theoretically derived relationships.

of PEG and hbPG.15 In addition, amphiphilic LHBCs with a hyperbranched carbosilane block were prepared by hypergrafting AB2 carbosilane monomers from multifunctional PS16 or PEG17macroinitiators. These works underline the general applicability of the hypergrafting strategy for chemically diverse monomers. The hypergrafting strategy can be applied to synthesize linear−hyperbranched graft copolymers (LHGCs) as well. These polymers consist of a linear backbone with hyperbranched side chains,18,19 which represent an interesting alternative to dendronized polymers20 (linear polymers with perfectly branched side chains) that are able to form cylindrical “nano-objects” due to the high steric demand of the side chains.21 The use of hyperbranched side chains is potentially advantageous, since they can be synthesized in a single reaction step by hypergrafting.22 In all these cases, the slow monomer-addition technique (SMA) (also called “semibatch process”) in the presence of a multifunctional initiator core was applied for the hypergrafting step. This enables control over molecular weight in a pseudochain-growth mechanism and ensures low polydispersities, which are crucial to gain high molecular precision of the overall polymer structure. Moreover, side reactions such as the formation of oligomers and cyclic side products can be suppressed under optimized reaction conditions.4 A number of theoretical works have derived relevant structural parameters of hyperbranched homopolymers. Already in 1952 Flory described the polycondensation of ABm monomers.23 After the first introduction of the term “hyperbranched polymer”,24 excellent works pioneering the theoretical discipline of AB2 polycondensation25−27 followed. It was found that for the polycondensation of ABm monomers the presence of multifunctional cores plays a major role to lower the PD (polydispersity index Mw/Mn) of the resulting hyperbranched polymers.28,29A recent review describes the overall progress in the field of kinetic theory of hyperbranched polymers comprehensively.30 In an important work by Radke, Litvinenko, and Müller, a general expression for the PD of hyperbranched polymers prepared via the self-condensing vinyl polymerization

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EXPERIMENTAL SECTION

Materials. All reagents were purchased from Acros Organics or Sigma-Aldrich and used as received, unless otherwise stated. Ethoxy ethyl glycidyl ether (EEGE), glycidol, and diglyme were freshly distilled from CaH2 prior to use. Instrumentation. 1H NMR spectra were recorded at 300 MHz on a Bruker AC300, and 13C spectra were recorded at 100 MHz on a Bruker Avance-II 400. All signals were referenced internally to residual proton signals of the deuterated solvents. For SEC measurements in DMF (containing 0.25 g/L of LiBr), an Agilent 1100 series was used as an integrated instrument including a PSS Gral column (106/104/102 Å porosity) and a RI as well as a UV detector. Calibration was achieved with poly(ethylene glycol) standards provided by PSS (Mainz). Synthesis of Linear Polyglycerol Macroinitiators (linPG). The monomer ethoxyethyl glycidyl ether (EEGE) was synthesized according to the literature.35 In a typical procedure, methoxyethanol was placed in a dry Schlenk flask, and the corresponding amount (0.9 equiv) of cesium hydroxide monohydrate and some milliliters of dry benzene were added. The suspension was stirred for 30 min under an argon atmosphere, then all solvents were removed under reduced pressure, and the initiator salt was dried for 4h at 80 °C in vacuo. EEGE and dry 1,4-dixoane (1:1 vol %) were added by syringe under vacuum to start the polymerization. After completion of the reaction, 1 mL of degassed methanol was added. The vessel was cooled to 40 °C, and a mixture of methanol/water/1 M HCl (8:1:1 vol %) was added to remove the protective groups. 1H NMR (300 MHz, CDCl3) shows the absence of the signals of the protective groups at δ [ppm] = 4.62 5824

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(br, 1H, OCHO) and 1.28−1.15 (br, 6H, 2 × CH3). The mixture was concentrated and precipitated into an excess of diethyl ether. Yield: 80−90%. 1H NMR (300 MHz, DMSO-d6): δ [ppm] = 4.25 (br, −OH), 3.72−3.25 (br, polyether backbone), 3.24 (s, 3H, CH3, initiator). Synthesis of Linear Polyglycerol-graf t-Hyperbranched Polglycerol (linPG-g-hbPG). LinPG was placed in a dry Schlenk flask, and a corresponding amount of cesium hydroxide monohydrate and some milliliters of dry benzene were added to deprotonate the hydroxyl groups by 10%. The suspension was stirred for 30 min under an argon atmosphere, then all solvents were removed under reduced pressure, and the initiator salt was dried overnight at 80 °C in vacuo. After suspending the initiator salt by adding a small amount of dry diglyme, a solution of glycidol in diglyme (15 vol %) was slowly added via a syringe pump overnight. After the addition of 1 mL of methanol at room temperature, the mixture was concentrated and precipitated into an excess of diethyl ether. Yield: 80−90%. 1H NMR (300 MHz, DMSO-d6): 4.87 (br, −OH), 3.78−3.25 (br, polyether backbone), 3.23 (s, 3H, −CH3, initiator). 13C NMR (100 MHz, MeOH-d4): detailed signal assignment of the hyperbranched repeat units was conducted according to the literature.4

Species i is converted to species j at a rate of xij. Let Gi(t) be the concentration of the polymer species with degree of polymerization DP = i at time t. In a slow addition process, the DP is changed in time by two processes: (i) It is increased by reaction of additional monomers with polymers of DP = i − 1. (ii) The DP is decreased by reaction of monomers with the polymers of DP = i, which then do not contribute to Gi any longer. Consequently, only the rates xi−1,i and xi,i+1 are nonzero. The distribution of the species at time t is then given by the solution of the set of n coupled differential equations, as given in eq 4 using Gi(t = 0) = G0i as the original condition. Gi̇ (t ) = −xi , i + 1Gi(t ) + (1 − δi0)xi − 1, iGi − 1(t )

This is the classical CME for the special case of slow monomer addition. The term (1 − δi0) accounts for the fact that G0 can only decrease during the process. Neglecting steric effects, the reaction rates can be assumed to be proportional to the number of reactive groups of the polymer. If the monomer functionality is given by m, the HGCs gains a number of m − 1 additional end groups for each monomer conversion. Taking into account that each free reactive B or B′ group of the initiator reacts with equal probability and that the reaction rate is proportional to the monomer concentration M and some rate constant k, the reaction rates are obtained as given in eq 5, where f represents the core functionality.

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RESULTS AND DISCUSSION A. Theoretical Considerations. Hyperbranched graft copolymers (HGCs) can by synthesized by hypergrafting (graf ting-f rom) of an ABm monomer from a multifunctional macroinitiator core. Other strategies like graf ting-to and graf ting-through have also been introduced recently.22,36 The hypergrafting strategy possesses the advantageous feature that only low molecular weight compounds have to be removed during purification, if monomer conversion is incomplete. The challenge of incomplete conversion of core functionalities can be overcome by using highly reactive cores, as described recently.22 The polydisperse Bf ̅ core contains a number-average of f ̅ B′ functionalities that can only react with the A group of the monomers to irreversibly form an AB′ bond. In analogy, a monodisperse macroinitiator core Bf possesses exactly f B groups. The number of attached ABm monomers equals the degree of polymerization (DP). These monomers form AB bonds in the side chains. For the addition of each ABm monomer, one core functionality B′ is consumed and (m − 1) new B functionalities are generated. The increase in the number of B groups is accompanied by the formation of branching points in the hyperbranched side chains. Similar to hyperbranched homopolymers, the degree of branching (DB) is the essential parameter to evaluate the “perfection” of branching compared to perfectly branched dendrimers. A low polydispersity index (PD = Mw/Mn) is crucial in order to guarantee maximum chemical precision of the targeted structures. Polydispersity Index. In the following, a general expression will be derived for the polydispersity of HGCs, prepared by hypergrafting from monodisperse and polydisperse macroinitiator cores Bf and Bf,̅ respectively, using the slow monomer addition (“semibatch process”) of ABm monomers. The reaction kinetics of a well-stirred mixture of molecular species is modeled by a chemical master equation (CME).37 In this system, the term “well stirred” implies that the concentrations of the molecules are considered to be consistently distributed. Equation 3 sketches a reaction system consisting of n different species with concentration Gi.

xi , i + 1 = kM(f + i(m − 1))

(5)

Important statistical parameters of polymerization reactions, such as the moments μn, molecular weight averages, and the PD of the resulting distributions, can now be calculated using a generating function method. The method is based on the theory of Markov branching processes (for details see Supporting Information), as it has been pointed out by Gordon38 and has been used in polymer science ever since.32 Defining the generating function ∞

ϕf (s , t ) =

∑ Gk(t )s f + k(m− 1)

(6)

k=0

one can calculate the moments of the molecular weight distribution by differentiating Φf(s,t). Taking the molecular weight of a monomer to be one, the moments are defined as μn = ∑i( f + i)nGi. For n = 0, 1, 2 the sums can be evaluated using eq 7. ∞

∑

Gi(t ) = ϕf (1, t )

(i = 0) ∞

∑

iGi(t ) =

(i = 0) ∞

∑ i 2Gi(t ) = i=0

1 ∂ ⎛ ϕ(s , t ) ⎞ ⎜ ⎟ m − 1 ∂s ⎝ s f ⎠

∂ ⎛ ϕ(s , t ) ⎞ 1 ⎜ ⎟ 2 (m − 1) ∂s 2 ⎝ s f ⎠

s=1

2

∂ ⎛ ϕ(s , t ) ⎞ ⎜ ⎟ ∂s ⎝ s f ⎠

+ s=1

1 (m − 1)2

s=1

(7)

For higher order moments, corresponding expressions can be derived in a straightforward manner. This calculation requires the knowledge of an explicit form of Φf(s,t), which can be acquired as a solution of the equation.

xij

Gi → Gj

(4)

(3) 5825

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= kM[(ϕ1)m − ϕ1]

considered to develop independently. This is obvious in the slow monomer addition process from a synthetic point of view. Accordingly, the equations derived for the moments of the distribution can be applied to each subset separately. The PD arising from slow monomer addition of ABm monomers to a polydisperse mixture of cores with different functionalities f is then given by m−1 PD = PDf + f̅ (12)

(8)

for Φ1(s,t). Φf(s,t) is then given by Φf = (Φ1) . Details on this derivation can be found in the Supporting Information. The resulting generating function is f

ϕ(s , t ) = G00s f [ekMt(m − 1) − (ekMt(m − 1) − 1)s m − 1]−f /(m − 1) (9)

Inserting eq 9 in eq 7 finally yields the desired expression for the polydispersity index M m−1 1+X PD = w = 1 + Mn f (1 + (m − 1)X )2

Here, PDf is the initial core polydispersity and f ̅ is the average core functionality. Details on the mathematical derivation can be found in the Supporting Information. Equation 12 states the key theoretical result of this work, demonstrating that low polydispersity of the initially used polymer core (PDf) is essential to maintain low polydispersity after grafting of the ABm monomers. Figure 2 displays the correlation of PD and the average core functionality f ̅ for different PDf values, using AB2 monomers.

(10)

with 1/X = (ekM(m−1)t − 1) = (m − 1)(Mn/f − 1), which in the limit of X → 0 or Mn → ∞ leads to the following important result: DP =

Mw m−1 =1+ Mn f

(11)

This is the same result as suggested by Hanselmann et al. (eq 2) on the basis of computer simulations for m = 2, 3 and f = 1− 12.33 With this systematic derivation, we demonstrate that eq 2 is valid for any arbitrary m and f value. However, one has to consider that to this point eq 10 is so far limited to monodisperse cores Bf. Figure 1 shows the development of

Figure 2. Correlation of the polydispersity (PD) of hyperbranched graft copolymers on the average core functionality f ̅ for the hypergrafting of AB2 monomers from polydisperse macroinitiator cores Bf.̅ PDf is the polydispersity of the core. This correlation is valid in the limit Mn → ∞ (cf. eq 11).

These parameters represent a typical synthetic reaction system, as it is found for the hypergrafting of glycidol (see below). The PD of the macroinitiator core significantly affects the PD value of the resulting HGC. For instance, if the macroinitiator possesses a PDf value of 1.3, at least five initiation sites Bf are necessary to obtain final polydispersities below 1.5 for the hyperbranched graft copolymers. For macroinitiators having PDf = 1.8, PDs around 2 are found even for average core functionalities of 10. In summary, for the slow monomer addition process, the PD value of the resulting polymers always increases compared to the PD value of the macroinitiators employed. This result is remarkably different from condensation processes, where a decrease in polydispersity was described for the addition of AB monomers to a Bf core, derived for star polymers long ago.39 Obviously, eq 11 (= eq 2 suggested by Hanselmann et al.33) for the case of monodisperse core molecules is the special case of eq 12 with PDf = 1 and f ̅ = f. Accordingly, eq 12 can not only be used to describe the hypergrafting process from polydisperse macroinitiator cores, but also for the simple case of hyper-

Figure 1. Correlation of the polydispersity index (PD) of hyperbranched graft copolymers on the core functionality f for the hypergrafting of ABm monomers from monodisperse cores Bf, applying the limit Mn → ∞ (cf. eq 11).

PD depending on f (monodisperse) for m = 2, 3, 4. It is demonstrated that for increasing monomer functionality m increasing PD values are obtained. Moreover, this plot emphasizes that at least three core functionalities are required for an AB2 monomer system (e.g., glycidol) to reach moderate to narrow molecular weight distributions with PD < 1.5. In synthetic works, mostly polydisperse macroinitiator cores Bf ̅ have been used for hypergrafting.12,22 Therefore, eq 11 has to be extended for this case. From a synthetic point of view, it can easily be assumed that core molecules do not react or interfere with each other. Hence, the polydisperse reaction mixture is split into separate monodisperse components that can be treated mathematically equal. Those subsets are defined by their respective (monodisperse) core functionality Bf and are 5826

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branched polymer synthesis by slow monomer addition of ABm monomers from Bf cores. Molecular Weight Distribution Function. To complete the picture, an explicit analytical solution for the molecular weight distribution {Gi} resulting from eq 4 for initial conditions G0i = Gi(t = 0) is derived. Since eq 4 is linear, the solution for arbitrary initial conditions Gi(t = 0) = G0i , i ∈  can be obtained by superposition of the solutions with the more simple initial condition Gi(t = 0) = δikG0k. However, the full solution with arbitrary initial conditions is presented. The solution of this set of equations can be acquired in three steps: (1) Laplace transformation of the master equation, (2) solution of the algebraic equations in the frequency domain, and (3) Laplace back-transform of the solution. Details on these calculations can be found in the Supporting Information. The resulting molecular weight distribution function turns out to be i

Gi(t ) =

∑ e −x t k

k=0

Lik (0)xk xi

k

∑ Gn0Xkn n=0

⎧1 n=0 ⎪ n−1 Xkn = ⎨ ⎪∏ (xj − xk)/xj n ≤ 1 ⎩ j=0

(13)

Here, Lik are the Lagrange base polynomials with nodes xl. i

x − xl xk − xl

(14)

Figure 3. Molecular weight distribution function for the hypergrafting of ABm monomers from monodisperse cores Bf for fixed m and variable f values (top) and fixed f and variable m values (bottom).

This function solves the CME (eq 4) for any given initial value G0i , i = 0, 1, 2, .... Figure 3 shows the molecular weight distribution function according to eq 13 for f = 6 and variable m values (Figure 3a) and for fixed m = 3 and variable f values (Figure 3b). Interestingly, a slight asymmetric broadening toward higher molecular weights for increasing m and decreasing f values is observed. With a few steps of reshaping, it can be seen that the calculated solution reduces to the expression given by Radke et al.32 for the special case of m = 2. B. Comparison of Theoretical Predictions and Experimental Results. Polydispersity. To verify the theoretical predications, a novel synthetic model system has been developed. To this end, the synthesis of linear−hyperbranched graft copolymers by hypergrafting of the AB2 monomer glycidol from polydisperse linear polyglycerol (linPG) macroinitiator cores using the slow monomer addition technique has been conducted (Scheme 2). The linPG macroinitiators have been prepared in the molecular weight range of polydisperse hyperbranched polyglycerol (hbPG) macroinitiators, which have been utilized in the synthesis of high molecular weight hbPGs by hypergrafting of glycidol in an earlier work by our group.8 In this manner, the molecular parameters such as achievable molecular weight, polydispersity, and degree of branching for the hypergrafting of glycidol from chemically similar polydisperse macroinitiators were investigated, independent of the macroinitiator topology. This is essential for comparison with the theoretical predictions, as topologies are not considered in this context. The use of polyglycerol as a macroinitiator guarantees the same reactivity of the macroinitiator and the side chains (B′  B, Figure 1), which is also one of the assumptions for the theoretical derivations.

The linPG macroinitiators have been synthesized by anionic ring-opening polymerization of the established oxirane monomer ethoxyethyl glycidyl ether (EEGE)35 using methoxyethanol as initiator and cesium as a counterion. Anionic polymerization leads to adjustable molecular weights and consequently adaptable f ̅ values for the synthetic model system. The acetal protective groups are stable under the highly basic reaction conditions during polymerization and can be removed conveniently under mild acidic conditions, providing linPG. The molecular weights of the macroinitiators were determined by 1H NMR spectroscopy and SEC (Table 1). SEC data were used for the nomenclature of the samples linPGn, where n states the degree of polymerization. The average core functionality can be calculated to be f ̅ = n + 1 due to the additional terminal hydroxyl group. The synthesis of linear polyglycerol-graf t-hyperbranched polyglycerol (linPG-g-hbPG) was performed by hypergrafting of glycidol under SMA conditions to control molecular weight, polydispersity, and degree of branching. The targeted molecular weight can conveniently be set via choosing the corresponding linPG/glycidol ratio. After optimization of the reaction parameters (temperature, counterion, dilution of the monomer solution, and injection time), polymer architectures with low PD values were obtained. Table 1 gives an overview of all polymers prepared and the characterization data. The molecular weights of the resulting linear−hyperbranched graft copolymers (LHGCs) were determined by SEC and 1H NMR spectroscopy. For all samples, we found monomodal molecular weight distributions in SEC (Figure S2, Supporting Information) with number-average molecular weights (Mn) between 1000 and 4000 g mol−1. Molecular weights have been kept low to ensure

Lik (x) =

∏ l=0 l≠k

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Scheme 2. Synthesis of Linear Polyglycerol (linPGn) Macroinitiators and Corresponding Hyperbranched Graft Copolymers (Linear Polyglycerol-graf t-Hyperbranched Polyglycerol, linPGn-g-hbPGm) by Hypergrafting of the AB2 Monomer Glycidol

linPGmacroinitiators. Nearly all PD values of the PEG-b-hbPG graft copolymers are approximately 1.2, which is in excellent agreement with the prediction from eq 12. In a work by Wilms et al., hyperbranched polyglycerol (hbPG) macroinitiator cores were utilized for the hypergrafting of glycidol.8 Two hbPG macroinitiators (Mn = 560 g mol−1, f ̅ = 7, PD = 1.32 and Mn = 1000 g mol−1, f ̅ = 13, PD = 1.40) with molecular parameters comparable to the presented linPG system have been used for the hypergrafting of glycidol. After hypergrafting, the authors found slightly increased PD values between 1.38 and 1.59 for degree of polymerization (DP) of glycidol between 54 and 224 and PDs up to 1.77 for DP of glycidol up to 324. Contrary to the novel linPG system, a high DP of polyglycerol is possible in this work due to a different reaction setup with intensive mechanical stirring. To sum up briefly, all polymers from the novel model system as well as from published works exhibit moderate to narrow PDs (mostly