Statistical Properties of Linear-Hyperbranched Graft

Home , Backbone chain

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B B BB B applied, where Φ is the angle between subsequent bonds n A + B B B and the bending constant K is chosen K = 10ǫ. Here B and throughout, the simulation units of length and en- F F F B ergy are the interaction parameter σ and ǫ of the WCA B B potential, and the time unit is τ = mσ/ǫ, where m is F FF B B B B the mass of one bead. ABm monomersp are modelled as dimers of two beads, one bead representing the A reactive FIG. 1. Schematic example of hypergrafting of AB2 group, and the other one all the B groups. We do not in- monomers on a linear (Ff ) backbone chain. clude explicit solvent. Since the beads repel each other, our system corresponds to a polymer in solvent under good solvent conditions. We carry out Molecular Dy- growth” process which allows for cluster-cluster aggre- namics simulations using the Velocity-Verlet algorithm gation. This generates chains with a fractal structure and a Langevin thermostat at temperature kBT = 0.5ǫ and Flory exponent ν 1/4. The second is a ”slow with the time step 0.005τ. The simulations were carried growth” process where≈ monomers are attached sequen- out using the open source program package ESPResSo tially to random sites of the chain, however, without [43], which we extended to allow for chemical reactions having to diffuse there first. This was found to result between monomers and the main chain. in dense chains where Rg scales logarithmically with the The monomers are created and equilibrated in a sepa- chain length, in accordance with the theoretical predic- rate reservoir and then added one by one in intervals of tion of Konkolewicz for short chains [25]. Juriju et al. 1000τ to the main system, which initially contains one [31] also investigated the effect of excluded volume inter- equilibrated backbone chain made of f beads F. This actions and found that they have almost no influence on is done by placing the monomers randomly on the sur- the results. face of a sphere centered at the center of mass of the In the present paper, we focus on linear-hyperbranched polymer, whose radius is 5σ larger than the largest dis- graft copolymers (LHGCs), which have recently been tance between the center and a chain bead (A, B, or F). proposed as an interesting alternative to standard hy- Fully periodic boundary conditions are applied such that perbranched polymers and dendronized polymers. Den- monomers cannot leave the simulation box. Once they dronized polymers have a linear backbone decorated with have diffused close to a chain bead, they may react with dendritic side chains [36–38], which gives them a cylin- the chain, thus making the polymer grow. In that case, drical shape. While they are interesting macromolecular a new harmonic bond is established between an A bead objects, their production is as cumbersome as that of and a B or F bead. dendrimers. LHGCs on the other hand also have a lin- Reactions may take place under the following condi- ear backbone, but use hyperbranched side chains instead tions [5, 14, 39–41]. They can be synthesized by hypergrafting The distance of the potential reaction partners from the backbone (see Fig. 1). In a recent paper [15], we • ((A,B) or (A,F)) is less than a critical distance have calculated the expected polydispersity index of such r =1.1σ. chains, taking into account the core dispersity, within c a rate equation theory. Here we will present off-lattice None of the beads involved in a reaction have al- Monte Carlos simulation of the growth of such chains • ready reached the maximum number of allowed re- under slow monomer addition, assuming that the poly- actions. A and F beads can react only once, B merization reaction is diffusion controlled, and analyze beads may react m times. the resulting topological and conformational properties. One of the partners belongs to the central chain. The paper is organized as follows. In the next section, • we introduce the simulation model and method. Then Reactions between free ABm monomers are not al- we will discuss the properties of the resulting chains, fo- lowed. cussing on statistical and topological properties in section If these conditions are fulfilled in a given time step, the III A and on conformational properties in section III B. monomers react with each other with a probability p. We We summarize and conclude in section IV. chose p = 1 for reactions with the backbone chain ((A,F) reactions), and p = 1 (”high reactivity”) or p = 0.01 (”low reactivity”) for (A,B) reactions. Since monomer II. SIMULATION MODEL pairs that meet the above conditions in a given time step may still have a distance less than rc in the subsequent We use a simple bead-spring model for the chains. time steps, the net reactivity (i.e. , the fraction of colli- Beads repel each other via a repulsive WCA potential sions that lead to bond formation) is not equal to p for [42] with diameter σ and energy prefactor ǫ, and are p < 1. In our simulations, we found that p = 0.01 re- connected by harmonic bonds with spring constant k = sults in a net reactivity of roughly r 1/3. (For p = 1, 2 ≈ 10ǫ/σ , resulting in the equilibrium distance r0 1.1σ. the reactivity is obviously r = 1.) In the present work, ≈ In addition, a bending potential V = K(1 cos(Φ)) is we take p to be the same for AB2 and AB3 monomers. b − 3

(a) (b) monomer functionality m=2 1

0.8

low reactivity (c) (d) 0.6

high reactivity 0.4

backbone conversion backbone functionality: 0.2 f=20 f=40 f=100 0 0 5 10 15 20 25 30 35 monomers per backbone atom FIG. 2. Simulation snapshot of hyperbranched polymers with backbone length f = 20 and monomer functionality m = monomer functionality m=3 2. The degrees of polymerization (i.e. the number of bound 1 monomers) are (a) DP=60, (b) DP=140, (c) DP=290. (d) shows a perfectly dendronized polymer of generation 3 on 0.8 the same backbone for comparison with the hyperbranched structures (DP=140). 0.6 low reactivity high reactivity However, since the monomers also have to move close to 0.4 each other before they can react, the resulting reaction backbone conversion backbone functionality: 0.2 f=20 rates may be different for AB2 and AB3 monomers as f=40 f=100 described below. 0 0 5 10 15 20 25 30 35 monomers per backbone atom III. RESULTS FIG. 3. Mean backbone conversion as a function of the We will now present the simulation results, focussing monomers per backbone unit for monomer functionality m = 2 (top) and m = 3 (bottom), different reactivities r = 1 first on the topological properties of the chains (i.e. , their ≈ architecture), and then on the resulting configurational (”high”) and r 1/3 (”low”), and different backbone functionalities f as indicated. The error bars (not shown) are properties (radius of gyration etc.). about twice the symbol size. For every choice of backbone functionality f (f = 20, 40, 100) and monomer functionality m (m =2, 3), 20- 25 independent simulation runs were carried out. For The backbone conversion is shown in Fig. 3 as a func- comparison, we have also simulated fully dendronized tion of the grafted monomers per backbone unit in the polymers with backbone functionalities f = 20, 40, 100 chain - which is in some sense a time axis in the poly- and monomer functionalities m = 2, 3. Typical snap- merization simulation. The curves for AB2 and AB3 shots of hyperbranched polymers and one dendronized monomers are very similar. In the initial regime, where polymer are shown in Fig. 2. monomers mainly react with backbone units, the backbone conversion increases linearly. Then, as monomers start interacting with already formed branching points, A. Statistical and topological properties of chains the curves deviate from linear growth and level off. For monomers with low reactivity, almost full conversion can We first address the important question how efficiently be reached in the course of the simulation. For highly re- the slow monomer addition process produces highly active monomers, a substantial fraction ( 20%) of back- branched molecules, i.e. , how close the resulting molec- bone reactive sites remain empty. ∼ ular structures are to perfectly branched, dendronized The degree of branching can be quantified by a measure polymers. To this end, we consider proposed by Hölter and Frey [44] The backbone conversion, i.e. , the average num- • m m (r 1) N ber of backbone groups that have reacted with a DB = r=1 − r , (1) m 1 P m rN monomer − r=1 r P the ”degree of branching” [44, 45], which quantifies where N is the number of monomers in a chain that • r the efficiency with which the full branching capa- serve as starting point for r branches. (Hence N0 is bilities of monomers are exploited in the process. the number of terminal units, N1 the number of linear 4

0.8

m=2 0.6 m=2 0.6

0.4 0.4

DB f=20, high reactivity f=20 f=40, high reactivity ANB f=40 f=100, high reactivity f=100 f=20, low reactivity 0.2 f=20, low reactivity 0.2 f=40 , low reactivity f=40 , low reactivity f=100, low reactivity f=100, low reactivity theory (parameter set s0) theory (parameter set s0) theory (parameter set s1) theory (parameter set s1) 0 0 0 5 10 15 20 25 30 35 0 5 10 15 20 25 30 35 monomers per backbone monomers per backbone

0.8

m=3 0.6 m=3 0.6

0.4 f=20, high reactivity f=20 0.4

DB f=40, high reactivity f=40 f=100, high reactivity ANB f=100 f=20, low reactivity f=20, low reactivity f=40, low reactivity 0.2 f=40, low reactivity 0.2 f=100, low reactivity f=100, low reactivity theory (parameter set s0') theory (parameter set s0') theory (parameter set s1') theory (parameter set s1') theory (parameter set s2') theory (parameter set s2') 0 0 0 5 10 15 20 25 30 35 0 5 10 15 20 25 30 35 monomers per backbone monomers per backbone

FIG. 4. Mean DB as a function of monomers per backbone FIG. 5. Mean average number of side branches ANB as a unit for monomer functionalities m = 2 (top) and m = 3 (bot- function of monomers per backbone for monomer functional- tom), different backbone functionalities f as indicated, and ities m = 2 (top) and m = 3 (bottom), different backbone different reactivities. Lines indicate theoretical predictions for functionalities f as indicated, and different reactivities. Lines different sets of rate parameters (cf. Eqs. (3-5)): kr ∝ (m − r) indicate theoretical predictions for the same parameter sets (s0 and s0’); k1/k0 = 0.45 (s1); k1/k0 = 0.5,k2/k0 = 0.3 (s1’); as in Fig. 4. k1/k0 = 0.45,k2/k0 = 0.27 (s2’), and kb = k0/m (all sets). Sets s0,s1,s1’, and s2’ are taken from Fig. 6. The behavior of the degree of branching DB and the average number of side branches ANB can be rationalized units etc.). DB quantifies the number of side branches within a simple rate equation approach proposed in Ref. in the molecule relative to the total number of branches [45], which we slightly modify to account for the effect of and is normalized such that it varies between 0 (linear backbone conversion. The resulting equations read chains) and 1 (perfectly branched dendronized chains). It is shown as a function of monomer units per backbone unit in Fig. 4. The curves look similar to those N˙F = ckbNF (3) for the backbone conversion (Fig. 3): They first increase − m and then level off. They are higher for AB2 monomers N˙ 0 = c kbNF + kr Nr (4) than for AB3 monomers, implying that it is less likely Xr=1 for AB3 than for AB2 monomers to fill all reaction sites. N˙ = c k −1 N −1 k N : 1 r m (5) Nevertheless, the total number of side branches per non- r r r − r r ≤ ≤ terminal monomer, quantified as Here c is the concentration of unreacted monomers in m solution, NB the number of unreacted backbone sites, r=1(r 1)Nr ANB = m − (2) and k gives the rate at which a free monomer reacts P N r r=1 r with a monomer that has already r branches. Using these P is higher for AB3 monomers than for AB2 monomers as equations, one can calculate DB and ANB numerically one would expect. This is shown in Fig. 5. Both DB and as a function of time and derive the limiting behavior for ANB increase slightly for less reactive monomers, but the large molecules. influence of reactivity on DB and ANB is much smaller In the late stages in the polymerization process, the than on the backbone conversion. contribution of B is negligible and the relative frac- 5

300 250 m=2,high reactivity m=2, low reactivity geometrical eﬀect: Once a monomer has grown one side 250 terminal units 200 terminal units linear units linear units branch, it is less accessible for free monomers and the 200 dendritic units dendritic units theory 150 theory probability to grow a second side branch drops. 150 100 100 Apart from the degree of branching, the Wiener index

number of units number of units 50 k1/k0= 0.5 50 k1/k0 = 0.45 W (T ) is another quantity that can be used to describe

0 0 the topology of a branched structure with the topology of 0 100 200 300 400 500 600 700 0 100 200 300 400 500 600 700 DP DP a tree T . It was ﬁrst introduced and applied in a chemical 300 300 m=3, high reactivity m=3, low reactivity context by Harry Wiener [46], but has numerous appli- 250 terminal units semidendritic units 250 terminal units semidendritic units linear units dendritic units linear units dendritic units cations in mathematics and other ﬁelds, too [47]. For a 200 theory 200 theory

k1/k0 = 0.4 k1/k0 = 0.5 tree consisting of N + 1 vertices (in our case monomers) 150 k /k = 0.27 150 k /k = 0.3 2 0 2 0 connected by N edges (bonds created by reactions), the 100 100

number of units 50 number of units 50 Wiener index is deﬁned as the sum over all ”distances”

0 0 s(vi, vj ) between any two vertices vi and vj , 0 100 200 300 400 500 600 700 0 100 200 300 400 500 600 700 DP DP 1 W = s(vi, vj ). (6) FIG. 6. Average number of terminal (N0), linear (N1), den- 2 Xij dritic (Nm) and semidendritic (for AB3 monomers: N2) units in the chains as a function of the degree of polymerization DP; (a) m = 2, high reactivity, (b) m = 2, low reactivity, (c) Here the ”distances” s(vi, vj ) are taken along the edges of m = 3, high reactivity, (d) m = 3, low reactivity. The back- the tree, and the length of one edge equals 1. The Wiener bone functionality in this example is f = 20. The evolution of index and the path lengths can be used to compare densi- the system is compatible with the effective rate theory, Eqs. ties of trees. Even though it can not be measured experi- (3)–(5) (black dashed lines) with rate parameters as given in mentally, it has been shown to correlate reasonably with the figures and kb = k0/m (the results are not sensitive to the properties such as density, viscosity and melting point choice of kb). [48–50]. For a linear chain consisting of N + 1 vertices, 1 the Wiener index is given by W = 6 N(N + 1)(N + 2) and hence scales as W N 3 [51]. It becomes smaller for tion of all other units reaches a steady state, i.e. , ∼ m branched molecules. For fully dendronized molecules, it N˙ / N˙ N / N , where N = DP is the 2 r r r ∝ r r r r=0 r scales as W N ln(N) [52, 53]. W can also be related degreeP of polymerization.P The solutionP becomes partic- to the first moment∼ of the distribution w(s) of distances ularly simple if the reaction rates take the simple intu- or strand lengths s [54]. With w(s) normalized as itive form kr (m r)kˆ, where the combinatorial factor (m r) accounts∝ for− the number of remaining available 1 − w(s)= 2 δ(s sij ), (7) reaction sites in an ABm monomer that has already re- N − Xi,j acted with r monomers. In that case, one obtains the limiting behavior DB m and ANB m−1 [45]. In → 2m−1 → m the Wiener index can be calculated according to our simulation model, we cannot expect k (m r)kˆ W = 1 N 2 s w(s). For a linear chain w(s) is given by r ∝ − 2 to hold, since the ”reaction sites” are on the same bead. P However, we can verify whether N is a linear function of 2 s r w(s)= 1 (8) DP at late stages of the polymerization process as pre- smax − smax dicted, and Fig. 6 shows that this is indeed the case. Extracting the rate constants kr of the theory from where smax = N 1 N is the longest strand. The these data, we can then calculate DB and ANB as a simulation results− for these≈ quantities are shown in Fig. function of the degree of polymerization per backbone 7 (Wiener index) and Fig. 8 (strand length distribution). (DP/f). This gives the lines shown in Figs. 4 and 5, which are in very good agreement with the simulation We find that the Wiener index roughly scales as W data except in the initial polzmerization process where M 2.3 with the molecular weight M = f + DP, hence∼ molecules are still very small. the scaling exponent is less than that of linear chains, We conclude that the simulation data can be described but higher than that expected for dendronized molecules. very well by the simple rate theory. One consequence is The monomer functionality m does not have an influ- that the degree of branching of the molecules obtained ence on W , and the influence of the monomer reactivity with this polymerization process mostly depends on the is also very small. The distributions w(s) are described ratio kr/k0 of rates with which monomers attach to inner by Eqn. (8) for DP = 0 (linear initiator chains). For monomers compared to terminal monomers. This ratio DP > 0, w(s) first displays an increase caused by the is more favorable if monomers have lower reactivity, since branched structures, followed by a decrease due to the the free monomers then have time to diffuse inside the finite length of the strands. For self similar structures, chain and are not captured by the first (usually terminal) the initial increase should be described by a power law chain segments they encounter. For AB3 monomers, the [54]. Such a behavior is not clearly identified here (in- rate k2/k0 is less favorable than the rate k1/k0. This is a sets in Fig. 8). This is not surprising, given the fact that 6

1e+08 3 f=20, high reactivity f=20, low reactivity DP=0 DP=120 f=40, high reactivity f=40, low reactivity DP=20 DP=300 2.5 1 1e+07 f=100, high reactivity f=100, low reactivity DP=60 DP=600

m=2 2 1.0 1e+06 s

max 0.1 1.5 0.02 0.1 0.5 1e+05 w(s) s

Wiener Index 1

1e+04 M2.3 0.5 f=20 1e+03 0 100 0 0.2 0.4 0.6 0.8 1 1.2

molecular weight M [M0] s/smax

1e+08 2.5 f=20, high reactivity f=20, low reactivity DP=0 DP=300 f=40, high reactivity f=40, low reactivity DP=100 DP=600 1e+07 f=100, high reactivity f=100, low reactivity 2 1 m=3 0.6 s0.7 1e+06 1.5

max 0.3 0.01 0.1 1e+05

w(s) s 1 Wiener Index

2.3 1e+04 M 0.5 f=100

1e+03 0 100 0 0.2 0.4 0.6 0.8 1 1.2

molecular weight M [M0] s/smax

FIG. 7. Wiener index of the hyperbranched polymers as FIG. 8. Averaged distribution of strand lengths of the hy- a function of the molecular weight M in units of monomer perbranched polymers for backbone length f = 20 (top) and mass M0 for monomer functionality m = 2 (top) and m = 3 f = 100 (bottom) at different degrees of polymerization DP (bottom) and different core functionalities f as indicated. At as indicated. DP = 0 corresponds to the bare backbone, with high molecular weight, the Wiener index displays a power law the distribution w(s) described by Eqn. 8. The plots show the behavior. data from simulations with high reactivity. Insets show the low s/smax regime of the same data on a double logarithmic scale, along with a power law function (black line) to give an idea of the behavior. Here, smax is the average length of the branched structures are grown on linear initiators, and longest strand. that the branched structures are not much larger than the backbone, especially for long backbones f = 100. Consistent with this, the decay of w(s) after the ini- ity f = 100 and total polymerization index DP = 600 . tial increase is dominated by the linear chain behavior For highly reactive monomers, the curve decreases mono- in this case. Interestingly, for f = 100 at high strand tonically and rapidly to a value close to zero. Since the lengths, the distributions increasingly deviate from the average side chain length must be 6 monomers, this im- linear chain behavior with growing degree of polymeriza- plies that molecules must contain a few long side chains tion, and a small shoulder develops at s/s 0.8 (Fig. max ≈ which compensate for the many very short chains. De- 8 (bottom)) . For comparison, we have also calculated creasing the monomer reactivity reduces the fluctuations the strand length distribution for ideal hyperbranched and leads to some compactification. The minimum of the chains, which were constructed based on the rate Eqs. distribution is shifted away from zero and the number of (3) – (5) (data not shown). The shoulder did not emerge side chains with 1-10 monomers increases. The inset of there. We conclude that it is most likely caused by the in- Fig. 9 shows the mass distribution (mass in monomer homogeneous distribution of branched side chains along units) along the backbone in the same chains. One can the backbone, which is analyzed below. clearly see that the mass accumulates at both ends of Last in this subsection, we study the distribution of the backbone. While the mass distribution is approxi- monomers on the side chains. The effect of monomer re- mately uniform in the middle region, the side chains that activity on the mass distribution is illustrated in Fig. 9, have grown from the ends of the backbone are about which shows the histogram of masses of side chains made twice as big on average. This effect is observed both of AB2 monomers in molecule with backbone functional- for highly reactive monomers and for monomers with re- 7

0.5 f=100, DP=600 m=2 chain simulation 0.4 m=3 m=2, low reactivity M0.588 m=3, low reactivity 10 0.3

σ 2/5 / DLA: M 20 g 16 f=100, m=2, DP=600 R m=2, backbone functionality f: 0.2 12 high reactivity f=20, high reactivity low reactivity 8 f=40, high reactivity number fraction 4 f=100, high reactivity 0 0.33 f=20, low reactivity 0.1 0 20 40 60 80 100 M backbone atom number f=40, low reactivity

monomers in side chains f=100 , low reactivity 1 0.0 10 100 0 5 10 15 20 25 30 molecular weight M [M0] size of side chain in monomer units

FIG. 9. Normalized distribution of side chain masses for molecules with core functionality f = 100, monomer function- chain simulation ality m = 2, and total degree of polymerization DP = 600. M0.588 One side chain hence contains 6 monomer units on average. 10 Inset shows distribution of polymer mass along the backbone

σ 2/5 of the chains. The units of the backbone chain are labelled / DLA: M g from 0 (one end) to 100 (other end). R m=3, backbone functionality f: f=20, high reactivity f=40, high reactivity f=100, high reactivity 0.33 duced reactivity. It can presumably be explained by a M f=20, low reactivity f=40, low reactivity reduction of screening close to the backbone ends. Side f=100 , low reactivity 1 branches grafted at the middle of the backbone chain are 10 100 surrounded by competing side branches that may also molecular weight M [M ] capture monomers, hence they are less accessible for dif- 0 fusing monomers than side branches grafted at the ends of the backbone. FIG. 10. Radius of gyration versus molecular weight (in monomer units) for monomer functionality m = 2 (top) and m = 3 (bottom), various backbone core functionalities f, and B. Configurational properties the two different choices of reactivity. Also shown for comparison are the results from simulations of linear chains. Dashed lines indicate slopes corresponding to different scaling behav- After having analyzed the topological properties of ior. the linear-hyperbranched chains generated by the hypergrafting process, we now study the configurational properties. Specifically, we examine the the gyration radius data collapse on two curves, one for high reactivity and and the hydrodynamic radius, and the amount of back- one for low reactivity. The only exceptions are the curves bone stretching induced by hypergrafting side chains. for f = 100 and low monomer reactivity, which appar- For ideal Gaussian chains, the radius of gyration ently do not reach the asymptotic limit for the molecular N weights under consideration. In the limit of large molecu- 1 R2 = (~r ~r )2 (9) lar weight, the two curves have the same slope in a double g 2N 2 i − j i,jX=1 logarithmic plot. The corresponding scaling exponent, ν = 0.38(1), lies between the exponent of DLA scaling (here the sum runs over all N beads of the molecules) (ν =0.4) and the exponent predicted by Konkolewicz et is directly related to the Wiener index, Eq. (6), via al. [25] (ν =0.33), and is clearly much smaller than the 2 2.3 (~ri ~rj ) s(vi, vj ). Inserting our result W N , Flory exponent for linear chains (ν =0.588). h − i ∝ ν ∼ this would suggest Rg √W /N N with ν = 0.15. Since the data shown in Fig. 10 were gathered in a sim- However, as we shall see∝ below, the∼ actual exponent ν is ulation of steadily growing chains, it is not clear whether much larger due to excluded volume effects. they reflect the properties of fully equilibrated chains. To Fig. 10 summarizes our data for the radius of gyration clarify this point, we have stored the architectures ob- Rg as a function of molecular weight for all backbone tained for selected degrees of polymerization and studied functionalities f, monomer functionalities m, and the two their configurational properties by separate equilibrium choices of monomer reactivity. The data ware gathered simulations without monomer addition steps. The results ”on the fly” during the simulation of the polymerization for Rg are shown in Fig. 11. They do not differ noticeably reaction. The results for m = 2 and m = 3 are essen- from the data taken in the polymerization simulation. tially identical. At late stages of the polymerization, all Hence, we can conclude that the polymers in the poly- 8

f=20, m=2 C chain simulation

10 ∝ M0.53 10 B

σ σ 0.4 / / ∝ M g h

R R m=2, backbone functionality f: A f=20, high reactivity f=40, high reactivity high reactivity f=100, high reactivity low reactivity f=20, low reactivity equilibrium averages f=40, low reactivity dendronized polymer f=100 , low reactivity 1 10 100 10 100

molecular weight M [M0] molecular weight M [M0]

FIG. 11. Radius of gyration versus molecular weight in FIG. 13. Hydrodynamic radius versus molecular weight (in monomer units for monomer functionality m = 2 and back- monomer units) for monomer functionality m = 2 various bone functionality f = 20 (top) and f = 40 (bottom). Small backbone core functionalities f, and the two diﬀerent choices symbols correspond to data that were collected during the of reactivity, compared to simulation results for linear chains. polymerization simulation, larger symbols to data that were Dashed lines indicate slopes corresponding to diﬀerent scaling obtained from separate, longer simulations. Also shown for behavior. The results for m = 3 are basically identical. comparison are data from simulations of fully dendronized polymers. increase and then reach a peak with values that can be

λ 0.7 0.7 as large as λ 5.5. The position and the height of the f=20 λ f=20 ∼ m=2 m=3 peak depend on the molecular details: It moves to higher 0.6 f=40 0.6 f=40 f=100 f=100 values of M and decreases if one increases f or reduces 0.5 0.5 the reactivity of the monomers. At large M, the diﬀer- 0.4 0.4 ential fractal dimensions decay and reach an asymptotic

0.3 effective exponent 0.3 value which is slightly below ν = 0.4. The asymptotic

effective exponent 10 100 10 100 molecular weight M[M0] value is universal and does not depend noticeably on m, molecular weight M[M0] f, and the monomer reactivity. Next we discuss the hydrodynamic radius R , which is FIG. 12. ”Effective exponents” (differential fractal di- h defined as the radius of a Stokes sphere with the same dif- mensions) λ = dln(Rg )/dln(M) versus molecular weight for m = 2 (left) and m = 3 (right). See text for explanation. fusion coefficient as the polymer and which we calculate via the approximate equation [57]

1 1 N 1 merization simulation are equilibrated despite the occa- = . (10) sional addition of one monomer. For comparison, we also Rh N(N 1) ~ri ~rj Xi=1 Xj6=i show data for dendronized polymers. We find that the − | − | latter are much more compact, and in addition, the scal- In the asymptotic limit, the scaling of Rh and Rg should ing of Rg with the molecular weight seems different. Our be the same. In the simulations, the results for Rh are data for the perfectly branched dendrimers are compat- very similar to those for Rg, therefore we only show the ible with previous dendrimer simulations [55, 56] that data from polymerization runs for m = 2 as an example 1/5 have suggested a scaling between Rg M (the Flory (Fig. 13). Much like in Fig. 10, the data collapse onto 1/3 ∝ prediction) and Rg M . The data for the hyper- two curves corresponding to high and low reactivity. The branched chains suggest∝ three different scaling regimes: two curves show a similar scaling behavior, which again An initial regime (A) with a scaling exponent of roughly differs distinctly from that of linear chains. We note that ν 0.4, a second regime (B) where Rg grows even the scaling of Rh for linear chains does not quite reach ≈ ν stronger as a function of M with an exponent ν 0.5, the asymptotic limit Rh M with ν =0.588 in our sys- and a third regime (C) where the exponent ν ≈0.4 is tem, the apparent exponent∝ (ν 0.53) is slightly smaller. recovered. ≈ This is in agreement with results≈ from other simulations To quantify this further, we have fitted fourth order and experiments [58]. In contrast, the scaling exponent polynomials to the double logarithmic data from the of R for the largest hyperbranched chains, ν 0.43(1), h ≈ polymerization runs and used them to extract local ”ef- is slightly larger than that measured for Rg. fective exponents”, i.e., the differential fractal dimen- The internal structure can be further investigated by sions λ(M) = d ln(Rg)/d ln(M). The results, shown in inspecting the ratio Rg/Rh. The value of Rg/Rh depends Fig. 12, confirm the nonmonotonic behavior of the effec- on the molecular architecture. For long ideal chains, it tive scaling. The differential fractal dimensions initially is known to be 1.5; experiments with linear polymers 9

1.4 1.4 16 dendronized 1.3 linear chain dendronized m=2, high react. 1.3 h high react. h linear chain m=3, high react. 1.2 high react. f=20 /R low react. /R m=2, low react. g g 1.2 low react. 14

R 1.1 R m=3, low react. 1 1.1 f=20, m=2 f=40, m=2 0.9 1 12

0 50 100 150 200 250 300 0 50 100 150 200 250 300 σ / molecular weight M [M0] molecular weight M [M0] e R 10

FIG. 14. Ratio Rg /Rh as a function of molecular weight in monomer units for hyperbranched chains with monomer 8 functionality m = 2 and backbone functionality f = 20 (left) equilib. averages: dendronized: m2, high react.: m=2, low react: and f = 40 (right). Data for linear chains and fully den- 6 dronized molecules (same backbone length) are also shown 0 100 200 300 400 500 600 700 for comparison. DP

35 give slightly smaller values [59]. Dendrimers have been m=2, high react. m=3, high react. f=40 reported to have ratios close to 1, star polymers with 30 m=2, low react. polydisperse arms have R /R 1.2 [60, 61]. Fig. 14 m=3, low react. g h ≈ shows the data for Rg/Rh from equilibrium simulations 25

for monomer functionality m = 2 and backbone length σ / e f = 20 and f = 40. Upon increasing the molecular R weight, one observes a distinct peak at M 50 100 20 and a subsequent slight decrease. A similar≈ peak− has been reported by Wang et al. [30] in simulations of hy- 15 perbranched polymers with a pointlike core. In our si- equilib. averages: dendronized: m2, high react.: m=2, low react: mulations, we observe that the peak is less pronounced 10 for longer backbone (core)chains. It almost vanishes at 0 100 200 300 400 500 600 700 f = 40 and completely disappears at f = 100 (not DP shown). One can rationalize the peak as follows: Starting from a linear chain, the polymer first develops into a star FIG. 15. Average end-to-end distance of the backbone chain shaped form with growing side chain number. With in- as a function of the degree of polymerization (the number of creasing size of side chains and degree of branching, the hypergrafted monomers) for backbone lengths f = 20 (top) and f = 40 (bottom). The black symbols represent the aver- i.e. overall structure becomes more spherical, , Rg/Rh ages from equilibrium simulations and the values from simu- decreases again. Dendronized molecules show a similar lations of dendronized polymers. behavior with an even more pronounced peak, whereas in linear polymers, Rg/Rh grows monotonously as a function of M. this curve will further flatten at much higher DP once Finally, we address the question whether and how the Re becomes comparable to the contour length, but this backbone chain stretches due to the growth of side chains. regime was not reached in the simulations. If monomers Naively, one might expect that the stiffness, i.e. , the per- have reduced reactivity, the range of the first regime is sistence length of the backbone chain, can be tuned in a slightly extended and Re increases. The slope of Re in controlled manner by hypergrafting branched side chains. the second high DP-regime does not seem to depend on However, extensive simulations of bottle-brush polymers the monomer reactivity. have indicated that the persistence length is no longer For comparison, we have also studied the end-to-end a well-defined concept for such chains [62]. The same distance of the corresponding dendronized molecules. should hold for linear-hyperbranched chains. Therefore, The behavior in the initial regime is quantitatively sim- we will not discuss the stiffness of the backbone here, ilar to that of hyperbranched chains. In the second but rather its elongation. The average end-to-end dis- regime, the slope is much larger. This underlines once tance Re is shown in Fig. 15 as a function of the number more the fundamental differences between dendronized DP of grafted monomers for backbones of length f = 20 molecules and hyperbranched molecules. and f = 40 and monomer functionality m = 2. One can clearly distinguish two regimes: In the initial regime up to DP f, where monomers attach directly to the back- ∼ IV. DISCUSSION AND CONCLUSION bone, Re increases sharply with DP. Then, the curves level off and continue to increase much more slowly, such that the additional stretching of Re relative to the end- We have studied the structural and conformational to-end distance of the bare backbone is only of order properties of linear-hyperbranched copolymers by com- 10−4 per additional grafted monomer. We expect that puter simulations of a simple, solvent-free, coarse-grained 10 spring-bead model. The model is designed to mimic are good reference systems. the synthesis of linear-hyperbranched copolymers in good We have specifically addressed the question, whether solvent under conditions of slow monomer addition. Our molecules can be made more compact by reducing the main results can be summarized as follows: monomer reactivity. We find that reducing the reactiv- The topological properties of the hyperbranched chain ity by a factor of three has relatively little influence on (the chain architecture) are intermediate between those the degree of branching and no influence on the scaling of linear and dendronized chains. The evolution of the properties of both structural and conformational proper- numbers of branching points with different coordination ties of the chains. It does, however, influence the prefac- numbers as a function total polymerization agreement is tor in the asymptotic power law for Rg and Rh. Hence in very good agreement to a simple rate theory proposed the chains are more compact, but their behavior does not by Hölter and Frey [45], despite the fact that the the- change qualitatively. ory neglects screening and excluded volume effects. The We have also studied the distribution of monomers on theory predicts an upper limit for the degree of branch- the side chains and found that it depends on the grafting ing, hence our results suggest that the dendronized limit point of the side chains on the core backbone. For long cannot be reached on principle. The Wiener index W is backbone chains, the side chains that have polymerized found to scale with the scaling exponent W M ω with at their ends are about twice as long as the ones in the ∝ ω 2.3, which is between the exponents for dendrimers middle, leading to a pom-pom like structure. This some- ≈ (ω = 2 with logarithmic corrections) and linear chains what unexpected result of the present study will also have (ω = 3). implications for the synthesis and resulting properties of Likewise, the behavior of the polymer size (as speci- such structures. fied by the radius of gyration Rg or the hydrodynamic Finally, we have examined the influence of the side radius Rh) as a function of molecular weight is inter- chains on the conformations of the backbone chain. As mediate between that of linear and dendronized chains. one would expect, linear backbone chains stretch out if The scaling exponent is found to be around ν 0.38, ≈ side chains are hypergrafted to them. However, the effect closer to the exponent of diffusion limited aggregation is much less pronounced than for dendronized chains. than to other exponents suggested in the literature (1/2, Since our simulations are done with an implicit solvent 1/3, 1/4). However, the asymptotic regime might not model, they do not include hydrodynamic interactions. have been reached, and we cannot exclude the possibil- In fluids at rest, hydrodynamics should not be important ity that the true exponent is ν = 1/3 as predicted by on the time scales relevant for slow monomer addition. Konkolewicz et al. [25]. At small molecular weights, the This might change if one applies flow externally, and it exponent is nonmonotonic and varies between 0.4 and might be possible to manipulate the synthesis process by 0.5. applying external flows, e.g. , in microfluidic setups. For We should note that these questions are not only of example, linear-hyperbranched copolymers might stretch academic interest. Molecular weight distributions of in flow at high Weissenberg numbers, which should facil- polymers are often measured by chromatography, and itate the grafting of monomers to inner side chains. This they are calibrated by comparison with linear polymers of will be the subject of future investigations. known length. Thus one implicitly assumes that the hydrodynamic radii of the target chains and the calibration chains behave similarly as a function of molecular weight. If the exponent ν deviates significangly for both architec- ACKNOWLEDGMENTS tures, the analysis becomes questionable and correction terms must be applied. This requires a good knowledge of We thank Christoph Schüll for inspiring discussions. ν in the target chain. Unfortunately, our results suggest The simulations were carried out on the High Perfor- that neither linear chains nor fully dendronized chains mance Computer Cluster Mogon at Mainz university.

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