The chain walking (CW) polymerization technique has the unique property of a movable catalyst synthesizing its own path by creating branch-on-branch structures. By successive attachment of monomers, the resulting architecture ranges from dendritic to linear growth depending on the walking rate, which is defined by the ratio of walking steps and reaction events of the catalyst. The transition regime is characterized by local dendritic sub-structures (dendritic blobs) and a global linear chain feature forming a dendritic bottle-brush. A scaling model for structures obtained by CW catalysis is presented and validated by computer simulation relating the extensions of CW structures to the catalyst’s walking ability. The limiting case of linear (low walking rate) and dendritic growth (high walking rate) is recovered, and the latter is shown to bear analogies to the Barabási–Albert graph and Bernoulli growth random walk. We could quantify the size of the dendritic blob as a function of the walking rate by using spectral properties of the connectivity matrix of the simulated macromolecules. This allows us to fit the numerical constants in the scaling approach. We predict that independent of the underlying chemical process, all CW polymerization syntheses involving a highly mobile catalyst ultimately result in bottle-brush structures whose properties depend on a unique parameter: the walking rate.

## I. INTRODUCTION

Polyethylene (PE) is one the most industrial synthesized and processed polymers with a wide variety of applications. Depending on the synthesis pathways, different polymer architectures, such as Low Density Polyethylene (LDPE), Low Linear Density Polyethylene (LLDPE), and High Density Polyethylene (HDPE) classified by their density and degree of branching, can be obtained.^{1,2} However, obtaining a precise and specific polymer architecture is more challenging. In 1995 Johnson *et al.*^{3} and in 1999 Guan *et al.*^{4} proposed a Pd(II)-*α*-diimine-based catalyst to polymerize ethylene and *α*-olefins in a one-pot setup in a variety of architectures depending mostly on the synthesis pressure. Another unique property of the catalyst is the ability to bind to the polymer and randomly walk along the structures introducing branch-on-branch structures^{5,6} given the name “chain walking” (CW) catalysis. In the course of history, it has been proposed and shown in the experiment^{7–11} and in simulation^{12–14} that the CW mechanism can produce rather linear chains for high pressure or even dendritic structures for low pressure. The highly branched transition state for ambient synthesis pressure has been correctly described as dendrigraft by Tomalia and Fréchet,^{15} but has been only termed hyperbranched by several authors^{16–20} without further addressing the underlying topology. As hyperbranched polymers differ significantly in their properties, there are at least two distinct universality classes for disordered dendrimers and randomly hyperbranched polymers depending on the preparation conditions.^{21,22} As we have shown in our previous studies,^{14,23–25} the CW polymerization technique operates in the linear to dendritic regime depending on the walking rate of the catalyst between successive monomer insertion events. The transition regime can be interpreted as a dendrigraft/dendritic bottle-brush tunable by the synthesized time, pressure, and temperature. Albeit the walking and polymerization process in the experiment is limited to tertiary carbons, quite recently, Figueiredo *et al.*^{26} mathematically investigated a similar process involving a random walk on the unbounded growing structure. Its “Bernoulli Growth Random Walk” (BGRW) shows similarities to the CW approach with one important exception that the mathematical treatment assumes unbounded functionality of the nodes (monomers). Both share the same transition process of evolving linearly to compact highly branched structures with a transition regime exhibiting a linear backbone grafted with dendritic sub-branches (Cayley-tree). Furthermore, it can be stated that the BGRW will produce the scale-free Barabási–Albert (BA) graph^{27,28} in the limit of the infinite walking rate,^{29} which we show in this work will also be the limiting case of the CW approach for a restricted trifunctional Barabási–Albert (r3BA) graph.

This paper is structured as follows: in Sec. II, the theoretical model is introduced describing the CW approach. In Sec. III, the simulation method is explained and the results for the investigated quantities in the ideal and in the excluded volume case are discussed in Sec. IV. Our conclusions are given in Sec. V. Further details on the simulation of BA and BGRW are presented in the Appendix.

## II. SCALING THEORY OF THE DENDRITIC BLOB-CHAIN-MODEL

The synthesis reaction of the CW catalyst can be considered as the Bernoulli process on a time step with an attachment event under probability *p* and a walking event with probability 1 − *p*. The primary parameter in the process of CW structure formation is the walking rate,

which we define as the number of time steps, which have been passed before the next reaction event happens. Hence, between two successive reaction events, the catalyst will perform a random walk of in average *τ* = *w* − 1 steps on the molecular structure. Note that the underlying topology is changing by every successfully performed insertion of the monomer, and no detachment of the catalyst is assumed. The branch-on-branch structures obtained in the experiment suggest^{3,4,12} that the walking of the catalyst and the insertion process of new monomers are restricted up to tertiary carbon. This results in the formation of highly branched molecules with primary, secondary, and tertiary carbon. Therefore, we restrict our discussion to maximum trifunctional monomers, but as we show below, the theoretical ansatz can be extended to arbitrary monomer functionalities *f*. For the limiting case *w* = 1, no walking and no branching occur, and the structure is a linear polymer chain. As shown in Ref. 14, for the limiting case of extensive walking *w* ≫ 1, the CW catalyst can reach any reactive site of the macromolecule, and attaching monomers leads to dendritic growth. These two limiting cases have to merge for intermediate values of the walking rate *w*. For low molecular mass and moderate walking rates, the catalyst explores the entire structure before adding a monomer, leading to dendritic growth in the sub-structure. At a characteristic length scale depending on *w*, not all monomers can be reached, and the catalyst will break-out from producing dendritic growth and create a new dendritic sub-structure. Hence, the CW structures for moderate values of *w* with high molecular molar mass can be thought of as dendritic blobs forming a linear chain on large scales, as sketched in Fig. 1. By comparing simulation with experimental findings, we can deduce a walking rate between 3 and 6 for our experimental setup in Ref. 14. As the catalyst performs a random walk on the structure, the average diffusive path between two reaction events is given by

where *T* denotes the diameter of the dendritic blob along the chemical structure, i.e., corresponds to the longest path between two reactive groups in units of monomers along the structure. For a dendrimer with mean functionality *f*, there is an exponential relation between the number of monomers, *g*, and the number of generations, *G*. On the other hand, the diameter, *T* (or thread length), is proportional to *G* times the length of the spacers *S* connecting two branching points. Thus, we can write

The last relation in Eq. (3) makes use of Eq. (2), which relates *T* to the walking rate *w* with a numerical prefactor *α*. If the degree of polymerization *N* is smaller than *g*, the dendritic topology dominates. As *N* exceeds *g*, the walker cannot explore the entire structure within *τ* steps, and the growth is no more isotropic with respect to the center of the existing structure. As the time of reaction, *t*, proceeds, an entire dendritic blob is added to the structure in *gτ* steps, leading to a linear global structure with dendritic sub-structures for *t* ≫ *gτ*. Hence, the resulting extension *R* of the CW structure resembles a linear chain composed of dendritic blobs of size *ξ*(*w*) instead of simple monomers,

where *ν* = 1/2 is the exponent in the ideal and *θ*-solvent case and reaches a value of $\nu \u22430.588\u224335$ in good and athermal solvent.^{30}

In the ideal case without excluded volume, the extension of the dendritic blob *ξ*_{0} is given by Gaussian relation^{31}

whereas the dendritic blob size for *θ*-solvent conditions *ξ*_{θ} yields^{21,32}

and the dendritic blob size for excluded volume structures *ξ*_{ev} in good solvent yields^{21,32,33}

Thus, the overall extension of the CW structure depends on the walking rate *w* and the solvent condition yielding in the ideal *R*_{0}, *θ*-solvent *R*_{θ}, and for the excluded volume *R*_{ev} case,

Note that the CW structures correspond to disordered dendritic bottle-brushes for sufficient high degrees of polymerization. However, the crossover toward a linear object can be shifted to extremely high molar masses even for moderate walking rates as the degree of polymerization for the dendritic sub-structure grows exponentially with *w*, see Eq. (3). We note that structures with larger walking rates obtained after very long reaction times will be extremely dense and correspond to so-called molecular objects that are stable in form and not dominated by fluctuations, as proposed by Schlüter *et al.*^{34}

## III. COMPUTATIONAL METHOD

The modeling of the chain walking catalyst mechanism is conducted in the framework of the Monte Carlo method applying the Bond Fluctuation Model^{35,36} (BFM) to simulate flexible polymeric structures in good solvent conditions as performed in our previous work.^{14} In this coarse-grained Monte Carlo method, polymers are modeled as connected cubes on a simple cubic lattice undergoing random movement by implicit solvent. A trial move of a monomer is implemented as follows: first, a random monomer is chosen, and second, a move direction along the principal lattice axis by one lattice unit is randomly chosen. The move for the selected monomer is accepted if the new lattice positions are vacant and all bond vectors connecting the structures are in the allowed set; otherwise, the move is rejected. For the present case, a set of 108 bonds with length $2,5,6,3$, and $10$ is allowed, and throughout this paper, we set the length of one lattice site to unity. The mean-squared bond length in the ideal case without correlation can be assumed to be $b2=798108$, whereas in the athermal solvent case, it is slightly extended as *b*^{2} ≃ 7.5. The bond vector set and the excluded volume condition are defined such that the local and global topology is preserved and cut-avoidance throughout the simulation procedure is ensured.^{36} One Monte Carlo step (MCS) is defined as one attempted Monte Carlo move per monomer in average to be the basic time unit.

The creation of the CW structures is carried out similar to our previous work:^{14} CW structures with degree of polymerization *N* ∈ [4; 9216] monomers for different chain walking rates *w* are created in a cubic *L* = 512 simulation box. We start with an initial configuration of *n* = 2 connected monomers placing the CW catalysts as “walker attribute” on one particular monomer on the initial structure. To account for the different walking rates *w*, an additional acceptance–rejection condition^{37} for the creation and attachment of a new monomer with a reaction probability *p* = 1/*w* has been applied. The reaction probability $p\u22080;1$ is compared against a randomly chosen uniformly distributed random number $\zeta \u22080;1$. If *ζ* < *p*, a new monomer may be added if two additional constraints can be satisfied: first, the monomer with the walker attribute is checked for functionality *f* < 3 ensuring a maximal monomer functionality limited to trifunctional branching unit similar to the chemical attachment process. Second, as all CW structures are grown under the excluded volume condition, an additional test is necessary to guarantee valid conformations within the BFM. Therefore, the vicinity of the walker monomer is checked for vacant sites in a shell of all permutations and sign combinations of the vector $P\xb12,0,0$ followed by a random selection of a possible placement out of the shell if available. If all of the above criteria are satisfied, a new monomer is added to the simulation (counter *n* is incremented) and connected to the monomer having the walker attribute followed by the movement of the walker attribute to the new monomer; otherwise, the trial addition is rejected. Here, we assume an instantaneous availability of the reactants within the time step of the simulation (hopping rate). Considering slower dynamics for the attaching monomer can be mapped to higher walking rates in the model. For *ζ* ≥ *p*, only the walker attribute randomly moves to the next connected neighbor without preference or bias with probability 1 − *p* corresponding to the walking of the catalyst. After this step, the structure is moved according to the BFM rules for 1000 MCS in both cases to equilibrate high density regions and promote the relaxation of the polymeric structure during growth. The algorithm repeats until the structure contains the desired number of monomers *n* ≡ *N*, and no detachment of the walker is considered. As stated above, low walking rates (*p* ≃ 1) leading to linear structures with mild branching, with the limiting case *w* = *p* = 1, where linear chain without branching is obtained. For high walking rates (*p* ≪ 1), the reaction events are rare, resulting in extensive walker movement and structures with high degree of branching. Note that there is no additional barrier for walking over trifunctional monomers as this will not affect the underlying topology. Implementing such a barrier only shifts the transition region to a higher molecular mass. In a simple sense, this can be associated with increasing the spacer length of the dendritic blob.

Within the creation algorithm, the excluded volume condition as well as cut-avoidance, and local and global topology conservation are satisfied at all times, leading to valid conformations within the BFM. The obtained CW structures have then been simulated according to the BFM rules for several MCS in order of magnitude 10^{7} to relax and equilibrate high density regions, followed by production runs in order of magnitudes 10^{6} − 10^{9} MCS (several Rouse times) for sampling equilibrium observables. At least 20 to 50 simulation runs for each pair (*N*, *w*) are performed to provide an ensemble average with respect to the random topology. Properties, which are based solely on the connectivity or on the creation process, have been calculated with at least 1000 creation runs. Throughout this paper, we use the short-hand notation $A\u2254A=AtE$ indicating the ensemble average $\cdots E$ of the time average $\cdots t$ of an observable $A$. For comparison, additional simulations of linear chains, perfect trifunctional dendrimers, and restricted trifunctional Barabási–Albert graph (r3BA), see the Appendix, are performed. Perfect dendrimers are generated for different generations *G* = 1, …, 10 with spacer lengths *S* = 1 and *S* = 2 between the branching units.^{14} The r3BA structures are generated according to the Barabási–Albert model with the exception that the new node can only be added if the parent node degree is less than three. The BFM algorithm, simulations, and processing of the data are performed by the C++ framework LeMonADE developed in our group,^{38} and the CW extension^{39} is available free of charge. Note that, throughout this work, the creation process of all CW, dendrimer, and r3BA structures both in the ideal and excluded volume case obeys the excluded volume condition in the creation process at all times to include the effect of steric hindrance during the simulated synthesis process.

## IV. RESULTS AND DISCUSSIONS

### A. Dynamic properties of the chain walker

First, we test the assumption made in our model that the average number of steps, *τ*, between two reaction events is proportional to the walking rate *w*. The inset in Fig. 2 shows the average number of steps *τ* the walker has traveled between two reaction events and confirms the overall assumption *τ* ∼ *w* − 1 under excluded volume conditions and for restricted insertion functionality at the monomer during the simulated growth, as described in Sec. III. In our simulations, we obtain a prefactor larger than unity. The reason for this is that the steric hindrance in the attachment process due to the excluded volume effect will reduce the effective reaction rate, i.e., not every attempted reaction event can be realized. Moreover, for larger structures, the catalyst will stay for longer times in the interior having a lower chance to spawn a reaction. Both effects increase the average number of walking steps, *τ*, with respect to *w*.

For random walks on fractals,^{40–42} one would expect an asymptotically power-law behavior between the mean-squared displacement along the structure and number of steps $d\tau 2\u223c\tau 2/dw$ related to the walking dimension *d*_{w} depending of the fractal object. This will be certainly true for the limiting case of linear structures with *d*_{w} = 2 yielding diffusive behavior $d\tau 2\u223c\tau $. On the other hand, the Bethe lattice (or infinite Cayley-tree) has no fractal dimension as the graph is unbound. Instead, for arbitrary functional perfect dendrimers, the asymptotic limit scales^{43} as $d\tau 2\u223c\tau 2$. However, the main difference between the CW structures and diffusion on an infinite Bethe lattice is that the random walk of the CW catalyst starts by definition on the boundary of the structure, while the ideal Bethe lattice is not bound, and thus, diffusion is considered to start in the center.^{43} Therefore, strong deviations from this ballistic relation are expected for finite-size structures.^{44–46} We focus on the mean average topological displacement *δ* during the creation process, which we define as the shortest path length along the structure between two reaction events in time *τ* averaged during the creation of the CW structure at a particular walking rate *w*. Note that the quantity *δ* measures the average path the walker is able to travel considering the dynamically extending CW structure and realistic constraints such as excluded volume and limited functionality for possible reaction events.

In Fig. 2, the mean average topological displacement *δ* is plotted for various walking rates *w* and degree of polymerization *N*. By construction for low *w* ≃ 1, the catalyst will not walk between successive reaction events and will attach monomers in a linear fashion with *δ*(*w* = 1) = 0. On the other hand, for higher walking rates *w* > 5, the distance increases with *δ* ∼ *w*^{1/2} but still remains next to the origin of the initial reaction event. As we will show below, even the small amount of traveling distance is sufficient to cross the overall structure as the dendritic structures have a small diameter proportional to thread length *T* = *G* · *S* ∼ ln(*N*).

Combining the findings from Fig. 2, we can conclude that the catalyst performs a diffusive walk along the structure

This result is in good agreement for all considered walking rates *w*, as shown in Fig. 3. Hence, this gives justification to use Eq. (2) instead of a ballistic expression expected for unbounded trifunctional Cayley-trees as investigated in Refs. 43 and 45. Thus, the dendritic blob can be expressed by the fundamental relations in Eqs. (2) and (3). We emphasize again the relatively small traveling distance the walker has to take on the dendritic structure even for high walking rates.

### B. Size of ideal CW structures

All structures obtained by CW polymerization as reflected by our simulation model can be considered as undirected acyclic tree graph, $G$, where the monomers are represented by vertices, $V$, connected by bonds related to the edges, $E$, of the graph. We define the (generalized) Rouse matrix, $M$, which describes the connectivity of a graph $G=(V,E)$ with *N* vertices $V={v1,\u2026,vN}$ and corresponding edges $E$. It is defined as an *N* × *N* matrix with the elements (*m*_{ij}) so that *m*_{ij} = −1 if bond ${i,j\u2260i}\u2208E$ exists and *m*_{ii} = deg(*v*_{i}) is the degree (functionality) of the vertex *v*_{i}. Here, *m*_{ij} denotes the matrix element at the *i*th row and *j*th column of the generalized Rouse matrix $M$.^{47,48} A simple example is given in Fig. 4.

We now consider that all bonds between branching points are flexible as realized by our BFM-model. Furthermore, we ignore excluded volume effects and consider the ideal statistics of the branched polymer first. The ideal radius of gyration $R02$ of such a generalized Gaussian structure is given by the sum of all non-zero *λ*_{k}th eigenvalue of the Rouse matrix by^{48,50}

with the ideal mean-squared bond length of the BFM $b2=798108$. Note that the eigenvalue *λ*_{1} = 0 is omitted in the summation as the translational degree of freedom does not contribute. Linear ideal chains scale as $R0,L2\u223cN$, whereas ideal dendrimers^{21,32} and Barabási–Albert trees^{51–53} exhibit a logarithmic growth $R0,D2\u223cln(N)$. We note that because of Eq. (10) being an exact result, there is no need to sample the conformations of ideal structures but using the eigenvalues of the Rouse matrix instead.

In Fig. 5, we show the ideal radius of gyration $R02$ for CW structures normalized by the linear chain behavior for various walking rates *w*. For low walking rates *w* ≃ 1, the obtained structures show global linear chain characteristics despite of some branching. On the other hand, for high walking rates *w* ≫ 1, the CW structures grow logarithmically. For intermediate walking rates, we can observe a crossover from dendritic to linear behavior. This corresponds to our theoretical argument given in Sec. II: As long as the number of monomers in the dendritic structure is smaller than the dendritic blob size given in Eq. (3), the catalyst can reach any reactive site during the walking time *τ*, thus giving rise to isotropic dendritic growth. After reaching *g* monomers, i.e., for reaction times beyond *τg*, the catalyst is not able to traverse the structure and creates subsequently new dendritic blobs after *τ* steps. Note that the CW dendritic structures are imperfect due to the stochastic process of attachment but still resemble the scaling of dendrimers. This can be directly observed in Fig. 5, where we have also displayed the results for perfect dendrimers with spacer lengths *S* = 1 and *S* = 2, respectively. Moreover, this corresponds to our previous study on random dendritic growth.^{21}

To emphasize the crossover between the two regimes in Fig. 5, the derivative of the ideal radius of gyration $dR02/dN$ with respect to the degree of polymerization *N* is shown. The crossover between the dendritic and the linear behavior can be clearly seen in Fig. 5 up to walking rates of *w* = 10. In accordance with Eq. (8a), linear polymer scaling $(R02\u223cN)$ for *N* > *g* is obtained up to *w* ≲ 10. Even higher walking rates *w* > 10 still show the transition region, but do not reach the linear plateau up to degrees of polymerization in order of 10 000. The shift of the crossover between dendritic and linear growth by increasing the walking rate toward very high molar masses is due to the exponential relation between the dendritic blob and the walking rate in Eq. (3). Therefore, the structures obtained at walking rates as high as 50 and 100 show the logarithmic relation $R02\u223cln(N)$ only, and thus mimic the behavior of pure perfect dendrimers and restricted Barabási–Albert graphs. We stress again that all structures are created under excluded volume conditions, and unlimited dendritic growth in three dimension is not possible due to steric hindrance. Therefore, all CW and r3BA structures still show similarities with dendrimers, but structural defects like missing branches or terminating leaves in the middle of the structures will occur. We note that additional restriction on the walking probability on the tertiary functional monomer only shifts the transition regime toward linear structures with less dendritic side-chains but does not change the dendritic bottle-brush behavior in the high molar mass limit.

### C. The dendritic blob

As the acyclic graph, $G$, has a topological center $C$ with maximum two vertices,^{54} we can define the topological density profile *c*(*s*) as the average number of monomers in distance *s* from $C$, see Fig. 4. The topological distance *s* is calculated by Dijkstra’s algorithm,^{49} providing the shortest path between the vertices. For the case of linear chains, we obtain *c*_{lin}(*s*) = 2, while for the case of trifunctional dendrimers, we obtain *c*_{dd}(*s*) = 3 · 2^{s−1}. The total number of monomers is given by

For asymptotically linear structures with local sub-branches as in the case of dendritic bottle-brushes, the degree of polymerization can be approximated by

where *c*_{0} is the average number of monomers from the center, and *s*_{B} is the length of the backbone. In Fig. 6, *c*(*s*) is displayed exemplary for *w* = 5. We can identify the crossover from disordered dendritic behavior for small *N* to the bottle-brush with constant plateau value *c*_{0} for high *N*, which reflects Eq. (12). As the repeating tree-like structure with backbone length *s*_{B} can be decomposed into *N*/*g* segments with *g* monomers and mean strand length *T* yields

Thus, *c*_{0} = *g*/*T* neglecting boundary effects at the ends of the bottle-brush.

Under the proven assumption of a random walk of the catalyst with Eq. (2), *T* ∼ *w*^{1/2}, and taking into account the exponential growth of the blob size with the walking rate, *g* ∼ exp(*T*), see Eq. (3), yields

where *a* denotes a numerical constant of order unity. In Fig. 7, the plateau value *c*_{0}(*w*) as obtained from the results displayed in Fig. 6 has been rescaled according to the theoretical prediction in Eq. (14) and is shown for various walking rates. These results are in very good agreement with Eq. (14) over a wide range of walking rates. Note that, for high walking rates, the plateau value of the CW structures is not yet fully developed, since the crossover to the bottle-brush regime is not completed. Thus, the apparent plateau values are underestimated and omitted from the fitting procedure for *w*^{1/2} > 3.75. The numerical constants *a* = 0.26 and *α* = 2.15 can be extracted from the plot.

We stress again that during the creation process of all CW structures, the excluded volume condition is strictly obeyed. The results for the ideal case (excluded volume is switched-off after the creation) are obtained using Eq. (10). As a consequence, the number of monomers inside a dendritic blob *g* and, thus, *c*_{0} are universal results that do not depend on the conformation statistics and solvent quality after synthesis but only on the synthesis conditions (creation process in simulations).

Using the Gaussian relation, see Eq. (5), for the ideal case, $\xi 02\u223cT$, we obtain from Eq. (14),

The numerical prefactor is set to unity, which defines the unit of the blob size. To obtain $\xi 02(g)$, we can make use of the results for the radius of gyration of CW structures in the limit of large walking rates where only the dendritic behavior is obtained, i.e., $R02(N|w\u226b1)=\xi 02(g)$. Using these results, Eq. (15) can be solved numerically. To do this, we use a best fit approximation for the ideal radius of gyration of a large dendritic structure using Eq. (5),

The highest walking rate *w* = 100 yields an excellent fit with the parameters *A*_{1} = 14.66, *A*_{2} = 0.073, and *A*_{3} = 1.17, as shown in the inset in Fig. 8. The numerical solution of Eq. (15) provides the number of monomers inside the dendritic blob with a simulation dependent prefactor. In Fig. 8, the predicted exponential relation for dendritic structures [see Eq. (3)] expected for the CW catalysis for dendritic bottle-brushes can be fitted by

with best fit parameter *B*_{1} = 7.0 with fixed *α* = 2.15. To conclude this part, we could prove the assumptions of the dendritic blob model and provide the numerical constants in simulations. We note that the walking rate and also the Kuhn length have to be determined from the experiment in order to use our results for quantitative predictions.

It is interesting to note that dendritic blobs of CW structures approach the behavior of r3BA trees with respect to the node degree distribution *P*(*f*). This is shown in Fig. 9. For low walking rate *w* ≃ 1, the majority of monomers are bifunctional resembling linear chain characteristics with a low amount of branching units. For walking rates *w* ≫ 1, *P*(*f*) converges to the distribution of the corresponding r3BA tree. The high number of trifunctional monomers with a low amount of two-functional spacer monomer gives rise to a highly branched CW structure. However, structural defects like dead leaves or reduced branching contrary to perfect dendritic structures are present, albeit the overall logarithmic scaling is maintained.

### D. The extension of the CW structures without and with excluded volume

Knowing the number of monomers inside the dendritic blob, *g*, and the extension of the ideal dendritic blob, *ξ*_{0}, we can construct a scaling plot according to Eqs. (4) and (5). In Fig. 10, we show the ideal extension of CW structures rescaled by the asymptotic result for *N* ≫ *g* as a function of the number of dendritic blobs for various walking rates. For sufficiently high number of blobs *N*/*g* ≫ 1, all CW structures can be described by the scaling ansatz, Eq. (4), to construct a master plot. If the number of monomers is not sufficient to form a dendritic blob *N*/*g* ≪ 1, the data do not collapse due to a non-vanishing intrinsic extension on logarithmic scale, and no unified master regime can be obtained.

We now turn to the results where the conformations of CW structures are sampled under good solvent conditions. The radius of gyration is calculated by

where $r\u20d7i$ and $r\u20d7CM=1N\u2211j=1Nr\u20d7j$ denote the position of the *i*th monomer and the center-of-mass, respectively. In Fig. 11, we show the radius of gyration scaled with respect to linear chain behavior for different structures in good solvent condition. Similar to Fig. 5, the CW structures show a global linear chain scaling $Rev,L2\u223cN2\u22c5\nu $ for low *w*, whereas a mixed logarithmic-power-law behavior for dendritic structures^{14,21}$Rev,D2\u223cN2/5\u22c5ln(N)4/5$ for high *w* can be observed. As before the transition regime can be directly observed also for the good solvent condition reflecting disordered dendritic bottle-brushes. This can be again clearly observed in Fig. 11 up to a value of *w* = 10 and in the simulation snapshots in Fig. 1. Combing Eqs. (5) and (7) provides a mean-field relation between the dendritic blob size in the good solvent and the ideal case according to^{21}

The radius of gyration in the good solvent for different walking rates can be scaled according to Eqs. (4) and (19), as shown in Fig. 12, by using the fit function for the ideal blob size $\xi 0,fit2$ and dendritic blob monomers *g*_{fit}, which is, of course, unchanged with respect to the solvent conditions. Similar to the ideal case in Fig. 10, the data collapse for high number of blobs *N*/*g* ≫ 1 indicating dendritic bottle-brush behavior. Albeit the data are in fair agreement with the theoretical model, the observed slight splitting of the values is attributed to the mean-field approximation for dendritic structures.

Figure 12 displays the scaling relation for dendritic bottle-brushes in the form

where *f*_{R}(*x*) denotes the scaling or crossover function with the asymptotic limit of *f*_{R}(*x* ≫ 1) ∼ *x*^{2ν}. Interestingly, the scaling is not only well obeyed in the asymptotic case where bottle-brushes are formed but also for the transition region down to the purely dendritic limit, *x* ≪ 1. Here, the scaling plot displays an approximate slope of 0.7. The reason for the somewhat better overall scaling in the case of good solvent conditions as compared to the ideal case is related to the fact that the extension of the dendritic blob follows a mixed power-law logarithmic relation with respect to the number of monomers, see Eq. (19) and Refs. 21 and 31, which may be approximated by an effective power-law behavior. By contrast, the ideal extension of the dendritic blob displays a purely logarithmic behavior, which cannot be covered by a single variate crossover function.

## V. CONCLUSION AND OUTLOOK

The polymerization strategy based on a catalyst with the unique property of walking along the structure between two successive synthesis events can lead to a new synthesis route for dendritic structures. We have identified the walking rate, defined as the rate between walking and reaction events, which is the principal parameter controlling the resulting macromolecular structure. For low walking rates, linear structures with a low amount of side-branching are obtained, while high walking rates lead to disordered dendrimers. The latter display all features of dendrimers but contain structural defects. In particular, we predict an exponential relation between the walking rate and the number of monomers in the dendritic structure. If the growth proceeds ultimately all structures will reach a linear growth regime, which leads to bottle-brush molecules consisting of dendritic sub-units, which we call dendritic blobs. However, for walking rates much larger than 10, this crossover is shifted to extremely high molecular weights.

We have compared the dendritic blob model, which is based on scaling arguments, with extensive Monte Carlo simulations. Here, the catalytic growth is performed under good solvent conditions. The conformations of the resulting structures were analyzed with and without excluded volume conditions. In particular, we could prove the central concept of our scaling approach, which is the structure and properties of the dendritic blob. In particular, we used methods based on the spectral analysis of the connectivity matrix of the macromolecules in order to determine the blob size as a function of the walking rate. The results are in very good agreement with the theoretically predicted exponential relation.

The crossover between dendritic and linear growth toward the bottle-brush regime can be analyzed in the dependence of the radius of gyration as a function of the molar mass under good solvent conditions. The theoretically predicted scaling can be confirmed by using the blob size as calculated based on the spectral method. Overall, we can state that the findings for the CW catalysis are in excellent agreement with structures ranging from linear to dendritic with a definite transition regime only depending on the walking rate. Furthermore, the CW catalysis exhibits similarities to the so-called Bernoulli growth random walk with the limiting case of the Barabási–Albert graph for infinite walking rates.

From the experimental point of view, the thickness of the bottle-brush, e.g., the size of the dendritic blob, can be adjusted by modulation of the catalyst’s walking rate through external stimuli, e.g., synthesis pressure or temperature. Larger walking rates will lead to thicker bottle-brushes. The overall extension of the bottle-brush will only dependent on the synthesis time as the formation of dendritic sub-structures is related to the amount of polymerized monomers yielding elongated macromolecules for high molar masses. The numerical constants of the dendritic blob model, which we have fixed for our coarse-grained simulation model, can be related to the Kuhn length and the number of walking steps taken on the structure. While the former can be obtained directly, the latter can be adjusted by comparison with the scaling results. Candidates can be the increase in molar mass with reaction time or an analysis of the radius of gyration as a function of the molar mass. Both aspects require advanced methods of mass spectroscopy. However, once these parameters are fixed for a given catalyst, we are able to predict structural and conformation properties of macromolecules obtained in CW catalysis.

An interesting aspect is the possible synthesis of very dense dendritic bottle-brushes, which would be obtained at higher walking rates above *w* = 10, which correspond to “molecular objects” with well-defined shape where conformation fluctuations play a minor role only. Walking catalysis could be an interesting approach since polymerization is monomer-wise, i.e., corresponds to “grafting-from” approaches in polymer brush terminology. Thus, it circumvents large entropic barriers in contrast to grafting dendritic sub-structures at once to a backbone molecule.

Albeit some aspects of the CW catalysis have been simplified, e.g., influence of the tertiary carbon as barrier, any synthesis with a walking catalysis will result in highly branched structures resembling linear chain behavior on global scale. Therefore, the chain walking concept provides a straightforward one-pot-synthesis strategy to obtain dendritic structures for novel applications.

## ACKNOWLEDGMENTS

This work was supported by the Deutsche Forschungsgemeinschaft (DFG) under Contract No. SO-277/13. We thank the Center for Information Services and High Performance Computing (ZIH) at TU Dresden for generous allocations of CPU and GPU time.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

## DATA AVAILABILITY

The data that support the findings of this study are openly available in Zenodo at https://doi.org/10.5281/zenodo.6521297, Refs. 55. Raw simulation data (79GiB) of this study are available from the corresponding author upon request. The simulation algorithm and evaluation tools of the CW structures are openly available in Zenodo at https://doi.org/10.5281/zenodo.6497097, Refs. 39. The C++ framework LeMonADE is openly available in Zenodo at https://doi.org/10.5281/zenodo.5061542, Refs. 38.

### APPENDIX: BGRW AND BA GRAPH

Similar to the CW model, quite recently, Figueiredo *et al.*^{26} introduce the model of Bernoulli Growth Random Walk (BGRW). This algorithm grows a tree by randomly adding one new leaf to the structure. Starting with *m*_{0} = 2 connected nodes, a walker is placed on one of them. According to an attachment probability $p=1w$, a new leaf *m* with one edge is added and connected to the node with the walker *m*_{w}. After successful attachment of the leaf, the walker attribute is moved to the added node. If the attachment is rejected, the walker attribute is randomly moved to the adjacent nodes of *m*_{w} on the current graph. The random attachment and movement procedure repeats until the graph contains the desired number of nodes *N* and then terminates.

By construction, the BGRW for *w* = 1 will produce linear structures, whereas for high *w* ≫ 1, a random tree will emerge converging to the Barabási–Albert model.^{29,56} The differences between the CW and the BGRW model are the limited degree/functionality of node *f* ≤ 3 and excluded volume property in CW and the unlimited but finite node degree with no spatial interaction in BGRW. The similarities between the CW and the BGRW are the transient and recurrence walking properties^{26}—the walker cannot reach all nodes and escapes from the structure. Similarly, the BGRW should provide insights into the structural relation and showing the same occurrence of structures made of dense trees connected along a linear backbone similar to dendritic bottle-brushes.

The Barabási–Albert model^{27} (BA) is constructed as follows: we start with *m*_{0} = 2 connected nodes, see Fig. 13. Then, a new node is added with *m* = 1 edges to randomly selected existing *i*th node with the preferential attachment probability $pi=fi\u2211j=1nfj$ against uniformly distributed random number $\zeta \u22080;1$, where *f*_{i} corresponds to the node degree and *n* is the number of all nodes in the current graph. The attachment is repeated until the desired number of nodes *N* is reached. The BA model with *m* = 1 will produce a scale-free network with graph diameter exhibiting logarithmic growth.^{51,53} We have performed 1000 simulations both for BGRW and BA without any excluded volume condition to obtain the observables. We measure the radius of gyration $R02$ of the obtained structure according to Eq. (10) with *b* = 1, as depicted in Fig. 14. Similar to the CW structures, the BGRW shows the same transition process between linear growth to a tree with logarithmic extension known for the Barabási–Albert model with *m* = 1.

As both regimes have to merge the same argument as for CW structures applies, there the walker explores the structure within *τ* = *w* − 1 steps on the logarithmic scale *δ* = ln(*w*) = ln(*τ* + 1), giving rise for an exponential relation between the performed steps and explored distance *τ* = exp(*δ*) − 1 on the compact graph (see Fig. 15). The difference to Eq. (9) in the CW model is based on the restricted node degree, and therefore, the higher mobility of the walker resulting in the approximation *τ* ∼ *δ*^{2} for CW structures in a feasible way.

Another indication for the transition process of the BGRW model from linear global structures to small BA trees is the node degree distribution depicted in Fig. 16. For the BA model, the exact solution is given as^{27,57–59}

with *f* as node degree and *m* as edge insertion on the existing graph following the power-law dependence *P*_{m}(*f*) ∼ *f*^{−3} for the scale-free BA network. For low walking rates, the BGRW shows linear chain behavior as a high amount of bifunctional nodes are present. Increasing the walking rate approaches the BA-distribution *P*_{1}(*f*) as a limiting case.

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^{13}C NMR