Introduction

Dendrimers are one unique class of branch macromolecules with well-defined branch-upon-branch structures, which have attracted intensive scientific interest due to potential applications like nanomedicine for drug delivery and gene therapy, catalysts, energy harvesters, and data storage [1,2,3,4,5,6,7,8,9,10,11]. Accurate knowledge of the structure is necessary for the applications of dendrimers. The pioneer theoretical analysis of de Gennes and Hervet suggested that dendrimers adopt a dense-shell structure; i.e., the density profile increases from the center to the periphery [12]. However, succeeding studies demonstrated that the structure of neutral dendrimers with flexible spacers should be described by the dense-core model due to the back folding of terminal groups, in which the maximum density occurs in the center of the molecule and terminal groups disperse throughout of the dendrimer [13,14,15,16,17,18,19,20,21,22]. There was and still is much research devoted to obtain dendrimers with dense-shell structure as a large number of peripheral-end groups and a larger hollow core are preferred for many applications.

Possible strategies to obtain dendrimers with terminal groups located at periphery include introducing charged terminal groups, grafting hydrophilic polymers to the terminal groups, or applying polyelectrolyte dendrimers [23,24,25,26,27,28,29,30,31]. Intuition suggests that rigid dendrimers might be promising candidates as rigid spacers cannot fold back. Indeed, a dense-shell structure was observed for dendrimers with rigid spacers like phenylene dendrimers with low generation (G ≤ 4) [32,33,34,35]. However, when generation is high, terminal groups were found at a short distance from the core [36,37,38,39,40]. The wide distribution of terminal groups in rigid dendrimers was explained by the branching angle effect: Due to the non-zero branching angle, a daughter spacer will grow laterally instead of linearly respect to the mother spacer, so after several branches, a terminal group might locate near the core of dendrimer (Fig. 1) [38, 39].

Fig. 1
figure 1

Schematic diagram of a rigid dendrimer and natural bond angles involved in the simulation. The functionality of branch points f = 3, spacer length P = 2, and generation G = 4. The arrows show an inward folding of rigid spacers due to the restriction of branching angles. The branching angles θ1 and θ2 are defined as the angles between the mother spacer and daughter spacers, which equal π/3 in the illustration

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It has been realized for that branching angle is a key parameter to descript a real tree’s form and probable species to which it might belong to [41, 42]. We can expect that branching angle might also play an important role to influence the properties of the tree-like polymers (dendrimers). To our knowledge, most studies of rigid dendrimers were carried out with certain branching angles [36,37,38,39,40]; the properties of stiff dendrimers with varying branching angles have not been well examined. In the present work, we apply a coarse-grained Monte Carlo (MC) simulation to investigate model stiff dendrimers with different branch points and reveal the effect of branching angle on the properties of rigid dendrimers like size and radial density profiles. The effects of spacer length and generation are also discussed.

The remainder of the paper is organized as follows. In Section 2, we introduced the MC simulation method and coarse-grained models of dendrimers. In Section 3, we firstly examined the properties of flexible dendrimers and then the properties of rigid dendrimers with different branching angles. The effect of spacer length and generation was also examined. A brief conclusion was given in Section 4.

Model and methods

The simulation was carried out on a three-dimensional simple cubic lattice using the bond fluctuation method (BFM) [43,44,45,46,47,48]. In BFM, each monomer occupies eight corners of an elementary cube of the lattice, and the squared bond lengths l2 between two successive monomers are chosen as 4, 5, 6, 9, or 10 in units of lattice spacing. There are in total 87 different bond angles between two sequential bonds. The excluded volume effect was well considered that each lattice site can be occupied by one monomer at once and no bond cross occurs. BFM is a good approximation of a continuum model and meanwhile keeps the advantages of lattice algorithms. It is especially suitable to study highly branched polymer-like dendrimers [45,46,47,48]. For details of the implementation, see our previous work [49].

The dendrimers are generated by a divergent growth method. When f spacers, each with P monomers, attached to a core monomer, they form a G = 0 dendrimer. A G > 0 dendrimer is built by growing f-1 spacers from the ends of a G-1 dendrimer during relaxation. As a coarse-grained model, the properties of the monomers as the core, branching units and the spacers are identical and the total number of the monomers is:

$$ N=1+ fP\frac{{\left(f-1\right)}^{G+1}-1}{f-2} $$
(1)

where f is the functionality of the branch points, P is the spacer length, and G is the generation of the dendrimer.

The bond rigidity was introduced by a cosine squared angle potential [49]:

$$ E/{k}_BT=K\times {\left(\cos \theta -\cos {\theta}_0\right)}^2, $$
(2)

where K is the bending constant, kB is the Boltzmann constant, T is the absolute temperature, θ is the angle between two consecutive bond vectors, θ0 is the natural bond angle, and its value depends on the position of the monomers (Fig. 1). This potential is suitable to study semiflexible polymers in BFM, and the persistent length can be estimated as \( {l}_p=l\sqrt{\pi K} \) [49]. In this study, the functionality of a rigid dendrimer was fixed as f = 3. For the core monomer and monomers in the middle of the spacers, the associated natural angles are fixed as 2π/3 and 0, respectively. At each branch points, the natural bond angles between a mother spacer and two daughter spacers and the angle between daughter spacers are defined as θ1, θ2, and θ3, respectively. The values of branching angles are chosen based on different trisubstituted benzenes (Table 1), and a rigid dendrimer is named according to the position of substituents on a benzene ring; e.g., the branch point of meta-meta rigid dendrimer can be viewed as a 1,3,5-trisubstituted benzene ring. For comparison, the properties of an imaginary para-para rigid dendrimer have been examined, in which the natural branching angles are set to 0. Though the actual branching angles cannot reach 0 due to the exclude volume effect between daughter spacers, it is helpful to illustrate the effect of small branching angles. It should be pointed that the properties of monomers as branching units are the same as other monomers and not influenced by the type of the rigid dendrimers. In our simulation, the dihedral angle is not considered, and the spacers can freely rotate according to the given branching angles. In our simulation, the distance is measured in the unit of lattice spacing which is set to be 1, and the time is measured in the unit of MC steps (MCs), in one MCs all monomers have attempted to move once on average. The Metropolis importance algorithm was employed to determine that the move is accepted or not [50].

Table 1 The illustration of examined branch types and corresponding branching angles based on trisubstituted benzenes. The branching angles (θ1 and θ2) are the natural angles between a mother spacer and two daughter spacers, and θ3 is the angle between two daughter spacers. The root-mean-squared (RMS) branching angle is defined as\( \sqrt{\overline{\theta^2}}=\sqrt{\left({\theta}_1^2+{\theta}_2^2\right)/2} \).
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Results and discussion

Properties of flexible dendrimers

Firstly, we examined the properties of flexible dendrimers. Figure 2 shows the radial density profiles of dendrimers as a function of distance from the center of mass of dendrimers with different generation. For low G (G < 5) dendrimers, the density profile of dendrimer ρ(r) monotonically decays to zero with increasing distance from the center. For high G (G ≥ 5) dendrimers, ρ(r) shows a broad plateau region and a local minimum valley between the center peak and the plateau region. For all dendrimers, terminal groups are found to delocalize throughout the dendrimer. Thus, the structure of flexible dendrimers should be described by the dense-core model [13,14,15,16,17,18,19,20,21,22].

Fig. 2
figure 2

The radial density profiles ρ(r) of flexible dendrimers (solid lines) and the last generation (dashed lines) as a function of distance r from the center of mass of dendrimers. The functionality f = 3, spacer length P = 4, generation G = 3~9 as the arrow indicated. Usually, systems relaxed 106 MCs from corresponding initial states and the properties were calculated with a time interval of 103 MCs in the following 107 MCs; 5 independent trajectories were averaged. For a better view, the density of the last generation instead of terminal groups is given, which is about 4 times larger than that of terminal groups

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Even to date, there are still arguments about the scaling relationship between the dendrimer size (the radius of gyration Rg) and the number of monomers N [15,16,17,18,19,20, 51,52,53]. Since N is determined by spacer length P, generation G, and functionality f, we examine these factors separately (Fig. 3). The change in spacer length P results a scaling Rg~N3/5, which is identical to the behavior of linear polymers in good solvents. If only generation G is varied, a scaling Rg~N1/3 was found. We found Rg~N1/5 by varying the functionality f. If we try to describe all the dependencies of Rg in a single scaling law, the best one seems to be:

$$ {R}_g\sim {(PG)}^{2/5}{N}^{1/5} $$
(3)

which was firstly proposed by Chen and Cui [19].

Fig. 3
figure 3

The radius of gyration Rg of flexible dendrimers as a function of monomer number N by varying the spacer length P, functionality f, and generation G, respectively. The simulation parameters are the same as those in Fig. 2

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We calculated the end-to-end distance R of spacers (the distance between two successive branch points) of each subgeneration g (Fig. 4a). For all dendrimers, R decreases monotonously with increasing g, which suggests that inner spacers are more stretched than outside ones due to the crowded environment. The end-to-end distance R of the inner-most spacers (g = 0) increases with increasing generation of dendrimers; e.g., R of g = 0 spacers of a G = 9 dendrimer is about 65% larger than that of a G = 3 dendrimer. Meanwhile, the spacers of the last generation are weakly stretched and seem insensitive to the generation of dendrimers. Figure 4b shows R of spacers as a function of a topological distance from the last generation (g′ = G − g). It was found that R of spacers with same g′ is similar and is slightly affected by the size of dendrimer (except the inner-most spacers, g′ = G). The stretching of middle spacers can be roughly described by a Gaussian distribution. Interestingly, Markelov et al. found that the segmental mobility of a dendrimer is also determined by a topological distance between the segment and periphery and is independent of G [54].

Fig. 4
figure 4

a The end-to-end distance R of spacers as a function of subgeneration g in dendrimers with indicated generation G. b R as a function of a topological distance from the last generation g′ = G − g. The spacer length P = 4 and functionality f = 3. The dot line was draw according to a Gaussian fit to guide eyes

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Properties of meta-meta rigid dendrimers

While studying properties of rigid dendrimers, we fixed the spacer length P = 3 and functionality f = 3. The generation was chosen as G = 6, which is high enough to reveal the branching angle effect. With a flexible dendrimer (the bending constant K = 0) as the initial state, a rigid dendrimer was obtained by gradually increasing K from 0 to 30kBT in 6 steps and relaxed 107 MCs at each step. Usually, the rigidity of a linear polymer can be characterized by the ratio between its persistent length (lp) and contour length (L). With the chosen bending constant, the ratio is about 5.6 for spacers and 0.8 for the chain from the center monomer to terminal monomers [49], suggesting local rigid spacers band globally semiflexible dendrimers.

Firstly, we examined the properties of meta-meta rigid dendrimers, of which the branching angles θ1 = θ2 = π/3. Figure 5 shows the snapshots of the initial and final states, respectively. When K = 0, the spacers are flexible, and overall structure of the dendrimer is compact (Fig. 5a). When K = 30kBT, the spacers become rod-like and connected together with given branching angles (as indicated by the cycle in Fig. 5b); the dendrimer expands and adopts a relative open structure (Fig. 5b).

Fig. 5
figure 5

Snapshots of meta-meta rigid dendrimers: (a) initial state (the bending constant K = 0); (b) final state (K = 30kBT). The functionality f = 3, spacer length P = 3, and generation G = 6. Monomers within the same subgeneration are displayed with the same color. The core monomer is blue and monomers of the last generation are red. The snapshots have been resized for clarity

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The density profiles of meta-meta rigid dendrimer are shown in Fig. 6. Both the densities of dendrimer ρ(r) and the last generation ρend(r) are much smaller than those of flexible dendrimer. The density of dendrimer ρ(r) displays a broad plateau in the core, but the location of the maximum is not so clear. The density suggests that terminal groups widely distribute throughout the meta-meta rigid dendrimer. This wide distribution can been explained by the branching angle effect: [38, 39] due to the restriction of branching angle, a daughter spacer will grow laterally instead of linearly respect to the mother spacer. At branch points, one half of the groups extend and the other fold back in towards the center of the dendrimer (as indicated in Fig. 1). After several generations, the terminal groups might approach the center of the mass of the dendrimer.

Fig. 6
figure 6

The radial density profiles of meta-meta rigid dendrimers (filled circles) and the last generation (filled triangles) as a function of distance r from the center of mass of dendrimers. The opened symbols are the densities of flexible dendrimers

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Effect of branching angle

According to the branching angle effect, we can expect that the value of branching angle plays a decisive role in the system of rigid dendrimers with fixed generation. We further examined rigid dendrimers with different branching angles. A relative open structure is observed for meta-para rigid dendrimer (Fig. 7a). The structure gradually becomes more compact with increasing branching angles from ortho-meta rigid dendrimer (Fig. 7b) to ortho-ortho rigid dendrimer (Fig. 7c). While in the rigid dendrimer with “zero” branching angle (para-para rigid dendrimer), a structure with a large hollow core and most terminal groups located at periphery is observed (Fig. 7d). We can find that the spacers of para-para rigid dendrimer are not strictly straight (Fig. 7d). One reason is that both daughter spacers cannot parallel to the mother spacers simultaneously due to the excluded volume effect. The other is that the applied bending constant was moderate and the global flexibility of dendrimer is preserved.

Fig. 7
figure 7

Snapshots of (a) meta-para, (b) ortho-meta, (c) ortho-ortho, and (d) para-para rigid dendrimers. The branching angles of given rigid dendrimers are shown in Table 1. The simulation parameters are the same as those in Fig. 5

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The root-mean-squared (RMS) branching angle \( \sqrt{\overline{\theta^2}}=\sqrt{\left({\theta}_1^2+{\theta}_2^2\right)/2} \) was applied to quantitatively characterize the branching angle effect. The scaled radius of gyration Rg/Rg,0 (Rg and Rg,0 are the radius of gyration of rigid and flexible dendrimers, respectively) decreases with increasing RMS branching angle (Fig. 8a): when \( \sqrt{\overline{\theta^2}}=0 \) (para-para rigid dendrimer), the scaled radius of gyration Rg/Rg,0 = 1.55, when \( \sqrt{\overline{\theta^2}}=2\pi /3 \) (ortho-ortho rigid dendrimer), Rg/Rg,0 = 0.86. Thus, the size of a rigid dendrimer can be larger or smaller than that of flexible dendrimers, which depends on the branching angle. Contrary to the size, the density of dendrimers increases with increasing RMS branching angle (Fig. 8b). Among all dendrimers, ortho-ortho rigid dendrimer has the densest core; the maximum density of the core is about 6 times larger than that of para-para rigid dendrimer. The density of terminal groups ρend(r) shows a similar trend which decreases with decreasing RMS branching angle (Fig. 8b). The peak of ρend(r) shifts further from the center of the dendrimer with decreasing RMS branching angle. Due to the bending of spacers as indicated in Fig. 7d, even in para-para rigid dendrimer (\( \sqrt{\overline{\theta^2}}=0 \)), ρend(r) in the core is very low but not zero.

Fig. 8
figure 8

a The scaled radius gyration Rg/Rg,0 of rigid dendrimers as a function of the root-mean-squared (RMS) branching angle \( \sqrt{\overline{\theta^2}} \). Rg,0 is the radius gyration of flexible dendrimers. b The radial density profiles ρ(r) of rigid dendrimers. The inset shows the density profiles ρend(r) of the last generation. The dotted lines are the results of flexible dendrimers

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The location of terminal groups of dendrimers is important for catalysis and molecular recognition applications. The spatial distribution of terminal groups was characterized by the proportion P(r) at a distance r to the center of mass of the dendrimer, which was calculated as:

$$ P(r)={N}_e(r)/{N}_e $$
(4)

where Ne(r) is the number of terminal groups at distance r. In rigid dendrimer with large RMS branching angle (ortho-ortho and meta-ortho rigid dendrimers), the terminal groups show an almost symmetry distribution (Fig. 9). The distribution gradually becomes asymmetric, and more terminal groups tend to locate further from the center of mass with decreasing RMS branching angle.

Fig. 9
figure 9

The proportion of terminal groups at a distance r from the center of mass of dendrimers. The dot line is the result of flexible dendrimer

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Effects of generation and spacer length

So far, we mainly focused on the effect of branching angle. The effects of generation and spacer length were further examined, and the results were briefly summarized as follows. According to the branching angle effect (Fig. 1), we can expect that a critical generation is required to find the terminal groups of a rigid dendrimer near the center. For meta-meta rigid dendrimers, this critical generation should be 5. The properties of meta-meta dendrimers are examined; both the densities of dendrimer ρ(r) (not shown) and terminal groups ρend(r) (Fig. 10a) are smaller than those of flexible ones. In the inner region, ρend(r) significantly increases with G. But, ρend(r) is non-zero near the center even when the generation is as low as G = 3. It might be explained by the fact that the bending constant is not very high. For the chain from the core monomer to one terminal monomer, the ratio between the persistent length and contour length is 1.4 when the bending constant K = 30kBT; i.e., the whole dendrimer still exhibits certain flexibility.

Fig. 10
figure 10

The radial density profiles of the last generation of meta-meta rigid dendrimers with given generation G (a) and spacer length P (b). The dot lines are the results of corresponding flexible dendrimers. The other parameters are the same as those in Fig. 5

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At first glance of Fig. 1, the spacer length should not influence the location of terminal groups. For a G = 5 meta-meta dendrimer, the terminal monomer always approaches the core monomer regardless the spacer length. We found that the densities of meta-meta dendrimers ρ(r) (not shown) and terminal groups ρend(r) significantly decreased with increasing spacer length (Fig. 10b). The reason is that the illustration of Fig. 1 is in a two-dimensional space. But in three dimensions, the spacers can freely rotate; thus, the chance of a terminal groups to approach center decreases with increasing spacer length due to large volume.

Conclusions

We have applied BFM to examine the effect of branching angles on the properties of rigid dendrimers. Dendrimers with flexible spacers were examined for comparison, which have a widely distributed terminal groups due to the back folding of spacers consistent with the prediction of dense-core model. According to the stretching of inner spacers, the topological distance from the last generation (g′ = G − g) might be a useful parameter to describe local properties for flexible dendrimers with different G.

Intuition suggests that rigid dendrimers might adopt a dense-shell structure with terminal groups locating at periphery. However, our simulation proved that the terminal groups of rigid dendrimers might locate near the center of molecule caused by the branching angle effect: Due to the restriction of a non-zero branching angle, a daughter spacer grows laterally with respect to the corresponding mother spacer. Thus, terminal groups might appear near the center for rigid dendrimers with large branching angles and high generations. For rigid dendrimers with fixed generation and spacer length, the properties such as size, density, and location of terminal groups could be tuned by adjusting branching angles. When the branching angle is small, we can obtain rigid dendrimers with an open structure, larger size, lower density, and more terminal groups at periphery.

Due to the restriction of the lattice nature of BFM, we have not applied a very large bending constant as which will reduce the Metropolis sampling efficiency. Certain flexibilities of the spacers and dendrimer are reserved; e. g., the terminal groups of rigid dendrimers with low G can still appear near the center. It might be possible to suppress the flexibility by applying off-lattice simulations like molecular dynamics simulation or dissipative particle simulation with a very large bending constant. But a small time interval is required to update the state of the system. We noticed that the shape of a tree-like body is affected by both the branching angles and the relative length of daughter spacers to mother spacer [41, 42]. It would be interesting to study rigid dendrimers with asymmetry branching like poly(L-lysine) dendrimers [55, 56] in the future.