Dendrimer molecules and aggregates are chemical structures with regular branching that underlies their physicochemical properties. Regular dendrimers have been studied both theoretically and experimentally, but the irregular intermediate structures between the dendrimers of neighboring generations have not. In the present work, dendrimer aggregates, both regular and intermediate, are investigated in terms of the information entropy approach. As found, the information entropy of the regular dendrimer asymptotically increases with the generation number; herewith, its maximal value equals 2. The intermediate structures have been studied for the growing dendrimer G1 → G2 → G3 → G4 with the tricoordinated building block. The plot of the information entropy of the growing dendrimer on the size has the frontier consisting of the lowest values that correspond to the regular and irregular structures described with the symmetrical graphs. Other intermediate structures have information entropies higher than the regular dendrimers. Thus, to move the system from one informationally stable state to another, its information capacity must be temporarily increased.

Fractals are mathematical objects made up of patterns manifesting self-similarity [

Chemical fractals and dendrimers have become the objects of theoretical research, including their treatment in terms of the graph and information theories [

Mathematical properties of the information entropies of dendrimers have been deeply studied by Chen et al. [

We briefly note that other theoretical approaches that are invoked to scrutinize the dendrimer growth and molecular dynamics techniques are some of the most efficient tools for this purpose [

In the present work, we use a simple dendrimer model to rationalize its information entropy in the context of physicochemical properties and, for the first time, consider imperfect dendrimers, intermediates between two generations of regular dendrimers.

We consider the dendrimer aggregate formed with identical building blocks, which are chemically indistinguishable and have the same coordination number,

Each block of the

According to the original Shannon approach, we select the particle _{i}_{i}_{1} vertices of the 1st shell, _{2} vertices of the 2nd shell, and … _{n}_{1} + 1 × _{2} + … + 1 × _{n}

The weights of the blocks from each shell (_{i}

To apply this approach to routinized calculations, expression (3) should be simplified to be more convenient. We present it in the following form (the transformation of Equation (3) is shown in

Two terms of the derived Equation (5) have a physicochemical (structural) interpretation. The term _{max}_{fract}

In the case of _{max}_{fract}

Two latter equations are the same as the expressions obtained for the linear polymer with the odd number of building blocks in terms of the general definition of information entropy and Equation (2) (

We used Equations (5)–(7) to scrutinize the behavior of the information entropy of the dendrimer upon the infinitely increasing generation number. As mentioned above, the generation numbers of the synthesized dendrimer molecules are not large, so assessing the upper bounds, _{∞}, primarily relates to the theoretical interest. On the other hand, the potential applications of dendrimers include information storage with nanopatterned materials [_{∞}, may also be used as waymarks in this direction.

We found that dependences, _{∞}, only numerically: 2 (_{∞} decreases with increasing

The asymptotic behavior of functions _{∞}) qualitatively relates to the experimental results. For example, the regularities of the processes of dendrimer formation have similar asymptotic behaviors such as velocity of the growth of silver dendrites upon precipitation [

Previous works on the topology of dendrimers have been devoted to the structures with fulfilled shells. However, the next generation of a dendrimer is not produced in one step, so that intermediate imperfect structures are formed when filling the shells. We consider, in general, the issue of intermediate structures between generations G

According to path I, the filling starts from any building block (due to their equivalence) of the outer shell number

Path II corresponds to another reactivity ratio; building blocks with a lower number of uncompensated valences have lower reactivity toward free blocks. Therefore, free blocks are attached to “bare” blocks of the outer shell (

The frontiers of lower _{front}_{i}_{Gn}, _{i}_{G(n + 1)} for _{Gn} < _{i}_{G(n+ 1)}. Function _{front}

Thus, the different reactivity of the terminal building blocks of the dendrimer is reflected with the type of the functions, _{front}

We propose that our theoretical findings will stimulate further studies of dendrimers in the aspects of their numerical modeling and material science. Symmetrical and homogeneous chemical systems manifest low information entropies, and vice versa [_{fract}_{max}

Thermodynamic entropy becomes a part of molecular dynamics studies on complex chemical systems; e.g., thermodynamic entropy of enzyme–substrate complexations [

We have used analytical expressions

For the first time, we have studied the irregular intermediates between the dendrimers of stepwise generations. We have found that size-dependencies,

We propose that the theoretical findings of this work will stimulate further studies of dendrimers regarding their material science applications [

The following are available online at

Conceptualization, D.S.; methodology, D.S.; validation, A.T. and I.S.; formal analysis, A.T.; investigation, D.S. and A.T.; resources, I.S.; writing—original draft preparation, D.S. and A.T.; writing—review and editing, I.S.; visualization, A.T.; supervision, D.S.; project administration, D.S.; funding acquisition, D.S. All authors have read and agreed to the published version of the manuscript.

This research was partly funded by the Council of the Grants of the President of RF, grant number MD-874.2021.1.3.

Not applicable.

Not applicable.

The work has been performed under the theme ‘Novel theoretical approaches and software for modeling complex chemical processes and compounds with tunable physicochemical properties’ (registration number AAAA-A19-119022290011-6, Russian Federation).

The authors declare no conflict of interest.

_{60}fullerene aggregation: Equilibrium thermodynamics approach

Klochkovskii, S.P. (private demonstration of the ammonia amalgam growth

_{60/70}(

_{60}fullerene

The dendrimer of generation

The dependences of information entropy of the dendrimers with various coordination numbers of building blocks,

The dependence of the limit information entropy, _{∞}, of dendrimers on the coordination number.

Schematic of two paths of transformation G1 → G2 (new added blocks are shown in red hereinafter, and the blocks are lettered according to the equivalence of their positions).

Information entropy upon dendrimer growth, G1 → G2 → G3 → G4, according to modes I (_{front}

The regular and intermediate structures, in which _{front}

The intermediate structures, in which _{front}