Insight into the Structure of Polybutylcarbosilane Dendrimer Melts via

Article pubs.acs.org/Macromolecules

Insight into the Structure of Polybutylcarbosilane Dendrimer Melts via Extensive Molecular Dynamics Simulations N. K. Balabaev,† M. A. Mazo,‡ and E. Yu. Kramarenko*,§,∥ †

Institute of Mathematical Problems of Biology, Keldysh Institute of Applied Mathematics RAS, Pushchino, Moscow Region 142290, Russia ‡ Semenov Institute of Chemical Physics RAS, Moscow 119991, Russia § Faculty of Physics, Lomonosov Moscow State University, Moscow 119991, Russia ∥ A.N. Nesmeyanov Institute for Organoelement Compounds RAS, Moscow 119991, Russia S Supporting Information *

ABSTRACT: Extensive molecular dynamics simulations of polybutylcarbosilane dendrimer melts were performed in a wide temperature range from 300 to 600 K. The melt macroscopic and structural characteristics were analyzed for the third up to the eighth generation dendrimers for the systems consisting of 8 and 27 dendrimer molecules in the simulation box. For every system, averaging was performed over 8 independent simulation runs and along equilibrium time trajectories of up to 5 ns. Calculated values of the thermal expansion coefficients, heat capacity, and self-diffusion coefficients are in a good agreement with experimental observations. Analysis of the molecular mass dependence of the gyration radius and shape factor, detailed radial density distributions of dendrimer structural units, mobility of the branching points, and intermolecular interaction energy allowed to shed light on the basics of distinction in behavior of low and high generation dendrimer melts and formulate the directions of further research.

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INTRODUCTION Dendrimers belong to a quite new class of polymer materials with unique architecture and properties.1−9 High monodispersity, an enormously large amount of functional groups in comparison with conventional macromolecules, and a variety of chemical compositions available make them very promising for numerous applications,10,11 some of them being already implemented, while most of their potential is believed to lie in the future. Dendrimer research is one of a few areas of polymer science where theory and computer modeling have been developing mainly in parallel with synthetic schemes. This was largely predetermined by a regular tree-like molecular structure of dendrimers which suggested itself to be modeled and studied by rigorous mathematical methods while the direct synthesis was nontrivial and time-consuming. Thus, these two directions have been developing hand in hand. On the first stages, the main attention was paid to conformational properties of isolated dendrimer molecules depending on generation, functionality of the branching points, and spacer lengths. The effects of limiting generation as well as the validity of two alternative models of single dendrimers, namely, dense-shell9,12 and dense-core9,13−19 models, were widely discussed. The density profile, increasing from the center of dendrimer to its periphery (dense-shell model), was obtained in the pioneering theoretical treatment of dendrimers by de Gennes and Hervet.12 Further computer simulations that enabled a direct insight into dendrimer structure have led to an alternative model.13−19 Namely, they have revealed a completely different © XXXX American Chemical Society

picture of spatial distribution of monomer units within dendrimer molecules. In particular, it has been shown that dendrimer density is maximum in the center and gradually decreases with the distance from the molecule center toward its exterior (dense-core model). Furthermore, it has been found that terminal groups are not localized at the periphery; instead, they are distributed throughout the whole dendrimer interior.13−19 Based on the obtained picture and developed simple models, the scaling relations for dendrimer meansquared radius of gyration were derived for various solvent conditions (good, theta, poor solvents)9 and were studied by extensive computer simulations.13−19 Since the number of monomer units in dendrimer molecule grows exponentially with generation, possibilities of computer modeling in the early stages of dendrimer development were limited to studies of isolated dendrimers within a coarsegrained approach.13−19 Modeling at the atomic level was performed for several first generations dendrimers.20−25 Experimental studies were in the beginning focused mainly on dilute solutions of low generation dendrimers7,18,26 since large-quantity synthesis was mainly carried out for the first generations where the unique properties of dendrimers justified the difficulty and cost of their preparation. Dendrimer behavior in concentrated solutions and melts has only started to be explored recently,27−35 and it is still not well Received: July 29, 2016 Revised: December 15, 2016

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DOI: 10.1021/acs.macromol.6b01639 Macromolecules XXXX, XXX, XXX−XXX

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Macromolecules comprehended even for low generations. The questions of conformational behavior of dendrimers as well as their interaction and interpenetration in concentrated solutions and melts remain to be addressed. Furthermore, high generation dendrimers acquiring all the specificity of dendritic architecture require special attention. New synthetic schemes allow now to obtain them in amounts necessary for comprehensive studies, and understanding of their solution and melt behavior could open new areas of their practical applications. In this respect, computer simulations could be of great benefit. Nowadays due to fast development of computer capacities, computer modeling rose to a new stage of development. It could now handle larger systems and become a powerful tool in solving new complex many-particle problems. Up to now there have appeared only a few papers on computer simulations of dendrimer melts. In particular, coarse-grained modeling was applied to study second to fifth generation polypropyleneimine dendrimer melts at 400 K,32 the fourth and fifth generation polypropyleneimine dendrimer solutions and melts at room temperature,33 and the second to fourth generation polyphenylene dendrimer melts at room temperature.34 Atomistic modeling of polycarbosilane dendrimers of the third, fifth, and sixth generations at 600 K was performed in ref 35. Nonequilibrium molecular dynamics simulations are reported for the shearing behavior of dendrimers in the melt in refs 36 and 37. In refs 38 and 39 molecular dynamics simulations of the third to the fifth generation dendrimers within united atom dendrimer model have been performed in a wide temperature range. This work is aimed to a development of an adequate atomistic melt model for various generation polycarbosilane dendrimers and to assess its capabilities in describing physical properties of real systems as well as to explore all the dendrimer melt main structural and dynamics features. Our choice is determined, first of all, by numerous experimental data collected over the past 10 years on these systems and an unusual behavior reported for high generation dendrimers. In particular, extensive experimental studies of poly(butylcarbosilane) dendrimers by various experimental techniques (viscometry, precision adiabatic vacuum and differential scanning calorimetry, small-angle dynamic light scattering, and atomic force microscopy) have shown that increasing dendrimer generation causes a qualitative change of the melt properties.29,40−45 Namely, while low generation dendrimers demonstrate liquid-like behavior, the high generation dendrimers acquire properties typical for solids. This transition occurs when switching from the sixth to the seventh generation in 3−3 dendrimers (the first number 3 stands for the core functionality while the second number 3 denotes the functionality of the branching points). Similar behavior was observed for the 4−3 and 4−4 polycarbosilane dendrimers of the same homologues series with the transition shifted by one generation with increasing functionality of the core and functionality of the branching point.41 Namely, it is located between the 5 and 6 generations for 4−3 dendrimers and between the 4 and 5 generations for 4−4 dendrimers. Furthermore, with increasing generation there appears a second relaxation transition in temperature dependence of the heat capacity of the dendrimer melts located at 450−500 K.41−43 The authors have supposed the formation of a physical network of entanglements in high generation dendrimer melts. Because of a rather short length of dendrimer spacers, these entanglements should differ from the classical ones typical for long linear macromolecules.41

It should be noted that the liquid-like to solid-like transition is observed for poly(butylcarbosilane) dendrimer generations which are far from the limiting one Gmax. Gmax describes a crossover between low generation dendrimers, for which defect-free synthesis is possible, and high-G dendrimers, where structural perfection is impossible because of packing constraints. The concept of the limiting generation is widely discussed, and the regimes of G > Gmax and G < Gmax were studied in detail in refs 46−48. It should be stressed that no specific interactions are expected in such systems which could lead to the melt gelation. The observed sharp transitions in relaxation mechanisms are waiting for a rigorous analysis. Because of the absence of any specific interactions, the microstructural reorganization in these systems could hardly be seen by traditional physical methods; in this respect, computer simulations could be competitive, giving some information on microscopic structure and dynamics. The present study covering for the first time a wide range of temperatures and dendrimer generations provides the first deep insight into these issues to draw a molecular picture of carbosilane dendrimer melts depending on generation and temperature. The paper is organized as follows. In the next section we present the model and the method of simulations. Then we study the static properties of the melts depending on generation and temperature; namely, we calculate some macroscopic melt characteristics such as melt density, thermal expansion coefficients, and heat capacity and analyze details of the melt structure, in particular, radial density distributions, interparticle interaction energy, radius of gyration, and shape factor for a single dendrimer. By comparing results obtained within the proposed model and those available from experiments, we confirm the reliability of the model. Additional studies of isolated dendrimers of the third up to the eighth generations in the same temperature range as for the melts allow us to elucidate the effect of the interparticle interactions on the conformational behavior of single molecules in melts. As a next step, we analyze for the first time the mobility of various dendrimer structuring units in melts. The obtained details on the melt microstructure and molecular mobility allow to draw some preliminary picture of the dendrimer behavior in melts.

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MODEL AND SIMULATION DETAILS Structure of a Dendrimer. Figure 1 shows a schematic representation of the molecular structure of the first generation carbosilane dendrimer under study. The 4−3 dendrimers are modeled; i.e., the core Si functionality is chosen to equal 4 while the functionality of the Si branching points is equal to 3.

Figure 1. Molecular structure of the first generation carbosilane dendrimer. B


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generation molecule, and 27G3 is for the melt of 27 dendrimer molecules of the third generation. All the equilibrium melt characteristics were calculated via time and ensemble averaging along the equilibrated conformation trajectories. Ensemble averaging was performed over eight independent system realizations for all temperatures and generations. Using both time and ensemble averaging allowed to cover time scales of up to 64 ns at the least.

Spacers consist of three methylene groups. The terminal segments are similar to the spacers comprising three methylene groups and a methyl terminal group. Methyl and methylene groups are treated as united atoms. All simulations presented in this work have been performed with the use of the PUMA software package.49,50 The AMBER force field51,52 is used in simulations with potentials accounting for bond stretching, bond bending, and dihedral angle rotation around the equilibrium values. Nonbonded interactions are expressed via the Lennard-Jones potential. Besides, electrostatic interactions arising from partial charges on Si atoms and methylene groups are also taken into account. Tables S1−S5 in the Supporting Information provide all the main potential parameters. The cutoff distance for LJ potential was set to 1.05 nm. The initial dendrimer structure with correct values of the bond length and valence angles and without atom overlap was constructed via a procedure described elsewhere.20 Equilibrium conformations of dendrimers from the third up to the eighth generations were obtained at T = 800 K. Modeling of Dendrimer Melts. Systems of 27 and 8 dendrimers for every generation (from G = 3 to G = 8) were obtained by multiplying a single dendrimer conformation and placing the molecules in a cubic box at an average low density of 0.1 g/cm3. The periodic boundary conditions were applied in X, Y, and Z directions. System equilibration was carried out at the constant high temperature of 800 K to provide good mixing. The box size was gradually decreased during this process until the melt density reached the value of 0.8 g/cm3. Then NPT ensemble simulation of the dendrimer melts was started at different temperatures T = 300, 350, 400, 500, and 600 K and atmospheric pressure. The systems were allowed to equilibrate so that an equilibrium density of the melt was reached at every temperature. Standard MD techniques with collisional thermostat53,54 and Berendsen barostat55 have been used for the system relaxation. Elementary integration step was 0.002 ps, and the total equilibration time was varied between 1 and 5 ns; the total simulation time reaches 8−13 ns. Achieving steady state was controlled by monitoring the system density, radius of gyration of single dendrimer molecules, and all contributions to the energy of the system. To compare dendrimer structural characteristics in melts and in dilute solutions, additional simulations of single dendrimer molecules from the third up to eighth generations have been performed at the same temperatures as for the melts. We have used the same interaction parameters for isolated dendrimers as for dendrimers in melts; it means that effectively we simulate isolated dendrimers in a vacuum. At room temperature single dendrimer conformations in a vacuum should roughly correspond to those of a dendrimer in a poor solvent. At enhanced temperatures the attractive part of the Lennard-Jones potential plays a minor role, and we could expect dendrimer swelling similar to that in a good solvent. Although in the latter case the effect of solvent quality on microscopic dendrimer characteristics could be somewhat different from that of temperature. In the following the systems under study are denoted as iGj, where “i” stands for the number of molecules in the simulation box (i = 1, 8, and 27 for an isolated dendrimer and for 8 molecules and 27 dendrimer molecules in the box, respectively) and “j” is the generation number (j takes the values from 3 up to 8). In particular, 8G3 refers to the melt of 8 dendrimers of the third generation in the box, 1G5 denotes a single fifth

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RESULTS AND DISCUSSION Structural Characteristics. Macroscopic Melt Characteristics vs Generation and Temperature. The first important characteristic of the dendrimer melt is its density, ρ. As an example in Figure 2 we plot the time dependences of ρ for the

Figure 2. Time evolution of the melt density for G7 systems containing 27 (solid lines) and 8 (dotted lines) dendrimers in the simulation box at various temperatures: black, 300 K; olive, 400 K; blue, 500 K; red, 600 K.

systems containing 8 and 27 dendrimers of the seventh generation. One can see that both systems reach the equilibrium. Simulations show that for a smaller generation and higher temperature less time is needed for the system to equilibrate. Larger system simulations are much more timeconsuming. One can see in Figure 2 that for 8G7 the equilibration time increased from roughly 0.2 ns at 600 K to more than 1.6 ns at 300 K. Equilibration of 27G7 melt takes even more time. Even at high temperature of 600 K the relaxation period proceeded up to 2.5 ns. The relaxation times calculated from the autocorrelation functions of the dendrimer gyration radius and asphericity (see Supporting Information, Figure S2) are much smaller than the equilibration time for all the systems. To check whether the system has reached the equilibrium state, we have additionally performed simulations at enhanced pressure of 0.1 GPa and then reducing it down to the atmospheric pressure value. Time evolution of the applied pressure and resulting density is shown in Figure S1. It has been shown that the density evolution follows the pressure change, and the density value returns back to the initial equilibrated value when the pressure is set at the atmospheric level. The obtained values of ρ for all the systems under study are summarized in Table 1. Up to the sixth generation, there is practically no difference in density for the melts containing 8 and 27 dendrimers at any temperature. However, for the seventh generation there is a clear increase of ρ in the case of 8 dendrimers. This fact can be explained by the effect of periodic boundary conditions imposing some restrictions on the type of C


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Macromolecules Table 1. Density of G-Generation Dendrimer Melts at Various Temperatures density, ρ (g/cm3) generation

no. of atoms in a dendrimer

no. of dendrimers in the box

no. of atoms in the box

T = 300 K

T = 350 K

T = 400 K

T = 500 K

T = 600 K

G3

269

G4

557

G5

1133

G6

2285

G7

4589 9197

7263 2152 15039 4456 30591 9064 61695 18280 123903 36712 248319 73576

1.015 1.015 1.016 1.016 1.015 1.021 1.006 1.020 0.999 1.018

G8

27 8 27 8 27 8 27 8 27 8 27 8

0.982 0.982 0.985 0.986 0.973 0.988 0.989 0.990 0.997 0.993 0.880 0.995

0.941 0.942 0.947 0.947 0.950 0.950 0.954 0.953 0.953 0.959 0.839 0.962

0.863 0.864 0.870 0.870 0.875 0.875 0.880 0.881 0.880 0.891 0.756 0.892

0.781 0.784 0.791 0.792 0.799 0.800 0.806 0.809 0.797 0.821 0.683 0.820

1.004

chosen force field. The change of the LJ potential form from 6−12 to 6−9 leads to even higher density values. This difference in ρ is believed not to affect the main conclusions of this work. More pronounced dependence of ρ on generation at T = 600 K in comparison with T = 300 K was mentioned in ref 35. Second, it is clearly seen in Figure 3 that the melt density decreases with increasing temperature for all dendrimer generations. The temperature dependence is stronger than generation dependence and is due to thermal expansion of the melt. The thermal expansion coefficients β are determined from the slope of ρ(T) dependences at every T; the obtained β values are summarized in Table 2. The coefficient β slightly

molecule packing in the system. It will be shown below that in contrast to low generation dendrimers demonstrating high degree of mixing, dendrimers of high generations become rather rigid and less interpenetrating. As a result, they tend to order to minimize the free volume between neighboring molecules. While a favorable hexagonal packing can be easily realized for the system of 8 dendrimers, in the case of 27 molecules in the simulation box this type of dendrimer ordering is broken due to translation of the simulation box, and only a less favorable cubic lattice arrangement is possible. This result clearly demonstrates that the size of the simulation box can affect the system behavior, and a careful approach is needed for the right choice of the system dimensions. Figure 3 shows the temperature dependence of the melt density calculated for the system of eight dendrimers for

Table 2. Coefficients of Thermal Expansion for Various Generation Dendrimers dendrimer melt 8G3 8G4 8G5 8G6 8G7 8G8

coefficient of thermal expansion 7.1 6.6 6.6 6.0 5.3 6.7

× × × × × ×

10−4 10−4 10−4 10−4 10−4 10−4

(T (T (T (T (T (T

= = = = = =

300 300 300 300 300 350

K)−10.5 × 10−4 (T = 600 K) K)−10.4 × 10−4 (T = 600 K) K)−9.8 × 10−4 (T = 600 K) K)−8.4 × 10−4 (T = 600 K) K)−9.0 × 10−4 (T = 600 K) K)−8.7 × 10−4 (T = 600 K)

increases with temperature but decreases with increasing generation. This behavior is quite expected because the lower generations are known to have a somewhat higher swelling capability (the change in volume of single dendrimer molecule upon solvent uptake is 30−40% for low generation dendrimers, and it drops down to 20% for G8 dendrimers56). Besides, the values of the calculated coefficients are close to those measured experimentally.57 The analysis of the temperature dependences of the system energy allowed us to estimate the heat capacity of the dendrimer melts. It was calculated as cp = dE/dT, where E = U + K, U and K being the total potential and kinetic energies of the melt, respectively. The average values obtained in the temperature range of 350−600 K equal 9, 19, 37, 73, 145, and 285 kJ/(mol K) for the third to the eighth generations, respectively, with the accuracy of about 1 kJ/(mol K). These values are in a surprisingly good agreement with the experimental data corresponding to the plateau of the experimental cp(T) dependences.41−43 Unfortunately, the presented simulation model does not allow to reproduce

Figure 3. Temperature dependence of the melt density for various generation dendrimers.

various G. First, one can see that the melt density only slightly increases with dendrimer generation. This increase is negligible at room temperature (there is a difference in density values in the third decimal place). A more pronounced change in densities is realized at higher temperatures, but even at T = 600 K it does not exceed 5%. A very weak ρ(G) dependence is in agreement with experimental observations.56 Indeed, the measured bulk density was reported to be a constant of 0.88 g/cm3 for G5−G7 dendrimers.40,56 A slight discrepancy in absolute values of ρ obtained in computer simulations at 300 K in comparison with the experimental data could result from the use of united atoms for CH2 and CH3 groups and from the D


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Figure 4. Radial density distributions of single dendrimer atoms (rose), atoms of other dendrimers (blue), and the total density distribution (black) around the center of mass of a dendrimer molecule calculated at T = 300 K (solid lines) and T = 600 K (dashed lines) for G = 3 (a) and G = 7 (b).

third, fifth, and sixth generations of carbosilane dendrimer melts at T = 600 K35 where a wrong conclusion on weakening of dendrimer interactions with generation was formulated (see discussion below). Another important feature of the dendrimer density distribution clearly seen in Figure 4b is the presence of a depletion layer at the dendrimer periphery, namely, the presence of the minimum in the total melt density located around 3.4 and 3.2 nm at T = 600 K and T = 300 K, respectively. Its indication appears already for G3 (see Figure 4a) and becomes more pronounced with increasing dendrimer generation. For large G the molecules become more particlelike, and the local density decrease at the dendrimer periphery is due to some holes formed between packing spheres. They can be clearly seen in the system snapshots (Figure S3). Comparing the radial density distributions for the systems of 27 and 8 dendrimers of high generations (see Figure S3), we could see again a pronounced difference in values. While the ρin functions within dendrimer core coincide rather well, a noticeable divergence is present in ρout and ρtotal densities due to different molecule packaging caused by finite size effects. We would like to stress that in spite of a pronounced decreasing relative dendrimer interpenetration, the length of the interpenetration layer itself where the density of both dendrimer own and foreign monomer units is nonzero changes to a much less extent for low and high generation dendrimers. It slightly decreases as 1.95, 1.90, 1.85, 1.80, and 1.60 nm for G4−G8 at 600 K and as 2.10, 2.125, 2.125, 2.05, and 1.90 nm for G4−G8 at 350 K. These values were obtained from the density distributions of dendrimer own and foreign monomer units; the boundaries of the interpenetration layer were estimated at the ρin and ρout density values of approximately 0.001 g/cm3. The change of the interpenetration layer length with generation is about 0.2 and 0.35 nm for 350 and 600 °C, respectively. It should be noted that while the boundary at which the ρin tend to zero shifts only by 0.05 nm with temperature, the shift of the boundary defined by zero ρout is 0.2−0.3 nm. Furthermore, we have calculated the intermolecular dendrimer interaction energy, U12, including van der Waals and Coulomb interactions reflecting the amount of intermolecular contacts. Its temperature dependence at various G is presented in Figure 5 for the system of eight molecules in the simulation box. It follows from Figure 5 that the interaction between neighboring dendrimers strengthens with increasing generation. The difference in interaction energy becomes larger at lower temperatures and increases with G. For instance, at 350 K the absolute value of U12 increases by 900−1100 kJ/mol from G3 to G4 and G5 dendrimers, but its increment reaches

correctly the temperature dependence of the heat capacity, perhaps, due to the united atom representation of the methyl and methylene groups of the dendrimer molecules and disregard of quantum effects. Intramolecular and Intermolecular Density Distributions. As the next step, a detailed structure of the melts, in particular, the spatial density distributions of all species, is studied. Let us analyze the radial distributions of (i) monomer units of a single molecule, (ii) foreign atoms, and (iii) the radial distribution of all monomer units in the system relative to the center of mass of a dendrimer. As an example, in Figure 4 we plot all three distributions for the dendrimers of the third and seventh generations for comparison. In this figure, ρin is the density distribution of dendrimer own monomer units, ρout is the density distribution of monomer units belonging to neighboring dendrimers, and ρall shows the total monomer unit density. Corresponding density distributions for the other generations are presented in Figure S3. For the third generation dendrimers, the single dendrimer density shows two maxima corresponding to the dendrimer dense core and the first layer Si branching centers. It smoothly decreases to zero at distances ranging from 0.7 up to 2.0 nm; i.e., the peripheral region is rather extended at both T = 300 K and T = 600 K. At the lower temperature dendrimer molecule becomes slightly denser; this fact is clearly seen from the shift of the first Si layer to the dendrimer center and increasing density in this maximum. One can see that G3 dendrimer molecules can penetrate deeply into each other. The foreign atoms are incorporated into the dendrimer up to its center, ρ is nonzero starting from 0.3 nm, and r < 0.3 nm corresponds to the narrow core region completely occupied by the core Si atom and surrounding CH2 groups and, thus, excluded for other atoms. The temperature effect is more significant for ρout rather than for ρin. Summation of single dendrimer density and foreign atom density provides a relatively homogeneous density distribution throughout the system with only a weakly pronounced minimum at the dendrimer periphery. For the seventh generation dendrimers, ρ(r) looks somewhat different. First, there appear peaks related to the intramolecular positions of massive Si branching centers up to the fifth layer. Second, the dendrimer becomes somehow less permeable for the foreign atoms. Indeed, for G7 the central part with the radius of 2 nm is densely filled with dendrimer own atoms and is unavailable for the others. The degree of relative dendrimer interpenetration, i.e., ratio between the interpenetration layer length and the dendrimer size, decreases with increasing generation number (see Figures S3). This tendency was previously observed in molecular dynamics simulations of the E


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8G6 system. They are calculated as the amount of Si atoms of a chosen branching number in a spherical layer at a distance r from the mass center of the dendrimer. One can see that the dendrimer has a layered structure especially pronounced for the dendrimer central region. Indeed, the maxima corresponding to the first three generations (at 0.55, 0.98, and 1.42 nm) are well separated. The peripheral layers (from the fourth to sixth) are better mixed together due to a larger amount of the terminal segments and their higher mobility (see the next section). Another interesting observation in Figure 6a is a higher probability to find terminal branching Si within the inner part of dendrimer than Si of the inner branching points. The left shoulder of the Si distribution becomes higher for higher Si branching numbers. For high generation dendrimers the enhanced amount of terminal segments causes a shift of the inner Si branching points out from the dendrimer center, and the nonzero density within the dendrimer is maintained via backfolding of the terminal groups. This tendency is clearly seen in Figure 6b demonstrating how the internal positions of Si atoms change with increasing dendrimer molecular mass. Single dendrimer simulations have also shown that for dendrimers of high generations the backfolding phenomenon is most pronounced for the terminal groups and outer branches while inner branches are well localized.9,58 In Figure 6b, we plot the distribution of Si belonging to the third layer for dendrimers of various generations. For D3 the Si3 atoms belong to the end groups. For higher generations Si3 are inner groups. Because of the possibility of backfolding, the Si atoms of terminal segments are delocalized throughout the whole molecule interior. The shift of the Si branching points from the center of dendrimer molecule and terminal group backfolding are observed for the other inner layers and the other dendrimer generations (see Figure S6). Radius of Gyration and Shape Factor. In addition to the detailed atom distributions discussed above, we have also analyzed such traditional integral dendrimer parameters as the average radius of gyration and the dendrimer shape as functions of the molecular mass and temperature. Dendrimer shape is characterized by the ratios R2/R1 and R3/R1 of principal moments of gyration tensor (shape factors). The obtained values of the gyration radii R1 + R2 and R3 are summarized in Table S6. The dependencies of the shape factors on the dendrimer generation for 8-dendrimer system are shown in Figure 7a (the results for 27-dendrimer system are presented in Figure S7). For comparison, we plot similar dependences for the single dendrimer in dilute solution in Figure 7b. One can see that

Figure 5. Dependence of intermolecular interaction energy, U12, on dendrimer generation for the system of eight molecules in the simulation box.

2000 kJ/mol and even exceeds 2300 kJ/mol going from G5 to G6 and from G6 to G7, respectively. Increase in the absolute intermolecular interaction energy at practically constant and even slightly decreasing length of interpenetration layers can result from some increase of the monomer unit density within this layer. The density distributions plotted in Figure 4 do not show any density increase at the dendrimer periphery simply because they display radially averaged values. On the contrary, the presence of holes in the melts of high generation dendrimers, clearly seen in system snapshots (Figure S4), decreases the average peripheral monomer density and appearance of the minimum in ρtot(r) function. One should mention that for low G there is no noticeable difference in energy values for 8 and 27 dendrimer systems (see Figure S5). However, starting from the sixth generation, some energy difference appears due to different packing of molecules in the simulation box. Because of restrictions imposed by periodic boundary conditions on the dendrimer ordering, the absolute value of the interaction energy of 27 dendrimer system is lower. Distribution of Si Atoms vs Temperature and Generation. Another important structural characteristic of dendrimer molecules in melts is the distribution of Si branching points of different topological branching numbers within the molecule depending on dendrimer generation and temperature. By topological branching number we mean the number of the Si branching layer or branching site in a path connecting the dendrimer central Si atom and the end group. The numbering starts from the central atom acquiring zero number. In Figure 6a, we plot the radial distributions of Si atoms belonging to different topological branching numbers for the

Figure 6. Distributions of Si branching points belonging to various layers for G6 dendrimers (a) and distributions of Si atoms of the third layer in dendrimers of various generations (b) at T = 600 K. In (a) ki is the number of the branching layer. F


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objects. However, the snapshots of the dendrimers clearly show that their shape is closer to spherical rather than cubic. The dependence of the gyration radius of an isolated dendrimer on its main structural parameters such as molecular mass, generation number, and the spacer length has been studied in many publications both theoretically and by computer simulations. According to the simple Flory-type calculations, the radius of gyration of a single dendrimer scales with molecular mass M and generation as (G2M)1/5, (G2M)1/4, and M1/3 for good, theta, and poor solvent conditions, respectively.9 Computer simulations of dendrimers with excluded volume mostly confirm the value of the scaling exponent n = 1/5;9,58 however, the values of n in between 0.31 and 0.35 were also found in a number of simulations.9,59 For poor solvent conditions shrinking of dendrimer and the scaling law Rg ∼ M1/3 were reported in refs 9, 58, and 59. The scaling behavior of gyration radius with dendrimer molecular mass was analyzed from the simulations of G3, G5, and G6 polycarbosilane melts at 600 K in ref 35 where the value of the scaling exponent n = 0.29 has been found. On the basis of this result, it has been concluded that dendrimers in bulk behave as the ones in a poor solvent. The value of n = 0.29 was also found from the atomistic modeling poly(propyleneimine) dendrimers32 and of poly(propyleneimine)-based dendrimers.24 Our simulations of both isolated dendrimer molecules and dendrimer melts allow to compare dendrimer swelling depending on environment. A log−log plot of the gyration radius vs molecular mass of polycarbosilane dendrimers for various temperatures is presented in Figure 8a while in Figure 8b Rg is shown as a function of MG2. One can see in Figure 8a that all the points lie roughly on one line (only a slight temperature dependence gives some point shift). Similar dependences have been obtained for the system of 27 molecules in the simulation box and isolated dendrimers (see Figure S8). The values of the scaling exponent extracted from the linear fit of the simulation data for log Rg(log M) functions for all the systems under study are presented in Table 3. It shows that at T

Figure 7. Dendrimer shape factors vs generation for eight-dendrimer system (a) and isolated dendrimer (b) at various temperatures.

while low generation dendrimers show some asymmetry, the dendrimer shape tends to spherical with increasing dendrimer generation at any temperature. Moreover, single dendrimers are less spherical than dendrimers in the melt; thus, interactions with the surrounding molecules cause shape distortion. A smooth transition from oblate to spherical shapes in going from low to high generations has previously been reported for single dendrimers.9 It was also found that in the melt the dendrimer molecules assume a more spherical shape as the generation number increases.32,35 It should be noted that the values of R2/R1 = 1 and R3/R1 = 1 do not necessarily evidence that the molecule adopts a spherical shape; the unity values of these ratios could be found, for instance, for cubically shaped

Table 3. Values of the Scaling Exponent of Rg(M) Dependences temp (K)

1Gj

8Gj

27Gj

600 350

0.3058 ± 0.0006 0.322 ± 0.001

0.2996 ± 0.0005 0.291 ± 0.003

0.2984 ± 0.0005 0.292 ± 0.002

= 600 K the value 0.3 of the scaling exponent applies with high accuracy both for melts and for single dendrimer molecule. At

Figure 8. Radius of gyration vs M (a) and vs MG2 (b) for the system containing eight dendrimer molecules in the simulation box at various temperatures. G


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Figure 9. Dendrimer density distributions for the melt and a single dendrimer molecule for generations 3 (a) and 8 (b) and various temperatures.

Figure 10. Time dependences of dendrimer center-of-mass displacements for G3 (a) and G6 (b) melts calculated at different temperatures.

T = 350 K its value for melts is somewhat smaller being equal to 0.29. At the same time for an isolated dendrimer n = 0.32, this value is close to 1/3, which is expected for poor solvent conditions. Indeed, simulated isolated dendrimers in a vacuum is expected to adopt conformations similar to those of single dendrimers in a poor solvent. We would like to stress a difference in the obtained value of the scaling exponent for dendrimer melt from that in a poor solvent. This contrast to a poor solvent behavior might be explained by a contribution of a swollen surface layer formed presumably due to penetration of neighboring dendrimer molecules (see the previous section). Some difference in conformational behavior between isolated dendrimers in vacuum (or in a poor solvent) and dendrimers in a melt also follows from Rg(MG2) dependence obtained. The log−log plot of Rg vs MG2 (Figure 8b) shows that except for G3 and G4, the Rg values of the other high generation dendrimers fall into one line with the slope of 0.2 predicted theoretically for dendrimers in a good solvent. In ref 35 the gyration radius of G3, G5, and G6 dendrimers in melts at 600 K was found to scale as (MG2)0.18. In our simulations, the scaling for low generation dendrimer also deviates from the one obtained for G5−G8 systems. It should be noted that the values of equivalent spherical radius calculated as R = (5/3)1/2Rg coincide well with those obtained experimentally. In particular, small-angle neutron and X-ray scattering studies of dilute solutions of polycarbosilane dendrimers in benzene which is a poor solvent for

polycarbosilane dendrimers have shown that Rg for G5, G6, and G7 are 2.0, 2.6, and 3.0 nm.56 According to our simulation results, the values of R vary as 1.9−2.1, 2.4−2.6, 3.1−3.2 nm in the temperature range 350−600 K. Comparison of the radial density distributions for isolated dendrimer molecules and dendrimers in melts (see Figure 9) shows that interaction of a molecule with the surrounding dendrimers in melts causes slight expansion of the monomer unit profile. Mobility of Dendrimer Segments. Apart from static properties of dendrimers, their dynamic behavior in melts is of great interest and importance. Its understanding could shed light on the basics of specific rheological response of high generation systems. We have examined for the first time the dynamics of a number of dendrimer structural units, in particular, molecular center-of-mass (CM), Si branching points of various generation layers as well as terminal CH3 groups. In Figure 10, we plot the time evolution of the center-ofmass position for G3 and G6 dendrimers calculated for 8G and 27G systems at various temperatures (the corresponding plots for the other generations are presented in Figure S9). First, one can see that the CM displacement increases considerably with temperature as expected. At T = 600 K the third generation dendrimers have enough time to move up to the distances exceeding the diameter of a single dendrimer molecule (Figure 10a), and the regime of dendrimer self-diffusion is reached at H


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Figure 11. Mean-squared displacement ⟨Δr2⟩ of the center of mass vs time (a) and the corresponding log−log plot (b) for G5 dendrimer melts at various temperatures. The dashed lines in (a) show linear interpolations of ⟨Δr2⟩ at large t while the ones in (b) indicate the slope of 1 for ⟨Δr2 − C(T)⟩ at large t: C(T) = 0.15, 1.5, 2.5, 10.0, and 10.0 for T = 300, 350, 400, 500, and 600 K, respectively.

with appropriate accuracy (see Figure 11a). For higher generation dendrimers there is also the regime where linear time dependence of Δr2 holds; However, the dendrimer CM displacements are smaller than their gyration radius; thus, we could not confidently call this regime as self-diffusion regime. However, from the relation Δr2 = Bt + C we have defined the slope of Δr2(t) dependences in the linear region. The values of the parameter B calculated for all the systems at every temperature are summarized in Table 4. We marked by bold

the scale of the simulation time. At lower temperatures, the CM shift of even the third generation dendrimers at the maximum simulation time is comparable but smaller than their Rg values. Second, dendrimer CM mobility in melt drops dramatically with an increase of dendrimer generation. In spite of similarly looking curves of Figures 10a and 10b, the scales of the Y-axes are completely different. G3 dendrimers move much faster than G6 dendrimers. Indeed, CM displacement of G6 dendrimer is only a few angstroms during the simulation time at any temperature. This distinction with generation is an indication of different mechanisms of movement of low and high generation dendrimers in melts. The third point which should be mentioned while discussing Figure 10 is the difference in CM displacements obtained for 8and 27-dendrimer systems. This phenomenon is a manifestation of the effects connected with system finite size. One could expect that it should be more pronounced for the smaller simulation box. Indeed, the system containing eight dendrimer molecules consists of only two dendrimer layers, and a displacement of a molecule in one direction causes a shift of the closest neighbor, which is translated due to periodic boundary conditions and, in turn, has an effect on the movement of the initial molecule. The higher the generation, the more rigid and less interpenetrable are the dendrimer molecules; thus, the difference in dynamics of smaller and larger system increases. This is clearly seen from comparison of curves presented in Figures 10a and 10b. At large time scales the mean-squared displacement of dendrimers becomes roughly proportional to time. It can be nicely described by the relation ⟨Δr2⟩ = Bt + C with the constant C appearing due to the presence of initial nondiffusive motion. For low generation systems at high temperatures the dendrimer has enough time to shift to the distances exceeding the dendrimer gyration radius; thus, one can consider that its movement is approaching diffusion-type regime at large t (see Figure 11b), and the molecular self-diffusion coefficients D could be estimated as 6D = B already from the presented data

Table 4. Values of the Coefficient B in the Dependence Δr2 = Bt + C for Polycarbosilane Dendrimers vs Generation at Various Temperaturesa B values at various temperatures, m2/s 350 K G3 G4 G5 G6

8.9 5.5 4.8 2.2

× × × ×

10−12 10−12 10−12 10−12

400 K

500 K

600 K

5.1 × 10−11 2.5 × 10−11 1.4 × 10−11 9.1 × 10−12

2.5 × 10−10 8.8 × 10−11 3.9 × 10−11 1.8 × 10−11

5.1 × 10−10 1.7 × 10−10 5.0 × 10−11 1.7 × 10−11

a

Values in italics correspond to dendrimer movement up to distances exceeding its Rg.

the values of B obtained for system trajectories where dendrimer CM diffused up to distances exceeding dendrimer gyration radius demonstrating dendrimer self-diffusion. In ref 60 the self-diffusion coefficients (SDC) for the third, fourth, and fifth generations of 3−3 polycarbosilane dendrimers were obtained by pulsed field gradient NMR. These dendrimers differ from the dendrimers under present study by the functionality of the central Si atom (3 vs 4); the other details of the dendrimer structure are the same. For melts the following SDC were obtained: 1.2 × 10−12, 3.4 × 10−13, and 1.7 × 10−13 m2/s for G(3)3, G(3)4, and G(3)5, respectively.60 Although the correspondence of the experimental and computation systems is rather indirect and slow relaxation of dendrimer melts at 300 K did not allow to extract SDC at room I


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Figure 12. Time evolution of the mean-squared displacements of CM, Si branching atoms and terminal CH3 groups for 27G5 melt at T = 300 K (a) and T = 600 K (b).

from the central Si atom and extend up to 2.1 nm from the center; thus, the possible displacement is around 1.6−1.8 nm, which exceeds by far the value where the dependence Δr2(t) saturates. The possible explanation of this discrepancy is that there is a fraction of branches in a “frozen” state which decreases considerably the mean displacement values. Of course, this speculation should be carefully checked and this is one of the aims of our ongoing activities. From the log−log plots of the MSD time dependences presented in Figure 13c,d, one can conclude that the diffusiontype motion of Si and CH3 groups is realized for long time periods at all temperatures under study; the slope equal to unity is shown by dashed lines. From the shifts of the curves along the Y-axis it is seen that the diffusion coefficients grow with temperature; furthermore, they are higher for terminal CH3 groups. Surprisingly, the plateau value of Δr2 reached at T = 600 K is lower than that at T = 500 K. This observation needs further investigation at longer length scales.

temperature with appropriate accuracy, however, we could see that the values of the parameter B obtained from simulations are in the same order of magnitude range as experimental SDC for the 3−5 generations of 3−3 polycarbosilane dendrimers.60 We have also studied mobility of Si branching atoms from different branching numbers as well as the terminal groups for all the melts under study. To this end, we have calculated time dependences of the mean-squared displacements (MSD) of Si atoms belonging to the same branching number with respect to their CM. The same has been done for the terminal CH3 groups. The analysis was performed for the time interval of 5 ns with the averaging along the trajectories of 13 ns and ensembles of eight independent trajectories. The corresponding plots for 27G5 melt at T = 300 and 600 K are presented in Figure 12. One can see that Δr2 increases with the branching atom number, which is quite natural. Si atoms belonging to peripheral layers have more freedom of movement. The highest MSD is demonstrated by the terminal groups. The temperature has a pronounced effect on the atom mobility. While at 300 K only oscillations of Si atoms around equilibrium positions are observed, at 600 K terminal CH3 groups as well as Si atoms of the peripheral layer can shift to distances comparable to the dendrimer size. It should be mentioned that the results presented in Figure 12 are obtained by averaging among all Si atoms belonging to the same branching layer. However, one could expect that the mobility of segments incorporated into the central region of dendrimers or localized deeply inside neighboring molecules is reduced. The assumption of the presence of groups with a reduced mobility is supported by the plots in Figure 13 where the radial MSD of terminal layer Si branching atoms and terminal CH3 groups in 8G4 melt calculated at large time scales (up to 10 ns) are presented. According to these graphs, the displacement of the terminal groups at 600 K after 10 ns is only 0.55 nm. However, the radial density distributions for the fourth generation dendrimer show that the terminal groups can penetrate deeply into the dendrimer core up to 0.3−0.5 nm

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OUTLOOK The detailed analysis of the melt structural characteristics showed that (i) molecular density profiles exhibit a gradual decrease from the dendrimer center toward the molecule periphery; (ii) Si branching atoms belonging to the outer layers are distributed throughout the whole molecule; i.e., the so-call backfolding phenomenon takes place; (iii) the radial profile of the total melt density relative to the dendrimer center of mass exhibits a minimum at the dendrimer periphery which becomes more pronounced with generation; (iv) the relative length of the layer where neighboring dendrimers interpenetrate drops considerably with generation but the decrease in its absolute value is minor; (v) the absolute value of the intermolecular interaction energy grows greatly with generation; (vi) the dendrimer molecules become more spherical with growing generation; furthermore, dendrimer spheres tend to order in melt, and dendrimers are more symmetrical in melts than in dilute solutions; (vii) the gyration radius of dendrimers in melt scales with the molecular mass as M−0.3 at high temperatures; J


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Figure 13. Time dependences of the mean-squared displacements of terminal CH3 groups (a) and terminal layer Si branching atoms (b) in 8G4 melt. Parts c and d are the corresponding log−log plots. The dashed lines show the slope equal to one.

the scaling exponent coincides with that found for an isolated dendrimer in a vacuum; at T = 350 K there is a difference in scaling exponents for dendrimers in melts and for isolated dendrimers, namely, n = 0.29 for melts and n = 0.32 for an isolated dendrimer; (viii) according to radial density distributions, dendrimer molecules are more swollen in melts rather than in dilute solutions in poor solvent at the same temperature; (ix) diffusion-type motion of Si and CH3 terminal groups is realized at all temperatures; the MSD saturates at the value much lower than the dendrimer average size. From these results the following picture of the dendrimer melt can be drawn. Low-generation dendrimers are fully interpenetrating in the melts at any temperature but retain a high mobility. With increasing generation, dendrimer molecules become denser and expel the foreign monomer units from their core region; thus, the interpenetration layer becomes localized at the dendrimer periphery. High-generation dendrimers comprise dense spheres tending to order in the melt, as a

result of dense sphere packing there appear regions in the melt with enhanced and reduced monomer unit densities. The enhanced density in the interpenetration layers causes considerable increase in dendrimer interaction energy with generation. Spherical shape of the molecules and high density of monomer units within molecules assume that the strongest interactions are realized in spherical segments formed by monomer units of the two nearest-neighboring dendrimer molecules. The monomer units being in between interpenetration segments are not subjected to intense intermolecular interactions (experience reduced impact of surrounding molecules). The presence of two distinct populations of dendrimer branches interacting to a different extent with the surrounding molecules could explain the experimental results on the temperature dependence of heat capacity exhibiting two transitions, the first one is realized at T = 180 K corresponding to the glass transition temperature, which is independent of dendrimer generation and the second one is observed only for K


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Macromolecules high generation dendrimer melts.41−43 The second transition could be connected with a drop in mobility of a part of dendrimer branches participating in the formation of interpenetration layer. Further insight into the system behavior is still needed for complete understanding of mechanism of dendrimer interactions leading to considerable change of the melt properties with generation. In particular, it is important to directly identify the presence of dendrimer branches with different mobility. Furthermore, study of the melt behavior under shear flow is one of the straightforward next steps. These are the aims of our ongoing studies.

temperatures; time evolution of the density for 8G5 and 27G5 melts at fixed temperature of 400 K and varying pressure; autocorrelation functions for the gyration radius and dendrimer asphericity; radial density distributions for all systems under study at T = 350 K and T = 600 K; snapshots of the dendrimer melt layers; dependence of intermolecular interaction energy, U12, on dendrimer generation for the system of 27 molecules in the simulation box; distance distributions of Si atoms belonging to various layers k from a central Si atom in dendrimers of various generations; dendrimer shape factors for the 27 molecule melt at various temperatures; gyration radius of isolated dendrimers vs molecular mass at various temperatures; time dependences of dendrimer center-of-mass displacements melts containing 8 and 27 dendrimers in the simulation box calculated at different temperatures (PDF)

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CONCLUSIONS We have performed extensive molecular dynamics simulations of melts of polybutylcarbosilane dendrimers with the functionalities of the core and branching Si atoms equal to 4 and 3, respectively. The spacers consisted of three CH2 groups and the terminal segments also comprised three methylene groups ended with the methyl (−CH3) group. In the simulation CH2 and CH3 were treated as united atoms. Dendrimers of the third through the eighth generations were modeled. Extensive molecular dynamics simulations of dendrimer melts were performed during 5 ns in a wide temperature range from 300 to 600 K. Furthermore, the melt structural characteristics were analyzed for the systems consisting of 8 and 27 dendrimer molecules in the simulation box. In addition to time averaging the ensemble averaging was performed over eight independent system realizations for all temperatures and systems. It has been shown that the model developed is reliable; it allows to describe adequately the properties of the real polycarbosilane melts. The following macroscopic characteristics of the dendrimer melts are in accordance with available experimental data: weak dependence of the melt density on dendrimer generation; values of the thermal expansion coefficients and self-diffusion coefficients. The values of the heat capacity are also close to the experimental ones at the plateau region near room temperatures, although the present model is not able to properly describe heat capacity increase with temperature in the high temperature range. A special attention has been paid to the finite-size effects which become progressively important with increasing dendrimer generation. It has been shown that in spite of a larger size of 27 dendrimer system; it appears to be less appropriate than the eight-dendrimer system for structural studies of high-generation melts because the periodic boundary conditions impose restrictions on the type of molecule ordering. On the other hand, melt dynamics could be better described on the basis of larger systems. Detailed analysis of the melt structural and dynamics characteristics gave some preliminary insight into distinction in behavior of low and high generation dendrimers and formulate the directions of further studies.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected] (E.Y.K.). ORCID

N. K. Balabaev: 0000-0001-7883-8119 E. Yu. Kramarenko: 0000-0003-1716-7010 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors are grateful to Prof. A. Muzafarov for many helpful discussions. This work was supported by the Russian Foundation for Basic Research (Grant No. 16-03-00669). Simulations have been performed on MSU supercomputers “Lomonosov” and computer cluster IMPB RAS.

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REFERENCES

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.macromol.6b01639. Parameters of the force field for the atoms; gyration radius and half shafts of an effective ellipsoid calculated for all iGj systems (i = 1, 8, 27 and j = 3−8) at various L


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