Orientational Mobility in Dendrimer Melts: Molecular Dynamics

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Orientational Mobility in Dendrimer Melts: Molecular Dynamics Simulations Denis A. Markelov,*,†,‡ Andrey N. Shishkin,† Vladimir V. Matveev,† Anastasia V. Penkova,† Erkki Laḧ deranta,§ and Vladimir I. Chizhik† †

St. Petersburg State University, 7/9 Universitetskaya nab., St. Petersburg 199034, Russia St. Petersburg National Research University of Information Technologies, Mechanics and Optics, Kronverkskiy pr. 49, St. Petersburg 197101, Russia § Laboratory of Physics, Lappeenranta University of Technology, Box 20, 53851 Lappeenranta, Finland ‡

S Supporting Information *

ABSTRACT: We have simulated the melt of poly(carbosilane) dendrimers using atomistic models and have reproduced the effect predicted by the analytical theory; i.e., orientational autocorrelation functions of a segment from the same layer (numbered from periphery) are practically identical and do not depend on dendrimer size. The frequency dependences of the dielectric and NMR relaxation were obtained and studied in detail. The main contribution to the maxima of these dependences is given by the pulsation process. It leads to a shift of the maxima to low frequencies for the core segment in comparison with the maxima for peripheral segments. The contribution of local reorientation can also be significant, and in some cases this contribution manifests as an additional maximum. The nontrivial scaling laws in the frequency dependences of dielectric permittivity and NMR relaxation rate averaged over all layers of a dendrimer macromolecule are found. A similar scaling law is observed in the experiments on NMR relaxation but is not described by the analytical theory.

I. INTRODUCTION Dendrimers are unique macromolecules with the regular treelike molecular architecture.1,2 Cascade synthesis technologies allow one to obtain dendrimers with monodisperse dimensions and structures. The chemical composition of dendrimers can flexibly be varied that provides specific properties of macromolecules, which are in demand in various areas of polymer chemistry, biology, and medicine.3−6 Various peculiarities of segmental reorientation in dendrimers are manifested in different experimental methods: dielectric relaxation, polarized luminescence, NMR relaxation, etc.7 The studies of local mobility in dendrimer melts have been carried out by dielectric relaxation, rheology, NMR relaxation, and others.8−19 In recent works,14,15 the mobility in the melted poly(propyleneimine) (PPI) dendrimers was investigated in a wide temperature range (120−400 K) by a few methods: dielectric spectroscopy, solid-state 2H NMR, and field-cycling 1 H NMR relaxometry. The comparison of the obtained results with ones for the melts of poly(propylene glycol) (PPG) © XXXX American Chemical Society

polymers was also made. Three reorientational processes was found: the main (α-) process (above glass transition temperature) and two β-processes (below the glass transition temperature). Characteristic times of β-processes are in accordance with the Arrhenius temperature dependence. One of these processes is similar to the process in the β-amorphous systems. The second β-process is atypical for polymeric systems: the width of its spectrum does not change with temperature and is characterized by a time constant which does not depend on the size of a dendrimer macromolecule. Therefore, the authors supposed that the existence of this process is a peculiarity of the dendrimer structure. The αprocess demonstrates the super-Arrhenius temperature dependence, which is typical for supercooled liquids or polymer melts.14 Received: July 12, 2016 Revised: November 11, 2016

A

DOI: 10.1021/acs.macromol.6b01502 Macromolecules XXXX, XXX, XXX−XXX

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Figure 1. Poly(butylcarbosilane) (PBC) dendrimer of the second generation (G2) (a), the structural formulas of PBC dendrimers (b), and an individual segment (c). rs is the vector directed along the segment (red arrow); rc1 and rc2 are vectors directed along the C−C bonds (green arrows) which are used for the calculation of the orthogonal vector rorto = [rc1∗rc2] (see details in the text). G is the number of a generation, g is the layer number beginning from the dendrimer core, and m = G − g is the layer number, which begins from the terminal layer.

supported by the results of the atomistic molecular dynamics simulations for poly(L-lysine) dendrimers in aqueous solution.44 Moreover, the simulation data allowed the authors to reproduce the experimental temperature dependences of 1/T1H for that dendrimer. The detailed consideration of the influence of semiflexibility effect on the frequency dependence of the NMR relaxation in dendrimers has been performed in the work,45 where the coarse-grained model of a dendrimer was simulated using Brownian dynamics with and without excluded volume interactions. It was established that if the semiflexibility effect was taken into account, the excluded volume interactions did not practically influence on 1/T1H. This conclusively explained the fact why the semiflexible viscoelastic model of dendrimers without excluded volume interactions described correctly the behavior of the functions which were manifested in the NMR relaxation. Besides, the influence of semiflexibility on the segmental reorientation in dendrimers has been studied theoretically in the limits of the Zimm−Rouse model.46 In this work the authors considered the averaged functions over all segments. Eventually, it has been shown that the orientation mobility of the individual segment of a dendrimer macromolecule in a solution is determined by three main processes:38,44 (i) the local mobility which very weakly depends on the position of the segment and the number of generations, G, in the dendrimer; (ii) the mobility of the branch, which originates from the selected segment, and therefore this motion is determined by the topological distance (along a chain) between the segment and the periphery (i.e., by the number of generations in the branch); (iii) the rotation of the dendrimer macromolecule as a

Usually, the experimental data on the segmental reorientation in dendrimers are processed/interpreted based on general conceptions of segmental mobility in polymers, i.e., without taking into account the specifics of the treelike structure of macromolecules.20−27 It is caused by the fact that until recent times the theoretical description (and understanding) of the orientational mobility in dendrimers was insufficiently developed. Computer simulation results were not analyzed in detail due to the superposition of several relaxation processes.28,29 The analytical theory30,31 based on the flexible viscoelastic model had significant discrepancies with the results of computer simulations28,29 and experimental data (for example, see NMR relaxation data in refs 26 and 32−34) for dendrimers in a solution. Hence, there are serious doubts about the applicability of the flexible viscoelastic model to dendrimer macromolecules. In the discussed area an important breakthrough was recently made that began with the works of Pinto et al.35,36 It was shown for poly(aryl ether) dendrimers in the aqueous solution that the maximum of the temperature dependence of the spin−lattice NMR relaxation rate, 1/T1, shifted to high temperatures with the displacement of an observed group from the terminal layer to the dendrimer core. These results indicated the decrease in the orientational mobility from periphery to a core. The shift of the maximum was not predicted both in the analytical theory31 and in computer simulations.37 Because of this motivation, the analytical theory was upgraded38 taking into account the semiflexibility between adjacent segments in the viscoelastic model.39−43 It was established31 that the improved model gave the correct trend in 1/T1, and the theory conclusions were B


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Macromolecules whole. Note that only the third process depends on G (i.e., on the size of a macromolecule) and cannot be described by the viscoelastic theory. It is expedient and convenient to characterize the orientational mobility in macromolecules using autocorrelation functions (ACF) P1 or P2: P1(t ) = ⟨u(t )u(0)⟩

P2(t ) =

3⎛ 1⎞ 2 ⎜⟨(u(t )u(0)) ⟩− ⎟ 2⎝ 3⎠

Herein we study the PBC dendrimer melts with the number of generations G = 2, 3, and 4 (G2, G3, and G4 in the abbreviated spelling). We limit our investigation to G4 because the G6 PBC dendrimers form network structures due to physical entanglements between macromolecules,51−53 and this effect could also appear in the G5 dendrimer melt. The paper consists of the following sections. In section II we briefly describe the model and simulation details. Section III contains the results and discussions of the calculated orientational ACFs and frequency dependences of functions manifested in the dielectric and NMR relaxations. In section IV the conclusions of the work are presented.

(1)

(2)

where ⟨u(t)u(0)⟩ is the scalar product of the unit vectors u in the time moments t and 0 for the selected vector r in a macromolecule, i.e., u = r/|r| (see Figure 1c). In the analytical theory, describing the segmental reorientation in dendrimers of different generations G, the P1 or P2 ACFs coincide for segments with the same topological distance from the periphery, i.e., for the identical number of a layer, m, if the layers are numbered from the terminal generation; see Figure 1a). For identical m the contribution of the third process (i.e., the rotation of a dendrimer macromolecule as a whole) gives the main difference in the P1 (or P2) functions calculated from computer simulations. The effect has been strictly shown only for a flexible dendrimer in a solution.30 In the case of a semiflexible dendrimer this was qualitatively demonstrated for coarse-grained model45 and atomistic model of lysine dendrimers.44 Thus, in contrast to the analytic theory, the computer simulations predict the significant contribution of the overall dendrimer rotation in a solution. It is of great interest to investigate systems in which the rotation is inhibited, and as a result, it insignificantly influences on the autocorrelation functions. In this connection, we suggest the study of the segmental reorientation in dendrimer melts. The “replacement” of the solvent molecules on the neighboring dendrimer macromolecules should dramatically slow down the rotational mobility due to the high viscosity of dendrimer melts. In this work, we examine the segmental reorientation in the melt of poly(butylcarbosilane) (PBC) dendrimer (see Figure 1) at 600 K by molecular dynamics (MD) simulations. The choice of the PBC dendrimer is caused by the idea to study the effect of the dendrimer topology on the segmental orientational mobility in melts. There are no specific interactions in this dendrimer, such as electrostatic interactions or hydrogen bonds. Moreover, because the Si atoms are branching points, the PBC dendrimer is more flexible than many others, such as poly(amidoamine) (PAMAM) dendrimer, PPI dendrimer, polylysine dendrimer, etc. The low local bending stiffness (semiflexibility) must decrease the influence of the overall dendrimer rotation because internal motions already leads to a strong decorrelation in autocorrelation functions that confirmed by Brownian dynamics simulation.45 Therefore, we can expect that the manifestation of the orientational mobility in the PBC dendrimer system is to be similar to the analytical theory predictions.38 We hope that the main conclusions of the work will be of general interest for the understanding of the properties of dendrimer systems in melts. We also hope that our study of the orientational mobility can be considered as the one of starting points for the investigation of other dendrimer melts because similar studies are nearly absent in the literature.47−50

II. MOLECULAR DYNAMICS SIMULATION DETAILS The melt of poly(butylcarbosilane) (PBC) dendrimers of different generations (G = 2, 3, and 4) has been studied at 600 K by molecular dynamics (MD) simulation. The functionalities of core, Fc, and other branching points, F, are equal to 4 and 3, respectively. The chemical structure of the dendrimer is shown in Figure 1b. Note that the number of generations starts at zero, i.e., G2 and G4 dendrimers contain 16 and 64 terminal groups, respectively. We use the numbering of dendrimer layers, m, from the periphery to the core; i.e., m = 0 corresponds to the terminal layer, and m = G corresponds to the core (see Figure 1a). The detailed description of the atomic model for PBC dendrimers, the simulation method, including the procedures of preparation and equilibration of a system, is given in our previous work.54 Here we briefly repeat the main details of the simulations. The cubic simulation box contains 27 dendrimer macromolecules. The periodical boundary conditions and the Gromos53a6 force field in Gromacs-4.5.5 package are used.55 Calculations were performed in the NPT ensemble using Berendsen barostat56 and V-rescale thermostat.57 The model of united atoms was used, and to obtain the partial charges of united atoms the Hartree−Fock method with the basis DNP (for instance, see ref 58) and the method of gradient correction (GGA)59−61 with the Purdue and Wang correction (PW91)62 were utilized. The RESP approximation was employed for the calculation of the charges.63 After the 20 ns equilibration run the MD simulations was continued during 50 ns to collect data for calculations of the system characteristics. To compare the dendrimer melt density with experimental data, which are known at room temperature,64 we cooled all systems to 300 K (see details in ref 54). For all dendrimers the differences between the simulated and experimental densities are less than 1% that confirms the correctness of our computer simulations. The additional data of the equilibration process are shown in the Supporting Information. An important parameter, characterizing the orientational mobility in a dendrimer, is the semiflexibility parameter q,38,40,41 associated with the cosine of the angle θ between two vectors rs (Figure 1c) of neighboring segments belonging to successive dendrimer layers. The calculations of the average angle of θ are presented in Table 1. It can be seen from the table that the angle θ increases from periphery to a core that demonstrates the grow of the semiflexibility parameter, q, to the dendrimer core; however, the difference of θ for identical m is insignificant. As result, the value averaged over all angles in a dendrimer, θav, does not depends on G and corresponds to q ≈ 0.35. Thus, the PBC dendrimer is more flexible than the poly(Llysine) dendrimer (q = 0.45−0.5)44 and more semiflexible than C


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no noticeable differences between P1 functions (for identical m) in contrast to the results of computer simulation of these functions for dendrimer solutions.44,65 Moreover, our result completely agrees with the conclusion of the analytical theory.38 However, there is an exception: the P1 function for a core segment of the G2 dendrimer (i.e., m = G = 2) decreases faster than the functions for the G3 and G4 generations. Note that a small splitting for the P1(t, m = G) functions is observed even in the analytic theory.38 We will show below that in the case of m = G = 2 the splitting effect is caused by the influence of the short time of the rotation of the G2 dendrimer as a whole. From the analytical theory the P1 function of a segment can be decomposed into the sum of normal modes of a macromolecule.66 Moreover, the analytical theory38 predicts that only two processes give main contributions to the P1 ACFs: (i) the small-scale mobility of a given segment which is reflected in “internal” spectrum of normal modes of a dendrimer and (ii) the pulsation of a branch which originates from the segment. Taking into account the above, it is necessary to add the mode describing the overall dendrimer rotation. In the case of a dendrimer the overall macromolecule rotation cannot be considered as an independent motion of inner modes. It is caused by the fact that the characteristic time of pulsation of the dendrimer branch as a whole is close to the time of overall dendrimer rotation (see, for instance, ref 37). Dendrimers do not have a rigid skeleton structure, and therefore we use the approach of refs 67 and 68; i.e., an additional mode is taken into account in the sum of the normal modes. Note that this approach was directly confirmed by data of computer simulations for flexible dendrimers.28,37 Thus, we approximate the P1 ACFs by three exponential terms (two inner modes and one overall rotation mode):

Table 1. Angle between the Segments Belonging to Neighboring Layers θ, grad G

m=0

m=1

m=2

2 3 4

109.1 109.0 107.6

112.1 111.9 112.6

114.4 115.1

m=3

θav, grad

117.4

110.1 110.6 110.6

the dendrimer of the coarse-grained model used in ref 45 (q ≈ 0.2).

III. RESULTS AND DISCUSSION Autocorrelation Functions P1 and P2. To directly compare our results with the conclusions of the analytical theory, we consider the segmental reorientation in dendrimer melts via the P1 autocorrelation function (ACF) for vectors rs/|rs| directed along segments (see rs in Figure 1c). The calculated time dependences of the P1 functions are shown in Figure 2a. One can see a very similar behavior of the P1 functions for the segments of the same layer (numbered from the periphery); i.e., these functions for the same m (but different G) practically coincide. To be more precise: no difference is observed in the area where the P1 functions decrease up to the 1/e level, but further the small splitting of the curves is observed. Thus, it can be concluded that there is

P1(t , m , G) = C in exp[−t /τin] + C br exp[−t /τm br] + Cr exp[−t /τr G]

(3)

where τin is the characteristic time of the internal spectrum, corresponding to the local small-scale mobility; τmbr is the pulsation time for a dendrimer branch consisting of m generations; τrG is the time of rotation of a dendrimer macromolecule of the G generation as a whole; Cin, Cbr, and Cr are the relative contributions, corresponding to these characteristic times (Cin + Cbr + Cr = 1). Naturally, the use of three normal modes instead of the complete mode set is a substantial simplification of the phenomenon, but the results of the analytical theory demonstrate31,38 that this approach is justified for the dendrimer. As in the previous works,37,44 we used the value of τr calculated from the finite slope of P1r(t,G) ACFs which correspond to the reorientation of the unit vector directed from the atom Si of a core to an atom Si of the terminal generation. The details of the calculation are presented in the Supporting Information. The results of the approximation of the P1 ACFs by eq 3 are shown in Figure 2b as well as in Table S1 (see Supporting Information). In general, our results are consistent with the conclusions of the analytical theory.38 The pulsating time, τmbr, exponentially increases with m and slightly depends on G. The relative contribution of the pulsation process is practically constant (about 0.6). The values of τin are similar for all cases; i.e., they are independent from m and G. An exception is only

Figure 2. Time dependences of the P1 functions for the individual segment of the PBC dendrimer melts at 600 K (a) and the contributions of different characteristic times to the P1 functions (b). m is the layer number counted from the terminal layer (see Figure 1a), τin is the averaged time of the internal spectrum, corresponding to the local small-scale mobility, τmbr is the pulsation time of a dendrimer branch consisting of m generations, and τrG is the time of rotation of a dendrimer macromolecule of the G generation as a whole. m is the layer number, which begins from the terminal layer. D


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the level 0.1 of the amplitude. Obviously, the P2 ACF decreases significantly faster than the P1 ACF, and therefore, the contribution of the overall dendrimer rotation mode with the maximal characteristic time to P2 practically disappears even for the core segments of the G2 dendrimer (G = m = 2). In the processes of nuclear magnetic relaxation, for example, in the spin−lattice relaxation 1H of the CH2 group, the orientation mobility of the vector, which is perpendicular to the direction of the segment, can play an important role. Therefore, we also consider the P2 ACFs for this vector which will be denoted P2orto (“orthogonal”). Following the work,44 for the calculation of the P2orto ACF we used the vector product of two vectors directed along the vectors of C−C bonds belonging to the segment (see Figure 1c). In Figure 3b, the P2orto ACF is shown. The P2orto ACF decreases several times faster than the “longitudinal” P2 functions, and as a result, differences between the functions with same m are practically absent up to 0.01. To describe the P2 ACFs in analytical form, it is convenient to establish the relation between the P1 and P2 functions. Typically two approaches are utilized: P2 ≈ P13 for short timescale motion and P2 ≈ P12 for long time-scale motion. We have checked both approximations and found that only P2 ≈ P13 is valid. The same result was established for dendrimer in a solvent.29,44 Thus, for the analytical representation and analysis of the P2 ACFs we will use the following relationship between the functions of P1 and P2:

the case of the terminal segments (m = 0): the dominant contribution (more than 0.8) corresponds to τin when the value of τin is about 2 times less than the one for m > 0. Apparently, for the “free tails” (terminal segments) the faster reorientations are realized. Note that in the analytical theory the time τin has the same value for inner and terminal segments. In framework of this work it is difficult to establish the origin of the difference between the theory and our simulations. We surmise that this is due to the fact that in the theory identical viscoelastic parameters are used for inner and terminal segments of a dendrimer. In a typical dendrimer a terminal segment and branching point (a terminal bead) differ from an inner segment. At least, the changes of viscoelastic parameters of terminal segments shift τin.69 Additionally, we add in the Supporting Information the data which can used for the discussion of this question in future. The contribution of the overall rotation process, characterized by τrG, is minimal for terminal segments and increases with the shift a segment to a dendrimer core, i.e., with the grow of m. This fact leads to the splitting of the P1 ACF at the same m. However, for most cases the splitting is insignificant with only the single exception for core segments of the G2 dendrimer (G = m = 2) when the ratio of τrG2 and τ2br times is close to 1 (see Figure 2b). Let us turn to the P2 function (see eq 2). In Figure 3a, the time dependences of the P2 functions for individual segments are presented. It can be seen that the P2 functions for the same m coincide even in more degree than the P1 functions. The difference between the functions is poorly distinguishable up to

P13 ≈ P2

(4)

P13

As an example, Figure 4 shows the time dependent and P2 functions for G4, and one can see that the functions almost coincide.

Figure 4. Time dependences of P13 and P2 functions for an individual segment of the G4 dendrimer. m is the layer number, which begins from the terminal layer.

Using eq 4 and the approximation functions of P1 (eq 3), we obtained characteristic times τ̃n and contributions C̃ n of normal modes to the function P2: ⎡

P2(t , m , G) =

∑ CiCjCk exp⎢−t i ,j,k

⎢⎣

1 1 1 ⎤⎥ = τi τj τk ⎥⎦

⎡ −t ⎤ ⎥ ⎣ τñ ⎦

∑ C̃ exp⎢ n

(5)

where indices i, j, k run from 1 to 3; C1 = Cin, C2 = Cbr, C3 = Cr; τ1 = τin, τ2 = τbr, τ1 = τr; τ̃n is the characteristic times of the nth mode which contribution is C̃ n. In Figure 5 the contributions of normal modes with different characteristic times to the P2 ACF for an individual segment of the G4 dendrimer are shown. It is important that namely the analytical form (eq 5) of the P2

Figure 3. Time dependences of the P2 ACFs for individual segments (a) and the P2orto ACFs for the unit orthogonal vector, rorto/|rorto|, of an individual segment (b) (rorto = [rc1∗rc2]; see Figure 1c). m is the layer number, which begins from the terminal layer. E


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Figure 5. Contributions of normal modes with different characteristic times (eq 5) to the P2 ACF for an individual segment of the G4 dendrimer. m is the layer number, which begins from the terminal layer.

function allows us to explain data in the section NMR Relaxation. Further, the analytical forms of the P1 and P2 ACFs will be used for the calculation and analysis of the frequency dependences of the dielectric permittivity and NMR relaxation rates. Dielectric Permittivity. The P1 ACF are manifested in the dielectric relaxation of electric dipoles oriented along the selected segment of a macromolecule. The segment between branching points in a dendrimer is very short, and therefore, the length of the segment is not significantly changed in the motion process (see Supporting Information). In this case the frequency dependence of the real and imaginary parts of the reduced complex permittivity (ε = ε* − iε**) for the selected polar segment can be obtained by the Fourier transform of the P1 ACF:66,70 ε′(ω) =

ε*(ω) − ε*(∞) = Re[2 Δε*

∫0

ε″(ω) =

ε **(ω) − ε **(∞) = Im[2 Δε **

∫0

∞

∞

P1(t ) eiωt dt ]

(6)

P1(t ) eiωt dt ]

(7)

Figure 6. (a) Frequency dependences of the imaginary parts of the permittivity, ε″, for individual segments. (b) The functions εav″ (lines) averaged over all segments of the dendrimer and the contributions to ε″av from each layer for G4 (symbols) (see eq 9). m is the layer number, which begins from the terminal layer.

of the maximum for m > 0 is close to (2π/τmbr). An additional maximum is observed for m > 1: this maximum corresponds to the local mobility of the segment with the characteristic time τin, and as result, its position does not depend on m. In experiments the frequency dependence of ε″, integrated over all segments of a macromolecule, is observed. Therefore, we consider the averaged function:

where Δε* and Δε** are the difference of the function values at zero and infinite frequencies for real and imaginary parts of the dielectric permittivity, respectively. Using the analytical form of the P1 ACFs (eq 3), we can rewrite eqs 6 and 7 as ε′(ω) =

C in 1 + (ωτin)2

+

C br 1 + (ωτm br)2

+

ε″av (ω , G) =

ε″(ω) =

C in(ωτin) 2

1 + (ωτin)

+

C br(ωτm ) br 2

1 + (ωτr G)2

1 + (ωτm )

G

+

G

∑ Nmε″(ω , m , G) m=0

(10)

where N and Nm are the numbers of segments in the dendrimer and mth layer, respectively. Figure 6b shows that the position of the maximum of ε″av in high frequency region does not depend on G. It is caused by the fact that the ε″av behavior in this region is determined by the contribution of the mode with τin. At low frequencies the splitting of the functions is observed that is caused by the increase in maximal pulsation time τGbr with G, and as a result, ε″av has a trivial slope ∼ ω1 at lower frequencies. In the intermediate region (between low frequencies and the high frequency maximum) the slope is proportional to ∼ω0.3. We ascertain that the region of that slope expands with G due to the superposition of the contributions of processes determined by τmbr with different values of m = 0, ..., G. The illustration the functions, corresponding to the contributions from each layer to ε″av for the G4 dendrimer, is presented in Figure 6b. Experimental data for PBC dendrimer are still absent in the literature. Therefore, we discuss here the results of experimental study of the dielectric relaxation for other dendrimer melts with

Cr (8)

br

1 N

Cr(ωτr ) 1 + (ωτr G)2 (9)

Different relaxation processes lead to the appearance of bends in ε′ and maxima in ε″. Because these effects are determined by the same processes, we focus the attention only on ε″. As it can be expected, the frequency dependences of the permittivity practically coincide for the same m and do not depend on G (Figure 6a). Small differences are observed at low frequencies that corresponds to the contribution of the rotation of a dendrimer macromolecule as a whole. The position of the ε″ maximum shifts to low frequencies with the grow of m that corresponds to the increase in τmbr and, moreover, the position F


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Macromolecules the same functionality of junctions (F = 3). In the work8 the melt of carbosilane (G1−G3) dendrimers with perfluorinated terminal groups were studied at temperatures above and below the glass transition. For all generations the frequency dependences at high temperatures have a maximum which could not be described taking in account a single relaxation process. The position of the maximum weakly depends on G. These results confirm our conclusions. In ref 14 the dielectric permeability of the melt of PPI dendrimers (with G = 1−5) was studied. At high temperatures three fairly wide and barely separated processes were observed in the frequency dependence ε″av. The detailed analysis of the data in a wide range of frequency and temperature showed that these processes were consistent with the theoretical prediction (local reorientations, breathing modes, and overall tumbling of the dendrimer). Thus, our simulation results are in qualitative agreement with the conclusion of ref 14. NMR Relaxation. In this section we consider the frequency dependences of functions which are used to characterize the phenomenon of nuclear magnetic relaxation. The nuclear magnetic relaxation is the process of the attainment of the equilibrium state of macroscopic magnetization in a system of nuclear magnetic moments, placed in a static magnetic field. The theory of NMR relaxation is well developed and is described in many books (see, for example, refs 71−74). For a long time the method of the nuclear magnetic relaxation has been successfully used for the investigation of the molecular mobility in matter. The magnetic relaxation processes are determined by the intensity of fluctuating internal electromagnetic fields, and therefore the investigation of these processes interweaves with the study of the nature and velocity of thermal molecular motion, which produces those fluctuating fields. The NMR relaxation is characterized by the high selectivity of information about the local molecular mobility in contrast, for example, to viscosity which reflects the integrated translational mobility in a system. In general, when the existence of several independent mechanisms of the relaxation is expected, the resulting rate of the spin−lattice (1/T1) or spin−spin (1/T2) relaxation is given by the formula

1 = T1,2

∑ k

frequencies we propose to consider our approach as an acceptable approximation (the precise consideration of the intermolecular contribution is associated with great difficulties). However, in the low-frequency region the noticeable contributions of intermolecular interactions or neighboring groups of atoms may be observed. In the indicated dipole−dipole approximation, the NMR relaxation rates for proton relaxation can be expressed through spectral densities, J(ω), by the following expressions:71,74−76 1 = A 0[J(ωH) + 4J(2ωH)] T1H (11) for the spin−lattice relaxation and A 1 = 0 [3J(0) + 5J(ωH) + 2J(2ωH)] T2H 2

(12)

for the spin−spin relaxation. Here ωH is the cyclic frequency of the hydrogen nuclei; A0 is the constant which does not depend on the frequency and temperature. The spectral densities can be found by Fourier transform of the P2 ACFs:71−74,77 J(ω) = 2

∫0

∞

P2(t ) cos(ωt ) dt

(13)

Strictly, for the calculation of the spectral density in the case of 1 H relaxation the P2orto ACF for the vector rorto should be used (see the definition of the vectors in Figure 1c). Because there is no an analytical form for the P2orto ACF, we use the P2 ACF for the vector directed along a segment, i.e. rs (see Figure 1c), for which the analytical form is presented by eqs 3 and 4. However, as it was shown in our work44 the qualitative picture is the same for P2orto and P2. For the following consideration it is convenient to use the reduced (dimensionless) NMR relaxation rates:38,44 ⎛ω⎞ 1 [R1H]ω ≡ ⎜ ⎟ , ⎝ A 0 ⎠ T1H

⎛ 2ω ⎞ 1 [R 2H]ω = ⎜ ⎟ ⎝ A 0 ⎠ T2H

(14)

Using the obtained form of the spectral density and eqs 11, 12, and 14, the frequency dependences [R1H]ω and [R2H]ω were obtained (Figure 7). As we expected, the functions are determined by the topological distance from the periphery (i.e., by m) and practically do not depend on G. The position of the [R1]ω maximum shifts to low frequencies with growing m (as well as the inflection region of [R2]ω). It is important to note that the trend of the maximum shift is in accordance with the semiflexible models,38,45 and it is absent in flexible models.31,45 These effects can be explained by the fact that the maximum position is determined by the contribution of the mode with τmbr in the semiflexible model and by the contribution of the mode with τin in the flexible approach (τmbr ≫ τin). However, in the case of the PBC dendrimer melt the shift is governed by a few factors. When m changes from 0 to 1 the [R1]ω maximum shifts due to growing τin in ≈2 times due to the increase in the semiflexibility parameter for inner segments compared with terminal segments. Note that for m = 0 the τin contribution is dominant, and therefore the position of the maximum corresponds to τin(m = 0)/3 (see eq 15). For m = 1 the contribution of the first process to P1 decreases (see Figure 2b), and the main contribution to the P2 is given by the cross component with τin (see Figure 5). For m = 2 and 3 both contribution of the modes with τmbr and τin are important. The pulsation time, τmbr, increases with growing m, but τin practically does not change. Therefore, (i)

1 T1,2, k

where T1,2,k is the relaxation time for the kth mechanism. Typically, a single relaxation mechanism dominates for the most relevant resonances of the 1H and 13C nuclei, used in the dendrimer investigations. Our calculations were made in the limits of the dipole−dipole relaxation mechanism for nuclei belonging to the same group. We believe that in the case of dendrimer melts this mechanism is the decisive factor in a fairly wide range of frequencies: from the high-frequency region to, at least, the middle one. This assumption is based on the analysis of the radial distribution functions which were obtained during the simulation. According to our data, the distance between the carbon atoms belonging to different branches of a dendrimer is not less than 0.38 nm, and therefore the minimal distance between hydrogen atoms attached to them is about 0.25 nm. Since the HH distance in the same molecular group of about 0.17 nm and the dipole relaxation strongly decreases with a distance (1/r6), the expected intermolecular contribution is less almost an order of magnitude than the intramolecular one. We are aware that there is a probability of the manifestation of the intermolecular contribution to the relaxation rates at low G


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Figure 7. Frequency dependences of the spin−lattice (a) and spin− spin (b) reduced relaxation rates for an individual segment. m is the layer number, which begins from the terminal layer.

Figure 8. Frequency dependences of (a) the reduced spin−lattice relaxation rates averaged over all segments in a dendrimer, [R1H]av, and (b) the main contributions of modes with similar characteristic times as τrot for G4.

for m = 2 one can see the broadening of the [R1]ω maximum and (ii) for m = 3 two maxima are observed because the ratio of the characteristic times becomes quite large: τ2br/τin ≈ 50. For illustration the characteristic frequencies (ωin, ωmbr) corresponded to τmbr and τin times are indicated for the G4 dendrimer in Figure 7a. The frequencies were calculated using the ratios (see, for example, ref 74): 0.616 ; τin(m = 0)/3 0.616 0.616 ; ωm br /3 = br −1 τm /3 2 + br

layers (m = 0, ..., G), and maximal pulsation time τbr m=G increases with the size of a dendrimer, i.e., τ4br > τ3br > τ2br. The contributions of the segments from different layers lead to the appearance of the slope ∼ ω0.5. It should be noted that for PPI dendrimers in melt the similar effect was observed but the slope of experimental dependences changes from 0.87 (G2) to 0.76 (G5).15 The difference in the slopes is caused by the fact that we used the P2 ACF for the vector rs (see Figure 1c) to calculate [R1H]ωav. In experiments the function P2orto ACF for orthogonal vector, rorto, manifests in the case of macromolecules. Remember that there is no an analytical form of P2 ACF for vector rorto, and therefore for the G4 dendrimer we calculate the [R1H]ωav function via the P2orto ACF by numerical methods (Figure 8a). The slope of [R1H]ωav is 0.79, which is very close to observed one in the experiments. Thus, the considered P2 ACF can be used for the qualitative comparison of our results with the NMR experiments. Certainly, for detailed comparisons it is necessary to use P2orto ACFs for vector rorto. It is also worth noting that the [R1H]ω function for m = 0 (for terminal segment) is smaller than [R1H]ωav functions in the left side of the maximum. The similar difference between [R1H]ω for terminal groups and [R1H]ωav was observed for PPI dendrimers in ref 14 where the authors used the partial deuteration of terminal groups of dendrimers. We suppose that these features are general for dendrimer melts. The influence of the overall rotation on the behavior of [R1H]av ω is noticeable at very low frequencies despite the direct contribution of the process to this function is small, e.g., 0.0013 for G4. The noticeable influence is caused by the fact that τrot ≈ τbr m=G, and the modes with similar characteristic times τ̃ = (1/

ωin(m = 0)/3 = ωin =

(

1 τin

τ1

)

(15)

Experimental studies of the NMR relaxation in a wide range of frequencies usually deal with averaged relaxation rates over a macromolecule (for instance, see ref 14). Taking into account this circumstance, we calculate the dependences averaged over all layers in accordance with expressions [R1]av ω =

1 N

G

∑ Nm[R1(m)]ω m=0

(16)

where N and Nm are the number of segments in a dendrimer and mth generation layer, respectively. We focus on the NMR relaxation of the 1H nuclei, but similar effects can be detected in the dependences for the 13C nuclei. In Figure 8 we present the [R1H]ωav and [R2H]ωav functions, and for comparison, the [R1H]ω and [R2H]ω functions for the terminal (m = 0) segment of the G4 dendrimer. At high frequencies the [R1H]ωav functions practically coincide, but at low frequencies some differences are observed. It is caused by the fact that averaging the functions in the bigger dendrimers was carried out over a larger number of H


Article

Macromolecules −1 −1 τbr and τ̃ = (2/τbr in P2 ACFs (see eq m=G + 1/τrot) m=G + 1/τrot) av 5) give the main contributions in [R1H]ω at the low frequencies. For illustration, these contributions are shown in Figure 8b. Besides we would like to note that the process with time τ̃ = −1 (2/τbr gives an additional significant contribution m=G + 1/τrot) in this frequency region (see Figure 8b). Similar conclusions can be formulated for frequency dependences of the spin−spin relaxation rate [R2H]ωav, and these functions are shown in Figure S6 of the Supporting Information. Finally, let us return to the slope (scaling) of the frequency dependences of [R1H]av and [ε″]av. No slope in these functions has been predicted in the analytical theory for a dendrimer structure. In our simulation we found that the scaling was present in both the NMR and dielectric relaxations. In particular, this fact is confirmed by experimental data for NMR relaxation.14,15 In the framework of our studies it is difficult to determine the origin of the difference between the theory and simulations. Nevertheless, we would like to draw the attention to this difference that is important for the development of the relaxation theory of dendrimer macromolecules.

broadening of the maximum and in the case of the big difference between τin and τmbr leads to the splitting of the maximum. The averaged functions εav″ and [R1]ωav over all segments in a dendrimer macromolecule demonstrate the different dependences on G only at low frequencies (see Figures 6b and 8a) that is caused by the contributions of the maximal pulsation (τbr m=G) and overall rotation (τrot) times. The averaging also leads to the superposition of τmbr contributions (m = 0, ..., G) that manifests in the slopes like ∼ω0.3 and ∼ω0.5 in the frequency dependences of εav″ and [R1]ωav, respectively (see Figures 6b and 8a). This effect is consistent with experimental data but is absent in theoretical predictions.

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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.6b01502. Justification of equilibration times; parameters of segments in different dendrimer layers; the calculation of overall rotation times of dendrimer macromolecules; the approximation of the P1 ACFs; the frequency dependences of 13C NMR relaxation rates (PDF)

IV. CONCLUSIONS To describe the orientation mobility in the melt of poly(butylcarbosilane) (PBC) dendrimers, we have adapted the data of our molecular dynamics simulations applying the model of united atoms.54 For these systems the orientation mobility behavior of a dendrimer segment the effect, which was earlier predicted by the analytical theory,31,38 has been reproduced for the first time. According to the theory, the segmental mobility is determined by a topological distance between the segment and periphery (which is characterized by index m) and is independent of the size of a dendrimer (i.e., independent of G). These features arise due to the decrease of the role of the overall rotational mobility of a dendrimer macromolecule in a melt (in contrast to the case of the behavior a dendrimer macromolecule in a solution). As we suggested in our previous works,37,44 namely this circumstance leads to the main difference in the results of the analytical theory and computer simulations for dendrimers in a solution. The results of the approximation of P1 ACFs also confirmed the conclusions of the theory that, in addition to the rotation of the dendrimer macromolecule as a whole, the orientation mobility of an individual segment is determined by two processes: (i) the small-scale mobility with the average time of the internal spectrum, τin, which is independent of G and m, and (ii) the pulsation of a branch, that originates in this segment, with the characteristic time τmbr, which depends only on the size of a branch (i.e., on m). We also found that even for relatively flexible carbosilane dendrimers, it is necessary to take into account the semiflexible effect (local bending stiffness)40 for the correct description of segmental reorientations. The obtained representations of P1 and P2 allow us to study in details the frequency dependence of the dielectric and NMR relaxations. It has been established that these dependences can be characterized by the contributions of two modes with the characteristic times τin and τmbr, and the main contribution corresponds to τmbr. As a result, the position of maximum of dielectric permittivity ε″ and reduced NMR relaxation rate [R1]ω is shifted to lower frequencies with increasing m. Besides, the significant contribution of the mode with τin causes the

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AUTHOR INFORMATION

Corresponding Author

*E-mail [email protected] (D.A.M.). ORCID

Denis A. Markelov: 0000-0002-1590-3582 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work is supported by the Russian Foundation for Basic Research (Grant No. 14-03-00926) and the Government of the Russian Federation (Grant 074-U01). The simulations have been performed by using the computer resources center of Saint Petersburg State University.

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REFERENCES

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Orientational Mobility in Dendrimer Melts: Molecular Dynamics

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