Orientational relaxation in semiflexible dendrimers.

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Orientational relaxation in semiflexible dendrimers Cite this: Phys. Chem. Chem. Phys., 2013, 15, 20294

Amit Kumar and Parbati Biswas* The orientational relaxation dynamics of semiflexible dendrimers are theoretically calculated within the framework of optimized Rouse–Zimm formalism. Semiflexibility is modeled through appropriate restrictions in the direction and orientation of the respective bond vectors, while the hydrodynamic interactions are included via the preaveraged Oseen tensor. The time autocorrelation function M(i) 1 (t) and the second order orientational autocorrelation function P(i) 2 (t) are analyzed as a function of the branch-point functionality and the degree of semiflexibility. Our approach of calculating M(i) 1 (t) is completely different from that of the earlier studies (A. Perico and M. Guenza J. Chem. Phys., 1985, 83, 3103; J. Chem. Phys., (i)

1986, 84, 510), where the expression of M1 (t) obtained from earlier studies does not demarcate the flexible dendrimers from the semiflexible ones. The component of global motion of the time autocorrelation function exhibits a strong dependence on both degree of semiflexibility and branch-point functionality, while the component of pulsation motion depends only on the degree of semiflexibility. But it is difficult to distinguish the difference in the extent of pulsation motion among the compressed (0 o f o p/2) and expanded (p/2 o f o p) conformations of semiflexible dendrimers. The qualitative exact (t) in the behavior of P(i) 2 (t) obtained from our calculations closely matches with the expression for P 2

earlier studies. Theoretically calculated spectral density, J(o), is found to depend on the degree of semiflexibility and the branch-point functionality for the compressed and expanded conformations of semiflexible dendrimers as a function of frequency, especially in the high frequency regime, where J(o) decays with frequency for both compressed and expanded conformations of semiflexible dendrimers. This decay of the spectral density occurs after displaying a cross-over behavior with the variaReceived 11th September 2013, Accepted 11th October 2013

tion in the degree of semiflexibility in the intermediate frequency regime. The characteristic area

DOI: 10.1039/c3cp53864h

flexible dendrimers record the maximum characteristic area. For the compressed conformations the rela-

increases with the increase in the semiflexibility parameter, where the expanded conformations of semitive increment of this area is considerably lower than that of the expanded conformations of

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semiflexible dendrimers.

1 Introduction Dendrimers comprise a hierarchic assembly of ‘‘branchesupon-branches’’ to form a tree-like network without any closed loops. Due to their extensively branched symmetric architecture dendrimers have been exploited in a wide range of potential applications with possible uses as catalysts or nanocarriers for drug, dye and metal nanoparticle delivery.1–7 To maximize the range of applicability, it is essential to probe the structure and characteristic molecular mobility including local dynamics and segmental motions of dendrimers. Mobility of dendrimers is experimentally studied by dielectric relaxation, nuclear magnetic resonance (NMR), birefringence measurements and polarized luminescence.1 NMR is a perfect tool to investigate the mobility of dendrimers, as mobility may be correlated to Department of Chemistry, University of Delhi, Delhi-110007, India. E-mail: [email protected]; Tel: +91 112766 7794

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various time scales in which several relaxation processes may be identified. This relaxation at large time scales is strongly dependent on their molecular size where the conformational details can be ignored. However, these time scales are much smaller than the time required for the typical relaxation process of the dendrimer, where dynamics occur on a local length scale, almost independent of the molecular size but strongly dependent on their local conformational detail. A widely accepted model-free approach was first proposed by Lipari–Szabo8,9 for the interpretation of NMR data, in which the decay of the orientational autocorrelation function, P(i) 2 (t), was mapped into mutually independent internal and global motions. Each internal fluctuation was described by a single characteristic time constant together with an order parameter, S2, which represented a measure of the restraint on internal motion. This approach was refined in various10 theoretical studies, where two different time-scales for internal motion were assumed.11 Other approaches included coupling between

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global and internal motions.12 However, such models were found to be consistent with the flexible polymers where the local motion follows two uncorrelated decay processes given by the local and global dynamics. Also they consisted of a finite number (sometimes large) of adjustable parameters that limited the theory’s predictive power and generality. Thus Lipari–Szabo’s theory was inadequate for dendrimers with rigid segments and short spacers. Experimental NMR data for such dendrimers were reproduced through a large number of adjustable fitting parameters. Description of the local dynamics of flexible dendrimers involves analytical theory13–16 and computer simulations.15,17 These approaches13–16 approximate the NMR data with more than two fitting parameters, neglecting the effect of hydrodynamic and excluded volume interactions. The excluded volume effects are more pronounced in dendrimers with short spacers where the long range interactions play a significant role in the orientational relaxation dynamics. Alternatively, simulations15–17 incorporate excluded volume interactions, but in the presence of non-preaveraged hydrodynamic interactions such calculations are time consuming demanding computational requirements. A recently developed generalized theoretical model18–20 of semiflexible dendrimers18,19 evaluates both conformational21,22 and dynamic properties of semiflexible dendrimers in the framework of optimized Rouse–Zimm formalism. Semiflexibility is implemented by topologically restricting the direction and orientation of the bond vectors. By tuning the bond orientation angles, this model18,19 captures a wide range of conformations, ranging from flexible23 to freely rotating,24,25 and dendrimers in good solvents with excluded volume interactions.26 The primary focus of this article is to evaluate the orientational relaxation parameters of semiflexible dendrimers. Of particular (i) interest are the orientational autocorrelation function, P2 (t), and the spectral density, J(o), which is the real part of the Fourier transform of P(i) 2 (t). Semiflexibility is incorporated by defining the bond correlation in terms of spherical harmonics Y m l (y, f) which imposes restrictions on the direction and orientation of the respective bond vectors through the angles y and f. Solvent mediated hydrodynamic interactions are modeled via the preaveraged Oseen tensor with an inverse scaling behavior r1. The present manuscript is organized in the following manner. In Section 2, the optimized Rouse–Zimm model for semiflexible dendrimers in a solvent is presented and the time autocorrelation function is evaluated followed by orientational relaxation parameters. In Section 3, orientational relaxation parameters are theoretically calculated as a function of degree of semiflexibility and branching topology of the dendrimer. Section 4 highlights the main results of work with concluding discussions.

2 Theory 2.1

Matrix building

The connectivity matrix [A] represents the architecture of the dendrimer, whose diagonal elements Aii give the number of bonds emanating from the i-th monomer, i.e., the branch-point functionality of the i-th monomer, fi, and the off-diagonal

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elements Aij are 1 if i and j are connected to each other, otherwise 0. Alternatively, the connectivity matrix A may be obtained through the N (N 1) incidence matrix G as24,25 [A] = [GUGT]

(1)

where the elements of the incidence matrix, Gij, are Gij = 1 if the bond vector lj starts at the i-th bead, Gij = +1 if the bond vector lj points to the i-th bead, and Gij = 0 otherwise. Semiflexibility is incorporated through the (N 1) (N 1) bond correlation matrix [U] in eqn (2), whose elements contain the average scalar product of bond vectors 1 li lj U ij ¼ (2) l2 From eqn (2), it is clear that semiflexibility is modeled through introducing the semiflexibility parameter as the normalized mean scalar product of the nearest neighboring bonds, i.e., hlilji, while the normalized mean scalar product of the non-nearest neighbors i and k is expressed in terms of product of nearest neighboring bonds as hlilki = hlilj1ihlj1lj2i hljnlkil2n

(3)

where ( j1, j2,. . .jn) denotes the unique shortest distance between the i-th and k-th bond vectors. Earlier studies modeled the normalized mean scalar product of the bond vectors for dendrimers24,25,27 through the cosine of the bond angle, y, assuming symmetrical rotational potential with unhindered bond rotations.28 However due to the presence of extensive steric crowding in dendrimers, the bond i depends not only on the bond i 1 through the bond angle y but also on the bond i + 1 through the bond orientation angle f. Thus the bond correlation of dendrimers must be expressed as a function of both y and f, which implies that bonds have hindered rotation, i.e., hcos fi a 0 (refer to ref. 28). The normalized mean scalar product of the neighboring bond vectors is usually defined in terms of the bond angle, y, and the orientational angle, f, where the bond correlations are formulated through the rotational matrix for the Cartesian coordinate system as defined by Flory.28 In a recent work,29 this definition of bond correlation is used to evaluate the bond correlation of the linear polymer chain by assuming hindered rotation of bonds for symmetrical rotational potential, i.e., (hcos fi a 0 and hcos fi a 0).28 This approach needs to be validated for branched polymers, where due to increased steric crowding, evaluating correlation between adjacent bond vectors becomes quite difficult. Based on the approach used in our model,18–22 the bond correlation for dendrimers may be approximated through norm malized spherical harmonics Y m l (y, f), such that Y l is equal to m 18,19 the complex conjugate of Y l , which accounts for the effect of both bond direction and orientation of the respective bond vectors. The model does not explicitly incorporate the excluded volume effect, which has a significant role in the orientational mobility of dendrimers.13,15 Our earlier studies19,20,22 confirm that the inclusion of semiflexibility in model dendrimers resembles the behavior of that of the excluded volume


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interactions. Hence, in this work the orientational mobility may not be significantly affected by neglecting the excluded volume effect. The hydrodynamic interaction matrix [H] measures the solvent-mediated preaveraged hydrodynamic interactions30,31 (HI) between the i-th and j-th monomers, which may be represented by the Oseen tensor as [H]ij = (di, j + zr hl/Riji (1 di, j))I

(4)

Eqn (4) is valid only for small perturbations from equilibrium and does not account for large deformations in shape. The mobility matrix [H] and the identity matrix [I] are three dimensional, while Rij = |Ri Rj| measures the separation between the centers of the i-th and j-th monomers. The reduced monomer friction coefficient is given by zr = z/6pZsl = a/l, which gives the effective monomer hydrodynamic radius in units of l and Zs, where Zs is the viscosity of the solvent. The value zr = 0.25 ensures that the matrix [H] is positive definite and does not show any instabilities related to the appearance of negative unphysical eigenvalues.32–34 2.2

governs the equation of motion for the time evolution of the spatial coordinates for dendrimers in solution.35,36 [H] and [A] are the hydrodynamic interaction matrix and the connectivity matrix respectively. The time evolution of the spatial coordinates of the dendrimer is represented by a diffusive Langevin equation in the framework of optimized Rouse–Zimm formalism,24,35–39 where the inertial term is neglected as the dynamics of dendrimers are overdamped. The time evolution of the spatial coordinate of the i-th bead, i.e., the vector Ri(t) is given by N X @Ri ðtÞ ¼ K z ½H Aij Rj ðtÞ þ f i ðtÞ @t j¼1

In eqn (10), the viscous force on the left-hand side is balanced by the intramolecular force and the random force fi (t) on the right-hand side. The random force is assumed to be a Gaussian white noise with zero mean, h fi(t)i = 0, and delta correlation h fi (t)fj (t 0 )i = 2kBTz[H1]ijdij d(t t 0 )

Formalism

In the optimized Rouse–Zimm24,35–39 formalism, dendrimers are represented by a simple bead-spring model, where the beads denote effective monomer units with coordinates Ri, connected via bonds depicted as simple harmonic springs. All bonds between monomers are assumed to be of equal mean square length l2, with the spring constant K = 3kBT/l2 and the friction coefficient z. The bond vector {li} (i = 1,2. . .N 1) connecting the i-th and i 1-th monomers is given by li = Ri Ri1

(5)

This expression may be recast in terms of the N (N 1) incidence matrix40 G X G Tij Rj (6) li ¼ j

The relaxation times measured by NMR are expressed in terms of the spectral density function J(o), defined as the cosine Fourier transform of the second rank orientational (i) autocorrelation function P2 (t)41,42 rffiffiffið 2 1 ðiÞ JðoÞ ¼ P ðtÞ cos otdt (7) p 0 2 where the second rank orientational autocorrelation function (i) P2 (t) associated with each bond vector li is defined as35,36 * + 3 ðl i ðtÞ l i ð0ÞÞ2 1 ðiÞ (8) P2 ðtÞ ¼ 2 li 2 ðtÞli 2 ð0Þ 2 (i) M 1 (t)

The time autocorrelation function characterizes the orientation of the i-th bond vector li(t) for polymer chains, which is defined as35,36 ðiÞ

M1 ðtÞ ¼

hl i ðtÞ l i ð0Þi hli 2 i

(i) M 1 (t)

may be expressed as a combined function of the eigenvalues and eigenvectors of the matrix product [HA], which

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(11)

The Langevin equation may be solved by decoupling into independent normal modes, X, by diagonalizing the matrix [HA],18 where the monomers’ coordinates, R, are transformed into normal coordinates X by Rj ðtÞ ¼

N X

Qjk X k ðtÞ

(12)

k¼1

where Q denotes the matrix of linearly independent normalized eigenvectors of [HA], defined as Q1[HA]Q = L

(13)

where L is a diagonal matrix with elements lk. The eigenvectors, Q, are also used for the separate diagonalization of matrix [A] through the congruent transformation QT[A]Q = U

(14)

where U is a diagonal matrix, whose elements mk are defined as the mean-square length of the k-th normal mode defined as hXk2i = l2mk1. The Langevin equation in terms of generic normal co-ordinates is thus @X k ðtÞ ¼ Klk X k ðtÞ þ f k ðtÞ z @t

(15)

where the relaxation rate of normal coordinates Xk is given by hXk(t)Xm(0)i = dkmhXk2i exp(tKlk/z)

(16)

The time autocorrelation function M (i) 1 (t) may be expressed in terms of normal modes by using the basic definition of the bond vector li as given in eqn (6) ðiÞ

(9)

(10)

M1 ðtÞ ¼

¼

hl i ðtÞ l i ð0Þi hl i 2 i E P D T G ij Rj ðtÞ G T is Rs ð0Þ

(17)

j;s

l2

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Substituting the value of position vectors from eqn (12), eqn (17) may be expressed as P Dh ðiÞ

M1 ðtÞ ¼

j;s;k;m

ij

E i h i Qjk X k ðtÞ G T is Qsm X m ð0Þ

P

l2 P

¼

GT

eqn (8). From eqn (6) and (12), the statistical average in eqn (8) assumes the form * + ðl i ðtÞ l i ð0ÞÞ2 li 2 ðtÞli 2 ð0Þ

k;m

ðiÞ

Dk DðiÞ m hX k ðtÞ X m ð0Þi

¼

j;s;k;m

h i

2 ðG T Þij Qjk X k ðtÞ ðG T Þis Qsm X m ð0Þ l4

P

l2 (18)

¼

ðiÞ

k;m;k0 ;m0

ðiÞ

ðiÞ

Dk DðiÞ m Dk0 Dm0 h½X k ðtÞ X m ð0Þ½X k0 ðtÞ X m0 ð0Þi l4 (21)

(i)

T (i) (i) where Dk = (GTQ)ik = (QTG)ki, D(i) m = (G Q)im and Dk Dm = (i) T T (Q GG Q)km. Substituting eqn (16), eqn. (18) may be simplified as

ðiÞ

M1 ðtÞ

Eqn (21) may be further simplified by substituting the expression for the second order correlation of normal modes, i.e., h[Xk(t)Xm(0)][Xk 0 (t)Xm 0 (0)]i. This ensemble average of normal modes may be expressed as nine terms resulting from the multiplication of two scalar products as

N ðiÞ P ðiÞ X k2 QT GG T Q kk expðslk tÞ hX 1 ðtÞX 1 ð0Þi QT GG T Q 11 þ

h½Xk ðtÞ Xm ð0Þ½Xk0 ðtÞ Xm0 ð0Þi ¼ 3 Xka ðtÞXma ð0ÞXka0 ðtÞXma 0 ð0Þ þ 6 Xka ðtÞXma ð0ÞXkb0 ðtÞXmb 0 ð0Þ

k¼2 l2

¼

(19)

where s = K/z and the first column of Q corresponds to the eigenvalues l1 = 0. In this case, the matrix elements in each row of the incidence matrix G have only one entry 1 and +1 respectively; therefore (QTGGTQ)(i) 11 = 0. Thus eqn (19) may be simplified as

ðiÞ M1 ðtÞ

T ðiÞ N X Q GG T Q kk ¼ expðslk tÞ mk k¼2 ¼

N X

(20)

with a a b. In eqn (22), each average can be expressed as the sum of the average of pairs of mode components36 as a Xk ðtÞXma ð0ÞXkb0 ðtÞXmb 0 ð0Þ ¼ Xka ðtÞXma ð0Þ Xkb0 ðtÞXmb 0 ð0Þ (23) þ Xka ðtÞXkb0 ðtÞ Xma ð0ÞXmb 0 ð0Þ þ Xka ðtÞXmb 0 ð0Þ þ Xkb0 ðtÞXma ð0Þ Thus by using eqn (16), (21) and (23), eqn (8) may be simplified by neglecting the cross terms as

ðiÞ

gk expðslk tÞ

ðiÞ

P2 ðtÞ ¼ 2

k¼2

N n X k;k0 ¼2

(i)

(i)

Here the superscript (i) in (QTGGTQ)kk denotes that M 1 (t) is determined for the bond vector i (refer to eqn 2.1 and 2.21 of (i) (i) ref. 36). In eqn (20), gk = (QTGGTQ)kk/mk, and the normalization N P ðiÞ gk ¼ 1. The time autocorrelation condition holds, such that k¼2

function is a linear combination of N decay modes, characterized by the index k. Each decay process has a weight of (i) (QTGGTQ)kk and tk = (slk)1 which represents the correlation or relaxation time. The difference between the earlier calcula(i) tions of Perico–Guenza35,36 and this work for M 1 (t) corresponds to the structural average over the whole dendrimer, (i) i.e., gk . The local orientational mobility of a single bond in the dendrimer may be measured through the orientational auto(i) (i) correlation functions P1 (t) and P2 (t). Following the same line (i) of approach to calculate M 1 (t), the second rank orientational (i) autocorrelation function, P2 (t), may be defined through

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(22)

ðiÞ ðiÞ o QT GG T Q kk QT GG T Q k0 k0

2 2 Xk Xk0 exp½sðlk þ lk0 Þt l4 ðiÞ 2 N h i2 QT GG T Q kk X ðiÞ ¼ 2 exp½2slk t ’ M1 ðtÞ 2 mk k¼2 (24) The spectral density of semiflexible dendrimers may be (i) evaluated by substituting the expression of P2 (t) from eqn (24) in eqn (7), which is given by rffiffiffiffiffiffiffiffiffiffi N

1 X ðiÞ 2 lk g (25) JðoÞ ¼ 2ps2 k¼2 k lk2 þ ðo=2sÞ2 From eqn (25), it is evident that the orientational relaxation parameters depend on the eigenvalues lk, the eigenvectors of the matrix [HA] and the eigenvalues mk of the matrix [QTAQ] through the second order orientational autocorrelation


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(i)

function P2 (t). Therefore, the accurate determination of the relaxation parameters depends on the matrices [A] and [H], (i) which provide a simple route to calculate P2 (t) and the spectral density, J(o), which are experimentally relevant.

3 Results and discussion The orientational relaxation dynamics may be quantitatively characterized by the second order orientational autocorrelation (i) function P2 (t), which may be evaluated from the eigenvalues l, the eigenvectors Q of the matrix product [HA] and the diagonal elements of matrix [QTAQ] and [QTGGTQ] (refer to eqn (24)). The dense asymmetric matrix [HA] is numerically diagonalized for dendrimers with varying degree of semiflexibility (f) to obtain the eigenvalues and the corresponding set of eigenvectors. From eqn (25), the reduced spectral density may be expressed as J ðo Þ ¼

N 1 X k¼1

ðiÞ

gk

2

lk lk2 þ ðo Þ2

(26)

,rffiffiffiffiffiffiffiffiffiffi 1 and the reduced frequency where J ðo Þ ¼ JðoÞ 2ps2

(i)

Fig. 2 Plot of structural average of the time autocorrelation function, M1 (t), for G = 8 semiflexible dendrimers with branch-point functionality fc = 3, f = 3 for the entire range of bond orientation angle, f (0 o f o p). Here solid lines denote the compressed conformations (0 o f o p/2) of semiflexible dendrimers, dashed lines represent the expanded conformations (p/2 o f o p) of semiflexible dendrimers and the dashed-dotted line denotes for flexible dendrimers (f = p/2). The different conformations of dendrimers are represented by increasing f values, i.e., f = 151, 301, 451, 601, 751, 901, 1051, 1201, 1351, 1501 and 1651.

o* = o/2s. 3.1

Segmental relaxation

The time autocorrelation function M 1(i)(t) is depicted in Fig. 1 and 2 as a function of time for both the compressed and the expanded conformations of G = 8 semiflexible dendrimers, which is compared with the flexible dendrimer in Fig. 2. In (i) Fig. 1, M 1 (t) is depicted for compressed (f = 301) and expanded (f = 1501) conformations of semiflexible dendrimers with varying branch-point functionality. Three different types of branch-point functionality are considered: (a) fc = 3, f = 3, (b) fc = 4, f = 4 and (c) fc = 10, f = 3, where fc and f are the branchpoint functionality of the core and that of the other branch (i) points respectively. In Fig. 2, M 1 (t) is plotted for all

Fig. 1 Plot of the structural average of the time autocorrelation function, M(i) 1 (t), for the compressed (f = 301, i.e., in black color) and expanded (f = 1501, i.e., in red color) conformations of G = 8 semiflexible dendrimers with variation in the branch-point functionality at the core and other branch junctions, i.e., (a) fc = 3, f = 3, (b) fc = 4, f = 4 and (c) fc = 10, f = 3.

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conformations of semiflexible dendrimers, i.e., with f A (0,p), for model (a) type semiflexible dendrimers. From Fig. 1, it is observed that the time autocorrelation function, M (i) 1 (t), decays with time for both compressed (black color in Fig. 1) and expanded (red color in Fig. 1) conformations of semiflexible dendrimers for all three types of dendrimers with different branch-point functionality. It may be noticed that at smaller times, the magnitudes of the time autocorrelation function of different model semiflexible dendrimers with different branch-point functionality are almost similar to each other. For larger times, the time autocorrelation function changes with the variation of the branch-point functionality for (a) and (b) type semiflexible dendrimers only. The time autocorrelation function for type (b) semiflexible dendrimers is lower in magnitude as compared to that for type (a) semiflexible dendrimers (see Fig. 1). From Fig. 2, it may be observed that M (i) 1 (t) is strongly dependent on the degree of semiflexibility, f, for the entire span of time. It may also be noted that at smaller times, the curves for M (i) 1 (t) of compressed (0 o f o p/2) and expanded (p/2 o f o p) conformations of semiflexible and flexible (f = 901) model dendrimers are very close to each other, while at larger times these curves are well demarcated as seen in the magnified plot of Fig. 2. The magnitude of M (i) 1 (t) for flexible dendrimers (f = 901) closely resembles that for compressed conformations of semiflexible dendrimers (for f = 751) as shown in Fig. 2. The relaxation spectrum of the dendrimer governs the (i) behavior of M 1 (t). This spectrum has two contributions: at (i) smaller times (time less than 1), M 1 (t) is controlled by the fast internal ‘‘pulsation’’ motion of the monomers, while at larger times, it is determined by the ‘‘global’’ motion of the entire (i) dendrimer. The pulsation motion (the initial slope of M 1 (t)) directly depends on the internal spectrum of the dendrimer, which is practically independent of the number of generations in the dendrimer.15 The global motion of the dendrimers

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exact Fig. 3 The comparison plot of the structural average of P(i) (t) (obtained through an earlier study36) for compressed (f = 301) and expanded (f = 1501) 2 (t) with P2 conformations of semiflexible dendrimers with branch-point functionality fc = 3, f = 3 and generation G = 8.

depends on both the internal spectrum and motion of subbranches relative to the immobile core. In Fig. 1, no change is observed in M (i) 1 (t) due to the pulsation motion for all types of branch-point functionalities. The pulsation motion is found to vary with the change in the degree of semiflexibility (Fig. 2), where the maximum initial slope is observed for compressed conformations and the minimum slope is observed for expanded conformations of semiflexible dendrimers, while the flexible model dendrimers (f = 901) are flanked between the compressed and expanded conformations of semiflexible dendrimers. However, it is difficult to compare the difference in the extent of pulsation motion among the compressed (0 o f o p/2) and expanded (p/2 o f o p) conformations of semiflexible dendrimers. The global motion component of (i) M 1 (t) exhibits a strong dependence on the degree of semiflexibility and the branch-point functionality of semiflexible dendrimers (only for type (a) and type (b)), where the global motions for both compressed and expanded conformations of semiflexible dendrimers are widely different from each other. But M (i) 1 (t) of dendrimers with different branch-point functionalities of the central core (i.e., type (a) and type (c) dendrimers) are indistinguishable from each other. M 1(i)(t) depends on the orientation of the respective bonds and the topological distance of the segment from the terminal branch point, i.e., the size of the corresponding sub-branch of the dendrimers.15 Thus, at smaller times, the numerical values of M (i) 1 (t) vary only with the change in f, while no effect is observed with the variation of the branch-point functionalities of semiflexible dendrimers. At larger times, the magnitude of (i) M 1 (t) strongly depends on both bond orientation angle and branch-point functionality (type (a) and type (b) only) of semiflexible dendrimers. This implies that varying the degree of

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semiflexibility affects the segment size of dendrimers, which is unaffected by the change in branch-point functionality at smaller times. At larger times, varying the branch-point functionality and the degree of semiflexibility affects the orientation of the bonds of dendrimers. However, the orientation of the bonds in the dendrimer remains unaffected by the change in the branch-point functionality of its core. The local orientational mobility of dendrimers may be evaluated in terms of the spectral density ( J(o)), which is basically the cosine Fourier transform of the second order (i) orientation autocorrelation function, P2 (t). Perico and 35,36 Guenza calculated the exact expression of the second rank orientation autocorrelation function for a polymer in terms of (i) the time autocorrelation function M 1 (t), which is expressed as

p 3 2 2 x arctan x (27) ðtÞ ¼ 1 3 x 1 Pexact 2 2 p (i)

(i)

where x = [1 (M 1 (t))2]1/2/M 1 (t) and the second rank orientation autocorrelation function is denoted by Pexact for differen2 (i) tiating from P2 (t) in eqn (24). Eqn (27) may be approximated35,36 in the limit of small and large times. The value of Pexact (t) is calculated from eqn (27) for the small time 2 and large time limits. The second rank orientational auto(i) correlation function, P(i) 2 (t), may also be approximated to P1 (t) within the rigid dumbbell approximation for polymeric chains13–15,17 (i)

(i)

(i)

P2 (t)E(P1 (t))3E(M 1 (t))3

(28)

where for any model polymer with a constant segment (i) length,35,36 the time autocorrelation function, M 1 (t), is approximately equal to the first rank orientational autocorrelation (i) (i) (i) function P1 (t) given by M 1 (t) E P(i) 1 (t). The value of P2 (t)


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calculated from eqn (24) and (28) is depicted as a function of time for the compressed (f = 301) and expanded (f = 1501) conformations of G = 8 semiflexible dendrimers. Semiflexible dendrimers with the branch-point functionality fc = f = 3 are compared with the exact expression of Pexact (t) in the limits35,36 2 of small (dashed-dotted lines in Fig. 3) and large (dashed lines in Fig. 3) times. It may be observed from Fig. 3 that the plot of Pexact (t)35,36 2 displays different behavior for both compressed (red line in Fig. 3) and expanded (black line in Fig. 3) conformations of semiflexible dendrimers in the limit of small and large times. Furthermore, the curves of Pexact (t) decay to 0 in the limit of 2 large times (dashed lines in Fig. 3), while those at small times show a similar behavior (dashed-dotted lines in Fig. 3) for both conformations of semiflexible dendrimers. At large times, Pexact (t) relaxes from a finite value (i.e., limt-0 Pexact (0) = 0.6) 2 2 for both conformations of semiflexible dendrimers (dashed lines in Fig. 3), while at small times, Pexact (t) relaxes from 1. 2 From Fig. 3, it may be noticed that P(i) 2 (t) obtained from eqn (24) and (28) decays to 0 at longer times for both compressed and expanded conformations of semiflexible dendrimers, implying (i) randomness in an isotropic system.43 P2 (t) decays instanta(i) neously for all times except t = 0, where P2 (0) = 1. This indicates (i) that P2 (t) values obtained from eqn (24) and (28) are qualitatively similar to the behavior of Pexact (t) (refer to Fig. 3). 2 At small times, the numerical values for Pexact (t) for com2 pressed and expanded conformations of semiflexible dendrimers are similar to those obtained from eqn (24). For all other times, these values lie between the magnitudes of P(i) 2 (t) obtained from eqn (24) and (28) for both conformations of semiflexible dendrimers. However the numerical magnitudes of P2(i)(t) obtained from eqn (24) are larger than those calculated within the rigid approximation (i.e., eqn (28)) for the entire range of time for both compressed and expanded conformations of semiflexible dendrimers. These differences in the (i) numerical magnitudes of P2 (t) are clearly prominent in the (i) intermediate time zone, where values of P2 (t) obtained from eqn (24) are much higher than those calculated from the rigid approximation for both compressed and expanded conformations of semiflexible dendrimers. 3.2

Spectral density

The spectral density evaluated from the cosine Fourier transform of the second order orientation autocorrelation function may be used to predict various NMR-relaxation parameters. In Fig. 4, the double logarithmic plot of the reduced spectral density, J(o*), is depicted as a function of the reduced frequency, o*, for the compressed (f = 301) and expanded (f = 1501) conformations of G = 8 semiflexible dendrimers with varying branch-point functionalities fc = f = 3 and 4. In the low frequency regime, the magnitude of the spectral density is found to be independent of frequency for all types of semiflexible dendrimers irrespective of the degree of semiflexibility and the branch-point functionality as observed from Fig. 4. It may be observed from Fig. 4 that the magnitudes of J(o*) show a strong dependence on both degree of semiflexibility and

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Fig. 4 Double logarithmic plot of spectral density, J(o*), as a function of adimensional frequency, o*, for compressed and expanded conformations of G = 8 semiflexible dendrimers with branch-point functionality fc = 3, f = 3 and fc = 4, f = 4.

branch-point functionality, where the numerical magnitudes of J(o*) are larger for semiflexible dendrimers with higher branch-point functionality and degree of semiflexibility, i.e., J(o*) has maximum values for semiflexible dendrimers with f = 1501 and fc = f = 4 and minimum values for semiflexible dendrimers with f = 301 and fc = f = 3. The dependence of the spectral density on frequency is clearly evident in the high frequency regime, where J(o*) decays with frequency as (o*)2 for the compressed and expanded conformations of semiflexible dendrimers for branch-point functionalities fc = f = 3 and 4. The decay of spectral density in the high frequency regime follows a cross-over behavior with the change in the degree of semiflexibility in the intermediate frequency regime (refer to the dotted line in Fig. 4), where spectral density curves for the compressed conformations of semiflexible dendrimers are distinct and well-separated from those of the expanded ones. From Fig. 4, it may be observed that the relative numerical values of spectral density are higher in semiflexible dendrimers with branch-point functionality fc = f = 4 compared to those with fc = f = 3. Unlike the behavior in the low frequency regime, values of the spectral density are higher for compressed conformations of semiflexible dendrimers compared to those for the expanded ones, due to the cross-over in the intermediate frequency regime. This implies that the degree of semiflexibility affects the overall nature of the spectral density of semiflexible dendrimers as compared to the change in branch-point functionality. In the high frequency regime, the magnitude of the spectral density decays with the power-law of frequency as (o*)2 after displaying a cross-over behavior with the variation in the degree of semiflexibility in the intermediate frequency regime (see the dotted line in Fig. 4). From Fig. 4, it is observed that the numerical values of J(o*) for semiflexible dendrimers with branch-point functionality fc = f = 4 are higher compared to those with fc = f = 3. The J(o*) values for the compressed conformations of semiflexible dendrimers are higher than

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Fig. 5 Semi-logarithmic plot of spectral density, J(o*), as a function of adimensional frequency, o*, for G = 8 semiflexible dendrimers with branch-point functionality fc = 3, f = 3 for the entire range of bond orientation angle, f (0 o f o p).

those for the expanded ones in the high frequency region, which implies that the crossover in the intermediate frequency region depends only on the degree of semiflexibility of dendrimers. The behavior of the spectral density as a function of frequency is depicted in Fig. 5 with varying degree of semiflexibility of G = 8 dendrimers with branch-point functionality fc = f = 3. From Fig. 5, it may be noted that the spectral density exhibits a similar trend for the entire range of frequency for all f. The spectral density exhibits a cross-over behavior in the intermediate frequency region for the compressed conformations of semiflexible dendrimers, i.e., 0 o f o p/2, in contrast to the expanded ones p/2 o f o p. It is also observed that the characteristic area under the curve strongly depends on the degree of semiflexibility, which is an important feature, as this area is independent of the correlation time.44 From Fig. 5, it may be noted that semiflexible dendrimers in the expanded conformation zone have the maximum characteristic area as compared to those in the compressed conformation zone and this characteristic area increases with increasing f. For the compressed conformations of semiflexible dendrimers, this characteristic area decreases with f, where a cross-over of spectral density occurs with varying degree of semiflexibility in the intermediate frequency region. But the relative change in the characteristic area is considerably lower than that of the expanded conformations of semiflexible dendrimers. Flexible dendrimers (f = p/2) have the smallest characteristic area compared to both compressed and expanded conformations of semiflexible dendrimers as shown in Fig. 5.

bond vectors. The behavior of the time autocorrelation function, (i) M 1 (t), and that of the second order orientational autocorrelation (i) function, P 2 (t), are analyzed as a function of the branch-point functionality, fc and f, and the degree of semiflexibility, f. Experimentally relevant quantities like the spectral density, J(o), are also evaluated from this study. (i) M 1 (t) is governed by two relaxation processes: the fast internal ‘‘pulsation’’ motion of the monomers at initial times and the ‘‘global’’ rotation of the dendrimer as a whole at longer times. The global motion of the dendrimer exhibits a strong dependence on both branch-point functionality and the degree of semiflexibility, while the pulsation motion varies only with the change in the degree of semiflexibility. Compressed conformations display the maximum rate of pulsation motion, while the minimum rate is observed for the expanded conformations of semiflexible dendrimers. But it is difficult to compare the difference in the extent of pulsation motion among the compressed (0 o f o p/2) and expanded (p/2 o f o p) conformations of semiflexible dendrimers. The global motions of the compressed and expanded conformations of semiflexible (i) dendrimers are widely different from each other. But M 1 (t) values of dendrimers with different branch-point functional(i) ities closely resemble each other. The behavior of P 2 (t) obtained from our calculations matches with the exact expression, Pexact (t), of earlier studies by Perico and Guenza.35,36 2 The dependence of the spectral density on frequency is clearly evident in the high frequency regime, where J(o*) decays with frequency as (o*)2 for the compressed and expanded conformations of semiflexible dendrimers with functionalities fc = f = 3 and 4. This decay of the spectral density occurs after displaying a cross-over behavior with the variation in the degree of semiflexibility in the intermediate frequency regime. The characteristic area increases with increasing f, and the expanded conformations of the semiflexible dendrimers (p/2 o f o p) record the maximum characteristic area. The relative increment in the characteristic area for the compressed conformations of semiflexible dendrimers is lower than that for the expanded ones. This theoretical approach of the orientational relaxation dynamics of semiflexible dendrimers provides a comprehensive understanding of the local and global dynamics of dendrimers, which may be useful for the analysis of experimentally relevant parameters for complex topologies.

Acknowledgements The authors gratefully acknowledge the University of Delhi for the financial support. A. Kumar acknowledges the University Grant Commission, India, for providing Senior Research Fellowship.

4 Conclusions This work predicts the orientational relaxation dynamics of semiflexible dendrimers in the theoretical framework of the optimized Rouse–Zimm model with preaveraged hydrodynamic interactions. Semiflexibility is incorporated through appropriate restrictions in the direction and orientation of the respective

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NMR relaxation of the orientation of single segments in semiflexible dendrimers.

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