Chain dynamics in a chiral C phase by deuteron spin relaxation study

ARTICLE IN PRESS

Solid State Nuclear Magnetic Resonance 28 (2005) 173–179 www.elsevier.com/locate/ssnmr

Chain dynamics in a chiral C phase by deuteron spin relaxation study Ronald Y. Donga,b,, J. Zhanga, C.A. Veracinic a

Department of Physics and Astronomy, University of Manitoba, Winnipeg, Man., Canada R3T 2N2 b Department of Physics and Astronomy, Brandon University, Brandon, Man., Canada R7A 6A9 c Dipartimento di Chimica e Chimica Industriale, Universita degli Studi di Pisa, via Risorgimento 35, 56126 Pisa, Italy Received 18 May 2005; received in revised form 12 July 2005 Available online 15 August 2005

Abstract Molecular reorientations and internal conformational transitions of an aligned chiral liquid crystal (LC) 10B1M7 are studied by means of deuterium spin-lattice relaxation in its smectic A (SmA) and smectic C* (SmC*) phase. The motional model which is applicable to uniaxial phases of many LCs is found to be adequate even when the phase is a tilted SmC* phase. The deuterium NMR spectrum in this phase cannot discern rotations of the molecular director about the pitch axis. The basic assumption is that the phase biaxiality is practically unobservable. However, the relaxation rates can be accounted for by the tilt angle between the molecular director and the layer normal in the SmC* phase. The tumbling motion appears to show a higher activation energy upon entering from the uniaxial SmA into the SmC* phase. r 2005 Elsevier Inc. All rights reserved. Keywords: Spin relaxation; Deuteron; Chiral phase; Liquid crystal

1. Introduction Chiral liquid crystals (LCs) have attracted much attention [1–6] because their ferroelectric, antiferroelectric and/or ferrielectric properties shown in tilted smectic phases can lead to potential technical applications. One consequence of molecular chirality from an asymmetric carbon site(s) is the occurrence of polar order, giving electric polarization in the layer due to the alignment of molecular lateral electric dipoles. The inlayer polarization can change synclinically (chiral SmC*) or anticlinically (SmCA*) as the molecules form a helical superstructure with its axis perpendicular to the layer planes. Among the different approaches to study molecular dynamics of LCs, deuterium spin relaxation measurements have been exploited successfully in achiral LC compounds [7–9]. In particular, the correCorresponding author. Department of Physics and Astronomy, Brandon University, 270 Eighteenth Street, Brandon, Man., Canada R7A 6A9. Fax: +1 204 728 7346. E-mail address: [email protected] (R.Y. Dong).

0926-2040/$ - see front matter r 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.ssnmr.2005.07.007

lated internal rotations within a flexible end chain(s) can best be studied by 2H NMR because of the site specificity of deuterium doublets observed in an aligned sample. However, specifically deuterated LC can often be expensive and time consuming to produce. 13C NMR spectrum also has the advantage of site specificity to some extent, but quantitative interpretation of relaxation rates can often be more problematic. Experimentally, site specificity is sometimes not easily achieved in chiral phases due to broad doublets observed in deuterium spectra [10–12] or broad aromatic carbon peaks [13–16] when dealing with ferroelectric phases. The reason for such line broadening effects is still an open question. While motional models for the interpretation of spin relaxation in uniaxial [17–19] and biaxial phases [20,21] are available, the biaxial phase studied theoretically thus far has a D2h symmetry such as observed in biaxial nematic. Thus, a suitable spin relaxation theory for chiral (C2v) phases is still lacking. As a consequence, quantitative analyses of spectral densities in chiral phases have been rather limited.

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Recently, a new approach based on some simplifying assumptions has been proposed to interpret the core dynamics in the SmC* phase of 8BEF5 [22]. Analyses of several sites in the same molecule have also been successful [23]. The present work aims to test the application of this approach on a deuterated achiral chain of 1-methylheptyl 40 -(4-n-decyloxy-benzoyloxy) biphenyl-4-carboxylate (10B1M7). This compound has been studied by us in the SmA phase at 15.1 and 46 MHz [11], and the decoupled model [24,25] was used successfully to gain motional parameters such as spinning (D:) and tumbling (D?) diffusion constants of the molecule, as well as internal jump rates. The present investigation extends the study to a new frequency (61.4 MHz) by measuring the deuterium spin-lattice relaxation in the SmA and SmC* phases of 10B1M7. For completeness and direct comparison with findings in the SmC* phase, we have reanalyzed the data in the SmA phase by using all three different Larmor frequencies. The new analysis should provide more reliable motional information for 10B1M7 in the SmA phase, but the aim of the present study is to carry out an analysis of the SmC* phase data. The paper is organized as follows. Section 2 summarizes the decoupled model to give the spectral densities of chain deuterons and its application in the chiral SmC* phase. Section 3 outlines the experimental method, and the last section gives the results and discussion.

2. Theory

motion has been given in the literature [24,25]. Four phenomenological jump constants k1, k2, kg and k0 g, which describe configurational transitions among different conformations are used to construct the transition rate matrix R. Suppose the carbon–carbon backbone of
a pentyl chain is in a configuration given by {ijklm}. The internal energy for a configuration is governed by the number of gauche linkages (Ng) in the chain, and Etg, the energy difference between a gauche linkage and the trans (t). The one-bond (k1) motion is given by {ijklm}-{ijklm0 }, while the two-bond motion is for {ijklm}-{ijkl0 m} and {ijkl0 m0 }. These two motions are called type-III motion by Helfand [28], while type-II motion consists of kink formation ðt t t t ! g t g tÞ described by a jump rate kg0 and gauche migration ðt t t g ! t g t tÞ described by kg. Here Etg used to fit the quadrupolar splittings, and the derived orientational order parameters of an average ‘‘conformer’’ in the SmA phase are those given previously [11]. Using the notation of Tarroni and Zannoni to describe the overall motion, the spectral densities of the Ci deuterons on the ~ field can be chain when the director n is along the B written as [7] 0 J ðiÞ m ðmo; 0 Þ ¼

N N X X 3p2 ðiÞ 2 X X X 2 ðqCD Þ d 2n0 ðyðiÞl N;Q Þd np ðyÞ 2 n n0 k¼1 l¼1 p ! h i ðiÞ;l ð1Þ ðkÞ exp iðnjN;Q Þ xl xl



N X X

0

2 d 2n0 0 ðyðiÞl N;Q Þd n0 p0 ðyÞ

l 0 ¼1 p0

It is well-known [7] that molecules with flexible side chain(s) can exist in many different conformational states, and on the NMR time scale the observed deuteron doublets also reflect a time average over these different conformations in addition to other motional processes. It is noted that these conformations are aligned differently by the nematic mean field [26]. The number (N) of possible conformations in the decyloxy chain generated by Flory’s rotameric model [27] is overwhelming, and we have followed our previous treatment by keeping only 683 conformational states, which possess relatively high probabilities of occurrence [11]. Furthermore, to solve the overall rotational dynamics of the molecule with many possible conformations it is assumed that an ‘average’ conformer exists, which has a particular rotational diffusion tensor. In principle, different conformations not only have their own order matrices, but also have their own rotation diffusion tensors [7]. The assumption is necessary to make the solution of the rotational diffusion equation tractable, and is a good approximation if the principal values of these diffusion tensors do not deviate much from each other. The decoupled model based on a master rate equation to describe the correlated internal

h i 0 ð1Þ ðkÞ exp iðn0 jðiÞ;l N;Q Þ xl 0 xl 0

!

h i 2 X ðbmnn0 Þj ða2mnn0 Þj þ jlk j  h i2 , j m2 o2 þ ða2 0 Þ þ jlk j mnn j

ð1Þ

where qðiÞ CD ¼ 165 kHz is the quadrupolar coupling ðiÞl constant, yðiÞl N;Q and jN;Q are the polar angles for the Ci–D bond of the conformer l in the molecular N frame fixed on the phenyl ring core, y is the angle between the ZN-axis (or para axis) and the long molecular axis (ZM) of 10B1M7, lk and ~ xðkÞ are the (negative) eigenvalues and eigenvectors from diagonalizing the R matrix, and
ða2mnn0 Þj =D? , the decay constants, and ðb2mnn0 Þj , the relative weights of the exponentials in the autocorrelation functions, are the eigenvalues and eigenvectors from diagonalizing the matrix of the rotational diffusion operator. A y value of 81 was used [11]. The rotational diffusion constants D: and D?, which appear in ða2mnn0 Þj , are for diffusive rotations of the molecule about its long axis, and about one of its short axes, respectively. The above equation, strictly speaking, is valid for a biaxial probe reorienting in a uniaxial environment. In the

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SmC* phase, molecules tilt uniformly in all layers, but the tilt rotates azimuthally when going from one layer to the next one to form a helicoidal structure. The observed quadrupolar doublet is found to be independent of the azimuthal angle on the NMR time scale. Furthermore, recent deuterium studies of another smectogen in its SmC* phase [22,23] appear to support the idea that deuterium spins relax in a macroscopically uniaxial medium, thereby unable to sense any possible phase biaxiality. The new approach also suggests that the observed spectral densities in the SmC* phase can be understood by taking into account the tilt angle b of the ~ field, molecular ‘‘director’’ with respect to the external B viz, one may write the spectral densities J m ðmo; bÞ as follows [29]: 

J 1 ðo; bÞ ¼ 32 cos2 b sin2 b J 0 o; 00

 þ 12 1  3cos2 b þ 4cos4 b J 1 o; 00
 þ 12ð1  cos4 bÞJ 2 o; 00 , ð2Þ

O O C10D21O

C

C

O O

5,6 9 87

4e+04

2e+04

C* C6H13 CH3

10

0e+00

3

4 1,2

-2e+04

-4e+04

ν (Hz) Fig. 1. A typical deuteron NMR spectrum of 10B1M7-d21 in its SmC* phase and its schematic structure. The peak assignment is same as given before [11].

80


 J 2 ð2o; bÞ ¼ 38ð1  cos2 bÞ2 J 2 2o; 00
 þ 12ð1  4 cos4 bÞJ 1 2o; 00

70

60

ð3Þ

in which the azimuthal angle f, describing the rotation of tilt in the helical structure, has been set to zero, and all biaxial spectral densities [21] have been ignored. It is believed that the lack of f dependence in quadrupolar doublets and spin relaxation rates is the basis of using Eqs. (1)–(3) in the SmC* phase of 10B1M7. The fitting of deuterium splittings in the SmC* phase was carried out as described for the SmA phase [11].

50 ∆ν (KHz)


 þ 18ð1 þ 6 cos2 b þ cos4 bÞJ 2 2o; 00

175

40

30

20

3. Experimental method 10

The chiral LC 10B1M7 has various mesophases at different ranges of temperature. The phase transition temperatures were reported before [6] for 10B1M7. The transition temperatures for our partially deuterated 10B1M7-d21 are about 51 lower, i.e. Isotropic 2 SmA 2 393K

373K

SmC 2 SmCferri 2 AFLC: 345K

341K

All deuterium spectra were carried out at 61.4 MHz on a Bruker AVANCE spectrometer. Temperature of the sample was controlled with a Bruker variabletemperature unit. Experiments were always performed after the sample was slowly cooled from the isotropic phase. Enough waiting time between each temperature change was taken to assure temperature stabilization and equilibration. Fig. 1 shows a typical deuterium spectrum in the SmC* phase of 10B1M7 and its schematic chemical structure. The NMR probe was

0

320

340

360

380

T(K) Fig. 2. Plot of quadrupolar splittings of 10B1M7-d21 versus the temperature. Solid downtriangles, diamonds denote C5,6 and C7 sites. Open uptriangles, circles, pluses, squares and crosses, stars denote C1,2, C3, C4, C8, C9 and C10, respectively. Solid curves are calculated splittings, starting from the top, C1,2, C3, C4, C5,6, C7, C8 and C9 sites.

always re-tuned for every temperature change to avoid slight de-tuning of the probe. The temperature gradient across the sample in the NMR probe was estimated to be better than 0.3 1C. The observed deuterium splittings are shown as a function of the temperature in Fig. 2. The Zeeman (T1Z) and quadrupolar (T1Q) spin-lattice relaxation times were measured simultaneously using a broadband Jeener–Broekaert pulse sequence [30] modified to subtract the equilibrium magnetization. The 2H

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176

901-pulse length was 2.6 ms, and relaxation delay was between 0.3 and 1.5 s depending on the T1 values. In every experiment, 15 partially relaxed spectra were collected. T1Z and T1Q were derived from the sum, SðtÞ, and difference, DðtÞ, of component areas in the doublet:

20

(a)

SmA

SmA

16

SmC* 12

SðtÞ / expðt=T 1Z Þ, DðtÞ / expðt=T 1Q Þ.

8

T 1 1Z ¼ J 1 ðo0 Þ þ 4J 2 ð2o0 Þ, T 1 1Q ¼ 3J 1 ðo0 Þ, where o0 =2p is the Larmor frequency. The spectral densities given by the above equations in the SmC* phase are in fact J m ðmo; bÞ. During the analysis of the relaxation data, especially in the SmC* phase, we noticed that the peaks for C1,2, C3, and C4 became progressively more unresolved upon decreasing temperature. This could lead to unacceptable errors when calculating areas to extract the T1Z and T1Q value. To ascertain the relaxation measurements for these sites, the peak deconvolution method based on the Bruker software has been employed to isolate C1,2, C3, and C4 peaks where possible in order to obtain their areas.

4

Spectral Density (S-1)

As noted in our previous work [11], the deuteron peaks for C1 and C2, as well as those of C5 and C6 do overlap to make them indistinguishable. The T1 value for the overlapped sites was measured as a single value. The spectral densities of motion are derived using the following equations:

0 14

(b)

SmA

SmA

12

SmC* 10

8

6

4

2

0

360

370

380

390

380

390

T (K)

4. Results and discussion Figs. 3 and 4 show the deuterium spectral densities J1(o) and J2(2o) versus the temperature at 61.4 MHz for deuterons of carbon sites 1 to 9 in 10B1M7, together with those reproduced from our previous study at 15.1 and 46 MHz in its SmA phase. The details for fitting the spectral densities in the SmA phase using Eq. (1) have been reported elsewhere, [11] and the global analysis of this phase has added an extra frequency and used nine temperatures. Arrhenius-type relations were used for the model parameters in Eq. (1), viz ? D? ¼ D0? exp½E D a =RT,

D

Djj ¼ D0jj exp½E a jj =RT, ki ¼ k0i exp½E ka i =RT , k

kg ¼ k0g exp½E a g =RT, 0

k0

k0g ¼ k0 g exp½E a g =RT,

(4)

Fig. 3. Plots of spectral densities versus the temperature in the SmA and SmC* phase of 10B1M7-d21 at 15.1 and 61.4 MHz (left panel) and 46 MHz (right panel). Closed and open symbols denote J1(o) and J2(2o), respectively. (a) uptriangles (15.1), diamonds (46) and squares (61.4) are for C1,2, while downtriangles (15,1), triangles (46), and circles (46) are for C9 deuterons, respectively, (b) uptriangles (15.1), diamonds (46) and squares (61.4) are for C3, while downtriangles (15,1), triangles (46), and circles (61.4) are for C8 deuterons, respectively. Solid and dashed lines denote calculated spectral densities for J1(o) and J2(2o), respectively.

where ki ¼ k1 or k2. The new fittings have produced global target parameters, which are summarized in Table 1 together with their error limits. The fitting quality factor Q is 3.7% after minimizing the meansquare percent deviation (F) with the routine AMEOBA [31]. Q is a bit larger than that obtained previously for just two frequencies. The calculated spectral densities are shown in Figs. 3–4 by solid (J1) and dashed (J2) curves. As seen in these figures, the fittings of two lower frequencies have not changed in a noticeable way after including an additional frequency in the fitting. Again, some systematic deviations between calculated and experimental spectral densities do exist for some sites

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(e.g. C1,2), and we believe that these discrepancies likely stem from many simplifying assumptions used in our decoupled model for a long chain molecule. The error limit for a particular global parameter was estimated by

varying the one under consideration while keeping all other global parameters identical to those for the optimized F value, to give a doubling in F. Some parameters are found not too sensitive in the fitting. Hence numbers in parentheses are for an increase of 50% in F. The jump constants and rotational diffusion coefficients in the SmA phase are plotted versus the reciprocal temperature in Fig. 5. These parameters seem to agree more or less with those values reported previously [11], although some variations in their temperature behaviors are seen. To fit the spectral densities at 61.4 MHz in the SmC* phase, we first fit the quadrupolar splittings in this phase using the additive potential (AP) method [32] in terms of

14 (a)

SmA

SmA

12 SmC*

10

177

8 6

10

17

10

10

2 0 10

10 (b)

SmA

14

SmA

8

10

9

D (s-1)

10 SmC*

k (s-1)

Spectral Density (s-1)

4

13

6 10

4

10

8

12

2

0 350

10

360

370

380

390

380

390

11

10 2.6

2.7

T(K)

7

2.8 1000/T

2.6

2.7

2.8

(K-1)

Fig. 5. (a) Plots of jump rate constants k1 (solid lines), k2 (dashed lines), kg (dotted lines) and k0g (dash-dotted lines) versus the reciprocal temperature, (b) plots of rotational diffusion constants D: (solid lines) and D? (dashed lines) versus the reciprocal temperature.

Fig. 4. Same as Fig. 3 with (a) uptriangles (15.1), diamonds (46) and squares (61.4) are for C4, while downtriangles (15.1), triangles (46), and circles (61.4) are for C7 deuterons, respectively, (b) uptriangles (15.1), circles (46) and squares (61.4) are for C5,6.

Table 1 Global parameters and their error limits in s1 for the SmA phase k01 Value Upper limit Lower limit

Value Upper limit Lower limit

6.98  1014 (9.54  1015) 1.62  1014

E ka 1 12.1 16.8 (3.85)

k02

E ka 2

k0g

E ag

4.28  1024 NA 2.47  1023

75.9 84.9 51.0

2.17  1018 7.11  1018 1.01  1018

35.5 37.9 31.7

k

k0g 0

E ag

D0?

? ED a

D0:

ED a

5.12  1019 2.38  1021 (2.06  1018)

53.2 (63.4) 41.0

2.63  109 5.45  109 1.36  109

13.6 15.7 11.3

3.77  1015 7.83  1015 2.26  1015

41.9 43.5 39.6

k 0

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30

0.8

0.7

25

0.6

Tilt angle (°)

Order Parameters

20

0.5

0.4

15

0.3 10

0.2

0.1

5

0.0 350

360

370

380

0 350

390

360

370

T(K) Fig. 6. (a) Plots of Szz (solid) and Sxx–Syy (open) versus the temperature in the SmA and SmC* phase of 10B1M7, (b) Plot of the tilt angle versus the temperature in the SmC* phase of 10B1M7.

12 0.14

10 0.12

8

0.10

0.08 6

λ

two adjustable parameters Xa and Xc, the interaction parameters of the ‘core’ and C–C bond, respectively. We define l ¼ X c =X a , which is varied with Xa at each temperature to give the theoretical splitting curves shown in Fig. 2. Note that the order matrix of each conformer has been obtained by the AP method, and the fittings in this phase are similar to those seen in the SmA phase [11]. The derived order parameters Szz and Sxx–Syy for the ‘average’ conformer are shown in Fig. 6, while Xa and l are plotted versus the temperature in Fig. 7. Szz and Xa are seen to decrease with decreasing temperature due to the onset of tilt angle in the SmC* phase. The average tilt angle (shown also in Fig. 6) used in Eqs. (2) and (3) is derived from several quadrupolar splittings of the chain in the SmC* phase when comparing them with their extrapolated values from the SmA phase[12]. Now Szz and Sxx–Syy are used to give the potential of mean torque in solving the rotational diffusion equation. In the SmC* phase, Szz values are those extrapolated from the SmA phase [22]. Xa and l are needed to evaluate the Peq(n), the equilibrium probability of the nth conformer. The Xa values used in the SmC* phase are also extrapolated from the SmA phase, while l as well as Sxx–Syy are those given in Figs. 6 and 7. A global target analysis of ten temperatures at 61.4 MHz is used to minimize F, yielding a Q value of 0.9% in this phase. The smaller Q is a result of fitting a single frequency. The calculated spectral densities based on Eqs. (1)–(3) are shown as

Xa (KJ / mol)

178

0.06 4 0.04

2 0.02

0 350 360 370 380 390

0.00 350 360 370 380 390

T(K)

Fig. 7. Plots of interaction parameters Xa and l ( ¼ Xc/Xa) versus the temperature.

solid and dashed curves in Figs. 3 and 4. The fittings in this phase appear to be reasonable, and the global target parameters are summarized in Table 2 together with their error limits. We note that several global parameters had very weak temperature behaviors, and their activation energies (i.e. Eka1, Eka2 and E D a ) were, therefore, set at zero. The number of variables for fittings in the SmC* phase was reduced to 8, as k0g was also fixed at 1.44  1037 s1. The uncertainty in this value was estimated by increasing it by a factor of 3, resulting in a doubling of F. The internal jump rates, and D: as well as D? in the SmC* phase are also plotted in Fig. 5. At the SmA–SmC* transition, all motional parameters appear to jump discontinuously. Some parameters, such as D? and kg, have much higher activation energies. The higher activation energy for tumbling motion in the SmC* phase of 10B1M7 is found to agree with the finding in the same phase of 8BEF5 [22]. This is an encouraging fact. Also, the gauche migration appears to be more hindered in this phase. The present study, therefore, suggests that the simple approach in extracting dynamical information in the chiral C phase is appropriate for the core and the chain deuterons, at least in two chiral mesogens. In summary, we believe that the usage of Eqs. (2) and (3) (albeit with many assumptions) to account for the spectral densities in the chiral C phase seems reasonable

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Table 2 Global parameters and their error limits in s1 for the SmC* phase

Value Upper limit Lower limit

k01

k02

E ag

k0g 0

E ag

D0?

? ED a

D0:

9.94  1012 5.58  1013 3.37  1012

1.02  1017 NA 4.02  1013

175.5 (181.8) 172.0

7.27  1017 1.51  1018 3.70  1017

30.6 32.6 28.4

2.46  1014 3.58  1014 1.62  1014

45.6 46.9 44.5

5.70  109 7.70  109 4.43  109

k

and biaxial spectral densities may not be an important factor in this phase as far as chain deuterons are concerned. In other words, we have demonstrated that molecules in the helical structure of a tilted phase appear to sense a macroscopically uniaxial phase symmetry to a first approximation. However, more NMR studies of this kind are needed to further confirm its generality.

Acknowledgments R.Y.D. would like to thank the NSERC of Canada for financial support, Canada Foundation of Innovation and Brandon University for the NMR facility. C.A.V. acknowledges financial support of Italian MIUR. References [1] M. Cˇepicˇ, B. Zˇeksˇ , Phys. Rev. Lett. 87 (2001) 085501. [2] A.D.L. Chandani, T. Hagiwara, Y. Suzuki, Y. Ouchi, H. Takezoe, A. Fukuda, Ferroelectrics 85 (1988) 99. [3] A. Dahlgren, M. Buivydas, F. Gouda, L. Komitov, M. Matuszczyk, St. Lagerwall, Liq. Cryst. 25 (1998) 553. [4] B. Jin, Z. Ling, Y. Takanishi, K. Ishikawa, H. Takezoe, A. Fukuda, M. Nakimoto, T. Kitazume, Phys. Rev. E 53 (1996) R4295. [5] K. Yoshino, Y. Fuwa, K. Nakayama, S. Uto, H. Moritake, M. Ozaki, Ferroelectrics 197 (1997) 1. [6] J.W. Goodby, J.S. Patel, E.J. Chin, J. Mater. Chem. 2 (1992) 197. [7] R.Y. Dong, Nuclear Magnetic Resonance of Liquid Crystals, Springer, New York, 1997. [8] E.E. Burnell, C. de Lange (Eds.), NMR of Orientationally Ordered Liquids, Kluwer Academic, Dordrecht, 2003. [9] R.Y. Dong, Ann. Rep. NMR Spectrosc. 53 (2004) 68. [10] D. Catalano, M. Cavazza, L. Chiezzi, M. Geppi, C.A. Veracini, Liq. Cryst. 27 (2000) 621.

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