7920
J. Phys. Chem. B 2001, 105, 7920-7927
31P
and 13C NMR Studies of a Liquid-Crystalline Cyclotriphosphazene Derivative: Orientational Characteristics and Contrasting Shielding Anisotropies for Inorganic and Organic Moieties Keiichi Moriya,† Toshiya Suzuki,† Shinichi Yano,† and Seiichi Miyajima*,‡,§ Department of Chemistry, Faculty of Engineering, Gifu UniVersity, Yanagido, Gifu 501-1193, Japan, and Institute for Molecular Science, Myodaiji, Okazaki 444-8585, Japan ReceiVed: NoVember 28, 2000; In Final Form: May 14, 2001
31P
and 13C NMR studies are reported for a smectic A liquid crystal of haxakis[4-dodecyl(4-biphenoxy)]cyclotriphosphazene. In an intense magnetic field, the long axes of the biphenylene fragments and the alkyl chains align along the magnetic field, and the cyclotriphosphazene (ctp) ring-normal points preferentially in the same direction. Orientational order parameters for the inorganic moiety (ctp fragment) and that for the organic moiety (biphenylene fragment) are determined independently from the 31P and 13C NMR spectroscopies, respectively. The two order parameters are revealed to take the same value, 0.45. The value is smaller than that for the conventional smectic A liquid crystals consisting of small organic molecules. Contrasting chemical shift anisotropies are revealed for the motionally averaged fragments: the least-shielding axis for the 31P nucleus lies on the ctp ring normal, while that for the aromatic 13C nucleus lies along the line connecting the two phenylene rings.
I. Introduction Polyphosphazenes have attracted attention as novel inorganic polymers having various functional properties.1,2 For example, the [-P(RR′)dN-]n bond-alternating inorganic linear chains with conjugated organic substituents, R or R′, with flexible chains exhibit unusual phase behaviors and offer model systems for side-chain effects on the polymer properties. One of the recent topics in phosphazene chemistry and physics is the discovery of a liquid-crystalline phase in cyclic trimers,3-5 and most recently the discovery of ferroelectric liquid crystals.6 In these compounds, which are called cyclotriphosphazene (ctp) derivatives (Figure 1), the organic aromatic groups and flexible alkyl chains were incorporated at the six substituent positions, R, of [-P(R2)dN-]3. Possible novel aspects of liquid-crystalline properties caused by the inorganic central skeleton are of considerable interest since most of the known liquid-crystalline compounds are organic ones, except for the growing group of metallomesogens.7 In this paper fundamental liquid-crystalline properties of a ctp derivative, hexakis[4-dodecyl(4-biphenoxy)]cyclotriphosphazene are analyzed by means of 31P and 13C NMR spectroscopies. The substituent R is -O-C6H4-C6H4-C12H25 for the present compound (Figure 1), and two liquid-crystalline phases, smectic A (SA) and smectic C (SC), have been reported with a transition sequence, crystal-406 K-SC-424 K-SA426 K-I.5 (The phase I denotes the isotropic liquid-phase hereafter.) The molecule contains twelve phenylene rings and six dodecyl chains, and the molecular weight amounts to 2157. It is of interest to know if the liquid-crystalline states consisting of such bulky molecules are different from the conventional liquid crystals of small organic molecules. This paper is devoted †
Gifu University. Institute for Molecular Science. § Present address: Miyajima Shoyu Co., Ltd., 2318 Funamiya, Karatsu 847-0062 Japan. ‡
Figure 1. Structural formula of cyclotriphosphazene (ctp) derivatives. The substituent R is -O-C6H4-C6H4-C12H25 for the present compound. The numbering scheme for the aromatic carbons is indicated.
to the analyses of fundamental physical properties such as basic molecular structure in the liquid-crystalline state, the orientational order parameters, and the chemical shift anisotropies, which are revealed from the NMR spectra in the SA phase. The orientational order parameters of the inorganic and organic moieties are deduced from each of the of 31P and 13C NMR experiments. Contrasting chemical shift anisotropies are revealed for inorganic and organic six-membered rings. Advanced analysis of the phase biaxiality in the SC phase will be reported in a subsequent paper. II. Experimental Section Proton-decoupled 31P and 13C NMR experiments were conducted with a JEOL GSX-270 spectrometer equipped with solid-state accessories working at 109, 68, and 270 MHz for 31P, 13C, and 1H nuclides, respectively. Experiments were made in a field-aligned condition in the liquid-crystalline state. Chemical shifts were measured relative to 85% H3PO4 aqueous solution for 31P, and to liquid tetramethylsilane for 13C, with positive values toward less shielding side.
10.1021/jp004299j CCC: $20.00 © 2001 American Chemical Society Published on Web 07/26/2001
NMR Studies of a LC Cyclotriphosphazene Derivative
J. Phys. Chem. B, Vol. 105, No. 33, 2001 7921 TABLE 1: Observed 13C Chemical Shifts Relative to TMS (Tetramethylsilane) phase (temp)
intensities
assignment
δ (ppm)
I (431 K)
ca. 6 2 2 2 2 2 1 1 ca. 6 4 4 2 1 0.5 0.5
inter CH2a ar CHb ar CH ar CH ar CH ar CCc ar CC ar CC (j ) 1) inter CH2 ar CH ar CH ar CC ar CC ar CC (j ) 1) ar CC (j ) 1)
30.64 122.53 127.58 128.45 129.42 138.55 142.05 151.18 28.00 141.21 145.09 177.68 182.62 187.5 195.2
SA (425 K)
Figure 2. 13C NMR spectra for a stationary sample. Spectrum (a) was recorded at 431 K in the I phase, and (b), at 425 K in the SA phase. The chemical shifts are measured relative to tetramethylsilane.
An angle-resolved 31P NMR experiment was made by using a laboratory-made spectrometer equipped with an iron magnet of 1.3 T (22 MHz for the 31P Larmor frequency), a goniometer, and a solenoidal 5 mm o.d. NMR coil. The sample tube was rotated about the axis perpendicular to the static field, and the angle dependence of the spectrum was recorded. In this experiment, the liquid crystal is located at an angle that is different from the initial stable state. Therefore, the experiment must be finished before the relaxation of the alignment takes place. The experiment at the magic-angle and at 90° orientations were repeated at the end of the experiment, to confirm that the relaxation did not take place during this experiment time. The total angular dependence experiment, including the confirmation, took 6 h. III. Results and Discussion III.1. Alignment-Induced Shifts in the 13C NMR Spectra. Figure 2 shows the 13C NMR spectra in the I phase (trace a) and in the SA phase (trace b) taken in a stationary condition and on cooling from the I phase. If the liquid-crystalline directors are randomly distributed in the external magnetic field, then the spectrum exhibits a powder pattern because the chemical shift interaction depends on the orientation of the principal axes of the interaction tensor relative to the direction of the external field, which is denoted by zLAB (the z-axis of the laboratory reference frame). The narrow individual peaks as shown in Figure 2 mean that the molecules are rotating rapidly and align in the magnetic field. The observed spectral data are summarized in Table 1. The individual peaks change their positions on entering the SA phase from the I phase. These alignment-induced shifts (AIS) are summarized in Table 2. The lines for the aliphatic carbons, found between 10 and 40 ppm in the I phase, exhibit diamagnetic AIS. The overlapped interior methylene peaks found at 30.64 ppm in I shift to 28.00 ppm in SA, exhibiting an AIS of -2.64 ppm. On the other hand, the lines assigned to the aromatic carbons, 120-160 ppm in the I phase, exhibit paramagnetic AIS. The four lines at 122.53, 127.58, 128.45, and 129.42 ppm with double intensity are assigned to the protonated aromatic carbons, C(2), C(3), C(6), and C(7), according to the numbering scheme shown in Figure 1. These lines appear at 141.21 and 145.09 ppm with quadruple intensity in SA because of overlap. The average AIS for the four lines is 16.16 ppm. The lines at 138.55 (double intensity) and 142.05 ppm in I are assigned to the aromatic nonprotonated carbons, C(4), C(5), and C(8). They appear at 177.68 (double intensity) and 182.62 ppm in SA. The
a Interior methylene carbons. b Aromatic protonated carbons. c Aromatic nonprotonated carbons.
TABLE 2: Experimentally Obtained 13C Average Alignment-Induced Shifts, AISexp, for Each Group of the Carbon Atoms and the Deduced Fragmental Orientational Order Parameters, Sf00 kinds of carbon atoms inter CH2a ar CHb (j ) 2, 3, 6, 7) ar CCc (j ) 4, 5, 8) ar CC (j ) 1)
average δ in average δ AISexp SA (ppm) in I (ppm) (ppm) 28.00 143.15 179.33 191.35
30.64 126.99 139.72 151.18
Sf00
-2.64 Smethylene ) 0.13 00 16.16 Sbiph ) 0.44 00 39.62 Sbiph ) 0.46 00 40.17
a Interior methylene carbons. b Aromatic protonated carbons. c Aromatic nonprotonated carbons.
average AIS is 39.62 ppm. The line at 151.18 ppm in I is for C(1), and it appears at 187.5 and 195.2 ppm as a 13C-31P dipolar doublet. The AIS is 40.17 ppm. Qualitatively, the features of AIS for both the aliphatic and the aromatic carbons are similar to the features found for conventional small-molecular weight organic liquid crystals of rodlike shapes.8 The diamagnetic AIS for the aliphatic carbons are caused by the alignment of the most-shielding axis (which is along the chain extension axis) along zLAB, and the paramagnetic AIS for the aromatic carbons are caused by the alignment of the less-shielding axis (long axis of the biphenylene core) along zLAB. The most-shielding axis for the aromatic carbons (the ring normal) points, preferentially, perpendicular to zLAB in the liquid-crystalline phases in a magnetic field, and a rapid rotation about the long-axis (namely, the axis along the chemical bond that connects the two rings) sharpens the spectral lines. Thus, the orientational characteristics of the present novel ctp liquid crystal show qualitatively similar features to the organic small-molecular-weight liquid crystals, as far as the organic moiety is concerned. III.2. 31P Dipolar Triplet Spectrum. Figure 3 shows the 31P NMR spectra in the I phase (trace a) and in the S phase A (trace b). A homonuclear dipolar triplet structure was observed in the SA phase. The spectral data are summarized in Table 3. Andrew and Bersohn showed in their classical work9 that rotation about the C3 axis simplifies the spectrum of a spin system consisting of three I ) 1/2 spins arranged on a regular triangle and gives a symmetric triplet structure with 1:2:1 intensities. The symmetric triplet is actually observed in the present compound. This fact means that the ctp ring is rotating rapidly about its C3 axis. The only difference from the case of Andrew and Bersohn is that, in the present case, the dipolar coupling is reduced further by the fluctuation of the C3 axis
7922 J. Phys. Chem. B, Vol. 105, No. 33, 2001
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Figure 4. Possible models for the alignment of a ctp director, zctp-DIR, in the magnetic field. Two models are considered: (a) zctp-DIR ⊥ zLAB, ctp-DIR namely, θctp || zLAB, namely, θctp 0 ) π/2, and (b) z 0 ) 0.
Figure 3. 31P NMR spectra for a stationary sample. Spectrum (a) was recorded at 431 K in the I phase, and (b), at 425 K in the SA phase. The signal positions are measured relative to 85% H3PO4 solution. Definition of the dipolar splitting width, 〈∆νLAB zz 〉, is shown as an inset.
TABLE 3: Experimental Results of 31P NMR phase (temp)
δ (ppm)
I (431 K) SA (425 K)a
10.14 14.24 20.22 26.19
a Symmetric triplet structure with a full spacing of 11.95 ppm corresponds to 1.306 kHz at the Larmor frequency of 109.365 MHz.
itself about its mean direction. The mean direction may be called the ctp director, and its direction is called the zctp-DIR axis, hereafter. To analyze the whole molecular structure in the liquidcrystalline state, it is necessary to know the orientation of the zctp-DIR axis relative to zLAB, because we know that the biphenylene axes of the organic moiety lie preferentially along the zLAB axis. Now the angle between the zctp-DIR and the zLAB axes is denoted by θctp 0 . The clear triplet structure in Figure 3b shows that θctp 0 takes a certain distinct value. If we define the dipolar splitting frequency as the frequency difference between the outer two peaks, as shown in an inset in Figure 3, then the splitting frequency is proportional to the second Legendre polynomial of θctp 0 , namely,
1 31 2 ctp ctp 〈∆νLAB zz ( P)〉 ∝ P2(cos θ0 ) ) (3 cos θ0 - 1) 2
(1)
We now try to determine θctp 0 experimentally. III.3. 31P NMR Angular Dependence Experiment for the Determination of the CTP Ring Orientation in the Field. The θctp may result from two factors: (i) the magnetic 0 shielding anisotropy and (ii) the steric factors. As to the shielding anisotropy, possible aromatic character10 may suggest that the most-shielding axis is along the ctp ring normal. Then the ring normal (zctp-FRAG) is expected to favor an orientation so that zctp-DIR lies in the plane perpendicular to zLAB, namely, θctp 0 )
π/2, as illustrated in Figure 4a. (Actually, it is shown in subsection III.7 that the most-shielding axis lies within the ctp plane, and the present model is proved invalid.) Steric factors may influence the orientation in a different way. The six biphenylene cores seem to align so that the biphenylene axes point preferentially along zLAB. A solid-state X-ray structural analysis has been made11 for hexakis(4-biphenoxy)cyclotriphosphazene, a prototypical compound of the present one without the six dodecyl chains. The resolved molecular structure suggests that steric factors favor that the biphenylene’s long axis and the ctp ring normal are in a parallel orientation, namely θctp 0 ) 0, as illustrated in Figure 4b. 31P To determine the value of θctp 0 , an angle-dependent NMR experiment was conducted. The field-aligned liquidcrystalline sample was prepared first, and then the sample tube was rotated about the axis perpendicular to zLAB. We denote the rotation angle Θ, and Θ was changed from 0 (the initial condition) to π/2. If θctp 0 ) 0 in the initial condition (Figure is equal to the rotation angle Θ throughout 4b), the angle θctp 0 the angular dependence experiment, and so the π/2 rotation causes θctp 0 to vary from zero to π/2 passing through the magic angle (ΘM). The dipolar splitting follows an equation,
〈∆νLAB zz (Θ)〉 〈∆νLAB zz (0)〉
) |P2(cos Θ)|
(2)
from eq 1, and the value of eq 2 varies as 1 f 0 f 1/2 when Θ changes from 0 to π/2. If, on the other hand, the molecules ctp-DIR axis align so that θctp 0 ) π/2 (Figure 4a), then the z distributes within the xLAByLAB plane. Therefore, when Θ is varied, the dipolar spectrum does not show clear dependence on Θ. The experimental result of the angle-resolved 31P NMR is shown in Figure 5. The spectrum quality is poor because of the low Larmor frequency and the poor field homogeneity, and so the triplet structure is not resolved. However, the trend of the dipolar splitting width is recognized. The spectrum is broadest at Θ ) 0 (Figure 5a), narrowest at Θ ) 54.7° (Figure 5b), and intermediate at Θ ) π/2 (Figure 5c). The observed sequence follows eq 2. We thus conclude that θctp 0 ) 0 in the initial condition, namely, the inorganic moiety of the molecule orients in the SA phase so that the ctp ring normal points preferentially along the zLAB axis. The gross molecular structure of the novel ctp liquid crystal can be obtained from the above analyses. The six biphenylene fragments orient and align in the magnetic field so that the biphenylene axes point preferentially along the magnetic field, and the dodecyl chains extend along this direction. This orientational feature is qualitatively similar to that of the usual
NMR Studies of a LC Cyclotriphosphazene Derivative
J. Phys. Chem. B, Vol. 105, No. 33, 2001 7923 the magnetic field, the observed chemical shift for the individual nucleus numbered as j is, generally, a motional average of the projection of the individual shielding tensor onto the z-axis of the laboratory frame (LAB), where zLAB is taken along the external field. The projection is calculated by using a spherical tensor δj-PAS(j) defined on the shielding principal axis system of the jth nucleus (j-PAS) and the Wigner rotation matrix D(2)(j-PASfLAB), such that12
1 LAB δ00 (j) + x3
〈δLAB zz (j)〉 ) -
1
)-
x23〈δ
(j) + δj-PAS 00
x3
x
2
LAB 20 (j)〉
2
∑
3 m)-2
j-PAS (2) δ2m (j)〈Dm0 (j-PASfLAB)〉 (3)
Figure 5. 31P NMR angular dependence measurement at a Larmor frequency of 22 MHz. The sample tube was rotated about the axis perpendicular to zLAB. The spectra were recorded at each rotation angle Θ. The figures show the spectra at (a) Θ ) 0°, (b) Θ ) 54.7°, and (c) Θ ) 90°.
The braces 〈 〉 represent the motional average. The spherical tensor is related with the Cartesian tensor by formal relationships,12
δj-PAS (j) ) 00 (j) ) δj-PAS 20
1 tr δ(j) ) - x3 δiso(j) x3
(4)
x32[δ
(5)
j-PAS (j) zz
- δiso(j)]
j-PAS (j) ) 0 δ2(1
(6)
1 j-PAS δ2(2 (j) ) [δj-PAS (j) - δj-PAS (j)] yy 2 xx
(7)
and
The Cartesian tensor elements are defined so that
(j) g δj-PAS (j) g δj-PAS (j) δj-PAS xx yy zz
Figure 6. Schematic molecular structure of the cyclotriphosphazene liquid crystal.
organic liquid crystals. As to the inorganic moiety, the ctp ring normal aligns along the magnetic field. Probable molecular structure in the liquid-crystalline state is illustrated in Figure 6. We next make quantitative analyses to obtain the orientational order parameters and the 31P and 13C chemical shift anisotropies. III.4. Theory for the NMR Spectra of Cyclotriphosphazene Liquid Crystals. When liquid crystals are aligned in
(8)
namely, the zj-PAS axis is taken along the most-shielding direction. The average value, δiso(j) ) trδ(j)/3, is called the isotropic chemical shift. The quantity 〈D(2)m0(j-PASfLAB)〉 contains the geometrical location of the j-PAS axes within a molecule, all the effects of molecular motions and orientational orders, and macroscopic alignment of the directors. Appropriate coordinate transformations are thus necessary to draw useful microscopic information from the experimental spectra. The j-PAS f LAB transformation is divided into a number of steps in our theoretical model by introducing additional coordinate systems as described in Figure 7. The most local coordinate systems are named f-fragment systems (f-FRAG), within which the internal atomic motions are neglected, and the atoms are fixed on this frame. For the present compound, the following fragments constitute the whole molecule: the ctp-FRAG (f ) ctp, the cyclotriphosphazene fragment), biph-FRAG, (f ) biph, the biphenylene fragment), and each of the methyl and the methylene fragments in the aliphatic chains. The coordinate axes for the f-FRAG are taken by considering the rotational property of each fragment. The zf-FRAG axis is taken along the easiest axis for rotation, and the yf-FRAG and xf-FRAG axes are taken along the principal axes of the moment of inertia on the plane perpendicular to zf-FRAG. We know from the analyses in the previous subsections that the biph-FRAG rotates mainly about the chemical bond that
7924 J. Phys. Chem. B, Vol. 105, No. 33, 2001
Moriya et al.
Figure 7. Definitions of the coordinate systems and the transformation sheme. The Wigner rotation matrices with the corresponding Eulerian angles are also shown.
The fragmental orientational order parameters are defined by the averaged rotational matrix for the f-FRAG f f-DIR transformation, f f f (2) 〈D(2) nq (f -FRAGff-DIR)〉 ) 〈Dnq (φ′ θ′ ψ′ )〉 f ) 〈exp(-inφ′f) d(2) nq (θ′ ) ×
exp(-iqψ′f)〉 (10)
Figure 8. Locations of the f-fragment axes for (a) ctp-FRAG and (b) biph-FRAG systems. The zctp-FRAG axis is taken along the C3 axis of the ctp ring, while zbiph-FRAG is along the long axis of the biphenylene fragment. The xf-FRAG and yf-FRAG axes are taken along the principal axes of the moment of inertia in the plane perpendicular to zf-FRAG.
connects the two aromatic rings and the ctp-FRAG rotates mainly about its C3 axis. These axes are taken as the zbiph-FRAG and the zctp-FRAG axes, respectively. The coordinate axes for the ctp-FRAG and biph-FRAG are illustrated in Figure 8. The zctp-FRAG axis is identical with the ctp ring normal, while the zbiph-FRAG axis is perpendicular to the aromatic ring normal. The first transformation thus gives 2
δf-FRAG (j) ) 2n
j-PAS (j) D(2) ∑ δ2m mn(Rjβjγj) m)-2
(9)
It is natural, next, to assume a molecular frame (a coordinate system determined by the three principal axes of moment of inertia of the whole molecule) in the analysis of the usual smallmolecular-weight liquid crystals.8 The orientational orders are determined by the statistical distribution of the molecular axes about their average directions, especially that of the molecular long axis about the “director”. In the present case, however, the molecule is extremely bulky, and the inorganic and the organic moieties may exhibit different motional characteristics or different orientational orders. Therefore, the coordinate system for the whole molecule is not assumed, and the orientational orders are considered for each of the fragments.
Now the f-DIR is the coordinate system for the directors (fdirectors) of the f-fragment, where the f-DIR axes are taken along the average directions of the f-FRAG axes. In the present treatment, the ctp and biph fragments are assumed to have their own director frames and have different orientational order parameters. For the ctp fragment (f ) ctp), the exponential factors containing the first Eulerian angle φ′f become zero except for the n ) 0 terms, because of the C3 symmetry of the fragment. For the biphenylene fragment, the xbiph-FRAG and the ybiph-FRAG axes are inequivalent. However, we make an assumption that the rotation about the long axis (zbiph-FRAG) is of higher symmetry than C3, and a similar average effect applies also to the biph fragment. Next, the factors containing the ψ′f also become zero except for the q ) 0 terms because of the uniaxial symmetry of the SA phase. Therefore, eq 10 is simplified to give (2) f 〈D(2) nq (f-FRAGff-DIR)〉 ) δn0δ0q〈d00 (θ′ )〉
(11)
A boldface is used to distinguish the Kronecker’s delta from the chemical shift. The only nonvanishing element (2) f f 〈D(2) 00 (f-FRAGff-DIR)〉 ) 〈d00 (θ′ )〉 ) 〈P2(cos θ′ )〉
≡ Sf00
(12)
is the orientational order parameter of the f-fragment. Here, P2 is the second Legendre polynomial. The averages in eqs 10-12 contain the rotations of each fragment about the f-FRAG axes and also the orientational fluctuation of the axes. Among these motional modes, the rotation about the zf-FRAG axis is special because this mode does not contribute to Sf00 but determines the effective symmetry of the rotating fragment. The above assumption of a rotational symmetry higher than C3 results in an average distribution of mass that is equivalent to a symmetric top rigid rotor. We now assume another reference frame on this matter for further analyses. This frame is named f-RIG, the f-fragment as a rigid body.
NMR Studies of a LC Cyclotriphosphazene Derivative
J. Phys. Chem. B, Vol. 105, No. 33, 2001 7925
TABLE 4: Typical 13C Chemical Shift Tensor Elements and the Eulerian Angles for the j-PAS f f-FRAG Transformations
a
kinds of carbon atoms
δj-PAS xx (ppm)
δj-PAS yy (ppm)
δj-PAS zz (ppm)
δiso (ppm)
Rj (rad)
βj (rad)
γj (rad)
AIScalc(Sf00)1) (ppm)
inter CH2a ar CHb ar CCc
51 215 221
38 145 160
15 17 21
35 126 134
0 π/3 0
0 π/2 π/2
0 0, π 0
-20 37 87
Interior methylene carbons. b Aromatic protonated carbons. c Aromatic nonprotonated carbons.
The f-FRAG f f-DIR transformation is now divided into two steps, f-FRAG f f-RIG and f-RIG f f-DIR. The second Eulerian angle βf for the first f-FRAG f f-RIG transformation is zero, and rigorous calculations are possible for the chemical shift tensors on the f-RIG system. This treatment is an example of the uncoupling approximation,13 which is often used in the theoretical treatments of the liquid-crystalline order. The rotation matrix for the f-FRAG f f-RIG transformation is
〈D(2) np (f-FRAGff-RIG)〉
)
δn0δ0p〈d(2) 00 (0)〉
) δn0δ0p
(13)
and the tensor element,
)
j-PAS (2) δ2m (j) Dm0 (Rjβjγj) ∑ m)-2
(j) δj-PAS 20
x
3 j-PAS δ (j) sin2 βj cos 2Rj (14) 2 22
is obtained by noting that only the n ) p ) 0 terms remain. The angle θ′f is equal to θf, and so (2) 〈D(2) 00 (f-FRAGff-DIR)〉 ) 〈D00 (f-RIGff-DIR)〉
(15)
holds. The tensor element on the f-DIR frame is 2
(j)〉 ) 〈δf-DIR 20
〈δf-RIG (j)〉〈D(2) ∑ 2p p0 (f-RIGff-DIR)〉 p)-2
) 〈δf-RIG (j)〉Sf00 20
(16)
from eqs 11 and 12. The final step of the transformations is that of the f-DIR system to the LAB system. This transformation describes the macroscopic orientation of the f-directors with respect to the laboratory frame, in which the zLAB axis is taken along the static field. Therefore, f-DIR (j)〉P2(cos θf0) 〈δLAB 20 (j)〉 ) 〈δ20
(17)
Let us now summarize the transformations and solve eq 3. Formally, eq 3 is rewritten as
〈δLAB zz (j)〉 ) )
x
2
2
2
x
1
(j) + δj-PAS 00
3
2
x13δ
j-PAS (j) 00
+
x23〈δ
f-RIG (j)〉 20
Sf00P2(cos θf0) (19)
or, in a Cartesian form,
{
j-PAS (j) - δiso(j)] P2(cos βj) + 〈δLAB zz (j)〉 ) δiso(j) + [δzz
}
1 j-PAS [δ (j) - δj-PAS (j)] sin2 βj cos 2R j Sf00P2(cos θf0) (20) yy 2 xx
AIS(j) ) 〈δLAB zz (j)〉 - δiso(j)
P2(cos βj) +
f f ) 〈d(2) 00 (θ )〉 ) S00
〈δLAB zz (j)〉 )
The alignment-induced shift (AIS), which is the difference in chemical shifts between the liquid-crystalline phase and the I phase, is expressed as follows,
2
(j)〉 ) 〈δf-RIG 20
and simplified by eqs 16 and 17, to give
2
f j-PAS (2) f δ2m (j) D(2) ∑ ∑ ∑ ∑ mn(Rjβjγj)〈Dnp (R 0γ )〉 3 q)-2 p)-2 n)-2 m)-2 f f f (2) f f f 〈D(2) pq (φ θ ψ )〉 Dq0 (φ0θ0ψ0) (18)
{
1 (j) - δiso(j)]P2(cos βj) + [δj-PAS (j) ) [δj-PAS zz 2 xx
}
(j)] sin2 βj cos 2Rj Sf00P2(cos θf0) (21) δj-PAS yy The procedure above can be applied to the formulations of other kinds of spectra. A case of nuclear dipolar structure is treated in subsection III.6. III.5. Orientational Order Parameter for the Organic Moiety. Now we make a quantitative analysis based on eq 21. Actually, we have not determined, experimentally, the chemical shift tensor elements nor the j-PAS f f-FRAG Eulerian angles for each carbon atom. Therefore, an empiricist’s approach is taken here. Table 4 shows the typical values for the 13C chemical shift tensor elements and the j-PAS f f-FRAG Eulerian angles for the aliphatic interior methylene carbon, the aromatic protonated carbon, and the aromatic nonprotonated carbon. These values are taken from the compilation of the data by Veeman14 and Duncan15 of the chemical shift tensors of organic ) 0 from subsection III.3, the compounds. Considering θbiph 0 parenthesized quantity, { }, in eq 21 is the AIS on the condition of perfect orientational order, Sf00 ) 1. Therefore, the AIScalcd (Sf00 ) 1) are calculated by eq 21 for each kind of the carbon atoms and shown in the last column of Table 4. The values of Sf00 are thus obtained from the experimental AIS values by
Sf00 )
AISexp(j) AIScalc(Sf00)1)
(22)
The results are shown in Table 2. From the experimental data of the interior methylene carbons, the order parameter for the typical methylene fragment, Smethylene ) 0.13, was obtained. 00 Two results were obtained for the biphenylene fragment: Sbiph 00 ) 0.44 from the experimental data of the protonated aromatic carbons, C(2), C(3), C(6), and C(7), and Sbiph 00 ) 0.46 from the data of the nonprotonated aromatic carbons, C(4), C(5), and C(8). By taking an average, Sbiph 00 ) 0.45 was concluded. The experimental data of C(1) are not used for the present analysis
7926 J. Phys. Chem. B, Vol. 105, No. 33, 2001
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because it has been difficult to determine the typical values of j-PAS f f-FRAG Eulerian angles from the compiled literature data. III.6. Orientational Order Parameter for the Inorganic Moiety. In this subsection the orientational order parameter for the ctp moiety is deduced from the observed 31P dipolar splitting frequency. An expression for the motionally averaged dipolar 31 splitting 〈∆νLAB zz ( P)〉, which is the observed quantity in the present experiment (see the inset in Figure 3), is obtained by a procedure similar to that taken for the formulation of the chemical shift in subsection III.4. Taking in mind that the dipolar interaction makes a traceless interaction tensor, and also that it is of cylindrical symmetry for the three-spin system arranged in a regular triangle, expressions similar to eqs 20 and 14 are obtained: 31 ctp-RIG 31 ctp 〈∆νLAB ( P)〉Sctp zz ( P)〉 ) |〈∆νzz 00 P2(cos θ0 )| (23)
and
〈∆νctp-RIG (31P)〉 ) ∆νj-PAS (31P) P2(cos βj(31P)) (24) zz zz Here, the principal axis system for the local dipolar interaction, j-PAS(31P), is taken so that the main interaction axis is along zj-PAS(31P). For the case of a regular triangle, the zj-PAS(31P) axis lies within the triangular plane, and so the second Eulerian angle, βj(31P), in the j-PAS(31P) to the ctp-FRAG transformation is π/2. If we define the dipolar splitting frequency tensor, ∆νj-PAS(31P), as the splitting between the outer two peaks observed when the external field is applied along each axis of the j-PAS(31P) frame, then the zz-element of the ∆ν j-PAS(31P) tensor is given by
∆νj-PAS (31P) ) 6νd(31P) zz
(25)
where
νd( P) ) 31
µ0γP2p 8π2rP-P3
(26)
Here, µ0 is the vacuum permeability, γP is the gyromagnetic ratio of a 31P nucleus, and rP-P is the 31P-31P distance. The dipolar frequency νd (31P) is calculated to be 0.978 kHz if we take the average value of the 31P-31P distance 0.272 nm, within the ctp ring, from the X-ray structural analysis of the related 31 compound.11 From the observed 〈∆νLAB zz ( P)〉 value of 1.306 ) 0, the order kHz in the SA phase, and also with θctp 0 parameter for the ctp fragment, Sctp 00 ) 0.45, was obtained. We thus obtained the orientational order parameters for both the organic and the inorganic moieties from the 13C and 31P NMR spectroscopies, respectively. An interesting feature is that the two-order parameters are found to take the same value, ctp Sbiph 00 ) 0.45 and S00 ) 0.45, though we made independent approaches to determine each value. This means that the present compound keeps “molecular identity” despite its extreme bulkiness. Another interesting feature is the smallness of the order parameter value. Considering the typical values of 0.7 for the SA liquid crystals of small-molecular-weight organic compounds, the less-ordered feature of the present compound is a novel aspect found in such a large molecule. The present molecule may be regarded as if the six organic liquid-crystalline
molecules, 4-dodecylbiphenyl, are fixed on the oxo-ctp ring at each end. It seems that the smectic A state sustains itself against large amplitude fluctuations because of this anchoring effect. III.7. 31P Chemical Shift Tensor. We next analyze the chemical shift tensor of a 31P nucleus. Figure 3 and Table 3 show that the center of gravity of the spectrum in the SA phase (Figure 3b) makes a paramagnetic (positive) shift compared with the spectrum in the I phase (Figure 3a). This change in δ, the 31P AIS, may be analyzed by eq 21, namely, the AIS is interpreted with the chemical shift elements together with Sctp 00 and θctp 0 obtained in the previous subsections. However, the three tensor elements cannot be obtained from the present experiment, because the number of the available experimental 31 quantities, δiso(31P) and 〈δLAB zz ( P)〉, is smaller than the number , δj-PAS , δj-PAS , Rj, and βj. of the unknown quantities, δj-PAS xx yy zz (31P) What we can do is to determine the two elements, δctp-RIG || 31P), of the motionally averaged δctp-RIG tensor on and δctp-RIG ( ⊥ the ctp-RIG frame, under cylindrical averaging expressed by eq 13. The anisotropy of the irreducible tensor on the ctp-RIG frame is obtained from eq 19 as
〈δctp-RIG (31P)〉 20
)
LAB 31 31 3〈δzz ( P)〉 - δiso( P) ctp 2 Sctp 00 P2(cos θ0 )
x
(27)
and, in a Cartesian form,
(31P)〉 〈δctp-RIG zz
- δiso( P) ) 31
31 31 〈δLAB zz ( P)〉 - δiso( P) ctp Sctp 00 P2(cos θ0 )
(28)
31 31 The experimental values, 〈δLAB zz ( P)〉 ) 20.22 ppm, δiso( P) ) ctp ctp-RIG 31 ) 0.45, and θ ) 0 give 〈δ ( P)〉 ) 10.14 ppm, Sctp 00 0 zz 32.78 ppm. The 〈δctp-RIG(31P)〉 tensor is thus
[
]
-1.18 0 0 -1.18 0 ppm 〈δctp-RIG(31P)〉 ) 0 0 0 32.78
(29)
This result means that the most-shielding axis exists within the ctp ring plane and that the least-shielding axis lies along the ctp ring normal, for a motionally averaged ctp ring. A tempting analogy to organic aromaticity for ctp may lead to a misleading assumption that the most-shielding axis lies along the ring normal.10 The present study showed that the assumption is invalid for the present compound. The feature of 〈δctp-RIG(31P)〉 is illustrated in Figure 9a. We revealed in subsection III.3 that the ctp ring orients in the magnetic field so that the ring normal points preferentially at the field direction, zLAB. By this orientation, the most-shielding axis tends to point perpendicular to zLAB. It is to be noted here that this orientation is favored from the aspects not only of the steric effect but also of the magnetic energy. III.8. 13C Chemical Shift Tensors and Contrasting Shielding Anisotropies for Inorganic and Organic Six-Membered Rings. The chemical shift tensors for the aromatic carbons were not obtained experimentally in the present work, but instead, typical values were taken from the literature data. Here we use the data in Table 4 to calculate the chemical shift tensors on a motionally averaged fragment, namely, on the biph-RIG frame.
NMR Studies of a LC Cyclotriphosphazene Derivative
J. Phys. Chem. B, Vol. 105, No. 33, 2001 7927 axes of the biphenylene fragments and the alkyl chains align along the magnetic field, and the ctp ring normal points preferentially at the same direction. The orientational order parameters for the inorganic and the organic moieties take the biph same value, Sctp 00 ) 0.45 and S00 ) 0.45, showing that the molecule behaves by keeping its total shape. In other words, the “molecular” identity is preserved despite its extreme bulkiness. The order parameter values are smaller than that of the conventional SA liquid crystals consisting of small organic molecules. Contrasting shielding anisotropies are revealed for the motionally averaged chemical shift tensors of the 13C and 31P nuclei: the least-shielding axis for the 31P nucleus is on the C3 axis of the ctp ring, while that for the aromatic 13C nucleus lies along the long axis of the biphenylene fragment. References and Notes
Figure 9. Chemical shift anisotropies of the inorganic and organic six-membered rings, under motional averaging. The least-shielding axes coincide with the axes for fragment rotation, zf-FRAG, for both cases. The most-shielding axis lies within the ring plane for the ctp fragment.
The anisotropy element of the irreducible tensor is given by eq 14, and the following chemical shift tensors are obtained:
[
]
107.5 0 0 107.5 0 〈δbiph-RIG(j;protonated)〉 ) 0 ppm (30) 0 0 163.0 for the aromatic protonated carbons, j ) 2, 3, 6, and 7, and
[
]
90.5 0 0 90.5 0 〈δbiph-RIG(j;nonprotonated)〉 ) 0 ppm 0 0 221.0
(31)
for the aromatic nonprotonated carbons, j ) 4, 5, and 8. Equations 30 and 31 show that the least-shielding axis for the aromatic carbons is along the zbiph-FRAG axis, the biphenylene long axis, for both the protonated and the nonprotonated carbons. On the other hand, the least-shielding axis coincides with the ring normal for the ctp fragment. The contrasting nature of the chemical shift tensors is illustrated in Figure 9. IV. Conclusion The present ctp derivative exhibits a molecular structure in the liquid-crystalline state as illustrated in Figure 6. If the SA liquid crystal is formed in an intense magnetic field, the long
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