19FMultipulse Nuclear Magnetic Resonance Study of Lithium Perfluoro-octanoate + Water Liquid Crystal Systems BY PETERG. MORRIS AND PETER MANSFIELD Department of Physics University of Nottingham Nottingham NG7 2RD AND GORDON J. T. TIDDY Unilever Research Port Sunlight Laboratory Port Sunlight Wirral Merseyside L62 4XN Received 3rd August 1978 Two modified MREV8 sub-cycles forming a partially permuted cycle (PP16) have been used in a multipulse study of the mesophases formed by the lithium perfluoro-octanoate (LiPFO) + water system. The lineshapes show characteristic axially symmetric chemical shift tensors which have been measured over a range of temperature (20-68 "C) and concentration (47.8-72 % LiPFO). The shift anisotropy in the hexagonal phase is halved in magnitude and opposite in sign to that observed in the lamellar phase indicating the presence of rapid rotational diffusion.A rotational study of an aligned hexagonal sample (47.8 % LiPFO 52.2 % H20)is used to demonstrate the effects of two dimensional averaging on a chemical shift tensor. The observed shift tensors are used to estimate values for the order parameter based on the known values of the 19Fchemical shift tensor elements for CF2 groups in Teflon at liquid nitrogen tempera- tures. When compared with chain dimensions estimated from X-ray data the values of S indicate the existence of rotational isomerisation for the surfactant chain. The molecular order generally increases with increasing concentration but a decrease is observed at the hexagonal/lamellar boundary.1. INTRODUCTION The development of multipulse techniques was originally directed towards the removal or the reduction of dipolar interactions in rigid solids so as to reveal the smaller and often more interesting chemical shift interactions which reflect the solid state electronic structure around resonant nuclei. Therefore much of the theoretical framework for multipulse experiments has been developed for an idealized rigid lattice although there has been some extension of the original theories to include spin-lattice relaxation time effects arising from molecular motions. However the basic premise that a certain reversibility of the spin system can be made to exhibit itself by application of appropriate r.f. pulse sequences has been assumed to be true in solids with rapid inter- and intra-molecular motions and such echo behaviour and multipulse line narrowing effects can readily be observed experimentally.It is therefore not too surprising that multipulse techniques can be applied in the study of liquid crystal systems; the potential for such experiments was realised by Losche and Grande.' More recently multipulse techniques in our laboratory have been used to investigate the dynamic properties of cesium perfluoro-octanoate (CsPFO) and ammonium perfluoro-octanoate (APFO) + D20 systems by observa- tion of the I9F resonance.2 The multipulse n.m.r. studies on liquid crystals are a small part of a wider series of studies by many groups using conventional n.m.r. techniques on both thermotropic and amphiphilic liquid Directly related to this work LIQUID CRYSTAL SYSTEMS are the 13Cproton double resonance experiments on a number of nematic liquid crystals6 and on lecithin bilayers.' The same technique has been used to study the 31P shift anisotropy also in lecithin bilayer~.~~~ In this paper we report results for lithium perfluoro-octanoate (LiPFO) + H20 liquid crystals obtained by 19F multipulse experiments.Three different liquid crystal phases occur in this system and were originally assigned complex hexagonal lamellar and reversed hexagonal structures." The complex hexagonal and reversed hexagonal structures are incorrect; these phases are now known to be a normal hexagonal phase and a second lamellar phase.'' From these results and using an extension of our previously developed theoretical model we have been able to obtain the order parameter S over the range 19.2-68.3 "C by recording and fitting the multipulse spectra.The values of S are consistent with a model for chain motion involving rapid rotation about the longitudinal chain axis and a wagging motion of the chain. The results demonstrate unequivocally that the order parameter changes sign at the hexagonal/lamellar phase boundary. 2. MULTIPULSE SEQUENCE The truncated spin interaction hamiltonian for a rigid solid of spins Zin a large static magnetic field Bois l3 where cc denotes the coordinate axis along which the static polarizing field is directed (usually the z axis). The first term is the dipolar interaction and Aij = -f(1 -3 cos2 0ij)/2r:j.The second term involves the chemical shift Siof the ith spin and a common offset from resonance Am. Normal 90" r.f. pulses applied along the x and y axis in the rotating reference frame alter the effective hamiltonian eqn (1) by simply changing a from its equilibrium value of z to either x or y. Over a properly chosen cycle (or subcycle) therefore the dipolar part of eqn (1) can be made to vanish identically (for ideal pulses) and a non-zero average hamitonian defined from the residues arising from the chemical shift and offset terms. The minimum number of orthogonal switched hamilton states of equal time weighting required to make the dipolar part vanish is three corresponding to a = x y and z. We shall call this the minimal subcycle.Fig. 1 shows various groupings of minimal subcycles to form a longer cycle. A reflection symmetry cycle is formed from two minimal subcycles placed back to back and has the property that second order terms in the expansion parameter T (pulse separation) of the effective spin hamiltonian vanish identically over the cycle. It has been shown14 that a larger cycle built up in such a way that all the permuta- tion states of the hamiltonian are cycled in three groups of reflection symmetry cycles removes correction terms up to third order in the average hamiltonian. This is true only for idealized 90" r.f. pulses. The effect of non-zero correction terms is to broaden the observed line width thus making the cycle less efficient.Non-idealized 90" pulses also broaden the line width by reintroducing lower order corrections which may not be removed by permu- tation symmetry. However in the case of the reflection symmetry cycle it is possible by using two similar cycles simultaneously to correct for finite r.f. pulse widths and r.f. inhomogeneity while maintaining second order interaction terms zero. Two such reflection symmetry cycles as shown in fig. 1 have been named the MREV8 P. G. MORRIS P. MANSFIELD AND G. J. T. TIDDY 39 I' I MREV8 c' I I I I I I I I I ,*-\ ,-. I I I I 8 '\ I q Yx XY z zTx XYZlZ xY YXJ ZZY YSTZl I +---121 I -12r ---_I 4 PP16 -1 * * I sub-cycle reflectiori symmetry cycle FIG.1 .-Multipulse timing sequence for the PP16 cycle used in liquid crystaI studies.P*a corre-sponds to 90" r.f. pulses applied along the a-axis in the rotating frame. S denotes the signal sampling pulse. The expected signal response at resonance is shown (solid lines). The broken lines corre- spond to a quadrature signal response. sequence.14-16 The chemical shift and offset terms remaining in the average hamil- tonian produce an effective scaling or reduction of the observed chemical shift of 3/dZ This cycle and its phase-corrected variant 16917 have been successfully employed for a number of years. We mentioned above the fully permuted cycle which removes all third order terms. A system which is intermediate to this is the partially permuted cycle (PP16). This comprises two MREVS cycles chosen so that two of the three possible permutation groups are covered in the cycle.It has been shownls that this cycle has a somewhat better line narrowing efficiency for small chemical shifts and offsets since the third order term is reduced to + of its value in the MREV8 cycle. The PP16 sequence is the cycle that we have used throughout this work and our previous work' in liquid crystals. We have found experimentally that this particular cycle is relatively easy to align and fairly stable in operation. We have therefore not found it necessary to use the phase compensated version of this cycle." The ideal chemical shift scaling factor for the PP16 cycle is 6The finite pulse width correc- tion to this factor is described elsewhere.18 3.CHEMICAL SHIFT THEORY As stated in Section 2 the truncated interaction hamiltonian for ''F nuclei com- prises two terms XDand Xcs.With the external magnetic field used here E 2.25 kG IXDI9 I%'csl and the single pulse spectra are characterized mainly by the dipolar interaction. However the multipulse sequence is designed to reduce the dipolar interaction to almost zero so that the dominant term in eqn (1) is in this case Xcs. For an external static field B the chemical shift interaction is Xcs = 2W:a(q')L43 (2) PA where p and q number groups of magnetically non-equivalent and equivalent nuclei respectively. All other symbols have their usual meaning. This can be expressed in terms of the irreducible spherical tensors Rik and T:; corresponding to the spatial and spin parts respectively.For a single spin eqn (2) becomes LIQUID CRYSTAL SYSTEMS ,water director d Fig. 2. Idealized representation of a LiPFO +water hexagonal phase. The first term in eqn (3) is simply the isotropic chemical shift Rft =3 trace (aaB)=cr and gives rise to a change in resonance frequency relative to the bare nucleus but does not lead to any line broadening. The 2nd rank tensor components occurring in eqn (3) are defined in the principal axes (PAX) system as RY$(PAX) = {cTZZ-CT) (44 RYZl(PAX) = 0 (4b) T:& =0 (4f1 where crxx oYyand oZzare the principal values of the chemical shift tensor y the magnetogyric ratio and I the spin displacement operators. In addition to the principal axes system defined w.r.t.the C-F bond it is useful in describing the motional and aggregate crystallite behaviour of liquid crystals in various phases to introduce a number of additional reference frames using simple tensor transformations operating on the spatial tensor components only. We define a molecular (MOL) frame in which the z axis coincides with the LiPFO molecular chain axis and in which there is rotation of the molecule about this axis. The two frames are related by the Euler angle QM = [0 w y(t)] through the trans- for mati on R;&(MOL) =29L,Mt (QM)RF&t(PAX). (5) M’ Fast rotation around one axis averages out the full chemical shift tensor making it axially symmetric.’’ In this case R::(MOL) =3 (011 -01) =3 ACT (64 P. G. MORRIS P.MANSFIELD AND G. J. T. TIDDY where 011 and oI are the principal components of the axially symmetric shift tensor. All other values of Rz2(MOL) for M + 0 vanish. From their measured values of ozz = -80 p.p.m. oyy= 10p.p.m. and oxx= 70 p.p.m. for rigid CF2groups in PTFE Garroway et al. have obtained the anisotropy da = 104.21 p.p.m. for a single rotating chain.20 This value is used below to obtain order parameters from the measured shift anisotropies. Depending on the liquid crystal phase we make further transformations similar to eqn (5) to go from the MOL frame to either the lamellar (LAM) frame or the hexagonal (HEX) frame. In the case of the LAM frame the Euler angle Q,(t) = [0,b(t),01 describes the director axis with respect to the MOL frame.For vibrational or wagging motion of the chain here regarded as rigid we consider later a model in which the angle p(t) undergoes a simple harmonic motion and derive an expression for the order parameter. In the case of the HEX frame the Euler angle Q,(t) = [~(t), 71/2 -/?(I) y(t)] describes the director axis (the surfactant cylinder axis) with respect to the MOL axis (see Fig. 2). The angles a(t) ~(t) describe the rotational diffusion of surfactant within the cylinder. Now the time average over the Wigner transformation coefficients for either LAM or HEX transformations defines the order parameter generally as == (9&[aL,H(f)l>av (7) which for the LAM frame gives s = (+[3 cos2P(t) -l]), = s (8) while for the HEX frame gives SH = -3(3[3 cos2P(t) -11) = -3s.(9) Note that S is the order parameter of the molecule relative to a direction normal to the surfactant/water interface. Finally to relate actual measurements we must transform the shift tensor to the laboratory (LAB) frame again using a transformation similar to eqn (5). The transformation from LAM to LAB employs the Euler angle a,, = (0 4 0) which describes the static distribution of crystallite directors with respect to the applied field. In this case the line position for the pth crystallite relative to a reference frequency cr) is p = -@) = tp + 3 S(a,P -ap;,(@). (10) u For an aligned sample the variation of line positions as a function of orientation is given by eqn (10). In the case of a non-aligned (poly-crystal) sample this must be spatially averaged over all 0 leading to the well-known lineshape expressionz1 where the sum is over all groups of non-equivalent nuclei and A is a normalization factor.The scaled values of shift tensor a and alsare given by GP -1 SI) -30 + 2s)q + +(I -s)ay (124 osl = +(I -s)oT + +(2 + S)of;.. (12b) This lineshape must be convoluted with 2 broadening function to account for relaxa- tion broadening and the residual dipolar interactions not fully removed by the multi- LIQUID CRYSTAL SYSTEMS pulse sequence. These effects which are in principle quite complicated are approxi- mated by a simple gaussian broadening'* so that the observed lineshape g(6) is given by +a g(6) =I- dS'f(6') exp { -(6 -6')2/2B2) where the characteristic broadening width B is to be determined experimentally.The transformation from HEX to LAB obeys a similar expression. However the hexagonal phase can be aligned such that the surfactant cylinders are distributed randomly in a plane normal to the alignment axis. For such a 2-D powder sample the transformation from HEX to LAB (Euler angle QLAB) can be effected by two con- FIG.3.-Euler angles for transformation from the HEX to LAB reference frame. secutive Euler rotations s1 =(0 n/2,n-d) ln2 =(0 0 0) where 0 is the angle between the normal to the director plane and Bo and 95 the azimuthal angle of the cylinder director axis; see fig. 3. For this case therefore the chemical shift line position rela- tive to the reference frequency w is given by 6 = a -1 S(q -al)[3 sin20 cos 2+ -(3 cos2 0 -I)].(14) 12 For the two special cases corresponding to 8 = 0" and 90" this reduces to +(all + a,) and +(q + 01) -+(a,,+ al) cos 295 respectively for S = 1. If the aligned sample is rotated about an axis perpendicular to the magnetic field then all the micelles will be parallel to each other with + = 0" and the lineshape func- tion g(0,d) will be given by g(0,d) = {[al sin28 + +(all+ al)cos201 -S}-t. (1 5) However if no such attempt is made there will be a random distribution over 4. Let the lineshape function for such a case be denoted by f(0,d) then the fraction of spins with line position in the range 6 to 6 + d6 will be given by m,6) = 44) d+ (16) P. G. MORRIS P. MANSFIELD AND G.J. T. TIDDY where n(4) will be a constant for a random distribution over 4. Thus From eqn (14) (S = 1) d6 = +(all+ aL)sin' 8 sin 24 d4 (18) so that f(8,6)oc I/[(all- aL)sin26 sin 241. (19) Substituting for 4in terms of 6 from eqn (14) this becomes The spectral function thus consists of a pair of delta functions at 6 = +(ali+ al) 6 = aI sin28 + +(all+ aL)cos28 which broaden into a doublet after convolution with the usual gaussian broadening function. For 8 = 90" the doublet peaks centre on aL and 3(all + aL)and as 8 is reduced to zero it collapses to give a singlet centred at 3bl + 01). 4. EXPERIMENTAL The samples of LiPFO + H20were prepared by mixing weighed constituents as reported previously.'0 All data were taken on the computer-controlled line narrowing spectrometer operating at 9.0 MHzwhich is described elsewhere." The multipulse spectra were recorded with the PP16 (partially-permuted) sequence operating with z = 6.4 ps.This achieved a true linewidth of 70 Hz for I9Fnuclei in a single crystal of CaF2 oriented with its [l 111 direction along the magnetic field at an offset of 500 Hz. Single pulse spectra were recorded by fourier-transforming the FID following one 90" r.f.pulse. Usually from 128 to 1024 spectra were accumulated to improve the signal-to- noise ratio. The sample temperature was regulated by blowing air of controlled temperature through the probe and was monitored by a copper/constantan thermocouple. Temperature stability during the experiments was estimated in most cases to be better than 0.5 "C.If no attempt is made to control temperature rapid sample heating occurs. This can be as much as 30 "C or so over the period of a typical multipulse experiment and is presumably the result of either unusually large dielectric absorption or ionic conduction. (Under similar conditions no noticeable temperature rise was apparent with water samples.) A two-dimensional powder sample of the hexagonal phase (47.8% LiPFO) was prepared by cooling from the isotropic phase in a field of wl T. At room temperature the sample preserved its alignment for a number of days without any noticeable change and could be rotated in the magnetic field without any trace of alignment relaxation. The accuracy of the angle setting when recording rotation spectra of the aligned samples was better than 0.5".5. RESULTS AND DISCUSSION HEXAGONAL MESOPHASE Typical multipulse spectra recorded at 20 "C for samples in the hexagonal meso- phase are presented in fig. 4. The different sign and reduced magnitude of the chemical shift anisotropy compared with lamellar phase spectra (see fig. 6) are clear. This indicates that the surfactant molecules are at right angles to the director and also undergo rapid rotational diffusion about this axis. The order parameter S was deter-mined by visual comparison of theoretical lineshapes generated by a computer and the LIQUID CRYSTAL SYSTEMS FIG.4.-19F multipulse spectra of LiPFO + water in the hexagonal phase referred to fluorine in CaFs at zero offset T = 20 "C.(a) 47.8% LiPFO SH = -0.21 ; (6)57.4% LiPFO SH= -0.25; (c)62.0%LiPFO SH= -0.29. experimental spectra. The solid curves in fig. 4 are the calculated lineshapes. Order parameters were determined also at higher temperatures and all the values are listed in table 1. The fact that only one spectral line is observed for the CF2 groups and that this can be fitted with a single negative order parameter provides good evidence in favour of TABLEOR ORDER PARAMETERS FOR LiPFO SAMPLES composition /"/o LiPFO temperature 1°C phase" order parameter isH,sL) 47.8 48.2 H -0.21 47.8 57.4 52.6 20.o isotropic H 0 -0.25 57.4 48.1 H -0.25 62.0 20.0 H -0.29 62.0 68.3 H -0.29 68.0 21.4 D 0.44 72.0 19.2 F 0.54 72.0 27.1 F 0.54 72.0 34.2 F 0.54 72.0 39.0 D 0.54 a H = hexagonal; F D = lamellar.P. G. MORRIS P. MANSFIELD AND G. J. T. TIDDY 45 the normal hexagonal structure for this phase.ll There is insufficient multipulse resolution to observe differences in the chemical shifts of the CF2 groups along the chain. Fig. 5 shows a series of multipulse spectra obtained from the two-dimensional powder sample (47.8% LiPFO). When 0 = 0" (magnetic field at right angles to the director) a singlet is observed fig. 5(a). This broadens into a doublet [fig. 5(b)]as the sample is rotated the maximum splitting occurring for 0 = 90" in agreement with the lineshape prediction of eqn (20). The doublet was barely resolved because of the small shielding anisotropy of this sample. Fig. 5(c) shows the spectrum of this kLnd o;o;o;oo4-j 0 oo 20 40 60 80 100 offset I p.p.rn FIC-5.-I9F multipulse spectra for a 47.8 % LiPFO sample measured for a range of temperatures.The reference compound is fluorine in C6F6 at zero offset. Curves (a)and (b)correspond to the two- dimensional powder averages in the hexagonal phase with the magnetic field normal to and in the plane of the cylinder directors respectively. Curve (c) shows the behaviour in the isotropic phase. (a) T = 19.5 oc e = 00; (6) T = 19.5 oc e = 900; (c) T = 52.6 oc. sample in the isotropic phase (at 52.6 "C) for comparison. Unfortunately we were unable to orient the more concentrated samples (which have higher shift anisotropies; see table 1). The anisotropic susceptibility of the amphiphiles means that the hexagonal meso- phase can be aligned in a magnetic field the minimum energy configuration being when the director (cylinder axis) is perpendicular to the field direction.This is the alignment direction expected from the shift anisotropy. If no attempt is made to spin the sample then the directors of separate crystallites will be distributed at random over the plane thus defined. Spinning the sample at right angles to the field direction however can lead to perfect alignment along the rotation axis.22 The single pulse spectra not shown here but obtained under the same experimental LIQUID CRYSTAL SYSTEMS conditions as for fig. 5 indicate a large angular variation of linewidth due to the un- averaged dipolar interaction ; this completely masks the effects of the two-dimensional averaging on the chemical shift tensor.Consequently no splitting of the CF2 line is observed in this case. LAMELLAR D AND F PHASES The multipulse spectra of both D and F phases (fig. 6) are very similar to those observed for the CsPFO and APFO + D20 lamellar systems.' 20 GO 60 80 100 offset 1p.p.m. FIG.6.-19F multipulse spectra for a 72% LiPFO sample referred to fluorine in CbF6at zero offset. Curve (a) corresponds to the lamellar F phase and (b) to the lamellar D phase. Note the reversed asymmetry of the CF2 peaks w.r.t. those of fig. 4. S = 0.54. (a) T = 19.2 "C,(6) T = 39 "C. Order parameters are listed in table 1. The 72.2 % LiPFO sample is particularly interesting since it is in the F phase at room temperature but undergoes a phase transition to the D phase at z 35 "C.Despite leaving the sample to equilibrate for over an hour there was no noticeable change in the spectrum on passing through the phase transition. The viscosity of the F phase is far higher than that of the D phase and it is remarkable that this is not reflected in a change of order parameter at the D/F transition. It suggests that an inter-layer cooperative effect is present which prevents the layers sliding over each other. It is worth emphasizing that all the spectra recorded were fitted with shift tensors scaled by a single order parameter S. This is strong evidence for the constancy of order parameters along the chain as has been observed for hydrocarbon surfac- and perhaps for the rigidity of the molecule.Certainly the order para- meters of the terminal CF2 groups (at C6 and C,) do not decrease by a factor of two or more as happens for hydrocarbon chain^.'^^^ The spectral width of the fluoro-methyl peak is identical to that found previously for CsPFO' and is relatively insensi- tive to variations of the order parameter. Unfortunately the results presented here do not contribute in great measure to the P. G. MORRIS P. MANSFIELD AND G. J. T. TIDDY detailed understanding of the nature of the chain disorder. However on the assump- tion of a rigid molecule executing a wagging motion (which would appear to be a reasonable model) it is possible to estimate the size of the angular fluctuations. Assuming a model in which p(t) [eqn (8) and (9)] varies harmonically over a range &Po then P(t) = Po cos wt (21) which when averaged over the motion gives an order parameter where J0(2p0)is a zero'th order Bessel function of the first kind.,' Note that this reduces to unity when Po is zero.A similar model has been employed by Seiter and Chan32 to describe motional averaging of dipolar splitting in hydrocarbon surfactant bilayers. Evaluation of this expression indicates that the amplitude of the motion varies from about 45"(S = 0.6) to 62"(S = 0.35). These results refer to the extrema. Calculations based on S = +(3 cos2PL-I) give for OL 31" (S = 0.6) and 41" (S =0.35). From the previously reported X-ray data" the surfactant layer dimensions are estimated to be 16.6 and 17.0 A for D and F phase samples in table 1.For an all- trans chain with S = 1 a value of 22.5 A can be calculated from known bond lengths. The X-ray data are consistent with an all-trans chain tilted at an angle BL = 41" (S = 0.35). This is clearly inconsistent with measured values of 0.44 and 0.54 (table 1) and shows that the chains cannot be in the all-trans conformation. On the basis of a rotational isomer model Shindler and Seelig25 have given an expression to calculate the effective chain length (L) from order parameters of CD2 groups (ScD) for hydrocarbon surfactants where n is the number of chain segments and i refers to the position of the group along the chain. The values of S in table 1 are a factor of two larger than values of ScD because they relate to the molecular axis rather than the CF bond.Chain lengths of 16.6 and 17.5 A are calculated from values in table 1 for the D and F phase samples at 20"' including an addition of 2 A for the head group and terminal CF group. These are in excellent agreement with the X-ray results. The magnitudes of the order parameters for the fluorocarbon chain are generally similar to values reported for hydrocarbon chains.2s30 Mely et ~1.~~9~~ observed little change in chain order parameters at the hexagonal/lamellar boundary for hydro- carbon soaps (apart from the factor of two due to change in alignment of chains w.r.t. the director). Our results indicate that the molecular order is higher in the hexagonal phase than in the lamellar phase. This suggests that the molecule is longer in the hexagonal phase (at the low-water boundary) a fact that is consistent with the increase in surfactant dimension estimated from X-ray results." 6.CONCLUSION Our results show that by fitting the spectra obtained using multipulse techniques to remove the dipolar broadening it is possible to measure the order parameter S over a range of concentrations and temperatures. This in turn has enabled us to confirm the hexagonal phase structure and the rapid translational diffusion of the amphiphiles 48 LIQUID CRYSTAL SYSTEMS around the cylinder surfaces corresponding to localized cylindrical rotation. Com-parison between chain dimensions estimated from X-ray results and values calculated for particular models using the S values indicates the presence of rotational iso- merisation for the fluorocarbon chain.We would like to thank Dr. N. Boden and his colleagues for communication of results prior to publication. A. Losche and S. 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