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Nuclear magnetic resonance in liquid crystals
CliEMlcAL PHYSICSLEITERS
Volume 10, numb& 1
NUCLEAR
MAGNETIC
RESCINANCE
IN LiQWD
1 July%971
CRYSTALS
C.C. SUNG DepmtmentofPhysics. The Ohio State University,CMumbus,Ohio 43210, USA Received 10 May 1971
The applicatibn of the theory of nuclear magnetic resonance to liquid crystals is reexaminexi and reasonable agreement is found between the present theory and experimental results.
On the basis of the theoretical work developed by De Cennes [l] and the Orsay Group [2] , Fincus [3] studied the relaxation time (Tr) of liquid crystals and obtained the field dependence (H) in good agreement with experimental results [4]. The predicted temperature (7’) dependence is unsatisfactory as shown by Doane and Visintainer [5] , who also raised a few additional questions: (1) why rotation ofH has no effect on Tl in smectic state in contrast to T2,
(2) why no discontinuity of T1 is obtained at nematic-smectic transition. Recently, Deane and Johnson [6] have generalized Pincus’ treatment and obtained an improved T dependence of Tl in the nematic state. We do not think, however, that the short range correlation time (collision time) should be neglected, as done in ref. [6], since the relaxation time T1 of liquid crystals is of the same order of ordinary liquids. Furthermore, this is important for the explanation of narrow linewidth. We point out, also, that the “viscosity” coefficient in the magnetic resonance theory should not be taken from hydrodynamical measurements. By employing the recent work of Saupe [7] and McMillan [83 we are able to show that a reasonable agreement exists between the present theory and experiment. We assume that intramolecular dipole-dipole interaction dominates the magnetic relaxation and that the relaxation times T1 and Tz are given by [9] *
(0 is the dipole-dipole where j sums over all particles with the intramolecular distance “a” from i, 9 of two spins distance ‘ir” apart and wo is the Larmor frequency. G is given by
interaction
(2) q(0,rJ) is a normalized spherical harmonic and (6,#) specify the direction of the vector rflj*co~ecting two spins at tj and rP 4 ‘1 in eq. (2)‘denotes the thermal average. In order to introduce the dynamical variable [ 1,2] forkfiu@nation, we rotate the z axis from the direction of Hto that of Mti) which is the instantaneous prefer&d direction [6,10],
35
Vohme_lO, nnn$~a 1
cm3fIcAL
PHYSCS
_-
LETTERS
.I.
1 July 1971
(3) where a, P and 7 are the EuIer’s angles. Dim @$y) = exp[-i&a] dmm@) exp[-imr] and dm,,,’ is explicitly given in ref. [l l] . Since only Q and j3 are needed to specify the transformation, y is chosen to be zero. The pres-
ent theory of liquid crystal can only handle the linear fluctuations [I] which enables us to set sin2p = 0, in eq. (3) and [2,10] l
coszfl =1 (4)
where K is the elastic constant, and q is the viscosity coeffcienc which, we want to emphasize, should not be taken from the hydrodynamical measurement.The so-called viscosity coeffkient g in eq. (4) refers only to the ro+ tation of one molecule which contains the two mxlear spins in question. How the translational motion of the molecule is hindered is not relevmt in eq. (4). Thus 9 in eq. (4) should not show the discontinuity nematic transition. In this case, 1) can be properly expressed by
at the smectic-
1)-m,
(5)
where r is the Maxwellian relaxation time orthe collision time. Eq. (5) is given by Landau and Lifshitz [ 1 l] in a phenomenologicai way, and its derivation is reviewed by Egelstaff [ 121. I’ is a short range correiative time which we have no reason to believe to be very much different from ordinary liquid, i.e., r * (d/v) eblT, where d is the intermolecular distance, V the average velocity, and b the activity energy. Whether or not this assumption is correct should be tested aith experimental data. Next, we discuss the time dependence of q(SZ’). The elongated molecules are free to rotate around n(n), but the rotation perpendicular to MH) is severely hindered. On the basis of this physical picture, the conditional probability fimction $@,Stb,t-to) can be written as [9]
5
~(s2’,SL;)J-f*) = G(cosB’-COSBb) z ’ m=_s exp[im(~‘-~~)lexp~-I~-~Ol/~ml
(6)
,
where I’;;;’ = JTZ~F-I according to the diffusion equation, and I? is taken the same as in eq. (4) for convenience. The possibility of diffusion of 0’ in eq. (6) may be incorporated by introducing a long relaxation time associated 8’. Ah the needed correlation functions in eqs. (1) and (2) are given by eq. (4), 2r,. (7j d~Y~(sz’,t,+t);~~~(~~~o)) eBiwr= iY~(s2,)12 1 +~z 03’ m
and
where fi and-Y? are assumed to be independent and cos 0 = (1 +r’, c&‘. given by the theory of Saupe (71 and McMiUan [8] _
36
lY$f(bo)12
the thermal average is
Vohune 10, number
1
CHEMICAL PHYSICS LErrERs
IY~i*= $ IY~(cosS)12
exp [ -T
gvo$(cosB)]dcmB
(J
exp [-9
1 July 1971
I$(eod)ldf2d)-’
,
(9)
0’
where g is the order parameter and VO a parameter. In our discussion above, time dependence of ea is ignored but can be incorporated in &@ by modifying the definition of T,, since n(rf) is assumed to rotate freely around the 2 axis. Tr in anphnses.
c
The important terms in eqs. (1) and (3) are according toeqs. (7) and (8)
__=p 1 8LJD 2 Jz(3c0Ge-02 T, 71 4
--
T 0 tj KK
35qjH + 2rl
(tie03sq2
1.
w
The first term is given in ref. [6 1. JZcj.(10) Ieads to several conchrsions different from those of ref. (61 in some respects: (a) T1 in liquid crystals is not much shorter than that of ordinary liquids, and we do not see any reason why PI (z lo-lo set) should be neglected. We notice that the second term is important to fit the experimental H dependence of q 1 = CIH -* + C2. The validity of this expression, however, is limited to small ho when the hydrodynamical approach of refs. [ 11 and [23 is correct, I?two 4 1. This limitation excludes the importance of magnetic energy in free ener (b) T dependence of K, 5i-T lYz I and -is sensitive to the parameter V. [7,8]. A qualitative study indicates that Tl is in agreement with ref. 5 . We note, however, that the experimental T1 may have more variety of T dependence [ 131. Also, -+7 lY212 f 1Y2I as given in ref. [63. A detailed study and related subject will be given elsewhere
t141. (c) At ,smectic-nematic
transition, g may or may not have a discontinuity depending on the choice of a parameter [83. lY~(cosB)j’ from eq. (9) thus may be continuous and a continuous T1 is obtained. At the isotopenematic transition, g has a discontinuity and so has {UT 12. This feature is confirmed in the data of TI by Blinc et al. [ 15,161. It would be interesting if there are experiments to correlate the second order phase transition with a continuous T1 and a first brderphase transition with a discontinuous Tl. We emphasize that q/K in eq. (10) should be continuous because of eq. (5). It should be pointed out that in ligbt scattering experiments [16) an equation similar to (4) is used where r amI I’ refer to the positions of two molecules. Since the diffusive mode in the problem is concerned with the motion of molecules, TJshould be taken from the hydrodynamical measurement, and a discontinuity is always expected at the amectic-nematic transition. T or the linewidth T2 in eq. (1) is dominated by Go or iI&(a$y)Y~(t?‘, $1 in q. (3)_ Thus eq. (7) gives I2 75 z WD2r 0 where I’,J is the very long collision time associated with 0’ as discussed above. JXG&‘~ % 1 as expetted, this expression is meaningless. T2 or the tinewidth of liquid crystals should have t&e features of a rigid lattice. This is indeed found experimentally [ 171, i.e., the line shape is a gaussian function with the second moment = 1G. The effect of the rotation ofH in smectic phase on the linewidth is given by
w&e ’ specifies the angle of the rotation of H and only m = 0 is important. In other words, the linewidth * Yz$B (a’)- l= r~;&)~l-l, as experimentally observed. Ibis feature again does not exist in Tt , which consists of two terms and makes the effect of tire rotation of H hard to observe (5,16l_ It is a great pleasure to thank Professor Korringa for hia valuable advice and criticism. 37
1 Jrlxy1971
-[I] P.G. de Gums, Coznpt.Ret& &ad. Sd (Paris) 266 (19S8) 15. [2] onrjr +p, J. C@IL Fhys. Sl(1969) 816. (3) P. Pina& solid statccommnn. 7 (1?6P) pis. [4] GH. l&own, J.W.Jhana adV.D. Neff. Critid Rev. Solid State Sci. 4 (1970) 303. [S] J-W. Doam antiJJ. Vi&~taincx, Phys. Rev. Letters 23 (1969) 421. [6] J.W. Doanc aad D.L. Joh+m, Chem. phus. Letters6 (1970) 291. [l] A. Saqm, 2. Nabufonid~ 15A (1960) 815. (61 W.E. McMillm, to lJ0pubwed [9] A. Abngmr, The principlesofmichu magnetism(ClarendonP&s, Oxford, 1961) p. 289. IlO] MB. ROE, Ehncntmy themy of angularmoma~tum(Wiley,New York, 1957). [ 111L.D. Landau ed EM Lifshitz, Tbeosy of elasticity CPergamon.London, 1959). (121 PA. Egdstdf, Rept. Rogr. Phys; 24 (1961) part 1. (131 R. Blinc, D.L. Hagenboom, D.E. O’ReiUy and EM. Peterson, whys. Rev. L&ten 23 (1969) 969. [14] cc. Sun& ‘0 be publi.lhcd. [IS] R.Blinc,K. Ebwuan, J. Pirs. M. ViEan and 1. Zupamcic. Phyc REV.Letters 25 (1970) 1327. (161 R. BIinc and V. Dhx+. Phys Letters 31A (1970) 10. (171 N. Bm, J.W. Dame, S.L.Amra and J.L. Feqason, J. Chem. Phys. 50 (1969) 1398.
Volume 10, numb& 1
NUCLEAR
MAGNETIC
RESCINANCE
IN LiQWD
1 July%971
CRYSTALS
C.C. SUNG DepmtmentofPhysics. The Ohio State University,CMumbus,Ohio 43210, USA Received 10 May 1971
The applicatibn of the theory of nuclear magnetic resonance to liquid crystals is reexaminexi and reasonable agreement is found between the present theory and experimental results.
On the basis of the theoretical work developed by De Cennes [l] and the Orsay Group [2] , Fincus [3] studied the relaxation time (Tr) of liquid crystals and obtained the field dependence (H) in good agreement with experimental results [4]. The predicted temperature (7’) dependence is unsatisfactory as shown by Doane and Visintainer [5] , who also raised a few additional questions: (1) why rotation ofH has no effect on Tl in smectic state in contrast to T2,
(2) why no discontinuity of T1 is obtained at nematic-smectic transition. Recently, Deane and Johnson [6] have generalized Pincus’ treatment and obtained an improved T dependence of Tl in the nematic state. We do not think, however, that the short range correlation time (collision time) should be neglected, as done in ref. [6], since the relaxation time T1 of liquid crystals is of the same order of ordinary liquids. Furthermore, this is important for the explanation of narrow linewidth. We point out, also, that the “viscosity” coefficient in the magnetic resonance theory should not be taken from hydrodynamical measurements. By employing the recent work of Saupe [7] and McMillan [83 we are able to show that a reasonable agreement exists between the present theory and experiment. We assume that intramolecular dipole-dipole interaction dominates the magnetic relaxation and that the relaxation times T1 and Tz are given by [9] *
(0 is the dipole-dipole where j sums over all particles with the intramolecular distance “a” from i, 9 of two spins distance ‘ir” apart and wo is the Larmor frequency. G is given by
interaction
(2) q(0,rJ) is a normalized spherical harmonic and (6,#) specify the direction of the vector rflj*co~ecting two spins at tj and rP 4 ‘1 in eq. (2)‘denotes the thermal average. In order to introduce the dynamical variable [ 1,2] forkfiu@nation, we rotate the z axis from the direction of Hto that of Mti) which is the instantaneous prefer&d direction [6,10],
35
Vohme_lO, nnn$~a 1
cm3fIcAL
PHYSCS
_-
LETTERS
.I.
1 July 1971
(3) where a, P and 7 are the EuIer’s angles. Dim @$y) = exp[-i&a] dmm@) exp[-imr] and dm,,,’ is explicitly given in ref. [l l] . Since only Q and j3 are needed to specify the transformation, y is chosen to be zero. The pres-
ent theory of liquid crystal can only handle the linear fluctuations [I] which enables us to set sin2p = 0, in eq. (3) and [2,10] l
coszfl =1 (4)
where K is the elastic constant, and q is the viscosity coeffcienc which, we want to emphasize, should not be taken from the hydrodynamical measurement.The so-called viscosity coeffkient g in eq. (4) refers only to the ro+ tation of one molecule which contains the two mxlear spins in question. How the translational motion of the molecule is hindered is not relevmt in eq. (4). Thus 9 in eq. (4) should not show the discontinuity nematic transition. In this case, 1) can be properly expressed by
at the smectic-
1)-m,
(5)
where r is the Maxwellian relaxation time orthe collision time. Eq. (5) is given by Landau and Lifshitz [ 1 l] in a phenomenologicai way, and its derivation is reviewed by Egelstaff [ 121. I’ is a short range correiative time which we have no reason to believe to be very much different from ordinary liquid, i.e., r * (d/v) eblT, where d is the intermolecular distance, V the average velocity, and b the activity energy. Whether or not this assumption is correct should be tested aith experimental data. Next, we discuss the time dependence of q(SZ’). The elongated molecules are free to rotate around n(n), but the rotation perpendicular to MH) is severely hindered. On the basis of this physical picture, the conditional probability fimction $@,Stb,t-to) can be written as [9]
5
~(s2’,SL;)J-f*) = G(cosB’-COSBb) z ’ m=_s exp[im(~‘-~~)lexp~-I~-~Ol/~ml
(6)
,
where I’;;;’ = JTZ~F-I according to the diffusion equation, and I? is taken the same as in eq. (4) for convenience. The possibility of diffusion of 0’ in eq. (6) may be incorporated by introducing a long relaxation time associated 8’. Ah the needed correlation functions in eqs. (1) and (2) are given by eq. (4), 2r,. (7j d~Y~(sz’,t,+t);~~~(~~~o)) eBiwr= iY~(s2,)12 1 +~z 03’ m
and
where fi and-Y? are assumed to be independent and cos 0 = (1 +r’, c&‘. given by the theory of Saupe (71 and McMiUan [8] _
36
lY$f(bo)12
the thermal average is
Vohune 10, number
1
CHEMICAL PHYSICS LErrERs
IY~i*= $ IY~(cosS)12
exp [ -T
gvo$(cosB)]dcmB
(J
exp [-9
1 July 1971
I$(eod)ldf2d)-’
,
(9)
0’
where g is the order parameter and VO a parameter. In our discussion above, time dependence of ea is ignored but can be incorporated in &@ by modifying the definition of T,, since n(rf) is assumed to rotate freely around the 2 axis. Tr in anphnses.
c
The important terms in eqs. (1) and (3) are according toeqs. (7) and (8)
__=p 1 8LJD 2 Jz(3c0Ge-02 T, 71 4
--
T 0 tj KK
35qjH + 2rl
(tie03sq2
1.
w
The first term is given in ref. [6 1. JZcj.(10) Ieads to several conchrsions different from those of ref. (61 in some respects: (a) T1 in liquid crystals is not much shorter than that of ordinary liquids, and we do not see any reason why PI (z lo-lo set) should be neglected. We notice that the second term is important to fit the experimental H dependence of q 1 = CIH -* + C2. The validity of this expression, however, is limited to small ho when the hydrodynamical approach of refs. [ 11 and [23 is correct, I?two 4 1. This limitation excludes the importance of magnetic energy in free ener (b) T dependence of K, 5i-T lYz I and -is sensitive to the parameter V. [7,8]. A qualitative study indicates that Tl is in agreement with ref. 5 . We note, however, that the experimental T1 may have more variety of T dependence [ 131. Also, -+7 lY212 f 1Y2I as given in ref. [63. A detailed study and related subject will be given elsewhere
t141. (c) At ,smectic-nematic
transition, g may or may not have a discontinuity depending on the choice of a parameter [83. lY~(cosB)j’ from eq. (9) thus may be continuous and a continuous T1 is obtained. At the isotopenematic transition, g has a discontinuity and so has {UT 12. This feature is confirmed in the data of TI by Blinc et al. [ 15,161. It would be interesting if there are experiments to correlate the second order phase transition with a continuous T1 and a first brderphase transition with a discontinuous Tl. We emphasize that q/K in eq. (10) should be continuous because of eq. (5). It should be pointed out that in ligbt scattering experiments [16) an equation similar to (4) is used where r amI I’ refer to the positions of two molecules. Since the diffusive mode in the problem is concerned with the motion of molecules, TJshould be taken from the hydrodynamical measurement, and a discontinuity is always expected at the amectic-nematic transition. T or the linewidth T2 in eq. (1) is dominated by Go or iI&(a$y)Y~(t?‘, $1 in q. (3)_ Thus eq. (7) gives I2 75 z WD2r 0 where I’,J is the very long collision time associated with 0’ as discussed above. JXG&‘~ % 1 as expetted, this expression is meaningless. T2 or the tinewidth of liquid crystals should have t&e features of a rigid lattice. This is indeed found experimentally [ 171, i.e., the line shape is a gaussian function with the second moment = 1G. The effect of the rotation ofH in smectic phase on the linewidth is given by
w&e ’ specifies the angle of the rotation of H and only m = 0 is important. In other words, the linewidth * Yz$B (a’)- l= r~;&)~l-l, as experimentally observed. Ibis feature again does not exist in Tt , which consists of two terms and makes the effect of tire rotation of H hard to observe (5,16l_ It is a great pleasure to thank Professor Korringa for hia valuable advice and criticism. 37
1 Jrlxy1971
-[I] P.G. de Gums, Coznpt.Ret& &ad. Sd (Paris) 266 (19S8) 15. [2] onrjr +p, J. C@IL Fhys. Sl(1969) 816. (3) P. Pina& solid statccommnn. 7 (1?6P) pis. [4] GH. l&own, J.W.Jhana adV.D. Neff. Critid Rev. Solid State Sci. 4 (1970) 303. [S] J-W. Doam antiJJ. Vi&~taincx, Phys. Rev. Letters 23 (1969) 421. [6] J.W. Doanc aad D.L. Joh+m, Chem. phus. Letters6 (1970) 291. [l] A. Saqm, 2. Nabufonid~ 15A (1960) 815. (61 W.E. McMillm, to lJ0pubwed [9] A. Abngmr, The principlesofmichu magnetism(ClarendonP&s, Oxford, 1961) p. 289. IlO] MB. ROE, Ehncntmy themy of angularmoma~tum(Wiley,New York, 1957). [ 111L.D. Landau ed EM Lifshitz, Tbeosy of elasticity CPergamon.London, 1959). (121 PA. Egdstdf, Rept. Rogr. Phys; 24 (1961) part 1. (131 R. Blinc, D.L. Hagenboom, D.E. O’ReiUy and EM. Peterson, whys. Rev. L&ten 23 (1969) 969. [14] cc. Sun& ‘0 be publi.lhcd. [IS] R.Blinc,K. Ebwuan, J. Pirs. M. ViEan and 1. Zupamcic. Phyc REV.Letters 25 (1970) 1327. (161 R. BIinc and V. Dhx+. Phys Letters 31A (1970) 10. (171 N. Bm, J.W. Dame, S.L.Amra and J.L. Feqason, J. Chem. Phys. 50 (1969) 1398.