- Email: [email protected]
Director fluctuations in a nematic liquid crystal probed using ALC spectroscopy
Physica B 289}290 (2000) 612}615
Director #uctuations in a nematic liquid crystal probed using ALC spectroscopy B.W. Lovett *, J.S. Stie{berger , S.J. Blundell , Th. JestaK dt , A. Ardavan , I.M. Marshall , F.L. Pratt , I.D. Reid Department of Physics, Clarendon Laboratory, University of Oxford, Parks Road, Oxford, OX1 3PU, UK RIKEN-RAL, Rutherford Appleton Laboratory, Chilton, Didcot OX11 0QX, UK Paul Scherrer Institute, Ch-5232 Villigen, Switzerland
Abstract We have investigated the molecular dynamics in the nematic liquid crystal 5CB using the ALC lSR technique. Our measurements are consistent with a change in the amplitude of director #uctuations at the nematic}isotropic transition and we develop a model to describe this. 2000 Elsevier Science B.V. All rights reserved. Keywords: Nematic; Director #uctuations
1. Introduction
2. Experimental details
Liquid crystals are a state of matter which do not "t in to the usual categories of simple solid (S), isotropic liquid (L) or gas (G). They have a symmetry which lies somewhere between that of a solid's rigid lattice and a liquid's dynamic isotropy, and fall into several distinct classes [1]. The liquid crystal phase with which we are concerned in this paper is the uniaxial nematic (N) phase, in which molecules have translational freedom, but are constrained in their orientation. The molecules are usually of an elongated rod-like shape and have random positions but orientations such that the long axis of each molecule tends to line up parallel to a common axis. The direction of this axis is de"ned by a unit vector known as the director, n( .
We carried out avoided-level-crossing muonspin-relaxation (ALC lSR) experiments on 4 -npentyl-4-cyanobiphenyl (5CB) (obtained from Merck Ltd., UK) using the ALC spectrometer at the Paul Scherrer Institute, Switzerland (PSI). Muon-spin-rotation measurements were also carried out on the GPD spectrometer at PSI. 5CB has a S}N transition at 243C and a N}L transition at 353C; its rod-like molecular structure is shown in Fig. 1. Muons were implanted into the sample and the time-integrated asymmetry (P ) in the number of ' decay positrons emitted forward and backward of the beam was recorded [2]. Interactions of the muon-spin with surrounding electronic and nuclear spins can cause a mixing of the muon's pure Zeeman substates at high magnetic #ux density (B) and thus result in a loss of polarization. This is an
* Corresponding author. Fax: #44-1865-272400. E-mail address: [email protected] (B.W. Lovett).
0921-4526/00/$ - see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 0 0 ) 0 0 2 9 4 - 5
B.W. Lovett et al. / Physica B 289}290 (2000) 612}615
Fig. 1. Chemical structure of 5CB.
Fig. 2. Four ALC resonances observed in 5CB in the liquid (L) and nematic (N) phases.
ALC resonance which is seen as a lower value of P . ' We record P as a function of B [2]. ' In Fig. 2, we show four such resonances which were observed in 5CB at four temperatures in the N- and L-phases (a background signal for each resonance has been subtracted). In our further analysis, we will assume that these are all *M"0 transitions (where the sum of muon and nuclear spin quantum numbers is preserved), since they can still be clearly seen well into the liquid phase [2].
3. Theoretical model and simulation of ALC spectra Once bonded to the molecule, the muon can couple to nearby electrons and nuclei through the
613
isotropic Fermi contact (A) and anisotropic dipolar (D) hyper"ne interactions. The position and shape of an ALC resonance depends on the values of A and D appropriate for the nuclear (k) and I I muonic (k) coupling to nearby electrons. We may probe director reorientational dynamics by looking at the e!ect of a time-varying anisotropic component to the muon coupling, which we assume to have the same symmetry as n( on a given molecule. Kreitzman et al. [3] developed an elegant theory which can be used to simulate the static spectra at high magnetic "eld. Here the eigenstates can be grouped into two non-interacting manifolds each having a di!erent electron spin quantum number and the muon sees a single e!ective "eld, around which it rotates. We use this theory and introduce dynamics by using the Monte-Carlo method of Tregenna}Piggott et al. [4] to simulate dynamic *M"0 resonances. We now develop a model to describe the types of liquid crystalline motion which are most relevant to the reorientational dynamics which are accessible to lSR. The model has two salient features. First, the spatial variation of the director orientation with respect to the z-axis is assumed to take on angles between zero and h with equal probability. It + takes account of the result of all the short axis rotations of the molecule which are rare and whose dynamics we have neglected in our calculations. Free energy arguments indicate that a small applied magnetic "eld aligns the molecules, so that their mean position is in the direction of the applied "eld (this is the FreH edericksz transition [5]). It does not, however, a!ect the expected range of angles for the director around this mean (this is controlled by the entropy of rod packing which has much larger free energy than any magnetic anisotropy). Thus we use zero "eld results [6] and the cone model of Wang and Pecora [5] to estimate h in the nematic + phase. Second, we introduce the thermal #uctuations of the director about its mean position (which is de"ned by an angle within our cone) by using thermodynamic arguments, following de Gennes and Prost [1]. They estimate that the angle b around the mean orientation into which molecules may roam has a mean square value given by
614
B.W. Lovett et al. / Physica B 289}290 (2000) 612}615
1b2+k ¹q /pK where q is an upper cuto! wave vector, K is a Frank elasticity constant [8] and ¹ is the temperature. We have here assumed the one-Frank-constant limit [1]. The correlation time q of a thermal #uctuation is related to the wavevector q by q &g/Kq where g is an e!ective viscosity. This equation can be integrated over wavevectors from zero to the cuto! to obtain the average correlation time q &3g/Kq. We thus obtain an expression for the amplitude of #uctuations for a given average correlation time:
k ¹ 3g . (1) 1b2+ p Kq The e!ective viscosity is slightly q dependent, but we neglect this to simplify calculations and use g"63 mPas (the value of the actual viscosity g [1] just below the N}L phase boundary and of the correct order of the e!ective viscosity de"ned by de Gennes [1] for certain q excitations). In our numerical simulations of the *M"0 ALC transition, we noticed that a dynamic modulation of the anisotropic hyper"ne coupling constant of q &10\ s has the largest resonant amplitude. We are therefore most interested in the #uctuations which have this correlation time and set q "10\ s. We can then obtain estimates of 1b2 by using the Frank constants K (¹) which are given in Ref. [8]. One further molecular motion, that of rotation around the long axis, is too fast to be observable on the muon timescale; it will merely result in an e!ectively uniaxial dipolar coupling. In simulations, the molecules start from one angle within our cone, and are then subject to #uctuations according to the mean square angle 1b2 described above. The angle j of each #uctuation was chosen according to an exponential probability distribution:
p(j)"
2 2 exp !j . 1b2 1b2
(2)
The situation is then as shown in Fig. 3, where the shaded region represents a characteristic range of director positions allowed for a muon attached to a molecule which initially points along the direction n( . The strong collision model was also as-
Fig. 3. Model for allowed director orientations in 5CB (see text).
sumed and the amplitude of each resonance was variable in the "tting procedure. We used results obtained at two temperatures in the nematic phase of 5CB (293C and 323C) and two in the liquid phase (373C and 503C). Muon Fermi contact hyper"ne constants for the four observed resonances were obtained by doing a transverse "eld muon-spin rotation experiment and correlating high "eld frequencies corresponding to transitions between levels in the electron-up and electron-down manifolds (the method is described in detail in Ref. [9]). These were assumed to be temperature independent and we used the following values: A "447.2, 460.3, 490.1 and I 493.6 MHz, which correspond to the four ALC resonances, in order of increasing magnetic "eld. Furthermore, we assumed that the only nuclear coupling was to a single proton, for which the dipolar coupling constant is &2 MHz. The "tting procedure was then carried out as follows: (1) h values were estimated for both nematic + curves by using measurements of the nematic order parameter P (see Ref. [1]) determined by SangstroK m et al. [7] in 5CB.
B.W. Lovett et al. / Physica B 289}290 (2000) 612}615 Table 1 Values of H and b used in the "tting procedure + ¹ (K)
H (rad) +
1b2 (rad)
29 32 37 50
0.721 0.721 0.698 0.419
0.01594 0.02318 0.05120 0.05335
(2) 1b2 values were estimated in the nematic phase by using Eq. (1). (3) The two sets of curves in the nematic phase were "tted by varying D and A . I (4) The low-"eld 503C resonances were "tted by varying h and 1b2, assuming an unchanged + D and A . The 373C low-"eld resonances were I then simulated by interpolating the values of 1h 2 from surrounding points and assuming + a linear ¹ dependence for 1b2 and normalizing to the value at 503C, whilst keeping other parameters "xed. (5) The high-"eld resonances in the two liquid spectra were "tted by using the same h and + 1b2 values as the low-"eld resonances and by varying D and A . I The "tted curves are shown in Fig. 2. We "nd D and A are relatively independent of temperI ature (A varies by less then 1% over the whole temperature range and D varies by less than 3%). I A more marked temperature dependence is seen in the values which we used for 1b2 and h , which + are shown in Table 1. Here one can see a distinct increase in 1b2 in the liquid phase, where molecules become much more free to #uctuate once the constraints of the molecular packing in the nematic phase are removed. Also, one can see a reduction in the cone angle h in the liquid phase. This is an +
615
unexpected result, since free energy arguments indicate that the liquid phase should be isotropic even in the presence of a moderate applied "eld. One possible explanation is that to use two angles is to over parameterize the data in the liquid phase and that once 1b2 becomes larger, h becomes less + meaningful. Indeed, the sum of h and 1b2 + behaves sensibly (i.e. becomes larger) as the nematic}liquid phase boundary is crossed.
4. Conclusion In this study, we have demonstrated that the lSR ALC technique can be used as a probe for detecting the change in molecular motion associated with a N}L phase transition. We obtain good "ts to our observed resonances in 5CB if we assume a cone model for the director distribution in the nematic phase and see evidence for more dynamic freedom of the director in the liquid phase.
References [1] P.G. de Gennes, J. Prost, The Physics of Liquid Crystals, OUP, New York, 1993. [2] M. Heming, E. Roduner, B.D. Patterson, W. Odermatt, J. Schneider, H. Baumeler, H. Keller, I. SavicH , Chem. Phys. Lett. 128 (1986) 100. [3] S.R. Kreitzman, E. Roduner, Chem. Phys. 192 (1995) 189. [4] P.L.W. Tregenna-Piggott, E. Roduner, S. Santos, Chem. Phys. 203 (1996) 317. [5] C.C. Wang, R. Pecora, J. Chem. Phys. 72 (1980) 5333. [6] V.K. FreH edericksz, A. Repiewa, Z. Phys. 42 (1927) 532. [7] D. SandstroK m, A.V. Komolkin, A. Malinak, J. Chem. Phys. 106 (1997) 7438. [8] M.J. Bradshaw, E.P Raynes, J.D. Bunning, T.E. Faber, J. Phys. 46 (1985) 1513. [9] E. Roduner, The Positive Muon as a Probe in Free Radical Chemistry, Springer, Berlin, 1988.
Director #uctuations in a nematic liquid crystal probed using ALC spectroscopy B.W. Lovett *, J.S. Stie{berger , S.J. Blundell , Th. JestaK dt , A. Ardavan , I.M. Marshall , F.L. Pratt , I.D. Reid Department of Physics, Clarendon Laboratory, University of Oxford, Parks Road, Oxford, OX1 3PU, UK RIKEN-RAL, Rutherford Appleton Laboratory, Chilton, Didcot OX11 0QX, UK Paul Scherrer Institute, Ch-5232 Villigen, Switzerland
Abstract We have investigated the molecular dynamics in the nematic liquid crystal 5CB using the ALC lSR technique. Our measurements are consistent with a change in the amplitude of director #uctuations at the nematic}isotropic transition and we develop a model to describe this. 2000 Elsevier Science B.V. All rights reserved. Keywords: Nematic; Director #uctuations
1. Introduction
2. Experimental details
Liquid crystals are a state of matter which do not "t in to the usual categories of simple solid (S), isotropic liquid (L) or gas (G). They have a symmetry which lies somewhere between that of a solid's rigid lattice and a liquid's dynamic isotropy, and fall into several distinct classes [1]. The liquid crystal phase with which we are concerned in this paper is the uniaxial nematic (N) phase, in which molecules have translational freedom, but are constrained in their orientation. The molecules are usually of an elongated rod-like shape and have random positions but orientations such that the long axis of each molecule tends to line up parallel to a common axis. The direction of this axis is de"ned by a unit vector known as the director, n( .
We carried out avoided-level-crossing muonspin-relaxation (ALC lSR) experiments on 4 -npentyl-4-cyanobiphenyl (5CB) (obtained from Merck Ltd., UK) using the ALC spectrometer at the Paul Scherrer Institute, Switzerland (PSI). Muon-spin-rotation measurements were also carried out on the GPD spectrometer at PSI. 5CB has a S}N transition at 243C and a N}L transition at 353C; its rod-like molecular structure is shown in Fig. 1. Muons were implanted into the sample and the time-integrated asymmetry (P ) in the number of ' decay positrons emitted forward and backward of the beam was recorded [2]. Interactions of the muon-spin with surrounding electronic and nuclear spins can cause a mixing of the muon's pure Zeeman substates at high magnetic #ux density (B) and thus result in a loss of polarization. This is an
* Corresponding author. Fax: #44-1865-272400. E-mail address: [email protected] (B.W. Lovett).
0921-4526/00/$ - see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 0 0 ) 0 0 2 9 4 - 5
B.W. Lovett et al. / Physica B 289}290 (2000) 612}615
Fig. 1. Chemical structure of 5CB.
Fig. 2. Four ALC resonances observed in 5CB in the liquid (L) and nematic (N) phases.
ALC resonance which is seen as a lower value of P . ' We record P as a function of B [2]. ' In Fig. 2, we show four such resonances which were observed in 5CB at four temperatures in the N- and L-phases (a background signal for each resonance has been subtracted). In our further analysis, we will assume that these are all *M"0 transitions (where the sum of muon and nuclear spin quantum numbers is preserved), since they can still be clearly seen well into the liquid phase [2].
3. Theoretical model and simulation of ALC spectra Once bonded to the molecule, the muon can couple to nearby electrons and nuclei through the
613
isotropic Fermi contact (A) and anisotropic dipolar (D) hyper"ne interactions. The position and shape of an ALC resonance depends on the values of A and D appropriate for the nuclear (k) and I I muonic (k) coupling to nearby electrons. We may probe director reorientational dynamics by looking at the e!ect of a time-varying anisotropic component to the muon coupling, which we assume to have the same symmetry as n( on a given molecule. Kreitzman et al. [3] developed an elegant theory which can be used to simulate the static spectra at high magnetic "eld. Here the eigenstates can be grouped into two non-interacting manifolds each having a di!erent electron spin quantum number and the muon sees a single e!ective "eld, around which it rotates. We use this theory and introduce dynamics by using the Monte-Carlo method of Tregenna}Piggott et al. [4] to simulate dynamic *M"0 resonances. We now develop a model to describe the types of liquid crystalline motion which are most relevant to the reorientational dynamics which are accessible to lSR. The model has two salient features. First, the spatial variation of the director orientation with respect to the z-axis is assumed to take on angles between zero and h with equal probability. It + takes account of the result of all the short axis rotations of the molecule which are rare and whose dynamics we have neglected in our calculations. Free energy arguments indicate that a small applied magnetic "eld aligns the molecules, so that their mean position is in the direction of the applied "eld (this is the FreH edericksz transition [5]). It does not, however, a!ect the expected range of angles for the director around this mean (this is controlled by the entropy of rod packing which has much larger free energy than any magnetic anisotropy). Thus we use zero "eld results [6] and the cone model of Wang and Pecora [5] to estimate h in the nematic + phase. Second, we introduce the thermal #uctuations of the director about its mean position (which is de"ned by an angle within our cone) by using thermodynamic arguments, following de Gennes and Prost [1]. They estimate that the angle b around the mean orientation into which molecules may roam has a mean square value given by
614
B.W. Lovett et al. / Physica B 289}290 (2000) 612}615
1b2+k ¹q /pK where q is an upper cuto! wave vector, K is a Frank elasticity constant [8] and ¹ is the temperature. We have here assumed the one-Frank-constant limit [1]. The correlation time q of a thermal #uctuation is related to the wavevector q by q &g/Kq where g is an e!ective viscosity. This equation can be integrated over wavevectors from zero to the cuto! to obtain the average correlation time q &3g/Kq. We thus obtain an expression for the amplitude of #uctuations for a given average correlation time:
k ¹ 3g . (1) 1b2+ p Kq The e!ective viscosity is slightly q dependent, but we neglect this to simplify calculations and use g"63 mPas (the value of the actual viscosity g [1] just below the N}L phase boundary and of the correct order of the e!ective viscosity de"ned by de Gennes [1] for certain q excitations). In our numerical simulations of the *M"0 ALC transition, we noticed that a dynamic modulation of the anisotropic hyper"ne coupling constant of q &10\ s has the largest resonant amplitude. We are therefore most interested in the #uctuations which have this correlation time and set q "10\ s. We can then obtain estimates of 1b2 by using the Frank constants K (¹) which are given in Ref. [8]. One further molecular motion, that of rotation around the long axis, is too fast to be observable on the muon timescale; it will merely result in an e!ectively uniaxial dipolar coupling. In simulations, the molecules start from one angle within our cone, and are then subject to #uctuations according to the mean square angle 1b2 described above. The angle j of each #uctuation was chosen according to an exponential probability distribution:
p(j)"
2 2 exp !j . 1b2 1b2
(2)
The situation is then as shown in Fig. 3, where the shaded region represents a characteristic range of director positions allowed for a muon attached to a molecule which initially points along the direction n( . The strong collision model was also as-
Fig. 3. Model for allowed director orientations in 5CB (see text).
sumed and the amplitude of each resonance was variable in the "tting procedure. We used results obtained at two temperatures in the nematic phase of 5CB (293C and 323C) and two in the liquid phase (373C and 503C). Muon Fermi contact hyper"ne constants for the four observed resonances were obtained by doing a transverse "eld muon-spin rotation experiment and correlating high "eld frequencies corresponding to transitions between levels in the electron-up and electron-down manifolds (the method is described in detail in Ref. [9]). These were assumed to be temperature independent and we used the following values: A "447.2, 460.3, 490.1 and I 493.6 MHz, which correspond to the four ALC resonances, in order of increasing magnetic "eld. Furthermore, we assumed that the only nuclear coupling was to a single proton, for which the dipolar coupling constant is &2 MHz. The "tting procedure was then carried out as follows: (1) h values were estimated for both nematic + curves by using measurements of the nematic order parameter P (see Ref. [1]) determined by SangstroK m et al. [7] in 5CB.
B.W. Lovett et al. / Physica B 289}290 (2000) 612}615 Table 1 Values of H and b used in the "tting procedure + ¹ (K)
H (rad) +
1b2 (rad)
29 32 37 50
0.721 0.721 0.698 0.419
0.01594 0.02318 0.05120 0.05335
(2) 1b2 values were estimated in the nematic phase by using Eq. (1). (3) The two sets of curves in the nematic phase were "tted by varying D and A . I (4) The low-"eld 503C resonances were "tted by varying h and 1b2, assuming an unchanged + D and A . The 373C low-"eld resonances were I then simulated by interpolating the values of 1h 2 from surrounding points and assuming + a linear ¹ dependence for 1b2 and normalizing to the value at 503C, whilst keeping other parameters "xed. (5) The high-"eld resonances in the two liquid spectra were "tted by using the same h and + 1b2 values as the low-"eld resonances and by varying D and A . I The "tted curves are shown in Fig. 2. We "nd D and A are relatively independent of temperI ature (A varies by less then 1% over the whole temperature range and D varies by less than 3%). I A more marked temperature dependence is seen in the values which we used for 1b2 and h , which + are shown in Table 1. Here one can see a distinct increase in 1b2 in the liquid phase, where molecules become much more free to #uctuate once the constraints of the molecular packing in the nematic phase are removed. Also, one can see a reduction in the cone angle h in the liquid phase. This is an +
615
unexpected result, since free energy arguments indicate that the liquid phase should be isotropic even in the presence of a moderate applied "eld. One possible explanation is that to use two angles is to over parameterize the data in the liquid phase and that once 1b2 becomes larger, h becomes less + meaningful. Indeed, the sum of h and 1b2 + behaves sensibly (i.e. becomes larger) as the nematic}liquid phase boundary is crossed.
4. Conclusion In this study, we have demonstrated that the lSR ALC technique can be used as a probe for detecting the change in molecular motion associated with a N}L phase transition. We obtain good "ts to our observed resonances in 5CB if we assume a cone model for the director distribution in the nematic phase and see evidence for more dynamic freedom of the director in the liquid phase.
References [1] P.G. de Gennes, J. Prost, The Physics of Liquid Crystals, OUP, New York, 1993. [2] M. Heming, E. Roduner, B.D. Patterson, W. Odermatt, J. Schneider, H. Baumeler, H. Keller, I. SavicH , Chem. Phys. Lett. 128 (1986) 100. [3] S.R. Kreitzman, E. Roduner, Chem. Phys. 192 (1995) 189. [4] P.L.W. Tregenna-Piggott, E. Roduner, S. Santos, Chem. Phys. 203 (1996) 317. [5] C.C. Wang, R. Pecora, J. Chem. Phys. 72 (1980) 5333. [6] V.K. FreH edericksz, A. Repiewa, Z. Phys. 42 (1927) 532. [7] D. SandstroK m, A.V. Komolkin, A. Malinak, J. Chem. Phys. 106 (1997) 7438. [8] M.J. Bradshaw, E.P Raynes, J.D. Bunning, T.E. Faber, J. Phys. 46 (1985) 1513. [9] E. Roduner, The Positive Muon as a Probe in Free Radical Chemistry, Springer, Berlin, 1988.