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Surface effects and dipolar correlations of confined and constrained liquids investigated by NMR relaxation experiments and computer simulations
Magnetic Resonance Imaging 19 (2001) 401– 404
Surface effects and dipolar correlations of confined and constrained liquids investigated by NMR relaxation experiments and computer simulations Farida Grinberg*, Rainer Kimmich Sektion Kernresonanzspektroskopie, Universita¨t Ulm, 89069 Ulm, Germany
Abstract Local order and molecular dynamics of liquids near surfaces strongly deviate from the behavior in the bulk. This in particular refers to liquid crystals above the bulk isotropization temperature. Transverse relaxation data of 5CB examined in porous glasses with different pore sizes are reported. A strong pore size effect was found. For the interpretation, a simple diffusion-adsorption computer simulation was carried out. Molecules can diffuse from the isotropic bulk part of the pore fluid to the ordered surface layer and vice versa. The residual dipolar correlation function is characterized by a slowly decaying tail owing to repeated returns of molecules to the surface. At each return the molecular orientation correlation is recovered as far as the surface sites visited have orientations correlated to the initial site. That is, molecular orientation is controlled by the “reorientation mediated by translational displacement” process considered in previous papers. © 2001 Elsevier Science Inc. All rights reserved. Keywords: Liquid crystals; Porous glasses; Dipolar correlations; NMR relaxation; Computer simulations
1. Introduction This study predominantly refers to liquid crystals confined in the void space of porous materials. There are important applications of such systems in new electro-optic technologies [1]. Confinements by pores and surface constraints strongly influence order and dynamics both below and above the bulk clearing point. Surface interactions give rise to orientational order even above the nematic-isotropic transition [2]. In this work, we examine this effect based on transverse relaxation experiments and computer simulations. Surface constraints are strongest in the near vicinity of the pore walls [2,3]. We therefore focused on materials with pore sizes of less than a few nanometers, i.e. systems that have attracted relatively little attention so far. The angular part of the dipole-dipole interaction of systems consisting of two spins 1⁄2 is described by time-dependent spherical harmonics. Nuclear magnetic relaxation is induced by stochastic modulations of these functions as a consequence of molecular reorientations. The analytical treatment of such surface effects is restricted to the simplest
geometries such as planar surfaces. More realistic void shapes and surface textures can numerically be treated with a suitable computer simulation program [4]. In the following, we will describe a procedure for the elucidation of the reduced dipolar correlation function.
2. Materials and instruments 4⬘-n-pentyl-4-cyanobiphenyl (5CB) was confined in porous silica glasses Bioran (mean pore dimension R ⫽ 5 nm), Vycor (Vyc, R ⫽ 2 nm), and controlled porous glasses (CPG-1.5, R ⫽ 1.5 nm; CPG-4, R ⫽ 4 nm). The bulk isotropization temperature (T NI ) of 5CB is 309.6 K. Attenuation curves of the transverse proton magnetisation were recorded at 90 and 400 MHz using high-power Bruker instruments. The transverse proton relaxation rates, T ⫺1 2 , were measured with the standard Hahn echo pulse sequence [5].
3. Computer simulations * Corresponding author. E-mail address: [email protected] (F. Grinberg).
The displacement and orientation of a molecule, henceforth called the “random walker”, was considered at or near
0730-725X/01/$ – see front matter © 2001 Elsevier Science Inc. All rights reserved. PII: S 0 7 3 0 - 7 2 5 X ( 0 1 ) 0 0 2 5 6 - 9
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F. Grinberg, R. Kimmich / Magnetic Resonance Imaging 19 (2001) 401– 404
a spherical surface of a given radius R at a time t. Within each cycle time, ⌬t ⬍⬍ t, the random walker performs a step of a fixed length ⌬l ⬍⬍ R. The position of the walker is described in a rectangular cartesian reference frame with the origin in the sphere center. Cartesion co-ordinates have been chosen in order to make the program generally applicable to surface symmetries other than spherical. Displacements along each of the co-ordinate axes are independently determined with the aid of a random-number generator. All sites in the interior of the surface sphere are equally accessible, whereas the outside space, i.e. radial distances from the center r兹x2 ⫹ y2 ⫹ z2 ⫽ ⬎ R ⫹ ⌬l, are prohibited. The random walker can be found in either of two possible states: a) it is “inside” the sphere, that is, the coordinates obey ⬍ R, or b) it is adsorbed “on” the surface, i.e. R ⱕ 7 ⱕ R ⫹ ⌬l. As long as the particle is “inside” the sphere, no preferential orientation exists, so that the average value of the spherical harmonics vanishes, 具Y 20 (t)典 ⫽ 0. That is, on the time scale the random walker needs to reach the surface, many isotropic molecular reorientations occur and no orientational correlation between the two states is retained. Since there is no restriction inside the sphere, the next migration step occurs with a probability 1. On the other hand, if the particle is “on” the surface, the current value of Y 20 (t) is determined due to the preferential orientation relative to the surface at the momentaneous position. In this preferential orientation of the molecule, it is assumed that the internuclear vector is oriented perpendicular to the surface. Taking the magnetic field along the z axis as usual, the spherical harmonics are calculated with polar angles relative to this direction. “On” the surface, the next migration step may occur with a predetermined probability W s⫺l ⬍ 1, provided that the step direction (which is again determined by the random number algorithm) complies to the confining geometry assumed. More specifically, steps toward the exterior or tangentially along the surface are not permitted. After n simulation cycles, the reduced correlation function to be evaluated is defined by G共t兲 ⫽
具Y 20共0兲Y 20共t兲典 , 具Y 220典
(1)
where 1 T3⬁ T
具Y 220典 ⫽ lim
冕
T
1 N3⬁ N
Y 220共t兲 dt ⫽ lim
0
冘Y N
2 20共m兲
m⫽0
(2)
1 T3⬁ N
冘Y N
⫽ lim
20共m
⫻ ⌬t兲Y 20共共m ⫹ n兲
m⫽0
⫻ ⌬t兲.
(3)
The limit “N 3 ⬁” practically means that N exceeds 108. All relevant time and length scales are given in terms of the predefined values of ⌬l and the diffusivity, D. The numerical algorithm furnishes a reduced dipolar auto-correlation function for diffusion between states with zero and finite values of the dipolar coupling constant. Note that this model is not identical to a two-site exchange model, where the two sites are characterized by two different local orientations. Rather we are dealing here with a process where the surface geometry is probed in a series of visits at different sites [5,6]. The values W s⫺l ⬍ 1 mean that the surface effectively slows down the particle displacement process by intermittent adsorption delays. The results of the simulations are shown in Fig. 1 for R ⫽ 10 nm, D ⫽ 1 ⫻ 10 ⫺10 m2/s, ⌬l ⫽ 0.1 nm and different values of W s⫺l as indicated in the plot. The t surf represents the time fraction 0 ⬍ t surf ⱕ 1 spent by a particle on the surface. With increasing t surf the correlation function decay slows down sharply. The correlation function generally reveals fast and slow attenuation components with relative intensities depending on W s⫺l . The slowly attenuating part may be well described by an exponential function G(t) ⬀ exp(t/ c ) as indicated by solid lines. The correlation time c was fitted to the correlation functions. The values are also given in the inset to Fig. 1. A graphical representation as a function of t surf can be found in Fig. 2.
4. Transverse relaxation Fig. 3 shows the experimental temperature dependences of T ⫺1 in 5CB confined in the four different porous glasses 2 examined. The relaxation rates depend on the pore size and, with the exception of 5CB in Bioran glass (R ⫽ 5 nm), are essentially higher than under bulk conditions. The bulk value above the clearing point is well below 100 s ⫺1 . When the temperature is decreased below T NI , the relaxation rate of 5CB in Bioran glass was found to increase abruptly. This behavior is in contrast to the finding in samples with smaller pores. Fig. 4 shows the dependence of the square root of the relaxation rates as a function of the pore radius at 313 K. With the exception of the Bioran sample, the data points can be well fitted with an exponential function.
and 具Y 20共0兲Y 20共t兲典 ⫽ 具Y 20共0兲Y 20共n ⫻ ⌬t兲典 ⫽ lim
T3⬁
1 T
冕
5. Discussion and conclusions
T
0
Y 20共t兲Y 20共t ⫹ n ⫻ ⌬t兲 dt
Relative to the bulk well above T NI , the transverse relaxation of 5CB confined in porous glasses is strongly
F. Grinberg, R. Kimmich / Magnetic Resonance Imaging 19 (2001) 401– 404
403
Fig. 1. Correlation functions, Eq. (1), obtained in the computer simulations for a sphere with R ⫽ 10 nm. The values of ⌬l and D were 0.1 nm and 1 ⫻ 10⫺10 m2/s, respectively. The values of W s⫺l are indicated in the plot. The values of t surf are the time fractions spent by a particle on the surface.
enhanced. This finding is interpreted by surface-induced orientational anisotropy of liquid crystalline molecules as already suggested by M. Vilfan et al. (see for instance Ref. [2] and references therein). The orientational anisotropy is stabilized by solid interfaces and thus prevents complete motional averaging of dipolar interactions. Diffusion between sites with different surface orientations and different levels of orientational anisotropy therefore produces a strong relaxation mechnism in the low-frequency range (kHz) which is relevant for transverse relaxation. This relaxation mechanism is corroborated by the results of our computer simulation, which demonstrates that the dipolar auto-correlation function slows down dramatically with the time spent on the surface. The computer simulation clearly reveals two characteri-
stic time scales of the correlation decay. The fast component occurring in times ⬍10⫺8 s (i.e. 公Dt ⬍ 1 nm) is attributed to initially adsorbed molecules leaving the surface and escaping into bulk with no return to the surface during the measuring interval. The slow component is connected with correlation times in the order of 10⫺6 (that is, 公Dt ⬇ 10 nm). This mechanism is attributed to molecules repeatedly returning to the surface, They thus effectively diffuse along the surface in a sort of hopping process. Eventually all correlation is lost after a time needed to visit all sites on the surface. The characteristic time of this process depends on the adsorbing and orienting properties of the surface which effectively slows down the diffusion along the surface. The experimental transverse relaxation data show that surface effects in relatively small pores are so strong that the liquid-nematic transition at the bulk clearing point is su-
Fig. 2. Fitted values of c as a function of t surf for the correlation functions shown in Fig. 1.
Fig. 3. Temperature dependences of the relaxation rates, T ⫺1 2 , in 5CB confined in the four different porous glasses.
404
F. Grinberg, R. Kimmich / Magnetic Resonance Imaging 19 (2001) 401– 404
Fig. 4. Square roots of the relaxation rates represented in Fig. 3 as a function of R at 313 K. The solid line is the fit of the exponential function, 公T ⫺1 ⫽ 公T ⫺1 2 2 (R30) exp(⫺R/ ), to the data points for R ⱕ 4 nm. The ⫺0.5 fitted values of 公T ⫺1 and ⫽ 2.67 nm, 2 (R30) and were 120.9 s respectively.
pressed. In the vicinity of T NI , a jumplike change of T ⫺1 2 , that is, a jumplike change of the order parameter, was observed only in Bioran glass with a mean pore dimension of 5 nm. Of course, it must be kept in mind that the transition properties do not depend only on the characteristic pore size but also on the specific void structure and surface interactions. The decrease of relaxation rates with increasing pore size is attributed to the averaging effect following from translational diffusion between regions with different surface orientation and different degrees of local anisotropies. The local order parameter induced by the surfaces is expected to decrease with the distance r from the wall as [2]
冉
R⫺r S共r兲 ⬀ exp ⫺
⫽ 0
冑
T* , T ⫺ T*
冊
(4)
(5)
where is a characteristic correlation length, 0 is of the order of the molecule length, T is the temperature and T* is about (T NI ⫺ 1) K. The transverse relaxation rates are proportional to the square of the (residual) dipolar couplings. In confined liquid crystals above T NI the square root of the relaxation rates then appears to be proportional to the local orientational order parameter induced by the surface, 公T ⫺1 ⬀ S. The 2 exponential dependence of 公T ⫺1 2 on R shown in Fig. 4 thus reflects the change of the average S with increasing R. A correlation length of 2.67 nm was fitted to the data. This value characterizes the variation of the surface-induced orientational anisotropy, and is reasonably close to the value ⬇ 5.4 nm evaluated from Eq. (5) for 0 ⬇ 0.65 nm (for 5CB) and T ⫽ 313 K. Acknowledgments We thank Professor Dr. M. Vilfan for fruitful discussions and the Deutsche Forschungsgemeinschaft for financial support. References [1] Crawford GP, Zumer S, editors. Liquid Crystals in Complex Geometries. London: Taylor & Francis, 1996. [2] Vilfan M, Vrbancic-Kopac N, Ziherl P, Crawford GP. Deuteron NMR relaxometry applied to confined liquid crystals. Appl Magn Reson 1999;17:329 – 44. [3] Grinberg F, Kimmich R. Pore size dependence of the dipolar-correlation effect on the stimulated echo in liquid crystals confined in porous glass. J Chem Phys 1996;105:3301– 6. [4] Valiullin R, Kimmich R, Fatkullin N. Le`vi walks of strong adsorbates on the surface: computer simulation and spin-lattice relaxation. Phys Rev E 1997;56:4371–5. [5] Kimmich R. NMR: Tomography, Diffusometry, Relaxometry. Heidelberg: Springer-Verlag, 1997. [6] Zavada T, Kimmich R. The anomalous adsorbate dynamics at surfaces in porous media studied by nuclear magnetic resonance methods. The orientational structure factor and Le`vi walks. J Chem Phys 1998;109:6929 –39.
Surface effects and dipolar correlations of confined and constrained liquids investigated by NMR relaxation experiments and computer simulations Farida Grinberg*, Rainer Kimmich Sektion Kernresonanzspektroskopie, Universita¨t Ulm, 89069 Ulm, Germany
Abstract Local order and molecular dynamics of liquids near surfaces strongly deviate from the behavior in the bulk. This in particular refers to liquid crystals above the bulk isotropization temperature. Transverse relaxation data of 5CB examined in porous glasses with different pore sizes are reported. A strong pore size effect was found. For the interpretation, a simple diffusion-adsorption computer simulation was carried out. Molecules can diffuse from the isotropic bulk part of the pore fluid to the ordered surface layer and vice versa. The residual dipolar correlation function is characterized by a slowly decaying tail owing to repeated returns of molecules to the surface. At each return the molecular orientation correlation is recovered as far as the surface sites visited have orientations correlated to the initial site. That is, molecular orientation is controlled by the “reorientation mediated by translational displacement” process considered in previous papers. © 2001 Elsevier Science Inc. All rights reserved. Keywords: Liquid crystals; Porous glasses; Dipolar correlations; NMR relaxation; Computer simulations
1. Introduction This study predominantly refers to liquid crystals confined in the void space of porous materials. There are important applications of such systems in new electro-optic technologies [1]. Confinements by pores and surface constraints strongly influence order and dynamics both below and above the bulk clearing point. Surface interactions give rise to orientational order even above the nematic-isotropic transition [2]. In this work, we examine this effect based on transverse relaxation experiments and computer simulations. Surface constraints are strongest in the near vicinity of the pore walls [2,3]. We therefore focused on materials with pore sizes of less than a few nanometers, i.e. systems that have attracted relatively little attention so far. The angular part of the dipole-dipole interaction of systems consisting of two spins 1⁄2 is described by time-dependent spherical harmonics. Nuclear magnetic relaxation is induced by stochastic modulations of these functions as a consequence of molecular reorientations. The analytical treatment of such surface effects is restricted to the simplest
geometries such as planar surfaces. More realistic void shapes and surface textures can numerically be treated with a suitable computer simulation program [4]. In the following, we will describe a procedure for the elucidation of the reduced dipolar correlation function.
2. Materials and instruments 4⬘-n-pentyl-4-cyanobiphenyl (5CB) was confined in porous silica glasses Bioran (mean pore dimension R ⫽ 5 nm), Vycor (Vyc, R ⫽ 2 nm), and controlled porous glasses (CPG-1.5, R ⫽ 1.5 nm; CPG-4, R ⫽ 4 nm). The bulk isotropization temperature (T NI ) of 5CB is 309.6 K. Attenuation curves of the transverse proton magnetisation were recorded at 90 and 400 MHz using high-power Bruker instruments. The transverse proton relaxation rates, T ⫺1 2 , were measured with the standard Hahn echo pulse sequence [5].
3. Computer simulations * Corresponding author. E-mail address: [email protected] (F. Grinberg).
The displacement and orientation of a molecule, henceforth called the “random walker”, was considered at or near
0730-725X/01/$ – see front matter © 2001 Elsevier Science Inc. All rights reserved. PII: S 0 7 3 0 - 7 2 5 X ( 0 1 ) 0 0 2 5 6 - 9
402
F. Grinberg, R. Kimmich / Magnetic Resonance Imaging 19 (2001) 401– 404
a spherical surface of a given radius R at a time t. Within each cycle time, ⌬t ⬍⬍ t, the random walker performs a step of a fixed length ⌬l ⬍⬍ R. The position of the walker is described in a rectangular cartesian reference frame with the origin in the sphere center. Cartesion co-ordinates have been chosen in order to make the program generally applicable to surface symmetries other than spherical. Displacements along each of the co-ordinate axes are independently determined with the aid of a random-number generator. All sites in the interior of the surface sphere are equally accessible, whereas the outside space, i.e. radial distances from the center r兹x2 ⫹ y2 ⫹ z2 ⫽ ⬎ R ⫹ ⌬l, are prohibited. The random walker can be found in either of two possible states: a) it is “inside” the sphere, that is, the coordinates obey ⬍ R, or b) it is adsorbed “on” the surface, i.e. R ⱕ 7 ⱕ R ⫹ ⌬l. As long as the particle is “inside” the sphere, no preferential orientation exists, so that the average value of the spherical harmonics vanishes, 具Y 20 (t)典 ⫽ 0. That is, on the time scale the random walker needs to reach the surface, many isotropic molecular reorientations occur and no orientational correlation between the two states is retained. Since there is no restriction inside the sphere, the next migration step occurs with a probability 1. On the other hand, if the particle is “on” the surface, the current value of Y 20 (t) is determined due to the preferential orientation relative to the surface at the momentaneous position. In this preferential orientation of the molecule, it is assumed that the internuclear vector is oriented perpendicular to the surface. Taking the magnetic field along the z axis as usual, the spherical harmonics are calculated with polar angles relative to this direction. “On” the surface, the next migration step may occur with a predetermined probability W s⫺l ⬍ 1, provided that the step direction (which is again determined by the random number algorithm) complies to the confining geometry assumed. More specifically, steps toward the exterior or tangentially along the surface are not permitted. After n simulation cycles, the reduced correlation function to be evaluated is defined by G共t兲 ⫽
具Y 20共0兲Y 20共t兲典 , 具Y 220典
(1)
where 1 T3⬁ T
具Y 220典 ⫽ lim
冕
T
1 N3⬁ N
Y 220共t兲 dt ⫽ lim
0
冘Y N
2 20共m兲
m⫽0
(2)
1 T3⬁ N
冘Y N
⫽ lim
20共m
⫻ ⌬t兲Y 20共共m ⫹ n兲
m⫽0
⫻ ⌬t兲.
(3)
The limit “N 3 ⬁” practically means that N exceeds 108. All relevant time and length scales are given in terms of the predefined values of ⌬l and the diffusivity, D. The numerical algorithm furnishes a reduced dipolar auto-correlation function for diffusion between states with zero and finite values of the dipolar coupling constant. Note that this model is not identical to a two-site exchange model, where the two sites are characterized by two different local orientations. Rather we are dealing here with a process where the surface geometry is probed in a series of visits at different sites [5,6]. The values W s⫺l ⬍ 1 mean that the surface effectively slows down the particle displacement process by intermittent adsorption delays. The results of the simulations are shown in Fig. 1 for R ⫽ 10 nm, D ⫽ 1 ⫻ 10 ⫺10 m2/s, ⌬l ⫽ 0.1 nm and different values of W s⫺l as indicated in the plot. The t surf represents the time fraction 0 ⬍ t surf ⱕ 1 spent by a particle on the surface. With increasing t surf the correlation function decay slows down sharply. The correlation function generally reveals fast and slow attenuation components with relative intensities depending on W s⫺l . The slowly attenuating part may be well described by an exponential function G(t) ⬀ exp(t/ c ) as indicated by solid lines. The correlation time c was fitted to the correlation functions. The values are also given in the inset to Fig. 1. A graphical representation as a function of t surf can be found in Fig. 2.
4. Transverse relaxation Fig. 3 shows the experimental temperature dependences of T ⫺1 in 5CB confined in the four different porous glasses 2 examined. The relaxation rates depend on the pore size and, with the exception of 5CB in Bioran glass (R ⫽ 5 nm), are essentially higher than under bulk conditions. The bulk value above the clearing point is well below 100 s ⫺1 . When the temperature is decreased below T NI , the relaxation rate of 5CB in Bioran glass was found to increase abruptly. This behavior is in contrast to the finding in samples with smaller pores. Fig. 4 shows the dependence of the square root of the relaxation rates as a function of the pore radius at 313 K. With the exception of the Bioran sample, the data points can be well fitted with an exponential function.
and 具Y 20共0兲Y 20共t兲典 ⫽ 具Y 20共0兲Y 20共n ⫻ ⌬t兲典 ⫽ lim
T3⬁
1 T
冕
5. Discussion and conclusions
T
0
Y 20共t兲Y 20共t ⫹ n ⫻ ⌬t兲 dt
Relative to the bulk well above T NI , the transverse relaxation of 5CB confined in porous glasses is strongly
F. Grinberg, R. Kimmich / Magnetic Resonance Imaging 19 (2001) 401– 404
403
Fig. 1. Correlation functions, Eq. (1), obtained in the computer simulations for a sphere with R ⫽ 10 nm. The values of ⌬l and D were 0.1 nm and 1 ⫻ 10⫺10 m2/s, respectively. The values of W s⫺l are indicated in the plot. The values of t surf are the time fractions spent by a particle on the surface.
enhanced. This finding is interpreted by surface-induced orientational anisotropy of liquid crystalline molecules as already suggested by M. Vilfan et al. (see for instance Ref. [2] and references therein). The orientational anisotropy is stabilized by solid interfaces and thus prevents complete motional averaging of dipolar interactions. Diffusion between sites with different surface orientations and different levels of orientational anisotropy therefore produces a strong relaxation mechnism in the low-frequency range (kHz) which is relevant for transverse relaxation. This relaxation mechanism is corroborated by the results of our computer simulation, which demonstrates that the dipolar auto-correlation function slows down dramatically with the time spent on the surface. The computer simulation clearly reveals two characteri-
stic time scales of the correlation decay. The fast component occurring in times ⬍10⫺8 s (i.e. 公Dt ⬍ 1 nm) is attributed to initially adsorbed molecules leaving the surface and escaping into bulk with no return to the surface during the measuring interval. The slow component is connected with correlation times in the order of 10⫺6 (that is, 公Dt ⬇ 10 nm). This mechanism is attributed to molecules repeatedly returning to the surface, They thus effectively diffuse along the surface in a sort of hopping process. Eventually all correlation is lost after a time needed to visit all sites on the surface. The characteristic time of this process depends on the adsorbing and orienting properties of the surface which effectively slows down the diffusion along the surface. The experimental transverse relaxation data show that surface effects in relatively small pores are so strong that the liquid-nematic transition at the bulk clearing point is su-
Fig. 2. Fitted values of c as a function of t surf for the correlation functions shown in Fig. 1.
Fig. 3. Temperature dependences of the relaxation rates, T ⫺1 2 , in 5CB confined in the four different porous glasses.
404
F. Grinberg, R. Kimmich / Magnetic Resonance Imaging 19 (2001) 401– 404
Fig. 4. Square roots of the relaxation rates represented in Fig. 3 as a function of R at 313 K. The solid line is the fit of the exponential function, 公T ⫺1 ⫽ 公T ⫺1 2 2 (R30) exp(⫺R/ ), to the data points for R ⱕ 4 nm. The ⫺0.5 fitted values of 公T ⫺1 and ⫽ 2.67 nm, 2 (R30) and were 120.9 s respectively.
pressed. In the vicinity of T NI , a jumplike change of T ⫺1 2 , that is, a jumplike change of the order parameter, was observed only in Bioran glass with a mean pore dimension of 5 nm. Of course, it must be kept in mind that the transition properties do not depend only on the characteristic pore size but also on the specific void structure and surface interactions. The decrease of relaxation rates with increasing pore size is attributed to the averaging effect following from translational diffusion between regions with different surface orientation and different degrees of local anisotropies. The local order parameter induced by the surfaces is expected to decrease with the distance r from the wall as [2]
冉
R⫺r S共r兲 ⬀ exp ⫺
⫽ 0
冑
T* , T ⫺ T*
冊
(4)
(5)
where is a characteristic correlation length, 0 is of the order of the molecule length, T is the temperature and T* is about (T NI ⫺ 1) K. The transverse relaxation rates are proportional to the square of the (residual) dipolar couplings. In confined liquid crystals above T NI the square root of the relaxation rates then appears to be proportional to the local orientational order parameter induced by the surface, 公T ⫺1 ⬀ S. The 2 exponential dependence of 公T ⫺1 2 on R shown in Fig. 4 thus reflects the change of the average S with increasing R. A correlation length of 2.67 nm was fitted to the data. This value characterizes the variation of the surface-induced orientational anisotropy, and is reasonably close to the value ⬇ 5.4 nm evaluated from Eq. (5) for 0 ⬇ 0.65 nm (for 5CB) and T ⫽ 313 K. Acknowledgments We thank Professor Dr. M. Vilfan for fruitful discussions and the Deutsche Forschungsgemeinschaft for financial support. References [1] Crawford GP, Zumer S, editors. Liquid Crystals in Complex Geometries. London: Taylor & Francis, 1996. [2] Vilfan M, Vrbancic-Kopac N, Ziherl P, Crawford GP. Deuteron NMR relaxometry applied to confined liquid crystals. Appl Magn Reson 1999;17:329 – 44. [3] Grinberg F, Kimmich R. Pore size dependence of the dipolar-correlation effect on the stimulated echo in liquid crystals confined in porous glass. J Chem Phys 1996;105:3301– 6. [4] Valiullin R, Kimmich R, Fatkullin N. Le`vi walks of strong adsorbates on the surface: computer simulation and spin-lattice relaxation. Phys Rev E 1997;56:4371–5. [5] Kimmich R. NMR: Tomography, Diffusometry, Relaxometry. Heidelberg: Springer-Verlag, 1997. [6] Zavada T, Kimmich R. The anomalous adsorbate dynamics at surfaces in porous media studied by nuclear magnetic resonance methods. The orientational structure factor and Le`vi walks. J Chem Phys 1998;109:6929 –39.