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Shear of Molecularly Confined Liquid Crystals. 2. Stress Anisotropy across a Model Nematogen Compressed between Sliding Solid Surfaces Joanna Janik,† Rafael Tadmor,‡ and Jacob Klein* Department of Materials and Interfaces, Weizmann Institute of Science, Rehovot 76100, Israel Received October 2, 2000. In Final Form: April 17, 2001 We have used a new design of the mica surface force balance (SFB), with extreme sensitivity in measuring normal and particularly shear or frictional forces between two surfaces sliding past each other, to measure the forces between two mica surfaces across a confined 4-cyano-4′-hexabiphenyl nematogen (6CB). In an earlier study (Langmuir 1997, 13, 4466, paper 1 of the series) we investigated the normal force-distance profiles and the orientation of the confined liquid crystal (LC). Here we extend this to study the shear forces Fs between the sliding mica surfaces across the 6CB nematogen as a function of orientation of the confined LC, the applied normal load Fn, the separation D of the mica surfaces, the shear velocity vs, and the relative shear direction of the confining surfaces (which may be varied using the new SFB design). Our results, where the shear forces are measured down to levels that are some orders of magnitude more sensitive than in earlier studies, show that the highly confined nematogen (D in the range from 16 to ca. 100 Å) behaves under shear in a quasi-solidlike fashion for all three orientations studied: planar, planar twisted, and homeotropic. There is a linear relation between Fs and Fn for each of the three orientations, with the effective friction coefficient (dFs/dFn) largest for the homeotropic orientation and lowest for the planar twisted one. Intriguingly, for the planar orientation a clear increase in the friction could be observed when the initial shear direction was changed by 90°. We attribute this to the effect of the initial shear in orienting the confined nematogen layer in the initial direction, so that subsequent sliding motion at right angles to this encounters greater resistance. We also find that within a range of some 40-fold in vs, there was little change in the shear force, in line with what is generally observed for solid-solid friction, and consistent with the fact that little relaxation in Fs is observed over macroscopic times on applying a step strain to the confined LC.
1. Introduction Hydrodynamic properties of liquid crystalline substances have been of great interest since soon after the discovery of this state of matter, and first attempts to find the anisotropy of the viscosity coefficient of a liquid crystal (LC) were undertaken as early as 1913.1 The anisotropy of the viscosity coefficient of an LC was determined only in 1935 by Miesowicz,2,3 and subsequently rheological properties of liquid crystals were investigated from the point of view of both basic knowledge and application. The formulation of Ericksen-Leslie-Parodi theory4-6 allowed the shear stress tensor to be expressed in terms of five independent viscosity coefficients, and experimental effort was directed to measuring all five Leslie coefficients, which was achieved by two independent groups at approximately the same time.7-9 Further studies examined the temperature and order parameter dependence of viscosity coefficients10-15 and extended the microscopic theory of the hydrodynamics of liquid crystals.14-16 * To whom correspondence should be addressed: W.I.S. or to: Physical and Theoretical Chemistry Laboratory, Oxford University, Oxford OX1 3QZ, U.K. (
[email protected]). † Affiliated also to: Department of Physics, Jagellonian University, Krakow, Poland. ‡ Present address: Department of Chemical Engineering, University of California at Santa Barbara, CA. (1) Neufeld, M. W. Phys. Z. 1913, 14, 645. (2) Miesowicz, A. Nature 1935, 136, 261. (3) Miesowicz, M. Nature 1946, 158, 27. (4) Ericksen, J. L. Arch. Ration. Mech. Anal. 1960, 4, 231. (5) Leslie, F. M. Arch. Ration. Mech. Anal. 1968, 28, 265. (6) Parodi, O. J. Phys. 1970, 31, 581. (7) Ga¨hviller, C. Phys. Lett. 1971, 36a, 311. (8) Ga¨hviller, C. Mol. Cryst. Liq. Cryst. 1973, 20, 301. (9) Orsay, L. C. G. Mol. Cryst. Liq. Cryst. 1971, 13, 187.
The anisotropic structural and optical properties of LC molecules, which have widespread practical applications, result from their rodlike structure. Corresponding asymmetry in the dynamic characteristics of the liquid crystalline state could also be of considerable interest: For example, in situations where surfaces may slide past each other in different relative directions across liquid crystals, such as anisotropy of shear or frictional forces. Indeed the conjunction of tribology and the liquid crystalline state has been examined in some detail.17 The possibility of using LC’s as lubricants was investigated by Fisher et al.,18 Cognard,19 and Lauer,20 and Cognard19 also reviewed extensively the use of LC’s in the nematic phase as friction modifiers (including the nematogen 5CB analogous to the materials used by us). Since the friction of rubbing surfaces frequently involves shear across gaps of molecular dimensions, the properties of LC’s confined to such dimensions are clearly relevant. Several studies have investigated the structural and interaction characteristics of (10) Diogo, A. C.; Martins, A. F. Mol. Cryst. Liq. Cryst. 1981, 66, 133. (11) Imura, H.; Okano, K. Jpn. J. Appl. Phys. 1972, 11, 1440. (12) Kneppe, H.; Schneider, F.; Sharma, N. H. Ber. Bunsen-Ges. Phys. Chem. 1981, 85, 784. (13) Kralj, S.; Zumer, S.; Allender, D. W. Phys. Rev. A 1991, 43, 2943. (14) Kuzuu, N.; Doi, M. J. Phys. Soc. Jpn. 1983, 52, 3486. (15) Osipov, M. A.; Terentjev, E. M. Phys. Lett. 1989, 134a, 301. (16) Fialkowski, M. Phys. Rev. E 1997, 55, 2902. (17) See for example: Tribology and the liquid crystalline state’; Biresaw, G., Ed.; ACS Symposium Series 441; American Chemical Society: Washington, DC, 1990. (18) Fisher, T. E.; Bhattacharya, S.; Salhher, R.; Lauer, J. L.; Ahn, Y. J. 1988, 31, 442. (19) Cognard, J. Tribology and the Liquid-Crystalline State; ACS Symposium Series 441; American Chemical Society: Washington, DC, 1990; p 1. (20) Lauer, J. L.; Ahn, Y. J.; Fischer, T. E. Tribology and the LiquidCrystalline State: ACS Symposium Series 441; American Chemical Society: Washington, DC, 1990); p 61.
10.1021/la001392q CCC: $20.00 © 2001 American Chemical Society Published on Web 08/11/2001
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Figure 1. Schematic summary of the different orientations adopted by 6CB in its nematic phase confined between (cylindrically curved) mica surfaces, from the results of paper 1.25 Shown schematically for each of the three orientations are the corresponding normal-force vs surface-separation profile and the refractive-index vs surface separation profile. In addition are shown the characteristic shapes of the interference fringes (FECO) at far-field surface separations associated with the respective orientations. (a) planar orientation: the force profile is repulsive, and the FECO have a characteristic shape resulting in two refractive indices (no (ordinary) and ne (extraordinary)); (b) planar twisted orientation: the force profile is repulsive, the FECO are regular doublets or singlets, from which only one refractive index of the average value of the ordinary and extraordinary refractive indices is measured; (c) homeotropic orientation: the force profile shows oscillatory forces (in paper 125 a single attractive well only was observed due to the use of single cantilever spring), while the FECO are regular doublets or singlets, from which only one refractive index is measurable.
different LC’s confined to interfacial layers orsusing the mica surface force balancescompressed to ultrathin layers between smooth solid surfaces.21 There has been far less work on the shear properties of such molecularly thin layers. In addition to the work cited above,18-20 Safinya, Israelachvili and co-workers22,23 probed the structure of a smectic LC (8CB) undergoing shear, down to surface separations of some 400 nm, while Ruths et al.24 measured the friction between mica surfaces across the same LC. In the first part25 of this study (henceforth paper 1) the structure and orientation of a nematic liquid crystal (4cyano-4′-hexylbiphenyl (6CB)ssee below) confined between two mica surfaces to molecular dimensions were investigated comprehensively. The effect of the relative orientation of the mica surface crystal lattice on the structure of the confined LC, the nature of the mica surface (extent of mica hydrophobicity and effect of humidity and water penetration, for example), and the issue of transitions between different LC orientations within the film were studied in detail in paper 1.25 Three possible orientations of the confined 6CB were found; planar, planar twisted, and homeotropic, and these are summarized schematically in Figure 1. (21) For references to recent work see: refs 24 and 25. See also: Moreau, L.; Richetti, P.; Barois, P. Phys. Rev. Lett. 1994, 73, 3556. (22) Idziak, S. H. J.; et al. Science 1994, 264, 1915. (23) Koltover, I.; et al. J. Phys. II 1996, 6, 893. (24) Ruths, M.; Steinberg, S.; Israelachvili, J. N. Langmuir 1996, 12, 6637. (25) Janik, J.; Tadmor, R.; Klein, J. Langmuir 1997, 13, 4466-4473.
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The nature of each orientation of the LC could be readily resolved from the force profiles and refractive index profiles, as indicated in Figure 1. The present work extends this study, and here the shear behavior of the confined 6CB in its different orientations (planar, planar twisted, and homeotropic) is investigated in detail. In particular, we examine possible effects of anisotropy induced by the LC orientation on the surface, by shearing the thin LC layers in different relative directions. The experimental technique is described in section 2, especially the modification which enables shear of the two surfaces in different relative directions; the results are presented in section 3 and discussed in section 4, where we pay attention to reorientation due to shear. Within the scatter we find that the shear force is proportional to the normal load for all three types of orientation, but also thatsfor the planar configurationssthe liquid crystal responds to shear differently when the surfaces slide past each other in two perpendicular directions. In this work we extend by some orders of magnitude the sensitivity and resolution of the shear forces measured across sheared molecularly thin LC layers. This enables us to probe in detail a regime of the confined LC that has not hitherto been investigated in the tribological studies, and we are able to examine for the first time the effect of different relative directions on the shear forces across such confined species. 2. Experimental Section 2.1. Materials. We use the same model LC as described in paper I, the nematogen 4-cyano-4′-hexylbiphenyl, 6CB. Its structure is given as follows.
This LC is crystalline below 14.5 °C, nematic in the range 14.5-29.4 °C, and isotropic at higher temperatures. At the temperature of our experimentss25.0 ( 0.2 °C throughoutsthe 6CB is in its nematic phase in the bulk, and there is evidence that it is in the dimerized form.26,27 2.2. Measuring Shear Forces. Forces between mica surfaces across the confined LC were determined using a surface force balance (SFB). The general principles and procedure have been described extensively; in extending the measurements to shear, we use a new version of the SFB,28 as shown and described in Figure 2. The liquid crystal is introduced between two curved mica surfaces (radius R ≈ 1 cm) glued in a crossed-cylinder configuration onto cylindrical lenses (see Figure 2), the upper of which is mounted on a piezoelectric tube (PZT) divided into 4 sectors A-D (Figure 3). By applying opposite voltages to sectors facing one another (i.e. A and C or B and D) one can obtain a shear motion in the direction of AC or BD (see Figure 3). Normal motion between the surfaces is obtained by applying a voltage to the inner sector E of the PZT or to a differential spring system on which the springs are mounted (Figure 2). The separation D of the surfaces is measured via fringes of equal chromatic order (FECO), and the normal forces Fn(D) between them are measured via the bending of the normal force spring (N in Figure 2), as described in detail elsewhere, for example in paper 1.25 The shear forces Fs between the surfaces, in response to an applied lateral motion of the top lens, are monitored by recording the bending of the shear springs S, to within (3 Å, via changes in the capacitance of an air-gap capacitor (Figure 2). We emphasize in particular the sensitivity and resolution of this new version of the SFB in measuring shear forces, which is considerably better than (0.1 µN in optimal cases, and of the (26) Leadbetter, A. J.; Richardson, R. M.; Colling, C. N. J. Phys. II 1975, 36, C1-37. (27) Ratna, B. R.; Shashidar, R. Mol. Cryst. Liq. Cryst. 1977, 42, 113. (28) First results using this version of the SFB (though without providing a description) were reported by: Eiser; et al. Phys. Rev. Lett. 1999, 85, 5076
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Figure 2. Schematic illustration of the surface force balance (SFB) used in the present study.28 Atomically smooth mica sheets are mounted on the crossed cylindrical lenses (top left inset), and their separation D is measured (to (1 Å) using multiple beam interference and controlled (to (1 Å) via a three stage mechanism terminating with a sectored piezoelectric tube (PZT) on which the top surface is mounted. Lateral motion in the x direction parallel to the lower surface, which is mounted on two orthogonal springs, is also provided by the PZT (to (1 Å). Parallelicity can be controlled via a feedback potential to the inner conducting surface of the PZT.32 The bending of the normal force spring N is monitored via changes in D, and that of the shear force spring S, ∆x, via changes in an air-gap capacitor (to (2 Å), top right inset. These provide a measure of normal and shear forces, respectively. The spring constant Ks of S is adjustable via a sliding clamp C (its value in the present study is set at Ks ) 68 N/m). The optimal sensitivity of this SFB in measuring shear forces between the surfaces is an order of magnitude better than in a previously described version,32 due mainly to improved signal-to-noise ratio arising from the present spring configuration. It is several orders of magnitude more sensitive in measuring shear stresses than scanning probe methods such as friction-force microscopy.32 order of (0.3 µN in the present experiments. As will be discussed in section 4, this enables us access for the first time to the shear properties of an extended confinement regime. The signal from the capacitor probe is recorded by an XY plotter or by a recording oscilloscope. The leaf springs (S in Figures 2 and 3) can only bend in one direction, at right angles to their length: The usual (standard) method of mounting the PZT (which has been used in all experiments reported to date29-32) has therefore been for the opposing sectors onto which equal and opposite voltages are applied (say sectors A and C) to be themselves at 90° to the leaf-springs, as shown in Figure 3a. With this standard mounting, the lateral motion resulting from opposing voltages on sectors A and C is then entirely at right angle to the shear springs, so that the full shear force is in the AC direction, and is reflected in the shear-spring bending (and the change in the air gap capacitance). To measure the shear response in two perpendicular directions, as in these experiments, the PZT was mounted in a nonstandard way with respect to the lower shear spring, as shown in Figure 3b. The two sets of sectors are now at an angle R = 45° with respect to the normal to the shear springs, as in Figure 3b. For this configuration, lateral forces in AC or BD directions both now have a component that can bend the shear springs. The shear forces F1 and F2 in the two perpendicular directions are now given by F1 ) FAC/(cos R), F2 ) FBD/(sin R), where FAC and FBD (the projection of F1 and F2 normal to the shear springs) are the measured forces when lateral motion is applied in the (29) Klein, J.; Kumacheva, E.; Mahalu, D.; Perahia, D.; Fetters, L. J. Nature 1994, 370, 634. (30) Klein, J.; Kumacheva, E.; Mahalu, D.; Perahia, D.; Warburg, S. Faraday Discuss. 1994, 98, 173. (31) Klein, J.; Kumacheva, E. Science 1995, 269, 816. (32) Klein, J.; Kumacheva, E. J. Chem. Phys. 1998, 108, 6996.
respective AC or BD directions. Note that the bending of the shear springs ∆xAC and ∆xBD are related to FAC and FBD as FAC = Ks∆xAC and FBD = Ks∆xBD, where Ks ()68 N/m) is the force constant of the shear spring S. Before each experiment the responses F°AC and F°BD in air contact, when the surfaces move rigidly together, was measured, enabling calibration of the applied shear motion and of the angle R ) tan-1(F°BD/F°AC). This is illustrated in Figure 3d-f: trace 3d shows the input voltage into either sectors A and C or B and D. Traces 3e and 3f show the response in shear spring bending for the respective inputs when the PZT is in the standard configuration (Figure 3a): The response to the potential on BD (3f) is, as expected, much smaller than for AC (3e), and the corresponding angle is R ) 3.2°, showing that the sectors AC are almost exactly at right angles to S. Traces 3e′ and 3f′ are for the nonstandard configuration (Figure 3b): the responses to potentials on AC (3e′) and on BD (3f′) are now almost equal, and the angle corresponding to the data of traces 3e′ and 3f′ is R ) 42.4°. Another approach to the issue of measurement in different relative sliding directions was developed as part of this study and is described here for completeness. The sectored PZT is mounted in either the standard (Figure 3a) or the nonstandard (Figure 3b) configuration, but the potential input to the PZT allows different pairs of voltages to be applied independently and simultaneously to both pairs of sectors. In practice the power supply was designed to apply voltages to the sectors A,C and B,D leading to lateral displacements ∆xAC ) ∆xtot. cos θ and ∆xBD ) ∆xtot. sin θ, respectively, so that the overall amplitude of the resulting motion is ∆xtot. and its angle with respect to the normal to the shear springs S is (R + θ), as illustrated in Figure 3c. Both the amplitude ∆xtot. and the angle θ can be independently varied. Figure 4 illustrates how the bending of the shear springs varies with θ when such a waveform is applied to the surfaces in
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Figure 3. Illustrating the different ways of applying lateral motion of the top surface relative to the shear springs S and to the lower surface. (a) Standard configuration: the direction of the PZT sectors A, C, along which lateral motion is applied, is approximately normal to the length of the shear spring S. Only the bending of S in this directionsthe bending directionsis measurable (the response to potentials applied across B and D is negligible). (b) In the nonstandard configuration the direction AC is approximately at 45° to the normal to S, and there are almost equal components of lateral motion in the bending direction of S when a potential is applied either to A.C or to B,D. (c) For either the standard or the nonstandard mounting (nonstandard shown here), application of simultaneous lateral motion ∆xAC ) ∆xtot. cos θ and ∆xBD ) ∆xtot. sin θ to the two sets of sectors results in a net motion as shown of amplitude ∆xtot. at angle (R + θ) to the bending direction of S. (d, e) Calibration runs for the different mountings, with the surfaces in rigid (nonsliding) contact: (d) Input potential to opposing PZT sectors. (e, f) Corresponding response of shear bending for potential inputs to AC, BD sectors, respectively, for PZT mounted in the standard configuration ((a) above), showing the large response for AC and the very weak one for BD (the AC direction is at an angle R ) 3.2° to the bending direction of S). (e′, f′) Corresponding response of shear bending for potential inputs to AC, BD sectors, respectively, for PZT mounted in the nonstandard configuration ((b) above), showing the approximately equal response for both AC and BD (the AC direction is at an angle R ) 42.4° to the bending direction of S).
Figure 4. Variation of the measured shear force Fmeasured determined from the bending of the shear spring S in response to simultaneous potential inputs A cos θ, A sin θ to sectors A,C, and B,D, respectively, as θ is varied. The mica surfaces are in semirigid contact across the LC, and the PZT is mounted in the nonstandard configuration as in Figure 3b or 3c. The variation of Fmeasured with θ is roughly sinusoidal: the broken curve is Fmeasured ∝ cos(θ + R), where the angle R ) 49.5° is determined independently, as described in the text. For a shear force varying isotropically with relative shear direction between the surfaces, an exactly sinusoidal variation of Fmeasured with cos(θ + R) is expected. semirigid contact across 6CB (we recall that R is fixed by the mounting of the sectored PZT, Figure 3c, and is determined independently as described above). The shear force Fs in a direction θ is related to the measured force Fmeasured (determined from the bending of S) as Fs ) (Fmeasured/ cos(R + θ)). For an isotropic shear force Fs is independent of θ,
and the spring-bending should follow a sinusoidal variation (the data in Figure 4 are indeed roughly sinusoidal with θ). However, any marked anisotropy of Fs with shear direction will be manifested as a deviation from this, i.e., (Fmeasured/cos(R + θ)) will not be constant. In the present paper the bulk of the shear results reported were obtained using the nonstandard mounting of the sectored PZT (Figure 3b), with lateral motion being applied in two orthogonal directions only (i.e. R ≈ 45°, and θ either 0° or 90° only). 2.3. Experimental Procedure. The surfaces were mounted and the apparatus calibrated in air, to establish the relative orientation of the mica surfaces,25 the zero of the contact position and the lateral motion response. A droplet of the 6CB was then placed between the mica surfaces, and a normal force profile and refractive index profile were measured to establish the orientation of the liquid crystal for a given contact position (see paper 1 and Figure 1). The surfaces were then compressed under a given (large) starting load Fn (and correspondingly small separation D) and a back-and-forth lateral motion, at velocity vs, was applied to the upper lens via potentials on sectors AC or BD (see Figure 3). The resulting shear force Fs, in the two perpendicular directions, was monitored as described. At the highest compressions and for the amplitudes of lateral sliding used, the friction force between the surfaces exceeded the shear force, so that the surfaces moved rigidly together and the lateral motion could be calibrated from the bending of the shear spring (see for example Figure 6, trace for D ) 16 Å). Following this, and while still sliding back and forth laterally, Fn was reduced and the surfaces separated (via the differential spring or via the PZT in the normal motion mode). The shear response at this new distance was measured simultaneously with the normal force and the distance.
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Figure 5. Trace of shear-spring bending response taken directly from experiment. The top surface is moved laterally back and forth with amplitude 560 nm and frequency 1 Hz, for the PZT mounted in the nonstandard configuration (Figure 3b). The LHS of the trace shows the surfaces under the highest load, and moving in rigid contact (the static friction exceeds the maximal shear force and there is no sliding). The load is progressively reduced about every 15 s (and the separation D progressively increases as indicated) while the top surface continues to move laterally. The procedure was repeated for a number of distances until the shear response was at a noise level. The protocol for measurements in two orthogonal directions is described later.
3. Results 3.1. Dependence of Shear Forces on the Normal Load. As described in paper 1,25 the orientation adopted by the confined 6CB depends on the relative orientation of the mica crystallographic axes and on the extent of hydrophobicity of the mica. Moreover, transitions from the planar twisted or homeotropic orientations to the planar one could occur. Therefore the actual orientation before each shear experiment was determined by measuring the normal force and far-field FECO pattern (and sometimes the refractive index profiles). A similar determination was carried out at the end of a shear experiment. Following this, as noted above, the surfaces were brought together to a normal load Fn and surface separation D, and a back-and-forth motion was applied to the top surface. During the course of the measurement the normal load (and the surface separation) were changed several times, enabling the normal force profile and the shear response to be measured simultaneously. Part of a set of raw data illustrating the procedure is given in Figure 5. Here a back-and-forth lateral motion of peak-to-peak amplitude of 560 nm was applied to the top surface at a frequency of 1 Hz, corresponding to an applied lateral motion with velocity vs ) 1.12 µm/s. The traces in Figure 5 show the shear-spring bending forces between the mica surfaces as the normal load is progressively reduced, and the separation D increases. At the higher loads the frictional forces are larger, as expected; once the shear response traces are fully developed for a given load, the load is reduced to a lower value and the measured shear force changes over a period of some seconds until it stabilizes at a new value characteristic of this lower load and larger D. A quantitative analysis of this relaxation process is probably not warranted in view of the large number of parameters that may be involved (the normal stress, the reduction in the stress at each point where the load is changed, the changing contact area as the normal load is reduced, and so on). However, we may attribute the rather long relaxation time for the shear force to stabilize as follows: each time the normal load is reduced, this leads to some
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Figure 6. Typical traces of shear-spring bendingsyielding Fssin response to a back-and-forth lateral motion of the top mica sheet for different distances between the surfaces taken from a different run to that of Figure 5). For D ) 16 Å the surfaces are still in a rigid contactsthe static friction exceeds the maximal shear force and there is no sliding. At lower loads they undergo some sliding (cf. D ) 27 Å). The fully developed sliding responsesfrom which Fs is extractedsoccurs in this run for D ) 51 Å and larger separations. For D ) 200 Å the signal is below the noise level in the trace.
reduction in the contact area on one hand, and a reduction in the shear stress due to weaker normal stresses between the nematogen molecules. The former effectsdue mostly to distortion off the glue layer underlying the micasis probably rapid. The long relaxation period seen in Figure 5 therefore, on the order of a few seconds, is probably due to rather slow rearrangements off the contacting nematogens to optimize their mutual orientations during shear at the lower normal stress. This time is many orders of magnitude longer than molecular relaxation times, probably due to the very slow dynamics of the highly confined LC molecules. At the highest loads (and lowest separations) in this trace the friction between the surfaces exceeds the maximum shear force applied by the lateral bending of the shear springs, and fully developed sliding does not occur. That is, some bending of the springs and some slipping of surfaces takes place, and it is not possible to assign a value for the sliding friction. At lower loads the shear force exceeds the friction, so that free sliding occurs and a value Fs characteristic of sliding is determined. All Fs values reported in this section are for fully developed sliding motion, as shown on a larger scale in Figure 6. Several typical shapes of shear force responses to the backand-forth sliding motion are shown in Figure 6 (taken from a run different from that of Figure 5). We note the resolution and sensitivity exhibited in these traces, which are essentially set by the noise level: for a resolution of δx ≈ (4 nm in the shear spring bending (typical of the traces in Figure 6), the resolution in the shear force is δFs ) Ksδx ≈ (0.3 µN as noted earlier. For D ) 16 Å, the friction exceeds the shear force throughout the motion, so that the surfaces remain rigidly in contact (no sliding) and the shear-spring bending merely reflects the applied back-and-forth lateral motion. At lower loads (higher D values) there is progressive sliding as the upper surface moves laterally: the well-developed plateaus at the higher D values indicate free sliding motion, once the shear force reaches the value corresponding to the plateau. The corresponding values of Fs are then given by the extent of the bending at the plateau (half the
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Figure 7. Normal force profiles before and after shear for the confined 6CB in a planar orientation, near the onset of repulsive interactions. Different symbols correspond to different runs. Open symbols denote measurements before shear, while the full ones were measured immediately following shear.
plateau-to-plateau amplitude) multiplied by the shearspring constant Ks. The main features of the traces given in Figure 6a-e are the same regardless of the particular orientation of the liquid crystal; i.e., there are no clear differences in the shape of the traces whether the 6CB was in a planar, planar twisted, or homeotropic orientation. It is important to know whether the shear measurements cause irreversible structural changes to the confined LC layers, and to check this, we used the normal force profile to examine the integrity of the confined LC layers before and after shear. Figure 7 shows such profiles for a 6CB layer in a planar orientation: within the scatter there is little variation in Fn(D), indicating that the nature of the confined (planar) layer remains unchanged (the normal force data are plotted here and in subsequent Fn(D) profiles as Fn/R, which yields the interaction energy in the Derjaguin approximation.33 While no such simple relationship applies32 to the shear force Fs, we nonetheless normalize by R so as to compare with the Fn/R data). For the case of confined layers that were initially in the planar twisted (Figure 1b) or homeotropic (Figure 1c) configurations, transitions could be induced by the shear, as described later. 3.2. Relaxation of Stress in Sheared Layers. By applying a different pattern of lateral motion, we examined the relaxation of the sheared LC layer. This is shown in Figure 8, where the top trace shows the applied lateral motion and the lower trace the corresponding shear forces between the surfaces confining the LC. When the applied motion stops (arrow) only slight relaxation occurs over the time of observation (∆Fs/Fs ≈ 0.02), showing that the LC layer can sustain a shear stress for extended periods, in a solidlike fashion. 3.3. Dependence of Shear Force on LC Orientation. Figure 9 shows the variation of the shear force Fs with normal load Fn for 6CB confined in a planar twisted orientation between the mica surfaces. In view of our later remarks about confinement-induced transitions between the different LC orientations, we note here that for the data of Figure 9 the 6CB remained in the twisted planar orientation throughout (both before and after the shear (33) Derjaguin, B. V. Kolloid Zeits. 1934, 69, 155.
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Figure 8. Response of the confined 6CB to shear (top trace) followed by a rest period when the lateral motion ceases. A slight relaxation can be observed, ∆Fs/Fs ∼ 0.02. The surface separation D ) 42 Å.
Figure 9. Variation of shear force Fs with normal load Fn between mica surfaces confining 6CB in the planar twisted orientation. The slope of the straight line is 0.54. Both forces are normalized with respect to the radius of curvature R.
run), as revealed by the nature of the FECO pattern at large separations (see section 3.4). All Fs values are taken from the plateau regime (i.e. fully developed sliding) of the respective shear force traces (see Figure 6). A single shear direction is used for this profile: the variation of Fs with Fn is linear, with a slope (dFs/dFn)planar twisted ) 0.54 ( 0.04. We note that the plot appears not to pass through zero, but that a finite (small) load is required to obtain a measurable friction force when the surfaces are made to slide. As will be seen subsequently, comparable linear dependencies are also observed for planar and homeotropic orientations of the confined 6CB when the shear is carried out at monotonically decreasing loads in a given direction. 3.4. Shear-Induced Transitions between Different LC Orientations. It has long been known that there is a coupling between the director field and the velocity field
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Figure 10. Photographs of fringes of equal chromatic order (FECO) in the far-field regime for the confined 6CB in the farfield regime (separation of a few µm). (a) Planar twisted orientation prior to shear. After some 3 h of shearing at surface separations close to contact (a few nm) the orientation in the whole field of view changed into the planar one. Picture b was taken (again in the far-field regime) just after the end of shearing following the transition from planar twisted to planar, showing the fringe pattern characteristic of planar orientation (Figure 1a and paper 1). Since the contrast in b is low, it is reproduced in b′ with guidelines to emphasize the fringe shape.
in liquid crystals,34,35 and shear of the confined 6CB layers might therefore be expected to orient them in the shear direction. For weak shear fields and strong anchoring such an orientation effect may be negligible; however, both in the homeotropic and in the planar twisted case under confinement (where the effects of the shearing surfaces are much more important than in the bulk), the reorientation into the planar one was observed after sufficient time following shear. In the case of the planar twisted orientation (Figure 1b) the transitionsinduced by shears to a planar orientation (Figure 1a) took place some 2-3 h from the start of the shear, in the conditions of our experiments. Prior to shear, the liquid crystal was stable in this orientation for at least 9 h. An example of this is given in Figure 10, where initial and final FECO patterns for the confined LC films are shown: there is a clear transition from the fringe pattern characteristic of the planar twisted (Figure 10a) to that characterizing the planar orientation (Figure 10b or b′). For the 6CB in the homeotropic orientation (Figure 1c), shear-induced reorientation into the planar one also took place, but after considerably longer time, some 36 h from the beginning of the shear, for typical shear rates and amplitudes used in the experiments. We emphasize that the transition between the different orientations (planar twisted to planar or homeotropic to planar) was quite marked, as it resulted in a qualitative change in the fringe pattern at large surface separations (Figure 10a vs 10b), so that we were in general able to determine unambiguously the orientation being examined. (34) De Gennes, P. G. The Physics of Liquid Crystals; Oxford University Press: Oxford, U.K., 1974. (35) Vertogen, G.; Jeu, W. H. d. Thermotropic Liquid Crystals, Fundamentals; Springer-Verlag: Berlin, Heidelberg, 1988; p 36. (36) Janik, J.; Moscicki, J. K.; Czuprynski, K.; Dabrowski, R. Phys. Rev. E 1998, 58, 3251. (37) Kim, M. G.; Park, S.; Cooper, S. M.; Letcher, S. V. Mol. Cryst. Liq. Cryst. 1976, 36, 143.
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3.5. Anisotropy of Shear Forces across Confined 6CB. Two protocols were used for inducing and measuring the directional anisotropy in the shear properties of the liquid crystal, using the nonstandard PZT mounting (Figure 3b). In the first (designated A), the surfaces are initially brought together under a large load to a given (small) D value. Back-and-forth shear is then applied in one directionshenceforth the aligning direction, which may be either AC or BD, Figure 3bsand Fs is then measured in that direction at progressively decreasing loads and correspondingly increasing D values, until it becomes immeasurably small. The surfaces are then brought together again, and the shear forces are measured in the direction perpendicular to the original aligning direction, as described in section 2.2. In the second protocol, (designated B), the surfaces are likewise brought together to a high initial compression, at surface separation D; the confined LC layer is then sheared back-and-forth in one of the shear directions (AC or BD). This first direction of shear determines the aligning direction, and Fs is first measured in this direction. It is then measured again at the same load and D value in a direction perpendicular to the aligning direction. The shear motion is then returned to the original (aligning) direction, and Fs is measured again (the value of Fs in the aligning direction is the mean of the two measurements in that direction). Finally, while the surfaces are moving laterally in the aligning direction, the normal load is decreased (while D increases) to a new value, and the procedure repeated. The difference between the two protocols is that, in A, the first Fs vs Fn profile sets the aligning direction, to be followed by a second profile at right angles to this direction; while in B, shear force values in the two perpendicular directions are measured point by point at the same load and D values. As seen below, within the scatter both protocols yield similar results. Figure 11 shows Fs vs Fn profiles for the 6CB confined in a planar orientation between the mica sheets, for shear in two orthogonal directions, for two different experiments (different pairs of mica sheets). Figure 11a shows results using protocol A, while Figure 11b is for results using protocol B. The two sets of profiles both show clearly the following features: (1) the shear forces Fs required to slide the surfaces at different loads Fn vary roughly linearly with the load, as indicated in Figure 9 (for planar twisted) where only a single direction of shear was used; (2) the value of Fs in the orienting direction is systematically lower than the value of Fs in the orthogonal direction, and the slope (dFs/dFn) is also lower; (3) within the scatter, both protocols lead to rather similar results. These slopes when Fs was measured in the aligning direction and in the direction orthogonal to it, were (dFs/dFn) ) 1 ( 0.2 and (dFs/dFn) ) 1.5 ( 0.2, respectively. Figure 12 shows the two orthogonal Fs vs Fn profiles (obtained using protocol B) for an experiment where, prior to the initial compression and aligning shear, the confined LC was in the planar twisted configuration (as revealed by the far-field FECO pattern). The profiles are similar to those obtained with the planar configuration above (Figure 11): for a given load Fn the Fs values are lower for the aligning direction, and the value of (dFs/dFn) is also lower for that direction. The magnitudes of the slopes in the aligning and perpendicular directions are 0.7 ( 0.15 and 1 ( 0.1, respectively. In this experiment we observed that although the starting configuration was planar twisted, the final configuration following the shear runs, as revealed by the FECO pattern at large surface separations, was characteristic rather of
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Figure 12. Variation of shear force Fs with normal load Fn between mica surfaces confining 6CB. Filled symbols are for shear forces measured in the orienting direction, while empty symbols are for a direction orthogonal to the aligning one. The data were measured using protocol B (see text). The orientation of the 6CB at the start of the shear experiment was planar twisted, but reverted to planar during the course of the measurements.
Figure 11. Variation of shear force Fs with normal load Fn between mica surfaces confining 6CB in the planar orientation. (a, top) Filled symbols are for shear forces measured in the orienting direction, while empty symbols are for a direction orthogonal to the aligning one (different symbols correspond to different runs). The data were measured using protocol A (see text). (b, bottom) These results are from a different experiment (different pairs of mica sheets). Filled symbols are for shear forces measured in the aligning direction, while empty symbols are for a direction orthogonal to the aligning one (different symbols correspond to different runs). The data were measured using protocol B (see text).
planar orientation. That is, during the shear measurements themselves the LC layer had undergone a transition to the planar state. We return to this point later. Figure 13 shows the variation of Fs with load Fn for 6CB confined in a homeotropic configuration between the mica surfaces (determined according to protocol A above). Though there is substantial scatter, the data do not appear to show a systematic difference between the values of Fs measured in the aligning direction, and its magnitude in the orthogonal direction. The effective slope here is (dFs/ dFn) ) 1.3 ( 0.2. We note that for both directions of shear Fs has a finite value even at negative values of the load, i.e., under tension. This is characteristic of shear between sliding surfaces in adhesive contact, as is the case for the mica surfaces across the homeotropic 6CB structure (see paper 1 and also Figure 1c). The inset shows both the normal force profile Fn(D) (full symbolssarrows indicate jumpsout) and the shear force profile Fs(D) (empty symbols) on a force vs distance plot. These results are considered later (section 4).
Figure 13. Variation of shear force Fs with normal load Fn between mica surfaces confining 6CB in the homeotropic orientation. Filled symbols are for shear forces measured in the orienting direction, while empty symbols are for a direction orthogonal to the aligning one. The data were measured using protocol A (see text). The inset shows the normal loads (filled symbols) and shear forces on a force vs distance profile. Arrows indicate jumps out.
3.6. Shear-Velocity Dependence of Shear Forces. In general we observed little systematic variation of Fs with applied sliding velocity vs. Figure 14 shows this for the case of planar twisted 6CB sheared at two widely differing velocities, vs: 1.2 µm/s and 30 nm/s, at different surface separations. We note that for the data of Figure 14 the LC remained in the planar twisted conformation throughout; i.e., it did not in the time of the measurements undergo the shear induced transitions noted in section 3.4.
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Figure 14. Variation of shear force Fs with sliding velocity vs between mica surfaces across 6CB at different surface separations (and different loads). The LC was in the planar twisted orientation throughout. Circles, vs ) 30 nm.s-1; crosses, vs ) 1.2 µm.s-1.
4. Discussion and Conclusions Our main findings reveal that, for all three orientations of the confined 6CB nematogen, the variation of Fs with Fn was roughly linear over the range of parameters in our experiments, and the magnitude of the slope (dFs/dFn) was in the range 0.5-1.5, depending on the particular LC orientation and the relative sliding directions. In addition we found a marked directional anisotropy in the magnitude of Fs: following initial alignment by shearing of the compressed LC in a given (aligning) direction, subsequent measured values of Fs in this direction were systematically lowersby some 20-30%sthan in a direction orthogonal to the aligning direction. To our knowledge this is the first indication of directional anisotropy in friction across LC lubricants. In what follows we consider these findings in more detail. Comparison of our results with previous studies of friction mediated by liquid crystals18-20,24 sheds light on the regime that we are probing in the present work. The most comprehensive work has been reported by Cognard,19 who studied friction of a range of LC’s, both on their own and in mixtures, using a standard tribometer. The magnitude of the friction coefficients for comparable LC’s in these earlier tribological investigations appears considerably smaller than the slopes (dFs/dFn) that we have determined for the 6CB in its nematic phase. Thus Cognard reported friction coefficients µ for the analogous nematogen 5CB in the nematic phase, and for 5CB mixed with 7CB (5CB has one CH2 group less on its alkyl tail, and 7CB has one more, than the 6CB). At high (macroscopic) sliding velocities values of µ in the range 0.03-0.1 were generally measured (the lowest values after extensive sliding times), though the value varied somewhat with the shear velocity. Cognard reported that the thickness of the lubricant was “polymolecular”, though in his experiments the lubricating layer thickness could not be measured, and at the velocities used a thick (i.e. .100 nm) hydrodynamic layer of LC could well be responsible for the efficient lubrication. The velocities used were macroscopic ones (order 10’s of cm.s-1 or even m.s-1), rather than the valuessof order µm.s-1s used in the present work. At the lowest shear velocities in Cognard’s report, where the surfaces might be expected to come closer together and where contacting asperities might be trapping the liquid crystal down to thicknesses comparable to those used in the present study, µ for the neat 5CB rose sharply to around 0.5 or more. While these observations are suggestive, it is difficult to compare directly the tribological study of Cognard and other macroscopic friction measurements17 to the present investigation, which is a single asperity one.
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The friction measurements using an SFB by Ruths et al.24 for 8CB in its nematic phase in a planar orientation between mica surfaces are for single asperity contact, and so resemble more the work described in paper 1 and in the present paper. They reported µ values of order 0.2, also appreciably smaller than our present values of (dFs/dFn) for the confined 6CB. However, the 8CB results24 were determined at normal loads that were some orders of magnitude higher than those used by us, of order mN to tens of mN as opposed to some µN to tens of µN in our experiments. In addition, the mica surfaces in the Ruths et al. work were throughout at their “hard wall” adhesive contact separation of ca. one 8CB dimer layer per surface. In contrast, the present study examined the shear response out to surface separations D well over 100 Å, comparable to (though less than) separations where normal forces between the confining surfaces are first detected. Thus our results provide a delicate probe of the shear properties of nematic LC’s confined to well-defined molecular dimensions. The general features of our results (we relate to the anisotropy later) show that, for the two planar orientations, the shear forces to slide the surfaces vary roughly linearly with the load, and pass through zero or close to it. This is reminiscent of solidlike friction, and the essential independence of the frictional force of the sliding velocity (within the parameters of our measurements), Figure 14, as well as the ability to sustain a shear stress for extended periods (Figure 8), is also consistent with this solidlike behavior. It suggests that layers of 6CB are pinned at either surface and that they are shearing past each other across an interfacial plane whose width is comparable with the thickness of a nematogen (ca. 4 Å), overcoming a drag that presumably represents the sliding friction. The slope (dFs/dFn) provides a measure of how this molecular friction varies with the compressive load. It may be viewed as the effective friction coefficient for the particular structure and mutual orientation of the sliding LC layers. If we attribute the drag opposing the sliding to dissipation arising from molecular friction within the sheared interfacial plane, then we might expect that the higher the degree of order in the confined layers, the higher the friction. This is because a higher degree of ordering is associated with a stronger nematic interaction between the molecules. This idea is consistent with the fact that for the shear of 6CB confined in the planar twisted orientation (Figure 9), where the degree of order (see Figure 1b) is less than that of the planar orientation (Figure 1a), the magnitude of (dFs/dFn)planar twisted ) 0.54 is also lower than its value for the confined planar orientation, Figure 11, where (dFs/dFn)planar ≈ 1. It is of interest that for a sliding experiment where the confined 6CB was originally in a planar twisted orientation but underwent a transition to the planar one during the course of the run, the magnitude of (dFs/dFn)planar ≈ 0.7 was intermediate between the two above values. For the homeotropic orientation, Figure 1c and Figure 13, there is a finite frictional resistance to sliding even at negative loads, characteristic of adhering surfaces. The larger magnitude of the slope (dFs/dFn)homeotropic ≈ 1.3 may reflect the fact that the nematogen molecules are being dragged at right angles to their length rather than sliding along their length. Such sliding would require shear of an interfacial plane whose thickness is comparable with the nematogen length (see Figure 1b) rather than its width as in the planar case, and so requires greater stress to overcome. These magnitudes of (dFs/dFn), all quite similar, are high relative to friction coefficient with traditional boundary lubricants (which are ca. 0.05 though at much
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Figure 15. Illustrating the relative shear orientations of an LC confined between sliding surfaces in the homeotropic and planar orientations, and the directions of the velocity vector v and the director (n), for the shear directions corresponding to the Miesowicz viscosity coefficients η1, η2, and η3.
higher normal stresses), and suggest that at these rather low loads, at least, these nematogens do not provide particularly efficient lubrication. Finally, we consider the directional anisotropy of the shear forces for the planar orientation. For this purpose it will be instructive to recall the possible directions of shear of the confined nematogen with reference to the three Miesowicz viscosity coefficients, as illustrated in Figure 15. We attribute the anisotropy of the shear forces in our study to the following. When the surfaces approach and are initially made to slide in one direction (the aligning direction), this aligns the planar-oriented 6CB molecules in the directionsrelative to the slidingsindicated in Figure 15b, and we presume that the 6CB layer in contact with the surfaces is then pinned strongly and essentially irreversibly in that (aligning) direction. When the shear direction is subsequently rotated by 90°, the nematogen remains pinned in the planar orientation in the aligning direction, by the layer in contact with each surface, and the relative motion of the surfaces is now as illustrated in Figure 15c. If we were dealing with shear of macroscopic gaps, we might expect the ratio of the effective viscosities in the two directions to be η2/η3. This ratio is, from several different studies,7,8,12,36,37 in the range 0.65-0.9. In the case of the nanometric gaps in our study, however, we do not have a shear response characteristic of viscosity (this is demonstrated by the fact that the shear force does not depend on velocity and that it does not relax in a liquidlike fashion, both of which are expected for viscous behavior). It is thus of interest that the ratio of the shear forces in the aligning direction and that orthogonal to it in the shear runs in Figures 11 and 12 are also in this range of values, as seen clearly in Figure 16. This observation suggests that the frictional factors controlling molecular shear in viscous flow are similar to those controlling it in the interfacial sliding (as in our study) for the same relative orientation of the molecules. We also note that for the homeotropic orientation (Figure 1c and Figure 13) there appears to be no anisotropy (within the scatter). This is consistent with the fact that there is no preferred direction relative to the orientation of the molecules (Figure 15a). In conclusion, using an SFB, we have measured the response to shear of confined layers of the nematogen 6CB, in the planar, twisted planar, and homeotropic orientations. Our investigations examined the shear behavior at normal and shear stresses that were some
Figure 16. Ratio of the shear forces in the aligning and orthogonal directions measured for 6CB confined in the planar orientation (taken from Figures 11 and 12), at different surface separations.
orders of magnitude weaker than that of earlier tribological studies, and so cannot directly be compared with those earlier works; they provide rather a delicate probe of the shear and mechanical properties of the confined 6CB layers from close to the point of onset of normal interactions between the confining surfaces. Our results reveal a linear relation between normal and shear forces and a solidlike response of the sheared films. In addition, by using a configuration of the SFB which enables measurement of the shear forces at different relative directions of sliding of the surfaces, we find an intriguing directional anisotropy in the shear/frictional forces. This suggests that once an alignment of the confined planar nematogen has been set by the initial shear direction, it remains fixed even when the direction of shear is subsequently changed. The relative magnitudes of the shear force in the aligned direction and that orthogonal to it are of order 0.7-0.9. This ratio is reminiscent of the ratio of the Miesowicz viscosity coefficients η2 and η3, which apply for the identical shear directions for bulk liquid being sheared parallel and normal to the director of the planar nematogen. This suggests that the molecular factors controlling the shear of these nanometer thick layers may have similar origins to the molecular friction modulating the bulk viscous behavior. Acknowledgment. We are grateful to Erika Eiser for her help to J.J. during the early stages of this project. We thank the Deutsche-Israel Program (DIP), the U.S.-Israel Binational Science Foundation (BSF), and the Minerva Foundation for support of this work. LA001392Q
