Sambles

M J Jory

P S Cann

J R Sambles

Surface-plasmon voltammetry using a gold grating

Keywords: 73.20Mf, 82.45Tv, 42.79Dj, 42.79Jq

HAL is

Introduction

The electrochemical response of the gold / electrolyte interface continues to be the subject of extensive research. [1][2][3][4][5][6][7][8][9][10][11][12][13] Cyclic voltammetry has been used to establish the current-voltage characteristics of the well-ordered surfaces associated with single-crystal electrodes. 1,2,4,6 Both the oxidation / reduction processes occurring at high anodic potentials and the more subtle changes that occur within the double-layer region have been studied, see for example Hamelin et al. 1 Additionally, several techniques have been used in combination with cyclicvoltammetry to gain further information regarding these processes including scanningtunneling microscopy (STM), [14][15][16][17][18] atomic-force microscopy (AFM), 19 the quartz crystal microblance (QCM), [20][21][22] reflectance studies, [START_REF] Kolb | Spectroelectrochemistry ed Gale R J[END_REF] ellipsometry [START_REF] Stedman | [END_REF][25][26][27] and surface-plasmon resonance (SPR) studies. [28][29][30][31][32][33][34][35][36][37][38] It is SPR-electrochemistry, which has now been recognised as a new and powerful tool for exploring not just electrochemistry but also bioelectrochemistry [37][38][39][40] that is of interest in this work.

A surface-plasmon polariton (SPP) is a coupled electromagnetic wave / longitudinal charge density oscillation that may be excited along the interface between a metal and a dielectric.

The associated enhanced optical fields decay exponentially away from the interface extending ~λ/10 into the metal and ~λ into the dielectric (where λ is the optical wavelength, typically ~400 nm in electrolyte.) Consequently, SPR-based techniques are sensitive to changes in the working electrode, the double-layer region and the charging of the diffuse layer. [28][29][30][31][32][33][34][35][36][37][38] The momentum of a surface-plasmon polariton (SPP) propagating along a metal / dielectric interface is greater than that of a grazing photon of the same frequency in the bulk dielectric. Consequently, it is not possible to optically excite SPR directly, it is necessary to enhance the in-plane momentum of the incident beam. Traditionally there are three configurations in which this momentum enhancement is achieved in order to study the electrochemical interface using SPR. The first two involve a prism of refractive index greater than that of the surrounding electrolyte. In one arrangement, the Otto geometry 41 , the prism is set so that one face is separated from the working electrode surface (usually gold or silver in the visible) by ~ 0.5 µm. 28,29 Having passed through the prism, light striking the prism / electrolyte interface beyond the critical angle is totally internally reflected. However, an evanescent wave is also produced that extends into the transmitting medium tunnelling across the narrow electrolyte gap and exciting SPR on the working electrode surface. This configuration allows the use of bulk, highly-ordered single-crystal working electrodes that are vital for studying fundamental, crystal-face sensitive, electrochemical processes using cyclic voltammetry. 1 However, this geometry has the disadvantages that, due to the small gap between prism and electrode, it is both difficult to fabricate and does not readily allow studies incorporating a flow of electrolyte as is often required in adlayer deposition, biosensing and / or cleaning of the electrode surface. It is because of the aforementioned reasons that the alternative Kretschmann-Raether prism 42 geometry is often used. Here an optically-thin (~45 nm thick) layer of gold is evaporated onto one face of a high-index prism (or on a glass substrate which is index matched to a prism). Now beyond the prism / electrolyte critical angle light tunnels through the gold film and excites an SPR on the outer electrode surface. This Kretschmann-Raether geometry overcomes the small gap fabrication and "trapped-electrolyte" problems of the Otto system and is much better suited to bio-sensing applications where the use of wellordered single crystal electrodes is not so crucial. 30,[33][34][35][36][37][38][39][40] However, the optically-thin metal film is very fragile and consequently does not allow extended electrochemical deposition cycles to be performed on the same electrode. In general, this geometry also necessitates the use of polycrystalline electrodes although there are exceptions. [START_REF] Kolb | Spectroelectrochemistry ed Gale R J[END_REF] The third geometry for SPR excitation is grating coupling. 43 The simplest example is a diffraction grating possessing a periodic surface with a well-specified groove depth and pitch.

The translational invariance of the grating surface is broken in one direction by the grooves. This means that momentum may be added to, or subtracted from the x-component of the momentum of incoming photons. (The 'x-direction' is defined as parallel to the plane of the grating and perpendicular to the direction of the grooves.) Conservation of momentum in the x-direction gives: 43

SPP

i el k mG k n = ± θ sin 0 (1)
where el n is the refractive index of the electrolyte, 0 k is the magnitude of the wavevector of the incident light in free-space, i θ is the angle-of-incidence,

2 1 0         + = el metal el metal SPP k k ε ε ε ε
is the surface-plasmon-polariton wavevector ( metal ε and el ε being the real parts of the optical permittivities of the metal and electrolyte, respectively), m is an integer and

g G λ π / 2 =
is the magnitude of the grating wavevector ( g λ being the pitch of the grating grooves.) When equation 1 is satisfied an SPR may be excited on the grating surface directly from the dielectric half-space (i.e. the electrolyte) without the constraints placed on the thickness of the metal electrode or the electrolyte-gap that are associated with the Kretschmann-Raether 42 and Otto 41 methods, respectively. As this grating configuration openly presents the entirety of a robust, bulk, working electrode to the electrolyte it may prove invaluable in the field of biosensing and opto-electrochemistry .

In this work we monitor the optical changes that occur at the electrochemical interface between a diffractive gold electrode and an acid electrolyte by using an acousto-optic differential reflectivity detection technique. 44 This detection system allows the position of the SPR, in wavelength, to be continuously monitored as cyclic voltammetry is performed. 45

Experimental

Electrode preparation

A clean, fused-silica disc is spin-coated 46 with a photoresist layer. Using a standard interferographic method a diffraction grating of groove pitch 600 nm and groove depth 20 nm is manufactured in the photo-resist. This groove pattern is then transferred to the underlying silica substrate by ion-beam etching using a low-pressure argon plasma. Full details of this method can be found elsewhere. [START_REF] Hutley | Diffraction Gratings[END_REF] A gold-film, at least 150 nm thick, is now evaporated onto the fused-silica diffraction grating.

In order to improve adhesion between the gold and the silica, the bare grating is first immersed in a solution of 0.01M dithiodipropionic acid in ethanol for 20 minutes. On removal from the solution the grating is allowed to dry in air and cleaned with chloroformsoaked cotton buds and lens tissue to remove the excess residue. The gold film is then deposited by evaporation in 10 -6 Torr.

The gold-coated grating is now partially immersed in a 0.02M H 2 SO 4 electrolyte (as shown in figure 1(a)) (This acid solution is prepared from 99.999% pure sulphuric acid and 18 MΩ-cm conductivity de-ionised water.) The grating is orientated so that its grooves lie vertically. A 99.999% pure gold wire used as the counter electrode and a standard Ag/AgCl reference electrode complete the cell. (All potentials are quoted with respect to Ag/AgCl.) Before any electrochemical measurements are taken, the electrolyte is purged of oxygen, for a period of one hour, by a continuous flow of argon gas. Cyclic voltammetry is then performed in the conventional way. 48a (A minimal argon flow through the electrolyte is maintained during all the measurements.)

Optical measurements

Light, whose wavelength has been selected from a white light source by an acousto-optic tuneable filter (AOTF) [START_REF] Gass | [END_REF] is now directed through the electrolyte and onto the working electrode (i.e. the gold diffraction grating) at an incidence angle of 14.3° (figure 1(b)). The grating has previously been orientated so that the optical plane of incidence (ABCD in figure 1(a)) is horizontal and lies perpendicular to the groove direction. Sweeping the AOTF drive frequency from 63 MHz to 125 MHz gives an optical wavelength scan from 775 nm to 425 nm. Using a photomultiplier tube (PMT) detector, the reflected intensity from the gold / electrolyte interface is monitored simultaneously. SPR excitation is observed as a minimum in the reflectivity at ~660 nm. 44 A frequency modulation is now added to the AOTF carrier signal resulting in a wavelength modulation of the optical beam. This, in turn, causes the Having set the phase of the lock-in amplifier appropriately, a computer program is now used to lock the wavelength of the optical beam (by adjusting the AOTF carrier frequency) to the position of zero differential i.e. the SPR position. With this system it is possible to measure the SPR position in wavelength to a precision of 0.0005nm. 44 While continuing to monitor the SPR wavelength, the potential of the working electrode is swept between -500 mV and 850 mV at 10 mV/s. Thus, a direct measure of SPR position in wavelength is obtained versus time as cyclic voltammetry is performed. A full explanation of this acousto-optic SPR technique can be found elsewhere. 44 Figure 2 shows a cyclic voltammogram obtained on a gold-coated diffraction grating-working electrode. Hydrogen evolution occurs on the positive sweep up to about -400 mV after which double-layer charging occurs. There is then a major peak in the current at 475 mV corresponding to adsorption of sulphate ions. On the negative sweep sulphate desorption occurs at 420 mV with hydrogen evolution re-commencing at -380 mV. This data is consistent with the conventional response of a polycrystalline gold electrode in weak sulphuric acid. 14,36,50,51 The periodic surface profile of the electrode (having a groove-depth : pitch ratio = 0.03) appears to have no effect on the electrochemistry of the interface. This data agrees well with that found by other authors on similar systems where the rate of change in SPR position with respect to potential is seen to increase as the electrode becomes more anodic especially when anion adsorption occurs. [30][31][32]51,52 Here, we are able to monitor the SPR position continuously as cyclic voltammetry is performed and therefore are able to observe the hysteresis in the optical response of the interface. Our optical data are also in very good agreement with the changes in electrode weight monitored using a quartz crystal microbalance (QCM) for the gold / H 2 SO 4 interface. 22 Other studies have shown that the electrode resistivity 53 exhibits a similarly-asymmetric response.

Having noted that points of inflection in the SPR / potential curve agree well with the positions of current peaks it is logical to explore the differential of the SPR position with respect to time. Figure 5 shows a comparison between

) / ( ) / ( 0 0 ′ ∆ = ∆ ∂ ∂ k k k k t spp spp
(open circles) and current (solid line) versus time. There is good agreement between the two curves.

Again we note that the large negative peak in current due to hydrogen evolution barely affects the SPR data. -600 -300 0 300 600 900 -10 )' (a.u.) )' (a.u.) We now compare the optical data acquired for electrode potentials beyond -300 mV (the open circles in figure 7) with the computer-modelled response of a capacitor (185 µF) and resistor (30 kΩ) in series (solid line). At these potentials, the current measurements obtained using conventional cyclic voltammetry are dominated by faradaic currents (i.e. hydrogen evolution as mentioned above). However, it is clear that the SPR responds only to the underlying capacitor-like charging of the double-layer. (It should be noted that for the purposes of this comparison, a slight negative offset (that is observed in both the electrical and optical data in figure 6) has been removed from the optical data in figure 7. We tentatively attribute this offset to low levels of dissolved oxygen or some other impurity in the electrolyte and / or hydrogen evolution at highly electroactive adatom sites.) Figure 8 shows the SPP wavevector response (averaged over 8 potential cycles) versus relative charge (charge being calculated by integrating current for potentials above -310 mV).

Initially, the SPR position (circles) increases at (2.93±0.02)× 10 -6 µC -1 as charge builds up on the electrode (arrow A in figure 8). (This rate is calculated by fitting a straight line to the data as shown in the figure .) Upon passing ~170 µC (~ 365mV) the gradient changes quickly to a value of (6.33±0.01)× 10 -6 µC -1 that is maintained until 273 µC (~ 500 mV) (arrows B and C).

After this point a gradient of (5.34±0.04)× 10 -6 µC -1 is maintained (arrow D) until the end of the positive sweep (850 mV). The average SPR response for the negative potential sweep is also shown (crosses). Initially the SPR position decreases at a rate of (5.18±0.02)× 10 -6 µC -1

(arrow E) and this value is maintained until the charge drops to ~210 µC (445 mV). For the remainder of the potential cycle below 333 mV a gradient of (-2.92±0.01)× 10 -6 µC -1 (arrow F) is recorded (the opposite of the charge build-up gradient). (Again the solid lines represent linear fits to the data.) An increase in SPR-sensitivity to charging of the electrode has previously been observed 32 and attributed to the re-structuring of the double-layer and / or the onset of anion adsorption upon reaching the point of zero charge. 31,32 The SPP wavevector response as a function of charge recorded in this work could be interpreted as indicating that anion adsorption is a two stage process, while desorption occurs in a single step. Moreover, the fact that the experimental data in figure 8 is multiple-valued (i.e. more than one value of ) / ( 0 k k spp ∆ for a given value of relative charge) suggests that the SPR is sensitive not only to the density of ions and water molecules in the ionic double-layer but also to the double-layer structure.

It has been shown that the application of a potential across the gold / acid interface results in modification of the optical response of the metal, the double-layer and the diffuse layer. [START_REF] Kolb | Spectroelectrochemistry ed Gale R J[END_REF]32 The low frequency fields applied during cyclic voltammetry are typically screened within one atomic layer of the electrode surface. [START_REF] Kolb | Spectroelectrochemistry ed Gale R J[END_REF]32 Forstmann et al 54 have shown that non-local optics must be used to accurately model the optical reflectivity response (known as the electroreflectance [START_REF] Mcintyre | Advances in Electrochemistry and Electrochemical Engineering[END_REF] ) of a gold electrode in both the visible and ultra-violet. However, in this present study the maximum photon energy used is 1.87 eV (equivalent to a wavelength of 662.15 nm). At such low photon energies the electroreflectance effect makes little contribution to the reflectivity of the interface with the predicted response using local and non-local optics being in good agreement. [START_REF] Kolb | Spectroelectrochemistry ed Gale R J[END_REF]54 Furthermore, Abeles et al have shown that changes in the metal electrode predominantly contribute to changes in the imaginary part of the SPP wavevector and do not affect the real part. 32 As described above, the differential detection technique used here only monitors the position, in wavelength, of the SPR -i.e. changes in the real part of the SPP wavevector-and so is insensitive to changes due to the electro-reflectance effect. This, along with the good agreement between our data and QCM studies (that are insensitive to charging of the metal) of the same system 22 where [START_REF] Pockrand | [END_REF] :

k k k SPP SPP ∆ + = mod (2) with ( )       -         - -         +         - = ∆ - λ π ε ε ε ε ε ε ε ε ε ε ε ε ε d k k el metal

Conclusions and future work

This work demonstrates that real time cyclic SPR voltammetry can be performed on polycrystalline-metal diffraction-grating electrodes. The periodic surface profile of the electrode gives the momentum enhancement required to allow the SPR to be excited along the electrochemical interface but does not appear to affect the voltammetric response of the system. Using an acousto-optic differential-detection technique, the SPR position, in wavelength, is readily monitored as cyclic voltammetry is performed.

In the double-layer charging region the SPR response is found to correlate strongly with charge. This is most clearly illustrated by plotting SPR position (in terms of relative change of wavevector) as a function of relative charge. The differential-detection system used here allows the resonant optical wavelength to be monitored to a precision of 0.0005nm. 44 This is equivalent to a sensitivity of ~8×10 -4 of a sulphate ion monolayer. By optimising the grating pitch and the refractive index contrast between the electro-deposited material and the bulk electrolyte it should be possible to improve this sensitivity by at least an order of magnitude.

Alternatively, the time-differential of the SPR position may be plotted as a function of electrode potential thereby producing an "optical voltammogram". This has provided a detailed comparison between the electrochemical and optical responses of the gold/ acid interface over the entire double-layer region.

As a first step, a simple theory has been used to model the electrochemical double-layer as an isotropic over-layer having a thickness of 2.15 nm. Maximum SPR shift, observed when the electrode is covered by a complete anion monolayer, would require a double and diffuse-layer refractive index of 1.43. This implies a 14.7% increase in ion concentration compared to that of the bulk electrolyte.

The wavelength at which the SPR is excited along the electrode surface may be selected by tuning the incident angle of the optical beam. Thus, the optical response of a chosen electrode / electrolyte combination could be characterised across the visible spectrum. However, with higher-energy photons as the optical probe it may become necessary to use non-local optics theory to provide a satisfactory model of the electrochemical interface.

It is clear that the evolution of hydrogen at the cathodic end of the potential cycles has very little effect on the SPR position. However, the SPP does seem to be sensitive to the resulting changes in surface structure of the electrode. It is well established that the groove profile of a metallic diffraction grating can be characterised by fitting angle-dependent reflectivity data (acquired using a monochromatic optical beam) to Chandezon theory. [START_REF] Chandezon | [END_REF] Characterising the diffraction grating groove profile in this way, before and after the electrode has been subjected to cycles in potential, may provide an elegant means of monitoring the electrochemically-induced changes in surface structure. Furthermore, comparison of the SPR response (at the cathodic end of the potential cycles) with the theoretical current-potential response of a simple capacitor / resistor network indicates that the SPP continues to respond to the under-lying double-layer charging and electrode-surface restructuring processes even when gas-evolution is occurring. Studying this aspect of the SPR electrochemical response as a function of electrolyte concentration (thereby changing the resistance and capacitance of the double-layer) may provide further insight into this phenomenon.

Having a thickness of 150 nm, these diffractive electrodes are far more robust than the traditional, optically-thin (<100 nm thick), planar films used in the Kretschmann-Raether technique. In addition, silica diffraction gratings can be replicated in plastic readily and cheaply. 59 Consequently, this diffraction grating electrode arrangement is well-suited to biosensor systems where it may be desirable to electrochemically control the binding strength of a particular antigen. 60 Atom etching a grating directly into a bulk single crystal electrode would provide, for the first time, a means of studying single-crystals via surface plasmon excitation in a configuration that overcomes the dielectric tunnel barrier problem particular to the Otto geometry.

Alternatively, the photoresist development process and / or the atom etch could be stopped before the photoresist layer has been completely removed so as to leave a periodic array of photoresist ridges on the electrode. Each ridge would then be separated from its nearest neighbours by an area of planar single crystal metal. This will allow SPR studies of specific, well-defined single-crystal faces via diffraction from the photoresist ridges. Further, it would be possible to electrochemically deposit metal adlayers between these ridges. The optical response of a periodic array of metal adlayers could then be explored by simply dissolving the photoresist. By depositing an electro-chromic material on a diffractive electrode it may also be possible to create electrochemically-tuneable photonic band-gaps.

Finally, combining the optical detection technique and cyclic voltammetry used here with an electrochemical quartz crystal microbalance 22 may provide a further means of distinguishing between electrode processes.

44 Figure 1 :

 441 Figure 1: (a). Experimental arrangement. ABCD is the (horizontal) plane of incidence. The grating grooves lie perpendicular to ABCD; (b). Experimental arrangement. In this work θ SPR =14.3 o .

Figure 2 .

 2 Figure 2. Cyclic voltammogram for a polycrystalline gold-diffraction grating electrode in 0.02M H 2 SO 4 electrolyte. The potential is cycled between -500 mV and 850 mV at 10 mV/s.

Figure 3 Figure 3 .Figure 4 .

 334 Figure 3 depicts electrode potential (solid line) and SPR position in wavelength (open circles)versus time. The SPR position clearly oscillates, as cyclic voltammetry is performed, with resonance wavelength amplitude 0.97 nm. These electrochemically-induced changes in SPR position may also be considered as relative changes in SPP wavevector (as is also shown in figure3) allowing ready comparison with the work of Abeles et al.32

Figure 4

 4 Figure4shows both current and SPR position versus potential. At the beginning of the upward potential sweep the SPP wavevector increases at a rate of ~3.0×10 -7 (∆k spp /k 0 )/mV (or in terms of resonant optical wavelength 1.4×10 -4 nm/mV). As the potential increases through 0 mV the gradient of the curve rises steadily until a point of

Figure 5 .

 5 Figure 5. Time dependence of current (solid line) compared with that of the differential of the relative change in SPP wavevector with respect to time (circles).

Figure 6 .

 6 Figure 6. Cyclic voltammogram (solid line), as shown in figure 2, compared with the differential of the relative change in SPP wavevector with respect to time (circles).

Figure 7 .

 7 Figure 7. Computer-modelled current-potential response of a capacitor (185 µF) and resistor (30 kΩ) in series (line). Differential of the relative change in SPP wavevector with respect to time (circles).

Figure 8 .

 8 Figure 8. Relative change in SPP wavevector (averaged over 8 potential cycles) versus relative charge. The circles and crosses correspond to data acquired during the positive and negative going potential sweeps, respectively. Linear fits to the data are also shown (solid lines).

  lead us to conclude that changes in SPR position are due to charging of the double and diffuse layer, anion adsorption and / or restructuring of the electrode surface. For the 0.02M electrolyte used in this work the double and diffuse layers combined have a thickness of no more than 2.15 nm, 48b approximately 0.003 of an optical wavelength in the electrolyte. Consequently, we now model the optical response of the electrochemical interface by the addition of an isotropic over-layer with thickness d=2.15 nm and optical permittivity overlayer ε . Here, the modified SPP wavevector becomes mod SPP k

  the ionic concentration of the double layer increases the local optical permittivity will increase above that of the bulk electrolyte resulting in a modification of the SPR position. If the bulk electrolyte is assumed to have an optical permittivity of 1.778 (refractive index 1.333) then the maximum shift in the SPP wavevector of over-layer permittivity of 2.04 (refractive index 1.428). Assuming that, to a first approximation, optical permittivity is proportional to density 57 then this is equivalent to an increase in ion concentration within the interfacial region of % to that in the bulk

Acknowledgements

The authors gratefully acknowledge the help of Mr. D.J. Jarvis who prepared the gold films and the EPSRC for financial support.


C R Evans

A J Davidson

C V Brown
email: carl.brown@ntu.ac.uk

N J Mottram

Static alignment states in a bistable azimuthal nematic device with blazed grating sidewalls

Keywords: Nematic liquid crystals, bistable states, polarisation, birefringence PACS keywords 42. Optics, 42.25.Ja Polarization, 42.25.Lc Birefringence, 42.70.Df Liquid crystals, 42.79.Kr Display devices, liquid-crystal devices, 77. Dielectrics etc, 77.84.Nh Liquids, emulsions, and suspensions, liquid crystals

Static alignment states in a bistable azimuthal nematic device with blazed grating sidewalls

C R Evans, a J Davidson, C V Brown,

Introduction

Conventional liquid crystal displays require continual electrical addressing to maintain even static images. Bistable nematic and cholesteric displays have the ability to retain a static image in the absence of any applied electrical power, and therefore have potential applications where low update rate is required for example in e-paper, e-billboards and for smart labels and smart cards in the retail arena [1] [2] [3] [4] [5].

In a nematic liquid crystal layer, azimuthal bistability (or multistability) refers to the situation where two (or more) distinct stable alignment states of the nematic n-director occur in the plane of the nematic layer, for example at different average angles of rotation in the layer. For the different states to remain stable under ambient conditions, i.e. with no applied electric field, there must be an energy barrier that prevents one of the states spontaneously transforming to the other state or vice versa. Methods of producing azimuthal bistability or multistability include: patterning the substrates that contain the nematic layer with a micro-scale surface bi-grating or with nano-scale gratings [6] [7]; arrays of posts within the layer or spanning the width of the layer [2] [8] [9]; and lateral confinement by dividing the layer into square well regions [8] [10] [11] [12] or channels [13] [14].

In a practical display device there must be a mechanism for switching between the bistable states in response to the electric fields produced by applying voltages to row or column electrodes. Switching between stable states in bistable nematic devices using the coupling to the flexoelectric polarisation has been investigated theoretically in the literature [15] [16] [17] [18]. In a previous experimental study of nematic channels by the current authors the switching between the bistable states was achieved using dual frequency addressing of a highly dispersive nematic liquid crystal [14].

The current authors have previously demonstrated stable azimuthal alignment states in the plane of a homogeneous layer of nematic liquid crystal that have been created by the action of one or more grating sidewalls. As long as the distance between the sidewalls (i.e. the width of the channels) is several times greater than the thickness of the layer then the occurrence and nature of these azimuthal alignment states are determined almost entirely by the morphology of sidewalls of the channels and not by the alignment properties of the upper and lower confining plates [13]. The nematic ndirector then remains in the plane of the layer and there is no observed out-of-plane tilt.

C.R. Evans et. al.

p. 3

Previously a periodic grating channel sidewall morphology has been investigated for which the profile of the grating is triangular and symmetrical within each period [13] [14]. In channels bounded by two symmetrical periodic grating sidewalls (either in phase or out of phase) two alignment states have been observed. These states did not form the basis for a practical display device because there was little optical contrast between the two bistable states when viewed in transmission (or in reflection) with polarised incident light and a crossed analyser. The motivation for current work was to investigate the director configurations when an asymmetry is introduced into the cell geometry, to consider the possible destabilising effect of this asymmetry, and to produce bistable states for which the average optical transmitted (or reflected) intensities are as distinct as possible. A practical display device that switches between these states would then be capable of displaying black and white images at high contrast. To achieve this aim a blaze asymmetry has been introduced within each period sidewall grating profiles.

Device details

The geometry used in the investigations is shown in figure 1. A nematic liquid crystal layer of thickness w = 22 µm was confined in the x-y plane by two glass substrates.

The inside surfaces of the substrates were coated with a 0.2 µm thick layer of SU8 photoresist which imparted degenerate planar alignment to the nematic liquid crystal (MicroChem Corp., MA, USA). The SU8 layer is indicated by the cross-hatch shading in figure 1(b). SU8 photoresist sidewalls further confined the nematic liquid crystal into channels running along the x-direction. The channel width was d = 100 µm, where d is defined in figures 1(a) and 1(b). The photolithography and process details for fabricating the device were the same as described in previous publications [12]. The ends of the channels were open and the commercial nematic liquid crystal E7 (Merck KGaA, Darmstadt, Germany) was filled into the channels by capillary action with both the material and the device at 80 °C, above the isotropicnematic phase transition temperature of 68 °C. After filling the device was cooled slowly at 0.5 °C per minute to room temperature.

The ratio of the channel width to the layer thickness, w/d, was in the regime where the layer thickness was small enough relative to the channel width so that the nematic ndirector remained in the x-y plane. At the same time the channel width was small enough relative to the layer thickness so that the interaction with the sidewalls determined the nature of the alignment states within the channel. This interaction overcomes the natural tendency of the nematic to form a Schlieren texture at the C.R. Evans et. al.

p. 4 SU8/glass substrates that would normally occur because of the planar degenerate alignment at the SU8 surfaces [11] [START_REF] Chandrasekhar | Liquid Crystals[END_REF].

The sidewalls were periodic gratings constructed from linear sections between the apex peaks and the troughs, as shown by the profile of the grating in the x-y plane in figure 1(a). The peak to trough amplitude of the grating was a = 20 µm and the grating pitch was p = 80 µm. Grating profiles with different blaze asymmetries were used in the investigation. The profiles of the gratings on either side of the channel are inverted relative to each other, i.e. if the channel width was reduced towards zero the two gratings would "fit together". Different degrees of asymmetry were introduced by creating channels in which the troughs had been laterally shifted in one direction towards the peaks by the distance b. When b = 0 µm (b/p = 0.0) each period of the sidewall gratings has a symmetrical triangular profile. The maximum amount of asymmetry in the investigation was with b = 40 µm (b/p = 0.5) which gives a highly blazed sawtooth profile.

Modelling the bistable alignment states

The n-director based Frank-Oseen nematic continuum theory [START_REF] Frank | [END_REF] [21] has been used to predict the nematic alignment states for the geometry depicted in figure 1.

Experimental observations indicate that the n-director remains in the x-y plane of the device [11] [14] and so the assumption is made that the n-director only varies in the lateral two dimensions and can be written in the form n = (cosθ, sinθ, 0) where θ(x, y) is the angle between the n-director and the x-direction defined in figure 1. On the SU8 boundaries there is degenerate planar anchoring of the n-director. In order for the easy direction to be parallel to the surface the surface free energy is written in the form

W surf = W sin 2 (θ -α(x))
where W is the anchoring strength and α(x) is local the angle of the grating surface with the x-axis [START_REF] De Gennes | The Physics of Liquid Crystals[END_REF] [START_REF] Rapini | [END_REF]. The governing equations were solved using a finite element package [24] to determine the resulting director profile for a given geometry. An isotropic elastic constant was used in the simulations, K = 1.75 × 10 -11 N, and relatively strong surface anchoring, W = 2 × 10 -5 N m -1 .

A potential disadvantage of using nematic continuum theory is that it does not allow the correct representation of nematic defects [START_REF] De Gennes | The Physics of Liquid Crystals[END_REF]. However, the present authors have previously found that modelling these types of geometries using n-director based theory and using the rigorous Q-Tensor theory give very similar results, even when dynamic reorientation occurs, provided that the defects remain close to the edges of the channels [14]. The alignment state shown in figures 2(a) and 2(b) will be referred to as the "horizontal" alignment configuration. The near-sidewall alignment for both of the director configurations is characterised by the nematic n-director smoothly following the contours of the grating. The variation in the orientation angle of the n-director θ (with respect to the x-direction) reduces exponentially with distance in the y-direction [25] from being at the tangent at the sidewall to being substantially parallel to the channel direction in the centre of the channel. Very similar mid-channel nearhorizontal alignment is predicted for both the symmetric zero blaze grating (b/p = 0.0) and for the high blaze asymmetric grating (b/p = 0.5) in figures 2(a) and 2(b) respectively.

The alignment state shown in figures 2(c) and 2(d) will be referred to as the "tilted" alignment configuration. The near-sidewall alignment for both of the director configurations is characterised by the nematic n-director pointing towards the vertex points near to the apexes of the peaks and in the troughs of the grooves leading to a high n-director orientation angle θ near to the surfaces. This near-grating alignment configuration was first investigated theoretically using Schwartz-Christoffel conformal mapping transformations for symmetric grating profiles in reference [26] and blazed grating profiles in reference [27]. For the symmetric zero blaze grating (b/p = 0.0) in figure 2(c) this type of near-surface alignment leads to a substantially vertical orientation of the n-director in the centre of the channel. For the high blaze asymmetric grating (b/p = 0.5) in figure 2(d) both of the grating surfaces promote a tilted n-director orientation with the tilt in the same direction and this leads to a substantially uniform tilted orientation in the centre of the channel. The symmetry here is critical, if the grating profiles were mirror images reflected about the centre of the channel [27] then they would promote tilted alignment in opposite directions, +θ from one side and -θ from the other side, leading to zero tilt horizontal alignment in the centre of the channel.

Figure 2 shows two stable states for each of the symmetric grating and the highly blazed grating sidewall geometries. For the symmetric grating there is a horizontal state (a) and a vertical state (c). The difference in average tilt angles for these two states is 90°, where the word "average" is used to refer to the n-director orientation for a significant proportion of the area of the channel. If these two states were viewed between crossed polarisors then they would give very similar optical transmission intensities. The theoretical maximum optical contrast that can be achieved between two azimuthal alignment states is when the two states differ in their net in-plane ndirector orientation angle by 45°. The blazed grating geometry in figures 2(b) and 2(d) is much closer to this, with θ ≈ 0°in the centre of the channel for the horizontal state (b) and θ ≈ 38°in the centre of the channel for the tilted state (d). The amount of blaze asymmetry that will give the optimum value of 45°in the tilted state is therefore likely to be between b/p = 0.0 and b/p = 0.5, but closer to the latter value. For all of the grating profiles shown in figure 3 the darkest texture with the minimum overall transmitted intensity occurs when the crossed polarisors are oriented at 0°/90°.The lightest texture having the maximum overall transmitted intensity occurs with the crossed polarisors at -45°/+45°. This observation is consistent with the ndirector orientation being horizontal for a significant proportion of the area of the device for all degrees of blaze asymmetry at the sidewall surfaces, as shown in figure 2(a) and 2(b). With the polarisors at 0°/90°light regions occur in the grooves of the sidewalls due to the tangential n-director orientation in the immediate vicinity of the walls and the associated elastic distortion of the n-director in these regions. When the polarisors are rotated these distortion regions give dark brushes which separate to one of the sides of the troughs, which side depends on the direction of rotation of the polarisors.

The optical textures in figure 4 correspond to the tilted alignment states shown in figures 2(c) and 2(d). For the most symmetrical grating profile, b/p = 0.0, in figure 4 the darkest texture is for polarisors at 0°/90°. For the symmetrical grating the ndirector alignment is close to vertical for a significant proportion of the channel which leads to an optical texture that is not so dissimilar from the corresponding horizontal state in figure 3 energy of nematic defect cores [START_REF] De Gennes | The Physics of Liquid Crystals[END_REF] is undefined in this theory, the subsequent energy integrations use a grid which excludes a small region around the defects. The horizontal state is predicted to contain the lowest elastic energy for sidewall gratings that are symmetrical and those that have a blaze asymmetry up to b/p = 0.43. There is a cross over at this asymmetry value and above b/p = 0.43 the tilted state is predicted to contain the lowest elastic energy. p. 9 value predicted by the theoretical simulations, at b/p = 0.43, is somewhat higher than the experimentally observed value. was the transmitted intensity of an unfilled device placed between crossed polarisors, and I para = 1.5 ± 0.1 Cd m -2 was the transmitted intensity of an unfilled device placed between parallel polarisors with broad band white light illumination.

Transmitted intensities of bistable alignment states

For the horizontal stable alignment state the minimum transmitted intensity occurs for all values of the blaze asymmetry when the crossed polarisers are oriented at 0°/90°, as shown in figure 8. There is good agreement between the theoretical curves in figure 8(a) and the results from observations in experimental devices in figure 8(b). This is consistent with the nematic n-director remaining oriented parallel to the channel and the x-axis for the majority of the area of the channel for all values of the blaze asymmetry.

The theoretical and experimental optical transmission results for the tilted stable This observation is consistent with the nematic n-director in the majority of the area of the channel tilting to higher values relative to the x-axis as the amount of blaze asymmetry is increased. The "ideal" tilt angle of 45°occurs for a blaze asymmetry of b/p = 0.3.

The minima in the experimental optical transmission curves in figure 9 amount of blaze asymmetry is not monotonic in an experimental device. In theoretical simulations small changes in grating shape can be made and the resolution of the mesh in the partial differentiation solver can be refined so that the results vary smoothly as a function of these small changes. A real experimental system does not necessarily always settle into the global minimum energy configuration. In the current device there are neighbouring bistable states along the same channel, inevitable small imperfections in the SU8 layers on the bounding plates (although the wall features themselves were sharply defined), and there is also some anchoring, albeit weak, on the upper and lower substrates that confine the layer into the x-y plane. The experimental values are the optical transmission values with the value for a blank device between crossed polarisors subtracted.

Conclusion and discussion

The theoretical maximum optical contrast that can be achieved between two azimuthal alignment states is when the two states differ in their net in-plane n-director orientation angle by 45°. Theoretical simulations and experiment measurements on devices that use sidewall-dictated alignment in nematic channels have shown that C.R. Evans et. al.

p. 11 bistability remains when an asymmetric, through a blazed grating, is introduced and that maximal contrast can be achieved using blazed grating sidewalls with a blaze asymmetry value of b/p = 0.3. In order to produce the tilted state the profiles of the gratings on either side of the channel must be inverted relative to each other.

In a practical device it would be possible to further increase the optical transmission contrast between the bistable horizontal and the tilted alignment states by reducing the amplitude and pitch of the grating as much as possible. This would have two beneficial effects. Firstly this would increase fill factor which is the area of the device that is covered by the electro-optical nematic layer as a fraction of the total area of the device which includes both the sidewalls plus the nematic layer. Secondly, a smaller amplitude sidewall grating would still dictate the stable nematic n-director orientation angles in the channel but the regions of distortion near to the grating peaks and troughs (which reduce contrast) would occupy a relatively smaller proportion of the active area of the device.

A further potential benefit of blaze asymmetry in the sidewall grating profiles is that the time for switching between the bistable states could be reduced. In a previous study of nematic channels, that was referred to in the introduction [14], there was one symmetric grating sidewall and one straight sidewall and switching between the bistable states was achieved using dual frequency addressing of a highly dispersive nematic liquid crystal. The switching mechanism involved the creation and collapsing of filament-like director reorientation (tilt-wall) loops with the associated motion and annihilation of surface defects along and close to the edge at the sawtooth sidewall.

This process would potentially be quicker along the shorter edges of a blazed grating, and using this effect in combination with reduced amplitude and reduced pitch gratings could improve switching speeds still further. An investigation of the effect of sidewalls with blaze asymmetry on switching mechanisms will be presented in a future publication.

Figure 2

 2 Figure 2 shows theoretical 2-dimensional n-director alignment configurations for two grating sidewall profiles, the symmetrical triangular profile with b/p = 0.0 in (a) and (c), and the blazed sawtooth asymmetrical profile with b/p = 0.5 in (b) and (d). In these figures the solid lines indicate the n-director orientation at different x-y positions in the channels. The director angle at the sidewalls is constrained by the strong anchoring to orient at the tangent to the grating surface. There are two distinct types of alignment configuration near to the grating boundaries of the channel that can be identified within the individual V-shaped grooves of the sidewalls.

4 .

 4 Optical textures of static bistable alignment states Experimental optical textures for horizontal and for tilted alignment states are shown in figure 3 and figure 4 respectively. These are bistable textures: in general both of the alignment textures can be observed in the same single channel separated by transition regions of elastic distortion. Temperature cycling the device up to above the nematicisotropic transition temperature of 68 °C and then back down to room temperature randomised the positions of the stable states. There were, however, small areas where one of the stable states remained pinned on a surface imperfection. The textures in figures 3 and 4 were recorded in transmission using a polarising optical microscope with a magnification of 500×. Each row shows the optical texture for a sidewall grating with a different degree of asymmetry starting from the left with a symmetrical triangular profile b/p = 0.0, and then increasing the asymmetry b/p = 0.1, b/p = 0.2, b/p = 0.3, b/p = 0.4, to the most asymmetrical sawtooth profile b/p = 0.5 on the right hand side. Each column shows the optical texture for five different orientations of the crossed polarisers: -45°/+45°, -22.5°/+67.5°, 0°/90°, +22.5°/-67.5°, and +45°/-45°where the first angle refers to the input polarisor orientation and the second angle to the crossed analyser orientation. The device remained static whilst the polarisors were rotated. As would be expected there is negligible difference in the optical texture recorded at -45°/+45°and the texture recorded at +45°/-45°for a given 2-dimensional director configuration. C.R. Evans et. al. p. 7

  figure 5 and figure 6 respectively. These textures were obtained using the expression )] , ( 2 [ sin 2 y x T = from the 2 dimensional n-director configurations n(x, y) calculated with Frank-Oseen nematic continuum theory. There is good agreement between the appearances of the theoretical optical textures and of the corresponding experimental textures for different values of the blaze asymmetry, as well as confirmation that each of the two states is stable for these values of blaze. For the horizontal states in figures

Figure 7 (

 7 Figure 7(a) shows the integrated elastic energies contained in the theoretically calculated 2 dimensional n-director configurations for the tilted alignment states (solid line) and the horizontal alignment states (dashed line) as a function of the amount of blaze, b/p. Nematic continuum theory was used in the calculations and, since the

Figure 7 (

 7 Figure 7(b) shows the frequencies of occurrence of the tilted (solid line) and horizontal (dashed line) states as a function of the amount of blaze, which was varied in the devices in the range b/p = 0 to 0.5 in steps of size b/p = 0.05. The frequency of occurrence at each value of blaze asymmetry was made by counting the number of periods of the sidewall channels that enclosed the vertical state, the number of periods that enclosed the tilted state, and then expressing each number as a percentage fraction of the total number of periods counted. For each value of the blaze asymmetry observations were made on up to 8 adjacent channels in the same device. Three measurements of the frequency of occurrence data are shown which are indicated (in the order in which the measurements were made) by circles, squares, and diamond symbols. Between each set of measurements the device underwent a thermal cycle up to the isotropic phase of the nematic liquid crystal.

Figures 8 (

 8 Figures 8(a) and 9(a) show the theoretical average transmitted intensities between crossed polarisors of the horizontal and the tilted alignment states respectively (figures 5 and 6). The intensities were calculated from the theoretical n-director alignment configuration at each blaze asymmetry value by averaging the value of )] , ( 2 [ sin 2 y x T =over an area that covered one period of the sidewall grating in the x-direction and across the channel from trough to trough of opposing grating sidewalls in the y-direction. The transmission through any regions covered by the sidewall grating was assumed to be zero.

  alignment state are shown in figures 9(a) and 9(b) respectively. The minimum of the theoretical curves shifts in figure 9(a) monotonically in the direction of negative input polarisor angles as a function of the amount of blaze asymmetry. The minimum transmitted intensity occurs at -5°for b/p = 0, -20°for b/p = 0.1, -41°for b/p = 0.2, C.R. Evans et. al. p. 10 45°(or -45°) for b/p = 0.3, 33°(or -57°) for b/p = 0.4, and 23°(or -67°) for b/p = 0.5.

  (b) shows the same trend of shifting in the direction of negative input polarisor angles as a function of the amount of blaze asymmetry. The minimum transmitted intensity occurs at -0°f or b/p = 0, -25°for b/p = 0.1, 25°(or -65°) for b/p = 0.2, 25°(or -65°) for b/p = 0.3, 30°(or -60°) for b/p = 0.4, and 25°(or -65°) for b/p = 0.5. The dependence on the

  The largest difference between optical transmission of the horizontal and tilted states occurs for b/p = 0.3 for the theoretical n-director configurations and for blaze asymmetry values between b/p = 0.2 and b/p = 0.4 in the experimental devices. For crossed polarisors oriented at 0°/90°and b/p = 0.3 the theoretical optical transmission is 0.076 for the horizontal state, 0.713 for the tilted state, and the ratio of these two values is 9.4. The experimental optical transmission for this polarisor orientation and grating asymmetry value is 0.27 ± 0.05 Cd m -2 for the horizontal state, 0.66 ± 0.05 Cd m -2 for the tilted state, and the ratio of these two values is 2.4 ± 0.6.

Figure 1

 1 Figure 1 The device geometry. (a) Shows a plan view of the structure and (b) shows a side view.

Figure 2

 2 Figure 2 Theoretical 2-dimensional director alignment configurations for: (a) and (b) the horizontal alignment state; (c) and (d) the tilted alignment state. The alignment configurations are shown for two grating sidewall profiles, symmetrical b/p = 0.0 in (a) and (c), and sawtooth (most asymmetrical) b/p = 0.5 in (b) and (d).

Figure 3

 3 Figure 3 Experimental optical textures for horizontal alignment states between crossed polarisers. Each column shows the optical texture for a sidewall grating with a different degree of asymmetry. Textures are shown for sidewalls where b/p = 0.0 (symmetrical), 0.1, 0.2, 0.3, 0.4, 0.5 (sawtooth profile, most asymmetrical). The rows show each of textures viewed at a different orientation of the crossed polarisers: -45°, -22.5°, 0°, 22.5°, and 45°(as shown to the left of the figure).

Figure 4

 4 Figure 4 Experimental optical textures for tilted alignment states between crossed polarisers. Each column shows the optical texture for a sidewall grating with a different degree of asymmetry. Textures are shown for sidewalls where b/p = 0.0 (symmetrical), 0.1, 0.2, 0.3, 0.4, 0.5 (sawtooth profile, most asymmetrical). The rows show each of textures viewed at a different orientation of the crossed polarisers: -45°, -22.5°, 0°, 22.5°, and 45°(as shown to the left of the figure).

Figure 5

 5 Figure 5 Theoretical optical textures for horizontal alignment states between crossed polarisers. Each column shows the optical texture for a sidewall grating with a different degree of asymmetry. Textures are shown for sidewalls where b/p = 0.0 (symmetrical), 0.1, 0.2, 0.3, 0.4, 0.5 (sawtooth profile, most asymmetrical).The rows show each of textures predicted for a different orientation of the crossed polarisers: -45°, -22.5°, 0°, 22.5°, and 45°(as shown to the left of the figure).

Figure 6

 6 Figure 6 Theoretical optical textures for tilted alignment states between crossed polarisers. Each column shows the optical texture for a sidewall grating with a different degree of asymmetry. Textures are shown for sidewalls

Figure 7 (

 7 Figure 7 (a) Elastic energies of the tilted (solid line) and horizontal (dashed line) alignment states as a function of the amount of blaze, b/p. (b) The frequencies of occurrence of the vertical (solid line) and horizontal (dashed line) states as a function of the amount of blaze, b/p. Three measurements of the frequency of occurrence data are shown (indicated by circles, squares, and diamond symbols) and the device was thermally annealed between each set of measurements.

Figure 8

 8 Figure 8 The transmitted intensity of horizontal alignment states as a function of polarisor angle. Data are shown for sidewalls where b/p = 0.0 (squares), 0.1 (circles), 0.2 (up triangle), 0.3 (down triangle), 0.4 (diamond), and 0.5 (crosses). The plots in figure (a) were generated from the theoretical textures shown in figure 5. The plots in figure (b) were measured using a photodiode on an experimental device exhibiting the textures shown in figure 2.

Figure 9

 9 Figure 9 The transmitted intensity of tilted alignment states as a function of polarisor angle. Data are shown for sidewalls where b/p = 0.0 (squares), 0.1 (circles), 0.2 (up triangle), 0.3 (down triangle), 0.4 (diamond), and 0.5 (crosses). The plots in figure (a) were generated from the theoretical textures shown in figure 6. The plots in figure (b) were measured using a photodiode on an experimental device exhibiting the textures shown in figure 3.

Figure 1 Figure 2 (

 12 Figure 1 (CREvans D363272PAP40015 figure01.eps)

Figure 7 (Figure 9 (

 79 Figure 7 (CREvans D363272PAP40015 figure07.eps)