PAPER www.rsc.org/softmatter | Soft Matter Dynamic charge separation in a liquid crystalline meniscus†‡ _ Tomasz Szymborski,a Olgierd Cybulski,a Iwona Bownik,a Andrzej Zywoci nski,a Stefan A. Wieczorek,a a ab a Marcin Fia1kowski, Robert Ho1yst* and Piotr Garstecki* Received 7th July 2008, Accepted 16th October 2008 First published as an Advance Article on the web 18th November 2008 DOI: 10.1039/b811561c Oscillating electric fields can sustain a macroscopic and steady separation of electrostatic charges. The control over the dynamic charge separation (dyCHASE) is presented for the example of circular menisci of thin, free standing smectic films. These films are subject to an in-plane, alternating radial electric field. The boundaries of the menisci become charged and unstable in the electric field and deform into pulsating, flower-like shapes. This instability ensues only at frequencies of the electric field that are lower than a critical one. The critical frequency is a linear function of the strength of the electric field. Since the speed of electrophoretic drift of ions is also linearly related to the strength of the field, the linear relation between critical frequency and the amplitude of the field sets a characteristic length scale in the system. We postulate that dyCHASE is due to (i) electrophoretic motion of ions in the liquid crystalline (LC) film, (ii) microscopic separation of charges over distances similar in magnitude to the Debye screening length, and (iii) further, macroscopic separation of charges through an electrohydrodynamic instability. Interestingly, the electrophoretic motion of ions couples with the macroscopic motion of the LC material that can be observed with the use of simple optical microscopy. Introduction In this paper we report macroscopic separation of electrostatic charge under the influence of an externally applied, oscillating electric field. Dynamic charge separation (dyCHASE) occurs only when we apply an electric field characterized by a frequency lower than a critical value, that is a function of the amplitude of the electric field. This frequency is directly linked to a lengthscale that is characteristic for the liquid crystalline (LC) materials that we use and—by virtue of our observations and simple calculations—appears to be closely related to the Debye screening length. We report on the observation of dyCHASE in a free standing thin film of a smectic LC, in which the electrophoretic transport of ions in the plane of the film couples with the motion of mass (meniscus), and can be readily visualized with the use of simple optical microscopy. Our method is simple in the sense that it uses optical microscopy to indirectly observe the electrophoretic motion of ions. The method takes advantage of (i) the relatively low viscous dissipation in a thin, free standing LC film, (ii) relatively low elasticity of the menisci, and (iii) an ease of observation of the thickness of the film in reflected light. Liquid crystals that are free from ionic impurities are good insulators. An electrostatic charge can be injected into LC samples e.g. by a high intensity laser field, chemical doping,1 a Institute of Physical Chemistry, Polish Academy of Sciences, Kasprzaka 44/52, 01-224 Warsaw, Poland. E-mail: garst@ichf.edu.pl; holyst@ptys. ichf.edu.pl b Department of Mathematics and Natural Science, College of Sciences, Cardinal Stefan Wyszynski University, 01-815 Warsaw, Poland † This paper is part of a Soft Matter issue highlighting the work of emerging investigators in the soft matter field. ‡ Electronic supplementary information (ESI) available: Additional images and data are presented. See DOI: 10.1039/b811561c 2352 | Soft Matter, 2009, 5, 2352–2360 applying a high electric field,2 or acquired from the surrounding atmosphere. Ionic impurities are unwanted in LC displays3 because the presence of charged impurities may lead to (i) screening of the electrodes,4 (ii) decrease of the quality of images via long-term image retention,5 (iii) flickering at low frequency due to the local change of the electric field by moving ions,6 or (iv) retention of the boundary of the image due to the distortion of transmission at the edges of the pixels.7 A long-term constant electric field applied to LC cells leads to the phenomenon of ‘image sticking’, caused by the leakage of current,8 which is also attributed to the presence of small concentrations of ionic impurities. Even if the material is free from charged impurities, they inevitably appear during the use of the material: either via tribocharging due to the rubbing of the bounding surfaces, or via ionization due to high electric field or UV illumination, or by contamination from the environment. The study of ion transport in LCs, and, in particular, a design for a simple method of visualization of the presence of ions in LC materials and of evaluation of their concentration is not only of basic but also of technological interest. In addition, the movement of ions sheds light on the local structure of the liquid crystal material.9,10 The transport of ions in ferroelectric smectic liquid crystals was studied from the point of view of their potential application in display technology with the emphasis on switching and bistability of smectic displays.11 Here, we provide a proof-of-concept demonstration that it is possible to visualize the presence and the motion of ions in LC materials with the use of optical microscopy. We use thin films spanning a round opening with the tip of a needle positioned in the center of the round rim, either slightly above the plane of the rim, or slightly below it. The films have nonuniform thickness; they either posses a meniscus of thicker material around the needle (when the needle pierces the film), or a freely floating ‘island’ of thicker LC material (when the needle remains This journal is ª The Royal Society of Chemistry 2009 underneath the film without piercing it). Meniscus and dislocations in free standing films have been described elsewhere.12–14 Importantly, for our experiment, the dislocations are circular and, approximately, centered around the vertical axis of the rim (and of the needle), and the change in the thickness of the film between the meniscus (or the ‘island’) and the region outside of the meniscus is large as it changes from a few micrometres to a few times the thickness of a single smectic layer (tens to a few hundreds of nanometres). This abrupt change in the thickness of the film causes a change in the electrical conductivity in the plane of the film which manifests itself via electrostatic charging of the boundary in oscillating electric fields and—in consequence—via an electro-hydrodynamic instability of the film. Our observations suggest that the parameters that control dyCHASE and the observed instability of the shape of the meniscus are: (i) the electrophoretic motion of ions in the film, (ii) the abrupt decrease in the conductivity of the film at the boundary of the meniscus or the island. In this report we discuss the role of the following effects: (i) dielectrophoresis, (ii) electrostatic torque acting on the polarization field of ferroelectric LCs, and (iii) electrostatic force on a charged boundary of the meniscus. We find that the electrophoretic force separates the positive and negative ions over a distance that is similar in magnitude to the Debye length.15 As a consequence of the change in the conductivity across the boundary, this separation localizes the charges on the dislocation lines, which subsequently leads to the instability of the shape of the meniscus in a radial electric field and consequently to the macroscopic separation of charge. Our experimental setup allows us to observe the coupling between the motion and separation of ions and the motion of the LC material with optical microscopy. Because the motion of the menisci in a LC film is subject to relatively small viscous resistance, we believe that it can be particularly useful for studying the dynamic (non-equilibrium) distribution of ions in soft matter systems. Materials and methods Experimental setup The experimental setup is illustrated in Fig. 1: a circular metal rim with an opening of diameter 5 mm was embedded in a hot stage. Both the rim and the stage were made of brass. The stage was mounted on an XYZ micromanipulator (Leica (XY) and ThorLabs (Z)). To control the temperature of the stage, a PID temperature controller (type 650, Unipan, Poland) was used, allowing the control of the temperature of the stage in the range from 0 to 99  C with an accuracy of 0.1  C. The temperature was monitored independently with two PT100 (Elfa, Poland) sensors. The first sensor was connected to the PID controller, and the second to a digital multimeter (196 System DMM, Keithley, USA). Below the circular opening, a stainless steel needle (diameter ca. 400 mm, with the tip sharpened to a radius of curvature of approximately 25 mm) was placed. The needle was mounted on its own heat stage and an XYZ micromanipulator (Leica). The temperature of the steel needle was controlled with a home-made electronic device, allowing the control with an accuracy of 0.01  C. A multimeter connected to the PT100 temperature sensor mounted on the needle stage was employed to monitor the temperature of the needle. This journal is ª The Royal Society of Chemistry 2009 Fig. 1 (a) A schematic illustration of our experimental setup. The metal frame on which we span the LC films was enclosed in a transparent screen-box to isolate the setup thermally and to reduce contamination from air. We observed the film with a microscope in reflected light. The frame and the needle were both equipped with temperature control units and with connections to the high voltage amplifier. (b) A photograph of the experimental setup; (c) schematic illustrations of a needle pierced through a film; (d) schematic illustrations of a needle under LC film; (e) pictures of the meniscus around the needle pierced through the 8CB film; (f) liquid crystalline island on freely suspended MHPPHBC film. A radial electric field between the needle and the circular rim was applied. The circular rim and the needle were connected to a high voltage amplifier (10/10B, Trek Inc.) controlled with Soft Matter, 2009, 5, 2352–2360 | 2353 a function generator (5062, Tabor Electronics Ltd). Square wave functions with duty cycle 50% were applied. Frequency (f) varied from single Hz to few kHz, while the peak potential difference (U) varied from tens of volts to 1 kV. Frequency and potential difference were monitored with a digital oscilloscope (54504A, Hewlett-Packard). Liquid crystalline films were prepared in the following way: first, the temperature of the stage and the rim was set to a few degrees above the melting point of the liquid crystal (the temperature of the needle was always set to the same value as the temperature of the stage). Then, a few milligrams of LC were placed on the metal rim. After the material has melted we span the film on the rim by sliding a metal spatula over the rim. The film was allowed to equilibrate for a few (circa 5) minutes. The liquid crystalline film was observed using a Nikon MSZ645 stereoscope equipped with a color CCD camera (Vido, AU-CC882SF) mounted on one of the eyepieces and connected to a PC via an analogue video card, and an LED light source, positioned in the second eyepiece. Both the camera and the LED were aligned with the optical axes with the use of home-made manipulators. All observations were carried out in reflected light: the light from the LED travelled through the stereoscope from one of the eyepieces onto the film; reflected from the film and came back through the stereoscope to the second eyepiece and the CCD chip. The stereoscope and the stage was placed in a transparent box in order to isolate the experimental setup thermally, to avoid the motion of the air above the film, and to prevent the film from absorbing impurities from the air. The box was equipped with a polyethylene sleeve to allow manipulation of the samples. Liquid crystals Four different liquid crystalline materials were investigated: 4octyl-40 -cyanobiphenyl (8CB), 40 -nitrophenyl-4-octyloxybenzoate (NPOB), methylbutyl 4-n-nonanoyloxy-biphenyl-40 -carboxylate (MBOBC) and 4-[4-(1-methylheptyloxycarbonyl)phenyl]-40 -[6(propanoyloxy)hexyloxy]biphenyl-4-carboxylate (MHPPHBC). The chemical structures of these compounds are shown in Fig. 2. For the sequence of phase transitions see Table S1 in the ESI.‡ Importantly, two of the LC materials (MBOBC and MHPPHBC) possess SmA* and SmC* ferroelectric phases in which the material exhibits a non-zero spontaneous polarization due to the dipole moment and chiral structure of the LC molecule. The remaining two compounds (8CB and NPOB) do not exhibit ferroelectric ordering and were examined in their SmA phases. All compounds were purchased from the Military University of Technology (Warsaw, Poland) and used as delivered. Experiments We used two variations of the experimental setup: (i) one in which the film was pierced by the needle, and (ii) the second in which the tip of the needle was placed slightly (250 mm) below the film. In the first case, the needle was wetted with a minute amount of LC material, and the meniscus around the needle was formed spontaneously at the instant of piercing the film. In the second case, a freely floating island was created above the needle. 2354 | Soft Matter, 2009, 5, 2352–2360 Fig. 2 The chemical structures of the liquid crystals used in the experiments. From top to bottom: 4-octyl-40 -cyanobiphenyl (8CB), 40 -nitrophenyl 4-octyloxybenzoate (NPOB), methylbutyl 4-n-nonanoyloxy-biphenyl-40 -carboxylate (MBOBC), 4-[4-(1-methylheptyloxy carbonyl)phenyl]-40 -[6-(propanoyloxy)hexyloxy]biphenyl-4carboxylate (MHPPHBC). Usually, the islands did not form spontaneously. In order to create an island, a low frequency (few Hz) and high amplitude (hundreds of volts) electric field was applied. It caused a disruption of the LC meniscus formed initially in the rim into many small islands. Then, either a DC or a high frequency ($ 100 Hz) electric field was applied to make the small islands merge into a single one. The single LC island formed above the needle because of dielectrophoresis: since the LC material has a higher dielectric constant than the air, it is pulled into the regions of higher intensity of the field—into the center of the rim. For each series of measurements, we typically set the potential difference to a constant value and slowly decreased the frequency, starting from a value of 1 kHz, until instability ensued. The shapes of the menisci (islands) were recorded using the CCD camera and analyzed. Results Inner electrode in contact with the film In the first type of experimental setup (the inner electrode piercing the film) we examined the behaviour of two liquid crystals: 8CB forming an SmA phase (at 30  C) and MHPPHBC forming an SmA* phase (at 85  C). We prepared the LC films and menisci as described in the previous section. Then, we applied an electric field of varying amplitude from 75 to 300 V per radius of the rim (2.5 mm) and of varying frequency (from 100 Hz to single Hz). For each set of parameters (f and U) we recorded the deformations of the LC film and later analyzed the micrographs. Our first observation is that for a given amplitude of the applied alternating electric field, the meniscus is stable for all This journal is ª The Royal Society of Chemistry 2009 frequencies higher than the critical frequency, fCR. For frequencies above fCR we did not observe any noticeable deformation of the shape of the boundary of the meniscus—it remained circular, as is depicted in Fig. 3a, d and g. For f z fCR the meniscus became unstable and its boundary began to deform periodically and followed the oscillations of the electric field (Fig. 3 b, c, e, f, h, and i). These deformations had the form of tips (or ‘fingers’) growing radially outwards from the boundary of the meniscus. As we decreased f further below fCR the amplitude of the deformations (the length of the fingers) grew (Fig. 4). At very low frequencies (few hertz with the exact value of f depending on the magnitude of the electric field) the length of the fingers became comparable to, or as long as, the radius of the rim. When the fingers reached the rim, the meniscus typically broke, and the whole system became disordered: small islands of thicker LC material were created and floated on the LC film. Fig. 4 Method of estimating the critical frequency for MHPPHBC liquid crystal (a). The graph shows the experimentally measured dependence of the amplitude of deformation of the LC island on the frequency of the applied electric field (plotted as a function of the reciprocal of the frequency). A linear fit allows us to determine the value of the critical frequency for the particular value of the electric field. Critical frequency plotted as a function of applied voltage for constant island diameter (ca. 2 mm) for MHPPHBC liquid crystals (SmA* phase) (b). Measurements were performed for the needle not in contact with the film. Critical frequency Fig. 3 The qualitative behaviour of the 8CB thick film subjected to an oscillating electric field is shown in parts (a) to (i). For a fixed potential difference U, there is a critical value of frequency above which the meniscus is stable (parts a, d and g). Below that frequency we observed instabilities (b, c, e, f, h, and i). Due to the sharp transition between the stable and unstable meniscus we were able to estimate the dependence of the value of the critical frequency on the applied voltage via a direct observation of the meniscus; we present this graph in part (j). This journal is ª The Royal Society of Chemistry 2009 The transition between a stable (immobile) meniscus (Fig. 3a, d and g) and an oscillating one (Fig. 3b, e, and h) was so sharp that the critical frequency could be easily determined by visual observation of the film while lowering the frequency by a single Hz at a time. We present the relation between such determined fCR and U in Fig. 3j. In order to introduce a more systematic method of determining fCR for the various LC materials that we used, and to measure how fCR depends on the difference, U, of the electrostatic potential between the electrodes, for each pair of values of f and U we recorded a video depicting the evolution of the shape of the meniscus. Then, by inspecting individual frames, we measured the minimum (DMIN) and the maximum (DMAX) distance between two points on the boundary of the meniscus located on opposite sides of the center of the rim. We defined the amplitude, A, of the deformation as A ¼ (DMAX  DMIN)/2. The amplitude grew monotonically—and roughly linearly—with Soft Matter, 2009, 5, 2352–2360 | 2355 the reciprocal of the frequency (A f 1/f) as shown in Fig. 4a. In order to determine the critical frequency fCR for each value of U, we plotted A as a function of 1/f, and then fitted a linear relation AFIT(1/f) ¼ A0 + a(1/f) to the experimental points with A0 and a being the fitting parameters. We then calculated the critical frequency as fCR ¼ a/A0. Inner electrode not in contact with the film When the electrode pierces the film, we observed that at a constant electric field (U ¼ const) there is a persistent whirling motion of the LC material. The circular area of the rim divided into a small number of ‘cells’ within which the LC material flew in a continuous circling motion (see Fig. S1 in the ESI‡). Although we did not measure the electrical current, we attribute this observation to electro-convection and to leakage or conduction of current through the LC film. When we withdrew the needle from the film (thus breaking the electrical circuit) and left it tens of microns below the LC material, with the field switched on, the convection stopped. In order to avoid problems associated with the leakage current at low frequencies and electro-convection, we performed experiments with the needle positioned below—that is not in contact with—the film. In these experiments we used MHPPHBC liquid crystal as MHPPHBC easily and reproducibly formed freely floating LC ‘islands’—circular regions of thicker film. We created single (ca. 1 mm in diameter) islands (Fig. S2a—see ESI‡) in steps. We first span the film that typically did not posses any islands. We then pulled LC material from the meniscus on the rim by applying a low frequency (3 Hz) and high amplitude (200 V) electric field. After a few cycles there were typically about ten small islands on the film. Then a high frequency (100 Hz) and high amplitude (300 V) electric field was applied to merge the islands into a single one by dielectrophoretic forces (see Fig. S2b in the ESI‡). Qualitatively, the behaviour of this system was similar to that when the needle was in contact with the film. The shapes of the islands depend on the applied voltage and frequency as shown in Fig. 5. For example, for U ¼ 400 V, the island was stable for all frequencies above fCR ¼ 20 Hz. For f < fCR the island became unstable and deformed in sync with the applied electric field. For U ¼ 600 V, fCR ¼ 30 Hz, and, for smaller frequencies, multiple ‘fingers’ showed. The amplitude of the deformations increased with increasing U and with decreasing f. Like in the case of films pierced by the needle, the critical frequency increased linearly with the amplitude of the applied electric field (Fig. 4b). Fig. 5 Deformation of the MHPPHBC (SmA* phase) liquid crystalline island. Lower voltage oscillation frequency is present at 20 Hz, whereas oscillation at 600 V is present at 30 Hz. Oscillations for higher voltage have higher amplitude thus are clearly visible. For low frequencies, in both cases, we observe deformations of the LC island. Qualitative analysis of the instability We observed similar phenomena in all the LCs that we tested (8CB, NPOB, MBOBC and MHPPHBC) and both in the smectic SmA and in the ferroelectric SmA* phases. In all these systems the menisci and islands were stable for frequencies larger than a critical frequency, and unstable below it (Fig. 3, 5, and 6). A study of a system similar to the one we study here (with a [4-(3-methyl-2chloropentanoyloxy)-40 -heptyloxybiphenyl ferroelectric LC)17 suggested that a low frequency electric field can be used to 2356 | Soft Matter, 2009, 5, 2352–2360 Fig. 6 Values of the critical frequencies, fCR, as a function of applied voltage, U, for the four different liquid crystals used in experiments. Measurements performed for the needle in contact with the film. This journal is ª The Royal Society of Chemistry 2009 partially lift the efficiency of screening imposed by the presence of ions in a ferroelectric LC material. In the absence of impurities a ferroelectric LC phase possesses a spatially uniform ground state with all the molecular dipoles aligned along a common direction, resulting in a macroscopic net polarization of the material. The presence of ions hinders the long-range order—the ions migrate to the boundaries of the unidirectional, polarized domains where they screen the charge that arises from the nonzero divergence of the polarization field at the grain boundaries. In ref. 17 the authors showed the effect of coupling between the direction of polarization of the ferroelectric LC and the direction of the dislocation vectors at a boundary of a meniscus or of an island. The film polarizes locally along the dislocation line. Application of a radial electric field (perpendicular to the direction of the dislocation, which is circular around the needle) applies a torque on the molecular dipoles, and—through the said coupling—a torque to the dislocation line. The expectation is thus that the field should destabilize the circular dislocation line into a flower-like pattern with extended segments of the dislocation lines oriented radially (along the direction of the electric field). However, due to the screening of the polarization grains, and the—resulting—short-range character of the dipolar order in the LC material, the interaction of the dislocation line with the radial electric field is too weak to overcome the line tension and to destabilize the circular shape of the dislocation. As the instability occurs only at low frequencies (few Hz) of the electric field, it was suggested17 that for a short interval after the event of switching the polarization of the electric field, the ionic impurities, resting at the boundaries of the grains of polarization, migrate to new positions of equilibrium, and, during this short interval the screening is partially lifted. Decreased efficiency of screening leads to a transient long-range ferroelectric ordering. As a consequence, during these short transients, the electric field acts with a torque on substantially large regions of the LC material polarized along the dislocation line, which leads to the observed instability of the shape of the meniscus. As we describe below, some of our observations are different from those reported in the paper of Ho1yst et al.17 As was suggested by the authors of the work cited, in LC samples containing ionic impurities the deformations of the film can be observed only at the moment of switching of the polarization of the applied electric field when the ions move and the efficiency of screening of the polarization field is partially lifted. Two of our observations suggest, however, an alternative explanation. First, we observe the fingering instability both in the ferroelectric and in the common smectic phases. Second, we observe that—as we lower the frequency below the critical value—the fingers continue to grow in length throughout the whole period of constant polarization of the electric field, rather than showing only for a short, and roughly constant in length, transient and then decaying. For very low frequencies (e.g. 1 Hz or lower), for the mechanism described in ref. 17 we would expect that after the reorganization of the ionic clouds in the LC material, the boundaries of the uniform polarization of the LC should again be screened and the fingers should shorten, leading to a circular meniscus. In fact, in ref. 17 the authors reported no instability in a constant electric field, while we do observe that in a DC electric field the meniscus is unstable. Further, the observation that when the needle is in contact with the film we observe the consistent This journal is ª The Royal Society of Chemistry 2009 whirling motion suggests that ionic impurities are constantly being created and transported between the electrodes. There are two other—than the torque on a polarization field— electric forces that can play a role in our system: (i) dielectrophoresis, and (ii) electrostatic force on the meniscus that is charged via an electrophoretic motion (and separation) of ionic impurities. Dielectrophoresis is expected to pull the liquid crystalline material towards the center of the rim, where the intensity of the electric field is greatest, and we use this effect to create single large islands of the LC material. This effect cannot be, however, attributed to the emergence of the flower instability. A consistent explanation of all our observations is based on the hypothesis that under the action of the oscillating electric field, at least a portion of the electrostatic charge present in the sample undergoes a microscopic separation which subsequently leads to dyCHASE and macroscopic patterns that we observe. Below we propose a simple model of this behaviour and compare it with the results of our experiments. Electric field in the plane of the film The ions in the LC film are dragged by an electrophoretic force and move at a speed u ¼ Ekm, where Ek is the component of the electric field in the plane of the film, and m is the electrophoretic mobility of the ions in the film. The amplitude of Ek depends on the radial position. In an idealized, cylindrical configuration, with an infinite needle positioned along the axis of an infinite, hollow, metallic cylinder (a cylindrically symmetric configuration) the electric field would be simply proportional to 1/r, where r is the radial distance from the axis of the needle. However, in the finite geometry of our experimental setup Ek depends differently on r: because the tip of the needle is positioned in the proximity of the plane of the circular rim, the planar component of the electric field first increases with increasing r, has a maximum and then decreases with increasing r. From our numerical calculations of the electric field (see an exemplary plot of Ek in Fig. S3 in the ESI‡) it follows that the maximum of Ek is located at a distance from the axis of the needle that is typically smaller than the radius of the meniscus, or an island d/2, where d is the diameter of the meniscus or the island. Throughout most of the variation in r, the magnitude of Ek does not change significantly (varies within a factor of 2). Separation of charges We do not know the chemical nature of the ions present in our samples. We start with an estimation of the electrophoretic mobility of hypothetical ions of different charge q ¼ 1, 2 and 3e (e ¼ 1.60217  1019 C) and of different hydrodynamic radii r ¼ 1, 5 and 10 Å respectively. The mobility m is given by m ¼ q/6prh, where h is the coefficient of viscosity of the medium. Because the coefficients of viscosity for the isotropic phases of the MHPPHBC, MBOB and NPOB liquid crystals are not available in the literature, in our estimates we assumed the value of the coefficient of viscosity for the isotropic phases of all our compounds as the same as of 8CB, h ¼ 25.84 mPa s at 314.1 K.18 Table 1 summarizes the results of this estimate. The values that we obtained compare well to the values reported in the literature7,11,19,20 for electrophoretic mobility of ions in LC (m z 1010 m2 V1 s1). Soft Matter, 2009, 5, 2352–2360 | 2357 Table 1 Electrophoretic mobility for different ion diameters and charges q (e) R (Å) m (m2 V1 s1) 1 2 3 1 5 10 3.291010 1.321010 0.981010 We now estimate the distance dions travelled by the ions within half the period of the oscillations of the electric field at the critical frequency. This distance is dions ¼ m(2fCR)1Ek, where for Ek—for the purpose of the order of magnitude estimate—we substitute the crude approximation that Ek ¼ U/Rrim. Because fCR was found to be linearly dependent on Ek, we can put: fCR ¼ (dfCR/dU)(U/Rrim)Rrim, and obtain dions ¼ (m/2Rrim)(dfCR/dU)1, with the value of the derivative (dfCR/dU) extracted from the linear fit to the experimentally determined relations of fCR(U). Table T2 in the ESI‡ contains the results of the calculation of the dions for all the series of experiments that we performed and all three estimated values of the electrophoretic mobility. In summary, for m ¼ 11010 m2 V1 s1, dions ranged from dions ¼ 50 nm for NPOB in the SmA phase (Fig. 6) to dions ¼ 460 nm for MHPPHBC in the SmA* phase (Fig. 5). The results of this estimation are similar in magnitude to the reported value of the Debye screening length (examined in free standing film prepared with a mixture of liquid crystals—CS1015), 0.7 mm.15 CS1015 is ferroelectric liquid crystal commercially available from Chisso Co. with SmC* phase in room temperature.15 Fig. 7 Critical interval of time t0 plotted as a function of the diameter of the island for MHPPHBC (SmA* phase). Measurements performed for the needle not in contact with the film for U ¼ 600 V. (or the diameter d of the meniscus). As seen, except for the biggest diameter, the differences between t0 for different island sizes are within the statistical errors. The slight increase in t0 observed with increasing diameter of the island can be attributed to the decrease in the amplitude of the radial component of the electric field with increasing r (see Fig. S3 in ESI‡). Rate of growth of the fingers At frequencies lower than critical, we divide the interval t ¼ (2f)1 between the switches of the polarization of the electric field into t ¼ t0 + tD where t0 ¼ (1/2fCR) is the time needed for separation of the charges and for charging of the boundary of the meniscus, and tD ¼ (1/2) (1/f  1/fCR) is the interval during which the (electrostatically charged) boundary of the meniscus deforms. In Fig. 8 we show the amplitude of the deformation of the island versus tD for Dependence of the critical frequency on the diameter of the meniscus The postulate that the distance over which the oppositely charged ions separate is related to the properties of the LC material (such as the Debye screening length), and not the geometry of the system, is supported by the observation that the value of the critical frequency does not depend on the diameter of the meniscus, d. In Fig. S4 (see ESI‡) we show the critical frequency measured for the system with the needle in contact with the film of 8CB LC. Measurements were performed in the SmA phase for four different diameters of the meniscus (from 1.19 mm to 1.7 mm). The critical frequency shows a monotonic— and approximately linear in U—increase for all series, and the slope of the experimental functions of fCR(U) does not depend systematically on the diameter of the meniscus. In Fig. 7 we show the critical interval of time—the interval needed for the ions to separate—t0 ¼ (1/2) (1/fCR) as a function of the diameter of the island. The measurements were performed with the needle not in contact with the film, and for different islands of diameters ranging from d ¼ 1.04 to 2.65 mm. The data presented in Fig. 7 suggests that t0 ¼ (1/2fCR) is constant within the approximation that the magnitude of the electric field does not depend appreciably on the radial position r 2358 | Soft Matter, 2009, 5, 2352–2360 Fig. 8 Amplitude of the deformation of the boundary of the meniscus vs. the interval between the switches of polarization minus the time needed for the ions to separate, (1/2)(1/f  1/fCR), for MHPPHBC liquid crystal (SmA* phase). Series of experiments were performed for different island diameters ranging from 1.04 mm to 3.04 mm. The needle was not in contact with the film and the applied voltage was U ¼ 600 V. As seen, the amplitude of the deformation increases with (1/2)(1/f  1/fCR). This journal is ª The Royal Society of Chemistry 2009 Fig. 9 Amplitude of the deformation of the boundary of the meniscus vs. the interval between the switches of polarization minus the time needed for the ions to separate. Experiments with MHPPHBC liquid crystal (SmA* phase). The needle was not in contact with the film. The applied voltage ranged from 100 V to 800 V. For clarity, we plot only five selected series of data. The inset shows the dependence of the rate of growth (S) on the amplitude of deformation as a function of the applied voltage (obtained by linear regression of series A(tD)). ferroelectric LC MHPPHBC in SmA* phase. We performed the measurements with the needle not in contact with the film and for LC islands of different diameters: from d ¼ 1.04 to 3.04 mm. Amplitude (A) of the deformation exhibits a monotonic increase as a function of tD. That is, the deformation never decreases as time progresses (with increasing tD). We note the apparent difference in the growth rate of the deformations for different diameters of the islands. In Fig. 9 we show a plot of the amplitude of deformation versus tD for a selected diameter of the island (d ¼ 1.93 mm), for the same compound but for different values of U. As before, the dependence of the amplitude of deformation, A, on tD shows a monotonic increase in tD and, with a good approximation, can be described by the relation A(tD) ¼ A0 + StD, where S is the rate of growth and A0 is a constant. For each value of voltage applied we calculated the quantity S from the linear fit to the data. As seen, the rate of growth is a linear function of the applied voltage U. Discussion To our knowledge this is the first experimental evidence of macroscopic separation of electrostatic charge induced by an oscillating electric field. We refer to this phenomenon as the dynamic charge separation (dyCHASE). Importantly, dyCHASE is sensitive both to the frequency and to the amplitude of the electric field. The feature that the transition from a relaxed distribution of charges to dyCHASE is sharp in frequency allows for an easy dynamic control of the phenomenon, and consequently to a control over—at least important characteristics of— charge distribution and over macroscopic behaviour of our system. We propose that dyCHASE can be explained with a model based on the electrophoretic motion of ions. We show that there This journal is ª The Royal Society of Chemistry 2009 exists a sharp frequency threshold (‘critical frequency’—fCR) below which the meniscus (or the island boundary) becomes unstable. This critical frequency is associated with the time that it takes for the ions to move (separate) electrophoretically over the Debye screening length. The above model finds confirmation in experimental results: the critical frequency fCR is a linear function of the voltage applied to the film, and it does not depend on the diameter of the meniscus or the island. Although our findings are different than the ones reported in ref. 17 we suppose that the effects of coupling between the directions of dislocations and polarization in ferroelectric smectic LCs exist and require further investigation. In our experiments we did not monitor the thickness of the LC islands (or menisci). It can be argued that the thickness should not affect neither the qualitative nor the quantitative behaviour because the forces that determine the motion (deformation) of the meniscus are both proportional to its thickness: (i) we can assume that the density of charge on the boundary is proportional to the content of the ionic impurities in the LC material, and, assuming a constant density of ionic impurities in the LC, the charge on the boundary should be proportional to the thickness of the meniscus, and (ii) the elastic forces (arising from the line tension of the dislocation lines) are also proportional to this thickness, as the number of dislocations at the boundary is. Also the viscous forces that oppose the motion of the boundary are proportional to the amount of LC material that is subject to motion (shear). The fact that we did not observe significant scatter in the values of critical frequency and the rate of growth of the fingers in repeated experiments with the same control parameters suggest that indeed the behaviour of the system does not depend appreciably on the thickness of the film. We were not able to discriminate the critical frequencies for the motion of cations and anions separately. If these differed substantially in their electrophoretic mobility our system could show substantial asymmetry between the positively and negatively charged ‘fingers’ and perhaps display a difference in the critical intervals for the positive and negative ions to reach the boundary of the meniscus and to deform it in the electric field. Our observations raise an important question about the microscopic details of the non-equilibrium distribution of charge for frequencies equal to and lower than the critical value that corresponds to oscillations of ions over the Debye length. Why do the charges macroscopically separate when the oscillations of ions exceed the threshold of the screening length? Can the observed behaviour shed light on the relation between ionic strength, screening lengths and relaxation time-scales for fluctuations in the local density of electrostatic charge? Finally, we can hypothesise that a similar methodology applied to heterogeneous systems characterized by a set of characteristic length- and time-scales (e.g. corresponding to different sizes of macro-ions present in the system) could exhibit selective, frequency modulated dyCHASE and allow independent control of the electrostatic interactions between the different charged components of the system. Acknowledgements This research was supported as a scientific project from the science budget of the Polish Ministry of Science and Higher Soft Matter, 2009, 5, 2352–2360 | 2359 Education (2005–2008 and 2007–2010) and by the SONS (SCALES) grant from the Ministry of Science and Higher Education (2006–2009). P.G. acknowledges financial support from the Foundation for Polish Science under the Homing fellowship and within the TEAM programme. R.H. acknowledges support from the Foundation for Polish Science under the MISTRZ grant. References 1 M. Funahashi and J. Hanna, Chem. Phys. Lett., 2004, 397, 319–323. 2 R. Bushby and O. R. Lozman, Curr. Opin. Solid State Mater. Sci., 2002, 6, 569–578. 3 K. Neyts, S. Vermael, C. Desimpel, G. Stojmenovik, R. van Asselt, A. R. M. Verschueren, D. K. G. de Boer, R. Snijkers, P. Machiels and A. van Brandenburg, J. Appl. Phys., 2003, 94, 3891. 4 R. S. Zola, L. R. Evangelista and G. Barbero, J. Phys. Chem. B, 2006, 110, 10186–10189. 5 G. Stojmenovik, S. Vermael, K. Neyts, R. van Asselt and A. R. M. Verschueren, J. Appl. Phys., 2004, 96, 3601–3608. 6 K. H. Yang, J. Appl. Phys., 1990, 67, 36–39. 7 G. Stojmenovik, K. Neyts, S. Varmael, A. R. M. Verschueren and R. v. Asselt, Jpn. J. Appl. Phys., 2005, 44, 6190–6195. 2360 | Soft Matter, 2009, 5, 2352–2360 8 S. Takahashi, J. Appl. Phys., 1991, 70, 5346–5350. 9 D. Adam, F. Closs, T. Frey, D. Funhoff, D. Haarer, P. Schuhmacher and K. Siemensmeyer, Phys. Rev. Lett., 1993, 70, 457–460. 10 M. Funahashi and J. Hanna, Phys. Rev. Lett., 1997, 78, 2184–2189. 11 K. Neyts and F. Beunis, Ferroelectrics, 2006, 344, 255–266. 12 J.-C. Geminard, R. Ho1yst and P. Oswald, Phys. Rev. Lett., 1997, 78, 1924–1927. 13 F. Picano, R. Ho1yst and P. Oswald, Phys. Rev. E, 2000, 62, 3747– 3757. 14 J. C. Geminard, R. Ho1yst and P. Oswald, Acta Phys. Polon. B, 1998, 29, 1737–1747. 15 J.-B. Lee, R. A. Pelcovis and R. B. Meyer, Phys. Rev. E, 2007, 75, 051701. 16 Z. Raszewski, J. Ke˛dzierski, P. Perkowski, J. Rutkowska, W. Piecek, _ J. Zieli nski, J. Zmija and R. Da˛browski, Mol. Cryst. Liq. Cryst., 1999, 328, 255–263. 17 R. Ho1yst, A. Poniewierski, P. Fortmeier and H. Stegemeyer, Phys. Rev. Lett., 1998, 81, 5848–5851. 18 J. Jadzyn, R. Da˛browski, T. Lech and G. Czechowski, J. Chem. Eng. Data, 2001, 46, 110–112. 19 H. d. Vleeschouwer, A. Verschueren, F. Bougrioua, R. v. Asselt, E. Alexander, S. Vermael, K. Neyts and H. Pauwels, Jpn. J. Appl. Phys., 2001, 40, 3272–3276. 20 L. O. Palomares, J. A. Reyes and G. Barbero, Phys. Lett. A, 2004, 333, 157–163. This journal is ª The Royal Society of Chemistry 2009