M Beaufils
email: m.beaufils@brgm.fr

N Hassan

C Galandrin

P Miraillet

B Le Hello

I Halfon

FAIR Geotechnical data to support BIM for Underground Infrastructure

Keywords: Geotechnical data, Building Information Modeling, Data interchange, Interoperability 1

In France, the MINnD Project aims at extending Building Information Modeling (BIM) capacities to describe infrastructures such as roads, bridges, railways, tunnels and geotechnics. The MINnD working group GT1-5 is dedicated to Geotechnics, especially in an underground infrastructure construction context. Such activity leads to the proposal of a conceptual data model and appropriate vocabularies to describe the geotechnical conditions. Those deliverables contribute to the specification of international standards for openBIM (as part of the buildingSmart International IFC Tunnel project) and in openGIS (Open Geospatial Consortium standards) in connection with existing practices (AFTES recommendation). MINnD GT1-5 then enables to organize the provision of geological and geotechnical data compliant with the FAIR principles (Findable, Accessible, Interoperable and Reusable). Such level of interoperability is necessary to envisage the definition of a Digital Twin of the environment of an Underground Infrastructure and its sustainable management.

INTRODUCTION

Infrastructure projects are always a team work. Different actors get to be involved in the same project to play different and various roles, such as the entity in charge of design or the one in charge of execution. Even in the same entity, different contributors for different topics (geotechnical, structure, drainage, landscape..etc) participate at the same project. Also, a complete project study is always done via different stages over time, providing progressive level of details at each step.

In France for example, four stages of geotechnical studies workflow are applicable: mission G1 to G4, where G1 aims of getting the main project context (geological, hydrogeological and ground related risks) till G4 with the construction follow up. Different actors may be in charge at each study level.

At each step of a project, geotechnical data may be collected, analyzed and interpreted. In the current practice of typical projects, each actor has its own method of data analysis and structuration. For instance, the same geotechnical property may be called with different names (e.g. undrained cohesion / undrained shear strength) and may be measured with different methods (e.g. vane test / UU triaxial test / derived from correlation).

In addition, many geotechnical properties (as for example drained cohesion and friction angle) are not the direct result of a simple measurement but are derived from more or less complex tests, involving different specimens, intermediate values and an interpretation, that are worth to be traceable. Regarding the data exchange, the same type of data can be provided with various formats: pdf, spreadsheet, etc. Such heterogeneity of formats is enforced by the existence of different softwares. For these reasons, geotechnical data are often not easily accessible and exchangeable between the different stakeholders of a given project, and also from a project to another one, and therefore are often used only once.

In addition, today the geotechnical model is becoming an essential part of the BIM model, especially for infrastructures like roads, bridges, railways, tunnels and underground projects. To be incorporated in a BIM model, it is essential that geotechnical properties and objects be clearly defined and described, and that their provenance (e.g measurement method or procedure) be tracked.

For these reasons, the geotechnical data exchange can be challenging and needs to develop interoperable formats, in order to avoid data loss and time consuming.

In 2016, FAIR principles have been introduced in order to describe the level of interoperability one user can have with numerical data. Four criteria were defined. They all together form the FAIR accronym.

-Findable: the capacity to discover existing data, -Accessible: the capacity to get data, -Interoperable: the capacity to understand the obtained data, -Reusable: the capacity to exploit the data.

In practice, setting data that match those FAIR principles require to take care of data description (Interoperability and Reusability criteria) but also set additional components to make data Findable and Accessible. These prerogatives enable to define the order in which those criteria shall be taken into account.

This paper presents the actions and results from the French MINnD GT1-5 project for geotechnical data standardization. The focus is on the compliance with the FAIR principles and the processes followed to reach them. More than technical solutions, it also opens discussion on geotechnical data exchange and sharing, and the possible role that organizations like national geological surveys can play.

INTEROPERABLE

The question of data description is a key topic in science and other domain. Shared language and vocabularies are necessary to prevent from misunderstanding the information and its meaning.

Standardization mostly rely on the identification and definition of agreed concepts to be used for data description. Such activity shall be driven by the requirements of the domain (i.e. geotechnics) and current practices.

The data structuration proposal from MINnD is based on two main inspiration:

-The AFTES guidelines that introduces three books for the furniture of geotechnical data. Book A, B and C for respectively the factual data (i.e. investigations), the modeling of the environment of the infrastucture and finally the design and construction methods, -

The French NF P 94500 standard that define the geotechnical missions and the level of knowledge that is expected at each stage of the project.

It introduces a distinguo between two families of features: objects and observations & measurements. The first one being dedicated to real or modeled objects, the second for describing a property or value that has been observed or measured.

Object identification and data modeling

Identification of the relevant concepts to have in the data model was based on the study of existing models built by the members of the group and current data exchange. The idea was to be able to describe each element geotechnical engineers use to represent or document.

One key statement was the co-existence of multiple accepted representations for the same kind of models. This early statement lead the group to focus on setting a conceptual model that would enable different kind of representations for the same concepts. In other words, idea was to be able to both describe and confront 2D crosssections, 2D maps and 3D models relying on the same terminology.

A distinction has been made between what are called the "observation supports" from which geotechnical investigations are made, and the "interpreted objects" which encompasses all used features in order to create a ground model.

They are respectively associated to Book A for the first and Book B or C for the second. In Book A, the focus is on the factual data or investigations. They can be either performed in-situ or through a proxy (ie. a sample that will be studied in a laboratory). This list covers all type of exploratory holes, such as cable percussion borehole and trial pit; in-situ analysis such as pressuremeter measurements; geophysics investigations like radar and geotechnical laboratory tests. The location of the test and the sampling location are important information to build a geomodel.

In Book B, which is dedicated to the geotechnical synthesis, the main elements are the components of the models, called the interpreted objects. The concept of a "Geotechnical Unit" for example, which is either a surface or a volume in which the mechanical behavior is modeled using the same values, is a fundamental concept for a geotechnical design, though it is not yet integrated in a standard.

Finally, Book C addresses the description of the designed solution. This is where objects describing the underground infrastructure (eg. bolts, tunnel sections) meet geotechnical information and recommendation.

Possible links can occur between different objects/elements, which should also be specified. For example, by specifying the link between a borehole and a layer, it could be always possible to identify the input data (borehole data, laboratory test results on a sample from this borehole), used in the analysis and affected to a certain soil layer.

In fact, an object does not only mean a singular numerical value, but also an overall context. By this way, designer can have easily a critical eye on data quality, which is very important in geotechnical design as it is not an exact science.

The figure below provides an overview of the observed supports and interpreted objects. Most of them are already defined in existing data models, especially from the ISO or the Open Geospatial Consortium (OGC). MINnD GT1-5 decided to reuse those existing concepts and look at the properties that enable to describe them.

Figure 2 Identified concepts and their associated properties for geotechnical data description and exchange

Properties, methods and other semantics

Geotechnical Engineering addresses several domains including geology, hydrogeology, geochemistry or geophysics.

A same word can be associated to different meanings from a topic to another. Such differences may lead to misunderstanding that could lead to bad information transmission.

In order to find interoperability, each term that is used shall be clearly defined. When the word by itself is not explicit enough or might be associated to different meaning, one should able to refer to a description of it.

The MINnD GT1-5 work leads to the identification of several terms including:

-Observed properties: the characteristics that are determined (eg. shear strength, permeability), -Methods and procedures: the processes and technics that enable to obtain data. This includes a large yet non exhaustive list of tests, analysis (generally associated to standards) or calculus that can be made and lead to the creation of result or knowledge, -Units of measures: to be necessarily associated to numerical values. In practice, the deliverable of that activity was a collection of spreadsheets that contain and organize all those terms, definitions and also the relationship between them. The final document was named Pro3, standing for Properties, Processus and Provenance.

It contains many references to existing standards or terminologies, such as the dictionary of geology, national and international standards, including Eurocodes and ISO.

REUSABLE

Using existing data, especially for other purposes than the one they were designed for is a very engaging act. It can only be done properly if the user has a sufficient knowledge or trust in the data or if he can access to the raw data.

Uncertainty expression and result contextualization

Uncertainty expression is a recurrent topic in geoscience and source of a lot of research. Several solutions have been designed to address it as depicted in Pérez-Díaz, L. et al (2020).

One targeted approach is to be able to qualify and quantify uncertainty. Technically the provision of such information would not be very different from expressing an observation or measurement. Yet, the main difficulty is concentrated on the description, i.e. qualification and quantification of the uncertainty itself, leading to a rare furniture of information regarding uncertainty expression.

As an alternative to explicit uncertainty expression, MINnD GT1-5 worked on the capacity to give the user the context in which the values of observations, measurements or interpretations were obtained. The rationale is that contextualization can enable a user to figure the quality and relevance of the result, especially for reuse for another purpose.

The ISO 19156 standard, also known as Observation & Measurement (O&M) was designed for that specific need. MINnD GT1-5 described its reuse in [START_REF] Beaufils | Geotechnical data standardization and management to support BIM for Underground Infrastructures and Tunnels[END_REF]. In practice, this means the results are not provided "all alone" but associated to some additional information:

-The unit of measurement, -

The date and location of the O&M, -

The observed property, -

The method or procedure that lead to the result, -

The link(s) to related observation(s). The latest is particularly interesting to explain the provenance of the O&M.

Provenance

Working collaboratively around BIM models must allow the pooling and capitalization of data set. The models must be able to evolve over time, throughout the life cycle of the infrastructure, and the model data (geological, geotechnical, etc.) must be reused in new projects, different from the initial project for which studies were conducted. For example, it is easy to imagine that geotechnical survey data used to size a building foundation can be reused in a new project located nearby: the same raw data from a survey can potentially be used, but the interpretation made by the geotechnical engineer may differ from one project to another.

The issue of traceability and quality of data in digital models is a key element for their subsequent use. Indeed, the BIM models carry information that have their own history, each one come from field acquisition, analysis and interpretation. Consequently, the characterization of the property of a model will have gone through the specific prism of the project with its share of hazards, uncertainties, and choices.

The MINnD GT1-5 working group conducts a reflection on this subject with the objective of guaranteeing future users of geological and geotechnical models the transparency necessary for the sharing of information within reusable models. Within this framework, the WG has produced a document called "Provenance" which proposes, for each property resulting from a geological or geotechnical model, to define the key elements allowing to trace its history from the acquisition of raw data to the achievement of a characteristic value for the given property.

Thus, for each characteristic property of a model are defined: the observation support of the raw data (survey, sensor, area of interest,...), the analysis/expertise method since the raw data to the qualified data for the needs of the project (statistical analysis, calculation, photo-interpretation...), the normative references (experimental standard, Eurocode...). The knowledge of this set allows to contextualize the data set and to realize a critical analysis of the model during its realization or its re-use. The capacity to track the provenance of the data offers several advantages: -Figuring the interpretation process followed by the modeller, -Assessing data quality, -Facilitating data update, especially interpretation when new data become available. In practice this capacity is made possible by doing links between accessible data.

ACCESSIBLE

The accessibility criteria is about the capacity to obtain the data and read its content. It addresses the question of the data availability (i.e. capacity to download a resource) but also what the user get and how.

Formats

The format in which the data are provided greatly influence the capacity of tools or users to understand the data and read their content. Some formats are simple to read with basic tools. They are also generally quite light, which can facilitate their sharing. Such formats include for example csv. AGS and the recent AGSi are derived from such formats, even if JSON encodings are also proposed [START_REF] Chadwick | Extension of the AGS Format to Incorporate Ground Model and Interpreted Data[END_REF].

Some use cases may require the use of more sophisticated formats.

Valuable reasons are the need of exchange of complex data, including complex geometries. STEP and IFC have been designed for that purpose and then enable to model different objects in complexed structures. Another reason might be to be able to certify the data content and then prevent from irregular edits.

All those formats here listed are yet file formats. Their use is limited to import/export. Other formats have been designed in order to facilitate exchange on the web. This includes XML format and their derivatives like GML supported by the OGC for the geospatial information. The DIGGS intiative intended to expose AGS data through this OGC protocols. Also data models from OGC like GeoSciML and GroundWaterML2 are designed to be exposed in these formats (Laxton, J. et al., 2012). The INSPIRE European Directive also recommend this encoding for the furniture the public environmental data.

API and web services

The development of the Internet and the spread of high speed connection largely contributed to the development of Information Systems and data exchange based on the technologies of the web. In telecommunications, 3G, 4G and 5G technologies for cellular networks also contribute to facilitate data synchronization on remote devices.

Application Programming Interface or API and web services enable to expose data on the web. In OpenGIS, standards from the OGC define protocols on how those data should be accessed and delivered. The most relevant but not exhaustive include: -OGC API Features for providing object description (eg. Geotechnical Unit, Fluid Body, …), -OGC SensorThings API for observations and measurements (eg. tests or interpretations descriptions and results). Such API have been successfully implemented by geoscience data providers, like national geological surveys or environmental organizations. They are also now accepted as Download Services for the conformance with the INSPIRE Directives [START_REF] Kotsev | Extending INSPIRE to the Internet of Things through SensorThings API[END_REF]. Then a demonstration of the possible connection between IFC and such API has been provided during the Beijing bSI Summit. The proposed approach was to have object description and properties provided with OGC API and geometries provided in IFC.

FINDABLE

Data you can envisage to use are at first data you can find. In the last decades, companies like Google demonstrated the importance of indexes and search engines, or just capacity to discover existing resources in a Big Data world [START_REF] Brin | The Anatomy of a Large-Scale Hypertextual Web Search Engine[END_REF]. The Findable deals this capability.

Identifiers, catalogs, indexes and metadata

Declaring the existence of a resource starts with recognizing it as a distinct element. This means the definition and association of a persistent and unique identifier for each resource. On the web, this is the definition of an Unique Resource Identifier (URI).

Identifiers are important for resource citation. For documents, Digital Object Identifier (DOI), that are in fact URI at first, are commonly used to enable someone to cite a paper and give access to it. Such identifiers shall be stable or persistent and not change during the data lifecycle.

Identifiers then enable to do the connections yet other components are necessary to find them. This is the purpose of indexes, catalogs and metadata.Metadata enable to get an overview of the data set. It also define parameters that will enable to find them. Once data have been associated to identifiers, solutions can be made to facilitate their discovery based on the metadata. Catalogs group and index content to make them ready for discovery.

In practice MINnD GT1-5 specified to associate identifiers to each geotechnical data: objects and their related observations, measurements and interpretations. This would enable to make the data citable and possibly findable and linked together.

The findability of the data is yet dependant of the data policy chosen by the data providers.

Geotechnical data sharing

As an example of making collective findable data, it could be cited the INSPIRE (Infrastructure for Spatial Information in the European community) directive which is an EU initiative specifically developed to establish an infrastructure for spatial information in Europe. The aiming of this directive is to make spatial or geographical information more accessible and interoperable for a wide range of purposes supporting sustainable development.

Figure 4 3D Geological model, Netherlands (GeoTop from TNO)

Another one is the possibility of sharing the geotechnical data across a wide user community: public local authorities, engineering companies, transport authorities, real estate developers, searchers, students, etc. A huge amount of geotechnical investigation data are collected in various civil engineering projects. Until now, most of the time, the data are of exclusive use for the given project often for reasons of confidentiality but as well for technical reasons: geotechnical factual data have not been collected and stored in an appropriate format, they come from different stages of investigation and from different companies and are not harmonized. Unfortunately, without a common lexical and data standard for the geotechnical tests and properties, it is often difficult, expensive and time-consuming to gather data coming from different sites, projects and operators [START_REF] Bachus | Standardizing Geodata Transfer and Storage, Deep Foundations[END_REF]. In addition, it is also a source of transcription errors. Therefore, at the end, geotechnical investigation data are often forgotten or even definitely lost.

The new FAIR standards created in the openBIM and MINnD projects make possible the use of the same data taxonomy and vocabulary for all the geotechnical investigation companies and enable the exchange and the recycling of the data for potential future projects or studies.

The Owners, mandating the ground investigations, will definitely find benefits of the standardization of geotechnical data. Indeed, they can load them in their own database or GIS, and have the ability of re-use historical data without having re-input the data, that ultimately reduces the cost of projects [START_REF] Deaton | What are the benefits of geotechnical Data Interchange[END_REF]. To get this result, it is necessary to specify in the technical requirements of a geotechnical investigation contract that the data must be provided in a standardized data interchange format and comply with various formats [START_REF] Deaton | What are the benefits of geotechnical Data Interchange[END_REF].

The National Geological Surveys also play a major role in the management, capitalization and sharing of the geoscientific knowledge. Therefore, the implementation of geotechnical investigation data in the existing national geoscientific databases, as well as other physical ground properties, becomes an obvious mission of these organizations.

In France, the BRGM (French geological survey) already capitalizes and shares the geological data in open access [START_REF] Nehlig | Où en est la connaissance du sous-sol national ?[END_REF], through its website InfoTerre™. A new project aiming to collect, store and share the geotechnical data is currently in progress. Its purpose is to allow the integration of geotechnical data coming from different sources owners thanks to dedicated web applications and to provide to the users various diffusion services: visualisation, data exports, targeted queries, edition of cross-sections, and at a longer term 3D geological models. Currently, the first step of this project, consisting in creating the geotechnical registers, including the list of soil properties, observation methods, units…, is on progress and of course is in line with MINnD GT1-5 work.

Another example is the BRO, the Dutch Key Register of the subsurface. It includes explorations data (some geotechnical and geophysical data), rights of use (mining and water acts), some infrastructures (boreholes, water wells, monitoring points), geological maps and also provides cross-sections and 3D geological models (van der [START_REF] Van Der Meulen | 3D geology in a 2D country: Perspectives for geological surveying in the Netherlands[END_REF].

Other countries are following the same trend and it is likely that in the next years, the geotechnical data be implemented in the national databases of the geological surveys all around the world.

OUTCOMES

International standardization

The work of the MINnD GT1-5 project was made in order to extend OpenBIM standards for geotechnics. As geotechnics is based on geosciences and aims at providing knowledge of the environment to help civil engineers to build sustainably, several standards are concerned: the one related to geosciences and the one related to civil engineering. Defining a sustainable standard imply to bridge the gap between those two worlds.

Interests from bSI to extend IFC to geotechnics start with the Port & Harbors use case. Such infrastructure is in strong interaction with its environment and needed additional concepts to describe "non-built" objects such as land and water. They were respectively called IFCSolidStratum and IFCWaterStratum.

In 2019, standardization activities were launched in order to extend IFC to address tunnel modelling. In that use case, description of the surbsurface is critical. A requirement analysis report was produced by the group and concluded that the available concepts in IFC for describing geotechnics were not sufficient to fulfil the requirements [START_REF] Rives | IFC-Tunnel Project[END_REF]. The report also recommended to connect or build from the existing work endorsed by the OGC, especially GeoSciML.

Those statements lead to the need for recasting IFC Geotechnics in order to target descriptions that would fulfil to geotechnical engineering practices. A subgroup dedicated to geotechnics was then created formed in the IFC Tunnel team.

The work from MINnD GT1-5 was then shared to the bSI team and is used as a basis of work for the IFC Tunnel proposal. In practice, several members intervene in both groups in order to build a sustainable proposal for IFC. Then standardization activities shall also happen on the OGC side to strengthen the required connection between OpenGIS and OpenBIM.

Shared and accessible registries

Once defined and organised, vocabularies have to be made accessible to users. Registers as defined by the INSPIRE Directive consist in code lists with their associated values. Registries are the tool to expose those registers.

In France the French Geological Survey (BRGM) is hosting such tool to share geoscience vocabularies to the community. The BRGM registry is freely accessible (https://data.geoscience.fr/ncl/) and contains registers in different domains of geoscience, including geology (lithology, chronostratigraphy, …) and hydrogeology. While this instance mainly target national vocabularies, another instance is also available on https://data.geoscience.earth/ncl/ for international vocabularies.

Figure 5 A snapshot of the BRGM registries and the associated tools

The vocabularies identified by MINnD GT1-5 have been proposed to feed those registries. The required Observed Properties, Procedures and Units of Measurements have then been added to the BRGM registries when they were not already available.

Such actions enable everyone to do reference to those vocabularies without ambiguity and build information systems from them.

Fostering the geoscience community

The MINnD work was mainly focused on defining standards for OpenBIM, yet it aimed at making a connection with existing knowledge and working groups. In the informatics domain, one target was to connect to existing standards for GIS oriented data (eg. the OGC or ISO). MINnD GT1-5 then contributed to strengthen the relationship between OGC and bSI to address the geotechnical data interoperability topic. One major outcome of that collaboration was the organization of the "Geotechnical Data Standardization Workshop" in January 2019.

Efforts were also made in order to create a continuity with geotechnical or tunnel engineering oriented groups. This includes the AFTES, CFGI, CFMS and CFMR which are respectively the French equivalent of ITA, IAEG, ISSMGE and ISRM.

CONCLUSION

The Working Group MINnD GT1-5 contributed to the definition of framework and solution for better geotechnical data exchange at the BIM era.

Following the path of OpenScience, the MINnD GT1-5 project aimed at defining geotechnical data description and management based on the FAIR principles.

The rationale is that having more qualitative geotechnical data available would help geotechnical engineers and more generally environmentalists to develop better geomodels and better manage the earth resources.

The main focus of the group was on the definition of right semantics and objects to express the richness of the data collected and interpreted. Such elements contribute to describe the data with the appropriate wordings and no ambiguity, thus its interoperability.

Efforts were also made to address the question of data quality. A solution was proposed to enable the users to express data provenance and measurement methods and finally make data reusable.

In addition to those technical aspects, the MINnD GT1-5 project also studied and discussed the necessary actions in order to introduce the proposed standards in the current practices.

The MINnD GT1-5 promotes the reuse of data models and technologies used by geological surveys and other environmental organisation. The main reasons are to defend an unity or at least interoperability in terms of data description in the geoscience community and to enable a possible cost and time saving for further projects. The rationale is then that the format to use for geotechnical investigation for a project shall not be different from the one geological surveys would provide on the web.

It is also worth noticing that the geotechnical survey companies, which are at the source of the data collection, play a major role in the standardization of the geotechnical data. Of course, after a period of appropriation of the data structuration, they will obviously find an internal benefit for a better capitalization of their data, traceability of the analysis and interpretation, and time saving for the data exchange with their clients or partners. In addition, the use of standardized format of geotechnical factual or interpreted data enables data exchange independently of the softwares that can be used in a project.

ACKNOWLEDGEMENT

The work of MINnD GT1-5 is born from the collaboration of several members from multiple French organizations (public, research, engineering, owners) including ANDRA, BRGM, CAN, COLAS, CETU, EDF, EGIS, Géolithe, GEOS and SETEC.

9.

Figure 1

 1 Figure 1 Identification of relevant concepts based on the study of the models made by the members. The one in brown are part of the solution and then are not addressed by the Geotechnical group.

Figure 3

 3 Figure 3 Importance of tracking data provenance. An example of the characterization of stress-strain relationship of the soil.


Ingo Dierking
email: ingo.dierking@manchester.ac.uk

Michel Mitov
email: mitov@cemes.fr

Mikhail A Osipov

Smectic layer instabilities in liquid crystals

Keywords: liquid crystal, polymer-stabilized liquid crystal, smectic, ferroelectricity, layer rotation, layer instability, horizontal chevron

Scientists aspire to understand the underlying physics behind the formation of instabilities in soft matter and how to manipulate them for diverse investigations, while engineers aim to design materials that inhibit or impede the nucleation and growth of these instabilities in critical applications. The present paper reviews the field-induced rotational instabilities which may occur in chiral smectic liquid-crystalline layers when subjected to an asymmetric electric field. Such instabilities destroy the so-named bookshelf geometry (in which the smectic layers are normal to the cell surfaces) and have a detrimental effect on all applications of ferroelectric liquid crystals as optical materials. The transformation of the bookshelf geometry into horizontal chevron structures (in which each layer is in a Vshaped structure), and the reorientation dynamics of these chevrons, are discussed in details with respect to the electric field conditions, the material properties and the boundary conditions. Particular attention is given to the polymer-stabilisation of smectic phases as a way to forbid the occurrence of instabilities and the decline of related electro-optical performances. It is also shown which benefit may be gained from layer instabilities to enhance the alignment of the liquid-crystalline geometry in practical devices, such as optical recording by ferroelectric liquid crystals. Finally, the theoretical background of layer instabilities is given and discussed in relation to the experimental data.

Introduction

Liquid crystals were discovered in 1888 by the botanist Friedrich Reinitzer, who unexpectedly observed "two melting points" for crystals extracted from the root of a carrot. The recognition of these mesomorphic states of matter by the scientific community required more than two decades. 1 Today, liquid crystals [START_REF] Collings | Introduction to Liquid Crystals: Chemistry and Physics[END_REF][START_REF] Chandrasekhar | Liquid Crystals[END_REF][START_REF] De Gennes | The Physics of Liquid Crystals[END_REF][START_REF]Physical Properties of Liquid Crystals[END_REF][START_REF]Handbook of Liquid Crystals[END_REF][START_REF] Blinov | Electrooptic Effects in Liquid Crystal Materials[END_REF] are commonly known by their applications as materials used in displays. [START_REF] Lueder | Liquid Crystal Displays[END_REF][START_REF] Wu | Reflective Liquid Crystal Displays[END_REF] But liquid crystals are much more than materials employed in image display; they also pave the way to switchable adaptive and nonlinear optics, addressable waveguides and sensor materials. [START_REF] Li | Liquid Crystals Beyond Displays[END_REF] At the same time, liquid crystals are ideal materials to study a wide range of fundamental phenomena in soft matter physics and material science, [START_REF] Hamley | Introduction to Soft Matter[END_REF][START_REF] Kleman | Soft Matter Physics[END_REF] such as phase transitions, chirality, [START_REF]Chirality in Liquid Crystals[END_REF] ferroelectricity in fluids, [START_REF] Bahr | Lagerwall Ferroelectric and Antiferroelectric Liquid Crystals[END_REF][START_REF] Musevic | The Physics of Ferroelectric and Antiferroelectric Liquid Crystals[END_REF] pattern formation, [START_REF]Pattern Formation in Liquid Crystals[END_REF] colloidal and polymer composites [START_REF]Liquid Crystals in Complex Geometries[END_REF][START_REF] Dierking | [END_REF][19][START_REF] Dierking | Chemistry of Carbon Nanotubes[END_REF][START_REF] Mitov | [END_REF] or defect structures and dynamics, 22,[START_REF]Dierking Textures of Liquid Crystals[END_REF] and living matter. [START_REF] Neville | Biology of Fibrous Composites-Development Beyond the Cell Membrane[END_REF][START_REF] Elicespergamon | Structural Biological Materials-Design and Structure-Property Relationship[END_REF][START_REF] Bouligand Ann | [END_REF][27] 1.1 The self-organized order of liquid crystals Liquid crystals are partially ordered fluids which are thermodynamically stable between the (isotropic) liquid and the (three-dimensionally ordered) crystal (Fig. 1). They combine the anisotropic properties of the crystalline phase with the positional disorder of molecules in a liquid. Numerous individual liquid crystal phases are distinguished. We will here limit our discussion to thermotropic liquid crystals, which are observed purely on change of temperature. Lyotropic phases, [START_REF] Neto | The Physics of Lyotropic Liquid Crystals[END_REF] i.e. those exhibited by a change of concentration of an appropriate solvent, are not covered in this review. The simplest thermotropic liquid crystal phase is the nematic, which exhibits only orientational order of the long axis of rod-like (calamitic) molecules. The two phases that this review is mainly focused on are the chiral fluid smectic phases SmA* and SmC*. In addition to orientational order of the long molecular axis along a preferred direction called the director n, these phases also exhibit a one dimensional positional order, with molecules arranged in a layered structure, which may realistically better be described as a one dimensional density distribution. In the (achiral) smectic A phase the director n, which also represents the optic axis of the phase, is oriented parallel to the smectic layer normal k, while in the (achiral) SmC phase the director is inclined to the layer normal by a temperature dependent angle θ, called the tilt angle. The respective structural features are summarised in Fig. 1. Fig. 1 Schematic scheme of the different states of matter (top). The liquid crystal states are observed between the three dimensionally ordered crystal and the (isotropic) liquid phase. Within the temperature interval of the liquid crystal state, several phases can be observed, which differ in their degree of order.

Ferroelectricity in chiral (fluid) smectic liquid crystals

The introduction of chirality into liquid crystals, either by making the mesogens themselves chiral or by adding chiral dopant molecules to an achiral host, leads to a wealth of novel phenomena, [START_REF] Symmetry | [END_REF] such as the formation of helical superstructures, the observation of novel phases and polar effects, such as ferroelectricity or the electroclinic effect, which is similar to the piezoelectric effect in solid state materials.

Meyer et al. [START_REF] Meyer | [END_REF] first demonstrated on the basis of symmetry considerations that every tilted chiral smectic phase can exhibit a spontaneous polarisation PS, i.e. exhibit pyroelectricity. The achiral SmC phase is of C2h symmetry and shows the symmetry elements of Fig. 2: (i) a two-fold rotation axis C2y, (ii) a mirror plane mxz and thus (iii) an inversion centre i. Applying inversion symmetry to an arbitrary permanent electric dipole moment results in a vanishing spontaneous polarization. The situation is different when the smectic C phase is composed of chiral molecules. In this case the mirror plane vanishes, and so does the inversion centre, while the two-fold rotation axis is retained. The SmC* phase is of C2 symmetry, and applying the only remaining symmetry element to a general dipole moment leads to a non-zero spontaneous polarization, PS = P0(k × n).

Fig. 2 Schematic illustration of the symmetry elements of the achiral smectic C phase. If the phase consists of chiral molecules, the mirror plane and the inversion center vanish, and a net dipole moment along the rotation axis is observed, which constitutes the spontaneous polarization of the ferroelectric smectic C* phase (reproduced by permission from ref. 23.

Copyright Wiley-VCH).

Introduction of chirality also leads to the observation of the electroclinic effect, 31,32 a deviation of the director from its equilibrium orientation by application of an electric field. The deviation occurs perpendicular to the electric field direction and is linear in field amplitude, i.e. also dependent on polarity. The electroclinic effect is strongly nonlinear with respect to temperature, diverging at the SmA*-SmC* phase transition. It is present in both phases, but can be most easily observed in the SmA* phase, as here it is not obscured by the ferroelectric switching of the SmC* phase. The electroclinic effect causes an induced tilt angle θind: where e is the material dependent electroclinic coefficient, E the electric field amplitude, and α(T-TC) the coefficient of the first term in the Landau expansion of the free energy density.

Within the context of smectic layer instabilities, both ferroelectricity and electroclinic effect, i.e. spontaneous and induced polarisations in chiral materials are of fundamental importance. None of the behaviour reported below has been observed in achiral liquid crystals.

Smectic C* layer and director configurations

The SmC* structure exhibits a surprising wealth of different director and smectic layer configurations, depending on chirality, boundary conditions and other applied external forces. The fundamental phenomena observed are shortly summarised below.

The helical bulk state

In the bulk state the spontaneous polarisation of the SmC* phase is cancelled on a microscopic scale, due to the formation of a helical superstructure, 33 as depicted in Fig. 3(a). When advancing along the direction of the smectic layer normal, the director precesses on a cone with a temperature dependent opening angle of twice the tilt angle, 2θ. Since the spontaneous polarisation is perpendicular to the director and lies in the smectic layer plane, PS cancels over the spatial period of one pitch of the helical superstructure. The pitch of the SmC* phase generally ranges from 100 nm to several tens of micrometers. If it is in the range of several micrometers, a periodic line pattern can easily be observed in polarising microscopy due to the helical superstructure (Fig. 3(b)), and the bulk SmC* phase should thus rather be called helielectric. 34 The direction of the line pattern is equivalent to the smectic layer plane, which is useful to determine the smectic layer orientation.

Surface influences on smectic layer geometries

Clark and Lagerwall 35 demonstrated that if a SmC* phase is confined to very thin cells with the gap approximately smaller than the pitch of the helix, the latter is unwound by surface interactions, resulting in a ferroelectric domain structure. This geometry is called the surface stabilised ferroelectric liquid crystal (SSFLC) state. In the ideal case the smectic layers are oriented perpendicular to the bounding substrates, in the so called bookshelf geometry schematically shown in Fig. 4(a). This allows switching of the spontaneous polarisation from up to down orientation, which is accompanied by a director reorientation of twice the tilt angle on the cone, giving maximum contrast of the device for appropriately oriented crossed polarisers. The respective zero field virgin ferroelectric domain structure of the bookshelf geometry is depicted in Fig. 4(b). Generally, the situation is somewhat more complicated with vertical chevron structures being formed. These have been intensively studied by Rieker and co-workers, [36][37][38] but also by Ouchi and coworkers. 39 Due to smectic layer contraction under the constraint of parallel anchoring when passing the SmA*-SmC* transition, smectic layers in the SmC* phase tilt with respect to the substrate normal, as schematically shown in Fig. 5(a). The layer tilt angle is often similar to the director tilt angle, δ ∼ θ, and tilt can occur in both directions. This leads to defects called zig-zags, 39 where regions of opposite layer tilt meet. Thin defect lines are of the type ⋘⋙, while thick lines are of type ⋙⋘. Chevron structures decrease device contrast and viewing angle characteristics (Fig. 5(b)). A texture example of zig-zag defects is shown in Fig. 5(c).

Electric field effects on smectic layers

Smectic layer straightening

It had long been thought that once formed, smectic layer structures are fixed. This view was revised through X-ray [40][41][42] as well as electro-optic 43 investigations, which indicated a reversible smectic C* layer straightening under application of an electric field in device geometry. Detailed time resolved synchrotron studies [44][45][46] confirmed the proposed behaviour of a reversibly decreasing chevron angle under applied electric fields of relatively small amplitude.

The investigations also indicated that the direction of the smectic layer normal may change in the plane of the cell substrate. The reason for smectic layer straightening lies in the fact that in a vertical chevron geometry the spontaneous polarisation is not parallel to the applied electric field vector, because the smectic layers are tilted with respect to the substrate plane (Fig. 5(a)). This exerts a torque on the smectic layers trying to straighten them in order to orient the spontaneous polarisation parallel to the applied electric field.

The horizontal chevron formation process

While for moderate electric field amplitudes the smectic layer straightening is a reversible process, for large amplitudes it can induce an irreversible rearrangement of smectic layers to form what has become known as 'horizontal chevrons', a term first coined by Jakli and Saupe. 47 The process of reorienting a vertical chevron structure (Fig. 5(b)) into a bookshelf structure (Fig. 4(a)) by application of large electric fields may in principle be achieved via two possible mechanisms: (i) smectic layers irreversibly straighten with the smectic layer normal k being parallel to the rubbing direction x. This implies that two bookshelf domains are being formed with director orientations of n = ±θ tilted with respect to the smectic layer normal k and thus also the rubbing direction x. (ii) The smectic layers reorient such that the layer normal k tilts with respect to the rubbing direction x by k = ±θ, while the director n in both ferroelectric domains points along the rubbing direction x.

As the substrate boundary conditions generally promote molecular alignment along the rubbing direction, the scenario most often observed is that of scheme (ii). During the irreversible smectic layer straightening process, a vertical chevron structure is transformed into a horizontal chevron structure.

The striped bookshelf texture of the SmA* phase 22,48 is one example of such a resultant smectic layer reorientation structure, with stripes basically oriented along the rubbing direction. Fig. 6(a) and (b) depict the schematic director and smectic layer configuration, and a typical texture photograph, respectively. Such structures can be induced by application of AC electric fields, preferably to thin cells, [49][50][51][52] and their periodicity is approximately equal to the employed cell gap. 47,50,53 The angle formed between the smectic layer normal k and the rubbing direction x is generally smaller than the induced director tilt angle θ. The observed periodicity, as well as the small layer inclination angles distinguish the striped bookshelf layer arrangement of the SmA* phase from horizontal chevron domains of the SmC* phase. The latter exhibit no periodicity, but nevertheless often elongated domains due to the influences of the rubbing direction. But via which process are horizontal chevrons actually formed?

One first has to realise that the zig-zag regions of vertical chevron structures are actually of the bookshelf-type, but with the layer normal inclined to the rubbing direction (Fig. 7(a)). As the smectic layers straighten for increasing electric field amplitude, these bookshelf-zones increase at the expense of the vertical chevron until the two domains of oppositely inclined bookshelf domains meet. The defect wall mediating between these two domains is of the vertical chevron type 55 (Fig. 7(b)). The smectic layer structure can easily be verified when viewing a helical sample between crossed polarisers at zero applied electric field condition (Fig. 7(c)), where the helix lines indicate that the smectic layers are inclined with respect to the rubbing direction, in opposite directions for the two respective domain types. It should also be noted that the horizontal chevron domain formation is not related to any surface memory effects, as domains of different shape and position are formed for consecutive experiments. 56 The horizontal chevron domain structure can thus be erased by heating into the isotropic phase. For samples with a suppressed helical superstructure, the polarisation up and polarisation down domains appear equivalent between crossed polarisers, mediated by a bright defect line. Application of a positive field switches both domains into the polarisation up state where one domain stays black, while the other switches to its bright state. Polarity reversal leads to a reversal of dark and bright domains. This behaviour is demonstrated in Fig. 8, together with a schematic of the smectic layer structure and the director orientations. The smectic layer structures can also be elucidated by small angle X-ray investigations on oriented samples (Fig. 8(c)), giving evidence for the discussed structures. 57 Already the textures of Fig. 8 imply a detrimental effect of the formation of horizontal chevrons on the electro-optic performance of any device, as both the polarisation up and down state consists of an equal distribution of dark and bright domains. Thus, no significant electro-optic response is observed on electric field polarity inversion, as demonstrated in Fig. 9. While each individual domain exhibits the well known bistable electro-optic response of a ferroelectric liquid crystal, although phase shifted by 180° for both domain types, the overall optic response of the horizontal chevron domain device tends to zero, independent of the position of the device between crossed polarisers. The dynamics of the formation process of horizontal chevron domains is discussed in detail in ref. 56 and summarised in Fig. 10. For increasing electric field amplitude, the time for the formation of horizontal chevrons strongly decreases, while the average domain width increases until saturation (Fig. 10(a)). Increasing the frequency of the applied electric field results in an increase of the time of formation with the domains formed exhibiting somewhat small average domain sizes (Fig. 10(b)). The reduced temperature below the SmA*-SmC* transition only has a pronounced effect on domain formation dynamics and size in the close vicinity of the transition with faster domain formation of larger size (Fig. 10(c)). It is presumed that the latter behaviour is related to a strong softening of the smectic layer structure, as it is generally observed that the smectic layer compressibility is largely enhance at phase transitions from the cholesteric phase 58,59 Fig

Smectic layer rotational instabilities

An effect not often talked about, possibly also due to its detrimental influences on the electrooptic properties of FLC devices, is the so called smectic layer rotation or smectic layer rotational instabilities. Application of asymmetric electric waveforms, as they are often used in addressing schemes, may lead to an in-plane rotation of smectic layers in mainly the SmC* phase, but due to the electroclinic effect also in the SmA* phase. This effect had first been reported by the Gothenburg group at the 13th International Liquid Crystal Conference Vancouver, Canada [START_REF] Andersson | presented at the 13th International Liquid Crystal Conference[END_REF] and was later elaborated on at other conferences. [START_REF] Skarp | presented at the 14th International Liquid Crystal Conference[END_REF][START_REF] Andersson | presented at the 4th International Conference on Ferroelectric Liquid Crystals[END_REF] A rotation of the smectic layer normal in the plane of the substrate by several hundreds of degrees had been reported for the application of a saw-tooth field to a high electroclinic material in the SmA* phase. Although strikingly beautiful, this work [START_REF] Andersson | presented at the 13th International Liquid Crystal Conference[END_REF][START_REF] Skarp | presented at the 14th International Liquid Crystal Conference[END_REF][START_REF] Andersson | presented at the 4th International Conference on Ferroelectric Liquid Crystals[END_REF] was never published and Fig. 11 depicts a texture example of their observed spiral instabilities after rotation of the smectic layers for several tens of minutes and hundreds of degrees. First reports of the smectic layer rotation in literature were published by the Osaka group on electroclinic SmA*, [START_REF] Nakayama | [END_REF] ferroelectric SmC*, 64,65 and antiferroelectric SmCA* phases, 66,67 which will be discussed in more detail below.

Dependence of the dynamics on external parameters

At first it should be noted that the in-plane rotation of smectic layers can be induced by any type of asymmetric electric waveforms. These can be amplitude asymmetric or time asymmetric without DC balance, time asymmetric with DC balance, or any combination thereof. 68 There has also been one report of smectic layer rotation by a pure DC field, 69 although it is not quite clear what constitutes the mechanism of this rotation, which could possibly also be related to ionic contamination. Secondly, the layer rotation can be continuous for weak anchoring conditions, or limited to an angle of approximately twice the tilt angle (in fact twice the horizontal chevron inclination 55 ) for strong anchoring. 70,71 Thirdly, it is only observed for chiral materials (see also below).

In the following we will examine the dependence of the rotation of smectic layers for strong anchoring conditions. A rotation time, tr, can be define which corresponds to the time needed to rotate a virgin horizontal chevron structure, obtained by application of sufficiently large electric fields into one, where all smectic layers are inclined to the same direction. For an applied electric field with DC bias, but qualitatively also for all other applied asymmetric fields, it is found that the rotation time exponentially decreases for increasing field amplitude, as depicted in Fig. 12(a). The frequency dependence of the rotation time is somewhat more complicated and depends on the applied bias (Fig. 12(b)). For small bias the rotation times increase and exhibit a maximum at approximately 100 Hz, the typical relaxation frequency of ions in a liquid crystal, before they decrease due to the faster director switching and vanishing of the ionic influences. Above the saturation frequency at about 5 kHz, director switching slowly ceases, and the layer rotation times increase again. For larger bias fields the ions quickly get trapped at the electrodes, and the rotation process becomes faster with increasing frequency, again until the director switching slowly ceases for several kHz. Approaching the SmC*-SmA* transition temperature from below, the rotation times decrease with increasing temperature, as shown in Fig. 12(c). It is presumed that this is the effect of two mechanisms, firstly a decrease in viscosity, which implies a faster layer rotation process and secondly, the softening of the smectic layers (decrease in layer compression modulus). It is anticipated that a similar effect would be observed for de Vries ferroelectrics, because the smectic layers are already "soft", as only a tilt ordering is needed to achieve the transition from SmA* to SmC*. At last, the rotation time decreases with increasing sandwich cell gap, which can be attributed to the vanishing influence of the boundary conditions for the bulk of the liquid crystal as the cell gap is increased (Fig. 12(d)). The latter behaviour hints towards the mechanism of the smectic layer rotational instability, 72 which is a permeation process of molecules across smectic layers. This can microscopically be observed through the motion of domain boundaries and deformations of formerly round air bubble enclosures toward an elongated shape. Applying an asymmetric electric field to a horizontal chevron structure causes molecules of the unfavoured domain orientation to permeate in the direction of rubbing, this translation takes place primarily at the domain wall to the favoured domain, thus causing a domain wall motion roughly in the direction perpendicular to the rubbing direction (see Fig. 13(a)). This can be illustrated by the textures with helix lines, which indicate the smectic layer direction (Fig. 13(b)), and is evidenced by small angle X-ray diffraction (Fig. 13(c)). Starting with a horizontal chevron structure, application of an asymmetric electric field causes a layer rotation until the sample is uniformly inclined with the layer normal making an angle with the rubbing direction equal to the director tilt angle. Reversal of the asymmetry causes a rotation in the opposite direction (through nucleation and domain growth) until the layer normal is now oppositely inclined to the rubbing direction. From the X-ray diagrams one can also see that during the rotation the smectic layer order parameter decreases, and increases again after the rotation process is completed. Note that to accomplish the whole reorientation process, molecules only need to translate over distances comparable to their molecular length.

Dependence of the dynamics on other parameters

The smectic layer rotational instability does not only depend on applied external conditions, but also on several parameters of the system itself. In the following, these will shortly be outlined and discussed. It was stressed above that the observed layer rotation only occurs in chiral materials. This can also be inferred by investigations of mixtures of the two left and right handed enantiomers of a FLC compound, 73 as depicted in Fig. 14(a). The general trend of a decreasing layer rotation time for increasing asymmetry ratio of an applied electric field is clearly observed for all concentrations. For the pure enantiomers with enantiomeric excess ee = -1 and ee = +1 the same rotation times are observed, only that the domains rotate in opposite direction. This is also the fastest reorientation process. As the enantiomeric excess becomes smaller and tends toward zero, ee → 0, the rotation time increases, seen to diverge exactly at ee = 0. 74 This clearly implies that chirality is needed to observe the smectic layer rotation. shows the asymmetry ratio dependence of the layer rotation times for several different alignment layers. 75 Quite a variation of rotation times is observed, which can be attributed to varying anchoring strengths of the substrates, increasing from polyimide used in commercial EHC cells, via polyimide from Dupont to Nylon 6/6. It should be noted that similar results are obtained when using the same substrate coating material and varying the number of rubbing cycles at constant pressure, or varying the rubbing pressure at a constant number of cycles. The anchoring strength increases with the number of rubbing cycles, as well as rubbing pressure. In both cases an increase of the smectic layer rotation times is observed.

Already the frequency dependence of the layer rotation times suggests a certain influence of ions on the speed of the rotation process. To study this influence, one can deliberately contaminate the liquid crystal through doping with an ionic dopant, 76 as it normally used for dynamic light scattering displays. The results are depicted in Fig. 14(c) and it is clear that the rotation time decreases as the ionic dopant concentration increases. A possible explanation of this behaviour could be electroconvective effects in the domain walls separating domains of opposite layer inclination. This would increasingly facilitate the permeation flow of molecules across adjacent smectic layers, decreasing the rotation time as the ionic concentration increases.

At last, the domain rotation time also depends on the nature of the phase transition (Fig. 14(d)), as shown for a homologous series of ferroelectric liquid crystals with largely varying phase sequences. 73,77 The shorter homologues exhibit a N*-SmC* transition (among a range of other transitions), for which the quickest rotation process is observed in the smectic C* phase. This seems to be independent of the molecular length. For a SmA*-SmC* transition, clearly longer domain rotation times are observed at the same reduced temperature in the SmC* phase. The smectic layer rotation process is significantly slower, especially when considering the doubling of the applied field amplitude. The medium length homologue D7 exhibits a phase sequence which incorporates a narrow TGBA* twist grain boundary phase. Here the rotation times are of intermediate magnitude, again if equivalent reduced temperatures in the smectic C* phase are considered. It is anticipated that this behaviour is due to the softening of the smectic layers, which appears to be larger approaching a N* phase than a SmA* phase. This is intuitively understandable, as the SmA* phase is a fluid smectic phase, just as SmC*, while in the N* phase all positional ordering vanishes. With the TGBA* phase being somewhat similar to N*, but additionally also exhibiting local smectic ordering, one would expect values to lie in between those of N* and SmA*, as observed.

Applicational aspects

Despite its detrimental influence on the electro optic performance, the smectic layer reorientation can nevertheless be employed for the benefit of ferroelectric liquid crystal devices. Some of these beneficial properties are outlined below, for example to obtain well aligned smectic layer configurations, possibilities for display damage recovery, or applications such as erasable optical recording, or simple display devices which only need voltages applied during image change, for example as switchable supermarket price tags.

Uniform alignment and texture damage recovery

A straight forward application of the formation process of horizontal chevrons in combination with the smectic layer reorientation is found for the generation of uniform alignment of ferroelectric liquid crystals. 78 As outlined above, smectic C* phases, when formed from their non-tilted smectic A* high temperature phase, most often exhibit (vertical) chevrons, which decrease the electrooptic performance, for example viewing angle and contrast ratio. Applying an electric field with sufficient amplitude changes the vertical into a horizontal domain structure. Subsequent application of an asymmetric AC field rotates all smectic layers into a uniform configuration with the smectic layer normal inclined to the rubbing direction by the amount of the tilt angle. Fixture of the crossed polarizers in an appropriate way the leads to a dark position for one voltage and a bright position for reversed polarity of the field.

Similarly, this procedure can be applied for display damage recovery. For nematic liquid crystals the uniformity of the director alignment is self-healing, or through applying pressure to the screen. This is not the case for smectic liquid crystals. Nevertheless the healing process of smectic layers can be aided by heating the device into the isotropic phase, then cooling it into smectic C*, and applying the above outlined procedure.

Simple low power displays

Using the smectic layer reorientation, simple displays can be envisioned which only need an applied voltage during changes made to the black and white image (on, off pixel). These could be employed for example in electronic price tags for supermarket products. The method of operation is straight forward. Switching between bright and dark is not via the normal director switching of an FLC, but rather through the reorientation of the smectic layers from one to the other inclination by applying electric fields of opposite asymmetry. Obviously, this will not be a high definition display, rather similar in information content to that used for wrist watches or pocket calculators, and also changing the respective information displayed will take seconds to minutes, but, unlike a TN display, no power is required to hold the information.

Optical recording

The applicability to employ the smectic layer reorientation in devices for optical recording has been demonstrated by the Yoshino group. [79][80][81][82] It can be used with materials of different phase sequence, namely N*-SmC* 79 and SmA*-SmC*. 80,81 In the former case the writing and erasure process is as follows: on cooling from the N* phase, one of the two possible smectic layer arrangements can be selected by selection of the asymmetry of the applied electric field. Erasure can be realised by heating the whole cell into the N* phase, which can simply be done via temperature control, but also locally by laser heating, or photoisomerization of a suitable dopant molecule. This allows for two layer arrangements, corresponding to a binary image. In the latter case, also grey-levels can be achieved. The recording process uses the fact that the layer rotation is much faster in the SmC* than in the SmA* phase. The cell is kept in the ferroelectric phase at temperatures below the SmC*-SmA* transition. Parts of the device are irradiated by a laser and heated into the SmA* phase. In the irradiated area no layer rotation will be observed, while in the non-irradiated areas the smectic layers can be rotated to any desired position, depending on the duration of asymmetric voltage application. The laser is then turned off and the whole sample reverts back to the SmC* phase. This optical recording process is schematically shown in Fig. 15(a) and demonstrated in Fig. 15(b). Erasure of the pattern can be achieved by a reversal of the writing process outlined above. The recorded pattern is stable against application of symmetric electric fields, and more complicated patterns can be designed. 82

Polymer-stabilized smectic layer structures

Since the smectic layer instabilities generally exhibit a detrimental effect on the electro-optic properties of a ferroelectric liquid crystal, ways may be designed to prohibit their occurrence all together. One of such ways is the so-named polymer-stabilization, which is based on the formation of a porous polymer network inside the volume of a liquid crystal, while largely maintaining the electro-optic properties. This approach is discussed in the following paragraphs.

Polymer-stabilized liquid crystals: main features

Polymer-stabilized LCs (PSLCs) [START_REF]Liquid Crystals in Complex Geometries[END_REF][START_REF] Dierking | [END_REF][19] are composite materials which are obtained by dissolving a small amount of a bifunctional photo-reactive mesogenic monomers (typically between 1 and 10 wt%) in a low molar mass LC, together with a photo-initiator. The gap of a sandwich-cell is filled with the mixture and the thin film is oriented in the desired director configuration. The polymer-network forming material is polymerized and crosslinked by UV light irradiation. After polymerization, the LC is trapped in the network which follows the mesophase order and acts as a template of the original texture, as it has been demonstrated by optical microscopy, scanning (or transmission) electron microscopy for a variety of mesophases. [START_REF]Liquid Crystals in Complex Geometries[END_REF][START_REF] Dierking | [END_REF][19][83][84][85] PSLCs have stimulated reflection on fundamental insightssuch as LC-polymer interactions, polymer percolation in an anisotropic fluid, template properties, phase transitions under random confinementand novel potential applications like mechanically stabilized FLC displays, 86,87 reflective displays, [START_REF] Wu | Reflective Liquid Crystal Displays[END_REF]88,89 heat repelling sheets or smart windows. 90

Suppression of smectic layer reorientation

Rotational instabilities may appear in textures of chiral smectic LCs when subjected to an asymmetric electric field as a consequence of smectic layer reorientations, as discussed above. This undesirable phenomenon can be prevented by the curing of a few per-cent of photopolymerizable monomers dispersed in the LC. [91][START_REF] Mitov | poster E2.O05, 16th Int. Liquid Crystal Conf[END_REF][START_REF] Mitov | [END_REF] In addition to stabilizing the smectic layer order, the upper stability of the LC order is considerably raised. By introduction of a polymer network, a template of the pristine director configuration is created, i.e. a memory state, now exerting a mechanical field on the liquid crystal. The velocity of layer reorientation is decreased when increasing the polymer network content, until the complete suppression of instabilities occurs. Optical microscopy textures do not drastically change after the polymerization process, which indicates that the network consists of smooth strands causing only small inhomogenities to the LC orientation (Fig. 16). Anisotropic patterns (stripe texture) are stored well above the clearing temperature of the liquid crystal before polymerization, due to molecules following the order of the template. Electro-optic properties are not drastically altered. 19 In the SmA* phase, there is no significant change in the induced tilt angle behaviour or magnitude upon polymerization, while the rise time is increased after curing. In the SmC* phase, the tilt angle is slightly increased and also the spontaneous polarization, while the rise time is not significantly increased (thus still being compatible with display technology requirements). Only if the polymerization is carried out in the isotropic phase, the polymer has no imparted order and prevents any bulk alignment, which affects the electro-optic characteristics. The network acts in two ways: first it gives a biased elastic torque counteracting a field of such symmetry as to cause a change from the templated layer direction; second, it introduces an increased effective viscosity counteracting the reorientation in both directions. 94 The smectic layer reorientation is strongly hindered by intra-layer polymerized networks as compared to inter-layer polymerization, which is in accordance with the above discussed permeation explanation of the layer reorientation. For the intra-layer polymerized samples with a polymer network formed basically perpendicular to the rubbing direction (at small layer tilt angles), the mass flow in a direction of rubbing is strongly hindered, whereas it is permitted for the polymer network formed along the smectic layer normal. 94,95

Polymer network influence on FLC parameters

The degree of elastic interaction between LC and polymer largely governs the electro-optic response of PSFLCs. The Landau-de Gennes potential incorporating an elastic coupling term, permits to describe the influence of a phase separated polymer network on the experimental properties of a FLC. 96 The elastic coupling between the network and the LC director increases linearly with increasing polymer content in PSFLCs cured in the SmA* phase. The network gives a contribution of 10-15% of the total free energy density. 97 With low concentrations of polymer, optical response and tilt angle remain fairly independent of concentration but as the concentration increases, switching speed and tilt angle may decrease dramatically. 98 By polymerizing at particular temperatures, characteristics of the PSFLC approach those exhibited by the FLC. 99 The rate of polymerization increases significantly as the order of the mesophase increases and the temperature decreases. The impact of the polymerization temperature on the electro-optic properties strongly depends on the choice of monomer. 99,100 Both the ferroelectric polarization and the optical response time increase with polymerization temperature in the ordered phases. FLC properties such as spontaneous polarization, threshold field and switching speed depend on the polymerization kinetics. Ferroelectric properties are less affected when the UV light irradiation conditions correspond to a low power and a long irradiation time. 101,102 The network may facilitate the so called Vshaped switching and shorten the response time. 103 A network formed at large tilt angle well into the SmC* phase tries to preserve the SmC* director orientation as the SmC*-SmA* transition is approached whereas polymerization in the SmA* phase results in a reduction of the polarization. 104 The dielectric strength at the SmC* to SmA* transition increase with increasing polymer concentration until saturation, and so do the relaxation frequencies in the SmC* phase. The Goldstone-mode within the SmC* phase is suppressed when the polymerization is carried out in the SmA* phase. 105 The long-term stability of the quasibookshelf texture is significantly improved as well as the contrast and multiplex addressing performances, by relieving the stress at the stripe interfaces. 106 Polymer stabilisation to increase mechanical stability of related devices retains the superior electro-optic properties of SmC* and SmA* phases, like very fast response times, little dependence on deviations from normal light incidence, high contrast or the ability for easy linear greyscale generation, or light modulation. The dynamic range of devices may even be increased through polymer stabilization. 107

Theory of smectic layer reorientation 7.1 Simple symmetry arguments

It is possible to describe qualitatively the layer rotation induced by the external electric field using some very generals symmetry arguments. Indeed, let us consider the SmA* layer with an electric field E applied parallel to the smectic plane. In this configuration the electric field specifies a polar direction in the plane of the smectic layer, and the axis of the layer rotation is expected to be parallel to this direction, which corresponds to the experiment. This is not surprising, as the electric field is the only vectorial physical quantity of the system. The rotation is characterized by the angular velocity ω and for symmetry reasons it is expected to be parallel to the electric field E.

However, the relationship between the external electric field and the angular velocity of layer rotation is not completely specified before the time reversal symmetry is also taken into consideration. One notes that the angular velocity ω changes sign under time reversal while the electric field does not. Thus in the linear approximation the angular velocity ω should be proportional to the time derivative of the electric field Ė which also changes sign under time reversal: ω = αĖ (1) One notes that in this simple equation the angular velocity ω is the so called pseudo-vector which does not change sign under space inversion while the electric field as well as its time derivative are polar vectors which do change sign under space inversion. This contradiction can be resolved if one assumes that the coefficient α is a pseudo-scalar and therefore the medium must be chiral. Then the product αĖ on the right hand side of eqn ( 1) is also a pseudo-vector which has the same transformation properties as ω. This conclusion again corresponds with existing experimental data which indicate that the layer rotation is observed only in chiral smectic phases.

Thus, from a general symmetry point of view, eqn (1) indicates that the rotation of chiral smectic layers in a time-dependent electric field is allowed by symmetry. The angular velocity of the layer rotation must change sign under time reversal of the external field, which indeed is observed experimentally.

It should be noted that the rotation is not expected to occur in a constant field which appears to be in contradiction with the experiment 69 where layer rotation in the SmC* phase has been observed in a DC electric field. One notes, however, that in this experiment the liquid crystal was doped by charged impurities which produced an electric current across the cell. Taking into account that the symmetry of electric current I is exactly the same as that of the time derivative of the electric field one can write down a similar linear relationship between the angular velocity and the electric current

ω = αI (2)
Thus it also follows from symmetry consideration that the rotation of chiral smectic layers can be induced by an electric current.

Eqn (1) and ( 2) are very general but they do not contain any information about a particular mechanism of rotation in chiral smectic phases. Such a mechanism for the SmA* phase is discussed in detail below. However, before considering the mechanism of the rotation, let us first investigate the consequences of these general equations.

One notes that in all experiments the electric field E(t) is periodic and thus the time derivative Ė changes signs during the period. According to eqn (1) this means that the layers are rotating in opposite directions for positive and negative values of Ė, respectively. The period of the electric field is usually very small, and thus only the constant part of the angular velocity is actually observed i.e., the zero-frequency component of ω. This zero frequency component is the average angular velocity ω0 which can be obtained by integration of eqn (1) over time (3) where t is the period of the field. One can readily see that for any periodic field the integral in eqn (2) vanishes, and thus, on average, there is no rotation.

However, one should also take into account that the orientation of smectic layers is always stabilized by some counter-torque, which stems from an interactions with the bounding glass plates. Thus the smectic layers are actually rotating only if the driving torque exceeds some threshold value that is during the part of the period when the time derivative Ė is sufficiently large. As a result the integral in eqn (4) does not vanish because one actually has to integrate only over these fractions of the total period when the derivative Ė is large enough. More detailed equations which describe the layer rotation in the SmC* phase are described in the following subsection which is based on the results obtained by Carlsson and Osipov in ref. 111.

General equations of motion for the layer rotation: thermodynamic approach

Let us consider a SmA* liquid crystal in the bookshelf geometry with an external timedependent electric field applied in a direction parallel to the smectic planes. It is well known that due to the electroclinic effect, the director n will deviate from the layer normal, and this deviation is specified by the tilt angle Θ which is the first natural dynamic variable of this system. The second dynamic variable is the angle γ between the layer normal k and the axes of the fixed coordinate system. The absolute value of the angular velocity of layer rotation is then equal to the time derivative of γ.

One notes that the two dynamic variables in the system i.e., the tilt angle Θ and the layer rotation angle γ are not equivalent from a thermodynamic point of view. Indeed, the free energy of the SmC* phase depends on Θ but is independent of γ. The dynamics of Θ is relaxational in nature, i.e., the tilt angle relaxes to its equilibrium value determined by the field. On the other hand, the change of γ is also associated with some energy dissipation. In this sense the angle Θ is analogous to the director n in the homogeneous nematic phase. In the latter case the free-energy density also does not depend on the orientation of the director, but the rotation of n is accompanied by a dissipation. In the SmA* phase subjected to an external electric field, the variables Θ and γ are coupled, and we will see that this coupling determines the rotation of the smectic layers.

The dynamics of such a system can be described by the general approach based on the Rayleigh dissipation function. The application of this approach to the study of the dynamics of nematic liquid crystals is discussed in detail by Vertogen and de Jeu. 108 In this approach the state of the system which is assumed to be close to thermodynamic equilibrium is specified by a set of macroscopic variables Xi = (Θ,γ) and their time derivatives Ẋi = ( , ). The properties of the system are described by the free-energy density functional F(Xi) and the dissipation function D(Ẋi). Then the dynamics of the system is described by a set of Euler-Lagrange equations:

In the general case, the dissipation function D can be expressed as a quadratic form of the time derivatives and

(5)

where γ1 is the rotational viscosity of the SmC* phase, λ is the rotational viscosity associated with the layer rotation and where the last term describes the dynamic coupling between the variables and .

Here, the free energy of the SmC* phase is given by the well known equation (see, for example, 109,110 )

where P is the spontaneous polarization, ε is the transverse dielectric constant and C is the pseudo-scalar coupling constant.

Minimization of the free energy with respect to Θ and P yields the following equations:

Expressing P from the second equation and substituting it into the first, one obtains the following expression for the derivative ∂F/∂Θ which enters one of the Euler-Lagrange equations:

One notes also that ∂F/∂γ = 0 because the free energy F does not depend on γ. Using these results, eqn (4) can be rewritten in the simple form:

γ1 + γ12 = 0 γ1 + γ12 = -a(T)Θ + εCE (9)
Here the first equation describes a relaxation of the tilt angle in a rotating layer while the second equation represents a counterbalance of the two generalized torques acting on the smectic layer normal.

One notes further, that there are three different time scales involved in eqn (9). The response time of the electroclinic effect τr = a(T)/γ1 falls within the submillisecond regime according to the experimental measurements. 64 At the same time the rotation of the smectic layers has been found experimentally to correspond to a time scale of several seconds. Thus the term γ12 in eqn ( 9) can be neglected and the equation can be approximated as γ1 = -a(T)Θ + εCE (10) This is the equation governing the dynamic behavior of the electroclinic effect in the system with fixed smectic layers.

The solution of eqn ( 10) can be substituted in to eqn (9) to obtain :

= κ (11) where κ = λ12/λ1.

In the next section we obtain explicit expressions for the angle of layer rotation using the particular form of the saw-tooth external electric field.

Rotation of the smectic layers caused by an asymmetric electric field

The asymmetric saw-tooth electric field, presented in Fig. 17, is assumed to be oscillating between the values +E0 and -E. The rise time τ1 is assumed to be shorter than the decay time τ2, and the asymmetry ratio of the field is introduced as h = τ1/τ2. The frequency of the field is f = (τ1 + τ2) -1 . Mathematically, the variation of the electric field within one period can be written in the form: (12) when 0 < t < τ1 and (13) when τ1 < t < τ2. Due to the fact that the rate of change of the electric field is much slower than the electroclinic response time, one can safely assume that during the switching the tilt will always adopt its equilibrium value for the given electric field. The rate of change of the tilt is then simply given by ( 14)

The angular velocity of layer rotation is now expressed as = σE (15) where σ = κεC/a0. Using the explicit expression for the saw-tooth electric field one obtains (16) when 0 < t < τ1 and (17) when τ1 < t < τ2.

The net rotation, Δγ, of the smectic layers during one period of the electric field is now calculated as (18) The value of Δγ calculated from eqn ( 16)-( 18) is obviously zero because the larger positive value of during the shorter time τ1 is exactly compensated by the smaller negative value of during the longer time interval τ2.

However, there must exist a stabilizing torque Γs which is related to the surface treatment of the glass plates surrounding the sample, and which is responsible for keeping the preferred orientation of the layers. In order for the layers to rotate, the driving torque must exceed some threshold, corresponding to the maximum possible value of the stabilizing torque Γ0. The stabilizing torque Γs has the nature of a friction torque and adopts the value needed to balance the driving torque as long as it is smaller then Γ0. However, if the driving torque exceeds a critical value, the stabilizing torque can no longer increase, but adopts its maximum value Γ0, always opposing the rotation of the layers. By adding the stabilizing torque with the proper sign to eqn (16), one obtains the final dynamic equations for the layer normal:

if and = 0 (20) if when 0 < t < τ1.

Similarly, Here E* = εCE/a0, d* = da0λ12/λ2Γ0, f* = fλ2/a0 and η = τ1/τ2. One notes that these non dimensional equations contain only three independent control parameters including the product of the frequency and the strength of the electric field f*E * 0, the asymmetry ratio η, and the sample thickness d*. It is also seen that only one combination of the material constants enters the final equations. This is the ratio. κ = λ12/λ1.

The dependence of the average angular velocity of smectic layer rotation is presented in Fig. 17 as a function of the product f*E * 0 for four different values of the asymmetry ratio η. One can readily see that there exists a lower value of f*E * 0 below which there is no layer rotation, corresponding to the threshold given by eqn (23) One can also see from the figure that the threshold decreases with the increasing asymmetry η. With the increasing f*E * 0 above the threshold the angular velocity increases linearly until the second threshold (determined by eqn (23b)) is reached and the angular velocity becomes saturated, independent of. f*E * 0.

The average angular velocity of the layers divided by κ is plotted as a function of the sample thickness in Fig. 18 One notes that the smectic layers rotate only if the layer thickness exceeds some threshold value d * tr which depends on η. Such a threshold has indeed been observed experimentally. It is interesting to note also that there exists an optimum sample thickness which corresponds to the largest possible angular velocity of layer rotation for given η and f*E * 0. This behavior can be explained qualitatively in the following way. Once the threshold d * tr is exceeded, the average angular velocity increases with the increasing sample thickness. This is due to the fact that the driving torque is a bulk effect which increases with increasing sample thickness. On the other hand, the counter-torque, acting only at the surfaces of the substrates, is independent of the sample thickness. As long as the second threshold (see eqn (23b)) is not exceeded, the layers rotate only during the interval τ1 of the total period of the electric field. At a certain value of the sample thickness, however, the second threshold is exceeded and the layers rotate in opposite directions during the intervals τ1 and τ2 within the same period. In this regime it is the difference between the driving torque and the counter-torque that is responsible for the net rotation of the layers. The relative difference between these torques during the intervals τ1 and τ2 decreases with increasing sample thickness. Thus, for a very thick sample the layers just oscillate back and forth, and the average rotation of the layers approaches zero.

Summary

This review summarizes the observation and behaviour of smectic layer rotational instabilities, or short, the smectic layer rotation, which can be observed by application of an asymmetric electric field to a chiral fluid smectic liquid crystal. Textures and small angle X-ray diffraction are used to elucidate the respective structures. The dependence of the layer rotation on a variety of parameters is detailed, such as the field asymmetry ratio, the applied field frequency and bias, the temperature approaching the SmC*-SmA* transition (thus the influence of the electroclinic effect), or the cell gap. Also other parameters are discussed, like the enantiomeric excess, varying alignment layers (thus indirectly the anchoring strength), the concentration of ions and the phase sequence of a specific material.

Since the smectic layer instabilities have a detrimental effect on most of the applications of ferroelectric liquid crystals, some ways for their avoidance, like polymer stabilization, are discussed, and even applications based on the layer rotation are proposed, such as optical recording. At last, a comprehensive theoretical description is developed, which accounts for all of the observations described.

Fig. 3 (

 3 Fig. 3 (a) Schematic representation of the helical superstructure of the smectic C* phase, due to a local cancellation of the spontaneous polarization, Ps. (b) Under appropriate boundary conditions, the signature of the periodic helicoidal structure appears by polarizing microscopy as a (periodic) stripe texture. The scale bar is approximately 25 μm.

Fig. 4 (

 4 Fig. 4 (a) In the bookshelf geometry the smectic layers are formed perpendicular to the substrate plates, with two possible positions of the spontaneous polarization, up or down. The latter are related with two positions of the director on the tilt cone. (b) In a surface stabilized ferroelectric liquid crystal the two director positions can be visualized by polarizing microscopy as two types of domains, dark and bright. The scale bar is approximately 25 μm.

Fig. 5 (

 5 Fig. 5 (a) Schematic of a vertical chevron device with the smectic layers being tilted with respect to the substrate plane by an angle, which is often about equal to the director tilt angle, δ ∼ θ. (b) Vertical chevrons can exist in two tilt directions. A defect wall (thick or thin) is observed where chevrons of different tilt direction meet. (c) In polarizing microscopy these defect walls are seen to make a zig-zag defect. The scale bar is approximately 50 μm (reproduced by permission from ref. 23. Copyright Wiley-VCH).

54

 54

Fig. 6 (

 6 Fig. 6 (a) In the chiral smectic A* phase layer straightening by application of a sufficiently large electric field can lead to stripe structure, where the contrast between different stripes depends on the induced tilt angle. (b) Demonstration of the stripe texture in polarizing microscopy, where the width of the individual stripes is often observed to be equal to the sandwich cell gap d (reproduced by permission from ref. 23. Copyright Wiley-VCH).

Fig. 7 (

 7 Fig. 7 (a) Model of a zig-zag defect in a vertical chevron structure. The regions of opposite chevron direction are of the bookshelf type. (b) Model of a horizontal chevron domain structure, as can be observed after application of a large electric field to a vertical chevron structure. The defect mediating the horizontal chevron domains of opposite inclination is of the vertical chevron type. (c) Polarizing microscopic example of a horizontal chevron structure. The smectic layers can easily be visualized through the equidistant helical line pattern (reproduced by permission from ref. 55. Copyright Taylor&Francis).

Fig. 8

 8 Fig. 8 Switching process of a horizontal domain texture (left), and respective schematic illustrating the director directions (right). (a) Zero field, (b) positive field and (c) negative field. Note that on polarity reversal bright and dark domains are interchanged. The bottom of the figure illustrates the formation of horizontal chevrons by time resolved synchrotron small angle X-ray scattering from an unoriented powder sample to the horizontal chevron structure exhibiting two domain types (t = 0 s to 20 s, left to right) (top: reproduced by permission from ref. 23. Copyright Wiley-VCH. Bottom: reproduced by permission from ref. 57. Copyright American Physical Society).

Fig. 9

 9 Fig. 9 Electro-optic response of a horizontal chevron structure. The two individual domains (dotted and dash-dotted lines) exhibit standard electro-optic FLC switching, though phase shifted by 180 degrees. The detrimental effect of the horizontal chevrons on the electro-optic response can be seen for the switching of the whole cell (dashed line), where the transmission is hardly modulated at all (reproduced by permission from I. Dierking, G. Scalia, Liq. Cryst. 27, (2000), 1059. Copyright Taylor & Francis).

  Fig. 10 Formation time tr and average domain width Wd of horizontal chevron structures formed as a function of (a) electric field amplitude E, (b) field frequency f and reduced temperature T -TC with respect to the SmA*-SmC* transition.

Fig. 11

 11 Fig. 11 Rotational smectic layer instabilities observed after rotating of the layers by several hundreds of degrees. Courtesy of Gunnar Andersson, Department of Physics, Chalmers University of Technology, Gothenburg, Sweden. The scale bar is approximately 100 μm (reproduced by permission from ref. 23. Copyright Wiley-VCH).

Fig. 12

 12 Fig. 12 Dependence of horizontal chevron reorientation times on applied external conditions of (a) electric field amplitude, (b) field frequency, (c) reduced temperature (Tc is the SmC*-SmA* transition temperature), and (d) cell gap. Experimental conditions are noted in the figure.

Fig. 13

 13 Fig. 13 Smectic layer permeation model for the domain wall motion of horizontal chevrons during the reorientation process from one to the other domain type of opposite layer inclination (top left). Only a small molecular displacement is needed for the rotation to be accomplished. Exemplary series of textures demonstrating the permeation reorientation model through the deformation of circular air bubbles (top right). The bottom row shows respective time resolved small angle synchrotron X-ray images of the reorientation process (t = 0 s to 50 s from (a) to (d)) (top: reproduced by permission from ref. 74. Copyright Taylor & Francis. Bottom: reproduced by permission from ref. 57. Copyright American Physical Society).

Fig. 14

 14 Fig. 14 Dependence of the layer reorientation time on several materials conditions as a function of electric field asymmetry. (a) Variation of enantiomeric excess, (b) different alignment layers with varying anchoring strength, (c) ionic content, and (d) variation of phase sequence within a homologous series of FLC compounds from N*-SmC* via N*-TGBA*-SmC* to SmA*-SmC*.

Fig. 15

 15 Fig. 15 Optical recording by use of the smectic layer reorientation. The top images show a schematic of the process with (a) and (b) the recording and (c) and (d) the erasing cycle. The bottom images show a respective experimental verification with (a) recording and (b) erasing (reproduced by permission from ref. 80. Copyright AIP Publishing).

Fig. 16

 16 Fig. 16 Optical textures of a polymer-stabilized ferroelectric liquid crystal: (a) at 0 min (left) and after 10 min. under an asymmetric field (right) in the SmA* phase. The smectic layer rotation is suppressed; (b) birefringent patterns in the stripe texture remain visible although most of the liquid crystal has undergone the transition to the isotropic state (reproduced from ref. 91).

Fig. 17

 17 Fig. 17 Average nondimensional angular velocity of the smectic layers divided by κ as a function of the product between the frequency and the amplitude of the electric field for four different values of the asymmetry ratio η = τ1/τ2. Here the nondimensional sample thickness d* is d* = 100.

  τ1 < t < τ2. Here we have taken into account that Γ0 is the surface torque, that is, the torque per unit area.Eqn(19) and(21) imply that there exist two threshold values of the electric field: and (23)Note that E1 < E2 when τ1 < τ2.The average angular velocity ω = 〈 〉 of the layers is then obtained as ω = fΔγ(24) where f is the frequency of the applied field and Δγ the net rotation during one period of the electric field. Using eqn (19)-(22) the average angular velocity of the smectic layer rotation as a function of the amplitude of the electric field E0 can be expressed as:ω = 0 (25) if E0 < E1, (26) if E1 < E0 < E2 and (27) if E2 < E0.These final equations can also be rewritten in the following non dimensional form: 111 ω = 0 if E

Fig. 18

 18 Fig. 18 Average nondimensional angular velocity of the smectic layers divided by κ as a function of the nondimensional sample thickness d* for four different values of the asymmetry ratio η = τ1/τ2 using the fixed value f*E * 0 = 0.005.

where he received his PhD in condensed matter physics. He is Director of Research at