Variational Theories for Liquid Crystals

E.G. Virga

• Publisher: Chapman and Hall/CRC
• Year: 2018
• Print ISBN:
  9780367826048
• eBook ISBN:
  9781351405645
• Topic: Mathematics
• Subtopic: Applied Mathematics
• Index: Mathematics

Essentially there are two variational theories of liquid crystals explained in this book. The theory put forward by Zocher, Oseen and Frank is classical, while that proposed by Ericksen is newer in its mathematical formulation although it has been postulated in the physical literature for the past two decades. The newer theory provides a better explanation of defects in liquid crystals, especially of those concentrated on lines and surfaces, which escape the scope of the classical theory. The book opens the way to the wealth of applications that will follow.

1

Physical background

This book mainly concerns the variational theories of liquid crystals, phrased in the language of modern continuum mechanics.

In this chapter I outline the phenomenological premisses upon which the following chapters are built. Here the development is merely descriptive; it aims to make the reader familiar with the most relevant features of this fascinating state of matter.

A fairly complete account of both chemical and physical properties of liquid crystals can be found in the books by Gray (1962), De Gennes (1974) and Chandrasekhar (1977), which have already become classic. I also refer the reader to the major review papers of Brown and Shaw (1957), Stephen and Straley (1974) and Ericksen (1976), as well as to the nice introductory booklet by Collings (1990). Two other papers, addressed to a broad learned public, deserve notice for their clarity, namely the article of Fergason (1964) and the collective work by the Orsay Group (1971).

1.1 Liquid crystals

Though the name liquid crystal is likely to be familiar to most readers, it might sound self-contradictory to those who first hear it. Most people intuitively associate the word crystal with something solid.

How can a crystal be liquid?

G. Friedel raised this issue in 1926 (cf. p. 1052 of Brown and Shaw (1957)), proposing to adopt the name mesomorphs or mesoforms instead of liquid crystals since, as we shall see shortly below, these substances are indeed to be regarded as intermediate states of matter. Nevertheless, these terms found little acceptance, like others such as paracrystals. The term liquid crystal has been widely used since O. Lehmann coined it in the late 19th century, maybe just because it sounds self-contradictory. Be that as it may; we shall stand on the side of tradition and call the substances treated in this book liquid crystals.

Liquid crystals were discovered in 1888 by F. Reinitzer and O. Lehmann. Reinitzer’s seminal paper appeared in Volume 9 (1888) of Monatshefte für Chemie (Reinitzer (1888)), and a translation into English has recently been published in Liquid Crystals (Reinitzer (1989)) to celebrate the centenary of the first paper. The attribution of the discovery has long been disputed. I do not venture any opinion about this slippery matter, which I leave to the historians of science. Several papers on the history of liquid crystals could help the interested reader to understand the wealth of the early literature, namely Kelker (1973), Kelker and Knoll (1989) and Sackmann (1989).

Reinitzer studied different derivatives of cholesterol which are solid at room temperature. In heating them up he observed peculiar behaviour: it appeared that these substances have, as it were, two melting points. Here is, in Reinitzer’s words, the description of the experiment he performed with cholesteryl benzoate:

With respect to the melting point, a significant deviation from Schulze’s data was noted. He found it to be 150-151°. However, despite continued careful purification I was able to find only 145.5° (corr. 146.6°). However, it struck me that the substance, in this case, melted not into a clear transparent but always into a cloudy, only translucent liquid, which I initially considered to be a sign of impurities, although both microscopic and crystallographic examinations of the compound revealed no sign of non-uniformity. Upon closer examination, it was then also noted that when it was heated to higher temperatures, the clouding suddenly vanished. This happens at 178.5°C (corr. 180.6). At the same time I found that the substance heated thus high displayed, upon cooling, colour phenomena quite similar to those already described for the acetate. This remarkable phenomenon of the presence of two melting points, if one may express it thus, and the appearance of the colour phenomenon were primarily what made me think that here and in the case of acetate physical isomerism must be present, for which reason I requested Professor Lehmann in A...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Dedication Page
• Contents
• Preface
• Acknowledgments
• 1 Physical background
• 2 Mathematical preliminaries
• 3 The classical theory
• 4 Instabilities
• 5 Drops
• 6 The new theory
• Index