Journal of the Indian Institute of Science- Volume 76: Number 4

Volume 76 July-August 1996 Number 4

Special Issue on lectures delivered on November 11,1995, at the Indian Institute of Science under the auspices ofthe

Jawahnrlal Nehru Centre for Advanced Scientific Research

Contents

Foreword

R. Chidambaram Materials response under static and dynamic high pressures

N. R. Krishnamurthy Laser-modulated colloids

A. J. Basu and A numerical study of axisymmetric vortex break- A. Khalili down

J. Chandrasekhar Exploring new structural motifs using computational methods

Madhav Gadgil Western Ghats: A lifescape

The Sawaharlal Nehru Centre for Advanced Scientific Research was established in 1989 to commemorate the centenary (1989) of Pandit Jawaharlal Nehru by the Government of India. The Centre maintains close links with the Indian Institute of Science. It as a number of distinguished scientists as Honorary Professors and Senior Fellows and has been recruiting full lime hculty. The Centre has a campus at Jakkur, I I km from the Indian Institute of Science campus. Units and Labora- tories devoted to certain chosen areas of research are already functioning.

At its Annual Faculty Meeling, lectures are delivered by its faculty covering different areas of research pursued by the Centre. These lectures are published in a special issue of the Journal of the Indian Institute of Science. This is the sixth such issue.

On behait of the Centre, I extend my thanks to Prof. M. S. Shaila, Edntor of the Journal, for kindly devoting a special issue for this purpose.

Bangalore

(C. N R. Rao) President

Jawaharlal Nehru Centre for Advanced Scientific Research

J I ndmn i n n Sci., Joly-Aug. 1996, 76, 437-463 O lndtan instttmc of Science

R. CHIDAMR4RAM High Prcssure Physm Dwision, Bhabhn Atornlc Research Centre (BARC), Boinbay 400 085, India

Recelvcd on Aprcl 30, !996.

Abstract

Studies on equation of statc and phase tranailions a1 high pressures have rlgnificantly contributed lo our basic understanding of condensed matter physlcs. High-pressure data on matemis also hnd important appi~catrnns in applted ncieace~. The dcveloprnents in first princcple theories and expenmental r r c h n l q u e are Irsted. The himi- la<ittrs and dlffcrmces in behaviour of matcirals under statlc and dynarnlc yrea,aiua are discussed The arlicle also describes thc current interplay hetween theoretlcai and experimental high-prcssare research w t h ~lhslrat ions from our own studws and empharis on Fulure xopz

Keywords. Equation of state, phase trans~lion, shock Hugonm, d~arnond anvil ccll

I. Introduction

The use of pressure to alter the physical and chemical states of materials is as fundamcn- tal as varying the chemical composition or temperature. Volume reduction obtained in materials by current high-pressure techniques is an order of magnitude higher than the change brought about by tempel-ature. This high compression, is achieved using static and dynamic pressures. Static high-pressure teclmiques have evolved from the use of hardened steel anvils to the current novel diamond anvil cell device'. Diamond anvil cell technology has brought about a revolution in terms of the range of pressures achieved and in the wide varrety of diagnostic techniques that can be employcd because diamonds act a excellent windows through which the sample can he viewed'-2. It is posaihle lo use a wide range of electromagnetic radiation to characterize the physical properties of samples in sits at high pressures. These newly acquircd capabilities have enabled researchers to exploit fully the psessure variable. On the othcr hand. dynamic high pressures are generated by introducmg a rapid impulse into a material tiirough the detonation of a high explosive, with the impact of a high-speed projectile or by the absorption of an intensc pulse of radiation3. Nigh-specd optical and electronic methods are necessary to measure certain dynamic variablcs which determine density, pressure and cncrgy. However, till recently, the diagnostic techniques used were mostly for carrying out measurements of continuum parameters. Unlike the static casc, the in-sit14 probing of samplcs during the passage of shock wave is cxtre~nely difficult. Nevertheless, Rash X- ray diffraction measurements by Johonson and ~ i t c h e i l ~ followed by the work of Whit-

T e a t of lecture dclivercd at the Annxnl Faculty Mceling of thc 5awah;ilal Nchru Ccntic for Advanced Scientific Research kt Bangtilure on Noveinher 1 I , 1995

438 R. CHIDAMBARAM

lock and Wark using laser-driven shocks has opened up the possibility of online examination of samples under shock. Currently, various laboraloi-ies arc seriously engaged in developing techniques to generate time-resolved optical data%nd to carry out X-ray diffraction measurements under shock conditions5.'. However-, the developnlenl of thesc techniques to obtain reliable data still remains a tough task. The limited time scales, the need for uniaxial configuration and destructive nature of the experiments severely limit such rncasurements which are widely employed in static high-pressure experiments. In static high-prcssure studies ineasurcments are carried out along the isotherms with a combination of ultrahigh-pressure and laser-heating techniques. I1 is possible to obtain isotherms of several thousand degrees Celsius at different pressures in materials using the diamond anvil Such controlled high-pressure temperature conditions can be realised for any amount of time. On the other hand, in shock experiments the measnre- ments are carried out mostly along the principal Hugoniots which represent the locii of all the points which can he reached by shocking a material from fixed initial condi- ' tionslu. In dynamic shock experiments the time of passage of the shock through the sample is short compared to the disassembly time of the sample. 'The time available for measurements thus varies from a few microseconds to nano seconds depending upon the method used for generation of shock. The shock cornprcssion 1s not a srmple one that occurs in static experiments. The material under shock compression is in Pact subjected to shear forces and temperature rise besides rise in pressure. The shock Hugoniot is different from the isotherm passing through the same initial state as shown in Fig. 1. The shock compression imparts higher internal energy to the material compared to the static case which consists of reversible and irreversible intcrnal energy components producing elastic compression and heating of the material, respectively. Thus, as shock conrpres- sion produces a large thermal component of pressure, the Hugoniot always lies above the isotherm (Fig. 1). The amount of compression produced in a single shock experiment is limited, as at higher shock strengths much of the energy goes into heating rather than to compression. As the Hugoniot traces a different path from an isotherm for the same phase change in a material the transition pressures in the two cases may not match and different phases can be encountered along an isotherm and i-Iugoniot. Comparcd to the stalic situation, shock compression is also accompanied by generation of defects m d dislocations. The role played by thcse defects in shock propagation is not well understood although they are known to act as a nucleation site for the gi-owth of new phase. The time duration of shock experiments is very small compared to the infinite time available in static experiments. The fast rise times and small duration of the shock coupled with the high strain rates that are characteristic of these experiments do not give suftlcienl time for the growth of a new phase; still, in many cases the polymorphic phase transitions observed to be exhibiting slow kinetics under static loading conditions are found to occur in short times under shock loading and only those phase transitions that have exlremely slow kinetics may not get deleclcd during the passage of shock in the material' ' . I 2 .

If is thus clear that in-situ data on.materials under shock conditions is the need of the hour to provide a microscopic understan?ing of the underlying physical phenomena and to verify the existence of steady state under dynamic situations". The reliabiiity of high-

MATERIALS RESPONSE UNDER STATIC P.ND DYNAMIC HIGH PRESSURES 43 9

': "Id Curve g:F:ressure achieved through various experimen

Siotrc Max pressure

Plston-cylinder 50 kbar Dialnond anvd cell 5 Mbar (X-ray diffraction)

v Dynamic

Gas gun 10 Mbar . 111 shock heating; z cold ~0n lp re~s ion Lasers 700 Mbar

FIG. 1. P-V dlagram for shock compression of cold Underground NE 50,000 Mbar material. Vertlcal lmes denote shock heating whrle the horizontal ones denote cold compression. * see Godwal" for details.

pressure data at large compressions is uncertain because the stress state at ultrahigh pressures is not well understood. Shear strength provides a basic description of a material's mechanical behaviour, but very little is known about this quantity at multimegabar pressures'4. In fact, upon compression, the stress state of all solids is nonhydrostatic when the sample has finite strength. If uncorrected, the presence of shear stresses can lead to systematic errors in physical quantities like bulk modulus and its pressure derivative". Shear strength effects can also introduce errors in measured pressures when secondary calibrants like ruby fluorescence or diffraction standards are used.

Not withstanding these difficulties, by careful experimentation and appropriate modelling, considerable amount of valuable information about behaviour of materials under static and dynamic high-pressure conditions has been obtained. The present article discusses material response under static and dynamic high pressures with interesting resuits on a variety of new phenomena encountered.

2. Present status of high pressures achieved

In Table T we summarise the present experimental situation for different kinds of experimental systems designed to generate high pressures. It is seen that static high- pressure techniques have reached pressures'6 higher than that at the centre of the Earth (360 GPa) and at the core-mantle boundary (150-200 GPa). Coupling ultrahigh pressures and laser-heating techniques we get close to producing pressure-temperature conditions existing at the centre of the Earth to the core-mantle boundary. Also pressures exceeding 4500 and 1000 GPa existing, respectively, at the centre of the planets Jupiter and Saturn have already been created in the laboratory using giant high-power lasers" developed for achieving inertially confined controlled thermonuclear fusion. There have been doubts about the viability of lasers for generation of high-pressure data because of the large error bars associated with determination of pressure from experimentally measured shock velocity. However, recent developments in the area of lases-driven shock waves point to the possibility of obtaining pressure values with accuracies of about 10%': It is also observed from Table I that pressures existing in stars like the white

dwarf can only be produced using nuclear explosions'%~hich however require large experimental configurations.

3. Physical phenomena of interest

Onc of the most important applications of high-pressure research is the study of pres- surc-volumotcmperature relationship of materials, usually known as thc equation of state (EOS). In fact-, EOS and phase stability are the most fundamental properties oh- tained from these investigations. Experimental and theoretical investigations of EOS are of immense importance to researchers in both basic and applied sciences. Its utility for meaningful interpretation of physical and chemical phenomena under pressure need not be emphasized. Questions like how the criticality factor for a fissionablc mass varies with compression call only he answered with the heip of EOS. It provides vital input for hydrodynamic calculalions in controlled fission-fusion research, in simulations or reac- tor accidents and in rock mechanical effccts of peaceful nuclear e ~ ~ l o s i o n s ~ " ~ ~ ~ . In geo- physics it helps to understand the structure of the Earth and in aslrophysics to unravel the mysteries of evolution of stellar bodies like white dwarfs, neutron m r s and black holes. In the basic sciences it provides the tesl lo the theoretical models of cohesion.

Pressure-induced phase transitions are a very interesting phenomenon and studies of these continue to be one of the most active areas of high-pressure research. With dramatic improvement in the experimental techniques employed in static and dynamic high-pressure research several structural, electronic and insulator-to-metal phase transitions have been observed in various materials22. Discovery of crystal-to-amorphous transition in various substances observed under pressure is an interesting example of pressure effec12'. Another exciting phenomenon obacrved only under dynamic: shock situations is melting and vaporization. It is hoped that it will be possible to reach pressures of 1000 GPa using diamond anvil cell (DAC) in the near future and we expect to obaerve pressure ionization of inner core electrons in static cxperiments. So far the combined pressure thermal ionization of core e!ectrons dominated the physical properties along the shock Hugoniots above 1000 GPa.

4. Experimental techniques

The High-pressure Physics Laboratory at BARC has heen engaged in thc studies of condensed matter under pressure for about 20 years. Various experimental facilities have becn built and used. These include four-probe rcsistancc mcasurernentsz4, angie- dispcrsive X-ray diffraction using WC anvils with Be gaskets2' and DAC-based full Bragg cone, energy-dispersive X-ray diffraction system (EDXRD) along with a ruby flourescence spectrometerz6 to measure the pressure. Several experimental studies have been carried out on the DAC-based EDXRD system using a white X-ray beam from a rotating anode X-ray generator. Thc diffracted beam is collected over the full Bragg cone by a conical slit and is cnergy analyscd using a large-area sen~iconductor (NPGe) detcc- tor and iudigenous microyrocesor-based multici~annel analyser. Later, a sinrilar EDXRD system" was built and installed at the Institute or Nuclear Physics, Novosibirsk, Russia, at ?he beam lines of synchrotron radiation source VEPP-3.

MATERIALS RESPONSE UNDER STATIC AND DYNAMIC HIGH PRESSURES 44 1

A variety of diamond cells have been fabricated a? Trombay. The first one to be made was on the design of Syassen and Holzapfel". A DAC, similar to MaeBel l type, was fabricated for the angle-dispersive diffraction studies using film method29. Another DAC which involved a combination of M a ~ B e l l and Bassett designs30 was coupled to an in- digenous Raman scattering facility3'. This cell has now been replaced by a Merrill- ~ a s s e t t ~ ~ - t ~ e cell for optical studies. Recently, a Raman setup was built for optical studies at high pressure33. This setup has been assembled using a 500-mm single-stage, double-pass scanning monochromator and a super notch filter for 514.4 nm line of Ar ion laser which permits the observation of Raman modes greater than or equal to 100 cm-'. The angle-dispersive X-ray diffraction system incorporating an imaging plate as an area detector has been c~rnrnissioned~~. The excellent features of an imaging plate like high sensitivity, dynamic range and linear response makes it most suitable for accurate EOS measurements and for the determination of the structure of high-pressure phases.

In order to study materials under dynamic shock pressures we have set up a compressed gas gun facility at Trombay3'. In this gun, the projectile, with the impactor mounted on the nose, flies through a barrel to impact the target material to generate a shock wave. The projectile could be accelerated up to 1.2 kmls to generate a peak pressure of about 40 GPa, depending upon the impedance of the impactor and the target.

5. Metallic hydrogen

As hydrogen is the most abundant element in the universe, and is the first entry in the periodic table, its metallization is not only a dramatic illustration of pressure-induced changes in bonding character, but also plays an important role in determining the internal state and evolution of the giant planets. At low pressures, hydrogen crystallizes as an insulating molecular solid and at extreme pressure conditions it forms a dense plasma fluid. Metallic hydrogen is expected to be a high-temperature s ~ ~ e r c o n d u c t o r ~ ~ .

Theoretical studies in the past predicted that under pressure insulating molecular hydrogen will undergo a band overlap transition to a molecular metallic phase before dis- sociating to form a monoatomic metallic solid. There are wide variations in the predicted pressures for these transitions depending upon the structure, molecular orientation and approximations employed in the theory. Local density approximation (LDA)- based calculations of c-axis-ordered hexagonal closed packed (hcp) phase predict closure of the band gap at 40 G P ~ ~ ' . The quasiparticle c a ~ c u l a t i o n s ~ ~ based on Green's function with screened Coulomb potential (GW) give the value I51 GPa. Lowering the symmetry increases this band gap at 150 G P ~ ~ ~ . Also, for the monoatomic metallic transition the theoretical predictions range from 250 to 400 GB~".

The central problem of hydrogen research at high pressure is the uncertainty of crystal structure. X-ray diffraction on solid hydrogen is the only direct measurement that can give a definite answer on the phase diagram. Hydrogen in the molecular form solidifies at 14 K in the hcp structure at ambient pressure (V/Vo = 1 with V, as normal volume).

442 R. CHIDAMBARAM

The past limit of 50 GPa in single-crystal X-ray diffractionZ studies has been raised to 120 G P ~ ~ ' . These measurements demonstrate that hydrogen remains in the hcp structure up to such pressures. The EOS measurements also reveal that the solid hydrogen is enormously compressible41. The experiments showed that the crystals became increas- ingly anisotropic with pressure and that the isotopic shift between hydrogen and deute- rium was much smaller than expected.

Dynamic shock experiments have been carried out on fluid hydrogen at Lawrence Livermore National Laboratory by Nellis and coworkers42. In these experiments, the EOS and electrical conductivity of fluid hydrogen was measured up to 200 GPa. The measured first shock temperatures up to 20 GPa were in excellent agreement with predictions based on molecular hydrogen. The second shock temperatures up to 90 GPa were lower than predicted for the moleculaj phase and were due to continuous dis- sociative phase transition above 20 GPa. The partial dissociation from the molecular to the atomic phase absorbs energy and lowers shock temperatures. In the electrical conductivity measurements high densities and low shock temperatures were obtained using reverberating shock to compress the liquid hydrogen. This avoided dissociation and enabled the measurements of density dependence of band gap closure as the samples were shock-compressed to small temperatures compared to the electronic band gap. In these experiments, the nearly cooled hydrogen provided the semiconducting fluid phase. Shock compression to a high density reduced the band gap while heating the hydrogen sufficiently excited the electronic carriers into the condllction band to produce measur- able electrical conductivity. The energy gaps measured were in the density of states of disordered fluid which is claimed to be in the metallic state. These results are important for understanding the physics of metallization of hydrogen and for understanding the isentropes of the interiors of Jupiter and Saturn. The determination of band gap from electrical conductivity measurements of a shocked fluid discussed above is also close to theiheoretical and static measurements for the solid phase.

Optical Raman measurement^^.^^ show that the hexagonal phase remains stable up to a pressure of 150 GPa (plpo = 9). At this pressure, the measured vibron frequency shows a discontinuity. The nature of this new phase is controversial, although its pressure is in the range of predictions for insulator-to-molecular metal phase transition. Further pressurisation up to 230 GPa indicated that molecular bonds are stable. Above 230 GPa Raman vibron disappeared4" The experimental findings further reveal that at around 250 GPa hydrogen begins to absorb visible light consistent with the hand gap dielectric model and fits to the index of refraction measurements. However, the ruby R1 peak used for pressure calibration was very weak and difficult to measure.

Disappearance of Raman vibron is consistent with molecwlar dissociation but not a necessary condition. The dissociation may also be due to pressure-induced absorption of an exciting laser or by fluorescence of a diamond and sample or loss of hydrogen to the anvils. These problems are being investigated by increasing the gasket hole and using a pressure marker in place of ruby. If there is hope for metallization of hydrogen then the pressure has to cross 300 GPa because the refractive index measurements indicate that direct gap closure requires pressure above 300 GPa.

MATERIALS RESPONSE UNDER STATIC AND DYNAMIC HIGH PRESSURES 443

6 . Equilibrium shock temperature

When the shock front propagating in a material is assumed to be a shaip discontinuity in stress, then, using the laws of conservation of mass, momentum and energy1', it can be shown that the undisturbed state is related to the shocked state at the shock front as follows:

Uspn = Pi(U, - Up) (1)

P,-Po = po(Us - Up) (2)

El-& = 112 ( P , + Po) (Vo-V) (3)

where U, is the shock-front velocity, Up, the particle velocity in the compressed region, P, E, p and V(= l ip) are, respectively, pressure, specific internal energy, density and specific volume, the suffixes 1 and 0 represent quantities in the shocked and the un- shocked regions, respectively.

If the equation of state E = E fP, V) of the material is also known, we can use (3) to write P as a function of V (Po is small and for all practical purposes it is neglected). The locus of all states (P, V) which can be obtained from an initial state (PO, Vo) is known as the Rankine-Hugoniot (RW) curve (Fig. 1). Measurement of two quantities U , and Up determines the shock Hugoniot. It is clear from (3) that the internal energy deposited in the compressed body is the area of the triangle OAC. If the compression is carried out isothermally at zero K to the same final volume V, the material will be at point B. The area of the curved triangle OBC represents the cold elastic energy. The difference between the areas of OAC and OBC represents the heat energy provided by the shock in the compressed sample. The contribution from defects created by the shock, such as vacancies and dislocations, is expected to be small and hence neglected. Measurement of shock temperature is still a burning problem with limited success for a few transparent materials. Information about shock temperatures is obtained from theoretical estimates. Generally, the total internal energy E and pressure P at a given volume are computed as a function of temperature and iterated self-consistently to satisfy eqn (3) to obtain the shock temperature. To illustrate it, a shock pressure of I 8 GPa will compress a sample of aluminium from a density of 2.78 to 3.3 g/cm3 and heat it to 974 K (Fig. 1).

After the shock wave has passed, the material will unload along the curved path ADO. In doing so, it does work on the surroundings, given by the area of the curved triangle ODAC. The difference in the areas OAC and ODAC represents the waste heat deposited on the terminal sample (Fig. 1). If the amount of waste heat exceeds the enthalpy of vaporization (or melting), the terminal sample will be in the vaporized (or molten) state. This interesting phenomenon results from using peaceful nuclear explosion discussed in the following section.

7. Pokhran experiment

India's first peaceful nuclear explosion experiment was carried our on May 18, 1974, in the Rajasthan desert at a place near Pokhran. The aim of the experiment was to study the

0 20 LO 60 80 1'

Time-msec FIG. 2. Computed propagation and cavity growrh in vertical d~rection for the Pokhran experiment.

explosion phenomenology, fracturing effects in rocks, ground motion and containment of radioactivity, etc., in the context of applications of peaceful nuclear explosion. In this experiment, a plutonium device of yield 12 kt equivalent of TNT was emplaced in a shale medium, at a depth of 107 m in a chamber at the end of an L-shaped hole. Upon detonation, the ground surface above the emplacement point rose with a velocity of 25- 30 m/s to form a dome 170 m in diameter and 34 m in height. There was no venting of radioactivity in the experiment. The resultant apparent crater, measured with respect to the preshot ground surface, had an average radius of 47 nl and a depth of 10 m. This, perhaps, is the only experiment which produced a crater (though shallow), and yet was completely contained from the radioactivity point of view.

Chidamharam and ~ a m a n n a " have explained the phenomenology of this experiment using computer modelling with an one-dimensional spherical symmetric rock mechanics computer code. The quick release of explosive energy of a nuclear device initiates various physico-mechanical processes in the geological medium, like vaporization, melting, crushing, fracture and motion of the surrounding rock. The reflection of stress waves at the free ground surface imparts additional kinetic energy to the rock medium. Most of the energy of the Pokhran device, as in any other nuclear explosive, was released in less than a microsecond. It has been shown by Chidambaram and coworkers2' that this re- sulted in 640 tons of rock, extending up to a radial distance of 6.2 m, which was shock- melted, from the criteria discussed earlier. At the vapour-liquid interface, the pressure is expected to be about 160 GPa. Computed wave propagation and cavity growth are depicted in Fig. 2. The final cavity radius in the horizontal direction is calculated to be about 28-29 m compared to the post-shot measured value of 30 m.

MATERIALS RESPONSE UNDER STATiC AND DYNAMIC HIGH PRESSURES 445

8. Interpretation of shock-wave propagation at atomistic levels5

In shock-wave experiments usually only two quantities are measured: the shock-wave velocity U, and the particle velocity Up due to the passage of the shock wave. Both these quantities are measured with respect to the uncompressed material ahead of the shock front. The interpretation of such data is based on two assumptions. First, the material compressed behind the shock front is assumed to be,in thermodynamic equilibrium. This means that stress and energy profiles of the shock wave are steady and enables thermally equilibrated region behind the shock front to propagate with the velocity Us. Second, the compressed material is assumed to have yielded completely so that the stresses may be assumed to be hydrostatic. An early three-dimensional (3-D) molecular dynamics (MD) calculation carried out by Tsai and B e ~ k e t t ~ ~ showed nonsteady behaviour and concluded that the RN jump conditions could not be used to analyse data from planar impact experiments. As the equations of motion solved in MD simulations explicitly obey the conservation laws, the test of the validity of RH relations in MD shock simulations is closely connected with the assumption of steady shock wave. In contrast to Tsai and ~ e c k e t t ' s ' ~ findings of nonsteady waves, Paskin and Dienes4' reported several MD calculations of shock waves in perfect Lennard-Jones crystals at nonzero initial temperature, where only steady waves were observed. A somewhat satisfactory resolution of these discoveries was attempted by Holian and straub4'. Their MD calculations of shock waves in perfect three-d~mensional soiids at nonzero initial temperatures reveal a transition in the nature of the asymptotic shock-wave structure as a function of shock strength. These authors concluded that the key to this transition from nonsteady to steady waves where RH relations are obeyed is the partial relaxation of compressive shear stress behind the shock front which accompanies small but perma- nent, transverse strains in atomic positions. Pulsed X-ray diffraction experiments in the past on LiF crystals with tens of nanosecond time resolution and shocked using explosive or gas gun drivers established that the compression of single crystals by shock waves is hydrostatic and therefore plastic on those time scales, but in a manner which somehow preserves single-crystal orientation4. However, later experimental measurements up to a few tens of kilobar support the view that the immediate response should be elastic and u n i a x i a ~ ~ ~ - ~ ~ . In these studies it was not possible to determine a transverse component of strain (although it was found in poly- crystalline samples of aluminium5') but the recent measurements of Whitlock and Wark5 reveal that definite compression was recorded in the inertially confined transverse direction ~ndicating the onset of plastic lattice response. It thus appears that steady shock conditions are attained for strong shocks or for weaker shocks at nonzero temperatures. ~ r i t z " using a relaxation time of about ps and a shock velocity of 7-10 kmls found that a typical sample thickness of 1 mm corresponds to 10 relaxation lengths. The MD calcuiations show that thermal equilibrium is established within 6 few tens of lattice parameters behind the shock fronti3. It is thus emphasized that at low pressures the time-dependent response of shock in condensed matter should be considered.

It is expected that there will be a large concentration of defects, like vacancies, dislocations, etc., in the shock-compressed materials4. In the description of propagation of

446 R. CHIDAMBARAM

shock in Meyers' models5-a modification of an earlier model by Smiths'-and ~ o g i l e v s k ~ ~ ' , dislocations are homogeneously nucleated at or close to the shock front by the deviatoric stresses in the material which in turn are relaxed by the generation of these dislocations. There is controversy regarding the speeds wilh which the dislocations move but the heating by them seems to be an important energy-dissipative mechanisms'. The dislocation density in recovered samples is typically 10"'-10" cmeZ at pressures up to about 10-20 GPa but seems to decrease in shock-loaded copper at still higher pressures due to increased residual ,temperature leading to annealing effects. The combination of residual and active measurements also seems to provide evidence for twical vacancy concentrations from 10.' tu 1 0 ' ~ in a pressure range up to few tens of GPa. The pressure correction needed for EOS data due to defects volumes is not well understood.

In the range of pressures from a few hundred kilobars to multi megabars the material behaviour is well understood using first-principles theories. In fact, incorporating their pred~ctions for total energy and pressure in RH relations has led to the determination of shock P-V curves in various materials in excellent agreement with experimental Hugo- niots measurements. The various predictions made by these theories are still being verified by current state-of-the-art experimentss%nd we suggest the reviews by Godwal et aL6' and Ross6' for details.

In the lower pressure region (hundred kilobars or so) the use of RH equations is best at suspect because of nonsteady shock. The use of perfect crystalline solids is far from reality as it is impossible to eliminate defect generation as the shock propagates. Al- though nonsteady shock behaviour is observed in MD sin~ulations the incorporation of defects in such attempts seems to be extremely difficult to know the changes brought about. Also the first-principles simulation based on coupled density MD approach lacks the required accuracy due to limitations on the number of atoms which can be simulated. Conditioned by these constraints we have thought of a model for shock propagation in solids in the regime of nonsteady waves45. We assume that an instantaneous response of materials to impact is uniaxial from its initial undisturbed state. We feel that relaxation mechanisms like phonon-mediated ones as noticed in MD simulations will drive the

, system to an equilibrium state which could be a nearly hydrostatically compressed state. The model is schematically depicted in Fig. 3. We assume that the difference in free energy between uniaxially compressed state and hydrostatic state appears as irreversible hekt and causes excitations of ionic and electronic degress of freedom and gives rise to shock temperature. In fact, we have carried out first-principles total energy calculations for Al using the energy band structure method. The computations were carried out in body-centered tetragonal (bct) phase to allow for uniaxial and hydrostatic states B and C as shown in Fig. 3. The details of such studies will be published elsewhere4'. Neverthe- less in Table I1 we compare the calculated temperatures with those obtained from the use of first-principles prescriptions for total internal energy and pressure in RH relations. The values seem to differ more for higher pressures. This is due to the fact that isotropic behaviour prevails at high shock strengths. Also our assumption of hydrostatic strain can be objectionable.

MATERIALS RESPONSE UNDER STATlC AND DYNAMIC HIGH PRESSURES 447

Time -+ Fro. 3. Schematic of shock compression model. Points A, B and C m free energy as tune graph represent uncompresscd, uniaxiaily compressed and hydrostatically compreasrd states

Table II Predictions of shock model for A1 ( T m e of relaxanon : pica second)

Pressure (GPa) Shack temperature (Degree K)

Nonequllibriurn Equilibnum model model

9.4 490 450 18 974 915 37 6 1044 980

9. Examples of phase transition

In the area of phase transitions a large number of studies have been made on different substances using various experimental tools mentioned above". The details of such studies have been discussed in various article^^^^^'. In the present article we will discuss a few examples for which the theory has provided the basic underlying mechanism and which otherwise is not available from the experiments.

Thorium crystallizes in the Fdce-centered cubic (fcc) structure at ambient conditions. Vohra and ~ k e l l a ~ ? using energy-dispersive X-ray diffraction with DAC found that i t undergoes an fcc-to-bct structural phase transition at 80 GPa (Fig. 4(a)). With the help of energy hand structure calculations, Rao et 0 1 . ~ ~ studied the 5-f band characteristic through the fcc-to-bct transition and analysed the angular momentum decomposition of occupied electron states, and the position of the 5-f band with respect to the Fermi level E,. We found that the 5-f band population increases with pressure, with the band con- taining more than one electron per atom at the volume fraction of 0.6. We also found from details of the energy band structure64 that this transition occurs when the bottom of the 5-f band falls below the Fermi level as shown in Fig. 4(b). Due to the proximity of this crossing to the fcc-bct transition, it is reasonable to conclude that the f-band occupation is central to this structural transition.

Several substances have been found experimentally to show crystal-to-amorphous phase transition under pressure. One such e ~ a m ~ l e ' ~ ~ ~ ' which initiated several studies on amorphization a2 Trombay is LiXS04. The EDXRD patterns of this compound ux~der increasing pressure are shown in Fig. 5(a-c). As the pressure is increased, a broad glass- like background increases with a simultaneous decrease in the intensity of Bragg lines.

- .: 1.5 d

fcc 1.0 0.8 0.6 0.L - .

Volume Fraction IVIV,,)

Volume fraction IV/V,) FIG. 4.(a) Axral ratto ( / a vr Viva for thorum Tlu: cia values correspond lo h a minimum or the total energy curves at varmus volume fractions. Circlet leprescnr experirnenral d d t ~ from Vobm and Ake11a6', (b) Posi- tion o f the 5-f band relatlve to Fermi cnergy Ef in thorium at various cornpres~ionb.

Energy (KeV) FIG. S(L+C) D~f iac t ion pattern, ol LxKSO, recorded at different prrssuiec. 5(d) i s recorded after thc rctea\e o f prenurc".

The diffraction pattcrn completely disappeared at about 13 GPa. The peaks which ale still seen are due to Au pressure marker and gasket. It is seen from Fig. 5(d) that the cryslalline form re-emerges on release of presaure after about 72 hours.

MATERIALS RESPONSE UNDER STATIC AND DYNAMIC HIGH PRESSURES 4 A 9

The cause of this transition is due to the fact that LiKSO, has a stuffed tridymite structure with three-dimensional networks built of six-membered rings of vertex-linked Li04 and SO4 tetrahedra with K in large open interstices. It is well known that coordination polyhedra, joined at vertices, have several configurations, with different relative orientations of polyhedra but with roughly the same energy. Also the possibility of different polyhedra having different compressibilities explains the abundance of phase transition caused by polyhedra tilting. This coupled with the fact that under pressure the tendency to have increased coordination, and kinetic constraints drive the system towards a kind of frustration resulting in amorphization. We refer to a recent reveiw article by Sharma and ~ i k k a " for an up-to-date account of the developments on amorphization.

Zn is unique among hcp metals, having unusually large cla axial ratio at ambient condition (c/o = 1.856). The anomaly in the variation of the axial ratio with pressure was first reported by Lynch and D r i ~ k a m e r ~ ~ around 7 FPa. They also reported an anomaly in electrical resistance in the same pressure range. Schulte et on the other hand, have found no anomaly in the axial ratio when studied using energy-dispers~ve X- ray diffraction up to a pressure of 32 GPa. The existence of anomaly in the axial ratio of Zn is thus controversial.

Meenakshi et ~ 1 . ~ ' calculated the change in the axml ratio of Zn under pressure from first-principles total energy calculations. The axial ratio at various compressions is obtained from m~nimization of total energy. It was noticed that volume dependence of the axial ratio changes slope around the relative volume of 0.92 (Fig. 6(a)). By careful analysis of the computed density of states as a function of compression, they came to the conclusion that the anomaly is related to the appearance of maximum in the density of states at the Fermi level.

Takemura6' has recently carried out angle-dispersive X-ray diffraction measurements on Zn for its axial ratio (c/o) variation with pressure using a DAC, synchrotron radia-

-' 1 c/o variation with VIV,, for Zn I

, ( 7

F!c. 6.(a) The cia varmion with VIV,, for Zinc The mild Ime IS cslcuiated fiom tolal energy calculations and is compared with ex erimenra? data iet\". (b) The failmg of energy level of L point In BZ below the Fermi level around i//K = 0.9'.

tion, and an imaging plate. It was revealed that the volume dependence of the c/o ratio changes the slope at V/V,, = 0.893. Also, in the recent past, Potzul rr ( ~ 1 . ~ " detected an anomaly in the Mossbauer spectrum of Zn at 6.6 GPa and 4.2 K. The anon~aly is accompanied by a drastic change or the lattice dynamics. The scalar-relalivistic linear- augmented plane wave calculation verified that the anomaly is related to the topological change of the Fermi surface around the L-symmetry point of the Brillouin zone. Thus the cause of the anomaly in the volume dependence of axial ratio is the electronic topological transition (ETT) as revealed in Fig. 6(b).

10. Symmetry systematics of pressure-induced phase transitions

The pressure increases the repulsive forces between neighbouring atoms and leads to an increase in the potential barrier. As the atoms have to overcome this barrier in order to diffuse it is thought that most pressure-induced phase transitions are diffu- sionless7'. Because of this, there exists a one-to-one correspondence between the positions of atoms in the parent and the product phases and is displayed in the interesting symmetry relationship. Based on this relationship, Gupta and Chidambaram: have classified pressure-induced phase transitions into four categories: (i) iso-symmetric transition, an example of which is alpha-to-beta phase change in molecular sub- stance resorcinol at a pressure of 0.5 GP~'~-", (ii) intersection group transition (for

Axial ratio (cia)

Fro. 7.(a) The fcc structure represented as a bct cell. (b) Total energy Eb,, a i thorum In the bct structure (relattve to that m the fcc phase), calculared as a function of axial ratio cia. The curves of various compresslona are as indicated in the legend.

MATERIALS RESPONSE UNDER STATIC AND DYNAMIC HIGH PRBSSURES 45 1

example, omega-to-beta transition7' in Zr at 32 GPa), (iii) order-disorder tran- ~ i t i o n ~ ~ and crystalline-to-amorphous phase changes65 fail in this class, and (iv) group- subgroup transition; for example, Th which undergoes kc-bct structural phase change63.64.

Thorium has a space group Fm3m, at ambient conditions. As discussed earlier, the high-pressure phase transition occurs to bct phase with space group I4/mmm. There is no volume change in this transformation. As seen in Fig. 7(a) the ambient fcc structure of Th can be represented in a bct unit cell with a, = a ~ d 2 , c, = a , and cJa, = 1.414, where at and c, are the lattice parameters for the bct phase. Figure 7(b) shows the behaviour of total energy as a function of axial ratio obtained by Rao et using the energy band method. The single minimum observed at normal volume shifts to a higher c/a ratio at high compressions. Symmetry systematics of various pressure-induced transitions observed by us under pressure are discussed by Gupta and Chidambaram''.

11. Predictions of phase transitions

The development of ab-initio total energy calculations and first-principles molecular dynamics simulations have been going through rapid expansions during the last decade, with the development of many diverse techniques to reduce the computational time^^'-'^. The aim of these calculations has been to predict accurately structures and related ground state properties of solids without using any information from the experiment. These include calculations of lattice constants of crystals; elastic, dielectric, and piezoe- lectric constants; phonon frequencies; pressures for transitions between different phases; and structures of complex crystals"-83. The other quantities not known experimentally have also been predicted, such as energies of nonequilibrium phases, the energy of solid along a continuous path connecting stable phases; eigenvectors of phonons; nonlinear elastic properties and structures of surfaces and interface^^^.^^. There are three main developments which have led to the growth of first-principles calculations. The first is the enormous increase in computational power which has made possible computations on real materials. The high accuracy achieved has enabled their detailed meaningful comparison with experimental measurements. Also, it has provided new tools for attacking the fundamental many-body quantum-mechanical problem. The second important development is the density-functional method for electron exchange and correlation. This method has made it feasible to calculate the above-mentioned ground state quantities with remarkably accurate results for real solids. In fact, it has become the starting point for almost all current first-principles calculations of total energies of solids. Many for- malisms like ob-initio pseudopotential method (AP)", generalized pseudopotential theory (GPT)'~, linear augmented plane wave (LAPW)" and linear muffin-tin orbital (LMTQ)" method are used for the calculation of total energies of solid as a function of pressure from details of band structure calcuiations. The recent development of techniques for direct minimization o i total energy using density-functional methodg0 further enhanced the speed of computation. Finally, there are significant new developments in experimental techniques and material preparation in ways never before realized. The most important advance is the ability to create multirnegabar pressures and explore the

452 R. CHIDAMBARAM

properties of matter over a wide range of d e n ~ i t i e s ~ " ~ ~ . In static high-pressure studies, the use of novel DAC incorporating excellent features of imaging plate two-dimensional area detector with rotating anode X-ray generator and at synchrotron facilities has enabled the detection of new phase transformations in solids at ultrahigh pre~sures~"~" . In the area of dynamic shock waves a new technique9' is used in which the phase transformation is inferred from the observation of a discontinuity in the measured sound velocity as a function of peak pressure in the shocked state. The detection is macroscopic in nature and the microscopic details of the transition are obtained either by comparison with the static measurements or from theoretical calculation^^^-'^'. The high-pressure data obtained thus provides a testing ground for the theory. In addition, there are new fields of research in which the electronic structures play a dominant role, yet are largely un- known experimentally. For example, semiconductor surfaces, where theoretical predictions have stimulated experiments and led to significant changes in understanding the nature of the surfacess5. Ab-initio total energy calculations have been very successful in predicting phase transitions under pressure prior to their confirmation by experiments. Silicon at ambient conditions is a tetrahedrally coordinated semiconductor with a diamond structure. In 1980, Yin and ohe en'^', using AP, showed that, starting with the atomic number of silicon, a precise estimate of the total energy of solid can be generated composed of specific arrangements of silicon cores of charge 4 embedded in a sea of itinerant valence electrons. The list of candidate structures used for various core arrangements consists of diamond, hexagonal diamond, white tin, simple cubic, bcc, hcp and fcc. Several other structures were also tried; however, the diamond structure was found to have the lowest energy. The calculated lattice parameter and bulk modulus (its pressure derivative) agreed with experiment to within 0.4 and 1 %, respectively. Yin and ohe en"' further showed that at reduced volumes (pressure 10 GPa) silicon should be metallic and stable in white tin form and in hexagonal close-packed structure at 40 GPa. Although the white tin transition was known, the stability of hexagonal close-packed structure was a prediction. Experiments were then carried out and the predicted structure was found at estimated pressure. When the calculations were extended to compute the lattice vibrational properties and the interactions of the valence electrons with the vibrat- ing core, it was found that simple hexagonal silicon should be superconducting at temperatures in the 5-10 Kelvin range and that hcp silicon would be superconducting around 4-5 Kelvin. This provided a successful prediction of the existence of high- pressure phases, their lattice constants, electronic and lattice properties, and supercon- ductivity. This was remarkable in view of the fact that the calculations required only atomic number, atomic mass and candidate structures as input. Many systems have been studied with similar success. For example, the simple hexagonal structure of germanium was successfully predicted to be stable around 84 GPa and the hcp structure around 105 GPa using total energy pseudopotential cal~ulations'~".'" and was experimentally ~erified"'~. McMahan and MoriartyIo6 predicted hcp-to-bcc phase transition in Mg at 50-57 GPa which was observed in DAC experiments at 50 GP~"'. The phenomenon of theoretical predictions and their subsequent verification by experiments was repeated in ~ r ' ~ and Pb'08. The theory has also played an important role in removing the controversies existing in various experiments. The interesting example i s found in Mo where the

controversy arising between static and dynamic high-pressure experiments is resolved from theory1'"

As part of our program to investigate materials for their phase stability at high pressures, variou.: studies have heen made and reported using energy band methods in the pastz2.'09. 111 the present article, we discuss a few recent examples which still continue to attract researchers.

Current first-principles calculations using ASA-LMTO~~ and full-potential LMTO"~ calculations show that Th is not an spd metal as believed earlier but kt has contribution to its bonding from 5-f band. This was verified by Rao et a1 "O by carrying out total energy calculations with 5-f electrons not contributing to metallic bonding by restrict- ing the muKin-tin-orbital expansion of the electron wave function up to angular momentum 1 = 2. This leads to the stabilization of the bcc phase at ambient conditions and continues to remain stable through the volume fraction of V/Vo = 0.4. Hence, Rao rr al."' were the first to show that the itinerant 5-f electrons were essential to stabilize the fcc structure. Thesc findings have been confirmed by Juhanrson and coworkers usmg full-potential LMTO calculation^'^^. The computed isotherm with and without inclusion of 5-f electrons is compared in Fig. 8 with the experimental data oC Vohra and Akella6'. It can thus be concluded that it is wrong to consider thorium as a traditional tetravalent d transition metal belonging to the same group of elements as Ti, Zr and Mf. The ohserved fcc structure and tile EOS is consistent with the view that Th should bc considered as an spdf metal belonging to heavier aclinide metal group (Pa-Pu).

The phase diagram of Ti, Zr and Hf has been of considerable interest in the recent past, both experimentally and theoretically. McQueen et al."' in shock experinlcnts noticed discontinuities in their shock velocity vs particle velocity plots at 17, 26, 40 GPa, respectively. Many speculations have been made to explain these discontinuities. McQuecn e l a1.l" assumcd that these were due to an alpha (hcp) to beta (bcc) transition, based on the observation of beta phase in shock-recovered samples of Ti. Kutsar and

6.0 1

454 R. CHIDAMBARAM

~ e r m a n " * associated these with the alpha-omega (omega, a :hree-atom simple hexagonal structure) as the omega phase was found by them in shock-recovered samples of Ti and Zr. carter1I3, on the other hand, thought that these could hc due to some electronic transition.

These controversies motivated Gupta et ol."%o carry out electronic structure calculations on these materials using ASA-LMTO method. For example, in Zr they correctly found the alpha structure as the most stable one at normal volume which continued to remain so till 5 GPa pressure where it transforms to an omega structure as compared to the experimental transition pressure of 2-6 GPa. Also, a new omega-to-beta structural transition at higher pressure (Fig. 9) was predicted. This was subsequently observed in static experiments at 30 GPa. Thus the cause of the shock discontinuity in Zr has been attributed to this transition.

Alpha-quartz form of SiOz exists at room temperature and up to pressures less than 7 GPa. With increase of pressure, it persists as a metastable structure, undergoes a transition to another metastable structure before making transition to an amorphous phase around 20 GPa (see Sharma and ~ i k k a ~ ~ for more details). The shock experiments show a break in the Hugoniot around 15-20 GPa and the recovered samples contain amorphous material. In shock waves there has been a controversy regarding the nature of the transition. In fact, it has been interpreted as alpha-quartz-Stishovite phase change.

In the recent past, first-principles electronic band structure calculation^"^ and molecular dynamic simu~ations"~ based on force fields derived from quantum mechanical studies have been made to understand the various aspects of this crystal to amorphous transition like change of bonding, coordination and driving mechanism, etc. Chelik- owsky et a1.115 and Di Pomponio and ~ o n t i n e n z a " ~ reproduced well the variations of VIVo, Si-0-Si and 04-0 angles as well as interpolyhedral 0--0 distance in alpha quartz as a function of pressure, respectively, using self-consistent pseudopotential method and LAPW technique. In fact, the interpolyhedral 0-4 distance has the short- est value of 2.7 .&. at 15 GPa in silicates. This is considered as the cause of the instability of the alpha-quartz structure because the compression beyond will be very expensive in energy. The calculated pressure vs ViVo curve by Somayazulu et al."' using molecular dynamic simulations is compared with experimental data in Fig. 10. 11 is seen that the agreement with the dara in the crystalline phase is excellent. At higher pressures, the EOS of the simulated amorphous phase is close to that of the shocked alpha-quartz. It is also revealed from Fig. 10 that the unloading path is much steeper which is similar to the observations of release wave measurements in the shock experiments on X-cut quartz.

The above examples show that theories play a powerful role in the predictions and interpretations of high-pressure phenomena. The introduction of linear band structure methods like LMTO and LAPW techniques for total energy calculations have enhanced their computational efficiencies by at least one order of magnitude. However, even when such methods are used, the calculations become extremely time consuming when the Hamiltonian matrix size increases beyond 300 by 300 (especially, when the periodic unit cell in the solid contains many atoms). Nevertheless, with the availability of supercom-

MATERIALS RESPONSE UNDER STATIC AND DYNAMIC HIGH PRESSURES 455

PRESSURE L GPa) 26.3 17.6 9.8 3 6 0

3.15 3.20 3.25 3.30 3.35

W.S. radius (a.u.) V/Vo

Fo. 9. Calculated stiuctural energy d~fferences for FIG. 10. Pressure-volume relation for quartz SiO, is Z P . shown with expenmental data"'.

puters and parallel-processing machines as an alternative, it is possible to carry out state-of-the-art electronic structure calculations for novel materials. In fact, we used the recently developed parallel-processing system at BARC referred to as BPPS"' to per- form electronic structure calculations on 0-Mgs2Zn4~ where O denotes the central site of packing unit, with Zn, Mg, Al and vacancy at 0. We employed the parallelized LMTO program"9 and the calculations in the bcc structure are related to 111 crystal ap- proximant to the Al-Zn-Mg quasicrystal. Based on total energy cal~ulations '~\t was found that packing units with A1 or Zn at the centers are more stable than those with empty centers, supporting the positron annihilation findings of Chidambaram and co- w o r k e r ~ ' ~ ' .

The density functional-based first-principles total energy calculations can determine the most stable structure from the chosen candidate structures, but might fail to locate the structure that has the lowest free energy. This problem has so far been attempted by intelligent search. However, using the density functional molecular dynamicsg0 approach it is possible to discover the equilibrium structure. Moreover, the Kohn-Sham equations- based electronic structure calculations can be applied mainly to pure crystals with ASA- LMTO and full-potential LMTO to be employed, respectively, for closed packed and less closed packed systems.

It is desired to have the first-principles electronic studies for complex materials in order to understand various physical phenomena. These include dynamics of materials, structure of solid surfaces and grain boundaries, structure of clusters, and defects and impurities in solids. In many of these complex tasks, the empirical potentials may not be reliable, or are difficult to construct. In the first-principles attempts to study some of these properties, for example, by employing supercells, the methods based on eigenvalue evaluation of the Hamiltonian matrix severely limits the system size. The number of ba-

456 R. CHIDAMBARAM

sis states required would increase linearly with the number of atoms, and the computational demands would increase at least as a cube of the total number of basis states. Hence, the first-principles molecular dynamic calculations would he impossible for more than ten atoms. The main drawback, inherent in the diagonalization and related schemes of finding matrix eigenvalues, is that one has to carry out the calculations for unoccu- pied electron states also, though not used anywhere in the self-consistent process. This may reduce the efficiency of computation by an order of magnitude. Alternatively, the direct minimization oftotal energy keeps track of occupied electron orbitals only. The basic idea is to consider the total energy as a function of all the wavefunction coefficients of occupied electron states, and ionic positions. Starting with any arbitrary set of trial electron orbitals, one can iteratively upgrade them to get the minimum of total energy. In principle, any minimization scheme can be employed, and the set of one-electron orbitals corresponding to the minimum total energy represent the Kohn- Sham orbitals. In order to study the dynamics of ions, their positions can be included as additional variables, and the structural relaxation can be treated as a global minimization problem, for example, by simulated annealing techniqueo2. Car and ~ a r r i n e l l o ~ ' treated the many-body quantum mechanical problem on the same footing by using coefficients of electron wave functions and ionic positions as though they formed a dynamical system and the corresponding phase space is explored by molecular dynamics simulations.

We have implemented the coupled density functional molecular dynamics scheme for simulations by the Car-Parrine!lo (C-P) method on B P P S ~ ~ ~ , " ~ . The program employs the easy space evaluation of various quantities in the calculations of total energy and force which leads to frequent transformations between reciprocal and direct spaces. Hence, the direct and inverse fast Fourier transforms (FFTs) are called often in each MD time step. Therefore, we parallelized the EFT routines and the results of a test simulation for Sn dimer were compared with Convex-220 machine to check the reproducibility on BPPS. By employing assembler-coded FFTs on BPPS with 8 nodes a speed up of 4 is achievedlZ4. The inclusion of soft potentials, commonly known as Vanderbilts potentials'", in C-P method and the use of separate heat bath for electrons have increased its applicability to complex solids and metallic ~ ~ s t e m s ' ~ " ' ~ ' .

First-principles simulations involving variable cells are currently being carried out to compute thermodynamic properties and phase stability of material^'^','^^. Attempts to locate the solid-liquid-solid boundary have been made for computing the change in physical properties of materials upon melting.

We have applied the C-P method to study the molten phase of sulphur and currently the first-principles simulations are being carried out to investigate the structure of its amorphous phase under pressure as there is controversy about its ~tructure' '~.

12. Conclusions

The extensive high-pressure data obtained using static and dynamic methods along with the development of theoretical means has contributed significantly to the understanding

MATERlAE S RESPONSF. UNDER STATIC ANI) DYNAMIC HIGH P R B S U K E S 457

of condensed matter physics. Thc need to have a proper agreement between theory and experiment has led to the incorporation of gradient c o r r ~ c t i o n ' ~ ~ , self-interaction terms and nonlocal effects"' in LDA. In facr, the generalized gradient correction of the exchange and correlation energy has been shown to correct qualitative errors of LDA in describing a high-pressure phase transition of Si02 in excclleut agreement with experiments"'. From the conclusions of recent reviewsn3~'" on rhls subject it is now well ac- cepted that the first-principles total energy calculations have proved to he a powerful means of providing valuable information on material behaviour under static and dynamic pressures. In fact, a long-ctanding goal of condensed matter physics, namely, prediction of phase transformations of real malerials From first-prmciples mici-oscopic quantum mechanical theory has been achieved. The theory has also been very successful in resolving controversies in different experimental data and on several occasions it has helped in reinterpretation of experiments. Some eflorts have been made lo generate the parameters of the empirical potentials by first-principles electronic stlucture calculations for the ionic configurations appropriate to MD simulations. The deviations of trajecto- ries obtained by MD and first-principles simulations for small sample size, for example, by C-P scheme, will indicate changes in the bonding characterisrics and help ro improvc the empirical potentials during the course of large-scale MD simulations. This is essential when making and breaking of bonds is involved or when core ~or~izarion occurs at ultrahigh pressures.

The precise DAC experiments ale eagerly awaited in the pressure range of 300 GPa to contirm the metallization of hydrogen. The optical and X--ray diffraction measurements with pico second time resolution shali provide an insight into the atomic- molecular processes governing the shocked state and because of the fast temporal nature of shock-wave loading, time-resolved measurements shall permit real-time examination of structural and chemical changes due to well-defined large conlpressions. The interpretation of such experiments by theory shall provide an insight into the microscopic mechanisms accompanying shock propagation in condensed matter. It appears that the goal of shock wave research to measure precisely the shock temperatures shall be ful- filled by finding proper sampie/window interface specially for opaque substances and with use of pm-second time-resolved optical spectroscopy135. With the availability of static pressures higher than those ar the centre of our planet, it is hoped that materials of geophysical interest, coupled with laser heating and synchrotron radiation, will be an active area of high-pressure research in the near future and will provide direct comparison between static and dynamic high-pressure data.

In short, research on substances at high pressures bas developed into an interdiscl- plinary area with important implication? for basic and applied sciences.

Acknowledgement

I am deeply grateful to Dr B. K. Godwal for numerous discussions on the contents of this article and for help in its preparation. I am also thankful lo Dr S. K. Sikka and other rncmbers of the High Pressuie Group; in particular, Drs S. C. Gupta and S. M. Sharma for helpful discussions.

458 R. CHIDAMBARAM

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J ladinn Ins t Set., July-Aug. 19LM.76. 4 6 5 4 7 5 0 Indian lnstllule of Sciencc

H. R. KRISHNAMURTHY Department of Physics, Indian lnstltute Of Science, Bangalore 560 012, and lawaharlal Nehru Centre for Advanced Scienlitic Research (JNCASR), Jakkur, Bangalare 560 061, India

Abstract

Interesting phenomena encountered in laser-modulated ci~llolds and recent progress in understanding them uslng density functional theory and simulational s t u d ~ s are r ev lewd Modulatmn of colloids by confinement, and other open questions are briefly touched on.

Keywords: Laser-modulated collards, laser-induced freezing, denslty functional theory, Monte-Carlo simulations of colloids.

1. Introduction

Colloidal suspensions' are systems of particles in solvents, with sizes much larger than atomic dimensions hut slill small enough that Brownian motion prevents their sedimen- tation due to gravity. Laser-modulated colloids are obtained by subjecting colloidal suspensions to standing wave patterns of an electromagnetic field obtained by interfering laser

The specific colloidal system' that I will consider in this talk consists of polyballs, which are spheres of entangled polystyrene chains, with a typical diameter 2R in the range 0.1-1 p. When suspended in water, the -KS04 end groups sticking out at the surface dissociate, leaving each polyball with a large negative charge Z*e (-1000 e ) . The cations released from the polyball and other (for example, salt) ions present in the sol- vent screen the Coulomb interaction between the polyballs, leading to an effective interaction, called the DLVO potential, given by'

Here K, the inverse (Debye) screening length, is given by:

( 2 )

where n, is the number density of the polyballs, and no, the number density of ions of type O. with charge z, which contributes to screening in addition to counterions.

"Text of lecture delivered at the Annual Faculty Meeting of the Jawahariai Nehru Centre for Advanced Sc~entific Research at Bangalore on November I I , 1995.

Beam splitter steering assernbl y

The crucial feature of the polyball system which makes it n wonderllrl experimental system is lbe fact that K can be tuned easily hy changing the salt concentration in the solvcnt or the volumc fraction of the polyballs: thus one can drive the system from d

weakly interacting ( K O , >> 1, where a , in the interparticle spacing) regime to slrongly interacting (small ~ a , ) regime. It is known' that in this process one can get phase transitions between liquid and crystalline (both bcc and fcc) phases of the colloidal system. One can even produce glassy phases with mixtures of polyhalls of difkrunt sizes'. Sub- jecting the system to laser modula~ions with wavelength about a, lcads to fascinating ordered structures of the polyballs, dubbed2 'optical matter'.

The typical experimental setup for generating optical matter is show11~ in Fig. I . The

dielectric susceptibility x of the poiyballs, given by the expression2 [(,I:-n$)/

( , I : / n : + z ) ] R ~ , where nl and n, are, respectively, the refractive indices of Lhe polyballs

(-1.58) and water (1.33L is large. The electric field of the laser modulation hence in- duces large dipole moments on the polyballs, leading to an effective potential

V,(rl) = - i X [ ~ ( r l ) ] Z on the ball (with its centre) at rl .V, is easily tuned to be compara-

LASER-MODULATED COLLOIDS 467

ble to or larger than the thermal or interaction energies. The ordered structures of optical matter2 then arise because the colloids! particles prefer to sit at the maxima of [.E(r)12. At the simplest level, for a potential of the form

v,(r1) = C < [ g j f ' ] exp[igjf).r] (3)

one will have an induced linear response in the density of the colloid given by

&p(r,) = z ~ ~ / , i t [ g ! ~ ) ] e x p [idf' .r] (4)

where S, is the structure factor of the polyball liquid. Thus, the polyball liquid develops modulations that mimic the potential. But more interesting is the nonlinear response that one can generate in the presence of a strong V,, strong correlations in the liquid, and a tuning of (gjf)) to the first peak of S, so that SE!,, is large. A typical example of such a

response is the phenomenon of laser-induced freezing which I discuss next.

2. Laser-induced freezing (LIF)

The phenomenon of LIF was first demonstrated by Chowdhury et al.' They subjected a 2-D polyball system (obtained by constraining the polyball system between glass plates such that only one layer of polyballs can he accommodated) to a 1-D laser modulation, with wave vectors g j f ) =+qo(O, I), with q , tuned to be at the first peak of the structure factor S, for the 2-D polyball liquid. They were able to demonstrate that turn- ing on the 1-D modulated V, generated a freezing of the polyball liquid into a 2-D triangular lattice.

It is not hard to understand this phenomenon as a nonlinear response in the context of a Landau-Alexander-Mctague In this theory the difference between the free energies of the liquid and the crystalline phases is expressed as a truncated power series in terms of the Fourier components of the order parameters at the smallest six reciprocal-lattice vectors of the triangular lattice (see inset of Fig. 2). One has, by symmetry,

PI =pa. p2 =p3 = ps =ps. Hence,

AF = ~ ( p : + 2 p : ) + ~ p , ~ i + C (p: +2p$)' +D ( p f +2p:)-2<pl (5 )

where the tuning of the wave vector of V , to q, makes it couple directly to the order parameter(~) p, (and p.,). For a typical set of parameters, B = -1, C = 112, D = 312, when

= 0, as A decreases there is a first-order transition from the liquid phase (with pi = 0) to the crystalline phase (with pl = pz # 0) at A 5 0.04. The full phase diagram obtained by minimising AF is shown in Fig. 2. The transition is continuous for large V,, which makes for a very interesting phase transition. However, whiie later studies by Loudiyi and Ackerson4 using direct observation and Monte Carlo simulations confirmed EIF, the nature of the transition was not explored carefully.

FIG 2. Phasc dragram in thc 1.andaii-Alexander- Eilctague theory. In the mudulared liquid phase, p, ;e 0, p. = il. In the (modulated) crystal phase p , + p, # 0. Pull line irldicatca a hoe of fira-order rranqitions, and the dashed line a linc of continuous

trans~lions. TCP, the lrtcrltical poim marked by x, \epulaIcs thc two. In .w The imallc\r s v e rlv of the trmnguiar lattice iilhelled 1-6 The rnodulatron V,, has wavc vectors 81 md 5.t =-g1 Their, by \yrnmctry, pi " pe, p2 = PI = PS = P,,.

3. Density functional theory (DFT) of laser-induced freezing

Thc DFT of freezing, pioneered by Ramakrishnan and ~ussoulf', has been extensively used' to study a variety of phenomena associated with the freezing of liquids, including defects in crystalsY, solid-solid interfacest0, phonons'l, etc. It has also been used, with considerable success, to understand/predict the phase diagram of the colloidal system from first prmciples'2.'3. So it is natural to attempt to describe LIF using this framework. harlier work in this direction".", using DFT plus the nonovcrlapping Guassian approximation fol- describing the periodic density modulations in the crystal, reached wrong conclusions about the continuous liquid 4 crystal transition suggested by the work of Chowdhury et al.? The conclusion was that "the symmetry gap between the fluid and the solid can never be bridged compietcly by rhe external constraint unless," (trivially,) "the external potential has the full symmetry ollhe solid."

We have d e ~ e l o ~ e d l ~ . ~ ' recently a DFT of LIF without uncontrolled approximations. We have shown definit~vely that, in the prescnce of an external potential 1.', of lower symmetry than the crystal, the transition from the modulated liquid to the (modulated) crystal can be made to change from first order to continuous wlth increasing V, via a tricritical point (TCP); but this happens only when a certain criterion (see later part) is satisfied by the wave vectors of V,. The wrong conclusion in the earlier worki4. was reached because the choice of the wave vectorc or V , used did not satisfy this criterion. Our DFT is not bound by the limitalions of the Landau-Alexander-Mctague theory (which, furthermore, as is clear from Fig. 2, has no stable crystalline phase for large V , ) . In addition, our DFT is a first principles theory. We have used it (see later part) to predict the parameters for the TCP and for obtaining continuous modulated liquid i bcc crystal transitions in the context of a 3-D colloid, subject to a carefully chosen 2-0 modulation.

To get a flavour of our DFT, it is enough to consider a simple version, whcre the order parameters {,, given by the Fourier components of the 'molecular field'

LASER-MODIJLATCD COLLOIDS 469

C(T) E ~ n [ ~ ( r ) l p ~ ] whcrc p(r ) is the per id ;c density in the rnndulaled liqllid or the crys-

tal, are dcterniined hy miniinising the I'm energy given (in the incompressible limit) by7, '

where

Here (0) are the reciprocal lattice vectors (rlv) of the crystal, and cg) is the direct correlation function of the liquid at G . The self-consistent equations for the order parameters obtained by minimising F, namely,

can bc solved essentially exactly numcrically for any finite set of ordcr parameters, even if large, using a technique developed by us''. ". Note that even iC is a finite set, all

the Fourier components pc of p(r), namely, pG =Ir'G.'e5('', are nonzero.

For purposes of understandmg the essential physics of LIF in 2-D it is enough to further simplify the theory by keeping only c$' -cia, i e., retain the order para-

meters only for the smallest set of riv. As is known'" the 2-D liquid freezes into a trian-

gular lattice when Vc = 0 with the smallest set o i rlv given by {g,(O)} = {(O, + l)qo,

( + & c ~ y , ) . Now, cons~der a I-D moduiated V , with wave vectors {e,(fl>={(~.+ l)qo).

Then the order parameters 4 ,,,, assume, by symmetry, only two values, kt for the nf(= 2)

veclors of the set {glf'}. and <,, for the nd( = 4) vectors of the complementary set

{gjd'} = {g,(O'}-{g~o} The free energy become?

where, $,(,-) - Z, cxp[ig:/'.p] and $,,(I.) - C, exp[igjd'.r]. Minlmising this and solving

the resulting self-consisten( equations, one gets ihe phase diagram16." shown in Fig. 3. There are two phasas: (i) the modulated liquid with <,t- 0 but <d = 0, and ('i) the

Titi. 3. DFT pliaic dhgiam ot a (mcornprcrnblc) 2-D system suh~ected to 1-U modulation, showng first- order (solid line) and eontinaous (il:whcd line) modulated hqusd i crystal tiltnst!ons. These are \epamtcd by the mcritm.l point (TCP). whicli is Ihc iirtrrseclion or lha Ti = O line and 7 ~ l i O lme (bee :ex:). Insel: Ti = 0 lmea of the truncated Landau thorics and that of the DFT The TCPz are marked by T (fourlh-order

!O mnca~ion), ' (8th- and 12th-order truocauons). and x (full DFT).

(modulated) crystalline phase with 5,# td # 0. As is known'', for V, = 0, the liquid i crystal transition is first order and takes place at cia = 0.857 (which corresponds to the

first peak height of the liquid structure factor S,,,,, = 7.14). Wlicn V,, is turned on, the transition remains Erst order, but moves to smaller valuea of r j2 ' , as indicated by the solid line in Fig. 3. Thus V , facilitates the liquid--crystal transition, which is the phenomenon of laser-induced freezing. This transition IS characterized by a discontinuous

change in 4, as we11 as by a discontinuous development of 5,). However, the jumps In

and 5, decrease with increasing V, and finally vanish at the TCP given by PV,, s 0.106

and cf" s 0.748. Thereatter, one has a coniinuous tranrition from the rtiodulated liquid to the (modulated) crystalline phase across the dashed line in Fig. 3. Qualitatively simi-

lar results are predicted by our theory for a 3 - 0 colloid, where, when V , = 0, thc liquid

freezes into a bcc crystal with the smallest rlv given by {gl"') ={(*I, *I, O)ql,/fi,

(0, &I, i l , )q , / f i . ( i - I , O ? r l , ) Now, if one turns on a 2-D modulation with wave

vectors chosen to be ig,(I)} = {(+I, i l , o)~,/&}, one gets a phase diagram qualitativciy

s~milar to Fig. 3 with the TCP at PV, E 0.22, cj2' z 0.55.

The general criterion for obtaining a TCP, and a phase diagram similar to that in Fig. 3 is that1"" the wave vectors of V , nust he so chosen thar c r y odd conlhinarion of

vectoi-s of the set {gjd'}= {g,("'}-{g:f)) cannot be written us un integer comhinatiorl of

vectors of the set {g,(f').

For, if this criterion i s satisfied, it can be verified easily by expanding the In@ term

in (9) that the Landau expansion for j?F in powers of 5<,, given k g 0 (with c, treated

nonperturbatively), has only even powers of &; the coefficients, which we can compute


numerically, are functions of five and c/Z1. As c;') and PV, increase, the coefficient of the second-order term in this Landau expansion, Tz, changes sign and becomes negative, leading to an instability with respect to the formation of &. In the region of the parameter space where the fourth-order coefficient, T4, is negative (but T6 is positive), the continuous transition is preempted by a first-order transition. The TCP thus arises when both T2 and T4 become zero. In the 2-D case, corresponding to Fig. 3, we also show in the figure the lines along which TZ = 0 (marked by dashes) and T4 = 0 (marked by small squares). To the left of the 7'4 = 0 line, where T4 is negative, the first-order transition

(solid line) preempts the continuous transition. The T4 = 0 line meets the T2 = 0 line at the tricritical point. In this way we obtain the precise location of the tricritical points quoted earlier. It is easy to verify that the above criterion is satisfied in the context of the modulations that we use. Note in particular that in the 3-D bcc case, a I-D set or any arbitrary 2-D set for { g j f ' ) picked out from {gjO'} would not satisfy the above criterion.

It is interesting at this point to compare the above phase diagram with that obtainable in the Landau-Alexander-Mctague t h e ~ r ~ ~ . ~ . ' . For this purpose we expanded the conventional Ramakrishnan-~ussouff7.8 density functional free energy in powers of the Fourier components of p(r)/p, for the wave vectors igjf'] and {g td ) } in the 2-D con-

text. We truncated the power series at different powers and found the phase diagram by minimizing the resulting free energy. We studied truncations up to the 12th order with the results shown in the inset of Fig. 3. In each of these cases we obtain a TCP as marked; however, the numbers are very different from, and converge very slowly to, those of the full density functional theory. More importantly, the continuous transition line eventually bends upwards for large enough V,, indicating that there is no stable crystal phase for large V , in these truncated Landau theories, as is also clear from Fig. 2. This is in contrast to the DFT result, where the critical line asymptotes to c/Z' E 0.5 for

large PV,.

We have also shown that if the above-stated criterion on the wave vectors of V , is satisfied, the TCP and the phase diagram are then robust with respect to the inclusion of more order parameters and of the effects of compressibility. For a more detailed discussion refer to Chakrabarti ct ol.'%nd chakrabarti17.

Finally, I would like to draw attention to our DFT calculation of laser-induced freezing in a real colioidal. For this purpose, we considered the same experimental system as Monovoukas and ~ a s t ~ ' for which the DFT phase diagram is known' in the absence of Ve. The liquid-state DCF for this system of charged colloidal particles with diameter 1334 A and surface charge 880 e was obtained using the rescaled mean spherical approximation of Hansen and ~ l a ~ t e r ~ ' . ' ~ for the modei DLVO potential in eqn (1). We focused our attention on the portion of the phase diagram where there is a first-order transition from the liquid to a bcc phase in the absence of V,. We took the modulation wave vectors { g j f ' } of the external potential to be along (+I, f l , ~ ) ~ o l f i as stated

. .

above. We did calculations retaining order parameters corresponding to 10, 20, and 50

472 H. R. KRISNNAMURTHY

0.006

0.005

0.004 0 +

0.003

CONTINUOUS 0'Oo2

0.001 0.0 0.05 0.10 0.15 0.20

1ST ORDER SURFACE

(a) PVe

Fio. 4. (a) Schematic phase dlagrarn of a ieallstic polybali suspension m the presence of an erlemal potential (b) Phase diagram for a realislic polyball suspension in thc +Pi/, plane for vvrmus values of n,. The sohd line j omng the open squares is the 1st order lme and that joinmg the filled circles is the continuous transition line. TCP denotes the tricritical point.

shells of the RLVs of the bcc lattice. We retained the three-body term^^^^ in the DFT in

the same spirit of Sengupta et al.12 with d3' = 0.23. In the 3-D parameter space of im-

purity concentration n,, volume fraction @, and PV,, the first-order and continuous transitions between the modulated liquid and the crystal now take place across surfaces, which meet in a line of TCP, as indicated schematically in Fig. 4a. We find that in the n,

range of (1.8-2.6) x mole/cm3, PV, is almost a constant -0.198 along the tricritical

line, as shown in Fig. 4b (compared with PV, = 0.22 obtained in the simplest theory).

4. Monte Carlo (MC) simulations of laser-modulated colloids

We have verified many of our conclusions discussed above using MC simulation studies of laser-modulated colloids, hut also obtained some surprising and interesting differ- e n c e ~ ' ~ . ' ~ . The simulations were conventional equilibrium MC simulations of 2-D poly-

balls with 2R = 1.07 p, n, = 1.81 x 1O7/cm2. The inter-ball potential was of the DLVO

form with & = 78, Z* = 7800 e . The laser-modulation potential was chosen to be - 43 ve cos(q,x) with q, = 2 n l ( a , a ) . The simulations were done in a -La, x(La,?) cell 2

with periodic boundary conditions for L = 6, 8, 10, 12 and 20, i .e . , for N = 36, 64, 100, 144 and 400 particles, with a (randomly distorted) triangular lattice as the starting configuration.

The results ~btained".'~ for the phase diagram in the (KUJ', PV, plane are shown in

Fig. 5. For PVe 5 0.2, the modulated liquid-lo-crystal transition is first order, and takes

(Modulated) ,*---- Cryslai

0- ! 0 I'

P'

place at a higher Ka, [lower (KU>)- ' ] for increasing SV,. This is clearly the phenomenon

of LIP. For higher valuc? of PV,, howevcr, the transition line bends back to lower (KO,)

[higher ( i r a ,%) - ' ~ values, i.e., to larger interaction strengths eventually saturating around

(m,J' : 0.1 1. The transition is also continuous for PV, t 0.25. The exlstencc of a cross-

over from first order to continuous transition with increasing PV,, and the stability of the

crystalline phase for large ( iruJ' , no matter how large is, are clearly consistent will1 our DFI' results.

However, there are two novel aspects to the simulational phase diagram (Fig. 5 ) as

compared lo the DFT phase diagram (Fig. 3). First, for . I1 >(?a,)-' > ,092, as in-

creases one gcts a laser-induced melting (LIM) transition! Second, for 0.066 c (KU,)-' < ,072, LIF is followed by an LIM transition to a reentrant-modulated liquid phase! These very interesting features deserve to be further explored using experimental, simulational and theoretical studies.

5. ModuPation of coLloids by confinement

There are other interesting ways to modulate colloids than by using larers; for example, by confining it between two glass plates with rcpulsivc charges on them. This leads to a layering phenomenon2%s the separation between thc glass plates is decreased. The tendcncy is especially strong when the separation is a (small) integral ~nultiple of the mean interparticle spacing a,,, as sketched schematically in Fig. 6. But each layer can bc liquid like within the laycr for large iru, (weak correlations between the particles.) For small KO,, i . ~ . , strong correlations, by analogy with what we have discussed, it seems clear that layering should he able lo induce crystalline order within the layers. This should result in a complicated and fascinating interplay between various competing ordering tenden- cies. Some glimpses of the possibiiitics are alrcady clear iii the work of Pieranski e! dZ3 where, in a wedge geometry, as a [unction of the ~eparalion the layers show alternately triangular and square ordering: 0 i lA i 2 O -+ 24 i .... .i nA i (n t 1 ) 13-, (n + I )

Ro. 6. Schematic diagram depicting the layering effecis in a colloidal system confined between rwu glasa plates.

A+ .... These phenomena need to be explored in much greater detail both experimentally and theoretically.

6. Concluding remarks

I conclude by mentioning other interesting issues in this area which I think are worth exploring in future research projects. The first obvious set of questions are about the critical phenomena associated with the continuous modulated liquid -+ crystal transition, in particular, its universality class. Of particular interest is light scattering from modulated colloids, especially the nature of the critical opalescence that will occur near the continuous transition from the modulated liquid to the crystal. Also for the 2-D case, complications arising from the two-dimensionality and the role of topological defects in this transition should be of interest. Another, largely unexplored class of phenomena has to do with quasiperiodic, incommensurate or random modulations. For example, can one get LIF f i e . , using laser modulations of lower dimensionality) into a quasicrystal? -

,I5 1- 1 / Specifically, can a 1-D modulation with .gif' ={(+I, O)q,z(+l, 0)q0 /z), where r 3 - 2

induce LIF into a pentagonal quasicrystal? Preliminary results from a Landau- Alexander-Mctague theory suggest that the answer is yes24: clearly this needs further exploring. Finally, there would be many interesting dynamical effects associated with the continuous liquid -+ crystal transition that would be worth exploring.

Acknowledgements

Much of what I have learnt about colloids has been through very enjoyable collaborative research projects involving Prof. A. K. Sood and Jaydeb Chakrabarti. 3 would also like to thank Sriram Ramaswamy, S.Sengupta, Chinmay Das, Rangan Lahiri and Yashodan Hatwalne for stimulating discussions. Computional resources for the research described


here were provided by the facilities at SERC, Indian institute of Science and the Com- puter lab at JNCASR.

References

BURNS, M. M., FOLIRNISK, 1. M, AND GOLovcnENKo, J. A.

CHOWDHURY, A , ACKERSON, 5. AND CLARK, N. A.

Louoln, K. AND ACKERSON. B I.

ALEXANDER, S. AND MCTAGIJE, J.

LANDAU, L. D. A N D LIPSHLTZ, E. M.

RAMAKRISHNAN, T. V. AND YUSSOUFF, M.

SrNGn, Y.

RAI LAKSHMI, M., KRISRNAMUKTHY, H. R. AND RAMAKRISHNAN, T. V.

SENOUPTA, S., KRISMKAMURTHY. H. R. AND RAMAKRISHNAN, T. V.

MAHATO, M. C., KRISHNAMURTHY, H. R. AND RAMAKRIS~~NAN. T V.

SBNGUPTA, S. AND SOOD, A. K.

SENGUPTA, S.

XU, H. AND BAUS, M.

BARRAT, I. L. AND XU, H. I.

CHAKRABARTI, J. , KRISHNAM~THY, H. R. AND SOOD, A. K.

CHAKRABARTI, I.

RAMAKRISHNAN, T. V.

CHAKRABARTI, J., KRISHNAMUKTHY, H. R. AND SOOD, A. K.

MONOVOIKAS, Y. AND GAST, A. P.

HANSEN, J. P. AND HAYIER, 1.8.

FEHR, T. AND LOWEN, H.

PIERRANSKI, P., SIREZELECKI, L. AND PANSU, B.

CHINMAY DAS AND KRISHNAMLIKNY, H. R.

In Solid stare physics ( E . Ehrenieich and D. Tornbull, eds), Vol. 51, p.1, 1991, Academic Press, and references cited therein.

Science, 1990,249,749.

Phys. Rev. Lett., 1985.55,833.

Ph)~sccaA, 1992,184, 26.

Phys. Rev. L e f t , 1978,41,702.

Statisrrcai physics Course of rheorerical physics, Val. 5, Part 1, 3rd edn, 1985, Pegamon Press.

Phys Rev B, 1979,19,2775.

Phys. Rep . 1991, 207, 351, and references cited therein.

Phys. Rev. B, 1988,37,1936.

Europhyr. Lett., 1994, 27, 587.

Phys Rev. A, 1991,43,4355.

P h y ~ . Rev. A, 1991.44, 1233.

Studies in the density functional theory of freezing, Ph. D. Thesis, Indian Institute of Science, Bangalore 560 012, India, 1991.

Phys. Len. A, 1986, 117, 127.

J . Phys.. Condens. Matter, 1990.2.9445.

Phys Rev. L a t , 1994, 73,2923.

Density funclivnol and computer simulation studies of colloidal suspensions, Ph. D. Thesis, Indian Institute of Science, Bangalore 560 012, India, 1994.

P h y ~ Rev. Lett., 1982,48,541.

Phys. Rev, Lett., 1995,75, 2232.

J . Coilozdlnterfoce Sci., 1989, 128. 533.

Mu1 Phys., 1982,25,651.

Phys Rev. E , 1995, 52.4016, and references cited therbin

Phys Rev Lett.. 1983.50.900.

Lnser-induced freezing of o colloid rnro decagonal quasicrystal, unpublished.

J Indian I m ? Sci., July-Aug. 1996, 76, 477-186 0 I n d m Institute oi'Scisnce

A. J. BASU* AND A. KHALILI ' *lawahar!a! Nehru Centre for Advanced Scientific Research, Jakkur, Bangalore 560 064, India, emall,nmrr@jnc rise ernet.in 'Max-Planck-Institute for Marine Microb~ology, 28239, Bremen, Germany, email:[email protected]. rmr-hwmen.de

Received on April 30, 1996.

Abstract

In this paper we look at the avlsymmetrlc flow of confined ma t ing fluid in a cylindrical container. The fluid is set in motion by the rotatlng bottom lid, and a vortex forms along the axis of the cylmder In some pieviaus analytical and experimental studies it was suggested that above a crmcal value of the upstream swirl angle a bubble-type vortex breakdown takes place along the axis, and below that it does not. However, m our present cornputatmns, where wc compute the swirl angle distributmn ewrywhcre in the flow-field, we find no evidence of strong dependence of axlsymmetric vortex breakdown an upstream swirl angle, since large sw~r i angles are seen even in cases where there is no vortex breakdown.

Keywords: Vortex breakdown, swxl angle

1. Introduction

Vortices exist in nature in nearly all conceivable spatial and temporal scales. From the quantized vortices in liquid helium (spatial dimension of about cm) to the galaxies (many light years in size), there are vortices in all sizes. They exist sometimes for a fleeting moment, but often for rather large periods of time. They could be life-giving (as in the case of the aortic valve, whose functioning depends critically on vortices', and at other times devastating (as in the case of hurricanes).

This dichotomy is also evident in the vortices that form over airplane wings; for example, in a delta-winged aircraft, two strong vortices are formed starting at the leading edge of each wing, and extending far behind the aircraft before they dissipate into the ambient. The parts of the vortices that are over the wing give additional lift to the aircraft, and stabilise it. But the trailing vortices behind a large aircraft can cause problems for smaller aircraft following it, as often happens in busy airports. Of great importance is the phenomenon cailed 'vortex breakdown" which is known to occur in such swirling flows; this refers to a sudden and explosive enlargement of the vortex core that is sometimes observed (for a review of the vortex breakdown phenomena, see ~ e i b o v i c h ~ ) . In particular, we will not want vortex breakdown to occur over the wings, but it is desirable

*Text of lecture delivered at the Annual Faculty Meeting of the Jawaharlal Nehru Cenire for Advanced Scientific Research at Bangalore on November I i , 1995.

178 A 1 RASU AND A. KHALSLI

that the aircraft trailing vortices undergo breakdown so that they do not turn into air- traffic hazards. Vortex breakdown is also of critical importance in awirling llows inside nozzles, diffusers and turbomachinery, in addition to tornadoes, etc., in nature.

This, ihen, is the primary motivation for studying vortcx breakdown. It was first reported by Peckhsrn and Atkinson3 in flow over delta wings al high angles of attack. There have been many subsequent sludies: analytical, experimental and numerical, and we will mention a few as we go along. There have been relatively fewer studies of vortex breakdown behind aircraft wings, presumably because of the difficulties involved. Vor- tex breakdown is known to occur in two main Forms: the bubble or axisymmetric type, and the spiral type4. We will be looking only at the bubble-type vortcx breakdown in [his paper.

The main analytical studies of this phenomenon can bc attributed to squirts, Benja- min6, and ~ a 1 1 ~ . An early theory of ~udwieg* which considered vortex breekdown to be a form of hydrodynamic instability has long been discredited, and we will not go into that here. Squire's Theory is based on treating the vortex breakdown phenomenon as standing waves, and he found that the swirl angie $4 (defined as the ratio of maximum swirl or azimuthal speed, to the axial speed in a rotating Row) is the most important parameter in determining whether vortex breakdown will occur or not. In h ~ s study involving in- viscid. axisymmetric rotating flow in an infinite domain, he found that a critical value for $between 45 and 50.2' exists for swirling flows where the axial velocity is constant, and the swirl angle bas the following distribution.

where r is the rad~al distance and B and C' are some constants. Benjamin, following a different approach involvir~g the critical state theory, arrived at the same conclusion about the existence of a critical upstream swirl angle. Hail's work consists of an analogy of the vortex breakdown phenomenon with boundary-layer separation, and, though interesting, we will not pursue it in this paper.

There have been some experimental verifications of Squire's theory involving the criticality of the upstream swirl angle. Harveyy did experiments in rotating pipe flow and measured maximum swirl angle ahead of the vortex breakdown bubble to be at 50.5', thereby validating Squire's theory. The distribution of the swirl angle measured just ahead of the bubble war found to be of the same form as analysed by Squire.

Axisymmetric vortex breakdown in rotating lid-cylinder geometry (Fig. 1) has received considerable attention ever sirice the flow-visualization experinlents of ~ s c u d i e r ' " who showed rather graphic examples of one, two and three vortex breakdowns as he changed thc rotation rate of the lid and the aspect ratio of the cylinder. In his experiments, a stationary cylinder is filled entirely by a viscous liquid, and a lid at one of the ends rotates at a constant angular velocity. What makes this geometry particularly interesting while numerically studying the vortex breakdown phenomenon is the preciseness with which the boundary conditions can be stated, and also that there are only two non-

AXESYMMETRIC VORTBX BREAKDOWN

Fro. I . A schematic of the vertlcal cross-sectmn of the rotating lid-cylinder geometry.

dimensional parameters governing h i s flow. There are excellent numerical simulations of this flow (most notably by Lopez1', among others) that are able to reproduce the experimentally observed features almost identically. In this paper, we attempt to put to test squire'ss theory about the existence of a criticai swirl angle by means of numerical simulations of vortex breakdown in this geometry.

Escudier'sl0 flow-visualization experiments initially received only a lukewarm re- ception as doubts were raised as to whether the recirculation regions seen in his studies were 'vortex breakdown' or not. Such doubts do not seem to exist anymore, but it just may be so that vortex breakdown as seen in this geometry is different from that in pipe flow, for example. Thus it is important to see whether such a critical value of swirl angle exists even in this flow.

2. Formulation and numerical scheme

In the following, we assume the fluid to be Newtonian and incompressible. The cylinder has a radius R and height H and is completely filled with some viscous fluid (Fig. 1). The lid at the bottom of the cylinder rotates with a uniform angular velocity a. The Navier-Stokes equations governing the flbw can be written in vector form as

which is to be solved along with the continuity equation

v.d=o.

480 A J . BASU AND A. IIHALIL.1

The pwmcte r s appearing in eqns (2)-(3) are defined as follows:

p = dcnsity of the fluid, p = nmlecuiar viscosity of the fluid, p = pressure,

? = the velocity vector, V = the gradient operator.

If we r~ondunensionalise using the reference length R and the reference velocity U = RSL, then we get the following nondimensional variables (denoted by *)

The nond~mens~onal form of the eqns (2)-(3) may then he wrltten as (atter d ropp~ng the *)

Here, Re is the Reynolds number defined as Re = pCW2ip. The Reynolds number. along w ~ t h the aspect ratio A =NIX, forms thc two governing parameter\ of this tlow.

Because of the geometry used here, we introducc a cylindrical coordinate system ( r ,

0, z) where r is the radial distance, 8, the azimuthal angle and 3 , the axial coordinate. Further, we assume the tlow to be axisym~nerric (which is a valid approximation for the moderate Reynolds numbers that we intend to study here), and so, using the equation of

continuity (5) we can introduce the streamfunction y, defined by

where u, v and ware the velocity compollents along I-, 0 and i, respectiveiy. Pressure can ROW he eliminated from eqn (4), resulting in the well-known 'vorticity-streamfuncrlon' equations

where 7 is the azimuthal component of vorticity def~ned as

AXISYMMETRIC VORTEX BREAKDOWN

so that dLV ! d~ dZv -rq=----+- a,-2 ,. a- azZ

The mirial conditions for this problem are

v, v, 7 = 0 at t < 0 for all rand z

v = TO at t = O for O < r < R , z = 0

and the boundary conditions are

1 y = 0, v = rQ, 17 =--- for O 5 r 5 R z= O

r rh2 '

The vorticity is evaluated at the boundaries lrom the computed solution of yr using a first-order accurate scheme for reaaons of stability". Equations (6)-(7) are solved using a high-order accurate compact finite-difference schemeI2. The scheme is up to s~xth- order accurate in the interior and up to third-order accurate at the boundaries. We use a second-order accurate central-difference scheme to solve the Poisson equation for y (8). A first-order accurate Euler time-integration scheme is used 20 integrate the equations of motion to steady state.

3. Restslts and discussion

In lhis ~ection, we look at the results obtained using the method above for fluid flow inside a cylindrical container (radius K = 1) with a rotating bottom lid. The fluid is at rest initially jt < 0) . At time r = 0, the lid is impulsively set in motion with a uniform angular velocity ( Q = 1 ) . An Ekman boundary layer develops on the rotating lid. This rotating boundary layer acts now as a centrifugal fan, throwing fluid radially outward m a spi- ralling motion and 'sucking' fluid into i t from above. A secondary meridio~lal circula- tion regime is then set up duc to thc existence of the solid walls. Abovc the lid, at first the fluid which is pumped out of the Ekman layer spirals up the cylindrical wall, estab- lishing a Stewartson layer until it reaches the top wall where it is turned and advected towards the central axis. The fluid then spirals down and is pumped back into the Ek- man layer.

In our numerical simulations, we have chosen an aspect ratio of 1.5, and three different Reynolds numbers, namely, Re = 1000, 1492 and 2000; these Reynolds numbers arc chosen because they lie, respectively, below, within and above the Reynolds number range where vortex breakdown occurs for A = 1.5 (see Fig. 7 of ~scudier" in this

A J BASU AND A. KHALILI

FIG 2 Temporal development ofslreainiunct!on contours in the flow at Re= 1492. Only sections rrom the central axis to the outw WBUS are SIIDWB. Tbe nond~mensiond tunes for the figures are. (0) 10, (b) 70. ( r ) 80, (4) 90, ( e ) 100, O 150, (8) 200, (h) 250, ii) 450. Them am 20 equlspaced contours befween y = 0 and 4 1 , and another 20 equispsced contours between y = 0 and 2 x 10.'. The positive strcamiines are shown ~n solid lines, wheiras the nagatwe ones are shown dashcd; thc zero streamlines *re shown by lung dashed liocs.

AXlSYMMETRlC VORTEX BREAKDOWN 483

context). These Reynolds numbers can be achieved, for instance, by vuriations of the angular velocity of the rotating lid. Thus, when we stark from rest and slowly increase SZ, flow develops inside the cylindrical container bul no voriex breakdown bubble can be seen till after Re = 1050 or so. As Rep-olds number is increased further, a bubble-type vortex breakdown occurs at the axis, but it disappears again around Re = 1900 or so. So, at Re = 1000, the breakdown bubble is yet to appear; at Re = 1492 there is a recircuia- tion region at the axis, and at Re = 2000 the bubble has disappeared. The Re = 1492 case was chosen specifically so that comparisons can be made with the visualizations of Es- cudier. The results presented here use a 61 x 91 uniform grid. Test computations using 81 x 121 grid showed no perceptible difference in any computed quantity.

We present the temporal development of the flow for the Re = 1492 case in Fig. 2. Since the flow is axisymmetric, in all the following figures we show only sections bounded by the central axis and the outer walls (equivalent to the region in the right half of Fig. 1). The streamfunctions are shown using equispaced contour levels on either side of iy= 0, so that the breakdown bubble can be seen easily (the flow in the bubble is very weak, and therefore will not be visible if the usual equispaced contours are drawn). Note that Fig. 2, etc., are not in uniform time-sequence; this was deliberately done so that important events during development of the flow can be captured. The breakdown bubble appears between (nondimensional) times t = 70 and t = 80. The bubble oscillates in size and shape (but never completely disappears) for considerable time before settling down to a steady state by t = 450, as can be seen from these figures.

In Fig. 3, we present the computed streamfunctions for the three Reynolds numbers after steady stares have been achieved in each case through numerical integration. Non- dimensional times to reach steady states are 250, 450 and 500, respectively, for the cases with Reynolds numbers 1000. 1492 and 2000 (a time-step of 0.025 was used in all

Fa. 3. Steady-state strenmfunctlon contours for: (a) Re = 1000, (b) Rc = 1492, and (c) Re = 2000. Sections from the ccntrai axis to the ourer walls are shown. There are ten equspaced contours each between p= 0 and -.01. and yi= 0 and 2 x 10.'. Positive contours are shown in solid lines, whereas the negative one are skown dashed.

A. J. BASU AND A. KHALILI

FIG 4. Computed distnhutioor <IF \wirl snglc $ ( m degrees) tn the flow Tor- (a) R e = 1000. (b) Re= 1442, (c) Rc = 2000. Secliona from the cenlrai axis to thc ootcr wails are shown The coniour levels are from $b = -90 to +90 a1 intervals of 10. Positive contours are shown In solid Ilnes, whereas the neganve one am shown dsshcd. The reglous between @ = -50 and -60 are shown filled m grey

cases). Figurc 3(b) for Re = 1492 can actually be compared with Fig. 3(d) of ~scudier ' s " flow visualization pictures. The similarities with the experimental flocv-visualization pictures confirm the accuracy of the present calculations. We can also see that there are perceptible waves in the flow-field for both Re = 1000 and Re = 2000, hut that thcre is no recirculation region at the axis (in conformity with Escudier's results). The recircu- lat~on zone in Fig. 3(b) is very weak, and has near-stagnant fluid as has been found in cxperiments too.

In Fig. 4, we have the computed swirl angle dibtributions over the entire ilow-field for the three different Reyuolds numbers. 'She swirl angle @ i s defined as

The regions between @=-50 and -60' are filled in grey in order to highlight them. From Squire's theory and also from various experiments in pipe flow, one should cxpect a region just ahead of the breakdown bubble where the absolute value of $ is greater than 50. From the figure, it is clear that though such a region for Re = 1492 exists, it is not close enough to the axis ol' rotation to be unambiguously called the 'upstream of thc bubble'. Moreover, one can see that such regions of high swirl exist even at Re = 1000 and 2000, and that these regions are closer lo the axis than that for Re = 1492. If the criticality condition of ~ ~ u i r b ' was necessary and sufficient for the existcnce of vorrex breakdown in this flow, one would expect no large swirl angles around the axis for Re = 1000 and 2000.

AXISIIMMETRIC VORTEX BREAKDOWN 485

radial distance 1 bubble radius FIG 5 . Cumpured dialr!bution of a w r l angle 4 ( ~ n degrees) at various dntar~ccs upstream of the vorren brcilk- down huhhle for R E = 1492

To compare with the results of ~ a r v e y ~ , we take a closer look at thc distribution of the swirl angle just upstream of the breakdown buhble. In Pig. 5, we present the distribution at different upstream distances as a function of the radial distance I-. Ar can be seen, :hc distribution (for small r) of the swirl angle g upsweam of the breakdown bubble changes very rapidly from the classical form given in eqn ( I ) close to the hubble, to a near-logarithmic ion111 a little distance away.

Thus it appears that a region of large swirl (swirl angle greater than 50") is not a cufficient condition lor vortex breakdown, although it may be a necessary one. W i l e this observation is vaiid for the rotating lid-cylinder geometry, we do not yet know whcther similar behaviour can be observed in swirling pipe flows or flow over deita wings.

4. Conclusion

We have computed the distribution of swir! angle over the e n t m flow-fieid inside a cy- hnfirical container with a rotating bottom lid. We lind rhat existence of ilpstrearn swirl angle greater than some critical value (as suggested by several investigators) is not a stfficicnt condition for the existence of a vortex breakdown bubble, even though it is liitely to be a necessary one. This finding polnos lo the poss~bili;y that the precise character of vortex breakdown may be very sensitive lo thc actual flow situation, especially on

486 A. L BASU AND A. KHALILI

the boundary conditions imposed, since the boundary condilions in the actual experiments (rotating pipe flow) and analytical solutions (infinite rotating disk: were different from that in the present numerical simulation.

Acknowledgements

The authors would like to thank Prof. Roddam Narasimha for illuminating discussions on this subject.

References

1. BBLLHOUSE, B. J. AND TALBOT, L.

2. LEIBOVICH, S.

3. PECKNAM, D. H. AND ATKINSON, S. A

4. SARPAKAYA, T.

5. SQUIRE, H. B.

6. BENJAMIN, T. B.

7. HALL. M. 6.

8. LUDWIEG, H.

9. HARVEY, J . K.

10. ESCUDIER, M. P.

11. LOPEZ, 1. M.

12. LELE, S. K.

I . Fluid Mech., 1969, 35, 721.

A Rev FIuidMech. , 1978,10,221-246.

Aaro R e p Coun. CP508, 1957.

J FluidMcch. , 1971,45,545-559.

Aero Depr ReportlOZ, IrnperialCollege, London, 1960

I . Fluid Mech., 1967,28,65-84.

A. Rev. FluidMech. . 1972.4, 195-218.

2. Fiugwos., 1962,10,242-249.

J . N u i d M e c h . , 1962,14,585-592

Exp. fluids, 1984.2, 189-196.

J N u i d M e c h . , 1990.22P.533-552.

J Compulationol Phys., 1992.103. 16-42.

J. Indian Insf. Sci., My-Aug. 1996,76,487-494. Q !ndian Inst~tute of Science

J. CHANDRASEKHAR Department of Organic Chemistry, Indian Institute of Sclence, Bangalore 560 012, and Jawaharlal Nehru Centre for Advanced Scientific Research, J d u r , Bangalore 560 064, India.

Received on April 30, 1996

Abstract

The potential of using computational methods to examine new molecular structural motifs is illustrated. Geomet- rical parameters, energetics, strain, electronic structures, frontier orbital separations, and esrimates of band gaps of polycyclic molecules with parallel sracks of C=C bonds have been obtained using AM$ calculations. Similar details have been derived for the cyclic pentaphenylene molecole, a fragment of Clo. The molecule is predicted to adopt a quinono~d form.

Keywords: Electronic effects, conformations. AM1 calculations.

Chemists have the privilege of creating, themselves, the objects of their interest. Usually, nature provides inspiration for the systems they choose to synthesise. Naturally occur- ring molecules and substances have enough variation and complexity to test the skills of chemists and often have the added advantage of having useful properties. Synthetic targets are also chosen on the basis of aesthetic appeal (e.g., dodecahedrane). The pleasure of being the first to make a molecule is undeniable. A chemist who is asked the reason for the monumental efforts expended in trying to make a highly strained molecule may well answer: "Because it is not there!" Everyday objects of the macroscopic world can also serve as goals (e.g., tinker toys provided the motivation for the synthesis of the series of molecules called n-staffanes).

There is increasing realisation that the scope of synthetic goals must be expanded. Insiead of trying to make specific molecules, the focus should be on properties. The de- cision of what to make should be as important as how to make. But how does one know the properties of a molecule before creating it first?

Computational chemistry of small molecules has reached sufficient levels of reliahil- ity to predict at least some molecular properties of general interest. The possible existence of a molecular entity in isolation as a long-lived species on the potential energy

T e x t of lecture delivered at the Annual Faculty Meeting of the Jawaharlal Nehru Centre for Advanced Scientific Research at Bangalore on November 11, 1995.

488 J CHANDRASEKBAR

surface can Re determined. The magnitude of strain energy with respect to related known nnolecules can be derived. Gcometries can be obtained to great precision. Key features of electronic structures can be elucidated. Many other moleclalar properties can also be cai- cuiated fairly accurateiy. The challenging problem of extending our knowledge of individual molecules Lo a collection of Lhcm in solution or in different periodic arrangements can also he approached using appropriate models. In effect, computational work can be used to design new synthetic targets. The methods can also be utilised to evaluate the thermodynamic likelihood of successful synthesis and perhaps cven the worthiness of thc effort.

The claims made above are not far-fetched. Many instances can be quoted in which chemically interesting predictions have been made using theory. The electronic structure of a complex system such as polyacetylene was debated and fairly accurately resolved by theoreticians' long before the material was made. The theoretical prediction of the existence and stability of Cso remained buried in a Japanese journal unsung, unhonoured and uncited for several years2. Occasionally, there is the happy instance of an experimental chemist who trusts an unusual theoretical prediction sufficiently enough to try and prove it in the laboratory. Two spectacular cases come to my mind; a dication derived from adamanlane with no hydrogen atoms a1 the bridgehead posilions3 and [ I . 1. l]propellane4. These remarkable structures were correctly predicted to be synthesisable. (It so happens that the experimental work was done by the same groups which did the cakolations.)

In this talk, I shall try to demonstrate thc usefulness of computational methods for analysing new structural motifs which are likely to have interesting chemical features. I shall discuss two classes of molecules. The first has a series of parallel C=C bonds. The 'second is a molecular fragment of C 7 ~ , whose possible independent existence as well as properties are evaluated.

2. Computational details

All calculations were done using the AM1 hamiltonianS with full geometry optimisation. Symmetry conslraints were imposed in a few cases to examine specific geometries of interest. A modified MOPAC package was used.

3. Wycyclic derivatives with stacked double bonds

The possible existence of allotropes of carbon was considered by Hoffmann and cowork- ers6 before the first report of Cso. While modelling various networks, the authors noted that parallel stacks of C=C double bonds (Fig. 1 ) have an interesting band structure. resulting from through-space interactions. Although the band gaps and bandwidlhs strongly depend on the inter-aikene distance, metallic behaviour was predicted for distances shortcr than 2.4 A. The conclusion was based on tight-binding calculations on stacks constructed from units such as A and B ~ .

It is indeed remarkable that through-space interactions can be as effective as direct conjugation in creating small band gaps. Experimental realisation of this idea is not straightforward since alkenes do not prefer'to stack in a parallel manner. AM1 caicula-

NEW STRUCTURAL MOTIFS

Fa. 1. Model of stacked olefines and structures A and B considered in Lhls arrangement.

tions on units of 5 or 10 ethylene molecules reveal strong repulsions between the closed shel! molecules, especially at the more interesting shorter distances. In principle, the ideas of crystal engineering can be used to align molecules in the desired fashion with appropriate substitution. Unfortunately, our control over noncovalent interactions in a large molecular assembly is far from complete. Further, large perturbations which lead to the preferred mode of packing may destroy the desired band structure.

An alternative is to hold the C=C units in place by covalent linkages in a poiycyclic framework. Two possible series of molecules, 1 and 2, appear attractive in this regard. The first series has cyclobutene rings fused to a ladderane skeleton (Fig. 2). The second set of molecules represent larger homolcgues, with endo-fused cyclobutenes on a norbornyl framework (Fig. 3).

As ma) be expected, the ladderane-fused system is highly strained. The computed hear of formation of Pa is 170.8 kcal/mol. Each subsequent expansion of the skeleton leads to a large increase of energy. Interestingly, the increase is roughly constant at 141 kcalimol, on going from l a to Ib 10 Ic and on to Id. On thermodynamic grounds, it wiH be a challenging task to make this series of molecules. But the goal is not altogether impossible, since ladderanes functionalised on the same side are already known8.

PIG. 2. Slruclures of stacked olefines covnlenrly linird to a ladderant framework

The important structural feature of 1 is the inter-double bond distance. In la , the calculated value is 2.83 A. The value increaser marginally in the larger derivatives. In both Ic and Id. the outer double bonds are 2.84 A apart. while the inner double honds have a separation of 2.87 A. These distances are in the region in which the band gaps are fairly large. The computed a MOs in the largest model :d are spread over a range of 1.5 eV,

2 2e Fio. 3. Structures of stacked olefines covalently linked to a norboinyi framework.

NEW STRUCTURAL MOTiFS

while the rt* MOs are in a range of 0.8 eV. The HOMO-LUMO gap is 9.5 eV, while the estimated So-S, transition energy using limited CI is around i eV.

T h e strai:~ in the iadderane series is evident in large bond lengths in the fused cy- clobutane rings. In the !arger derivatives, these distances are as large as 1.68 A. As a result, many s MOs interact very strongly with the a MOs, both in thc filled and un- filled manifolds.

492 1. CHANDRASEWIAR

The fused norbornyl derivatives are more interesting from several points of view: They are far less strained and hence must be amenable to synthesis on thermodynamic grounds. Elaboration of the skeleton in 2 increases the heat of formation in the range of 115 kcalimol. More importantly, the C=C bonds are much closer to each other. While the distance is 2.72 A in 2b (due to strong filled orbital repulsions), it becomes succes- sively shorter in the larger derivatives. The inner double bonds in 2e arewithin 2.48 A

\of each other. This reduction in distance is not a consequence of any attractive interaction between the alkenes. It is more a result of the angular preferences in the ffised norbornyl skeleton. The double bonds lie on a curved surface (Fig. 4). From the computed curvature, the cycle is expected to come full circle in about 44 units! Interestingly, the double bonds of the ladderane series, 1, are curved in the reverse fashion (Fig. 5). A strategy to construct a very long polymeric chain of roughly parallel C=C double bonds is to intersperse the two structural motifs 1 and 2!

The shorter inter-olefin distance in 2e leads to a large spread of then and n* 'hands'. They vary over 3 and 1.5 eV, respectively. The widths are comparable in magnitude to those computed for long conjugated chains of C=C bondsg. The frontier orbital separation is 8.3 eV in 2e. An estimate of the 'optical hand gap' obtained via limited CI calculations is around 3 eV, again comparable to that calculated for linearly conjugated poiy- enes. Based on computed and experimental results on models of polyenes and polyacety- Pene, the band structure of a large oligomer of 2 is likely to be similar to that of polyacetylene. Hence, such systems would be attractive synthetic targets as a model organic semiconductor, although perhaps not as a metal.

4. Cyclic pentaphenylene

The molecular and electronic structure as well as many aspects of chemical reactivity of C7' can be understood in terms of a simple bonding picture. The molecule can be visualised as being built of two corannulene rings held together by a central pentaphenyl belt". Corannulene is a well-studied aromatic molecule. But the properties and even the possible existence of the central unit as an independent molecular entity has not been explored. The closest systems considered are beltenes, with edge-fused hexagons1', and polyphenylenes, both as oligomers and as an infinite chain.

The optimised structure of cyclic pentaphenylene, (C&)s, 3, is remarkable in many ways. The D s ~ symmetric structure is a true minimum on the potential energy surface. The benzene rings are highly distorted to boat forms (Fig. 6). The estimated extra strain relative to linearly fused pentapheny19 is about 100 kcal/mol. The approximate diameter of the ring (6.7 A) is quite similar to the equatorial diameter of C70 (6.9 A). However, 3 is unique in one respect. The computed bond lengths reveal that the benzene rings are in the quinonoid form. While the two central C-C distances in the ring are 1.35 A, four other ring distances are 1.46 A. Further, the inter-ring distance is 1.36 A. In contrast, the rings have more uniform bond lengths in the pentaphenyl belt of C70 as well as in poiyphenylene at the same level of theory.

The preferred quinonoid form leads to a small gap between the frontier orbitals in 3. Previous calculations have shown that polyphenylene has indeed a small hand gap if it

NEW STRUCTURAI. MOTIFS

adopts rhe quinonoid form". Geomelry optimisations have also revealed that the corresponding form does not exist as an independent minimum for polyphenyiene9. The la]-ge angular distortion in the cyclic structure reverses the geometric preference.

Cyclic pentaphenylene is expected to have interesting niolecular properties. Inter- molecular interactions are also likely to he significant in this system. 41 would be an excellent, though somewhat difficult, synthetic target. It may be visualised as perhaps the smallest possible molecular tubule. Planned synthesis of larger cyclic polyphenylenes woilld also !x of' interest.

Acknowledgements

This work would not have been possible without the enthusiastic participation of Nisha Mathew, M. N. Jagadeesh, A. Rathna and Anindita Makur in the computational work.

1. CHANDRASEKHAR

References

1. SALEM, L.

3 BREMER, M., v a n SCRLEYER, P. R.. SCHOR K., KAUSCH, M. AND

SCHINDLER, M.

4 WIBERG, K. B. AND WALKER. F. H

5. DEWAR, M. J. S., ZOEBISCH, E. G., HEALY, E. F. AND SEWART, J. J. P.

6 HOFFMA~N, R., HUGHBANKS.T. AND KERTESZ, M

7. MERZ, K. M., HOFFMANN, R. AND BALABAN, A. T.

8. MEHW, 6.. VISWANATH. M. B., SASmY, 6. N., JEWUIS, E. I)., REDDY, D. S. K. AND KUNWAR, A. C.

9. CIU, C. X., KERTESZ, M. AND JIANO, Y.

11. KOHNKE, F. H., MATHIAS, J.P.AND STODDARD, I. F.

The mu!rcirLr orbi rd rhenr? 01' l ' w j u ~ o l r d r~armm!.~, 1966, Benjamin.

K a ~ a k a (Kyoto), 1970.15, 854-863: Cl i rm Abslr . 197 1, 74, 7.5698~.

A t i ~ e ~ . . Chem lnr . Edn En$ , lYX7.36,761-763

J Am Clicm. Soc.. 1982, 104,5239-5240.

I . Am Chem. Soc , 1985, 1lJ7.3902-3900.

.I. Am. Chcni Sor , 1983. 105.4831-4832

J Am. Cheni. Soc , 1987,109,6742-6751.

Anpex,. Chsm 11zf. Edn Engl . lYY2,31,14XX-1489

J Phys C h o n . . 1990.94, 5172-5179.

Fallerme So. Techno!.. 1995.3,hRl-705

Ad?. Morer, 1989.28, 1103-1110.

. I . Phys Chrm. , 1992.96, 679-685.

J . indran ins1 Sri.. Juiy-Aitg i Y Y 6 , 76, 495-504 D lndlan Institute oiSclencc

MADHAV GADGIL Czr~tic far Ecoioglcal Scicnccs. indian inirilote of Sclcncc, Hangaiore 560012, Indid. and Bmdiveriity Unit Jawaharlal Nehru Cenlrc for Aduan~cd Scient~fic Reacaich. lakkur. Bangdore 560 064. Indin.

Abstract

Yhc patterns of distribution of biological d~versity ovcr the hill chain of Western Ghats are moulded by manifold natural factors as well as human interventions. There are many surprises in store when one looks at them in detatl-for instance, the inverse conelatian between bird and woody plant spews diversity in the Uttara Kannnda disuict. Humans have not only extensively transformed natural communities into plantations and pat lo other uses, but also have greatly depleted biomass and divers~ty levels in areas remaining under forest cover. The extent of such depletion of biomass may be about 50% over the last half century for Uttara Kannada. This has also led to lass of specles, especially the mole delicate eve~gieen species incapable afcoppicing.

Mountains are the treasure troves of biological diversity in the world today for a variety of reasons. Their topography promotes environmental heterogeneity. The annual rainfall, for instance, ranges from as much as 8000 rnm in the southwestern corner of upper Nilgiris to a mere 500 mm in the Moyar gorge just 30 km to its east. in contrast, the annual rainfail spans a range of nq more than 800 to I000 mm over hundreds of kilometers across the Deccan plateau. Mountains also create isolated habitats far away from other similar habitats, promoting local speciation. Thus distinct species of Rhododendrons occur on the higher reaches of Western Ghats and Himalayas, with a large gap in the disisibutjon or the genus in hetween. Finally niounrains are less hospitable to human occupation and therefore retain much larger areas under natural or semi-natural biological communities. This ia why the Western Ghats and the Eastern Himalayas are today the most significant repositories of India's biodiversity1.

The Western Ghats and Eastern Himalayas are regarded not only as biological treasure troves. hut are also considered as two o i the world's 18 biodiversity hot spots'. The hot spots are defined as biodiversity-rich areas that are under a high degree of threat. Indeed the landscape of Western Ghats and Eastern Himalayas is today being rapidly transformed in many ways leading to an erosion of their biodiversity. Given the current imperative of conserving world's biodiversity, it is imporrant to understand the lifescape, i.e., patterns of distribution of biodiversity-over these mountains, and the processes underlying ongoing changes in these patterns. Along with several collaborators and ,,I

496 MADNAV GADGIL

50 Distance

FIG. 1. An east-west crass section through the Uttara Rannada district indicating topography, annual rainfall and natural vegetation types. Also indicated are mean and standard deviation of numbers of woody plant species in 2400 mi belt transects.

students, I have been investigating the lifescape of the Western Ghats over the last two decades. This article reviews some of the recent results, based in particular on the work carried out jointly with R.J. Ranjit Daniels, M.D. Subash Chandran, Shri Niwas Singh and N.V. Joshi.

2. Diversity gradients

Levels of biological diversity vary along many gradients; for instance, as one passes from temperate regions towards tropics, or from islands to continents. Three kinds of gradients are well known for levels of diversity of flowering plants along the hill chain of Western Ghats'." The diversity of plant species increases as one travels south across the 1600 km length of the Ghats from river Tapi to Kanyakumari. This has been related to the southward increase in the number of rainy days. Diversity of plant species also increases as one crosses the Western Ghats from the east to the west, in conjunction with an increase in the total rainfall. Figure 1 depicts the results of a study of 38 belt transects of 600 x 4 m, in the district of Uttara Kannada (13'55'N-l5"32'N lat and 74'5'E to 75'5' E long) near the centre of the Western Ghats. These transects sample dry deciduous, moist deciduous and evergreen forest types along gradient of increasing rainfall. The number of woody plant species in the 2400 m2 sample increases from 23.22 i 4.2 to 36.33 i 13.16 to 50.3 ? 5.97'. The third known gradient is the increase in the number of plant species with an increase in the mean temperature from higher ele- vations to the coastal plains.

But our studies have revealed further complexities. DanielsS censussed birds by walking along the centre of the 600 m long transect strip at a pace of -10 m every 2 min during the morning hours of 0800-1000. A limit of 100 m on either side was set to rec- ord birds sighted and/or heard on the transect. Birds thus recorded on these systematic transects represented about 3040% of the total bird species in a locality. As mentioned above, all woody plants were also surveyed along a strip of 4 m width. All individual

WESTERN GHATS : A LiFESCAPE 497

Table B he correlation matrix of attributes of 38 samples of natursi vegetation in Uttarra Kannada district. Sim- ple correlations are to the left Of and below the diagonal, partial carrelstions to the right of and above the diagonal

.----;3 Partial conelations

Woody p l Woody pl Vertical Canopy CV of Tree Bird species species species strofificafion density canopy denslfy richness richness diversity dcnrirv

Woody plant - 0.73* 0.28 0.53 0.57* 0.06 0.17 sp. richness Woody plant O X * - -0.09 -0.i4 -0.29 -0.06 -0.35' sp. diversity Vertical 0.52* 0.44' - 0.00 -0.27 -0.06 -0.24 stratification Canopy 0.70* 0.62* 0,56* -0.64* 0.46* 0.15 density CV canopy -0.3 i -0.37* -0.49" -0.77* - -0.14 0.07 density Tree density 0.56* 0.49* 0.45* 0.83* -0.69* - -0.28 Bird species -0.31 -0.44* -0.39* -0.36' 0.37' -0.43* - richness

Simple correlations f- * and' indicate that values of simpleipartial correlation coefficients are srgnifieantly different from 0 at p < 0.01 and 0.05 level. respectively.

plants were identified and assigned to one of the following seven height classes: 0-1 m (seedlings), 1-2 m (shrubs), 2-4 m (understory) and several canopy layers at 4-8, 8-16, 16-32 and > 32 m. These height categories were used to assign vertical stratification to the vegetation. Canopy density was recorded every 5 m at 120 points as follows : 0 when there was none overhead, 1 when canopies from adjacent trees barely met, 2 when the canopies overlapped with the sky still showing through and 3 when the sky was no longer visible through the overhead foliage. Density of trees above 30 cm girth at 130 cm above ground was estimated in 10 quadrats of 10 x 10 m at 50 m intervals.

Woody plant species diversity was computed as eH, where H i s the Shannon-Weaver

index-4 p,.lnp,, p; being the proportion of plants belonging to ith species. Vertical stratification is computed as 112 p:, where pi is the proportion of plants in the ith height class. Canopy cover is the average of canopy density scores. Tree density is expressed as the number of trees with a girth > 30 cm at breast height per 1000 m2. As Table I brings Out, bird species richness is negatively correlated with woody plant species diversity as well as with vertical stratification, canopy density and tree density. Bird species richness is however significantly correlated positively with the coefficient of variation of canopy density. Since the various attributes of vegetation strongly correlate with each other, we performed a partial correlation analysis to determine the relative significance of different vegetational attributes with path analysis technique. This analysis reveals that the predominant influence is that of woody plant species diversity. The more diverse natural

vegezalion supports a lower number of bird species, while the bird species richness in- crcascs with patchiness of the tree cover.

The explanation for the lower level of bird spccles ilchness in the structuraily anore complex and diverse vegetation may lie in the fact h i the less complcx and diverse, drier, vnorc deciduous vegetation has been more effectively coloniscd from 3 pool of bird species of the adjacent tracts of peninsulas India. The evergreen forests of Western Ghats, on the other hand, constitute a relatively reatrictcd habital 14and (64750 km') at a considerable distance (1500 km) from the large tracts of the evergreen vegetation of Eastern Himalayas and Southeast Asia. in consequence, the evergreerl forest bird fauna of the Westcrn Ghats is rather impoverished, especially with respect to largc-sized birds, fruit eaters and above all sedentary insectivores such as babblers and laughing thrushes.

3. Impact of man

These natural gradients of biological diversity are today greatly affected by a whole range of human interventions. These include overharvest of natural vegetation leading to a rcdnction in standing biomass or regression to lower successions? stages. as well as conver~ion o r natural vegetation to man-made plantations, cultivalion or aquaculture. Shri Niwas Singh has attempted to systematically sample the ongoing lransformnlions in Kumtn and S m i taluks of Uttara Kannada district along a series of transects. Figure 2 summarizea his conclusions. Along the coast there ia a tendcncy lo convert sand dunes Ir)

Fra 2. Ongoing piitrernr ot iandscape rranifnrmat~on In Uttaril Kannada d:stncl

WESTERN GHATS : A LIFBSC'APE 499

coconut orchards either dircctly or via paddy crtltivittion. At the same t i n e mangrove forests have been giving way lo brackish water paddy cultivation. and more recently prawn farms. In the hilis the characteristic Myistica swamps, a special vegetation asso- ciation of narrow wet valleys is giving way to nrccanut orchards, either directly or via paddy fields. The natural evergreen l'oresr is being opened up and reduced to scrub and then converted to Acacia u~o.iatlij%rniis plantations by the forest department. in more exposed localities it may end up as a laterite hardpan with scanty shrub and grass growth. Where land is under private control the evergreen ibrest may be converted to so- called betta or bena lands. Retta lands are maintained under more open, deciduous tree growth to supply leaf manure to arecanut orchards; bena lands are maintained as grass- lands for grazing livestock through annual fires.

But human impact on bird communities is less drastic. Daniels undertook 20 transects in man-made plantations along with 38 transects in the more natural vegetation (Table 11). His studies showed that the plantations were poorer in number and diversity of plant species, in vertical stratification as well as in canopy density. However, planta-

Table I1 Attributes of natural vegetation (n = 38) and man-made plantations ( n = 20) of Uttara Kannada district

Atrribute Vescrution type Mean SD Minimum Marimum Smtasticui rlgnificnnce

No. of woody Plantations 11.89 5.99 1.00 23.00 S plant species* Natural vegetation 40.58 13.76 17.00 64.00

piant* Plantations 3.70 2.08 1.00 7.90 S species diversity Natural vegetation 15.52 7.57 2.50 32.70

Vertical stratiti- Plantations 3.26 1.24 1.50 5.69 S

cation Natural vegetation 4.34 0.85 2.25 5.59

Canopy densay plantations 1.23 0.57 0.26 2.21 S

Natural vegetatlan 2.85

C. V. of canopy planfalions 0.66 0.38 0.26 1.69 . NS density Naturhl vegetation 0.57 0-27 1.69

Tree dmsit? plantations 114.44 94.37 21.00 305.00 s N~~~~~~~~~~~~~~~~ 56.87 23.99 23.00 119.00

Bird species Plantations 17.33 4.84 10.00 31.00 NS richness Natural vegetation 18.66 7.10 4.00 31.00

- ~p -~

S: Plantations andnatural vegetation differ significantly a tp < 0.01 NS: Not significant by t test *per 2400 rn2 + eX', H'= Shannon-Weaver diversity index

IiEp,', where p is the proportion of individual plants in the ith height class Mean of 120 points scored as 0, I , 2 and 3 Coefficient of vanatian for the 120 scores per 1000 m2: plants 2 30 cm GBA

Per transect 600 x 200 m. Plantations end natural vegetation are significantly different at p c 0.01 (t-test) not significant

500 MADHAV GADOIL

rions had higher tree densities and mere patchy canopy. But notably enough there was no significant difference amongst the plantations and natural vegetation in the richness of the bird species. This may be related to the fact that the bird communities of the plantations are very similar to those of drier, more deciduous forests which harbour a richer species pool6.

4. Response to disturbance

One of the least understood aspects of ecological change in India is the slow attrition of . biomass in biological communities retaining forest physiognomy. In the district of Uttata

Kannada, for example, the forest cover has declined from around 70 to 60% of laud over the last 50 years. There is however no assessment of the change in levels of biomass or diversity in the 60% of the remaining land that continues under forest cover. We have examined from this perspective 30 of the 600 x 4 m transects located in natural evergreen forest sites7.

When plotted in the phase space of number of plants per transect and the proportion of deciduous plants, these 30 transects segregate into two clusters of 20 and 10 (Fig. 3). It turns out that the cluster of 20 points includes sites significantly further from the nearest villages, with fewer foot paths, and lower incidences of grazing and fire. The

Distance from nearest village

0

J 0 0.2 0.4 0.6 0.8 7.0

Proportion of deciduous plants

FIG. 3. Dauibution of low (4 and high disturbance (e) mnsects according to the fraction of individuals of de ckiuous species (x-axis) and the rota1 number ofindividual plants (y-axis) in each uaoseca.

WESTERN GHATS : A LIFESCAPE 501

Low t i i ~ h

0.213 0.32Ra 0.2* 0.171 0.242"* 0.148 0.131 0.11 0.153 0.184 0.058 0.059

-0 0

Low High

Species richness 48.8 i 5.6 35.7 -L 11.9 Spccies turnover 0.65 2 0.1 0.73 i 0.07 Number of 694 -L 136 379 t 135

piantsi2400 M'

All differences are statistically significant at I% level

*Statistically significant a1 5%. **Statistically significant at 1 %.

species composition of the two clusters of sites is also significantly more distinctive than woufd be expected on the basis of chance alone. These two clusters may therefore be compared to assess the differences between sites with low and high level of disturbance. The density of plants in sites with high levels of disturbance is roughly half that in sites with low levels of disturbance. There is no consistent difference in the distribution of individuals amongst different height classes, presumably because local villagers selec- tively remove smaller trees while commercial harvests tend to remove larger ones from the more disturbed sites (Table 111). This suggests that the plant biomass loss from the more disturbed sites may be around 50%, probably over the last 50 years when the pace of forest exploitation has picked up with eradication of malaria, establishment of plywood and paper industries and a spurt in population growth. We may then estimate the rate of attrition of forest biomass at around 1.7% per year.

We have also looked at the differences in two indices of diversity, species richness, or number of species at a given site and species turnover, or the proportion of unshared species amongst any pair of sites with low or high levels of disturbance (Table IV). The sites with low disturbance level are significantly richer in species, but have lower levels of unshared species. However, both these differences are no more than what would be expected on the basis of the fact that sites with higher levels of disturbance have much lower plant densities.

But a more careful look at the species composition of the two sets of sites revdals many differences (Table V). Expectedly, there is preponderence of evergreen species on sites with low levels of disturbance, and deciduous species on sites with high levels of disturbance. Most interestingly sites with high levels of disturbance are significantly deficient in species with poor coppicing power in demand by plywood industry. Table VI provides a list of sets of species which are characteristic of the two types of sites.

5. Management regimes

Given the significance of human impact, an important applied question pertains to bow these relate to various management regimes. We have attempted to look at this issue

502 MADBAV (~AwGli .

Table V Number of species

Attrihule All LD HI> Only LD On!" ill) LD uird s i r m srtrs rilvr s ~ t e s srtrs 111) silr.5 .... -- -

Deciduous 33 23 31 2 I0 2 1 Everereen 121 107 65 56 14 51 Human demand 51 44 35 1') 10 25 Coppicing possible 136 110 95 41 26 60 Used for plywood 36 30 23 13 6 17 Used for plywood and 27 21 21 6 6 13

cappicmg possible Used for plywood but 9 9 2 7 0 2

non-coppicing Coppicingposs~ble 109 89 74 35 20 54

but no plywood use

Total 525 433 346 179 92 254

LD: low dirtuibance: IID: high disturbmce.

through a study of 29 one ha plots located in Kunata and Siddapur t:ilnks of Uttara Xannada district under a variety of situationsx. Figure 4 depicts these in rtie phase qsace of number of trees of more than 30 cm girth at breasl height vcm11s numher of species. The best sites lie in reserve forests on steep, inaccessibie slopes of rivcr Sharavathy. One of the most depleted 4tes is a so-called minor forest, an accessible reservc forest site which is left open to harvest by viilagers without any authority to regulate. In contrast, two of the three village forest sites where local comrnuniries have a measure of regula tory authority are in better condition. Most notablc are some of the sacred groves. In spite of being highly accessible some of thcse support diverse tree communities harbour-ing the few remaining stands of Ilipter-oriwpus and Myristiru swamp asso- ciates. Thls suggests that active participation by local communities couk1 be a very significant step towards long-term conservation of the biologicai diversity of thc Wehtern Ghats.

Table VIa Characteristically exelusircly on site, with low

levels of disturbances (in 5 7 out of 20 tmnsects)

hyrigun; xordneri Myrisrrco malobonco lfolignrna grahamii I Industrial Colophyl l~m elotum Dysoxyion mulahnr-icum Poiyaithio frograns

Gnerum ulo Dimpetdun; gelinoides Village use Elaeocarpns sp. N~o i i i s ra zcylanrra

Tahie V l h Chnracteris?icallr. exclusively on highly disturbed sites (in 2 4 out of 10 trancrcts)

- 5 f e e p inaccessible r i ve r volley slopes

* Sacred grove

* Reserve forest

a Private forest

Village forests

Open access

" Minor forest "

0 100 250 400 550 700 850

Tree density

Acknowledgement

I am grateful to my many s fude~~ts and collaborators, in particular R.J. Ranjit Daniels, N.V. Joshi, Shri Niwas Singh alld M.D. Subash Chandran for the ideas, information and '

insights they have provided This work has been financially supported by the Ministry of Environment and Forests, Government of India, and made possible by the cooperalion of the Forest Department, Government of Karnataka.

References

M. AND MEHER-HOMII. V M. Ecologi~al dlvcrsity. In Consirvatron I" de\,eloping coun- fries - Prohiem mid pr-ospecrs (J. C Daniels and J. S . Serrao, (eds) Proc. Ce,ripnrrry Stminor of the Bombay Natural His- tory Sor.ieiy. pp. 175- 198, 1990, Burnbay Natural History Society and Oxford Universtly Press, Bombay

The b~od ivers i t~ challenge:expanded hot spots analysis, The Environmrnrobst. 1990, 10. 243-256.

5. DANIELS, R. J. R., JOSHI, N. V. AND GADGIL, M.

6. DANIELS, R. I. R., HEGDE, M. AND GADGIL, M.

7 . DANIELS, R. I. R., GADOIL, M. AND JOSHI, N.V.

MADHAV GADGIL

Wet evergreen forests of the Western Ghats ojlndio . Ecol- ogy, stmcture, floristic composition and successron, 1988, Institute Francam de Pondicherry, Pondicherry.

A conservation strategy for the birds of the Uttara Kannada district, Ph. D. disserlation, Indian Institute of Sclence. Banga- lore. Indla, 1989.

On the relationship between bird and woody plant species diversity in the Uttara Kanriada district of south India, Proc. Natn. Acad Sci. USA, 1992.89,5311-5315.

Birds of the man-made ecosystems: the plantations. Proc. Indian Aead. Sci. (Anim Sci.), 1990.99.79-89.

Imoact of human extraction on trooical humid lorests in the Western Ghats in Uttaia Kannada, Sonth India, 1995.31.866-874.

Appl. Ecol..

Vegeiotional changes in rho evergreen forest bell of Uttara Kannodo district of Karnntaka State, Ph.D. dissertation, Kar- nataka University, Dharwad, India, 1993.

Published by N. M. Malwad, Executive Editor, Journal of the Indian Institute of Science Bangalore 560 012: Typeset by Creative Literati Pvt. Ud., Bangaioie, TeL-cum-Fax: 546 2698; ~ i i n t e i at Lotus Printers, Bangalore, Tel.: 338 0167; Fax: 335 7378

Journal of the Indian Institute of Science- Volume 76: Number 4

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