Multilevel modular mesocrystalline organization in red coral

American Mineralogist, Volume 95, pages 242–248, 2010

0003-004X/10/0203–242$05.00/DOI: 10.2138/am.2010.3268

242

a n i e l

i e l z e u f

* n

i c o l e

l o q u e t

o m i n i q u e

h a t a i n

r a n ç o i s e

o n n e t é

a n i e l

e r r y

o a q u i m

a r r a

B o u

a n

D w a r

m. s

t o l p e r

Centre Interdisciplinaire de Nanoscience de Marseille, CNRS-Aix-Marseille University, 13288, Marseille, France

Institut de Ciències del Mar, Passeig Maritim de la Barceloneta 37-49, 08003 Barcelona, Spain

Division of Geological and Planetary Sciences, California Institute of Technology, MC 170-25, 1200 East California Boulevard, Pasadena,

California 91125, U.S.A.

B s t r a c t

Biominerals can achieve complex shapes as aggregates of crystalline building blocks. In the red

coral skeleton, we observe that these building blocks are arranged into eight hierarchical levels of

similarly (but not identically) oriented modules. The modules in each hierarchical level assemble into

larger units that comprise the next higher level of the hierarchy, and consist themselves of smaller,

oriented modules. EBSD and TEM studies show that the degree of crystallographic misorientation

between the building blocks decreases with decreasing module size. We observe this organization

down to a few nanometers. Thus, the transition from imperfect crystallographic order at millimeter

scale to nearly perfect single crystalline domains at nanometer scale is progressive. The concept of

“mesocrystal” involves the three-dimensional crystallographic organization of nanoparticles into a

highly ordered mesostructure. We add to this concept the notion of “multilevel modularity.” This

modularity has potential implications for the origin of complex biomineral shapes in nature. A multi

level modular organization with small intermodular misorientations combines a simple construction

scheme, ruled by crystallographic laws, with the possibility of complex shapes. If the observations we

have made on red coral extend to other biominerals, long-range crystallographic order and interfaces

at all scales may be key to how some biominerals achieve complex shapes adapted to the environment

in which they grow.

Keywords:

Biomineral, mesocrystal, crystallography, calcite, EBSD, hierarchical organization,

modularity,

Corallium rubrum

, complex shape

n t r o

D u c t i o n

Biominerals often display morphological, chemical, and crys

tallographic patterns at length scales ranging from the nano- to

the macroscale (Lowenstam and Weiner 1989; Mann 2001). As

such, they correspond to the definition of “complex systems”

(Grimm et al. 2005). Either individually or in combination with

each other, these patterns can provide information on the mecha

nisms that produce complex structures. An important feature

of biominerals is that they can achieve “beautifully sculpted”

shapes (Weiner et al. 2005) using crystal symmetry-dependent

blocks, but it is unclear how these shapes are formed from such

building blocks. One possibility is that a disordered amorphous

precursor phase that can be “molded into any shape” (Weiner et

al. 2005) plays an important role in generating a wide variety of

forms (Pecher et al. 2009; Weiner et al. 2005). Another is that

complex shapes can emerge from particular arrangements of

crystalline units during growth (Towe 2006). In this article, we

demonstrate that red coral (

Corallium rubrum

) acts as an astute

crystallographer, assembling its skeleton as a delicate arrange

ment of a hierarchy of crystals with well-defined orientations

relative to their near- and far-field neighbors.

x p e r i m e n t a l

m e t h o

D s

Electron backscatter diffraction (EBSD) using a SEM enables quantitative

measurements of the crystallographic orientation of crystal domains as small as

200 nm (Prior et al. 1999), and the calculation of misorientation axes and angles

between any two data points. EBSD patterns were obtained at Caltech on a LEO

1550VP SEM equipped with a HKL technology “channel 5” EBSD system using

an accelerating voltage of 20 kV, a probe current of 2 nA, and a working distance

of 14 mm. Samples were prepared by conventional polishing using diamond paste

with grit sizes down to 1 or ¼

m, followed by a final polish with colloidal silica.

Collection times of ~0.3 s under the SEM beam were sufficient to generate good

quality EBSD patterns for the Mg-calcite red coral skeleton. A 200

100 grid

with 4

m spacing between points generated an orientation map in <2 h. In the

orientation map, each pixel corresponds to a crystallographic orientation character

ized by three Euler angles (

2) transformed into a RGB color code using

the relations red = 255

/180, green = 255

/180, blue = 255

/180. The lattice

orientations of the Mg-calcite crystals making up the red coral skeleton (Grillo

et al. 1993) were determined with a lateral diameter of the diffracting volume of

~200 nm and an absolute angular resolution of ±0.5°. EBSD patterns with a mean

angular uncertainty ≥1° were discarded and plotted as white dots on the map; on

the orientation map, 42.6% of the points were indexed. Stereographic projections

(upper hemisphere) of {

hkil

} planes and the corresponding pole density/contour

diagrams were automatically generated.

A SEM equipped with a forescatter detector generates images of texture

of polycrystalline samples with a submicrometer spatial resolution (Prior et al.

1999). SEM orientation contrast (OC) imaging is based on signal contrast between

materials with differences in crystallographic orientations. Sharp contrast changes

correspond to sudden changes in crystallographic orientations. This technique

detects small changes in orientation (<0.5°) unseen by EBSD (Prior et al. 1999).

It is also a high spatial resolution technique since domains about 0.25

m wide

can be resolved with a field emission source. OC images are not quantitative (in

* E-mail: vielzeuf@cinam.univ-mrs.fr

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contrast to EBSD), but they do allow the characterization of small crystallographic

changes in polycrystalline materials.

Other more widely used techniques (SEM and TEM) and the provenance of

the biological material are described in Vielzeuf et al. (2008). Results presented

here are based on the study of about 20 different samples of red coral.

e s u l t s

In this section, the crystallographic structure of the red coral

skeleton is presented from the largest to the smallest unit. In the

figures, features of interest are labeled (e.g., “

” for herringbone

unit, etc.); in the text, they are referred to by the figure number

followed by the label (e.g., Fig. 1b [

]).

Herringbone units and strips (10

–2

–10

–3

Observation of polished surfaces of red coral cut perpendicu

lar to its main axis under a reflected light microscope shows a

regular alternation of a few millimeters long, darker and brighter

strips (Figs. 1a and 1b [

]). A dark strip indicates that the op

tical axes of the Mg-calcite crystals that compose the skeleton are

parallel or perpendicular to the axes of the polarizer or analyzer,

while any other orientation generates bright strips. A rotation of

45° reverses the pattern. The alternation of bright and dark strips

in a radial arrangement is shown in Figure 1a. It is important to

note that crystals in a single strip are, from a crystallographic

point of view, almost similarly oriented over distances of a few

millimeters. The coupling of a dark and a bright strip forms a

composite crystallographic superstructure that we refer to as the

herringbone unit (Figs. 1b [

] and 1c). Some herringbone units

reach the crenulated rim of the skeleton (Fig. 1a), and each of

these units coincides with a concave outward portion of the rim

(Fig. 1a [

]); the wavy pattern at the rim is the expression in

cross section of longitudinal grooves running along the skeleton

(Grillo et al. 1993; Vielzeuf et al. 2008). The herringbone units

display a roughly constant width, ca. 200–300

m (Figs. 1a and

1b). Thus, the increase in diameter of the skeleton is accommo

dated not by an increasing width of the herringbone units but

by an increasing number of units (Fig. 1a). New units appear at

macroscopic dislocations (Figs. 1a and 1b [

]).

Figure 2b shows an EBSD orientation map with three her

ringbone units, superposed upon a reflected light microphoto

graph (Fig. 2a). The EBSD study was performed on a different

sample than those shown in Figure 1; however, as indicated by

the scale bars, the width of the EBSD map is about twice the

width of Figure 1c. The orientation map shows the alternation

of blue and pink bands coinciding with the bright and dark strips

of the photograph, respectively. Pole figures (Figs. 2c and 2d)

indicate that the

axes of the crystals of Mg-calcite are almost

parallel to the plane of observation Xy

(tilt < ~6°) in both types

of strips. However, most importantly, their orientations in that

plane systematically differ: in the blue strips, the

axes are

oriented ~ –22° (relative to the

y axis shown in Fig. 2a), while

they are oriented ~ +22° in the pink strips. The histogram of

the crystallographic orientation of the

axes in the Xy

plane

in the blue strips (Fig. 2e) shows a unimodal distribution and a

peak width at half height of ~30°. The distribution for the pink

strips is bimodal (Fig. 2e), but with a comparable peak width at

half height if it is treated as a unimodal distribution. We do not

ascribe important significance to this bimodality since it could

be related to statistical bias.

Spindles (10

–4

Observation at higher magnification indicates that the bright/

dark strips seen in reflected light are not perfectly homogeneous,

but are composed of elongated and irregular spindle-shape sub

units (Fig. 1b). These spindles are 100 to 300

m long and 10

to 50

m wide; some of them are contoured in Figure 1c [

The angle between spindles in the bright and dark strips is in

general close to 45°, but shows a large dispersion of ±20°. It is

this arrangement that results in the visually obvious herringbone

pattern in Figure 1c. The combination of EBSD map (Fig. 2b),

pole figures (Figs. 2c and 2d), and reflected light images show

that the elongation of the spindles is more or less parallel to the

optical axis of the crystals. Histograms of distribution of crystal

lographic orientations can be retrieved from zones corresponding

to a single spindle (Figs. 2b and 2e). For a given set of EBSD data,

the histogram of orientations of the

axes of a strip is obviously

the external envelope of the local histograms of the spindles that

constitute the strip. The histogram for a single spindle displays

a significantly lower spread of orientations than the strip as a

whole (Fig. 2e) and a remarkable similarity of crystallographic

orientation with a peak width at half height of only ~15°, i.e.,

about half that observed for the strip as a whole. Normalization

of the data to generate identical areas under the strip or spindle

distribution curves does not affect the width at half height of the

distributions. Other EBSD maps (not shown here) performed

on the same sample with a better spatial resolution (2 and 1

spacing between points) confirm this hierarchy of statistic distri

butions. Furthermore, the OC image of spindles shown in Figure

1d is consistent with this observation: dark and bright lineations

(emphasized by yellow lines in Fig. 1d) indicate a preferential

crystallographic orientation within each spindle.

We noted earlier that the surface of the red coral skeleton is

crenulated and shows a wavy pattern (~300

m wavelength).

The surface of each wave is not smooth but, in a self-similar

fashion, made of numerous tree-like smaller (~30

m) micro

protuberances (Grillo et al. 1993; Vielzeuf et al. 2008). Similarity

of orientation and size, together with other EBSD observations

(not shown here) suggest that microprotuberances are the surface

expression of the spindles.

Lozenges (10

–5

OC images at higher magnification (Fig. 1e) show that

spindles are not single crystals. Instead, they are made of 5

wide, 10

m long lozenges (Fig. 1e [

]) separated by narrow,

dark, porous zones (Fig. 1e [

]). In spite of being made of

separate sub-units, the crystallographic orientation within each

spindle remains nearly constant as indicated by the alignment of

the long axis of the lozenges and the parallelism of dark/bright

elongated bands (Fig. 1e [

]). In both the spindles and the loz

enges, EBSD showed that the long axes of these bands coincide

with the

axes of calcite crystals (Figs. 1d and 1e [

]).

Fibers (10

–6

The alternation of dark and bright elongated bands (<1

wide, 5–10

m long) in the lozenges (Fig. 1e [

]) indicate that

crystallographic misorientations still exist within each lozenge.

However, these misorientations may be very small since minute

differences can be detected by OC imaging (<0.1° according to

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i G u r e

Crystallography of

Corallium rubrum

. (

) Mosaic photograph of a section of skeleton under reflected light microscope (polarized

and analyzed light) with long-range crystallographic order and herringbone units.

: dislocation,

: trough on the wavy surface.

(

) Enlargement

. The herringbone units (

) are made of the association of a dark (

) and a bright (

) strip. New herringbone units appear at dislocations (

The wavy black dashed line represents a past growth surface and indicates the polarity of the structure. (

) Internal structure of the herringbone:

each strip is made of elongated spindles (

). Spindles in two adjacent strips in a herringbone unit form an angle of ~45° (see also

). Black arrows

indicate both the elongation of the spindles and the orientation of the

axes of calcite crystals. (

) SEM observation of spindles. Differences in gray

levels are due to orientation contrast; the limit between two adjacent spindles corresponds to a change in crystallographic orientations.

yellow lines

and black arrows underline the

axis orientations in different spindles. Note the drastic change of orientations on both sides of the herringbone axis

shown as a heavy black line. The white dashed lines are traces of growth rings; they are orthogonal to crystallographic orientations. (

) Internal

structure of the spindles (orientation contrast SEM image). Spindles are made of lozenges (

). These lozenges are reminiscent of sections of calcite

scalenohedra and are separated by dark zones indicative of porosity (

). Each lozenge shows elongated dark/bright bands (

) corresponding to

crystalline fibers. Black arrows indicate the

axis orientations. (

) Crystalline fibers (

) observed under the SEM (secondary electrons). Fibers

are made of a piling of submicrometer units (

). (

) A crystalline fiber in a focused ion beam foil observed under the TEM, dark-field image. (

)

Submicrometer units under the TEM in a focused ion beam foil. Variations in gray levels are due to variations in crystallographic orientations.

The darkest zones between the crystals are indicative of porosity (

). (

) HRTEM image showing lattice fringes and the internal structure of

submicrometer units composed of 2–5 nm similarly oriented nanodomains. Variations in adsorption contrast indicate the presence of nanopores

(

). Some characteristic interplane distances are indicated in nanometers. Inside frames in

, and

are drawn for relative scale purpose

and do not represent exact locations of enlargements.

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Prior et al. 1999). SEM and TEM observations (Figs. 1f [

] and

1g [

]), respectively, indicate the existence of crystalline fibers

(see also Vielzeuf et al. 2008) whose dimensions (ca. 1

m wide,

ca. 10

m long) are identical to the elongated bands observed in

the lozenges. Thus, we infer that the OC bands in the lozenges

correspond to crystalline fibers.

Submicrometer units (10

–7

In a previous SEM and TEM study, Vielzeuf et al. (2008)

showed that the micrometer-scale crystalline fibers in the red

coral do not correspond to single crystals, but are themselves

made of the piling of 200–500 nm units (Fig. 1f [

]). This con

clusion is consistent with atomic force microscopy observations

(Dauphin 2006). These submicrometer units are clearly identified

on focused ion beam foils observed with a TEM (Fig. 1h [

]).

In the dark-field images, the boundaries of the submicrometer

units are commonly marked by thin black porous spaces (Fig. 1h

[

]). Differences in diffraction contrast from one submicrom

eter unit to the other (Figs. 1g and 1h) can be ascribed to slight

crystallographic misorientations.

Nanodomains (10

–8

–10

–9

TEM electron diffraction patterns on selected areas in the

submicrometer units, such as the one shown as an inset in

i G u r e

EBSD study. (

) Reflected light microphotograph of a section of red coral skeleton normal to the axis (compare to Fig. 1a);

, and

as in Figure 1b. (

) Electron back-scattered diffraction orientation map: 200

100 grid with a 4

m interval between points, spatial resolution

~200 nm. For relative scale and polarity purpose, the wavy dark-blue dashed line represents a fossil growth surface. The black arrows correspond

to the orientation of the

axes of the calcite crystals in the different strips. The dark-blue and dark-orange rectangles are the selected areas for the

blue and pink spindle histograms in

. (

and

) Pole figures of the {0001} planes in the blue and pink strips. Density plot, stereographic projection,

half width 10°, cluster size 5°, contour line interval 2.5. (

) Histograms of crystallographic orientation of the

axes in the Xy

plane of crystals in

the blue and pink strips (entire map) and in a blue and pink spindle (selected areas shown in

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Figure 1i, indicate that each unit diffracts as a single crystal.

However, TEM images show that the submicrometer units are

not single crystal, but are made of an assemblage of 2–5 nm

nanodomains (Vielzeuf et al. 2008). Variations in absorption

contrast and the presence of free crystalline surfaces suggest

the presence of an intimate network of nanopores (Vielzeuf et

al. 2008). High-resolution TEM imaging (Fig. 1i) confirms the

presence of nanograins. Some of them display a rhombohedral

shape. Most importantly, the parallelism of lattice fringes across

adjacent domains demonstrates that the crystallographic axes of

all domains are nearly parallel.

i s c u s s i o n

a n

i m p l i c a t i o n s

Figure 3 summarizes the different levels of crystallographic

arrangement in the red coral skeleton. From this hierarchy of

crystallographic structures, a simple pattern emerges: imperfectly

similarly oriented (ISO) nanograins combine into submicrometer

units; the ISO submicrometer units combine into fibers; ISO

fibers make up the lozenge units; ISO lozenges are arranged

into spindles; the ISO spindles comprise the millimetric strips;

the strips joined side by side form herringbone units; and finally,

the juxtaposition of herringbone units plus the addition of new

units at macroscopic dislocations lead to the radiating structure

of the red coral skeleton. Each entity is composed of modules

and is at the same time a modular part of a larger module with

remarkable similarity of crystallographic orientations. Histo

grams of crystallographic orientation of

axes (Fig. 4) can be

used to present a Russian nesting doll-like organization that spans

over seven orders of magnitude. We have already seen that the

histogram of orientation in the millimeter-scale strips (Fig. 2e) is

the envelope of the histograms of the spindles that constitute the

strip. In turn, the spindle histogram is necessarily the envelope

of the histograms of the lozenges that constitute the spindles,

and so on. Observations with different analytical techniques

from the reflected light microscope to the TEM suggest that the

degree of misorientation decreases with the decreasing size of

the crystallographic modules. This is demonstrated by EBSD

measurements for the strip and the spindle units. Our attempts

to quantify the degree of misorientation in the smallest units by

doing EBSD maps with a better spatial resolution (0.2

m spac

ing between points) failed, in part as a result of beam damage

of the sample. However, qualitative observations of the smallest

units as described earlier are in agreement with the interpretation

of increasing order with decreasing size of the modules. Thus,

if our interpretation is correct, the red coral skeleton shows a

progressive transition from imperfect to almost perfect crystalline

domains down through a hierarchical crystallographic structure.

This multilevel modular arrangement preserves long-range crys

tallographic ordering (though imperfect), scattering properties

close to single crystals and, at the same time, it allows the pres

i G u r e

Summary and conceptual interpretation of seven levels of crystallographic hierarchy in the skeleton of

Corallium rubrum

. The

macroscopic radial arrangement of herringbone units (as in Fig. 1a) representing an eighth level is not shown here. For clarity, some angles and

lengths are not to scale. Note the preservation of crystallographic order (

axis) over long distance. Possible slight misorientations between the

constituting units are not shown.

i G u r e

Modular organization illustrated through a schematic

composite histogram of crystallographic orientations. Each unit is the

external envelope of various sub-units and at the same time a sub-unit

of a larger envelope. The degree of misorientation in the sub-unit is

lower than in the unit. Note that the distribution in the blue and pink

strip overlap. Not to scale.

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ence of porosity and crystal interfaces at all scales.

A “mesocrystal” is a recent concept defined as a super

structure made of almost perfectly aligned crystalline particles

(Cölfen and Antonietti 2005; Meldrum and Cölfen 2008). This

concept arose from observations of natural and synthetic materi

als (Penn and Banfield 1998, 1999). In these examples, primary

particles aligned, docked, and fused to form oriented chains by

a mechanism called “oriented attachment” (Penn and Banfield

1998). This mechanism was an elaboration of the concept of

“particle aggregation” developed for colloids (Bailey et al. 1993;

Privman et al. 1999); it applies in biomineralogy as shown by

experiments in the CaCO

system (Cölfen and Antonietti 2005;

Imai et al. 2006; Oaki et al. 2007). An important point is that

although they are made of distinct nanoparticles, mesocrystals

display scattering properties of single crystals (Cölfen and An

tonietti 2005). Most experimental studies related to mesocrystals

describe a single level of organization, i.e., nanograins assem

bling into a mesocrystal. However, recently, in a study dealing

with the growth of calcite in an organic gel matrix, Oaki et al.

(2007) observed structures made of a hierarchy of nanocrystals,

submicrometric and micrometric units, each exhibiting rhombo

hedral habits. They concluded that the resultant calcite could be

regarded as a hierarchical self-similar structure.

Our observations are consistent with most aspects of mesocrys

tals as described in these references, and thus we infer that each of

the hierarchical levels we have observed in the red coral skeleton

would be appropriately described as mesocrystalline. However,

some observations presented here are new and may help refining

the concept of mesocrystals in biomineral systems. For example,

contrary to previous examples of mesocrystals but in agreement

with the experimental observations of Oaki et al. (2007), the red

coral skeleton displays not a single stage but rather a multilevel

crystallographic hierarchy. This is a new feature of natural me

socrystals. Further studies will be required to determine if, as we

expect, this is a general feature of biominerals.

Mesocrystals have been viewed as examples of crystallization

that does not proceed through ion-by-ion attachment but rather

by attachment of modular building blocks (Cölfen and Antonietti

2005). Our observations only describe a hierarchical organization

and not necessarily a crystallization mechanism. Given the way in

which a red coral skeleton grows (i.e., via layer-by-layer addition

of particles at the surface, as shown by chemical patterns and the

presence of annual growth rings (Marschal et al. 2004; Vielzeuf

et al. 2008), it seems likely that only the smallest hierarchical

scales may have been assembled by the addition of separately

built modules. In the case of the red coral, we have a modular

organization but not a modular construction [see Baldwin and

Clark (1997) for an interesting analogy with modularity in the

design of complex engineering systems]. A major difference

between inorganic mineral structures (e.g., rocks) and biomin

eral structures (e.g., skeletons), is that the latter are evolved by

organisms to fulfill particular functions. The skeleton of the red

coral must respond to competing demands: anchor the colony on

a rocky sea floor; support and grip the living tissues; allow the

growth of the organism; achieve mechanical strength against sea

currents; and adapt its shape in a way that favors both the access

of the polyps to the nutrients and the removal of metabolic waste

in the sea water. The modular crystallographic organization in

the red coral and its ability to achieve complex morphologies

overcome the poor mechanical properties of calcite. For instance,

microprotuberances, the morphological surface expression of

the crystallographic spindles as discussed earlier, end up acting

as self-blocking structures between growth rings in the skeleton

(Vielzeuf et al. 2008). Thus, in the red coral, morphologically

complex, mechanically resistant structures result from a construc

tion scheme ruled in great part by crystallographic principles.

Colonies of red coral display a wide range of morpholo

gies indicating that they are able to adapt to surrounding

hydrodynamic motion. We noted above that the similarity of

crystallographic orientation between different modules in each

level is imperfect. Such misorientations may not be viewed as

“imperfections” but rather as degrees of freedom that, together

with the polydispersity of the modules and the presence of

interfaces at all levels, allow morphological adjustments. The

precise nature of the interfaces between the crystalline units is

not yet known. However, Vielzeuf et al. (2008) noted the pres

ence of porosity and organic matter at all scales, indicating that

the red coral skeleton is a composite organic/inorganic material.

Thus, both inorganic/inorganic and organic/inorganic types of

interfaces are possible. It has been suggested that the formation

of biogenic calcium carbonates is directed by an ordered template

of macromolecules (Cuif and Dauphin 2005; Pouget et al. 2009).

Whether the hierarchical modular organization presented here is

also directed by a hierarchical organization of macromolecules

remains to be determined.

In nature, the existence of an amorphous phase that could be

“molded into any shape” has been put forward as a key factor of

morphological control in biominerals (Weiner et al. 2005). We do

not challenge the existence and the importance of an amorphous

phase as a precursor in crystalline biominerals, but we doubt that

it plays a major role in the final morphology of the red coral.

Instead, we consider that the presence of interfaces at all scales

is a key property of biominerals to achieve all kinds of shapes

and adapt to the environment. How amorphous precursor phase

may influence or relate to the multiscale interface model that we

propose here is not yet known.

Developing nanomaterials with controlled hierarchical

structures, crystalline morphology, orientation, and surface

architecture remains a challenge in materials science, and there

is still much to learn from biomineral archetypes (Cölfen 2003).

The multiscale interface model presented here may prove use

ful to develop new strategies to design complex-shaped three-

dimensional crystalline synthetic materials.

c k n o w l e

D G

m e n t s

This work has been supported by Centre National de la Recherche Scienti

fique (CNRS), Institut National des Sciences de l’Univers (INSU) through grants

ECLIPSE 2005, INTERRVIE 2009, and by Centre Interdisciplinaire de Nanosci

ence de Marseille (CINaM) through internal grants to D.V. We thank C. Henry for

his support, Chi Ma for his assistance with EBSD and OC imaging at Caltech, and

C. Vanni, W. Saikali, C. Dominici, and T. Neisius for their help with the FIB foil

preparation and observation at CP2M-Marseille. We made the TEM observations

at the INSU TEM facility at Marseille. We thank S. Nitsche and D. Chaudanson

for their supervision during the sessions. One of us (D.V.) is grateful for a Caltech

financial support to this project. Reviews by three anonymous reviewers as well

as editorial handling by Nita Sahai are gratefully acknowledged.

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