Higgins M.D. Quantitative textural measurement in igneous and metamorphic petrology

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Grain spatial distributions and relations

6.1 Introduction

Grains in many rocks are not distributed randomly in space, but organised

into clusters, layers and chains (Figure 6.1). Such spatial patterns can be

defined in a number of ways: by the presence of grains as well as by their

size, shape, orientation and mineral associations. Such non-random distribu-

tions of grains, termed patterns or packings, have been observed in meta-

morphic, igneous and sedimentary rocks (Rogers et al., 1994, Jerram et al.,

2003). Despite the fact that quantitative investigation of such structures in a

rock was started in 1966 (Kretz, 1966a), there have been relatively few quanti-

tative studies since that time. Many terms have been used to describe nonuni-

form grain distributions: in igneous rocks grain clusters are referred to as

clumps, clots or glomerocrysts, whereas in metamorphic petrology other

textural terms are used. Another important textural aspect of a rock is the

spatial association of minerals, or lack thereof. This may reflect nucleation,

growth or mineral breakdown processes, but has been much less studied than

the actual crystal positions.

Patterns can be imposed on a rock during its formation by externally

varying conditions or can develop spontaneously. For instance, during deposi-

tion of sediments variations in the flow regime or sediment source can produce

layers; these are clearly imposed externally. Some layering in igneous rocks

may result from similar sedimentary processes (Naslund & McBirney, 1996).

Layering in metamorphic rocks can reflect layering of a sedimentary protolith,

and deformation of dykes and enclaves. Layers or other structures are referred

to as templates (Ortoleva, 1994). In some homogeneous materials patterns can

arise spontaneously, or existing templates can be amplified and modified.

These effects are called ‘geochemical self-organisation’ (Ortoleva, 1994).

A classic example of this effect is Liesegang layering. Hence, grain clusters,

199

chains and layers appear to be common phenomena, with varying origins, and

should yield petrogenetic information.

6.2 Brief review of theory

6.2.1 Nucleation of crystals

Nucleation occurs when it is energetically favourable to create a new phase, or

more crystals of an existing phase (see Section 3.2). If nucleation occurs

spontaneously within a phase it is termed homogeneous nucleation. The posi-

tion of the nuclei will be random, provided that the temperature and composi-

tion of the liquid are constant. If, however, a region of the liquid is depleted

in the crystal components, for example by crystallisation of an adjacent crystal,

then nucleation may be suppressed there. This may produce megascopic

patterns by geochemical self-organisation (Ortoleva, 1994).

Nucleation can also occur on existing phases, termed heterogeneous nuclea-

tion. The position of the nuclei of the new crystals will obviously depend on the

Ordered

Random Layered

Clustered Chains/Foams Broken la

ers

Figure 6.1 Schematic grain spatial distributions in two dimensions. Such

patterns may be defined by the presence of grains of a phase, their orientation

and mineral associations or by differences in grain size.

200 Grain spatial distributions and relations

location of the host materials. It is unlikely to be random, unless the host

grains are very small and initially randomly distributed. It is more likely that

the nuclei will be clustered. If the crystals do not move significantly during

solidification, this may be visible in the final rock. It is debatable if any magma

is ever completely free of crystals, hence heterogeneous nucleation may be the

normal situation.

Minerals can also form by the breakdown of other solid phases. If two

minerals are formed by such reactions in the solid state then they will be

spatially associated.

6.2.2 Geochemical self-organisation

Many periodic geological structures are thought to originate by variations in

externally controlled parameters. For example, sedimentary layering may

form by variations in the mineralogy and/or size of the particles being depos-

ited. The variation in externally controlled parameters is considered to be a

‘template’ which forces the structure on the rocks. However, patterns can also

arise spontaneously without the intervention of a template: this vast subject is

termed geochemical self-organisation (Ortoleva, 1994, Jamtveit & Meakin,

1999). A number of patterned structures in geology may be produced by this

effect (Ortoleva, 1994):

layers and spots in igneous rocks

metamorphic diff erentiation and layering

oscillatory zoning in crystals

Liesegang banding and layering

diagenetic patterns

geodes, concretions, agates and orbicules

Geochemical self-organisation may seem to violate the second law of thermo-

dynamics: that all systems evolve spontaneously towards a state of increased

disorder. However, although geochemical self-organisation is clearly a local

increase in order, it must be compensated by a greater decrease in order of the

whole system.

Self-organised patterns originate by the dissipation of free energy and are

hence called dissipative structures. The initial unordered system is subject to

random variations of thermal origin. This ‘noise’ can be decomposed into

periodic fluctuations of all possible symmetries, in the same way that any wave

can be Fourier transformed into an infinite number of harmonic waves. When

the system is close to equilibrium none of these patterns dominates and the

system appears completely unordered. However, when the system evolves

6.2 Brief review of theory 201

away from its equilibrium state certain patterns may be amplified by positive

feedback loops and an ordering developed. If the structure is observable and

stable it is termed a dissipative structure. Such patterns may be comprised of

layers, spots, spirals, etc. The patterns form by reaction–transport: this

involves solution of some grains, transport of material and growth of other

grains. Some patterns are only maintained by a continuous supply of free

energy; however, the occurrence of dissipative structures in old rocks indicates

that they can be frozen in place and observed long after the activity of the

system has ceased. It should be emphasised that dissipative structures can form

only in systems that are maintained away from equilibrium.

A classic example of patterning is Liesegang banding. In the simplest

situation this consists of bands richer in a phase in a two-dimensional

system. Three-dimensional Liesegang layers also occur, but have been little

studied (Boudreau, 1995). Two different origins have been proposed

(Chacron & L’Heureux, 1999). In the pre-nucleation model, nucleation and

growth of a crystal depletes the surrounding area in that component, inhibit-

ing nucleation of further crystals. As the reaction front moves away, higher

concentrations are encountered and nucleation is enabled again, starting

the production of a new layer. However, Liesegang bands can also form

after nucleation from a homogeneous material by competitive particle

growth and coarsening (see Section 3.2.4). If a small region has a local

increase in crystal size, then the concentration of the component in the liquid

will decrease. As material diffuses in from adjacent regions the concentration

there will decrease and hence the crystals will dissolve. The process can

produce a series of bands with different crystal sizes and abundances. Both

processes have been combined in a one-dimensional model by Chacron

and L’Heureux (1999). However, observations of the actual formation of

Liesegang bands and layers suggest the post-nucleation model is more

important (John Ross, quoted in Boudreau, 1995). The post-nucleation

process has been proposed for the origin of the famous ‘inch-scale’ layering

of pyroxenes in anorthosite in the Stillwater intrusion (Boudreau, 1995).

Philpotts and Dickson (2002) have observed millimetre-scale layering in

the roof zone of a thick basalt flow. It consists of sheets of ophitic

plagioclase–pyroxene clusters separated by sheets of residual liquid. They

suggest an origin by cycles of nucleation and that such crystal sheets are

separated by gravity along planes of weakness. It is also possible that the

whole structure originates from self-organisation. I have observed that in

some igneous rocks well-developed layering passes laterally into broken layers

and finally clusters. Such behaviour is also predicted by self-organisation

processes (Krug et al., 1996).

202 Grain spatial distributions and relations

6.2.3 Grain aggregation and agglutination

Ideally, a grain will sink or rise in a Newtonian fluid at a speed that is related to

the density contrast and size by Stokes’ law. However, in most geological situa-

tions the concentration of grains is sufficiently high that grains interact and will

gather into clusters, if there is sufficient time (see the review in Schwindinger,

1999). Such clusters will move as if they were made of a single, larger grain, and

will hence descend faster than each of the original grains. The c lusters can form by

vertical and horizontal grain convergence: a large grain will descend faster than

smaller grains and sweep such grains from in front of it. Grains may move

laterally to reduce differences in liquid velocity on either side of the grain – this

is similar to the ‘Bagnold effect’ that produces a concentration of particles in the

centre of a conduit. Grain aggregates may be preserved in volcanic rocks, but it

seems unlikely that such structures can be observed in cumulate plutonic rocks.

In some aggregates euhedral crystals touch along crystal faces (Vance,

1969). One of the best-known examples is that of olivine crystals from the

1959 Kilauea Iki eruption (Figure 6.2; Schwindinger & Anderson, 1989). Most

crystal pairs are united by faces with the same indices, and larger faces are

more commonly in contact than smaller ones. The whole structure resembles a

twinned crystal except that it is not a growth form. The term synneusis

(‘swimming together’) has been used to refer both to nonoriented aggregates

of crystals, and such aligned crystal groups. Schwindinger (1999) considered

that such aligned groups cannot agglutinate during settling, but must be pro-

duced during shear flow or turbulence. The driving force of this process is the

reduction of the total surface (interfacial) energy of the mineral grains: contacts

{021}-{010}

{110}-{110}

{021}-{021}

{010}-{010}

021

110

010

101

001

120

Figure 6.2 Synneusis of olivine grains from the 1959 Kilauea Iki eruption

(from Schwindinger 1999).

6.2 Brief review of theory 203

between similar grains will have a lower surface energy than crystal–magma

contacts. This process has also been proposed for the origin of some garnet

porphyroblasts in metamorphic rocks (Prior et al., 2002).

6.2.4 Magma mixing

If several magmas or crystal mushes are introduced into a conduit or chamber

they will have a tendency to mix. The vigour and duration of the mixing will

determine if the resulting material appears to be homogeneous on a micro-

scopic scale or if the two magmas can still be recognised in blocks and out-

crops: the two cases are commonly referred to as magma mixing and magma

mingling. If the original magmas differ in their crystal contents then the

resulting magma may have a non-random distribution of crystals. Of course,

measurements of crystal positions must be done at a higher resolution than

that of the expected structure. In some rocks the results of mixing are clearly

revealed by compositional or isotopic studies. However, if mineral composi-

tions are constant for all grains, then an origin of crystal clusters by incomplete

mixing may not by easy to determine.

6.2.5 Grain packing

Grains in some sedimentary rocks, and crystals in some cumulate igneous

rocks, are packings of hard particles (Rogers et al., 1994). Theoretical packing

of different sized and shaped particles is very complex; hence this subject has

been explored using numerical models of simple shapes, commonly spheres.

These studies have been concerned mainly with the overall packing density,

that is its efficiency, rather than the actual positions of the particles. Jerram

et al.(1996) used one such study to establish a reference texture for nearest

neighbour distances.

6.3 Methodology

The position of a grain has several measurable parameters.

The simplest is its centre, which can be approximated by the centre of mass and

specified by a single triplet of x, y and z values.

The position of the surface of the grain must also be considered: for instance, two

large grains may be distant, but touching . Many more data are needed to specify the

grain surface – many x, y, z triplets.

The nature of a grain’s neighbours: certain mineral pairs may be touching or

spatially associated more frequently than would be expected for a random texture.

204 Grain spatial distributions and relations

Most analytical methods assume that the sample is homogeneous. It is parti-

cularly important to avoid materials like vesicular lavas, which do not meet

this criterion.

6.3.1 Three-dimensional analytical methods

The centre and surface of a grain can be determined precisely using three-

dimensional methods, if the surface of the grain can be separated from adjoining

grains by physical or analytical means. In the earliest study of crystal positions a

sample was dismantled using a chisel and the crystal positions measured with a

ruler (Kretz, 1969). However, such a laborious study has not been repeated. The

extent of individual grains can be determined by serial sectioning, if their

concentration is low enough that few crystals are touching, or if grain bound-

aries are clear (see Section 2.2.1). In transparent materials, such as volcanic

glasses, the extent of crystals can be measured directly with a regular or confocal

microscope (see Section 2.2.2). X-ray tomography can be used to determine the

extent of grains if adjoining grains can be separated (see Section 2.2.3).

These direct methods are the only ones that can be used for grains with

complex shapes. They are also the only ones that can be used to identify

‘chains’ of crystals (e.g. Philpotts et al., 1999). However, it is not always

possible to separate grains using current 3-D methods. Some two-dimensional

methods are much better at separating adjoining crystals and hence may have

to be used (see Section 2.7).

6.3.2 Section and surface analytical methods

In a section the centre of the grain intersection is not the true centre of the

grain, and is therefore referred to as the centroid. For simple shapes the

centroid is the projection of the centre of the grain onto the section. This is

not the case for grains with complex shapes and they cannot be examined

successfully in two dimensions.

Grain outlines can be acquired easily from rock surfaces, slabs and sections,

using a variety of techniques described in Section 2.5. Such images can be

processed manually or automatically to produce classified mineral images (see

Section 2.6). Some methods require that the grain outlines are obtained from

the classified images. The centroid of the grain intersection can be determined

as the centre of mass.

If two grains touch in a section, then clearly they touch in three dimensions.

However, it is not possible to determine from a section if adjacent, but

separate, grains actually touch out of the section plane.

6.3 Methodology 205

6.3.3 Extraction of spatial distribution parameters

6.3.3.1 Nearest-neighbour distance (‘big’ R)

One of the simplest ways of describing spatial distribution patterns uses the

mean nearest-neighbour distance calculated from the centres of the grains

(Jerram et al ., 1996, Jerram et al., 2003). This is compared to the mean

nearest-neighbour distance from a random distribution of points with the

same population density. This is a method that can be applied to any 2-D

textures where a grain or crystal can be defined by a grain centre co-ordinate.

In principle, the method could be extended into 3-D, where the 3-D nearest

neighbour is compared to a random 3-D distribution of points, but as yet very

few 3-D data-sets of rock textures exist and this has not been tested. However,

it is possible to compare the 2-D spatial distribution pattern to existing known

reference textures where their 3-D spatial pattern is known (Jerram et al., 1996,

Jerram et al., 2003). So far this has been limited to isotropic, homogeneous

materials and it is not clear if strongly foliated rocks, or sections with a

significant porosity, can be analysed in the same way.

The positions of grain centres are generally determined when the other

textural parameters are determined. A simple procedure goes through the list

of N crystal centroid positions and finds the distance to the nearest neighbour,

r (see Appendix). The mean nearest-neighbour distance is

¼ð1=NÞ

The mean nearest neighbour for a random distribution of points with the same

population density (number of crystals per unit area), N

¼ 1=ð2

ﬃﬃﬃﬃﬃﬃﬃ

The ratio of these parameters, ‘big’ R describes the distribution of the grains.

R ¼ r

In principle, for a distribution of random points R ¼ 1. For clustered points

R < 1 and for ordered points R > 1. However, crystals are not points: each

occupies a certain volume in space and area in a section. Hence, crystals

cannot be coincident like points. To enable the comparison of ‘big’ R values

Jerram et al.(1996) calculated the random sphere distribution line (RSDL,

Figure 6.3). This represents the 2-D ‘big’ R values calculated from 2-D sections

taken through 3-D random distributions of spheres with different porosities

up to the minimum porosity: that for random close packed spheres with a

porosity of 37%. The RSDL is calculated using a combination of computer

206 Grain spatial distributions and relations

generated sphere models and a naturally generated model for random close

packing (Jerram et al., 1996). Vectors for common processes have been estab-

lished, like mechanical compaction, grain overgrowth and better sorting.

This method has considerable advantages as it is simple to apply. However,

the validity of the reference models for more realistic crystal shapes and

variable crystal sizes has not been tested. Also, the method gives only a single

parameter that does not describe cluster size. More elaborate methods will be

described in the following section.

If many or all crystals touch in three dimensions then the resulting frame-

work will have an important effect on the rigidity of the material. Hence there

is much interest in determining the minimum amount of crystals that is neces-

sary to make such a framework. Jerram et al.(2003) used nearest-neighbour

diagrams to determine if crystals in a rock form a touching framework. This

technique has the advantage that it can be applied to sections; however, more

data are needed before the method can be verified fully.

6.3.3.2 Cluster analysis

The goal of cluster analysis is to quantify the size and characteristics of clusters

of grains in a rock. Clustering can reflect packing processes, the distribution of

abundance of matrix (porosity) (%)

R =

mean distance to nearest-neighbour

distance for random distribution

1.0

2.0

Ordered

Clustered

Random sphere distribution line

Better

sorting

Mechanical

compaction

Overgrowth

100

Figure 6.3 Nearest neighbour distance determined from sections through

materials (Jerram et al., 1996). The porosity is eq ual to 100  abundance in %

of the phase analysed. The random sphere dist ribution line is indicated on the

graph representing the spatial distributions of randomly packed spheres up to

37% porosity which is the minimum possible for random close packed

spheres.

6.3 Methodology 207

original nucleation sites or other processes (Jerram & Cheadle, 2000). The

first attempt to quantify clustering sizes and distributions used the nearest-

neighbour distance technique (see Section 6.3.3.1; Jerram et al., 1996). More

recently two, more complicated, techniques have been developed: complete

linkage hierarchical cluster analysis (CLHCA) and density linkage cluster

analysis (DLCA). These methods are ideally applied to the materials in

which the complete, 3-D position of each grain is known. Jerram and

Cheadle (2000) applied these methods to sections and tested performance

with comparative 2-D sections through known 3-D synthetic textures (SAS

package – see Appendix).

CLHCA was found to give the clearest results. It uses the distance between

the grain centres. These are agglomerated to give the most compact groups.

The result is a dendrogram (tree diagram) that links the grains and groups with

lines. The distance between groups gives the ‘height’ in the tree. However, it is

difficult to get direct quantitative information from a dendrogram – any

distribution will give a dendrogram, including a random distribution.

Another way of presenting the data is a graph of cluster frequency (number

of clusters) against cluster size (number of grains in the cluster). At a cluster

number of one all grains are included; at a cluster size of one grain the number

of clusters equals the number of grains. This is a smooth curve for a random

distribution, but has gentle steps for a clustered distribution. A further refine-

ment can be made by normalising the data to a random distribution; devia-

tions are then easier to see. This can be done by calculating the distance

between clusters and subtracting the expected distance for a random distribu-

tion. A graph of this residual distance against cluster frequency gives the

characteristics of the dominant clusters (Figure 6.4).

DLCA uses a different approach: clusters are defined on the basis of the

amount of material within a certain radius. Hence this method considers the

whole volume of the grain, and n ot just its centre. Clearly, this is a better approach

if grains have a high concentration and many are touching. The technique can be

applied easily to images with pixels or voxels. In many studies we can only observe

grains in section, hence we are faced with the problem of calculating a 3-D

parameter from 2-D data. Jerram and Cheadle (2000) tested this problem with

a synthetic 3-D distribution, as before. They did not achieve good results, but this

may reflect the weakness of the test distribution and not the method.

6.3.3.3 Touching crystals

Analytical methods discussed so far are only concerned with the position of the

grain centre. They are used to reveal processes that acted early on grains.

Conversely, the distance between the edges of the grains will reflect later

208 Grain spatial distributions and relations