Here is a little bit more detail as to what I put in my thesis - click on the thumbnails for associated illustrations. [Note: some of the pictures, particularly maps, are far larger than the screen, so you might have to download and then resize them]

Understanding of the kinetics of metamorphic processes has lagged behind the advances made in the subject through equilibrium thermodynamics, the main stumbling block being an inadequate knowledge of the nucleation process.

For a new mineral to grow as a result of a prograde metamorphic reaction, a number of processes must operate in concert. First of all, nuclei of the new mineral must appear. New material must be added to the embryo crystal coupled with dissolution of reactant phases, both processes occurring by means of reactions on the mineral surfaces. These chemical species must be transported through the intergranular medium from reactant to product. Each of these processes has with it an associated energy barrier, and the slowest of these processes is known as the rate determining step. If a large energy barrier or overstep is required to be overcome for nucleation, the possibility of metastable reactions occurring instead of stable ones is accessed.

The current state of knowledge concerning reaction kinetics was discussed in Chapter 2 with a view to seeing what potential advances can be made in this study and what methods are available to obtain them. In this study, it has been decided to measure the systematic variation between porphyroblast size and heating rate observed in metamorphic rocks of known thermal and chemical history from the metamorphic aureole beneath the Rustenburg Layered Suite of the Bushveld Complex, South Africa to quantify the critical overstepping for nucleation, and an extensive suite of samples of andalusite-biotite-staurolite/cordierite bearing pelites showing this variation is examined. Central to this line of enquiry is the use of a textural parameter, b, a measure of the distribution of porphyroblast sizes in a population. Large porphyroblasts are observed to result from slow heating rates whilst faster heating gives rise to more hornfelsic textures. By using a forward model that simulates porphyroblastic nucleation and growth, the results of crystal size distribution (CSD) analysis for samples of known heating rate, when combined with a reasonable estimate for either the reaction rate or critical overstepping of reaction, allow the value of the other to be determined. Initial estimates for the overstepping have been obtained using a value for the forward reaction rate determined experimentally by Schramke et al, 1987, American Journal of Science, 287, 517-559.

	Diagram showing how the texture observed in the rock is the product of how different heating rates cause nucleation and growth rates to interact
	The trade-off between volumetric (negative) and surface and strain (positive) free energy terms involved in adding matter to an embryo nucleus shows that a critical size, rc, must be reached before the growing nucleus is stabilized

The field area has been chosen because it acts as a 'natural laboratory' where the aureole rocks have been subjected to a single episode of metamorphism in the form of a thermal pulse without the complications of deformation. Field relationships show that the floor of the Rustenburg Layered Suite and the Pretoria Group metapelites dip concordantly at 15° in the Penge area to the north of Burgersfort. Further north, the dips rise to approximately 40°, whilst south of Burgersfort, dips are shallow and undulose. The tight constraints imposed by the geology in the Penge area allow precise models of emplacement to be constructed.

	Schematic diagram showing the rocks of the study area in relation to the Bushveld Complex
	The location of the study area within the Bushveld Complex, South Africa
	General map of sampling localities from the Eastern Aureole of the Bushveld Complex
	Sampling localities in the Hoogenoeg - Malipsdrif area
	Sampling localities in the Annesley - Wimbledon - Putney area
	Sampling localities in the Burgersfort - Longsight area

Chapter 4 presents a petrological discussion of the samples with the goal of defining a reaction sequence and relative timings of mineral growth which is compared with the predictions of equilibrium thermodynamic modelling. Pelitic lithologies are examined in terms of grade of metamorphism and in terms of Formation which allows similarities and differences to be highlighted. This is important since the intrusion gently cuts down into the stratigraphy in a northerly direction, exposing lower stratigraphic levels to higher grades of thermal metamorphism.

	Large chloritoid crystals and a rare flake of biotite (upper right) in thin section DLB299c, one of the lowest grade rocks studied from the Timeball Hill Formation. The width of this field of view is approximately 1 mm
	Slightly higher up-grade in the Timeball Hill Formation. Giant chiastolite crystals in a highly graphitic pelite matrix from Hoogenoeg Andalusite Mine from which insitu CSD analysis was possible. Sheet of A5 paper is approximately 20 cm wide
	Thin section of the above lithology. Large euhedral porphyroblasts of chiastolite (colourless) and staurolite (yellow) are noted in a highly graphitic matrix of biotite, quartz, plagioclase and scattered porphyroblasts of garnet. Field of view approximately 20x20 mm
	Strongly banded sample DLB120a. The staurolite to the right of the picture and the biotite to the left display inclusion patterns that reflect the variation in graphite content of the bands. This field of view is approximately 25 mm wide
	Replacement of chloritoid by quartz and biotite coarser than that found in the matrix in a typical sample from Annesley. Tiny spindles of chloritoid can just be identified. Field of view approximately 1 mm wide
	View of M32 [field of view 1 mm]. In addition to biotite, large anhedral andalusite and smaller euhedral staurolite crystals contain lath shaped inclusion free areas which indicate the former positions of chloritoid crystals. The chloritoid [marked with a red x] has been overgrown by both species
	Hexagonal cordierite pseudomorphs in thin section DLB9b. Field of view [4 mm] shows the relationship between one of these pseudomorphs and the matrix and other minerals in the rock
	Tiny isolated blebs of cordierite (lightest grey ovals) in sample DLB9b that have withstood alteration. Scale bar approximately 100 microns
	Core region of andalusite typical of samples from DLB383 and 384. Field of view 10 mm width shows pink cores which are sometimes developed
	This plate shows the distinct enrichment in Fe and Mg in a core region with certain lath shaped areas showing uniformly low enrichment; the back-scattered electron image CP picks out the flakey texture revealed in the birefringence. A fissure can be distinguished; the bright mineral inside it being chlorite
	Enlargement in cordierite poikiloblast size with decreasing metamorphic grade in rocks of the Silverton Formation. Scale bars are 1 mm in the left hand column and 500 microns in the right
	Cyclic twinning in cordierite crystals of sample DLB33. The width of the crystal is about 1.5 mm

Geothermobarometric results are presented in Chapter 5 which allow constraints to be placed on the thickness of intrusion and peak metamorphic temperatures in the outer aureole, both of which have implications for the degree of lateral thinning of the Rustenburg Layered Suite which in turn affect the heating rates experienced by rocks of the outer aureole. Results were obtained from both conventional garnet-biotite thermometry and the use of Thermocalc in 'average PT' mode. It is seen in this chapter that samples from the outer part of the aureole suitable for textural analysis have reached peak metamorphic temperatures of 550 to 600° at approximately isobaric conditions of 3 kbar, whilst inner aureole temperatures reach up to 700°.

	Manganese profiles across a range of differently sized garnets in sample 7DLH21a. Note that the largest garnets in general have higher central Mn contents which implies that they grew first and thus that central Mn content can be used as a measure of the relative ages of the crystals in the sample
	Summary of garnet--biotite pair thermometry of a variety of pelitic samples from the eastern Bushveld metamorphic aureole. Locations are arranged in terms of geographical area, decreasing in grade towards the right. Abbreviations are as follows: Bhatt-HW, Bhattacharya [1992] using the data of Hackler and Wood [1989]; Hodges/Spear, Hodges and Spear [1982]; Holdaway [1997]; HP98, Holland and Powell [1998]. Error bars of 30° are applied to all points
	Temperature estimates for the eastern aureole of the Bushveld computed using the garnet-biotite thermometer

CSD and spatial analysis requires interpretation of a 2-D slice through a volume of rock. Chapter 6 considers the assumptions made in this process, for example the under-sampling of the small grains within a population and the distortion of true grain size created by taking a randomly orientated slice through a crystal. These effects combine to cancel one another out which means that for the larger size classes in a population the 2-D CSD is a good representative of the actual 3-D CSD. Methods of obtaining precise CSD results - tracing at scales ranging from thin-section to rock face, image digitization and a summary of the method of Cashman and Ferry, 1988, Contributions to Mineralogy and Petrology, 99, 401-415 - are presented. This method involves the fitting of a b parameter that describes the gradient of the log-linear portion of a plot of crystal size against population density, where such a straight line segment is present. If growth rates are assumed constant and linear, this kind of CSD shows the increase of nucleation rate with respect to time.

	Digitization of a sparse population of garnets traced from thin section 3DLB84a. The height of this area is about 25 mm
	Digitization of a dense population of chiastolite crystals from rockface DLA7. NIH image numbers each crystal and measures their dimensions and position. The area was scanned from a sheet of A3 paper about 40 cm in height
	Diagram showing how the data plotted on a histogram and a cumulative frequency graph. Notice the gradient of the second graph defines the 'Population Density Function'
	Diagram showing how the b parameter, the slope of the best fit line, is determined from actual data
	Frequency histogram showing that the variations in crystal size represent changes in nucleation rate through time. The crystals that nucleated first are the largest, followed by an (exponential) increase in nucleation rate relative to growth rate. The smallest size classes show a fall in nucleation rate towards the end of the reaction

Chiastolite porphyroblasts in the outer aureole where rates of heating have been slow show b parameters of approximately 5 cm, and maximum crystal diameters in excess of 10 mm. Rocks from inner aureole locations where heating has been rapid typically have b greater than 50 and crystal diameters in the order of 1 mm. Further b parameters have been obtained for populations of garnet, cordierite, staurolite, biotite and chloritoid. In some CSD plots, the numbers of crystals in the smaller size classes falls, which indicates a decrease in the rate of nucleation. This is suggested to arise from the influence of multivariant equilibria, in which the position of equilibrium shifts during reaction causing an effective decrease in overstepping and hence nucleation rate.

The spatial distributions of the populations of crystals subject to CSD analysis do not show any significant departure towards ordering which would be expected if slow diffusion of nutrients required for crystal growth caused zones of depletion to form in which further nucleation was suppressed.

Field evidence, thermobarometric constraints and a postulated duration of intrusion of 75,000 years [Cawthorn and Walraven, 1998, Journal of Petrology, 39(9), 1669-1687] have allowed detailed modelling of the thermal pulse leaving the Rustenburg Layered Suite which is shown to have thinned in an easterly direction from 8 to 6 km. Heating rates have been quantified and vary from less than 0.1° per thousand years in outer aureole locations approximately 3 km stratigraphically below the contact to rates greater than 1° per thousand years in the highest grade locations that were suitable for CSD analysis approximately 1.5 km stratigraphically below the contact.

	Map showing the locations from which CSD data for various minerals was collected. Lithological abbreviations as follows; Vsq, Steenkampsberg Quartzite; Vlq, Lakenvalei Quartzite; Vv, Vermont Formation; Vm, Magalieberg Quartzite; Vsi, Silverton Formation; Vd, Daspoort Quartzite; Vt, Timeball Hill Formation
	Cross-section for the Penge area showing the locations from which chiastolite CSD data was collected
	CSD plots for part of the data from Annesley Andalusite Mine. In common with Hoogenoeg Andalusite Mine the distributions reveal large maximum crystal sizes and low b parameter
	Composite CSD plots for samples on the Annesley-Wimbledon-Putney traverse. Crystal size decreases as b parameter increases as the igneous intrusion is approached, exactly the same trend as the Hoogenoeg-Malipsdrif traverse
	The variation of b parameter with distance from intrusion for samples of the Wimbledon-Annesley traverse
	Map of the eastern metamorphic aureole of the Bushveld Complex showing how the results of CSD analysis expressed in terms of the b parameter vary both spatially around the aureole within crystals of the same mineral, and between different minerals. Andalusite b parameter shown in black, garnet in red, cordierite in blue and other minerals in green.
	Variation of log b parameter with stratigraphic distance below contact for all chistolite samples. Notice that those furthest below the contact have the lowest b parameters - slow heating has caused growth to predominate over nucleation and hence larger crystals are formed. Samples with smaller symbol and dotted error bars show results that do not fit with the general trend

The CSD results obtained have been compared with those obtained using a numerical model [NuGrow] for the nucleation and growth of andalusite porphyroblasts [Waters, 1989, Terra Abstracts, 1, 297] which generate synthetic b parameters and oversteppings when rates of heating and reaction are supplied. Initial results in which simulations were performed used calculated heating rates, but assumed the forward rates of reaction experimentally determined by Schranke et al. (1987) which show that a temperature overstepping of between 5 and 10° is required to initiate andalusite nucleation. The estimated degree of overstepping is tied to the rate of forward reaction - higher oversteppings will be required if slower reaction rates are assumed.

	Example of how a thermal pulse will cause different heating rates and peak temperatures to be recorded at various distances from the contact. This model has been calculated for samples on the Wimbledon-Annesley traverse. P.I.F. represents the Penge Iron Formation
	Plot of temperatures estimated at various points on the Wimbledon-Annesley traverse. The straight line indicates the projected temperatures at intervening points on the traverse. Notice the estimates in the P.I.F. are quite different to the rest of the data. Error bars of 25° are shown
	Graph to show the relationship between heating rate and b parameter, based on the plot of b parameter versus distance from intrusion
	Results of NuGrow with the input parameters of Schramke et al 1987. A contoured plot of b parameters is produced, with observed b parameters produced at reasonable values of heating rate and critical overstep. The abbreviation LK refers to sample data from Longsight Cutting and Krugers Post Andalusite Mine

The second half of Chapter 8 deals with an associated study [Waters, D. J. and Lovegrove, D. P., 2002. Assessing the extent of disequilibrium and overstepping of prograde metamorphic reactions in metapelites from the Bushveld Complex aureole, South Africa. Journal of Metamorphic Geology, 20, 135-149] in which the critical overstepping is evaluated using independent methods with the same suite of samples. Detailed microtextural study of graphite poor andalusite-staurolite-biotite hornfelses from the Upper Timeball Hill horizon 2.8 km stratigraphically below the base of the Rustenburg Layered Suite shows that the observed sequence of reaction is different to that predicted by equilibrium thermodynamic modelling. Chloritoid persists far beyond its predicted breakdown temperature to remain present into a period of simultaneous growth of andalusite, staurolite and biotite. The appearance (and subsequent disappearance) of cordierite during this interval is also inconsistent with the predictions of equilibrium thermodynamics and implies a delay in the nucleation of andalusite approaching 100° with respect to its predicted equilibrium temperature of formation. Replacement of early chloritoid crystals at constant volume by a range of chemically different minerals indicates mobility of chemical species through the intergranular medium, and that processes on the mineral surfaces themselves are rate limiting.

Diagram illustrating the potential reactions that could occur in Upper Timeball Hill lithologies. Green dotted lines indicate the positions of univariant equilibria, faint grey lines indicate the boundaries of divariant assemblage fields at equilibrium. Reactions which form andalusite are shown in pink, staurolite in yellow and cordierite in blue; those petrographically determined to be responsible for the production of these minerals in Upper Timeball Hill rocks are additionally highlighted in red

The predicted reactions for the formation of andalusite, staurolite and cordierite by chloritoid breakdown all have low entropy change of reaction. Formation of these minerals occurs once these lower entropy reactions have been overtaken in terms of reaction affinity by reactions predicted to be metastable with significantly higher entropy change, whence nucleation may begin. Such affinity calculations imply that the nucleation of andalusite requires a critical overstepping of over 4 kJ/mol porphyroblast formed, a value which translates to a 40° overstepping of the equilibrium temperature of the chlorite breakdown reaction. On a per gram atom oxygen basis, andalusite requires about 0.8 kJ for nucleation, whilst staurolite and cordierite require significantly lower oversteps of about 0.4 and 0.2 kJ respectively.

Diagram showing the variation of reaction affinity as a function of temperature which allows the evaluation of the critical overstepping for nucleation of andalusite, staurolite and cordierite.

Differences between the amount of overstepping estimated by these two independent approaches may be explained by large differences between reaction rates determined experimentally and those operating in natural metamorphic environments. The coupling of nucleation rate, crystal growth rate and grain size distributions suggests that reactions occur at rates at least an order of magnitude slower than those measured in the laboratory by Schramke et al, 1987, and are comparable with those determined from Sr isotopic analysis of metamorphic terranes by Baxter and De Paulo, 2000, Science, 1411-1414.

Diagram showing how the interaction of reaction rate and overstepping affects the b parameter as modelled by NuGrow. Petrological evidence and thermodynamic data indicate that natural reaction rates are at least an order of magnitude slower than laboratory estimates. WW represents the reaction rate constant determined by Walther and Wood 1984; SKL represents that of Schramke et al. 1987

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