The role of background electrolytes on the kinetics and mechanism of calcite dissolution

The influence of background electrolytes on the mechanism and kinetics of calcite dissolution was investigated using in situ Atomic Force Microscopy (AFM). Experiments were carried out far from equilibrium by passing alkali halide salt (NaCl, NaF, NaI, KCl and LiCl) solutions over calcite cleavage surfaces. This AFM study shows that all the electrolytes tested enhance the calcite dissolution rate. The effect and its magnitude is determined by the nature and concentration of the electrolyte solution. Changes in morphology of dissolution etch pits and dissolution rates are interpreted in terms of modification in water structure dynamics (i.e. in the activation energy barrier of breaking water-water interactions), as well as solute and surface hydration induced by the presence of different ions in solution. At low ionic strength, stabilization of water hydration shells of calcium ions by non-paired electrolytes leads to a reduction in the calcite dissolution rate compared to pure water. At high ionic strength, salts with a common anion yield similar dissolution rates, increasing in the order Cl⁻ < I⁻ < F⁻ for salts with a common cation due to an increasing mobility of water around the calcium ion. Changes in etch pit morphology observed in the presence of F⁻ and Li⁺ are explained by stabilization of etch pit edges bonded by like-charged ions and ion incorporation, respectively. As previously reported and confirmed here for the case of F⁻, highly hydrated ions increased the etch pit nucleation density on calcite surfaces compared to pure water. This may be related to a reduction in the energy barrier for etch pit nucleation due to disruption of the surface hydration layer.     

Specific effects of background electrolytes on the kinetics of step propagation during calcite growth

The mechanisms by which background electrolytes modify the kinetics of non-equivalent step propagation during calcite growth were investigated using Atomic Force Microscopy (AFM), at constant driving force and solution stoichiometry. Our results suggest that the acute step spreading rate is controlled by kink-site nucleation and, ultimately, by the dehydration of surface sites, while the velocity of obtuse step advancement is mainly determined by hydration of calcium ions in solution. According to our results, kink nucleation at acute steps could be promoted by carbonate-assisted calcium attachment. The different sensitivity of obtuse and acute step propagation kinetics to cation and surface hydration could be the origin of the reversed geometries of calcite growth hillocks (i.e., rate of obtuse step spreading < rate of acute step spreading) observed in concentrated (ionic strength, IS = 0.1) KCl and CsCl solutions. At low IS (0.02), ion-specific effects seem to be mainly associated with changes in the solvation environment of calcium ions in solution. With increasing electrolyte concentration, the stabilization of surface water by weakly paired salts appears to become increasingly important in determining step spreading rate. At high ionic strength (IS = 0.1), overall calcite growth rates increased with increasing hydration of calcium in solution (i.e., decreasing ion pairing of background electrolytes for sodium-bearing salts) and with decreasing hydration of the carbonate surface site (i.e., increasing ion pairing for chloride-bearing salts). Changes in growth hillock morphology were observed in the presence of Li⁺, F⁻ and , and can be interpreted as the result of the stabilization of polar surfaces due to increased ion hydration. These results increase our ability to predict crystal reactivity in natural fluids which contain significant amounts of solutes.     

Cronobacter sakazakii还原作用对针铁矿晶体结构的影响

厌氧条件下,Cronobacter sakazakii以乙酸钠作为电子供体,针铁矿中Fe(Ⅲ)作为电子受体进行生命活动,其新陈代谢过程伴随Fe(Ⅲ)的还原。细菌增殖和稳定生长过程中不停还原针铁矿并大量累积Fe(Ⅱ);当细菌衰亡时,Fe(Ⅱ)的产生随之减缓;细菌的活动停止时,Fe(Ⅱ)不再积累并最终保持稳定。同步辐射XRD测试表明,微生物还原作用后针铁矿出现了一系列新衍射峰：4.8、6.03、6.13、6.84、7.7和11.4 峰,可能形成具层状结构的新物相。在XANES图谱中Fe主吸收峰向低能量方向移动1 eV,边前峰峰位中心向低能量方向移动且峰面积减小,表明Cronobacter sakazakii的异化Fe(Ⅲ)还原作用使针铁矿中Fe氧化态降低,矿物晶体结构发生了变化。     

The effect of Fe-oxidizing bacteria on Fe-silicate mineral dissolution

Acidithiobacillus ferrooxidans are commonly present in acid mine drainage (AMD). A. ferrooxidans derive metabolic energy from oxidation of Fe²⁺ present in natural acid solutions and also may be able to utilize Fe²⁺ released by dissolution of silicate minerals during acid neutralization reactions. Natural and synthetic fayalites were reacted in solutions with initial pH values of 2.0, 3.0 and 4.0 in the presence of A. ferrooxidans and in abiotic solutions in order to determine whether these chemolithotrophic bacteria can be sustained by acid-promoted fayalite dissolution and to measure the impact of their metabolism on acid neutralization rates. The production of almost the maximum Fe³⁺ from the available Fe in solution in microbial experiments (compared to no production of Fe³⁺ in abiotic controls) confirms A. ferrooxidans metabolism. Furthermore, cell division was detected and the total cell numbers increased over the duration of experiments. Thus, over the pH range 2–4, fayalite dissolution can sustain growth of A. ferrooxidans. However, ferric iron released by A. ferrooxidans metabolism dramatically inhibited dissolution rates by 50–98% compared to the abiotic controls.

Two sets of abiotic experiments were conducted to determine why microbial iron oxidation suppressed fayalite dissolution. Firstly, fayalite was dissolved at pH 2 in fully oxygenated and anoxic solutions. No significant difference was observed between rates in these experiments, as expected, due to extremely slow inorganic ferrous iron oxidation rates at pH 2. Experiments were also carried out to determine the effects of the concentrations of Fe²⁺, Mg²⁺ and Fe³⁺ on fayalite dissolution. Neither Fe²⁺ nor Mg²⁺ had an effect on the dissolution reaction. However, Fe³⁺, in the solution, inhibited both silica and iron release in the control, very similar to the biologically mediated fayalite dissolution reaction. Because ferric iron produced in microbial experiments was partitioned into nanocrystalline goethite (with very low Si) that was loosely associated with fayalite surfaces or coated the A. ferrooxidans cells, the decreased rates of accumulation of Fe and Si in solution cannot be attributed to diffusion inhibition by goethite or to precipitation of Fe–Si-rich minerals. The magnitude of the effect of Fe³⁺ addition (or enzymatic iron oxidation) on fayalite dissolution rates, especially at low extents of fayalite reaction, is most consistent with suppression of dissolution by interaction between Fe³⁺ and surface sites. These results suggest that microorganisms can significantly reduce the rate at which silicate hydrolysis reactions can neutralize acidic solutions in the environment.     

Characterization of synthetic nanocrystalline mackinawite: crystal structure, particle size, and specific surface area

Iron sulfide was synthesized by reacting aqueous solutions of sodium sulfide and ferrous chloride for 3 days. By X-ray powder diffraction (XRPD), the resultant phase was determined to be primarily nanocrystalline mackinawite (space group: P4/ nmm) with unit cell parameters a = b = 3.67 Å and c = 5.20 Å. Iron K-edge XAS analysis also indicated the dominance of mackinawite. Lattice expansion of synthetic mackinawite was observed along the c-axis relative to well-crystalline mackinawite. Compared with relatively short-aged phase, the mackinawite prepared here was composed of larger crystallites with less elongated lattice spacings. The direct observation of lattice fringes by HR-TEM verified the applicability of Bragg diffraction in determining the lattice parameters of nanocrystalline mackinawite from XRPD patterns. Estimated particle size and external specific surface area (SSA_ext) of nanocrystalline mackinawite varied significantly with the methods used. The use of Scherrer equation for measuring crystallite size based on XRPD patterns is limited by uncertainty of the Scherrer constant (K) due to the presence of polydisperse particles. The presence of polycrystalline particles may also lead to inaccurate particle size estimation by Scherrer equation, given that crystallite and particle sizes are not equivalent. The TEM observation yielded the smallest SSA_ext of 103 m²/g. This measurement was not representative of dispersed particles due to particle aggregation from drying during sample preparation. In contrast, EGME method and PCS measurement yielded higher SSA_ext (276-345 m²/g by EGME and 424 ± 130 m²/g by PCS). These were in reasonable agreement with those previously measured by the methods insensitive to particle aggregation.     

Kinetics of crystal nucleation in ionic solutions: Electrostatics and hydration forces

The heat of precipitation, the mean crystal size and the broadness of crystal size distribution of barium sulfate precipitating in aqueous solutions of different background electrolytes (KCl, NaCl, LiCl, NaBr or NaF), was shown to vary at constant thermodynamic driving force (supersaturation) and constant ionic strength depending on the salt present in solution. The relative inversion in the effect of respective background ions on the characteristics of barite precipitate was observed between two studied supersaturation (Ω) and ionic strength (IS) conditions. The crystal size variance (β²) increased in the presence of background electrolytes in the order LiCl < NaCl < KCl at Ω = 10^3.33 and IS = 0.03 M and KCl < NaCl < LiCl at Ω = 10^3.77 and IS = 0.09 M. At a given Ω and IS the respective size of barite crystals decreased with increasing β² in chloride salts of different cations and remained constant in sodium salts of different anions.We suggest that ionic salts affect the kinetics of barite nucleation and growth due to their influence on water of solvation and bulk solvent structure. This idea is consistent with the hypothesis that the kinetic barrier for barium sulfate nucleation depends on the frequency of water exchange around respective building units that can be modified by additives present in solution. In electrolyte solution the relative switchover between long range electrostatic interactions and short range hydration forces, which influence the dynamics of solvent exchange between an ion solvation shell and bulk fluid, results in the observed inversion in the effect of differently hydrated salts on nucleation rates and the resulting precipitate characteristics.     

The role of magnesium in the crystal growth of calcite and aragonite from sea water

The seeded precipitation (crystal growth) of aragonite and calcite from sea water, magnesium-depleted sea water, and magnesium-free sea water has been studied by means of the steady-state disequilibrium initial rate method. Dissolved magnesium at sea water levels appears to have no effect on the rate of crystal growth of aragonite, but a strong retarding effect on that of calcite. By contrast, at levels less than about 5 per cent of the sea water level, Mg has little or no effect on calcite growth. Extended crystal growth on pure calcite seeds in sea water of normal Mg content resulted in the crystallization of magnesium calcite overgrowths, containing 7–10 mole % MgCO₃ in solid solution. This suggests that the rate inhibition by Mg is due to its incorporation within the calcite crystal structure during growth, which causes the resulting magnesian calcite to be considerably more soluble than pure calcite. The standard free energy of formation of 8.5 mole% Mg calcite calculated on this assumption is in good agreement with independent estimates of magnesian calcite stability.From the work of Katz (Geochim. Cosmochim. Acta37, 1563–1586, 1973), Plummer and Mackenzie (Amer. J. Sci. 273, 515–522, 1974), and the present paper, it can be predicted that the most stable calcite in Ca-Mg exchange equilibrium with sea water contains between 2 and 7 mole%MgCO₃ in solid solution. Likewise, calcites containing more than 8.5 mole% MgCO₃ are less stable, and those containing less than 8.5 mole% MgCO₃ are more stable than aragonite plus Ca and Mg in sea water.     

Catchments as reactors: a comprehensive approach for water fluxes and solute turnover

Sustainable water quality management requires a profound understanding of water fluxes (precipitation, run-off, recharge, etc.) and solute turnover such as retention, reaction, transformation, etc. at the catchment or landscape scale. The Water and Earth System Science competence cluster (WESS, http://www.wess.info/) aims at a holistic analysis of the water cycle coupled to reactive solute transport, including soil–plant–atmosphere and groundwater–surface water interactions. To facilitate exploring the impact of land-use and climate changes on water cycling and water quality, special emphasis is placed on feedbacks between the atmosphere, the land surface, and the subsurface. A major challenge lies in bridging the scales in monitoring and modeling of surface/subsurface versus atmospheric processes. The field work follows the approach of contrasting catchments, i.e. neighboring watersheds with different land use or similar watersheds with different climate. This paper introduces the featured catchments and explains methodologies of WESS by selected examples.     

The effect of oxalate on the dissolution rates of oligoclase and tremolite

The effect of oxalate, a strong chelator for Al and other cations, on the dissolution rates of oligoclase feldspar and tremolite amphibole was investigated in a flow-through reactor at 22°C. Oxalate at concentrations of 0.5 and 1 mM has essentially no effect on the dissolution rate of tremolite, nor on the steady-state rate of release of Si from oligoclase. The fact that oxalate has no effect on dissolution rate suggests that detachment of Si rather than Al or Mg is the rate-limiting step. At pH 4 and 9, oxalate has no effect on the steady-state rate of release of Al, and dissolution is congruent. At pH 5 and 7, oligoclase dissolution is congruent in the presence of oxalate, but in the absence of oxalate Al is preferentially retained in the solid relative to Si.Large transient “spikes” of Al or Si are observed when oxalate is added to or removed from the system. The cause of the spikes is unknown; we suggest adsorption on feldspar surfaces away from sites of active dissolution as a possibility. Solutions in the reactors are undersaturated with respect to both gibbsite and kaolinite, so neither the spikes nor the incongruent dissolution can be explained by formation of a secondary precipitate.The rate of dissolution of tremolite is independent of pH over the pH range 2–5, and decreases at higher pH. The rate of dissolution of oligoclase in our experiments was independent of pH over the pH range 4–9. Since the dissolution rate of these minerals is independent of pH and organic ligand concentration, the effect of acid deposition from the atmosphere on the rate of supply of cations from weathering of granitic rocks should be minor.     

Epiphyte-grazer relationships in seagrass meadows: Consequences for seagrass growth and production

Studies of seagrass meadows have shown that the production of algal epiphytes attached to seagrass blades approaches 20% of the seagrass production and that epiphytes are more important as food for associated fauna than are the more refractory seagrass blades. Since epiphytes may compete with seagrasses for light and water column nutrients, excessive epiphytic fouling could have serious consequences for seagrass growth. We summarize much of the literature on epiphytegrazer relationships in seagrass meadows within the context of seagrass growth and production. We also provide insights from mathematical modeling simulations of these relationships for a Chesapeake BayZostera marina meadow. Finally we focus on future research needs for more completely understanding the influences that epiphyte grazers have on seagrass production.     

Consequences of crystal shape and fabric on anisotropic permeability in magmatic mush

Crystals that form an interconnected porous network can become preferentially oriented both prior to and during compaction of magmatic mush. This introduces anisotropy in the melt pore-space that can reduce permeability in the direction of compaction and in turn decrease melt flux and compaction rate. Using a number of grain-scale numerical models, the consequences of end-member magmatic fabrics on the directional dependence of permeability are tested over a range in melt fraction from 22 to 77%. As the crystal aspect ratio (i.e. ratio of long to short axis length) increases from 2 to 10, isotropic permeability decreases by a factor of 2 and 5 for randomly oriented prolate and oblate-shaped crystals, respectively, at a melt fraction of 22%. With a flattening fabric, permeability is reduced in the compaction direction no more than approximately a factor of 2 relative to the isotropic permeability at the same melt fraction and crystal shape for both oblate and triaxial prisms. However, permeability is enhanced in directions orthogonal to the compaction direction. For example, permeability is enhanced up to a factor of 11 relative to the isotropic permeability at a melt fraction of 22% for oblate prisms with a ratio of the long to short axis length of 10. Anisotropy in permeability increases as the melt fraction decreases and the crystal aspect ratio increases. Ratios of the principal permeabilities are sufficiently large based on the realistic crystal shapes tested here to warrant including anisotropic permeability into macroscale melt segregation models including those for compaction.     

The crystal structure of natural and synthetic holtedahlite

Summary The crystal structure of synthetic holtedahlite, Mg₁₂(PO₃OH, PO4)(PO₄)₅(OH,O)₆,P31m,a = 11.186(3),c = 4.977(1) Å,Z = 1, has been refined toR = 0.033 for 718 observed reflections. Natural holtedahlite, Mg₁₂(PO₃OH, CO₃)(PO₄)₅(OH, O)₆,a = 11.203(3),c = 4.977(1) Å, was refined toR = 0.031 for 1202 observed reflections. The structure contains pairs of face-sharing Mg-octahedra linked by edge-sharing to form double chains alongc. Hydrogen phosphate groups on three-fold axes are partially replaced by carbonate groups in natural holtedahlite. Structural similarities with ellenbergerite are pointed out.

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Die Kristallstruktur von natürlichem und synthetischem Holtedahlit

Zusammenfassung Die Kristallstruktur von synthetischem Holtedahlit, Mg₁₂(PO₃OH,PO₄)(PO₄)₅(OH,O)₆,P31m,a = 11,186(3),c = 4,977(1) Å,Z = 1, wurde für 718 beobachtete Reflexe aufR = 0,033 verfeinert. Natürlicher Holtedahlit, Mg₁₂(PO₃OH, CO₃)(PO₄)₅(OH, O)6,a = 11,203(3),c = 4,977(1) A wurde für 1202 beobachtete Reflexe aufR = 0,031 verfeinert. Die Struktur enthält Paare von über eine Fläche verknüpften Mg-Oktaedern, die weiter durch Kantenverknüpfung Doppelketten nachc bilden. Saure Phosphatgruppen auf dreizähligen Achsen sind im natürlichen Holtedahlit teilweise durch Karbonatgruppen ersetzt. Strukturelle Ähnlichkeiten zu Ellenbergerit werden aufgezeigt.

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真实自回避行走中晶体生长界面结构的分形行为

晶体生长过程实际上就是生长基元从周围环境中不断地通过界面而进入晶格座位的过程。一般认为,研究生长基元以何种方式以及如何通过界面进入晶格座位是晶体生长界面结构研究中的关键。在生长基元以分子或者原子的微粒子形式在生长环境中进行无规游走的前提下,本文运用真实自回避行走(TSAW)模型,通过重整化群思想来研究晶体生长界面结构的分形行为。研究发现:晶体生长界面结构的分形行为与生长基元的游走路径形态密切相关,并且在理想状况下真实自回避行走与标准Koch曲线的分形维极为接近。     

The crystal structure of 2O brittle mica: Anandite

Summary Anandite has an approximate formula of Ba(Fe³⁺, Fe²⁺)₃[Si₂(Fe³⁺, Fe²⁺, Si)₂O_10–x(OH)_x] (S, Cl) (OH), withx=0–1, and belongs to the 2 O brittle mica group. It is orthorhombic; space groupPnmn;a=5.468(9) Å,b=9.489(18)Å,c=19.963(11) Å;Z=4.The structure was determined from 3dim. Weissenberg-data, starting with an approximate structure in the pseudo space groupCcmm. Least squares refinement resulted inR=0.061 for 409 photometric intensities, andR=0.131 for all 853 observedhkl-reflexions.The iron of the tetrahedral layer is concentrated in one of the two crystallographically different kinds of tetrahedra. The basal oxygen rings of the tetrahedral layer form approximate hexagons and have not the ditrigonal configuration of the common micas. This peculiarity is considered to be a consequence of the size and charge of the barium ion. The role of OH in the common micas is played partly by S^2– and Cl^– in anandite.

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Die Kristallstruktur des 2 O Sprödglimmers Anandit

Zusammenfassung Anandit hat die ungefähre Formel Ba(Fe³⁺, Fe²⁺)₃[Si₂(Fe³⁺, Fe²⁺, Si)₂O_10–x(OH)_x] (S, Cl) (OH) mitx=0–1 und gehört zur 2O Sprödglimmergruppe. Er ist rhombisch; RaumgruppePnmn; a=5,468(9) Å,b=9,489(18) Å,c=19,963(11) Å;Z=4.Die Struktur wurde aus Weissenberg-Daten bestimmt, wobei mit einer approximativen Struktur in der PseudoraumpruppeCcmm begonnen wurde. Die Verfeinerung nach der Methode der kleinsten Quadrate führte für 409 photometrierte Reflexe aufR=0,061 und für alle 853 beobachtetenhkl-Reflexe aufR=0,131.Der Eisengehalt der Tetraederschicht ist in einer der beiden kristallographisch verschiedenen Tetraederarten konzentriert. Die basalen Sauerstoffringe der Tetraederschicht bilden annäherungsweise Sechsecke und haben nicht die ditrigonale Konfiguration der gewöhnlichen Glimmer. In Anandit spielen S^2– und Cl^– teilweise die Rolle der Hydroxylgruppen in den gewöhnlichen Glimmern.

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The crystal structure of loparite: a new acentric variety

Mineralogy and Petrology - The crystal structure of a new structural variety of loparite (Na0.56Ce0.21La0.14Ca0.06Sr0.03Nd0.02Pr0.01)Σ=1.03(Ti0.83Nb0.15)Σ=0.98O3 from the Khibiny alkaline...     

Distribution of Fe among octahedral sites and its effect on the crystal structure of pumpellyite

Two pumpellyites with the general formula W ₈ X ₄ Y ₈ Z ₁₂O_56-n (OH) _n were studied using ⁵⁷Fe Mössbauer spectroscopic and X-ray Rietveld methods to investigate the relationship between the crystal chemical behavior of iron and structural change. The samples are ferrian pumpellyite-(Al) collected from Mitsu and Kouragahana, Shimane Peninsula, Japan. Rietveld refinements gave Fe(X):Fe(Y) ratios (%) of 41.5(4):58.5(4) for the Mitsu pumpellyite and 46(1):54(1) for the Kouragahana pumpellyite, where Fe(X) and Fe(Y) represent Fe content at the X and Y sites, respectively. The Mössbauer spectra consisted of two Fe²⁺ and two Fe³⁺ doublets for the Mitsu pumpellyite, and one Fe²⁺ and two Fe³⁺ doublets for the Kouragahana pumpellyite. In terms of the area ratios of the Mössbauer doublets and the Fe(X):Fe(Y) ratios determined by the Rietveld refinements, Fe²⁺(X):Fe³⁺(X):Fe³⁺(Y) ratios are determined to be 22:14:64 for the Mitsu pumpellyite and 27:8:65 for the Kouragahana pumpellyite. By applying the Fe²⁺:Fe³⁺-ratio determined by the Mössbauer analysis and the site occupancies of Fe at the X and Y sites given by the Rietveld method together with chemical analysis, the resulting formula of the Mitsu and Kouragahana pumpellyites are established as Ca₈(Fe _0.88 ²⁺ Mg_0.68Fe _0.77 ³⁺ Al_1.66)_Σ3.99(Al_5.67Fe _2.34 ³⁺ )_Σ8.01Si₁₂O_42.41(OH)_13.59 and Ca₈(Mg_1.24Fe _0.65 ²⁺ Fe _0.46 ³⁺ Al_1.66)_Σ4.01(Al_6.71Fe _1.29 ³⁺ )_Σ8.00Si₁₂O_42.14(OH)_13.86, respectively. Mean Y–O distances and volumes of the YO₆ octahedra increase with increasing mean ionic radii, i.e., the Fe³⁺→Al substitution at the Y site. However, change of the sizes of XO₆ octahedra against the mean ionic radii at the X site is not distinct, and tends to depend on the volume change of the YO₆ octahedra. Thus, the geometrical change of the YO₆ octahedra with Fe³⁺→Al substitution at the Y site is essential for the structural changes of pumpellyite. The expansion of the YO₆ octahedra by the ionic substitution of Fe³⁺ for Al causes gradual change of the octahedra to more symmetrical and regular forms.     

The kinetics of nucleation and crystal growth and scaling laws for magmatic crystallization

Magmatic crystallization depends on the kinetics of nucleation and crystal growth. It occurs over a region of finite thickness called the crystallization interval, which moves into uncrystallized magma. We present a dimensional analysis which allows a simple understanding of the crystallization characteristics. We use scales for the rates of nucleation and crystal growth, denoted by I _m and Y _m respectively. The crystallization time-scale _c and length-scale d _c are given by (Y _m ³ /I _m )^–1/4 and (·) _m ^1/2 respectively, where is thermal diffusivity. The thickness of the crystallization interval is proportional to this length-scale. The scale for crystal sizes is given by (Y _m /I _m )^1/4. We use numerical calculations to derive dimensionless relationships between all the parameters of interest: position of the crystallization front versus time, thickness of the crystallization interval versus time, crystal size versus distance to the margin, temperature versus time. We assess the sensitivity of the results to the form of the kinetic functions. The form of the growth function has little influence on the crystallization behaviour, contrary to that of the nucleation function. This shows that nucleation is the critical process. In natural cases, magmatic crystallization proceeds in continously evolving conditions. Local scaling laws apply, with time and size given by =(Y ³/I)^–1/4 and R=(Y/I)^1/4, where Y and I are the rates at which crystal are grown and nucleated locally. is the time to achieve crystallization and R the mean crystal size. We use these laws together with petrological observations to infer the in-situ values of the rates of nucleation and growth. Two crystallization regimes are defined. In the highly transient conditions prevailing at the margins of basaltic intrusions, undercoolings are high and the peak nucleation and growth rates must be close to 1cm^–3· ^–1 and 10^–7cm/s, in good agreement with laboratory measurements. In quasi-equilibrium conditions prevailing in the interior of large intrusions, undercoolings are small. We find ranges of 10^–7 to 10^–3 cm^–3 s^–1 and of 10^–10 to 10^–8cm/s for the local rates of nucleation and growth respectively.