Pressure solution at grain-to-grain contacts

0016 7037,7X~O901-13X3102.00/0

Geochimica et Cosmochimica Acta, Vol. 42. PP. 1383 to 1389. 0 Pergamon Press Ltd. 1975 Printed m Greal Britain

Pressure solution at grain-to-grain contacts* PIERRE-YVESF. ROBIN Department of Geology and Erindale College, University of Toronto, Mississauga, Ontario, Canada (Received 4 January

1978; accepted in revised form 3 May 1978)

Abstract-Whereas much petrographic evidence for pressure solution. in sedimentary rocks has been accumulated since Sorby’s work, its thermodynamic justification has never been clearly established, and has been challenged by some authors. Difficulties disappear when it is recognized that in the most general case migration of chemical components proceeds down chemical-potential gradients rather than down concentration gradients. Along a grain contact a chemical-potential gradient is produced by variations in contact pressure and by local variations in Helmholtz energy of the solid. For example, in a sand made up of even-sized spherical grains buried 5OOm, the ratio of the diameter (D) of the grains to that of a spherical elastic contact circle (a) is D/a 2 26. The chemical potential at the center of such an elastic contact is = 14 kcal mol-’ higher than in surrounding pore water saturated with respect to quartz. For comparison, at a temperature of 30°C saturation with respect to amorphous silica rather than quartz raises the chemical potential by only 1.6 kcal mol-‘. If the diameter of the contact circle has enlarged to e.g. a = D/5, the chemical potential at its center is still 0.5 kcal mol- ’ greater than that of free quartz under hydrostatic pressure. The corresponding potential gradients are the driving force of pressure solution. The concept of pressure solution thus does not contradict any thermodynamic principle; in particular, it does not require that the chemical component of the solid have a smaller partial volume in solution than in the solid state. Petrographic and experimental evidence can therefore be accepted without reservation. INTRODUCTION

WITHIN a buried granular aggregate, pressure on the boundary of a grain is not uniformly equal to the lithostatic pressure. Stresses higher than lithostatic occur where the grain is in contact with its neighbours. On the other hand, the ‘free’ portion of its boundary is merely subjected to the pressure of the pore fluid, which is generally less than lithostatic. Pressure solution is the process whereby the substance of the grain is allegedly dissolved at the highpressure points and may be precipitated on the free portion of its boundary. Chemical components of the solid are assumed to move along the contact boundary into the free pore water. Diffusion proceeds along the contact through an adsorbed fluid-boundary film, and/or through the lattice of inert crystals such as clay minerals sandwiched between the two grains (WEYL, 1959), and/or along the disordered region between the two neighbouting lattices (ELLIOTT, 1973). Starting with SORBY(1863a,b), pressure solution has been called upon to explain many compaction and deformation features of sedimentary and metamorphic rocks. Recent reviews of the subject have been given by ELLIOTT(1973), DEELMAN(1974), MOTHER(1976) and SPRUNTand NUR (1977). The connection between pressure solution and the phenomenon known in metals and ceramics as Coble Creep is reviewed by MCCLAY (1977) and ROBIN (sub* This paper was presented at the Second Symposium on Water-Rock Interaction, Strasbourg, France, August 1977. The present version is an expansion and modification of what appeared in the Symposium proceedings (ROBIN, 1977).

mitted for publication). Experimental evidence (SPRUNT and NUR, 1977; DE BOER et al., 1977) as well as petrographic evidence for pressure solution have generally been accepted as convincing. However ELLIOTT(1973, p. 2646) notes that “. . the theoretical understanding of pressure solution is in a much more unhappy situation”. Problems with existing theories led DEELMAN(1974, p. 237) to deny any “. . theoretical foundation [to] the concept of pressure solution”. The purpose of this work is to show that pressure solution has in fact a sound thermodynamic basis, and that simple models can lead to straightforward quantitative evaluations. EXISTING THERMODYNAMIC

THEORIES

Authors of the existing models of pressure solution have generally required, or tried to prove that the local equilibrium concentration of the substance of the dissolving grains at points within the contact boundary increases with an increase in pressure normal to the boundary. Only then, according to Fick’s law of diffusion would the dissolved chemical component be transported from the high-pressure to the low-pressure portion of the boundary. SORBY(1863a) had studied experimentally the effect of pressure on the solubility of various salts. He found that, depending on whether their crystallihe volume was larger or smaller than their partial volume in solution, their solubility appeared to be increased or decreased, respectively, by an increase in pressure. SORBY (1863a,b) consequently suggested (cautiously and, to his credit, not v&y definitely) that such considerations of partial volumes were important in explaining pressure solution phenomena in rocks. A work by RIECKE (1895) next gave rise to a still often quoted ‘justification’ of pressure solution called Riecke’s principle (e.g. BILLINGS,1972,p. 428; PETTIJOHN, 1975, p. 243; SPENCER, 1977; pp. 212, 225). Riecke’s actual work had little to do with pressure solution (e.g. TURNER and

1383

P.-Y. F. ROBIN

1384

V~IKHOOGEN, 1960. p. 476; VOLL. 1960). Riecke only studied the equilibrium of a homogeneously stressed solid in contact with a fluid. Furthermore. Riecke did not discuss solubility or its variations; he only considered a fluid offixed composition, in particular the case of a melt of same com. position as the fluid.* Following Sorby’s suggestions, WEYL(1959) states that pressure solution will occur only if pressure enhance the solubility of the grains. Weyl assumes that a positive ‘pressure coefficient of solubility’ obtains for most crystalsolvent combinations, as is indeed the case for both quartz and calcite in water. WEYL (1959, p. 2024) further speculates that minerals with a negative coefficient “. . will have a rather strange behavior. for at points of stress the force of crystallization will manifest itself and will further increase the stress.. [this] will result in the growth of needles in the direction of maximum principal stress”. DEELMAN (1974. 1975) argued on the contrary that the partial molar volume of the grain-forming substance is generally expected to be larger in the disordered dissolved state than in the crystalline state. And indeed solubility data such as for quartz only prove that the combined volume of [dissolved silica + associated surrounding water] is less than that of [crystalline quartz] + [free water]. ELLIOTT (1973). following earlier work by CORRENS (1949). attempted, in effect. to prove that the pressure coefficient of solubility was always positive. However, a critical examination of Elliott’s derivation substantiates Deelman’s objection, which can be written with the usual thermodynamic formalism. For a one-component phase, in this case the solid s. the molar internal energy of which is a function only of temperature. X and of a hydrostatic pressure, P. (&l’/2P)r = t;,,

(I)

where p‘ is the chemical potential of its unique component and v. is its molar volume (e.g. DENBIGH,1971. p. 102). ELLIOTT (1973) considered the equilibrium between the solid and what he called the ‘diffusion path’, that is, this part of the system where significant diffusion can occur. In the context of pressure solution. the diffusion path is the boundary between two grains. Elliott assimilated the matter within the diffusion path to a fluid solution, assumed to be ideal for simplicity. The chemical potential of the component of the solid at a point within the fluid is then given, in terms of its concentration, by p’ = n’ + RT In c.

(2)

Elliott assumed that Jo’. which he called “the chemical potential at the standard state” (p. 2647). was inde~ndent of the pressure on the diffusion path at that point. Elliott thus reasoned that if the ‘diffusion path’ is in chemical equilibrium with the directly adjacent solid. variations in pressure on the solid along the diffusion path are matched by variations in concentrations within the path, i.e. combining (I) and (2) (c?In cjaP), = B;lR T. (3) * Even within its restricted scope. Riecke’s analysis is unsatisfactory in several respects, and in direct contradiction with what is usually stated as Riecke’s principle. Riecke assumed that a stressed solid was characterized by a unique chemical potential, P. whereas Grear (1877) had already demonstrated that this was not the case. Also. Riecke envisaged that a solid could be in contact and in ~uilibrium along all of its surface area with one fluid at pressure P, while the solid itself was under a different, nonhydrostatic stress. No description of how this could be mechanically achieved was given. Thus Riecke did not recognize that fluids in contact with the various faces of a single stressed solid would reach different equilibria, depending on the pressure across their particular interface (GIBIB, 1877).

Since R. Tand ii’ are positive, equation (3) expresses that solubility always increases with pressure. However, pressure at a point of the boundary is the same as that on the surface of the adjacent solid. There is for the diffusion path ‘fluid’, at constant composition, a relation equivalent to equation (I)+ specifically @!-f/dP)T., = V’,

W

where v’ is the partial molar volume of the component of the solid in the ‘fluid’. For Elliott’s ideal solution, p’ varies therefore as (&!ll’/dPfr,c= V’ (4b) and the correct dependence of equilibrium concentration. c, on pressure is obtained by combining (1). (2) and (4): (d In c/aP)r = (V’ - P’)/R 7:

(5) If B‘ < i7r. c will decrease with an increase in P, as Le Chatelier’s rule would predict. DURNEY(1972) and DE BOER(1977) avoided the above dificulty by assuming implicitly that the Ruid which was in equilibrium with a face of the solid under compressive normal stress un. was itself at a constant pressure P,. One can conceive of an imaginary experiment in which a porous inert matrix applies a stress un on a face of a solid while water at pressure P, is free to circulate through that matrix and to reach chemical equilibrium with the solid along that face. A concentration (DURNEY, 1972) or activity gradient (DE BOER, 1977) in the fluid would thus correspond to a gradient of en along the face of the solid. However, such an experiment is impossible to realize physically because the fluid at pressure P, cannot reach the solid at points where the inert matrix actually contacts the solid with a normal stress higher than P,. This hypothetical experiment is thus an inadequate simulation of pressure solution. De Boer’s conclusion is that WEYL’S (1959) model is correct. Does pressure solution only occure when P‘ exceeds v’? Are observations better explained by non-thermodynamic theories (ISTAPENKO,1968; DEELMAN,1974, l975)? The

answer to both questions is negative. THE DRIVING

FORCE

OF DIFFUSION

The main difficulties with these analyses stem from misapplication of Fick’s law. In one dimension, Fick’s law is commonly written f = -D(dc/dx),

(6)

where J is the flux of the diffusing chemicai component, D is its diffusion constant, and (&/ax) is its concentration gradient. Although convenient, equation (6) hides the fact that the driving force of diffusion is not the concentration gradient but the chemical potential gradient (EINSTEIN,1905; see also e.g. DENBIGH, 1971, pp. W-87). In the absence of gravity or other external body-force field, equilibrium with respect to possible transfer of a chemical component requires that its chemical potential be equalized throughout. This remains true even when pressure is not uniform, as d~onstrated, for example, in GIBFS’ (1877) discussion of osmotic equilibrium. Spontaneous diffusion toward higher chemical potentials would thus violate the laws of thermodynamics. In the presence of pressure or temperature gradients Fick’s law should

therefore

be rewritten

J = -M(dtri;3x).

(7)

Pressure solution at grain-to-grain

Two problems thus arise when applying Fick’s law to pressure solution. Firstly, P varies along the diffusion path. Therefore the more general equation (7) rather than (6) should be used. Secondly, equations such as (2) assume that a grain boundary region can be treated as an homogeneous phase, an assumption which is not generally justified. For example, even if the boundary region is wide and continuous enough to be able to define a macroscopic concentration, c, at any point within it, this concentration varies across the interface as well as along it. For an ideally simple boundary it would vary from c = l/P’ sufficiently far into the grain, down to a minimum in the middle of the boundary region, back to c = l/p within the grain on the other side. Therefore, depending on where one draws the limits of that boundary region, one gets a different value of its average concentration. As GIBES (1877) demonstrated in his discussion of capillarity, such non-uniformity does not invalidate his conclusions on definition and equality of chemical potentials at equilibrium. In ceramic materials the process of sintering in presence of a fluid phase is very similar to that of pressure solution. That chemical potential, rather than concentration gradient, is the driving force of diffusion transfer during sintering has been recognized by ceramicists (KINGERY, 1959). However, in order to do rate calculations, Kingery goes on to identify chemical potential gradient and concentration gradient. This latter part of his analysis is therefore similar to that of ELLIOTT (1973) discussed above, and is subject to the same criticism.

CHEMICAL

POTENTIAL

CONTACT

ALONG

A

1385

contacts

Fig. 1. Sketch of a hypothetical contact between two spherical grains of equal diameters, D. The diameter of the contact area is n. The pressure in the pores is P,.

The pressure P across the contact varies with position along that contact. rate, the solid dissolves into the adjacent boundary region in an attempt to maintain the chemical potential to its equilibrium level. The actual values and variations of the chemical potentials within the boundary region are then set by flow processes which are designated, rather than described, by the terms dissolution and diffusion. Except for the substitution of chemical potentials for concentrations, the process is the same as that described by WEYL (1959). However, even without any knowledge of solubility or of its pressure coefficient, it is possible to evaluate chemical potentials for

simple models of grain boundaries. Gibbs’ equation for stressed solids

The unstressed grain, under a hydrostatic pressure P, and at the same temperature as the sediment, is a convenient choice for reference state, for which, by convention,

BOUNDARY

/lo = F, + P, vi = 0,

Consider a contact between two spherical grains (Fig. 1). Free water surrounding the grains is at a pressure P,. Along the contact the normal pressure P varies as a function of position, r. Many distributions of P are possible; three examples are shown in Fig. 2. If the rate of diffusion is very small the chemical potential of the component of the solid in the contact boundary corresponds to local equilibrium (no dissolution, no growth at that point); it varies along the contact. If diffusion down the resulting chemical-potential gradient occurs at a significant

(l?)

(8)

where F is the molar Hehnholtz energy of the solid. If P is the pressure at a point along the contact, it is convenient to define the quantity P’ = P - P, at that point. Integration of equation (1) at constant temperature would then give, neglecting changes in Ps, /l

=

PV;.

(9)

As pointed out earlier, however, equation (1) is strictly valid only if the solid is submitted to hydrostatic pressure. GIBBS (1877) studied the equilibrium with

T p’

Fig. 2. Three of many possible distributions of pressure p’ = P - P,.

along a contact

between

two grains.

P.-Y. F. ROBIN

1386

respect to growth or dissolution of a stressed solid in contact with a fluid at pressure P. For a one component solid, Gibbs’ equation 387 gives the necessary chemical potential in the fluid as p = F + P t’ where F and v’ are molar Helmholtz energy and volume in rhe strrxwd statr. For our choice of reference state. this equation becomes

p = (F - F,) + P(V’ - Vb) + Pv;, /I = AF + PAii + P’V;.

(lOa)

(lob)

The strain terms AF + PA\t7 are typically small but are not always negligible compared to P’vO. For example. right at the periphery of the contact, the strain energy may be quite large whereas P’pG = 0 (because P’ = 0). However, equation (9) is a good approximation at the center of the contact, where p is maximum, and it will be used in the rest of this paper. Use of equation (IO) would require a complete elastic analysis which is generally very difficult. Pressure

und force

along a contact

The pressure f’ can be integrated over the area of a contact to give an ‘effective force’ F’ across that contact. F’ is a forcelgrain contact and is a function of depth of burial. grain size and grain packing. The total weight of a vertical column of rock is supported along its base by an average pressure. P,. which is the lithostatic pressure. Let us imagine that the actual surface along which P, is applied is chosen such that it only goes through the pore fluid and along grain contacts. P, will be made up of P, acting on the whole area. and of the vertical components of the ‘effective forces’ F’ at the contacts; i.e. F’ must contribute the difference between P, and P,. Given grains of equal diameter, D, in simple cubic packing arrangement (e.g. PETTIJOHN. 1975. Fig. 3--37. case I), we have, therefore. F’ = D’(P,

- P,)

= D2 P,..

where P,, is the eficriou prrssurr, as commonly defined in soil and rock mechanics (e.g. JAEGER and COOK. 1976. $8.8). For hexagonal or cubic closest packing. F’ = ~

I

2,. ;-

D’P,

= 0.354D2P,.

For the present purpose it is sufficient and convenient to take an intermediate value F’ = ~$3 D= P,, = 0.393 D2 P,,.

(11)

Calling a the diameter of a circular contact area, the average value of P’. (P’), across that contact is (P’)

or. using equation

= n]li F’.

(I l), (P’)

= ;

; ii

2P,

(12)

As mentioned earlier. there are many possible distributions of P’ along the contact. even within the constraints of axial symmetry and of monotonic decrease from the center. The three examples shown in Fig. 2 correspond respectively to: (a) a parabolic variation of P’ with distance from the center (e.g. WEYL. 1959); (b) a purely elastic contact between two spherical surfaces (e.g. LANDAU and LIFSHITZ, 1959, p. 30); (c) a constant P’. For any distribution the maximum of P’. at the center of the contact is given by pm,, = 171x (p’).

(13)

where 111is a numerical coefficient. Its value is 2, 1.5 and I. respectively. in each of the three cases of Fig. 2. For the numerical calculations which follow. 17)will simply be assumed to have the intermediate value of 1.5. Combining equations (9). (12) and (13) (with IN = 1.5) yields the value of the chemical potential at the center of the contact: AL”,,,= au;,

D =p..

i’, \u

(14)

Assuming that the pore fluid is in hydrostatic equilibrium, it is a sufficient approximation to take P, to be increasing linearly with burial depth. e.g. at a rate of IO4 Pa/m. Equation (14) may therefore be rewritten as D‘2 /(“1.,X= 7.5 X IO” I;\0 -- 1, ! (1 I

with z in m.

(15)

Variations in packing, or in P’ distribution along a contact could change this result by a factor of two or three. To estimate the magnitudes involved, let us consider a quartzose sand buried SOOm. Let us first assume that the grains are perfectly spherical and that they are in spherical elastic contact (Hertz’s problem, e.g. LANDAU and LIFSHITZ. 1959, p. 30). The diameter of the contact circle would then be given by D/a = 26 (Appendix). For quartz, v,‘ = 23.11 cm3 mol-’ = 0.5523 x 10-5cal Pa-’ mol-‘. Equation (15) then gives the chemical potential at the center to be &ax = 13.X kcal mol-‘. If. for example D = I mm (a coarse sand), then a = 39 {irn. The difference in equilibrium chemical potential between the center of the contact and its periphery, 20pm away. is therefore 13.8 kcal mol-‘. If the corresponding potential gradients can drive silica toward the periphery at a geologically significant rate. pressure solution occurs. The spherical elastic contacts are perhaps a good model for the load concentrations at the onset of pressure solution. Later on. the diameter of the contact may increase to, say D/5. In this case the chemical potential at the center of the contact still exceeds that of quartz at P, by pm._ = 518 cal mol- ‘. For D = I mm, the distance of difffusion along the contact is 0.1 mm.

Pressure solution at grain-to-grain contacts The chemical-potential differences calculated above may be compared with the differences brought about by supersaturation of the solution at constant pressure. For example, the ratio of amorphous silica solubility to that of quartz at 30°C is about 15 (e.g. HOLLAND,1967). Given that dissolved silica is chiefly in the electrically neutral form, H,Si04, one may assume an activity coefficient close to 1. Therefore Ap = RTln 15 = 1.63 kcal mol-‘. DE BOERet al. (1977) in fact, report observing the precipitation of amorphous silica in their pressure solution experiments on quartz sand. DISCUSSION Maximum possible supersaturation

In the above discussion, as in the models of WEYL (1959) the pore water is assumed to be in solution equilibrium with quartz at P,. This corresponds to a situation in which crystallization along the pore walls is more efficient than dissolution and diffusion along the contacts. It is possible however that crystallization of quartz would be slow compared to solution and diffusion. In the limiting case, there would be no crystallization in the pores, for example as a result of the free boundaries of the grains not permitting any nucleation and growth of new quartz; on the other hand rapid dissolution and diffusion would lead to a un$orm chemical potential along the contacts. The pressure distribution would be that of Fig. 2c. In effect, the pore fluid would then be in equilibrium with the ‘contact quartz’ rather than with quartz at hydrostatic P,. This limiting case gives the maximum value of supersaturation of the pore fluid to be expected from pressure solution alone. It is sufficient to set m = 1 (instead of 1.5) in equation (13) and rewrite equation (15) accordingly as: 02 PC,,, = 5 x 103. u;, 2. a

0

Thus, at a 500 m burial depth, P,,,.~= 5.52 kcal mol-’ for D/a = 20, 1.38 kcal mol-’ for D/a = 190.345 kcal mol-’ for D/a = 5, etc. In practice, at high supcrsaturation levels, amorphous or microcrystalline forms of silica would crystallize fast. It may therefore be more appropriate to examine this situation the other way around, with the saturation level in the pores controlling the value of D/a. Stepwise marginal dissolution and crushing

Clearly then, the details of the progress of pressure solution along a contact may be quite complicated. They depend on the relative rates of dissolution and diffusion along the boundaries, and on the level of chemical potential maintained in the pore water. In some cases, diffusion along the contacts could be too slow whereas dissolution of the grains at the strained contact periphery could be quite efficient. Contact areas would then decrease in size. rather than enlaree._.

1387

Ultimately, the remaining contacting quartz could be crushed mechanically. This is the mechanism of ‘pressure solution by marginal dissolution and crushing’ described by WEYL(1959, Fig. 2) and OSTAPENKO (1975). It is probably inappropriate to describe this model as a non-thermodynamic one (OSTAPENKO,1968, 1975) because, after all, the dissolution of quartz at the periphery of the contact is still due to a high strain energy, AF + PAV in equation (lob). Whether that strain energy is stored elastically or in the form of a high density of lattice dislocations does not alter the basic thermodynamic equation. In practice, it is doubtful that such mechanism would occur in definite cycles, as implied by WEYL (1959) and OSTAPENKO(1975). Crushing of quartz along the periphery of the contact could be occurring at any time (due to the stress concentration at points of this periphery), while the center of the contact would remain intact. The two grains may then regain contact outward of the crushed zone. This would, firstly, decrease the pressure to be supported by the otherwise shrinking contact area, and thus stop the inward propagation of the crushing zone. Secondly, it would ‘trap’ such zone of crushed quartz along the contact. This crushed quartz is, in effect, silica in a higher energy form than the surrounding quartz. It is therefore likely to be eliminated, for example by growth of one of the two grains at its expense. An explanation of microstylolitic contacts between quartz grains may well involve such crushing as one step of the process. Crystallization pressure

It is also possible for the pore water to be supersaturated in excess of the equilibrium with the ‘contact quartz’. For example the pore water may be percolating from a greater depth, where it equilibrated with quartz at higher temperature and/or with more heavily loaded grain contacts; it may also have become supersaturated through the various other mechanisms reviewed by PEITIJOHN(1975, p. 243). It is reasonable to expect silica to precipitate first along the walls of the pores. However, if sufficient supersaturation is maintained, crystallization could also occur along the contact, pushing the grains apart. WEYL (1959, p. 2021) reviewed the experimental evidence for such process. The maximum contact pressure, P’, which a given supersaturation may overcome in this manner is given by equation (9) in which, it is recalled, the reference level for chemical potential (i.e. p = 0) is that of free quartz at P,. Textures of some chalcedony-cemented sandstones can perhaps be interpreted as resulting from such a process. Pressure solution in other environments

While pressure solution between grains has been emphasized in the present work, its mechanism is also called upon to explain other phenomena in rocks. Dissolution of material along stylolite seams (chiefly

1388

P.-Y. F. ROBIN

but not solely in carbonate

rocks, PETTIJOHN,1975,

p. 340) is perhaps the best known example.

The chemical potential of e.g. calcite within the stylolite seam is controlled by the maximum compressive stress, er, acting perpendicularly to the stylolite. The chemical potential in pores of the rock (where the pore pressure P, < a,) or in veins perpendicular to the stylolites (in which case P, = aj) is set by P,. The driving force for diffusion is therefore the potential difference Ap = (ot - P,)v. The process is very similar to that of diffusion creep, particularly boundary diffusion creep (COBLE,1963). Stylolite formation differs from ‘Coble creep’ in two ways. First, the rock is probably often an open system, with a net gain or loss of material, brought in or taken away by fluid flow. Second the transfer distances evidenced by stylolites are much larger than the dimensions of single crystals. That diffusion transfer over relatively large distances can occur at near-surface temperatures may well be related to the high water content and to the presence of clays and micas serving as ‘avenues’ for diffusion. Much of the discussion by WEYL (1959, pp. 2016-2019) on the likely effect of clays and micas on stylohte formation remains quite convincing to the present author. A recent, general review of diffusion flow in rocks (theory and petrographic evidence) is given by ROBIN (submitted

for publication).

GIBBSJ. W. (1877) On the equilibrium of heterogeneous substances. Transactions of the Connecticut Academy 3, 108-248 and 343-524. (Also appeared in: The scientific papers of J. Willard Gibbs, Vol. one. Thermod~narnj~,.s, Longmans, Green and Cy, 1906, pp, 55-353). HOLLANDH. D. (1967) Gangue minerals in hydrothermal deposits. In Geochemistry of Hydrothermal Ore Deposits (ed. H. L. Barnes), pp. 382436. Hoft, Rinehart and Winston. JAEGERJ. C. and CIXK N. G. W. (1976) Fundamenrals o/ Rock Mechanics, 2nd ed. Chapman and Half. KINGERYW. D. (1959) Densification during sintering in the presence of a liquid phase-I. Theory. J. Appt. Phys. 30, 301-306. LANDAUL. D. and LIF~HITZE. M. (1959) Theory o/Elasticify. Pergamon Press. M&LAY K. R. (1977) Pressure solution and Cobfe creep in rocks and minerals: a review and a conference report. J. &of. Sot. Lond. 134, 57-75. MWHER S. (1976) Pressure solution as a deformation mechanism in Pennsylvanian conglomerates from Rhode Island. J. Geol. 84. 355363. OSTAPENKOG. T. (1968) Recrystallization of minerals under stress. Geokhimiya 2. 234237 (Transl. Geochem. Int. 5, 183- 186.) OSTAPENKOG. T. (1975) Theories of focal and absolute chemical potential, their experimental testing and application of the phase rule to the systems with nonhydrostatically stressed solid phases. Geokhimiya 3, 460-471. (Transl. Geochem. Int. 12, 126135.) PETTIJOHNF. J. (1975) Sedimentary Rocks. 3rd ed. Harper and Row. RIECKEE. (1895) Ueber das Gfeichgewicht zwischen einem festen. homogen deformirten K&per und einer ffiissigen Phase, insbedondere ifber die Depression des Schmelzpunktes durch einseitige Spannung. Ann. Physik 54,

Acknowledgemmrs-This work was supported by the National Research Council of Canada. R. C. FLETCHER. 73 f-738. M. M. KIMBERLEY. A. LASAGAand W. M. SCHWERDTNER ROBINP.-Y. F. (1977) Thermodynamic analysis of pressure reviewed the manuscript and suggested numerous improvesolution at grain-to-grain contacts. Procgs. of rhe Second ments. .$mPosium on Water-Rock Interaction. Strasbourg. France, Section IV, 12% f37. ROBIN P.-Y. F. (submitted) Theory of metamorphic segregation and related processes. Submitted to Geoc~i~. REFERENCES Cosmochim.

BILLINGSM. P. (1972) Structural

Geology,

3rd ed. Prentice-

Half. BOERR. B. DE (1977) On the thermodynamics of pressure solution-interaction between chemical and mechanical forces. Geochim. Cosmochim. Acta 41, 249-256. BOER R. 8. DE, NAGTEGAALP. J. C. and DUYVISE. M. (1977) Pressure solution experiments on quartz sand. Geochim.

Cosmochim.

Acta 41, 257-264.

COBLER. L. (1963) A model for boundary diffusion controlled creep in polycrystaffine material. J. Appt. Phys. 34, 1679-1682.

CORRENSC. W. (1949) Growth and dissolution of crystals under linear pressure. Discuss. Fur&y Sot. 5, 16791782. DEELMAN J. C. (1974) Granulo-mechanical aspects of fithification-H. N. Jahrbuch Geol. Paliiont. Abh. 147,

Acta.

SPENCERE. W. (1977) Introduction to the Structure of the Earth. McGraw-Hill. SORBY H. C. (1863a) The Bakerian lecture. On the direct correlation of mechanical and chemical forces. Ray. Sot. Land. Proc. 12, 538-550. SORBY H. C. (1863b) ober Kafkstein-Gesch~ebe mit Eindriicken. Jamb. Miner~~~~e, 801-807. SPRUNTE. S. and NUR A. (1977) Destruction of porosity through pressure solution. Geophysics 42. 726-741. TURNERF. J. and VERH~~GENJ. (1960) fgneous and metcrmorphic petrology, 2nd ed. McGraw-Hill. VOLLG. (1960) New work on petrofabrics. Geol. J. (Liverpool and Manchester) 2, 502-597. WEYL(1959) Pressure solution and the force of crystaffization-a phenomenologicaf theory. J. Geuphys. Res. 64, 200

I -2025.

237-268.

DEELMANJ. C. (1975) ‘Pressure solution or indentation? Geology 3, 23-24. DENBIGHK,. (1971) The Principles of Chemical Equilibrium. 3rd ed. Cambridge University Press. DURNEYD. W. (1972) Solution-transfer. an important geological deformation mechanism. Nature 235, 3 15 3 17. ELLIOTTD. (1973) Diffusion flow laws in metamorphic rocks. Bull. Geol. Sot. Am. 84, 2645-2664. EINSTEINA. (1905) Uber die von der molekufarkinetischen Theorie der Warme geforderte Bewegung von in ruhenden Fftissigkeiten suspendierten Teifchen. Annal. Phys. 17, 549560.

APPENDIX: DIAMETERS OF HERTZIAN CONTACTS BETWEEN GRAINS The two grains are assumed to be in elastic contact and to be perfectly spherical. That is, their ‘packing radius’. D/2. as used in equation (I 1). is identical to their stress-free radius of curvature at the contact, as used in the elastic analysis [equation (Al)]. The diameter of the circle of contact is then (e.g. LANDAUand LIFSHITZ,1959. p. 30) .,1,3 T?,i _ ,,I\ -“I (Al) a=

L---9

DF’ J’

Pressure solution al grain-t~grain where E and Y are respectively Young’s modulus and Poisson’s ratio. Using equation (11) and a gradient of P, with burial depth of lO*Pa/m. f’ = ~$5 D2

x 10’~.

with z in m.

(A21

Combining (AI) and (A2) gives D -_= a

C

(A31

1389

contacts

For quartz we can take equivalent isotropic elastic constants to be E = 8.7 x IO’O Pa (0.87 Mbar) and Y = 0.375. in which case: D/a = 2.05 :-li3. For a burial depth of i = 500m. D/a = 26.