Glikin A.E. Polymineral-Metasomatic Crystallogenesis

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3.3 Physicochemical Model 101

1. The isotherm line is a region, where the pure metasomatic processes take place,

which, as was shown in Sect. 3.2, can proceed with the volume deficit or excess,

if crystal compositions are above or below the point F, respectively. Trajectories

of changing the solution and crystal compositions coincide with the isotherm

and are directed toward each other. The point where the trajectories meet cor-

responds to an equilibrium state of the system, with its position in the isotherm

depending upon the initial quantitative proportion of the phases. If a large crystal

undergoes replacement in a little volume of solution, the point is located in the

neighborhood of the crystal composition; if the ratio is inversed, the point lies

within the neighborhood of the initial composition of solution F. Thus, there

should be expected a direct dependence of replacement rate upon increasing the

difference between compositions of solution and crystal.

2. Sector aFb delimits compositions of solutions obtained if spontaneous

co-crystallization of both substances takes place in the initially homogeneous

solution F due to lowering the temperature. (Apparently, a region situated

beyond the boundaries of aFb sector can be reached if the system contains an

excess of the solid component A or B, which undergoes dissolution.) Here solu-

tions are supersaturated with each substance; supersaturation for the each of the

components can be derived from the distance between the coordinates of the

initial and the final figurative points on the abscissa and the ordinate.

Proportions of components in solid and liquid phases vary with time. Nevertheless,

the process direction and position of the final figurative point can be determined

with the aid of the described diagram with any prescribed accuracy only in the case

of quasi-equilibrium or equilibrium process.

(a) If a system keeps a state of quasi-equilibrium, any moment a new layer having

a composition, which is in equilibrium with the solution composition, grows on

a crystal. Diffusion rate within the crystal under these conditions is negligible.

To determine a pathway of the process starting from the initial point (F, Fig. 3.7) it

is necessary to set up an absolute mass of the crystallizing phase (the less the value,

the more precise are the calculations). Then, proportion of isomorphic components

in the crystallizing phase can be determined from the initial scale on the isotherm,

and their absolute quantities are calculated. Then, these amounts are subtracted

from the quantities of dissolved components, thus, allowing to evaluate a new com-

position of the solution. After that, the isotherm corresponding to the new composi-

tion of solution is plotted via interpolation, and a composition of the solid phase,

which is in equilibrium with this solution, is estimated with an aid of the solidus

isoline grid. Then the new absolute value of composition of precipitating solid

phase is set again, and the process is repeated till desirable isotherm is obtained.

Appropriate values of the solution composition and total composition of the solid

phase are marked on each isotherm.

Figure 3.7 represents trajectory F–l of the altering solution composition, which

coincides with the trajectory of changing composition of the crystal surface layer,

while the trajectory of the solid-phase total composition coincides with the line F–s,

102 3 Formation of Mixed Crystals in Solutions

which is located below the previous one, as the solution is enriched with more-soluble

component A and the crystal – with less-soluble component B. It is evident, that the

crystal acquires a zonal structure, and, as a whole, the system remains unbalanced.

(b) If the system is in equilibrium state, it differs from the previous case in having

a sufficiently slow rate of precipitation, which is compatible with diffusion rate

in the crystal. This ensures the crystal uniformity. Under these conditions the

system is in equilibrium, and the trajectories F–l and F–s described above coin-

cide with the intermediate line F–e

Plotting the line F–e

involves estimating the compositions of the products formed

via isothermal interaction between the quasi-equilibrium solution and crystal. It is

to be noted that this interaction is a monocrystalline volume-excess replacement. In

this case it is conveniently to use the isotherms plotted during the calculation of

quasi-equilibrium trajectories.

Total proportion of component contents in the solid phase and composition of

the initial solution correspond to the points of intersection of the isotherm and the

lines F–s and F–l, respectively. Then n = 3–5 of control dots are selected randomly

within the interval delimited by the intersection points. Then the changes in con-

centrations of the components contained in the solution are calculated for each

control point with the aid of the diagram of mutual solubility: a

transfers from the

crystal into the solution, while b

precipitates from the solution. Conjectural

amounts of components A′

= A − a

and B′

= B + b

and their ratios A′

/B′

in the

solid phases are calculated for all control points from the absolute amounts of com-

ponents A and B in the precipitated substance, which have been previously derived

in the course of plotting the quasi-equilibrium trajectories. Concurrently, propor-

tions A″

/B″

of the components are estimated with the aid of the scale of the solid-

phase compositions positioned along the isotherm. The point of the solid-phase

composition scale having the same values of A′

/B′

and A″

;

″ is the required

point of the equilibrium line F–e

. Also, it can be determined as an intersection of

the auxiliary graphs A′

/B′

= f ′(n) and A″

/B″

= f ″(n).

Thus, the assembly of trajectories e

′–e

depicted in Fig. 3.7 together with the

isotherms makes up a nomogram, which allows to determine a pathway of equilib-

rium crystallization and thermodynamic supersaturation in any figurative point

under particular supercooling conditions. It is to be noted that the above estimation

technique is valid only for examining a spontaneous crystallization; seed crystalli-

zation (including the seeds, which are in the equilibrium with the initial solution)

changes proportion of components in the system.

specific final supercooling, arbitrary precipitation rate, and absence of equilib-

rium between the crystal surface layer and the solution. Ranges of divergence

of the trajectories F–l and F–s inside the aFb region have not been determined

for real processes. Such divergences may be considerable at sufficiently strong

supercoolings and growth rates. The data obtained by us showed a 30% and

greater divergence of crystal compositions from the equilibrium values in

3.3 Physicochemical Model 103

(Co,Ni)(NH

)

(SO

)

·7H

O–H

O and (Pb,Ba)(NO

)

–H

O systems with super-

cooling ranging from 2 to 20°C and at about 2°C, respectively.

This divergence induces the component interchange between the crystal and solu-

tion, i.e., it determines a metasomatic component of the crystal-formation process.

The crystal becomes enriched with a less-soluble component, so the reaction pro-

ceeds with the volume excess. The reaction must be accompanied by formation of

replacement blocking layers including specific autoepitaxial excrescences, and thus

result in lowering the growth rate and convergence of F–l and F–s trajectories.

These effects become obvious if growth rates of the faces, rates of metasomatic

interchange, and solid-state diffusion interchange along the dislocation channels or

cracks with attached excrescences are commensurable. Combination of autoepitax-

ial excrescences and growth layers on K

(S,Cr)O

crystals (Kasatkin 1993) can be

accounted for by an intensive metasomatic reaction proceeding at the stage of

growing the faces.

Divergence of trajectories at relatively high crystallization rate reflects lowering

the rate of capturing a more-soluble component. At the same time, the solution

composition trajectory flattens having position of the initial point fixed (e.g.,

F–l

and F–l

in Fig. 3.7), and the limiting position of the trajectory is the line paral-

lel to the abscissa (F–a), which corresponds to a zero supersaturation with the

component A and crystallization of the pure component B; this conclusion is appli-

cable for initial solutions of any composition. The trajectory of crystal composition

leaves the initial point and flattens as well, moving toward the abscissa (F

–s

and

–s

corresponding to the trajectories F–l

and F–l

) and coinciding with it if crys-

tallization of the pure component B takes place.

Whether this limit is attainable still remains unclear; however, the above expla-

nation may help in interpreting formation of pure minerals having high isomorphic

capacity under unsterile natural conditions.

It is to be noted that the common consequence of the three phenomena described

above is unavailability of solution compositions below the equilibrium line during

spontaneous crystallization (e.g., for the initial solution F, compositions below

F–e

line are unattainable).

3. Sector a′Fb″ delimits the region of the substance dissolution. The solutions are

undersaturated with both substances; degree of undersaturation can be estimated

from distances between coordinates of the initial and final figurative points in

the abscissa and ordinate for each component separately. Proceeding the dissolu-

tion is somewhat analogous to precipitation in the sector aFb. However, these

processes differ, as dissolution includes no analog to a spontaneous growth, and

the system is heterogeneous in the initial point.

(a) Equilibrium dissolution of the substance crystallized spontaneously in

accordance with the equilibrium trajectory e′

–F (F–e

is its continuation)

occurs in the reversed sequence along the F–e′

trajectory. So, determination

of undersaturation degree and analysis of the process development can be

advantageously made with the aid of a growth nomogram described above.

104 3 Formation of Mixed Crystals in Solutions

The equilibrium state is maintained due to sufficiently slow dissolution of sub-

stance. As long as the solution composition changes with the process development,

the crystal continues entering into metasomatic reaction with the solution, and so

the surface layer will be also changing until the equilibrium with the solution is

reached. Thus, to attain a total equilibrium the dissolution rate must be lower than

the rate of evening-out the concentration differences within the crystal.

In contrast to growth process (aFb sector), metasomatic component in this case

is characterized by volume deficit, and so must have a considerably higher rate.

(b) Quasi-equilibrium dissolution takes place when matter interchange within the

crystal bulk is negligibly small, and dissolution is slow enough to maintain equi-

librium between the solution and the crystal surface.

In contrast to the growth processes, the particular case given above is meaningless.

On one hand, the composition of the initial solid phase may be considered as homo-

geneous. When the volume of an equilibrium layer tends to a minimum (similarly

to the case described in point 2(a) of this section), its bulk composition remains

almost the same, and quantitative estimations of the process would not differ from

the nonequilibrium case (see below). On the other hand, if a crystal has zoned

structure formed in the course of quasi-equilibrium growth, sequential dissolution

of the crystal growth layers is impossible due to the principal differences in mor-

phology between the growth and dissolution surfaces.

metasomatic interaction between the crystal and solution. The crystal composition

remains the same and the corresponding trajectory coincides with the solidus iso-

composite that crosses the initial point; in this example it is F–s′ trajectory. At any

moment solution is enriched with both components in constant proportion corre-

sponding to their ratio in the crystal, and the trajectory of changing the solution

composition is a half-line starting from the initial point, which in the example

discussed is F–l′ trajectory making the angle of 45°C with the coordinate axes.

In accordance with initial conditions, the solid phase dissolves completely at 40°C.

In this case, the points l′ and e

′ should coincide. At lower temperatures solution

compositions depend upon efficiency of the metasomatic component and should be

located between the trajectories F–e

′ and F–l′, whereas the crystal compositions

are to be positioned between the lines F–e

′ and F–s′.

4. Sectors aFb′ and a′Fb comprise different regions of metasomatic replacement pro-

ceeding at varied temperature. Figure 3.9 represents four types of replacement

occurring in the initial solution F and corresponding to the following final points:

, M

, and M

, where M

and M

correspond to cooling, and M

and M

– for

heating; M

and M

show replacement in the crystal enriched with the component

A; M

and M

indicate replacement in the crystal enriched with the component B.

(a) Replacement at rising temperature (regions a

Fb′ and a′Fb

) is analogous to iso-

thermal substitution, which corresponds to line a

(see point 1 of this section),

but with some complications caused by indefinite degree of concurrence with

ordinary dissolution.

3.3 Physicochemical Model 105

Fig. 3.9 Polythermal isomorphic replacement of crystals in the initial solution F

106 3 Formation of Mixed Crystals in Solutions

One of the complications is an arbitrary configuration of the composition trajectory

of solution interacting with the crystal that undergoes replacement, which may

affect a zoned structure and shape of the produced pseudomorphs.

For example, transition of the system to state M

occurring during replacement of

initially pure K-Cr alum may proceed, generally with volume deficit, along any path-

ways within the limits of quadrangle Fi

′b

. If the trajectories coincide with the

extreme lines F–b

–M

and F–i

′–M

, their sectors b

–M

and F–i

′ correspond to the

periods of isothermic replacement with volume deficit. When the process proceeds

along the trajectory F–b

–M

, the produced pseudomorph will bear some signs of the

initial crystal dissolution, while if it assumes F–i

′–M

trajectory, the signs of

the pseudomorph dissolution will appear. Location of the process trajectory within

the limits of quadrangle Fi

indicates absence of dissolution independent stage.

It can be considered that replacement is accompanied by extra dissolution, similar to

metasomatic component accompanying the dissolution within the sector a′Fb′.

On one hand, configurations of the trajectories are determined by size and com-

position of the protocrystal, and on the other hand, by the ratio between the rates of

heating and metasomatic reaction. If the protocrystal contains (A,B)-component,

the right line of the delimiting quadrangle shifts into the region of dissolution a′Fb′

(e.g., along the line Fb

′b).

Another complication occurs in the region a′Fb

. The main analysis of the process is

similar in to the previous one, the exception being that a′Fb

region is divided into two

parts. In b

part and the adjoining sector of the isotherm Fb

the process proceeds

with volume excess, while in a′Fv

part the process proceeds with volume deficit.

(b) Replacement at lowering temperature (regions aFa

and bFb

) is also complicated

in comparison with isothermal process or replacement at rising temperature. It is

caused by the indefinite character of combination with the normal growth.

One of the complications (as in the case of replacement when the system is

heated) consists in arbitrary appearance of the trajectory of liquidus composi-

tion corresponding to a specific shape and zonal structure of pseudomorphs.

For example, the system transition from the state F to the state M

during

replacement of crystal B can take any pathway within the limits of the quadran-

gle Fb

′i

′M

. When the trajectory occupies the extreme position F–b

–M

, the

replacement is preceded by autoepitaxial growth of the pure phase A over

the initial crystal B, and the replacement proceeds according to volume-excess

isothermal mechanism. Naturally, shielding layer A causes additional com-

plexities, but they are not considered at present. Another extreme trajectory

F–i

′–M

corresponds to the initial stage of volume-excess isothermal replace-

ment followed by growth of the pure phase B over a pseudomorph. Composition

of the protocrystal, its mass, and cooling and replacement rates determine con-

figuration of the trajectory.

One more complication, which is also similar to the above replacement with

heating, consists in dividing the region aFa

into two parts. Within the part a

and

in the adjoining sector of isotherm Fa

the process is volume-deficient, while in the

part aFv

it is volume-excessive.

3.3 Physicochemical Model 107

3.3.3 Heterogeneous Metastable Equilibrium

Between Crystals and Solutions

Heterogeneous metastable equilibria between crystals and supercooled solutions

are peculiar states of systems containing isomorphic components (Glikin 1996a).

Lowering temperature to a considerable degree can prevent metasomatic replace-

ment and bring the whole system into an unusual metastable state, when the phe-

nomena inducing the salting-out are compensated by supercooling.

Indeed, replacement (salting-out) originated from dissolution of the initial crys-

tal, which can be terminated by supercooling the system. The required degree of

supercooling depends upon solution composition and nature of the crystal undergo-

ing replacement. The value of supercooling can be defined (Fig. 3.10) as a differ-

ence between the equilibrium temperature of the initial solution (e.g., 25°C for the

point F) and the temperature of the isotherm passing through the intersections

between a composition isoline of the crystal undergoing replacement and the lines

Fa and Fb. For example, for solution F, it is the point a when the crystal A is being

replaced or the point b when the crystal B is being replaced.

Hereinafter, solutions and crystals coexisting in the state of heterogeneous

equilibrium supercooling, and the diagram elements corresponding to the above

solutions and crystals (including the lines Fa and Fb) will be referred to as meta-

equilibrium elements.

Examination of the diagram represented in Fig. 3.10 can furnish the following

conclusion: at a given supercooling degree two solid phases with different isomor-

phic compositions can exist simultaneously in meta-equilibrium state with the solu-

tion. In contrast to the composition of a crystal being in thermodynamic equilibrium

with solution (without any supersaturation), compositions of both two phases are

enriched with one of the extreme member of the series. In the solution F super-

cooled by 5°C, the corresponding crystals are q

and q

′, while at supercooling of

10°C they are q

and q

′, and at 15°C supercooling they correspond to q

and q

′,

respectively (Fig. 3.10). In the phase diagram these crystals correspond to two

“ternary” crosspoints of the following lines: isotherm of the specified supercooling,

two meta-equilibrium lines (aF and bF), and two isocomposites (S

At the same time, the supercooled solution is metastably supersaturated in com-

parison with crystals having any intermediate composition within the sector aFb,

but it is undersaturated in comparison with any crystals corresponding to the other

regions aFa

and bFb

of extreme compositions (Fig. 3.9). Such interpretation gives

lines Fa and Fb a physical meaning, i.e., they mark conditions of heterogeneous

metastable equilibrium and separate regions of crystal compositions, which can be

supersaturated or undersaturated compared to the solution F.

It is essentially to distinguish supercooled solutions under conditions of hetero-

geneous metastable equilibrium from solutions being in the state of metastable

supersaturation. The latter are similar to the metastable supersaturated solutions in

binary systems and are homogeneous, i.e., they contain no seeds for which solution

is supersaturated. Meta-equilibrium crystals contained in the solution can be

108 3 Formation of Mixed Crystals in Solutions

Fig. 3.10 Equilibria existing between supercooled solution F and the crystals having different

compositions (%A): q

= 59, q

′ = 45, q

= 72, q

′ = 39, q

= 90, q

′ = 28

3.3 Physicochemical Model 109

considered as inert bodies, similar to the vessel walls. Heterogeneous meta-equi-

libria are unattainable in binary systems.

It is interesting to consider a mechanism of meta-equilibrium crystallization of

a supercooled solution in a way, analogous to that used for examination of the

above model of quasi-equilibrium crystallization discussed in Sect. 3.3.2. Diffusion

resistance effect should be ignored. By definition, composition of the solid phase

precipitating in this way at any moment of the process corresponds to a state of

meta-equilibrium with saturated solution. Compositions of both metastable phases

are determined from the graph at the specified supercooling, and trajectories of

changes in phase compositions are defined in the same way: quantities of both

isomorphic components leaving the initial solution are equal to their amounts con-

tained in the given mass of the precipitating solid phase, which is minimized in

accordance with required calculation accuracy. The operation is performed repeat-

edly for all the compositions obtained in every particular step; it is obvious that

degree of supercooling decreases with each subsequent step, and the compositions

of precipitating crystals change accordingly. Graphically, in the diagrams, the proc-

ess termination corresponding to thermodynamic equilibrium of the crystal surface

and the solution coincides with intersection of trajectories of solid and liquid phases

of the final isotherm.

The above estimations (Glikin 1996a) are similar to those described in Sect.

3.3.2 for quasi-equilibrium systems (Glikin 1995), the difference being the method

of determining the solid-phase composition. The examples are shown in Fig. 3.11.

Composition of the initial solution F corresponds to that of the equilibrium crystal

containing equal contents of both components − 50% of A and 50% of B (same as

for the examples described in Sect. 3.3.2). There are three cases possible (Fig.

3.11): (a) the crystals are enriched with A-component (ternary points of the isocon-

centrate aF, e.g., q

; Fig. 3.11a), (b) crystals are enriched with B-component

(ternary points of bF, e.g., q

′; Fig. 3.11b), and (c) both crystals precipitate simul-

taneously (ternary points of aF and bF, e.g., q

and q

′ respectively; Fig. 3.11c; rates

of precipitation of both phases correlate so that both the crystal compositions are in

meta-equilibrium at any moment). Trajectories of phase compositions for the initial

supercoolings equal to 5, 10, and 15°C are represented in the figures.

Iteration procedure applied, for example, to a diagram plotted for supercooling

of 15°C (Fig. 3.11a, b) shows that at the first step the solid phases have the follow-

ing compositions (%A): a −S

= 91 (ternary point q

), b − S

′= 28.5 (q

′). Precipitation

0.5 g of crystals would cause the compositions of the solutions to reach points F

and F

′, i.e., crosspoints of the new isoconcentrates F

and F

, F

′a

′ and F

′b

′, and

the new isotherms, as in each case, supercooling decreases by the value of a differ-

ence between the temperatures of the previous and subsequent isotherms. The

second step consists in precipitating the crystals having the following composi-

tions: a − S

3 − Δ

= 87.5 (q

), b − S′

3i − Δ

= 33 (q

′) and plotting the new points for

solution compositions and drawing the new isoconcentrate lines. Continued itera-

tions allow to find the final points f

and f

′.

Iterations used in the last example (Fig. 3.11c) involve similar steps of finding

compositions of crystals enriched with different components. It is to be noted that

Fig. 3.11 Proceeding the quasi-equilibrium crystallization in a supercooled solution of two isomorphic

components A and B a, b – precipitating the crystals enriched with components A (a) and B (b); c

concurrent depositing the crystals of the two types. F – experimental solution; Ff

– trajectories of

changing the solution compositions (hereinafter i = 1, 2, 3 correspond to the initial supercoolings

5, 10, and 15°C); S

– trajectories of changing the compositions of the crystal surface zones,