Concerning the Influence of Surface Charge on the Rate of Growth of

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Concerning the influence of surface charge on the rate of growth of surfaces during crystallization Frank K. Crundwell Cryst. Growth Des., Just Accepted Manuscript • DOI: 10.1021/acs.cgd.6b00939 • Publication Date (Web): 29 Aug 2016 Downloaded from http://pubs.acs.org on September 10, 2016

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Crystal Growth & Design

Concerning the influence of surface charge on the rate of growth of surfaces during crystallization. Frank K. Crundwell CM Solutions (Pty) Ltd, Building T5, Pinelands Office Park, 1 Ardeer Road, Modderfontein, 1609, South Africa (email: [email protected]; www.cm-solutions.co.za) Abstract: The electrically charged nature of crystal surfaces is a property that is not frequently considered in the analysis of the kinetics of growth of crystals, even though such charge has been measured for many salts and minerals. The purpose of this contribution is to consider a possible mechanism for the development of this surface charge under conditions applicable to crystal growth, and to examine the implications of this proposed mechanism. In particular, necessary conditions for stoichiometric growth and the stability of growth are developed. The implication of the requirement for the electrical stability of the surface is that the rates of deposition of ions are dependent on the surface potential difference across the Stern layer. It is proposed that this dependence is a Boltzmann-like distribution, from which a kinetic equation for rate of growth of crystals surfaces is derived. This equation is consistent with the chemical thermodynamics, and has limiting forms that exhibit ‘orders’ with respect to saturation that fall between first and second order, in agreement with the experimental data. This agreement suggests that the potential-dependent dehydration of ions during attachment might be the rate-controlling process in crystal growth.

Synopsis: It is argued that the surface charge has a determining effect on the kinetics of the growth of crystals. Conditions for stability and stoichiometry and a rate law are derived based of this mechanism that is consistent with experimental data for the growth of crystals.

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Concerning the influence of surface charge on the rate of growth of surfaces during crystallization. Frank K. Crundwell CM Solutions (Pty) Ltd, Building T5 Pinelands Office Park, 1 Ardeer Road, Modderfontein, 1609, South Africa (email: [email protected])

Crystal growth; kinetics; rate of growth; potential difference; surface charge; precipitation

ABSTRACT

The electrically charged nature of crystal surfaces is a property that is not frequently considered in the analysis of the kinetics of growth of crystals, even though such charge has been measured for many salts and minerals. The purpose of this contribution is to consider a possible mechanism for the development of this surface charge under conditions applicable to crystal growth, and to examine the implications of this proposed mechanism. In particular, necessary conditions for stoichiometric growth and the stability of growth are developed. The implication


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of the requirement for the electrical stability of the surface is that the rates of deposition of ions are dependent on the surface potential difference across the Stern layer. It is proposed that this dependence is a Boltzmann-like distribution, from which a kinetic equation for rate of growth of crystals surfaces is derived. This equation is consistent with the chemical thermodynamics, and has limiting forms that exhibit ‘orders’ with respect to saturation that fall between first and second order, in agreement with the experimental data. This agreement suggests that the potential-dependent dehydration of ions during attachment might be the rate-controlling process in crystal growth.

1.

Introduction

Crystallization is a fundamental reaction of chemistry that finds application in many industrial processes and scientific procedures. Crystallization and precipitation are used to separate and purify salts, present products in their final form, and prepare crystals for study. For example, in the refining and recycling of platinum metals1, slight differences in the solubility products of ammonium salts are used to purify these metals.

Crystals grow by the attachment of building blocks, such as ions, atoms and molecules, from the growth medium onto the surface2. More recently, arguments have been put forth that these building blocks also include clusters of molecules2,3. Growth is envisaged as occurring at “kinks” (see Figure 1), where the building blocks have about half the number of neighbours it would have in the bulk of the crystal4,5,6. The growth of a crystal face might occur by direct attachment of building blocks to the kink from the growth medium, or first by attachment to the bare surface followed by diffusion across the surface to a position at a kink site2,4,7,8,9, , although current work


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favours the notion that attachment first occurs on the bare surface11,12. These possible mechanisms of growth are illustrated in Figure 1.

Figure 1. Schematic illustration of some of the possible mechanisms of crystal growth.

Kinks represent the edge of a terrace. As building blocks attach and are incorporated at a kink site, the terraces advance, resulting in the overall growth of the crystal. Terrace edges themselves form by a variety of possible mechanisms. The mechanism that is most commonly advocated is that due to screw dislocations13, because this mechanism allows the surface to grow indefinitely without the discontinuity that will occur when a layer is complete and a new terrace is required. The mechanism of two-dimensional (2D) nucleation proposes that the surface of a crystal grows by the deposition of islands on the surface around which terraced layers form6. An alternative mechanism that is similar to the 2D nucleation mechanism proposes that clusters of material deposit on the surface and crystallize, resulting in multiple terrace edges for further growth of the surface12. For example, it has been proposed that the growth of ZnS occurs first by the formation of a Zn3S3 cluster, which is then deposited on the growing surface by “condensation” 3,14. However, other researchers, such as Gebauer et al.15, consider that the question of the role of these pre-nucleation clusters in the growth of crystals remains open.


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The rate of growth of crystals surfaces is might be dependent on the rate of incorporation of a particle into a kink site and hence on the density of kink sites on the surface4,5. Other steps, such as those represented by the free energy barriers to the attachment or dehydration, might also be important4,5. However, most models of crystallization are based on the description of the physical phenomena of 2D-nucleation, surface diffusion, and incorporation6. The predominance of these mechanisms in the different regions of supersaturation is shown in Figure 250,51.

Figure 2. Regions of dominance of possible mechanisms of crystal growth50,51.

A phenomenon that these models do not account for is the electrical charge on the surface of the crystal. It might be argued that surface charge is already accounted for by insisting that stoichiometric deposition occurs. In other words, stoichiometry requires charge neutrality, and vice versa, and beyond this electrical charge is irrelevant to crystallization. By necessity this argument leads to the prediction that the mineral surface is neutral. However, this argument is flawed because the charged nature of surfaces of most minerals has been experimentally


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determined by various measurements of the zeta potential or electrophoretic mobility16. Even highly soluble minerals such as KCl, NaCl and LiCl in saturated solutions have charged surfaces17. Indeed, Miller and Yalamanchili17 reported the surface charge of 21 chloride, iodide, fluoride and bromide salts in saturated brines. Clearly the surfaces of salts have electrical charge, and neither the source of this charge, nor its effect on crystallization has been accounted for.

Perhaps even more interesting is the observation of surface charge in the growth of single crystals from molten salts18,19,20. On the other hand, changes in potential difference at the surface also give rise to morphological changes in ice crystals, and are implicated in the formation of hopper crystals21. Industrial experience has been that gypsum forms on the surface of anodes in copper electrowinning, but not on the cathodes. These observations of the influence of surface charge on crystallization hint that surface charge (equivalently, surface potential) might be an important variable to consider in the modelling of crystallization.

The purpose of this paper is to consider how the proposition that surface charge is an important physical phenomenon that needs to be taken into account leads to an alternative mechanism for the growth of crystal surfaces. It will be demonstrated that these considerations give rise to conditions for stoichiometric growth of the surface and for surface stability. It is shown that in order to meet this stability criteria, the dependence of the rate of deposition on charge can be determined, and from that a new rate law for crystallization can be developed. This rate law is tested for several situations to demonstrate the utility of the proposed rate law.


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2.

Surface charge and interfacial potential

The excess surface charge, σ , at a particular time is proportional to the difference in the concentration of anions and cations on the surface, and is given by Eq (1):

σ = F (ν + n+ −ν − n− )

(1)

where n+ and n− represent the surface concentration of cations and anions, respectively, ν + and

ν − represents the charge number on the cation and anion, respectively, and F is the Faraday constant. (Please refer to the list of symbols.)

In a solution containing only the constituent ions of the growing crystal, the surface charge is a result of a slight imbalance between the number of cations and anions on the surface. The surface charge developed in this manner is not static, but is part of the dynamics of depositing anions and cations onto the surface. This imbalance in surface concentration is estimated to be of the order of 10-7 mol/m2 (estimated by assuming a potential difference across the surface of about 0.1 V and an interfacial capacitance of 0.2 F/m2, which means that the difference in surface concentration is equal to 0.1 V x 0.2 F/m2 / 96500 C/mol = 2x10-7 mol/m2).

The surface charge contemplated in Eq. (1) results in a potential difference across the interfacial region, as shown in Figure 3. Since the zeta potential is measured in concentrated brines of up to


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4 M 17, this potential difference occurs across the Stern layer, not across the diffuse GouyChapman layer. The Stern layer is a region that lies between the surface and the outer Helmholtz plane (ohp), where the ohp represents the plane of closest approach of hydrated ions. Consequently, the region between the surface and the ohp is a charge-free dielectric composed of solvent molecules. The Stern layer is usually modelled as a capacitor16,22,23, given by Eq. (2):

∆φ =

σ Cd

(2)

∆φ , the difference in the Galvani (outer) potential between the surface and the ohp (as shown in

Figure 3) is the independent variable in interfacial studies52, and is referred to as the surface potential difference in the rest of this paper. Cd represents the capacity of the double layer.


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Figure 3. Structure at the electrified interface of a salt particle in a concentrated solution. The surface potential difference in the potential difference between the surface and the outer Helmholtz plane, that is, across the Stern layer. Water dipoles at the surface are highly directional, and the permittivity is consequently significantly lower than the bulk solution due to the high electric field (in excess of 109 V/m)52.

The combination of Eqs. (1) and (2) leads to Eq. (3):

∆φ =

F (ν + n+ −ν − n− ) Cd

(3)

This equation can be transformed into a more useful form by taking derivatives of both sides of Eq. (3) with respect to time and recognizing that the rate of deposition of cations, r+ , is equal to dn+ dt (and similarly for the anions). These actions mean that the change in surface potential difference with time can be expressed in terms of the rates of deposition of anions and cations, given by Eq. (4):

d ∆φ F (ν + r+ −ν − r− ) = dt Cd

(4)

The symbols r+ and r− represent dn+ dt and dn− dt , respectively.


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Eq. (4), the first result of this paper, is significant because it indicates that the surface potential difference, ∆φ , is determined by the difference in the rates of deposition of cations and anions. No specific chemical reaction has been assigned yet; as a result, Eq. (4) applies equally to crystallization as it does to dissolution22 to 32, except that in the case of dissolution it is the rate of removal of ions from the surface. Eq. (4) expresses that the maintenance of the charge is dependent on the continued deposition of anions and cations; in this sense it is dynamic.

Eq. (4) indicates that the potential difference is also dynamic in the sense that it will change with time unless steady state conditions are achieved, that is, unless d ∆φ / dt approaches zero. An indication of such a change in the surface potential difference can be obtained from the change in zeta potential with time, as shown in Figure 4 for the quartz surface (ref. 16, p. 283) and the ZnS surface (ref. 54). (Note that zeta potential is not ∆φ , but the potential difference between the solution bulk and the ohp. The relationship between zeta potential and ∆φ is discussed in refs. 32 and 53.)

Traditionally, such changes in zeta potential have been interpreted as being caused by exogenous factors, such as dissolution of the glass vessel16. However, Eq. (4) indicates that such changes in the zeta potentials are endogenous, that is, an integral part of the formation or dissolution of the surface.


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Acknowledgement of changes in surface properties with time raises the question of whether a stable condition can be achieved. In other words, does steady state ever occur, and if so what conditions govern such surface stability. It is therefore important to establish the conditions for steady state and the criteria for stability at this steady-state point. Prior to discussing the topic of stability, the question of the steady-state deposition of the ions is considered.

-25.0

Zeta potential, mV

(a) -30.0

-35.0

-40.0 0

50

100

150

200

Time, h 35 30

Zeta potential, mV

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25 20 15 10 5

(b)

0 0

5

10

15

Time, h


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Figure 4. Change in zeta potential with time for (a) quartz16 and (b) zinc sulfide54. Although these minerals are probably undergoing dissolution during these tests32,53, the underlying cause is the same.

3.

Steady-state and stoichiometric growth of the surface

At steady-state (with respect to the surface potential) there are no changes with time, that is, the left-hand side of Eq. (4) is zero. This yields Eq. (5):

0 = ν + r+, s −ν − r−, s

(5)

where the subscript s refers to the rate at steady state of the deposition of the respective ion.

The relationship given by Eq. (5), the second result of this paper, is the condition for stoichiometric deposition of ions on the surface. That Eq. (5) is the stoichiometric condition might be seen from the overall reaction for the formation of the crystal, which is represented by Eq. (6).

ν − M ν + + ν + Aν − → M ν − Aν +

(6)


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The rates of deposition of the cations and anions are governed by the stoichiometric relationship given in Eq. (7):

r+

ν−

=

r−

ν+

(7)

Eq. (7), which is the statement of stoichiometry for the crystallization reaction, is equivalent to Eq. (5), which is derived from the proposed model of the surface potential difference at steadystate. Importantly, this analysis indicates that stoichiometric growth of the crystal surface is not guaranteed except at steady state, where the steady state refers to the establishment of a point where the surface potential difference does not change with time. In other words, stoichiometric growth is only guaranteed when steady-state with respect to the surface potential difference is achieved. This generic model of the growth of a crystal surface suggests that the necessary condition for the stoichiometric growth of a crystal is that the surface potential difference remains constant.

4.

Stability of the stoichiometric point

The existence of a steady-state solution does not imply that crystal growth will occur in a stoichiometric manner. Certainly, the non-stoichiometric dissolution of component ions from a solid surface is well known33, and presumably, non-stoichiometric deposition might similarly occur. Consequently, it is important to establish the conditions required for stability at the point of steady-state growth.


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The stability of the point at which stoichiometric deposition of a crystal occurs is dependent on the functional relationship of the rates of deposition of the cations and the anions on the surface potential difference, ∆φ . Stability can be analysed in terms of deviation from steady state. The subtraction of Eq. (5) from Eq. (4) yields Eq. (8).

d ∆φ% F (ν + r%+ −ν − r%− ) = dt Cd

(8)

The symbols with the overscript “~” represent the deviation of that value from its steady-state value. For example, r%+ is equal to r+ − r+ ,s , where r+ ,s refers to the value of r+ at steady state.

The stoichiometric point is stable if any deviation from the steady-state condition counteracts that deviation. In other words, when the surface potential is above the steady state value, d ∆φ% dt is negative, so that with time the surface potential moves back towards the steady state

value. If, on the other hand, the surface potential is below the steady state value, d ∆φ% dt is positive, again moving the surface back to the steady state value with time. This is expressed by Eq. (9). d ∆φ% < 0 when ∆φ% > 0 dt d ∆φ% > 0 when ∆φ% < 0 dt

(9)

These conditions can be expressed by Eq. (10).


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ν + r%+ −ν − r%− < 0 when ∆φ% > 0 ν + r%+ −ν − r%− > 0 when ∆φ% < 0

(10)

Consequently, the surface potential difference at steady state, ∆φs , is stable if ν + r%+ > ν − r%− when ∆φ < ∆φs and ν + r%+ < ν − r%− when ∆φ > ∆φs . These conditions demonstrate that in order for growth of a crystal surface to be stable, the rate of deposition of cations must be higher than that of anions when the potential difference is less than the steady state point, and vice versa. These conditions, which represent the third result of this paper, are illustrated graphically in Figure 5, where the independent variable is the surface potential difference.

Figure 5. Illustration of the requirements for stability with respect to surface potential difference. If the potential difference is less than the steady state value (the point at which the lines cross), the rate of deposition of the cations must be higher than rate of deposition of anions for steady state to be stable. Likewise, if the potential difference is higher than the steady state value, the rate of deposition of anions must be higher than cations for steady state to be stable.


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5.

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Potential difference as a hidden variable in crystallization

An implication of the stability analysis is that in order for the steady-state operating point to be stable the underlying functions for the deposition of cations and anions must be a function of the surface potential difference. This is significant because it indicates that the surface potential difference is a hidden variable in crystallization. That the surface potential difference, ∆φ , is a hidden variable is a somewhat surprising result, and represents the fourth result of this paper.

Four results have been presented so thus far: (i) the surface potential difference (or, equivalently, charge) develops dynamically according to Eq. (4); (ii) stoichiometric deposition of cations and anions occurs at steady state with respect to the surface potential difference; (iii) in order for the stoichiometric point to be stable in crystallization, the conditions ν + r%+ > ν − r%− when ∆φ < ∆φs and

ν + r%+ < ν − r%− when ∆φ > ∆φs must hold; and (iv) that for stability to be achieved, the rates of deposition of both cations and anions must be dependent on the surface potential difference.

The condition for stoichiometric deposition does not require that the surface potential be zero; the condition is the rate of change of surface potential is zero. The surface potential difference will stable at a value that satisfies the stoichiometric condition. Thus, if the functional dependence of r+ and r− with respect to ∆φ are known, or can be inferred, an expression for ∆φ at stoichiometric growth can be derived. Indeed, this will also allow the derivation of a rate equation for the rate of crystal growth as a function of concentrations in the bulk solution.


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Inferring a functional form for the dependence of r+ and r− on ∆φ is the topic of the next section. After this, a rate equation is derived.

6.

Functional dependence of ionic deposition on surface potential difference

Not only must the rate of deposition of anions and cations be dependent on potential, r+ must decrease with increasing ∆φ , and r− must increase with increasing ∆φ , as sketched in Figure 5. This is in line with expectations based on an understanding of electrostatics: lowering the surface potential difference from the steady state value will result in a more negatively charged surface that will accelerate the deposition of cations. Likewise, increasing the surface potential difference from the steady state value will accelerate the deposition of anions.

However, the effect of the surface charge on the rate of deposition of ions is more than just the electrostatic work to move an ion from one position to another in an electric field. At the very least the ion needs to break bonds in the liquid phase (such as the bonds with the solvent) and form new chemical bonds with the surface. Recent work in dissolution, a field sufficiently closely related to crystallization that it has been frequently argued that dissolution is simply the reverse of crystallization34, has suggested that the functional dependence on surface potential difference might be a Boltzmann distribution, similar to that found for the electrochemical deposition of metals24 to 32.


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Since the surface potential difference critically affects the rate of approach of an ion to the surface, the free energy barrier for an ion to attach to the surface, given by Eq. (11), explicitly incorporates surface potential difference22,23.

∆G # = ∆Gc# ± 0.5ν ± F ∆φ

(11)

∆G # represents the total Gibbs free energy of the activation barrier, ∆Gc# represents the Gibbs free energy due to chemical contributions, and the term 0.5 F ∆φ represents the effect of the surface potential difference across the interface on the Gibbs free energy of the activation barrier. The factor 0.5 arises because the activated state (transition state) occurs halfway between the surface and the outer Helmholtz plane. The sign of this second term on the right-hand side of Eq. (11) is dependent on the sign of the depositing ion. If the ion is positively charged, the sign in Eq. (11) is negative, and vice versa.

Should the charges of the ions be unequal, as in the case of say MgF2, the charge numbers, ν may be required in Eq. 11. If the ion is positively charged, ν ± is ν + and vice versa. Our previous work on dissolution24 to 32 suggested that the values of both ν + and ν − be taken as equal to one, but this requires further analysis. Certainly, the mathematics of the analysis that follows in the rest of this paper is significantly more complex if ν + and ν − are not equal to each other. Thus,


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we limit the discussion to salts that are 1:1, 2:2, 3:3, etc, and the symbol z is adopted for this purpose.

The rate of a depositing cation on the salt surface, given by Eq. (12), is dependent on the concentration of the cation at the ohp, and is exponentially dependent on the activation barrier by Eyring’s transition state theory35,36,37,38.

rate = κ

 ∆Gc#   ∆Gc#  k BT k BT  z + F ∆φ  z+ M [ M z + ]exp  − = κ [ ]exp  −  exp  −  h h  2 RT   RT   RT 

(12)

The symbols are defined as follows: κ is the transmission coefficient, kB represents Boltzmann’s constant, h is Planck’s constant, R is the gas constant, T is the temperature. Note that the concentration at the ohp is close to the concentration in the bulk for solutions with ionic strengths commonly found in crystallizing systems.

Eq. (12) can be written more compactly as Eq. (13) in which the symbol y represents the factor

s z+ F ∆φ / 2 RT and the symbol k+ represents the term κ k BT h exp ( − ∆Gc# RT ) . s rate = k+ [ M ν + ]exp ( − y )

(13)


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7.

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Kinetics of stoichiometric growth of the crystal surface

In this section, two expressions are derived: (i) for the surface potential difference, and (ii) for the rate of stoichiometric growth of a crystal surface. The analysis in this and the following sections is restricted to the case in which the charges of the cations are equal to the charges of anions, that is, the reaction is M z + + A z − → MA where z+ = z− . The derivation of the case where the charges are unequal may require the charge number in Eq. (11), and if so requires numerical solution.

The rate of deposition of the component ions of the salt MA onto the surface can be represented as two independent ‘partial reactions’. The rates of these partial reactions are linked by their mutual dependence on the potential difference and by Eq. (4). These individual ‘partial reactions’ for the crystallisation of a solid MA are given by Eqs. (14) and (15).

M z+

⇔

≡ M z+

(14)

Az−

⇔

≡ Az−

(15)

The symbol ≡ M z + represents the M z + cation deposited on the surface, and ≡ A z − represents the Az − anion deposited on the surface. The anionic and cations charges are considered equal in this

section.


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In describing the deposition of ions from the medium, it is important to account for both deposition and removal. Yau et al. 39 measured the arrivals and departures of “particles” from a particular step site using atomic force microscopy with slow scanning axis disabled and the fast axis oriented perpendicular to the step direction. Using this technique, they observed 25 arrivals at a particular site, but also 22 departures.

The rates of these two partial reactions, including the reverse reaction for each of them, are given by Eqs. (16) and (17) 24 to 32:

s r r+ = k+ [M z + ]exp ( − y ) − k+ exp( y) s r r− = k− [ Az − ]exp ( y ) − k− exp(− y)

s

(16) (17)

s

The symbols k+ and k− represent the respective rate constants for the deposition of cations and

r

r

anions onto the surface, respectively, k+ and k− represent the respective rate constants for the removal of cations and anions from the surface, respectively, and the […] brackets represent the concentrations in solution (at the ohp).

The first term on the right-hand side of Eqs. (16) and (17) represents the rate of deposition of the respective ion, while the second term represents the rate of removal of the ion.


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Page 22 of 45

Eqs. (16) and (17) clearly meet the requirements that r+ must decrease with increasing ∆φ , and r− must increase with increasing ∆φ . The second term of the right-hand side of both equations is required to correctly describe the situation as equilibrium is approached. In the next section, these equations will be used, together with the steady-state criterion, to derive a rate expression for the growth of a crystal surface.

Eqs. (5), (16) and (17) represent three equations in the three unknowns: r+ , r− and y. (Note that the factor y incorporates the surface potential difference, ∆φ ). These three equations can be easily solved by first substituting Eqs. (16) and (17) into Eq. (5) to yield an equation in only y, and then to substitute this expression back into either Eq. (16) or Eq. (17). The first step yields the expression for y, given by Eq. (18).

s r  ν + k+ [ M z + ] + ν − k −  r  2 y = F ∆φ / RT = ln  s z − [ ] k A + k ν ν  − − + + 

(18)

Eq. (18) represents an expression for the surface potential difference as a function of the concentrations in the bulk solution. Crundwell32, 53 has shown that Eq. (18) successfully describes the surface potential of the interface between the solid and water for several minerals.

The substitution of this expression into Eq. (16) yields Eq. (19) after some rearrangement.


22

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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


r = k1

([ M ([ M

z+

z+

] + k2

][ A z − ] − k 4

) ([ A 1

2

z−

)

] + k3

The symbols in Eq. (19) are defined as follows: r is given by

(

s s k + k − / z + z−

)

1

2

r

s

)

1

(19)

2

r+ r− = = r , k1 is given by z− z+

r

s

; k2 is given by k − z − / k + z + ; k3 is given by k + z + / k − z − , and,

k4 , is given by

r r s s k+ k− / k+ k− .

(

)

Eq. (19) represents the predicted form of the rate expression for the growth the solid phase MA at conditions both close to and far from equilibrium. This model is wide in scope, since it predicts the rate for crystallization and dissolution at conditions both close to and far from equilibrium, while at the same time predicting the development of surface potential difference at steady state conditions.

8.

Equilibrium

Before discussing the limiting forms of Eq. (19) it is important to examine the equilibrium conditions. At equilibrium the net rate of growth of the surface is zero. Eq. (20) is the result of setting the left-hand side of Eq. (19) to zero.

[ M z + ][ A z − ] = k4

(20)


23


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Page 24 of 45

Eq. (20) is simply the thermodynamic expression for the solubility constant for the overall reaction, that is, for the reaction M z + + A z − → MA . The model predicts the correct equilibrium expression at zero rate, even though it was not derived from a statement of the overall reaction, but from the individual partial reactions. Therefore, this model is in agreement with the thermodynamics at the level of deposition of individual ions (Eqs. 14 and 15) and at the level of the overall reaction. The parameter k 4 can be therefore be identified with the solubility product, K sp .

9.

Limiting forms of the proposed rate law

If crystallization occurs from an equivalent solution, then the concentrations [ M z + ] and [ A z − ] are both equal to the concentration of the salt, c. This means that the rate of growth of the crystal surface is given by Eq. (21).

r = k1

(c

2 1

− k4

)

( c + k 2 ) 2 ( c + k3 )

1

2

(21)

Eq. (21) predicts that the rate of growth ranges between first order and second order with respect to the concentration of the salt, depending on the values of k2 and k3 relative to c. Experimental “orders” with respect to the salt concentration are frequently found to range between one and two (as discussed earlier when Figure 2 was presented) in general agreement with this expression.


24

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Eq. (21) has four limiting forms:

(i)

growth close to equilibrium (specifically, when both c k 2 and c >> k3 );

(iii) partial equilibrium due to M z + (specifically, when only c >> k3 ); and. (iv) partial equilibrium due to Az − (specifically, when only c >> k 2 ). Partial equilibrium is the situation in which either the deposition of the cations or anions reaches equilibrium before the other one does. Thus, if the second term on the right-hand side of Eq. (16) is very small compared to the first term, the situation arises in which the deposition of the cations is not near equilibrium conditions, while the deposition of the anions might be near equilibrium. This mismatched condition is referred to as partial equilibrium.

Each of these limiting forms is discussed briefly in turn.

(i) Growth close to equilibrium (when both c k 2 and c >> k3 ) The rate of growth under these conditions leads to the rate expression given by Eq. (23).

k   r = k1  c − 4  c  

(23)

This expression is first order in the concentration of the salt.

(iii) Partial equilibrium due to dissolution and deposition of M ν + (when only c >> k3 ) The rate of growth under these conditions is given by Eq. (24).

(

r = k1 k 2 −1/ 2 c1.5 − k 4 c −1/ 2

)

(24)

This expression has an “order” of one and a half, midway between that of the limiting cases (i) and (ii).

(iv) Partial equilibrium due to dissolution and deposition of Aν − (when only c >> k 2 ) The rate of growth under these conditions is given by Eq. (25).

(

r = k1 k3 −1/ 2 c1.5 − k 4 c −1/ 2

)

(25)

Like the previous case, this expression also has an “order” of one and a half, between that of cases (i) and (ii).


26

Page 27 of 45

Thus, Eq. (21) gives rises to limiting forms that exhibit “orders” with respect to the concentration of the salt that fall between one (linear) and two (parabolic), as shown in Figure 6. The results in Figure 6 are plotted as a function of relative supersaturation, s = c ce − 1 , as a comparison with the usual way in which the driving force for crystallization is presented, although it must be pointed out that the driving force for the theory presented here is not supersaturation, but is the absolute values of the concentrations themselves. (Note: the calculation of supersaturation is done as a transformation or mapping of the concentration.)

The model admits the possible change in order from parabolic to linear as the concentration) increases. This is illustrated in Figure 7, which is plotted against supersaturation for ease of comparison with other crystallization studies.

Rate of growth, nm/s

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Parabolic 'Order' 1.5 Linear

Relative supersaturation, s

Figure 6. The three limiting forms of Eq. (21) for the rate of growth of a crystal as a function of relative supersaturation.


27

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Page 28 of 45

Linear growth

Parabolic growth

Relative supersaturation, s

Figure 7. The prediction of the proposed model showing the change from parabolic to linear growth with increasing supersaturation.

10.

Comparison with experimental data

As mentioned in the previous section, Eq. (21) gives rise to limiting forms that are either parabolic or linear. Indeed, experimentally, the “order” of growth of a crystal with respect to relative supersaturation also varies between values of one and two, that is, linear and parabolic. The purpose of this section is to demonstrate that Eq. (21) is able to describe the experimental data from these two limiting cases. The results for the crystallization of BaSO4 and CaCO3

10

,

shown in Figure 8, demonstrate the parabolic rate expression, in accordance with Eq. (22), is followed. Nielsen and Toft9 present data for several other systems that follow the parabolic rate equation.

At the other end of the range of the limiting forms proposed here is the linear rate expression. Bennema40determine the rate of crystal growth at low supersaturations for potassium chlorate and potassium aluminium sulfate (alum) in an attempt to distinguish between mechanisms. He


28

Page 29 of 45

showed that the rate of growth is linear with supersaturation. Klein Haneveld41 showed that the rate of crystal growth of KCl is also linear. The results for KCl are shown in Figure 9.

14 BaSO4 Nielsen and Toft, 1984 CaCO3 Model

12 Rate of growth, nm/s

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


10 8 6 4 2 0 0

0.2

0.4

0.6

0.8

1

c, mM

Figure 8. The rate of growth of BaSO4 and CaCO3 as a function of the concentration of the salt, showing the parabolic kinetics. The data was reported by Nielsen and Toft9, while the lines represent Eq. (22) fitted to the data. The parameters are as follows: (a) BaSO4 k1 ( k2 k3 ) 2.47x107 nm/M2s; k4 9.99x10-11 M2 (b) CaCO3 k1 ( k2 k3 )

−1

2 1.45x107

−1

2

nm/M2s k4 9.10x10-9 M2.


29


2.5


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 30 of 45

KCl Model

2.0 1.5 1.0 0.5 0.0 0.0

5.0

10.0

15.0

Relative supersaturation x 104, s

Figure 9. The rate of growth of KCl showing linear kinetics. The points represent data reported by Klein Haneveld41, while the line represents Eq. (23) fitted to the data. The parameters are as follows: k1 1.84x102 nm/Ms k4 2.17x102 M2.

The combination of Figures 8 and 9 demonstrate that the experimental evidence supports the proposal theory. On the other hand, the kinetics of crystal growth of calcite (CaCO3) have been described by both parabolic and linear rate expressions. Reddy and Nancollas42 found that the rate of crystal growth is parabolic and is described by Eq. (22). However, the crystallisation of CaCO3 is dependent on the concentrations of both Ca2+ and CO32- ions, and hence Eq. (19) might be considered a more appropriate expression. The data of Inskeep and Bloom43 is shown as the points in Figure 10, while the model represented by Eq. (19) is shown as the line. The fitted parameters indicate that k2 and k3 are close to zero, which means that the form of the model is similar in concept to Eq. (23) that was used to fit the linear data for KCl. However, the fitted 1

form shown in Figure 10 is not linear, because the denominator (given by [Ca 2+ ] 2 [CO32− ]

1

2

if


30

Page 31 of 45

k2 and k3 are close to zero as they are in this case) affects both terms in the numerator of Eq. (19).

Depending on the conditions chosen for the study, such as temperature and concentration, it appears that calcite might exhibit different kinetics. Interestingly, and in line with the model presented here, Romanek et al.44 report that the rate of growth of aragonite (another form of CaCO3) has an order of about 1.6, close to the limiting forms represented by Eqs. (24) and (25).

30.0

Rate x 107, mol/m2/s

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


25.0 20.0 15.0 10.0 5.0

Inskeep and Bloom Model Eq. 19

0.0 0.0

10.0

20.0

30.0

40.0

[Ca2+][CO3+] x 108, M2

Figure 10. The rate of growth of CaCO3. The points represent data reported by Inskeep and Bloom43, while the line represents Eq. (19) fitted to the data. Parameters are as follows: k1 4.80x10-11 mol/Mm2s; k2 0.0; k3 0.0; k4 1.51x10-8 M2.


31


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

11.

Page 32 of 45

Implications of the proposed theory

Any further discussion of the mechanism of crystal growth proposed here requires a deeper consideration of Eqs. (16) and (17), or the irreversible form given as Eq. (13). The functional form is that of the Boltzmann distribution as found in electrochemistry. Indeed, the deposition of an ion at an electrode during electro-crystallization follows a form similar to Eq. (13). Crundwell31 discussed the modern theories underpinning the removal or deposition of a metal ion at an electrode, and argued, following on the work of Gileadi45, Gileadi et al.46 and Taylor47, that no electron transfer actually takes place between the metal ion in the solution and the solid electrode. This is because of the model of metallic bonding, that the metal consists of metal ions in a “sea of electrons” 48. Deposition of an ion from solution therefore involves dehydration and incorporation onto the surface of the electrode. The metal ion is not itself reduced, but neutralizes excess electronic charge at the surface45, which is similar to “normal” crystallization.

If the model suggested by Gileadi45 holds, and the work of Taylor46 seems to support it, then Eq. (13) describes the effect of the potential on releasing the ion from its hydration sheath and allowing it to deposit on the surface. The differences between “normal” crystallization and electro-crystallization are two-fold: (i) that a counter-ion must also deposit in “normal” crystallization, and (ii) that, as argued in the section leading to Eq. (4), it is the rates of deposition of both cations and anions that results in the surface potential difference across the


32

Page 33 of 45

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interface in “normal” crystallization, whereas a potential difference is applied externally in electro-crystallization. These differences do not necessarily negate the line of argument presented here.

Furthermore, if Eq. (19) successfully describes the rate of growth of crystals, and arguments of the preceding section suggests there is evidence in favour of this contention, then the critical step in the growth of crystals is not incorporation at kink sites or surface diffusion as envisaged by Burton, Carrera and Frank13, but the dehydration and attachment of the ions to the surface under the influence of the surface potential difference. While kinks and dislocation certainly exist and may support the models of Burton, Carrera and Frank13, it might be that the rate-determining step is the dehydration and attachment of the ions to the surface under the influence of the surface potential difference. It is this surface potential difference that has not been examined previously in the context of crystallization.

12.

Further tests of the proposed theory

Two tests of the theory proposed in this paper are suggested. The first is to test the prediction of the theory that the surface charge is determined by the rates of depositions of cations and anions, as described by Eq. (18). Miller and Yalamanchili17 showed using laser doppler electrophoresis that a wide range of soluble salts, including KCl, LiCl, NaCl, RbCl, CsCl, LiF, NaF, KF, RbF, CsF, LiBr, NaBr, RbBr, KBr, CsBr, LiI, NaI, KI, RbI, CsI, NaI.2H2O, exhibit charged surfaces in saturated brines. They confirmed these electrophoretic measurements using flotation tests in saturated brines and the observations of particle-particle interactions, phenomena that are also


33


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Page 34 of 45

dependent on surface charge. (Note that the flotation steps used by the potash industry for the separation of KCl and NaCl, which has been performed for the last fifty years, must be done at saturation, or else these salts dissolve.) The theory presented here suggests that further characterization of the surface would be fruitful.

The second test of the proposed theory arises from the argument of the previous section. If dependence on surface potential given in Eq. (13) is valid, then the crystallization of salts will bear some of the characteristics found for the electrodeposition of metal ions. The role of water is critical in stabilizing ions in solution, and crystallization requires the removal of this water. If dehydration is a significant factor in metal deposition at an electrode, one would expect there to be a correlation between rates of metal dissolution/deposition at metal electrodes and a measure of the rate of metal-ion hydration. Such a correlation was reported by Vijh and Randin49 between the exchange current density and the self-exchange rate water in the hydration sphere of the corresponding ion in sulfate solutions. The exchange current density is a function of the rate

s constant k + in Eq. (13) and hence represents the rate constant for metal deposition at an electrode. The original data presented by Vijh and Randin’s 49 is shown in Figure 11, together with data from other solution chemistries. The essential argument here is that the rate of metal deposition and the removal of water from the hydration sheath are intimately related.


34

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Exchange current density, A/cm2

Page 35 of 45

1.E+01 1.E+00 1.E-01 1.E-02 1.E-03 1.E-04 1.E-05 1.E-06 1.E-07 1.E-08 1.E-09 1.E-10 1.E+04

Perchloric Sulfate 1.E+05

1.E+06

1.E+07

1.E+08

1.E+09

1.E+10

Exchange rate of water in hydration sphere of aqueous metal ion, 1/s

Figure 11. The correlation between the rate of electrodeposition and the self-exchange of water from the hydration sphere of the metal ion. Data from Vijh and Randin49 and Crundwell31.

If Eq. (13) represents the underlying functional dependency on surface potential, then a similar correlation between the rates of crystal growth and the rate of dehydration should be apparent. The data from Nielsen10 for the rate of growth of oxalate crystals as a function of the selfexchange rate of water in the hydration sphere of the cation is shown in Figure 12. While there is probably not enough data to draw a definitive conclusion, the trend is certainly the same, and the results are promising, justifying further investigation. (Incidentally, it should be mentioned that a similar correlation between rate of dissolution and the rate of water self-exchange is well known, as shown by Crundwell31.)


35


1.E-07 1.E-08

Rate of growth, m/s

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 36 of 45

Sr

1.E-09

Ba

Ag

Co 1.E-10

Ca

1.E-11 Mg 1.E-12 1.E-13 1.E+04

1.E+06

1.E+08

1.E+10

Exchange rate of water in hydration sphere of aqueous metal ion, 1/s

Figure 12. The correlation between the rate of crystal growth of various oxalates and the selfexchange of water from the hydration sphere of the metal ion. Data from Nielsen (ref 10, Table 4).

The dehydration step for the deposition of the cation onto the surface as envisaged here is illustrated in Figure 13. For illustration purposes, the surface is assumed to be negatively charged, which directs the water dipoles away from the surface. The water dipoles of the cation are directly towards the cation. In order for the cation to deposit, at least five water dipoles need either to be removed from the cation hydration sheath or to be removed from the crystal surface (which also requires dipole reorientation because of the change in electrical field between the surface and the ohp). Thus, dehydration also required the reorientation of water dipoles, both of which are affected by the surface potential difference, given by Eq. (13).


36

Page 37 of 45

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Figure 13. Dehydration of a cation as it deposits on the surface, showing the removal and reorientation of the water dipoles. The electrical field occurs between the surface and the ohp.

Conclusions

13.

The argument presented in this paper started with the rather simple concept that the charge of a growing crystal face is important, and that this charge is due to the component ions themselves. By following the implications of this concept, we argued that: (i)

Surface charge arises from the deposition of anions and cations onto the surface (see Eq. (1));

(ii)

Changes in the surface potential difference with time occur because of changes in the relative rates of deposition of ions and cations (see Eq. (4));

(iii)

Stoichiometric deposition does not imply a neutral surface charge (see Eq. (18));

(iv)

Stoichiometric deposition only occurs once steady state with respect to the surface potential difference has been achieved (see Eq. (5));


37


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

(v)

Page 38 of 45

For a crystal to grow, and continue to grow, in a stable manner, the rates of deposition of the anions and cations must both be dependent on surface potential difference in opposite ways;

(vi)

The deposition reactions for anions and cations are both dependent on surface potential difference, which reveals the surface potential difference at the surface as a hidden variable not yet investigated in the context of crystallization;

(vii) By postulating that the deposition of anions and cations follows a Boltzmann-type dependence on surface potential difference (as in electrochemistry), a rate law for the kinetics of growth during crystallization was derived. (viii) This rate law admits several of limiting forms, between parabolic and linear limits. (ix)

This rate law is consistent with the chemical thermodynamics for the overall reaction.

(x)

This rate law is consistent with the experimental data.

The arguments presented here justify further investigation of the mechanism of crystallization presented here. Tests of the mechanism have been suggested.

Nomenclature s s k+ , k−

Rate constants for deposition of ions onto the surface, mol/s/m2

r r k+ , k−

Rate constants for removal of ions from the surface, mol/s/m2

n+, n-

Surface concentration of cations and anions, respectively, mol/m2


38

Page 39 of 45

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


r+ , r−

Rates of cation and anion deposition onto the surface, respectively, mol/s/m2

c

Concentration of salt, mol/m3

Cd

Differential capacity of the Stern layer, farads/m2

F

Faraday’s constant, C/mol

h

Planck’s constant

kB

Boltzmann’s constant

MA

Salt composed of cations and anions of equal charge.

R

Universal gas constant, J/mol/K

s

Relative supersaturation, unitless

T

Absolute temperature, K

y

z+ F ∆φ / 2 RT or z− F ∆φ / 2 RT where z+ = z− , unitless

ν + ,ν −

Charge number on cation and anion, respectively, unitless

ν±

Either ν + or ν − , as appropriate, unitless

σ

Excess surface charge, C/m2

[…]

Concentration in solution, mol/m3

∆φ

Potential difference across Stern layer, V

κ

Transmission coefficient in transition state theory

∆G #

Total (chemical and electrical) Gibbs free energy of the activation barrier,

∆Gc#

Gibbs free energy of the activation barrier due to chemical contributions

~

The deviation of a value from its steady-state value

≡ M z+

M z + cation deposited on the crystal surface

≡ Az−

A z − anion deposited on the surface

References


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Page 40 of 45

1. Crundwell, F.K.; Moats, M.; Ramachandran, V.; Robinson, T.G.; Davenport, W.G. Extractive Metallurgy of Nickel, Cobalt, and the Platinum Group Metals. Elsevier, Oxford, UK, 2011.

2. Vekilov, P. Crystal Growth and Design 2007, 7, 2796-2810. 3. Luther III, G.W.; Theberge, S.M.; Rickard, D.T.. Geochimica et Cosmochimica Acta 1999, 63, 3159–3169. 4. Zhang, J.-W.; Nancollas, G.H. Rev. Mineralogy and Geochemistry 1990, 23, 365-396. 5. Brady, P.V., and House, W.A. (1996) Surface controlled dissolution and growth of minerals. In: Brady, P.V. Physics and Chemistry of Mineral Surfaces. CRC Press, Boca Raton, FL, 1996, pp 225-305.

6. DeYoreo, J.J.; and Vekilov, P. Reviews in Mineral Geochem. 2003, 54, 57-93. 7. Nancollas, G.H. Adv. Colloid and Interface Science 1979, 10, 215-252. 8. De Yoreo, J.J.; Gilbert, P.U.; Sommerdijk, N.A.; Penn, R.L.; Whitelam, S.; Joester, D.; Zhang, H.; Rimer, J.D.; Navrotsky, A.; Banfield, J.F.; Wallace, A.F.; Michel, F.M.; Meldrum, F.C.; Cölfen, H.; Dove, P.M. Science. 2015, 349(6247)

9. Nielsen, A.E.; Toft, J.M. J. Crystal Growth 1984, 67, 278-288. 10. Nielsen, A.E. J. Crystal Growth 1984, 67, 289-310. 11. Bennema, P. J. Crystal Growth 1967, 1, 278-286.


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12. Gliko, O.; Neumaier, N.; Pan,W.; Haase, Fischer, I.M.; Bacher, A.: Weinkauf, S.; Vekilov, P.G. J. Am. Chem. Soc. 2005, 127, 3433-3438.

13. Burton, W.K.; Cabrera, N.; Frank, F.C. Phil. Trans. Roy. Soc. London 1951, A243, 299-358.

14. Lewis, A.E. Hydromet. 2010, 104, 222-234. 15. Gebauer,D., Kellermeier, M, Gale, J.D.,, Bergstrom, L. and Colfen, H. Chem. Soc. Rev. 2014, 43, 2348-2371.

16. Hunter, R.J. Zeta potential in colloid science - Principles and applications. Academic Press, 1981.

17. Miller, J.D.; Yalamanchili, M. R. Langmuir 1992, 8, 1464-1469. 18. P. Somasundaran. Encyclopaedia of Surface and Colloid Science, 2nd edition, Volume 3, CRC Press, 2006. pp 1925-1930.

19. Conway, B.E.; Qian, S.Y. Electrochimica Acta 1989, 34, 1785-1798. 20. Ukshe, E.A.; Bukun, N. G.; Leikis, D.I.; Frumkin, A.N. Electrochimica Acta 1964, 9, 431-439.

21. Libbrecht, K.G.; Tanusheva, V.M. Phys. Rev. Lett. 1998, 81, 176-179. 22. Bockris, J.O’M., Reddy, A.K.N. Modern Electrochemistry. Vol. 2. Plenum Press, New York, 1970.


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53. Crundwell, F.K. Minerals Engineering, 2016, 95, 185–196. 54. Nicolau, Y.F., Manard, J.C. J. Colloid Interfac. Sci. 1992, 148 551-570.


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Table of Contents Use Only - Graphic and Synopsis Concerning the influence of surface charge on the rate of growth of surfaces during crystallization. Frank K. Crundwell

Synopsis: It is argued that the surface charge has a determining effect on the kinetics of the growth of crystals. Conditions for stability and stoichiometry and a rate law are derived based of this mechanism that is consistent with experimental data for the growth of crystals.


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Concerning the Influence of Surface Charge on the Rate of Growth of

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