Matriz de análisis de los diarios de campo

Crystals, at their most basic, can be described as a solid composed of a regular, repeating arrangement of atoms or molecules. The study of crystal dissolu- tion is of high importance in many areas of nature science, such as geology and the carbon cycle,100,101 but also has huge importance in industries af- fecting peoples everyday lives such as in oral drug delivery102,103 _{to food}104

and construction.105 _{Consequently, methods for studying crystal dissolution}

are equally important, with a need for reliable techniques that can provide quantitative information on the kinetics and mechanisms.

1.3.1

Mechanisms for Crystal Dissolution

Dissolution is governed by two competing processes, mass transport and sur- face phenomena,106 _{as shown in Figure 1.6. These processes occur in series}

so, when measuring dissolution rates, only the rate limiting step, the slower of the two, can be observed. The former is the movement of material from the interfacial region to the bulk solution, this occurs in three separate ways:

1. Convection, very generally, is defined as the physical movement of a fluid. This can be from natural causes, such as temperature gradients or density differences resulting from a reaction. Examples of artifcial introduced convection include the movement of solution under a mechanical force, such as stirring, or a flow cell. Convection can be purposefully added in the form of a rotating disk or flow cell in order to overwhelm the contribution from natural convection with a highly controlled and defined form, Increasing convection is also used to increase the overall mass transport rate, in order to access surface kinetics of some crystal systems. However, convection is usually discounted from experiments by keeping the solution still and at constant temperature or conducting each experiment on a short time scale.

2. Diffusion is the motion of particles along a concentration gradient, acting to homogenize the solution. Fick mathematically described diffusion

as a function of diffusional flux (j), defined as the moles of material diffusing through a unit area in one second, this is Fick’s first law of diffusion.

j =−DB δ[B]

δx (1.3)

where [B] is the concentration of diffusing species B and DB is the diffusion

coefficient for B. Each species has a constant diffusion coefficient. A more useful form of this equation gives the change of concentration of B at a point in space. This is Fick’s second law and is used to describe the diffusion component in the time dependent 3D FEM models used in this thesis.107

∂[B] ∂t =DB ∂2_[B] ∂x2 (1.4)

Diffusion is highly important to understand and quantify as it cannot be re- moved under experimental conditions.

3. Migration, which is movement of ions under an applied potential, is worth discussion here despite most crystal dissolution studies not requiring a potential to be applied. However, some techniques, such as channel flow cells and SECM are dependent upon the use of electrodes for detection or control and therefore migration must be considered. In this thesis, crystal dissolution is not dependent on an electrochemical reaction, however, an electric field is still present and is described as migratory flux, jm

jm ∝ −u[B] ∂φ

∂x (1.5)

where φ is the applied potential and u is the ion mobility.108 Ion mobility is dependent not only on the characteristics of the ion but also on the solution. H+moving in aqueous solution has a mobility of 3.62×10−7m2s−1V−1 at 298 K,109 _{and can move under the Grotthuss mechanism, in which rather than}

a single pacticle moving, a proton will attach to a water molecule and force another off, then that one forces a proton from a second water molecule in a ’bucket chain’ style.110 _{However, most ions have a mobility on the order of}

10−8.111 _{Migration carries the ion current through the bulk solution, however,}

Figure 1.6: Diagram of mass transport processes: A) Migration, particles moving under the influence of an electric field B) Diffusion, particles moving down a concentration gradient C) Convection, particles moving under the influence of an external force, in this case stirring. D) Surface processes: surface diffusion and adsorption/desorption.

tron transfer. In the large majority of cases migration can be neglected by increasing the ionic strength of the solution used.

Surface phenomena includes processes such as surface diffusion and the detachment of ions or molecules at the crystal surface, this detachment or movement of ions on a surface is often treated as following the terrace ledge kink model112_{. This model provides the basis to consider the processes of}

adatoms desorbing, surface vacancies forming and step edges retreating. The relative thermodynamic probability of each of these processes will then vary between crystal systems.

If one is interested in surface kinetics, the experimental setup must be capable of making mass transport sufficiently high and well-defined to enable

the kinetics of the surface processes to be measured. Additionally, due to the dependence of surface process rates on the surface morphology, different sites on the same crystal surface may have different dissolution characteristics.113

This means the most informative studies are those which have a means of controlling and defining the surface in question.

1.3.2

Defects in Crystal Structure

There is a considerable body of work on the topic of dissolution from dis- locations and defects, which can seriously alter and also control dissolution in crystals.114 Many types of defect exist, such as point defects, dislocations, stacking faults and grain boundaries.115,116 Point defects are single atom de- fects arising from breaks in a crystalline materials crystal structure, such as a missing atom or the wrong atom. Dislocations and stacking faults are more substantial defects which cause large section of a crystal to shift away from the crystalline structure. Grain boundaries tend to be formed when the many faces of the crystal are crystallising at the same time, when the faces grow towards each other they form a rough edge where they meet. A full awareness of the affect of defects in crystal structures is vital to understanding more complex dissolution kinetics as dissolution can partly or almost entirely dependent on defect density.117,118