a theoretical and numerical multiscale framework … · polymers and (d) low-molecular-mass...

AUTHOR’S POST PRINT (Romeo Colour: Green) Journal for Multiscale Computational Engineering (ISSN: 1543-1649), 9 (2): 149–174 (2011).

DOI: 10.1615/IntJMultCompEng.v9.i2.20 Publisher version available at

http://www.begellhouse.com/journals/61fd1b191cf7e96f,510397234a1cbb30,007f3e5b72098e9b.html

A THEORETICAL AND NUMERICAL MULTISCALE

FRAMEWORK FOR THE ANALYSIS OF PATTERN FORMATION IN

PROTEIN CRYSTAL ENGINEERING

Marcello Lappa CTC

Via Salvator Rosa 53, San Giorgio a Cremano (Na), 80046, Italy

ABSTRACT

The relevance of self-organization, pattern formation, nonlinear phenomena and non-equilibrium behavior in a wide range of problems related to macromolecular crystal engineering, calls for a concerted approach using the tools of statistical physics, thermodynamics, fluid-dynamics, nonlinear dynamics, mathematical modeling and numerical simulation, in synergy with experimentally oriented work. The reason behind such a need is that in many instances of relevance in this field one witnesses interplay between molecular and macroscopic-level entities and processes. Along these lines, two models are defined here and discussed in detail, one dealing with issues of complex behavior at the microscopic level and the second referring to the strong nonlinear nature of macroscopic evolution. Such models share a common fundamental feature, a group of equations, strictly related, from a mathematical point of view, to the ‘kinetic conditions’ used to model mass transfer at the crystal surface; model diversification then occurs on the basis of the desired scale length, i.e. according to the level of detail required by the analysis ('local' or 'global'). If the "local" evolution of the crystal surface is the subject of the investigation (distribution of the local growth rate along crystal face, shape instabilities, onset of surface depressions due to diffusive and/or convective effects, etc, i.e. all those factors dealing with the "local" history of the shape) the model is conceived to provide "microscopic" and "morphological" details. For this specific case a ‘kinetic-coefficient-based’ moving boundary numerical (CFD) strategy is carefully developed on the basis of Volume-of-Fluid methods (also known as Volume Tracking methods) and Level-set techniques, which have become popular in the last years as numerical techniques capable of modeling complex multi-phase problems as well as for their capability to undertake a fixed-grid solution without resorting to mathematical manipulations and transformations. On the contrary, if the size of the crystals is negligible with respect to the size of the reactor (i.e. if they are small and undergo only small dimensional changes with respect to the overall dimensions of the cell containing the feeding solution), the shape of the crystals is ignored and the proposed approach relies directly on an algebraic formulation of the nucleation events and on the application of an integral form of the mass balance kinetics for each protein crystal. The applicability and the suitability of the different sub-models are discussed according to some worked examples of practical interest. Pattern formation in these processes is described here with respect to crystal shapes, nuclei spatial discrete arrangements and the convective multicellular structures arising as a consequence of buoyancy forces, thus enriching the discussions with some interdisciplinary flavor.

current e-mail address: [email protected], [email protected]

Submitted to the International Journal for Multiscale Computational Engineering 2

1. INTRODUCTION By injecting energy into a system, typically a homogeneous equilibrium state becomes unstable above

a certain threshold; as a result of this instability space-time patterns emerge above this threshold (under certain idealizations it is natural to consider the process of pattern formation as the “life” of a dynamical system).

Pattern formation is a subfield of nonlinear dynamics in spatially extended systems. Although the latter term is often used narrowly to describe nonlinear systems with not too many degrees of freedom, in general it may be applied to describe more or less everything that happens in the Universe. Thus this statement can hardly be used as a definition.

More precisely, the field of pattern formation focuses on systems where the nonlinearities conspire to form organized spatial patterns (Lappa, 2010). The study of pattern formation also falls under the broader rubric of non-equilibrium phenomena, which are observed in not just physical (see, e.g., Lappa, 2004 and 2007), but also chemical and biological systems (the subject of the present article). Indeed the fact that strikingly well-ordered and similar patterns are found across disciplines is an important impetus for research in this field.

In general, studies of pattern formation use a common set of fundamental concepts to describe how non-equilibrium processes cause structures to appear. The theoretical starting point is usually a set of deterministic equations of evolution, typically in the form of nonlinear partial differential equations. While much progress toward understanding pattern dynamics has been made in recent years, however, fundamental challenges remain, especially in the field of macromolecular crystal engineering.

Because of the large size and complexity of protein molecules and the weakness of the bonding forces between them, it is often believed that the experimental principles and methodologies (and associated mathematical models) traditionally used for inorganic crystal growth are of little help in protein crystallization and related problems of pattern formation (Rosenberger, 1986; McPherson, 1990).

Macromolecules are extremely complex physical-chemical systems whose properties vary as a function of many environmental influences (see, e.g., Chou and Shen, 2009). The perplexing difficulties that arise in the crystallization of macromolecules in comparison with conventional small molecules stem from the fact that they are very sensitive to their environment and if exposed to sufficiently severe conditions may denature and/or degrade. Thus common methods for the crystallization of conventional molecules are unsuitable and destructive. They must be supplanted with more gentle and restricted techniques. Because proteins are sensitive and labile macromolecules that readily loose their native structures, the only conditions that can support crystal growth are those that cause little or no perturbation of the molecular properties. For these reasons they are constantly maintained in a thoroughly hydrated state at or near physiological pH and temperature, i.e. crystals are grown from a solution to which they are tolerant. This is called the mother liquor. Because complete hydration is essential for the maintenance of structure, protein crystals are always bathed in the aforementioned solvent. The strategy employed to bring about crystallization is to guide the system very "slowly" toward a state of reduced solubility by modifying the properties of the mother liquor.

Since it is expected that growth is driven by the phenomena occurring in a thin zone surrounding the crystal (the so-called "depletion zone", see, e.g., Otàlora and Garcìa-Ruiz, 1997) where the concentration of the nutrients is lowered due to solute absorption, many authors have focused on "what happens close to the crystal". Several experimental studies have shown that crystal growth rate is sensitive to very large amount of parameters: e.g., temperature, solution composition, supersaturation, impurities, convection etc. Moreover it usually depends on the crystallographic orientation (Coriell et al., 1998) and on crystal habit changes with the level of supersaturation (Pusey et al., 1986; Monaco and Rosenberger, 1993). Morphological stability of crystal facets and thus crystal quality also depend on supersaturation and impurities (Coriell et al., 1998). All these aspects are usually known as "surface kinetics" of the growth process. A comprehensive review of the previous fundamental protein crystal growth and morphology

…a multiscale framework for macromolecular crystal engineering 3

experimental studies has been given for instance in Monaco and Rosenberger (1993), Otàlora et al. (2001), and Vekilov and Chernov (2002). A big effort has been made also to understand nucleation kinetics (see, e.g., Galkin and Vekilov, 1999 and references therein).

In general, macromolecular crystallization is a matter of searching, as systematically as possible, the ranges of the individual parameters that impact upon crystal formation, finding a set or multiple sets of these factors that yield some kind of crystals, and then optimizing the variable sets to obtain the best possible crystals for X-ray analysis.

In light of the arguments given above also “ensemble behaviors” (when many crystals are grown in a given reactor) are an important aspect of the problem. In fact, crystallization at a macroscopic scale is characterized by the interplay of different phenomena: transport in liquid phase, nucleation and ensuing growth, competition among different crystals, sedimentation and convection, etc. Understanding of this interplay requires a global analysis that would consider all the relevant phenomena simultaneously in order to track system evolution. Such global analysis may support optimization of growth techniques, i.e. it may help to discern interrelations between various "macroscopic" parameters of interest.

To summarize, the crystallization process occurs at two distinct length scales: on the microscopic length scale of the crystal interface, the process is governed by the above-mentioned "surface incorporation kinetics"; and, on the macroscopic scale, the process is controlled by the transport of nutrients available in liquid phase and by the competition for growth of "interacting" crystals ("ensemble behaviors"). In view of the arguments above, the analysis of pattern formation at both microscopic and macroscopic scales plays a crucial role in furthering our knowledge of the overall macromolecular crystal growth.

As outlined before, while efficient numerical methods have been developed and made available for the scientific community in the case of growth of inorganic substances, the modeling of the above-mentioned aspects in the case of growth of protein crystals is still an open task. There is still a disappointing lack of mathematical models and numerical methods able to provide a comprehensive description of the macroscopic ensemble behaviors as well as of all those factors dealing with the "local" history of the crystal surface. Aim of the present article is to present relevant mathematical models and tools along these lines (towards this end, relevant material is elicited from several recent and relevant papers of Lappa and coworkers). The objective, in particular, is to develop an integrated numerical framework for macromolecular crystal engineering that is able to relate the evolution of the crystallization process to the input parameters according to the desired level of detail.

2. MATHEMATICAL MODEL AND NUMERICAL METHOD The classical procedure for inducing proteins to separate from solution and produce a solid phase is to

gradually increase the level of saturation of a precipitant agent. Protein precipitants fall into four broad categories: (a) salts; (b) organic solvents; (c) long chain

polymers and (d) low-molecular-mass polymers and non-volatile organic compounds. For a specific protein, the precipitation point (or solubility minima), however, is usually critically

dependent on the pH, temperature, the chemical composition of the precipitant and the properties of both the protein and the solvent. Just as proteins may be driven from solution at constant pH and temperature by the addition or removal of salt, they can similarly be crystallized or precipitated at constant ionic strength by changes in pH or temperature.

By these techniques a ‘suitable’ limited degree of supersaturation can be achieved. In very concentrated solutions, in fact, the macromolecules may aggregate as an amorphous precipitate. This result is to be avoided if possible and is indicative that supersaturation has proceeded too extensively or too swiftly.

From a mathematical point of view and in the specific case of mass crystallization from a supersaturated solution one must generally accomplish at least two things simultaneously: (a) determine the concentration fields of organic substance and precipitant in the liquid phase and (b) determine the position


of the interface between the solid and liquid phases. According to the technique used to address (a) and (b), in principle the numerical procedures able to solve these problems can be divided into different groups.

As anticipated in the introduction, if the size of the crystals is negligible with respect to the size of the reactor, i.e. if they are small and undergo only small dimensional changes with respect to the overall dimensions of the cell containing the feeding solution, a "macroscopic" approach is required. On the contrary, if the "local" evolution of the crystal surface is the subject of the investigation (distribution of the local growth rate along crystal face, shape instabilities, onset of surface depressions due to diffusive and/or convective effects, etc, i.e. all those factors dealing with the "local" history of the shape) the approach must take into account "microscopic" details.

For the latter case, in principle, transformed co-ordinate systems would be required to account for the position of the crystallization front. This technique for instance has been used by Noh et al. (1998). They investigated the growth behaviors of a precipitate particle in a supersaturated solution under the effect of ‘a priori’ (well-defined) imposed ‘ambient’ flows. The particle was assumed to have initially a spherical shape. The Stokes flow approximation was assumed to simplify the model. Numerical solutions were obtained through numerically generated orthogonal curvilinear co-ordinate system, automatically adjusted to fit the boundary shape at any instant. The initially spherical particle was found to evolve into a peach-like shape if posed in a uniform streaming flow.

This approach to the problem takes the point of view that the interface separating the bulk phases is a mathematical boundary of zero thickness where interfacial conditions are applied. Difficulties arise however when this technique is employed since in this case in the vicinity of the crystallization front (phase change), conditions on mass flux, velocity, and solubility evolution have to be accounted for. This effectively rules out the application of a fixed-grid numerical solution, as deforming grids or transformed co-ordinate systems are required. Moreover the presence of many crystals in the same domain can be hardly handled.

On the contrary, a versatile and robust method would require a fixed-grid technique and therefore standard solution procedures for the fluid flow and species equations, without resorting to mathematical manipulations and transformations. This purpose can be obtained by introducing a phase-field variable which varies in space and time to characterize the phase of the material.

Furthermore, it is worthwhile to stress how this approach leads to "build" methods that can handle the case of microscopic (shape) description as well as the macroscopic behavior of many interacting crystals. For both cases, the solid mass stored in the generic computational cell can be modeled by assigning an appropriate value of the "phase variable" to each mesh point (=1 computational cell filled with solid crystal, =0 feeding solution and 0<<1 for a computational cell containing both liquid and solid phases).

Therefore the key element for the method is its technique for adjourning . Upon changing phase, the -value of the cell must be adjusted to account for mass release or absorption, this adjustment being reflected in the protein (solute) concentration distribution in the liquid phase as either a source or sink. The modeling of these phenomena leads to the introduction of a group of equations, strictly related, from a mathematical point of view, to the ‘kinetic conditions’ used to model mass transfer at the crystal surface and to the level of detail required by the analysis (macroscopic or microscopic).

From a physical point of view, on a molecular level crystal growth consists of the successive addition of ‘growth units’ or ‘building blocks’ to a lattice. Growth units can consist of single molecules or possibly of clusters of molecules. In solution, growth units are typically solvated, i.e. are surrounded by highly regularly arranged shells of solvent molecules that interact with the growth unit in a bond-like fashion.

Protein crystallization does not fit any of the traditional (inorganic) models without major modification. The ‘attachment rates’ in protein crystallization are considerably lower than in most inorganic systems (this slow interface kinetics has been interpreted in terms of the low symmetry and the small binding energies involved in protein crystallization).

Surface attachment kinetics at the crystal surface depend on the local value of solubility and on a coefficient (kinetic coefficient) having the dimension of a velocity [cm/s] whose value may be different


according to the local orientation of the crystal surface (surface-orientation-dependent growth); using mass balance (see, e.g., Rosenberger, 1986 and Pusey et al., 1986), and assuming a linear dependence of the growth rate by the interface supersaturation (see, e.g., Lin et al., 1995 and Lappa, 2003), one obtains:

S

SCn

n

C

C

D cs

csLcsCP

ˆ/

(1)

where Ccs is the concentration of the protein at the crystal surface, D is the related diffusion coefficient,

S is the solubility (its value is function of the local concentration of the precipitant agent and/or of temperature), Pand C are the protein mass density and the total mass density in the crystal, L is the total

density of the solution, is the kinetic coefficient and n̂ is the unit vector perpendicular to the crystal

surface pointing into the liquid. The dependence on n̂ takes into account anisotropic growth. This situation

occurs for crystalline proteins which have high anisotropy (preferred orientations) in either their surface energy or atomic attachment kinetics.

Whenever protein in solute phase and solid crystal co-exist in equilibrium (saturation condition): Ccs =S (2) In a saturated solution, two states exist in equilibrium, the solid phase, and one consisting of molecules

free in solution. At saturation, no net increase in the proportion of solid phase can accrue since it would be counterbalanced by an equivalent dissolution. Thus ‘crystals do not grow from a saturated solution’. The system must be in a non-equilibrium, or supersaturated state to provide the thermodynamic driving force for crystallization. The solution is said to be supersaturated when the solute content is greater than S, and the degree of supersaturation is defined by = Ccs/S. Further details on these dynamics are given below on the basis of the excellent theoretical analysis of McPherson (1990).

In practice, if no solid is present, as a supersaturation condition is reached (Ccs>S), then solute will not immediately partition into two phases, and the solution will remain in the supersaturation state. The solid state does not necessarily develop spontaneously as the saturation limit is exceeded because energy, analogous to the activation energy of a chemical reaction, is required to create the second phase, the stable nucleus of a crystal or a precipitate. Thus, a kinetic or energy barrier allows conditions to proceed further and further from equilibrium, into the zone of supersaturation.

Figure 1: Phase diagram showing protein solubility as a function of precipitant concentration and regions for the occurrence of “typical” phenomena when the system is in supersaturated conditions. The solid line represents the saturation curve. The supersaturation region lies above this curve and is, in turn, demarcated by a boundary separating the metastable region (where only already existing deposits can grow) from the labile region (where crystal nuclei can spontaneously form and grow).


On a phase diagram (Figure 1), the line indicative of saturation is also a boundary that marks the requirement for energy–requiring events to occur in order for a second phase to be established, the formation of the nucleus of a crystal or the nonspecific aggregate that characterizes a precipitate.

Once a stable nucleus has formed in a supersaturated solution, it will continue to grow until the system regains equilibrium. While non-equilibrium forces prevail and some degree of supersaturation exists to drive events, a crystal will grow or precipitate continue to form.

It is important to understand the significance of the term “stable nucleus”. Many aggregates or nuclei spontaneously form once supersaturation is achieved, but most are, in general “not stable”. Instead of continuing to develop, they redissolve as rapidly as they form and their constituent molecules return to solution. A “stable nucleus” is a molecular aggregate of such size and physical coherence that it will enlist new molecules into its growing surfaces faster than others are lost to solution; that is, it will continue to grow so long as the system is supersaturated.

In classical theories describing crystal growth of conventional molecules, the region of supersaturation that pertains above saturation is further divided into what are termed the “metastable” region and the “labile” region as shown in Figure 1. By definition, stable nuclei cannot form in the metastable region just beyond saturation. If, however, a stable nucleus or solid is already present in the metastable region, then it can and will continue to grow.

Solution must by some means be transformed or brought into the metastable state whereby its return to equilibrium, forces exclusion of solute molecules into the solid state, the crystal. As long as Ccs<S, more solid material will dissolve if any. If, on the other hand, Ccs >S, material will condense on any material already existing and augment its size.

The labile region of greater supersaturation is discriminated from the metastable in that stable nuclei can spontaneously from. Moreover, because they are stable they will accumulate molecules and thus deplete the liquid phase until the system reenters the metastable, and ultimately, the saturated state.

In practice, an important point, shown graphically in Figure 1, is that there are two regions above saturation, one of which can support crystal growth but no formation of stable nuclei, and another one where new crystals can be formed.

In terms of phase diagram, ideal crystal growth would begin with nuclei formed in the labile region but just beyond the metastable. There, growth would occur slowly, the solution, by depletion, would return to the metastable state where no more stable nuclei could form, and the few nuclei that had established themselves would continue to grow to maturity free of defect formation.

At this stage, diversification of a model for the treatment of these aspects is needed according to the desired scale length. The mathematical conditions accounting for the surface kinetics, in fact, must behave as differential equations in the case of microscopic approach and as integral balance laws if the analysis is carried out at macroscopic scale (investigation of the "ensemble behavior").

2.1. Macroscopic Analysis and Integral Formulation of the Kinetic Conditions If the shape of the crystal and its morphological time-evolution are not an essential part of the problem,

the mathematical approach can rely directly on the application of an integral form of the kinetic balance law for each protein crystal. The shape of the crystals is ignored. Each solid particle is considered in terms of its mass and position within the reactor. Accordingly, to account for the kinetic condition, eq. (1) is re-written as an integral condition to be applied along the frontier of the computational cell containing the solid particle. The concentration at the crystal surface (Ccs) is assumed to be equal to the value of the concentration C computed along the frontier of the computational cell containing the crystal. Of course Ccs is not constant along the frontier; it has a different value according to the side where it is computed, i.e. the integral form of eq. (1) takes into account the relative direction of the different faces, which induces varying conditions for each face. In particular, for each face Ccs is expressed as average value ( average

iC ) with


respect to the concentration C~ of the cell accommodating the crystal and that of the neighbor cell sharing

the face in question neighbouriC .

Integrating over the frontier of the computational cell equation (1) reads:

dA

S

CCdAnCD cs

L

csCP 1ˆ

(3)

now n̂ is the unit vector perpendicular to the computational cell frontier pointing into the generic

neighbor cell. In discretized form:

i

i

neighbouri

L

neighbouri

CP

i

i

i

neighbouri

AS

CCCC

An

CCD

12

~

2

~

~

(4)

The concentration C~

satisfying the integral form of eq. (1) is obtained from eq. (4), then the mass stored in the computational cell is updated according to the equation:

dAS

CC

t

M cs

L

csCP 1

dAS

CC

vtcs

L

cs

P

C 111

(5)

The volume [cm3] of the crystal mass M stored in a grid cell in fact can be computed as:

Pstored

Mv

v

vstored

(6)

where P is the protein mass density in the crystal and v is the volume of the computational cell. Hence (n superscript denotes time step):

i L

neighbouri

P

Cnn CC

vt

2

~1

11i

neighbouri A

S

CC

1

2

~ (7)

with C

~ satisfying eqs. (4). According to equations (4) and (7) the growth velocity is not directly imposed but it results from

internal conditions related to solute transport and incorporation kinetics. If the protein (solute) concentration is locally depleted, correspondingly, the solid mass stored in the computational cell grows and the phase variable is increased; on the other hand if mass stored in the cell begins to re-dissolve, protein is released in solute phase and the local value of protein concentration is increased.

The shape of the crystals is ignored while sacrificing little in accuracy for the macroscopic description of the spatio-temporal behavior. Evidence of the reliability of the present assumptions is provided by the good agreement between experimental and numerical results (see, e.g., Lappa et al., 2003) as will be shown in Sect. 3.

It is worthwhile to stress how the technique discussed above for the case of crystals modeled only in terms of mass and position, allows a simple and efficient treatment of the nucleation phenomena.


As explained in the initial part of Sect. 2, when there is no pre-existing deposit, it is generally found that the concentration of protein has to be greater than S to create one spontaneously, say

(8) where is the supersaturation limit (>1); once nuclei are created, precipitation can continue if 1<

and vice-versa material can come back to the solute condition if 0<<1. According to the crystallization criteria introduced by Henisch and Garcìa Ruiz (1986) and Henisch

(1991) (algebraic model), once the solid particles are formed it can be assumed that they are in equilibrium with protein and salt in liquid phase. This assumption is supported by the well-known fact that "at the supersaturation required for nucleation, crystals typically grow extremely rapid" (Galkin and Vekilov, 1999). If the supersaturation limit is exceeded, the amount of supersaturated protein in liquid phase is posed in solid without any time delay (Lappa and Castagnolo, 2003; Lappa et al., 2002; Lappa and Carotenuto, 2003). Thus the solute content of the solution will have to be decremented from the original C value to the solubility value.

If nand n =(C/S) n

vSCM nograin )(

, n+1 11 )(

n

P

n

P

ograin SM

v

and Cn+1 =S n (9)

This nucleation model has been also successfully used by Piccolo et al. (2002) and Carotenuto et al.

(2002) to elucidate their experimental results.

2.2 Microscopic Analysis and Differential Formulation of the Kinetic Conditions If a detailed description of the local evolution of the crystal is required (distribution of the local growth

rate along crystal face, growth habit simulation, onset of surface protuberances due to diffusive and/or convective effects, etc.), the simple algebraic model defined in the preceding subsection is no longer applicable.

As already explained to a certain extent in the introduction, crystal growth morphology results from an interplay of crystallographic anisotropy and growth kinetics, the latter consisting of interfacial processes as well as (diffusive and convective) transport of solute in the external environment (the mother liquor).

Before embarking in the treatment of the numerical methods expressly conceived for protein crystal growth (Lappa, 2003 and 2004), the text below provides a short and focused review of other possible approaches that have been proposed in the literature to model (with various levels of approximation) such phenomena. Indeed, some interesting variants have been elaborated over the years by other researchers in a variety of fields. Even though most of such “variants” are not strictly related to the production of protein crystals, the interested reader will take advantage of such additional references and eventually use them for the elaboration of new methods (driven by some synergetic “digestion” of the approach suggested here together with those proposed in similar fields by other authors).

A deductive approach is followed with models of growing complexity being treated as the discussion progresses (ranging from the simple algebraic formulation described in Sect 2.1 up to complex approaches based on coupled systems of partial differential equations).

Let us start from the consideration that for many evolving faceted surfaces, a facet velocity law can be observed (see, e.g., Kolmogorov, 1940 and van der Drift, 1967), assumed (Pfeiffer et al., 1985; Shangguan and Hunt, 1989), or derived (Gurtin, and Voorhees, 1998; Watson et al., 2003; Watson and. Norris, 2006) which specifies the normal velocity of each facet, often in configurational form which depends on the geometry of the facet (in general this velocity depends on both orientation and curvature; in some situations


the growth depends only on the orientation of the facets and this is called the van der Drift model, after Drift, 1967; in other circumstances, see Taylor, 1993, the growth is controlled mainly by curvature).

In general, a baseline considered as an important reference in the literature for crystal morphology is the so-called “equilibrium shape”, which results from minimizing the anisotropic surface free energy of a crystal under the constraint of constant volume (this equilibrium shape is given by the classical Wulff construction; Wulff, 1901). If the shape is determined on the basis of an anisotropic interfacial kinetic coefficient rather than the anisotropic surface free energy, however, it is referred to as the “kinetic Wulff shape” (by definition, the kinetic Wulff shape is the asymptotic shape approached by a crystal that is growing under conditions of pure interface control governed by an anisotropic kinetic coefficient under the condition of uniform concentration and/or temperature, i.e. with a normal growth velocity V(n) that

depends only on the angle of the unit surface normal n̂ , see, e.g., Peng et al., 1999b).

In this way, the dynamics of a continuous, two-dimensional surface can be concisely represented by a discrete collection of such velocities, and overall computational complexity reduced to that of a system (ODE) of ordinary differential equations (the resulting system, in particular, is known as a Piecewise-Affine Dynamic Surface, or PADS).

Local geometric growth models able to deal with equilibrium forms containing both facets and rough regions have been also elaborated (e.g., Wettlaufer et al., 1994).

The numerics involved in the direct geometric simulation of an arbitrary PADS is straightforward for one-dimensional surfaces, requiring nothing beyond traditional ODE techniques except simple geometric translation between facet displacement and edge displacement, and a small surface correction associated with typical coarsening events. Consequently, such simulations accompany many of the above facet treatments of facet dynamics, and have also been independently repeated in the literature (Wild et al., 1990; Thijssen et al., 1992; Paritosh et al., 1999; Zhang and. Adams, 2002 and 2004).

However, in two or three dimensions, any code must be able to deal with a family of topological events that alter the neighbor relations between nearby facets. This has led many researchers over the years to introduce additional levels of complication to the PADS approach discussed above to expressly take into account topological events (e.g., Norris and Watson, 2007 and 2009). Such numerical methods generally use a three-component facet/edge/junction storage model, which by naturally mirroring the intrinsic surface structure allows both rapid simulation and easy extraction of geometrical statistics and can be applied to the simulation of surfaces with arbitrary symmetry groups.

Practically speaking, however transport of solute (the nutrient) is important during at least some stage of a crystal growth process. Crystal morphology, therefore, tends to be influenced by the shape and geometry properties of the container in which a crystal is grown. The growth velocity is not a mere function of the orientation of the facets since the growth velocity changes in time owing to changes in the concentration and in the supersaturation level. The situation is even more complicated because the transport processes in the mother liquor themselves can become unstable (fluid-dynamic instabilities). Furthermore, plane facets can be replaced in time by rough surfaces if the degree of supersaturation is too strong (which leads typically to the growth of solid mass in the form of amorphous precipitate).

At this stage a clear distinction must be introduced about the two broad classes of methods which have appeared so far in the literature for crystal growth: geometric and non-geometric.

Geometric models pertain to the category of methods (PADS and related variants) discussed above; they are appropriate when the interfacial growth velocity may be determined solely by local interfacial parameters, decoupled from diffusional or other external influences. In particular, following the general concepts introduced by Taylor et al. (1992), a model can be defined “geometrical” if the normal velocity at an interfacial point depends on the shape and position of the interface and not on field variables modified by the interface motion or transport in the liquid bulk (thus, a model is geometric in the sense that only the shape and shape-dependent quantities of the interface determine the motion).

Unlike geometric models, non-geometric methods generally treat growth or surfaces with anisotropy introduced into the interfacial conditions coupling the crystal to its external environment.


As a result, the resulting shapes can be cellular or dendritic, but can also exhibit corners and facets related to the underlying crystallographic anisotropy.

Such complex crystal morphologies are very difficult to model because of the need to track a sharp crystal-nutrient interface in detail (see the discussion at the beginning of Sect. 2 about the approach proposed by Noh et al., 1998).

Within the last decade, however, powerful phase-field models have been developed to handle simultaneously, at least in principle, all of the above phenomena. These methods are based on a diffuse crystal interface (rather than a sharp interface) and incorporate an order parameter to indicate the phase. Evolution of the phase field and the transport fields (temperature, solute) is based on partial differential equations (PDE).

As a first relevant example of methods pertaining to this category it is worth mentioning Yokoyama and Sekerka (1992), who investigated the combined effect of the anisotropy of surface tension and interface kinetics, and demonstrated the formation or suppression of corners by means of an asymptotic analysis.

McFadden et al. (1993) studied the inclusion of anisotropic surface free energy and anisotropic linear interface kinetics in phase-field models for the solidification of a pure material; the formulation was described for a two-dimensional system with a smooth crystal-melt interface and for a surface free energy that varies smoothly with orientation (in which case a quite general dependence of the surface free energy and kinetic coefficient on orientation can be treated).

Taylor and Cahn (1994) compared four surface motion laws for sharp surfaces with their diffuse interface counterparts by means of gradient flows on corresponding energy functionals; in particular, the energy functionals were defined to give the same dependence on normal direction for the energy of sharp plane surfaces as for their diffuse counterparts. In a later work (Taylor and Cahn, 1998) they elaborated an analysis of the motion of diffuse interfaces including non-differentiable and non-convex gradient energy terms. The problem of non-differentiability was resolved by using nonlocal variations to move entire facets or line segments with orientations having such facets or line segments in the underlying Wulff shape. The problem of ill-posedness was resolved by using varifolds (infinitesimally corrugated diffuse interfaces) constructed from the edges and corners in the underlying Wulff shape, or equivalently by using the convexification of the gradient energy term and then reinterpreting the solutions as varifolds. This was justified on a variational basis.

Kazaryan et al. (2000) reported the effect of the anisotropies in the grain boundary energy and grain mobility on the grain shape by using generalized phase-field model for a polycrystalline model.

Eggleston et al. (2001) developed a phase field model for highly anisotropic interfacial energy. The method was developed for two-phase systems with anisotropic interfacial energy and allowed for anisotropies sufficiently high that the interface had corners or missing crystallographic orientations. The method was based on a regularization enforcing local equilibrium at the corners and allowing corners to be added or removed without explicitly tracking their location. Numerical simulations for various degrees of anisotropy were performed and they showed excellent agreement with analytical equilibrium shapes for a wide variety of initial conditions.

Loginova et al. (2001) and Provatas et al. (1999) developed adaptive gridding finite element methods, in which only the vicinity of the interface is divided into a fine mesh while most of the space far from the interface has a coarse mesh. Plapp and Karma (2000) also developed a multi-scale method by using a random-walk algorithm for the far-field solution.

Uehara and Sekerka (2003) simulated facet formation by the phase field model by introducing a function having a narrow minimum to express the anisotropy of the kinetic coefficient.

In the VOF (volume of fraction) method developed by Lappa (2003), the surface attachment kinetic conditions were directly used to introduce a group of differential equations for the protein concentration at the crystal surface and the evolution of the solid mass displacement.

The details of such approach (which is of particular interest in the context of this article) are carefully reported in the following.


Unlike the algebraic model developed in Sect 2.1, the crystals are no longer modeled as "single" computational cells. Each crystal occupies a significant portion (i.e. a large number of computational cells) of the computational domain and protein concentration must satisfy the differential kinetic condition (1) along its surface ( 0 ,0<<1):

n̂ (10)

as an example, for two-dimensional problems ),(ˆ n with

22

/

yxx

22

/

yxy

(11)

since y

C

x

C

n

C

, (hereafter the subscript ‘cs’ is omitted) eq. (1) can be written as

)1/(/,

SCCDy

C

x

CLCP (12)

)1/(/ SCC LCP represents the mass exchange flux between solid and liquid phase (i.e. crystal

and protein solution). The mass stored in computational cells that are undergoing phase change can be computed according to the equation:

sSCCt

MLCP

)1/(/, (13)

where s is the ‘reconstructed’ portion of the crystal surface (by definition perpendicular to the

interface normal vector n̂ ) ‘bounded’ by the frontier of the control volume (computational cell) located

astride the crystal surface. Therefore the phase field equation reads

0

t

, if 0

dv

sSCC

t P

csLcsCP

)1/(/, if 0 , 0<<1 (14)

with C satisfying eq. (12). Equations (12), (13) and (14) behave as ‘moving boundary conditions’, their solution being strictly

associated to the computational check on the value of and its gradient.

In practice, the unit vector n̂ can be computed from the gradient of a smoothed phase field

, where

the transition from one phase to the other takes place continuously over several cells (4 or 5). The smoothed

phase field

is obtained by convolution of the unsmoothed field with an interpolation function.


Obviously, depending on the interface’s orientation, concentration gradients must be discretized by forward or backward schemes. For this reason eq. (12) in discretized form reads:

>0, >0: (15a)

yxSCD

yCxCCDC

Ln

jiCP

nji

njiL

njiCPn

ji

/////

////1

,

11,

1,1

1,1

,

<0, >0: (15b)

yxSCD

yCxCCDC

Ln

jiCP

nji

njiL

njiCPn

ji

/////

////1

,

11,

1,1

1,1

,

>0, <0: (15c)

yxSCD

yCxCCDC

Ln

jiCP

nji

njiL

njiCPn

ji

/////

////1

,

11,

1,1

1,1

,

<0, <0: (15d)

yxSCD

yCxCCDC

Ln

jiCP

nji

njiL

njiCPn

ji

/////

////1

,

11,

1,1

1,1

,

1,njiC is computed from equation (15) iterating up to reach convergence, then the phase variable is

updated using equation (14):

yx

sSCCt

P

njiL

njiCPn

jin

ji

)1/(/ 1

,1

,,

1,

(16)

According to equations (15) and (16), the considered phenomena are driven by the attachment kinetic

condition, i.e. the existing deposit grows if protein concentration is larger than S and the mass exchange is proportional to the local value of the orientation-dependent kinetic coefficient; in the opposite situation, i.e. in case protein concentration becomes smaller than S, the deposit begins to re-dissolve.

As outlined before, in eq. (16) s is the ‘reconstructed’ portion of the solid wall ‘bounded’ by the frontier of the control volume (computational cell) located astride the crystal surface. The determination of s requires a well defined ‘interface-reconstruction’ technique (the shape of the crystal for a fixed time is not known a priori and must be determined as part of the solution). A reconstruction is a geometrical approximation of the true solid/liquid interface. Usually the interface can be approximated by a straight line of appropriate inclination in each cell. Various conditions can be used to determine the reconstruction, one of which is a condition that the straight lines connect at cell-faces. With this condition, the resulting interface representation becomes continuous. However, for strong distortions and topology changes similar to those involved in organic crystal growth, this condition becomes virtually unenforceable. If non-connecting straight lines (each line is determined independently of the neighbor lines and their ends need not necessarily connect at the cell-faces, see, e.g., Gueyffier et al., 1999) are used, this guarantees maximum robustness and simplicity, and allows extension to three dimensions, while sacrificing little in accuracy.

In practice, two conditions have to be used for the straight line in cell (i,j), which always yield an

unambiguous solution: a) it is perpendicular to the interface-normal-unit vector n̂ ; b) it delimits a solid

area ‘matching’ the given nji, for the cell.


At this stage, it is worth highlighting some analogies/differences of the technique defined by eqs. (10)-(16) with respect to the aforementioned geometric models as well as with respect to the classical VOF methods appeared in the literature (mainly used to simulate the dynamics of liquid-liquid or liquid-gas systems) on which it is based.

There is no doubt that it shares with the aforementioned PADS methods the interface-orientation analysis, that in those methods is needed to select the relevant growth velocity for the considered crystal face. In classical VOF methods for fluid-fluid systems, in general, the interface-orientation analysis is needed to compute the surface tension force perpendicular to the interface separating the fluid-phases (expressed as a corresponding volume force, which can be included in the momentum equation).

In practice, here it is needed for the orientation-dependent kinetic coefficients, but also for the discretization of the protein concentration gradients perpendicular to the surface. Hence, unlike PADS methods that do not take into account what happens in the external medium, with the present method the growth velocity is affected by both intrinsic crystal growth kinetics and transport of nutrients in the external phase (the concentrations appearing in the second member of eqs. (15) have to be computed on the basis of additional mass balance equations governing the diffusion and convective transport of solute in the mother liquor, which will be discussed in Sect. 2.3).

Even if they are not specifically related to the problems considered here, it is also worth citing the variants elaborated by Hong et al. (2006), Ma et al. (2006) and Chen et al. (2008).

As a possible alternative to the phase field and VOF strategies illustrated in detail above, a different approach can be also used known as « Level-set » (Lappa, 2004).

This method, first introduced by Osher and Sethian (1988), is conceptually similar to a phase-field model in that the solid–liquid interface is represented as the zero contour of a level set function, (r,t), which has its own equation of motion. The movement of the interface is taken care of implicitly through an advection equation for (r,t). Unlike the phase-field model, however there is no arbitrary interface width introduced in the level set method; the sharp-interface equations can be solved directly and, as a result, no interface reconstruction techniques are required (Kim et al., 2000).

With this technique the goal is to compute and analyze the subsequent motion of under a velocity field q (Osher and Fedkiw, 2002). This velocity can depend on position of the interface of the organic crystal () and the external physics (for the case under investigation it depends on the surface incorporation kinetics, i.e. on the mechanisms of attachment of solute molecules to the growing solid surface, i.e. eq. (1)). The interface is captured for later time as the zero level set of the function (r,t), i.e.,

0),()( trrt .

In practice the level set function is defined as the signed normal distance from the solid–liquid interface such that is positive in the liquid phase, negative in the crystal, and zero at the solid/feeding-solution interface:

(r,t)>0 for r (r,t)=0 for r=(t) (17) (r,t)<0 for r

Thus, the interface is to be captured for all later time, by merely locating the set (t) for which

vanishes (at this stage it is clear that there is a very simple relationship between and the phase field variable used by the VOF method: =0 for >0 and =1 for 0).

With the Level-set technique, the interface motion is analyzed by convecting the values (levels) with the velocity field q:

0

qt

(18)


where q is the desired velocity on the interface, and is arbitrary elsewhere.

Actually, only the normal component of q is needed:

qqn so eq. (18) becomes

0

Ft

(19)

Integrating Eq. (19) for one time step results in moving the contours of along the directions normal

to the interface according to the velocity field F, which varies in space. F is constructed to be an extension of the interface velocity, qn , such that F=qn for points on the interface and the lines of constant F are normal to the interface (see eq. (21)). Thus, advecting according to Eq. (19) moves the front with the correct velocity.

After solving Eq. (19) for one time step, the level set function will no longer be equal to the distance away from the interface. It is necessary to reinitialize to be a signed distance function. This step is accomplished by solving

01)(

sqn (20)

),()0,( trr

Where is an artificial time. Following Osher and Fedkiw (2002), in order to define the distance in a

band of width around (t), eq. (20) is solved only for = O(). The basic level set method concerns a function (r,t) which is defined throughout space. Clearly this is

useless if one only cares about information near the zero level set. The local level set method defines only near the zero level set. In practice, eq. (19) is solved in a neighborhood of (t) of width mr, where m is typically 5 or 6. Points outside of this neighborhood need not be updated by this motion. The function is reinitialized to be signed distance to (t), only near the boundary, smoothly extending the velocity field qn off of the front (t) (see eq. (21)) and solving equation (19) only locally near the interface (t), thus lowering the complexity of this calculation by an order of magnitude (Osher and Fedkiw, 2002). This makes the cost of level set methods competitive with the VOF technique.

With regard to the smooth extension of the quantity qn on (t) to a neighborhood of (t), this step can in principle be accomplished by various possible strategies.

One way of extending the normal velocity off the interface is extrapolation in the normal direction, following characteristics that flow outward from the interface, such that the velocity is constant on rays normal to the interface. This method, introduced by Malladi et al. (1995), works particularly well when no other information is available except for what is known on the interface–as is the case here.

An alternative way to formulate this construction is as a pair of linear Hamilton-Jacobi equations (see Chen et al., 1997; Zhao et al., 1998; Peng et al., 1999a, Osher and Fedkiw, 2001):

0)(

FsqnF

(21)

),()0,( trqrF n


Here designates a pseudo-time for the relaxation of the equations to steady-state at each moment of

time t. Again, this equation is solved only for = O() in order to extend qn to be constant in the direction normal to the interface in a region of width .

An additional alternative way of computing extension velocities was also developed by Adalsteinsson and Sethian (1999).

For the case considered in this article qn is given by eq. (22) below with C at the crystal surface satisfying eq. (1).

1

S

Cnq (22)

It is worthwhile to stress how the function qn is not given "a priori" but has to be computed at any

instant as part of the problem (it dynamically changes during the growth process according to the residual protein available in liquid phase, and according to the steepness of the concentration gradient at the solid/liquid interface that in turn may be changed by the motion of the surrounding fluid).

Along these lines, following the same approach undertaken after the illustration of the phase-field (VOF) approach, a short review of other relevant approaches (developed by other investigators in the framework of the level-set strategy) is provided below.

We have explained before as the situation in which the growth rate (velocity) of each facet depends only on the crystallographic orientation of the facet (this is the so-called interface-limited growth and is common in, for example, the chemical vapor deposition of diamond) is an idealization of the growth process originally introduced by van der Drift. For such simplified cases the reader may consider the level-set method developed by Osher and Merriman (1997). A similar formulation has been used by Taylor et al. (1992). Using this formulation, under general assumptions, Osher and Merriman (1997) were able to prove that an initial interface asymptotically approaches the Wulff shape. Earlier, Soravia (1994) proved this result by a slightly different approach (the reader is also referred to the interesting developments provided by Merriman et al., 1994). A similar result was obtained by Peng et al. (1998). They also explored the connection between motion of faceted interfaces and motion of shocks for conservation laws. More recent interesting contributions are due to Russo and Smereka (2000) and Smereka et al. (2005)

Most recently, Burger et al. (2007) have investigated anisotropic growth with curvature-dependent energies. They have considered the two dominating growth modes, namely attachment–detachment kinetics and surface diffusion. The level set formulations have been given in terms of metric gradient flows, discretized by finite element methods in space and in a semi-implicit way as local variational problems in time. Finally, the constructed level set methods have been applied to the simulation of faceting of embedded surfaces and thin films.

Despite some problems (see Lappa, 2005b,c for a review), level-set methods offer important geometrical advantages. A drawback of the VOF, in fact, is that it is hard to compute accurate interface normals and curvatures because of the discontinuity in the reconstructed interface is not smooth or even continuous, lowering the accuracy of the geometrical information. Since is continuous, the calculation of the interface normal and curvature in the level-set approach is natural, easy and accurate, in contrast to the corresponding calculations in the VOF method, that require unphysical smoothing of .

There are also other limitations in the VOF approach. As explained before, the proper use of these models also requires, in fact, that an asymptotic analysis is performed in order to obtain a mapping between the function and the sharp-interface, i.e. a very accurate interface-reconstruction technique is required. Moreover, computationally, the grid spacing must be small enough to resolve the interfacial region. The interface is, in fact, defined by the condition 0<<1 and is, therefore, associated with a somewhat arbitrary thickness (the width depends on the resolution of the computational mesh).


These reconstruction procedures are very laborious. On the contrary, the level-set computational approach has the capability to track the motion of the interface without resorting to mathematical manipulations and complex reconstruction techniques. Also, shape changes, corner and cusp development, topological merging and breaking are naturally obtained in this setting without the elaborated machinery (see Chang et al., 1996; Sussman and Ohta, 2007).

2.3 Governing Field Equations

In general, the specific algorithm used to capture the interface must be supplemented with the

equations governing mass transport in the liquid phase. These are the continuity, Navier-Stokes and species equations, that in conservative form read :

0 V (23)

ViCCgiCCg

VVVpt

V

goNaClNaClNaClgoprotein

1

21

)()(

(24)

CDCVt

C 21 (25)

saltsaltsaltsalt CDCVt

C 21

(26)

where is the kinematic viscosity, V and p are the velocity and pressure and C(o) and Csalt(o) are the

initial value of protein and precipitant agent (salt) respectively. In the momentum equation the Boussinesque approximation has been used to model the buoyancy forces: protein and salt are the solutal expansion coefficients related to organic substance and salt respectively. Note that in case solubility modulation is induced by temperature control, eq. (26) must be replaced by the energy equation.

Assumptions invoked in the development of equations for this continuum model also include: laminar flow, Newtonian behavior of the phases (this implies that solids, should be treated as highly viscous fluids), constant phase densities (and Boussinesque approximation).

Moreover the solid phase (crystal) is assumed to be non-deforming and free of internal stress, while the multiphase region (region where phase change occurs) is viewed as a porous solid characterized by an isotropic permeability by analogy with the enthalpy methods (see, e.g., Voller and Prakash, 1987; Bennon and Incropera, 1987; Lappa and Savino, 2002).

The species equations (25) and (26) must be solved throughout the computational domain including both the solid and liquid phases. The presence of the term (1-) ensures in fact that the solid phase is impermeable to protein and precipitant in solute phase.

An important problem with the fixed-grid solution procedures described in Sect. 2.2, is accommodating the zero velocity condition, which is required as a liquid region turns to solid. Various methods can be used in principle to ‘switch off’ velocities in computational cells that are becoming solid (or ‘switch on’ velocities in the reverse case). Velocities may simply be set to zero. Another possible approach is based on the viscosity. If the viscosity of a cell is driven to a very large value as the solid mass stored in this cell increases, the increasing viscosity will provide the necessary coupling between the physical state of the material in the cell and the momentum equations, thereby driving velocities to zero in cells that are solid. A


third possible approach (porosity approach) requires that computational cells that are undergoing a phase change are modeled as a pseudo-porous media, with the porosity being a function of ranging between 0 (fully liquid cell) and 1 (fully solid cell). For the present case of macromolecular organic crystal growth, this assumption is supported by the fact that solid formation occurs as a ‘permeable’ crystalline matrix which coexists with the liquid phase. The term –V/ in equation (24) is the Darcy term added to the momentum equation to eliminate convection in the solid phase. Here permeability is assumed to vary according to the Carman-Kozeny equation (Bennon and Incropera, 1987).

2

3)1(

(27)

In the pure solid (=1) and pure liquid (=0), equation (27) reduces to the appropriate limits, namely =0 and = respectively. In practice the effect of is as follows: in full liquid elements 1/ is zero and has no influence; in elements that are changing phase, the value of will dominate over the transient, convective and diffusive components of the momentum equation, thereby forcing them to imitate Carman-Kozeny law; in totally solid elements, the final large value of 1/ will swamp out all terms in the governing equations and force any velocity predictions effectively to zero.

Since the momentum equation is valid throughout the entire domain, explicit consideration need not be given to boundaries between solid, multiphase and liquid regions.

3. RESULTS AND EXAMPLES

Table 1: Lysozyme Properties

Dlys [cm2 s-1] 10-6

DNaCl [cm2 s-1] 10-5

[cm2 s-1] 8.6310-3

c [g cm-3] 1.2

lys [g-1 cm3] 0.3

NaCl [g-1 cm3] 0.6

P [mg ml-1] 820

[Å s-1] O(10)

pH 4.5

In order to shed further light on the techniques described in section 2.1 and 2.2 and, in particular, on

the related field of applications, in the present section, some "reference" cases are discussed and elucidated. Such discussion may help the Reader in discerning both the advantages and limitations of the microscopic and macroscopic approaches, respectively.

Hen egg white lysozyme is used as model protein, being a well-characterized and widely-used molecule (see Table 1 for the properties).

Lysozyme has become over the years a very important "paradigm" model protein for fundamental research. Usually it is selected as the protein to crystallize because its solubility curve S = S(CNaCl) is perfectly known and because of the current experience of organic crystal growers in producing gel reinforced crystals to be used as "seeds" for surface studies.

The precipitant agent is NaCl. Two ‘typical’ cases are considered in the present article. In the first case (Figures 2-4), the growth of a small number of lysozyme seeds (size 1 [mm]) is investigated under the effect of weak buoyancy convection induced by a small body acceleration (g=10-4 go, this is the typical


value of residual gravity occurring on orbiting space platforms, e.g., the Space Shuttle or the International Space Station, where researchers try to optimize crystal growth in the absence of gel). In the second case (Figures 5-8), the formation of a multi-crystal pattern is investigated in the framework of a counter-diffusion technique and a gellified configuration.

3.1 Morphological Evolution of Seeds in the Presence of Fluid Convection

The procedure usually employed in studies for morphological evolution foresees that at the beginning a

relatively large number of nuclei is obtained by means of a separate method (in general the counter-diffusion technique); then, those displaying better quality and size are placed in a new small reactor expressly conceived and equipped for surface analysis (in situ, nonintrusive monitoring techniques). At this stage they are called “seeds.” Particular care is devoted to avoid further nucleation phenomena, i.e. the formation of new nuclei, that would spoil interferometry data.

For the sake of simplicity for the present example the initial shape of the specimens is supposed to be quadrate (i.e. a horizontal layer of fluid confined between two parallel walls with a periodic array of evenly spaced square bodies in the interior) and the kinetic coefficient is supposed to be the same for the different sides of the crystal.

Relevant non-dimensional geometrical parameters are the aspect ratio A=H/L (see Figure 2) of the reactor and the parameter = /d (ratio of the crystals size and related distance).

Following typical experimental procedures the seeds are supposed to be fixed (e.g. by glue) to the bottom of the reactor (see Figure 2).

The present example focuses on the case of N seeds (N=6) located one away from the other at a distance of the order of the crystal size (=O(1)) and considering a fixed value of the aspect ratio of the reactor (A=H/L=3).

The condition =O(1) is investigated since (owing to the small size of the reactors used for morphological analyses) this is the typical situation occurring in this kind of studies (Lappa, 2005a). Growth is obtained from a super-saturated solution with

6, i.e. the metastable region of the phase

diagram in Figure 1 is considered (where existing deposits can grow but no new nuclei can form). Numerical simulations are carried out within the framework of the approach described in Sect. 2.2

y

x

g

d

l

H

L

Figure 2: Sketch of the growth reactor with the protein seeds and of the relative direction of the residual g.

As shown by the simulations, as seeds grow from the solution, each crystal depletes the concentration

of the growth units that are incorporated into the crystal lattice producing a concentration-depleted zone around it. In this region, the solute concentration changes continuously from the concentration at the crystal face to the concentration in the bulk of the solution. The concentration profile in the concentration-depleted


zone varies with time as the crystal grows and is controlled by the balance between the flow of growth units towards the crystal face and the rate of incorporation of these growth units into the crystal lattice. The kinetics of incorporation at the crystal surface are linked to the bond configuration of the crystallographic structure as discussed in Sect. 2 (for the present case of isotropic growth, the kinetics of incorporation are simply linked to the value of the kinetic coefficient and to the supersaturation level), while the flow towards the crystal face is highly dependent on the symmetry of the concentration pattern, which turn out to be crucial for the overall crystal-growth process.

With regard to the latter aspect it is important to highlight how convective effects and related pattern formation process play a critical role in determining the deformation of the concentration distribution around the growing crystals.

The formation of halos due to the lowered solute concentration around growing seeds, in fact, has the effect of producing density gradients in these areas. These in turn (under the effects of gravity) result in the onset of convection.

In addition to these arguments one must keep in mind that there is an effect related to the fact that, when some crystals grow in the same reactor competition for growth occurs due to superposition and intersection of the related "depletion zones". The coupling between convective "pluming phenomena" and the overlapping of the depletion zones makes the concentration pattern very complex. It is worthwhile to highlight that when more than one crystal is considered, the process is mainly controlled by these aspects.

At the beginning (Figure 3a) each crystal is characterized by its own rising solutal plume. In this early phase of the growth process, i.e. when depletion zones are separated, the growth dynamics of the different seeds can be regarded as independent. Two convective cells are generated around each crystal. After this initial transient behavior, however, the depletion zones for two growing consecutive seeds intersect and overlap (Figure 3b). At this stage mutual interference occurs. Crystals, in fact, absorb protein from "common" regions.

The delicate evolutionary equilibrium among crystal growth (absorption process), solution depletion phenomena and competition among several seeds is coupled to significant and intriguing adjustments in the convective roll pattern inside the reactor.

A very complex multicellular structure is created above the interacting crystals. This convective field undergoes interesting subsequent transitions to different regimes as time passes.

For the case N=6, at the beginning the mode of the multicellular field is m=12 (Figure 3a) and 6 rising convective jets occur above the seeds (hereafter the crystals will be referred to as crystal "i" with i=1N respectively starting from the left side of the reactor).

For t= 8.3104 [s] (see Figure 3b), a large depletion zone surrounding all the crystals is formed and the

mode is decreased to m=8; for t= 1.35105 [s] (Figure 3c) the depleted lighter fluid is transported upwards by two solutal rising jets only and the m=8 mode is taken over by a new regime with m=4. The surviving solutal plumes are located approximately at x=H/4, and x=3H/4 respectively whereas the onset of pluming phenomena at x=H/2 is prevented.

For t>1.35105 [s] (Figures 3d and 3e) the structure with four vortex cells seems to be quite stable without further transitions.

Interestingly, Figures 3 also show that the ‘depth’ of the face protuberances is proportional to the size of the crystals (competition for growth in the intermediate regions and corner effects are responsible for the tooth-shape of the crystals). Moreover the increase of volume is more pronounced for the crystals at the extremities of the reactor (i=1 and i=6) than for the central crystals (i=3 and i=4).

Possible explanations of the above trends can be outlined on the basis of the local growth rate distributions plotted in Figures 4.

In agreement with earlier results (Lappa, 2003) these figures show that corners and edges of the crystals are more readily supplied with solute than the center of sides (this is responsible for the morphological instability and the presence of a macroscopic depression around the center of the faces shown in Figures 3).


48 4847 4744 45

49 49

a)

4949 49

46 46

40 41

26

43 42

26

46 46

36 37

b)

40 40

36 3635 35

4631 31

43 43

38 38

27 2738

c)

28 28

28 28

26 26

29 29

13 1319 19

d)

5 5

6 6

66

e)

Figures 3: Snapshots of growing crystals (A=3, N=6), concentration distribution and velocity field under microgravity conditions (g=10-4 go): (a) t= 1.0x104 [s], (b) t= 8.3x104 [s], (c) t=1.35x105 [s], (d) t=2.075x105 [s], (e) t=8.3x105 [s] (level 12.8x10-2 [g cm-3], level 506x10-2 [g cm-3], C=6.4x10-4 [g cm-3] ) .


a)

b)

c)

d)

e)

Figures. 4: Snapshots of growth rate distribution, A=3, N=6: (a) t= 1.0x104 [s], (b) t= 8.3x104 [s], (c) t=1.35x105 [s], (d) t=2.075x105 [s], (e) t=8.3x105 [s].

As discussed before, incorporation of the solute into the crystal causes a local depletion in

concentration and a solutal concentration gradient to form between the bulk solution and the growth interface. As expected, the growth rate is always lower at the center than at the corners where the


convective flows (and the associated shear stresses) are the strongest, and hence, the interfacial concentration gradients the steepest.

Figures 4 also allow to point out that the difference in gradient steepness (i.e. local growth rate) between face corner and face center increases as time passes; this explains why the ‘depth’ of the face depressions is proportional to the volume of the samples. Moreover the convection effects tend to increase the local growth rates for the crystals at the extremities.

Despite the macroscopic analogies and similarities pointed out above, however each crystal exhibits different morphological evolution and different growth history according to its position within the reactor.

For instance the final "ridge" (slope of the upper side) of the crystals is inclined with respect to the vertical direction according to their position within the reactor: the ridge is inclined inwardly for i=1 and i=6, and outwardly for i=2,3,4,5 (Figure 3e). "Edge effects" are very evident in Figure 4e.

In practice the characters of the final samples change depending on the local fluid-dynamic conditions under which they are operated. In addition to the effect associated to the corners more readily supplied with solute, there is an effect due to the local convective pattern around the crystal. Hence, there is a critical and remarkable “link” between crystal and fluid-dynamic pattern formation and related evolutionary progress. The cross-comparison frame by frame of Figures 3 and 4, in particular, elucidates that the convection effect results in higher local growth rates near the surface where the flow is directed downwards and lower local growth rates near the surface where the fluid is carried upwards.

It is very important to point out how the correspondence elucidated above between surface growth rates and roll pattern evolution is an indirect evidence of the fact that for the case under investigation, mass transport in the bulk and surface attachment kinetics are competitive as rate-limiting steps for growth (in the case of phenomena strictly governed by the kinetic barrier in fact the effect of convection on the overall growth process is almost negligible).

3.2 Periodic Precipitation: Propagation of Nucleation and Local Growth Laws If the problem under investigation is characterized by the formation of new crystals and by multiple

nucleation phenomena, the morphology-oriented method used in Sect. 3.1 is no longer suitable to a comprehensive analysis of the system. For instance this occurs in the case of crystal grown by a counter-diffusion technique. In this case a range of supersaturations is established in different locations along the protein solution during the process leading to different crystallization conditions in the same experiment (pertaining to both the metastable and the labile regions in Figure 1).

The case of a gellified protein chamber is considered for the present example. This chamber is put in contact with an additional chamber containing the precipitant agent (salt). Therefore the configuration under investigation simply consists of a protein solution and a precipitant agent solution placed one above the other and separated by the gel interface. The crystallization cell is a chamber whose length and height are L and H respectively; the interface is placed at yh (see Figure 5). At the initial time, both solutions are at constant concentration respectively. As time passes salt diffuses in the protein chamber through the gel interface lowering the solubility and starting nucleation and ensuing growth phenomena.

The following properties and operating conditions are used: CNaCl(o)=1410-2 [g cm-3], L=1 [cm], H=4 [cm], width=0.1 [cm], average height of the protein chamber h=1.95 [cm], initial protein concentration C(o)

=1410-2 [g cm-3]. The process is simulated in the context of the approach described in Sect. 2.1. Simulations show that some "bands" of crystals are produced in the gel, .i.e. discrete nucleation events

separated by certain time (and thus space) intervals occur in the protein chamber. These bands are not spatially uniform (see Figures 7; in these figures the salt chamber is located on the top of the protein chamber). According to the evolution of the protein concentration field (Figures 6) nucleating particles deplete their surroundings of protein which causes a drop in the local level of supersaturation such that the nucleation rate falls in the neighborhood, leading naturally to a spacing between regions of nucleation that


gives rise to the alternate presence of depleted zones and crystals. This behavior exhibits interesting analogies with the well-known so-called "Liesegang patterns" (see, e.g., Henisch and Garcìa-Ruiz, 1986 and Lappa et al., 2002) for which occurrence of periodicity is found not only parallel but also perpendicular to the direction of the diffusion (see, e.g., Büki et al., 1995).

Figure 5: Sketch of the reactor used for the counter-diffusion technique

For the present case, in particular, the spacing among different solid particles and the size of the

particles vary according to the distance from the origin of the imposed concentration gradient (the gel interface between the salt and protein chambers). The space distance among near solid particles, in fact, tends to decrease uniformly towards the interface of the protein chamber. Correspondingly, the size of these particles is minimum at the interface and increases towards the bottom of the chamber up to a constant value in the bulk. According to such results, the “density of particles” defined as ratio of the number of particles in a fixed reference volume and the reference volume, is maximum near the interface and almost constant in the bulk. These behaviors are shown in Figures 9.

The models and methods proposed in Sect. 2.1 exhibit remarkable capabilities to predict and elucidate experimental observations at macroscopic level (Figure 8) and to identify cause-and-effect relationships. Both experimental observation and numerical modeling show comparable features of the phenomena under investigation. The consistency of model predictions with experimental data suggests that the proper rate-controlling steps have been taken into account, and that simplifications do not influence much the system behavior.

Though the model in Sect. 2.1 is oriented to the description of macroscopic behaviors, however it is also able to provide some interesting information dealing with the crystal growth as a function of local conditions established in the protein chamber. As an example Figure 9c shows the growth law for a reference crystal located in the bulk of the protein chamber (x =0.4 [cm], y=0.9 [cm]); the growth is

compared with the evolution of local supersaturation. This crystal is formed at t7x104 [s] and reaches its

final size after almost 105 seconds. A "measure" of the average growth rate is also possible. The growth rate (computed over the time interval required to reach the 90% of the final mass of the crystal) is 1.27x10-9 [g/s]. As previously pointed out, however, the shape of the crystal is ignored and for this reason only an "integral" growth rate (accounting for the total mass variation) can be computed for each crystal.


1

4 512

18 18Level A19 0.9918 0.9417 0.8916 0.8315 0.7814 0.7313 0.6812 0.6311 0.5710 0.529 0.478 0.427 0.366 0.315 0.264 0.213 0.162 0.101 0.00

Salt chamberInterface

g

a)

Salt chamber Interface

a)

3 3

2

6 68

13 1316 16

20

Level A20 0.9919 0.9418 0.8917 0.8316 0.7815 0.7314 0.6813 0.6312 0.5711 0.5210 0.479 0.428 0.367 0.316 0.265 0.214 0.163 0.102 0.051 0.00


g

b)


b)

2

3 3

4 4

5 5

7 7

10 10

13 13

1616

1514 17 15

Level A20 0.9919 0.9418 0.8917 0.8316 0.7815 0.7314 0.6813 0.6312 0.5711 0.5210 0.479 0.428 0.367 0.316 0.265 0.214 0.163 0.102 0.051 0.00


g

c)


c)

Figures 6 Figures 7

Non-dimensional protein concentration contour lines (Figures 6) and distribution of the crystals (Figures 7)–(zero-g conditions, C(o) =410-2 [g cm-3]) Time=8.00103 [s], 7.2104 [s], 1.08105 [s].


Figure 8: Experimental distribution of the crystals – (gellified configuration, C(o)=410-2 [g/cm3]) Time=1.73105 [s].

a)


0 5 10 15 20Distance from the gel interface [mm]

0

50

100

150

200

Num

ber

of s

olid

par

ticl

esNumerical results

Experimental results

b)

25000 50000 75000 100000 125000 150000Time (s)

0.00

1.00

2.00

3.00

4.00

5.00

Seed mass (g x e5)

Supersaturation

90 %

c)

Figures 9: Comparison between numerical and experimental results (gellified configuration, C(o) =410-2 [g cm-3]): (a) solid particle size distribution, (b) density of particles distribution, (c) computed "growth law" of a reference crystal (x = 0.4 [cm], y=0.9 [cm]) located in the bulk of the protein chamber.

In view of the above arguments the macroscopic approach has proven to be a very efficient and robust method to be used when a very large number of interacting crystals is involved in the growth process (this situation often occurs in the case of the aforementioned counter-diffusion techniques). In this case the model predicts the main features of the process, providing spatial and size distributions of crystals that are in good agreement with the experimental observations and very interesting information about the "ensemble behavior".


It important to point out that a local detailed description of the morphology evolution of all the particles in Figures 7 according to the method elucidated in section 2.2 would require very dense grids (of the order of 107 points) and therefore a prohibitive computational time.

Let us conclude this discussion by observing that such macroscopic approach, in principle could be applied to other phenomena, which even if not strictly related to the field of protein crystal engineering, however, fall under the general heading of “Liesegang patterns” mentioned before.

Other applications can be found in a variety of crystal growth processes in reaction–diffusion systems (e.g., Lagzi and Ueyama, 2009 and references therein), in the field of metallurgy (ingot crystallization), directional crystallization of melt, crystallizing polymers, petrological studies, oscillating phenomena in the minerals of igneous, hydrothermal and metamorphic rocks (see the recent work of Cherepanova et al., 2009 and all references therein)

For other interesting theoretical developments and numerical methods related to the complex dynamics of Liesegang Patterns, the reader may consider Chernavskii et al. (1991), who elaborated an interesting model explaining the appearance of non-homogeneous precipitation patterns in the form of bands, taking into account effects of supersaturation, competition between nucleation and droplet growth kinetics as well as redistribution of matter between particles of precipitant.

Relevant contributions have been also provided by Hebert (1992), Rácz (1999), Izsák and Lagzi (2003), and George and Varghese (2002 and 2005).

The article of George and Varghese (2002), in particular, deserves some additional discussion. They, in fact, investigated theoretically the mechanism of rhythmic pattern formation in reaction–diffusion systems by resorting to moving-boundary concepts similar to those illustrated in Sect. 2.2.

This idea has been further developed in George and Varghese (2005) in which the boundary that separates the two reacting species was treated as a front virtually migrating as the diffusion proceeds into the gelatinous medium. The existing time law, space law, and width law typical of Liesegang patterns in gelatinous media were revisited accordingly and reformulated on the basis of such a moving boundary assumption.

4. CONCLUSIONS In general, pattern formation can be regarded as a truly interdisciplinary science. Regular patterns arise

everywhere in nature and virtually every technological process involves pattern formation at some stage. The similarity in fundamental mechanisms and the accompanying mathematics brings together scientists from many disciplines, such as, mathematics, fluid-dynamics, material science, chemistry, biology, medicine, etc.

The deep question of whether universality classes exist for patterning behavior is still unanswered. The characterization of dynamics that are complex in both space and time is far from complete for pattern forming systems.

Problems of this type have recently witnessed a renewed interest in view of their relevance in several fields. This particular progression is linked to the fact that the study of pattern formation, as illustrated before, stands at the intersection of many branches of science, which make it a multi-domain field of investigations.

In this article a rich mathematical protocol has been described in detail which can significantly help the investigators in simulating macromolecular crystal growth at macroscopic and microscopic length scales. Such integrated theoretical and numerical strategy has been based on a fixed common formulation of the crystal surface kinetics.

Some relevant examples (the morphological evolution of a limited number of protein seeds subjected to the influence of natural convection, and the nucleation and ensuing growth of a large number of crystals in gel, respectively), have been discussed to show how the proposed numerical approach can handle a wide spectrum of situations and cases.


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