Chemistry Everyday for Everyone
Arcade Games for Teaching Crystal Growth Juan Manuel García-Ruiz* Instituto Andaluz de Ciencias de la Tierra, Laboratorio de Estudios Cristalográficos, CSIC-Universidad de Granada, Av. Fuentenueva, 18002-Granada, Spain
In spite of the ubiquity and importance of crystals, very little attention is paid in undergraduate and graduate studies to the subject of how they form, a matter still considered rather difficult and obscure (1). Here is described a simple way to introduce this matter, which is very basic yet powerful enough to help teachers explain the growth behavior of crystals from their mother solutions. When teaching crystal growth from solutions, remember that all we expect is to segregate a solid phase from the mother liquid and to order its basic components, filling the space with a certain periodicity; in other words, to make a molecular tessellation of the space (2). A useful and simple model for this phenomenon is the so-called “wall and bricks” analogy, which is in fact used by many teachers of crystallography to explain K symmetry groups. Imagine the growth of a twodimensional crystal just like the building of a wall using bricks or clusters of bricks with a given point symmetry. Obviously the wall is the analogue for a well-ordered crystal lattice and the bricks are the asymmetric growth units (or their clusters) that move randomly in the fluid phase surrounding the crystal. As shown in Figure 1, there are several steps involved in the flow of a growth unit (3) from the bulk solution toward the crystal surface; but for crystals growing from solutions, for example large biological macromolecules, they can be reduced to two main consecutive steps: 1. The transport of growth units from the bulk solution towards the crystal face; 2. The interactions taking place at the crystal–fluid interface until the growth units move themselves into a lattice position minimizing the reticular energy and contributing to forming a perfect crystal.
Using the wall-and-bricks analogy, it becomes clear that the rate of transport toward the crystal surface plays an important role. Let such a rate, J, be the number, N, of bricks per unit time furnished to the wall-maker. For low values of J, say, one brick per unit time, the wall-builder has enough time to find the right location for the brick and he may add the brick with the maximum efficiency to create the required wall pattern. As we increase J, supplying several bricks per unit time, the builder sometimes places them in the right position but sometimes is forced to place them with a certain mismatch. A degree of disorder will arise, and clearly as J increases so does this disorder. Vacancies, dislocations, and boundary grains will appear: the wall might be divided into a number of regions, M, made up of bricks that are perfectly arranged as regards the pattern but slightly disoriented with respect to the neighboring regions. This number M (which is related to the term mosaicity used by crystallographers [4 ]) increases with J. At the limit, for very large J values, our despairing wall-builder will give up and the bricks will just form a pile: an amorphous phase will be created. Thus, it is clear *Email:
[email protected] that to obtain a good single crystal its growth units must be supplied slowly. In the laboratory, this is currently achieved by ensuring that mass transport is controlled by a slow mechanism termed diffusion. When a crystal of a given volume grows within a much larger volume of solution, it removes solute molecules from the neighboring volumes of solution. This leads to a depletion of the concentration of solute molecules in the vicinity of the growing crystal Ci, while the concentration farther away from the crystal remains at the concentration of the bulk solution C (Fig. 1). This difference in concentration in the space (∆C = C – Ci) provokes the formation of a concentration gradient ∆C/∆x. This disequilibrium tends to disappear over time because the random movement of molecules provokes a directional flow of the particles following Fick’s first law of diffusion:
JD = {D ∆C ∆x where D is termed the diffusion constant, which takes values on the order of 10{5 cm2 s{1 for small molecules and one or two orders of magnitude smaller for large macromolecules. To ensure that diffusion takes control of the mass transport, other mechanisms such as convection must be prevented. There are only a few ways to prevent (or more exactly, to reduce) convective flow: namely, either working within porous media, with high viscosity fluids or very thin capillary volumes, or alternatively performing the experiments at low gravity in space. This is one reason why crystal growth in gels (5) is a technique of increasing interest and why several space agencies are currently performing crystal growth experiments in space to get protein crystals that are as perfect as possible (6 ). We can approximate the mass transfer rate dM/dt from the bulk
Figure 1. The main steps in the flow of growth units from the bulk of the solution toward the bulk of the crystal. (a) flow toward the crystal face; (b) surface diffusion and adhesion to the crystal face; (c) heat dissipation; (d) backwards solvent flow. At the level of this introductory model to crystal growth, steps c and d can be neglected. δ is the thickness of the diffusion layer. Adapted from Nyvlt (2).
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Chemistry Everyday for Everyone
(at concentration C) to the crystal surface (at concentration Ci) by the relation
dM = k w C – C d i dt
d
where kd is the ratio between D and the thickness of the diffusion layer δ (Fig. 1). Now the question is, how slow should the mass transfer be in order to grow a perfect crystal? Let’s continue with the wall-builder analogy and the answer appears evident: slow enough to give the wall-builder time to place each arriving brick in the right position. Certainly, a smart student may realize that the answer also depends on the skill of the bricklayer, and he would be right. In fact, this degree of skill represents the second important process in the problem, the kinetics of the incorporation of the growth units onto the crystal surface. That is, the ability of the growth units to jump on the crystal surface from one position to another. As soon as a growth unit lands on the crystal surface, it starts to move, looking for the location with highest bonding energy. This is like the work of the bricklayer, who must look for the right position to locate the bricks in order to create the perfect tile. It is again clear that, in creating this perfect tile, the faster the bricklayer can find the right location, the faster the bricks can be supplied to him. The rate of transport of the growth units onto the crystal surfaces depends on considerations of energetics. Once a growth unit lands on the crystal surface it becomes attached to the other molecules on the surface with an energy that is proportional to the number of shared bonds. This bonding energy may be high or low, depending on the landing position. There are positions termed K (kink) or semi-crystal positions because when a growth unit reaches one of these kinks it shares half its bonding energy with the crystal (position 3 in Fig. 2). Therefore, if the arriving growth unit lands on one of these positions it has a high probability of remaining there. If it arrives at a position of lower attachment energy (such as position 1 or 2 in Fig. 2), it will desorb and move to better locations on the surface such as 3, 4, or 5. Thus, the longer the time a growth unit is allowed to move on the surface, the higher its probability of reaching the most energetically favorable position (the one making the perfect crystal). The
Figure 2. The surface of a growing crystal. Different positions available for a landing growth unit. A two-dimensional nucleus is also shown in the upper left corner.
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mass deposition controlled by these surface processes can be expressed by the relation
dM = k w C – C r s e dt
r
where Ce is the equilibrium concentration and k r is a kinetics coefficient that depends on surface roughness. Note that Ci is controlled by the flow of units from the bulk solution. Therefore, we can conclude that the compromise between the kinetics of the processes taking place on the crystal surface and those of the transport of growth units toward this surface determines the quality of the growing crystal. For a computer simulation of this problem see refs 7 and 8. Thus, crystals growing at a rate controlled by mass diffusion will contain a low density of defects, have reduced mosaicity, and presumably will diffract X-rays with better resolution. To further illustrate the topic, I suggest using the wellknown arcade-type computer game called Tetris. In the classical Tetris game the player is challenged to make a perfect wall with point group symmetry P4mm (eliminating color) by stacking different types of brick clusters flowing at a constant rate from the upper part of the screen. By using rotation and translation movements, the player must orient the building blocks toward the positions that create a perfect wall in the lower part of the screen. As the game advances and the player demonstrates enough skill to make a perfect wall, the building blocks fall at higher rates. The game ends when the player is unable to control the flow of building blocks; that is, when a wall with a high density of defects fills the screen. The player’s skill represents the kinetics of surface processes; the rate at which the bricks fall down represents the transport kinetics through the bulk of the solution; and the clusters of bricks are analogous to the different degrees of aggregation of the precritical clusters moving toward the crystal surface. Playing Tetris, the student may understand that the faster the transport of units towards the crystal face, the poorer the quality of the growing crystal. The teacher may then explain, using the above equations, that such a transport depends on how large the difference is between the concentration in the solution and the equilibrium concentration, a parameter named supersaturation. Vacancies and structural defects can be easily visualized. Students can also realize that the higher the complexity and oddity of the growth units the more difficult it is to find the right location on the surface of the crystal. They can notice that when the complexity of the crystal surface increases (which usually occurs as the game advances), to search the best location on the surface is more difficult. For instance, by the Tetris analogy, we can consider a typical question: why are proteins and other biological macromolecules reluctant to crystallize and why do we try to grow them in space (6 ) or in capillaries (9). We know that all organic and inorganic crystals, whether of low or high molecular weight, grow according to the same physical principles and under the same chemical laws. The obvious difference between crystals of small and large molecules is related to the size of the lattice cell of the structure, because the number of atoms in the asymmetric unit of the large macromolecule is huge compared with that of the small molecules. However, the main difference when trying to create crystal lattices made up of asymmetric units of either small
Journal of Chemical Education • Vol. 76 No. 4 April 1999 • JChemEd.chem.wisc.edu
Chemistry Everyday for Everyone
or large molecules arises from the large amount of solvent contained in the crystals of the biological macromolecules. It follows that the energy density in such a crystal is very small because of the small number of weak contact points between the molecular groups forming the building crystal. Thus, when we consider a growth unit moving toward the surface of a macromolecular crystal searching for the right position on the surface (lowest energy location), there are many other positions where the growth units might land that are energetically almost as favorable as the small number of specific sites through which an ordered array is formed. The slow translational diffusivity of proteins together with the low probability of properly orienting large molecules with respect to the lattice are the reasons why they are difficult to grow fitting an ordered array. In addition, the propensity of the biomolecular solutions to polymerize in the mother solution increases the number of different growth units and their complexity, making difficult the right arrangement on the crystal surface. In other words, proteins are bad Tetris players.
Literature Cited 1. Hulliger, J. Angew. Chem. Int. Ed. Engl. 1994, 33, 143–162. 2. Holden, A.; Morrison, P. Crystals and Crystal Growing; MIT Press: Cambridge, MA, 1982. 3. Nyvlt, J.; Söhnel, O.; Matuchová, M.; Broul, M. The Kinetics of Industrial Crystallization; Chemical Engineering Monographs, Vol. 19; Elsevier: Amsterdam, 1985; p 177. 4. Helliwell, J. R. J. Cryst.Growth 1988, 90, 259–272. 5. Henisch, H. K. Crystal Growth in Gels and Liesegang Rings; Cambridge University Press: Cambridge, 1988. 6. Giegé, R.; Drenth, J.; Ducruix, A.; McPherson, A.; Saenger, W. Prog. Cryst. Growth Charact. 1995, 30, 237–281. 7. García-Ruiz, J. M.; Otálora, F. Physica A 1991, 178, 415–420. 8. Otalora, F.; García-Ruiz, J. M. Cryst. Prop. Prep. 1991, 36– 38, 686–695. 9. García-Ruiz, J. M.; Moreno, A.; Otálora, F.; Rondón, D.; Viedma, C.; Zautscher, F. J. Chem. Educ. 1998, 75, 442–446.
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