Meeting Report
Everyone is familiar with granular materials—sand, powders, and collections of even larger particles. Yet these materials behave differently from other forms of matter such as gases, solids, and liquids, leading some scientists to believe that granularity may represent an additional state of matter. This is more than an academic argument. Understanding granular materials carries implications in fields ranging from civil engineering to pharmaceutical manufacturing.
The speakers in a symposium held at the Academy on January 23, 2008, and sponsored by the Soft Materials Discussion Group, represent a cross-section of research on granular materials. Bruno Hancock of Pfizer provided an overview of segregation and handling issues for granular pharmaceutical ingredients. Mark Shattuck of the City College of New York took the discussion to a more theoretical level, describing the behavior of granular "gases" and "crystals" under various stresses. Paul Chaikin of New York University presented an amazing finding in an unusual granular system—containers of M&M's candies.
Granular materials as drug-delivery systems
In ancient Roman times, the philosopher Lucretius noted, "One can scoop up poppy seeds with a ladle as easily as if they were water and, when dipping the ladle, the seeds flow in a continuous stream." Lucretius was describing a phenomenon known to ancient farmers and tradesmen that continues to fascinate scientists today: particles falling within a particular range of size and density behave like fluids. Indeed, flow performance, along with the maintenance of a uniform composition and good compression properties, is critical in the formulation of pharmaceuticals. These characteristics are often achieved by "granulating" the particles in a formula, which means "agglomerating primary particles to form uniform larger composite particles that are suitable for further processing," says Bruno Hancock.
Most pharmaceuticals today are produced as powders before being formulated into liquid or solid dosage forms. The most popular pharmaceutical dosage form, the tablet, presents the most challenges for formulators. Tablets are mixtures of numerous excipients that impart color and flavor; facilitate stability, absorption, and compaction; or that serve as lubricants, coatings, and binders.
Poorly granulated powders can lead to inconsistent distribution of active ingredients in pharmaceutical tablets.
Granulation is essential for avoiding the particle segregation that plagues industrial-scale powder processing during manipulation steps. When ungranulated or poorly granulated powders are discharged into bins, finer particles tend to settle more slowly and cling to the walls of the bin. This effect is known as dusting. Since dusts contain unusually high levels of active pharmaceutical ingredient (API), tablets produced late in a manufacturing run may contain as much as 20% more active ingredient than those at the beginning of the run.
Dusting segregation may be compounded by fluidized segregation, where air entering the bin on emptying carries smaller, lighter particles upward and causes larger, heavier particles to segregate near the opening.
A properly granulated formulation may segregate somewhat by size, but that will not be a problem provided that granules are of uniform composition.
Properties and performance
Factors affecting the design of a new pharmaceutical dosage form include performance (bioavailability, uniformity, active ingredient release rate), chemical and physical stability, and manufacturability. All are related to the powder formulation, especially how it behaves during manipulation.
Pharmaceutical granules are held together through physical interactions (Van der Waals and electrostatic forces, mechanical interlocking, liquid or solid bridges) and by virtue of the granules' size, shape, density, elasticity, hardness, and chemistry. Granules possessing uniform shape and density flow easily and tend not to segregate by size or weight.
A powder's resistance to shear, flow, deformation, or other force depends on the number and strengths of inter-particle interactions, the degree of consolidation or packing, and the existence of compressive forces pushing the particles together. Particles not experiencing these forces readily yield to shear, meaning they flow and mix well.
Ultimately, Hancock and others would like to be able to predict how granules and bulk powders will behave without having to do large scale experiments. Nevertheless, physical and mathematical descriptions of powder and granule behavior notwithstanding, pharmaceutical formulation currently relies heavily on empiricism—"doing the experiment."
Shaken, not stirred: granular equilibrium
Experimentation is often the only way to arrive at answers for more theoretical systems such as particles and colloids. "If you want to know if ordinary fluids are going to be a gas, liquid, or solid, all you need to know is the temperature and pressure," notes Mark Shattuck of City College. "With granular systems you can't tell because ordinary temperature doesn't play a role in determining the state. It doesn't matter if the grains are hot or cold."
Shattuck's research, which aims at predicting how and when granular systems will behave as fluids or solids, holds significant importance for materials handling in pharmaceuticals, chemicals, and other industries. It may also explain, for example, why the Mars Rover craft became stuck for a month in a dune composed of one of the most common granular materials—sand.
Analogs of molecular systems
A hallmark of granular systems is their loss of translational energy during collisions. Where a bouncing ball loses energy each time it hits the ground, energy is conserved in a dynamic system consisting of two water molecules colliding.
For molecules, temperature or internal energy are determined by their kinetic energy. For grains of sand in motion, the internal energy, as measured by temperature, is on the order of 10−28 of the translational energy. Nevertheless, viewing granules and even larger particles as molecules can yield some interesting findings.
The physical states associated with shaken particles are analogous to equilibrium systems.
It is possible to construct granular systems in which all the energy is contained within translational degrees of freedom, with almost none available to vibration, rotation, or other factors. Such a system requires a constant input of energy. Because the physical states associated with shaken particles are analogous to equilibrium systems, physicists call them non-equilibrium steady-states (NESSs).
But aren't motionless assemblies of particles or granules, without energy input, at equilibrium? Yes, says Shattuck, but the molecular analogy is with a molecular system at absolute zero, where molecules are not moving at all. "That is the real equilibrium state of granular systems."
Structurally identical
Granular NESSs are structurally and behaviorally identical to molecular systems at equilibrium. Shattuck showed a brief movie where particles in a nearly two-dimensional enclosure begin in a highly ordered state, but under the influence of heavy shaking suddenly transform into a disordered state, similar to how dry ice changes discontinuously from a solid to a gas. According to Shattuck, this was the first observation of a first-order phase transition in non-equilibrium systems.
Shattuck unreservedly refers to these states as crystalline and gaseous, but a traditional chemist might wonder if these labels are more allegorical than factual, since granules cannot really form gases or crystals in the traditional sense. Shattuck responds that "it's only a difference in scale. It's a gas with a small number of molecules. This is what a similar experiment on 50 argon atoms or carbon dioxide molecules in a small box would look like."
A new thermodynamics?
Since the particles' kinetic energy on shaking is not the same as their temperature, it becomes difficult to discuss granular systems in terms of thermodynamic energy functions, particularly with respect to G, the Gibbs free energy. In chemical systems G describes the spontaneity of a chemical reaction in energy units. In granular systems Shattuck has discovered a G-like quantity which, despite the presence of entropy (S) lacks the units of energy, and therefore cannot be a true measure of free energy. "The analogy with G is not as strong as with gas and crystalline states."
It all comes back to kinetic and thermodynamic temperatures, which are the same for molecules but can be vastly different for particles. Without a true temperature, G is undefined, and in fact Shattuck does not even have a name for the quantity he is observing. The dynamic factor that comes closest is shaking amplitude, but as Shattuck points out, amplitude is not a thermodynamic function.
He believes the mysterious quantity may be related to energy flow or flux, but he cannot prove it until he obtains a camera fast enough to track individual particles in a shaken system. "Part of the fun of this is we feel like we've gone back to the 1850s, and are trying to re-discover thermodynamics—but for non-equilibrium systems."
Colloids and candies
Like granularity, packing problems have been recognized for millenia. A biblical quote exhorts dealers to "pack down" and "shake" bushels of grain so as to provide a fair measure.
Ellipsoidal M&M's candies are an interesting physical system and taste good too.
After spheres, the simplest systems to study are ellipsoids—shapes with one constrained dimension, or as Paul Chaikin referred to them, "squashed spheres." Packing density determinations for ellipsoids lack the mathematical rigor of spheres', which makes them a challenging physical system. "I like the kind of problem where you can't find the answer in any way except by making a model, putting things together, and seeing how they work," Chaikin says. His particles of choice, ellipsoidal M&M's candies, serve the dual function of an interesting physical system and one of Chaikin's favorite lunchtime foods.
But before launching into his work on ellipsoids work, Chaikin illustrated the packing behavior of colloids, particles of up to about 1 micrometer in size which are suspended by the movement of water molecules. Colloids exist as liquids (lowest density), solids, and crystals (highest density).
Colloids are rich experimental systems because of their wide range of sizes and shapes, and the possibility of adding chemical functionality to their surfaces. Chaikin described a system of DNA-modified colloidal chains that may be coaxed to "self-replicate" in solution by attracting particles carrying complementary DNA sequences.
Packing densities for spheres
Packing with spheres is a very old problem that has been appreciated for thousands of years but only recently explained mathematically.
In face-centered cubic (FCC) packing, the most efficient way to stack spheres, two-dimensional layers are arranged such that the spheres fit into interstices of layers immediately above and below (think oranges stacked at fruit stands). The theoretical packing fraction of spheres in FCC is approximately 0.74.
FCC packing requires deliberate organization. If ball bearings are simply poured into a container and shaken to settle them, they arrange in a structure known as random close packing, with a packing density of about 0.64.
An experiment by the botanist Stephen Hales in 1727 illustrated the implications of random close packing. Hales cooked peas in a closed cylindrical container, and noted the deformation in the peas caused by their softening and swelling. The deformation patterns, which corresponded to the presence of close neighbors during packing, signified a configuration analogous to an amorphous rather than a crystalline state.
Chaikin illustrated this point using couscous rather than peas. After the grain had absorbed water and swelled, he replaced the excess water with ink, revealing contact surfaces, some of which formed in an icosohedral pattern, with pentagonal faces and five-fold symmetry. As Chaikin pointed out, icosahedra and pentagons frustrate the formation of crystals because a crystal lattice inverts through the origin to itself while pentagons invert to their mirror image.
Sweet stuff
M&M's are an ideal system for packing experiments because they have regular shapes with very low size and dimensional variability, and they are ellipsoids, the geometric shapes closest to spheres. Chaikin expected that random close packing of M&M's would be no better than for spheres (0.64).
In an early experiment with M&M's, one of Chaikin's students discovered that the candies packed to a density of approximately 0.69, which is denser than spheres. Chaikin was surprised because a coordinate change along one axis should not change the packing fraction.
Chaikin set out to prove, through computer simulation, that the results could not be correct. He began by simulating the two-dimensional FCC packing of spheres in a box. When he changed the aspect ratio of the particles to 1.91 (the same as M&M's), the volume of the box fell by an identical amount, 1.91. Therefore, he concluded that the packing density of ellipsoids had to be identical to that of spheres.
The virtual experiment, repeated using random packing, gave apparently similar results, but did not account for torques acting on the non-spherical particles.
Chaikin obtained very different results when he repeated the virtual experiment for random packing. The figure on the left looks reasonable enough, but when the aspect ratio is changed, the resulting ellipsoids appear to be oddly suspended in air. Chaikin concluded that the torques acting on the M&M's, and the additional degrees of freedom in the candies due to one flattened axis, cause the M&M's to rotate and fill volumes in ways that spheres could not.
According to theoretical predictions for spheres and ellipsoids, random close packing of M&M's should result in an average of 10 touching neighbors based on the relation Z = 2d where Z equals the coordination number and d is the number of degrees of freedom. This is what Chaikin found through another experiment on M&M's in random close packing. Here paint was poured into randomly packed M&M's and the number of contact points determined by counting more than 1000 particles.
Packing of hard spheres and ellipsoids has numerous implications for grains, the structures of liquids, solids, and glasses, freezing transition, flow and plasticity of sand, optimization of data storage/transmission, and the design of opto-electronic materials. For example, shapes of nano-sized components might be skewed towards a favorable packing density.
Packing, particle segregation, and granular thermodynamics all reflect the unusual properties of granular materials and the very real situations that are caused by the behavior of such materials.
Open Questions
Can blend uniformity in pharmaceutical formulations be promoted through the use of excipients, or only by blending?
What roles if any do particle size and property distributions of APIs play in achieving uniform granulation?
What are the units for the mysterious G-like factor in NESS granular systems?
Are there other properties of granular systems that are thermodynamic-like?
What other shapes exhibit denser packing in the random state than in the crystalline state?