The Scale of Materials Design

How many potential materials might there be? For those of us trying to compute our way to the next technological breakthrough, it’s important to grasp the scale of the problem. The first step is to define the term material.

The scale of materials

The objects that we interact with every day are composed of about 1023 atoms. Rather than describe all those atoms, however, materials scientists classify a material’s structure into different length scales.

Materials Scales
Materials exhibit structure on many length scales, each of which influences the overall properties.

At the fundamental scale, solid objects are composed of atoms that repeat in a pattern defined by a unit cell, much like a three-dimensional M.C. Escher tessellation.[1] The positions of atoms in the unit cell constitutes the crystal structure, whereas the elemental identity of the atoms define the chemistry. We will refer to some combination of crystal structure and chemistry as a chemical compound. In other words, a compound is a specific arrangement of atoms with chemical identities (e.g., Si, Na, Fe, etc.) in a repeating box.

A material encompasses much more than a compound, and includes additional structure at the micro- and macro- scales. At the microstructure level, the compound’s unit cells are arranged into a regular pattern within a grain. The properties of the grains and their boundaries as well as defects and impurities within the grain can have a large effect on a material’s properties. Materials can also form composites of multiple compounds at this scale. A large part of materials science concerns engineering microstructure to achieve certain properties without changing the fundamental compound(s). Similarly, the structure of the material at the macro scale – such as their surfaces – can further influence properties.

For the purposes of counting, we will restrict ourselves to counting compounds, not materials. This restriction anticipates that when we screen materials computationally, we will model all 1023 atoms of the material as one infinitely repeating unit cell. The number of possible materials would be much bigger than our estimation of the number of compounds.

Estimation method I: A googol of compounds in a box

To count compounds, we might arrange atoms in a box representing the unit cell. Let’s position 30 atoms into a 10x10x10 grid of points, selecting one of 50 elements from the periodic table[2] for each of the 30 atoms.

Packing materials in a box
Tossing 30 atoms into a 10x10x10 box using 50 possible elements gives a googol of possibilities!

It’s now straightforward to calculate the number of compound combinations. The number of potential choices of grid points for our 30 atoms is 1000C30, or about 1057, representing the possible crystal structures. Next, we assign elements to each of those atoms to get chemistry. Since we are independently assigning one of 50 elements to 30 atoms, there are 5030 ways to pick, or about 1051 possibilities. Multiplying those two numbers together, we obtain a result of about 10108 different compounds, or over a googol (10100)!

A googol is a truly unimaginable number. It is not the number of grains of sand on all the beaches on earth (~1021), or the number of hair widths to the sun (~1016). It is even more than the number of known atoms in the universe (~1080). There is no way we can compute all the combinations of 30 atoms on a 10x10x10 grid.

Estimation method II: Modify the known crystal prototypes

You wouldn’t estimate the space of potential 100,000-word novels by counting all the possibilities for words randomly strewn across pages.[3] Like a novel, a legitimate chemical compound obeys rules and patterns[4][5]. A more reasonable guess at the number of compounds might apply a few rules:

  • Rather than be arranged arbitrarily on grid points, the atoms in compounds tend to recur in the same crystal structures. New materials are very likely to reuse an old arrangement of atoms, but with different elements substituted in.
  • Known compounds, or at least those that are not nightmarish to synthesize, tend to be composed of only a few distinct elements, generally 5 or less per compound.

Applying these rules completely changes our estimation. As of this writing, the Inorganic Crystal Structure Database has classified about 6500 crystal structure prototypes, or known arrangements of atoms in a box. In terms of chemistry, if we choose 5 element compounds from a set of 50 elements, that’s 505 or 312 million combinations per prototype[6]. Multiplying those numbers together, we get about 2 trillion compounds.

At this point, we could start splitting hairs. Certainly, we don’t know all the crystal structure prototypes yet. However, even if we’ve found only 1% of crystal structures so far, we are only changing our final number by a factor of 100 (to 200 trillion). The story is similar if we want to try more than 50 elements or more than 5 element compounds. We could instead go the other way, and add further heuristic rules that reduce the estimation down from a trillion. The order of trillions of possible permutations is perhaps a reasonable middle ground, keeping in mind that our goal is just a basic estimation.

Narrowing it down

The current state-of-the-art in high-throughput computations can test about 10,000 to 100,000 materials candidates, depending on the complexity of the materials involved. So, even materials scientists armed with supercomputers must first narrow down about 1,000,000,000,000 possibilities down to 10,000 using chemical intuition[7]. Computations can narrow down that list to maybe 100. Even fewer of those will see experimental success, and fewer still will be commercialized.

Our estimate for the number of chemical compounds is not too far from the number of humans that have ever lived (~100 billion). Identifying the handful of technologically relevant compounds, then, is truly liking picking out the Da Vincis, Shakespeares, and Beethovens amongst the materials world.

Footnotes:
[1] Many materials do not fall neatly into this description, in particular polymers and glasses. (11/20/2013)
[2] Although there are more than 100 elements on the periodic table, many of them are rare, radioactive, synthetic, or extremely expensive. 50 is a nice round number to use.
[3] However, Jorge Luis Borges already performed a very similar thought experiment in a short story titled “The Library of Babel” (4/25/2014)
[4] For example, structures tend to display symmetry; symmetric configurations tend to form an extremum in the crystal structure energy landscape which is often a minimum energy point.
[5] For examples of rules exhibited by novels, check out this wonderful article by Brian Hayes on early work by Markov. (11/26/2013)
[6] Not all prototypes support 5 elements, but we are giving them the benefit of the doubt…
[7] Hopefully, informatics will assist with this process in the future.
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