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Self-assembling nanoparticles may be key to new materials
Like cheerleaders forming a human pyramid, particles, too, can assemble themselves into intricate patterns. In a new study, researchers at the University of Michigan found that an object’s shape greatly affects how it responds to crowding and that, with a properly designed shape, tiny material building blocks known as nanoparticles could self-assemble into predictable larger structures simply by being forced to share space with neighbors. The study, which appeared in the July 27 Science, could help researchers design new materials.
The investigators ran computer simulations of 145 different particles having idealized polyhedral shapes. (A polyhedron is a solid formed by planar faces.) When stacked closely with identically shaped particles, most of those polyhedrons assembled into a crystal lattice or a crystal-like arrangement. Study co-author Sharon Glotzer, a Michigan professor of chemical engineering, materials science and physics, and her colleagues had previously found that some particle shapes naturally self-assemble. Yet the new simulations showed that such behavior is the rule, not the exception.
Moreover, some of the shapes displayed an impressively coordinated assembly process. A pyramid shape with a square base joined into “supercubes” of six pyramids each, which then formed a larger cubic lattice. The researchers also found that the collective behavior of a given particle type is far from random. In fact, two numbers all but foretell the outcome (result). A number called the isoperimetric quotient, which roughly captures a particle’s shape, and a measure called the coordination number, which describes how many neighbors a particle has, predicted 94 percent of the time which crystalline form a polyhedron would take. The relation between shape and self-assembly could be used to make nanoparticles exhibit a specific collective behavior.
“This is sort of a holy grail of materials research: to just look at a building block and be able to say, ‘Oh yes, I know all the kinds of crystal structure that would be stable with this,’” Glotzer says. “This study allows us to take a first step in that direction.”
(I) The practical value of geometry lies in the fact that we can abstract and illustrate physical objects by drawings and models. For example, a drawing of a circle is not a circle, it suggests the idea of a circle. In our study of geometry we separate all geometric figures into two groups: plane figures whose points lie in one plane and space figures or solids. A point is a primary and starting concept in geometry. Line segments, rays, triangles and circles are definite sets of points. A simple closed curve with line segments as its boundaries is a polygon.
The line segments are sides of the polygon and the end points of the segments are vertices of the polygon. A polygon with four sides is a quadrilateral. We can name some important quadrilaterals. Remember, that in each case we name a specific set of points. A trapezoid is a quadrilateral with one pair of parallel sides. A rectangle is a parallelogram with four right angles. A square is a rectangle with all sides of the same length. The regular polyhedra are a part of geometric study chiefly in antiquity. They have a symmetrical beauty that fascinates men of all ages. The first question concerning regular polyhedra is: how many different types are there? Thanks to the ancient Greeks we know that there are exactly five types of polyhedra. All objects in their view are composed of four basic elements, earth, air, fire and water. They believed that the fundamental particles of fire had the shape of tetrahedron, the air particles had the shape of octahedron, particles of water - the icosahedron, and particles of the earth - the cube. The fifth shape, the dodecahedron, they reserve for the shape of the universe itself.
Plane geometry is the science of the fundamental properties of the sizes and shapes of objects and treats geometric properties of figures. The first question is: under what conditions two objects are equal or congruent in size and shape. Next: if figures are not equal, what significant relationship may they possess to each other and what geometric properties can they have in common? The basic relationship is shape. Figures of unequal size but of the same shape, that is, similar figures have many geometric properties in common. If figures have neither shape nor size in common, they may have the same area, or, in geometric terms, they may be equivalent, or may have endless other possible relationships.
(II) Geometry is the science of the properties, measurement and construction of lines, planes, surfaces and different geometric figures. What do we call “constructions” in our study of geometry? Ruler-compass constructions are simply the drawings, which we can make when we use only a straightedge and a compass. For a ruler you ought to use an unmarked straightedge because measurement has no role in ruler-compass constructions. We measure segments in terms of other segments and angles in terms of other angles. It seems only natural that we find areas indirectly as well. How does a person find the area of a floor? Does he take little squares one foot on a side, lay them out over the entire floor and thus decide that the area of a floor is 100 square feet, for this is indeed the meaning of area? Of course, he does not. He measures the length and width, quantities usually quite simple, and then multiplies the two numbers to obtain the area. This is indirect measurement, for we find the area when we measure lengths. The dimensions we take in the case of volume are the area and the length or the height. Greek mathematicians are the founders of indirect measurement methods. Their contribution to this subject are formulae for areas and volumes of particular geometric shapes, that we use nowadays. Thus, thanks to the Greeks we can find the area of any one single triangle when we take the product of its base and half its height. We also know due to them, that the “areas of two similar triangles are to each other as the squares of corresponding sides”. In other words, even the very common formulae of geometry, which we owe to the Greeks, permit us to measure areas and volumes indirectly, when we express these quantities as lengths.
We ought not to undervalue this contribution of the ancient Greek mathematicians. Their formulae for areas and volumes represent a great practical and important result. However, this type of indirect measurement is not the only one of interest to the Greeks. They measure indirectly the radius of the Earth, the diameter of the Sun and Moon, the distances to the Moon, the Sun, some planets and stars.