Elastic Modulii, Stiffness, and Thermal Fluctuations

In a post on Metamodern, I’ve written briefly about the stiffness of various materials and why this property matters. Here, I'd like to be a bit more quantitative, and to note some of the simplifications made in the post.

Why stiffness matters to (nano)mechanical engineers

Stiffness determines how much a component will deform in response to force, or equivalently, how much elastic energy is required to distort a component into a given shape. In mechanical engineering for nanotechnology, this is of special importance: Stiffness limits the amplitude of thermal fluctuations, because the probability that a machine of a distorted shape decreases exponentially as the elastic energy increases. In a material, “stiffness” means elastic modulus, and is measured in force per unit area. In a specific structure, “stiffness” means spring constant, and is measured in force per unit distance. In a structure of a specific size and shape, the stiffness is proportional to the elastic modulus of the material of which it is made. (As I note below, however “elastic modulus” is not as simple as this discussion suggests.)

Stiffness limits thermal fluctuations

To be more specific, the amplitude of thermal fluctuations along a given coordinate in a in a linear system has a Gaussian distribution with an r.m.s. value of (kBT/ks)1/2, where kBT is the characteristic thermal energy at a temperature T, and ks is the spring constant, which is proportional to the elastic modulus of the materials of which the machine is made. Quantum mechanics slightly increases the fluctuation amplitude as a consequence of the zero-point energy. For nanoscale components at ordinary temperatures, the correction is a tiny fraction of a percent, and even measuring it at temperatures near absolute zero is a challenge. Nanosystems discusses this topic in more detail, and I’m sure there are ample discussions on the web.

There are several kinds of elastic modulus

Speaking as if there were only one elastic modulus is a simplification. In reality, there are several kinds of elastic modulus in isotropic materials, and several more kinds in crystals, and in inhomogeneous or nonlinear materials, the concept of “elastic modulus” itself becomes ill-defined. Granta Design has provided an excellent tutorial that explains Young’s modulus, shear modulus, bulk modulus, Poisson’s ratio, and how they are related. When I speak of an “elastic modulus” without qualification, I mean Young’s modulus, the property that determines (for example) how much a rod will stretch or bend in response to an applied force. Shear modulus becomes important chiefly in thicker, more chunky shapes. Fortunately, in typical strong materials, the values of the shear and Young’s modulus are roughly proportional, hence Young’s modulus by itself is a good basis for comparing materials for mechanical applications.