8. Transport Properties 80 8 Transport Properties 8.1 Planets have Transport Thermodynamics tells you about equilibrium: the state (phase) that a medium prefers to have at a specified P and T. However, planets are continuously driven out of equilibrium: Heat flows from one place to another, materials flow and sometimes react chemically, electrical currents flow, etc. So it is of great importance to understand the transport properties of planet-forming materials. 8.2 The Nature of Microscopic Transport Properties These properties have a far less secure theoretical framework than the thermodynamic properties. Moreover, they are often harder to measure. They are also far more susceptible to “dirt” effects (minor constituents or imperfections). On the other hand, our expectations (the desired accuracy with which we need to know the parameters) are often less. Here we wish to focus on three questions: Is the process important? How does it work at the microscopic level? How does it scale to other (unmeasured) thermodynamic conditions? We deal much later in the course with macroscopic transport, specifically fluid flow, such as convection, but we do deal here with parameters that are input into fluid dynamics, e.g., viscosity. 8.3 Which Transport Processes? Diffusion of Heat is governed by an equation of the form (8.1) (Fourier’s law) where F is the heat flux (energy per unit area per unit time), k is called the thermal conductivity and κ is called the thermal diffusivity. For constant material properties, the resulting continuity equation ∇.F=-ρCp ∂T/∂t becomes the well-known diffusion equation (assuming no internal heat sources) ∂T∂t=κ∇2T (8.2) and κ accordingly has units of (length)2/time. The existence of a continuum equation is contingent on a short mean free path for the entity carrying heat (phonons, photons). So this equation can also describe radiative transfer but only in the limit just described. (In practice, this limit works fine for radiation inside planets and works poorly in atmospheres. However, it should be noted that radiative “conductivity” is highly temperature-dependent and so the diffusivity cannot be brought outside the gradient operator as has been done in the above equation. The correct equation is then ). By dimensional analysis, there is a characteristic diffusion length (κt)1/2 which describes the distance within which a heat disturbance will modify its surroundings in an elapsed time t.8. Transport Properties 81 Diffusion of composition is governed by an equation which has the same underlying physical origin ( a random walk): J = −ρD∇C (8.3) where J is a mass flux (mass per unit area per unit time), D is a diffusivity and C is a concentration of a diffusing species (which could even be the only species in the case of self-diffusion... e.g., as in diffusion of isotopes.) The differential equation for time evolution has the same form as for heat. Diffusion of Momentum is governed by the divergence of the stress tensor. The constitutive equation for the stress tensor depends on the nature of the material and can in principle very complicated (unlike the preceding examples of diffusion). For an incompressible Newtonian fluid, it has the form: σij=η(∂ui∂xj+∂uj∂xi) ; η≡ρν (8.4) where σ is the deviatoric stress tensor, u is the fluid flow, η is the dynamic viscosity (sometimes written as µ, though this symbol is more often reserved for shear modulus) and ν (nu) is the kinematic viscosity and has units of diffusivity, i.e., (length)2/time. “Deviatoric” means that the mean stress state (the pressure term) is omitted. We will not make explicit use of this formula until later in the course. Although there exist fluid dynamical problems where a diffusion-like equation (similar to those mentioned earlier) arises, we are usually concerned with quasi-steady state applications of the viscous stress. The reason is that most problems where viscosity is of great interest (e.g. mantle convection) are also problems where steady state has been established (i.e., no net force). Electrical Conductivity is defined by the equation J =σE = −σ∇V (8.5) where J is the current density, E is the electric field and V is the electric potential. As we shall see in due course, the electrical conductivity is most often seen in planetary problems as a “magnetic diffusivity”, describing the Ohmic decay of electrical currents and associated magnetic field. 8.4 A Simple View of these Transport Coefficients (Motivated by the Kinetic Theory of Gases). Recall that for a diffusive process, if the single diffusion step has length a and takes a typical time τ, then after N steps (elapsed time Nτ), the distance gone is N1/2a. This is what is meant by a random walk. For a diffusivity D, we thus must have Dt = a N when t = Nτ⇒ D = a2/τ= av (8.6) where v = a/τ is the velocity of the entity that is diffusing. For heat transport in an insulator or diffusion in a liquid or viscosity in a liquid, v should naturally be identified as the sound speed, and a as the distance between atoms. So we get the result that8. Transport Properties 82 κ~ D ~ν~ (10−8cm)(106cm / s) ~ 10−2cm2/ s (8.7) This is actually correct to within an order of magnitude, for materials where the transport is done by atoms and there is no strong bonding of the atoms to particular sites. It is also roughly true for heat diffusion in insulating solids, where the heat is actually carried by phonons. (The mean free path a is in that case best thought of as the scattering distance for phonons. If the material were perfectly harmonic then it would have “infinite” thermal conductivity!) Electrons are more efficient carriers of heat (or electrical current) and will dominate if they are free to move, so metals typically have higher thermal diffusivities than that given above by an order of magnitude or two. 8.5 Does Diffusion Matter? In the age of the solar system. a diffusivity of 0.01cm2/sec implies a possible diffusion distance of (0.01cm2/ s).(4.5x109yr).(3.15x107sec/ yr) ≈ 400km (8.8) and this is small compared to the size of most planets. So diffusion of heat in insulators is often unimportant on the planetary scale. An example of where it would matter is in small bodies (e.g., asteroids or small icy satellites or Kuiper belt objects). Even