2.4 Transport processes

A system is in ``equilibrium'' if its properties do not change spontaneously over time. (In case there are external fields - such as gravity - the material properties may vary in space, otherwise they will be independent also of position: the system is not only in equilibrium but also homogeneous.)

In particular, in equilibrium the local energy density, i.e. the energy contained in a volume element $dV$ divided by that $dV$, as well as the momentum density and the particle or mass density remain constant. These densities, which refer to the three conserved quantities of mechanics, play a special role in what follows.

Again considering the dilute gas, and assuming there are no external fields, the densities are

$$\rho_{E} \equiv \frac{E}{V} = \frac{1}{V} \sum_{i=1}^{N} \fr... ...{v}_{i} ,\;\;\; {\rm and} \;\;\; \rho_{m} \equiv \frac{Nm}{V}$$ (2.41)

By looking more closely we would see that these local densities are in in fact not quite constant; rather, they will fluctuate about their mean. In other words, there will be a spontaneous waxing and waning of local gradients of those densities. Also, the experimentor may intervene to create a gradient. For instance, one might induce in a horizontal layer of a gas or liquid a certain $x$-velocity $u_{x} \neq 0$, and thereby a momentum $p_{x}$. The difference between the momenta in adjacent layers then defines a gradient of the momentum density. If we left the system to itself this gradient would decrease and eventually vanish. The property governing the speed of this decrease is called viscosity.

If a physical quantity - such as mass, or momentum - is conserved, any local change can only be achieved by a flow into or out of the space region under consideration; we are then speaking of transport processes whose speeds are governed by the respective transport coefficients - such as the viscosity $\eta$, the heat conductivity $\lambda$ (for the energy transport) and the diffusion constant $D$ (for mass transport).

In real experiments aimed at determining these transport coefficients the respective gradient is artificially maintained. Thus there will be a continuing flow of momentum, energy or matter in the (reverse) direction of the gradient - which is obviously a non-equilibrium situation. However, by a careful setup we can keep at least these flows and the local densities and gradients constant in time. This is called a stationary non-equilibrium situation.

DEFINITION OF TRANSPORT COEFFICIENTS
In order to define the transport coefficients $\eta$, $\lambda$ and $D$ we consider the basic experimental setup.

$\bullet$

Viscosity: To measure the coefficient of momentum transport $\eta$ we generate a laminary flow by placing a gas or fluid layer between two horizontal plates and moving the upper plate with constant velocity $u_{0}$ to the right. In this manner we superimpose a systematic $x$-velocity onto the random thermal motion of the molecules.

The magnitude of the thermal speed is of the order $10^{3} m/s$; by adding a ``shear velocity'' $u_{x}(z)$ of some centimeters per second the local equilibrium is not considerably disturbed. Thus we may assume that we have still a Maxwell-Boltzmann distribution of velocities at any point in the fluid, with the same value of $\langle E \rangle$ (bzw. $kT$) everywhere.

Yet by imposing a velocity gradient we have slightly perturbed the equilibrium; a certain amount of $x$-momentum will flow against the gradient - in our case downwards - so as to reestablish equilibrium. The amount of momentum flowing down through a unit of area per unit of time is called flow density $\vec{j}_{p}$. In a first (linear) approximation this flow will be proportional to the imposed velocity gradient, and the coefficient of proportionality is, by definition, the transport coefficient $\eta$:

$$j_{p}= -\eta \frac{d u_{x}(z)}{dz}$$ (2.42)

This defining equation for $\eta$ is often called Newton's law of viscous flow. The parameter $\eta$ is known as viscosity.

$\bullet$

Thermal conductivity: To generate a gradient of energy density we may place the substance between two heat reservoirs with different temperatures. if we use the same simple geometry as for the viscosity then the temperature (or energy) gradient will have only the one non-vanishing component $dT/dz$ which accordingly produces a flow of energy in the counter direction. The coefficient $\lambda$ of thermal conduction is defined by (Fourier's law),

$$j_{E}= -\lambda \frac{d T(z)}{dz}$$ (2.43)

$\bullet$

Diffusion constant: Even in a homogeneous gas (or a fluid, or a solid, for that matter) the individual molecules will alter their positions in the course of time. This process is known as self diffusion. It is possible to ``tag'' some of the particles - for instance, by giving them a radioactive nucleus. The density of the marked species - call it $1$ - may then once more have a gradient that will be balanced by a particle flow (Fick's law):

$$j_{1}= -D \frac{d \rho_{1}(z)}{dz}$$ (2.44)

MEAN FREE PATH
We will presently attempt to calculate the above defined transport coefficients from the microscopic dynamics of the molecules. In a simple kinetic treatment the elementary process determining the various flows is the free flight of a gas particle between two collisions with other molecules. Thus the first step should be an estimation of the average path length covered by a particle between two such collisions.

Let $P(x)$ be the - as yet unknown - probability that a particle meets another one before it has passed a distance $x$. Then $1-P(x)$ is the probability for the molecule to fly a distance $x$ without undergoing a collision. But the probability of an encounter within an infinitesimal path element $dx$ is given by $p =\rho \sigma^{2} \pi dx$, where $\sigma$ is the diameter of the particles. ($p$ equals the fraction of the ``target area'' covered by other particles.) Thus the differential probability for a collision between $x$ and $x+dx$ is

$$dP(x)=[1-P(x)]\rho \sigma^{2}\pi dx$$ (2.45)

which upon integration yields

$$P(x)= 1- e^{-\rho \sigma^{2}\pi x}$$ (2.46)

The probability density $p(x) \equiv dP(x)/dx$ for a collision within that interval is

$$p(x)=\rho \sigma^{2}\pi e^{-\rho \sigma^{2}\pi x}$$ (2.47)

The mean free path is then the first moment of this density:

$$l \equiv \langle x \rangle \equiv \int_{0}^{\infty} dx x p(x) = \frac{1}{\rho \sigma^{2}\pi }$$ (2.48)

EXAMPLE: The diameter of a $H_{2}$ molecule is $\sigma \approx 10^{-10}m$ ($= 1$ Ångstrøm). According to Loschmidt (and the later but more often quoted Avogadro) a volume of $22.4 \cdot 10^{-3}m^{3}$ contains, at standard conditions, $6.022 \cdot 10^{23}$ particles; the particle density is thus $\rho = 2.69 \cdot 10^{25} m^{-3}$, and the mean free path is $l=1.18 \cdot 10^{-6}m$.

Applet Hspheres: Start

$\parbox{360pt}{ \mbox{}\\ {\bf Simulation: Collision rate in a dilu... ...$ with the empirical one $Z_{ex}$ \\ \vspace{2ex} {\small [Code: Hspheres]} }$



ESTIMATING THE TRANSPORT COEFFICIENTS
In the case of a dilute gas we can actually find expressions for the transport coefficients. Assuming once more the basic geometric setting used in most experiments, we suppose that the gradient of the respective conserved quantity has only a $z$ component, leading to a flow that is also directed along the $z$ axis.

• Viscosity: In the course of the random (thermal) motion of the particles the small systematic term $u_{x}(z)$ is carried along. On the average, an equal number of particles will cross the interface between two horizontal layers from below and above. However, since the upper layer has a larger $u_{x}$ (assuming $du_{x}/dz > 0$, it will gradually slow down, while the lower layer will take up speed. The flow of momentum due to this mechanism can be estimated as follows:
  The mean number of particles crossing the unit of area from below and from above, respectively, is $\rho \langle v \rangle /6$. Each of the molecules carries that systematic $u_{x}(z)$ which pertains to the layer of its previous collision, i.e. $u_{x}(z \pm l)$. The momentum flow density is therefore
  

  $$j_{\vec{p}} = \frac{1}{6} \rho \langle v \rangle \left[ m u(... ...} \rho \langle v \rangle \left[- 2 m \frac{du}{dz} l\right]$$ (2.49)

  
  
  Comparing this to the defining equation for the viscosity $\eta$ we find
  

  $$\eta = \frac{1}{3} \rho l \langle v \rangle m$$ (2.50)

  
  
  It is a somewhat surprising consequence of this formula that, since $l \propto 1/\rho$, the viscosity $\eta$ is apparently independent of density! Maxwell, who was the first to derive this result, had to convince himself of its validity by accurate experiments on dilute gases.
• Thermal conductivity: A similar consideration as in the case of momentum transport yields, with $j_{E} = - \lambda dT/dz$,
  

  $$\lambda = \frac{1}{3} \rho l \langle v \rangle m c_{V}$$ (2.51)

  
  
  where $c_{V} \equiv C_{V}/M$ is the specific heat ($C_{V}$.. molar heat capacity; $M$..molar mass).
• Diffusion: Again we assume a gradient of the relevant density - in this case, the particle density of some species $1$ - along the $z$ direction. Writing $j_{1} = - D d\rho_{1}/dz$ and using the same reasoning as before we find
  

  $$D = \frac{1}{3} l \langle v \rangle$$ (2.52)