PHYS3410 BIOPHYSICS II
Lecture Notes

Section I: Thermodynamics of Biological Systems

Lecture 7 : The Phenomenological Equations

The dissipation function can be evaluated only when the flows and forces are known. However, sometime it is possible to measure the flows, sometimes the forces. So, correlation between the flows and forces would useful. Fourier established linear relationship between heat flow and temperature gradient (1811). During the first half of 19^th century Ohm showed that the electric current is proportional to the electromotive force and Fick proved that the rate of diffusion of matter is determined by the negative gradient of concentration.

(Note that in all these cases force is proportional to velocity rather than acceleration. How can this be reconciled with Newton first law? This law only holds for phenomena with no friction)

However, this nice simple relationship did not seem to hold, when more than one flow and force were present in the experimental system. The thermodynamic study of coupling between different phenomena was pioneered by Kelvin in 1854: when several flows are present, they are not independent of each other. In many cases interference between the flows can give rise to additional, cross, phenomena.

Examples:

· Seebeck effect (heat flow due to a gradient in temperature gives rise to a gradient in electric potential).

· Peltier effect (electric current gives rise to isothermal transport of heat)

· Sorret effect (diffusion and heat conduction)

· Electro-osmosis and streaming potential (coupling between volume flow and electric current)

In these cross phenomena, the flow of one type is coupled to the thermodynamic force of another type. In general, the flow can be represented by a set of linear phenomenological equations. The choice of a force conjugate to each flow is restricted by the requirement that the product J_iF_i should have the dimensions of entropy production or decrease in free energy with time

J₁ = L₁₁F₁ + L₁₂F₂ +……..+L_1kF_k                

J₂ = L₂₁F₁ + L₂₂F₂ +……..+L_2kF_k

J_k = L_k1F₁ + L_k2F₂ +……..+L_kkF_k                                                                              

      (7.1)

The coefficients L_jk are the phenomenological coefficients: generalised admittances

The diagonal coefficients L_kk give rise to the usual relation between the flows and their conjugate forces. The cross coefficients L_jk (j¹k) give rise to the interference phenomena. The above equations imply that the dependence of flows on the non-conjugated forces is also linear. The linear relationship was found to only apply for slow processes not too far removed from equilibrium.

The linear form of the phenomenological relations makes it possible to re-write them as:

      _(7.2)

The coefficients R_ik are generalized resistances. The resistances are related to admittances by:

           (7.3)

where [L_ik] is the cofactor of term L_ik and |L| is the determinant of the matrix of the coefficients (7.1).

Example: two flows and two forces

J₁ = L₁₁F₁  +  L₁₂F₂                                           F₁ = R₁₁J₁  +  R₁₂J₂

J₂ = L₂₁F₂  +  L₂₂F₂                                                        F₂ = R₂₁J₂  +  R₂₂J₂

|L| = L₁₁L₂₂ - L₁₂L₂₁

Onsager's Theory

Despite the simple linear dependence, the system of several forces and flows soon yields a large number of coefficients to be determined. Fortunately, Lars Onsager (1903 - 1976) was able to show that the matrix of phenomenological coefficients is symmetric, if the forces and flows are chosen in the right way:

 L_ik = L_ki  (i¹k)        (7.4)

This relationship is sometimes referred to as the Fourth Law of Thermodynamics. The eqn. (7.4) plays central role in the thermodynamics of irreversible processes and predicts correlation between flow phenomena of great interest and importance.

The proof is based on statistical mechanics and depends on validity of microscopic reversibility. That is: equations of motion of molecules are invariant under a time reversal (t = -t) for systems close to equilibrium.

For two forces and two flows:

s = J₁F₁ + J₂F₂

Thus the Second Law demands that for irreversible processes d_iS>0, restricting the magnitudes of phenomenological coefficients. Since either of the forces may be made to vanish, then:

        (7.5)

Further, eqn. (7.4) remains positive only if:

   (7.6)

which restricts magnitudes of the coupling coefficients L₁₂ and L₂₁:

       (7.7)

Equivalent restrictions also apply to the resistance coefficients when the phenomenological equations are written in the form of eqn. (7.2).

The cross-coupling of flows leads to possibility that some flows may take place in a direction opposed to that of their conjugate forces. Some J_iX_i can be negative, but the sum must remain positive.

Coupling of chemical reactions and vectorial flows

The thermodynamic flows and forces are of different tensorial order:

Chemical reactions are scalar, flow of heat and matter are vectors, viscous phenomena are tensors of second order.

               (7.8)

To satisfy the tensorial order of the flows, L_vs and L_sv must be vectors. In isotropic systems with symmetries, coupling between scalar and vector flows is not feasible, as such quantities would change sign with reflection of coordinates. Thus, in isotropic system, only coupling between flows and forces of equal tensorial order is possible. This aspect of flow/force coupling was first pointed out by Prigogine, based on consideration by Curie.

Thus, in symmetrical membrane diffusional flows cannot be directly coupled to chemical reactions taking place in the membrane. However, in asymmetrical membrane such coupling may be possible.