BENG 221: Mathematical Methods in Bioengineering Lecture 14 Brownian Motion and Diffusion References Einstein, A. Investigations on the theory of the Brownian movement. Dover Publications Inc., 1956 (translation from the 1905 original). Crank, J. The mathematics of diffusion. Oxford University Press, Oxford, 1980 (2nd. ed). Purcell, E.M. “Life at Low Reynolds Number,” Lyman Laboratory, Harvard University, June 1976. Also American Journal of Physics, vol. 45, pp. 3-11, 1977. http://en.wikipedia.org/wiki/Brownian_motionhttp://en.wikipedia.org/wiki/Diffusion_equationBENG 221 Lecture 14 M. Intaglietta Diffusion Diffusion occurs when a system is not in equilibrium and random molecular motion causes the system to become uniform in its properties, such as the distribution of material, thermal energy and charge. We tend to assume that diffusion is associated with cooling or heating, and the mixing of fluids, however diffusion is seldom the dominant mechanism. Bulk motion of the fluid is the cause of most of the mixing effects that we experience in liquids and gases. While diffusion is important in the detail of these processes, convection is what we most often experience. Were it not for convective motions in the air it would take one year to smell our feet after taking off our shoes due to molecular diffusion. It is the stirring of milk in the coffee that mixes it; molecular diffusion would take so long that the drink would long spoil and evaporate before it was mixed. The most common example of diffusion in our everyday experience, is the diffusion of heat in a solid. However, at the microscopic level, cells are fundamentally dependant on the process of diffusion to absorb and reject materials from and into the environment, particularly in the immediate proximity of their membranes where convection become very small as a consequence of the no slip condition imposed by solid boundaries to the motion of their surrounding fluid. A common form of diffusion is viscosity, which determines how momentum diffuses in fluid flow. The diffusion of momentum generally a small effect compared to the effects due to fluid inertia. In breathing and swimming we are not particularly aware of the consequent drag effect due to viscosity. However cellular organism live in a world where viscous diffusion is the dominant effect (see “Life at Low Reynolds Number”, Purcell 1976). Viscosity is a fundamental hydrodynamic property of fluids that determines the existence of boundary layers, which are the critical element for avoiding metal to metal contact all rotating mechanism and thus maintaining the integrity of their surfaces. Diffusion processes are irreversible and do not run backwards. Diffusion processes are intimately related to entropy, irreversibility, and probability. Brownian motion The motion of a particle suspended in a viscous fluid results from fluctuating forces which are the consequence of collisions with molecules of the fluid. As an example a sphere of 1 µm in diameter in air is subjected to 1016 collisions per second. The details of Brownian movement cannot be predicted exactly, however, we may assume that the events (collisions, displacements, etc.) are random. Therefore even though we cannot know the details of the phenomenon, we can determine the average behavior.Theory for one dimensional displacement Let us analyze the displacements of a particle of mass m along the x axis as a consequence of the action of a random force X acting in the x direction. Both the magnitude and sign of X are random. The velocity of the particle dx/dt and the acceleration d2x/dt2 are opposed by a frictional force F, where each of these terms is defined by: 22;;dx d xxxFfxdt dt===&&&& The frictional force F opposes the action of the force X, where the frictional coefficient f is given by: f = 6πηR 1 where η is the viscosity of the medium in which the particle moves and R is the radius of the particle (Stokes law). According to Newton's second law we obtain the equation of motion: Xfx mx−=&&& 2 We assume that at t = 0 the particle is at the origin x = 0, and we wish to determine what will be the average distance <x> that the particle has moved from its staring point as time progresses. A convenient variable that allows tracking the distance from the origin independently of the direction of the motion along the x axis is to measure x2, and if we let y = x2, the derivatives of this function are: 2222; 2 2dy d yxxxxxdt dt==+&&&& 3 The averages of these functions are: 2222; 22dy d yxxxxxdt dt<>=< >< >=< + >&&&& 4 If we multiply equation 2 by x and take the time average we obtain: =xXfxxmxx<>−<> <>&&& 5 Comparing equations (5) and (3) we note that we can introduce the variable y into the equation of motion by simple substitution and obtain: 2211222()dy d yxX f m xdt dt<>−<>= < >−<>& 6 The kinetic theory of gases indicates that each molecule has an average kinetic energy <KE> given by:23122)KE m x kT<>=<>=& 7 where k is Boltzmann's constant whose value is 1.38 x 10-16 ergs/degree and T is the absolute temperature. We make the assumption that the particles in our system have the same kinetic energy as that of a gas, and furthermore since the particle under consideration moves only in one direction it must have one third of the total average kinetic energy. The average velocity squared is then obtained from (7) to be: 2kTxm<>=& 8 Equation (6) can now be integrated by making the substitution given in (8) and noting that the term < xX > = 0 since X varies at random. By making the substitution: dzydt=<> 9 we obtain the first order equation: 2dzmfzkTdt+= 10 whose solution is: ()12tmfkTzKef−=+ 11 where K1 is a constant of integration. For microscopic particles the term m/f is very small and the exponential term vanishes. It is also shown that this term, which has the units of time, is a measure of the time required by a particle to reach its final velocity when acted upon by the force X. Utilizing the definitions of z and x we obtain: 22dkTzxdt f=<>= 12 and integrating we obtain: 222kTxtKf<>= + 13 where K2 is a constant equal to zero since x = 0 at t = 0. This result can be extended to motion of a particle in three dimensions, since equation (13) holds for each of