130 Chapter 4. Random walks, friction, and diffusion[[Student version, December 8, 2002]]
- Stokes: Foramacroscopic (many nanometers) sphere of radiusamoving slowly through a
fluid, the drag coefficient isζ=6πηa(Equation 4.14), whereηis the fluid viscosity.
[In contrast, at high speed the drag force on a fixed object in a flow is of the form−Bv^2 for
some constantBcharacterizing the object and the fluid (see Problem 1.7).] - Fick and diffusion: The flux of particles alongˆxis the net number of particles passing
from negative to positivex,per area per time. The flux created by a concentration gradient
isj = −Ddc/dx(Equation 4.18), wherec(x)isthe number density (concentration) of
particles. (In three dimensionsj=−D∇c.) The rate of change ofc(x, t)isthenddct=Dd
(^2) c
dx^2
(Equation 4.19).
- Membrane permeability: The flux of solute through a membrane isjs=−Ps∆c(Equa-
tion 4.20), wherePsis the permeation constant and ∆cis the jump in concentration across
the membrane. - Relaxation: The concentration difference of a permeable solute between the inside and
outside of a spherical bag decreases in time, following the equation−d(∆dtc) =
(
APs
V)
∆c
(Equation 4.21).- Nernst–Planck: When diffusion is accompanied by an electric field, we must modify Fick’s
law to find the electrophoretic flux:j=D
(
−ddcx+kBqTEc)
(Equation 4.23).- Nernst: If an electrical potential difference ∆Vis imposed across a region of fluid, then each
dissolved ion species with chargeqcomes to equilibrium (no net flux) with a concentration
change across the region fixed by ∆V =−kBqT∆(lnc)orV 2 −V 1 = −^58 q/emVlog 10 (c 2 /c 1 )
(Equation 4.25). - Ohm: The flux of electric current created by an electric fieldEis proportional toE,leading
to Ohm’s law. The resistance of a conductor of lengthdand cross-sectionAisR=d/(Aκ),
whereκis the conductivity of the material. In our simple model, each ion species contributes
Dq^2 c/kBTtoκ(Equation 4.26). - Diffusion from an initial sharp point: SupposeN molecules all begin at the same loca-
tion in 3-dimensional space at time zero. Later we find the concentration to bec(r,t)=
N
(4πDt)^3 /^2 e
−r^2 /(4Dt)(Equation 4.27).
Further reading
Semipopular:
Historical: (Pais, 1982,§5)
Finance: (Malkiel, 1996)
Intermediate:
General: (Berg, 1993; Tinoco Jr.et al.,2001)
Polymers: (Grosberg & Khokhlov, 1997)
Better derivations of the Einstein relation: (Benedek & Villars, 2000a),§2.5A–C, (Feynmanet al.,
1963a,§41)
Technical:
Einstein’s original discussion: (Einstein, 1956)