Diffusion and Tanspot 10. Fiction and the Langevin Equation Now let s elate the phenomena of ownian motion and diffusion to the concept of fiction, i.e., the esistance to movement that the paticle in the fluid expeiences. These concepts wee developed by Einstein in the case of micoscopic motion unde themal excitation, and macoscopically by Geoge Stokes who was the fathe of hydodynamic theoy. Langevin Equation Conside the foces acting on a paticle as we pull it though a fluid. We pull the paticle with an extenal foce fext, which is opposed by a dag foce fom the fluid, fd. The dag o damping acts as esistance to motion of the paticle, which esults fom tying to move the fluid out of the way. f d v A dag foce equies movement, so it is popotional to the velocity of the paticle vdx/ dt x and the fiction coefficient is the popotionality constant that descibes the magnitude of the damping. Newton s second law would say the acceleation of this paticle is popotional to the sum of these foces: ma f d f ext. Now micoscopically, we also ecognize that thee ae timedependent andom foces that the molecules of the fluid exet on a molecule (f). So that the specific molecula details of solute solvent collisions can be aveaged ove, it is useful to think about a nanoscale solute in wate (e.g., biological macomolecules) with dimensions lage enough that its position is simultaneously influenced by many solvent molecules, but is also light enough that the constant inteactions with the solvent leave an unbalanced foce acting on the solute at any moment in time: f () t f () t. Then Newton s second law is ma f f f t. i i d ext The dag foce is pesent egadless of whethe an extenal foce is pesent, so in the absence of extenal foces (fext=0) the equation of motion govening the spontaneous fluctuations of this solute is detemined fom the foces due to dag and the andom fluctuations: d ma f f t mx x f ( t) 0 Andei Tokmakoff 5/3/2017
This equation of motion is the Langevin equation. An equation of motion such as this that includes a time-dependent andom foce is known as stochastic. Inseting a andom pocess into a deteministic equation means that we need to use a statistical appoach to solve this equation. We will be looking to descibe the aveage and oot-meansquaed position of the paticle. Fist, what can we say about the andom foce? Although thee may be momentay imbalances, on aveage the petubations fom the solvent on a lage paticle will aveage to zeo at equilibium: f ( t) 0 Equation seems to imply that the dag foce and the andom foce ae independent, but in fact they oiginate in the same molecula foces. If the molecule of inteest is a potein that expeiences the fluctuations of many apidly moving solvent molecules, then the aveaged foces due to andom fluctuations and the dag foces ae elated. The fluctuation dissipation theoem is the geneal elationship that elates the fiction to the coelation function fo the andom foce. In the Makovian limit this is f ( t) f ( t) 2 k T ( t t) Makovian indicates that no coelation exists between the andom foce fo t t > 0. Moe geneally, we can ecove the fiction coefficient fom the integal ove the coelation function fo the andom foce 1 dt fr fr( t) 2kT To descibe the time evolution of the position of ou potein molecule, we would like to obtain an expession fo mean-squae displacement x 2 (t). The position of the molecule can be descibed by integating ove its time-dependent velocity: mean-squae displacement in tems of the velocity autocoelation function t t 2 x t dt dt x t x t 0 0 ( ) ( ) ( ) t x() t dtx ( t), so we can expess the 0 Ou appoach to obtaining x 2 (t) stats by multiplying eq. by x and then ensemble aveaging. d m x x xx x f () t 0 dt Fom eq., the last tem is zeo, and fom the chain ule we know d ( xx) x d x dx x dt dt dt Theefoe we can wite eq. as 2
d m xxxx xx0 dt Futhe, the equipatition theoem states that fo each tanslational degee of feedom the kinetic enegy is patitioned as 1 2 kt mx 2 2 d So, m xx xx kt dt Hee we ae descibing motion in 1D, but when fluctuations and displacement ae included fo 3D motion, then we switch x and kt 3 kt. Integating eq. twice with espect to time, and using the initial condition x = 0, we obtain 2kT 2 m x t exp t 1 m To investigate this equation, let s conside two limiting cases. Fom eq. we see that m/ζ has units of time, and so we define the damping time and investigate time scale shot and long compaed to τc: C m / 1) Fo t C, we can expand the exponential in eq. and etain the fist thee tems, which leads to kt m 2) Fo t C, eq. is dominated by the leading tem: 2kT 2 x t (long time: diffusive) 2 2 2 2 x t v t (shot time: inetial) In the diffusive limit the behavio of the molecule is govened entiely by the fluid, and its mass does not matte. The diffusive limit in a stochastic equation of motion is equivalent to setting m 0. We see that τc is a time-scale sepaating motion in the inetial and diffusive limits. It is a coelation time fo the andomization of the velocity of the ballistic diffusive 3
paticle due to the andom fluctuations of the envionment. Fo vey little fiction o shot time, the paticle moves with taditional deteministic motion xms = vms t, whee oot-mean-squae displacement xms = x 2 1/2 and vms comes fom the aveage tanslational kinetic enegy of the paticle. Fo high-fiction o long times, we see diffusive behavio with xms~t 1/2. Futhemoe, by compaing eq. to ou ealie continuum esult, x 2 = 2Dt, we see that the diffusion constant can be elated to the fiction coefficient by D kt (in 1D) This is the Einstein fomula. Fo 3D poblems, we eplace kt with 3kT in the expessions above and find D3D = 3kT/ How long does it take to appoach the diffusive egime? Vey fast. Conside a 100 kda potein with R = 3 nm in wate at T = 300 K, we find a chaacteistic coelation time fo andomizing 12 velocities of τc 3 10 s, which coesponds to a distance of about 10 2 nm befoe the onset of diffusive behavio. We can find othe elationships. Noting the elationship of x 2 to the velocity autocoelation function in eq., we find that the paticle velocity is descibed by 2 tm / 2 t/ C x x( ) v e x v e x x v v t v x which can be integated ove time to obtain the diffusion constant. kt vx vx( t) dt D 0 This expession is the Geen Kubo elationship. This is a pactical way of analyzing molecula tajectoies in simulations o using paticle-tacking expeiments to quantify diffusion constants o fiction coefficients. 4
Fiction and Viscosity How is the micoscopic fiction oiginating in andom foces elated to macoscopic expeimental obsevables that measue a fluid s esistance to moving an object? ζ is elated to the dynamic viscosity of the fluid and factos descibing the size and shape of the object (but not its mass). Viscosity measues the esistance to shea. A fluid is placed between two plates of aea a sepaated along z, and one plate is moved elative to anothe by applying a foce along x. Since the velocity of the fluid at the inteface with a plate is taken to be the velocity of the plate (noslip bounday conditions: v ( z 0) 0 ), this sets up a velocity gadient along z. The elationship between the shea velocity gadient and the foce is dv x x fx a dz whee η, the dynamic viscosity (kg m 1 s 1 ), is the popotionality facto. shea stess f x a Shape Mattes A sphee, od, o cube with the same mass and suface aea will espond diffeently to flow. Stokes detemined the elationships between dag coefficient and fluid viscosity. Specifically, consideing the case whee a sphee of adius R is esisted by lamina flow of the fluid, one finds that the dag foce on the sphee is fd 6Rv and the viscous foce pe unit aea is entiely unifom acoss the suface of the sphee. This gives us Stokes Law 6R 5
Hee R is efeed to as the hydodynamic adius of the sphee, which efes to the adius at which one can apply the no-slip bounday condition, but which on a molecula scale may include wate that is stongly bound to the molecule. Combining eq. with eq. gives the Stokes Einstein elationship fo the tanslation diffusion constant of the sphee 1 D tans k T 6R One can obtain a simila a Stokes Einstein elationship fo oientational diffusion of a sphee in a viscous fluid. Relating the oientational diffusion constant and the dag foce that aises fom esistance to shea, one obtains V=4πR 3 /3 is the volume of the sphee. Reynolds Numbe D ot k T 6V The Reynolds numbe is a dimensionless numbe that indicates whethe the motion of a paticle in a fluid is dominated by inetial o viscous foces. 2 inetial foces R viscous foces When R 1, the paticle moves feely, expeiencing only weak esistance to its motion by the fluid. If R 1, it is dominated by the esistance and intenal foces of the fluid. Fo the latte case, we can conside the limit m 0 in eq., and find that the velocity of the paticle is popotional to the andom fluctuations: vt ( ) f( t) /. Hydodynamically, fo a sphee of adius moving though a fluid with dynamic viscosity η and density ρ at velocity v, v v( dv / dz) R Using pictue above: R 2 2 ( dv/ dz) Conside fo an object with adius 1 cm moving at 10 cm/s though wate: R 10 3. Now compae to a potein with adius 1 nm moving at 10 m/s: R =10 2. 1.. J. ene and R. Pecoa, Dynamic Light Scatteing: With Applications to Chemisty, iology, and Physics. (Wiley, New Yok, 1976), pp. 78, 91. 2. E. M. Pucell, Life at low Reynolds numbe, Am. J. Phys. 45, 3-11 (1977). 6
Dag Foce in Hydodynamics The dag foce on an object is detemined by the foce equied to displace the fluid against the diection of flow, 1 2 2 fd Cv d a This expession takes the fom of a pessue (tem in backets) exeted on the coss-sectional aea of the object along the diection of flow, a. Cd is the dag coefficient, a dimensionless popotionality constant that depends on the shape of the object. In the case of a sphee of adius : a=π 2 in the tubulent flow egime ( R 1000 ) Cd = 0.44 0.47. Detemination of Cd is somewhat empiical since it depends on R and the type of flow aound the sphee. The dag coefficient fo a sphee in the viscous/lamina/stokes flow egimes (R <1) is Cd 24 / R. This comes fom using the Stokes Law fo the dag foce on a sphee f 6v and the Reynolds numbe R vd. d 7
Readings 1. R. Zwanzig, Nonequilibium Statistical Mechanics. (Oxfod Univesity Pess, New Yok, 2001). 2.. J. ene and R. Pecoa, Dynamic Light Scatteing: With Applications to Chemisty, iology, and Physics. (Wiley, New Yok, 1976). 8