36 Flows and Transport in Liquids
36.0.1 Diffusion and Brownian Motion
- particles are moving in soft materials all the time→thermal fluctuations, Brownian motion
- in addition flows, friction→hydrodynamics, mechanical properties
- elasticity, rheology
a) Continuity
flux \(\vec{j}=ρ\vec{n}\) Consider a volume surrounded by a surface \(S\) flow through surface:
\[\begin{equation} \int_S\vec{j}\cdot\vec{n}\,\mathrm dS \end{equation}\]
material in volume, e.g., particle density:
\[\begin{equation} \int_V f \,\mathrm dV \frac{\mathrm d}{\mathrm dt}\int f \, \mathrm dV=\int_V\frac{\mathrm df}{\mathrm dt}\, \mathrm dV \end{equation}\]
conservation of matter:
\[\begin{equation} \int_S\vec{j}\cdot\vec{n}\,\mathrm dS=-\int_V\frac{\mathrm df}{\mathrm dt}\,\mathrm dV \end{equation}\]
Due to Gauss’s law
\[\begin{equation} \int_S\vec j \cdot \vec n \,\mathrm dS=\int_V \mathrm{div}\,\vec j \,\mathrm dV \end{equation}\]
and thus \[\begin{equation} \int_V \left(\frac{\mathrm df}{\mathrm dt}+\vec\nabla \cdot \vec j\right)\mathrm dV=0 \end{equation}\]
This yields the continuity equation
\[\begin{equation} \frac{\mathrm df}{\mathrm dt}+\vec \nabla\cdot\vec j = 0. \end{equation}\]
For the density \(n=f\), \(\vec{j}=n\cdot\vec{v}\)
\[ \frac{\partial n}{\partial t}+\vec \nabla(n\vec v)=0 \]
for \(n=\mathrm{const.}\) it is \(\vec{\nabla}\cdot\vec{V}=0\), \(\vec{\nabla}\cdot\vec{j}=0\) incompressible
Diffusion
\(N(x)\), \(N(x+\Delta x)\), \(0.5⋅N(x)\) to the right, \(0.5⋅N(x+\Delta x)\) to the left net number \(-1⁄2 (N(x+\Delta x)-N(x))\) thus
\[ j=\frac{-\tfrac 1 2 (N(x+\Delta x)-N(x))}{A \,\tau} \]
With the number density of particles \(n(x)≡N(x)⁄Aτ\) one obtains
\[\begin{equation} j=-\frac 1 2 \frac{n(x+\Delta x) \, A \, \Delta x -n(x)\,A\,\Delta x}{A \,\tau}= -\frac 1 2 \frac{(\Delta x)^2}{\tau}\frac{n(x+\Delta x)-n(x)}{\Delta x} \end{equation}\]
For \(Δx→0\) it is
\[\begin{equation} j=-D \frac{\partial n}{\partial x} \end{equation}\]
with \(D=(Δx)^2/2τ\). In 3d, this is
\[\begin{equation} \vec j=-D\vec \nabla n \end{equation}\]
This is Fick’s first law (1855, Fick, empirical). flux from a gradient is due to fluctuations in the velocity, diffusion max. entropy to vanish the gradient
Combination with the continuity equation
in 1d:
\[\begin{equation} \frac{\mathrm dn}{\mathrm dt}=-\frac{\partial j}{\partial x} \frac{\partial j}{\partial x}=-D\frac{\partial ^2n}{\partial x^2}=-\frac{\partial n}{\partial t} \end{equation}\]
Fick’s second law:
\[\begin{equation} \frac{\partial n}{\partial t}=D\nabla^2n \end{equation}\]
\(n f\) could be different quantities
| flux | transport property | gradient |
|---|---|---|
| particles | diffusivity | particle density |
| charge | conductivity | potential |
| liquid | permeability | pressure |
| momentum | viscosity | momentum density |
| energy | heat conductivity | temperature |
36.0.2 Smoluchowski-diffusion in an external potential
Smoluchowski-diffusion in an external potential
\[\begin{equation} F=-\frac{\mathrm dU}{\mathrm dx} \end{equation}\]
velocity of the particle
\[\begin{equation} v=-\frac 1 \xi \frac{\partial U}{\partial x} \end{equation}\]
\(ξ\)is the friction coefficient. For a sphere \(ξ=6πηR\). flux due to flow with \(v\), \(j_v=nv\)
\[\begin{equation} j=-D\frac{\partial n}{\partial x}-\frac n \xi \frac{\partial U}{\partial x} \end{equation}\]
in steady state \(j=0\) and
\[\begin{equation} n=n_0\,\mathrm{exp}\left({-\frac{U}{k_{\mathrm B}T}}\right) \end{equation}\]
\[\begin{equation} \begin{aligned} 0&=-D\frac{\mathrm d}{\mathrm dx}\left(n_0\,\mathrm{exp}\left({-\frac{U}{k_{\mathrm B}T}}\right)\right)-\frac 1 \xi\frac{\mathrm d U}{\mathrm d x}n_0\,\mathrm{exp}\left({-\frac{U}{k_{\mathrm B}T}}\right)+\frac{D}{k_{\mathrm B}T}\frac{\mathrm d U}{\mathrm d x}n_0\,\mathrm{exp}\left({-\frac{U}{k_{\mathrm B}T}}\right)\\ &=-\frac 1 \xi \frac{\mathrm d U}{\mathrm d x}n_0\,\mathrm{exp}\left({-\frac{U}{k_{\mathrm B}T}}\right) \end{aligned} \end{equation}\]
With the diffusion coefficient \(D=k_B T/ξ\) one has
\[\begin{equation} j=-\frac 1 \xi \left(k_{\mathrm B}T \frac{\partial n}{\partial x}+n\frac{\partial U}{\partial x}\right)=-\frac 1 \xi n\frac{\partial}{\partial x}(k_{\mathrm B}T\cdot \ln(n)+U), \end{equation}\]
since \[\begin{equation} \frac{\mathrm d}{\mathrm d x}\ln(n(x))=\frac 1 n \frac{\mathrm du}{\mathrm dx} \end{equation}\] and thus \[\begin{equation} \frac{\mathrm du}{\mathrm dx}=n\cdot \ln(n). \end{equation}\]
Here, \(k_B T⋅\ln{n}+U\) is the chemical potential.
\[\begin{equation} j=-\frac 1 \xi n \frac{\partial \mu}{\partial x} \end{equation}\]
The chemical potential is constant in equilibrium, extend potential for ?? \(\mu=\mathrm{const}\) .
\[\begin{equation} U=-k_{\mathrm B}T \ln(n(x))+\mathrm{const.} \end{equation}\]
With Fick’s second law
\[\begin{equation} \frac{\partial n}{\partial t}=-\frac{\partial j}{\partial x}=\frac 1 \xi \frac{\partial}{\partial x}\left(k_{\mathrm B}T\frac{\partial U}{\partial x}+n\frac{\partial U}{\partial x}\right) \frac{\partial n}{\partial t}=\vec \nabla\left(D\vec \nabla n+\frac{\vec{F_\mathrm{ext}}}{\xi}n\right) \end{equation}\]
This is the Smoluchowski equation. More general: Fokker—Planck equation
\[\begin{equation} \frac{\partial p(x,t)}{\partial t}=-\frac{\partial}{\partial x}\left[\frac{1}{\xi (x,t)}p(x,t)\right]+\frac{\partial^2}{\partial x^2}\left[D(x,t)p(x,t)\right] \end{equation}\]
\(D=k_B T/ξ\) gives a connection between thermal position fluctuations and viscous friction (fluctuation—dissipartion) \(D=k_B T/6πηR\) for sphere, e.g., protein: \(R\equiv 3 \, \mathrm{nm},\, D=100 \,\mathrm{\mu m^2/s},\, \eta=0.7\, \mathrm{mPas}\). Solution of the diffusion equation
\[\begin{equation} \frac{\partial}{\partial t}n(\vec r, t |\vec{r_0}, t_0)=D\nabla^2n(\vec r, t |\vec{r_0}, t_0) \end{equation}\]
initial condition \(n(\vec r,t→t_0 |\vec{r_0 },t_0 )=δ(\vec r-\vec{r_0 } )\)
solution:
\[\begin{equation} n(\vec r,t|\vec{r_0},t_0)=\frac{1}{(4\pi D (t-t_0))^\frac 3 2}\mathrm{exp}\left(-\frac{(\vec r-\vec r_0)^2}{4D(t-t_0)}\right) \end{equation}\]
This is the Green’s function for the diffusion equation for the given boundary conditions. It propagates the initial conditions through the ??? space. solution for any initial condition can be found
\[\begin{equation} n(\vec r, t\to 0)=f(\vec{r_0}) \\ n(\vec r, t)=\int\mathrm d^3r_0\,n(\vec r,t|\vec{r_0},t_0)\,f(\vec{r_0}) \end{equation}\]
properties: e.g., particle initially at \(x_0\) in 1D, initial distribution \(δ(x_0 )\)
\[\begin{equation} \langle x \rangle=\int_{-\infty}^\infty x \frac{1}{\sqrt{4\pi D(t-t_0)}}\mathrm{exp}\left(-\frac{(x-x_0)^2}{4D(t-t_0)}\right)\mathrm dx \end{equation}\]
\(⟨x⟩=x_0\) ,does not depend on time, particles do not change their initial position on average \[\begin{equation} \langle (x-x_0)^2\rangle =\int_{-\infty}^\infty (x-x_0)^2\,n(x,t)\,\mathrm dx \end{equation}\]
\(⟨(x-x_0 )^2⟩=2Dt\) in 1D
\(⟨r^2⟩=6Dt\) in 3D
mean squared displacement= ̂width of the probability distribution increases linearly in time
\(⟨r^2⟩=2dDt\) ,\(d\) is the dimension in more complex diffusion scenarios: \(⟨r^2⟩=cDt^α\) with \(α<1\) for subdiffusion and \(α>1\) for superdiffusion Application: fluorescence recovery after photobleaching Molecules are photophysically/chemically bleached in a spatial region with high intensity layer. Diffusion causes fluorescent molecules to diffuse in again, ?? bleached come out The “hole” is filling up and the dynamics is determined by the diffusion coefficient. initial distribution after the bleach is \(w(x,t_0 )\)
\[\begin{equation} n(x,t_0)=\Theta(-a-x)+\Theta(x-a) \end{equation}\]
Heaviside step function
\[\begin{equation} \Theta(x)=\begin{cases} 0 & x<0 \\ 1 & x\ge0 \end{cases} \end{equation}\]
\[\begin{equation} \frac{\partial n(x,t)}{\partial t}=D\frac{\partial^2}{\partial x^2}n(x,t) \end{equation}\]
boundary condition \(\lim_{x→∞} n(x,t) =0\) solution with Green’s function \[\begin{equation} n(x,t|x_0,t_0)=\frac{1}{\sqrt{4 \pi D(t-t_0)}}\mathrm{exp}\left(-\frac{(x-x_0)^2}{4D(t-t_0)} \right) \end{equation}\]
\[\begin{equation} \begin{split} n(x,t) &=\int_{-\infty}^\infty \mathrm dx_0\,n(x,t|x_0,t_0)\,\left[\Theta(-a-x)+\Theta(x-a)\right]\\ &= \int_{-\infty}^{-a}\mathrm dx_0 \frac{1}{\sqrt{4 \pi D(t-t_0)}}\mathrm{exp}\left(-\frac{(x-x_0)^2}{4D(t-t_0)} \right)\\ &+\int_{a}^{\infty}\mathrm dx_0 \frac{1}{\sqrt{4 \pi D(t-t_0)}}\mathrm{exp}\left(-\frac{(x-x_0)^2}{4D(t-t_0)} \right) \end{split} \end{equation}\]
\[\begin{equation} \mathrm{erf}(x)=\frac{1}{\sqrt \pi}\int_{-x}^x\mathrm{exp}\left(-t^2\right)\,\mathrm dt=\frac{2}{\sqrt \pi}\int_0^x\mathrm{exp}\left(-t^2\right)\,\mathrm dt \end{equation}\]
solution: \[\begin{equation} n(x,t)=1+\frac 1 2 \left(\mathrm{erf\left[\frac{x+a}{2 \sqrt{D(t-t_0)}}\right]-\mathrm{erf}\left[\frac{x-a}{2 \sqrt{D(t-t_0)}}\right]}\right) N(t,t_0)=c_0\int_{-a}^a\mathrm dx\,n(x,t) \end{equation}\]
if there is a concentration of \(c_0\) fluorescent molecules
\[\begin{equation} N(t,t_0)=\frac{\sqrt{D(t-t_0)}}{a\sqrt \pi}\left(\mathrm{exp}\left[-\frac{a^2}{D(t-t_0)}\right]-1\right)+1+\mathrm{erf}\left[\frac{a}{\sqrt{D(t-t_0)}}\right] \end{equation}\]
Extension
Diffusion can also occur on a spherical surface ?? to a point ?? surface is ?? doing rotational diffusion
\[\begin{equation} \frac{\partial n(\vec r,t|\vec{r_0},t_0)}{\partial t}=D\nabla^2n(\vec r, t|\vec{r_0},t_0) \end{equation}\]
with \(|\vec{r_0} |=|\vec r |=1\) so no radial ??
\[\begin{equation} \frac{\partial n(\Omega,t|\Omega_0,t_0)}{\partial t}=D_\mathrm{rot}\left[\frac{1}{\sin \vartheta}\frac{\partial}{\partial \vartheta}\left(\sin \vartheta\frac{\partial}{\partial \vartheta}\right)+\frac{1}{\sin^2 \vartheta} \frac{\partial^2}{\partial \varphi^2}\right]n(\Omega,t,\Omega_0, t_0) \end{equation}\]
The eigenfunctions of the operator on the right are sperical harmonics and the solution may be expressed in terms of \(Y_l^m\).
\[\begin{equation} Y_l^m=N\,\mathrm{exp}\left(im\varphi\right)\,P_l^m(\cos \theta) p(\Omega,t|\Omega_0,t_0)=\sum_{l=0}^\infty\sum_{m=-l}^{+l}A_{lm}(t,\Omega_0,t_0)Y_l^m(\Omega) \end{equation}\]
insert into diffusion equation
\[\begin{equation} \sum_{l=0}^\infty\sum_{m=-l}^{+l}\frac{\partial A_{lm}}{\partial t}Y_l^m=-\sum_{l=0}^\infty \sum_{m=-l}^l l(l+1)\frac{1}{\tau_R}A_{lm}Y_l^m \end{equation}\]
with orthonormality
\[\begin{equation} \begin{aligned} \frac{\partial A_{lm}}{\partial t}&=-l(l+1)\,\tau_R^{-1}\,A_{lm} A_{lm}(t|\Omega_0,t_0)=\mathrm{exp}\left(-l(l+1)\frac{t-t_0}{\tau_R}\right)\,a_{lm}(\Omega_0) p(\Omega,t|\Omega_0,t_0)\\ &=\sum_{l=0}^\infty\sum_{m=-l}^l \mathrm{exp}\left(-l(l+1)\frac{t-t_0}{\tau_R}\right)\,a_{lm}(\Omega_0)Y_l^m \end{aligned} \end{equation}\]
\(a_{lm}\) from \(p(Ω,t_0 |Ω_0,t_0 )=δ(Ω-Ω_0)\) and
\[\begin{equation} \delta(Ω-Ω_0 )=\sum_{l=0}^{\infty} \sum_{m=-l}^{l} Y_l^m (Ω_0 ) Y_{lm} (Ω) \end{equation}\]
\[\begin{equation} a_{lm}(\Omega_0)=Y_{lm}^*(\Omega_0) p(\Omega,t|\Omega_0,t_0)=\sum_{l=0}^\infty \sum_{m=-l}^l\mathrm{exp}\left(-\frac{l(l+1)}{\tau_R}\right)\,Y_{lm}^*(\Omega_0)\,Y_{lm}(\Omega) \end{equation}\]
for example projection on z-direction (NMR/diel.)
\[\begin{equation} P_3=P_0\cos(\theta),\,l=1 \langle P_3 \rangle=\frac{1}{4 \pi}\int \mathrm d\Omega\,P_0 \cos(\theta) \to 0 \end{equation}\]
?? correlation:
\[\begin{equation} \begin{split} \langle P_3(t)P_3^*(t_0) \rangle&=P_0^2\int\mathrm d\Omega\int\mathrm d\Omega_0\cos(\theta)\cos(\theta_0)\,p(\Omega,t|\Omega_0,t_0)\,p_0(\Omega_0)\\&=\frac{4 \pi}{3}P_0^2 \sum_{m=-l}^l\mathrm{exp}\left(-l(l+1)\frac{t-t_0}{\tau_r}\right)|C_{10lm}|^2 \end{split} \end{equation}\]
\[\begin{equation} C_{10lm}=\int \mathrm d\Omega\,Y_{10}^*(\Omega)\,Y_{lm}(\Omega)=\delta_{l1}\delta_{m0} \langle P_3(t)P_3^*(t_0)\rangle = \frac{4 \pi}{3}\,P_0^2\,\mathrm{exp}\left(-\frac{2(t-t_0)}{\tau_r}\right) \end{equation}\]
application in dielectric spectroscopy or NMR for example, \(A_lm→C_lm exp(-t/τ_l )\) rotational diffusion of Janus particles
\[\begin{equation} P_3=P_0\cos(\vartheta) \end{equation}\]
\(l=1\),Legendre projection on z-axis, NMR, DS
\[\begin{equation} \begin{aligned} \langle P_3(t)P_3^*(t_0)\rangle&=\frac{4 \pi}{3}\,P_0^2\,\mathrm{exp}\left(-\frac{2(t-t_0)}{\tau_r}\right) \tau_l=\frac{D_{1}}{D_\mathrm{rot}l(l+1)} n(\vartheta,\varphi,t)\\ &=\sum_{l=0}^\infty\sum_{m=-l}^lC_{lm}\,Y_l^m(\vartheta,\varphi)\,\mathrm{exp}\left(-\frac{t}{\tau_l}\right) \end{aligned} \end{equation}\]
quadratic angular displacement
\[\begin{equation} \langle |\hat u (t)-\hat u_0|^2\rangle = 2-2\langle\cos (\theta)\rangle \end{equation}\]
u ̂is a unit vector
\[\begin{equation} \frac{\mathrm d}{\mathrm dt}p(\theta,t)=D_r\nabla^2p(\theta,t) 2 \pi \int_0^\pi p(\theta, t)\sin(\theta)\,\mathrm d \theta=1 \langle \cos(\theta)\rangle=2 \pi \int_0^\pi p \sin(\theta)\cos(\theta)\,\mathrm d \theta \end{equation}\]
\[\begin{equation} \begin{split} \frac{\mathrm d}{\mathrm d t}\langle \cos(\theta)\rangle&=2 \pi \int_0^\pi\frac{\partial p}{\partial t} \sin(\theta)\cos(\theta)\,\mathrm d \theta \\ &=2 \pi D_r\int_0^\pi\frac{1}{\sin(\theta)}\left[\frac{\partial}{\partial t}\left(\sin(\theta)\frac{\partial p}{\partial \theta}\right)\right]\sin(\theta)\cos(\theta)\,\mathrm d \theta \end{split} \end{equation}\]