%!TEX program = xelatex
% miko:title: Complete Soft Matter
% miko:summary: A concise guide to polymers, colloids, surfactants, liquid crystals, Brownian motion, elasticity, diffusion, rheology, and ionic soft matter.
% miko:category: softmatter
% miko:type: complete
% miko:origin: ai
% miko:tags: softmatter
% miko:difficulty: graduate
% miko:keywords: soft matter, polymers, colloids, liquid crystals, Brownian motion, rheology, surfactants

\documentclass[a4paper,11pt]{article}
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\title{Complete Soft Matter}
\author{Miko-pedia AI}
\date{\today}

\begin{document}
\maketitle
\tableofcontents

\section{What is Soft Matter?}
\subsection{Core ideas}
Soft matter includes polymers, colloids, foams, gels, membranes, liquid crystals, and biological materials. Energies are often comparable to $k_BT$, so entropy, fluctuations, interfaces, and slow collective dynamics dominate. Mesoscopic length scales make continuum and statistical descriptions meet.

For review, be able to identify soft-matter systems, compare thermal and mechanical energy scales, and explain why fluctuations and dissipation are central. Keep the physical question visible: identify the degrees of freedom, the conserved quantities, the approximation being made, and the observable that would be measured.

\subsection{Mathematical spine}
\[
k_BT\simeq4.1\,\mathrm{pN\,nm}\quad(300\,\mathrm K),\qquad \mathrm{Pe}=\frac{vL}{D}
\]

\subsection{Worked example}
A 1 \mum polystyrene bead in water at 300 K has thermal energy $k_BT\approx4.1\times10^{-21}\,\mathrm J$. The gravitational potential energy over 1 \mum is $mg\Delta z\sim(4\times10^{-15}\,\mathrm{kg})(10\,\mathrm{m/s^2})(10^{-6}\,\mathrm m)=4\times10^{-20}\,\mathrm J$, about 10 times $k_BT$. Thus thermal fluctuations are significant but do not completely overwhelm gravity for micron-scale colloids.

\subsection{Problems with Solutions}
\paragraph{Problem 1} Estimate the Péclet number for a 100 nm nanoparticle moving at $v=10\,\mu\mathrm{m/s}$ in water ($D\sim10^{-9}\,\mathrm{m^2/s}$).
\emph{Solution.} $\mathrm{Pe}=vL/D=(10\times10^{-6})(100\times10^{-9})/10^{-9}=1$. The particle is at the crossover between diffusion-dominated and advection-dominated transport.

\paragraph{Problem 2} At what length scale does thermal energy equal the bending energy of a lipid bilayer with bending rigidity $\kappa_b=20\,k_BT$?
\emph{Solution.} The bending energy for a patch of area $A$ and curvature radius $R$ is $E\sim\kappa_b A/R^2$. Setting $E\sim k_BT$ with $A\sim R^2$ gives $R\sim\sqrt{\kappa_b/k_BT}\,\xi$ where $\xi$ is the molecular size. More directly, the characteristic length is the bilayer thickness $\sim 5\,\mathrm{nm}$, where thermal and bending energies are comparable.

\paragraph{Section summary}
Soft materials are structured, fluctuating, and easily deformed.

\section{Soft Matter Solutions}
\subsection{Core ideas}
Solutions of polymers, surfactants, and colloids are controlled by concentration, solvent quality, excluded volume, osmotic pressure, and interactions. Dilute, semidilute, and concentrated regimes have different length scales and response.

For review, be able to use volume fraction, osmotic pressure, radius of gyration, and screening length to classify solution behavior. Keep the physical question visible: identify the degrees of freedom, the conserved quantities, the approximation being made, and the observable that would be measured.

\subsection{Mathematical spine}
\[
R_g\sim aN^\nu,\qquad \Pi\simeq ck_BT\quad(\mathrm{dilute})
\]

\subsection{Worked example}
A dilute solution of polystyrene in toluene has polymer concentration $c=10^{-3}\,\mathrm{g/cm^3}$. The monomer molecular weight is $M_0=104\,\mathrm{g/mol}$ and degree of polymerization $N=10^4$. The monomer concentration is $c_m=c/M_0\approx9.6\times10^{-6}\,\mathrm{mol/cm^3}=9.6\,\mathrm{mol/m^3}$. In the dilute limit the osmotic pressure is $\Pi\approx c_mRT\approx(9.6)(8.314)(300)\approx2.4\times10^4\,\mathrm{Pa}=0.24\,\mathrm{atm}$.

\subsection{Problems with Solutions}
\paragraph{Problem 1} A polymer has $N=10^5$ monomers of size $a=0.5\,\mathrm{nm}$. Estimate its radius of gyration in a good solvent ($\nu\approx0.588$).
\emph{Solution.} $R_g\sim aN^\nu=(0.5\,\mathrm{nm})(10^5)^{0.588}\approx0.5\times10^{2.94}\,\mathrm{nm}\approx0.5\times870\,\mathrm{nm}\approx435\,\mathrm{nm}$.

\paragraph{Problem 2} At what concentration does a polymer solution become semidilute if $R_g\approx50\,\mathrm{nm}$ and monomer volume is $v_m=a^3=(0.5\,\mathrm{nm})^3$?
\emph{Solution.} The overlap concentration is $c^*\sim N/R_g^3\sim10^5/(50\,\mathrm{nm})^3\sim10^5/(1.25\times10^5)\,\mathrm{nm^{-3}}\sim0.8\,\mathrm{nm^{-3}}$. In mass units this is roughly $c^*\sim0.01\,\mathrm{g/cm^3}$.

\paragraph{Section summary}
Soft-matter solutions are governed by entropy, interactions, and concentration.

\section{Elastic Soft Matter}
\subsection{Core ideas}
Gels, elastomers, membranes, and polymer networks store elastic energy while allowing large deformation. Entropic elasticity explains rubber-like behavior; bending and stretching energies control sheets, shells, and membranes.

For review, be able to distinguish entropic and energetic elasticity, write simple elastic free energies, and interpret moduli. Keep the physical question visible: identify the degrees of freedom, the conserved quantities, the approximation being made, and the observable that would be measured.

\subsection{Mathematical spine}
\[
F_{\rm el}\sim \frac12G\gamma^2V,\qquad \kappa_b\int(2H-C_0)^2\,dA
\]

\subsection{Worked example}
A rubber band with shear modulus $G=0.5\,\mathrm{MPa}$, cross-sectional area $A=1\,\mathrm{mm^2}$, and unstretched length $L_0=10\,\mathrm{cm}$ is stretched to $L=15\,\mathrm{cm}$ ($\gamma=0.5$). The elastic free energy is $F_{\rm el}\sim\frac12G\gamma^2V=\frac12(0.5\times10^6)(0.25)(10^{-7})\approx6.25\,\mathrm J$. The restoring force is $F\sim GA\gamma=(0.5\times10^6)(10^{-6})(0.5)=0.25\,\mathrm N$, about the force needed to hold a 25 g mass.

\subsection{Problems with Solutions}
\paragraph{Problem 1} A spherical vesicle of radius $R=10\,\mu\mathrm m$ has bending rigidity $\kappa_b=20\,k_BT$. What is the energy cost to deform it into an ellipsoid with semi-axes $a=12\,\mu\mathrm m$, $b=10\,\mu\mathrm m$, $c=8\,\mu\mathrm m$? (Assume area is conserved.)
\emph{Solution.} The mean curvature changes from $H_0=1/R=0.1\,\mu\mathrm{m^{-1}}$ to a value of order $H\sim0.1\,\mu\mathrm{m^{-1}}$ still. The bending energy is $E_b\sim8\pi\kappa_b\approx8\pi(20k_BT)\approx500\,k_BT\approx2\times10^{-18}\,\mathrm J$.

\paragraph{Problem 2} A gel cylinder of radius $r=5\,\mathrm{mm}$ and height $h=20\,\mathrm{mm}$ is compressed by 10%. If $G=10^4\,\mathrm{Pa}$, what force is required?
\emph{Solution.} The strain is $\gamma=0.1$. The stress is $\sigma=G\gamma=10^3\,\mathrm{Pa}$. The force is $F=\sigma A=(10^3)\pi(5\times10^{-3})^2\approx0.079\,\mathrm N$.

\paragraph{Section summary}
Soft elasticity often comes from entropy and geometry.

\section{Surfaces and Surfactants}
\subsection{Core ideas}
Interfaces cost free energy because molecules at surfaces have different environments. Surface tension drives droplets, capillary rise, wetting, and emulsions. Surfactants lower interfacial tension and self-assemble into micelles and membranes.

For review, be able to use Young-Laplace pressure, contact angles, capillary length, and critical micelle concentration. Keep the physical question visible: identify the degrees of freedom, the conserved quantities, the approximation being made, and the observable that would be measured.

\subsection{Mathematical spine}
\[
\Delta p=\gamma\left(\frac1{R_1}+\frac1{R_2}\right),\qquad \gamma_{SV}=\gamma_{SL}+\gamma_{LV}\cos\theta
\]

\subsection{Worked example}
Water at 20^{\circ}C has surface tension $\gamma=72.8\,\mathrm{mN/m}$. A water droplet of radius $R=1\,\mathrm{mm}$ has internal excess pressure $\Delta p=2\gamma/R=2(0.0728)/10^{-3}=145.6\,\mathrm{Pa}$. For a bubble in water (two surfaces), $\Delta p=4\gamma/R=291\,\mathrm{Pa}$. This Laplace pressure is small for millimeter drops but reaches $\sim1\,\mathrm{atm}$ for $R\sim0.3\,\mu\mathrm m$ droplets.

\subsection{Problems with Solutions}
\paragraph{Problem 1} A capillary tube of radius $r=0.5\,\mathrm{mm}$ is dipped in water ($\gamma=72.8\,\mathrm{mN/m}$, $\theta=0^{\circ}$). How high does the water rise?
\emph{Solution.} $h=2\gamma\cos\theta/(\rho gr)=2(0.0728)(1)/(1000\times9.8\times5\times10^{-4})=0.0297\,\mathrm m=2.97\,\mathrm{cm}$.

\paragraph{Problem 2} A spherical oil droplet of radius $R=2\,\mu\mathrm m$ has surface tension $\gamma=30\,\mathrm{mN/m}$. What is the energy released if two such droplets coalesce into one?
\emph{Solution.} Initial area $A_i=2\times4\pi R^2=32\pi\,\mu\mathrm{m^2}$. Final radius $R_f=2^{1/3}R\approx2.52\,\mu\mathrm m$, final area $A_f=4\pi R_f^2\approx79.5\,\mu\mathrm{m^2}$. Wait: $V_f=2V_i$ so $R_f=2^{1/3}R$. $A_f=4\pi(2^{1/3}R)^2=2^{2/3}A_i\approx1.587\times(2\times4\pi R^2)\approx1.587\times32\pi$. Actually $A_f=4\pi(2^{1/3}\times2\,\mu\mathrm m)^2=4\pi\times5.04\,\mu\mathrm{m^2}\approx63.3\,\mu\mathrm{m^2}$. Initial area was $2\times4\pi(2)^2=100.5\,\mu\mathrm{m^2}$. The change is $\Delta A=-37.2\,\mu\mathrm{m^2}$. Energy released is $\gamma|\Delta A|\approx30\times10^{-3}\times37.2\times10^{-12}\approx1.1\times10^{-12}\,\mathrm J$.

\paragraph{Section summary}
Interfacial energy shapes soft materials.

\section{Liquid Crystals}
\subsection{Core ideas}
Liquid crystals have orientational order without full crystalline order. The director field describes average molecular alignment; elastic distortions, defects, and coupling to fields produce rich optical and mechanical behavior.

For review, be able to define nematic order, use the director, identify splay/twist/bend distortions, and understand birefringence. Keep the physical question visible: identify the degrees of freedom, the conserved quantities, the approximation being made, and the observable that would be measured.

\subsection{Mathematical spine}
\[
Q_{ij}=S\left(n_in_j-\frac13\delta_{ij}\right),\qquad F=\frac12\int(K_1(\nabla\cdot\bm n)^2+K_2(\bm n\cdot\nabla\times\bm n)^2+K_3|\bm n\times\nabla\times\bm n|^2)dV
\]

\subsection{Worked example}
A nematic liquid crystal in a 1 cm$^3$ cell has splay elastic constant $K_1=6\,\mathrm{pN}$. If the director rotates by $\Delta\theta=10^{\circ}$ over a distance $d=10\,\mu\mathrm m$, the splay energy density is $\frac12K_1(\nabla\cdot\bm n)^2\sim\frac12K_1(\Delta\theta/d)^2\approx\frac12(6\times10^{-12})(0.174/10^{-5})^2\approx9\times10^{-4}\,\mathrm{J/m^3}$. The total energy is $E\approx9\times10^{-4}\times10^{-6}=9\times10^{-10}\,\mathrm J$, about $2\times10^8\,k_BT$.

\subsection{Problems with Solutions}
\paragraph{Problem 1} A twisted nematic cell has pitch $p=20\,\mu\mathrm m$ and $K_2=4\,\mathrm{pN}$. What is the twist energy per unit area for a cell of thickness $d=10\,\mu\mathrm m$?
\emph{Solution.} The twist angle changes by $\pi$ over distance $d$ (half pitch). Energy per area $f=\frac12K_2(\partial_z\theta)^2d=\frac12K_2(\pi/d)^2d=\frac12K_2\pi^2/d=\frac12(4\times10^{-12})\pi^2/(10^{-5})\approx2\times10^{-6}\,\mathrm{J/m^2}$.

\paragraph{Problem 2} Estimate the Frank elastic energy of a $+1$ disclination in a nematic with $K\sim5\,\mathrm{pN}$ and core radius $r_c=5\,\mathrm{nm}$, observed in a film of radius $R=10\,\mu\mathrm m$.
\emph{Solution.} $E\sim\pi K\ln(R/r_c)\approx\pi(5\times10^{-12})\ln(2000)\approx\pi(5\times10^{-12})(7.6)\approx1.2\times10^{-10}\,\mathrm J$.

\paragraph{Section summary}
Liquid crystals combine fluidity with orientational order.

\section{Brownian Motion and Thermal Fluctuations}
\subsection{Core ideas}
Brownian motion is random motion driven by thermal collisions and opposed by viscous drag. The same microscopic fluctuations that cause diffusion also determine dissipation through fluctuation-dissipation relations.

For review, be able to derive mean-square displacement, use Stokes-Einstein, and connect Langevin and diffusion descriptions. Keep the physical question visible: identify the degrees of freedom, the conserved quantities, the approximation being made, and the observable that would be measured.

\subsection{Mathematical spine}
\[
\langle\Delta r^2(t)\rangle=2dDt,\qquad D=\frac{k_BT}{6\pi\eta a}
\]

\subsection{Worked example}
A 100 nm polystyrene bead in water at 300 K ($\eta=10^{-3}\,\mathrm{Pa\,s}$) has diffusion coefficient $D=k_BT/(6\pi\eta a)=(4.1\times10^{-21})/(6\pi\times10^{-3}\times50\times10^{-9})\approx4.4\times10^{-12}\,\mathrm{m^2/s}$. In 1 s, the root-mean-square displacement is $\sqrt{\langle\Delta r^2\rangle}=\sqrt{6Dt}=\sqrt{6(4.4\times10^{-12})}\approx5.1\,\mu\mathrm m$. In 1 ms this drops to $0.16\,\mu\mathrm m$.

\subsection{Problems with Solutions}
\paragraph{Problem 1} A protein of radius $a=3\,\mathrm{nm}$ diffuses in cytoplasm ($\eta\approx2\times10^{-3}\,\mathrm{Pa\,s}$). What is its diffusion coefficient at 310 K?
\emph{Solution.} $D=k_BT/(6\pi\eta a)=(1.38\times10^{-23}\times310)/(6\pi\times2\times10^{-3}\times3\times10^{-9})\approx3.8\times10^{-11}\,\mathrm{m^2/s}$.

\paragraph{Problem 2} How long does it take for a 1 \mum bead to diffuse across a 100 \mum cell?
\emph{Solution.} $\tau\sim L^2/(6D)$. With $D\sim4.4\times10^{-13}\,\mathrm{m^2/s}$ for $a=0.5\,\mu\mathrm m$, $\tau\sim(10^{-4})^2/(6\times4.4\times10^{-13})\sim3.8\times10^3\,\mathrm s\approx1\,\mathrm{hour}$.

\paragraph{Section summary}
Thermal noise is measurable motion at soft-matter scales.

\section{Variational Principle in Soft Matter Dynamics}
\subsection{Core ideas}
Many soft-matter dynamics follow from minimizing free energy while dissipating energy through viscosity or friction. Gradient flows, Onsager's principle, and hydrodynamic couplings provide systematic equations for slow variables.

For review, be able to write a free energy functional, compute functional derivatives, and connect thermodynamic forces to dissipative fluxes. Keep the physical question visible: identify the degrees of freedom, the conserved quantities, the approximation being made, and the observable that would be measured.

\subsection{Mathematical spine}
\[
\partial_t\phi=\nabla\cdot M\nabla\frac{\delta F}{\delta\phi},\qquad \dot F\le0
\]

\subsection{Worked example}
Consider phase separation described by the Cahn-Hilliard free energy $F=\int[\frac12(\nabla\phi)^2+\frac14(\phi^2-1)^2]dV$. The chemical potential is $\mu=\delta F/\delta\phi=\phi^3-\phi-\nabla^2\phi$. For a flat interface at $x=0$ with boundary conditions $\phi(\pm\infty)=\pm1$, the equilibrium profile satisfies $\partial_x^2\phi=\phi^3-\phi$, giving $\phi(x)=\tanh(x/\sqrt{2})$. The interfacial width is $\xi=\sqrt{2}$ in these units, or about 1 nm--1 \mum in physical systems.

\subsection{Problems with Solutions}
\paragraph{Problem 1} Show that the Cahn-Hilliard dynamics $\partial_t\phi=\nabla^2\mu$ conserves total "mass" $\int\phi\,dV$.
\emph{Solution.} $\frac{d}{dt}\int\phi\,dV=\int\partial_t\phi\,dV=\int\nabla^2\mu\,dV=\oint\nabla\mu\cdot d\bm A=0$ for periodic or no-flux boundaries.

\paragraph{Problem 2} For the Ginzburg-Landau free energy $F=\int[\frac12K(\nabla\phi)^2+\frac12r\phi^2+\frac14u\phi^4]dV$ with $r<0$, find the two homogeneous minima.
\emph{Solution.} Setting $\delta F/\delta\phi=0$ gives $r\phi+u\phi^3=0$, so $\phi=0$ or $\phi=\pm\sqrt{-r/u}$. The nonzero solutions are the stable minima when $r<0$.

\paragraph{Section summary}
Soft dynamics often relax down free-energy gradients.

\section{Diffusion and Permeation in Soft Matter}
\subsection{Core ideas}
Transport through gels, membranes, porous media, and polymer networks combines diffusion, partitioning, binding, and hydrodynamic resistance. Permeability depends on both mobility and solubility.

For review, be able to use Fick's laws, diffusion times, permeability, and boundary partitioning. Keep the physical question visible: identify the degrees of freedom, the conserved quantities, the approximation being made, and the observable that would be measured.

\subsection{Mathematical spine}
\[
\bm J=-D\nabla c,\qquad \partial_tc=D\nabla^2c,\qquad t_D\sim L^2/D
\]

\subsection{Worked example}
Oxygen diffuses through a hydrogel membrane of thickness $L=100\,\mu\mathrm m$ with $D=10^{-9}\,\mathrm{m^2/s}$. The characteristic diffusion time is $t_D=L^2/D=(10^{-4})^2/10^{-9}=10\,\mathrm s$. The permeability is $P=KD/L$ where $K$ is the partition coefficient. If $K=0.3$ (gels exclude some solute), $P=0.3(10^{-9})/10^{-4}=3\times10^{-6}\,\mathrm{m/s}$.

\subsection{Problems with Solutions}
\paragraph{Problem 1} A drug patch releases molecules through skin of thickness $L=0.5\,\mathrm{mm}$ with $D=10^{-10}\,\mathrm{m^2/s}$. How long does it take for concentration to reach 90% of steady state at the far side?
\emph{Solution.} For diffusion across a slab, the time to reach fraction $f$ is $t\sim-\ln(1-f)L^2/(\pi^2D)$. For $f=0.9$, $t\sim2.3(2.5\times10^{-7})/(9.87\times10^{-10})\approx580\,\mathrm s\approx10\,\mathrm{min}$.

\paragraph{Problem 2} A spherical cell of radius $R=10\,\mu\mathrm m$ has glucose consumption rate $q=10^{-3}\,\mathrm{mol/(m^3\,s)}$. If external concentration is $c_0=5\,\mathrm{mM}$ and $D=10^{-9}\,\mathrm{m^2/s}$, what is the central concentration?
\emph{Solution.} Solving $\nabla^2c=-q/D$ with spherical symmetry gives $c(r)=c_0+q(R^2-r^2)/(6D)$. At $r=0$, $c(0)=5\times10^{-3}+10^{-3}(10^{-5})^2/(6\times10^{-9})=5\times10^{-3}+1.67\times10^{-5}\approx5.02\,\mathrm{mM}$. Diffusion is fast enough to keep the cell nearly uniform.

\paragraph{Section summary}
Diffusion sets the timescale for soft-matter transport.

\section{Rheology and Flow}
\subsection{Core ideas}
Rheology studies how materials deform and flow. Newtonian fluids, viscoelastic fluids, gels, yield-stress materials, and shear-thinning suspensions are distinguished by stress-strain-rate relations and memory.

For review, be able to read flow curves, use Maxwell and Kelvin-Voigt models, define viscosity and moduli, and interpret Deborah number. Keep the physical question visible: identify the degrees of freedom, the conserved quantities, the approximation being made, and the observable that would be measured.

\subsection{Mathematical spine}
\[
\sigma=\eta\dot\gamma,\qquad \tau\dot\sigma+\sigma=\eta\dot\gamma,\qquad G^*(\omega)=G'(\omega)+iG''(\omega)
\]

\subsection{Worked example}
A Maxwell fluid has viscosity $\eta=1\,\mathrm{Pa\,s}$ and relaxation time $\tau=0.1\,\mathrm s$. At shear rate $\dot\gamma=10\,\mathrm{s^{-1}}$, the steady-state shear stress is $\sigma=\eta\dot\gamma=10\,\mathrm{Pa}$. In oscillatory shear at $\omega=1\,\mathrm{rad/s}$, the storage modulus is $G'(\omega)=\eta\tau\omega^2/(1+\tau^2\omega^2)=1(0.1)(1)/(1+0.01)=0.099\,\mathrm{Pa}$ and the loss modulus is $G''(\omega)=\eta\omega/(1+\tau^2\omega^2)=1/(1.01)=0.99\,\mathrm{Pa}$. The fluid is loss-dominated at this frequency.

\subsection{Problems with Solutions}
\paragraph{Problem 1} A Kelvin-Voigt material has $G=10^4\,\mathrm{Pa}$ and $\eta=10^3\,\mathrm{Pa\,s}$. What is its creep compliance after $t=1\,\mathrm s$ under constant stress $\sigma_0=10^3\,\mathrm{Pa}$?
\emph{Solution.} $J(t)=\frac{1}{G}(1-e^{-t/\tau_r})$ where $\tau_r=\eta/G=0.1\,\mathrm s$. At $t=1\,\mathrm s$, $J(1)=10^{-4}(1-e^{-10})\approx10^{-4}\,\mathrm{Pa^{-1}}$. The strain is $\gamma=J\sigma_0=0.1$.

\paragraph{Problem 2} At what frequency does a Maxwell fluid with $\tau=1\,\mathrm{ms}$ have $G'=G''$?
\emph{Solution.} Setting $G'=G''$ gives $\tau\omega=1$, so $\omega=1/\tau=10^3\,\mathrm{rad/s}$, or $f\approx160\,\mathrm{Hz}$.

\paragraph{Section summary}
Rheology connects microscopic relaxation to macroscopic flow.

\section{Ionic Soft Matter}
\subsection{Core ideas}
Charged soft matter includes polyelectrolytes, electrolytes, charged colloids, membranes, and biological macromolecules. Electrostatic interactions are screened by mobile ions; double layers and osmotic effects control stability and forces.

For review, be able to use Debye length, Poisson-Boltzmann ideas, electrophoresis qualitatively, and charge regulation concepts. Keep the physical question visible: identify the degrees of freedom, the conserved quantities, the approximation being made, and the observable that would be measured.

\subsection{Mathematical spine}
\[
\lambda_D=\left(\frac{\epsilon k_BT}{\sum_i n_iq_i^2}\right)^{1/2},\qquad \nabla^2\phi=-\rho(\phi)/\epsilon
\]

\subsection{Worked example}
A 1:1 electrolyte (NaCl) in water at 300 K with concentration $c=100\,\mathrm{mM}=10^{26}\,\mathrm{m^{-3}}$ has Debye length
\[
\lambda_D=\sqrt{\frac{\epsilon k_BT}{2nc^2}}=\sqrt{\frac{(80\times8.85\times10^{-12})(4.1\times10^{-21})}{2(10^{26})(1.6\times10^{-19})^2}}\approx\sqrt{\frac{2.9\times10^{-30}}{5.1\times10^{-12}}}\approx0.96\,\mathrm{nm}.
\]
The electrostatic interaction between two charged surfaces is screened beyond this distance.

\subsection{Problems with Solutions}
\paragraph{Problem 1} A charged colloid of radius $a=50\,\mathrm{nm}$ has surface potential $\psi_0=50\,\mathrm{mV}$ in 10 mM NaCl. Estimate the surface charge density using the Debye-Hückel approximation.
\emph{Solution.} In the Debye-Hückel limit, $\sigma=\epsilon\psi_0(1+\kappa a)/(\kappa a^2)$ where $\kappa=1/\lambda_D$. For 10 mM, $\lambda_D\approx3\,\mathrm{nm}$ and $\kappa a\approx16.7$. Thus $\sigma\approx(80\times8.85\times10^{-12})(0.05)(17.7)/(16.7\times(5\times10^{-8})^2)\approx0.015\,\mathrm{C/m^2}$, corresponding to about 1 charge per 10 nm$^2$.

\paragraph{Problem 2} By what factor does the Debye length change when the salt concentration increases from 1 mM to 100 mM?
\emph{Solution.} Since $\lambda_D\propto c^{-1/2}$, the ratio is $\lambda_{D,100}/\lambda_{D,1}=\sqrt{1/100}=0.1$. The Debye length shrinks by a factor of 10.

\paragraph{Section summary}
Mobile ions screen and reshape electrostatic interactions in soft matter.

\end{document}
