Past exam of the soft condensed matter course of the University of Cambridge / 2012

Original paper: Public original, physical PDF pages 228–231.

Use the squared three-dimensional endpoint displacements, not their arithmetic mean. The mean squared displacement estimate and corresponding diffusion coefficient areThe factor six is for three-dimensional Brownian motion. For an isolated no-slip sphere, the Stokes–Einstein relation gives the Brownian radius estimate from endpoint displacements:Here room temperature is taken as $298\mathrm{K}$; choosing $293\mathrm{K}$ gives $55\mathrm{nm}$, a much smaller difference than the sampling uncertainty. Under independent increments with a normal distribution, $\mathrm{sd}(\hat{D})/D=\sqrt{2/(3n)}\simeq 0.37$ for $n=5$. The sampling uncertainty of an endpoint diffusivity estimate therefore makes this a rough radius estimate. It assumes negligible drift and hydrodynamic coupling between probes, the supplied bulk viscosity, and actual three-dimensional tracking. Localization noise adds to the squared displacements and would bias this inferred radius downward if uncorrected.

For the three-dimensional worm-like chain, define the bending coefficient ${\kappa }_b$ by $F_b=({\kappa }_b/2)\int |{\partial }_s\mathbf{t}{|}^{2}ds$. Its persistence length is $l_p={\kappa }_b/(k_{B}T)$. Holding that coefficient and the polymer contour length $L$ fixed givesThe temperatures must be absolute. The end-to-end distance of a worm-like chain obeysFor a long flexible chain, $L\gg l_p$, the typical size is $R_{\mathrm{rms}}\simeq \sqrt{2L{l}_p}$, so warming contracts the chain, with $R_{\mathrm{rms}}(T_2)/R_{\mathrm{rms}}(T_1)\simeq \sqrt{T_1/T_2}$. The finite-length formula also decreases as $l_p$ decreases, but the rod limit $L\ll l_p$ changes very little. A large segment count alone does not guarantee $L\gg l_p$. If the material's bending coefficient itself depends on temperature, replace the first ratio by $[{\kappa }_b(T_2)/{\kappa }_b(T_1)]T_1/T_2$; the sign then needs that additional material information. This calculation concerns a generic physical filament, with no biomolecular application.

For monovalent mobile ions, the linearized Debye–Hückel screening length isHere $n_i$ is a number density in ${\mathrm{m}}^{-3}$, not a molarity. Use the question's idealized neutral-water convention, $c_{+}=c_{-}=10^{-7}\mathrm{mol}{\mathrm{L}}^{-1}$, and $T=293.15\mathrm{K}$. Both ion species contribute:For added salt concentration $c_s$ in $\mathrm{mol}{\mathrm{L}}^{-1}$, the ionic strength in this model is $I=c_s+10^{-7}\mathrm{mol}{\mathrm{L}}^{-1}$. With the numerical molarity $I_M=I/(1\mathrm{mol}{\mathrm{L}}^{-1})$,The water contribution is negligible at that salt level. The factor $10^3$ converts molarity into $\mathrm{mol}{\mathrm{m}}^{-3}$; omitting it or counting only one ion species gives a wrong screening length. Exact water autoionization and activities vary with temperature, so the first estimate uses the simplified neutral $10^{-7}\mathrm{M}$ concentrations implied by the question. At the higher salt level, ion-size effects, nonideal activities and changes in permittivity limit the numerical accuracy of this continuum estimate.

A synthetic fluid membrane is a thin, laterally fluid sheet whose components can rearrange in its plane while the sheet resists changes of area and curvature. Synthetic amphiphiles can assemble into two opposed layers with solvent-compatible groups facing outward. Their fluid in-plane response gives no equilibrium static shear modulus, but does not imply zero membrane bending rigidity or zero area-compression modulus. A thin membrane can therefore bend easily while remaining difficult to stretch. These notes cover generic synthetic interfaces.

TODO: biological, biofluid and biomolecular membrane applications are user-deferred.

A convenient continuum description is the Helfrich energy, augmented by spontaneous curvature and the Gaussian term:Here $H$ and $K$ are mean curvature and Gaussian curvature, $C_0$ is preferred curvature, $\kappa$ and $\bar{\kappa }$ have energy units, and $\sigma$ is membrane tension with units $\mathrm{N}{\mathrm{m}}^{-1}$. Leaflet asymmetry can favor nonzero $C_0$. At fixed topology the integrated Gaussian term is constant for a closed surface by the Gauss-Bonnet theorem. A symmetric tensionless spherical sheet has mean-curvature bending energy $8\pi \kappa$, independent of its radius in this ideal model. Edge energy can favor closure, while fixed area and enclosed volume constrain the resulting shapes.

For a nearly flat sheet, its small-slope elastic membrane energy becomesWith $h_{\mathbf{q}}=A^{-1}\int h(\mathbf{x})e^{-i\mathbf{q}\cdot \mathbf{x}}{d}^{2}x$, the equipartition theorem gives the membrane fluctuation spectrum under tension:Bending suppresses short wavelengths, whereas positive tension suppresses the longest undulations. Their crossover is $q\sim \sqrt{\sigma /\kappa }$. Measuring the spectrum can separate these two contributions, provided finite image resolution, projected versus true area, and external confinement are accounted for. The drawn modes below are synthetic calculations, not an image of a sample.

For a clean liquid interface at fixed temperature and composition, surface tension is the reversible free-energy cost per added area, $\gamma =(\partial F/\partial A{)}_{T,V,N}$, so $dF=\gamma dA$. Its units are both $\mathrm{J}{\mathrm{m}}^{-2}$ and $\mathrm{N}{\mathrm{m}}^{-1}$. Molecules at an interface have a different environment from those in the bulk; creating more interface changes their interactions and entropy. For constant $\gamma$, the interfacial free energy is $\gamma A$ up to a reference constant. This is not generally identical to the internal surface energy: the interfacial entropy density is $-d\gamma /dT$, giving internal-energy density $\gamma -Td\gamma /dT$.

A line of length $\ell$ in one interface transmits a tangential force $\gamma \ell$, perpendicular to the line. A soap film has two interfaces, giving force $2\gamma \ell$ on a sliding rim. Quasistatic extension by $dx$ creates total area $2\ell dx$, recovering that force from $dF=2\gamma \ell dx$. Uniform positive surface tension favors minimum area at fixed volume, hence an isolated spherical droplet when gravity is negligible. curvature gives the Young–Laplace equation, with outward normal chosen consistently:A spherical liquid drop has $\Delta p=2\gamma /R$; a thin soap bubble with two similar interfaces has approximately $4\gamma /R$. The capillary length $\sqrt{\gamma /(\Delta \rho g)}$ compares gravity with capillarity. surface tension also restores capillary waves, rather than providing a bulk static shear modulus.

Interfacial contamination and added surfactants can change $\gamma$ substantially; a nonuniform value produces tangential Marangoni stress. A rim-force measurement should therefore control temperature, cleanliness and the number of interfaces. Capillary-rise methods additionally need the contact angle, because the vertical component is proportional to $\gamma \cos \theta$. An independent frame-force measurement can distinguish a contact-angle change from a true change of surface tension.

A moving or externally forced colloid disturbs the surrounding fluid, changing the velocity of other colloids even without direct contact. These hydrodynamic interactions are long-ranged at small Reynolds number. In an unbounded incompressible Newtonian liquid, the Stokes flow response to a point force is the Stokeslet:Its Oseen tensor isThis is a mobility tensor, with units velocity per force, rather than a stress tensor. At separation $r$ the response parallel to the line of centres is $1/(4\pi \eta r)$; the perpendicular response is $1/(8\pi \eta r)$. Flow is not isotropic even though the solvent is. The disturbance decays as $1/r$, so treating all drag forces as independent can miss correlated motion.

For well-separated equal spheres of radius $a$, the leading hydrodynamic mobility matrix givesThe first term is the isolated-sphere self mobility; the remaining terms couple distinct particles. At thermal equilibrium the fluctuation-dissipation theorem also predicts correlated short-time increments, $⟨\Delta r_{i\alpha }\Delta r_{j\beta }⟩\simeq 2k_{B}T{\mu }_{i\alpha ,j\beta }\Delta t$ with the configuration locally fixed. Thus correlated Brownian displacements can reveal hydrodynamic coupling without an applied force.

The point-force approximation requires $r\gg a$ and negligible inertia. It diverges at the origin and cannot replace the finite self mobility. Near-contact lubrication, walls, many-body corrections, and a viscoelastic solvent require more complete mobility laws. A force-free particle can instead have a faster-decaying leading disturbance, so the $1/r$ claim specifically concerns a net-force-driven object.

The three solvent conditions describe effective interactions between segments of a generic polymer. In a good solvent, solvent contacts favor a swollen, self-avoiding coil with positive excluded volume. At a theta solvent condition, leading pair repulsion and attraction cancel; the chain has approximately ideal random-walk statistics. In a poor solvent, effective attraction contracts the chain, eventually producing a dense polymer globule for a sufficiently long chain.

For $N$ statistical segments of length $a$, the radius of gyration has the scalingThe good-solvent power $3/5$ is the Flory approximation; the asymptotic three-dimensional exponent is closer to $0.588$. In the ideal freely jointed chain, the more precise result is $R_g^2\simeq N{a}^2/6$, whereas the end-to-end mean square is $N{a}^2$. Scaling notation suppresses those geometry-dependent factors. The globule power follows from a volume proportional to $N$, and presumes a dense collapsed state away from the theta crossover. The dilute condition excludes overlap with other chains; it does not eliminate contacts between distant segments of the same chain.

Use polymer volume fraction $\varphi$ horizontally and the Flory interaction parameter $\chi$ vertically. Lower $\chi$ gives better solvent quality. The simplest Flory–Huggins solution theory assigns the dimensionless mixing free energy per lattice siteThus $\chi <1/2$ gives positive excluded volume and a good-solvent coil, $\chi =1/2$ is the long-chain theta condition, and $\chi >1/2$ is the poor-solvent side. It is the net two-segment interaction that vanishes at theta, not every molecular force.

At fixed good-solvent quality, the polymer overlap volume fraction ${\varphi }^{\ast }\sim N{a}^3/R_g^3$ separates isolated coils from a semidilute polymer solution. Above overlap, many chains interpenetrate, but there is still solvent on the segment scale. As $\varphi$ approaches order unity the solution is concentrated; these concentration-regime boundaries are crossovers, not first-order phase transitions. The overlap fraction increases from order $N^{-4/5}$ in a strong good solvent toward order $N^{-1/2}$ at theta. The crossover can be expressed as ${\varphi }^{\ast }\sim N^{-4/5}(1-2\chi {)}^{-3/5}$ while $1-2\chi \gg N^{-1/2}$; it saturates at the theta scaling when that condition fails.

On the poor-solvent side there can instead be coexistence of polymer-poor and polymer-rich phases. The binodal follows a common tangent of $f$; the spinodal followsThe critical point of a Flory–Huggins polymer solution isFor finite $N$, ${\chi }_c$ lies above $1/2$. Therefore poor solvent does not automatically mean demixing at every concentration, and an overlap line must not be mistaken for a coexistence boundary. The original figure uses the stated ideal model for an illustrative chain length; it does not assert a measured phase boundary. A vertical temperature axis would additionally require a material-specific relation $\chi (T)$.

Overlap means roughly one coil per coil volume. With segment volume $a^3$, the corresponding polymer overlap volume fraction isFor a strong good solvent, $R_g\sim a{N}^{3/5}$, and henceor about 0.063% by volume in the unit-prefactor scaling convention. The segment length cancels. A coil has no sharply defined boundary, so using $4\pi R_g^3/3$ or retaining a size amplitude changes an order-one prefactor. This calculation is a scaling estimate, not an exact volume-fraction measurement.

The correlation length in a semidilute solution is the concentration blob size. Below it a chain retains isolated good-solvent statistics; beyond it concentration fluctuations are screened and many chains interpenetrate. It is a local scale, not the total chain radius. If a blob contains $g$ segments,Eliminating $g$ gives the good-solvent concentration-blob scalingUsing the supplied segment size and concentration,Here $g\simeq 316$ and $N/g\simeq 32$, so there are many segments within each blob and several dozen blobs per chain. Also $\varphi /{\varphi }^{\ast }\simeq 16$, consistent with a semidilute estimate rather than isolated coils. The cancellation of $N$ from $\xi$ is physical: once chains are long enough, the local concentration sets the mesh scale. As throughout the scaling argument, numerical amplitudes are not determined by the given data.

In a dilute solution, separate coils diffuse through solvent with long-ranged hydrodynamic coupling among their segments. The Zimm model gives whole-chain diffusionwhere $R_H$ is a hydrodynamic radius and ${\eta }_s$ the solvent viscosity. The coil continually changes shape while its centre of mass performs long-time Brownian motion; treating it as a rigid sphere is only a scaling analogy.

At high concentration, solvent-mediated hydrodynamic interactions are screened on sufficiently large scales and times. If chains are not entangled, additive local or blob friction gives Rouse model behaviour, $D\propto N^{-1}$ at fixed concentration. For long entangled chains, neighboring chains prevent rapid transverse escape. In the reptation tube picture the chain moves mainly along its contour until it renews the surrounding tube. Contour diffusion scales as $N^{-1}$, the tube-renewal time as $N^3$, and the random-coil size squared as $N$. ThereforeBoth cases ultimately give diffusive centre-of-mass motion, but entangled concentrated chains move much more slowly and retain constraints for longer. The inequality $\varphi \gg {\varphi }^{\ast }$ establishes strong overlap, not by itself entanglement: the latter also requires sufficiently many segments relative to an entanglement length. No numerical entanglement scale or concentrated-solution diffusion coefficient is supplied, so only the conditional regimes and scaling exponents can be fixed here.

In linear viscoelasticity, stress is a linear functional of past strain increments: scaling the deformation scales the response, and responses to small deformations add. For a time-translation-invariant material, Boltzmann superposition writes $\sigma (t)={\int }_{-\infty }^{t}G(t-s)\dot{ϵ}(s)ds$, where $G$ is a relaxation modulus. The material combines reversible storage with delayed or dissipative response; the description is restricted to small deformations about the chosen state.

The Maxwell model places an elastic spring and viscous dashpot in series. They carry the same shear stress $\sigma$, and their shear strains add:Differentiating givesHere $ϵ$ is dimensionless shear strain, $\sigma$ shear stress in pascals, $G_0$ the spring shear modulus, $\eta$ dashpot viscosity in $\mathrm{Pa}\mathrm{s}$, and $\tau$ the relaxation time. The first term describes changing elastic strain; the second is viscous flow. In this model the material flows at long times and has zero static shear modulus.

A relaxation modulus is measured by imposing a step of strain and then holding that strain fixed. Let its amplitude be ${ϵ}_0$, so the initial stress jump is ${\sigma }_0=G_0{ϵ}_0$. For $t>0$, $\dot{ϵ}=0$ and the Maxwell model givesThe initial stress in the wording must be understood as that produced by the initial deformation. If an external apparatus instead maintains $\sigma ={\sigma }_0$, the stress does not decay: the experiment is creep, considered below. Merely specifying an initial stress without a subsequent constraint would not uniquely prescribe the relaxation experiment.

Choose complex amplitudes proportional to $e^{i\omega t}$ and define $\tilde{\sigma }=G^{\ast }(\omega )\widetilde{ϵ}$. The Maxwell model givesRationalizing the denominator yields the storage modulus and loss modulus:For $\omega \tau \ll 1$, $G^{\prime \prime }\simeq \eta \omega$ dominates and $G^{\prime }\simeq G_0{\omega }^2{\tau }^2$ is small: the material has time to flow. For $\omega \tau \gg 1$, $G^{\prime }\to G_0$ and $G^{\prime \prime }\simeq G_0/(\omega \tau )$: the dashpot scarcely deforms during a cycle and the response is mainly elastic. At $\omega \tau =1$, $G^{\prime }=G^{\prime \prime }=G_0/2$, and $G^{\prime \prime }$ is maximal. The stress leads strain by $\delta =\arctan (G^{\prime \prime }/G^{\prime })$, ranging from nearly $\pi /2$ to nearly zero.

For a fixed strain amplitude ${ϵ}_a$, dissipation per cycle is $\pi G^{\prime \prime }{ϵ}_a^2$. A peak in $G^{\prime \prime }$ therefore concerns that normalization. At fixed stress amplitude ${\sigma }_a$, it is instead $\pi {\sigma }_a^2/(\eta \omega )$, so it has no finite-frequency maximum in this ideal model. Reversing the Fourier time convention reverses the imaginary-part sign, while the physical dissipated energy remains positive.

A creep experiment applies a small step of stress and holds it constant while recording strain. For $\sigma ={\sigma }_{0}H(t)$, the series spring jumps by ${\sigma }_0/G_0$ and the dashpot subsequently flows at constant rate ${\sigma }_0/\eta$. The creep compliance is thusUsing the stipulated effective stress and strain substitutions givesOn linear axes the model is a straight line of positive intercept $A=k_{B}T/(\pi a{G}_0)$ and slope $A/\tau$. On logarithmic axes it has an approximate elastic plateau for $t\ll \tau$ and slope one for $t\gg \tau$, within the time window where the continuum approximation applies. The long-time slope is $6D$ with $D=k_{B}T/(6\pi \eta a)$, agreeing with the Stokes–Einstein relation.

There is an essential zero-lag convention. A literal lag displacement $\Delta \mathbf{r}(t)=\mathbf{r}(t_0+t)-\mathbf{r}(t_0)$ has $⟨|\Delta \mathbf{r}(0){|}^2⟩=0$. The nonzero intercept above is the spring-compliance extrapolation of the idealized microrheology of a Maxwell fluid, after unresolved solvent or inertial transients. It must not be interpreted as an instantaneous finite physical displacement at exactly zero lag. The stated mesh condition motivates a continuum probe response; quantitative use is more reliable for $a\gg m$ than merely $a>m$.

For a thermally elastic network, estimate one elastically active element per mesh volume. Each contributes an elastic free-energy scale of order $k_{B}T$, giving the entropic network modulusat $T=298\mathrm{K}$. Mesh size alone would not determine the modulus of an arbitrary bending-dominated or athermal network; this is the entropic scaling assumption needed for the estimate.

The formal elastic intercept in the given model is consequentlyThe corresponding three-dimensional root-mean-square displacement is about $0.20\mu \mathrm{m}$. The temperature cancels from this estimate because the assumed modulus is itself proportional to $T$. This is the extrapolated plateau from the preceding part, not a nonzero exact zero-lag displacement. Here $m/a=1/2$, so a continuum estimate is only approximate.

In passive microrheology, track dilute inert probe spheres by video microscopy and average squared lag displacements over starting times and probes. An isotropic two-dimensional image gives $⟨\Delta r^2{⟩}_{3d}=(3/2)⟨\Delta r^2{⟩}_{2d}$. Alternatively, dynamic light scattering probes ensemble displacement statistics; the original optical microrheology study compared inferred storage and loss moduli with mechanical measurements.^[ref]

Remove drift, calibrate spatial scale and frame timing, estimate localization noise using an immobilized-bead control, and account for motion blur. Keep probes away from walls and check that changing probe radius or tracer concentration does not change the inferred bulk response unexpectedly. Noise alone can generate a false short-time intercept, while adsorption or confinement can mimic elasticity. No numerical long-time slope is fixed because a numerical viscosity is not supplied.