Past exam of the soft matter and biological physics course of the University of Cambridge / 2014 / Paper 1 / 2

Consider generic synthetic filaments and films. buckling occurs when compression makes a transverse deformation lower the energy. For a straight filament of bending rigidity $B=EI$ under compressive force $P$,Pinned ends admit $y=A\sin (\pi s/L)$, giving ${\delta }^{2}F=A^{2}L[B(\pi /L{)}^4-P(\pi /L{)}^2]/4$. The first instability is therefore $P_c={\pi }^{2}B/L^2$. True clamping raises this to $4{\pi }^{2}B/L^2$; a clamped-free rod has ${\pi }^{2}B/(4L^2)$. boundary conditions are part of the threshold, not a minor correction. Above threshold geometric nonlinearities and end constraints determine the buckled amplitude.

For buckling of a synthetic film, let the bending modulus be $\kappa$ and the isotropic in-plane tension be $\sigma$. The small-slope quadratic energy isA mode of wave number $q$ has stiffness $\kappa q^4+\sigma q^2$. tension stabilizes it; compression corresponds to $\sigma <0$ and makes modes with $q^2<|\sigma |/\kappa$ unstable. A finite sheet has a smallest allowed $q$, giving a finite threshold. A supporting foundation adds positive stiffness and can select a nonzero wrinkle wavelength.

Approaching either threshold softens a fluctuation mode: its thermal variance is proportional to $k_{B}T$ divided by modal stiffness. The apparent quadratic-theory divergence is rounded by nonlinear geometry, finite size and imperfections. Experimentally vary end shortening or load, measure the transverse profile, and control attachment stiffness and initial curvature. Load control and displacement control need not trace the same postbuckling response. Biological applications are outside this generic mechanics note.

single-molecule force spectroscopy measures force and extension of an individual tether while controlling position or load. In an atomic-force arrangement, a synthetic chain connects a cantilever probe to a translated substrate. For the cantilever compliance correction, a deflection sensor gives $d$, a calibrated spring constant gives $F=k_{c}d$, and substrate displacement $Z$ gives chain extension $x=Z-d$ after geometric offsets are removed. Additional linkers and instrument compliance must be included if appreciable. Thermal deflection calibration uses $k_c⟨d^2⟩=k_{B}T$ only after isolating the cantilever's calibrated thermal contribution.

An optical-tweezers implementation detects a tethered bead's displacement from a calibrated trap centre, with $F=k_{\mathrm{trap}}\Delta x$ in the harmonic regime. Detector calibration, drag and trap linearity determine the usable force and frequency range. These instruments measure individual force–extension curves and distributions; averaging many curves can hide multiple tether populations.

For a synthetic freely jointed chain with Kuhn length $b$,A strongly stretched worm-like chain instead has $1-x/L\simeq \sqrt{k_{B}T/(4F{\ell }_p)}$ before backbone stretching becomes important. Fits can estimate contour and persistence lengths only in their appropriate regimes. Hysteresis and pulling-rate dependence test whether an apparent mechanical transition is quasistatic or activated.

Control single versus multiple attachments, subtract nonspecific adhesion, vary pulling speed, calibrate temperature and stiffness, and correct drift. A rupture peak is not a universal bond strength independent of loading history. A primary nonbiological example measured individual polydimethylsiloxane chains by atomic-force methods; its flexibility interpretation combines those measurements with molecular calculations.^[ref]

TODO: user-deferred biological kinetic mechanism and its biochemical pattern-forming applications. The remainder treats generic two-field reaction-diffusion physics only.

Let $\mathbf{u}=(u,v)$ obey ${\partial }_t\mathbf{u}=\mathbf{f}(\mathbf{u})+\mathrm{diag}(D_u,D_v){\nabla }^2\mathbf{u}$. At a uniform steady state, write its reaction Jacobian as $J=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)$. Without diffusion stability requires $a+d<0$ and $\Delta =ad-bc>0$. A Fourier perturbation evolves under $J-q^2\mathrm{diag}(D_u,D_v)$, whose determinant isIts trace becomes more negative with increasing $q$. A stationary diffusion-driven instability therefore requireswith an allowed domain mode in the resulting negative-determinant band. Equal diffusivities cannot satisfy this while $a+d<0$. The physical mechanism is that transport can destabilize coupled fields even when the local kinetics relaxes a spatially uniform perturbation. Faster diffusion of an inhibitory response is one possible realization.

The selected wave numbers depend on growth rates and boundaries; nonlinear saturation determines the final stripes, spots or other states. Linear instability alone neither supplies a biochemical mechanism nor uniquely predicts a final finite-amplitude pattern.