Past exam of the quantum condensed matter course of the University of Cambridge / 2005 / Section A / A1

A nematic liquid crystal has long-range orientational order without a three-dimensional positional lattice. Its nematic director $\mathbf{n}$ identifies the preferred molecular axis, with $\mathbf{n}$ and $-\mathbf{n}$ equivalent for apolar molecules. The scalar nematic order parameter $S=⟨(3{\cos }^2\theta -1)/2⟩$ is zero for isotropic orientations and one for perfectly parallel axes. The tensor $Q_{ij}=⟨u_i{u}_j-{\delta }_{ij}/3⟩$ describes alignment without choosing a sign for the director.

Ordering balances orientational entropy against interactions and packing. For long rods, alignment can increase available translational configurations by reducing mutual excluded volume; thermotropic materials additionally have temperature-dependent energetic preferences. Spatially varying alignment costs Frank elastic energy through splay, twist and bend. Surface anchoring and electric or magnetic fields select the director in a finite specimen.

Crossed polarizers reveal birefringence and defect textures, while scattering tests positional order independently. A particularly direct response experiment is the Fréedericksz transition: confine a planar-aligned layer between anchored plates, apply a transverse field and monitor the transmitted polarization as the director tilts above an elastic threshold. This tests orientational elasticity and dielectric anisotropy rather than presuming that an optically anisotropic fluid has crystalline positional order.

A lipid bilayer self-assembles from amphiphiles: hydrophilic head groups face the surrounding water, while hydrocarbon tails form a hydrophobic interior. In a fluid membrane, molecules diffuse laterally within each membrane leaflet. Closing the bilayer into a vesicle avoids an exposed edge. The interior is a substantial barrier to isolated ions and many polar solutes; membrane proteins provide selective transport pathways in a cell membrane.

The Helfrich energy describes long-wavelength bending:Here $H=(c_1+c_2)/2$ and $K=c_1{c}_2$ are mean curvature and Gaussian curvature, $C_0$ is spontaneous curvature, ${\kappa }_b$ the bending modulus, $\bar{\kappa }$ the Gaussian-curvature coefficient and $\sigma$ the membrane tension. Bending is comparatively inexpensive, whereas stretching the molecular area strongly is costly. Leaflet asymmetry can favor a nonzero spontaneous curvature.

In micropipette aspiration, controlled suction draws part of a large vesicle into a narrow pipette. Pressure and geometry determine membrane tension, and the aspirated length measures apparent area change. At low tension, smoothing thermal undulations supplies much of that area; stronger tension probes actual stretching. Tracking vesicle shape fluctuations provides a complementary estimate of the bending modulus.

Here the complex viscoelastic modulus is a stress-to-strain response coefficient, rather than a complex number's absolute value. Use strain $\gamma (t)=\mathrm{Re}({\gamma }_0{e}^{i\omega t})$ and stress $\tau (t)=\mathrm{Re}[G^{\ast }(\omega ){\gamma }_0{e}^{i\omega t}]$, withThe storage modulus $G^{\prime }$ measures the elastic component; the loss modulus $G^{\prime \prime }$ measures dissipation. The mean power per volume is $\omega {\gamma }_0^2{G}^{\prime \prime }/2$, and the energy dissipated per cycle is $\pi {\gamma }_0^2{G}^{\prime \prime }$, fixing the sign convention.

A Hookean solid has $G^{\ast }=G_0$, while a Newtonian fluid has $G^{\ast }=i\omega \eta$. The Maxwell model interpolates with $G^{\ast }=G_{0}i\omega {\tau }_R/(1+i\omega {\tau }_R)$, giving $G^{\prime }=G_0(\omega {\tau }_R{)}^2/[1+(\omega {\tau }_R{)}^2]$ and $G^{\prime \prime }=G_0\omega {\tau }_R/[1+(\omega {\tau }_R{)}^2]$. It is fluid-like at low frequency and elastic on times short compared with relaxation time ${\tau }_R$.

A rheometer imposes small-amplitude oscillatory shear and resolves the torque amplitude and phase relative to the imposed motion. The torsional measurement of a complex shear modulus makes the conversion from torque and twist to $G^{\ast }$ explicit. A frequency sweep locates relaxation scales; keeping the strain small tests the linear-response assumption.

Wetting concerns how a liquid covers a solid in contact with another phase, such as its vapor. For an ideal flat homogeneous surface, balance of the three interfacial tensions gives Young's equationwhere the contact angle $\theta$ is measured through the liquid. Small angle means favorable contact; partial wetting gives a finite angle. The spreading coefficient $S={\gamma }_{SV}-{\gamma }_{SL}-{\gamma }_{LV}$ compares a covered surface with a dry one. Positive $S$ favors complete wetting, while negative $S$ permits a finite-angle droplet in the partial-wetting regime.

A sessile-drop experiment images the tangent at the contact line on a clean level substrate. Varying drop volume slowly distinguishes advancing and receding angles; their difference is contact-angle hysteresis from pinning and heterogeneity, so a single observed angle need not equal the ideal equilibrium value. A capillary-rise experiment gives a second consequence: the Young–Laplace equation and hydrostatic balance predict $h=2{\gamma }_{LV}\cos \theta /(\rho gr)$ in a narrow circular tube when meniscus-volume corrections are negligible. A wetting liquid rises, whereas a sufficiently nonwetting one is depressed.