Past exam of the MPhil in micro- and nanotechnology enterprise course of the University of Cambridge / 2022 / Paper 2 / Section A(iii)

TODO — user-deferred biology. This part belongs to the explicitly biological molecular-system question and is preserved without a solution under the requested nonbiology scope.

TODO — user-deferred biology. This part belongs to the explicitly biological molecular-system question and is preserved without a solution under the requested nonbiology scope.

TODO — user-deferred biology. This part belongs to the explicitly biological molecular-system question and is preserved without a solution under the requested nonbiology scope.

Symmetric wetting means that both film boundaries prefer the same block: A at both, or B at both. Asymmetric wetting means opposite preferences, such as A at the substrate and B at the free surface. A neutral boundary has approximately equal interfacial free energies against A and B and therefore imposes no first-order preference between them.

These terms compare the substrate–polymer and atmosphere–polymer interfaces, not the volume fractions or molecular weights of the two blocks. A film can have one neutral boundary and one selective boundary; saying only that its top surface is neutral does not make both interfaces neutral. Surface preference competes with chain stretching and the energetic cost of A/B contacts in selecting a block copolymer morphology.

Symmetric wetting and an A-preferring substrate imply an A-preferring free surface as well. Parallel lamellae satisfy both preferences. For equal-width A and B domains, a thickness $2L_0$ contains two repeats, with terminal A half-domains:The sequence is written from substrate to atmosphere. Its total thickness is $2L_0$, the total A and B thicknesses are each $L_0$, and both interfaces contact A. A diagram using five full half-period domains would instead give $2.5L_0$ and would not match the stated thickness.

The quarter-period end domains place an A-domain centre at each boundary in this ideal symmetric-lamella model. Real surface reconstruction and non-equal block fractions can change their detailed widths without changing the key requirement of identical terminations.

An asymmetric film with a B-preferring substrate must have an A-preferring atmosphere interface. A parallel stack with half-integer thickness $2.5L_0$ can terminate correctly at both surfaces. For equal block widths, one explicit sequence from bottom to top isThe thicknesses add to $2.5L_0$, with $1.25L_0$ of each block. B contacts the substrate and A contacts the atmosphere, as shown in the accompanying schematic. The alternating widths in the interior reflect half-period domains, while the terminal domains are half of an interior domain in this idealized construction.

With identical A-selective boundaries, stress-free parallel stacks favour approximately integer multiples of $L_0$. An arbitrary imposed thickness generally cannot preserve both the bulk period and those terminations everywhere.

A small mismatch can be accommodated by compressing or stretching the lamellae. A larger mismatch can favour terraces, islands or holes with locally commensurate heights, separated by steps of roughly $L_0$. For example, a mean height $\overline{h}=2.35L_0$ can be assembled from regions of heights $2L_0$ and $3L_0$, with area fraction $f=0.35$ at the larger height:The cartoon shows this volume-conserving alternative. Which response wins depends on elastic stretching energy, surface energies, lateral constraints, thickness mismatch and equilibration time. If the substrate and processing permit it, dewetting or a different domain orientation can also occur; an arbitrary thickness does not prove that a single specific terrace pattern must form.

Neutralizing the top boundary removes its preference for A, but the substrate still prefers A. The original integer-period constraint from two A-selective interfaces is therefore relaxed. The film can adjust its top termination without paying the same surface penalty, and thickness frustration can be reduced.

A possible morphology is an A-rich wetting layer near the substrate with lamellae turning towards a more perpendicular arrangement in the upper film, as schematically illustrated. A predominantly parallel stack is also possible if the substrate anchoring and thickness commensurability dominate. Neutrality of the top alone does not uniquely determine the orientation.

If both boundaries were neutral, perpendicular lamellae could expose comparable A and B areas without a selective-surface penalty and would avoid the same parallel-stack thickness constraint. That is a different boundary condition from the one given here. A correct sketch must retain the A preference at the bottom rather than silently replacing it with a second neutral interface.

Positive dielectrophoresis acts up the gradient of the squared RMS field, towards stronger-field regions; negative dielectrophoresis acts down that gradient, towards weaker-field regions. The sign is the sign of $\Re \tilde{K}(\omega )$.

The force is not generally tangent to an electric field line. Field lines show the direction of the field itself, whereas $\nabla |{\mathbf{E}}_{\mathrm{RMS}}{|}^2$ describes how its strength changes in space. A spatially uniform field gives zero dipole-order dielectrophoretic force even when it is large. Reversing the field each half-cycle does not reverse this time-averaged force, because the relevant intensity-like quantity is quadratic.

At fixed material response, frequency and field-strength gradient,Thus the force is proportional to particle volume. Doubling the radius multiplies the force by eight, not two. This scaling assumes that the particle is small enough for the dipole approximation and does not substantially distort the prescribed field distribution. The mechanical drift speed need not scale with volume, because the hydrodynamic drag also changes with particle size.

For ordinary lossless dielectrics with positive real permittivities,The denominator is positive, soThe particle is more polarizable than the displaced medium and is driven towards the region where its polarization lowers the field-plus-source energy most effectively. If the two permittivities match, this ideal dipole contrast vanishes.

Under the same positive, real-permittivity assumptions,The particle is less polarizable than the surrounding medium and moves towards weaker-field regions. The sign describes the polarizability contrast, not a negative net electric charge. Conducting or lossy media require the real part of the full complex Clausius-Mossotti factor; a comparison of only the real dielectric constants is then insufficient.

The narrow electrode tip concentrates field lines and produces the strong-field region. For positive dielectrophoresis, both particles move towards that region; for negative dielectrophoresis, both move away. Particle 1 is approximately on the central symmetry line, so its arrow is approximately horizontal. Particle 2 lies above it, so a qualitative gradient arrow points towards or away from the high-field tip rather than simply following its local field line.

The source drawing makes particle 2 approximately twice the radius of particle 1. With the explicitly stipulated equal gradient magnitudes and identical material,Accordingly, the second force arrow should be much longer in either sign case. The exact ratio should remain expressed through the drawn radius ratio because the figure supplies no calibrated particle dimensions. Exact off-axis force directions would require solving the electrode boundary-value problem; the drawing supports a qualitative strong-field/weak-field distinction.

Different materials have different frequency-dependent polarization and conduction. For an $e^{i\omega t}$ convention, a useful relative complex permittivity is ${\widetilde{ϵ}}_r={ϵ}_r-i\sigma /({ϵ}_0\omega )$. Substituting the particle and medium responses into $\tilde{K}$ determines whether each population undergoes positive or negative dielectrophoresis and with what magnitude.

Choose a frequency at which one population has $\Re \tilde{K}>0$ and another has $\Re \tilde{K}<0$. A nonuniform electrode field then sends them towards different regions, allowing collection in separate channels or trapping one while the other passes. Alternatively, different positive-force magnitudes can produce different deflections in a flowing suspension. A crossover where $\Re \tilde{K}=0$ can selectively suppress the force on one material.

The useful frequency depends on the surrounding medium as well as the particle. An unchanging, real-permittivity model would not by itself give frequency-selective separation; dielectric dispersion or conductivity contrast is needed. Particle size affects force magnitude, and heating, electro-osmotic flow and surface polarization can complicate a real separation.