One consistent five-postulate formulation is the following; different textbooks group the same content differently.

The position-dependent probability density is $|\psi {|}^2$, and its flux is the probability current. These distinguish the wave amplitude from the measurable particle flow in the following scattering problem.

A c-axis wurtzite ZnO wire has crystallographically polar basal ends, conventionally Zn-terminated $(0001)$ and O-terminated $(000\bar{1})$, and approximately nonpolar prismatic side facets such as $\{10\bar{1}0\}$. Crystallographic polarity is not identical to net surface charge in a growth solution: protonation, adsorbates and ionic speciation can change the latter.

For face-selective electrostatic growth inhibition, an oppositely charged additive adsorbs preferentially to a particular face and obstructs incorporation there. Blocking the axial end lowers axial growth relative to radial growth and yields a smaller aspect ratio; blocking sidewalls suppresses radial growth and favours long thin wires.

A primary hydrothermal experiment at pH 11 found negatively charged basal top surfaces and positively charged sidewalls. Positively charged Cd-containing complexes inhibited the top and favoured platelets; negatively charged Al-containing complexes inhibited sidewalls and favoured large aspect ratios. These signs are condition-specific, not fixed by the formal Zn/O termination alone.^[ref]

Cooling changes the catalyst state that is being measured. Crossing Au–Ga phase boundaries can cause solidification, segregation and intermetallic formation. Solubility changes can expel Ga or precipitate additional GaAs after the intended growth interval. The droplet contact angle and interfacial shape also change as temperature and composition change.

A cooled catalyst–wire interface can consequently have a different composition, phase assemblage, geometry and even last-grown crystal segment from the active growth interface. Continued precursor supply during cooling can add another uncontrolled history. An ex situ observation of a solid catalyst does not prove that growth occurred by a solid-catalyst mechanism, and its final geometry cannot simply be substituted into a nucleation model for the liquid growth stage.

Use in situ microscopy, diffraction or spectroscopy under representative growth conditions, or compare controlled interruption and quench protocols. Quenching reduces some rearrangement but does not guarantee a perfectly frozen interface. The Au-catalysed GaAs composition experiment provides a direct example of measuring catalyst state during growth instead of inferring it from the cooled residue.^[ref]

For the common $[111]$-axis zinc-blende GaAs wire, a representative cross-section is a faceted hexagon with low-index sidewalls, often described by $\{110\}$-type facets. The threefold crystallographic symmetry of a polar $[111]$ direction can make alternate facets inequivalent, so triangular or truncated-triangular sections are also possible. The zinc-blende label alone does not fix a unique hexagon: growth direction, termination, twins, surface chemistry and kinetic facet selection also matter.

The accompanying sketch shows a representative faceted section rather than a circular cylinder. Facets that advance slowly in their normal direction tend to remain exposed, so a kinetic growth shape need not be an equilibrium Wulff shape.

A simple cause of tapering is continued radial vapor–solid growth on sidewalls while the tip grows axially. A lower wire segment has been exposed to precursors for longer than a segment formed near the end of growth. For a roughly constant radial growth speed $v_r$, its radius gains about $v_r[t_{\mathrm{end}}-t_{\mathrm{formation}}(z)]$, giving a larger base. Substrate-fed species, changing catalyst size and droplet wetting can also contribute. Tapering is not proof that the original VLS droplet was wider at every earlier stage.

Zinc blende and wurtzite differ in their stacking of tetrahedrally bonded bilayers: an ideal cubic sequence is ABCABC, while the hexagonal sequence is ABAB. During vapor–liquid–solid growth, each new interfacial nucleus can choose between continuing the existing sequence and changing it.

The selected phase can be the one with the smaller nucleation barrier, rather than the lower bulk free energy alone. A nucleus at the droplet–wire–vapour triple line has edge and interfacial energies different from a nucleus fully inside the liquid–solid interface. Schematically, its free energy combines a favourable chemical-potential term with new edge/interface costs and, when appropriate, a stacking penalty. Supersaturation changes the relative importance of these terms; the droplet shape and contact angle change which nucleation sites are available.

Control is therefore possible through temperature, III/V supply ratio, precursor fluxes, catalyst composition, droplet volume/contact angle and wire diameter. Monitoring growth rather than only inspecting cooled specimens helps determine which parameter actually changes the interface. A nucleation model for Au-catalysed III–V wires explicitly showed how triple-line nucleation can favour wurtzite under suitable interface energies and supersaturation.^[ref] There is no universal rule that one phase always corresponds to the larger temperature or smaller diameter regardless of catalyst and interface geometry.

An Au-containing seed absorbs gallium and forms a liquid Au–Ga alloy under the selected growth conditions. The supplied binary diagram indicates which temperature and Ga fraction lie in the liquid field; alloying allows a liquid catalyst at temperatures far below the melting point of pure Au. Vapour precursors supply Ga and As, which enter or react through the droplet and precipitate GaAs at the liquid–solid interface when the growth chemical potentials make incorporation favourable.

The droplet remains near the wire tip as interface nucleation and step propagation extend the nanowire. Its footprint helps select the wire diameter, though sidewall growth can change the final profile. The catalyst is a reservoir and reaction interface, not a stoichiometric constituent consumed once per GaAs unit.

Low As concentration makes the Au–Ga diagram a useful first approximation to catalyst state, but does not mean that arsenic supply or chemical potential is irrelevant. A binary phase diagram alone cannot calculate supersaturation with respect to ternary GaAs formation, and nanoscale interfacial energies can shift phase boundaries. In situ composition measurements have directly distinguished substantial Ga content from the very small As content of Au-seeded catalyst droplets.^[ref]

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.

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.

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.

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.

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.

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.