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The midpoint potential $E_m$ characterizes the redox couple under specified solution conditions: it is the potential at which oxidized and reduced forms have equal activities, or equal concentrations in the corresponding formal-potential convention. For a reversible cyclic voltammogram it is estimated by $(E_{pa}+E_{pc})/2$. It approximates the formal redox potential when the usual reversibility and diffusion assumptions apply; it is not the potential halfway along a scan chosen arbitrarily.

Make heterogeneous electron transfer fast relative to delivery of reactant, usually by applying sufficient overpotential so that its surface concentration approaches zero. Use excess supporting electrolyte to suppress migration. For purely diffusional transient transport, suppress uncontrolled convection; for a rotating disk electrode, controlled convection supplies the bulk while diffusion across the thin interfacial boundary layer limits the current. A limiting current that no longer increases with driving potential is the characteristic test; a large current alone is not sufficient.

All three marked points are at strongly oxidizing potentials, where X is depleted at the electrode: $c_X(0)\simeq 0$. Assuming only X initially in bulk, $c_{X^{+}}(\infty )=0$ and $c_X(\infty )=c_b$. Oxidation produces X${}^{+}$ at the interface.

For equal diffusion coefficients, the two concentration profiles are complementary, $c_X+c_{X^{+}}=c_b$. More generally equality of their interfacial molar fluxes requires $D_X{c}_X^{\prime }(0)=-D_{X^{+}}{c}_{X^{+}}^{\prime }(0)$, so the product surface concentration need not equal $c_b$. A and B are approximately steady hydrodynamic profiles; C is a time-dependent diffusion profile.

A one-dimensional probability density has units $L^{-1}$, while $D$ has units $L^2{T}^{-1}$. With no other length scale, dimensional analysis requiresNormalization gives ${\int }_{-\infty }^{\infty }F(\xi )d\xi =1$ and unbiased diffusion gives an even $F$. Dimensional arguments alone do not specify $F$.

For ordinary Fickian diffusion, impose ${\partial }_{t}n=D{\partial }_x^{2}n$ with $n(x,0)=\delta (x)$. Substitution of the similarity form yieldsA decaying symmetric solution obeys $F^{\prime }=-\xi F/2$, so $F=C{e}^{-{\xi }^2/4}$. Normalization fixes $C=1/\sqrt{4\pi }$:This heat kernel integrates to one, has variance $2Dt$, and tends to the initial Dirac delta distribution as $t\downarrow 0$.

The conventional lattice-defect description is vacancy-mediated diffusion: an ion moves into a neighbouring vacancy and leaves a vacancy behind. sodium chloride is another familiar example. In a real alpha-alumina specimen the dominant species and transport path depend on oxygen chemical potential, dopants and grain boundaries, so this label should not be read as a universal microscopic assignment for every aluminium and oxygen diffusion experiment. A vacancy mechanism is different from an interstitial ion moving between otherwise empty interstitial sites.

Taking the imaginary part of the complex potential gives the streamfunctionThe corresponding velocity field isPut $\zeta =z{e}^{i\alpha /2}=\xi +i\eta$. Then $w=\beta {\zeta }^2$ and $\psi =2\beta \xi \eta$. Thus the streamlines are the previous rectangular hyperbolas rotated clockwise by $\alpha /2$. Their separatrices have angles $\theta =-\alpha /2+n\pi /2$; the extension axis is $\theta =-\alpha /2$ and the compression axis is perpendicular to it. The middle panel of the figure shows this rotation.

For pure advection, a marked fluid element obeysConsequently the dye circle becomes an ellipse with semiaxes $R{e}^{2\beta t}$ and $R{e}^{-2\beta t}$. Its area isStretching in one principal direction exactly compensates compression in the other. Equivalently the flow-map Jacobian determinant is one, as required by incompressibility. Negligible diffusion is needed to regard the sharp dye boundary as a material curve.

Write $\eta =H{e}^{\sigma t}{J}_0(\alpha r)$ using the Bessel function of the first kind. The wall boundary conditions at $z=\pm h$ selectBecause $J_0^{\prime }(s)=-J_1(s)$, the sidewall no-penetration boundary condition requireswhere $j_{1,n}$ are the positive zeros of $J_1$. These radial eigenfunctions automatically preserve layer volume becauseThe constant radial mode $\alpha =0$ cannot change the interface height in a closed tank: the Laplace equation and the impermeable lids then give zero vertical velocity, and a uniform displacement would change each layer's volume.

The kinematic boundary condition fixesThus the interface values are ${\varphi }_1(0)=\sigma \eta \coth (\alpha h)/\alpha$ and ${\varphi }_2(0)=-\sigma \eta \coth (\alpha h)/\alpha$. Substitution in the normal stress condition gives the finite-depth Rayleigh-Taylor dispersion relation:soHere $\mathcal{A}$ is the Atwood number. Every allowed mode has a positive growth rate and is subject to Rayleigh-Taylor instability. Since $d[\alpha \tanh (\alpha h)]/d\alpha =\tanh (\alpha h)+\alpha h{\mathrm{sech}}^2(\alpha h)>0$, shorter radial scales grow faster. The limits areThere is no fastest finite wavenumber in the ideal sharp-interface model without surface tension, viscosity or diffusion. The zero-initial-velocity condition uses both exponential branches: an initial mode $H{J}_0({\alpha }_{n}r)$ evolves as $H\cosh ({\sigma }_{n,+}t)J_0({\alpha }_{n}r)$. If the mass density ordering were reversed the same calculation would give oscillatory stable modes; equal mass densities give neutral modes.

For an inviscid flow, the force per unit volume is $\mathbf{F}=-\nabla p-\rho g\hat{\mathbf{z}}$. The gravitational acceleration has zero curl. Using the curl product rule,The supplied two-dimensional vorticity equation therefore becomesThe out-of-plane vorticity has no stretching term in this two-dimensional setting. The source is baroclinic vorticity generation: differently directed mass density and pressure gradients generate rotation even from zero initial vorticity.

For an explicit sign check, set ${\omega }_y={\partial }_{z}u-{\partial }_{x}w$, with $z$ upward. A thin smoothed interface has $\rho \simeq \bar{\rho }(z-\eta (x,t))$, with ${\bar{\rho }}^{\prime }>0$. If $P(x,t)$ denotes the total pressure evaluated on the interface, then within the thin layer $p_x+{\eta }_x{p}_z\simeq P_x$ andIntegrating through that layer and taking its thickness to zero gives the vortex sheet strength $\gamma =u_2-u_1$ and its linear evolution:For a growing Fourier mode, the potentials from the preceding calculation with $\alpha$ replaced by $k$ giveThus the right-hand descending slope of a light-fluid crest produces positive ${\omega }_y$, while the left-hand ascending slope produces negative ${\omega }_y$. Their induced velocity raises the light-fluid crest and lowers the heavy-fluid trough, increasing the slope and generating more vorticity. This is the positive feedback underlying Rayleigh-Taylor instability.

One can close this mechanism quantitatively. For $\eta =H(t)\cos (kx)$, the kinematic boundary conditions give $\gamma =2\dot{H}\coth (kh)\sin (kx)$. Combining with the sheet-generation equation givesFor $kh\ll 1$, the lids confine motion throughout both layers and horizontal redistribution is slow, yielding $\sigma \sim \sqrt{\mathcal{A}gh}k$. For $kh\gg 1$, motion is localized within depth $k^{-1}$ on each side. Less fluid must accelerate at shorter wavelength, giving $\sigma \sim \sqrt{\mathcal{A}gk}$. Since $k_n=n\pi /b$, the ideal growth rate increases with $n$ with no short-wave cutoff.

The sharp-interface result requires $k\delta \ll 1$ for the smoothed layer. When $k\delta$ becomes order one, it no longer applies. A smooth adverse mass density gradient has a local overturning time of order $[g{\bar{\rho }}^{\prime }/\bar{\rho }{]}^{-1/2}$ rather than the unbounded sharp-interface rate. Finite molecular diffusion also relaxes small-scale mass density perturbations on a time of order $(D{k}^2{)}^{-1}$, and viscosity, if included, damps small-scale velocity. Weak diffusion motivates the thin-layer picture; it does not justify extrapolating the discontinuous dispersion relation to arbitrary $n$.

The bacteria disperse by diffusion with coefficient $D_B$. Their additional chemotaxis flux isIt points up the nutrient concentration gradient, and its negative spatial derivative enters the balance equation. The positive $N_0$ prevents singular sensitivity at zero nutrient concentration; increasing the background nutrient concentration reduces the drift response to a fixed gradient. The logistic growth equation term $rB(1-B/K)$ represents net reproduction with low-population growth rate $r$ and carrying capacity $K$.

The nutrient undergoes diffusion with coefficient $D_N$. Its production term $a{B}^2/(B_0^2+B^2)$ is an increasing, bounded response to the bacteria, reaching half its maximum $a$ at $B=B_0$. It has the algebraic form of a Hill equation with exponent two; the equation alone does not specify a molecular mechanism. The term $dBN$ removes nutrient at a rate proportional to both bacteria abundance and nutrient concentration.

The full fluxes areThe homogeneous Neumann boundary conditions make both fluxes vanish at the endpoints, so neither bacteria nor nutrient crosses the boundaries. The initial conditions specify their starting spatial distributions. This model includes nutrient production by bacteria, rather than a separate external supply term.

Write $b=1+u$, $n=n_{\ast }+v$ and defineThe first-order perturbation equations areThe homogeneous Neumann boundary conditions select Fourier modes proportional to $\cos (m\pi x)$, with squared wavenumber $q_m=(m\pi {)}^2$ for $m=0,1,\dots$. For a mode $e^{\sigma t}(U,V)\cos (m\pi x)$, the eigenvalue problem isIts trace is $-[1+\mu +(d_b+d_n)q]<0$. Therefore a growing mode exists exactly when its determinant is negative. SetThe dispersion relation isThe constant mode has $P(0)=\mu >0$; very short wavelengths also decay because $P(q)\sim d_b{d}_n{q}^2$. For a negative interval at positive $q$, the upward-opening quadratic polynomial must have its minimum below zero at positive $q$. This givesIn particular, $\boxed{b_0>1}$ is necessary, since otherwise $\eta \le 0$ and every coefficient of $P(q)$ is positive. When $b_0>1$, a locally increased population must increase net nutrient production, so attraction to that nutrient reinforces the population increase. This feedback must overcome both diffusion and local relaxation. The calculation is an instance of chemotactic instability with logistic growth and saturating production.

The displayed continuous-wavenumber inequality is necessary on the finite domain, but an allowed mode must actually lie in the unstable band:Equivalently, the exact discrete threshold isEquality gives a marginal zero eigenvalue rather than positive growth. At the unrestricted onset, $q_c=\sqrt{\mu /(d_b{d}_n)}$, so $k_c=(\mu /(d_b{d}_n){)}^{1/4}$. On the finite interval, Discrete Neumann mode selection first destabilizes the mode minimizing the discrete threshold; it need not attain $k_c$ exactly.

To predict the early form of a perturbation, use the larger growth eigenvalueFor $\eta >0$, the square root is real. Choose $m_{\mathrm{fast}}$ maximizing ${\sigma }_{+}(q_m)$ among allowed modes with positive growth. The corresponding perturbation has wavelength $2/m_{\mathrm{fast}}$ in $x$, or $2L/m_{\mathrm{fast}}$ in $X$. Its eigenvector satisfiesso bacteria and nutrient peaks initially coincide. A mode with larger $m$ has finer spatial structure. The two signs of $\cos (m\pi x)$ are both allowed, exchanging peaks and troughs.

The illustration holds $d_b=0.002$, $d_n=0.04$, $\mu =0.5$, $b_0=2$ and $\lambda =2.5$ fixed, giving $n_{\ast }=1$ and $\eta =0.3$, and varies ${\chi }_b$. Crossing the discrete threshold makes a nonuniform Fourier mode grow; stronger chemotaxis widens the unstable band and can favor finer structure. Increasing either diffusion coefficient raises the threshold of each fixed mode. Varying $\lambda$ can have a nonmonotone effect: with $s=1+b_0^2$,is maximal at $\lambda =\mu s$, where $n_{\ast }=1$. Additional production beyond that point weakens sensitivity enough to reduce this feedback. Changes in $L$ also change the allowed dimensional wavenumbers $m\pi /L$ and all three scaled transport coefficients. These are predictions of linear stability analysis: eventual pattern amplitude, competition between growing modes and possible merging require the nonlinear system. The threshold is a necessary condition for spontaneous pattern growth from arbitrarily small perturbations, rather than a claim that every finite-amplitude nonuniform state is excluded below it.

Bicoid and Nanos are opposed maternal inputs with different principal molecular outputs: Bicoid directly helps activate anterior zygotic transcription, whereas Nanos prevents inappropriate posterior translation of maternal hunchback messenger RNA. They cooperate with, rather than replace, other maternal and zygotic regulators. It is misleading to describe them as two otherwise equivalent transcription-factor gradients.

The first shared feature is spatial control of a maternally produced transcript. During oogenesis, bicoid messenger RNA is supplied from the maternal germline and concentrated at the anterior of the oocyte through cis-acting localization information and transport/anchoring machinery. Its translation after egg activation establishes an anterior source of Bicoid protein. Specific antibody detection showed nuclear accumulation and an approximately declining anterior-to-posterior distribution, with mutant embryos providing specificity controls.^[ref] Nuclear localization is functionally relevant because Bicoid is a homeodomain DNA-binding protein, acting on enhancers in the syncytial blastoderm before cellularization.

Nanos is also maternally supplied, but its effective source is posterior. Oskar-organized germ plasm provides a posterior localization and translational environment for nanos messenger RNA. RNA retained elsewhere is normally translationally repressed, preventing a broadly distributed transcript population from producing an inappropriate embryo-wide Nanos signal. Posterior localization should not be reduced to directed transport of every molecule: live imaging of endogenous nanos messenger RNA demonstrated diffusion and entrapment as an important mechanism.^[ref] Thus transcript localization, local translation and repression outside the source together determine where Nanos protein is produced. Bicoid and Nanos gradients are protein/activity distributions, not claims that the entire maternal RNA pool is itself a matching exponential gradient.

Their main readouts occur at different regulatory levels. Bicoid binds enhancer DNA and contributes concentration-dependent activation of anterior targets, including zygotic hunchback gene expression. Different target responses depend on binding-site organization, other transcription factors, chromatin accessibility and time available for transcription. A nucleus does not merely read one Bicoid concentration in isolation: maternal Hunchback, gap-gene interactions and temporal integration sharpen or modify its response. Bicoid also represses translation of caudal messenger RNA in the anterior by recognizing RNA regulatory information, helping generate an opposing Caudal protein distribution from broadly available maternal RNA. Reporter and RNA-binding experiments establish this second, post-transcriptional role of a homeodomain protein.^[ref]

Nanos is a zinc finger RNA-binding protein rather than a sequence-specific transcriptional activator of posterior gap genes. In cooperation with Pumilio and other repression machinery, it acts on Nanos response elements in the Hunchback 3′ untranslated region, reducing maternal Hunchback production posteriorly. Repression can involve inhibition of translation and regulated RNA stability; measuring protein alone would not distinguish their contributions. Response-element mutation and transfer into reporters test whether regulation depends on a particular RNA sequence rather than on posterior position alone. Such experiments showed that the quality and number of response elements influence Nanos-dependent regulation.^[ref]

The distinction between maternal and zygotic Hunchback explains how the two inputs fit together. Maternal hunchback messenger RNA is broadly present, but posterior Nanos prevents it from giving broadly high Hunchback protein. Anteriorly, where Nanos is low, maternal Hunchback remains available and Bicoid activates additional zygotic hunchback gene transcription. Posteriorly, suppression of maternal Hunchback removes an inhibitory influence on abdominal gap-gene programmes, while Caudal and other inputs support posterior patterning. Bicoid's repression of anterior Caudal is complementary: an anterior transcriptional activator also helps keep a posterior-promoting protein out of the wrong region. These interactions build distinct regulatory environments at the two ends, rather than assigning each end with one independent positive instruction.

An idealized concentration model makes the difference between a gradient and its readout explicit. With a localized Bicoid source, uniform diffusion coefficient $D$ and first-order loss $k$, a one-dimensional steady-state approximation away from the source givesA hypothetical target responding at threshold $B_{\ast }$ has boundary $x_{\ast }=\lambda \log (B_0/B_{\ast })$. If the source amplitude doubles without changing the other parameters, the boundary shifts by $\lambda \log 2$. This is a model prediction, not a fitted measurement from the exam. Source-distribution changes, nuclear trapping, protein lifetime, non-steady dynamics and enhancer regulation can invalidate its simplest assumptions. The expression makes clear why source dosage can move a boundary without changing the identity of the target enhancer. In the classic tests, genetic changes in Bicoid source amount or distribution altered the protein gradient and displaced anterior fate-map positions, supporting an instructive concentration-dependent role.^[ref]

Nanos positional activity can instead be read through the fraction of Hunchback transcripts repressed at a given position. Its effect depends on protein concentration, RNA-element affinity and abundance of its partners. Calling it a morphogen is compatible with an indirect transcriptional outcome: its graded activity is translated first into an opposing Hunchback distribution, which then influences zygotic transcription. That does not imply that Nanos supplies the same number of independently interpreted concentration thresholds as Bicoid, or that it alone specifies every abdominal identity.

Both inputs are necessary in the normal regulatory context because they correct different problems. Without maternal Bicoid, normal anterior zygotic activation and anterior repression of Caudal fail, causing major anterior-pattern defects. Without Nanos, maternal Hunchback persists posteriorly and interferes with abdominal development despite the presence of Bicoid anteriorly. The decisive genetic test is that reducing maternal Hunchback can suppress the posterior-pattern defect of Nanos mutants: this places much of Nanos's abdominal function in removal of an inhibitor rather than direct activation of every posterior target.^[ref] Such suppression does not restore every Nanos-dependent function, particularly its separate roles in germ-cell development.

A further discriminating design would compare localized wild-type RNAs with localization-defective but equally expressed versions; use separate enhancer and 3′-UTR reporters; quantify target RNA and protein simultaneously; and test target-site mutants. Ectopic posterior Bicoid should impose inappropriate anterior-like regulatory activity where competent targets remain, while ectopic anterior Nanos should repress its susceptible maternal targets rather than operate as a conventional DNA-binding anterior repressor. Appropriate controls include RNA abundance, protein stability, maternal genotype and developmental stage. Their complementarity is therefore an anterior transcriptional programme reinforced by selective translational repression at both ends, with Nanos additionally protecting a posterior environment in which abdominal patterning can proceed.

TODO: Practical weapon-fabrication and optimization instructions are not provided. The general metallurgy and diffusion analysis are retained in the preceding parts; this manufacturing subpart remains explicitly limited.

Assume one temperature-independent activation energy over this interval and an unchanged dopant-controlled carrier population. Write the Arrhenius equation as $D=D_0\exp [-Q/(k_{B}T)]$. The two measurements giveUsing $D(T)=10^{-13}\exp [(Q/k_B)(1/1000-1/T)]$ yieldsThe Nernst-Einstein relation includes the factor $1/T$: $\sigma =n{q}^{2}D/(k_{B}T)$. For fixed $n$ and a fixed correlation correction,Using conductivity proportional to $D$ alone would miss the temperature factor. Carrier association, a changing vacancy population or a changing diffusion correlation factor would invalidate this simple extrapolation.

At equilibrium at room temperature the phases are pure Ag and pure Cu, with mass fractions $0.80$ and $0.20$. A representative solidification structure contains primary Ag-rich grains or dendrites surrounded by a finer two-phase eutectic aggregate. Subsequent equilibrium solute rejection modifies both constituents as they cool.

The $42.1\%$ primary-$\alpha$ and $57.9\%$ eutectic-constituent fractions calculated at the eutectic arrest describe how the liquid froze. They are not the final pure-Ag and pure-Cu phase fractions. For example, just below the arrest the total copper-rich phase fraction is $(20-9)/(92-9)=11/83=0.1325$; it rises to $0.20$ as solubility is lost on further equilibrium cooling. The distinction separates a microstructural constituent from a thermodynamic phase.

Faster solidification usually refines the dendrite and eutectic spacings. If solid-state diffusion cannot keep up, composition gradients, supersaturated solids and delayed precipitation can persist. Their exact fractions then depend on the thermal history; faster cooling does not change the alloy's overall $20\%$ copper inventory. Complete long-term equilibration would restore the same equilibrium phase compositions and mass fractions.

Material creep is time-dependent permanent deformation under sustained stress. Its rate is often appreciable at a large homologous temperature $T/T_m$, because diffusion and obstacle-escape processes become rapid. In an activated regime a useful form iswith additional elastic, geometric and temperature prefactors absorbed into $A$ for the chosen approximation. The activation energy belongs to the rate-controlling process: bulk diffusion, grain-boundary diffusion and dislocation motion need not have the same value.

Boundary sliding can accommodate grain-shape changes during diffusional material creep; it is not a reason to identify either diffusion field as dislocation glide. The supplied map's A/B/C placement, including the boundary between B and C, determines these assignments. Its numerical contours apply to the material state used to construct that map, not to arbitrary grain sizes.

Lower the operating temperature or stress, suppress diffusion, and remove or stabilize the paths that accommodate material creep. Increasing the melting temperature can reduce $T/T_m$ at a specified operating temperature. Chemistry that reduces diffusivity or impedes dislocation climb can reduce the material creep rate; stable precipitates or dispersoids can resist glide and climb, provided they do not dissolve or coarsen rapidly in service.

Large grains reduce both Nabarro-Herring and Coble creep, according to their $d^{-2}$ and $d^{-3}$ dependencies. Directional solidification can reduce boundaries normal to the dominant tensile stress, while a single crystal removes grain-boundary transport and sliding from the ordinary polycrystal mechanism inventory. It does not remove bulk diffusion or dislocation material creep.

This is why room-temperature grain-refinement strengthening is not automatically a high-temperature material creep strategy: fine grains create more boundary transport paths. Any protective microstructure must remain stable at the service temperature, and the design must also avoid stress concentrations and unwanted thermal loading.

At fixed transformed volume, an equilibrium solid-state precipitate seeks a low total free energy, including the orientation-dependent interfacial energy and elastic strain energy associated with misfit and applied stress. Low-energy interface orientations occupy large faces. The Wulff construction captures surface-energy anisotropy when strain and other contributions can be neglected; a solid-state precipitate generally also pays coherency strain energy, and its optimum shape can change with size.

During formation, growth kinetics and diffusion may prevent the equilibrium shape being reached. Coherent, semicoherent and incoherent interfaces have different bonding and defect structures, so their energies and attachment rates differ. The disc shape is consistent with much cheaper broad faces than edges, but morphology alone does not uniquely measure the energies or prove equilibrium.

At turbine operating temperatures, grain boundaries provide paths for rapid diffusion, grain-boundary sliding and cavity formation, encouraging creep rupture. A single-crystal turbine blade removes these boundaries, especially boundaries unfavourably oriented across the tensile load. It retains dislocations and can still creep; eliminating boundaries is useful in this high-temperature regime even though fine crystal grains can strengthen other materials at lower temperatures.

Use directional solidification of the cast alloy through a controlled thermal gradient. A suitably oriented seed or a narrow grain selector lets one crystal survive and grow into the blade mould while suppressing competing nucleation. A spiral selector is a common way to exclude competing crystal grains. The blade is generally oriented near a $⟨001⟩$ axis for the intended anisotropic properties. crystal grain selection is experimentally verified by orientation mapping, rather than assuming a directionally elongated casting is automatically a single crystal.^[ref]

At $227^{\circ }\mathrm{C}\simeq 500\mathrm{K}$, $1000/T\simeq 2$. Reading the carbon line gives $D_C\sim 2\times 10^{-15}{\mathrm{m}}^2{\mathrm{s}}^{-1}$ to graphical accuracy. With $t=60\mathrm{s}$, the characteristic diffusion length isIf “distance” means a one-coordinate root-mean-square displacement, use $\sqrt{2Dt}\sim 0.49\mu \mathrm{m}$; the three-dimensional radial root-mean-square is $\sqrt{6Dt}\sim 0.85\mu \mathrm{m}$. The graph does not justify precision between these conventions. Equal diffusion distance in equal time requires equal diffusion coefficients. The copper line reaches this diffusivity around $1000/T\simeq 0.96$, giving $T_{\mathrm{Cu}}\sim 1.0\times 10^3\mathrm{K}$, about $770^{\circ }\mathrm{C}$. Carbon moves interstitially through ferrite, whereas copper self-diffusion is substitutional and vacancy-mediated, with a much higher effective activation barrier.

At $700^{\circ }\mathrm{C}$ the undercooling below the eutectoid temperature is small, so the free-energy driving force for forming pearlite is weak and nucleation is difficult. At $300^{\circ }\mathrm{C}$ the driving force is larger, but carbon diffusion is slow, limiting the coupled growth of ferrite and cementite. Their competition produces the nose of the isothermal transformation diagram near $550^{\circ }\mathrm{C}$ and $1\mathrm{s}$. A roughly linear quench from $740$–$800^{\circ }\mathrm{C}$ must pass left of this nose, requiring a rate of order $2\times 10^2$–$3\times 10^2\mathrm{K}{\mathrm{s}}^{-1}$, then continue below the martensite-start region near $240^{\circ }\mathrm{C}$. The tangent to the full start curve, rather than just the nose, refines this estimate; a true continuous-cooling diagram would be preferable. Martensite forms by a coordinated lattice-shape change without long-range carbon partitioning, so its formation is a displacive transformation. The tetragonal-to-monoclinic transformation of zirconia is a nonmetallic example.