Diffusion and phase transformations

Atoms redistribute over time

Diffusion describes atomic transport associated with random motion and thermodynamic driving forces. In a simple constant-diffusivity description, a characteristic distance scales with √(Dt). D has units of area per time and depends on material state and temperature.

A solid can change composition locally without melting. In a vacancy mechanism, an atom exchanges with an unoccupied lattice site; the vacancy moves the other way. An interstitial species can move between available positions between the normally occupied sites. Repeated jumps redistribute material. Defects, boundaries and interfaces develops the distinction between a missing site, a pore and a moving atom.

Individual paths can turn back and need not point along the net transport direction. At a common temperature, atoms still move when the average composition is uniform: equal opposing transfers can give no net compositional flux. A concentration profile describes many atoms averaged over a region, not the itinerary of one atom.

A jump requires access to a higher-energy configuration. Thermal activation and the availability of suitable sites influence how often it occurs. In a declared range, a model D = D₀ exp[−Q/(RT)] is often useful: D₀ has units m²/s, Q is an effective molar activation energy in J/mol, R is the gas constant in J/(mol·K), and T is absolute temperature. The exponential argument is dimensionless. A fitted Q can include more than one microscopic contribution; changing phase or mechanism can invalidate a single fit.

A flux has a direction, an area and a material basis

Let c be the amount of one specified solute per unit local material volume, in mol/m³. Define J as the net amount crossing unit area per unit time in the positive x direction, in mol/(m²·s). Negative J means net transfer toward decreasing x; it does not mean a negative amount of material exists.

For a homogeneous single phase at uniform temperature, with a declared constant positive diffusivity and a valid concentration-gradient description, Fick’s law is J = −D dc/dx. The derivative means the local slope of the concentration profile. If the profile is a straight line over a distance ℓ, use J = −D(c_right − c_left)/ℓ. The minus sign makes this model’s flux point down the concentration slope.

For example, with D = 1.00 × 10⁻¹⁴ m²/s, c_left = 10000 mol/m³, c_right = 10500 mol/m³ and ℓ = 10.0 µm, the slope is +5.00 × 10⁷ mol/m⁴. Thus J = −5.00 × 10⁻⁷ mol/(m²·s). The net solute moves left. Specifying only D would not have supplied that flux: the profile and boundary conditions also matter.

Concentration is not the most general thermodynamic driving variable. At fixed temperature and pressure, chemical potential describes the change in Gibbs free energy per small amount of a component added while the other component amounts are held fixed. Composition, stress and other fields can affect it. In a nonideal or coupled system, concentration alone need not specify the direction or coefficient. The simple Fick model below is explicitly restricted rather than made a law for every alloy and phase.

A balance on a thin fixed slice explains why a profile changes: more solute entering than leaving increases the amount stored there. With no reaction or bulk motion, this gives ∂c/∂t = −∂J/∂x; when D is constant it becomes ∂c/∂t = D∂²c/∂x². The first equation states conservation. Initial and boundary conditions are needed in addition to it; a single profile snapshot is not a complete time evolution.

A characteristic length is a scale, not a furthest atom

For the following estimates define ℓ_D = √(Dt). This definition fixes the convention used in the arithmetic. It is a characteristic scale in a constant-D diffusion model, not a sharp penetration edge, the distance travelled by every atom or a specified completion fraction. Other definitions, such as a chosen concentration contour or root-mean-square displacement in a stated dimension, introduce factors and boundary dependence.

Use a fictional stable single-phase material with D_A = 1.00 × 10⁻¹⁴ m²/s at its supplied state A. After 1.00 h = 3600 s, ℓ_D = √(3.60 × 10⁻¹¹ m²) = 6.00 µm. Reaching ℓ_D = 12.0 µm requires t = ℓ_D²/D_A = 14400 s = 4.00 h. This reproduces the original fourfold-time logic on an explicit unit basis.

At a different supplied temperature/state B, suppose independent evidence instead gives D_B = 4.00 × 10⁻¹⁴ m²/s for the same species and path. Within the unchanged model, a 12.0 µm scale now takes 3600 s. Heating can change mobility, but this comparison does not establish that every transformation becomes faster at every higher temperature: phase stability, supersaturation, available sites and interfaces may also change.

A limited time-varying extension replaces Dt by the diffusion exposure Θ = ∫D(t)dt. The integral means add diffusivity multiplied by elapsed time; Θ has units m². This extension applies when D changes uniformly in time while the same single-phase linear spatial problem and boundary/initial definitions remain applicable. Then use ℓ_D = √Θ. It is not a general transformation clock when concentration, phase or path evolution changes the operator or its boundaries.

Paths and mechanisms matter

Bulk, boundary and surface transport can differ. Phase changes can alter composition and diffusivity while a process proceeds. The square-root scaling is a useful estimate for its model, not an exact treatment of every moving interface or multicomponent system.

A boundary or a dislocation core can provide a faster path than the surrounding lattice for a particular species and state. That path occupies only part of a specimen and must connect the relevant locations. A tracer tail deep along a boundary is not evidence that the entire grain interior has reached that composition. Transfer from a fast path into the interior can still be slow.

For a bounded parallel-path illustration, let a fraction φ of the cross-sectional area be connected fast channels and the remainder ordinary lattice. Stipulate steady one-dimensional linear concentration profiles in both paths, the same length and end concentrations, constant coefficients and no exchange between them. Fluxes add by area: J_average = −[(1 − φ)D_lattice + φD_fast]Δc/ℓ. With φ = 0.0100 and D_fast = 100D_lattice, the coefficient in brackets is only 1.99D_lattice. The much larger channel value cannot simply be assigned to the whole cross-section.

This parallel construction is an original ideal geometry, not a universal formula for grain-boundary transport. Actual boundary orientation, segregation, width, connectivity and leakage into grains can require a spatial model. The retained normalised figure shows the original fixed-D scaling only; it does not establish those missing paths.

Redistribution can feed a new phase without defining its stability

A phase has a material structure and composition state. Two regions with different solute concentrations need not be different phases: a single solid solution can have a smooth composition gradient. Conversely, an already formed second phase can have a composition different from the matrix and require additional solute as it grows.

For an existing solute-rich precipitate or layer, the surrounding material may have to supply solute over a distance. The atoms must also attach and rearrange at the interface. Diffusion control means transport is the dominant limitation in the declared model; interface control means the local attachment/migration response is limiting. Both processes can contribute. Neither a high D nor a thermodynamic preference alone specifies a growth rate.

The following example assumes that phase evidence and the permitted interface compositions have already been supplied, and that an interface already exists. It does not calculate nucleation. A phase diagram or a suitable thermodynamic description constrains possible boundary compositions; D does not determine them. Energy and the meaning of stability and Equilibrium tendencies and transformation rates provide the energetic and kinetic preparation.

Diffusion problems with a growing phase are moving-boundary problems: the location where boundary compositions apply changes as the transformation proceeds. Some ideal solutions grow approximately with √t, but their prefactors depend on compositions, geometry and other conditions. Do not relabel ℓ_D as a phase-front position just because both can involve a square root. Solidification, nucleation and growth develops its separate cooling/growth context.

Account for the solute consumed by an advancing interface

Use an original planar solid-state model at a fixed supplied temperature and pressure. The existing solute-rich phase β is to the left, the matrix α to the right, and x increases rightward. Let s be the interface position and v = ds/dt its positive growth speed into α. Assume fixed cross-sectional area, no material volume change or bulk convection, and no solute stored on the interface itself. All concentrations use the same mol of solute per m³ of the relevant phase.

At the interface, stipulate c_β = 20000 mol/m³ and c_α = 10000 mol/m³, compatible with the assumed phase relation. Replacing a thin slice A ds of α by β needs the extra solute (c_β − c_α)A ds. It does not need c_βA ds supplied from outside, because the slice of α already contained c_αA ds.

Let J_β and J_α be the signed solute fluxes immediately on the left and right sides, both measured positive toward +x. In time dt, net delivery to the interface is (J_β − J_α)A dt. Equating it with the extra demand gives (c_β − c_α)v = J_β − J_α. Area cancels for the speed; it is needed again for a total amount. This is a local conservation condition, not by itself an interface-attachment law.

For one supplied snapshot, the α-side profile rises from 10000 mol/m³ at s to 10500 mol/m³ at s + 10.0 µm. With D_α = 1.00 × 10⁻¹⁴ m²/s, J_α = −5.00 × 10⁻⁷ mol/(m²·s). Stipulate negligible flux on the β side, J_β = 0. Then v = [0 − (−5.00 × 10⁻⁷)]/10000 = 5.00 × 10⁻¹¹ m/s = 0.0500 nm/s. Leftward solute delivery is compatible with a front moving right into the matrix.

The concentration discontinuity between α and β is not a slope to put into one Fick equation across zero distance. Apply the relevant transport law within each phase, and match them with an interface condition. The figure enlarges the small α-side concentration variation in its own separate plot so that the two-phase jump is not mistaken for that gradient.

The snapshot determines an instantaneous compatible speed under the supplied assumptions. It does not demonstrate that the profile, D or interface compositions remain fixed for the next 600 s. Faster transport can also expose an attachment limitation. A complete prediction must evolve the profile and interface consistently, or use a separately supplied finite record.

A finite record can test a growth account

Now take a separate original interval record for the same two interface compositions and a cross-sectional area A = 2.00 mm². The table supplies the net interfacial influx J_net = J_β − J_α, constant within each listed interval. It is a declared model input, not an extrapolation of the previous single snapshot or a measured specification of a named material.

The first 300 s supplies Δn₁ = A J_net Δt = (2.00 × 10⁻⁶)(4.00 × 10⁻⁷)(300) = 2.40 × 10⁻¹⁰ mol. The second supplies 1.20 × 10⁻¹⁰ mol. Total extra delivery is 3.60 × 10⁻¹⁰ mol = 0.360 nmol, where 1 nmol = 10⁻⁹ mol.

With the same constant composition difference, Δs = Δn/[A(c_β − c_α)] = 18.0 nm. The first interval advances 12.0 nm and the second 6.00 nm. Multiplying the earlier snapshot’s 0.0500 nm/s by 600 s instead would give 30.0 nm; that assumes a maintained flux not supplied by this record. The two clocks cannot silently be interchanged.

If a separate phase relation changes c_β to 15000 mol/m³ while c_α remains 10000 mol/m³, the same stipulated net delivery and area would support 36.0 nm, provided the other assumptions still hold. A smaller extra demand per volume increases the compatible advance. Such a changed phase relation needs its own thermodynamic and kinetic evidence; it is not a correction inferred from D.

A useful evidence set combines spatial concentration and phase identification, time-resolved profiles or interfacial flux accounts, actual temperature history, interface position and specimen geometry. A square-root fit alone can be compatible with several transport or thermal problems. A concentration gradient alone does not prove a new phase, and agreement with this conservation model does not establish a unique microscopic mechanism or a treatment schedule.

Worked examples

Try the reasoning

1. Does D have units of m/s?

   Show the answer and reasoning

   No.

   Its units are m²/s.

2. Can a grain boundary provide a different transport path?

   Show the answer and reasoning

   Yes.

   Its structure differs from the crystal interior.

3. A different homogeneous phase has D = 4.00 × 10⁻¹⁵ m²/s. Its supplied linear profile rises from 7000 to 9000 mol/m³ over 4.00 µm as x increases. Find J and the signed amount through area 0.500 mm² in 120 s if this flux is maintained.

   Show the answer and reasoning

   J = −2.00 × 10⁻⁶ mol/(m²·s); the signed amount is −1.20 × 10⁻¹⁰ mol, or −0.120 nmol toward +x (net transfer is toward −x).

   Use the positive slope 2000/(4.00 × 10⁻⁶), then the minus sign in Fick’s law. Amount is JAt with the area converted to 0.500 × 10⁻⁶ m². The maintained-profile condition is part of this separate case.

4. In the same length convention, state A reaches 6.00 µm in 3600 s. A new state has four times D and a target length of 12.0 µm. Is its required time four times greater?

   Show the answer and reasoning

   No. It is again 3600 s under the unchanged model and boundary interpretation.

   Time scales as length squared divided by D. Doubling length supplies a factor four, but quadrupling D cancels it. This algebra cannot decide whether a real new temperature preserves the phase and boundary assumptions.

5. For the restricted time-only D model, hold D_A = 1.00 × 10⁻¹⁴ m²/s for 900 s, then D_B = 4.00 × 10⁻¹⁴ m²/s for 2700 s. Find Θ, ℓ_D and equivalent time at D_A. Would reversing these two complete holds necessarily give the same transformed microstructure?

   Show the answer and reasoning

   Θ = 1.17 × 10⁻¹⁰ m², ℓ_D ≈ 10.8 µm, and equivalent D_A time is 11700 s. Equal exposure does not generally establish the same transformed microstructure.

   Add D_A × 900 and D_B × 2700 before taking the square root. Reversing intact holds preserves this scalar sum in the restricted linear problem. Phase formation, supersaturation, nucleation and changing paths can retain history outside it.

6. In the stipulated independent parallel-path geometry, only 0.500% of area is fast channels and D_fast = 200D_lattice. Stipulate steady one-dimensional linear profiles in both paths, with the same length and end concentrations. Find the area-average flux coefficient relative to D_lattice. Does one fast channel make the whole specimen 200 times faster?

   Show the answer and reasoning

   The coefficient is 1.995D_lattice, about twice rather than 200 times the lattice value.

   φ = 0.00500, so (1 − φ) + 200φ = 0.995 + 1.000. This result uses the declared parallel connectivity and absence of exchange; it is not a general polycrystal diffusivity formula.

7. In a new instantaneous interface record, J_β = +1.00 × 10⁻⁷ and J_α = −2.50 × 10⁻⁷ mol/(m²·s). Keep the 10000 mol/m³ composition difference and the same sign convention. Find v. Does adding the two signed fluxes give the net interfacial supply?

   Show the answer and reasoning

   v = +3.50 × 10⁻¹¹ m/s = +0.0350 nm/s. The required supply is J_β − J_α, not their sum.

   Rightward delivery from the left and leftward delivery from the right both feed the interface. Subtract the right-side outward flux: 1.00 × 10⁻⁷ − (−2.50 × 10⁻⁷) = 3.50 × 10⁻⁷. This new case no longer stipulates zero β-side flux.

8. A report shows a broadened solute profile and a quantity fitting √t. It declares both a new phase and a unique diffusion-controlled growth mechanism. Which additional evidence would discriminate those claims?

   Show the answer and reasoning

   Identify the structure/phase and its composition, track the actual interface and spatial profiles over time, and test the temperature, transport and attachment/boundary assumptions.

   A single-phase profile can broaden without a second phase forming. Several transport or thermal moving-boundary models can give a square-root trend. Evidence must address the phase identity and the rate-limiting route separately; the fit alone does not select them.

What to carry forward

• Diffusion times often scale with distance squared.
• Identify the path and state controlling D.
• Use a signed flux on a declared species, area and concentration basis; D alone supplies no rate of delivery.
• A diffusion length is a defined characteristic scale, not a sharp front or proof of completed transformation.
• A growing phase needs a solute and interface account as well as energetic permission.
• At the specified moving interface, extra demand is c_β − c_α, and net supply is J_β − J_α.
• Integrate the actual finite record; temperature, phase, paths and boundaries can invalidate a reused scalar clock.
• A concentration profile or scaling fit does not uniquely identify phase or mechanism.