Two crystals, two different phases — a new kind of seam
In guide 2 you met the grain boundary: the seam between two crystals of the same phase — same structure, same spacing — that merely sit at different orientations, like floor tiles laid at different angles. This last guide takes the harder step. An interphase boundary joins two crystals of different phases — different crystal structures, or different compositions, or both. Now the two sides do not merely point different ways; their very lattices disagree. Think of two sheets of wallpaper whose repeating motifs are printed at slightly different spacings: even if you rotate them into best alignment, the patterns still drift out of step as you march across. That drift is the new problem this guide is about.
We measure the disagreement with a single number, the lattice misfit, written delta. If one phase has spacing a-matrix and the other has a-particle along the matching direction, then delta = (a-particle minus a-matrix) divided by a-matrix. A misfit of delta = 0.02 means the two spacings differ by 2 percent — one lattice gains a whole extra plane on the other roughly every 50 planes (1 divided by 0.02). Almost everything that follows is a story about how an interface copes with delta: whether it can swallow the mismatch by stretching, or must confess it with defects.
Coherent: every plane shakes hands across the seam
The cleanest way to join two phases is to make their atoms line up one-for-one, so lattice planes run continuously from one side straight into the other with no break. This is a coherent interface, and it is the lowest-energy interphase boundary there is — often only tens of mJ per square metre, in the same ballpark as a coherent twin. The only cost is chemical: the atoms across the boundary are of different kinds, so the bonds there are a little unhappy, but the geometry is seamless. Every plane on one side finds a partner plane on the other and shakes hands.
But if the two natural spacings differ by delta, forcing them to match plane-for-plane means straining one lattice (or both) elastically until they fit — the interface is seamless only because the crystals are stretched to make it so. That stored elastic strain is called coherency strain, and it is the hidden price of a coherent interface. Here is the crucial bookkeeping: the chemical interface energy grows with the boundary area (like r^2 for a particle of radius r), but the coherency strain energy grows with the whole strained volume (like r^3). Volume outruns area. A tiny particle can afford to strain itself perfectly coherent; a big one cannot. Coherency is a small-particle luxury.
Semicoherent: hand out a few misfit dislocations
When the misfit or the particle grows too large, straining the whole lattice becomes ruinously expensive, and the interface strikes a bargain. Over most of its area it keeps a good coherent match, but every so often it inserts an extra half-plane on the more crowded side — an edge dislocation lying in the boundary. This is a semicoherent interface, and those periodic defects are misfit dislocations. They are ordinary dislocations, storing the usual line energy you met last rung, but here they earn their keep: each one absorbs one plane of mismatch, so the patches of crystal between them can relax back to a near-perfect fit.
The spacing between misfit dislocations follows a beautifully simple rule: D is approximately b divided by delta, where b is the dislocation's Burgers vector. Put numbers on it. Take a misfit of delta = 0.04 (4 percent) and a Burgers vector of about b = 0.25 nm. Then D is roughly 0.25 nm divided by 0.04 = about 6 nm — a misfit dislocation every 25 planes or so (since 1 divided by 0.04 = 25). The larger the misfit, the more closely spaced the misfit dislocations, until at very large delta they crowd together so tightly there is no good-fit crystal left between them at all — which is the doorway to the next case.
- A brand-new nucleus is tiny, so it strains itself elastically into perfect registry — fully coherent, lowest interface energy.
- As it grows, coherency strain energy (scaling with volume, r^3) climbs faster than interface energy (scaling with area, r^2).
- At a critical size it becomes cheaper to swap some of that elastic strain for a periodic array of misfit dislocations — the interface goes semicoherent.
- If the misfit is very large or the orientation relationship is lost entirely, matching breaks down completely and the interface goes incoherent.
Incoherent, and the whole energy ladder
When the two lattices are too different to match at all — very large misfit, or crystals with no simple orientation relationship between them — you get an incoherent interface. No plane crosses continuously; the atoms at the boundary are packed with the same disordered, high-energy mismatch you find in a high-angle grain boundary. Its energy is correspondingly high, typically 0.5 to 1 J per square metre. In fact an incoherent interphase boundary and a high-angle grain boundary are close cousins in structure: both are regions where matching has simply given up, and both cost about the same in energy per area.
THREE WAYS TWO PHASES CAN MEET (misfit delta grows left -> right)
(a) COHERENT (b) SEMICOHERENT (c) INCOHERENT
planes cross 1:1, mostly matched, an extra no matching at all,
lattice strained to half-plane every ~1/delta like a high-angle
swallow the misfit planes (spacing D=b/delta) grain boundary
A A A A A A A A A A A A A A A A A A A
| | | | | | | | | | | | | | | | | | |
==+=+=+=+=+=+== ==+=+=T=+=+=T=+== ==~=~=~=~=~=~== <- interface
| | | | | | | | | | | | | \ |/ \| /
B B B B B B B B B B B B B B B B B
cost: ELASTIC cost: a little strain cost: disordered
coherency strain + misfit-dislocation broken bonds
(grows with VOLUME) cores (T = extra plane) everywhere
energy LOWEST < energy MIDDLING < energy HIGHEST
ENERGY LADDER (mJ per m^2, order-of-magnitude):
coherent twin ~10s < coherent interphase < semicoherent
< incoherent ~ high-angle GB ~500-1000 < free surface ~1000-3000Epitaxy: growing one crystal on another
The most technologically important interphase boundary of all is the one you build on purpose. Epitaxy is growing a crystalline film so that it inherits the orientation of the crystal beneath it — the substrate acts as a template and the film lines up with it. If the film is thin enough it strains itself to match the substrate exactly, staying fully coherent; this pseudomorphic, strained layer is the coherent case of the previous sections, deliberately engineered. But as the film thickens, the stored coherency strain (growing with volume) eventually outweighs the cost of relaxing, and beyond a critical thickness the film relieves itself with misfit dislocations and goes semicoherent — the very same crossover, now controlled by film thickness instead of particle size.
Real numbers make it concrete. Silicon has a = 5.43 angstrom and germanium a = 5.66 angstrom, a misfit of (5.66 minus 5.43) divided by 5.43 = about 0.042, or 4.2 percent — big enough that a Si-Ge layer on silicon stays coherent only up to a critical thin film thickness of a few nm before misfit dislocations set in; engineers exploit that strained layer to speed up transistors. Contrast GaAs (a = 5.653 angstrom) on AlAs (a = 5.661 angstrom): a misfit of only about 0.14 percent, so nearly perfect that you can stack hundreds of alternating layers as a defect-free superlattice — the basis of semiconductor lasers and LEDs.
Two-dimensional materials rewrite this rule in a lovely way. When you stack sheets like graphene held together only by weak van der Waals bonds, there are no strong bonds across the gap demanding registry — so a layer can sit on a mismatched or rotated partner without paying coherency strain at all. Instead of misfit dislocations you get a moire superstructure: the two slightly-off lattices beat against each other to make a large-scale interference pattern, exactly like two overlaid window screens. This van der Waals epitaxy of two-dimensional materials sidesteps the whole misfit penalty, and the resulting moire patterns are themselves now prized for the new physics they create.
Interfacial energy is a force: grain growth, wetting, and precipitates
Every boundary in this rung — surface, grain boundary, twin, interphase boundary — costs energy per unit area, and a system left alone will lower its total energy by shrinking its total boundary area. That single fact is a mechanical driving force. Its plainest expression is grain growth: a curved grain boundary can lower its energy by straightening, so boundaries migrate toward their centre of curvature, big grains swallow small ones, and the average grain size creeps upward on annealing. The boundary between two soap bubbles bulges the same way and for the same reason — surface tension pulling area down.
The same tension governs wetting. Where a second phase touches a grain boundary, the surface tensions pull like three ropes at a knot and settle at a balance angle, the dihedral angle phi, set by gamma-gb = 2 times gamma-interface times cos(phi divided by 2). If the interphase energy is low enough the angle goes to zero and the second phase spreads right along the boundaries, wetting them — sometimes helpfully (liquid-phase sintering), sometimes disastrously (a brittle film that embrittles a whole metal). The geometry of a microstructure is, at heart, a tug-of-war between competing interfacial energies.
This is also why phases are so fussy about how they sit relative to one another. A precipitate does not appear in a random orientation; it adopts a specific orientation relationship with its matrix and grows on a particular habit plane — precisely the choices that make the largest, lowest-energy coherent or semicoherent interface possible. The plate-shaped habit of martensite, the disc shapes of coherent precipitates, the flat facets on an incoherent particle: all are the shapes that minimise interfacial energy for the matching available. The interphase boundary, far from being a passive seam, quietly dictates where new phases form, what shape they take, and how a material's whole microstructure is built.