The definition of coherent boundary in physical metallurgy is as such '  İt is a planar boundary, which separates  one part of the crystalline region from the other part as such that  there is a one to one correspondence between the  lattice planes at the interface even though  one has lattice parameter mismatch that compensated by bending and twisting the planes in the transition region having a finite thickness. 

The extra-stored energy associated with the mismatch strain may be partially by released through  the formation of dislocations network along the boundary, thereby creating semi-coherent boundary. 

There is close connection between some type of  stacking fault and the so-called  ''coherent''  boundary of two twin crystals whose orientations differ by 180. degrees.  Such as

ABC/AB/ACBACBA......    Similarly   ABCAB/ABCABC can be regarded to a first approximation as a single- layer coherent twin.

Tarik, Thank you so much for your response! So as long as there is a one to one correspondence between lattice planes at the interface then the boundary is coherent. This is easy to visualize for a twin boundary, however, for other sigma types is there any other way of determining coherence without visual inspection? Is there a way to know if a boundary is coherent based assuming you know the misorienation between the crystals as well as the boundary plane (all 5 GB parameters known)?

The answer is affirmative but I don't have that mathematical connection at hand. For crystalline systems there are many constrains as such that the interface should be definitely a cleavage plane.  That  means it should be defined by  hkl  Miller indices. Neighboring crystal planes should also meet at the atomic positions lying on the so-called at the interface or boundary plane. I should play  around may be I can find some simple  connection! 

In a sigma-boundary you have of course coherency when you consider the coincidence-lattice.

A coherent boundary does not necessarily have to be flat.

Dear John,  this last statement  you have made is   really great.  if  one allows to somewhat  bending and/or twisting  the lattice planes.  Then we have a coherent boundary but having appreciable long range strain field associated with,  which could be  relived in practice by the formation of  mixed dislocations having edge and screw components. (W. T. Read, The General Grain Boundaries in Dislocations in Xtals, 19539. 

For a sigma boundary you would have 'coincidence' between CSL lattice points in both grains, however, wouldn't you only have 'coherence' if the lattice planes coincide at the boundary plane? I wish there was an example (other than Sigma3) that I could refer to for comparison. 

For the Sigma 3 you would need the boundary plane to be 111 for coherence. So coherence is highly dependent on the interface plane.

Do you know of any examples in literature of a curved coherent boundary? Just curious since it is hard to picture. I assume that for the curved boundary coherence would likely only be established for certain sections of the boundary, since it would be rare for the entire curve to be coherent.

H. Mykura provides a listing see " Appendix - A checklist of Cubic Coincidence Site Lattice Relations", which is in "Grain-Boundary Structure and Kinetics" - papers presented at the 1979 ASM Materials Science Seminar 15-16 September 1979 Milwaukee, Wisconsin. 

However, maybe it does not give the information you are after (but see below), because, as remarked by V. Randle in "Microtexture Determination and its applications", 1st edition (not second !), pub The Institute of Materials, 1992, "...........Therefore, although the S-value (read Sigma Value) tells you how many sites in the two lattices could theoretically coincide, this is not necessarily the number that actually will coincide at the boundary. The true number of coinciding sites at the boundary will depend upon where it is 'placed' between the two lattices."

In this section Randle gives two references (worth checking out ?).

147 P.J. Goodhew , T.Y Tan and R.W. Baluffi, Acta Met , vol. 26, 1978 p557.

117 S.R. Ortner and V. Randle, Scripta Met. vol. 23, 1989, p. 1903.

So your question could have been framed as follows :-  given the various CSL relationships, (e.g. as enumerated by H. Mykura), what are the actual  boundary planes that give the maximum possible number of coincident sites?  This may have been answered in the literature already, and maybe there are some computations of this given in those papers.

I've not thought too deeply about this, but H Mykura in his various Tables also gives a listing of the 'twinning planes' associated with each of the CSLs he has computed. If we imagine a sort of twinning shear creating lattice 2 from lattice 1, then his listing of such twin planes could be these boundary planes of maximum coincidence, that you seek, but I could have the wrong picture.

As other people have pointed out in any case this sort of implies a flat 2D boundary between such CSL related grains. The reality will probably be quite different. One can imagine a boundary between such lattices being at least stepped, e.g. an area of the 'twin plane', then regions with curvature (varying normal for the GB plane). Also if you have do have region in the boundary of good lattice matching then there might a complex adjacent structure with in effect dislocations or atom displacements occurring if the overall boundary energy is lowered by such small accommodations.

Attached is a paper from Prof. Tony Rollet's group at Carnegie Mellon, Table 1 should be of help to you, as well as the paper's references and its content in general. The paper is one that came out when many people were talking about grain boundary engineering in metals. With the availability of tools to sequentially map the 3D grain structure, work like that attached became possible.

 Thank you so much for your detailed response. I am looking into that literature now, but as a side note I do not believe that maximum 'coincidence' is the same thing as 'coherence.' From what I have gathered coherence implies matching lattice planes meeting at the interface. This does not pertain to coincidence which is related to the coincident site lattice and matching points in both crystals. I do believe there are special interface planes which will allow for more CSL points, but do these maximum CSL planes relate to coherency?

Jason,

When we talk about (full) coherence generally we mean that the lattice planes are more or less undisturbed passing from lattice A to lattice B, (as well as A and B both being matrix, e.g. coherent twin, A could be matrix and B could be a precipitate/second phase), so the spacing of the planes match within a few percent, or they are some exact multiple of each other. Small mismatches can be accommodated by a little lattice distortion, or occasional/regular interface dislocations (moving towards partial coherence, i.e. with greater disregistry between the two sets of lattice planes). The drop in energy due to to that lattice matching is often seen to assist the nucleation and growth of precipitates, which on further growth might lose that coherence for various reasons.  Such lattice coherence has been cited for observed cases of epitaxial growth, we always look for that to explain planar morphologies of ppts.

In TEM studies you can sometimes see that coherence 'directly' if you diffract simultaneously from  the matrix and the precipitate, that's something that can be seen between and austenitic steel matrix (FCC) and M23C6 carbide in some cases, i.e. if you orient the zone axis correctly, e.g. using a double tilt or a tilt-rotate specimen holder, then you get the diffraction spots from the carbides  lying inside (on the same reciprocal vectors) as those from the matrix.

So although the M23C6 lattice is ~ 3 times the size of that of the matrix we still term the above precipitation as coherent. 

Do people use the term coherent for the higher CSL boundaries where the number of coincident sites is relatively low ? Sorry I don't know, it seems to be stretching the concept quite a bit certainly at the higher end. I think you need to see what terms are used by those active in the field, past and present. I wouldn't be too surprised if they used coherence to describe case where there is the higher degree of matching, then talk about drop in coherence higher up the CSL series. (You could ask prof. Tony Rollet for an opinion, or read those GBE papers to check out the terminology.)

I can not recall that I have seen people using the term "coherent" when talking about CSL boundaries but it is a way to apply the definition that in a coherent boundary the lattice planes pass more or less undisturbed over the boundary. Observe also that such a definition only consider the orientation of the two crystals and does not involve the orientation of the boundary. That is why we introdyce the two extremes twist and tilt boundary for the sigma-type boundaries.

Dear  Professors  Butler and Agren, as a physical metallurgist we some how  have relaxed the definition of coherency by allowing some reasonable distortions (small mismatch strain in the range of  0.1-0.05%  )otherwise perfectly planar LP   to  include  the Coincidence Site Lattice boundaries in the same category.  Even though  this type of boundary may have transient existence but still it is important for the initiation of  Stranski-Krastanow islanding (Q-Dots) on the rough surfaces of the hetero-epitaxially strained thin films.  In the final  stage where  the overall system reaches to the non-equilibrium stationary state,  the CSL interface may be  transformed  into the  semi-coherent type boundary   by the formation of dislocation array at the interface to accommodate the large stored strain energy associated with long range  distortion of lattice planes (The Strain Relaxation Phenomena SRP)) by the dislocation core energies. In fact this happens completely spontaneously and suddenly like the twin boundary formation at low temperatures (i.e., massive transformations).. 

Please see the following paper for the initial stage of strain relaxation phenomena, which shows how  the relation of SRP compensates the increase in the surface free energy due to QD  formation.

Thanks John, it's as well to keep in mind that the existence of a CSL relationship between adjacent lattices makes no comment at all on the orientation of the actual boundary plane between the two lattices, as per the quote from Val Randle's book given in a previous post.

[There's basic introduction to some of these ideas re boundaries in Chapter 3 of 'Recrystallisation and related annealing phenomena' by F.J. Humphreys and M. Hatherly, pub. by Pergamon. Chapter 3 is entitled 'The structure and energy of grain boundaries']

I tried to attach the Acta Met paper cited in the above post by Goodhew et al that Prof. Goodhew has kindly provided to me from his personal archive. Unfortunately I think it is too big. However, Prof. Goodhew is happy to supply copies of the many papers he has authored/co-authored by emailing him at [email protected]

That one cited above  is number 15 in his list.

The following book:

Grain Boundaries - From theory to Engineering, By Louisette Priester has been extremely helpful for me. Just thought I would pass it along. Thanks for the great discussion everyone!