As shown in our paper

https://www.researchgate.net/publication/233802377_A_continuum_model_for_initiation_and_evolution_of_deformation_twinning

the number of twins depends on two competing factors: i) the bulk energy of the crystal including the surface energy of the twin boundaries, ii) the elastic energy of the side boundary layers. The number of twins can be determined by minimizing the total energy. This number turns out to depend on the overall strain applied to the crystal.  

Article A continuum model for initiation and evolution of deformation twinning

One shouldn't forget any imperfections in crystalline solids such as  self-interstitial,   dislocations, twins, stacking faults and grain boundaries are not thermostatically equilibrium defects or better to say they are athermal defects therefore their spontaneous decimation is a natural process. Of course whether this takes place immediately or  not is a kinetics problem, which  strictly depends on the temperature fluctuation and its  localization at the proper place and time!  Mostly it may be treated by Fokker-Planck type equation in practice (Physical Kinetics, Landau & Lifshitz, p.427)

Of course,  the formation of twins in cold work grains or single crystals;  the generation and rearrangements of  dislocations into  polygonization or  low angle grain boundaries in deformed  solids can reduced the stored strain energy of the closed system.   But whether they can occur spontaneously as natural isothermal processes depend upon the constrains on the system.  If the system would be enclosed by rigid inflexible walls (isochoric system) then the negative variations in  the global Helmholtz free energy  with the decrease in the stored elastic energy play the dominate role, and the strain recovery becomes a natural process. On the other hand if the closed system with stored  strain energy would be under the constant external stressing by surface tractions  and body forces, the global Gibbs free energy plays dominant role (isobaric system) , and then the  strain recovery by any types of   isothermal processes  become unnatural , and it can never take place in nature without external manipulations.Best Regards

Unfortunately most people in the literature missing the point that the elastic strain energy density [ better to say; the stress:strain tensors inner double product ] enters into the Gibbs free energy characteristic function with a negative sign, which is just the opposite case for the Helmholtz free energy characteristic function where it appears with a positive sign. (SEE:  OUR RECENT PAPER ON THE IRREVERSIBLE THERMODYNAMICS OF DEFORMED SOLIDS IN RG)  

Note:  As a summary for the natural processes:

dE =T dS + sigma: d Strain,     dE < 0    adiabatic isochoric, 

dF =- S dT +  sigma : d Strain   d F < 0   isothermal isochoric, 

dG =- SdT - Strain : d Sigma     dG  < 0  isothermal isobaric        where strain and sigma are second order tensor quantities,   ' : '  corresponds to double inner product. 

Dear Jiansheng Wei,

I think the above answers cover the fundamentals of the set problem. So my humble contribution is only a geometrical remark: supposing that twinning occurs by preference along certain twinning planes, the possible maximum of number of possible twins in one grain will certainly be determined by its size. The maximally observed number of twins will be related to effects linked with  entropy and energy.

@Khanh Chau L, Tarik Ömer Oğurtani,  P. Van Mourik

Thanks for the reference and direct answers.

 Reading the paper by Khanh Chau L now, all thought a little bit hard to digest.

Physical Kinetics. Hard to find, even I can find it, I guess it will be well above me. But, still tried to understand the text you put here, thank you, Tarik Ömer Oğurtani.

Here I'd like to put more background information here.

This question comes from what I observed in a magnesium alloy. when twins present them in a high density, the strength goes up significantly. Then the question comes natrually, how can I get as many as possible twins into one grain. But before that, I want to konw is there a solid limit which cannot be break through.

So, I like konw any further ideas on this.

"the strength goes up significantly" generally by introducing any impediment to dislocation movement and coalescence. However, there usually is a break-even point where the positive effects of defects will be overcome by the fracture mechanics and crack propagation. It also depends on the materials system you use. The only assumption I could make by your use of the term "strength" is that you mean yield strength and in poly-crystalline substances. Is that correct?

My thing is identifying twins through real time XRD Microscopy. So, I'd be interested in defining a "twin", Nano structurally. Its length, strain field around it, its 3D nature, its composition, its electrical/magnetic/semiconducting/superconducting characteristics, its effect on macroscopic and microscopic properties, etc.

It would seem to be an impossible task to create just "twins" in any substance, wouldn't it? As noted earlier, there are competing defect types based on the energy requirements and the entropy state.

Ravi Ananth

Yes that is what I mean. But the UTS increase too, though not as much as YS.

The even point idea is logical, but could it be possible that the twinning happens during high temperature deformation, where recrystallization follows? That way, twinning still has it's positive effect, but cracks fade away.

@Igor Altenberger

Thanks for the answer.

I guess it works for most of the alloys overall.

Have you encountered any anomaly one this pattern? For example, I got many twins in a alloy with fairly small grain size and low SFE, but at high temperature compared with the melting point.

Indeed, twinning can happen during high temperature deformation when dynamic recrystallization occurs. At this stage, twins can form in different manners depending on deformation conditions (température, strain rate and strain) as well as chemistry alloys (with low or high stacking fault energy). The topology (density and orientations) as well as the location (with respect to the overall substructure) of the resulted twins formation play an important role into the performance of the metal. As meaning that negative effect (e.g.fracture) can infortunately occur due to strain concentration at this point. Many papers exist in the literature that describe in detail these mecanisms, specifically for magnesium alloys.

For your guidance, see the following reference (and others therein):

Three-dimensional effects of twinning in magnesium alloys, Scripta materialia 100 (2015) 9-12

As far as I  recall plastic deformation by twinning becomes predomiant micro-mechanism at low temperatures below a well defined threshold level compared to the slipping  which invonves dislocations and disclinations. Both mechanisms of plastic deformation leave their own finger prints at the surface. The slip leaves a series of steps, whereas twinning leaves small but well define smooth region of the crystal deformed. Twinning becomes most important for the HCP systems (such Ti ,  Zn, Zr, Rh etc.) because they have very small number of slp systems, where even at room temperature twinning can be observed. For BCC metals alloys twinnin observed very low temperature (at Liquid Nitrogen temperature and below). For the FCC copper the deformation twin was observed at 4K at high stress levels.

Just an extra remark: if you increase the number of twins per grain, at some point the twins will start to interact and other phenomena like secondary and ternary twinning can start to step in as well. This will make the analysis possible a lot more complex than simply stating that one should maximize the number of twins. An important parameter is also the width of the twins: if you have wide twins, they might actually decrease the flow stress of the material (due to secondary twinning or dislocation movement as the twin might have a more suitable crystal orientation compared to the original grain for dislocation movement). In some cases twins can actually grow and start covering most of the original grain, possibly annihilating the original hardening effect, e.g. in titanium (see links).

http://oro.open.ac.uk/29554/

https://inis.iaea.org/search/search.aspx?orig_q=RN:45024547

@Frederik Coghe

Thanks for the links.

It will be complicated for sure, but right now I just want to focus on introducing as many twins as possible into one grain and see how they can be related to the mechanical properties.

Decreasing flow stress does seem a problem. I admit that it baffled me for some time, how a softening twinning give rise to a higer strength. But what I want is to produce twins at high termperatures during forging or exrusion, then use them as a strenthening tool at room temperature. Do you think it works this way?

@Jiansheng:

The idea itself is definitely not bad: the Bazinski-mechanism might indeed help you in strengthening the material (it is kind of a Hall-Petch effect by splitting the grain into multiple regions by sessile twin boundaries). The problem is of course the many other mechanisms that might lead to a softening effect, added to the fact that contrary to the regular Hall-Petch effect, you will not necessarily have in increase in failure strain. Anyway, good luck with the research!

Dear M.L. Fares

Thanks for the link, working on it right now.

Adding some results I recently got.

How many exact? I have no idea, but right now, I have put at leat 20 twins(more like lines) into one grain, if some of them are intercepted by others, maybe more.

Of course, there are materials which can posess more tiwns, and ways potentially boost the number in the same material. Still welcome any suggestions.

I will try to get a higher number in the coming days.Anyone who are interest on this, maybe we can talk about it.

Hi

I just sent you a message (via researchgate) wherein an example of image for a binary Ni-Nb alloy (obtained by EBSD) showing various twins formed in one or two adjacent grains. Note an interisting point that these twins are differently oriented in the same grains.

Just to satisfy your curiosity and broaden your imagination I would like to remind you that a twin in its ultimate finite size is no different than a stacking fault. And remember that a stacking fault is a fault comprised of one atomic layer. So one may say from this perspective that the number of twins one can create in a grain is the number of stacking faults that can be created. Knowing that a SF is a one atomic layer fault you do the calculation for a given crystal size then... you may find it amusing!

And there is just a system pretty much like this. Read the following article:

Materials Research Letters, 'Ultrastrong Mg Alloy via Nano-spaced Stacking Faults'

W. W. Jian, G. M. Cheng, W. Z. Xu, H. Yuan, M. H. Tsai, Q. D. Wang, C. C. Koch, Y. T. Zhu & S. N. Mathaudhu.

But I know your question was not based on this approach referring to the equivalency of a SF and twin at the scale of atomic layers. And at this point I must remind you that there is a limiting grain size below which creation of twins becomes impossible. Actually for any system finding this treshold grain size should interest you more.

And as a final remark, one has to lower the stacking fault energy level to increase the population of twins that can be created.

Fun reply, I like it, and it may in some way enlightened me for the future work.

I read the paper, and yes I do not mean to produce twins in that way. Nice reminder!

Grain size, yes, it is tempting to find the fine line. But right now, I go the other way around. Big grain size, not a problem right now. When I get there, or, at least half way, I will come around and try to eliminate the twin instead.

SFE, I am working on mangesium alloy, so I am on the right track I guess.

Thanks for the answer.

Here adding a article to give a sample answer to my quesiton.

Nanotwinned diamond with unprecedented hardness and stability

Like Ali Arslan Kaya said, the twin size can be extremely fine and hence twin number in one grain could be ridiculously big.

What I want is a real record in paper like the one in the article above, but in alloys, Al, Cu, Ti or alike, sigle crystals are also welcomed if you think it is related to my question.