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Question: Is the local energy dictating dislocation emission constant for single crystal?

Kejie Zhao's picture

Hi everyone, in my size dependence study, I find the local energy dictating dislocation emission is almost constant for varied sized samples, in given directions of single crystal. I don't know this is an interesing finding, or just a common sense. Will you give me some suggestion, Thank you!

Kejie

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Dear Kejie,

0. This is interesting--the way you phrase it.

1. I would like to know the following:
(i) What technique do you use for size-dependence study? Analytical? Computational? If the latter, which method?
(ii) What do you mean by "in given directions of single crystal"? Do you mean the planes and directions of dislocation motion/activity such as the closed packed planes? Or do you mean: "any arbitrary direction"?
(iii) What is the expression for the "local energy dictating dislocation emission" do you use? What is it local to? The core of a dislocation source? If so, what model do you use for the same? Does it allow finite energy at the core? Or a singularity?

2. Now, for some speculative thinking:

If a finite point-force acts at a single point in a volume of a homogeneous isotropic elastic solid, it clearly has size dependence. The question is, what if there is a dislocation source instead of the point-force. Say, a steady-state dislocation source...

In analytical model, there should be a singularity at the source.  But does the strength of that singularity depend on size? In infinite-extent models, it won't. Analytical models are only for the infinitely large regions.

But any finer FEM study for finite sized specimens should bring out a size dependence for both stress and strain. If so, even the local values of the strain energy density (SED) would depend on the specimen size. (When you say "energy", do you have SED in mind?) But does this then mean that the FEM analysis would be fine enough that the *local* values of SED predicted by the FEM model for the points near the dislocation source would also bring out the subtle differences due to the far-off boundaries? I am doubtful.

3. To answer your question. By common sense, the local energy dictating dislocation emission, at the core of the source, is not expected to be even finite, let alone a constant, in analytical models. That would be so even if one would expect the global energy over the entire single crystal to be finite. But please remember, this "common" sense itself is based only on a particular type of model--the analytical one.

Now, if you have some sophisticated reasoning to show otherwise in an analytical model, then I would be happy to know about it. After all, in the real crystal, there are only finite-sized atoms each of which has only a finite energy associated with it. So, there is no reason why the local energy *must* become infinite. Further, this is not a static case but a dynamic case. A time-average could be finite, and if so, it could even be a constant.

What do you think?

Kejie Zhao's picture

Dear Ajit R. Jadhav,

Thanks a lot for the insightful thoughts, and sorry for my sloppy description. I will try to state my question detailly. Also you may find more information in my paper.

Regarding on your inquiry:

i) What technique do you use for size-dependence study? Analytical? Computational? If the latter, which method?

I did a molecular dynamics study on the size effect of incipient yield strength of nano-voided single crystal copper, under uniaxial tension. So it is the latter.

(ii) What do you mean by "in given directions of single crystal"? Do you mean the planes and directions of dislocation motion/activity such as the closed packed planes? Or do you mean: "any arbitrary direction"?

In my study, I considered two crystallline systems: [100]-[010]-[001] and [-110]-[111]-[11-2]. The uniaxal loading direction is along [100] and [-110] direction respectively. So the given direction I metioned should be [100] and [-110] orientations. I don't know how to insert an image in my comment, so I attached a results summary after my topic (attachment for Ajit R. Jadhav.pdf ). In the file, Fig.1 and Fig.2 are the calculated size effect of incipient yield strength for two void volume fractions considered, in two crystalline systems.  The x-axis represents the model size, and y-axis is the strength value. We can see an apparent size effect for given void volume fraction.

(iii) What is the expression for the "local energy dictating dislocation emission" do you use? What is it local to? The core of a dislocation source? If so, what model do you use for the same? Does it allow finite energy at the core? Or a singularity?

For intepreting the observed size effect, we studied local/atomic energy (kinetic+potential) where partial dislocations firstly nucleated and emittd on the void free surface. This study is based on my thoughts that the atomic energy represents the movement ability of atom itself,  so the dislocation could be emitted unless local energy reach some critical value. In this sense, we calculate the maximum local energy (corresponds to the point where dislocation nucleation firstly) for different sized samples. The results are also listed in attachment file, Fig. 3 and Fig. 4 for [100]-[010]-[001] and [-110]-[111]-[11-2] systems respectively. It is found the maximum local energy is almost constant when dislocations are firstly initiated.  Of course the value in various directions should be different by considering the structure constitution in special directions.

So I wonder my result "the local energy on dislocation nucleation is constant" is an intrinsic feature of materials or not. I don't know whether I give you a clear description of my question. But as my limited knowledge in materials science, there may be some fatal errors in my thoughts. Please point them and wish to hear from you.  Thanks!

kejie

Dear Kejie,


Thank you very much for your clarification, and also for taking all the trouble about the attachment here.

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OK. So you do MD.

Given that, all of my previous comments related to FEM now become inapplicable. In fact, given the situation you model with MD, much of my above thinking also becomes inapplicable. What you study here is the emission of partial dislocations from a central multi-vacancies void, and not a classical source of dislocations such as, e.g. the Frank-Reed source.

Also, note, I have almost no background in MD proper. So, all that I can offer you are some common-sense observations. So, my comments below could be too naive, even actually misleading.

With that qualification in place, I notice that you use periodic boundary conditions. As you yourself note in the paper on page 5, L_x is maintained to be greater than the potential cut-off radius. Since the source of the partial dislocations--the hole--is at the center of the unit cell, therefore, naively thinking, it would seem that no size effects could at all make an appearance in your model. This conclusion directly follows from your method and the model. Since the potential cannot go out far enough from the dislocation source to the boundary, therefore, speaking in reverse, no size effects could at all propagate from the boundary to the source, and it is at the source that you measure local stress (because the maximum stress is what you measure and the maximum stress would exist at the hole).

Am I correct in my thinking? (Please note, I could very easily be wrong here because I don't know MD!)

Note, by the actual physics of the situation, one *would* have expected a size effect. It's just that the methodology of MD which drops it.

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Another point. What you report is the yield stress, but you simulate a fairly big void and only few number of atoms in the matrix.

Some time back, there was a discussion here at iMechanica about what is appropriate in this kind of nano context: yield stress (a materials property) or compliance (a structural mechanics concept). I do not know if any conclusion was reached or not, but certainly, there *was* such a debate. You may want to check it.

Kejie Zhao's picture

Dear Jadhav,

Thanks a lot for your discussion. All your comments related to FEM and analytical model are great helpful for my further study.

Due to my poor background on materials sciense, I dont quite follow you on some points. I will try to clarify my misleading statement of my model and method

With that qualification in place, I notice that you use periodic boundary conditions.

Right. I applied periodic boundary condition (PBC) on three directions to simulate periodical arrayed voids, more in MD, to simulate the bulk behaviors . It also means that there is no free boundaries except for the void free surface.

As you yourself note in the paper on page 5, L_x is maintained to be greater than the potential cut-off radius.

Right. L_x being maintained to be greater than the potential cut-off distance (0.495nm), so to eliminate the interactions from the same atom in neighboring periodic cells. Actually this is a necessity in MD, because among the cut-off distance, there are only several atoms (the lattice constant for FCC cu is 0.3615 nm), MD can not applied in so small model. Note that the "potential" I used here is the atomic interaction potential (introducted in page 3~4).

 Since the potential cannot go out far enough from the dislocation source to the boundary, therefore, speaking in reverse, no size effects could at all propagate from the boundary to the source, and it is at the source that you measure local stress (because the maximum stress is what you measure and the maximum stress would exist at the hole).

I am sorry I dont quite follow you on this part. I think the "potential" we used may be different, is it?

I found the point where maximum local stress set is exactly the dislocation nucleated and emitted from. And I thought it is the local stress dominating the dislocation emission, am I right? But using local stress to interprete observed size effect may be farfecthed, because these two are both measurements not the physical characteristic (actual physics of the situation), in other words, who explain who, may be a question.

I appreciate your point on "yield stress", I also think the application of contiuum concepts on nano-scaled study may be discussable. Will hear your points on it later.

Kejie

Dear Kejie,

As you know, I have almost no background in MD. So, your replies help me learn it! Thanks for the same.

By way of a reply to your comment above, here are a few questions: What is the meaning of the term "potential cut-off distance"? Why do you need it in the theory? What would happen if the distance were to be zero? Infinite?

Putting the concern in mathematically precise terms: Does the potential have finite support? If so, does it extend all the way to the domain boundaries? The support won't extend beyond the domain boundaries, say to infinity, because the domain simply repeats, right?

The reason I raise these questions is that in the standard QM, the wavefunction of any particle like electron, proton etc. (even a photon) is spread everywhere, i.e. up to infinity, in any direction.

-----

On another count, I want to note that I made an error when I wrote a comparison of yield stress and compliance in my above post. This was a slip of the mind! (That is, something like the slip of the tongue!) The two (yield stress and compliance) cannot be compared, really speaking. Reading the other parts of your paper in a hurry, in place of yield stress, somehow, I started reading Young's modulus. I then even wrote accordingly here in my above comment. Sorry. I now stand corrected.

Kejie Zhao's picture

Dear Jadhav,

Thanks a lot for thoughtful comments. I see a basic concept in MD method, potential, should be firstly clarificated.

Regarding on your questions,What is the meaning of the term "potential cut-off distance"? Why do you need it in the theory? What would happen if the distance were to be zero? Infinite?

We know the main ingredient of a simulation is a model for the physical system. For a MD simulation, this amounts to choosing the potential, a function V(r1,r2,...rN) of the positions of nuclei, representing the potential energy of the system when the atoms are arranged in that specific configuration. This function is translationally and rotationally invariant, and is usually constructed from the relative positions of the atoms with respect to each other, rather than from the absolute positions. Then, forces are derived as the gradient of the potential with respect to atomic displacements.  The simplset choice for the potential V is to write it as a sum of pariwise interactions. Here we take the famous Lennard-Jones potentials for example: V(r)=4e*[(sigma/r)^12-(sigma/r)^6], among which the parameters e and sigma are determined by experiments for different materials, and r represents the distance between atoms. So the potential is strongly repulsive at shorter distance, passing through 0 at r=sigma and increasing steeply as r is decreased further. So according to the expression above, the potential has an infinite range. However, in practical applications, it is customary to establish a cutoff radius Rc and disregard the interactions between atoms separated by more than Rc. This results in simpler programs and enormous savings of computer resource. We can say Rc also as "potential truncation".

Of course the pairwaise potentials have been proven questionable in many materials, and numeours many-body potentials, like EAM (embedded atom method) .etc, have been developed for specific materials. However, the basic principles are almost the same. And the EAM potentials cut-off radius for FCC single crystals is often between the second and the third nearest neighbor distances.  May it give you some help on it. Thanks again. 

Kejie

Dear Kejie,

Thanks.

1. Your reply makes it clear that the overall potential function chosen by you is defined piecewise in such a way that while the individual pair potentials may be subject to a cut-off distance, since all these local potentials overlap, the overall potential function always has support everywhere in the entire region. (Please correct me if I am wrong here, because most everything given below depends on it.)

2. Given this fact, domain-size effects should be present in the results, at least in principle.

Yet, your simulation study shows otherwise. Such a result seems surprising because the repeating cell of the domain (under the periodic boundary condition assumption) is only tens/hundreds of atoms wide--not millions and billions of them.

Relying on the details of the MD technique given by you above, I would suppose that, it is this particular potential function which seems to fall short in delivering on the anticipated size effect.


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3. The maximum stress for (or during) dislocation emissions does not seem to be the most appropriate parameter in order to see whether the anticipated size effect comes up in the simulation or not.

It would be better to undertake separate, simpler studies, e.g., a study of the effect of the location of a single vacancy on the local stress experienced at the center of an otherwise contiguous single crystal (i.e. a domain without the central hole). In such a study, the variation in the local stresses at the center-point may be plotted as a single-atom vacancy is systematically brought in, one lattice position at a time, from the region boundary to the center of the domain.

Then, as the next step, a hole may be introduced at the center, and, then, a single vacancy once again brought in systematically to the domain center (i.e. the hole surface).

Simpler studies such as these may first be undertaken to see how the MD computational model behaves with the selected potential function. These data may then be used for comparison with the more complex situation such as dislocation emissions.


Best,

Ajit R. Jadhav

Kejie Zhao's picture

Dear Jadhav,

Thank you for the further attention. Concering on your discussion,

1. Your reply makes it clear that the overall potential function chosen by you is defined piecewise in such a way that while the individual pair potentials may be subject to a cut-off distance, since all these local potentials overlap, the overall potential function always has support everywhere in the entire region. (Please correct me if I am wrong here, because most everything given below depends on it.)

Right, atoms are allways interacted with their neighboring atoms (within cut-off distance), so maybe we can say the potential spread all over the sample.

2. Given this fact, domain-size effects should be present in the results, at least in principle.

Yet, your simulation study shows otherwise. Such a result seems surprising because the repeating cell of the domain (under the periodic boundary condition assumption) is only tens/hundreds of atoms wide--not millions and billions of them.

I am sorry I have not followed your thought, the fact in 1 could support the presence of domain-size effect. The domain-size effect is exactly the size effect on the yield strength metioned in my paper? Also, the simulated unit cell have included 20 thousands to 7 millions atoms when L_x increased from 24a0 to 140a0 (seen the detail in the attachmentfile Fig.1).

Relying on the details of the MD technique given by you above, I would suppose that, it is this particular potential function which seems to fall short in delivering on the anticipated size effect.

Yes, you are right. The materials behaviors is almost determined by chosen potential function. To eliminate the dependence of chosen potential function, we have re-calculated the size effects by employing another EAM potential, and similary size effects, dislocation emission and defect patterns are observed (seen page 11~12 in my paper). So we believe it is not the special potential function dominated the observed phenomenon.

It would be better to undertake separate, simpler studies, e.g., a study of the effect of the location of a single vacancy on the local stress experienced at the center of an otherwise contiguous single crystal (i.e. a domain without the central hole).

I am sorry I dont quite understand this point. On my superficial understanding: the location of maximum local stress is well known for a voided material, the linear elastic contiuum model has given the solution. Also for the practicality consideration, this study may be not applicable in a unit cell including millions of atoms, is it right?

However, I maybe misinterprete your points. Please correct them if I am wrong. Thank you!

Regards, 

Kejie

Dear Kejie,

Your points are in italics.

1. Right, atoms are allways interacted with their neighboring atoms (within cut-off distance), so maybe we can say the potential spread all over the sample.

The overall potential is spread all over the domain because (i) there are *overlaps* in between the local, pair-potentials, and (ii) the pair potentials are defined at every pair in the domain.


2. Also, the simulated unit cell have included 20 thousands to 7 millions atoms when L_x increased from 24a0 to 140a0

But, the length of the *side* still is only 140 a0, right? Not millions of a0. So, it still is, comparatively, a very small sample. (It's decidedly a nano-sized sample, not a micro- or meso-sized one.). For such a small sample, the absence of the domain size effect *is* surprising, even if the property in question is only mechanical--not optical or so.

Afer all, even an ordinary (meter-scale) structural truss with a similar number of members would show a mechanical size effect.

3. the location of maximum local stress is well known for a voided material, the linear elastic contiuum model has given the solution.

Is it for finite-sized domain? Under periodic boundary conditions? I think the absence of a suitable analytical solution precisely is the reason why you need to have as much supporting simpler data as is possible--just to see how your computational *method* behaves (apart from the specific model you have for study).

Note that in a finite-sized domain, the maximum stress will not be a constant--it will depend on the domain size. This is unlike the analytica solutions in infinitely large domains. Since the values of the maximum stress would vary, you need to simplify the problem as much as possible.

Now, in simplifying everything, first, anyone would remove the central hole--i.e. start with a plain crystal. (I didn't mention it earlier separately because I thought this much was obvious.) I then suggested adding a vacancy because that way you could see how far the distortion in the stress/strain field it produces goes. Note, I spoke of a single-atom vacancy because your model inherently carries that atomic level granularity. With this data behind you, you would have some baseline to compare a more complicated defect like dislocation.

Instead of any such background study, if you directly speak of as complex a process as emission of dislocations and also size effects taken with it, then, the first question to strike is: What do you compare your results with, esp. if you come up with a negative result (*no* size effect)? You see, there is no *outside* data (data external to your current simulation) to compare your negative result with. No empirical data coming from physical experiment. No analytical solution (with dislocations). Not even computational results by *other* methods. And you present a result that goes against the anticipation. Now, if you didn't have trials by other computational methods, for comparison then, with your already working software system, you could have at least tried more simple MD trials. One could then get a somewhat better judgment of the overall picture. That's what I suggested.

So, overall, the apparent lack of comparative data was the main reason behind suggesting those simpler trials. Take it or leave it.

Ajit

Kejie Zhao's picture

Dear Ajit,

Thanks a lot for your patient suggestion. I see the "domain-size effct" is the effect on the local enrgy or max stress, but not on the yield strength.

Yes, you are right. Maybe the first quetion I should clarify is the dependence of max stress in finite domain size, by simplying my models. This issue is different from the analytical model, and would be interesting in my further study. Also you metioned the comparative date with MD study, I particularly agree with you on this point. Actually MD simulated results are mostly often effcted by the chosen potential function, the algorithm, the technique details and so on. Many trival things in calculation maybe lead to the reverse. So it is better to have a experiemental result to test the reliability the simulation date. If can not do this, maybe we should check the simulation results at the simplest conditions.

Best,

Kejie

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Ajit R. Jadhav

 

Dear Kejie,

You have reported that the maximum stress during dislocation emission is independent of the size of the repeating domain (i.e. the unit cell) under the periodic boundary condition.

Earlier in this thread, I had concluded that the maximum stress ought to depend on the size of the region in any finite-sized domain. Thus, I argued that there should be a size dependency, at least in principle.

However, on second thoughts, thinking more carefully about the dynamic situation you study, I think that there exists enough room to suppose otherwise.

The logic in support of your results is that since the sample is so small, there cannot be any significant work-hardening. Hence, any emitted dislocation would undergo almost unhindered motion (i.e. the dislocation motion would experience the simplest constant drag condition). In such a case, the maximum stress near the central hole simply cannot go above the local value required for dislocation emission. In other words, what you are modeling is that which is technically called the elastic-perfectly plastic material.

However, I would be still uncomfortable as to how this reasoning gels with another fact, namely, that you don't have a free-surface boundary but a periodic one.

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This case study seems to be an easy situation but apparently isn't so. In any case, it would be difficult to have any form of common sense about it--whether true or a misleading one. As such, I would welcome comments from others on all the above discussed issues.


Ajit R. Jadhav

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