iMechanica - Journal Club November 2007: Surface Effects on Nanomaterials - Comments

Yes, it's always hard for computer scientists...

Adrian S. J. Koh — Mon, 10 Dec 2007 03:22:09 -0500

Dear Prof. Upmanyu,

Thanks for your comprehensive review of the validity of MD potentials. From my presentations @ various conferences and even for my PhD oral defense, I almost always face the question of validity of MD simulations. It has indeed been an Achilles' heel. I have passionately defended its validity for both bulk properties, to atomistic interactions but usually left the questioners half-convinced. I think the bane of computational scientists is to defend the value of modeling and simulations - Why are we spending so much time developing various models, from quantum to atomistic to continuum simulations, and even thinking how to couple the multiple length- and time-scales together when most phenomena can now be observed experimentally?

IMHO, in order for computational physicists to add value to research, they must make their modeling and simulation works forward-looking. That is, using existing experimental results simply to benchmark and verify our model, and then, extend the use of our model to something unobservable at this present state of technology. For instance, it is difficult to fabricate nanowires smaller than 1.0nm, we should use our models (boldly) to predict mechanical, thermal, electrical, optical and magnetic properties at this size scale. This, I think, has been done by a couple of excellent groups and is the way to go for computational physicists.

Thanks for your reply, I now have a much better understanding on my stuff :).

Cheerios,

Adrian KSJ

non-linear effects

mupmanyu — Fri, 07 Dec 2007 04:42:24 -0500

Adrian:

Your point regarding suitability of interaction potentials is certainly valid. It boils down to how well we capture electronic effects via effective nuclei-nuclei interaction frameworks. At the very least, we are interested in getting qualitative trends right, and for this reason alone most good potentials are fit to DFT data, in addition to experiments. This transfer of data is often forgotten in most opinions that dismiss the validity of inter-atomic potentials, and the need for more explicit ab-initio frameworks. Usually, the fit is to bulk data, and therefore its validity is always questionable for scenarios where surface effects become important. But, there is no reason why the data has to be limited to bulk data. You can always perform a DFT calculation on a assumed surface structure, and try to fit your potential to parameters associated with the surface structure. Then, it not a big jump in trying to predict the physics of other surface structures, or combinations thereof.

If you feel that in a particular system the surface can induce size effects whose origin is electronic in nature, and ignored by the fitting database (i.e. your intuition), there are no easy answers. More accuracy in the inter-atomic physics means you lose the ability to determine effects at larger length scales. Searching for global mininum in morphology becomes difficult, and you have to rely on experiments (never a bad thing) or start with an assumed structure (surface facets, roughness) - again, relying on your intuition (can be good or bad!). You also run into problems with meaningful statistics on time dependent phenomena (if quantum MD is your poison of choice, or for that matter, even MD!), or effect of local perturbations due to deformation and surface chemistry.

As a way around, free electron models of nanowires with radii of the order of the Fermi length in the system have been employed to study how the wires neck (see work by the U of Arizona group - Stafford/Goldstein/Burki), and might offer a framework to study these systems.

While I am on this topic, I must mention that there is definitely a perception in the scientific community that if you are doing DFT, you are doing something very precise, or that it is an exact calculation. To get more insight into this claim, I have always asked my students to see how a DFT calculation works. Is it immune to any assumptions/approximations for what is essentially solving an N-body quantum mechanical problem? What kind of assumptions are made, and how do they impact the problem at hand?

To answer your question on surface effects as linear combination of a surface and a core, the two don't contribute independent. The surface stress will always induce a strain in the core (the bulk sans the surface). At sufficiently small sizes, we have found that the core modulus in copper nanowires can no longer be assumed to be the bulk modulus for that system, as it is now stiffer. Non-linearities in response of the strained core become important as well, and sometimes even dominate the size effect of the core+surface system. We expect this to be a general response. See http://dx.doi.org/10.1103/PhysRevB.71.241403

PRB paper on size-dependent elastic properties of ZnO nanofilms

Xiaodong Li — Thu, 06 Dec 2007 22:01:57 -0500

Following up this topic, I just realized that Xi Chen's group had already published an excellent PRB paper on size-dependent elastic properties of ZnO nanofilms. His paper has nice results in the thickness range less than 8 nm. The size dependence of elastic modulus of ZnO nanofilms is investigated by using atomistic simulations. The strain energy and elastic stiffness of the surface and interior atomic layers, as well as interlayer interactions, are decoupled. The surface stiffness is found to be much lower than that of the interior layers and bulk counterpart, and with the decrease of film thickness, the residual tension-stiffened interior atomic layers are the main contributions of the increased elastic modulus of nanofilms.

Size dependency of the elastic modulus of ZnO nanowires

Xiaodong Li — Thu, 06 Dec 2007 11:57:06 -0500

Guofeng Wang and I just published a peper in APL about surface effects. Relation between the elastic modulus and the diameter (D) of ZnO nanowires was elucidated using a model with the calculated ZnO surface stresses as input. We predict for ZnO nanowires due to surface stress effect: (1) when D>20 nm, the elastic modulus would be lower than the bulk modulus and decrease with the decreasing diameter, (2) when 20 nm>D>2 nm, the nanowires with a longer length and a wurtzite crystal structure could be mechanically unstable, and (3) when D<2 nm, the elastic modulus would be higher than that of the bulk value and increase with a decrease in nanowire diameter.

Has anyone paused and thought about...

Adrian S. J. Koh — Thu, 06 Dec 2007 04:23:25 -0500

Hi Harold,

Thanks for your reply. Yes, I agree it would be interesting to observe the temperature-dependency of surface elasticity. It would appear to be two opposing poles between temperature and surface elasticity as higher temperature would result in surface pre-melting and therefore, logically reducing its stiffness. This would surely soften a nanowire that is made up of a large proportion of surface atoms, as opposed to strenghtening it.

Now back to why I titled my post as such... but I was recently thinking... computational mechanicians have been using many-body potentials like EAM, MEAM, FS, SC, QSC, TB-SMA freely in their analysis. But has anyone paused to think about the validity of these potentials, at a very small size scale such at 1.0nm? Can we still interpret the simulation results fairly and correctly at these scales? Does verification of the potential based on stacking fault energy, vacancy formation energy, elastic constants, bulk modulus and thermal constants be sufficient to support the validity of using these potentials at an extremely small size scale? There must be a lower limit as to how small can MD go before its validity becomes a question mark.

I'm probably a little condescending here but I think it could be time for computational mechanicians to pause and think about this.

Cheerios,

Adrian KSJ

more on surface elasticity

Harold S. Park — Tue, 27 Nov 2007 21:45:16 -0500

Hi Adrian:

Sorry for the late response due to travel. Regarding this comment:

"I currently, have several pertinent questions on surface elasticity in
mind: Is surface elasticity, like its bulk counterpart, a unique value
for a specific material, or is the surface elasticity size-dependent
or, curvature-dependent, or even temperature-dependent? Are surface
elasticity always larger than the bulk elasticity, are there
exceptions? Can the effective elasticity of a surface-atom dominated
nanowire be computed simply as the weighted-average of the surface and
bulk elasticities?"

I have a couple thoughts. (1) In the Zhou and Huang paper I posted above, they demonstrate that the surface can be either stiffer or softer than the bulk depending on the surface orientation. (2) You raised a great point regarding surface elasticity as a function of temperature. In my opinion, it definitely is, though I have not seen it quantified anywhere. (3) I also do not feel that the elasticity (effective stiffness) of nanowires can be computed through a weighted average of the bulk and surface elasticities, and that the relationship is a nonlinear function of some sort. Moneesh Upmanyu, who has also done a lot of work on this, might have some better insights.

Surface effects (size effects) need studies/supports

Xiaodong Li — Sat, 24 Nov 2007 14:32:11 -0500

Thanks so much for the exciting discussions. I very much like the point how to tune surface to get new functionalities. I think there is still a lot we need to work on. I hope we can promote this topic and also attract funding agencies' attention. This topic needs more fundamental research which in turn advances the true applications of nanomaterials.

Surface Effects - Bridging the Small & Really Small Scales

Adrian S. J. Koh — Mon, 19 Nov 2007 03:17:54 -0500

Dear Harold,

Thanks for proposing this topic for the JClub this month.

I have been performing MD simulations for metallic nanowires/nanorods for my entire PhD tenure, and came to gradual realization that surface effects is THE core phenomenon giving small, single-crystalline nanostructures their puzzling, unique and sometimes exceptional characteristics. Much have been said about size and interface effects for nanomaterials but that would only apply for polycrystalline nanomaterials that contain constituent grains - applicable for larger (in a relative sense) nanomaterials in excess of 10nm. Hence, while the size and interface effects characterize discrete atomic behavior in nanomaterials, it is the surface effects that determine their behavior down to quantum, sub-atomic levels.

While I concede that, due to the small vacancy formation energy relative to its cohesive energy in metallic bonding, a single-crystalline, perfect nanostructure is physically difficult to fabricate, one could not say that this will always be the case (there are, in fact, promising emerging techniques to fabricate near defect-free metallic nanowires). The unique properties at this size scale is too attractive and interesting for researchers to forgo its research, simply based on the reason that such materials are currently difficult to fabricate.

There exist very good potentials for modeling metallic d-bonds. One candidate is the potentials derived from the Tight-Binding Second Moment Approximation (TB-SMA) Method (a good exposition of this method could be found in The Physics of Metals, J. Friedel, 1969, pp. 340-408). The benchmark TB-SMA potential was proposed by Finnis and Sinclair, and further verified by Ackland, Finnis & Vitek for its validity for all band fillings. Simplified expressions for the cohesive pair functional was subsequently proposed by Sutton and Chen, and parametrized for FCC metals, alloys and corrected for Quantum Effects by Kimura et al. It is based on these excellent works that MD simulation of small metallic nanowires becomes a reliable tool in providing a glimpse into the fascinating world of sub-10nm metallic nanowires/nanorods.

As metallic bonds are non-directional, surface atoms would be reasonably taken to be the atoms lying within one lattice constant from the outermost atomic layer, where the coordination number is less than 12 (for FCC metals). This makes metallic nanowires with a thickness dimension of 1.0nm or less fully surface-atom dominated (i.e. 100% surface atoms). As Harold has mentioned in his blog, surface stress and elasticity has led to many new physics to be discovered in nanomaterials. I have some works relating to surface effects leading to enhanced dynamic and transport properties, which I shall with-hold its details here as the works are in the midst of review.

I currently, have several pertinent questions on surface elasticity in mind: Is surface elasticity, like its bulk counterpart, a unique value for a specific material, or is the surface elasticity size-dependent or, curvature-dependent, or even temperature-dependent? Are surface elasticity always larger than the bulk elasticity, are there exceptions? Can the effective elasticity of a surface-atom dominated nanowire be computed simply as the weighted-average of the surface and bulk elasticities?

There are, in my view, deep and intriguing questions to ask in the field of nanoscale surface effects. These questions, I would venture to say, are currently in the realm of the computational physicists/scientists. I await the day where we could quantify surface stress and elasticity in the laboratory.

Cheerios,

Adrian KSJ

surface chemistry

Harold S. Park — Fri, 16 Nov 2007 18:13:38 -0500

Moneesh:

Thanks for raising a great point - the effects of surface chemistry. I first noticed this in terms of oxide layers on metal nanowires which will obviously change the scale and magnitude of the surface effect (and induce charge effects, etc); however, it seems clear that while having adsorbed atoms of some sort on the surface will make the bonding more bulk-like for the surface atoms, the surface stress will still exist, due to the absorbate-substrate interactions. An experimental reference to this can be found here (http://link.aps.org/abstract/PRB/v73/e235409).

The surface chemistry issue is a good one to explore; however, it's worth noting that there is still considerable disagreement on the elastic properties of nanomaterials due to surface stress/elastic effects. Moreover, my personal feeling is that even if you have passivated surface layers of some sort, you still should be able to define an effective change in stiffness that results.

self-consistent reference

mupmanyu — Fri, 16 Nov 2007 13:19:03 -0500

Harold:

I had a specific comment regarding the work that you referenced on Si nanowires. The chemistry of the surface is crucial - it is hydrogen-passivated. To motivate that study, we continue to think of the microscopic origin of size effect as some modification to the environment of the surface atoms, relative to a reference. What if I passivate the surface with an environment that is quite nearly that in the bulk, as is the case in H-passivated Si? Would I still see a surface effect? The authors answer that in the affirmative, although the size dependence of the Young's modulus is considerably reduced. Microscopic considerations become necessary to understand the origin of this surface effect, and it is not so easy to decouple the effect of several interactions on the surface (H-H, Si-Si, Si-H, their relaxed configuration, and eventually the change with strain). Another interesting observation is that the lateral facets do not passivate equally. Therefore, besides the obvious geometric anisotropy due to dimensionality, the interplay of the surface chemistry with the distribution of facets becomes important for the overall elasticity.

The paper is an excellent example of how chemistry and morphology conspire to induce interesting size effects. It is qualitatively different here compared to what we have found in Cu nanowires (http://dx.doi.org/10.1103/PhysRevB.71.241403). The much larger surface stresses induce a net relaxation that can drive the bulk well past its linear elastic response. Then, the issue becomes that of defining a surface elastic tensor relative to this non-linearly strained bulk. The situation is analogous to defining the segregation of dirt at a defect in the bulk (say, Cottrell atmosphere around a dislocation). If the amount of dirt is conserved, the segregation changes the bulk concentration. The segregation is defined with respect to the bulk, which itself becomes a moving target. Larche-Cahn is an excellent place to start on frameworks to tackle such issues in a self-consistent manner.

To broaden the discussion a bit, morphology can be size dependent as well, especially in supramolecular filamentous assemblies. As example, our work on ropes of carbon nanotubes show that they twist over nanoscopic scales, quite like a twisted n-ply cable (http://dx.doi.org/10.1016/j.carbon.2005.05.040). The degree of the twist is strongly size dependent, and has a major impact on the axial load-bearing characteristics of these nanostructures.

Re: residual stress in nanowire bending

Xiaodong Li — Wed, 14 Nov 2007 18:33:46 -0500

Thanks a lot Harold for pointing out this. Sorry for my delay in replying your post (I was traveling). In general, residual stresses exist in synthesized thin films/coatings. We still do not (experimentally) if this is true for nanowires. A novel experimental design is needed to study this. From my previous studies, my group did not take this into account. However, I think this is a good subject that needs to be investigated from both experiment and modeling.

Surface effects/self-similar objects ...

Phani K. Nukala — Wed, 14 Nov 2007 15:37:46 -0500

Dear Lee:
I thought I just posted a response, but then my comp went crazy. Anyway, If I understand you correctly, this may be what you are referring to.
First, let $d$ be the dimension of a body ${\cal B}$, which can have
Riemannian metric defined on it (It is not important here for the discussion).
All we are saying is that the body can be parametrized by $d$ parameters. For example, a surface S in \Re^3 can be parametrized by $X: (u,v) \in \Re^2 \rightarrow S \subset \Re^3$. Now, consider a characteristic dimension $L$ in the body. Above a lower cutoff length scale, any of the macroscopic (thermodynamically extensive) properties such as mass scale as $L^d$ times a thermodynamically intensive (specific) quantity such as density (specific density). This is valid in the thermodynamic limit.

Now, here is the big difference. The size effects that we are discussing here
(such as the surface effects, confinement effects and/or nonlocal effects
etc...) refer to how the so-called intensive/specific quantities such as
Young's modulus change as the system sizes approach or fall below the lower cutoff length scale. In this limit, one must pay attention to how the
thermodynamic limit (N,V \rightarrow \infty, but N/V is constant) is taken.
Typically, the surface contribution term, which vanishes in the thermodynamic limit, does not vanish below this lower cutoff length scale. Consequently, we start seeing size effect dependence in these so-called thermodynamically intensive quantities when the characteristic dimension approaches/below lower cutoff length scale. This is true for Youngs modulus, specific resistivity/conductivity, specific energy and many others. Size effects we are referring here deal mainly with these quantities.

Now, on the other hand, the paper you are referring to deals with fractal
structures, wherein the mass scales as $L^D$ times density, where $D$ is the fractal dimension of the object. Here the scaling laws refer to how the
extensive quantities such as volume and area scale as the length scale of
observation is changed. Note that such length scale changes have no effect on the intensive quantities such as specific conductivity say. These are
appropriate in obtaining the scaling laws of materials such as foam and other
fractal structures.

In summary, the typical size effects such as surface effects enter when the
characteristic length scale is below a lower cutoff length scale wherein
thermodynamic limit can't be taken because of the existence of finite size
system effects. Consequently, these size effects influence the so-called
intensive quantities. On the other hand, the scaling laws due to fractal
nature of objects do not change the intensive quantities. However, the
apparent system size effect comes due to the scaling of extensive quantities
such as the volume, mass etc. Hope I am clear, although I might have rambled along the way.

reply to Phani's comments

Yonggang Huang — Sun, 11 Nov 2007 09:57:22 -0500

Phani,

Sorry about the late reply.

There are different definitions for curvatures in the differential geometry. For example, there are different curves through the same point on the surfuce along the same direction, and they have different values of curvature. The tensor b and tensor B are connected with the curves of normal curvatures (i.e., the curvatures of the normal sections of the undeformed and deformed surfaces.)

We are not interested in the curvatures of different curves lying in the surface. Instead, we are only interested in the straight distance between two neighboring atoms on the surface, which are required for the computation of energy via the Brenner potential. Such a straight distance can be obtained from the difference between the 2nd fundamental forms b and B, via the Taylor expansion.

In summary, the calculation of straight distance needs only the difference between the 2nd fundamental forms. We have called this difference the curvature, but its name is not critical.

reply to Pradeep and Harold on the curvature effect

Yonggang Huang — Sun, 11 Nov 2007 08:11:26 -0500

Pradeep and Harold,

Thanks for your comments. I am sorry about the late reply.

I want to distinguish two curvature effects for nanowires. One is due to the overall bending of nanowires, which we can call the "global" curvature effect. Every atom in the nanowire will experience this "global" curvature effect once the nanowire is subjected to bending.

The other is the "local" curvature effect because the surface atoms of the nanowire are on a curved surface. Even without any overll bending these surface atoms behave differently from those inside the nanowire because (1) they have different environments (e.g., number of bonds); and (2) they are on a curved surface.

Of course these two curvature effects are coupled once the nanowire is subjected to bending.

Carbon nanotue is an example, for which all atoms are surface atoms. The JMPS paper (An atomistic-based finite-deformation shell theory for single-wall carbon nanotubes, JMPS) presents a way to account for both curvature effects.

Rui, Thanks for your

Pradeep Sharma — Thu, 08 Nov 2007 11:45:24 -0500

Rui,

Thanks for your comments.....there is actually a rich discussion on non-affine deformation in polymers and amorphous materials. One can prove that significant non-affine deformations lead to a "non-local effect". The paper by DiDonna and Lubensky actually hints towards it (recall that this paper was discussed during Jan 2007 J-club issue). I have unpublished notes showing this as well. Your intuitiion is correct. This nonlocal-nonaffine deformation is linked with gradation of properties. As far as surface effects are considered, I have to disagree here....when both effects are present (say close to the surface) it often may not be easy to distinguish between the two but here is a simple physical arguement that can help differentiate the two and underscore why polymers have low surface energy related effects:

surface quantities (such as energy or stress) are related to the difference in the behavior of atoms close to the surface as compared with the "rest" (e.g. bulk in cases of large structures). The difference is in coordination number, charge distribution etc. In a crystalline solid, a free surface present a drastic difference since there is an abrupt change in coordination number, symmetry etc. In an amorphous material, statistically, the environment (even near the surface) changes much less (due to the randomness) compared to the bulk. This results in lower surface excess quantities. Non-affine deformations are however present and the correlation length of the non-affine deformations essentially produces the size-effect (the same correlation length in crystalline materials is negligible but surface effects are not).

Incidentally, there is a nice paper that appeared recently which provides an overview of some aspects of deformation and size-effects in amorphous materials.

Journal Club November 2007: Surface Effects on Nanomaterials

Harold S. Park — Mon, 29 Oct 2007 18:35:58 -0400

Nanoscale materials, including thin films, quantum dots, nanowires, nanobelts, etc – are all structurally unique because they have a relatively high ratio of surface area to volume ratio. This increase in surface area to volume ratio is important for nanomaterials because wide and unexpected variations in mechanical and other physical properties, such as thermal, electrical and optical, have been found to scale in some proportion to increase in surface area to volume ratio. The critical impact of this is that standard continuum relations, which do not account for size-dependence or the discrete nature of atomistic surfaces, are no longer valid at the nanometer length scale.

Mechanically speaking, there are two distinct but critical effects due to nanoscale free surfaces. The first effect is that of surface stress. Surface stresses exist in nanomaterials due to the fact that atoms lying at the material surfaces have a different bonding configuration as compared to bulk atoms. Because these atoms are therefore not at an energy minimum, surface stresses exist which serve to cause these atoms to deform in order to find their minimum energy configuration. A very good review and introduction to surface stresses is found (http://dx.doi.org/10.1016/0079-6816(94)90005-1), while an interesting effect of surface stresses on nanomaterials can be found (http://dx.doi.org/10.1038/nmat977).

Surface elasticity is another effect that occurs due to the lack of bonding neighbors for surface atoms. Again because surface atoms have a different bonding environment than atoms that lie within the material bulk, the elastic properties (stiffness in particular) of surfaces differ from those of an idealized bulk material, and the effects of the difference between surface and bulk elastic properties become magnified as the surface area to volume ratio increases with decreasing structural dimension. In particular, recent theoretical work has found that at the nanoscale, the type of surface ({100} vs. {110} vs. {111}) that the material exposes can significantly alter the elastic properties as compared to the bulk material (http://dx.doi.org/10.1063/1.1682698)
(http://dx.doi.org/10.1016/j.jmps.2005.02.012).

The above mentioned issues of surface stress and surface elasticity lead to many new physics and phenomena in nanomaterials that are not well-understood, and thus not predictable or as yet controllable. To initiate the discussion, I present problems and applications where surface effects are important and which seem amenable to a mechanics investigation, either experimental, theoretical, or computational. These issues are:
(1) How the Young’s modulus of nanomaterials depends upon size, surface orientation, crystal orientation, and geometry (http://dx.doi.org/10.1103/PhysRevB.69.165410).
(2) Why do NEMS exhibit dramatic decreases in quality (Q)-factors with increasing surface area to volume ratio (http://physicsworldarchive.iop.org/index.cfm?action=summary&doc=14%2F2%2...).
(3) How to best incorporate surface effects into multiscale computational models such that realistic studies of nanomaterials and nanomaterial design can be performed?
(4) Nanoscale sensing (force, mass, pressure) requires the accurate prediction of resonant frequencies. In light of the preceding discussion on surface elastic effects, and considering recent experimental results (http://dx.doi.org/10.1063/1.2388925) indicating that the stiffness of the substrate nanomaterial does not remain constant due to adsorbate/substrate interactions, how can predictive computational and theoretical models (http://dx.doi.org/10.1063/1.359562) be developed to understand and predict the results of nanoscale sensing experiments?
(5) How are the mechanical properties of surface-dominated nanomaterials altered when coupled physics phenomena, such as thermoelasticity, are considered (http://dx.doi.org/10.1103/PhysRevB.72.113404).