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# Journal Club November 2007: Surface Effects on Nanomaterials

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continuum relations, which do not account for size-dependence or the discrete nature of atomistic surfaces, are no longer valid at the nanometer length scale.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).

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cammarataPSS1994.pdf | 1.74 MB |

chenJAP1995.pdf | 612.16 KB |

cuenotPRB2004.pdf | 51.96 KB |

diaoNM2003.pdf | 720.6 KB |

dingrevilleJMPS2005.pdf | 386.42 KB |

pathakPRB2005.pdf | 189.48 KB |

roukesPW2001.pdf | 16.04 MB |

tamayoAPL2006.pdf | 95.21 KB |

zhouAPL2004.pdf | 236.9 KB |

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## Comments

## Modeling Surface/Interface Effects in Micro/Nano Systems

I would like to point out that recently we could formulate a higher-order theory that can successfully incorporate the surface/interface effects on the scale-dependent yield strength and strain hardening rates in micro/nano systems (doi:10.1016/j.ijplas.2007.09.005). This paper is in press in the International Journal of Plasticity.

Here is the abstract of this paper:

The effect of the material microstructural interfaces increases as the surface-to-volume ratio increases. It is shown in this work that interfacial effects have a profound impact on the scale-dependent yield strength and strain hardening of micro/nano-systems even under uniform stressing. This is achieved by adopting a higher-order gradient-dependent plasticity theory [Abu Al-Rub, R.K., Voyiadjis, G.Z., Bammann, D.J., 2007. A thermodynamic based higher-order gradient theory for size dependent plasticity. Int. J. Solids Struct. 44, 2888–2923] that enforces microscopic boundary conditions at interfaces and free surfaces. Those nonstandard boundary conditions relate a microtraction stress to the interfacial energy at the interface. In addition to the nonlocal yield condition for the material’s bulk, a microscopic yield condition for the interface is presented, which determines the stress at which the interface begins to deform plastically and harden. Hence, two material length scales are incorporated: one for the bulk and the other for the interface. Different expressions for the interfacial energy are investigated. The effect of the interfacial yield strength and interfacial hardening are studied by analytically solving a one-dimensional Hall–Petch-type size effect problem. It is found that when assuming compliant interfaces the interface properties control both the material’s global yield strength and rates of strain hardening such that the interfacial strength controls the global yield strength whereas the interfacial hardening controls both the global yield strength and strain hardening rates. On the other hand, when assuming a stiff interface, the bulk length scale controls both the global yield strength and strain hardening rates. Moreover, it is found that in order to correctly predict the increase in the yield strength with decreasing size, the interfacial length scale should scale the magnitude of both the interfacial yield strength and interfacial hardening.

Rashid K. Abu Al-Rub, Ph.D. Texas A&M University

## Nanoscale sensing

Recently, to my knowlege, nano-scale sensing has been attracted for gaining insight into various physical phenomena. For instance, nano-scale or micro-scale resonators have allowed various research groups to detect the quantum state of resonators [Ref. A. Naik et al.,

Nature,443, p193 (2006)]. In a very recent year, Roukes group at Caltech reported the zepto-gram mass sensing of molecules by NEMS resonator [Ref. Yang et al.,Nano Lett.,6, p586 (2006)]. In such a case, they detected inorganic molecules such as Xe, N2, etc., so that the detection principle is the contribution of molecular mass into change of resonant frequency. However, this detection principle may not be directly applicable to biomolecular detection by using NEMS resonators. As mentioned above, the surface effect may play a role in resonant frequency shift, if the thickness of biomolecular layer becomes comparable to resonator's thickness [see Ref. A.K. Gupta et al.,PNAS,103, p13362]. Further, as stated above, the surface stress effect may affect the resonant frequency shift. However, as far as I know, the rigorous model for explaning such effect has not been well provided yet. I just recently considered the simple model which allows me to explain how DNA-DNA interactions affect the resonance for DNA adsorption on a resonator's surface [see Ref. K. Eom et al.,PRB,76, 113408 (2007); preprint is available at http://www.imechanica.org/node/1620].As far as I know, the understanding how molecular interactions affect the dynamic behavior of a resonator has been still lack. I think that understanding such issue in biomolecular detection by using a resonator is still open.

## Dear Kilho: Thanks for your

Dear Kilho:

Thanks for your comments, and for sharing your work - it is definitely timely and elegant. I tend to agree with you regarding the surface effects on the resonator shifts for following reasons:

1. The fundamental resonant frequency of the sensing element (say a nanowire) is definitely impacted by surface stress. The dependency of the shift is likely to be either positive or negative on a variety of factors, but there definitely is a shift.

2. There is also adsorbate-substrate bonding that changes both the local elastic properties (due to the new bonding energy/stiffness), and thus also the local surface stress.

3. Most models analytic models that I have seen so far use pair potentials to describe the adsorbate/substrate interactions - as is well known, these do not work when applied to metals and semiconductors, i.e. metals show a positive tensile stress that induces contraction of the nanostructures, while pair potentials such as LJ predict a negative surface stress, i.e. a metal that is modeled using LJ will expand rather than contract.

## Re: Modeling provides insights into surface effects

Great! The theme of this month's J-Club is very timely. We had excellent discussions on experimental nanomechanics in May 2007. This month's J-Club will provide insightful information/mechanisms about surface/size effects. I hope discussions here will further stimulate our interests in the fundamemntal understanding of the mechanisms that govern the surface/size effects. Suggestions from the modeling side on futher experimental work are welcome.

## Dear Xiaodong: Thank you -

Dear Xiaodong:

Thank you - I really enjoyed the Jclub you did on experimental building blocks of nanostructures. As an experimentalist, I had a few questions for you, and for general discussion:

1. The two main ways to obtain the elastic properties of nanostructures (wires, nanotubes, etc) are bending and resonance. Do you have any insights and opinions into which one is easier to perform, or more accurate, i.e. believable?

2. The resonance of nanostructures is generally obtained via electrostatic actuation. Is there any evidence that the Young's modulus that is measured is altered significantly by the electric field? See the following link:

(http://dx.doi.org:10.1063/1.2358848)

## Calibration is the key for experimental nanomechanics

Thanks a lot Harold. Experimentally measuring the elastic properties of nanomaterials is VERY challenging. Tons of papers have been published on the synthesis of nanomaterials, but only very few papers about experimental measurements are available in literature. I think that bending and resonance are all difficult to carry out. Calibration is the key to get reliable results. Cross-checking with different techniques is greatly needed to get joint reliablity of measured results. For instance, a detailed calibration procedure for AFM three-point bending has been described in the following paper.

Hai Ni, Xiaodong Li, and Hongsheng Gao, "Elastic Modulus of Amorphous Silicon Oxide Nanowires," Applied Physics Letters, 88(2006) 043108.

Electric field may affect surface electron density and configuration, leading to different surface stresses. This may affect the apparent elastic modulus of a nanomaterial (like a nanowire).

I think that experimental and modeling should team up. I very much appreciate this month's J-Club and your great efforts.

## more on experiments

Xiaodong,

I agree that modeling and experiment should team up to lend greater insights to nanomechanical behavior and properties. Along those lines, another thing I was wondering - when you are performing bending tests on the nanowires, do you have any insights or ideas as to the state of stress (residual due to synthesis or due to surface stress effects, etc) in the nanowires prior to bending? In general, is it well-known what effect the CVD synthesis process, or other synthesis processes have on the nanowire state of stress. Thanks for your comments.

## Re: residual stress in nanowire bending

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.

## Harold, Nice comments.

Harold,

Nice comments. Are the two aspects (surface stress and surface elasticity) related? Do they have to be accounted for separately?

Another question is, besides the strain, should one account for the effect of curvature? For nanowires and nanobelts, the deformation induced by curvature is important. This is not accounted for in the Cauchy-Born rule.

## surface stress, elasticity and curvature

Young:

Surface stress and surface elasticity are related. There is a very nice work by Shenoy that discusses this (http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=PRB...), and I will summarize. The surface elasticity can be written like Hooke's law, except for surfaces, with a surface stiffness tensor. However, the elastic constants that comprise the surface stiffness tensor vary in curvature (as a function of strain) depending on whether relaxation due to surface stresses is accounted for first. Another way of looking at this is that not accounting for surface stress effects (which cause the crystal to relax) means that the surface elastic constants will be evaluated from a configuration that is not the minimum energy configuration. Thus, the two are strongly related.

A more interesting question from my perspective is whether using the traditional continuum definitions of surface stress and surface energy, one can predict the amount of relaxation due to surface stress a nanostructure undergoes. Typically what has been done is that the surface elastic constants are evaluated using MD simulations; of course, the MD simulations can predict the minimum energy configurations, but what about in general for larger, arbitrary nanostructures?

With regards to the curvature, I don't think this has been strongly considered in nanowire modeling, to the best of my knowledge. I would suspect that it plays a larger role in nanowires than in nanotubes, for example, due to the fact that the surface elastic properties and surface stresses would lead to greater strain gradients than in nanotubes, though the disclaimer is that this is speculation on my part.

## curvature effect

Harold,

Thanks for the explanation the aspects of surface stress and surface elasticity.

As for the curvature effect, nanotubes display strong curvature effect. For examples, the membrane theories for carbon nanotubes based on the Cauchy-Born rule cannot predict the important effect of bending and curvature, not can they give the buckling load.

The recent work by Wu et al. (An atomistic-based finite-deformation shell theory for single-wall carbon nanotubes, JMPS) clearly shows the importance of the curvature effect. The essential idea is to extend the Cauchy-Born rule, which only accounts for the effect of strain, to include the effect of surface curvature. In other words, the Cauchy-Born rule involves strain as input (and stress as output), while this approach involves both strain and curvature as input (and stress and moment as output).

## comments

Young and Harold,

I hope I have understood your discussions correctly; here are a comment on the curvature effect and some general thoughts:

the theory of surface elasticity has been developed (to some extent in parallel) by two groups of people: a more physically based perspective based on Gibbs notions (Cahn, Cammarata, Johnson, Muller, Ibach, Larche and many others) and one based on rigorous mechanics axioms (Murdoch and Gurtin, Steigmann and Ogden). Ironically, some of the confusing elements that appear to exist in the contemporary mechanics literature in regards to surface elasticity are perhaps due to a lack of a "clean and clear" reconciliation between the two thought processes. I don't believe that either of the frameworks are at odds with each other (just different flavors).

In both formalisms, far as a I understand, curvature naturally enters the picture and plays a key role. Young: as you pointed out, curvature of course plays an important role in nanotubes but is of quite a different nature than the role of curvature in surface elasticity.

When teaching surface elasticity in a graduate course I heavily rely on two papers: Cammarata, R.C., "Surface and Interface Stress Effects in Thin Films,"

Prog. Surf. Sci.46 (1994): 1.and continuum theory of elastic material surfaces, Archs Rat

ME Gurtin, AI Murdoch - Mech. Anal, 1975.

Recently there have been some nice review articles on this subject also which I mentioned in a previous post:

Elastic effects on surface physics, Surface Science Reports, Volume 54, Issues 5-8, August 2004, Pages 157-258Pierre Müller and Andrés Saúl

Surface thermodynamics revisited, Surface Science Reports, Volume 58, Issues 5-8, September 2005, Pages 111-239A.I. Rusanov

## Free access to articles cited in journal club?

Harold,

By citing only articles that are not freely accesible, you are limiting journal club discussions to academic members only. In my case I can't order articles at work (industry) as they are not related to current projects, and I do not have access to a library having the articles you cite. Please either provide a more self-contained information about the topic including some formulas/equations or give at least some freely accessible citations. Same comment for Rashid: if your article is under press and you deem it interesting for the present discussion, why do you not put a link to a preprint so we can read it?

Thanks, Mihai

## j-club articles

Dear Mihai:

I have attached the papers to the original J-club post.

Regards,

Harold

## Second fundamental form ...

Dear Yonggang,

Through your post, I became curious about your article in JMPS and looked through it briefly. I am not sure I completely understand/agree with the definition of curvature as defined there. Curvature is an intrinsic concept of surfaces and is defined through the Levi-Cevita connection. The second fundamental form however defines only the normal curvature of lines on the surfaces (manifolds). For paths on surfaces, there is also geodesic curvature defined through parallel ttransport. Both these components define the curvature of a path. So, I guess my question is how the difference in second fundamental form is equal to curvature tensor. It may be an adequate quantity for small deformations. But since the title of your paper says finite deformation, I am not sure that the difference in second fundamental form is a correct definition of curvature.

Best Regards

Phani Nukala.

## What about the geometry?

The Euclid space is familiar to all of us; we define the stress, Young’s modulus in this space. However, the true space in the world is not the Euclid space, it is Riemann space, which is not regular as the Euclid space. Therefore, the definitions of these physical qualities may not proper. For example, the D denotes the dimension of the space. As a plane D=2 and for space, D=3. The D of the true space is about 2-3. Therefore, as the scale is into nanometer, D will close to 2, then the surface effect is very important. However, for the large materials (bulk materials), D=3. As our definition of the Young’s modulus is in Euclid space, which ignored that the D could not equal to 3. Therefore, the Young’s modulus of nanomaterials (D=2) could not be correctly predicted by the definition in D=3. Therefore, the large difference between the Young’s modulus in large scale and it is in nano scale may induced by the wrong definition. A different model, which will consider the D changing as the scale, should be proposed.

Lee

Plz see http://www.nature.com/nmat/journal/v4/n6/full/nmat1408.html

## bulk vs. surface stress and modulus

Dear Lee:

If I understand your question, the dimensionality of nanostructures has been taken into account by many researchers. See for example the Dingreville paper that I posted above. In equation (1), the surface energy density is defined as the energy/area; note that this differs from our traditional bulk strain energy density, which is energy/volume. Thus, when you take derivatives of the bulk strain energy density, you get stress, or force/area. If you take derivatives of surface energy density, you get surface stress, or force/length. Similarly, if you take another derivative of the bulk strain energy density, you get the modulus; if you do the same for the surface energy density, you get the surface modulus.

The idea is that because surface atoms have a different energy than bulk atoms due to their different bonding environment, if you decompose the total energy of the system into the integral over bulk + the integral over surface, the surface term will become dominant as the material gets smaller. This has been extremely useful for finite element modeling of surface effects on nanostructures, see here: (http://imechanica.org/node/737)

## Is there uniform definition?

Thank you, Harold! I of course agree with what you say. However, is there a uniform expression for there definitions? As you say, the surface energy density is defined as the energy/area in nanoscale; but it is energy/volume our traditional bulk strain energy density. In now time, people want do the muti-scale computations. How could they define the surface energy density in computation? energy/volume or energy/area ? Therefore, I think a more comprehensive definition of the surface energy density may be more accurate and easy to use. That is to say, define it considering the D (dimension of the space). Then the energy/volume or energy/area will be included in this definition. It will also easily for us to do the muti-scale computations and give a true description of the world (as D=2.5). Maybe a joke, it is quit like the Grand Unification Theory, ^_^.

Lee

## range of surface effect

Interesting discussions!

One question I have is: When does the surface effect become important? For nanowires, for example, what diameters are we talking about? a few nanometers, tens of nanometers, or hundreds of nanometers? Is there a critical dimension? I would also expect different ranges for different materials, from metals to ceramics (e.g., Si) and to polymers. I had a little experience with thin films, for which the surface effect becomes important when the thickness is down to nanoscale (see Surface effects on thin film wrinkling ). Based on experimental measurements of polymer thin films, we found that the surface properties start to affect the elastic modulus when the film thickness is less than 40 nm. Here the surface properties include surface energy, surface stress, surface modulus, and a surface layer thickness. For polymers, the molecular structures near the surface differ from the bulk, which gives a surface layer of roughly 1-2 nm based on MD simulations. For metals or Si, I speculated this thickness to be much smaller, and thus the surface effect would only become important for much thinner films.

RH

## on the range of surface effect

Rui,

Recently my student and I looked at the emergence of size-effect in different materials. Paper # 37 on my website has a graph (Figure 2) which compares the range of two size-effect mechanisms for different materials (including one polymer and one amorphouse materials). We find that surface effects are negligible in polymers and amorphous materials and the large size-effect observed in these is due to non-affine deformations. In contrast, for crystalline metals and ceramics, surface effects are dominant but as you point out (and shown in the figure of our paper), such size effects are confined to nanoscale.

## non-affine deformation or nonlocal interactions

Pradeep,

Thanks for sharing your paper. Apparently you have a deep understanding on the surface mechanics. I must admit that I have not followed much of the previous works by Muller and Gurtin as you suggested in a previous comment. In particular, I don't quite understand what you call "non-affine" deformation (I mean, literally) in polymers and amorphous materials. I understand your paper a little better, where you considered two effects, one for surface energy effect and the other for nonlocal interactions. May I assume that the effect due to nonlocal interactions is indeed non-affine deformation? If so, I would understand the nonlocal interactions or non-affine deformation within the framework of continuum strain-gradient elasticity. Personally, however, I would rather believe that the material properties are graded (thus physically heterogenous) near the surface instead of strain gradient (or mechanically heterogeneous). We know that the deformation of strain relies on the choice of a reference state, which is nontrivial (at least for me) at the surface, especially if you treat it at the atomistic or molecular scales. Finally, as discussed above by Harold, the surface effect is far more than just the surface energy effect. My experience with polymer thin films is that we had to include surface modulus and surface stress for the least.

RH

## Rui, Thanks for your

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.

## the range of surface effects - an open question, i think

Dear Rui:

You raise a very good, and possibly unquantifiable question (at least as of November, 2007!). First off, I think most people would agree that the surface effect is different for FCC metals than semiconductors, like silicon. For example, recent first-principles studies of <100> silicon nanowires (http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=PRB...)

indicate that the size effect might only be appreciable less than about 10 nm. Furthermore, the surfaces of silicon expand rather than contract as FCC metals do. I have done calculations on FCC gold indicating that nanowires with cross sections greater than 10 nm still experience compressive strains due to surface stresses on the order of 0.5%, which cannot be neglected. Other theoretical works (see the Zhou paper posted above) indicate that the surface itself ({100} vs. {110} vs. {111}) plays a critical role in both the magnitude and sign of the surface effect. Some of the experimental papers (see the Cuenot work) I posted above show size effects on the modulus out to 50-80 nm thickness. Unfortunately, the experimental papers did not indicate the surface structure, or the axial orientation of the bulk. Clearly, while the size-effect is accepted, its range has not been quantified.

It might be nice to get an experimental perspective here to balance all the theoretical work as to what length scales surface effects are expected to be significant, and how different materials might be expected to behave across multiple length scales.

## reply to Pradeep and Harold on the curvature effect

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.## reply to Phani's comments

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.

## Surface effects/self-similar objects ...

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.

## self-consistent reference

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.

## surface chemistry

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.

## Surface Effects - Bridging the Small & Really Small Scales

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

## more on surface elasticity

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.

## Has anyone paused and thought about...

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

## non-linear effects

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

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

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

## Surface effects (size effects) need studies/supports

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.

## Size dependency of the elastic modulus of ZnO nanowires

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) whenD>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) whenD<2 nm, the elastic modulus would be higher than that of the bulk value and increase with a decrease in nanowire diameter.## PRB paper on size-dependent elastic properties of ZnO nanofilms

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.