Promoting Statistics of Distributions in Nanoscience: The Case of

Aug 7, 2012 - ... Jixi Li , Jennifer M. Bui , Calvin K. Yip , Thibault Mayor , Hongbin Li , Hao Wu , Jörg A. ... Sascha Vongehr , Shaochun Tang , Xia...
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Promoting Statistics of Distributions in Nanoscience: The Case of Improving Yield Strength Estimates from Ultrasound Scission Sascha Vongehr, Shaochun Tang, and Xiangkang Meng* Institute of Materials Engineering, National Laboratory of Solid State Microstructures and College of Engineering and Applied Sciences, Nanjing University, Jiangsu, People's Republic of China ABSTRACT: Statistics often reveals unexpected errors. This is accepted and crucial in mature fields like medicine and experimental high energy particle physics, but usually underappreciated in still relatively novel areas. This work presents our strongest case in support for elevating the status of statistics in nanotechnology. Ultrasound fracturing can analyze tensile strengths of elongated nanostructures. It is rapidly growing in popularity. We calculate the location and shapes of the statistical distributions of fragment lengths and find that the usually employed analysis underestimates yield strengths. In unfortunate special cases, the true yield strength can be up to four times as high. Numerical simulations with generic assumptions still predict true values that are roughly twice as large as the result of the usually applied formulas. We suggest a different data analysis which removes the systematic error. It additionally provides an improved estimate of the statistical error, which combines the variation in yield strengths with the method’s measurement accuracy. A detailed reanalysis of previously published data supports our interpretations. The general discussion of the involved systematic and statistical errors calls for caution. Further confirmation by, for example, micro manipulation, is needed before ultrasound fracturing analysis can be trusted.

1. INTRODUCTION Knowledge of the statistics of random processes can predict resulting statistical distributions. This often reveals unexpected and at times surprisingly large systematic errors while simultaneously improving the statistical accuracy of an analysis method. This important aspect of the scientific method is almost as central to modern medicine and particle physics as their own specific theories, but it must be expected to lack proper status in still relatively novel fields like nanotechnology, which barely went though the pioneering phase. We have previously called attention to this by providing examples in the subfield of nano clusters.1,2 The present work similarly discovers large systematic errors and advertises the relevance of statistical errors and the shape of statistical distributions, this time in the field of nanowires. We emphasize that the discovered errors are not embarrassing faults of the implicated researchers, who in our honest opinion did as well as pioneers in any new field can possibly do. The main message is that errors due to missing statistics are natural for an emerging field; they must be expected to be common; and indeed one can easily find them once one is aware of this. We hope that presenting an example involving nanowires attracts the attention of a wide audience interested in nanotechnology. Nanowires are considered vital building blocks for nano electromechanical systems as both interconnects and active components because of their excellent electrical and mechanical properties. The mechanical characteristics of nanowires differ greatly from those of the corresponding bulk materials. Pure bulk metals with high conductivities such as Ag, Cu, and Al are © 2012 American Chemical Society

soft. Nevertheless, their nanowires can be strong, often having several times the bulk material’s strength.3,4 Research on metallic nanowires suggests that twin boundaries strengthen the wires,5−7 but also dislocation starvation8 has been shown to render nano sized elongated objects extremely strong. High flexibility and tensile strength are desired to increase the lifetime of devices; controlled synthesis of nanowires is therefore an area of very active research. The emergence of simple one-pot wet synthesis methods has enabled a wide range of scientists to embark on this research, but many lack sophisticated means to investigate the strength of individual structures via micro manipulation. A relatively recent idea9 is to apply ultrasound-induced scission to nanowires in order to infer their yield strength from fragment sizes. This method is available to almost every university laboratory. A liquid suspension of the nanowires, tubes, or filaments is ultrasonicated at frequencies of about 20 kHz and with powers exceeding 10 W/cm2, which ensures bubble cavitations.10 Near the collapsing cavities, the extended nanostructures are broken down along their lengths L, resulting in shorter fragments. The fragments are shortened further, but the applied forces are proportional to the square of the fragment’s length. Therefore, a quite strict short length cutoff exists, called the limit length Llim. Below this limit, the corresponding tension is not sufficient to break the fragment Received: June 13, 2012 Revised: July 29, 2012 Published: August 7, 2012 18533

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further. Llim is the shortest length that still can break (its fragments are of course yet shorter). The sample is ultrasound irradiated for between five to fifteen hours. Such long times ensure that the limit length is indeed reached and only few long fragments remain. The final (terminal) fragments’ lengths are measured under a scanning electron microscope (SEM). The forces vary in strength very quickly along with the ultrasound frequency. They are most of the time far below the plastic deformation regime. If the structure happens to be drawn into a position where an imploding cavity applies large forces, then these are only applied very briefly. The structure is not held for extended times in elastic elongation near the transition regime into plastic deformation. Thermal fluctuations are therefore neglected. When the tension happens to reach the plastic deformation limit anywhere along the nanostructure, the slight formation of a neck renders the cross-section thinner at that point, which is the location where it will later break. Because of this dependence on the initial deformation, the technique is sensitive to the yield strength, at least in cases such as the metal nanowires which many researchers concentrate on and which we also focus on in this work. The cavities oscillate and will in some later ultrasound cycle reach similarly high flow velocities around the structure. Then the forces pull on the already thinner, weakened part, and the structure is then even a little longer (recall that forces are proportional to L2). The structure is pulled apart further. Tension is thus applied repeatedly for many but very short times. Hence, once a structure is weakened, it will break, perhaps an hour later. The method is remarkably indifferent toward factors like the choice of fluid, as long as it has a low viscosity. This is likely due to different effects counteracting each other. For example, a slightly higher viscosity increases the drag but lowers the implosion velocity. The method is therefore robust and the following formula9 is sufficient for deriving the tensile yield strength σY from the observed fragments’ lengths, which are usually assumed to distribute around Llim: L lim = αd σ Y

previously published data and will afterward reproduce the previously published, uncorrected result precisely. This will strongly support our analysis, because it crucially depends on the here discovered, skewed shape of the terminal distribution.

2. THEORETICAL 2.1. Initial Distribution and Diameter Dependence. The fluid close to a collapsing vapor bubble rushes toward the bubble faster than the fluid further away from it. The velocity, v, which can reach up to 2.5 km/s, falls off with the square of the inverse distance s from the bubble center, i.e., ν(s) = ν(R) (R/s)2, where R < s is the bubble radius. The maximum radius of the pulsating bubbles is on the order of 10 to 100 μm, but scission is most likely at the high velocities reached near the smaller bubbles,12 thus one assumes R ≈ 10 μm. In such an inhomogeneous velocity field, elongated structures orient toward the center of the bubble. The long major axis is aligned in parallel with the bubble’s radial direction. Let us henceforth assume a solid nanowire for simplicity. As the bubble collapses further, the difference in fluid flow velocities along the oriented wire increases and the drag friction pulls along the wire’s length L = s2 − s1. The drag on any section of the wire is proportional to the fluid’s relative velocity ṽ(s) at the section’s location s.13 The difference in (relative) velocities between the two ends of the wire therefore tries to pull the wire apart and this difference increases with length: v(s2) = v(s1+L). The wires are most likely to break when they are close to the receding cavity wall, i.e., s1 ≈ R, because the velocities are largest there. Since we are interested in the resulting small fragments after long times, we look at lengths that are already shorter than the cavity radius: L ≪ R. The difference in velocities therefore increases proportional to the length of the wire: v(s1) − v(s2) = v(R)2L/R. There is no need to consider the true initial distribution of longer wires, which means that the “initial distribution” n0(L) as defined in this work contains all of the initial wires that satisfy L ≪ R, combined with all of the fragments from longer wires once they fulfill the same criterion. This initial distribution may never exist at any certain point in time, but this fact is irrelevant for the terminal distribution. The drag integrates along the length, which makes the total tension F proportional to the square of the length: F ∝ L2. The tensile yield strength σY is the force at the point where the wire yields divided by the area of the wire’s cross section, thus Flim ∝ σYd2. This suffices to understand eq 1 as far as we need, especially that the yield strength depends on the slope of the limit length versus diameter plot, because the terminal aspect ratio obeys Llim/d ∝ (σY)1/2. 2.2. Terminal Range and General Systematic Error (GSE). The fluid’s forces, while pulling into opposite directions, balance each other and maximize at the fluid stagnation point s* = (s1s2)1/2, which is where the wire therefore breaks. The shorter of the two resulting fragments has a length of Lsmall = s* − s1. The condition s1 ≈ R leads to Lsmall = [R(R + L)]1/2 − R. The relation L ≪ R simplifies this further to Lsmall≈L/2. This has an important impact on the terminal length distribution nT of the smallest fragments. It is usually assumed that after very long times, one finds fragments with lengths that are evenly distributed around the limiting length. However, even at the limit length, the fragments still break, and if a fragment breaks, then it is replaced by two fragments of half the previous length. For example, even if the fragments’ lengths are already distributed inside the range Llim/2 to 2Llim, all those between one and two Llim will slowly disappear, each being replaced by

(1)

where d is the wire’s diameter and the proportionality constant α is roughly equal to 0.7/(MPa)1/2. Mega Pascal (i.e., MPa = 106 N/m2) is the convenient unit for nanowire yield strengths. Equation 1 is the definition of the short length limit, which is consistently used in the literature and consistent with our above definition, namely, it is the shortest length that still can break. The accuracy of the method is limited by the distribution of the sizes of small fragments, fluctuations in yield strength, and conceivable unknown systematic errors. These unknown errors are presently large enough to render the accuracy of α irrelevant. The above introduced the status quo. We will now reconsider relevant details and investigate the effect of the ultrasound scission on the statistical distribution n(L) = dN/dL of the wires’ lengths, where N counts the number of wire fragments of a certain length L. The time evolution of such distributions has been investigated for long polymers11 and also carbon nanotubes,12 but the terminal distribution nT(L) after very long times has not yet been established. We establish the locations and shapes of terminal distributions, which suggest a new data analysis procedure. It results in much larger yield strengths. The statistical scattering of the data due to the fractionation statistics is removed from the interesting (physical) variations, for example that of the tensile yield strength among the wires. We will apply the method to 18534

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With a usual terminal range around 200 nm, these wires have been only 1600 nm long and belonged to the initial distribution n0 as far as our discussion is concerned (L ≪ R). These wires have a 43 = 64 times larger impact on the terminal distribution than those short wires that never split because they were in the terminal range from the start. Therefore, and counterintuitively with broad initial distributions n0 especially, it is the shape of their decreasing long-length tail that dominates the terminal distribution (see blue curve in Figure 1). This bias of the final

two fragments with a length inside the terminal range, which is centered around the median value Λ = 3/4 Llim: ⎤ ⎡1 terminal‐range = ⎢ L lim , L lim ⎥ ⎦ ⎣2 ⎤ ⎡ 1 1 = ⎢Λ − L lim , Λ + L lim ⎥ ⎦ ⎣ 4 4

(2)

For a critical evaluation of the relevance of systematic and statistical errors, we consider a general systematic error (GSE) as follows: Most conveniently, one measures the whole length of a wire fragment between the tips of the often rounded ends. These “end-caps” come into existence when the mother fragment is pulled apart. They are “pulled out” of the ends of the two fragments as they start to depart from each other and stretch the still attached material in the “neck” between them, especially so for metal nanowires. The sum of the lengths of the fragments is therefore larger than the original wire, i.e., 2Lsmall > L. This could be approximately compensated by the substitution: Lsmall = Lobserved − d/2

(3)

We take the magnitude of this error as an educated guess of the scale of likely present systematic errors that may arise during the SEM image analysis (operator measuring not quite from tip to tip) or neglected complications concerning fluid flow and drag at the wires’ ends. The absolute size of the GSE provides a reasonable, cautionary large estimate of further, unknown systematic errors that should be expected to additionally afflict ultrasound fractionation analysis. 2.3. Fragment Length Distributions and Systematic Yield Strength Error. Given nanorods from a size selecting synthesis with lengths that happen to be tightly distributed around two or four Llim for example, the above considerations predict that the observed terminal lengths cluster tightly around Llim/2, not Llim. We write “tightly”, because wires in a range of lengths dL lead to half-length fragments in a range (dL)small = (dL)/2 that is also only half as wide after one breakage, a quarter as wide after two, and so on. This “focusing” counteracts the naturally assumed broadening of the width of the length distributions during ultrasound fracturing. Mistaking Llim/2 for Llim and applying eq 1 in this case will underestimate the tensile yield strength σY ∝ L2lim by a factor of 4. In light of this, reports of strengths that are approximately a quarter of the theoretical yield strength14 look suspicious, but we better consider less special initial conditions. Given generic initial conditions, one might expect that the fragments distribute evenly inside the terminal range or otherwise symmetrically around its middle. If so, the usual procedure will confuse the limit length with the median Λ. The latter results, via eq 1, in a calculated tensile yield strength that is by a factor of (4/3)2 ≈ 1.8 smaller and thus only 56% of the actual strength. However, even assuming that a broad terminal distribution is observed in the SEM, for example due to inhomogeneous wires breaking only more or less close to s*, the distribution is always skewed by two effects: There are 2 smaller fragments for every breakage [i.e., N → 2N] and the “focusing”; the latter being the already mentioned fact that these fragments from wires in a range of lengths dL will be found in a range that is also only half as wide [i.e., dL → (dL)/ 2]. For distributions dN/dL, this implies a total factor of 4 per split. Consider for instance those wires which had to split three times in order for their fragments to land in the terminal range.

Figure 1. Simulated terminal length distributions of a fractionation that repeatedly halves (here up to four times) until fragments are equal to or shorter than the limit length at 300 nm. The red hatched curve is from initial lengths around 440 nm. The blue curve is from a very broad initial distribution (standard deviation of 1000 nm).

size distribution toward Llim/2 cannot be refuted with some ad hoc “smearing”, if this smearing does not involve the fracture mechanism at lengths that are only a few doublings of the limit length. Smearing effects on longer lengths merely present a broader initial distribution as defined in section 2.1. We have simulated the fractionation statistics for generic initial distributions of lengths that one may well expect either right from the beginning (say when testing nanorods) or after some time of ultrasound irradiation. Starting with n0, the distribution nk after the kth split still equals n0 below Llim/2. Above Llim, it equals 4kn0(2kL), which initially grows fast with k and obviously swamps the fraction of wires that were very short from the start. The terminal distribution nT at large k is therefore completely inside the terminal range: k

n(kL) =

∑ 4in(20 L); i

i=0

L lim /2 ≤ L ≤ L lim

(4)

For example, assuming Llim to be 300 nm and starting with a standard normal distribution n0 of lengths L = (440 ± 80)nm, the terminal distribution appears to be completely inside the terminal range between 150 to 300 nm after only twice splitting all wires above the limit length into halves (red hatched curve in Figure 1). The maximum is at 220 nm, which is just below the median at 225 nm. The distribution is not symmetric around the maximum but biased toward yet smaller lengths, putting the average at ⟨L⟩ = ∫ (L*nT)dL = 218 nm (97% of median). If this average is taken as the limiting length, which is effectively done by a linear regression in the length versus diameter plot, then the thereby calculated yield strength will be only 53% of the true value. A wire diameter of d ≈ ⟨L⟩/10 corresponds to a usual terminal aspect ratio like often observed with silver nanowires.9 This renders the GSE to be about 10 18535

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nm, which translates into a shift of only 6%. Even if this were fortuitously correcting upward, the calculated yield strength would be still only 58% of the true value. Generic initial conditions with broad initial distributions lead to terminal distributions that peak further below their medians. The broader the initial distribution, the less the initial average matters. With the same parameters as above but a standard distribution of 1 μm, one obtains the characteristic shape of terminal distributions (blue curve in Figure 1) that results from long wires which lead to broad initial distributions. As expected, it resembles a section that was cut out of the right tail of a Gaussian bell curve. An initial log-normal distribution instead of a Gaussian for example does not appreciably change this result. It has an average ⟨L⟩ of 205 nm, which is only 91% of the median. The usual analysis would lead to a systematic underestimation of the yield strength by 2.1 (only 47% of the true value). The peak is very close to 1/2Llim, so an analysis that emphasizes the peak rather than the average will underestimate the tensile yield strength indeed by a factor of 3 to 4 even under quite generic initial conditions. Most reported analyses effectively calculate the average. Summarizing the simulations, generic initial distributions let ⟨L⟩ fall between 97% and 91% of the median Λ. A rough but reasonable expectation is therefore ⟨L⟩ ≈ 94%Λ, which leads to results that are only half the true yield strength. 2.4. Comparison with Experiment and Improved Analysis. The terminal lengths spread through the whole terminal range. This apparent statistical error is neither the proper statistical error of the ultrasound fracturing method nor the variation in the yield strengths among the wires. In fact, all of the above considerations assumed that there are no statistical errors of this nature at all. Nevertheless, the considerations are valid for each separate subpopulation of any particular yield strength for example. Hence, the effect of statistical errors of this nature generally equals a smoothing of the terminal length distribution, which will therefore “leak out” of the average terminal range. A “leakage” on the order of the GSE or more should be naturally expected. Figure 2 shows published data for silver nanowires [Huang et al.,9 Figure 3A]. After extended ultrasound irradiation, the lengths of fragments had been plotted versus their diameters. The gray linear regression (labeled “original”) is the unaltered published estimate for the terminal aspect ratio a = Llim/d. The distribution of the data falls well in between a lower slope (lower yellow wedge) and the double of that slope (upper yellow wedge), as predicted from the diameter dependent terminal range eq 2. The yellow wedge areas are due to the statistical error of those two slopes, the just mentioned “leakage”, which turns out to be indeed comparable to the GSE. How did our analysis produce the yellow wedges? The lowermost slope b of the (Llim/2d)-wedge is the linear regression through the ten smallest aspect ratios (fixed at the coordinate origin). The uppermost edge B of the (Llim/d)wedge is similarly the regression through the ten largest aspect ratios with one obvious outlier removed. No more input is required. The other two boundaries of the yellow wedges are simply the slopes 2b = 11.43 and B/2 with B = 14.29. Interval [b,2b] corresponds to the shortest terminal ranges and [B/2,B] to the longest. The top of any terminal range is its limit length, thus:

Figure 2. Lengths vs diameters of wire fragments after sonication. The gray line is the usual method’s estimate of the aspect ratio a = Llim/d as originally published. Our analysis only uses the lower and uppermost slopes b and B. The two yellow wedge-shaped areas are the resulting broadened boundaries of the terminal ranges. The terminal range at 450 nm is indicated in blue. The published ratio a cuts even below the median Λ, far below Llim.

The tensile yield strength is therefore σY = (337.4 ± 8.7)MPa, which equals double the bulk yield strength of silver (170 MPa). The new result is consistent with that nanowires are usually stronger than the bulk material. This supports our analysis, but much better support comes from the ability to understand and reproduce the published result as follows. Equation 5 constrains the median to be Λ = (9.64 ± 1.07)d. The usual method confuses the average ⟨L⟩ ≈ 94%Λ (see end of Section 2.3) with the limit length. Entering 94%Λ as if it is Llim into eq 1, reproduces the uncorrected yield strength: σY = (168 ± 6.3)MPa is precisely the published (169 ± 4)MPa. Reproducing this closely depends crucially on the predicted skew of the terminal distribution! Varying the 94% somewhat (it can vary ±3% absolute, but this is not a standard deviation; we forego confusion with yet another type of error, although it would further strengthen our case) leaves the published value inside of the improved error range around the reproduced value, but omission of the skew does not come close to the published value.

3. DISCUSSION For clarity, let us briefly recapture the suggested data analysis method: Long fractionation times and observing many fragments can reliably reveal the position of the lower edge of the terminal distribution, which is half of the limit length, Llim/2. In the plot of the aspect ratios L/d, the lower edges of all terminal distributions corresponds to the lower bound b. The upper edge of the terminal distribution is the upper bound on the limit length, corresponding to B. The limit length is given by the left equality in eq 5. We used an absolute error ± (2b−B)/2 for simplicity and removed a discussion about the fraction of data points to include when establishing the edges of the distribution. It is presently not worthwhile to establish any of the errors in more detail as long as they are on the order of the GSE. Nevertheless, future research will likely change this situation, and then the suggested analysis method allows further improvement of the

L lim /d = (2b + B)/2 ± (2b − B)/2 = 12.86 ± 1.43 (5) 18536

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can be further improved led us to discussing whether diameter dependent yield strengths can be explored. In final conclusion: Ultrasound induced scission analysis of the tensile yield strengths is an important method that is applicable to nano structures but that should be handled with more caution. Detailed investigations should include verification by other methods like micromanipulation. For now, it can be improved markedly with the consideration of the involved statistical distributions.

accuracy: The old procedure’s accuracy is always affected by the terminal range’s width 1/4 Llim, which is not reduced by observing more wires. The uncertainty in the position of the boundaries of the terminal distribution however decreases with the observation of more fragments, as a truly statistical error of course should do. With that uncertainty diminishing, only the true variation in yield strengths among different wires and the ultrasound fractionation’s intrinsic accuracy remain in the statistical error. However, there is not as yet enough experience with ultrasound scission analysis to confidently report errors that are smaller than the GSE. Many potential issues could be discussed. For example, the system is usually cooled in an ice bath in order to avoid thermal degradation, but bubble cavitations can lead to high local temperatures. Perhaps there are further systematic errors related to thermal fluctuations which would likely slightly decrease the observed yield strength further. Especially worrying is diameter dependence, which people start investigating. In our experience, plotting various samples’ average limit sizes ⟨Llim⟩ versus average diameters ⟨d⟩ easily results in an apparently diameter-dependent σY, because the sample batches’ different synthesis conditions lead to different strengths while producing different average diameters. Plotting a single batch’s fragment lengths and diameters instead, d-dependence disappears. If true d-dependence should ever appear, will it be the physics of the nanowires or that of the ultrasonic cavitations? Strength should decrease with larger d, because the wires are stronger than bulk material, but limit lengths approach the bubble radius R. At very small d, nanostructures usually exhibit interesting size dependence too, but very small d lead to very small Llim, which may well have other method-related issues. A true d-dependence of σY, if present, may not show up because of the diameter-dependent issues of the method. After all, opposing effects balancing each other seems characteristic for ultrasound scission, as alluded to in the introduction and additionally discovered in the focusing of the length distribution’s width for instance.

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AUTHOR INFORMATION

Corresponding Author

*Phone: +86-025-8368-5585; fax: +86-025-8359-5535; e-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was jointly supported by the PAPD, the Fundamental Research Funds for the Central Universities, the National Natural Science Foundation of China, and the State Key Program for Basic Research of China.

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REFERENCES

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4. CONCLUSIONS We have discussed the distribution of fragment lengths that results from ultrasound-induced scission of nanowires. Theoretical considerations and numerical simulations predicted terminal distributions of lengths. Knowledge of their positions and shapes removed a large systematic error that otherwise substantially underestimates the wires’ tensile yield strength. Results are usually only half of the true strength, but deviations can go up to a factor of 4 in special situations or even in the case of broad initial distributions when the peak of the distribution is interpreted as the limit length. The terminal distribution is skewed toward short lengths. An improved data analysis method was suggested: It removes the large systematic error that is due to the position of the terminal range (I); it is independent of, and thus never biased by, the possibly uncertain skew (II), and it does not let the width of the terminal range affect the reported statistical variation (III). Applying this new analysis to previously published data confirmed the validity of this new method. Especially, the skewed shape of the terminal distribution, which derives from the fractionation statistics, was used to reproduce the published, uncorrected result precisely. Without the discovered and calculated skew, we would not be able to reproduce the uncorrected result and could not claim a full understanding. Discussing the involved errors once more and how the statistics 18537

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