DENG, et al.
A Statistical Approach to Quantifying the Elastic Deformation of
Nanomaterials †
Xinwei Deng , V. Roshan Joseph † , Wenjie Mai ‡ , Z. L. Wang ‡ , C. F. Jeff Wu † †
H. Milton Stewart School of Industrial and Systems Engineering, ‡ School of Materials Science and Engineering, Georgia Institute of Technology, Atlanta, GA
Abstract
empirical statistical modeling technique to identify the effects of such factors. After filtering out such effects, we can estimate the elastic modulus based on the physical law more efficiently. The article is organized as follows. In Section 2, we give a brief description of the measurement method and the model used in Mai and Wang (2006), which provides the motivation of our proposed approach. Section 3 is devoted to building and fitting appropriate statistical models through model selection. Estimate of the elastic modulus is obtained from the selected model as well. To accommodate the correlation structure in the bending curves, we propose a more flexible model with general error structure in Section 4. Some discussions and future research directions are given in Section 5.
Accurate estimation of elastic modulus of certain nanomaterials such as Zinc Oxide nanobelt is important in many applications. In this work, a statistical modeling technique called SPAR is proposed to account for the various errors introduced during measurement. SPAR can automatically detect and remove the systematic errors and therefore, can give an accurate and precise estimate of the elastic modulus. KEY WORDS: nanostructure, profile adjustment, statistical modeling, model selection, correlation structure
1 Introduction
2 Existing Method of Experimentation and Modeling
One-dimensional (1D) nanomaterials, such as carbon nanotubes, semiconductor nanowires, and oxide nanobelts, are the fundamental building blocks for constructing nanodevices and nanosystems that exhibit superior performances. Mechanical behavior of 1D nanomaterials is one of the most important properties that dictate their applications in nanotechnology. For example, carbon nanotubes are found to be the strongest and stiffest materials on earth in terms of tensile strength and elastic modulus. They can be promising building blocks for future Nano Electro Mechanical Systems (NEMS). One approach of quantifying the mechanical property such as elastic modulus of 1D nanomaterials is based on atomic force microscope (AFM) and nanoindenter. A common strategy is to deform a 1D nanostructure that is supported at the two ends by using an AFM tip, which pushes the 1D nanostructure at some locations. Then the elastic modulus can be determined through the quantified force-displacement curves. However, the accuracy of this measurement is limited by noise factors such as the size of tip, the accuracy of positioning the AFM tip, and the surface roughness of the 1D nanomaterials. This article proposes a new approach to estimate the elastic modulus through statistical and physical modeling. Mai and Wang (2006) proposed a new approach for quantifying the elastic deformation behavior of a 1D nanostructure. The approach is based on an AFM contact mode continuous scan of a ZnO nanobelt that is supported at the two ends. A quantitative fitting of the elastic bending shape of the nanobelt as a function of the bending force provides a reliable method for estimating the elastic modulus of the nanobelt. However, the nanobelt may shift or deform during measurement, leading to bias in the estimation of the elastic modulus. The small size of the nanostructure makes their manipulation rather difficult. Moreover, the level of allowable tolerance on measurement errors for nanomaterials decreases since noise or bias becomes much larger in comparison to the smaller scale of nanomaterials. The presence of these noise factors and other unknown factors makes the observed curve between the deflection and force look quite different from the free-free beam model (FFBM) derived using the physical law. Since the FFBM does not account for the various noise factors, we use an
The ZnO nanobelts used for the study were prepared by physical vapor deposition. A silicon substrate is prepared with long and parallel trenches caved at its surface by nanofabrication. The trenches are about 200 nm deep and 1.25 um wide. Long ZnO nanobelts were manipulated across the trenches over many periods. The morphology and dimensions of the nanobelt were captured by a scanning electron microscope (SEM) and AFM. The SEM image gives the width of the trench and the length and width of the nanobelt, and the AFM image gives the thickness of the nanobelt. The measurement was made by scanning the nanobelt along its length direction using an AFM tip in contact mode at a constant applied force. A series of bending images of the nanobelt were recorded by changing the magnitude of the contact force from low to high.
(a)
(b)
Figure 1: (a) The AFM image profiles of the suspended NB under different load forces in contact mode. (b) The normalized AFM image profile by subtracting the profile acquired at 78 nN (nano Newton) from the profiles in (a).
The profiles of a suspended nanobelt along the length direction under different contact forces are shown in Figure 1(a). As shown in Figure 1(a), the image profiles of a nanobelt (denoted by NB) recorded the deflection of all the points along
1
its length under different applied forces. Each curve was obtained by averaging ten consecutive measurements along its length under the same loading force. The curves in Figure 1(a) are not smooth due to a small surface roughness (around 1 nm) of the nanobelt. In addition, the as-attached nanobelt on the trenches is not perfectly straight, possibly due to initial bending during the sample manipulation. Figure 1(a) indicates that there are some noise factors affecting the deflection curves. Mai and Wang (2006) suggested a free-free beam model (FFBM) to quantify the elastic deflection. The diagram of the FFBM is shown in Figure 2.
poorly with a lot of noise, then subtracting this profile to normalize the data can result in poor estimation of E from the FFBM. Recall that the deflection v in the FFBM in (1) is a linear function of the applied force F given the distance x . The reason for normalizing the data by subtracting the image profile at a low applied force is to eliminate the initial bias, i.e., the effect of the surface roughness and initial bending of the nanobelt. However, if some systematic biases occurred during the measurement, normalizing the data may not be enough for obtaining a good fitting based on the FFBM. For example, the deflection profiles obtained at applied force F = 235, 248 and 261 nN lie above on those obtained at lower force F = 209 and 222 nN. This is inconsistent with the model equation in (1) because the deflection is expected to increase with force. The FFBM itself cannot explain this phenomenon. This pattern still persists in the normalized profiles in Figure 1(b), where the profile is normalized by subtracting the profile at F = 78 nN from the original profiles in Figure 1(a). Therefore, the MW method may not be a proper way to fit the profile data of NB. It requires a more general model to identify other factors besides the initial bias. To overcome these problems, we propose a general regression model that directly incorporates the initial bias and potential systematic biases introduced during measurement. Then we use variable selection to identify terms associated with the systematic biases and then adjust the profiles by subtracting these terms from the original profiles. When a final model is chosen, we estimate the elastic modulus E based on the FFBM. We call the method sequential profile adjustment by regression (SPAR).
Figure 2: The schematic diagram of the free-free beam model (FFBM).
In this model, the two ends of the nanobelt can freely slide. When a concentrated load force F is applied at the contact point x of the AFM tip away from the end A , the deflection of nanobelt at x is determined by
v = −
Fx 2 (L − x)2 , 3 E IL
(1)
L is the width of trench, and I is the moment of inertia given by wh3 / 12 for the rectangular beam, where w and h are respectively the width
where E is the elastic modulus,
and thickness of nanobelt. The elastic modulus E can be estimated from the image profile data based on the FFBM. Figure 3 shows an illustrative example of FFBM profiles, which are symmetric and perfectly smooth but does not account for the noise factors in the measurements.
3 General Model and Model Selection 3.1 General Model As shown in Figure 1(a), suppose there are K image profiles, i.e., the nanobelt is scanned sequentially under K different applied forces F1 , F2 , …, FK . The experimenter usually changes the magnitude of applied force F from low to high, i.e., F1 < F2 1)+ +δK−1(x)I(k >K−1)+ε(x,Fk), (3) where I (⋅) is an indicator function and ε ( x, Fk ) is the error 2
RMSE = {∑ i =1 ∑ k =1 (v( xi , Fk ) − vˆ( xi , Fk )) 2 / df }1/2 , (5) n
term. Specifically, when the force Fk is applied to make the
K
AFM tip in contact with the nanobelt, the proposed approach models the deflection as
and df is the degrees of freedom in the corresponding model.
k , i.e., some of the δ k ’s may be zero. We therefore use a
selection, where
v( x, Fk ) = β ( x) Fk + δ 0 ( x) + δ1 ( x) +
+ δ k −1 ( x) + ε ( x, Fk ). Alternatively, we can use the Bayesian information criterion In reality, there may or may not be a deformation at stage (BIC) to select δ k ( x) into the model at each step of the
variable selection technique to identify the significant δ k ’s and include only them in the final model.
BIC = ∑i=1 ∑k =1 (v( xi , Fk ) − vˆ( xi , Fk ))2 / σ 2 + p log N, (6)
3.2 Model Selection
p is the number of parameters, and N is the number of
n
K
observations in the corresponding regression model. If 2
3.3 Example
the model (3) is a linear regression with K+1 parameters 1 / E, δ 0 ( xi ), δ1 ( xi ), , δ K −1 xi , i.e.,
where
knowledge
of
the
2
FFBM.
ε i = (ε ( xi , F1 ), , ε xi , FK
T
In ,
In the image profiles of NB, the deformation is recorded at n = 161 points along the length of the nanobelt under K=15 different applied forces. The length of NB is L = 1252 nm and the moment of inertia in the FFBM is I = 8216510 nm4. Figure 4 shows the model selection results using the proposed method. The δ k ( x ) is sequentially selected into the model in the
⎞⎛ 1 / E ⎞ ⎟⎜ ⎟ ⎟⎜ δ0 ( xi ) ⎟ + ε i, (4) ⎟⎜ ⎟ ⎟⎜ ⎟ ⎠⎝ δ K −1 ( xi ) ⎠
γ ( xi ) = x ( L − xi ) / (−3IL) 2 i
incorporates the
ε ( xi , Fk )
error
in
(6) is not available, an estimate σˆ can be obtained from the data.
The general model (3) considers all potential bias factors. In reality, it is likely that only a few of them contribute toward the deformation on the nanobelt. So it is important to find significant δ k ’s and build an appropriate model. Given the distance xi ,
⎛ v( xi , F1 ) ⎞ ⎛ γ ( xi )F1 1 0 ? ⎜ ⎟ ⎜ ⎜ v( xi , F2 ) ⎟ = ⎜ γ ( xi )F2 1 1 ? ⎜ ⎟ ⎜ ⎜ ⎟ ⎜ ⎝ v( xi , FK ) ⎠ ⎝ γ ( xi )FK 1 1 ?
σ2
following order:
δ 6 ( x)
and
δ 0 ( x) , δ12 ( x) , δ10 ( x) , δ 8 ( x) , δ 9 ( x)
δ 2 ( x) .
,
It can be seen that after adding three or
four terms, the decrease of RMSE starts to level off while the corresponding BIC value starts to increase. By considering both criteria, we take three δ k terms to build the final model. Thus,
the vector
represents the
the chosen model is
error occurred at distance xi under applied force Fk . The
v(x,Fk) =β(x)Fk +δ0(x)+δ10(x)I(k >10)+δ12(x)I(k >12)+ε(x,Fk),k =1, ,K.(7)
model (3) considering all xi is an over-parameterized liner model with parameters 1 / E , δ 0 (x), δ1 (x), ? , δ K −1 x) , where
x = ( x1 , x2 , , xn ) , δ k (x) = (δ k ( x1 ),… , δ k ( xn )) for k = 0,… , K − 1 . To find a proper model, we need a model selection strategy (Miller, 2002). In our situation, however, it is not appropriate to implement variable selection among all nK+1 covariates associated with the parameters 1/E, δ0 (x), δ1 (x),? , δ K −1 x) . Recall that δ0 (x) is interpreted as
Figure 4: Forward model selection using RMSE and BIC on the NB data.
the initial bias effect from the surface roughness and initial bending and δ k ( x) as the systematic bias effect from applied
Here not only the initial bias
δ 0 ( x)
δ10 ( x) and δ12 ( x)
is significant, the
force Fk for k ≥ 1 . It is thus more reasonable to keep each
systematic biases
δ k (x) as a whole parameter set in model selection. It can make
role in modeling the data. To get more insight on the selected δ k ’s in (7), at each stage of the selection, we define an adjusted
the selected model more interpretable from the physical perspective. Starting with the model including only the FFBM, we use forward selection to add one δ k ( x) at a time. To begin with, we assume that the errors
ε (xi , Fk )
deflection For
also play an important
v( xi , F ) adj as v( xi , F ) minus the selected δ k ’s. example,
at
stage
3,
v(x, Fk )adj =v(x, Fk ) −δ0(x) −δ12(x)I(k >12) −δ10(x)I(k >10),k =1, , K.
are independent with a
Note that the systematic bias
δ12 ( x)
introduces the
normal distribution N (0, σ ) . Then the estimation of parameters can be easily calculated by the maximum likelihood estimation (MLE). In each step of the forward selection, we select δ k ( x ) that has the smallest root mean square error
profiles at F = 235, 248 and 261 nN. Similarly,
(RMSE) of the corresponding model, where
to F15 . Figure 5 shows the changes of five adjusted deflection
2
deflection into the image profiles starting from F13 , i.e., the brings in deformation on the profiles under
profiles under applied force
3
δ10 ( x) only applied force F11
F11 to F15 as the three δ k
selected model and lead to a better fitting of the NB data.
terms are sequentially selected into the model. The original five profiles are shown in Figure 5(a). When δ 0 ( x) is selected into the model at stage 1 of selection, it adjusts the initial bias among the five image profiles. In Figure 5(b), the adjusted deflection v( xi , Fk ) adj = v( xi , Fk ) − δ 0 ( x) looks closer to the FFBM, but the inconsistent pattern shown in Figure 1 still remains. Note that the inconsistent pattern appears between the profiles under F11 = 209 nN, F12 = 222 nN and those under F13 = 235 nN, F1 4 = 2 4 8 nN, F1 5 = 2 6 1 nN. At stage 2 of the selection,
δ12 ( x)
estimate of
δ12 ( x)
Figure 6: Estimates of
is selected into the model. It further adjusts
δ12 ( x)
estimate of and
δ10 (x)
δ10 ( x)
from the selected model
of NB.
the profiles under the applied force F1 3 , F1 4 , and F1 5 . From Figure 5(c), we can see that the adjusted deflection v( x, Fk )adj = v( x, Fk ) − δ 0 ( x) − δ12 ( x) I (k > 12) is to push
The image profiles of the adjusted deflection v ( x, F ) adj based on the selected model is shown in Figure 7. Compared with the original profiles in Figure 1, the adjusted deflection profiles are more consistent with the FFBM as shown in Figure 2. With three selected δ k ’s in the model, the selected model (7) can provide
the profiles obtained at force F13 , F14 , and F15 to lie below
F12 . The inconsistency no more reliable and precise estimation of the elastic modulus E. longer exists in Figure 5(c). Therefore, adding δ12 ( x) can those obtained at force F11 , and
remove the inconsistent pattern in the NB data. At stage 3 of the selection, δ10 ( x) is chosen into the model. It can adjust the five image profiles at the applied force from F11 = 209 nN to F15 = 261 nN. As shown in Figure 5(b), to adjust the inconsistency among these five profiles, it is likely that the adjusted deflections have been pushed downwards too much. From Figure 5(d), we can see that adding δ10 ( x) into the model is to pull all five profiles upwards and make the adjusted deflection v(x, F)adj = v(x, Fk ) −δ0 (x) −δ12 (x)I (k >12) −δ10 (x)I (k >10) a
Figure 7: The image profiles for the adjusted deflection of NB.
better fit to the FFBM.
Stage 0: original profiles
(a) Fitting: MW
Stage 1: adjusted by δ 0
(c) Residuals: MW Stage 2: adjusted by
δ0
and
δ12
Stage 3: adjusted by δ 0 ,
δ12
δˆ10 ( x)
To gauge the performance of the selected model using SPAR, we compare it with the MW method. The fit and residual plots from these two approaches are shown in Figure 8. The fitting in Figure 8(a) is obtained by adding the initial profile back to the fitted normalized data from the MW method. The residuals from the MW method show some systematic patterns, which indicates that the model needs improvement. No systematic pattern is observed in the residuals based on SPAR. Clearly, the
are shown in
Figure 6. We can see that the opposite shapes of δˆ12 ( x ) and
δˆ10 ( x )
(d) Residuals: SPAR
Figure 8: Comparison of two methods on the NB data.
and δ10
Figure 5: Illustration of the adjusted deflection profiles under applied force from F11 = 209 nN to F15 = 261 nN.
The two estimates δˆ12 ( x ) and
(b) Fitting: SPAR
in Figure 6 help remove the inconsistent pattern of the
4
selected model performs better. It removes the inconsistent pattern discussed above, while the MW method does not recognize this pattern. The residuals from the selected model are also much smaller. Table 1 summarizes the estimation results using the two methods. Clearly SPAR gives a more precise estimate of the elastic modulus E. The 95% confidence interval of E from SPAR is (99.97, 103.07) and that from MW method is (91.24, 97.44). The non-overlapping of intervals suggests that one of the estimates can be misleading or wrong. Because SPAR incorporates the initial bias and adjusts the inconsistent pattern in the profiles, it is expected to provide more accurate estimate of the elastic modulus than the MW method. To further verify this point, we perform SPAR using only half of the profiles of NB, i.e., the profiles under applied force F = 78, 105, 131,…, 201 nN. The estimate of the elastic modulus Eˆ = 102.67 and the 95% confidence interval (100.55, 104.79) are similar to those using SPAR with all the profiles of NB. This shows that SPAR can give a more reliable estimate even using half of the profiles.
describing the error structure within the deflection curve over xi ’s for the applied force Fk . The resulting covariance matrix
Σ has a diagonal block structure since the errors between different deflection curves are considered to be independent. The
σ2
term is used to quantify the error variation for each deflection curve obtained from averaging 10 consecutive measurements.
= τ k2 R, where τ k2 is the error variance of the curve obtained at Fk , and R = ( rij ) n×n is the Next we model Σ k as Σ k
correlation matrix with rij quantifying the correlation between two deflections at the distance xi and x j obtained under the same applied force. We use the Gaussian correlation function (Santner et al., 2003) to model the correlation matrix R with
rij as rij = exp(−θ ( xi − x j ) 2 ). Here each Σ k uses the
same θ since we assume that different deflections share the same correlation pattern.
Table 1: Comparison of estimates with the NB data.
RMSE MW Method 0.86 SPAR 0.37
1 / Eˆ 1.06e-02 9.85e-03
se(1/E) 1.77e-04 7.63e-05
Eˆ 94.34 101.52
se(E) 1.58 0.79
4.1 Parameter Estimation The proposed model in (3) is a general linear model with the Σ constructed above, i.e., error structure Y = Xβ + ε, ε ∼ N (0, Σ) . The parameters in this model
4 Modeling with General Error Structures
are (β, τ 2, θ ) , where τ 2 = (τ 1 , ? ,τ K . We consider the maximum likelihood estimation (MLE) for these parameters. For 2
The deflection of nanobelt is a continuous and smooth phenomenon. For a given force F, the nanobelt is scanned using AFM tip along the distance x. Therefore, the deflection obtained at the distance x = x i should be positively correlated with
notational convenience, denote Y = ( Y1, ? , Y k
deflection curves, which are obtained by using different applied forces, their correlation is much smaller. So we can consider them to be independent in the modeling. As shown in the residual plots in Figure 8(d), the residuals have systematic patterns along the distance x . In particular, for a given force Fk , if the residual at xi is large, then the residual at
l(β, τ2,θ ) = −
imposing some error structure is warranted. Note that each profile curve was obtained by averaging 10 consecutive measurements along its length under the same loading force. The aim of taking average is to reduce the measurement error due to sources like equipment accuracy. There are still errors after taking average. By incorporating this prior information and correlation structure, we can build a more general error structure as follows. The model in (3) can be written as a linear regression model Y = Xβ + ε , where the responses vector is
parameters in the log-likelihood function. The parameters τ 2 and
θ
−1
are involved in every matrix inverse Σ k . It requires
intensive computations to estimate these parameters by directly maximizing the log-likelihood function. Instead we propose an iterative algorithm to efficiently estimate parameters from
l (β, τ 2,θ ) . First observe that given τ 2 and generalized least squares estimate
β =(1/ E,δ0(x1),? ,δ0 xn),δ1(x1),? ,δ1 xn),? ,δK−1 x1),? ,δK−1 xn)) , and T
X is the corresponding model matrix. Assuming that the error
K
diagonal
block
is its
−1
ε = (ε ( x1, F1 ),? , ε xn , F1 ),? , ε x1, FK ),? , ε xn , FK )) follows a normal distribution, i.e., ε ∼ N (0, Σ) , we consider the model Y = Xβ + ε with a more general error structure Σ the
θ , the MLE of β
⎡K ⎤ ⎡K ⎤ β = ⎢∑ (XTk Σ-k1Xk )⎥ ⎢∑ (XTk Σ-k1Yk )⎥ . ⎣ k =1 ⎦ ⎣ k =1 ⎦
T
of
(8)
= τ k2 R + σ 02I . Suppose the number of δ k ( x) in the model is m, 0 ≤ m ≤ K , then there are ( nm + 1) + K + 1
parameters
consists
1K ⎡⎣log | Σk | +(Yk − Xkβ)T Σ-k1(Yk − Xkβ)⎤⎦ , ∑ 2 k=1
where Σ k
Y = (v( x1 , F1 ), ? , v xn , F1 ),? , v x1 , FK ),? , v xn , FK ))T ,
which
, where
T
a distance close to xi is also likely to be large. It indicates that
vector
T
Y k ∈ R as Y k = (v( x1 , Fk ),? , v xn , Fk )) . Similarly, T n× p denote X = ( X1, ? , X k , where X k ∈ R is the part of the model matrix X corresponding to Y k , and p is the dimension of β . The log-likelihood function can be written as n
those obtained near xi . On the other hand, between two
the
2
For given
submatrics
(9)
β and θ , maximizing the log-likelihood function
l (β, τ ,θ ) in (8) is equivalent to maximizing each lk (β, τ 2,θ ) individually, where 2
Σ k + σ 2I, i = 1,… , K , where Σ k is an n × n matrix 5
1 lk (β, τ2,θ ) = − ⎡⎣log | Σk | +(Yk − Xkβ)T Σ-k1(Yk − Xkβ)⎤⎦. 2 Since
τ k2
underlying model is Y = Xβ + ε, ε ∼ N (0, Σ) . Since the true value of the elastic modulus E is unknown, we compare the
(10)
standard error of the estimate Eˆ . For convenience, denote Eiid as the elastic modulus parameter E in the model with iid error, and E gen as the corresponding one in the model with general
appears only in lk (β, τ 2, θ ) of the log-likelihood
function (8),
τ k2
can be individually estimated by maximizing
error structure. In the case of NB1: Eˆ gen = 114.84 which is
lk (β, τ 2,θ ) . Obviously, optimization with only one parameter
different from Eˆ iid = 101.52 in Table 1. This difference can
is easy. For given β and τ , estimating θ is also a one-parameter optimization problem by maximizing the 2
be explained by the fact that the residuals in Figure 8(d) have a clear pattern of positive correlation. The 95% confidence interval (111.34, 118.34) of Egen is disjoint with the 95% confidence
log-likelihood function l (β, τ 2, θ ) in (8). Thus, if some initial
estimates of β and θ are available, we can obtain the MLE of parameters through the following iterative algorithm:
interval (99.97, 103.07) of E iid . Although the length of the confidence interval of E gen is larger, it is a more reasonable
ˆ. Step 0. Obtain initial estimates θˆ and β
estimate because it incorporates the correlation structure in the deflection profiles.
ˆ , update τ 2 = (τ 2 ,? ,τ 2 , i.e., Step 1. Given θˆ and β 1 K 2 2 − 2lk (β, τ k ,θ ) . τˆ k = arg min 2
5 Discussions and Conclusions
τk
2 2 Step 2. Given τ垐 2 = (τ 1 , ? ,τ K
and βˆ , update
θ
, i.e.,
In this article, we report a new method called SPAR to estimate the elastic modulus of nanobelts through statistical modeling and analysis. It can identify significant bias factors using model selection, and incorporates them into the fitted model. It can automatically remove the initial bias, and adjust the inconsistent pattern caused by the systematic biases. Therefore, it can give more precise and reliable estimation of the elastic modulus. Due to the small size of nanomaterials, the noise and bias during measurement become relatively large compared with the actual scale of nanomaterials. Taking account of bias factors into the theoretical model through statistical modeling can greatly improve the estimation of the interested mechanical property. Here we propose the SPAR method and study its performance for data arising from a specific experiment on nanobelts. Generally, in the field of quantifying mechanical properties of 1D nanomaterials, SPAR can have broad applications. For example, San Paulo et al. (2005) studied the mechanical elasticity of single and double clamped nanowires. The deflection of nanowires is measured by the controlled application of different normal forces with AFM. There exists initial bias due to the growth of nanowires. The systematic bias can also occur during the measurement under different applied forces. Therefore, SPAR can be used to get better estimate of elastic modulus.
K
θˆ = arg m in ∑ − 2 l k ( β , τ 2k , θ ). θ
k =1
Step 3. Given θˆ and τˆ 2 , update β using (10). Step 4. Go to Step 1 until convergence. To obtain the initial estimates ordinary least squares estimate for Since
θ
θˆ
and βˆ , we use the
β as βˆ = ( XT X) −1 XTY .
is the parameter in the Gaussian correlation function,
we can take a relative large value as the initial estimate θˆ (Santner et al. 2003).
4.2 Illustration Now we apply the proposed general error structure Σ to the selected model (7). We define the generalized
Y = Xβ and Σ1/ 2 ˆ are estimates of β , Σ . satisfies Σ1/ 2 Σ1/ 2 = Σ . Here βˆ , Σ ˆ ) , where residual e = Σ −1/ 2(Y − Y
Clearly, Figure 9(a) shows that the selected model fits the data well using the general error structure. Moreover, the generalized residuals in Figure 9(b) look much more random with the one in Figure 8(d). Therefore, the selected model is more appropriate than the one with independent error structure.
References Mai, W. J. and Wang, Z. L. (2006). Quantifying the elastic deformation behavior of bridged nanobelts. Applied Physics Letters 073112. Miller, A. (2002) Subset Selection in Regression, CRC Press, New York.
(a) fitting
San Paulo, A., Bokor, J., Howe, R. T., He, R., Yang, P., Gao, D., Carraro, C. and Maboudian, R. (2005). Mechanical elasticity of single and double clamped silicon nanobeams fabricated by the vapor-liquid-solid method. Applied Physics Letters 053111.
(b) generalized residual
Figure 9: Performance of the selected model using general error structure for NB.
Santner, T .J., Williams, B. J., and Notz, W. (2003), The Design and Analysis of Computer Experiments, Springer, New York.
To compare the efficiency of the estimates of E in the selected model with different error structures, we assume that the
6
Nanomaterials †
Xinwei Deng , V. Roshan Joseph † , Wenjie Mai ‡ , Z. L. Wang ‡ , C. F. Jeff Wu † †
H. Milton Stewart School of Industrial and Systems Engineering, ‡ School of Materials Science and Engineering, Georgia Institute of Technology, Atlanta, GA
Abstract
empirical statistical modeling technique to identify the effects of such factors. After filtering out such effects, we can estimate the elastic modulus based on the physical law more efficiently. The article is organized as follows. In Section 2, we give a brief description of the measurement method and the model used in Mai and Wang (2006), which provides the motivation of our proposed approach. Section 3 is devoted to building and fitting appropriate statistical models through model selection. Estimate of the elastic modulus is obtained from the selected model as well. To accommodate the correlation structure in the bending curves, we propose a more flexible model with general error structure in Section 4. Some discussions and future research directions are given in Section 5.
Accurate estimation of elastic modulus of certain nanomaterials such as Zinc Oxide nanobelt is important in many applications. In this work, a statistical modeling technique called SPAR is proposed to account for the various errors introduced during measurement. SPAR can automatically detect and remove the systematic errors and therefore, can give an accurate and precise estimate of the elastic modulus. KEY WORDS: nanostructure, profile adjustment, statistical modeling, model selection, correlation structure
1 Introduction
2 Existing Method of Experimentation and Modeling
One-dimensional (1D) nanomaterials, such as carbon nanotubes, semiconductor nanowires, and oxide nanobelts, are the fundamental building blocks for constructing nanodevices and nanosystems that exhibit superior performances. Mechanical behavior of 1D nanomaterials is one of the most important properties that dictate their applications in nanotechnology. For example, carbon nanotubes are found to be the strongest and stiffest materials on earth in terms of tensile strength and elastic modulus. They can be promising building blocks for future Nano Electro Mechanical Systems (NEMS). One approach of quantifying the mechanical property such as elastic modulus of 1D nanomaterials is based on atomic force microscope (AFM) and nanoindenter. A common strategy is to deform a 1D nanostructure that is supported at the two ends by using an AFM tip, which pushes the 1D nanostructure at some locations. Then the elastic modulus can be determined through the quantified force-displacement curves. However, the accuracy of this measurement is limited by noise factors such as the size of tip, the accuracy of positioning the AFM tip, and the surface roughness of the 1D nanomaterials. This article proposes a new approach to estimate the elastic modulus through statistical and physical modeling. Mai and Wang (2006) proposed a new approach for quantifying the elastic deformation behavior of a 1D nanostructure. The approach is based on an AFM contact mode continuous scan of a ZnO nanobelt that is supported at the two ends. A quantitative fitting of the elastic bending shape of the nanobelt as a function of the bending force provides a reliable method for estimating the elastic modulus of the nanobelt. However, the nanobelt may shift or deform during measurement, leading to bias in the estimation of the elastic modulus. The small size of the nanostructure makes their manipulation rather difficult. Moreover, the level of allowable tolerance on measurement errors for nanomaterials decreases since noise or bias becomes much larger in comparison to the smaller scale of nanomaterials. The presence of these noise factors and other unknown factors makes the observed curve between the deflection and force look quite different from the free-free beam model (FFBM) derived using the physical law. Since the FFBM does not account for the various noise factors, we use an
The ZnO nanobelts used for the study were prepared by physical vapor deposition. A silicon substrate is prepared with long and parallel trenches caved at its surface by nanofabrication. The trenches are about 200 nm deep and 1.25 um wide. Long ZnO nanobelts were manipulated across the trenches over many periods. The morphology and dimensions of the nanobelt were captured by a scanning electron microscope (SEM) and AFM. The SEM image gives the width of the trench and the length and width of the nanobelt, and the AFM image gives the thickness of the nanobelt. The measurement was made by scanning the nanobelt along its length direction using an AFM tip in contact mode at a constant applied force. A series of bending images of the nanobelt were recorded by changing the magnitude of the contact force from low to high.
(a)
(b)
Figure 1: (a) The AFM image profiles of the suspended NB under different load forces in contact mode. (b) The normalized AFM image profile by subtracting the profile acquired at 78 nN (nano Newton) from the profiles in (a).
The profiles of a suspended nanobelt along the length direction under different contact forces are shown in Figure 1(a). As shown in Figure 1(a), the image profiles of a nanobelt (denoted by NB) recorded the deflection of all the points along
1
its length under different applied forces. Each curve was obtained by averaging ten consecutive measurements along its length under the same loading force. The curves in Figure 1(a) are not smooth due to a small surface roughness (around 1 nm) of the nanobelt. In addition, the as-attached nanobelt on the trenches is not perfectly straight, possibly due to initial bending during the sample manipulation. Figure 1(a) indicates that there are some noise factors affecting the deflection curves. Mai and Wang (2006) suggested a free-free beam model (FFBM) to quantify the elastic deflection. The diagram of the FFBM is shown in Figure 2.
poorly with a lot of noise, then subtracting this profile to normalize the data can result in poor estimation of E from the FFBM. Recall that the deflection v in the FFBM in (1) is a linear function of the applied force F given the distance x . The reason for normalizing the data by subtracting the image profile at a low applied force is to eliminate the initial bias, i.e., the effect of the surface roughness and initial bending of the nanobelt. However, if some systematic biases occurred during the measurement, normalizing the data may not be enough for obtaining a good fitting based on the FFBM. For example, the deflection profiles obtained at applied force F = 235, 248 and 261 nN lie above on those obtained at lower force F = 209 and 222 nN. This is inconsistent with the model equation in (1) because the deflection is expected to increase with force. The FFBM itself cannot explain this phenomenon. This pattern still persists in the normalized profiles in Figure 1(b), where the profile is normalized by subtracting the profile at F = 78 nN from the original profiles in Figure 1(a). Therefore, the MW method may not be a proper way to fit the profile data of NB. It requires a more general model to identify other factors besides the initial bias. To overcome these problems, we propose a general regression model that directly incorporates the initial bias and potential systematic biases introduced during measurement. Then we use variable selection to identify terms associated with the systematic biases and then adjust the profiles by subtracting these terms from the original profiles. When a final model is chosen, we estimate the elastic modulus E based on the FFBM. We call the method sequential profile adjustment by regression (SPAR).
Figure 2: The schematic diagram of the free-free beam model (FFBM).
In this model, the two ends of the nanobelt can freely slide. When a concentrated load force F is applied at the contact point x of the AFM tip away from the end A , the deflection of nanobelt at x is determined by
v = −
Fx 2 (L − x)2 , 3 E IL
(1)
L is the width of trench, and I is the moment of inertia given by wh3 / 12 for the rectangular beam, where w and h are respectively the width
where E is the elastic modulus,
and thickness of nanobelt. The elastic modulus E can be estimated from the image profile data based on the FFBM. Figure 3 shows an illustrative example of FFBM profiles, which are symmetric and perfectly smooth but does not account for the noise factors in the measurements.
3 General Model and Model Selection 3.1 General Model As shown in Figure 1(a), suppose there are K image profiles, i.e., the nanobelt is scanned sequentially under K different applied forces F1 , F2 , …, FK . The experimenter usually changes the magnitude of applied force F from low to high, i.e., F1 < F2 1)+ +δK−1(x)I(k >K−1)+ε(x,Fk), (3) where I (⋅) is an indicator function and ε ( x, Fk ) is the error 2
RMSE = {∑ i =1 ∑ k =1 (v( xi , Fk ) − vˆ( xi , Fk )) 2 / df }1/2 , (5) n
term. Specifically, when the force Fk is applied to make the
K
AFM tip in contact with the nanobelt, the proposed approach models the deflection as
and df is the degrees of freedom in the corresponding model.
k , i.e., some of the δ k ’s may be zero. We therefore use a
selection, where
v( x, Fk ) = β ( x) Fk + δ 0 ( x) + δ1 ( x) +
+ δ k −1 ( x) + ε ( x, Fk ). Alternatively, we can use the Bayesian information criterion In reality, there may or may not be a deformation at stage (BIC) to select δ k ( x) into the model at each step of the
variable selection technique to identify the significant δ k ’s and include only them in the final model.
BIC = ∑i=1 ∑k =1 (v( xi , Fk ) − vˆ( xi , Fk ))2 / σ 2 + p log N, (6)
3.2 Model Selection
p is the number of parameters, and N is the number of
n
K
observations in the corresponding regression model. If 2
3.3 Example
the model (3) is a linear regression with K+1 parameters 1 / E, δ 0 ( xi ), δ1 ( xi ), , δ K −1 xi , i.e.,
where
knowledge
of
the
2
FFBM.
ε i = (ε ( xi , F1 ), , ε xi , FK
T
In ,
In the image profiles of NB, the deformation is recorded at n = 161 points along the length of the nanobelt under K=15 different applied forces. The length of NB is L = 1252 nm and the moment of inertia in the FFBM is I = 8216510 nm4. Figure 4 shows the model selection results using the proposed method. The δ k ( x ) is sequentially selected into the model in the
⎞⎛ 1 / E ⎞ ⎟⎜ ⎟ ⎟⎜ δ0 ( xi ) ⎟ + ε i, (4) ⎟⎜ ⎟ ⎟⎜ ⎟ ⎠⎝ δ K −1 ( xi ) ⎠
γ ( xi ) = x ( L − xi ) / (−3IL) 2 i
incorporates the
ε ( xi , Fk )
error
in
(6) is not available, an estimate σˆ can be obtained from the data.
The general model (3) considers all potential bias factors. In reality, it is likely that only a few of them contribute toward the deformation on the nanobelt. So it is important to find significant δ k ’s and build an appropriate model. Given the distance xi ,
⎛ v( xi , F1 ) ⎞ ⎛ γ ( xi )F1 1 0 ? ⎜ ⎟ ⎜ ⎜ v( xi , F2 ) ⎟ = ⎜ γ ( xi )F2 1 1 ? ⎜ ⎟ ⎜ ⎜ ⎟ ⎜ ⎝ v( xi , FK ) ⎠ ⎝ γ ( xi )FK 1 1 ?
σ2
following order:
δ 6 ( x)
and
δ 0 ( x) , δ12 ( x) , δ10 ( x) , δ 8 ( x) , δ 9 ( x)
δ 2 ( x) .
,
It can be seen that after adding three or
four terms, the decrease of RMSE starts to level off while the corresponding BIC value starts to increase. By considering both criteria, we take three δ k terms to build the final model. Thus,
the vector
represents the
the chosen model is
error occurred at distance xi under applied force Fk . The
v(x,Fk) =β(x)Fk +δ0(x)+δ10(x)I(k >10)+δ12(x)I(k >12)+ε(x,Fk),k =1, ,K.(7)
model (3) considering all xi is an over-parameterized liner model with parameters 1 / E , δ 0 (x), δ1 (x), ? , δ K −1 x) , where
x = ( x1 , x2 , , xn ) , δ k (x) = (δ k ( x1 ),… , δ k ( xn )) for k = 0,… , K − 1 . To find a proper model, we need a model selection strategy (Miller, 2002). In our situation, however, it is not appropriate to implement variable selection among all nK+1 covariates associated with the parameters 1/E, δ0 (x), δ1 (x),? , δ K −1 x) . Recall that δ0 (x) is interpreted as
Figure 4: Forward model selection using RMSE and BIC on the NB data.
the initial bias effect from the surface roughness and initial bending and δ k ( x) as the systematic bias effect from applied
Here not only the initial bias
δ 0 ( x)
δ10 ( x) and δ12 ( x)
is significant, the
force Fk for k ≥ 1 . It is thus more reasonable to keep each
systematic biases
δ k (x) as a whole parameter set in model selection. It can make
role in modeling the data. To get more insight on the selected δ k ’s in (7), at each stage of the selection, we define an adjusted
the selected model more interpretable from the physical perspective. Starting with the model including only the FFBM, we use forward selection to add one δ k ( x) at a time. To begin with, we assume that the errors
ε (xi , Fk )
deflection For
also play an important
v( xi , F ) adj as v( xi , F ) minus the selected δ k ’s. example,
at
stage
3,
v(x, Fk )adj =v(x, Fk ) −δ0(x) −δ12(x)I(k >12) −δ10(x)I(k >10),k =1, , K.
are independent with a
Note that the systematic bias
δ12 ( x)
introduces the
normal distribution N (0, σ ) . Then the estimation of parameters can be easily calculated by the maximum likelihood estimation (MLE). In each step of the forward selection, we select δ k ( x ) that has the smallest root mean square error
profiles at F = 235, 248 and 261 nN. Similarly,
(RMSE) of the corresponding model, where
to F15 . Figure 5 shows the changes of five adjusted deflection
2
deflection into the image profiles starting from F13 , i.e., the brings in deformation on the profiles under
profiles under applied force
3
δ10 ( x) only applied force F11
F11 to F15 as the three δ k
selected model and lead to a better fitting of the NB data.
terms are sequentially selected into the model. The original five profiles are shown in Figure 5(a). When δ 0 ( x) is selected into the model at stage 1 of selection, it adjusts the initial bias among the five image profiles. In Figure 5(b), the adjusted deflection v( xi , Fk ) adj = v( xi , Fk ) − δ 0 ( x) looks closer to the FFBM, but the inconsistent pattern shown in Figure 1 still remains. Note that the inconsistent pattern appears between the profiles under F11 = 209 nN, F12 = 222 nN and those under F13 = 235 nN, F1 4 = 2 4 8 nN, F1 5 = 2 6 1 nN. At stage 2 of the selection,
δ12 ( x)
estimate of
δ12 ( x)
Figure 6: Estimates of
is selected into the model. It further adjusts
δ12 ( x)
estimate of and
δ10 (x)
δ10 ( x)
from the selected model
of NB.
the profiles under the applied force F1 3 , F1 4 , and F1 5 . From Figure 5(c), we can see that the adjusted deflection v( x, Fk )adj = v( x, Fk ) − δ 0 ( x) − δ12 ( x) I (k > 12) is to push
The image profiles of the adjusted deflection v ( x, F ) adj based on the selected model is shown in Figure 7. Compared with the original profiles in Figure 1, the adjusted deflection profiles are more consistent with the FFBM as shown in Figure 2. With three selected δ k ’s in the model, the selected model (7) can provide
the profiles obtained at force F13 , F14 , and F15 to lie below
F12 . The inconsistency no more reliable and precise estimation of the elastic modulus E. longer exists in Figure 5(c). Therefore, adding δ12 ( x) can those obtained at force F11 , and
remove the inconsistent pattern in the NB data. At stage 3 of the selection, δ10 ( x) is chosen into the model. It can adjust the five image profiles at the applied force from F11 = 209 nN to F15 = 261 nN. As shown in Figure 5(b), to adjust the inconsistency among these five profiles, it is likely that the adjusted deflections have been pushed downwards too much. From Figure 5(d), we can see that adding δ10 ( x) into the model is to pull all five profiles upwards and make the adjusted deflection v(x, F)adj = v(x, Fk ) −δ0 (x) −δ12 (x)I (k >12) −δ10 (x)I (k >10) a
Figure 7: The image profiles for the adjusted deflection of NB.
better fit to the FFBM.
Stage 0: original profiles
(a) Fitting: MW
Stage 1: adjusted by δ 0
(c) Residuals: MW Stage 2: adjusted by
δ0
and
δ12
Stage 3: adjusted by δ 0 ,
δ12
δˆ10 ( x)
To gauge the performance of the selected model using SPAR, we compare it with the MW method. The fit and residual plots from these two approaches are shown in Figure 8. The fitting in Figure 8(a) is obtained by adding the initial profile back to the fitted normalized data from the MW method. The residuals from the MW method show some systematic patterns, which indicates that the model needs improvement. No systematic pattern is observed in the residuals based on SPAR. Clearly, the
are shown in
Figure 6. We can see that the opposite shapes of δˆ12 ( x ) and
δˆ10 ( x )
(d) Residuals: SPAR
Figure 8: Comparison of two methods on the NB data.
and δ10
Figure 5: Illustration of the adjusted deflection profiles under applied force from F11 = 209 nN to F15 = 261 nN.
The two estimates δˆ12 ( x ) and
(b) Fitting: SPAR
in Figure 6 help remove the inconsistent pattern of the
4
selected model performs better. It removes the inconsistent pattern discussed above, while the MW method does not recognize this pattern. The residuals from the selected model are also much smaller. Table 1 summarizes the estimation results using the two methods. Clearly SPAR gives a more precise estimate of the elastic modulus E. The 95% confidence interval of E from SPAR is (99.97, 103.07) and that from MW method is (91.24, 97.44). The non-overlapping of intervals suggests that one of the estimates can be misleading or wrong. Because SPAR incorporates the initial bias and adjusts the inconsistent pattern in the profiles, it is expected to provide more accurate estimate of the elastic modulus than the MW method. To further verify this point, we perform SPAR using only half of the profiles of NB, i.e., the profiles under applied force F = 78, 105, 131,…, 201 nN. The estimate of the elastic modulus Eˆ = 102.67 and the 95% confidence interval (100.55, 104.79) are similar to those using SPAR with all the profiles of NB. This shows that SPAR can give a more reliable estimate even using half of the profiles.
describing the error structure within the deflection curve over xi ’s for the applied force Fk . The resulting covariance matrix
Σ has a diagonal block structure since the errors between different deflection curves are considered to be independent. The
σ2
term is used to quantify the error variation for each deflection curve obtained from averaging 10 consecutive measurements.
= τ k2 R, where τ k2 is the error variance of the curve obtained at Fk , and R = ( rij ) n×n is the Next we model Σ k as Σ k
correlation matrix with rij quantifying the correlation between two deflections at the distance xi and x j obtained under the same applied force. We use the Gaussian correlation function (Santner et al., 2003) to model the correlation matrix R with
rij as rij = exp(−θ ( xi − x j ) 2 ). Here each Σ k uses the
same θ since we assume that different deflections share the same correlation pattern.
Table 1: Comparison of estimates with the NB data.
RMSE MW Method 0.86 SPAR 0.37
1 / Eˆ 1.06e-02 9.85e-03
se(1/E) 1.77e-04 7.63e-05
Eˆ 94.34 101.52
se(E) 1.58 0.79
4.1 Parameter Estimation The proposed model in (3) is a general linear model with the Σ constructed above, i.e., error structure Y = Xβ + ε, ε ∼ N (0, Σ) . The parameters in this model
4 Modeling with General Error Structures
are (β, τ 2, θ ) , where τ 2 = (τ 1 , ? ,τ K . We consider the maximum likelihood estimation (MLE) for these parameters. For 2
The deflection of nanobelt is a continuous and smooth phenomenon. For a given force F, the nanobelt is scanned using AFM tip along the distance x. Therefore, the deflection obtained at the distance x = x i should be positively correlated with
notational convenience, denote Y = ( Y1, ? , Y k
deflection curves, which are obtained by using different applied forces, their correlation is much smaller. So we can consider them to be independent in the modeling. As shown in the residual plots in Figure 8(d), the residuals have systematic patterns along the distance x . In particular, for a given force Fk , if the residual at xi is large, then the residual at
l(β, τ2,θ ) = −
imposing some error structure is warranted. Note that each profile curve was obtained by averaging 10 consecutive measurements along its length under the same loading force. The aim of taking average is to reduce the measurement error due to sources like equipment accuracy. There are still errors after taking average. By incorporating this prior information and correlation structure, we can build a more general error structure as follows. The model in (3) can be written as a linear regression model Y = Xβ + ε , where the responses vector is
parameters in the log-likelihood function. The parameters τ 2 and
θ
−1
are involved in every matrix inverse Σ k . It requires
intensive computations to estimate these parameters by directly maximizing the log-likelihood function. Instead we propose an iterative algorithm to efficiently estimate parameters from
l (β, τ 2,θ ) . First observe that given τ 2 and generalized least squares estimate
β =(1/ E,δ0(x1),? ,δ0 xn),δ1(x1),? ,δ1 xn),? ,δK−1 x1),? ,δK−1 xn)) , and T
X is the corresponding model matrix. Assuming that the error
K
diagonal
block
is its
−1
ε = (ε ( x1, F1 ),? , ε xn , F1 ),? , ε x1, FK ),? , ε xn , FK )) follows a normal distribution, i.e., ε ∼ N (0, Σ) , we consider the model Y = Xβ + ε with a more general error structure Σ the
θ , the MLE of β
⎡K ⎤ ⎡K ⎤ β = ⎢∑ (XTk Σ-k1Xk )⎥ ⎢∑ (XTk Σ-k1Yk )⎥ . ⎣ k =1 ⎦ ⎣ k =1 ⎦
T
of
(8)
= τ k2 R + σ 02I . Suppose the number of δ k ( x) in the model is m, 0 ≤ m ≤ K , then there are ( nm + 1) + K + 1
parameters
consists
1K ⎡⎣log | Σk | +(Yk − Xkβ)T Σ-k1(Yk − Xkβ)⎤⎦ , ∑ 2 k=1
where Σ k
Y = (v( x1 , F1 ), ? , v xn , F1 ),? , v x1 , FK ),? , v xn , FK ))T ,
which
, where
T
a distance close to xi is also likely to be large. It indicates that
vector
T
Y k ∈ R as Y k = (v( x1 , Fk ),? , v xn , Fk )) . Similarly, T n× p denote X = ( X1, ? , X k , where X k ∈ R is the part of the model matrix X corresponding to Y k , and p is the dimension of β . The log-likelihood function can be written as n
those obtained near xi . On the other hand, between two
the
2
For given
submatrics
(9)
β and θ , maximizing the log-likelihood function
l (β, τ ,θ ) in (8) is equivalent to maximizing each lk (β, τ 2,θ ) individually, where 2
Σ k + σ 2I, i = 1,… , K , where Σ k is an n × n matrix 5
1 lk (β, τ2,θ ) = − ⎡⎣log | Σk | +(Yk − Xkβ)T Σ-k1(Yk − Xkβ)⎤⎦. 2 Since
τ k2
underlying model is Y = Xβ + ε, ε ∼ N (0, Σ) . Since the true value of the elastic modulus E is unknown, we compare the
(10)
standard error of the estimate Eˆ . For convenience, denote Eiid as the elastic modulus parameter E in the model with iid error, and E gen as the corresponding one in the model with general
appears only in lk (β, τ 2, θ ) of the log-likelihood
function (8),
τ k2
can be individually estimated by maximizing
error structure. In the case of NB1: Eˆ gen = 114.84 which is
lk (β, τ 2,θ ) . Obviously, optimization with only one parameter
different from Eˆ iid = 101.52 in Table 1. This difference can
is easy. For given β and τ , estimating θ is also a one-parameter optimization problem by maximizing the 2
be explained by the fact that the residuals in Figure 8(d) have a clear pattern of positive correlation. The 95% confidence interval (111.34, 118.34) of Egen is disjoint with the 95% confidence
log-likelihood function l (β, τ 2, θ ) in (8). Thus, if some initial
estimates of β and θ are available, we can obtain the MLE of parameters through the following iterative algorithm:
interval (99.97, 103.07) of E iid . Although the length of the confidence interval of E gen is larger, it is a more reasonable
ˆ. Step 0. Obtain initial estimates θˆ and β
estimate because it incorporates the correlation structure in the deflection profiles.
ˆ , update τ 2 = (τ 2 ,? ,τ 2 , i.e., Step 1. Given θˆ and β 1 K 2 2 − 2lk (β, τ k ,θ ) . τˆ k = arg min 2
5 Discussions and Conclusions
τk
2 2 Step 2. Given τ垐 2 = (τ 1 , ? ,τ K
and βˆ , update
θ
, i.e.,
In this article, we report a new method called SPAR to estimate the elastic modulus of nanobelts through statistical modeling and analysis. It can identify significant bias factors using model selection, and incorporates them into the fitted model. It can automatically remove the initial bias, and adjust the inconsistent pattern caused by the systematic biases. Therefore, it can give more precise and reliable estimation of the elastic modulus. Due to the small size of nanomaterials, the noise and bias during measurement become relatively large compared with the actual scale of nanomaterials. Taking account of bias factors into the theoretical model through statistical modeling can greatly improve the estimation of the interested mechanical property. Here we propose the SPAR method and study its performance for data arising from a specific experiment on nanobelts. Generally, in the field of quantifying mechanical properties of 1D nanomaterials, SPAR can have broad applications. For example, San Paulo et al. (2005) studied the mechanical elasticity of single and double clamped nanowires. The deflection of nanowires is measured by the controlled application of different normal forces with AFM. There exists initial bias due to the growth of nanowires. The systematic bias can also occur during the measurement under different applied forces. Therefore, SPAR can be used to get better estimate of elastic modulus.
K
θˆ = arg m in ∑ − 2 l k ( β , τ 2k , θ ). θ
k =1
Step 3. Given θˆ and τˆ 2 , update β using (10). Step 4. Go to Step 1 until convergence. To obtain the initial estimates ordinary least squares estimate for Since
θ
θˆ
and βˆ , we use the
β as βˆ = ( XT X) −1 XTY .
is the parameter in the Gaussian correlation function,
we can take a relative large value as the initial estimate θˆ (Santner et al. 2003).
4.2 Illustration Now we apply the proposed general error structure Σ to the selected model (7). We define the generalized
Y = Xβ and Σ1/ 2 ˆ are estimates of β , Σ . satisfies Σ1/ 2 Σ1/ 2 = Σ . Here βˆ , Σ ˆ ) , where residual e = Σ −1/ 2(Y − Y
Clearly, Figure 9(a) shows that the selected model fits the data well using the general error structure. Moreover, the generalized residuals in Figure 9(b) look much more random with the one in Figure 8(d). Therefore, the selected model is more appropriate than the one with independent error structure.
References Mai, W. J. and Wang, Z. L. (2006). Quantifying the elastic deformation behavior of bridged nanobelts. Applied Physics Letters 073112. Miller, A. (2002) Subset Selection in Regression, CRC Press, New York.
(a) fitting
San Paulo, A., Bokor, J., Howe, R. T., He, R., Yang, P., Gao, D., Carraro, C. and Maboudian, R. (2005). Mechanical elasticity of single and double clamped silicon nanobeams fabricated by the vapor-liquid-solid method. Applied Physics Letters 053111.
(b) generalized residual
Figure 9: Performance of the selected model using general error structure for NB.
Santner, T .J., Williams, B. J., and Notz, W. (2003), The Design and Analysis of Computer Experiments, Springer, New York.
To compare the efficiency of the estimates of E in the selected model with different error structures, we assume that the
6
