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Experimental evidence on the role of hydrophobic silane coating on
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Si stamps in nanoimprint lithography
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Alborz Amirsadeghi1, Lance Brumfield1, Junseo Choi1, Emily Brown1, Jae Jong Lee2,
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Sunggook Park1,*
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1
Mechanical Engineering Department and Center of Bio-Modular and Multi-scale Systems,
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Louisiana State University, USA
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Nano-Mechanical Systems Research Division, Korea Institute of Machinery and Materials,
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104 Sinseongno, Yuseong-Gu, Daejeon 305-343, Korea
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Contents
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1. Demolding theory
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1.1. Energy balance in demolding
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1.2. Dependence of L on work to break adhesion
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1.3. Dependence of L on work to break adhesion ∆𝑼 and 𝑸𝒇
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1.4. Implementation of the dependence of L on demolding work
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2. Curve fitting results
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References
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Page 1 of 7
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1. Demolding theory
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1.1. Energy balance in demolding
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The schematics of the initial stage and partially debonded stage of the fiber-matrix in
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composites are given in Figure S.1. The only difference between demolding in NIL and
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debonding in fiber-matrix composite systems is that the adhesion term in NIL also includes the
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contribution from horizontal interfaces of the resist/stamp system while adhesion and friction
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are operative at the vertical (sidewall) interfaces in both processes.
Applied stress 
Applied stress 
Bonded
Interface
Matrix
or stamp
z
Debond
length, L
Fiber
or resist
2a
 / Vr
 / Vr
(a)
(b)
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10
2a
Figure S.1. Schematics of the (a) initial stage and (b) partially debonded stage of the fibermatrix in composites.
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According to the theoretical work on fiber-matrix debonding energy by Sutcu and
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Hillig,1 equilibrium demolding occurs over an incremental distance d𝐿 if the external work, i.e.
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work of demolding, is just equal to the summation of the local increase in the elastic energy,
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work to break adhesion between the interfaces prior to sliding and incremental dissipative
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energy during sliding by friction. The energy balance yields,
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d𝑊𝑡
d𝐿
=
d(∆𝑈)
d𝐿
+
2𝑉𝑟 𝛤𝑑
𝑎
+
d𝑄𝑓
(1)
d𝐿
where 𝑊𝑡 , ∆𝑈 , 𝛤𝑑 and 𝑄𝑓 are the external work, stored elastic energy, interfacial debond
Page 2 of 7
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energy and frictional dissipation, all of which are presented in the units of energy per unit area.
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𝑉𝑟 is the volume fraction of the resist in the patterned region and 𝑎 is the half of the pattern
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width. Unless the pattern width a is changed, 𝑉𝑟 does not change with the stamp depth or, in
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other word, the demolding length L and is thus constant. Also, 𝛤𝑑 is constant which has a
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characteristic value for a pair of bonding materials. Integration of equation (1) over the
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demolding length 𝐿 (equal to the stamp depth) will provide the following energy balance,
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𝑊𝑡 = ∆𝑈 +
2𝑉𝑟 𝛤𝑑
𝐿 + 𝑄𝑓
𝑎
(2)
𝑆𝑡𝑜𝑟𝑒𝑑
𝑇𝑜𝑡𝑎𝑙
𝑊𝑜𝑟𝑘
𝐹𝑟𝑖𝑐𝑡𝑖𝑜𝑛𝑎𝑙
(𝑑𝑒𝑚𝑜𝑙𝑑𝑖𝑛𝑔 ) = ( 𝑒𝑙𝑎𝑠𝑡𝑖𝑐 ) + ( 𝑡𝑜 𝑏𝑟𝑒𝑎𝑘 ) + (
)
𝑑𝑖𝑠𝑠𝑖𝑝𝑎𝑡𝑖𝑜𝑛
𝑒𝑛𝑒𝑟𝑔𝑦
𝑎𝑑ℎ𝑒𝑠𝑖𝑜𝑛
𝑤𝑜𝑟𝑘
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1.2. Dependence of L on work to break adhesion
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Equation (2) clearly shows that the work to break adhesion has a linear dependence on 𝐿, with
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the slope related to the pattern dimensions (𝑉𝑟 and 𝑎) and the interfacial debonding energy 𝛤𝑑 .
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1.3. Dependence of L on work to break ∆𝑼 and 𝑸𝒇
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According to Sutcu and Hillig,1 the changes in the elastic energy per unit area (∆𝑈/𝑎)
and frictional dissipation per unit area (𝑄𝑓 /𝑎 ) are given by,
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∆𝑈
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1
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𝑎
𝜎3
= 12𝐸
𝑟 𝜏𝑟
) +
𝜌
2
𝑉𝐸
( 𝑉𝑠 𝐸𝑠 ) −
𝑟
𝑉𝑟 𝐸𝜏𝑟 𝜎𝑟𝑟
𝜎𝜏𝑟
𝐸𝑟
𝜎𝑟 2
6𝑉𝑠 𝐸𝑠 𝐸𝑟
[( 𝜏𝑟 ) −
𝐿
𝜎𝑠𝐼 𝐸
𝑟
𝜎𝑟 2
1
[(2𝜏𝑟 ) − 𝜌2 ] +
𝑟
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𝜌2
𝜏𝑒2 𝜗𝑉𝑟 3
6𝐺𝑠 𝜌
𝜏
(2 − 𝜏𝑒) +
𝑟
𝜏𝑒 𝜏𝑟 𝜗𝑉𝑟
4𝐺𝑠
]
𝜎𝑟
𝜏
(2 − 𝜏𝑒 ) ( 𝜏𝑟 −
𝑟
𝑟
(3)
and
𝑄𝑓
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𝑎
= (𝑎) (𝜎 +
𝐸𝑠
𝜏
𝐿
2
𝐿 2 4𝜏𝑟2 𝑉𝑟 𝐸 𝐿
(
3𝑉𝑠 𝐸𝑠 𝐸𝑟 𝑎
) 𝐸𝑟 (𝑎 + 𝜌) − (𝑎)
𝑟
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where,
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∆𝑈: the increase in stored elastic energy
Page 3 of 7
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𝐿 𝜗𝜏𝑟2 𝑉𝑟
𝐺𝑠
+ 𝜌 ) − (𝑎 )
(4)
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a: half width of a stamp structure
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L: the demolding length, which is equal to the stamp depth.
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𝑉𝑟 𝑎𝑛𝑑 𝑉𝑠 : volume fraction of resist and stamp, respectively.
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𝐸𝑟 , 𝐸𝑠 , 𝐸: the resist, stamp, and composite elastic moduli
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𝐺𝑠 : Stamp shear modulus which is related to Es by 𝐺𝑠 = 2(1+𝑠
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Poisson’s ratio
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𝜎: applied stress
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𝜎𝑟𝑟 𝑎𝑛𝑑 𝜎𝑠𝑟 : the residual axial stresses in the resist and stamp, respectively, which satisfy
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𝐸
𝑠)
where s is the stamp
𝑉𝑟 𝜎𝑟𝑟 + 𝑉𝑠 𝜎𝑠𝑟 = 0 for the uncracked resist/stamp region
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𝜏𝑟 : frictional shear stress at the cracked interface between in the resist and stamp.
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𝜏𝑒 : elastic shear stress given by
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𝜗: a convenience parameter given by 𝜗 = −
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𝜌: the stress decay parameter given by 𝜌2 =
2 log 𝑉𝑟 +𝑉𝑠 (3−𝑉𝑟 )
2𝑉𝑠2
4𝐸𝐺𝑠
𝑉𝑠 𝐸𝑠 𝐸𝑟 𝜗
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In our experiments, we used stamps with the same grating patterns but with different
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stamp depths. Thus, the parameters related to the pattern dimensions, a, 𝑉𝑟 and 𝑉𝑠 , can be
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considered as constant. 𝜏𝑒 is defined as the elastic shear stress at the start of the coherent region
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and will exponentially increase as it moves further into the coherent region in the z direction by
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𝜏(𝑧) = 𝜏𝑒 𝑒𝑥𝑝[−𝜌(𝑧 − 𝐿)/𝑎]. 𝜏𝑟 is the frictional shear stresses at the cracked region and can
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be approximated as a constant over L. Experimentally, 𝜏𝑟 can be measured using friction force
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microscopy. Debonding at the sidewall of the stamp/resist interface can only proceed if the
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level of 𝜏𝑒 reaches a characteristic limiting shear stress value 𝜏𝑑 . The relationship between 𝜏𝑑
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and 𝛤𝑑 is given by 𝜏𝑑 − 𝜏𝑓 = √4𝐺𝑠 𝛤𝑑 /(𝑎𝜗) .
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The requirement that the axial stress be continuous in the matrix and the fiber at the end
Page 4 of 7
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of the slip region where z = L, leads to the relation between 𝜏𝑒 and 𝜏𝑓 .
1 𝑉𝑠 𝐸𝑠
𝜏𝑒 = 𝜌𝜏𝑓 [ 2𝜏
𝑟
(𝜎 +
𝑉𝐸
𝑟
𝐸𝜎𝑠𝑟
𝐸𝑠
𝐿
)−𝑎]
(5)
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Assume that the applied external stress (σ) and the residual axial stresses in resist and stamp in
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the patterned region (𝜎𝑟𝑟 , 𝜎𝑠𝑟 ) are constant during the demolding process, only 𝜏𝑒 in Equation (3)
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is a function of L. Combining Equation (3) and (5), it is easily seen that ∆𝑈/𝑎 is a third-order
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polynomial function of L in the following form:
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∆𝑈
𝑎
= 𝐴′ + 𝐵′𝐿 + 𝐶′𝐿2 + 𝐷′𝐿3
(6)
where 𝐴′ > 0 and 𝐷′ > 0. On the other hand, 𝑄𝑓 /𝑎 is presented in the form of
𝑄𝑓
𝑎
= 𝐵′′𝐿 + 𝐶′′𝐿2 + 𝐷′′𝐿3
(7)
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where 𝐷′’ < 0. Also, the requirement that the demolding work does not become negative with
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increasing L provides an additional condition of 𝐷′ + 𝐷′′ > 0.
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1.4. Implementation of the dependence of L on demolding work
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The case which is first considered is the condition where the imprint stamp has
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structures with a low aspect ratio resulting in relatively low magnitude of the elastic storage
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energy (∆𝑈). When the stamp surface is not coated with an anti-adhesive silane molecule, then
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both the interfacial debonding energy (𝛤𝑑 ) and frictional dissipation (𝑄𝑓 ) are high in magnitude.
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Thus, the contribution of the elastic energy to the 3rd order polynomial function is subtracted
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by that of the frictional dissipation which is also a 3rd order polynomial function, leading to
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weakening the 3rd order polynomial dependence on L. As a result, the contribution of adhesion
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dominates in the demolding work and accordingly the demolding work will follow a linear
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dependence on L. When the stamp surface is coated with an effective anti-adhesive coating, the
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most significant change is a decrease in 𝛤𝑑 compared to the decrease in ∆𝑈 and 𝑄𝑓 . As a result,
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the demolding force should show a 3rd order polynomial function of L.
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When the imprint stamp has high aspect ratio structures, a significant amount of elastic
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energy (∆𝑈) needs to be stored in the resist before the interface is broken and the frictional
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dissipation occurs. Thus, the magnitude of the elastic storage energy dominates over 𝛤𝑑 and 𝑄𝑓
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and the demolding work will follow a 3rd order polynomial function of L irrespective of the
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application of an anti-adhesive coating.
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In order to verify the different dependence of L on the demolding work by experiments,
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the depths of the stamp structures are designed to be low enough so that the elastic storage
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energy does not dominate the demolding work.
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2. Curve fitting results
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Figure S.2 shows the results of fitting the demolding work vs. stamp depth curve with
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with two different functions: one is the linear function and the other is a simple cubic function
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of 𝐿. It should be noted that satisfactory fitting results with the adjusted R2 values higher than
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0.999 were obtained with the linear fit for the bare Si stamp and with a simple cubic fit for the
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F13-TCS coated stamp. For the F3-TCS coated stamp, neither a linear fit nor a simple 3rd order
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polynomial fit with L does not provide satisfactory results both visually and with adjusted R2
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values higher than 0.999, indicating that the contribution of adhesion to the demolding work is
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not negligible.
F13-TCS
W demolding=7.13+7.32L
F3-TCS
2
Adj. R =0.99971
No silane
Linear fit for F13-TCS
Linear fit for F3-TCS
Edemolding=5.75+4.13L
Linear fit for no silane
Demolding work / mJ
40
35
30
2
Adj. R =0.99164
25
20
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Edemolding=4.8+2.83L
10
2
Adj. R =0.98385
45
5
0
19
1
2
3
4
3
W debonding=7.31+0.36222L
F13-TCS
2
F3-TCS
Adj. R =0.97934
No silane
Cubic Fit of F13-TCS
Cubic Fit of F3-TCS
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Cubic Fit of No Silane W debonding=5.75+0.20578L
40
Demolding work / mJ
45
5
35
30
2
Adj. R =0.99584
25
20
15
10
W debonding=4.8+0.14157L
5
2
Adj. R =0.99931
0
1
2
3
4
Stamp depth, L / m
Stamp depth, L / m
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3
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Figure S.2. The fitting results of the demolding work vs. stamp depth curve with two
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different functions: (a) a linear function of 𝑊𝑑𝑒𝑚𝑜𝑙𝑑𝑖𝑛𝑔 = 𝑊𝑑𝑒𝑚𝑜𝑙𝑑𝑖𝑛𝑔,0 + b ∙ 𝐿 and a simple
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cubic function of 𝑊𝑑𝑒𝑚𝑜𝑙𝑑𝑖𝑛𝑔 = 𝑊𝑑𝑒𝑚𝑜𝑙𝑑𝑖𝑛𝑔,0 + b′ ∙ 𝐿3 . In the figure is also included the fitting
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result and adjusted R2 value for each curve fitting.
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References:
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1
M. Sutcu and W. B. Hillig, Acta Metallurgica Et Materialia 38 (12), 2653 (1990).
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