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Cu solder joints
Materials Science & Engineering A 777 (2020) 139080
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Materials Science & Engineering A journal homepage: http://www.elsevier.com/locate/msea
A modified constitutive model of Ag nanoparticle-modified graphene/ Sn–Ag–Cu/Cu solder joints Y.D. Han a, b, Y. Gao a, b, H.Y. Jing a, b, J. Wei c, L. Zhao a, b, L.Y. Xu a, b, * a
School of Materials Science and Engineering, Tianjin University, Tianjin, 300350, PR China Tianjin Key Laboratory of Advanced Joining Technology, Tianjin, 300350, PR China c Singapore Institute of Manufacturing Technology, 71 Nanyang Drive, Singapore b
A R T I C L E I N F O
A B S T R A C T
Keywords: Solder joints Nanoindentation Strengthening stress Ag-graphene nanosheets Constitutive model
In this study, Sn–Ag–Cu solder alloys and Sn–Ag–Cu solder alloys reinforced with 0.1 wt% Ag-graphene nano sheets (Ag-GNSs) by mechanical mixing (H for abbreviation) and ball milling (Q for abbreviation), which were referred as SAC, H/0.1Ag-GNSs, and Q/0.1Ag-GNSs, respectively, were used to form solder joints. The creep behavior of the above solder joints was investigated by conducting nanoindentation tests. A method for calcu lating the strengthening stress generated by the load transfer and orientation of the Ag-graphene nanosheets was proposed. The method considers the geometry and grain data of the Ag-graphene nanosheets, which were ob tained through scanning electron microscopy and electron back scattering diffraction. Considering other strengthening stresses generated by the dislocation strengthening, fine grain strengthening, Orowan strength ening for intermetallic compounds and metal matrix nanocomposites, and strengthening stress generated by Aggraphene nanosheets, a modified constitutive model was proposed to investigate the constitutive behavior for creep performance of solder joints formed by SAC, H/0.1Ag-GNSs and Q/0.1Ag-GNSs. The results show good agreement with the experimental data.
1. Introduction The selection and application of solder alloys in the field of electronic packaging is extremely important [1–3]. Sn–Pb solder alloys were widely used owing to their easy handling, good wettability with copper, and other advantages [4–7]. However, lead can no longer be degraded in the environment and remains toxic once it has been discharged into the environment for a long time, which can cause irreparable harm to human health and the environment [8]. Both Restriction of Hazardous Substances (RoHS) and Waste Electrical and Electronic Equipment (WEEE) legislations have prohibited the use of lead in various industries based on the above-mentioned reasons [9–12]. Since then, lead-free alternative solder alloys have been widely used in academic research and engineering practice owing to their lower environmental impact and superior properties compared to Pb-based solder alloys [13–17]. Various lead-free solder alloys have been devel oped and applied, such as Sn-Ag [18], Sn-Cu [19], Sn-Bi [20], Sn-Zn [21], Sn-Au [22], Sn-Ni [23], Sn-Sb [24], Sn-Sb-Cu [25], Sn-Zn-Bi [26], Sn-Zn-Ag [27], and Sn–Ag–Cu [28]. Among a variety of lead-free solder alloys, Sn–Ag–Cu solder alloys have become the most promising
and widely used lead-free solder alloys owing to their optimized per formance and good reliability [29–31]. However, investigations have found that Sn–Ag–Cu solder alloys also have some disadvantages, such as a high melting temperature, thick intermetallic compounds, and insufficient oxidation resistance characteristics, which can lead to poor reliabilities of solder joints [32–35]. To solve these problems, it is a more effective way to add alloying elements or micro- or nanoparticles to the solder alloys. El-Daly et al. [36] added Zn to Sn–Ag–Cu solder alloys and found that, unlike the ductility of the joints, the mechanical strengths of the solder joints, such as the yield strength and ultimate tensile strength, improved. Yang et al. [37] used Ni-coated carbon nanotubes (Ni-CNTs) to reinforce Sn–Ag–Cu solder alloys and found the ultimate tensile strength of the solder joints formed by Sn–Ag–Cu solder alloys reinforced with 0.05 wt% Ni-CNTs was the highest, but it decreased with increasing Ni-CNTs content. Graphene is a popular choice for strengthening phases owing to its excellent mechanical and thermal properties [38]. However, Sn–Ag–Cu solder alloys reinforced with graphene also have disadvantages such as an inhomogeneous distribution of graphene and poor connection strength to metal substrates [39]. Previous studies have shown that the
* Corresponding author. School of Materials Science and Engineering, Tianjin University, Tianjin, 300350, PR China. E-mail address: [email protected] (L.Y. Xu). https://doi.org/10.1016/j.msea.2020.139080 Received 7 August 2019; Received in revised form 6 January 2020; Accepted 6 February 2020 Available online 11 February 2020 0921-5093/© 2020 Elsevier B.V. All rights reserved.
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Materials Science & Engineering A 777 (2020) 139080
addition of Ag nanoparticles to graphene can enhance the joint strength and inhibit the growth of the IMC layer, thereby improving the perfor mance of the joint [40,41]. However, this improvement is most effective when the Ag-graphene content reaches 0.1 wt% [42]. Mechanical mixing and ball milling were used to add alloying ele ments or micro- or nanoparticles to Sn–Ag–Cu solder alloys. Li et al. [43] reinforced Sn–Ag–Cu solder alloys with CeO2 nanoparticles by me chanical mixing. Chen et al. [44] reinforced Sn–Ag–Cu solder alloys with TiC nano-reinforcement via ball milling. Creep deformation is the dominant mechanism in the deformation of solder joints[45]. The creep properties of solder joints in electronic devices cannot be studied by traditional uniaxial tensile or compression methods owing to the small structure of the joints. Nanoindentation technology can solve this problem, and it has been widely used in the study of the creep behavior of solder joints. Many different creep models were proposed to describe the creep behavior of Sn–Ag–Cu solder joints. Garofalo-Arrhenius hyperbolic sine law [46] and Dorn power law [47] are the most widely used models, but they are all empirical formulas. They cannot explain the relationship between creep behavior and microstructure. Thambi et al. [48] used a modified constitutive model considering Orowan strengthening stress to study the creep behavior of Pb-free solder alloys. Dutta et al. [49] used a microstructure-based constitutive model to determine the creep behavior of SnAg-based sol der. Gong et al. [50] improved the Dorn power law by considering the effect of Ag3Sn IMCs to study the creep behavior of Sn–Ag solder. However, only the effects of partial strengthening mechanisms on creep behavior have been studied in the available literature, and there has been no comprehensive investigation of the effects of various strength ening mechanisms such as fine grain strengthening, dislocation strengthening, and strengthening phase on creep behavior. The creep properties and strengthening mechanisms of solder joints formed by Ag-graphene reinforced Sn–Ag–Cu solder alloys need to be investigated. The new model can better describe the actual stress conditions existing inside the solder joints due to the existence of all strengthening mech anisms and the creep behavior during service. Since the presence of strengthening stress causes the nominal stress to be greater than the actual stress, the new model takes this into account, so that the new model is closer to the actual creep behavior of the solder joints compared with traditional models which means that the new model can provide an effective prediction method to prevent the solder joints from failing during service and minimize the creep failure. Accordingly, in the present study, the creep behavior of solder joints formed by Sn–Ag–Cu solder alloys and Sn–Ag–Cu solder alloys rein forced with 0.1 wt% Ag-graphene nanosheets (Ag-GNSs) by mechanical mixing and ball milling was investigated by nanoindentation. A method for calculating the strengthening stress generated by the orientation and load transfer of Ag-graphene nanosheets based on the shear-lag model and Rosen theory is proposed. The method involves the geometry of Aggraphene nanosheets, which was obtained through scanning electron microscopy (SEM). Considering the strengthening stress generated by Ag-graphene nanosheets and other strengthening stress, a modified constitutive model was proposed. Electron back scattering diffraction (EBSD) was used to obtain the grain size and orientation angle, which was required for the modified constitutive model.
2. Material procedure 2.1. Material processing In this study, the solder matrix containing 96.5Sn-3.0Ag-0.5Cu sol der alloys with a particle size of 25–45 μm (the actual composition of the 96.5Sn-3.0Ag-0.5Cu was listed in Table 1) was supplied by Shenzhen Fitech Co. Ltd (China). Graphene nanosheets with an average diameter and thickness of about 0.5–2 μm and 5–25 nm, respectively, were sup plied by XFNANO Material Tech (China). Graphene was mixed with sodium lauryl sulfate and then ultra-sonicated in dimethylformamide for 2 h. A silver nitrate solution with a concentration of 0.06 mol/mL was poured into the above mixture, which was then ultra-sonicated for 30 min. The mixture was heated at 70 � C for 1 h, filtered, and washed with distilled water and alcohol. Through this process, the Ag-graphene was successfully prepared. Sn–Ag–Cu solder alloys and Sn–Ag–Cu solder alloys reinforced with 0.1 wt% Ag-graphene nanosheets (Ag-GNSs) by mechanical mixing (H for abbreviation) and ball milling (Q for abbreviation) which were referred as SAC, H/0.1Ag-GNSs and Q/0.1Ag-GNSs, respectively, were used to form solder joints. The above solder alloys were placed on a 5 � 10 � 10 mm H62 brass sheet and rosin, then heated on Torrey Pines HP40A-2 heating platform by a fixed heating procedure as shown in Fig. 1 to form solder joints. Then, the solder joints were cut along the across section, and the samples without defects were prepared by ultrasonic cleaning with ethyl alcohol, rough grinding, mechanical polishing, argon ion polishing by GATAN Ilion þ II, and etching by a methanol solution containing 10 vol % hydrochloric acid for 10–12 s. The processed samples were used for further research. 2.2. Microstructural analysis The morphology of Ag-graphene was observed by a ZEISS SUPRA 55 field emission scanning electron microscope equipped with an X-Max20 energy dispersive spectroscopy detector. The grain data, such as grain size and grain misorientation, were obtained by the same SEM device above with a symmetry electron backscattered diffraction detector. 2.3. Nanoindentation test The nanoindentation tests were conducted on a nanoscale micro mechanical system from Keysight Aligent Nano Indenter G200 U8920A equipped with vibration isolation, a laser heater, and a circulating water cooling system. Indentation point selection and indentation data collection in the nanoindentation are all controlled by Nanosuite soft ware. The tests was conducted in load-control model with constant _ strain rate (P=P ¼ 0:05s 1 , in which P_ ¼ dP=dt is the loading rate and P
Table 1 The composition of 96.5Sn-3.0Ag-0.5Cu. Composition
Content/wt. %
Composition
Content/wt. %
Sn Pb Bi Sb Cu Zn Ag
Bal 0.0075 0.0048 0.0108 0.4787 <0.0001 2.9017
Al As Au Cd Fe Ge Ni
<0.0001 <0.0025 <0.0001 <0.0002 0.0005 <0.0010 0.0011
Fig. 1. The heating procedure to form solder joints. 2
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Materials Science & Engineering A 777 (2020) 139080
Fig. 2. Stress condition when the average length of Ag-graphene nanosheets is (a) less than lc , (b) equal to lc , (c) greater than lc .
Fig. 4. The cross-section A in the representative volume element (RVE) taken from composites.
Fig. 5. An equivalent non load-carrying length δ at each end of the Aggraphene nanosheet.
3. Strengthening stress in Ag-nanoparticle modified graphene/ Sn–Ag–Cu solder alloys 3.1. Strengthening mechanisms According to the literature, the good creep resistance of Agnanoparticle-modified graphene/Sn–Ag–Cu solder alloys fabricated via mechanical mixing and ball milling can be attributed to the following five mechanisms: dislocation strengthening, fine grain strengthening, Orowan strengthening for intermetallic compounds (IMCs) and metal matrix nanocomposites (MMNCs), and Ag-graphene strengthening [53–58]. Therefore, the comprehensive stress is determined by the following formula according to the Clyne method [59,60]: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�ffiffiffi 2 2 σ p ¼ ðσ dislocation Þ2 þ σfine grain þðσOrowan IMCs þ σOrowan MMNCs Þ2 þ σgraphene
Fig. 3. A representative volume element (RVE) taken from composites and the definition of azimuth θ.
(1)
is the constant prescribe load)[51]. The indenter used for the nano indentation tests, which were conducted at 22 � C, was the Berkovich indenter. The tests were conducted in the circulating water cooling system, which also ensured that the experimental temperature did not change. The maximum load is determined on the basis that the inden tation can cover all phases in the material under a maximum load, which can ensure that the overall creep properties of the material rather than that of single phase are obtained and that the indentation is deep enough to avoid surface effects. The samples were loaded for 300 s after the load reached the maximum value to achieve full creep and determine the creep properties. Five nanoindentation tests for each experimental condition were conducted to avoid errors caused by environmental factors and minimize instrument errors [52].
The strengthening stress generated by the dislocation strengthening can be calculated by the following formula [61]: pffiffi (2) σ dislocation ¼ αMGb ρ
ρ¼
2θ ub
(3)
Where α is 0.5, Mis the Taylor factor (5 for Sn–Ag–Cu solder joints) [62], G is the shear modulus of Sn–Ag–Cu solder joints, which is 15.3 GPa [63], b is the Burger’s vector (0.317 nm for Sn–Ag–Cu solder joints [63], ρis the dislocation density determined by Eq. (3), θ is the misorientation 3
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Materials Science & Engineering A 777 (2020) 139080
Fig. 6. The typical morphology of Ag-graphene in solder joints formed by (a) H/0.1Ag-GNSs and (b) Q/0.1Ag-GNSs.
Fig. 7. The morphology of the intermetallic compound in solder joints formed by (a) SAC, (b) H/0.1Ag-GNSs and (c) Q/0.1Ag-GNSs.
angle obtained by EBSD data, uis the unit length (100 nm) of the measured point. The strengthening stress generated by the fine grain strengthening can be calculated by the Hall–Petch relationship:
σ fine
grain
1 ¼ K pffiffiffi d
Burger’s vector (0.317 nm for Sn–Ag–Cu solder joints), λ is the inter particle spacing, r is the mean radius of grain, which is calculated from the EBSD data, dIMCs is the average intermetallics diameter calculated from the EBSD data, f is the volume fraction of IMCs, Nv is the number of precipitates per unit volume. The strengthening stress generated by the Orowan strengthening for MMNCs can be calculated by the following formula [65]:
(4)
where K is the Hall–Petch parameter (8.42 for Sn–Ag–Cu solder joints [42]), d is the average grain size of samples which is calculated by EBSD data. The strengthening stress generated by the Orowan strengthening for IMCs can be calculated by the following formula [64]: IMCs
¼
0:84MGb λ 2r
� �12
f ¼π
d3IMCs 6
6f � Nv
¼
0:13Gb dg 3 ln 2 2b 1 �3 � 6 7 dg 4 1=2V 15
(8)
r
pffiffiffiffiffi dg ¼ wl
(5)
(9)
where G is the shear modulus of Sn–Ag–Cu solder joints which is 15.3 GPa, b is the Burger’s vector (0.317 nm for Sn–Ag–Cu solder joints), dg is the equivalent average diameter of Ag-graphene, w and l are the width and length of Ag-graphene nanosheets, respectively, and Vr is the vol ume fraction of Ag-graphene. The strengthening stress generated by the Ag-graphene strength ening would be discussed and proposed in the next section.
=
λ ¼ dIMCs
π
MMNCs
=
σ Orowan
σ Orowan
(6) (7)
where M is the Taylor factor (5 for Sn–Ag–Cu solder joints), G is the shear modulus of Sn–Ag–Cu solder joints, which is 15.3 GPa, b is the 4
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Materials Science & Engineering A 777 (2020) 139080
Fig. 8. The average size of the grains and IMCs and the orientation angle of solder joints formed by (a) (d) (g) SAC, (b) (e) (h) H/0.1Ag-GNSs and (c) (f) (i) Q/0.1AgGNSs, respectively.
3.2. Strengthening stress by Ag-graphene nanosheets (Ag-GNSs)
stressσYS , and the corresponding length is its critical length lc which can calculated as follows:
Based on Shear-lag model and Gao’s analysis [66], the stress con dition of an Ag-graphene nanosheet is related to the length of the Ag-graphene nanosheet. The stress at the midpoint of the Ag-graphene nanosheet increases with increasing length; however, this increase is not infinite. It has been proven in various studies [54–56,58] that the maximum stress at the midpoint of Ag-graphene nanosheets is its yield
lc ¼
tσ YS
τm
(10)
where t is the thickness of Ag-graphene nanosheets. If the average length of the Ag-graphene nanosheets is continuously 5
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Materials Science & Engineering A 777 (2020) 139080
Fig. 8. (continued).
increased over the critical length, the stress at the midpoint of the Aggraphene nanosheets will remain the same as the yield stress. The three stress conditions discussed above are shown in Fig. 2(a-c). If the length of Ag-graphene nanosheet is less than the critical length, the average tensile stress σg in the Ag-graphene nanosheet can be calculated as follows [66]: � � Z 1 l l l l < lc ; σ g ¼ ¼ ϕ τm σ g ðxÞdx ¼ ϕσ g (11) l 0 2 t
where ϕ is a parameter determined by the mean value theorem. If the length of the Ag-graphene nanosheet is greater than the critical length, the average tensile stress σ g in the Ag-graphene nanosheet can be calculated as follows [66]: Z 1 l 1 (12) l � lc ; σg ¼ σ g ðxÞdx ¼ ½ϕσ YS lc þ σYS ðl lc Þ� l 0 l where ϕ is a parameter determined by the mean value theorem. 6
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Materials Science & Engineering A 777 (2020) 139080
However, Ag-graphene nanosheets are randomly distributed in the composite. This means that each Ag-graphene nanosheet makes a certain angle with the load direction. Considering the orientation dis tribution of the Ag-graphene nanosheets, the stress model should be modified according to the fiber-reinforced composites orientation stress model to adapt it to the actual situation. A representative volume element (RVE) is taken from composites as shown in Fig. 3. The definition of the azimuth θ is introduced here for subsequent calculation. The azimuth θ is the angle between the longi tudinal direction of the Ag-graphene nanosheet and the loading direc tion. The azimuth θ varies from 0 to 90� . Assuming that Pg is the sum of the load carried by all Ag-graphene nanosheets taken by cross-section A, as shown in Fig. 4, the following formula can be obtained according to the force balance equation: Z π2 Pg ¼ nc ðθÞσ g wt cos θdθ (13)
Table 2 All parameters used in modified constitutive model. Strengthening Stress
σdislocation
σhall
petch
Parameters
SAC
H/0.1AgGNSs
Q/0.1AgGNSs
Units
α
0.5
non-DIM
M
5
non-DIM
G
15.3
GPa
b
0.317
θ
47.4688
K
8.42
nm 22.2718
50.4089
�
MPa �
pffiffiffiffiffiffi μm
d
2.4043
0.9693
1.0769
μm
σOrowan
IMCs
λ
7.5745
2.5484
2.6248
μm
2r
2.4043
0.9693
1.0769
μm
σorowan
MMNCs
dg
1.5494
1.7423
Vr
0.3%
non-DIM
Eg
1.0
TPa
30
GPa
σgraphene
σYS τm
35.6
l
1.5592
t
0.8
μm
0
wherenc ðθÞ is the azimuth-density distribution of Ag-graphene nano sheets intercepted by the cross-section A, and it can be calculated by Eq. (14): � nc ðθÞ ¼ nv ðθÞ lg ðθÞ L (14)
MPa 1.7985
μm nm
Specific parameters will be explained and calculated below. L is the length of the RVE. nv ðθÞ, which is calculated by Eq. (15), is the azimuth-density distribution of Ag-graphene nanosheets in the RVE: nv ðθÞ ¼ Nf ðθÞ ¼
VRVE Vr f ðθÞ Vg
(15)
where Nis the total number of Ag-graphene nanosheets in the RVE, VRVE is the volume of the RVE, Vg is the volume of an Ag-graphene nanosheet, fðθÞ is the standardized statistical probability function for random dis tribution of Ag-graphene nanosheets azimuth angles. lg ðθÞ is the effective length of the Ag-graphene nanosheet in the load direction and is calculated by the formula below: lg ðθÞ ¼ ðl
The concept of an equivalent non load-carrying length δ at each end of the Ag-graphene nanosheet here is introduced for subsequent calcu lations. As Fig. 5 shows that the stress in this distance range δ is less than the average tensile stressσ g calculated based on Eq. (11) or Eq. (12). According to Rosen [67], for rod-like fiber reinforcements, δ can be calculated by the following formula:
h) curves of three solder joints under
2 df 6 δ¼ 4 1 2
1
0 � α2 B @V r 1 2
312
C Ef 7 1A 5 cosh Gb
=
Fig. 9. The load versus displacement (P 100 mN.
(16)
2δÞcos θ
1
� � 1 þ ð1 φÞ2 2ð1 φÞ
(17)
where Ef is the Young’s modulus of rod-like fiber reinforcement, Gb is the shear modulus of matrix and φ is a constant. Considering the geometric similarities between rod-like fibers and Ag-graphene nanosheets and as the constant α2 ≪1 for Ag-graphene nanosheets, δ can be rewrote as follows 20
1
t 6B δ ¼ 4@V r 1 2 2
312
C Eg 7 1A 5 cosh Gm
1
� � 1 þ ð1 φÞ2 2ð1 φÞ
(18)
=
where Eg is the Young’s modulus of Ag-graphene and Gm is the shear modulus of solder joints. Substituting Eq. (18) into Eq. (16), lg ðθÞ can be obtained as follows: Fig. 10. The displacement versus holding time (h joints under 100 mN.
t) curves of three solder
7
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Materials Science & Engineering A 777 (2020) 139080
Fig. 11. The comparison between the fitted curve and the experimental data of solder joints formed by (a) SAC, (b) H/0.1Ag-GNSs and (c) Q/0.1Ag-GNSs.
Fig. 12. The creep strain rate versus holding time (_εH h t curves shown in Fig. 11(a-c).
Fig. 13. The creep stress versus holding time (σ H t) curves obtained from h t curves shown in Fig. 11 and P h curves shown in Fig. 9.
t) curves obtained from
8
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Materials Science & Engineering A 777 (2020) 139080
Fig. 14. A comparison between the modified constitutive model and experimental data of solder joints formed by SAC, H/0.1Ag-GNSs and Q/0.1Ag-GNSs in steadystate creep under (a) 80 mN, (b) 100 mN and (c) 120 mN, respectively.
σdislocation σfine
grain
σOrowan
IMCs
σOrowan
MMNCs
σgraphene σe
20
H/0.1Ag-GNSs
Q/0.1Ag-GNSs
20.98
14.37
21.62
5.43
8.55
8.11
3.94
12.90
13.16
0
0.71
0.64
0
38.61
42.61
22.02
44.23
50.39
9 > 312 > > > > � � = E 1 þ ð1 φÞ2 C g7 1A 5 cosh 1 cos θ Gm > 2ð1 φÞ > > > > ;
6B t4@V r 1 2
Solder Joints
80 mN β
n
β
n
β
n
SAC
4.1612E19 1.7949E12 2.2516E14
7.2992
4.3949E20 2.2566E16 2.1259E13
7.9554
4.0648E19 5.8838E15 8.5543E16
7.6659
H/0.1AgGNSs Q/0.1AgGNSs
8 > > > > > AVr < nc ðθÞ ¼ 1 wt > > > > > :
1
=
8 > > > > > < lg ðθÞ ¼ l > > > > > :
SAC
Table 4 The parameters required for the modified constitutive model of solder joints formed by SAC, H/0.1Ag-GNSs and Q/0.1Ag-GNSs under different loads.
(19)
Substituting Eq. (15) and Eq. (19) into Eq. (14), nc ðθÞ can be obtained as follows:
100 mN
4.5101 5.3483
120 mN
6.3410 4.7931
5.6316 6.0810
9 > 312 > > > > � � = E 1 þ ð1 φÞ2 C g7 1A 5 cosh 1 f ðθÞcos θ Gm > 2ð1 φÞ > > > > ; 1
20 t 6B 1 2 4@V r l =
Table 3 The strengthening stress generated by the corresponding strengthening mechanism.
(20) According to Blumenthal [68],fðθÞ ¼ sin θ. Substituting Eq. (20) into Eq. (13) can calculate Pg and Ag-graphene-reinforced stressσgraphene consid ering orientation is obtained as follows:
9
Y.D. Han et al.
σgraphene ¼
Materials Science & Engineering A 777 (2020) 139080
Pg A
8 > > > > > σ g Vr < ¼ 1 3 > > > > > :
σe ¼ σ 9 > 1 312 > > > > � � = 1 þ ð1 φÞ2 C Eg 7 1A 5 cosh 1 Gm > 2ð1 φÞ > > > > ;
20 t 6B 1 2 4@V r l
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�ffiffiffi 2 2 ðσ dislocation Þ2 þ σ fine grain þðσ Orowan IMCs þ σ Orowan MMNCs Þ2 þ σ graphene
(21)
(24) By substituting Eq. (24) into Eq. (23), the modified constitutive model at a fixed temperature can be rewritten as the following equation:
=
According to Rosenφ ¼ 0:9 [67], and σ g in Eq. (21) need to be
�
ε_ ¼ β σ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�ffiffiffiffi �n 2 2 ðσ dislocation Þ2 þ σfine grain þ ðσ Orowan IMCs þ σOrowan MMNCs Þ2 þ σgraphene
4.2. Determination of the parameters used in the modified constitutive model
determined. Substituting the calculation of σg according to Gao’s analysis into Eq. (21), the Ag-graphene-reinforced stressσ graphene considering orientation is obtained as follows:
8 > > > > > ϕVr τm > > > > :
¼ � Vr 1 3l
From the above discussion, the parameters used in the modified constitutive model are the length, diameter and thickness of Ag-
Pg A
t l
� V r 1=2
20 6B 1 2 4@V r
� Eg 1 Gm
9 > 312 > > > > � � = E 1 þ ð1 φÞ2 C g7 1A 5 cosh 1 ; l < lc Gm > 2ð1 φÞ > > > > ; 1
=
σgraphene ¼
�1=2 � ½ϕσYS lc þ σYS ðl
lc Þ�cosh
1
(22)
� � 1 þ ð1 φÞ2 ; l � lc 2ð1 φÞ
graphene, the size of the grains and IMCs, and the orientation angle. The Ag-graphene used as a strengthening phase is single-layer Aggraphene; moreover, the thickness of Ag-graphene does not change during the preparation of the samples, so the thickness of Ag-graphene is 0.8 nm according to its specification. As the morphology of Ag-graphene changes during the preparation of samples, the length and diameter of Ag-graphene need to be determined from the SEM image. Fig. 6 shows the typical morphology of Ag-graphene in the solder joints formed by H/ 0.1Ag-GNSs and Q/0.1Ag-GNSs. The lengths of Ag-graphene in the solder joints formed by H/0.1AgGNSs and Q/0.1Ag-GNSs are 1.5592 and 1.7985 μm, respectively. The equivalent diameters of Ag-graphene in the solder joints formed by H/ 0.1Ag-GNSs and Q/0.1Ag-GNSs are 1.5494 and 1.7423 μm, respectively. The morphology of the intermetallic compounds in solder joints is shown in Fig. 7. The average size of the grains and IMCs and the orientation angle is obtained from the EBSD data which is shown in Fig. 8(a-i). According to the EBSD data in Fig. 8(a-c), the average size of the grains in the solder joints formed by SAC, H/0.1Ag-GNSs, and Q/0.1AgGNSs is 2.5199 μm, 1.2338 μm, and 1.3395 μm, respectively. Based on the EBSD data in Fig. 8(d-f), the average size of the IMCs in solder joints formed by SAC, H/0.1Ag-GNSs and Q/0.1Ag-GNSs is 0.8756 μm, 0.7636 μm and 0.8696 μm, respectively. As shown in Fig. 8(g-i), the misorientation angle in the solder joints formed by SAC, H/0.1Ag-GNSs, and Q/0.1Ag-GNSs is 47.4688� , 22.2718� , and 50.4089� , respectively.
Many previous studies have proven that ϕ ¼ Vr [55,56,58,69]. 4. The modified constitutive model for Ag-nanoparticle modified graphene/Sn–Ag–Cu/Cu solder joints 4.1. The effective stress The creep constitutive model at a fixed temperature can generally be simplified to the following equation:
ε_ ¼ βσn
(25)
(23)
where ε_ is the strain rate in the steady-state creep, β is a comprehensive material constant considering many factors such as the activation en ergy, Boltzmann constant, and shear modulus, σ is the stress in the steady-state creep, and n is the stress exponent. However, owing to the existence of various strengthening mecha nisms, σ does not represent the true stress of the material in the steadystate creep. In fact, it is generally larger than the real stress in the ma terial. Effective stress is put forward to reflect the true stress conditions in the material. By substituting Eqs. (2)–(9) and Eq. (22) into Eq. (1), the effective stress σ e is obtained:
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All the parameters used in the modified constitutive model are listed in Table 2.
5.2. Verification of the modified constitutive model The stress in the Ag-graphene nanosheets should be determined before verifying the reliability of the model. According to Section 3.2, the stress in Ag-graphene can be determined by comparing the Aggraphene length and critical length. The critical length of Ag-graphene nanosheets is 674.16 nm based on Eq. (10), which is smaller than the average length of Ag-graphene nanosheets obtained by processing by ball milling and mechanical mixing. Therefore, the stress in the Aggraphene nanosheet gradually increases from the end face and to a stable valueσYS , as shown in Fig. 2(c). According to the above discussion, the modified constitutive model for solder joints formed by SAC is as follows: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi �n � 2 ε_ ¼ β σ ðσ dislocation Þ2 þ σ fine grain þ ðσOrowan IMCs Þ2 (29)
5. Results 5.1. Creep behavior Fig. 9 shows the load versus displacement (P h) curves of three solder joints under 100 mN. The maximum depth of the indentation of Q/0.1Ag-GNSs is 5667.36 nm, which is smaller than that of SAC (5960.89 nm) and H/0.1Ag-GNSs (5808.91 nm). Q/0.1Ag-GNSs’ indentation depth is 95% of SAC’s. As for H/0.1Ag-GNSs’ the value is 97%.These findings also indicate that the surface deformation decreases in the following order: Q/0.1Ag-GNSs solder joints > H/0.1Ag-GNSs solder joints > SAC solder joints. Fig. 10 shows the displacement versus holding time (h t) curves of three solder joints under 100 mN. After a holding time of 300 s under 100 mN, all the samples show obvious creep deformation. Furthermore
�
ε_ ¼ β σ
Furthermore, the modified constitutive model for solder joints formed by H/0.1Ag-GNSs and Q/0.1Ag-GNSs is as follows:
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�ffiffiffiffi �n 2 2 ðσ dislocation Þ2 þ σfine grain þ ðσ Orowan IMCs þ σOrowan MMNCs Þ2 þ σgraphene
the displacement in the holding regime of the three materials reflects their creep resistance. That of solder joints formed by SAC, H/0.1AgGNSs and Q/0.1Ag-GNSs is 1286.39 nm, 1260 nm and 1214.09 nm, respectively. It can be seen that the displacement in the holding regime of the solder joints formed by Q/0.1Ag-GNSs is 94% of that of the solder joints formed by SAC. As for solder joints formed by H/0.1Ag-GNSs, the value is 98%. It can be seen that the surface deformation decreases in the following order: Q/0.1Ag-GNSs solder joints > H/0.1Ag-GNSs solder joints > SAC solder joints. These findings are in line with the above results. In addition, the h t curves can be well fitted by the following empirical equation [70]: h ¼ hi þ aðt
ti Þ1=2 þ bðt
ti Þ1=4 þ cðt
ti Þ1=8
The strengthening stress generated by the corresponding strength ening mechanism can be calculated separately based on the relevant data in Table 1, which is listed in Table 3. Furthermore, the parameters required for the modified constitutive model of solder joints formed by SAC, H/0.1Ag-GNSs, and Q/0.1Ag-GNSs under different loads are listed in Table 4. Fig. 14(a-c) show a comparison of the predictions of the modified constitutive model and experimental data of solder joints formed by SAC, H/0.1Ag-GNSs, and Q/0.1Ag-GNSs at steady-state creep stage under different loads. A conclusion can be made that the modified constitutive model predictions are in good agreement with the experi mental data, reflecting the feasibility of the model in predicting and quantifying the creep behavior of solder joints formed by Ag-graphene nanosheets reinforced Sn–Ag–Cu solder alloys.
(26)
where h is the displacement in the holding regime, and hi ; ti ; a; b and c are fitting parameters. Eq. (26) can perfectly fit all our h t data withR2 > 0:99. Fig. 11(a-c) shows the comparison between the fitted curve and the experimental data. The creep strain rate ε_ H can be calculated by the following equation [71]: h_ h
ε_ H ¼ ¼
1 dh h dt
6. Conclusion In this study, a method for calculating the strengthening stress considering the orientation and load transfer of Ag-graphene nanosheets based on the shear-lag mechanism and Rosen theory is introduced. The effective stress is calculated according to the Clyne method considering the strengthening stress generated by Ag-graphene nanosheets and other strengthening stresses generated by the dislocation strengthening, fine grain strengthening, and Orowan strengthening for IMCs and MMNCs. A modified constitutive model is obtained by substituting the effective stress into the simplified creep constitutive model. The modified constitutive model establishes a connection between creep behavior and microstructure, making the modified constitutive model more realistic. By applying the modified constitutive model to the creep behavior of solder joints formed by SAC, H/0.1Ag-GNSs, and Q/0.1Ag-GNSs, we found that the modified constitutive model suited well agreement with nanoindentation experimental data which proves the feasibility of the modified constitutive model. Therefore, the modified constitutive model is capable of describing the true stress inside the solder joints during the creep process and predicting and quantifying the creep behavior of solder joints formed by Ag-graphene nanosheets reinforced Sn–Ag–Cu solder alloys. The modified constitutive model can better describe and predict the creep behavior of solder joints formed by Ag-graphene nanosheets reinforced Sn–Ag–Cu solder alloys than the traditional model, which also means that the modified constitutive model can
(27)
Fig. 12 shows the creep strain rate versus the holding time (_εH t) curves obtained from the h t curves shown in Fig. 11. It can be seen from Fig. 12 that as the holding time t increases, the creep strain rate ε_ H decreases rapidly then slowly and steadily, which means it enters the steady-state creep. The creep stress σ H for the Berkovich indenter can be calculated by the following equation [72]:
σH ¼
P 24:5h2
(30)
(28)
Fig. 13 shows the creep stress versus holding time (σ H t) curves obtained from the h t curves shown in Fig. 11 and P h curves shown in Fig. 9. It can be seen from Fig. 14 that as holding time t increases, the creep stress σ H decreases rapidly then slowly and steadily.
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minimize the creep failure of solder joints formed by Ag-graphene nanosheets reinforced Sn–Ag–Cu solder alloys during use.
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Author contribution statement Yongdian Han: Methodology, Conceptualization, Writing - review & editing, Yu Gao: Resources, Data curation, Writing - original draft, Hongyang Jing: Validation, Supervision, Wei Jun: Investigation, Formal analysis, Lei Zhao: Supervision, Lianyong Xu: Project administration, Funding acquisition Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgment The authors acknowledge the research funding by National Natural Science Foundation of China (Grant No. 51974198). References [1] A.A. Ibrahiem, A.A. El-Daly, Investigation on multi-temperature short-term stress and creep relaxation of Sn-Ag-Cu-In solder alloys under the effect of rotating magnetic field, Microelectron. Reliab. 98 (2019) 10–18. [2] S. Xu, A.H. Habib, A.D. Pickel, M.E. McHenry, Magnetic nanoparticle-based solder composites for electronic packaging applications, Prog. Mater. Sci. 67 (2015) 95–160. [3] Y. Gao, C. Zou, B. Yang, Q. Zhai, J. Liu, E. Zhuravlev, C. Schick, Nanoparticles of SnAgCu lead-free solder alloy with an equivalent melting temperature of SnPb solder alloy, J. Alloys Compd. 484 (2009) 777–781. [4] K. Suganuma, Advances in lead-free electronics soldering, Curr. Opin. Solid State Mater. Sci. 5 (2001) 64. [5] R.A. Islam, Y.C. Chan, W. Jillek, S. Islam, Comparative study of wetting behavior and mechanical properties (microhardness) of Sn–Zn and Sn–Pb solders, Microelectron. J. 37 (2006) 705–713. [6] M. Wang, J. Wang, H. Feng, W. Ke, Effect of Ag3Sn intermetallic compounds on corrosion of Sn-3.0Ag-0.5Cu solder under high-temperature and high-humidity condition, Corrosion Sci. 63 (2012) 20–28. [7] W.R. Os� orio, L.C. Peixoto, L.R. Garcia, N. Mangelinck-No€ el, A. Garcia, Microstructure and mechanical properties of Sn–Bi, Sn–Ag and Sn–Zn lead-free solder alloys, J. Alloys Compd. 572 (2013) 97–106. [8] A. Wierzbicka-Miernik, J. Guspiel, L. Zabdyr, Corrosion behavior of lead-free SACtype solder alloys in liquid media, Arch. Civ. Mech. Eng. 15 (2015) 206–213. [9] A.M. Erer, S. Oguz, Y. Türen, Influence of bismuth (Bi) addition on wetting characteristics of Sn-3Ag-0.5Cu solder alloy on Cu substrate, Engineering Science and Technology, Int. J. 21 (2018) 1159–1163. [10] L.M. Satizabal, D. Costa, P.B. Moraes, A.D. Bortolozo, W.R. Os� orio, Microstructural array and solute content affecting electrochemical behavior of Sn Ag and Sn Bi alloys compared with a traditional Sn Pb alloy, Mater. Chem. Phys. 223 (2019) 410–425. [11] G. Ren, M.N. Collins, On the mechanism of Sn tunnelling induced intermetallic formation between Sn-8Zn-3Bi solder alloys and Cu substrates, J. Alloys Compd. 791 (2019) 559–566. [12] E. Dalton, G. Ren, J. Punch, M.N. Collins, Accelerated temperature cycling induced strain and failure behaviour for BGA assemblies of third generation high Ag content Pb-free solder alloys, Mater. Des. 154 (2018) 184–191. [13] Q.B. Tao, L. Benabou, L. Vivet, V.N. Le, F.B. Ouezdou, Effect of Ni and Sb additions and testing conditions on the mechanical properties and microstructures of leadfree solder joints, Mater. Sci. Eng., A 669 (2016) 403–416. [14] X. Chen, F. Xue, J. Zhou, Y. Yao, Effect of in on microstructure, thermodynamic characteristic and mechanical properties of Sn–Bi based lead-free solder, J. Alloys Compd. 633 (2015) 377–383. [15] Z. Tan, S. Xie, L. Jiang, J. Xing, Y. Chen, J. Zhu, D. Xiao, Q. Wang, Oxygen octahedron tilting, electrical properties and mechanical behaviors in alkali niobate-based lead-free piezoelectric ceramics, J. Materiom. 5 (2019) 372–384. [16] A.E. Hammad, Enhancing the ductility and mechanical behavior of Sn-1.0Ag-0.5Cu lead-free solder by adding trace amount of elements Ni and Sb, Microelectron. Reliab. 87 (2018) 133–141. [17] L. Yang, Y. Zhang, J. Dai, Y. Jing, J. Ge, N. Zhang, Microstructure, interfacial IMC and mechanical properties of Sn–0.7Cu–xAl (x¼0–0.075) lead-free solder alloy, Mater. Des. 67 (2015) 209–216. [18] H. Kang, M. Lee, D. Sun, S. Pae, J. Park, Formation of octahedral corrosion products in Sn–Ag flip chip solder bump, Scripta Mater. 108 (2015) 126–129. [19] T. Maeshima, H. Ikehata, K. Terui, Y. Sakamoto, Effect of Ni to the Cu substrate on the interfacial reaction with Sn-Cu solder, Mater. Des. 103 (2016) 106–113.
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Contents lists available at ScienceDirect
Materials Science & Engineering A journal homepage: http://www.elsevier.com/locate/msea
A modified constitutive model of Ag nanoparticle-modified graphene/ Sn–Ag–Cu/Cu solder joints Y.D. Han a, b, Y. Gao a, b, H.Y. Jing a, b, J. Wei c, L. Zhao a, b, L.Y. Xu a, b, * a
School of Materials Science and Engineering, Tianjin University, Tianjin, 300350, PR China Tianjin Key Laboratory of Advanced Joining Technology, Tianjin, 300350, PR China c Singapore Institute of Manufacturing Technology, 71 Nanyang Drive, Singapore b
A R T I C L E I N F O
A B S T R A C T
Keywords: Solder joints Nanoindentation Strengthening stress Ag-graphene nanosheets Constitutive model
In this study, Sn–Ag–Cu solder alloys and Sn–Ag–Cu solder alloys reinforced with 0.1 wt% Ag-graphene nano sheets (Ag-GNSs) by mechanical mixing (H for abbreviation) and ball milling (Q for abbreviation), which were referred as SAC, H/0.1Ag-GNSs, and Q/0.1Ag-GNSs, respectively, were used to form solder joints. The creep behavior of the above solder joints was investigated by conducting nanoindentation tests. A method for calcu lating the strengthening stress generated by the load transfer and orientation of the Ag-graphene nanosheets was proposed. The method considers the geometry and grain data of the Ag-graphene nanosheets, which were ob tained through scanning electron microscopy and electron back scattering diffraction. Considering other strengthening stresses generated by the dislocation strengthening, fine grain strengthening, Orowan strength ening for intermetallic compounds and metal matrix nanocomposites, and strengthening stress generated by Aggraphene nanosheets, a modified constitutive model was proposed to investigate the constitutive behavior for creep performance of solder joints formed by SAC, H/0.1Ag-GNSs and Q/0.1Ag-GNSs. The results show good agreement with the experimental data.
1. Introduction The selection and application of solder alloys in the field of electronic packaging is extremely important [1–3]. Sn–Pb solder alloys were widely used owing to their easy handling, good wettability with copper, and other advantages [4–7]. However, lead can no longer be degraded in the environment and remains toxic once it has been discharged into the environment for a long time, which can cause irreparable harm to human health and the environment [8]. Both Restriction of Hazardous Substances (RoHS) and Waste Electrical and Electronic Equipment (WEEE) legislations have prohibited the use of lead in various industries based on the above-mentioned reasons [9–12]. Since then, lead-free alternative solder alloys have been widely used in academic research and engineering practice owing to their lower environmental impact and superior properties compared to Pb-based solder alloys [13–17]. Various lead-free solder alloys have been devel oped and applied, such as Sn-Ag [18], Sn-Cu [19], Sn-Bi [20], Sn-Zn [21], Sn-Au [22], Sn-Ni [23], Sn-Sb [24], Sn-Sb-Cu [25], Sn-Zn-Bi [26], Sn-Zn-Ag [27], and Sn–Ag–Cu [28]. Among a variety of lead-free solder alloys, Sn–Ag–Cu solder alloys have become the most promising
and widely used lead-free solder alloys owing to their optimized per formance and good reliability [29–31]. However, investigations have found that Sn–Ag–Cu solder alloys also have some disadvantages, such as a high melting temperature, thick intermetallic compounds, and insufficient oxidation resistance characteristics, which can lead to poor reliabilities of solder joints [32–35]. To solve these problems, it is a more effective way to add alloying elements or micro- or nanoparticles to the solder alloys. El-Daly et al. [36] added Zn to Sn–Ag–Cu solder alloys and found that, unlike the ductility of the joints, the mechanical strengths of the solder joints, such as the yield strength and ultimate tensile strength, improved. Yang et al. [37] used Ni-coated carbon nanotubes (Ni-CNTs) to reinforce Sn–Ag–Cu solder alloys and found the ultimate tensile strength of the solder joints formed by Sn–Ag–Cu solder alloys reinforced with 0.05 wt% Ni-CNTs was the highest, but it decreased with increasing Ni-CNTs content. Graphene is a popular choice for strengthening phases owing to its excellent mechanical and thermal properties [38]. However, Sn–Ag–Cu solder alloys reinforced with graphene also have disadvantages such as an inhomogeneous distribution of graphene and poor connection strength to metal substrates [39]. Previous studies have shown that the
* Corresponding author. School of Materials Science and Engineering, Tianjin University, Tianjin, 300350, PR China. E-mail address: [email protected] (L.Y. Xu). https://doi.org/10.1016/j.msea.2020.139080 Received 7 August 2019; Received in revised form 6 January 2020; Accepted 6 February 2020 Available online 11 February 2020 0921-5093/© 2020 Elsevier B.V. All rights reserved.
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addition of Ag nanoparticles to graphene can enhance the joint strength and inhibit the growth of the IMC layer, thereby improving the perfor mance of the joint [40,41]. However, this improvement is most effective when the Ag-graphene content reaches 0.1 wt% [42]. Mechanical mixing and ball milling were used to add alloying ele ments or micro- or nanoparticles to Sn–Ag–Cu solder alloys. Li et al. [43] reinforced Sn–Ag–Cu solder alloys with CeO2 nanoparticles by me chanical mixing. Chen et al. [44] reinforced Sn–Ag–Cu solder alloys with TiC nano-reinforcement via ball milling. Creep deformation is the dominant mechanism in the deformation of solder joints[45]. The creep properties of solder joints in electronic devices cannot be studied by traditional uniaxial tensile or compression methods owing to the small structure of the joints. Nanoindentation technology can solve this problem, and it has been widely used in the study of the creep behavior of solder joints. Many different creep models were proposed to describe the creep behavior of Sn–Ag–Cu solder joints. Garofalo-Arrhenius hyperbolic sine law [46] and Dorn power law [47] are the most widely used models, but they are all empirical formulas. They cannot explain the relationship between creep behavior and microstructure. Thambi et al. [48] used a modified constitutive model considering Orowan strengthening stress to study the creep behavior of Pb-free solder alloys. Dutta et al. [49] used a microstructure-based constitutive model to determine the creep behavior of SnAg-based sol der. Gong et al. [50] improved the Dorn power law by considering the effect of Ag3Sn IMCs to study the creep behavior of Sn–Ag solder. However, only the effects of partial strengthening mechanisms on creep behavior have been studied in the available literature, and there has been no comprehensive investigation of the effects of various strength ening mechanisms such as fine grain strengthening, dislocation strengthening, and strengthening phase on creep behavior. The creep properties and strengthening mechanisms of solder joints formed by Ag-graphene reinforced Sn–Ag–Cu solder alloys need to be investigated. The new model can better describe the actual stress conditions existing inside the solder joints due to the existence of all strengthening mech anisms and the creep behavior during service. Since the presence of strengthening stress causes the nominal stress to be greater than the actual stress, the new model takes this into account, so that the new model is closer to the actual creep behavior of the solder joints compared with traditional models which means that the new model can provide an effective prediction method to prevent the solder joints from failing during service and minimize the creep failure. Accordingly, in the present study, the creep behavior of solder joints formed by Sn–Ag–Cu solder alloys and Sn–Ag–Cu solder alloys rein forced with 0.1 wt% Ag-graphene nanosheets (Ag-GNSs) by mechanical mixing and ball milling was investigated by nanoindentation. A method for calculating the strengthening stress generated by the orientation and load transfer of Ag-graphene nanosheets based on the shear-lag model and Rosen theory is proposed. The method involves the geometry of Aggraphene nanosheets, which was obtained through scanning electron microscopy (SEM). Considering the strengthening stress generated by Ag-graphene nanosheets and other strengthening stress, a modified constitutive model was proposed. Electron back scattering diffraction (EBSD) was used to obtain the grain size and orientation angle, which was required for the modified constitutive model.
2. Material procedure 2.1. Material processing In this study, the solder matrix containing 96.5Sn-3.0Ag-0.5Cu sol der alloys with a particle size of 25–45 μm (the actual composition of the 96.5Sn-3.0Ag-0.5Cu was listed in Table 1) was supplied by Shenzhen Fitech Co. Ltd (China). Graphene nanosheets with an average diameter and thickness of about 0.5–2 μm and 5–25 nm, respectively, were sup plied by XFNANO Material Tech (China). Graphene was mixed with sodium lauryl sulfate and then ultra-sonicated in dimethylformamide for 2 h. A silver nitrate solution with a concentration of 0.06 mol/mL was poured into the above mixture, which was then ultra-sonicated for 30 min. The mixture was heated at 70 � C for 1 h, filtered, and washed with distilled water and alcohol. Through this process, the Ag-graphene was successfully prepared. Sn–Ag–Cu solder alloys and Sn–Ag–Cu solder alloys reinforced with 0.1 wt% Ag-graphene nanosheets (Ag-GNSs) by mechanical mixing (H for abbreviation) and ball milling (Q for abbreviation) which were referred as SAC, H/0.1Ag-GNSs and Q/0.1Ag-GNSs, respectively, were used to form solder joints. The above solder alloys were placed on a 5 � 10 � 10 mm H62 brass sheet and rosin, then heated on Torrey Pines HP40A-2 heating platform by a fixed heating procedure as shown in Fig. 1 to form solder joints. Then, the solder joints were cut along the across section, and the samples without defects were prepared by ultrasonic cleaning with ethyl alcohol, rough grinding, mechanical polishing, argon ion polishing by GATAN Ilion þ II, and etching by a methanol solution containing 10 vol % hydrochloric acid for 10–12 s. The processed samples were used for further research. 2.2. Microstructural analysis The morphology of Ag-graphene was observed by a ZEISS SUPRA 55 field emission scanning electron microscope equipped with an X-Max20 energy dispersive spectroscopy detector. The grain data, such as grain size and grain misorientation, were obtained by the same SEM device above with a symmetry electron backscattered diffraction detector. 2.3. Nanoindentation test The nanoindentation tests were conducted on a nanoscale micro mechanical system from Keysight Aligent Nano Indenter G200 U8920A equipped with vibration isolation, a laser heater, and a circulating water cooling system. Indentation point selection and indentation data collection in the nanoindentation are all controlled by Nanosuite soft ware. The tests was conducted in load-control model with constant _ strain rate (P=P ¼ 0:05s 1 , in which P_ ¼ dP=dt is the loading rate and P
Table 1 The composition of 96.5Sn-3.0Ag-0.5Cu. Composition
Content/wt. %
Composition
Content/wt. %
Sn Pb Bi Sb Cu Zn Ag
Bal 0.0075 0.0048 0.0108 0.4787 <0.0001 2.9017
Al As Au Cd Fe Ge Ni
<0.0001 <0.0025 <0.0001 <0.0002 0.0005 <0.0010 0.0011
Fig. 1. The heating procedure to form solder joints. 2
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Fig. 2. Stress condition when the average length of Ag-graphene nanosheets is (a) less than lc , (b) equal to lc , (c) greater than lc .
Fig. 4. The cross-section A in the representative volume element (RVE) taken from composites.
Fig. 5. An equivalent non load-carrying length δ at each end of the Aggraphene nanosheet.
3. Strengthening stress in Ag-nanoparticle modified graphene/ Sn–Ag–Cu solder alloys 3.1. Strengthening mechanisms According to the literature, the good creep resistance of Agnanoparticle-modified graphene/Sn–Ag–Cu solder alloys fabricated via mechanical mixing and ball milling can be attributed to the following five mechanisms: dislocation strengthening, fine grain strengthening, Orowan strengthening for intermetallic compounds (IMCs) and metal matrix nanocomposites (MMNCs), and Ag-graphene strengthening [53–58]. Therefore, the comprehensive stress is determined by the following formula according to the Clyne method [59,60]: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�ffiffiffi 2 2 σ p ¼ ðσ dislocation Þ2 þ σfine grain þðσOrowan IMCs þ σOrowan MMNCs Þ2 þ σgraphene
Fig. 3. A representative volume element (RVE) taken from composites and the definition of azimuth θ.
(1)
is the constant prescribe load)[51]. The indenter used for the nano indentation tests, which were conducted at 22 � C, was the Berkovich indenter. The tests were conducted in the circulating water cooling system, which also ensured that the experimental temperature did not change. The maximum load is determined on the basis that the inden tation can cover all phases in the material under a maximum load, which can ensure that the overall creep properties of the material rather than that of single phase are obtained and that the indentation is deep enough to avoid surface effects. The samples were loaded for 300 s after the load reached the maximum value to achieve full creep and determine the creep properties. Five nanoindentation tests for each experimental condition were conducted to avoid errors caused by environmental factors and minimize instrument errors [52].
The strengthening stress generated by the dislocation strengthening can be calculated by the following formula [61]: pffiffi (2) σ dislocation ¼ αMGb ρ
ρ¼
2θ ub
(3)
Where α is 0.5, Mis the Taylor factor (5 for Sn–Ag–Cu solder joints) [62], G is the shear modulus of Sn–Ag–Cu solder joints, which is 15.3 GPa [63], b is the Burger’s vector (0.317 nm for Sn–Ag–Cu solder joints [63], ρis the dislocation density determined by Eq. (3), θ is the misorientation 3
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Fig. 6. The typical morphology of Ag-graphene in solder joints formed by (a) H/0.1Ag-GNSs and (b) Q/0.1Ag-GNSs.
Fig. 7. The morphology of the intermetallic compound in solder joints formed by (a) SAC, (b) H/0.1Ag-GNSs and (c) Q/0.1Ag-GNSs.
angle obtained by EBSD data, uis the unit length (100 nm) of the measured point. The strengthening stress generated by the fine grain strengthening can be calculated by the Hall–Petch relationship:
σ fine
grain
1 ¼ K pffiffiffi d
Burger’s vector (0.317 nm for Sn–Ag–Cu solder joints), λ is the inter particle spacing, r is the mean radius of grain, which is calculated from the EBSD data, dIMCs is the average intermetallics diameter calculated from the EBSD data, f is the volume fraction of IMCs, Nv is the number of precipitates per unit volume. The strengthening stress generated by the Orowan strengthening for MMNCs can be calculated by the following formula [65]:
(4)
where K is the Hall–Petch parameter (8.42 for Sn–Ag–Cu solder joints [42]), d is the average grain size of samples which is calculated by EBSD data. The strengthening stress generated by the Orowan strengthening for IMCs can be calculated by the following formula [64]: IMCs
¼
0:84MGb λ 2r
� �12
f ¼π
d3IMCs 6
6f � Nv
¼
0:13Gb dg 3 ln 2 2b 1 �3 � 6 7 dg 4 1=2V 15
(8)
r
pffiffiffiffiffi dg ¼ wl
(5)
(9)
where G is the shear modulus of Sn–Ag–Cu solder joints which is 15.3 GPa, b is the Burger’s vector (0.317 nm for Sn–Ag–Cu solder joints), dg is the equivalent average diameter of Ag-graphene, w and l are the width and length of Ag-graphene nanosheets, respectively, and Vr is the vol ume fraction of Ag-graphene. The strengthening stress generated by the Ag-graphene strength ening would be discussed and proposed in the next section.
=
λ ¼ dIMCs
π
MMNCs
=
σ Orowan
σ Orowan
(6) (7)
where M is the Taylor factor (5 for Sn–Ag–Cu solder joints), G is the shear modulus of Sn–Ag–Cu solder joints, which is 15.3 GPa, b is the 4
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Fig. 8. The average size of the grains and IMCs and the orientation angle of solder joints formed by (a) (d) (g) SAC, (b) (e) (h) H/0.1Ag-GNSs and (c) (f) (i) Q/0.1AgGNSs, respectively.
3.2. Strengthening stress by Ag-graphene nanosheets (Ag-GNSs)
stressσYS , and the corresponding length is its critical length lc which can calculated as follows:
Based on Shear-lag model and Gao’s analysis [66], the stress con dition of an Ag-graphene nanosheet is related to the length of the Ag-graphene nanosheet. The stress at the midpoint of the Ag-graphene nanosheet increases with increasing length; however, this increase is not infinite. It has been proven in various studies [54–56,58] that the maximum stress at the midpoint of Ag-graphene nanosheets is its yield
lc ¼
tσ YS
τm
(10)
where t is the thickness of Ag-graphene nanosheets. If the average length of the Ag-graphene nanosheets is continuously 5
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Fig. 8. (continued).
increased over the critical length, the stress at the midpoint of the Aggraphene nanosheets will remain the same as the yield stress. The three stress conditions discussed above are shown in Fig. 2(a-c). If the length of Ag-graphene nanosheet is less than the critical length, the average tensile stress σg in the Ag-graphene nanosheet can be calculated as follows [66]: � � Z 1 l l l l < lc ; σ g ¼ ¼ ϕ τm σ g ðxÞdx ¼ ϕσ g (11) l 0 2 t
where ϕ is a parameter determined by the mean value theorem. If the length of the Ag-graphene nanosheet is greater than the critical length, the average tensile stress σ g in the Ag-graphene nanosheet can be calculated as follows [66]: Z 1 l 1 (12) l � lc ; σg ¼ σ g ðxÞdx ¼ ½ϕσ YS lc þ σYS ðl lc Þ� l 0 l where ϕ is a parameter determined by the mean value theorem. 6
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Materials Science & Engineering A 777 (2020) 139080
However, Ag-graphene nanosheets are randomly distributed in the composite. This means that each Ag-graphene nanosheet makes a certain angle with the load direction. Considering the orientation dis tribution of the Ag-graphene nanosheets, the stress model should be modified according to the fiber-reinforced composites orientation stress model to adapt it to the actual situation. A representative volume element (RVE) is taken from composites as shown in Fig. 3. The definition of the azimuth θ is introduced here for subsequent calculation. The azimuth θ is the angle between the longi tudinal direction of the Ag-graphene nanosheet and the loading direc tion. The azimuth θ varies from 0 to 90� . Assuming that Pg is the sum of the load carried by all Ag-graphene nanosheets taken by cross-section A, as shown in Fig. 4, the following formula can be obtained according to the force balance equation: Z π2 Pg ¼ nc ðθÞσ g wt cos θdθ (13)
Table 2 All parameters used in modified constitutive model. Strengthening Stress
σdislocation
σhall
petch
Parameters
SAC
H/0.1AgGNSs
Q/0.1AgGNSs
Units
α
0.5
non-DIM
M
5
non-DIM
G
15.3
GPa
b
0.317
θ
47.4688
K
8.42
nm 22.2718
50.4089
�
MPa �
pffiffiffiffiffiffi μm
d
2.4043
0.9693
1.0769
μm
σOrowan
IMCs
λ
7.5745
2.5484
2.6248
μm
2r
2.4043
0.9693
1.0769
μm
σorowan
MMNCs
dg
1.5494
1.7423
Vr
0.3%
non-DIM
Eg
1.0
TPa
30
GPa
σgraphene
σYS τm
35.6
l
1.5592
t
0.8
μm
0
wherenc ðθÞ is the azimuth-density distribution of Ag-graphene nano sheets intercepted by the cross-section A, and it can be calculated by Eq. (14): � nc ðθÞ ¼ nv ðθÞ lg ðθÞ L (14)
MPa 1.7985
μm nm
Specific parameters will be explained and calculated below. L is the length of the RVE. nv ðθÞ, which is calculated by Eq. (15), is the azimuth-density distribution of Ag-graphene nanosheets in the RVE: nv ðθÞ ¼ Nf ðθÞ ¼
VRVE Vr f ðθÞ Vg
(15)
where Nis the total number of Ag-graphene nanosheets in the RVE, VRVE is the volume of the RVE, Vg is the volume of an Ag-graphene nanosheet, fðθÞ is the standardized statistical probability function for random dis tribution of Ag-graphene nanosheets azimuth angles. lg ðθÞ is the effective length of the Ag-graphene nanosheet in the load direction and is calculated by the formula below: lg ðθÞ ¼ ðl
The concept of an equivalent non load-carrying length δ at each end of the Ag-graphene nanosheet here is introduced for subsequent calcu lations. As Fig. 5 shows that the stress in this distance range δ is less than the average tensile stressσ g calculated based on Eq. (11) or Eq. (12). According to Rosen [67], for rod-like fiber reinforcements, δ can be calculated by the following formula:
h) curves of three solder joints under
2 df 6 δ¼ 4 1 2
1
0 � α2 B @V r 1 2
312
C Ef 7 1A 5 cosh Gb
=
Fig. 9. The load versus displacement (P 100 mN.
(16)
2δÞcos θ
1
� � 1 þ ð1 φÞ2 2ð1 φÞ
(17)
where Ef is the Young’s modulus of rod-like fiber reinforcement, Gb is the shear modulus of matrix and φ is a constant. Considering the geometric similarities between rod-like fibers and Ag-graphene nanosheets and as the constant α2 ≪1 for Ag-graphene nanosheets, δ can be rewrote as follows 20
1
t 6B δ ¼ 4@V r 1 2 2
312
C Eg 7 1A 5 cosh Gm
1
� � 1 þ ð1 φÞ2 2ð1 φÞ
(18)
=
where Eg is the Young’s modulus of Ag-graphene and Gm is the shear modulus of solder joints. Substituting Eq. (18) into Eq. (16), lg ðθÞ can be obtained as follows: Fig. 10. The displacement versus holding time (h joints under 100 mN.
t) curves of three solder
7
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Materials Science & Engineering A 777 (2020) 139080
Fig. 11. The comparison between the fitted curve and the experimental data of solder joints formed by (a) SAC, (b) H/0.1Ag-GNSs and (c) Q/0.1Ag-GNSs.
Fig. 12. The creep strain rate versus holding time (_εH h t curves shown in Fig. 11(a-c).
Fig. 13. The creep stress versus holding time (σ H t) curves obtained from h t curves shown in Fig. 11 and P h curves shown in Fig. 9.
t) curves obtained from
8
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Materials Science & Engineering A 777 (2020) 139080
Fig. 14. A comparison between the modified constitutive model and experimental data of solder joints formed by SAC, H/0.1Ag-GNSs and Q/0.1Ag-GNSs in steadystate creep under (a) 80 mN, (b) 100 mN and (c) 120 mN, respectively.
σdislocation σfine
grain
σOrowan
IMCs
σOrowan
MMNCs
σgraphene σe
20
H/0.1Ag-GNSs
Q/0.1Ag-GNSs
20.98
14.37
21.62
5.43
8.55
8.11
3.94
12.90
13.16
0
0.71
0.64
0
38.61
42.61
22.02
44.23
50.39
9 > 312 > > > > � � = E 1 þ ð1 φÞ2 C g7 1A 5 cosh 1 cos θ Gm > 2ð1 φÞ > > > > ;
6B t4@V r 1 2
Solder Joints
80 mN β
n
β
n
β
n
SAC
4.1612E19 1.7949E12 2.2516E14
7.2992
4.3949E20 2.2566E16 2.1259E13
7.9554
4.0648E19 5.8838E15 8.5543E16
7.6659
H/0.1AgGNSs Q/0.1AgGNSs
8 > > > > > AVr < nc ðθÞ ¼ 1 wt > > > > > :
1
=
8 > > > > > < lg ðθÞ ¼ l > > > > > :
SAC
Table 4 The parameters required for the modified constitutive model of solder joints formed by SAC, H/0.1Ag-GNSs and Q/0.1Ag-GNSs under different loads.
(19)
Substituting Eq. (15) and Eq. (19) into Eq. (14), nc ðθÞ can be obtained as follows:
100 mN
4.5101 5.3483
120 mN
6.3410 4.7931
5.6316 6.0810
9 > 312 > > > > � � = E 1 þ ð1 φÞ2 C g7 1A 5 cosh 1 f ðθÞcos θ Gm > 2ð1 φÞ > > > > ; 1
20 t 6B 1 2 4@V r l =
Table 3 The strengthening stress generated by the corresponding strengthening mechanism.
(20) According to Blumenthal [68],fðθÞ ¼ sin θ. Substituting Eq. (20) into Eq. (13) can calculate Pg and Ag-graphene-reinforced stressσgraphene consid ering orientation is obtained as follows:
9
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σgraphene ¼
Materials Science & Engineering A 777 (2020) 139080
Pg A
8 > > > > > σ g Vr < ¼ 1 3 > > > > > :
σe ¼ σ 9 > 1 312 > > > > � � = 1 þ ð1 φÞ2 C Eg 7 1A 5 cosh 1 Gm > 2ð1 φÞ > > > > ;
20 t 6B 1 2 4@V r l
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�ffiffiffi 2 2 ðσ dislocation Þ2 þ σ fine grain þðσ Orowan IMCs þ σ Orowan MMNCs Þ2 þ σ graphene
(21)
(24) By substituting Eq. (24) into Eq. (23), the modified constitutive model at a fixed temperature can be rewritten as the following equation:
=
According to Rosenφ ¼ 0:9 [67], and σ g in Eq. (21) need to be
�
ε_ ¼ β σ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�ffiffiffiffi �n 2 2 ðσ dislocation Þ2 þ σfine grain þ ðσ Orowan IMCs þ σOrowan MMNCs Þ2 þ σgraphene
4.2. Determination of the parameters used in the modified constitutive model
determined. Substituting the calculation of σg according to Gao’s analysis into Eq. (21), the Ag-graphene-reinforced stressσ graphene considering orientation is obtained as follows:
8 > > > > > ϕVr τm
¼ � Vr 1 3l
From the above discussion, the parameters used in the modified constitutive model are the length, diameter and thickness of Ag-
Pg A
t l
� V r 1=2
20 6B 1 2 4@V r
� Eg 1 Gm
9 > 312 > > > > � � = E 1 þ ð1 φÞ2 C g7 1A 5 cosh 1 ; l < lc Gm > 2ð1 φÞ > > > > ; 1
=
σgraphene ¼
�1=2 � ½ϕσYS lc þ σYS ðl
lc Þ�cosh
1
(22)
� � 1 þ ð1 φÞ2 ; l � lc 2ð1 φÞ
graphene, the size of the grains and IMCs, and the orientation angle. The Ag-graphene used as a strengthening phase is single-layer Aggraphene; moreover, the thickness of Ag-graphene does not change during the preparation of the samples, so the thickness of Ag-graphene is 0.8 nm according to its specification. As the morphology of Ag-graphene changes during the preparation of samples, the length and diameter of Ag-graphene need to be determined from the SEM image. Fig. 6 shows the typical morphology of Ag-graphene in the solder joints formed by H/ 0.1Ag-GNSs and Q/0.1Ag-GNSs. The lengths of Ag-graphene in the solder joints formed by H/0.1AgGNSs and Q/0.1Ag-GNSs are 1.5592 and 1.7985 μm, respectively. The equivalent diameters of Ag-graphene in the solder joints formed by H/ 0.1Ag-GNSs and Q/0.1Ag-GNSs are 1.5494 and 1.7423 μm, respectively. The morphology of the intermetallic compounds in solder joints is shown in Fig. 7. The average size of the grains and IMCs and the orientation angle is obtained from the EBSD data which is shown in Fig. 8(a-i). According to the EBSD data in Fig. 8(a-c), the average size of the grains in the solder joints formed by SAC, H/0.1Ag-GNSs, and Q/0.1AgGNSs is 2.5199 μm, 1.2338 μm, and 1.3395 μm, respectively. Based on the EBSD data in Fig. 8(d-f), the average size of the IMCs in solder joints formed by SAC, H/0.1Ag-GNSs and Q/0.1Ag-GNSs is 0.8756 μm, 0.7636 μm and 0.8696 μm, respectively. As shown in Fig. 8(g-i), the misorientation angle in the solder joints formed by SAC, H/0.1Ag-GNSs, and Q/0.1Ag-GNSs is 47.4688� , 22.2718� , and 50.4089� , respectively.
Many previous studies have proven that ϕ ¼ Vr [55,56,58,69]. 4. The modified constitutive model for Ag-nanoparticle modified graphene/Sn–Ag–Cu/Cu solder joints 4.1. The effective stress The creep constitutive model at a fixed temperature can generally be simplified to the following equation:
ε_ ¼ βσn
(25)
(23)
where ε_ is the strain rate in the steady-state creep, β is a comprehensive material constant considering many factors such as the activation en ergy, Boltzmann constant, and shear modulus, σ is the stress in the steady-state creep, and n is the stress exponent. However, owing to the existence of various strengthening mecha nisms, σ does not represent the true stress of the material in the steadystate creep. In fact, it is generally larger than the real stress in the ma terial. Effective stress is put forward to reflect the true stress conditions in the material. By substituting Eqs. (2)–(9) and Eq. (22) into Eq. (1), the effective stress σ e is obtained:
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All the parameters used in the modified constitutive model are listed in Table 2.
5.2. Verification of the modified constitutive model The stress in the Ag-graphene nanosheets should be determined before verifying the reliability of the model. According to Section 3.2, the stress in Ag-graphene can be determined by comparing the Aggraphene length and critical length. The critical length of Ag-graphene nanosheets is 674.16 nm based on Eq. (10), which is smaller than the average length of Ag-graphene nanosheets obtained by processing by ball milling and mechanical mixing. Therefore, the stress in the Aggraphene nanosheet gradually increases from the end face and to a stable valueσYS , as shown in Fig. 2(c). According to the above discussion, the modified constitutive model for solder joints formed by SAC is as follows: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi �n � 2 ε_ ¼ β σ ðσ dislocation Þ2 þ σ fine grain þ ðσOrowan IMCs Þ2 (29)
5. Results 5.1. Creep behavior Fig. 9 shows the load versus displacement (P h) curves of three solder joints under 100 mN. The maximum depth of the indentation of Q/0.1Ag-GNSs is 5667.36 nm, which is smaller than that of SAC (5960.89 nm) and H/0.1Ag-GNSs (5808.91 nm). Q/0.1Ag-GNSs’ indentation depth is 95% of SAC’s. As for H/0.1Ag-GNSs’ the value is 97%.These findings also indicate that the surface deformation decreases in the following order: Q/0.1Ag-GNSs solder joints > H/0.1Ag-GNSs solder joints > SAC solder joints. Fig. 10 shows the displacement versus holding time (h t) curves of three solder joints under 100 mN. After a holding time of 300 s under 100 mN, all the samples show obvious creep deformation. Furthermore
�
ε_ ¼ β σ
Furthermore, the modified constitutive model for solder joints formed by H/0.1Ag-GNSs and Q/0.1Ag-GNSs is as follows:
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�ffiffiffiffi �n 2 2 ðσ dislocation Þ2 þ σfine grain þ ðσ Orowan IMCs þ σOrowan MMNCs Þ2 þ σgraphene
the displacement in the holding regime of the three materials reflects their creep resistance. That of solder joints formed by SAC, H/0.1AgGNSs and Q/0.1Ag-GNSs is 1286.39 nm, 1260 nm and 1214.09 nm, respectively. It can be seen that the displacement in the holding regime of the solder joints formed by Q/0.1Ag-GNSs is 94% of that of the solder joints formed by SAC. As for solder joints formed by H/0.1Ag-GNSs, the value is 98%. It can be seen that the surface deformation decreases in the following order: Q/0.1Ag-GNSs solder joints > H/0.1Ag-GNSs solder joints > SAC solder joints. These findings are in line with the above results. In addition, the h t curves can be well fitted by the following empirical equation [70]: h ¼ hi þ aðt
ti Þ1=2 þ bðt
ti Þ1=4 þ cðt
ti Þ1=8
The strengthening stress generated by the corresponding strength ening mechanism can be calculated separately based on the relevant data in Table 1, which is listed in Table 3. Furthermore, the parameters required for the modified constitutive model of solder joints formed by SAC, H/0.1Ag-GNSs, and Q/0.1Ag-GNSs under different loads are listed in Table 4. Fig. 14(a-c) show a comparison of the predictions of the modified constitutive model and experimental data of solder joints formed by SAC, H/0.1Ag-GNSs, and Q/0.1Ag-GNSs at steady-state creep stage under different loads. A conclusion can be made that the modified constitutive model predictions are in good agreement with the experi mental data, reflecting the feasibility of the model in predicting and quantifying the creep behavior of solder joints formed by Ag-graphene nanosheets reinforced Sn–Ag–Cu solder alloys.
(26)
where h is the displacement in the holding regime, and hi ; ti ; a; b and c are fitting parameters. Eq. (26) can perfectly fit all our h t data withR2 > 0:99. Fig. 11(a-c) shows the comparison between the fitted curve and the experimental data. The creep strain rate ε_ H can be calculated by the following equation [71]: h_ h
ε_ H ¼ ¼
1 dh h dt
6. Conclusion In this study, a method for calculating the strengthening stress considering the orientation and load transfer of Ag-graphene nanosheets based on the shear-lag mechanism and Rosen theory is introduced. The effective stress is calculated according to the Clyne method considering the strengthening stress generated by Ag-graphene nanosheets and other strengthening stresses generated by the dislocation strengthening, fine grain strengthening, and Orowan strengthening for IMCs and MMNCs. A modified constitutive model is obtained by substituting the effective stress into the simplified creep constitutive model. The modified constitutive model establishes a connection between creep behavior and microstructure, making the modified constitutive model more realistic. By applying the modified constitutive model to the creep behavior of solder joints formed by SAC, H/0.1Ag-GNSs, and Q/0.1Ag-GNSs, we found that the modified constitutive model suited well agreement with nanoindentation experimental data which proves the feasibility of the modified constitutive model. Therefore, the modified constitutive model is capable of describing the true stress inside the solder joints during the creep process and predicting and quantifying the creep behavior of solder joints formed by Ag-graphene nanosheets reinforced Sn–Ag–Cu solder alloys. The modified constitutive model can better describe and predict the creep behavior of solder joints formed by Ag-graphene nanosheets reinforced Sn–Ag–Cu solder alloys than the traditional model, which also means that the modified constitutive model can
(27)
Fig. 12 shows the creep strain rate versus the holding time (_εH t) curves obtained from the h t curves shown in Fig. 11. It can be seen from Fig. 12 that as the holding time t increases, the creep strain rate ε_ H decreases rapidly then slowly and steadily, which means it enters the steady-state creep. The creep stress σ H for the Berkovich indenter can be calculated by the following equation [72]:
σH ¼
P 24:5h2
(30)
(28)
Fig. 13 shows the creep stress versus holding time (σ H t) curves obtained from the h t curves shown in Fig. 11 and P h curves shown in Fig. 9. It can be seen from Fig. 14 that as holding time t increases, the creep stress σ H decreases rapidly then slowly and steadily.
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minimize the creep failure of solder joints formed by Ag-graphene nanosheets reinforced Sn–Ag–Cu solder alloys during use.
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Author contribution statement Yongdian Han: Methodology, Conceptualization, Writing - review & editing, Yu Gao: Resources, Data curation, Writing - original draft, Hongyang Jing: Validation, Supervision, Wei Jun: Investigation, Formal analysis, Lei Zhao: Supervision, Lianyong Xu: Project administration, Funding acquisition Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgment The authors acknowledge the research funding by National Natural Science Foundation of China (Grant No. 51974198). References [1] A.A. Ibrahiem, A.A. El-Daly, Investigation on multi-temperature short-term stress and creep relaxation of Sn-Ag-Cu-In solder alloys under the effect of rotating magnetic field, Microelectron. Reliab. 98 (2019) 10–18. [2] S. Xu, A.H. Habib, A.D. Pickel, M.E. McHenry, Magnetic nanoparticle-based solder composites for electronic packaging applications, Prog. Mater. Sci. 67 (2015) 95–160. [3] Y. Gao, C. Zou, B. Yang, Q. Zhai, J. Liu, E. Zhuravlev, C. Schick, Nanoparticles of SnAgCu lead-free solder alloy with an equivalent melting temperature of SnPb solder alloy, J. Alloys Compd. 484 (2009) 777–781. [4] K. Suganuma, Advances in lead-free electronics soldering, Curr. Opin. Solid State Mater. Sci. 5 (2001) 64. [5] R.A. Islam, Y.C. Chan, W. Jillek, S. Islam, Comparative study of wetting behavior and mechanical properties (microhardness) of Sn–Zn and Sn–Pb solders, Microelectron. J. 37 (2006) 705–713. [6] M. Wang, J. Wang, H. Feng, W. Ke, Effect of Ag3Sn intermetallic compounds on corrosion of Sn-3.0Ag-0.5Cu solder under high-temperature and high-humidity condition, Corrosion Sci. 63 (2012) 20–28. [7] W.R. Os� orio, L.C. Peixoto, L.R. Garcia, N. Mangelinck-No€ el, A. Garcia, Microstructure and mechanical properties of Sn–Bi, Sn–Ag and Sn–Zn lead-free solder alloys, J. Alloys Compd. 572 (2013) 97–106. [8] A. Wierzbicka-Miernik, J. Guspiel, L. Zabdyr, Corrosion behavior of lead-free SACtype solder alloys in liquid media, Arch. Civ. Mech. Eng. 15 (2015) 206–213. [9] A.M. Erer, S. Oguz, Y. Türen, Influence of bismuth (Bi) addition on wetting characteristics of Sn-3Ag-0.5Cu solder alloy on Cu substrate, Engineering Science and Technology, Int. J. 21 (2018) 1159–1163. [10] L.M. Satizabal, D. Costa, P.B. Moraes, A.D. Bortolozo, W.R. Os� orio, Microstructural array and solute content affecting electrochemical behavior of Sn Ag and Sn Bi alloys compared with a traditional Sn Pb alloy, Mater. Chem. Phys. 223 (2019) 410–425. [11] G. Ren, M.N. Collins, On the mechanism of Sn tunnelling induced intermetallic formation between Sn-8Zn-3Bi solder alloys and Cu substrates, J. Alloys Compd. 791 (2019) 559–566. [12] E. Dalton, G. Ren, J. Punch, M.N. Collins, Accelerated temperature cycling induced strain and failure behaviour for BGA assemblies of third generation high Ag content Pb-free solder alloys, Mater. Des. 154 (2018) 184–191. [13] Q.B. Tao, L. Benabou, L. Vivet, V.N. Le, F.B. Ouezdou, Effect of Ni and Sb additions and testing conditions on the mechanical properties and microstructures of leadfree solder joints, Mater. Sci. Eng., A 669 (2016) 403–416. [14] X. Chen, F. Xue, J. Zhou, Y. Yao, Effect of in on microstructure, thermodynamic characteristic and mechanical properties of Sn–Bi based lead-free solder, J. Alloys Compd. 633 (2015) 377–383. [15] Z. Tan, S. Xie, L. Jiang, J. Xing, Y. Chen, J. Zhu, D. Xiao, Q. Wang, Oxygen octahedron tilting, electrical properties and mechanical behaviors in alkali niobate-based lead-free piezoelectric ceramics, J. Materiom. 5 (2019) 372–384. [16] A.E. Hammad, Enhancing the ductility and mechanical behavior of Sn-1.0Ag-0.5Cu lead-free solder by adding trace amount of elements Ni and Sb, Microelectron. Reliab. 87 (2018) 133–141. [17] L. Yang, Y. Zhang, J. Dai, Y. Jing, J. Ge, N. Zhang, Microstructure, interfacial IMC and mechanical properties of Sn–0.7Cu–xAl (x¼0–0.075) lead-free solder alloy, Mater. Des. 67 (2015) 209–216. [18] H. Kang, M. Lee, D. Sun, S. Pae, J. Park, Formation of octahedral corrosion products in Sn–Ag flip chip solder bump, Scripta Mater. 108 (2015) 126–129. [19] T. Maeshima, H. Ikehata, K. Terui, Y. Sakamoto, Effect of Ni to the Cu substrate on the interfacial reaction with Sn-Cu solder, Mater. Des. 103 (2016) 106–113.
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