Cu solder joints

Journal Pre-proof A modified constitutive model of Ag nanoparticle-modified graphene/Sn–Ag–Cu/Cu solder joints Y.D. Han, Y. Gao, H.Y. Jing, J. Wei, L. Zhao, L.Y. Xu PII:

S0921-5093(20)30168-4

DOI:

https://doi.org/10.1016/j.msea.2020.139080

Reference:

MSA 139080

To appear in:

Materials Science & Engineering A

Received Date: 7 August 2019 Revised Date:

6 January 2020

Accepted Date: 6 February 2020

Please cite this article as: Y.D. Han, Y. Gao, H.Y. Jing, J. Wei, L. Zhao, L.Y. Xu, A modified constitutive model of Ag nanoparticle-modified graphene/Sn–Ag–Cu/Cu solder joints, Materials Science & Engineering A (2020), doi: https://doi.org/10.1016/j.msea.2020.139080. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2020 Published by Elsevier B.V.

1

A modified constitutive model of Ag

2

nanoparticle-modified graphene/Sn-Ag-Cu/Cu solder

3

joints

4

Y.D. Han1,2, Y.Gao1,2, H.Y. Jing1,2, J. Wei3, L. Zhao1,2, L.Y. Xu1,2*

5

1. School of Materials Science and Engineering, Tianjin University, Tianjin 300350, PR China

6

2. Tianjin Key Laboratory of Advanced Joining Technology, Tianjin 300350, China

7

3 Singapore Institute of Manufacturing Technology, 71 Nanyang Drive, Singapore *

8

9

Email: [email protected]

Abstract

10

In this study, Sn-Ag-Cu solder alloys and Sn-Ag-Cu solder alloys reinforced with

11

0.1 wt% Ag-graphene nanosheets (Ag-GNSs) by mechanical mixing (H for

12

abbreviation) and ball milling (Q for abbreviation), which were referred as SAC,

13

H/0.1Ag-GNSs, and Q/0.1Ag-GNSs, respectively, were used to form solder joints.

14

The creep behavior of the above solder joints was investigated by conducting

15

nanoindentation tests. A method for calculating the strengthening stress generated by

16

the load transfer and orientation of the Ag-graphene nanosheets was proposed. The

17

method considers the geometry and grain data of the Ag-graphene nanosheets, which

18

were obtained through scanning electron microscopy and electron back scattering

19

diffraction. Considering other strengthening stresses generated by the dislocation

1

strengthening, fine grain strengthening, Orowan strengthening for intermetallic

2

compounds and metal matrix nanocomposites, and strengthening stress generated by

3

Ag-graphene nanosheets, a modified constitutive model was proposed to investigate

4

the constitutive behavior for creep performance of solder joints formed by SAC,

5

H/0.1Ag-GNSs and Q/0.1Ag-GNSs. The results show good agreement with the

6

experimental data.

7 8 9

Key words: solder joints, nanoindentation, strengthening stress, Ag-graphene nanosheets, constitutive model

1 2

3

1. Introduction

4

The selection and application of solder alloys in the field of electronic

5

packaging is extremely important [1-3]. Sn-Pb solder alloys were widely used owing

6

to their easy handling, good wettability with copper, and other advantages [4-7].

7

However, lead can no longer be degraded in the environment and remains toxic once

8

it has been discharged into the environment for a long time, which can cause

9

irreparable harm to human health and the environment [8]. Both Restriction of

10

Hazardous Substances (RoHS) and Waste Electrical and Electronic Equipment

11

(WEEE) legislations have prohibited the use of lead in various industries based on the

12

above-mentioned reasons [9-12].

13

Since then, lead-free alternative solder alloys have been widely used in

14

academic research and engineering practice owing to their lower environmental

15

impact and superior properties compared to Pb-based solder alloys [13-17]. Various

16

lead-free solder alloys have been developed and applied, such as Sn-Ag[18],

17

Sn-Cu[19], Sn-Bi[20], Sn-Zn[21], Sn-Au[22], Sn-Ni[23], Sn-Sb[24], Sn-Sb-Cu[25],

18

Sn-Zn-Bi[26], Sn-Zn-Ag[27], and Sn-Ag-Cu[28]. Among a variety of lead-free solder

19

alloys, Sn-Ag-Cu solder alloys have become the most promising and widely used

20

lead-free solder alloys owing to their optimized performance and good reliability

1

[29-31]. However, investigations have found that Sn-Ag-Cu solder alloys also have

2

some disadvantages, such as a high melting temperature, thick intermetallic

3

compounds, and insufficient oxidation resistance characteristics, which can lead to

4

poor reliabilities of solder joints [32-35].

5

To solve these problems, it is a more effective way to add alloying elements or

6

micro- or nanoparticles to the solder alloys. El-Daly et al.[36] added Zn to Sn-Ag-Cu

7

solder alloys and found that, unlike the ductility of the joints, the mechanical strengths

8

of the solder joints, such as the yield strength and ultimate tensile strength, improved.

9

Yang et al.[37] used Ni-coated carbon nanotubes (Ni-CNTs) to reinforce Sn-Ag-Cu

10

solder alloys and found the ultimate tensile strength of the solder joints formed by

11

Sn-Ag-Cu solder alloys reinforced with 0.05 wt% Ni-CNTs was the highest, but it

12

decreased with increasing Ni-CNTs content.

13

Graphene is a popular choice for strengthening phases owing to its excellent

14

mechanical and thermal properties [38]. However, Sn-Ag-Cu solder alloys reinforced

15

with graphene also have disadvantages such as an inhomogeneous distribution of

16

graphene and poor connection strength to metal substrates [39]. Previous studies have

17

shown that the addition of Ag nanoparticles to graphene can enhance the joint strength

18

and inhibit the growth of the IMC layer, thereby improving the performance of the

19

joint [40, 41]. However, this improvement is most effective when the Ag-graphene

20

content reaches 0.1 wt% [42].

21

Mechanical mixing and ball milling were used to add alloying elements or

1

micro- or nanoparticles to Sn-Ag-Cu solder alloys. Li et al.[43] reinforced Sn-Ag-Cu

2

solder alloys with CeO2 nanoparticles by mechanical mixing. Chen et al.[44]

3

reinforced Sn-Ag-Cu solder alloys with TiC nano-reinforcement via ball milling.

4

Creep deformation is the dominant mechanism in the deformation of solder

5

joints[45]. The creep properties of solder joints in electronic devices cannot be studied

6

by traditional uniaxial tensile or compression methods owing to the small structure of

7

the joints. Nanoindentation technology can solve this problem, and it has been widely

8

used in the study of the creep behavior of solder joints. Many different creep models

9

were proposed to describe the creep behavior of Sn-Ag-Cu solder joints.

10

Garofalo-Arrhenius hyperbolic sine law [46] and Dorn power law [47] are the most

11

widely used models, but they are all empirical formulas. They cannot explain the

12

relationship between creep behavior and microstructure. Thambi et al. [48] used a

13

modified constitutive model considering Orowan strengthening stress to study the

14

creep behavior of Pb-free solder alloys. Dutta et al. [49] used a microstructure-based

15

constitutive model to determine the creep behavior of SnAg-based solder. Gong et al.

16

[50] improved the Dorn power law by considering the effect of Ag3Sn IMCs to study

17

the creep behavior of Sn-Ag solder. However, only the effects of partial strengthening

18

mechanisms on creep behavior have been studied in the available literature, and there

19

has been no comprehensive investigation of the effects of various strengthening

20

mechanisms such as fine grain strengthening, dislocation strengthening, and

21

strengthening phase on creep behavior. The creep properties and strengthening

1

mechanisms of solder joints formed by Ag-graphene reinforced Sn-Ag-Cu solder

2

alloys need to be investigated. The new model can better describe the actual stress

3

conditions existing inside the solder joints due to the existence of all strengthening

4

mechanisms and the creep behavior during service. Since the presence of

5

strengthening stress causes the nominal stress to be greater than the actual stress, the

6

new model takes this into account, so that the new model is closer to the actual creep

7

behavior of the solder joints compared with traditional models which means that the

8

new model can provide an effective prediction method to prevent the solder joints

9

from failing during service and minimize the creep failure.

10

Accordingly, in the present study, the creep behavior of solder joints formed by

11

Sn-Ag-Cu solder alloys and Sn-Ag-Cu solder alloys reinforced with 0.1wt%

12

Ag-graphene nanosheets (Ag-GNSs) by mechanical mixing and ball milling was

13

investigated by nanoindentation. A method for calculating the strengthening stress

14

generated by the orientation and load transfer of Ag-graphene nanosheets based on the

15

shear-lag model and Rosen theory is proposed. The method involves the geometry of

16

Ag-graphene nanosheets, which was obtained through scanning electron microscopy

17

(SEM). Considering the strengthening stress generated by Ag-graphene nanosheets

18

and other strengthening stress, a modified constitutive model was proposed. Electron

19

back scattering diffraction (EBSD) was used to obtain the grain size and orientation

20

angle, which was required for the modified constitutive model.

1

2. Material procedure

2

2.1 Material processing

3

In this study, the solder matrix containing 96.5Sn-3.0Ag-0.5Cu solder alloys with

4

a particle size of 25–45 µm (the actual composition of the 96.5Sn-3.0Ag-0.5Cu was

5

listed in the Table 1) was supplied by Shenzhen Fitech Co. Ltd (China). Graphene

6

nanosheets with an average diameter and thickness of about 0.5–2 µm and 5–25 nm,

7

respectively, were supplied by XFNANO Material Tech (China). Graphene was mixed

8

with sodium lauryl sulfate and then ultra-sonicated in dimethylformamide for 2 h. A

9

silver nitrate solution with a concentration of 0.06 mol/mL was poured into the above

10

mixture, which was then ultra-sonicated for 30 min. The mixture was heated at 70 °C

11

for 1 h, filtered, and washed with distilled water and alcohol. Through this process,

12

the Ag-graphene was successfully prepared.

13

Table 1 The composition of 96.5Sn-3.0Ag-0.5Cu Composition

Content/wt. %

Composition

Content/wt. %

Sn

Bal

Al

＜0.0001

Pb

0.0075

As

＜0.0025

Bi

0.0048

Au

＜0.0001

Sb

0.0108

Cd

＜0.0002

Cu

0.4787

Fe

0.0005

Zn

＜0.0001

Ge

＜0.0010

Ag

2.9017

Ni

0.0011

1

Sn-Ag-Cu solder alloys and Sn-Ag-Cu solder alloys reinforced with 0.1wt%

2

Ag-graphene nanosheets (Ag-GNSs) by mechanical mixing (H for abbreviation) and

3

ball milling (Q for abbreviation) which were referred as SAC, H/0.1Ag-GNSs and

4

Q/0.1Ag-GNSs, respectively, were used to form solder joints. The above solder alloys

5

were placed on a 5 × 10 × 10 mm H62 brass sheet and rosin, then heated on Torrey

6

Pines HP40A-2 heating platform by a fixed heating procedure as shown in Fig. 1 to

7

form solder joints.

8 9

Fig. 1 The heating procedure to form solder joints

10

Then, the solder joints were cut along the across section, and the samples without

11

defects were prepared by ultrasonic cleaning with ethyl alcohol, rough grinding,

12

mechanical polishing, argon ion polishing by GATAN Ilion+ II, and etching by a

13

methanol solution containing 10 vol.% hydrochloric acid for 10–12 s. The processed

14

samples were used for further research.

1

2.2 Microstructural analysis

2

The morphology of Ag-graphene was observed by a ZEISS SUPRA 55 field

3

emission scanning electron microscope equipped with an X-Max20 energy dispersive

4

spectroscopy detector. The grain data, such as grain size and grain misorientation,

5

were obtained by the same SEM device above with a symmetry electron backscattered

6

diffraction detector.

7

2.3 Nanoindentation test

8

The nanoindentation tests were conducted on a nanoscale micromechanical

9

system from Keysight Aligent Nano Indenter G200 U8920A equipped with vibration

10

isolation, a laser heater, and a circulating water cooling system. Indentation point

11

selection and indentation data collection in the nanoindentation are all controlled by

12

Nanosuite software. The tests was conducted in load-control model with constant

13

& P =0.05s-1 , in which P& = dP dt is the loading rate and P is the strain rate ( P

14

constant prescribe load)[51]. The indenter used for the nanoindentation tests, which

15

were conducted at 22 °C, was the Berkovich indenter. The tests were conducted in the

16

circulating water cooling system, which also ensured that that the experimental

17

temperature did not change. The maximum load is determined on the basis that the

18

indentation can cover all phases in the material under a maximum load, which can

19

ensure that the overall creep properties of the material rather than that of single phase

20

are obtained and that the indentation is deep enough to avoid surface effects. The

21

samples were loaded for 300 s after the load reached the maximum value to achieve

1

full creep and determine the creep properties. Five nanoindentation tests for each

2

experimental condition were conducted to avoid errors caused by environmental

3

factors and minimize instrument errors [52].

4

3. Strengthening stress in Ag-nanoparticle modified

5

graphene/Sn-Ag-Cu solder alloys

6

3.1 Strengthening mechanisms

7

According

to

the

literature,

the

good

creep

resistance

of

8

Ag-nanoparticle-modified graphene/Sn-Ag-Cu solder alloys fabricated via mechanical

9

mixing and ball milling can be attributed to the following five mechanisms:

10

dislocation strengthening, fine grain strengthening, Orowan strengthening for

11

intermetallic compounds (IMCs) and metal matrix nanocomposites (MMNCs), and

12

Ag-graphene strengthening [53-58]. Therefore, the comprehensive stress is

13

determined by the following formula according to the Clyne method [59, 60]:

14

σ p = (σ dislocation ) + (σ fine− grain ) + (σ Orowan− IMCs + σ Orowan−MMNCs ) + (σ graphene )

15

The strengthening stress generated by the dislocation strengthening can be

2

16

2

σ dislocation = α MGb ρ ρ =

18

20

2

(1)

calculated by the following formula [61]:

17

19

2

Where

α

2θ ub

is 0.5, M is the Taylor factor (5 for Sn-Ag-Cu solder joints)[62], G

is the shear modulus of Sn-Ag-Cu solder joints, which is 15.3GPa[63], b is the

(2)

(3)

1

Burger’s vector(0.317 nm for Sn-Ag-Cu solder joints[63], ρ is the dislocation

2

density determined by Eq. (3), θ is the misorientation angle obtained by EBSD data,

3

u is the unit length (100 nm) of the measured point.

4 5 6

The strengthening stress generated by the fine grain strengthening can be calculated by the Hall–Petch relationship:

σ fine − grain = K

1 d

7

where K is the Hall–Petch parameter (8.42 for Sn-Ag-Cu solder joints[42]), d is

8

the average grain size of samples which is calculated by EBSD data.

9 10 11

The strengthening stress generated by the Orowan strengthening for IMCs can be calculated by the following formula [64]:

σ Orowan − IMCs =

0.84 MGb λ − 2r 1

12

 π  λ = d IMCs   6f 

13

d IMCs 3 f =π Nv 6

14

where M is the Taylor factor (5 for Sn-Ag-Cu solder joints), G is the shear

15

modulus of Sn-Ag-Cu solder joints, which is 15.3GPa, b is the Burger’s vector

16

(0.317nm for Sn-Ag-Cu solder joints), λ is the interparticle spacing, r is the mean

17

radius of grain, which is calculated from the EBSD data, dIMCs is the average

18

intermetallics diameter calculated from the EBSD data, f is the volume fraction of

19

IMCs, N v is the number of precipitates per unit volume.

20 21

(4)

2

The strengthening stress generated by the Orowan strengthening for MMNCs can be calculated by the following formula [65]:

(5)

(6)

(7)

σ Orowan− MMNCs =

1

0.13Gb 1   3  1 d g  − 1  2Vr   

ln

dg 2b

2

d g = wl

3

where G is the shear modulus of Sn-Ag-Cu solder joints which is 15.3GPa, b is

4

the Burger’s vector (0.317nm for Sn-Ag-Cu solder joints), d g is the equivalent

5

average diameter of Ag-graphene, w and l are the width and length of

6

Ag-graphene nanosheets, respectively, and

7

Ag-graphene.

8 9

(9)

Vr

is the volume fraction of

The strengthening stress generated by the Ag-graphene strengthening would be discussed and proposed in the next section.

10

3.2 Strengthening stress by Ag-graphene nanosheets

11

(Ag-GNSs)

12

Based on Shear-lag model and Gao’s analysis [66], the stress condition of an

13

Ag-graphene nanosheet is related to the length of the Ag-graphene nanosheet. The

14

stress at the midpoint of the Ag-graphene nanosheet increases with increasing length;

15

however, this increase is not infinite. It has been proven in various studies [54-56, 58]

16

that the maximum stress at the midpoint of Ag-graphene nanosheets is its yield

17

stress σ YS , and the corresponding length is its critical length lc which can calculated

18

as follows:

19

(8)

lc =

tσ YS

τm

(10)

1

where t is the thickness of Ag-graphene nanosheets.

2

If the average length of the Ag-graphene nanosheets is continuously increased

3

over the critical length, the stress at the midpoint of the Ag-graphene nanosheets will

4

remain the same as the yield stress. The three stress conditions discussed above are

5

shown in Figs. 2(a-c).

6

7 8

Fig. 2 Stress condition when the average length of Ag-graphene nanosheets is (a) less than l c , (b) equal to l c , (c) greater than l c .

9 10 11

If the length of Ag-graphene nanosheet is less than the critical length, the average tensile stress σ g in the Ag-graphene nanosheet can be calculated as follows [66]: l < lc , σ g =

12

13

1 l l l σ g ( x ) dx = ϕσ g   = ϕ τ m ∫ 0 l t 2

where ϕ is a parameter determined by the mean value theorem.

14

If the length of the Ag-graphene nanosheet is greater than the critical length, the

15

average tensile stress σ g in the Ag-graphene nanosheet can be calculated as follows

16

[66]:

(11)

1 2

l ≥ lc , σ g =

1 l 1 σ g ( x ) dx = ϕσ YS lc + σ YS ( l − lc )  ∫ 0 l l

where ϕ is a parameter determined by the mean value theorem.

3

However, Ag-graphene nanosheets are randomly distributed in the composite.

4

This means that each Ag-graphene nanosheet makes a certain angle with the load

5

direction. Considering the orientation distribution of the Ag-graphene nanosheets, the

6

stress model should be modified according to the fiber-reinforced composites

7

orientation stress model to adapt it to the actual situation.

8

9 10

Fig. 3 A representative volume element (RVE) taken from composites and the

11

definition of azimuth θ .

12

(12)

1

A representative volume element (RVE) is taken from composites as shown in

2

Fig. 3. The definition of the azimuth θ is introduced here for subsequent calculation.

3

The azimuth θ is the angle between the longitudinal direction of the Ag-graphene

4

nanosheet and the loading direction. The azimuth θ varies from 0 to 90°.

5 6

Fig. 4 The cross-section A in the representative volume element (RVE) taken from

7

composites.

8

Assuming that Pg is the sum of the load carried by all Ag-graphene nanosheets

9

taken by cross-section A, as shown in Fig. 4, the following formula can be obtained

10

according to the force balance equation: π

11

Pg = ∫ 2 nc (θ ) σ g wt cos θ dθ

12

where nc (θ ) is the azimuth-density distribution of Ag-graphene nanosheets

13

intercepted by the cross-section A, and it can be calculated by the Eq. (14):

0

(13)

nc (θ ) = nv (θ )

1

l g (θ ) L

2

Specific parameters will be explained and calculated below.

3

L is the length of the RVE. nv (θ ) , which is calculated by Eq. (15), is the

4

azimuth-density distribution of Ag-graphene nanosheets in the RVE:

nv (θ ) = Nf (θ ) =

5

VRVE Vr f (θ ) Vg

6

where N is the total number of Ag-graphene nanosheets in the RVE, VRVE is the

7

volume of the RVE, Vg is the volume of an Ag-graphene nanosheet, f (θ ) is the

8

standardized statistical probability function for random distribution of Ag-graphene

9

nanosheets azimuth angles.

10 11 12

(14)

(15)

lg (θ ) is the effective length of the Ag-graphene nanosheet in the load direction and is calculated by the formula below:

lg (θ ) = ( l − 2δ ) cos θ

13

The concept of an equivalent non load-carrying length δ at each end of the

14

Ag-graphene nanosheet here is introduced for subsequent calculations. As Fig. 5

15

shows that the stress in this distance range δ is less than the average tensile

16

stress σ g calculated based on Eq. (11) or Eq. (12).

(16)

1 2

Fig. 5 An equivalent non load-carrying length δ at each end of the Ag-graphene

3

nanosheet.

4 5

According to Rosen [67], for rod-like fiber reinforcements, δ can be calculated by the following formula: 1

)

2   df  Ef 2 −1 2 −1 1 + (1 − φ ) 2 δ = (1 − α ) Vr − 1   cosh  2  Gb   2 (1 − φ ) 

(

6 7

where E f is the Young’s modulus of rod-like fiber reinforcement, Gb

8

modulus of matrix and φ is a constant.

(17) is the shear

9

Considering the geometric similarities between rod-like fibers and Ag-graphene

10

nanosheets and as the constant α 2  1 for Ag-graphene nanosheets, δ can be

11

rewrote as follows

)

1

12

2   Eg  2 t  −1 2 −1 1 + (1 − φ ) δ =  Vr − 1   cosh  2 Gm   2 (1 − φ ) 

13

where E g is the Young’s modulus of Ag-graphene and Gm is the shear modulus of

14

solder joints.

15

(

Substituting the Eq. (18) into the Eq. (16), lg (θ ) can be obtained as follows:

(18)

1   2 1 + (1 − φ )2  E 1 −     g 1 − lg (θ ) = l − t  Vr 2 − 1  cos θ   cosh  G 2 1 − φ ( )   m       

follows:

AVr nc (θ ) = wt

4

1   2   Eg  2   t  −12 −1 1 + (1 − φ ) 1 − V − 1 cosh   f (θ ) cos θ   r   Gm    2 (1 − φ )   l  

)

(

5

According to Blumenthal [68], f (θ ) = sin θ . Substituting Eq. (20) into Eq. (13)

6

can calculate Pg and Ag-graphene-reinforced stress σ graphene considering orientation

7

is obtained as follows：

σ graphene = 8 =

A

1   2   Eg  2  t  −1 2 −1 1 + (1 − φ ) 1 − V − 1 cosh     r   Gm   3  l  2 (1 − φ )   

Ag-graphene-reinforced stress σ graphene considering orientation is obtained as follows：

σ graphene =

Pg A 

1  2 2   E   g −1 1 + (1 − φ ) 2 − V − 1 cosh   , l < lc   r   Gm    2 (1 − φ )  t   

ϕVrτ m  l  12

3

= Vr 3l 13 14

)

(

σ gVr 

Substituting the calculation of σ g according to Gao’s analysis into Eq. (21), the

10

(20)

Pg

According to Rosen φ = 0.9 [67], and σ g in Eq. (21) need to be determined.

9

11

(19)

Substituting Eq. (15) and Eq. (19) into Eq. (14), nc (θ ) can be obtained as

2 3

)

(

1

(

−1

)

12 2   Eg    t  −1 2 −1 1 + (1 − φ ) l + l − l cosh , l ≥ lc ϕσ σ     ( ) 1 − (Vr − 1)   YS c YS c  Gm    2 1 − φ ( )    l    

(22) Many previous studies have proven that ϕ = Vr [55, 56, 58, 69].

(21)

1

4. The modified constitutive model for

2

Ag-nanoparticle modified graphene/Sn-Ag-Cu/Cu

3

solder joints

4

4.1 The effective stress

5 6

The creep constitutive model at a fixed temperature can generally be simplified to the following equation：

7

ε& = βσ n

8

where ε& is the strain rate in the steady-state creep, β is a comprehensive material

9

constant considering many factors such as the activation energy, Boltzmann constant,

10

and shear modulus, σ is the stress in the steady-state creep, and n is the stress

11

exponent.

(23)

12

However, owing to the existence of various strengthening mechanisms, σ does

13

not represent the true stress of the material in the steady-state creep. In fact, it is

14

generally larger than the real stress in the material. Effective stress is put forward to

15

reflect the true stress conditions in the material.

16

By substituting Eqs. (2-9) and Eq. (22) into Eq. (1), the effective stress σ e is

17

obtained:

18

σe = σ −

19 20

(σ dislocation ) + (σ fine− grain ) + (σ Orowan − IMCs + σ Orowan− MMNCs ) + (σ graphene ) 2

2

2

2

(24)

By substituting Eq. (24) into Eq. (23), the modified constitutive model at a fixed temperature can be rewritten as the following equation:

 ε& = β  σ − 

2

4.2 Determination of the parameters used in the modified

3

constitutive model

(σ dislocation ) + (σ fine− grain ) + (σ Orowan− IMCs + σ Orowan− MMNCs ) + (σ graphene ) 2

2

2

2

  

n

1

4

From the above discussion, the parameters used in the modified constitutive

5

model are the length, diameter and thickness of Ag-graphene, the size of the grains

6

and IMCs, and the orientation angle.

7

The Ag-graphene used as a strengthening phase is single-layer Ag-graphene;

8

moreover, the thickness of Ag-graphene does not change during the preparation of the

9

samples, so the thickness of Ag-graphene is 0.8 nm according to its specification. As

10

the morphology of Ag-graphene changes during the preparation of samples, the length

11

and diameter of Ag-graphene need to be determined from the SEM image. Fig. 6

12

shows the typical morphology of Ag-graphene in the solder joints formed by

13

H/0.1Ag-GNSs and Q/0.1Ag-GNSs.

14 15

Fig. 6 The typical morphology of Ag-graphene in solder joints formed by (a)

16

H/0.1Ag-GNSs and (b) Q/0.1Ag-GNSs.

17

The lengths of Ag-graphene in the solder joints formed by H/0.1Ag-GNSs and

(25)

1

Q/0.1Ag-GNSs are 1.5592 and 1.7985 µm, respectively. The equivalent diameters of

2

Ag-graphene in the solder joints formed by H/0.1Ag-GNSs and Q/0.1Ag-GNSs are

3

1.5494 and 1.7423 µm, respectively.

4 5

The morphology of the intermetallic compounds in solder joints is shown in the Fig. 7.

6

7 8

Fig. 7 The morphology of the intermetallic compound in solder joints formed by

9

(a)SAC, (b)H/0.1Ag-GNSs and (c)Q/0.1Ag-GNSs.

10

The average size of the grains and IMCs and the orientation angle is obtained

11

from the EBSD data which is shown in Figs. 8(a-i).

1

2

3

1

2

3 4

Fig. 8 The average size of the grains and IMCs and the orientation angle of solder

5

joints formed by (a)(d)(g)SAC, (b)(e)(h)H/0.1Ag-GNSs and (c)(f)(i)Q/0.1Ag-GNSs,

6

respectively.

7

According to the EBSD data in Figs. 8(a-c), t the average size of the grains in the

8

solder joints formed by SAC, H/0.1Ag-GNSs, and Q/0.1Ag-GNSs is 2.5199 µm,

1

1.2338 µm, and 1.3395 µm, respectively. Based on the EBSD data in Figs. 8(d-f), the

2

average size of the IMCs in solder joints formed by SAC, H/0.1Ag-GNSs and

3

Q/0.1Ag-GNSs is 0.8756 µm, 0.7636 µm and 0.8696 µm, respectively. As shown in

4

Figs. 8(g-i), the misorientation angle in the solder joints formed by SAC,

5

H/0.1Ag-GNSs, and Q/0.1Ag-GNSs is 47.4688°, 22.2718°, and 50.4089°,

6

respectively. All the parameters used in the modified constitutive model are listed in

7

Table 2.

8

Table 2 All parameters used in modified constitutive model. Strengthening Stress

σ dislocation

Parameters

SAC

H/0.1Ag-G NSs

Q/0.1Ag-G NSs

Units

α

0.5

non-DIM

M

5

non-DIM

G

15.3

GPa

b

0.317

nm

θ

47.4688

22.2718

50.4089

°

MPa • µ m

σ hall − petch

σ Orowan − IMCs

σ orowan − MMNCs

8.42

K

d

2.4043

0.9693

1.0769

µm

λ

7.5745

2.5484

2.6248

µm

2r

2.4043

0.9693

1.0769

µm

1.5494

1.7423

µm

dg

Vr

0.3%

non-DIM

σ graphene

Eg

1.0

TPa

σ YS

30

GPa

τm

35.6

MPa

l t

1

5. Results

2

5.1 Creep behavior

1.5592

1.7985 0.8

µm nm

3

Fig. 9 shows the load versus displacement ( P − h ) curves of three solder joints

4

under 100 mN. The maximum depth of the indentation of Q/0.1Ag-GNSs is 5667.36

5

nm, which is smaller than that of SAC (5960.89 nm) and H/0.1Ag-GNSs (5808.91

6

nm). Q/0.1Ag-GNSs’ indentation depth is 95% of SAC’s. As for H/0.1Ag-GNSs’ the

7

value is 97%.These findings also indicate that the surface deformation decreases in

8

the following order: Q/0.1Ag-GNSs solder joints > H/0.1Ag-GNSs solder joints >

9

SAC solder joints.

1 2

Fig. 9 The load versus displacement ( P − h ) curves of three solder joints under

3

100mN.

4

Fig. 10 shows the displacement versus holding time ( h − t ) curves of three solder

5

joints under 100 mN. After a holding time of 300 s under 100 mN, all the samples

6

show obvious creep deformation. Furthermore the displacement in the holding regime

7

of the three materials reflects their creep resistance. That of solder joints formed by

8

SAC, H/0.1Ag-GNSs and Q/0.1Ag-GNSs is 1286.39 nm, 1260 nm and 1214.09 nm,

9

respectively. It can be seen that the displacement in the holding regime of the solder

10

joints formed by Q/0.1Ag-GNSs is 94% of that of the solder joints formed by SAC.

11

As for solder joints formed by H/0.1Ag-GNSs, the value is 98%. It can be seen that

12

the surface deformation decreases in the following order: Q/0.1Ag-GNSs solder

13

joints > H/0.1Ag-GNSs solder joints > SAC solder joints. These findings are in line

14

with the above results.

1 2

Fig. 10 The displacement versus holding time ( h − t ) curves of three solder joints

3

under 100mN.

4 5

6

In addition, the h − t curves can be well fitted by the following empirical equation [70]:

h =hi + a ( t − ti ) + b ( t − ti ) + c ( t − ti ) 12

14

18

7

where h is the displacement in the holding regime, and hi , ti , a, b and c are fitting

8

parameters. Eq. (26) can perfectly fit all our h − t data with R 2 > 0.99 . Figs. 11(a-c)

9

shows the comparison between the fitted curve and the experimental

10

data.

(26)

1 2

Fig. 11 The comparison between the fitted curve and the experimental data of solder

3

joints formed by (a)SAC, (b)H/0.1Ag-GNSs and (c)Q/0.1Ag-GNSs.

4

The creep strain rate ε&H can be calculated by the following equation [71]:

5

ε&H =

h& 1 dh = h h dt

6

Fig. 12 shows the creep strain rate versus the holding time ( ε&H − t ) curves

7

obtained from the h − t curves shown in Fig. 11. It can be seen from Fig. 12 that as

8

the holding time t increases, the creep strain rate ε&H decreases rapidly then slowly

9

and steadily, which means it enters the steady-state creep.

10 11

Fig. 12 The creep strain rate versus holding time ( ε&H − t ) curves obtained from h − t

12

curves shown in Fig. 11(a-c).

(27)

1 2

The creep stress σ H for the Berkovich indenter can be calculated by the following equation [72]:

σH =

3

P 24.5h 2

4

Fig. 13 shows the creep stress versus holding time ( σ H − t ) curves obtained from

5

the h − t curves shown in Fig. 11 and P − h curves shown in Fig. 9. It can be seen

6

from Fig. 14 that as holding time t increases, the creep stress σ H decreases rapidly

7

then slowly and steadily.

8 9

Fig. 13 The creep stress versus holding time ( σ H − t ) curves obtained from h − t

10

curves shown in Fig. 11 and P − h curves shown in Fig. 9.

11

5.2 Verification of the modified constitutive model

12

The stress in the Ag-graphene nanosheets should be determined before verifying

13

the reliability of the model. According to Section 3.2, the stress in Ag-graphene can be

14

determined by comparing the Ag-graphene length and critical length. The critical

15

length of Ag-graphene nanosheets is 674.16 nm based on Eq. (10), which is smaller

16

than the average length of Ag-graphene nanosheets obtained by processing by ball

(28)

1

milling and mechanical mixing. Therefore, the stress in the Ag-graphene nanosheet

2

gradually increases from the end face and to a stable value σ YS , as shown in Fig. 2(c).

3 4

According to the above discussion, the modified constitutive model for solder joints formed by SAC is as follows:

 

ε& = β  σ −

5 6

(σ dislocation ) + (σ fine− grain ) + (σ Orowan− IMCs ) 2

2

2

  

n

(29)

Furthermore, the modified constitutive model for solder joints formed by

7

H/0.1Ag-GNSs and Q/0.1Ag-GNSs is as follows:

8

 ε& = β  σ − 

(σ dislocation ) + (σ fine− grain ) + (σ Orowan− IMCs + σ Orowan−MMNCs ) + (σ graphene ) 2

2

2

2

n

 (30)  

9

The strengthening stress generated by the corresponding strengthening

10

mechanism can be calculated separately based on the relevant data in Table 1, which

11

is listed in Table 3. Furthermore, the parameters required for the modified constitutive

12

model of solder joints formed by SAC, H/0.1Ag-GNSs, and Q/0.1Ag-GNSs under

13

different loads are listed in Table 4.

14

Table 3 The strengthening stress generated by the corresponding strengthening

15

mechanism. SAC

H/0.1Ag-GNSs

Q/0.1Ag-GNSs

σ dislocation

20.98

14.37

21.62

σ fine − grain

5.43

8.55

8.11

σ Orowan − IMCs

3.94

12.90

13.16

σ Orowan −MMNCs

0

0.71

0.64

σ graphene

0

38.61

42.61

σe

22.02

44.23

50.39

1 2

Table 4 The parameters required for the modified constitutive model of solder joints

3

formed by SAC, H/0.1Ag-GNSs and Q/0.1Ag-GNSs under different loads. Solder Joints SAC

80mN

β

100mN n

β

120mN n

β

n

4.1612E-19 7.2992 4.3949E-20 7.9554 4.0648E-19 7.6659

H/0.1Ag-GNSs 1.7949E-12 4.5101 2.2566E-16 6.3410 5.8838E-15 5.6316 Q/0.1Ag-GNSs 2.2516E-14 5.3483 2.1259E-13 4.7931 8.5543E-16 6.0810 4

Figs. 14(a-c) show a comparison of the predictions of the modified constitutive

5

model and experimental data of solder joints formed by SAC, H/0.1Ag-GNSs, and

6

Q/0.1Ag-GNSs at steady-state creep stage under different loads. A conclusion can be

7

made that the modified constitutive model predictions are in good agreement with the

8

experimental data, reflecting the feasibility of the model in predicting and quantifying

9

the creep behavior of solder joints formed by Ag-graphene nanosheets reinforced

10

Sn-Ag-Cu solder alloys.

1

2 3

Fig. 14 A comparison between the modified constitutive model and experimental data

4

of solder joints formed by SAC, H/0.1Ag-GNSs and Q/0.1Ag-GNSs in steady-state

5

creep under (a)80mN, (b)100mN and (c)120mN, respectively.

6

6. Conclusion

7

In this study, a method for calculating the strengthening stress considering the

8

orientation and load transfer of Ag-graphene nanosheets based on the shear-lag

9

mechanism and Rosen theory is introduced. The effective stress is calculated

10

according to the Clyne method considering the strengthening stress generated by

11

Ag-graphene nanosheets and other strengthening stresses generated by the dislocation

1

strengthening, fine grain strengthening, and Orowan strengthening for IMCs and

2

MMNCs. A modified constitutive model is obtained by substituting the effective

3

stress into the simplified creep constitutive model. The modified constitutive model

4

establishes a connection between creep behavior and microstructure, making the

5

modified constitutive model more realistic.

6

By applying the modified constitutive model to the creep behavior of solder

7

joints formed by SAC, H/0.1Ag-GNSs, and Q/0.1Ag-GNSs, we found that the

8

modified constitutive model suited well agreement with nanoindentation experimental

9

data which proves the feasibility of the modified constitutive model. Therefore, the

10

modified constitutive model is capable of describing the true stress inside the solder

11

joints during the creep process and predicting and quantifying the creep behavior of

12

solder joints formed by Ag-graphene nanosheets reinforced Sn-Ag-Cu solder alloys.

13

The modified constitutive model can better describe and predict the creep behavior of

14

solder joints formed by Ag-graphene nanosheets reinforced Sn-Ag-Cu solder alloys

15

than the traditional model, which also means that the modified constitutive model can

16

minimize the creep failure of solder joints formed by Ag-graphene nanosheets

17

reinforced Sn-Ag-Cu solder alloys during use.

18

ACKNOWLEDGMENT

19

The authors acknowledge the research funding by National Natural Science

20

Foundation of China (Grant No. 51974198).

1 2 3 4

References:

5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39

[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. Alloy. Compd. 484(2009) 777-781. [4] K. Suganuma, Advances in lead-free electronics soldering, Curr. Opin. Solid St. M. 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, Corros. Sci. 63(2012) 20-28. [7] W.R. Osório, L.C. Peixoto, L.R. Garcia, N. Mangelinck-Noël, A. Garcia, Microstructure and mechanical properties of Sn–Bi, Sn–Ag and Sn–Zn lead-free solder alloys, J. Alloy. Compd. 572(2013) 97-106. [8] A. Wierzbicka-Miernik, J. Guspiel, L. Zabdyr, Corrosion behavior of lead-free SAC-type 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, an International Journal 21(2018) 1159-1163. [10] L.M. Satizabal, D. Costa, P.B. Moraes, A.D. Bortolozo, W.R. Osório, 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. Alloy. 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, Materials & Design 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 lead-free solder joints, Materials Science and Engineering: A 669(2016) 403-416.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42

[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. Alloy. 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, Journal of Materiomics (2019). [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 (x=0–0.075) lead-free solder alloy, Materials & Design 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, Materials & Design 103(2016) 106-113. [20] B.L. Silva, G. Reinhart, H. Nguyen-Thi, N. Mangelinck-Noël, A. Garcia, J.E. Spinelli, Microstructural development and mechanical properties of a near-eutectic directionally solidified Sn–Bi solder alloy, Mater. Charact. 107(2015) 43-53. [21] M.F.M. Nazeri, A.B. Ismail, A.A. Mohamad, Effect of polarizations on Sn–Zn solders alloys in alkaline electrolyte, J. Alloy. Compd. 606(2014) 278-287. [22] M.L. Huang, Y.C. Yang, Y. Chen, C. Dong, Microstructure and mechanical properties of Sn-rich Au-Sn solders designed using cluster-plus-glue-atom model, Materials Science and Engineering: A 664(2016) 221-226. [23] Y. Xiao, Y. Zhang, K. Zhao, S. Li, L. Wang, J. Xiao, L. Liu, Ultrasound-assisted soldering of alumina using Ni-foam reinforced Sn-based composite solders, Ceram. Int. 43(2017) 14314-14320. [24] M. Dias, T.A. Costa, B.L. Silva, J.E. Spinelli, N. Cheung, A. Garcia, A comparative analysis of microstructural features, tensile properties and wettability of hypoperitectic and peritectic Sn-Sb solder alloys, Microelectron. Reliab. 81(2018) 150-158. [25] P. Šebo, P. Švec, D. Janičkovič, E. Illeková, Y. Plevachuk, Interface between Sn–Sb–Cu solder and copper substrate, Materials Science and Engineering: A 528(2011) 5955-5960. [26] A.A. El-Daly, Y. Swilem, M.H. Makled, M.G. El-Shaarawy, A.M. Abdraboh, Thermal and mechanical properties of Sn–Zn–Bi lead-free solder alloys, J. Alloy. Compd. 484(2009) 134-142. [27] T. Luo, Z. Chen, A. Hu, M. Li, Study on melt properties, microstructure, tensile properties of low Ag content Sn–Ag–Zn Lead-free solders, Materials Science and Engineering: A 556(2012) 885-890. [28] J. Hah, Y. Kim, P. Fernandez-Zelaia, S. Hwang, S. Lee, L. Christie, P. Houston, S. Melkote, K. Moon, C. Wong, Comprehensive comparative analysis of microstructure of Sn–Ag–Cu (SAC) solder joints by traditional reflow and thermo-compression bonding (TCB) processes, Materialia 6(2019) 100327. [29] L.Y. Xu, S.T. Zhang, H.Y. Jing, L.X. Wang, J. Wei, X.C. Kong, Y.D. Han, Indentation Size

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42

Effect on Ag Nanoparticle-Modified Graphene/Sn-Ag-Cu Solders, J. Electron. Mater. 47(2018) 612-619. [30] M. Yang, H. Ji, S. Wang, Y. Ko, C. Lee, J. Wu, M. Li, Effects of Ag content on the interfacial reactions between liquid Sn–Ag–Cu solders and Cu substrates during soldering, J. Alloy. Compd. 679(2016) 18-25. [31] J. Yoon, B. Noh, B. Kim, C. Shur, S. Jung, Wettability and interfacial reactions of Sn–Ag– Cu/Cu and Sn–Ag–Ni/Cu solder joints, J. Alloy. Compd. 486(2009) 142-147. [32] Z.L. Ma, H. Shang, A.A. Daszki, S.A. Belyakov, C.M. Gourlay, Mechanisms of beta-Sn nucleation and microstructure evolution in Sn-Ag-Cu solders containing titanium, J. Alloy. Compd. 777(2019) 1357-1366. [33] H.Y. Jing, H.J. Guo, L.X. Wang, J. Wei, L.Y. Xu, Y.D. Han, Influence of Ag-modified graphene nanosheets addition into Sn–Ag–Cu solders on the formation and growth of intermetallic compound layers, J. Alloy. Compd. 702(2017) 669-678. [34] R. Sayyadi, H. Naffakh-Moosavy, Physical and mechanical properties of synthesized low Ag/lead-free Sn-Ag-Cu-xBi (x = 0, 1, 2.5, 5 wt%) solders, Materials Science and Engineering: A 735(2018) 367-377. [35] A.A. El-Daly, A.E. Hammad, Enhancement of creep resistance and thermal behavior of eutectic Sn–Cu lead-free solder alloy by Ag and In-additions, Materials & Design 40(2012) 292-298. [36] A.A. El-Daly, H. El-Hosainy, T.A. Elmosalami, W.M. Desoky, Microstructural modifications and properties of low-Ag-content Sn–Ag–Cu solder joints induced by Zn alloying, J. Alloy. Compd. 653(2015) 402-410. [37] Z. Yang, W. Zhou, P. Wu, Effects of Ni-coated carbon nanotubes addition on the microstructure and mechanical properties of Sn–Ag–Cu solder alloys, Materials Science and Engineering: A 590(2014) 295-300. [38] C. Tu, L. Hong, T. Song, X. Li, Q. Dou, Y. Ding, T. Liao, S. Zhang, G. Gao, Z. Wang, Y. Jiang, Superior mechanical properties of sulfonated graphene reinforced carbon-graphite composites, Carbon 148(2019) 378-386. [39] L. Xu, X. Chen, H. Jing, L. Wang, J. Wei, Y. Han, Design and performance of Ag nanoparticle-modified graphene/SnAgCu lead-free solders, Materials Science and Engineering: A 667(2016) 87-96. [40] L. Xu, L. Wang, H. Jing, X. Liu, J. Wei, Y. Han, Effects of graphene nanosheets on interfacial reaction of Sn–Ag–Cu solder joints, J. Alloy. Compd. 650(2015) 475-481. [41] L.Y. Xu, S.T. Zhang, H.Y. Jing, L.X. Wang, J. Wei, X.C. Kong, Y.D. Han, Indentation Size Effect on Ag Nanoparticle-Modified Graphene/Sn-Ag-Cu Solders, J. Electron. Mater. 47(2018) 612-619. [42] Y.D. Han, Y. Gao, S.T. Zhang, H.Y. Jing, J. Wei, L. Zhao, L.Y. Xu, Study of mechanical properties of Ag nanoparticle-modified graphene/Sn-Ag-Cu solders by nanoindentation, Materials Science and Engineering: A 761(2019) 138051. [43] Z.H. Li, Y. Tang, Q.W. Guo, G.Y. Li, Effects of CeO2 nanoparticles addition on shear properties of low-silver Sn–0.3Ag–0.7Cu-xCeO2 solder alloys, J. Alloy. Compd. 789(2019) 150-162. [44] G. Chen, H. Peng, V.V. Silberschmidt, Y.C. Chan, C. Liu, F. Wu, Performance of Sn–3.0Ag– 0.5Cu composite solder with TiC reinforcement: Physical properties, solderability and

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42

microstructural evolution under isothermal ageing, J. Alloy. Compd. 685(2016) 680-689. [45] A. Wattanakornphaiboon, R. Canyook, K. Fakpan, Effect of SnO2 reinforcement on creep property of Sn–Ag–Cu solders, Materials Today: Proceedings 5(2018) 9213-9219. [46] E.H. Amalu, N.N. Ekere, Modelling evaluation of Garofalo-Arrhenius creep relation for lead-free solder joints in surface mount electronic component assemblies, J. Manuf. Syst. 39(2016) 9-23. [47] A.A. El-Daly, A.M. El-Taher, S. Gouda, Development of new multicomponent Sn–Ag–Cu–Bi lead-free solders for low-cost commercial electronic assembly, J. Alloy. Compd. 627(2015) 268-275. [48] J. Thambi, A. Schiessl, M. Waltz, K. Lang, U. Tetzlaff, Modified constitutive creep laws with micro-mechanical modelling of Pb-free solder alloys, J. Electron. Packaging 139(2017). [49] I. Dutta, D. Pan, R.A. Marks, S.G. Jadhav, Effect of thermo-mechanically induced microstructural coarsening on the evolution of creep response of SnAg-based microelectronic solders, Materials Science and Engineering: A 410-411(2005) 48-52. [50] J. Gong, C. Liu, P.P. Conway, V.V. Silberschmidt, Modelling of Ag3Sn coarsening and its effect on creep of Sn–Ag eutectics, Materials Science and Engineering: A 427(2006) 60-68. [51] C. Su, E.G. Herbert, S. Sohn, J.A. LaManna, W.C. Oliver, G.M. Pharr, Measurement of power-law creep parameters by instrumented indentation methods, J. Mech. Phys. Solids 61(2013) 517-536. [52] Y.D. Han, H.Y. Jing, S.M.L. Nai, L.Y. Xu, C.M. Tan, J. Wei, Creep mitigation in Sn–Ag–Cu composite solder with Ni-coated carbon nanotubes, Journal of Materials Science: Materials in Electronics 23(2012) 1108-1115. [53] Z. ZHANG, D. CHEN, Consideration of Orowan strengthening effect in particulate-reinforced metal matrix nanocomposites: A model for predicting their yield strength, Scripta Mater. 54(2006) 1321-1326. [54] X.C. Tang, L.Y. Meng, J.M. Zhan, W.R. Jian, W.H. Li, X.H. Yao, Y.L. Han, Strengthening effects of encapsulating graphene in SiC particle-reinforced Al-matrix composites, Comp. Mater. Sci. 153(2018) 275-281. [55] M. Rashad, F. Pan, A. Tang, M. Asif, Effect of Graphene Nanoplatelets addition on mechanical properties of pure aluminum using a semi-powder method, Progress in Natural Science: Materials International 24(2014) 101-108. [56] S.J. Yan, S.L. Dai, X.Y. Zhang, C. Yang, Q.H. Hong, J.Z. Chen, Z.M. Lin, Investigating aluminum alloy reinforced by graphene nanoflakes, Materials Science and Engineering: A 612(2014) 440-444. [57] S. Spigarelli, C. Paoletti, A new model for the description of creep behaviour of aluminium-based composites reinforced with nanosized particles, Composites Part A: Applied Science and Manufacturing 112(2018) 346-355. [58] M. Wang, Y. Zhao, L. Wang, Y. Zhu, X. Wang, J. Sheng, Z. Yang, H. Shi, Z. Shi, W. Fei, Achieving high strength and ductility in graphene/magnesium composite via an in-situ reaction wetting process, Carbon 139(2018) 954-963. [59] A. Sanaty-Zadeh, Comparison between current models for the strength of particulate-reinforced metal matrix nanocomposites with emphasis on consideration of Hall–Petch effect, Materials Science and Engineering: A 531(2012) 112-118.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32

[60] C. Cui, X. Cui, X. Li, K. Luo, J. Lu, X. Ren, J. Zhou, C. Fang, R. Farkouh, Y. Lu, Plastic-deformation-driven SiC nanoparticle implantation in an Al surface by laser shock wave: Mechanical properties, microstructure characteristics, and synergistic strengthening mechanisms, Int. J. Plasticity 102(2018) 83-100. [61] Nix WD, Gao H. Indentation size effects in crystalline materials: a law for strain gradient plasticity. J Mech Phys Solids 1998;46(3):411–25 [62] L. Liu, Y. Yao, T. Zeng, A micromechanical analysis to the elasto-viscoplastic behavior of solder alloys, Int. J. Solids Struct. 159(2019) 211-220. [63] Y.D. Han, H.Y. Jing, S.M.L. Nai, C.M. Tan, J. Wei, L.Y. Xu, S.R. Zhang, A modified constitutive model for creep of Sn–3.5Ag–0.7Cu solder joints, Journal of Physics D Applied Physics 42(2009) 125411. [64] J. Gong, C. Liu, P.P. Conway, V.V. Silberschmidt, Modelling of Ag3Sn coarsening and its effect on creep of Sn–Ag eutectics, Materials Science and Engineering: A 427(2006) 60-68. [65] Dieter GE. Mechanical metallurgy. third ed. New York (NY): Mc Graw-Hill; 1986. p. 212–20 [66] C. Gao, B. Zhan, L. Chen, X. Li, A micromechanical model of graphene-reinforced metal matrix nanocomposites with consideration of graphene orientations, Compos. Sci. Technol. 152(2017) 120-128. [67] B.W. Rosen. Tensile failure of fibrous composites., Aerospace Sciences Meeting, 1964. [68] Y.T. ZHU, W.R. BLUMENTHAL, T.C. LOWE, The tensile strength of short fibre-reinforced composites, J. Mater. Sci. 32(1997) 2037-2043. [69] S. Xiang, X. Wang, M. Gupta, K. Wu, X. Hu, M. Zheng, Graphene nanoplatelets induced heterogeneous bimodal structural magnesium matrix composites with enhanced mechanical properties, Sci. Rep.-UK 6(2016). [70] X. Liu, Q. Zhang, X. Zhao, X. Yang, L. Luo, Ambient-temperature nanoindentation creep in ultrafine-grained titanium processed by ECAP, Materials Science and Engineering: A 676(2016) 73-79. [71] Z. Cao, X. Zhang, Nanoindentation stress–strain curves of plasma-enhanced chemical vapor deposited silicon oxide thin films, Thin Solid Films 516(2008) 1941-1951. [72] C. Guo, Y. Fang, F. Chen, T. Feng, Nanoindentation creep behavior of electrodeposited Ni-P nanoglass films, Intermetallics 110(2019) 106480.

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 interests 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. ☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: