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Effects of interfacial properties on the ductility of polymer-supported metal films for flexible electronics
International Journal of Solids and Structures 47 (2010) 1830–1837
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Effects of interfacial properties on the ductility of polymer-supported metal films for flexible electronics W. Xu, T.J. Lu *, F. Wang MOE Key Laboratory for Strength and Vibration, Xi’an Jiaotong University, Xi’an 710049, PR China
a r t i c l e
i n f o
Article history: Received 26 June 2009 Received in revised form 8 February 2010 Available online 19 March 2010 Keywords: Metal film Flexible substrate Ductility Cohesive zone model Interface
a b s t r a c t Polymer-supported metal films as interconnects for flexible, large area electronics may rupture when they are stretched, and the rupture strain is strongly dependent upon the film/substrate interfacial properties. This paper investigates the influence of interfacial properties on the ductility of polymer-supported metal films by modeling the microstructure of the metal film as well as the film/substrate interface using the method of finite elements and the cohesive zone model (CZM). The influence of various system parameters including substrate thickness, Young’s modulus of substrate material, film/ substrate interfacial stiffness, strength and interfacial fracture energy on the ductility of polymersupported metal films is systematically studied. Obtained results demonstrate that the ductility of polymer-supported metal films increases as the interfacial strength increases, but the increasing trend is affected distinctly by the interfacial stiffness. Ó 2010 Elsevier Ltd. All rights reserved.
1. Introduction Large area, flexible electronic products are attractive due to their flexibility and deformability. Flexible electronics is an emerging technology built upon metal films supported by flexible polymer substrates, with which electronic functional elements and interconnects can be fabricated. With the development of flexible electronic technology, electronic products such as flexible (rollable) displays, printable thin film solar cells and electronic (artificial) skins will be improved (Suo et al., 2005). Flexible electronic products are mainly consisted of small functional islands and metal interconnects, both supported by polymer substrates. When the polymer substrate is stretched, the stiff functional islands experience relatively small strains in comparison with those in metal interconnects (Lacour et al., 2005; Li and Suo, 2006). The deformability of the metal interconnects is therefore vital to the practical applicability of flexible electronics. The aim of the present research is to investigate theoretically the stretchability of metal interconnects (in the form of metal films) that are supported by a polymer substrate. Different from its bulk counterpart, a free-standing metal film with high strength (e.g., single-crystal gold film) could only sustain a small strain (about 1%) before rupture due to necking and strain localization during stretching (Pashley, 1960). When the metal film is bonded to a polymer substrate, however, the situation becomes distinctly different. It has been demonstrated both by experimental * Corresponding author. Tel.: +86 29 82665600; fax: +86 29 83234781. E-mail address: [email protected] (T.J. Lu). 0020-7683/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijsolstr.2010.03.017
measurements and numerical simulations (Li et al., 2005; Li and Suo, 2007; Lu et al., 2007) that strain localization and necking are retarded in polymer-supported metal films due mainly to the constraint of the polymer substrate, leading to a significantly larger rupture strain of the metal film. Furthermore, the rupture strain is modulated by the adhesion between the film and the substrate (Xiang et al., 2005). When subjected to a tensile strain, a weakly bonded film would easily debond from the substrate and become free-standing, resulting in a smaller rupture strain, while a well bonded film can retard strain localization and necking more effectively. Recent experimental results show that a well bonded metal film can attain a rupture strain as high as 50% (Lu et al., 2007). The aforementioned simulations and experiments assume the metal film is a ductile continuum, without accounting for the potential influence of its microstructure on rupture. This simplification may overestimate the ductility of polymer-supported metal films for neglecting the grain boundary fracture mode, as the grain boundary is traditionally a weak link in metals where intergranular cracks may easily form (Xiang et al., 2005). For example, when brittle impurities precipitate in the grain boundary, the adhesion between two adjacent grains decreases, causing intergranular cracks to form that may reduce the ductility of polycrystalline metals (Suo et al., 2005). Although both the grain boundaries and the film/substrate interface can significantly influence the ductility of metal films supported by polymer substrates, existing numerical and experimental studies have focused on the film/substrate interface (Li and Suo, 2007; Lu et al., 2007; Xiang et al., 2005) and grain boundary (Zhang and Li, 2008) separately. To address this deficiency, the
W. Xu et al. / International Journal of Solids and Structures 47 (2010) 1830–1837
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present study models the metal film as an array of uniform grains and employs the cohesive zone model (CZM) to study the combined effects of film/substrate interface and grain boundary on the fracture behavior of polymer-supported metal films. 2. Computational model 2.1. Experimental observation To understand the microstructure of a typical polymer-supported metal film, copper film was deposited on a 125 lm thick Ò polyimide (Kapton by Dupont) substrate with the direct-current (DC) magnetron sputter-deposition system. The substrate was ultrasonically cleaned in ethanol before deposition. The sputtering power and the bias voltage were set as 120 W and 80 V, respectively. The base pressure was 1 107 Torr and the working gas (Ar) pressure was 2.3 103 Torr. As depicted in Fig. 1, the copper film with an average thickness of about 3.2 lm, is composed of many columnar grains with an average size of about 0.5 lm, and the grain boundaries are mostly perpendicular to the polyimide substrate. Based on this observation, we simulate the grain morphology of a polymer substrate-supported metal film system with the finite element method (FEM), as follows. 2.2. Cohesive zone model The cohesive zone model has been widely utilized to simulate fracture processes in material systems such as fiber/matrix interfacial cracking in composite materials. To specify the model, numerous traction–separation (T–S) laws have been proposed (Chandra et al., 2002). It has been established that whilst the peak value and area of the T–S curve are vital for capturing the interface separation behavior, its precise shape is of lesser significance (Freund and Suresh, 2003). Consequently, for simplicity, the bilinear T–S law (Chandra et al., 2002; Ghosh et al., 2000; Li and Chandra, 2003; Lin et al., 2001) shown in Fig. 2 is selected for the present study, which is specified by the peak traction stress rm, the corresponding separation displacement uc, the maximum separation displacement um, and the interfacial fracture energy C (consisted of Ca and Cb) that represents the area underneath the T–S curve. To distinguish the tensile T–S law from the shear one, let the superscript ‘‘n” represents the normal direction and ‘‘s” denotes the shear direction of an interface. The constitutive relation of the bilinear T–S law (Chandra et al., 2002) may thence be expressed as:
Fig. 2. Typical bilinear traction–separation law of cohesive zone model.
Tn ¼
s
T ¼
8 n < rm dn
ðd 6 dc Þ
dc
: rnm 1d dn d 1dc 8 n n r < dm uums ds c
ðd > dc Þ ðd 6 dc Þ
m
: rnm
n 1d um d 1dc usm
ds
ð1Þ
ðd > dc Þ
Here, Tn and Ts are the traction stress components; dn, ds and d are separately the normal, shear and total non-dimensional displacement jump defined as
dn ¼
un ; unm
ds ¼
us ; usm
d¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðdn Þ2 þ ðds Þ2
ð2Þ
and dc, defined by uc/um, is a characteristic length beyond which the interface starts to soften either in the pure opening or the pure shearing mode. For simplicity, it is assumed that the value of dc in the pure opening mode is identical to that in the pure shearing mode. Fig. 2 shows either Tn or Ts with respect to un or us corresponding to the pure opening mode (us = 0) or the pure shearing mode (un = 0) of interfacial separation. Since the maximum value of Tn is rnm while that of jTsj is rsm , the interfacial fracture energies in the two directions can be expressed as:
Cn ¼ s
C ¼
Z Z
unm
T n dun ¼
0 usm
0
1 n n r u 2 m m
1 T du ¼ rsm usm 2 s
ð3Þ
s
As the load is increased beyond a critical value, the interface begins to soften and degrade, namely, the interface is now in the damaged (or softening) state. Typically, damage is initiated when a certain criterion is satisfied. In the present study, inspired by the bilinear law of Fig. 2, the quadratic nominal stress criterion is adopted to characterize the interfacial damage, described as:
n 2 hT i
rnm
Fig. 1. Cross-sectional morphology of Cu film on polyimide substrate.
þ
Ts
rsm
2 ¼1
ð4Þ
where h i represents the Macaulay bracket defined by hxi = 1/ 2(x + jxj), with the usual interpretation that a pure compressive deformation or stress state does not initiate damage. It is assumed that interfacial damage in a film/substrate system occurs when Eq. (4) is satisfied and two damage variables, Dn and Ds, may be used to represent the extent of interfacial damage in the opening and shearing directions, respectively. Once Eq. (4) is satisfied and the load is further increased, the value of Dn (or Ds) increases monotonically from 0 (corresponding to damage initiation) to 1 (corresponding to complete failure). According to Fig. 2, the damage variables can be expressed as:
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D¼
W. Xu et al. / International Journal of Solids and Structures 47 (2010) 1830–1837
um ðu uc Þ ðuc u um Þ uðum uc Þ
ð5Þ
where, for convenience, the superscript of D has been dropped. With reference to Fig. 2, the interfacial stiffness in the softening n s state, kd (with normal component kd and shear component kd ), decreases during damage evolution, which can be linked to the damage variable by (see Appendix A):
kd ¼
1 Dn
0
0
1D
s ko
ð6Þ
Here, ko is the vector of initial interfacial stiffness with normal comn s ponent ko and shear component ko : n
ko ¼
ko
s ko
!
0 rn 1 m n
u ¼ @ rsc A
ð7Þ
m
usc
For the bilinear T–S law adopted in the present study, built upon previous studies (Balzani and Wagner, 2008), a single damage b based on total displacement D is introduced (i.e., variable D qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi D ¼ hun i2 þ ðus Þ2 Þ, as:
b ¼ Df ðDmax Dc Þ D Dmax ðDf Dc Þ
ð8Þ
where Dc and Df 2C=T eff c denote the total displacement at damage initiation and complete failure, T eff c is the effective traction at damqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n 2 s 2ffi eff T c þ T c ), and Dmax denotes the age initiation (i.e., T c ¼ maximum total displacement ever experienced during the loading
b ko . history. Accordingly, Eq. (6) can be simplified to kd ¼ 1 D Note that, with the linkage of D via Eq. (6) to the ratio of current to initial interfacial stiffness, the damage variable describing the extent of interfacial damage has a meaning physically equivalent to that introduced in the continuum damage mechanics (CDM) for engineering materials. In CDM, D is linearly proportional to the ratio of current Young’s stiffness E of the material to its initial value E0, i.e., D = 1 E/E0, if the damage is isotropic; for anisotropic damage, a tensor D is typically used (Chow and Lu, 1989). It is worth mentioning that the peak traction stresses rnm and rsm are also called the cohesive strengths (Chandra et al., 2002). Strictly speaking, although their magnitudes are similar, the cohesive strength is different from the interfacial strength. In practice, however, the cohesive strength is generally treated as the same as the interfacial strength when evaluating the interfacial property of a film/substrate system (Li and Suo, 2007). The same approach is followed in the present study. The mechanical properties of grain boundaries in polycrystalline materials have also been modeled using cohesive zone models (Sfantos and Aliabadi, 2007; Sheldon et al., 2007; Tello and Bower, 2008). Since the precise cohesive parameters of grain boundaries are at present unavailable from existing experimental studies of film/substrate systems, following previous researches (Sfantos and Aliabadi, 2007; Wu and Wei, 2008), the peak traction stress of the copper film grain boundary, rnm G , is selected to be the same as the yield strength of bulk copper rYf . Also, according to the relationship between stiffness and Young’s modulus (Balzani and Wagner, 2008), the interfacial stiffness of the grain boundary is selected n;s as kG h=Ef ¼ 10. In the sections to follow, subscripts ‘‘I” and ‘‘G” are utilized to represent the mechanical parameters associated with the film/substrate interface and the film grain boundary, respectively.
2.3. Computational model The commercially available finite element code ABAQUS is employed to simulate the effects of interfacial properties on the ductility of a polymer-supported copper film. In view of the film morphology shown in Fig. 1, a two-dimensional (2D) FEM model is employed as shown in Fig. 3(a). The film and the substrate have thickness h and H, respectively, and it is assumed that the columnar grains in the film have the same size. Under uniaxial tensile stressing, the film/substrate lamination is taken to deform under plane strain. Fig. 3(b) depicts a representative unit cell of the film/substrate lamination. The vertical displacement is set as zero along the bottom of the substrate, whereas a uniform displacement of u/2 is applied to both sides of the film/substrate system. The nominal strain of the system is then u/d, with d denoting the grain width of 4h (Fig. 3(b)). Both the metal film and the polymer substrate are modeled as elastic–plastic solids, with their true stress versus strain curves fitted using the power-law hardening laws, as:
r¼
8 < Ee : rY
e rY =E
N
e 6 rY =E e > rY =E
ð9Þ
where E is the Young’s modulus, N is the strain hardening exponent, and rY is the yield strength. For the present system, the following material properties taken from Li and Suo (2007) are used: N f ¼ 0:02; rYf ¼ 100 MPa and Ef = 100 GPa for the copper film; N s ¼ 0:5; rYs ¼ 50 MPa and Es = 0.5–8 GPa for the polymer substrate. Both the film and the substrate are meshed using quadrilateral plane strain elements. The film/substrate interface and film grain boundaries are each modeled by a single layer of four-node cohesive zone elements, which shares nodes with the neighboring elements in the film and in the substrate. The regions near the interface and the grain boundary are meshed densely, while sparse mesh is adopted far away from the interface and grain boundary (Fig. 3c).
3. Results and discussion In practice, the ductility of a metal film/polymer substrate system may be affected by many factors, such as grain size d, intergranular fracture energy CG, intergranular adhesive strength n;s rm G and stiffness kGn;s ; substrate Young’s modulus Es and thickness H; film/substrate interfacial fracture energy CI, interfacial n;s strength rn;s m I and stiffness kI . The normalized rupture strain er of the substrate-supported metal film can thence be expressed as:
1
0
n;s n;s C B er B CG Ef rn;s CI Ef rn;s Es H dC m G kG h m I kI h C ¼fB ; ; ; ; ; ; ; ; 2 Y 2 Y Ef Ef hC e0 B Ef h A @ rYf h rf rf h rYf |ffl{zffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} effects of substrate effects of grain boundary
effects of film=substrate interface
ð10Þ where e0 denotes the rupture strain of the corresponding freestanding film. To improve the ductility of a polymer-supported metal film, increasing the adhesion of grain boundaries and decreasing the grain size may be ideal choices (Zhang and Li, 2008). However, for a given metal film such as Cu widely used in electronics engineering, improving the adhesion of the grain boundaries may be challenging and inefficient. Optimizing the properties of the substrate and the film/substrate interface is perhaps more realistic and meaningful.
W. Xu et al. / International Journal of Solids and Structures 47 (2010) 1830–1837
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Fig. 3. (a) Computational model for polymer-supported metal film under uniaxial tension; (b) unit cell in the computational model; and (c) representative finite element mesh.
3.1. Intergranular fracture affected by polymer substrate properties A free-standing metal film ruptures under tensile stressing due mainly to intergranular fracture. The rupture strain e0 of a freestanding Cu film has been calculated to be approximately 1.008% based on the normalized intergranular fracture energy of .h i CG Ef rYf 2 h ¼ 100, which agrees well with experimental measurements (Pashley, 1960). The use of a polymer substrate could retard intergranular fracturing in the metal film because of its constraint effects. It has been demonstrated that an elastomer substrate having a Young’s modulus varying in the range of 1–200 MPa could increase the rupture strain of the metal film (Li et al., 2004). Compared with the elastomer substrate, however, a stiffer polymer substrate such as polyimide has more widespread applications in flexible electronics.
In order to study the effects of polymer substrate separately, the film is taken to be bonded to the substrate directly so that there is no need to use cohesive elements between the film and the substrate. As the external strain increases beyond a critical value, damage and rupture occurs along the grain boundary. When the film/substrate system is stretched up to the rupture strain er, the whole grain boundary ruptures and channel cracks form along the film thickness. Fig. 4 plots the normalized rupture strain as a function of normalized Young’s modulus of the polymer substrate for selected film thicknesses. The rupture strain increases with increasing Young’s modulus, indicating that the mismatch of Young’s modulus between the film and substrate materials plays an important role in restraining grain boundary cracking. The substrate constraint related to the mismatch could be explained by the energy release rate of the grain boundary, which is modulated by the Dundurs parameters (Beuth, 1992). When a film is deposited on
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For all subsequent calculations, it is assumed that H/h = 9. Furthermore, the film/substrate interface is modeled using cohesive zone elements to quantify the effects of film/substrate interfacial properties on grain boundary rupture.
3.2. Effect of interfacial strength
rmn;s
I
For simplicity, the film/substrate interface is assumed to have identical properties along its normal and shear directions, i.e., n rm I ¼ rsm I ¼ rsm I and knI ¼ ksI ¼ kIn;s . The fracture energy of the film/substrate is selected according to Kamiya et al. (2007), .h i rYf 2 h ¼ 250. The predictions plotted in Fig. 6 demwith CI Ef
a relatively compliant substrate, the energy release rate may easily exceed the fracture toughness of the film, making the film vulnerable to channel cracking and failure (Tsui et al., 2005). Thus, the stiff polyimide substrate provides a stronger constraint on cracking than softer substrates such as silicone and elastomer. In all subsequent FEM simulations, polyimide with Young’s modulus of 8 GPa is selected as the substrate material. Fig. 5 plots the normalized rupture strain er/e0 as a function of the normalized substrate thickness H/h. Two distinct stages are identified, separated by a normalized substrate thickness of H/ h 8 (transition thickness). When H/h < 8, the rupture strain increases sharply with increasing substrate thickness, whereas the rupture strain is nearly constant when H/h > 8. This trend holds even if the normalized interfacial energy is varied as shown in Fig. 5. Therefore, for a metal film supported by polymer substrate, the transition substrate thickness plays an important role in restraining film cracking and the selection of substrate thickness considerably larger than the transition thickness is unnecessary.
onstrate that the rupture strain increases with increasing interfacial strength for a given interfacial stiffness, which is consistent with the results reported previously (Li and Suo, 2007). Note that the gradient of rupture strain increase is strongly dependent upon the interfacial strength. Initially, as the interfacial strength increases, the gradient of rupture strain increase also increases. However, when the interfacial strength exceeds a critical value, Y i.e., rn;s m I rf ¼ 2:0, the trend is reversed. For a given value of interfacial strength, the gradient is also affected by interfacial stiffness, increasing as the interfacial stiffness is increased especially n;s Y rf > when the interface strength is relatively high, e.g., rm I 2:0. This indicates that, in addition to the interfacial strength, the interfacial stiffness exerts noticeable effects on the rupture strain of a stretched metal film bonded to a polymer substrate. The influence of interfacial strength on film rupture strain could be explained as follows. When subjected to a uniaxial tensile load, the metal film ruptures along the grain boundary by separation of adjacently bonded grains. This separating process is dominated by the relative displacement between the two adjacent grains. When the relative displacement exceeds a critical value, the rupture (damage) process initiates on the free surface of the film and spreads towards to the film/substrate interface along the grain boundary till the whole grain boundary ruptures. Meanwhile, the presence of a polymer substrate retards the separating process so that a larger rupture strain of the metal film may be achieved. However, if the interfacial strength is sufficiently weak to sustain only a small externally applied stress, the film/substrate interface
Fig. 5. Rupture strain plotted as a function of normalized substrate thicknesses for various normalized intergranular fracture energy with normalized Young’s modulus of substrate Es/Ef = 0.08; e0 is the rupture strain of a free-standing metal film .h i rYf 2 h ¼ 50. with normalized intergranular fracture energy of CG Ef
Fig. 6. Rupture strain plotted as a function of normalized interfacial strength for various normalized interfacial stiffnesses, with film/substrate interfacial fracture .h i rYf 2 h ¼ 250; e0 is the rupture strain of a free-standing metal film energy CI Ef .h i with normalized intergranular fracture energy of CG Ef rYf 2 h ¼ 230.
Fig. 4. Rupture strain plotted as a function of normalized substrate Young’s modulus for.various thicknesses with normalized intergranular fracture h substrate i energy CG Ef rYf 2 h ¼ 100; e0 is the rupture strain of a free-standing metal film.
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fails before the film ruptures completely. Once the metal film loses constraint from the substrate, the metal film ruptures at a strain smaller than that achieved when the interfacial strength is larger. Thus, the film rupture strain can be increased by improving the interfacial strength, as experimentally observed by previous reports (Lu et al., 2007; Xiang et al., 2005). n;s
3.3. Effect of interfacial stiffness kI
As depicted in Fig. 6, the rupture strain can be enhanced dramatically by increasing the interfacial strength when the interfacial stiffness is relatively large. To highlight the influence of interfacial stiffness, Fig. 7 plots the rupture strain as a function of interfacial stiffness for selected values of interfacial strength. At n;s Y rf ¼ 0:5—1:5, the relatively low interfacial strengths, e.g., rm I film rupture strain slightly decreases with increasing interfacial stiffness. When the interfacial strength is increased beyond a critical value, however, a high interfacial stiffness leads to a larger rupture strain. Consequently, the results of Fig. 7 suggest that there exists a critical interfacial strength beyond which increasing the interfacial stiffness is beneficial for enhanced film rupture strain. The present FEM simulations demonstrate that the ductility of a polymer-supported metal film is influenced significantly by both the stiffness and strength of the film/substrate interface. The effects could be interpreted as follows. Under a given externally applied tensile strain, a high interfacial stiffness induces high interfacial stresses. If the interfacial strength is relatively low, the high interfacial stresses result in interfacial damage more easily, reducing the constraint from the substrate to the metal film and hence the interfacial stiffness. As a result, damage and rupture along the grain boundary in the metal film tend to occur Y earlier n;s rf ¼ 2:5, than that if the interfacial strength is high, e.g., rm I leading to a smaller film rupture strain. Experimentally, it has been found (Lu et al., 2007) that a thin interlayer between the metal film and the polymer substrate, such as Cr or Ti, could increase considerably the ductility of the film. It is believed that the presence of the interlayer not only enhances the interfacial strength but also increases the interfacial stiffness. In microelectronics industry, metal films are typically bonded on substrates with glue (e.g., epoxy), so that the glue acts as one form of the interlay layer. As the stiffness of Ti (or Cr) is much greater than
Fig. 7. Rupture strain plotted as a function of normalized interfacial stiffnesses for various normalized interfacial strengths with film/substrate interfacial fracture .h i rYf 2 h ¼ 250; e0 is the rupture strain of a free-standing metal film energy CI Ef .h i with normalized intergranular fracture energy of CG Ef rYf 2 h ¼ 230.
that of the glue, improving the stiffness of the film/substrate interface could be realized by choosing suitable interlayer materials. In practice, taking Cu film bonded on a polyimide (PI) substrate as an example, Cu would diffuse into PI when it is deposited (Liang et al., 2005), thus a diffusion layer consisted of Cu and PI would form between the film and the substrate. The diffusion layer may be considered as another form of the interface. Accordingly, the interfacial stiffness of a film/substrate system without any interlayer between them would depend upon the stiffness of both film and substrate materials. It is of significance to obtain the exact value of the interfacial stiffness of the system. However, there is no such data reported in the literature according the authors’ best knowledge. Future work regarding the interfacial stiffness is clearly needed. 3.4. Effect of interfacial fracture energy CI As shown in Fig. 2, the interfacial fracture energy CI consists of two parts, CaI and CbI . Whilst CaI is dominated by interfacial strength and stiffness, CbI is dominated by energy dissipation during interface softening. With the interfacial stiffness fixed, Fig. 8 shows how the variation of CaI or CbI affects the film rupture strain. At the initial point, the normalized interfacial fracture energy .h i CI Ef rYf 2 h has a value of 250, of which the first part .h i CaI Ef rYf 2 h ¼ 12:5. When CaI is fixed and CbI is increased (the increment denoted by DCbI in Fig. 8), the resulting increase in the rupture strain is negligible. In sharp contrast, the increasing of rupture strain becomes substantial if CI is fixed and CaI is increased. The first part of the interfacial fracture energy CaI is mainly dominated by the strength and stiffness of the film/substrate interface. The results of Fig. 8 demonstrate that the film rupture strain is mainly affected byCaI , implying that the interfacial strength and stiffness play a more important role than the total interfacial fracture energy. In practice, in order to improve the ductility of a polymer-supported metal film, efforts should be devoted on enhancing both the strength and stiffness of the film/substrate interface.
Fig. 8. Rupture strain plotted as a function of interfacial fracture energy increment. .h i rYf 2 h ¼ 250, At the initial point, the normalized interfacial fracture energy CI Ef .h i of which the first part C1I Ef rYf 2 h ¼ 12:5; e0 is the rupture strain of a freestanding .h
CG Ef
metal i
film
rYf 2 h ¼ 230.
with
normalized
intergranular
fracture
energy
of
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3.5. Outlook
Acknowledgements
A multitude of assumptions and simplifications are introduced in the present model. One of these is to assume that the grain size along the tensile direction is identical, and in view of the morphology of Cu film shown in Fig. 1, all the grains are columnar whilst it has been reported that equiaxed grains can also enhance the ductility (Thompson, 2000). Furthermore, the influence of various types of defect in metals, such as pre-existing micro-cracks and local debondings (Fig. 9), is not accounted for in our model. In addition to the defects, impurities may segregate and make the grain boundaries brittle. As pre-existing defects and impurities may reduce the rupture strain of the metal film, further research is needed to address these issues. Another important aspect of the problem that has been ignored by the present model is the size effect. It is well known the flow stress of the film continuously increases with decreasing grain size, which can be expressed by the classic Hall–Petch relation (Hommel and Kraft, 2001; Niu et al., 2007; Yu and Spaepen, 2004). As the mechanical properties of metal films with sub-micron or nano-scale grain sizes are considerably different from their bulk counterparts, the effects of grain size on the ductility of metal films should be considered in future research. Besides the grain size effects, film thickness has similar effects to that of grain size, and grain morphology and grain boundary diffusion (Gao et al., 1999) may also play an important role in the yield strength and ductility of metal films. Thus further researches should consider these factors.
This work was supported by the National Basic Research Program of China (2006CB601201, 2006CB601202), the National Natural Science Foundation of China (10632060, 10825210 and 50701034), the National 111 Project of China (B06024), the National High Technology Research and Development Program (2006AA03Z519).
4. Concluding remarks The method of finite elements has been utilized to simulate the effects of interfacial properties on the ductility of polymer-supported metal films, with the cohesive zone model introduced to characterize both the film/substrate interface and the grain boundaries within the film. The influence of various system parameters including substrate thickness, Young’s modulus of substrate material, film/substrate interfacial stiffness together with strength and interfacial fracture energy on the ductility of the film/substrate system has been systematically studied. Obtained results demonstrate that the ductility of the metal film/substrate lamination can not only be enhanced by improving the interfacial strength but also be modulated by the interfacial stiffness. To attain a large rupture strain of the polymer-supported metal film, efforts should be devoted not only to enhancing the interfacial strength but also to increasing the interfacial stiffness.
Fig. 9. Defects such as micro-cracks and local debonding in as-deposited Cu film on polymer substrate.
Appendix A. Relation between damage and reduction of interfacial stiffness Consider first the damage variable Dn corresponding to the pure opening mode of interfacial separation. In terms of the separation displacement un ðunc un unm Þ, Eq. (5) can be rewritten as:
1 Dn ¼
unc unm un un unm unc
ðA:1Þ
Combination of Eqs. (7) and (A.1) leads to: n
ð1 D
n Þko
rn unc unm un rnm unm un ¼ m n n ¼ n n n n n uc
u um uc
u um uc
ðA:2Þ
From the T–S relation shown in Fig. 2, we have:
rn unm un ¼ rnm unm unc
ðA:3Þ
and hence:
rnm ¼
unm unc n r unm un
ðA:4Þ
Substitution of Eq. (A.4) into Eq. (A.2) results in: n
ð1 Dn Þko ¼
rn ¼ n ¼ knd n n n n n u u um u um uc
rn unm unc unm un
ðA:5Þ
Similarly, for the pure shearing mode of interfacial separation, it can be shown that: s
s
ð1 Ds Þko ¼ kd :
ðA:6Þ
References Balzani, C., Wagner, W., 2008. An interface element for the simulation of delamination in unidirectional fiber-reinforced composite laminates. Engineering Fracture Mechanics 75, 2597–2615. Beuth, J.L., 1992. Cracking of thin bonded films in residual tension. International Journal of Solids and Structures 29, 1657–1675. Chandra, N., Li, H., Shet, C., Ghonem, H., 2002. Some issues in the application of cohesive zone models for metal–ceramic interfaces. International Journal of Solids and Structures 39, 2827–2855. Chow, C.L., Lu, T.J., 1989. On evolution laws of anisotropic damage. Engineering Fracture Mechanics 34, 679–701. Freund, L.B., Suresh, S., 2003. Thin Film Materials: Stress, Defect Formation and Surface Evolution. Cambridge University Press, Cambridge. Gao, H., Zhang, L., Nix, W.D., Thompson, C.V., Arzt, E., 1999. Crack-like grainboundary diffusion wedges in thin metal films. Acta Materialia 47, 2865–2878. Ghosh, S., Ling, Y., Majumdar, B., Kim, R., 2000. Interfacial debonding analysis in multiple fiber reinforced composites. Mechanics of Materials 32, 561–591. Hommel, M., Kraft, O., 2001. Deformation behavior of thin copper films on deformable substrates. Acta Materialia 49, 3935–3947. Kamiya, S., Furuta, H., Omiya, M., 2007. Adhesion energy of Cu/polyimide interface in flexible printed circuits. Surface & Coatings Technology 202, 1084–1088. Lacour, S.P., Jones, J., Wagner, S., Li, T., Suo, Z.G., 2005. Stretchable interconnects for elastic electronic surfaces. Proceedings of the IEEE 93, 1459–1467. Li, H., Chandra, N., 2003. Analysis of crack growth and crack-tip plasticity in ductile materials using cohesive zone models. International Journal of Plasticity 19, 849–882. Li, T., Suo, Z., 2006. Deformability of thin metal films on elastomer substrates. International Journal of Solids and Structures 43, 2351–2363. Li, T., Suo, Z., 2007. Ductility of thin metal films on polymer substrates modulated by interfacial adhesion. International Journal of Solids and Structures 44, 1696– 1705.
W. Xu et al. / International Journal of Solids and Structures 47 (2010) 1830–1837 Liang, T.X., Liu, Y.Q., Fu, Z.Q., Luo, T.Y., Zhang, K.Y., 2005. Diffusion and adhesion properties of Cu films on polyimide substrates. Thin Solid Films 473, 247–251. Li, T., Huang, Z.Y., Suo, Z., Lacour, S.P., Wagner, S., 2004. Stretchability of thin metal films on elastomer substrates. Applied Physics Letters 85, 3435–3437. Li, T., Huang, Z.Y., Xi, Z.C., Lacour, S.P., Wagner, S., Suo, Z., 2005. Delocalizing strain in a thin metal film on a polymer substrate. Mechanics of Materials 37, 261–273. Lin, G., Geubelle, P.H., Sottos, N.R., 2001. Simulation of fiber debonding with friction in a model composite pushout test. International Journal of Solids and Structures 38, 8547–8562. Lu, N.S., Wang, X., Suo, Z.G., Vlassak, J., 2007. Metal films on polymer substrates stretched beyond 50%. Applied Physics Letters 91, 221909. Niu, R.M., Liu, G., Wang, C., Zhang, G., Ding, X.D., Sun, J., 2007. Thickness dependent critical strain in submicron Cu films adherent to polymer substrate. Applied Physics Letters 90, 161907. Pashley, D.W., 1960. A study of the deformation and fracture of single-crystal gold films of high strength inside an electron microscope. Proceedings of the Royal Society of London Series A – Mathematical and Physical Sciences 255, 218–231. Sfantos, G.K., Aliabadi, M.H., 2007. A boundary cohesive grain element formulation for modelling intergranular microfracture in polycrystalline brittle materials. International Journal for Numerical Methods in Engineering 69, 1590–1626.
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Sheldon, B.W., Bhandari, A., Bower, A.F., Raghavan, S., Weng, X.J., Redwing, M., 2007. Steady-state tensile stresses during the growth of polycrystalline films. Acta Materialia 55, 4973–4982. Suo, Z., Vlassak, J., Wagner, S., 2005. Micromechanics of macroelectronics. China Particuology 3, 321–328. Tello, J.S., Bower, A.F., 2008. Numerical simulations of stress generation and evolution in Volmer–Weber thin films. Journal of the Mechanics and Physics of Solids 56, 2727–2747. Thompson, C.V., 2000. Structure evolution during processing of polycrystalline films. Annual Review of Materials Science 30, 159–190. Tsui, T.Y., McKerrow, A.J., Vlassak, J.J., 2005. Constraint effects on thin film channel cracking behavior. Journal of Materials Research 20, 2266–2273. Wu, B., Wei, Y.G., 2008. Simulations of mechanical behavior of polycrystalline copper with nano-twins. Acta Mechanica Solida Sinica 21, 189–197. Xiang, Y., Li, T., Suo, Z.G., Vlassak, J.J., 2005. High ductility of a metal film adherent on a polymer substrate. Applied Physics Letters 87, 161910. Yu, D.Y.W., Spaepen, F., 2004. The yield strength of thin copper films on Kapton. Journal of Applied Physics 95, 2991–2997. Zhang, Z., Li, T., 2008. Effects of grain boundary adhesion and grain size on ductility of thin metal films on polymer substrates. Scripta Materialia 59, 862–865.
Contents lists available at ScienceDirect
International Journal of Solids and Structures journal homepage: www.elsevier.com/locate/ijsolstr
Effects of interfacial properties on the ductility of polymer-supported metal films for flexible electronics W. Xu, T.J. Lu *, F. Wang MOE Key Laboratory for Strength and Vibration, Xi’an Jiaotong University, Xi’an 710049, PR China
a r t i c l e
i n f o
Article history: Received 26 June 2009 Received in revised form 8 February 2010 Available online 19 March 2010 Keywords: Metal film Flexible substrate Ductility Cohesive zone model Interface
a b s t r a c t Polymer-supported metal films as interconnects for flexible, large area electronics may rupture when they are stretched, and the rupture strain is strongly dependent upon the film/substrate interfacial properties. This paper investigates the influence of interfacial properties on the ductility of polymer-supported metal films by modeling the microstructure of the metal film as well as the film/substrate interface using the method of finite elements and the cohesive zone model (CZM). The influence of various system parameters including substrate thickness, Young’s modulus of substrate material, film/ substrate interfacial stiffness, strength and interfacial fracture energy on the ductility of polymersupported metal films is systematically studied. Obtained results demonstrate that the ductility of polymer-supported metal films increases as the interfacial strength increases, but the increasing trend is affected distinctly by the interfacial stiffness. Ó 2010 Elsevier Ltd. All rights reserved.
1. Introduction Large area, flexible electronic products are attractive due to their flexibility and deformability. Flexible electronics is an emerging technology built upon metal films supported by flexible polymer substrates, with which electronic functional elements and interconnects can be fabricated. With the development of flexible electronic technology, electronic products such as flexible (rollable) displays, printable thin film solar cells and electronic (artificial) skins will be improved (Suo et al., 2005). Flexible electronic products are mainly consisted of small functional islands and metal interconnects, both supported by polymer substrates. When the polymer substrate is stretched, the stiff functional islands experience relatively small strains in comparison with those in metal interconnects (Lacour et al., 2005; Li and Suo, 2006). The deformability of the metal interconnects is therefore vital to the practical applicability of flexible electronics. The aim of the present research is to investigate theoretically the stretchability of metal interconnects (in the form of metal films) that are supported by a polymer substrate. Different from its bulk counterpart, a free-standing metal film with high strength (e.g., single-crystal gold film) could only sustain a small strain (about 1%) before rupture due to necking and strain localization during stretching (Pashley, 1960). When the metal film is bonded to a polymer substrate, however, the situation becomes distinctly different. It has been demonstrated both by experimental * Corresponding author. Tel.: +86 29 82665600; fax: +86 29 83234781. E-mail address: [email protected] (T.J. Lu). 0020-7683/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijsolstr.2010.03.017
measurements and numerical simulations (Li et al., 2005; Li and Suo, 2007; Lu et al., 2007) that strain localization and necking are retarded in polymer-supported metal films due mainly to the constraint of the polymer substrate, leading to a significantly larger rupture strain of the metal film. Furthermore, the rupture strain is modulated by the adhesion between the film and the substrate (Xiang et al., 2005). When subjected to a tensile strain, a weakly bonded film would easily debond from the substrate and become free-standing, resulting in a smaller rupture strain, while a well bonded film can retard strain localization and necking more effectively. Recent experimental results show that a well bonded metal film can attain a rupture strain as high as 50% (Lu et al., 2007). The aforementioned simulations and experiments assume the metal film is a ductile continuum, without accounting for the potential influence of its microstructure on rupture. This simplification may overestimate the ductility of polymer-supported metal films for neglecting the grain boundary fracture mode, as the grain boundary is traditionally a weak link in metals where intergranular cracks may easily form (Xiang et al., 2005). For example, when brittle impurities precipitate in the grain boundary, the adhesion between two adjacent grains decreases, causing intergranular cracks to form that may reduce the ductility of polycrystalline metals (Suo et al., 2005). Although both the grain boundaries and the film/substrate interface can significantly influence the ductility of metal films supported by polymer substrates, existing numerical and experimental studies have focused on the film/substrate interface (Li and Suo, 2007; Lu et al., 2007; Xiang et al., 2005) and grain boundary (Zhang and Li, 2008) separately. To address this deficiency, the
W. Xu et al. / International Journal of Solids and Structures 47 (2010) 1830–1837
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present study models the metal film as an array of uniform grains and employs the cohesive zone model (CZM) to study the combined effects of film/substrate interface and grain boundary on the fracture behavior of polymer-supported metal films. 2. Computational model 2.1. Experimental observation To understand the microstructure of a typical polymer-supported metal film, copper film was deposited on a 125 lm thick Ò polyimide (Kapton by Dupont) substrate with the direct-current (DC) magnetron sputter-deposition system. The substrate was ultrasonically cleaned in ethanol before deposition. The sputtering power and the bias voltage were set as 120 W and 80 V, respectively. The base pressure was 1 107 Torr and the working gas (Ar) pressure was 2.3 103 Torr. As depicted in Fig. 1, the copper film with an average thickness of about 3.2 lm, is composed of many columnar grains with an average size of about 0.5 lm, and the grain boundaries are mostly perpendicular to the polyimide substrate. Based on this observation, we simulate the grain morphology of a polymer substrate-supported metal film system with the finite element method (FEM), as follows. 2.2. Cohesive zone model The cohesive zone model has been widely utilized to simulate fracture processes in material systems such as fiber/matrix interfacial cracking in composite materials. To specify the model, numerous traction–separation (T–S) laws have been proposed (Chandra et al., 2002). It has been established that whilst the peak value and area of the T–S curve are vital for capturing the interface separation behavior, its precise shape is of lesser significance (Freund and Suresh, 2003). Consequently, for simplicity, the bilinear T–S law (Chandra et al., 2002; Ghosh et al., 2000; Li and Chandra, 2003; Lin et al., 2001) shown in Fig. 2 is selected for the present study, which is specified by the peak traction stress rm, the corresponding separation displacement uc, the maximum separation displacement um, and the interfacial fracture energy C (consisted of Ca and Cb) that represents the area underneath the T–S curve. To distinguish the tensile T–S law from the shear one, let the superscript ‘‘n” represents the normal direction and ‘‘s” denotes the shear direction of an interface. The constitutive relation of the bilinear T–S law (Chandra et al., 2002) may thence be expressed as:
Fig. 2. Typical bilinear traction–separation law of cohesive zone model.
Tn ¼
s
T ¼
8 n < rm dn
ðd 6 dc Þ
dc
: rnm 1d dn d 1dc 8 n n r < dm uums ds c
ðd > dc Þ ðd 6 dc Þ
m
: rnm
n 1d um d 1dc usm
ds
ð1Þ
ðd > dc Þ
Here, Tn and Ts are the traction stress components; dn, ds and d are separately the normal, shear and total non-dimensional displacement jump defined as
dn ¼
un ; unm
ds ¼
us ; usm
d¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðdn Þ2 þ ðds Þ2
ð2Þ
and dc, defined by uc/um, is a characteristic length beyond which the interface starts to soften either in the pure opening or the pure shearing mode. For simplicity, it is assumed that the value of dc in the pure opening mode is identical to that in the pure shearing mode. Fig. 2 shows either Tn or Ts with respect to un or us corresponding to the pure opening mode (us = 0) or the pure shearing mode (un = 0) of interfacial separation. Since the maximum value of Tn is rnm while that of jTsj is rsm , the interfacial fracture energies in the two directions can be expressed as:
Cn ¼ s
C ¼
Z Z
unm
T n dun ¼
0 usm
0
1 n n r u 2 m m
1 T du ¼ rsm usm 2 s
ð3Þ
s
As the load is increased beyond a critical value, the interface begins to soften and degrade, namely, the interface is now in the damaged (or softening) state. Typically, damage is initiated when a certain criterion is satisfied. In the present study, inspired by the bilinear law of Fig. 2, the quadratic nominal stress criterion is adopted to characterize the interfacial damage, described as:
n 2 hT i
rnm
Fig. 1. Cross-sectional morphology of Cu film on polyimide substrate.
þ
Ts
rsm
2 ¼1
ð4Þ
where h i represents the Macaulay bracket defined by hxi = 1/ 2(x + jxj), with the usual interpretation that a pure compressive deformation or stress state does not initiate damage. It is assumed that interfacial damage in a film/substrate system occurs when Eq. (4) is satisfied and two damage variables, Dn and Ds, may be used to represent the extent of interfacial damage in the opening and shearing directions, respectively. Once Eq. (4) is satisfied and the load is further increased, the value of Dn (or Ds) increases monotonically from 0 (corresponding to damage initiation) to 1 (corresponding to complete failure). According to Fig. 2, the damage variables can be expressed as:
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D¼
W. Xu et al. / International Journal of Solids and Structures 47 (2010) 1830–1837
um ðu uc Þ ðuc u um Þ uðum uc Þ
ð5Þ
where, for convenience, the superscript of D has been dropped. With reference to Fig. 2, the interfacial stiffness in the softening n s state, kd (with normal component kd and shear component kd ), decreases during damage evolution, which can be linked to the damage variable by (see Appendix A):
kd ¼
1 Dn
0
0
1D
s ko
ð6Þ
Here, ko is the vector of initial interfacial stiffness with normal comn s ponent ko and shear component ko : n
ko ¼
ko
s ko
!
0 rn 1 m n
u ¼ @ rsc A
ð7Þ
m
usc
For the bilinear T–S law adopted in the present study, built upon previous studies (Balzani and Wagner, 2008), a single damage b based on total displacement D is introduced (i.e., variable D qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi D ¼ hun i2 þ ðus Þ2 Þ, as:
b ¼ Df ðDmax Dc Þ D Dmax ðDf Dc Þ
ð8Þ
where Dc and Df 2C=T eff c denote the total displacement at damage initiation and complete failure, T eff c is the effective traction at damqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n 2 s 2ffi eff T c þ T c ), and Dmax denotes the age initiation (i.e., T c ¼ maximum total displacement ever experienced during the loading
b ko . history. Accordingly, Eq. (6) can be simplified to kd ¼ 1 D Note that, with the linkage of D via Eq. (6) to the ratio of current to initial interfacial stiffness, the damage variable describing the extent of interfacial damage has a meaning physically equivalent to that introduced in the continuum damage mechanics (CDM) for engineering materials. In CDM, D is linearly proportional to the ratio of current Young’s stiffness E of the material to its initial value E0, i.e., D = 1 E/E0, if the damage is isotropic; for anisotropic damage, a tensor D is typically used (Chow and Lu, 1989). It is worth mentioning that the peak traction stresses rnm and rsm are also called the cohesive strengths (Chandra et al., 2002). Strictly speaking, although their magnitudes are similar, the cohesive strength is different from the interfacial strength. In practice, however, the cohesive strength is generally treated as the same as the interfacial strength when evaluating the interfacial property of a film/substrate system (Li and Suo, 2007). The same approach is followed in the present study. The mechanical properties of grain boundaries in polycrystalline materials have also been modeled using cohesive zone models (Sfantos and Aliabadi, 2007; Sheldon et al., 2007; Tello and Bower, 2008). Since the precise cohesive parameters of grain boundaries are at present unavailable from existing experimental studies of film/substrate systems, following previous researches (Sfantos and Aliabadi, 2007; Wu and Wei, 2008), the peak traction stress of the copper film grain boundary, rnm G , is selected to be the same as the yield strength of bulk copper rYf . Also, according to the relationship between stiffness and Young’s modulus (Balzani and Wagner, 2008), the interfacial stiffness of the grain boundary is selected n;s as kG h=Ef ¼ 10. In the sections to follow, subscripts ‘‘I” and ‘‘G” are utilized to represent the mechanical parameters associated with the film/substrate interface and the film grain boundary, respectively.
2.3. Computational model The commercially available finite element code ABAQUS is employed to simulate the effects of interfacial properties on the ductility of a polymer-supported copper film. In view of the film morphology shown in Fig. 1, a two-dimensional (2D) FEM model is employed as shown in Fig. 3(a). The film and the substrate have thickness h and H, respectively, and it is assumed that the columnar grains in the film have the same size. Under uniaxial tensile stressing, the film/substrate lamination is taken to deform under plane strain. Fig. 3(b) depicts a representative unit cell of the film/substrate lamination. The vertical displacement is set as zero along the bottom of the substrate, whereas a uniform displacement of u/2 is applied to both sides of the film/substrate system. The nominal strain of the system is then u/d, with d denoting the grain width of 4h (Fig. 3(b)). Both the metal film and the polymer substrate are modeled as elastic–plastic solids, with their true stress versus strain curves fitted using the power-law hardening laws, as:
r¼
8 < Ee : rY
e rY =E
N
e 6 rY =E e > rY =E
ð9Þ
where E is the Young’s modulus, N is the strain hardening exponent, and rY is the yield strength. For the present system, the following material properties taken from Li and Suo (2007) are used: N f ¼ 0:02; rYf ¼ 100 MPa and Ef = 100 GPa for the copper film; N s ¼ 0:5; rYs ¼ 50 MPa and Es = 0.5–8 GPa for the polymer substrate. Both the film and the substrate are meshed using quadrilateral plane strain elements. The film/substrate interface and film grain boundaries are each modeled by a single layer of four-node cohesive zone elements, which shares nodes with the neighboring elements in the film and in the substrate. The regions near the interface and the grain boundary are meshed densely, while sparse mesh is adopted far away from the interface and grain boundary (Fig. 3c).
3. Results and discussion In practice, the ductility of a metal film/polymer substrate system may be affected by many factors, such as grain size d, intergranular fracture energy CG, intergranular adhesive strength n;s rm G and stiffness kGn;s ; substrate Young’s modulus Es and thickness H; film/substrate interfacial fracture energy CI, interfacial n;s strength rn;s m I and stiffness kI . The normalized rupture strain er of the substrate-supported metal film can thence be expressed as:
1
0
n;s n;s C B er B CG Ef rn;s CI Ef rn;s Es H dC m G kG h m I kI h C ¼fB ; ; ; ; ; ; ; ; 2 Y 2 Y Ef Ef hC e0 B Ef h A @ rYf h rf rf h rYf |ffl{zffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} effects of substrate effects of grain boundary
effects of film=substrate interface
ð10Þ where e0 denotes the rupture strain of the corresponding freestanding film. To improve the ductility of a polymer-supported metal film, increasing the adhesion of grain boundaries and decreasing the grain size may be ideal choices (Zhang and Li, 2008). However, for a given metal film such as Cu widely used in electronics engineering, improving the adhesion of the grain boundaries may be challenging and inefficient. Optimizing the properties of the substrate and the film/substrate interface is perhaps more realistic and meaningful.
W. Xu et al. / International Journal of Solids and Structures 47 (2010) 1830–1837
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Fig. 3. (a) Computational model for polymer-supported metal film under uniaxial tension; (b) unit cell in the computational model; and (c) representative finite element mesh.
3.1. Intergranular fracture affected by polymer substrate properties A free-standing metal film ruptures under tensile stressing due mainly to intergranular fracture. The rupture strain e0 of a freestanding Cu film has been calculated to be approximately 1.008% based on the normalized intergranular fracture energy of .h i CG Ef rYf 2 h ¼ 100, which agrees well with experimental measurements (Pashley, 1960). The use of a polymer substrate could retard intergranular fracturing in the metal film because of its constraint effects. It has been demonstrated that an elastomer substrate having a Young’s modulus varying in the range of 1–200 MPa could increase the rupture strain of the metal film (Li et al., 2004). Compared with the elastomer substrate, however, a stiffer polymer substrate such as polyimide has more widespread applications in flexible electronics.
In order to study the effects of polymer substrate separately, the film is taken to be bonded to the substrate directly so that there is no need to use cohesive elements between the film and the substrate. As the external strain increases beyond a critical value, damage and rupture occurs along the grain boundary. When the film/substrate system is stretched up to the rupture strain er, the whole grain boundary ruptures and channel cracks form along the film thickness. Fig. 4 plots the normalized rupture strain as a function of normalized Young’s modulus of the polymer substrate for selected film thicknesses. The rupture strain increases with increasing Young’s modulus, indicating that the mismatch of Young’s modulus between the film and substrate materials plays an important role in restraining grain boundary cracking. The substrate constraint related to the mismatch could be explained by the energy release rate of the grain boundary, which is modulated by the Dundurs parameters (Beuth, 1992). When a film is deposited on
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For all subsequent calculations, it is assumed that H/h = 9. Furthermore, the film/substrate interface is modeled using cohesive zone elements to quantify the effects of film/substrate interfacial properties on grain boundary rupture.
3.2. Effect of interfacial strength
rmn;s
I
For simplicity, the film/substrate interface is assumed to have identical properties along its normal and shear directions, i.e., n rm I ¼ rsm I ¼ rsm I and knI ¼ ksI ¼ kIn;s . The fracture energy of the film/substrate is selected according to Kamiya et al. (2007), .h i rYf 2 h ¼ 250. The predictions plotted in Fig. 6 demwith CI Ef
a relatively compliant substrate, the energy release rate may easily exceed the fracture toughness of the film, making the film vulnerable to channel cracking and failure (Tsui et al., 2005). Thus, the stiff polyimide substrate provides a stronger constraint on cracking than softer substrates such as silicone and elastomer. In all subsequent FEM simulations, polyimide with Young’s modulus of 8 GPa is selected as the substrate material. Fig. 5 plots the normalized rupture strain er/e0 as a function of the normalized substrate thickness H/h. Two distinct stages are identified, separated by a normalized substrate thickness of H/ h 8 (transition thickness). When H/h < 8, the rupture strain increases sharply with increasing substrate thickness, whereas the rupture strain is nearly constant when H/h > 8. This trend holds even if the normalized interfacial energy is varied as shown in Fig. 5. Therefore, for a metal film supported by polymer substrate, the transition substrate thickness plays an important role in restraining film cracking and the selection of substrate thickness considerably larger than the transition thickness is unnecessary.
onstrate that the rupture strain increases with increasing interfacial strength for a given interfacial stiffness, which is consistent with the results reported previously (Li and Suo, 2007). Note that the gradient of rupture strain increase is strongly dependent upon the interfacial strength. Initially, as the interfacial strength increases, the gradient of rupture strain increase also increases. However, when the interfacial strength exceeds a critical value, Y i.e., rn;s m I rf ¼ 2:0, the trend is reversed. For a given value of interfacial strength, the gradient is also affected by interfacial stiffness, increasing as the interfacial stiffness is increased especially n;s Y rf > when the interface strength is relatively high, e.g., rm I 2:0. This indicates that, in addition to the interfacial strength, the interfacial stiffness exerts noticeable effects on the rupture strain of a stretched metal film bonded to a polymer substrate. The influence of interfacial strength on film rupture strain could be explained as follows. When subjected to a uniaxial tensile load, the metal film ruptures along the grain boundary by separation of adjacently bonded grains. This separating process is dominated by the relative displacement between the two adjacent grains. When the relative displacement exceeds a critical value, the rupture (damage) process initiates on the free surface of the film and spreads towards to the film/substrate interface along the grain boundary till the whole grain boundary ruptures. Meanwhile, the presence of a polymer substrate retards the separating process so that a larger rupture strain of the metal film may be achieved. However, if the interfacial strength is sufficiently weak to sustain only a small externally applied stress, the film/substrate interface
Fig. 5. Rupture strain plotted as a function of normalized substrate thicknesses for various normalized intergranular fracture energy with normalized Young’s modulus of substrate Es/Ef = 0.08; e0 is the rupture strain of a free-standing metal film .h i rYf 2 h ¼ 50. with normalized intergranular fracture energy of CG Ef
Fig. 6. Rupture strain plotted as a function of normalized interfacial strength for various normalized interfacial stiffnesses, with film/substrate interfacial fracture .h i rYf 2 h ¼ 250; e0 is the rupture strain of a free-standing metal film energy CI Ef .h i with normalized intergranular fracture energy of CG Ef rYf 2 h ¼ 230.
Fig. 4. Rupture strain plotted as a function of normalized substrate Young’s modulus for.various thicknesses with normalized intergranular fracture h substrate i energy CG Ef rYf 2 h ¼ 100; e0 is the rupture strain of a free-standing metal film.
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fails before the film ruptures completely. Once the metal film loses constraint from the substrate, the metal film ruptures at a strain smaller than that achieved when the interfacial strength is larger. Thus, the film rupture strain can be increased by improving the interfacial strength, as experimentally observed by previous reports (Lu et al., 2007; Xiang et al., 2005). n;s
3.3. Effect of interfacial stiffness kI
As depicted in Fig. 6, the rupture strain can be enhanced dramatically by increasing the interfacial strength when the interfacial stiffness is relatively large. To highlight the influence of interfacial stiffness, Fig. 7 plots the rupture strain as a function of interfacial stiffness for selected values of interfacial strength. At n;s Y rf ¼ 0:5—1:5, the relatively low interfacial strengths, e.g., rm I film rupture strain slightly decreases with increasing interfacial stiffness. When the interfacial strength is increased beyond a critical value, however, a high interfacial stiffness leads to a larger rupture strain. Consequently, the results of Fig. 7 suggest that there exists a critical interfacial strength beyond which increasing the interfacial stiffness is beneficial for enhanced film rupture strain. The present FEM simulations demonstrate that the ductility of a polymer-supported metal film is influenced significantly by both the stiffness and strength of the film/substrate interface. The effects could be interpreted as follows. Under a given externally applied tensile strain, a high interfacial stiffness induces high interfacial stresses. If the interfacial strength is relatively low, the high interfacial stresses result in interfacial damage more easily, reducing the constraint from the substrate to the metal film and hence the interfacial stiffness. As a result, damage and rupture along the grain boundary in the metal film tend to occur Y earlier n;s rf ¼ 2:5, than that if the interfacial strength is high, e.g., rm I leading to a smaller film rupture strain. Experimentally, it has been found (Lu et al., 2007) that a thin interlayer between the metal film and the polymer substrate, such as Cr or Ti, could increase considerably the ductility of the film. It is believed that the presence of the interlayer not only enhances the interfacial strength but also increases the interfacial stiffness. In microelectronics industry, metal films are typically bonded on substrates with glue (e.g., epoxy), so that the glue acts as one form of the interlay layer. As the stiffness of Ti (or Cr) is much greater than
Fig. 7. Rupture strain plotted as a function of normalized interfacial stiffnesses for various normalized interfacial strengths with film/substrate interfacial fracture .h i rYf 2 h ¼ 250; e0 is the rupture strain of a free-standing metal film energy CI Ef .h i with normalized intergranular fracture energy of CG Ef rYf 2 h ¼ 230.
that of the glue, improving the stiffness of the film/substrate interface could be realized by choosing suitable interlayer materials. In practice, taking Cu film bonded on a polyimide (PI) substrate as an example, Cu would diffuse into PI when it is deposited (Liang et al., 2005), thus a diffusion layer consisted of Cu and PI would form between the film and the substrate. The diffusion layer may be considered as another form of the interface. Accordingly, the interfacial stiffness of a film/substrate system without any interlayer between them would depend upon the stiffness of both film and substrate materials. It is of significance to obtain the exact value of the interfacial stiffness of the system. However, there is no such data reported in the literature according the authors’ best knowledge. Future work regarding the interfacial stiffness is clearly needed. 3.4. Effect of interfacial fracture energy CI As shown in Fig. 2, the interfacial fracture energy CI consists of two parts, CaI and CbI . Whilst CaI is dominated by interfacial strength and stiffness, CbI is dominated by energy dissipation during interface softening. With the interfacial stiffness fixed, Fig. 8 shows how the variation of CaI or CbI affects the film rupture strain. At the initial point, the normalized interfacial fracture energy .h i CI Ef rYf 2 h has a value of 250, of which the first part .h i CaI Ef rYf 2 h ¼ 12:5. When CaI is fixed and CbI is increased (the increment denoted by DCbI in Fig. 8), the resulting increase in the rupture strain is negligible. In sharp contrast, the increasing of rupture strain becomes substantial if CI is fixed and CaI is increased. The first part of the interfacial fracture energy CaI is mainly dominated by the strength and stiffness of the film/substrate interface. The results of Fig. 8 demonstrate that the film rupture strain is mainly affected byCaI , implying that the interfacial strength and stiffness play a more important role than the total interfacial fracture energy. In practice, in order to improve the ductility of a polymer-supported metal film, efforts should be devoted on enhancing both the strength and stiffness of the film/substrate interface.
Fig. 8. Rupture strain plotted as a function of interfacial fracture energy increment. .h i rYf 2 h ¼ 250, At the initial point, the normalized interfacial fracture energy CI Ef .h i of which the first part C1I Ef rYf 2 h ¼ 12:5; e0 is the rupture strain of a freestanding .h
CG Ef
metal i
film
rYf 2 h ¼ 230.
with
normalized
intergranular
fracture
energy
of
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3.5. Outlook
Acknowledgements
A multitude of assumptions and simplifications are introduced in the present model. One of these is to assume that the grain size along the tensile direction is identical, and in view of the morphology of Cu film shown in Fig. 1, all the grains are columnar whilst it has been reported that equiaxed grains can also enhance the ductility (Thompson, 2000). Furthermore, the influence of various types of defect in metals, such as pre-existing micro-cracks and local debondings (Fig. 9), is not accounted for in our model. In addition to the defects, impurities may segregate and make the grain boundaries brittle. As pre-existing defects and impurities may reduce the rupture strain of the metal film, further research is needed to address these issues. Another important aspect of the problem that has been ignored by the present model is the size effect. It is well known the flow stress of the film continuously increases with decreasing grain size, which can be expressed by the classic Hall–Petch relation (Hommel and Kraft, 2001; Niu et al., 2007; Yu and Spaepen, 2004). As the mechanical properties of metal films with sub-micron or nano-scale grain sizes are considerably different from their bulk counterparts, the effects of grain size on the ductility of metal films should be considered in future research. Besides the grain size effects, film thickness has similar effects to that of grain size, and grain morphology and grain boundary diffusion (Gao et al., 1999) may also play an important role in the yield strength and ductility of metal films. Thus further researches should consider these factors.
This work was supported by the National Basic Research Program of China (2006CB601201, 2006CB601202), the National Natural Science Foundation of China (10632060, 10825210 and 50701034), the National 111 Project of China (B06024), the National High Technology Research and Development Program (2006AA03Z519).
4. Concluding remarks The method of finite elements has been utilized to simulate the effects of interfacial properties on the ductility of polymer-supported metal films, with the cohesive zone model introduced to characterize both the film/substrate interface and the grain boundaries within the film. The influence of various system parameters including substrate thickness, Young’s modulus of substrate material, film/substrate interfacial stiffness together with strength and interfacial fracture energy on the ductility of the film/substrate system has been systematically studied. Obtained results demonstrate that the ductility of the metal film/substrate lamination can not only be enhanced by improving the interfacial strength but also be modulated by the interfacial stiffness. To attain a large rupture strain of the polymer-supported metal film, efforts should be devoted not only to enhancing the interfacial strength but also to increasing the interfacial stiffness.
Fig. 9. Defects such as micro-cracks and local debonding in as-deposited Cu film on polymer substrate.
Appendix A. Relation between damage and reduction of interfacial stiffness Consider first the damage variable Dn corresponding to the pure opening mode of interfacial separation. In terms of the separation displacement un ðunc un unm Þ, Eq. (5) can be rewritten as:
1 Dn ¼
unc unm un un unm unc
ðA:1Þ
Combination of Eqs. (7) and (A.1) leads to: n
ð1 D
n Þko
rn unc unm un rnm unm un ¼ m n n ¼ n n n n n uc
u um uc
u um uc
ðA:2Þ
From the T–S relation shown in Fig. 2, we have:
rn unm un ¼ rnm unm unc
ðA:3Þ
and hence:
rnm ¼
unm unc n r unm un
ðA:4Þ
Substitution of Eq. (A.4) into Eq. (A.2) results in: n
ð1 Dn Þko ¼
rn ¼ n ¼ knd n n n n n u u um u um uc
rn unm unc unm un
ðA:5Þ
Similarly, for the pure shearing mode of interfacial separation, it can be shown that: s
s
ð1 Ds Þko ¼ kd :
ðA:6Þ
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