Wrinkling of the metal–polymer bilayer: the effect of periodical distribution of stresses and strains

Abstract

The viscoelastic wrinkling of aluminium films formed on silicon substrates with polystyrene interlayers (aluminium–polystyrene bilayers) is studied under thermal annealing. Compressive thermal stresses induced by the difference in thermal expansions of the bilayer and the silicon substrate are shown to lead to wrinkling instability of the Al film. The effects of annealing conditions and film thickness on the wrinkling regularities are investigated. The evolution of wrinkles, which has a three-stage character, is found to be controlled by the sign and value of in-plane stresses and those normal to the metal–polymer interface. On increasing temperature and duration of annealing, lateral extension of the aluminum–polystyrene bilayer inhibits wrinkle growth and results in smoothening of the film surface.

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1. Introduction

In recent decades, wrinkling of thin films on compliant substrates has attracted a lot of attention because it offers a straightforward means for formation of large-area, well ordered structures that could find a variety of applications in science and technology.^1–7 Depending on the substrate material, elastic wrinkling of a thin film can be accompanied by elastic, viscous or viscoelastic deformation of the substrate. The latter case is most promising in terms of controlling the formation of wrinkle patterns, since the substrate's compliance favours wrinkling at comparatively low stresses, while its elasticity ensures stabilization of the resulting wrinkle pattern in a certain stage of its formation.

The main regularities of viscoelastic wrinkling of multilayers were studied theoretically and experimentally. Thus, investigations of wrinkling dynamics by numerical simulations showed that the evolution of wrinkle patterns involves three stages: initial growth, coarsening and equilibration.^8,9 They are characterized by a gradual increase and saturation of the wrinkle wavelength and amplitude, resulting in the formation of equilibrium wrinkle patterns. Experimental studies revealed more complicated peculiarities of wrinkle evolution in a metal–polymer bilayer film. A number of wrinkling types characterized by different wrinkle growth and strain behaviour were found for a number of annealing temperatures, polymer-layer thicknesses and molecular weights, with five distinct wrinkling modes associated with the observed morphological changes being distinguished.¹⁰ Certain studies also reported that gradual wrinkle coarsening was followed by spontaneous formation of wrinkles having longer wavelengths, with the initial wrinkles remaining among the coarser ones.¹¹

Wrinkling of thin films on compliant substrates is accompanied by considerable changes in the stress-strain state of the system. Specifically, both analytical calculations and finite-element analyses revealed that, in contrast to the uniform plane strain distribution in the case of flat films, there is periodic distribution of stresses and strains in the wrinkled oxide films on metal substrates over the interface during their thermal oxidation.^12–14 In particular, wrinkling produces regions of tensile stress normal to the interface at the crests and corresponding compressive stress in the valley regions. This periodical stress variation was justified by piezo-spectroscopic stress measurements.^12,15 However the effect of the periodical stresses and strains on wrinkle evolution in the oxide/metal system has not yet been studied in detail because of continuously increasing thickness and density of an oxide film.

In this paper, viscoelastic wrinkling of an aluminum–polystyrene bilayer film on a silicon substrate is studied in terms of the periodical stress distribution. Large differences in thermal expansion of aluminum and silicon makes it possible to initiate compression of the metal–polymer bilayer. A thin film of native oxide protects aluminum against thermal oxidation that allows annealing in ambient atmosphere. Polystyrene (PS) has sufficiently low glass transition (105 °C) and flow (160 °C) temperatures,¹⁶ providing for considerable variation in its viscoelastic properties within this temperature range.

2. Experimental

Al–PS bilayer films were deposited onto 400 μm Si (100) substrates that were preliminary subjected to ion cleaning in an Ar atmosphere. Then the substrate was spin coated with 3 μm thick polystyrene layer, followed by magnetron sputtering of thin Al film. The thicknesses of the Al films were 50 and 100 nm. Al/PS/Si compositions were annealed in ambient conditions at temperatures of 100, 110, 150 and 180 °C. Annealing duration was varied from 5 seconds to 40 hours.

Surface morphology of the films was examined at room temperature using a Solver HV atomic force microscope (AFM) and a Quanta 200 3D scanning electron microscope (SEM). The strain of wrinkled metal films was estimated from analysis of AFM-images using the following equation:

where S₀ is the projected surface area, ΔS is the difference between the actual area and the projected area, ν is the Poisson's ratio of Al film.

The residual stress in the as-deposited Al films was determined by measuring changes in the curvature of the substrate with and without the metal film using the laser scanning technique, in a Tencor FLX-2320 apparatus. The technique was described in detail elsewhere.¹⁷ For the purpose of stress measurements, the films deposited on circular Si wafers with a diameter of 100 mm were used. The stress was calculated from Stoney's equation:

where E_s is the Young's modulus of the substrate, h_s is the substrate thickness, h is the Al film thickness, ν_s is the Poisson's ratio of the substrate, R is the radius of the curvature of the substrate with the Al film, and R₀ is the radius of the curvature of the substrate before deposition of the film. The silicon wafer with the polystyrene layer was assumed as a substrate. The Young's modulus and the Poisson's ratio of Si were used for stress estimations because of small thickness of the PS layer. In view of the low reflectivity of polystyrene, all the measurements were performed using the backside of the Si substrate. The same technique was used to determine thermal stresses developed in the films during heating to 200 °C and the following cooling to room temperature.

3. Results

3.1 Annealing at 110 °C

Annealing at the temperature 100 °C, which is lower than the glass transition temperature of polystyrene, does not result in any changes in surface morphology of the Al films. At higher temperatures wrinkles appear on the film surface, which wavelength and height depend on the temperature and duration of annealing.

At the temperature 110 °C deformation of a 100 nm thick Al film after annealing for 1.5 min becomes visible as single hillocks on the film surface with a diameter of 2–5 μm and a height of up to 30 nm (Fig. 1a). A disordered island-like pattern of wrinkles with an average wavelength and a height of 17.5 μm and 100 nm, respectively, is formed on increasing annealing duration to 3 min (Fig. 1b). A longer annealing results in transformation of the island-like pattern into the labyrinth one (Fig. 1c). The evolution of the wrinkles involves three stages (Fig. 2, curve 1). The first stage continues during 10 hours and is characterized by the nearly constant wavelength (λ), with the wrinkle height (A) being increased. At the second stage (annealing for 10–30 hours) λ increases and A slowly grows too. In the course of annealing for 30 to 40 hours (third stage), both the characteristics remain constant, i.e. the wrinkles form an equilibrium pattern.

Fig. 1 AFM-images of temporal evolution of surfaces of 100 nm thick Al films after annealing at 110 °C for 1.5 (a) and 3 (b) min as well as 1 (c) and 40 (d) hours.

Fig. 2 Wavelength (a) and height (b) of wrinkles on 100 nm thick Al films as a function of annealing time at the temperatures 110 (1), 150 (2) and 180 °C (3).

It should be noted that local ordering of the wrinkles is observed on increasing annealing duration. As a result, the chaotic labyrinth wrinkle pattern (Fig. 1c) transforms into the partially ordered zigzag one (Fig. 1d) that is characterized by straight segments of wrinkles parallel to each other. In addition, stripe pattern of wrinkles oriented normal to the borders of the specimen is formed near its edges (Fig. 3a). Moreover, delamination of the Al film from the polystyrene is observed near the specimen borders (Fig. 3b). Wrinkles on the polystyrene surface are clearly visible in these areas that confirm coherent deformation of the polymer layer with the metal film. Dimensions of the wrinkles on the polymer surface are the same as for wrinkles on the metal film.

Fig. 3 Wrinkles near the specimen's edge after annealing at the temperature 110 °C for 30 hours (a), delamination of the Al film near the specimen's edge (b), lateral extension of the Al–PS bilayer outwards the Si substrate after annealing at the temperature 150 °C for 40 hours (c).

Analysis of wrinkle profiles with AFM allowed revealing temporal dependence of strain of the Al films on annealing (Fig. 4, curve 1). An increase in film strain is observed at the first stage of wrinkle evolution so as it attained a maximum value of 0.33% after annealing for 10 hours. The second stage is characterized by decreasing the strain with time. Finally, the film strain keeps constant at the third stage due to the formation of the equilibrium wrinkle pattern.

Fig. 4 Temporal evolution of strain of a 100 nm thick Al film upon annealing at the temperatures 110 (1), 150 (2) and 180 °C (3).

3.2 Annealing at 150 °C

An increase in annealing temperature up to 150 °C leads to faster evolution of wrinkles on the Al films. In this case, a well-pronounced wrinkle pattern forms already after annealing for 15 s (Fig. 5a), with the initial wrinkle wavelength decreasing to 15.3 μm. The temporal evolution of wrinkles at a temperature of 150 °C is also characterized by three different stages. The duration of the stages is however substantially smaller compared to annealing at 110 °C and there is a considerable increase in the maximum values of the wrinkle wavelength and height (Fig. 2, curve 2).

Fig. 5 AFM-images of wrinkle patterns on Al films measuring 100 (a–c) and 50 nm (d) in thickness after annealing at the temperature 150 °C for 15 s (a), 5 min (b) and 40 hours (c, d).

Within the two initial stages, λ and A vary similarly to the case of annealing at 110 °C. In the third stage of evolution (annealing for more than 10 hours), the wavelength also remains constant, while the wrinkles combine to form the larger ones resulting in two-level wrinkle patterns (Fig. 5c). No complete merging of the wrinkles is observed until the end of the experiment, i.e. the initial wrinkles remain among the coarser ones. This process is accompanied by decreasing the height of initial wrinkles down to 590 nm after annealing for 40 hours.

In addition, annealing at a temperature of 150 °C for 10 hours and longer results in smoothening of the wrinkles near the film edges. Moreover, the lateral expansion of the Al–PS composition and its extension outwards the Si substrate become visible (Fig. 3c).

In contrast to the annealing at a temperature of 110 °C, the strain of the Al films increases not only at the first stage of wrinkle evolution but at the second stage too, reaching its maximum value (0.49%) after annealing for 5 hours (Fig. 4, curve 2). The strain rate in the second stage is however lower than that in the first stage. The third stage of wrinkle evolution is characterized by decreasing strain of the Al film that continues until the experiment is terminated.

The evolution of wrinkles on 50 nm Al films is similar to that in thicker films. The wrinkling process also begins after annealing for 15 s and is characterized by three stages, which durations are the same as for the films with the thickness of 100 nm. On the other hand, in the case of the thinner films, the wavelength of wrinkles is considerably lower (Fig. 5d). So, the maximum values of λ and A are 8.7 μm and 700 nm respectively.

3.3 Annealing at 180 °C

An increase in the temperature up to 180 °C results in the formation of a labyrinth wrinkle pattern on Al films already within 8 s of annealing (Fig. 6a). Due to fast wrinkling, it is impossible to reveal the first stage of the wrinkle evolution; moreover, the wrinkle wavelength and height are observed to increase from the very beginning of the experiment (Fig. 2, curve 3). As a result, already in 1 min of annealing the wavelength and height are as large as 16.8 μm and 700 nm, respectively (Fig. 6b). Longer annealing times give rise to wrinkle coarsening (Fig. 6c) and formation of a two-level wrinkle pattern (Fig. 6d) same as in the experiments carried out at the temperature 150 °C. As shown in Fig. 4, the strain of the Al films increases with time during initial 5 minutes and drops thereafter.

Fig. 6 AFM-images of wrinkle patterns on 100 nm thick Al films after annealing at the temperature 180 °C for 8 s (a), 1 min (b), 1 hour (c) and 40 hours (d).

4. Discussion

The wrinkle patterns observed on the Al films under study result from biaxial compressive stresses developing in the films in the course of heating up to the annealing temperature. The substrate curvature measurements performed using a laser scanning technique before and after film deposition showed that as-deposited Al films were characterized by a residual tensile stress of about 50 MPa. During heating, the differences in thermal expansion of the Al film, the polystyrene sublayer and the Si substrate cause a mismatch in their strain values. Since the total thickness of the metal–polymer bilayer is two orders of magnitude smaller than that of the substrate and all the components are rigidly bonded with one another, biaxial stresses arise in the film and the sublayer that strain them so as to fit their lateral dimensions to substrate dimensions. The coefficients of thermal expansion of aluminum and polystyrene are larger than that of silicon, therefore, the stress is compressive and its value can be estimated from the following expression

σ = MΔαΔT (3)

where M is the biaxial elastic modulus of the film, Δα is the difference in coefficients of thermal expansion of the film and the substrate, ΔT is the temperature change. Using appropriate values (Δα_Al-Si = −21.6 × 10⁻⁶ °C⁻¹; M_Al = 106 GPa), we obtain that on heating up to a temperature of 100 °C compressive thermal stress in Al films can reach ∼180 MPa. Actually, experimentally measured compressive stress at this temperature does not exceed 60 MPa that is due to both residual tensile stress in the as-deposited films and relaxation processes in the metal–polymer bilayer at elevated temperatures.

The latter value of thermal stress is insufficient to induce wrinkling of Al films at the temperatures below the polystyrene glass transition temperature. This is due to the fact that at such temperatures polystyrene is in elastic state, while to produce coherent deformation of the metal–polymer bilayer a much higher stress is necessary. Wrinkling of an elastic film on an elastic substrate is known to occur only after exceeding the critical compressive stress:^18,19

where and are the plane-strain moduli of the film and the substrate, respectively. Using appropriate values for the Al film Ē_Al ≈ 79 GPa and the polystyrene sublayer Ē_PS ≈ 3.4 GPa, we find that a stress of more than 5 GPa is necessary for elastic wrinkling of the Al–PS bilayer, while the measured thermal stress at a temperature of 100 °C is two orders of magnitude lower than this value.

When heated above 105 °C, polystyrene changes from a glassy state to a rubbery state, with its elastic modulus and shear viscosity decreasing sharply, and becomes sufficiently compliant to allow coherent deformation with the Al film. Deformation of polymers in the rubbery state is governed by their relaxation properties and changes slowly with time at a constant applied stress. This is due to the fact that the strain is produced by partial disentanglement of the polymer chain molecules rather than the changes in interatomic distances, as is the case for elastic deformation, which allows high reversible strains (hundreds of percent) to develop. There is a spectrum of relaxation times for rearrangement of different structural elements of a polymer (links, units of a chain, etc.). Because the rubbery state of a polymer is intermediate between the glassy state and the fluidic one, wrinkling of the Al film above the glass transition temperature of polystyrene is governed by both elastic and viscous deformation of the sublayer.

According to the nonlinear model of wrinkling the bilayer thin film consisting of an elastic layer on a viscoelastic layer,⁸ the critical stress for viscoelastic wrinkling is given by

where h and H are the thicknesses of the film and the sublayer, respectively, and is the plane-strain modulus of the polystyrene sublayer in the rubbery state. For the Al/PS system being considered (h = 0.1 μm, H = 3 μm, Ē_f = 79 GPa, ν_sl = 0.38 and E^R_sl ≈ 0.3 MPa),^16,20 eqn. (5) gives a value of σ_w ∼ 20 MPa that is two orders of magnitude lower than the critical stress for elastic wrinkling and sufficiently lower than the measured compressive stress in the film. Thus, we could argue that it is the glassy to rubbery state transition of polystyrene, which gives rise to wrinkling in the Al films.

While the theoretical models of wrinkling assume that instability of a thin film begins as simultaneous periodical buckling throughout its surface, wrinkling of the Al films studied here was observed to start in a random manner, apparently on irregularities at the film–sublayer interface. Since the interface is not perfectly flat, its local curvature produces a bending moment giving rise to bending the film accompanied by coherent deformation of the polystyrene sublayer that manifests itself as a hillock on its surface (Fig. 1a). In addition, this results in formation of depressions around the hillock. Tension of the film in the buckled surface areas provides for partial relaxation of compressive stress. Stress gradients arising at the perimeter of the hillock cause gradual merging of the hillocks into wrinkles.

Wrinkling results in periodical distribution of in-plane stresses in the film as well as of normal stresses acting across the wavy interface (Fig. 7). The sign and the value of these stresses govern the following evolution of wrinkles. The stresses normal to the interface (σ_i) vary from a maximum tensile value near the interface to zero at the top of the film in the crest regions and from a maximum compressive value to zero in the valley regions.^12,15 The in-plane stresses (σ_p) are location sensitive and non-uniform through the film thickness.²¹ In the valley regions they are compressive throughout the film thickness. In the crest areas the stresses are compressive at the bottom of the film near the interface but can become tensile at the top after exceeding a critical value of film curvature.

Fig. 7 Stress distribution in the film and at the film–substrate interface.

The values of the normal stress across the interface at the crests of wrinkles can be represented with substantial accuracy by using a soap-film analogy in which the normal stress is given by the Laplace equation:¹²

where σ_r is the residual biaxial stress in the film that can be estimated from Hooke's law using the value of elastic strain of the Al films and R is the effective radius of curvature at the crests of the wrinkle calculated from AFM-images. In turn, the values of the in-plane stress at the top of the wrinkle crest can be estimated using the expression of the classical beam theory for the case of symmetrical beam bending:

The estimations performed using eqn (6) and (7) are shown in Fig. 8.

Fig. 8 Normal stresses across the film–sublayer interface (a) and in-plane stresses at the crests of wrinkles (b) as a function of annealing time at the temperatures 110 (1), 150 (2) and 180 °C (3). The results are presented for a 100 nm thick Al film.

Eqn (6) implies that the values of the normal stresses across the metal–polymer interface are governed by competition between variations of the wrinkle curvature and the residual stress. At the beginning of wrinkling, when island-like wrinkle patterns are being formed, the radius of curvature decreases more rapidly than does the residual stress. This results in a fast increase in the normal stress across the interface, which is most pronounced at the temperature 110 °C (Fig. 8a, curve 1), in which case wrinkling is relatively slow. Relaxation of the normal stresses occurs via deformation of the polymer layer and favours an additional increase in the wrinkle height.

In the course of further annealing (in the first stage of wrinkling), decreasing of the radius of wrinkle curvature slows down, and the variation of stresses normal to the film/sublayer interface is to a great extent governed by relaxation of the residual thermal stresses. This gives rise to a decrease in the value of σ_i and, as a consequence, in the speed of polymer extrusion from the valley regions to the crests (indicated by arrows in Fig. 7). The latter, in turn, leads to reduction of rising the wrinkle height (Fig. 2b).

The duration of the first stage of wrinkling is determined by the sign of the in-plane stress that acts at the crests of wrinkles. In this stage, an increased height involves relaxation of the in-plane stress (Fig. 8b). The second stage of wrinkle evolution begins, when the stress becomes tensile and a further increase in the height of wrinkles is energetically unfavourable because it causes the rising of bending energy of the film. Therefore, relaxation of the tensile in-plane stress occurs by means of wrinkle coarsening, i.e. by simultaneous increasing their wavelength and height.

At the second stage, there is slow relaxation of both the in-plane and normal stresses (Fig. 8). When the stresses approach zero, the third stage begins, where evolution of wrinkles depends on the annealing temperature. An equilibrium wrinkle pattern is formed at a temperature of 110 °C. In the case of annealing at 150 and 180 °C, the third stage of wrinkle evolution is characterized by gradual decreasing the wrinkle height, with the wavelength being constant. This effect is concerned with lateral expansion of the Al/PS bilayer and will be considered below.

The wavelength and the height of wrinkles are determined by correlation between variations of the elastic energy of film compression and the sum of the elastic energy of film bending and the substrate strain energy. Longer wrinkle wavelengths are energetically unfavourable due to an increase in the substrate strain energy, while shorter wavelengths result in higher bending energy of the film. Film thickening leads to increasing its bending energy and, as a consequence, to longer wavelengths. Our experiments showed that wrinkle wavelength depends linearly on the thickness of the Al films (film thickening from 50 to 100 nm results in a twofold increase in λ) that is in good agreement with theoretical predictions.^8,9

An increase in annealing temperature has a great impact on the parameters of wrinkles because of increasing both compliance of the polymer sublayer and the value of residual thermal stresses. The former increases via a decrease in polystyrene shear viscosity, which occurs above its glass transition temperature. An analysis of the curves presented in Fig. 4 shows that for the wrinkling starting at 110, 150 and 180 °C, the strain rates of the Al/PS bilayer are equal to 5.2 × 10⁻⁷, 5.8 × 10⁻⁶ and 6.4 × 10⁻⁵ s⁻¹, respectively. Since shear viscosity is inversely proportional to strain rate,²² it would be expected that a 30–40 °C temperature increase could result in an order of magnitude higher shear viscosity of the polystyrene sublayer. This leads to essential shortening of the two initial stages of wrinkle evolution. At a temperature of 110 °C wrinkles are observed only after annealing for 3 min, while at 180 °C a developed wrinkle pattern forms already after 8 s. In turn, a higher value of thermal stress results in increasing maximum strain of the films (from 0.3% at a temperature of 110 °C to 0.5% at 180 °C) and higher wrinkles.

As mentioned above, a decrease in the wrinkle height at the third stage of evolution at temperatures of 150 and 180 °C is related to the lateral expansion of the Al–PS composition and its extension outwards the Si substrate. A similar effect was studied earlier and shown to be an alternative mechanism of relaxation of the thermal stress.^22,23 The lateral expansion begins at the early stages of wrinkle evolution but it is a slower process than wrinkling. Therefore, an initial relaxation of compressive stress occurs primarily by means of wrinkling of the film. At this point, the lateral expansion of the film that is accompanied by viscous flow of the polymer sublayer manifests itself only near specimen borders and provides for stress relaxation mainly in one direction. The stress component normal to the border keeps unrelaxed that leads to formation of stripe wrinkle pattern oriented parallel to the specimen border (Fig. 3). This is in good agreement with results of scaling analysis and two-dimensional numerical simulations of wrinkling of uniaxially and biaxially stressed films.⁹ The lateral expansion not only provides for stress relaxation but also allows essential decreasing the bending energy of the film. This mechanism is therefore more effective; with time it inhibits the growth of wrinkles and even smoothes them, i.e. reduces the wrinkle height. Eventually, viscous flow of the polymer sublayer would flatten the film surface, as it appears from theoretical predictions.^22,24

5. Summary

The features of wrinkling of an elastic Al film formed on a viscoelastic polystyrene sublayer have been studied under thermal annealing within the temperatures 100–180 °C. It has been found out that wrinkling of the Al–PS bilayer occurs at the temperatures exceeding that of the polystyrene glass transition. It has been demonstrated that evolution of wrinkle patterns is governed by the periodic distribution of in-plane stresses in the film and normal stresses at the wavy interface. The former stresses are responsible for increased wrinkle wavelengths, while the latter control the rate of polymer layer relaxation and, hence, the rate of increase in the wrinkle height. Temporal height and wavelength variations exhibit three-stage behaviour. Higher annealing temperatures give rise to increased thermal stresses and decreased elastic modulus and shear viscosity values of polystyrene, which results in an increased strain rate of the Al/PS bilayer and shorter initial stages of wrinkle evolution. At higher annealing temperatures and longer times, in addition to wrinkling, the bilayer undergoes deformation via its lateral expansion. It has been concluded that it is viscous flow of the polymer sublayer which provides for extension of the Al/PS bilayer outwards the Si substrate, which gradually smoothes the wrinkles.

The work is supported by the Siberian Branch of the Russian Academy of Sciences (grant no. III.23.1.3).

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