Article pubs.acs.org/Langmuir
Viscoelastic Leveling of Annealed Thin Polystyrene Films Etienne Rognin,*,†,‡ Stefan Landis,† and Laurent Davoust‡ †
CEA-LETI-Minatec Campus, 17 Rue des Martyrs, 38054 Grenoble Cedex 9, France SIMAP/EPM, Grenoble Institute of Technology, 1130 Rue de la Piscine, 38400 Saint-Martin-d’Hères, France
‡
ABSTRACT: Theoretical and experimental work on nanoscale viscoelastic flows of polystyrene melts is presented. The reflow above the glass transition temperature (Tg) of a continuous patterned film is characterized. Attention is paid to the topographical consequences of the flow rather than to the temporal description of the leveling of the film. In the framework of capillary wave theory, it is shown that only the shortest spatial wavelengths of the topography exhibit an elastic behavior, while long waves follow a viscous decay. The threshold wavelength depends on the surface tension, on the elastic plateau modulus, and, for ultrathin films, on the film thickness. Besides, for polystyrene, this threshold is a nanoscale parameter and weakly depends on the temperature of annealing. Experiments are conducted on polystyrene 130 kg/mol submicrometer films. The samples are embossed using thermal nanoimprint technology and then annealed at different temperatures between Tg + 10 °C and Tg + 50 °C. The smoothed topographies of the films are measured by atomic force microscopy and compared to a single-mode Maxwell leveling model and a more elaborated model based on reptation theory.
■
Bousfield.12 More recent developments in microfluidics were described by Jäckle13 and Henle and Levine.14 Experimentally, the leveling of thin polymer films near the glass transition temperature (Tg) has been used to investigate the surface mobility of polymer chains.21−23 However, apart from dewetting and instability phenomena, little work on the topographical impact of the viscoelastic relaxation of patterns above Tg has been published. In fact, studies in that field come from the nanoimprint community for whom elastic recovery is a major technological issue.15−17 Ding et al.18,19 and Jones et al.20 studied the decay of imprinted polymer patterns during annealing. Although they were able to capture with great accuracy the temporal evolution of the pattern, they focused only on periodical and simple line−space patterns. In this article, we present theoretical and experimental work on viscoelastic leveling of thin polymer films. As an extension of previous studies conducted by our group,24−26 capillary wave theory coupled with various rheological models is used here to predict the reflow of arbitrary shaped films. This work focuses on the topographical consequences of the elastic nature of the flow rather than on the temporal description of the leveling dynamics. Experiments are conducted on polystyrene 130 kg/ mol submicrometer films. The samples are embossed using thermal nanoimprint technology and then annealed at different temperatures between Tg + 10 °C and Tg + 50 °C. Above Tg, the elastic modulus of the polymer decreases from its glassy value to its rubbery value, and the material exhibits a finite viscosity. Then surface tension tends to flatten the protrusions of the surface, causing a viscoelastic flow of the material. The smoothed topographies of the films are measured by atomic
INTRODUCTION Many applications in optics, microelectro-mechanical devices, or microfluidics require three-dimensional (3D) nanostructures that are difficult or expensive to manufacture with standard nanolithography techniques. Although sequential etching processes or numerous lithography steps can be employed for such a purpose,1 other methods are needed to realize fast and cost-effective complex shaping. Techniques for manufacturing 3D shapes based on a polymer reflow process have been developed to tackle this issue. In the field of optical devices, microlenses have been fabricated using reflow for more than two decades.2 Various topographies can be created with this technique combined with capillary instabilities,3−5 or simply by smoothing an initial pattern.6 Finally, higher-aspect ratio features can be made by grayscale lithography, followed by a selective reflow step.7 Nevertheless, the reliability and development of these processes require a comprehensive characterization and modeling of reflow phenomena on a submicrometer scale. From a theoretical point of view, the leveling dynamics of a thin film has been extensively reviewed by Oron et al.8 and Craster and Matar.9 In the field of microelectronics, Stillwagon and Larson10 studied the leveling of a thin film previously spincoated onto a nonflat substrate. However, lubrication theory used as a starting point in their work is very efficient for predicting the long-scale behavior of the liquid film but totally unable to account for the reflow of the shortest features. If the film is continuous, in other words, if there are no dewetted areas, or far from triple contact lines, another approach based on capillary wave theory can be chosen. First, quantitative developments applied to thin films came from the paint industry when Orchard11 derived the conditions for brush marks to relax before the paint dries. A significant contribution for viscoelastic fluids was later made by Keunings and © 2014 American Chemical Society
Received: March 9, 2014 Revised: May 7, 2014 Published: May 21, 2014 6963
dx.doi.org/10.1021/la5009279 | Langmuir 2014, 30, 6963−6969
Langmuir
Article
⎛ τ (x ) = ⎜x − ⎝
force microscopy (AFM), and it is compared to the theoretical modeling developed in this paper.
■
THEORY Linear Viscoelasticity. In the framework of continuum mechanics, the problem of a leveling liquid film at a very low Reynolds number is defined by the continuity equation for mass conservation, the Stokes equation for momentum transport, and a set of boundary conditions.9 The closure of the momentum equation by an expression of stresses as a function of strains or strain rates is a delicate issue for realistic fluids such as polymer melts. An extensive study of rheological models for polymers as well as experimental results can be found in the book by Ferry.27 Here we report only basic ideas that are useful for this work. At low strain rates, a fluid may behave like an ideal Newton’s fluid. The shear stress is proportional to the strain rate, and the linear coefficient is called the dynamic viscosity. However, at both low strains and strain rates, a realistic fluid material, such as a polymer melt, stores a part of the mechanical energy and dissipates the remaining part into heat. If a stress is instantaneously applied to such a material, it flows but also stores some elastic energy. If the stress is stopped, the elastic energy is relaxed by shape deformation, which can be harmful in nanoimprinted processes for example. Viscoelastic fluids can be considered in another way. If an oscillating shear stress of frequency ω is applied on the fluid, the strain is not in quadrature phase (+π/2), as it would be for a Newtonian fluid. It is neither in phase, like a Hookean spring. In fact, both the phase and the proportionality coefficient depend on the frequency of the oscillation. We can define a complex viscosity, written η(ω), by σt(ω) = η(ω)γ(̇ ω)
τC =
η0 =
1
τ (x ) dx 1 − iωτ(x)
Nx 4 2ν τC for 0 < x < 16 N
(6)
ρRT τC 3Me
η(ω) = η0
(7)
∫0
1
3x 2 dx 1 − iωτCx 2
(8)
Evolution Model for the Topography. The leveling of the topography of a thin film can be modeled by a capillary wave approach. In this theory, the topography of the film is expanded in terms of surface waves. Under the assumption of no slip at the bottom interface and when surface tension is the only driving force, the leveling dynamics is constrained by a dispersion relation, in other words, a necessary condition that binds wavevector k and complex frequency ω:14,24
(1)
iωη(ω)h0 = f (kh0) γ
(9)
where h0 is the average film thickness, γ is the surface tension, k = |k|, and f is a dimensionless function of the normalized wavevector kh0: kh0 tanh kh0 − f (kh0) =
(
2+2
(
kh0 cosh kh0
kh0 cosh kh0
2
)
2
)
(10)
Although the expression of f seems elaborate, its shape is quite simple, which can be seen in Figure 1. The function exhibits two asymptotic limits. For kh0 ≪ 1, f ∼ (kh0)4/3; this domain is known as the thin film or long wave regime. On the other hand, for kh0 ≫ 1, f ∼ kh0/2; this is the thick film or bulk regime. The problem is now to find an explicit expression of ω as a function of k. Newtonian Case. If the material is a Newtonian fluid [η(ω) is constant], the relation dispersion has only one root on the imaginary axis, which means that each spatial mode is purely damped. The decay time τ(k) for each mode of wavevector k is
(3)
where ρ is the density, R is the gas constant, T is the temperature, Me is the average molecular mass between entanglements (for polystyrene, Me ≈ 17 kg/mol), and τ(x) is a relaxation time spectrum of the form τ (x ) =
π 2kBT
and the model can be recast in
With regard to polystyrene melts, Majeste et al. gave a comprehensive review of the form of η(ω).a In the entangled regime, they summed up the contributions of different relaxation mechanisms. However, for low shear frequencies, the following term accounting for reptation motion prevails:
∫0
ζ0REE 2Ne 2N 2
where ζ0 is the monomeric friction coefficient, REE is the endto-end distance, N is the degree of polymerization, Ne is the average number of repeat units between entanglements, kB is Boltzmann’s constant, and ν is an adjustable parameter. In the end, eq 3 represents a Maxwell model with a continuous relaxation spectrum, accounting for the different relaxation times in the reptation process as well as tube length fluctuations. If ν = 0, it corresponds to a pure reptation. In this case, we have a simple relation between the zero-shear viscosity η0 and the terminal time τC, in the limit ω → 0:
28
ρRT Me
(5)
and where τC is the terminal relaxation time, given by
where σt is the shear stress and γ̇ the strain rate. Transposed in the time domain, the stress would be a convolution of the history of the strain rate with a viscosity kernel. For example, the shear behavior can be modeled by a spring of elasticity G put serially with a dashpot of viscosity η0. For a stress σt, the total strain is the sum of the elastic strain σt/G plus the viscous strain −σt/iωη0 (i is the imaginary unit). If we note τ = η0/G the relaxation time, we get σt/γ = (−iωη0)/(1 − iωτ). In terms of strain rates, we finally get the Maxwell model: η0 η(ω) = (2) 1 − iωτ
η(ω) =
2 ν ⎞ 2ν ⎟ τC for
Viscoelastic Leveling of Annealed Thin Polystyrene Films Etienne Rognin,*,†,‡ Stefan Landis,† and Laurent Davoust‡ †
CEA-LETI-Minatec Campus, 17 Rue des Martyrs, 38054 Grenoble Cedex 9, France SIMAP/EPM, Grenoble Institute of Technology, 1130 Rue de la Piscine, 38400 Saint-Martin-d’Hères, France
‡
ABSTRACT: Theoretical and experimental work on nanoscale viscoelastic flows of polystyrene melts is presented. The reflow above the glass transition temperature (Tg) of a continuous patterned film is characterized. Attention is paid to the topographical consequences of the flow rather than to the temporal description of the leveling of the film. In the framework of capillary wave theory, it is shown that only the shortest spatial wavelengths of the topography exhibit an elastic behavior, while long waves follow a viscous decay. The threshold wavelength depends on the surface tension, on the elastic plateau modulus, and, for ultrathin films, on the film thickness. Besides, for polystyrene, this threshold is a nanoscale parameter and weakly depends on the temperature of annealing. Experiments are conducted on polystyrene 130 kg/mol submicrometer films. The samples are embossed using thermal nanoimprint technology and then annealed at different temperatures between Tg + 10 °C and Tg + 50 °C. The smoothed topographies of the films are measured by atomic force microscopy and compared to a single-mode Maxwell leveling model and a more elaborated model based on reptation theory.
■
Bousfield.12 More recent developments in microfluidics were described by Jäckle13 and Henle and Levine.14 Experimentally, the leveling of thin polymer films near the glass transition temperature (Tg) has been used to investigate the surface mobility of polymer chains.21−23 However, apart from dewetting and instability phenomena, little work on the topographical impact of the viscoelastic relaxation of patterns above Tg has been published. In fact, studies in that field come from the nanoimprint community for whom elastic recovery is a major technological issue.15−17 Ding et al.18,19 and Jones et al.20 studied the decay of imprinted polymer patterns during annealing. Although they were able to capture with great accuracy the temporal evolution of the pattern, they focused only on periodical and simple line−space patterns. In this article, we present theoretical and experimental work on viscoelastic leveling of thin polymer films. As an extension of previous studies conducted by our group,24−26 capillary wave theory coupled with various rheological models is used here to predict the reflow of arbitrary shaped films. This work focuses on the topographical consequences of the elastic nature of the flow rather than on the temporal description of the leveling dynamics. Experiments are conducted on polystyrene 130 kg/ mol submicrometer films. The samples are embossed using thermal nanoimprint technology and then annealed at different temperatures between Tg + 10 °C and Tg + 50 °C. Above Tg, the elastic modulus of the polymer decreases from its glassy value to its rubbery value, and the material exhibits a finite viscosity. Then surface tension tends to flatten the protrusions of the surface, causing a viscoelastic flow of the material. The smoothed topographies of the films are measured by atomic
INTRODUCTION Many applications in optics, microelectro-mechanical devices, or microfluidics require three-dimensional (3D) nanostructures that are difficult or expensive to manufacture with standard nanolithography techniques. Although sequential etching processes or numerous lithography steps can be employed for such a purpose,1 other methods are needed to realize fast and cost-effective complex shaping. Techniques for manufacturing 3D shapes based on a polymer reflow process have been developed to tackle this issue. In the field of optical devices, microlenses have been fabricated using reflow for more than two decades.2 Various topographies can be created with this technique combined with capillary instabilities,3−5 or simply by smoothing an initial pattern.6 Finally, higher-aspect ratio features can be made by grayscale lithography, followed by a selective reflow step.7 Nevertheless, the reliability and development of these processes require a comprehensive characterization and modeling of reflow phenomena on a submicrometer scale. From a theoretical point of view, the leveling dynamics of a thin film has been extensively reviewed by Oron et al.8 and Craster and Matar.9 In the field of microelectronics, Stillwagon and Larson10 studied the leveling of a thin film previously spincoated onto a nonflat substrate. However, lubrication theory used as a starting point in their work is very efficient for predicting the long-scale behavior of the liquid film but totally unable to account for the reflow of the shortest features. If the film is continuous, in other words, if there are no dewetted areas, or far from triple contact lines, another approach based on capillary wave theory can be chosen. First, quantitative developments applied to thin films came from the paint industry when Orchard11 derived the conditions for brush marks to relax before the paint dries. A significant contribution for viscoelastic fluids was later made by Keunings and © 2014 American Chemical Society
Received: March 9, 2014 Revised: May 7, 2014 Published: May 21, 2014 6963
dx.doi.org/10.1021/la5009279 | Langmuir 2014, 30, 6963−6969
Langmuir
Article
⎛ τ (x ) = ⎜x − ⎝
force microscopy (AFM), and it is compared to the theoretical modeling developed in this paper.
■
THEORY Linear Viscoelasticity. In the framework of continuum mechanics, the problem of a leveling liquid film at a very low Reynolds number is defined by the continuity equation for mass conservation, the Stokes equation for momentum transport, and a set of boundary conditions.9 The closure of the momentum equation by an expression of stresses as a function of strains or strain rates is a delicate issue for realistic fluids such as polymer melts. An extensive study of rheological models for polymers as well as experimental results can be found in the book by Ferry.27 Here we report only basic ideas that are useful for this work. At low strain rates, a fluid may behave like an ideal Newton’s fluid. The shear stress is proportional to the strain rate, and the linear coefficient is called the dynamic viscosity. However, at both low strains and strain rates, a realistic fluid material, such as a polymer melt, stores a part of the mechanical energy and dissipates the remaining part into heat. If a stress is instantaneously applied to such a material, it flows but also stores some elastic energy. If the stress is stopped, the elastic energy is relaxed by shape deformation, which can be harmful in nanoimprinted processes for example. Viscoelastic fluids can be considered in another way. If an oscillating shear stress of frequency ω is applied on the fluid, the strain is not in quadrature phase (+π/2), as it would be for a Newtonian fluid. It is neither in phase, like a Hookean spring. In fact, both the phase and the proportionality coefficient depend on the frequency of the oscillation. We can define a complex viscosity, written η(ω), by σt(ω) = η(ω)γ(̇ ω)
τC =
η0 =
1
τ (x ) dx 1 − iωτ(x)
Nx 4 2ν τC for 0 < x < 16 N
(6)
ρRT τC 3Me
η(ω) = η0
(7)
∫0
1
3x 2 dx 1 − iωτCx 2
(8)
Evolution Model for the Topography. The leveling of the topography of a thin film can be modeled by a capillary wave approach. In this theory, the topography of the film is expanded in terms of surface waves. Under the assumption of no slip at the bottom interface and when surface tension is the only driving force, the leveling dynamics is constrained by a dispersion relation, in other words, a necessary condition that binds wavevector k and complex frequency ω:14,24
(1)
iωη(ω)h0 = f (kh0) γ
(9)
where h0 is the average film thickness, γ is the surface tension, k = |k|, and f is a dimensionless function of the normalized wavevector kh0: kh0 tanh kh0 − f (kh0) =
(
2+2
(
kh0 cosh kh0
kh0 cosh kh0
2
)
2
)
(10)
Although the expression of f seems elaborate, its shape is quite simple, which can be seen in Figure 1. The function exhibits two asymptotic limits. For kh0 ≪ 1, f ∼ (kh0)4/3; this domain is known as the thin film or long wave regime. On the other hand, for kh0 ≫ 1, f ∼ kh0/2; this is the thick film or bulk regime. The problem is now to find an explicit expression of ω as a function of k. Newtonian Case. If the material is a Newtonian fluid [η(ω) is constant], the relation dispersion has only one root on the imaginary axis, which means that each spatial mode is purely damped. The decay time τ(k) for each mode of wavevector k is
(3)
where ρ is the density, R is the gas constant, T is the temperature, Me is the average molecular mass between entanglements (for polystyrene, Me ≈ 17 kg/mol), and τ(x) is a relaxation time spectrum of the form τ (x ) =
π 2kBT
and the model can be recast in
With regard to polystyrene melts, Majeste et al. gave a comprehensive review of the form of η(ω).a In the entangled regime, they summed up the contributions of different relaxation mechanisms. However, for low shear frequencies, the following term accounting for reptation motion prevails:
∫0
ζ0REE 2Ne 2N 2
where ζ0 is the monomeric friction coefficient, REE is the endto-end distance, N is the degree of polymerization, Ne is the average number of repeat units between entanglements, kB is Boltzmann’s constant, and ν is an adjustable parameter. In the end, eq 3 represents a Maxwell model with a continuous relaxation spectrum, accounting for the different relaxation times in the reptation process as well as tube length fluctuations. If ν = 0, it corresponds to a pure reptation. In this case, we have a simple relation between the zero-shear viscosity η0 and the terminal time τC, in the limit ω → 0:
28
ρRT Me
(5)
and where τC is the terminal relaxation time, given by
where σt is the shear stress and γ̇ the strain rate. Transposed in the time domain, the stress would be a convolution of the history of the strain rate with a viscosity kernel. For example, the shear behavior can be modeled by a spring of elasticity G put serially with a dashpot of viscosity η0. For a stress σt, the total strain is the sum of the elastic strain σt/G plus the viscous strain −σt/iωη0 (i is the imaginary unit). If we note τ = η0/G the relaxation time, we get σt/γ = (−iωη0)/(1 − iωτ). In terms of strain rates, we finally get the Maxwell model: η0 η(ω) = (2) 1 − iωτ
η(ω) =
2 ν ⎞ 2ν ⎟ τC for
