pubs.acs.org/Langmuir © 2009 American Chemical Society
Molecular Dynamics Simulations of Nanoimprinting Lithography Jan-Michael Y. Carrillo and Andrey V. Dobrynin* Polymer Program, Institute of Materials Science and Department of Physics, University of Connecticut, Storrs, Connecticut 06269 Received May 29, 2009. Revised Manuscript Received August 4, 2009 We have performed coarse-grained molecular dynamics simulations of molding and replication of nanometer-size objects. Nanoimprinting of hemispherical particles was modeled as a three-step process: (1) a mold was created by pressing a hard hemispherical particle (master template) into a polymeric film; (2) a polymeric film was cross-linked fixing a negative image of a master mold into a polymeric film; (3) a polymeric mold was pressed into a monomeric liquid replicating an original master. The quality of the replication process was analyzed by comparing the shape of the replica with the shape of the master. It is shown that deformation of a polymeric stamp during the replication process (Step 3) is a result of optimization of the surface energy of the mold-liquid interface and the elastic energy of the polymeric mold. The relative deformation, ε, of the replica is a function of the universal parameter γ/(GR0), where γ is the surface energy of the polymer-liquid interface, G is the shear modulus of the polymer network, and R0 is the radius of the master. In the case of small deformations, this function reduces to ε γ/(GR0).
1. Introduction Nanofabrication is a set of techniques that allows fabrication of functional structures with nanoscale pattern sizes.1-10 These methods are traditionally classified either as “bottom up” or “top down” approaches. The “bottom up” approach utilizes the interactions between polymers, colloidal particles, or surfactants to assemble two- or three-dimensional structures. The “top down” approach uses molding, embossing, or various methods of lithography for the processing of patterned features with lateral and vertical dimensions of e100 nm. The commercial implementation of the nanofabrication includes production of the integrated circuits, information storage devices, miniaturized sensors, microfluidic devices, photonic bandgap crystals, imaging agents, and drug-delivery vectors.7,8,10,11 During the past decade, soft lithography1,5,11-15 has emerge as an important tool for low cost pattern replication on the micrometer and nanometer scales. In soft lithography, a master is *Corresponding author. (1) Euliss, L. E.; DuPont, J. A.; Gratton, S.; DeSimone, J. Chem. Soc. Rev. 2006, 35, 1095–1104. (2) Lu, Y.; Chen, S. C. Adv. Drug Delivery Rev. 2004, 56(11), 1621–1633. (3) Gadegaard, N.; Mosler, S.; Larsen, N. B. Macromol. Mater. Eng. 2003, 288 (1), 76–83. (4) Bruder, J. M.; Monu, N. C.; Harrison, M. W.; Hoffman-Kim, D. Langmuir 2006, 22(20), 8266–8270. (5) Gates, B. D.; Xu, Q. B.; Stewart, M.; Ryan, D.; Willson, C. G.; Whitesides, G. M. Chem. Rev. 2005, 105, 1171–1196. (6) Geissler, M.; Xia, Y. N. Adv. Mater. 2004, 16(15), 1249–1269. (7) Chou, S. Y.; Krauss, P. R.; Renstrom, P. J. Appl. Phys. Lett. 1995, 67(21), 3114–3116. (8) Chou, S. Y.; Krauss, P. R.; Renstrom, P. J. Science 1996, 272(5258), 85–87. (9) Guo, L. J. J. Phys. D: Appl. Phys. 2004, 37(11), R123–R141. (10) Bratton, D.; Yang, D.; Dai, J. Y.; Ober, C. K. Polym. Adv. Technol. 2006, 17 (2), 94–103. (11) Xia, Y. N.; Whitesides, G. M. Annu. Rev. Mater. Sci. 1998, 28, 153–184. (12) Zhao, X. M.; Xia, Y. N.; Whitesides, G. M. J. Mater. Chem. 1997, 7(7), 1069–1074. (13) Barbero, D. R.; Saifullah, M. S. M.; Hoffmann, P.; Mathieu, H. J.; Anderson, D.; Jones, G. A. C.; Welland, M. E.; Steiner, U. Adv. Funct. Mater. 2007, 17, 2419–2425. (14) Rolland, J. P.; Maynor, B. W.; Euliss, L. E.; Exner, A. E.; Denison, G. M.; DeSimone, J. M. J. Am. Chem. Soc. 2005, 127(28), 10096–10100. (15) Maynor, B. W.; Larue, I.; Hu, Z.; Rolland, J. P.; Pandya, A.; Fu, Q.; Liu, J.; Spontak, R. J.; Sheiko, S. S.; Samulski, R. J.; Samulski, E. T.; DeSimone, J. M. Small 2007, 3(5), 845–849.
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molded and replicated using an elastomeric material. The traditional materials used for this technique are poly(dimethylsiloxane) (PDMS) and perfluoropolyether-based elastomers (PFPEs). These polymers have such low surface energies (γPFPE = 12 16 mJ/m2 and γPDMS = 23 mJ/m2) that enable replication of harvestable, scam-free objects or nanoparticles.1,5,11,12,14 Soft lithography starts with the preparation of a master template, typically an etched silicon wafer formed by using advanced lithographic techniques, which is pressed into a photocurable liquid PFPE or PDMS film.1,5,6 Once the liquid precursor has completely wetted the master template, it is photochemically cross-linked and subsequently peeled away to generate a precise mold with nanoscale features. This mold is then used to replicate features from a variety of materials. This lithography technique can replicate pattern features as small as 5 nm with aspect ratios of 20 (height to width). It was shown that this technique is also successful in replicating the naturally occurring supramolecular objects, such as micelles and viruses.15 While there is a substantial amount of experimental data available,1,5,6 the theoretical and computational studies of the soft lithography process are lagging behind.16 It is understood that minimization of the surface energies is one of the necessary requirements that have to be satisfied, in order to improve the method resolution limit.1,5,6 However, when dealing with a soft object such as elastomeric molds and nanoparticles, the effect of the interfacial interactions is more complex, as it changes both adhesion and shape, i.e., contact areas. For soft objects, the Young’s relation17 determining the force balance on the interface no longer holds. The ability of elastomers to change their shapes requires simultaneous optimization of the surface and elastic energies to achieve equilibrium.17-19 This phenomenon is called “elastic wetting”.17 The concept of the elastic wetting was applied (16) Yu-Su, S. Y.; Thomas, D. R.; Alford, J. E.; Larue, I.; Pitsikalis, M.; Hadjichristidis, N.; DeSimone, J. M.; Dobrynin, A. V.; Sheiko, S. S. Langmuir 2008, 24(21), 12671–12679. (17) de Gennes, P.-G.; Brochard-Wyart, F.; Quere, D. Capillarity and Wetting Phenomena; Springer: New York, 2002. (18) Lau, A. W. C.; Portigliatti, M.; Raphael, E.; Leger, L. Europhys. Lett. 2002, 60(5), 717–723. (19) Joanny, J.-F.; Johner, A.; Vilgis, T. A. Eur. Phys. J. E 2001, 6, 201–209.
Published on Web 08/24/2009
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by Yu-Su et al.16 to rationalize the data on block copolymer micelle replication (molding). Using a scaling analysis of the elastic and surface energy contributions to the system free energy, they calculated a diagram of molding and transfer (mold-particle separation) conditions for rubber particles as a function of the substrate surface energy γs and particle deformation, ε. The surprising finding of this study was the existence of the narrow envelop of parameters where the replication (molding) is possible. We will expand on this work and further study how both interfacial and elastic properties of materials involved in the soft lithography process influence the shape reproducibility. To achieve this goal we have performed coarse-grained molecular dynamics simulations20 of the nanoimprinting lithography. The molecular dynamics simulations consisted of three separate simulation steps. During the first step, a mold (stamp) was created by pressing a hard hemispherical particle (master template) into a polymeric film. Then the polymeric film was cross-linked and relaxed fixing a negative image of the master template into the polymeric film. The prepared polymeric mold was pressed into a monomeric liquid to reproduce and replicate the original master. Molecular dynamics simulations were performed at temperatures above the glass transition temperature for polymers. Hence only cross-links maintained the mold structure. To establish an effect of the cross-link density on the replica deformation, the shape of the replica was compared with the master mold at different cross-link densities and master mold sizes. In addition, we compared replicas obtained by using monomeric liquids that are both poor and good solvents for the polymer networks. Analysis of our simulation data shows that the final shape of a replica is the result of optimization of the surface energy of the mold-liquid interface and the elastic energy of the polymeric stamp. It turns out that there is a simple relationship that couples the elastic deformation of the mold-replica interface with interface surface energy. This relation generalizes the Laplace equation17 for the stability of the curved interface as a function of the excess pressure, the interfacial energy, and the interface radius of curvature to the case of the deformable elastic objects.
2. Molecular Dynamics Simulations of Nanoimprinting Lithography We have implemented a three-step coarse-grained molecular dynamic simulation procedure to model nanoimprinting of hemispherical nanoparticles. The first step involved the creation of the mold by pressing a rigid hemispherical nanoparticle, which was made of overlapping beads with diameter σ and was attached to a planar substrate, into a polymeric thin film consisting of Nch chains with a degree of polymerization of N = 32 (see Figure 1 a). The top and the bottom substrates were modeled by a periodic, hexagonally packed lattice of spheres with diameter σ. The system was periodic in the x-y directions. The radius of the hemispherical nanoparticle, R0, and the system sizes are listed in Table 1. In our coarse-grained molecular dynamics simulations all particles in the system interacted through the truncated-shifted Lennard-Jones (LJ) potential:20 8 >
:
!12
σ rij
!6 3 σ 12 σ 65 rercut þ rcut rcut 0 r > rcut ð1Þ
(20) Frenkel, D.; Smit, B. Understanding Molecular Simulations; Academic Press: New York, 2002.
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Figure 1. Mold and replica creation. (a) A substrate with attached rigid hemispherical nanoparticle is pressed at a constant force into a polymeric film consisting of polymer chains with a degree of polymerization Np = 32. (b) A polymeric film is crosslinked at an average cross-link density, Fc, and annealed to relax cross-link bonds. (c) A cross-linked mold is pressed into a monomeric film at a constant force replicating a hemispherical nanoparticle. Table 1. System Sizes R0 [σ]
Nch
Lx [σ]
Ly [σ]
22.60 18.47 14.33 11.30
5632 3104 1472 696
34.00 28.00 22.00 17.00
33.77 27.71 21.65 16.45
Table 2. LJ-Interaction Parameters mold-mold
mold-liquid
liquid-liquid
system
εLJ [kBT]
rcut [σ]
εLJ [kBT]
rcut [σ]
εLJ [kBT]
rcut [σ]
1 2 3
1.5 1.5 1.5
2.5 2.5 2.5
1.0 1.0 1.0
2.01/6 2.01/6 2.01/6
1.5 1.0 0.5
2.5 2.5 2.5
where rij is the distance between ith and jth beads, and σ is the bead diameter chosen to be the same regardless of the bead type. The cutoff distance, rcut = 2.5σ, was chosen for polymer-polymer and polymer-bottom substrate interactions, and rcut = 21/6σ for all other pairwise interactions. The value of the LJ interaction parameter for the polymer-bottom substrate pair was set to 1.5 kBT. Table 2 summarizes the interaction parameters between the remaining interacting pairs used in simulations. The connectivity of beads in polymer chains and the cross-link bonds were modeled by the finite extension nonlinear elastic (FENE) potential:20 1 r2 UFENE ðrÞ ¼ - kspring Rmax 2 ln 1 2 Rmax 2
! ð2Þ
with the spring constant kspring = 30 kBT/σ2 and the maximum bond length Rmax = 1.5σ. Simulations were carried out in a constant number of particles and temperature ensemble. The constant temperature was maintained by coupling the system to a Langevin thermostat implemented in LAMMPS.21 In this case, the equation of motion of ith particle is
m
R dνBi ðtÞ ¼ FBi ðtÞ -ξνBi ðtÞþFBi ðtÞ dt
ð3Þ
where νBi(t) is the ith bead velocity, and FBi(t) is the net deterministic force acting on the ith bead with mass m. FBR i (t) is the (21) Plimpton, S.J. J. Comp. Phys. 1995, 117, 1–19. http://lammps.sandia.gov.
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stochastic force with zero average value ÆFBR i (t)æ = 0 and R 0 0 (t)F B (t )æ = 6ξk δ-functional correlations ÆFBR i i BTδ(t - t ). The friction coefficient ξ was set to ξ = m/τLJ, where τLJ is the standard LJ time τLJ = σ(m/εLJ)1/2. The velocity-Verlet algorithm20,21 with a time step of Δt = 0.005τLJ was used for integration of the equations of motion (eq 3). All simulations were performed using LAMMPS. To create a negative master image, the top substrate was pressed into a polymeric film with a constant force 1.0 kBT/σ acting on each bead forming a top substrate. The simulation run was continued for 5000τLJ steps, which was well above the time needed for the volume of the system to equilibrate. This initial process is illustrated in Figure 1a. The second step involved cross-linking of a polymeric film in the presence of the master. During this procedure, neighboring monomers were randomly cross-linked by FENE bonds if they fell within a 1.5σ cutoff distance from each other. One cross-link bond was allowed per each monomer, thus creating a randomly cross-linked network. The elastic properties of the network were varied by changing the number of cross-links per simulation cell. After completion of the cross-linking process, a system was relaxed by performing NVT molecular dynamics simulation runs lasting 104 τLJ (see Figure 1b). The third simulation run was initiated with pressing a cross-linked mold into a monomeric liquid at a constant force of 1.0 kBT/σ acting on each bead forming a top substrate. The simulation continued for 2.5 104 τLJ steps and the last 5000 τLJ steps were used for the production runs to obtain system averages. Figure 1c illustrates the setup for the final step of the replication process. We have performed simulations of three different systems (see Table 2) to elucidate the effect of the LJ interactions on the master reproducibility. In order to obtain a shear modulus of the cross-linked network forming a polymeric mold, we have performed a set of simulations of the three-dimensional (3-D) periodic networks with the same density of cross-links per unit volume and the same monomer density as in our simulations of the molding process. The networks were uniaxially deformed by changing√the initial box size L0 along the x direction to λL0 and to L0/ λ in the y and z directions.22-25 Under such deformation, the system volume remained the same. This deformation was achieved by a series of small affine deformations {xi, yi, zi} f {Δλxi, yi/(Δλ)1/2, zi/(Δλ)1/2} until the desired strain was obtained. The molecular dynamics simulation proceeded during the constant-volume deformation process such that the network was allowed to adjust its conformations. The stress σx in the direction of the strain was evaluated from the simulations through the pressure tensor Pij as follows:22-24 3 1X σx ¼ Pxx Pii 2 2 i
ð4Þ
Figure 2 shows the simulation results for the stress deformation curves of the deformed networks. The network modulus was obtained by fitting the network stress to the stress-deformation function of the uniaxially deformed network σx = G(λ2 - λ-1).26 The results of this fitting procedure are summarized in Figure 3. (22) Grest, G. S.; Putz, M.; Everaers, R.; Kremer, K. J. Non-Cryst. Solids 2000, 274(1-3), 139–146. (23) Rottach, D. R.; Curro, J. G.; Budzien, J.; Grest, G. S.; Svaneborg, C.; Everaers, R. Macromolecules 2007, 40(1), 131–139. (24) Svaneborg, C.; Grest, G. S.; Everaers, R. Polymer 2005, 46(12), 4283–4295. (25) Rottach, D. R.; Curro, J. G.; Budzien, J.; Grest, G. S.; Svaneborg, C.; Everaers, R. Macromolecules 2006, 39(16), 5521–5530. (26) Rubinstein, M.; Colby, R. H. Polymer Physics; Oxford University Press: New York, 2003.
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Figure 2. Dependence of the stress σx on λ2 - 1/λ, where λ is the relative elongation along the x-axis, for affinely deformed networks with different cross-link densities Fc.
Figure 3. Dependence of the network shear modulus G on the cross-link density Fc. The dashed line corresponds to the equation Gσ3/kBT = a(Fcσ3)2 þ bFcσ3 with the numerical coefficients a = 3.98 and b = 0.38.
As one can see, there are two distinct regimes in the modulus dependence on the cross-link density. At low cross-link densities, the network modulus is proportional to kBTFc, where Fc is the density of cross-links. However, as the number of cross-links increases, the cross-linking bonds themselves start supporting the network stress. In this regime of large cross-link densities, the network modulus G scales quadratically with the density of crosslinks (see Figure 3). There are two possible explanations of this quadratic dependence. This quadratic dependence of the network modulus on the cross-link density could be a crossover regime between tetrafunctional and trifunctional network behavior.26 In the case of the low cross-link density each cross-link bond can be effectively considered as a tetrafunctional junction point. However, when the number of bonds between cross-links becomes on the order of unity, each cross-linking bond forms two trifunctional junctions, changing the network functionality to three. Another possible explanation is that this could be a nonaffine deformation regime for which the network shear modulus is known to have a quadratic dependence on the monomer density. This is usually the case when the distance between cross-links is on the order of the persistence length of the network forming chains.27 For our (27) MacKintosh, F. C. Elasticity and dynamics of cytoskeletal filaments and their networks. In Soft Condensed Matter Physics in Molecular and Cell Biology; Poon, W. C. K., Andelman, D., Eds.; Taylor & Francis: New York, 2006; pp 139-155.
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Figure 4. Dependence of surface energy γ on the average system density F. Filled squares correspond to System 1 (see Table 2), open circles represent System 2, and half-filled circles denote System 3.
system, the chain’s persistence length is on the order of the bond length, thus the crossover to the nonaffine deformation regime could occur when the number of bonds between cross-links becomes on the order of unity. Unfortunately, the detailed study of the elastic properties of the network in this regime is beyond the scope of this paper and will be addressed in a future publication. The surface energy of the mold-liquid interface was evaluated by performing molecular dynamics simulations of the 3-D periodic system of two phase-separated monomeric liquids with the same set of LJ interaction parameters as our original systems. For these simulations, the box size was 8.38σ 8.38σ Lz and the number of particles was equal to 2000. The size of the simulation box along the z direction, Lz, was varied between 40σ and 30σ to obtain the dependence of the interfacial energy on the average system density F. The surface energy then was obtained by integrating the difference of the normal PN(z) and tangential PT(z) to the interface components of the pressure tensor.28-30 Note that, in our simulations, the z direction was normal to the interface. γ¼
1 2
Z
Lz =2 -Lz =2
ðPN ðzÞ -PT ðzÞÞdz
ð5Þ
In eq 5, the factor 1/2 accounts for the presence of two interfaces per simulation box. Figure 4 summarizes the results for dependence of the surface energy on the average component densities and different values of the interaction parameters used in our simulations.
3. Results and Discussion In Figures 5 we showed the variations of the shape change of the replica in comparison with the shape of the original mold. This figure shows that the difference between the original mold with the master and the replica increases as the mold modulus decreases and at the same time the size of the master decreases as well. The large mold modulus restricts large displacement of the polymeric strands forming a mold network to optimize the number of contacts between the mold and the mold-filling monomeric liquid. In order to quantify the errors during the molding process in Figure 6, we plot the dependence of the difference of the replica height and the height of the original mold normalized by the size of the master, R0. (28) Simmons, V.; Hubbard, J. B. J. Chem. Phys. 2004, 120(6), 2893–2900. (29) Buhn, J. B.; Bopp, P. A.; Hampe, M. J. Fluid Phase Equilib. 2004, 224(2), 221–230. (30) Buhn, J. B.; Bopp, P. A.; Hampe, M. J. J. Mol. Liq. 2006, 125(2-3), 187– 196.
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As expected, the error in the replication process decreases as the cross-link density (value of the modulus) increases. One can identify two different scaling regimes in dependence of the relative replica deformation Δhmax/R0 on the mold cross-link density, which corresponds to the regimes of small and large mold deformations. Let us now establish the effect of the liquid-mold interactions on the replica quality. Figure 7 shows the snapshots of the molecular dynamics simulations of the replica shape obtained during the final step of the replication process. In the case when a monomeric liquid is a poor solvent for the mold network, there is no liquid protrusions inside the network. This is true for any cross-link densities considered in our simulations. However, when the interaction parameters for the mold-liquid interactions are selected in such a way that monomeric liquid is a good solvent for the mold network, the monomeric liquid swells the mold. The swelling is significant for weakly cross-linked mold, but becomes less pronounced as the cross-link density increases. For the highest cross-link density, the shape of the mold remains almost unperturbed. Thus, there are two factors that determine the quality of the replica: the liquid-mold interfacial energy and the mold elastic modulus. Below we will present a simple scaling model, which accounts for both these effects. Consider the deformation of the master mold after injection of the monomeric liquid. In our simulations, the monomeric liquid was injected into the mold at a constant pressure, which is generated by a constant force acting on each bead of the top substrate and pointing toward the bottom substrate. Since the magnitude of this force is the same as that during the mold preparation process, at equilibrium, the volume of the liquid filling the stamp cavity is equal to the volume of the original hemisphere, and the deformation of the mold occurs at a constant volume. To simplify calculations we will represent a deformable cavity by a barrel (see Figure 8). Let us assume that λ is the deformation parameter of the height of the master mold (barrel) interface along the z-axis, λ = h/R0 = (1 - Δhmax/R0) and the mold shape deformation occurs at a constant volume such that the radius, √ R, of the deformed mold in the xy-plane is equal to R = R0/ λ (see Figure 8). Upon liquid injection, the surface area of the master mold-liquid interface changes. This change in the interfacial energy generates a stress in the mold network. Thus, a new shape of the master mold is obtained by optimizing surface and elastic energies. The elastic energy of a uniaxially deformed cavity in a polymeric network can be estimated as ! 2 1 3 λ ð6Þ þ Felast GR0 2 λ where G is the shear modulus of the mold network. The change in surface energy of uniaxially deformed cavity is pffiffiffi Fsurf γAþγhRγAþγR0 2 λ ð7Þ where γ is the surface energy of the mold-liquid interface, and A is the area of our simulation box. In evaluating a surface energy contribution we have approximated a shape of deformed master mold by a barrel with height h and radius R (see Figure 8). Combining elastic and surface energy contributions together, we obtain the total free energy of the deformed master mold. ! 2 pffiffiffi 1 3 λ þ þγR0 2 λ ð8Þ F GR0 2 λ DOI: 10.1021/la9019266
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Figure 5. Evolution of the replica shape (open circles) in comparison with the shape of the original master (filled circles) as a function of the master size, R0, and the elastic modulus, G, of the mold network.
Figure 7. Snapshots of the replica shapes for different cross-link densities and mold-liquid affinity. The choice of the parameters for the systems shown in the first row corresponds to poor solvent conditions for the mold network-liquid interactions. The second row represents the case of a miscible mold network-liquid system. Figure 6. Dependence of the relative deformation Δhmax/R0 of the
replica height on the cross-link density Fc for molds with radii R0 = 11.30σ (circles), R0 = 14.33σ (hexagons), R0 = 18.47σ (triangles), and R0 = 22.6σ (rhombs). Inset shows the definition of the length scales.
The equilibrium value of the deformation parameter of the mold boundary is obtained by minimizing eq 8 with respect to 13248 DOI: 10.1021/la9019266
parameter λ. This results in the following expression describing mold deformation: λ -3=2 -λ3=2 ≈
γ GR0
ð9Þ
Note that eq 9 provides a scaling relation between system parameters neglecting numerical coefficients. It follows from Langmuir 2009, 25(22), 13244–13249
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process is G Figure 8. Schematic representation of cavity deformation.
Δhmax γ R0 R0
ð10Þ
This equation can be viewed as a generalization of the Laplace equation17 to the case of the elastic interface. The left-hand-side of the equation represents the elastic force per unit area generated by the mold deformation. At equilibrium, this force is balanced by the capillary force at the liquid-mold interface (see right hand side of eq 10).
4. Conclusions
Figure 9. Dependence of the replica deformation Δhmax/R0 on the parameter γ/GR0 for the systems with different cross-link densities Fc, LJ interaction parameters and mold radii R0 = 11.30σ (circles), R0 =14.33σ (hexagons), R0 =18.47σ (triangles), and R0 =22.6σ (rhombs). Filled symbols are for System 1, open symbols are for System 2, and half-filled symbols for System 3 in Table 2. Solid line is the best fit to the equation cγ ~/GR0 = (1 Δhmax/R0)-3/2 - (1 - Δhmax/R0)3/2 with numerical coefficient c~ = 1.33.
the last equation that the mold deformation increases with increasing the surface energy of the network-liquid interface, γ, but it decreases as the network becomes harder (shear modulus G of the network increases). In Figure 9, we combined our results for replica shape deformations for systems with different degrees of cross-linking, master sizes, and different values of the interaction parameters. As one can see the data points collapse into one universal curve given by the eq 9. The values of the network shear modulus G and the surface energy of the network-liquid interface for this plot were obtained from Figures 3 and 4. This confirms our conjecture that the shape of the replica is controlled by the master mold elasticity and the surface energy of the mold-replica interface. It is important to point out that the problem of deformation of the mold interface is similar to the problem of void formation in polymeric adhesives.31 In the limit of small deformations, the analysis of eq 9 shows that the stress produced in the mold during the replication (31) Shull, K. R.; Creton, C. J. Polym. Sci., Part B: Polym. Phys. 2004, 42(22), 4023–4043.
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We have presented results of the molecular dynamics simulations of the nanoimprinting lithography. Our simulations showed that the quality of the replica obtained during the reproduction process depends on the mold elastic modulus, G, the interfacial energy of the injected into the mold liquid, γ, and on the size of the master, R0 (see eqs 9 and 10). Thus, in order to achieve the desired replica quality, one has to optimize all these parameters simultaneously. We can also look at the eq 10 as a condition determining a quality of the replica. Let us assume that it is required to reproduce features of the original (master) with accuracy (replica shape deformation, ε = Δhmax/R0) better than 10%. In this case, the ratio of the parameters γ/ GR0 should be smaller than 0.1, γ/GR0 < 0.1. This imposes conditions for the feature size R0 that can be reproduced by the molding process using materials with fixed γ and elastic modulus, G. For example, for polymeric networks with Young’s modulus E = 10 MPa, in order to reproduce features with size R0 = 20 nm, it will require one to use liquids that have surface energies with polymeric network γ e 20 mN/m2. These calculations provided an estimate for the surface-energy range, which allows molding and replication of objects with desired accuracy. The range depends on the particle size, shape, and mechanical properties. (By doubling the feature size, R0 = 40 nm, the threshold surface energy will also double in value.) We can generalize our findings by providing the following simple rules for nanoimprinting lithography: (1) the mold deformation (error in the replica shape, ε = Δhmax/ R0) is inversely proportional to the shear modulus of the polymeric stamp (mold) (harder stamps are more difficult to deform); (2) the error in the replica shape increases as the surface energy of the mold-liquid interface increases (the deformation of the sample leads to decrease of the mold-liquid contact area); (3) the deformation (error) of the replica is also inversely proportional to the size of the master, which indicates that one will need a larger value of the mold shear modulus, G, to achieve the same quality for reproduction of the smaller features. We hope that our computer simulations will inspire experimental studies in this direction.
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