Polycarbonate as an Elasto-Plastic Material Model for Simulation of the Microstructure Hot Imprint Process

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1 Sensors 013, 13, ; doi: /s OPEN ACCESS sensors ISSN mdpi.com/jornal/sensors Article Polcarbonate as an Elasto-Plastic Material Model for Simlation of the Microstrctre Hot Imprint Process Birtė Narijaskaitė 1, Ardas Paleičis 1, Rimdas Gaids, Giedris Janšas 1, * and Rokas Šakals 1 1 International Stdies Centre, Kanas Uniersit of Technolog, A. Mickeičias Gatė 37, Kanas 4444, Lithania; s: birte.narijaskaite@kt.lt (B.N.); ardas.paleicis@kt.lt (A.P.); rokas.sakals@kt.lt (R.Š.) Department of Engineering Graphics, Kanas Uniersit of Technolog, Kęstčio Gatė 7, Kanas 4405, Lithania; rimdas.gaids@kt.lt * Athor to hom correspondence shold be addressed; giedris.jansas@kt.lt; Tel./Fa: Receied: 14 Jne 013; in reised form: 13 Agst 013 / Accepted: 15 Agst 013 / Pblished: Agst 013 Abstract: The thermal imprint process of polmer micro-patterning is idel applied in areas sch as manfactring of optical parts, solar energ, bio-mechanical deices and chemical chips. Polcarbonate (PC), as an amorphos polmer, is often sed in thermoforming processes becase of its good replication characteristics. In order to obtain replicas of the best qalit, the imprint parameters (e.g., pressre, temperatre, time, etc.) mst be determined. Therefore finite element model of the hot imprint process of lamellar periodical microstrctre into PC has been created sing COMSOL Mltiphsics. The mathematical model of the hot imprint process incldes three steps: heating, imprinting and demolding. The material properties of amorphos PC strongl depend on the imprint temperatre and loading pressre. Polcarbonate as modelled as an elasto-plastic material, since it as analed belo the glass transition temperatre. The hot imprint model as soled sing the heat transfer and the solid stress-strain application modes ith thermal contact problem beteen the mold and polcarbonate. It as sed for the ealation of temperatre and stress distribtions in the polcarbonate dring the hot imprint process. The qalit of the replica, b means of lands filling ratio, as determined as ell.

2 Sensors 013, Keords: elasto-plastic; hot imprint; finite element method; polcarbonate 1. Introdction Noadas microstrctres hae a er ide range of applications. The are sed for beam shaping, splitting and steering [1], in optical interconnections [], optical teeers [3], mltiphoton spectroscop [4], lithographic fabrication of photonic crstals [5], object contoring [6], biological microscop [7], measrement of moing object [8], characteriation of micro optical elements, sstems ith CD and DVD [9], in X-ra microscop [10], etc. Therefore the qalit of prodced microstrctres is er important. E-beam lithograph (EBL) is the most commonl sed techniqe for micro- or nanolithograph. The sage of EBL for pattering offers man adantages, hich proide an ideal lithographic platform for the MEMS (NEMS) fabrication. On the other hand, this technolog is better sited for prototping, bt not for mass prodction [11]. In mass prodction ell-knon conentional technologies, like injection molding, injection compression molding and hot embossing are sed. The are etensiel emploed in micro scale replication. Hot embossing is the most poplar manfactring process, and is ell sited for prodcing dedicated microstrctres ith high aspect ratios and small distortions [1,13]. Hot embossing has seeral adantages oer other replication processes, inclding relatiel lo costs for embossing tools, a simple process, and high replication accrac for small featres. Hoeer man scientists highlight the folloing hot imprint process problems: filling ratio of microstrctre [14]; non-niform mold imprint [15]; adhesion beteen mold and polmer [16]; srface roghness [17]; long ccle time [18]. Hoeer, it is not enogh to optimie the parameters of hot imprint process. It is also necessar to take into accont the different technological eqipment, materials, etc. This reqires ne methods and areas for improement in order to achiee better qalit of replicas. The reasons for these problems can be detected sing nmerical simlation. The literatre analsis shos that there is not enogh information abot polcarbonate material behaior in mechanical hot imprint processes. The imprint process inoles finite temperatre-dependent deformation of a thermoplastic material. As a reslt, it is costl to establish eperimentall the process behaior for these materials, leading to a critical need for improed simlation capabilities. The material behaior is highl sensitie to ariations of temperatre and strain rate. Frthermore, nmerical simlation is essential to psh the limits of hot imprinting to smaller length scales here high precision is critical. The polmethl methacrlate (PMMA) is most nmericall analed at different temperatres and state of the material. Therefore, an elastoplastic polcarbonate material model is designed and analed b the finite element method in this paper. It is necessar to inestigate the stress and strain state in each step of the process. The filling ratio of the polmer in the mechanical hot imprint process is stdied based on nmerical simlation and

3 Sensors 013, eperimental research. Heating, imprinting, and demolding steps in the hot imprint process are inestigated in detail b a nmerical simlation std. Therefore the aim of this paper is to create finite element model (FEM) of a nickel mold hot imprint into polcarbonate, hich corresponds to the eperiment conditions.. Finite Element Model of Mechanical Hot Imprint Process The finite element model of the nickel mold hot imprint process into polcarbonate near the glass transition temperatre, hich is created ith COMSOL Mltiphsics 3.5a, is presented in this section. Man scientists diide the mechanical hot imprint process into for steps: heating, imprint, cooling and demolding. Hoeer e are making assmption that after the imprint step the temperatre of polmer decreases er qickl. Therefore the cooling step as not analed separatel. The scheme of the modified hot imprint process, hich consists of three steps: heating, imprinting and demolding, is presented in Figre 1. Figre 1. Diagram of the mechanical hot imprint process. Temperatre & Pressre Pressre Temperatre Heating Press Demolding Time Figre. The hot imprint process modeling. Mechanical design Materials science Mathematical formlation and of the problem into to copled initial bondar ale problems Heat transfer Contact Mechanical bondar ale problem Thermal bondar ale problem Formlation of the finite element model based on these to bondar ale problems and their copling Mechanical FEM formlation Elasto-plastic FEM Copling FEM formlation -D FEM model HOT IMPRINT Thermal FEM formlation RESULTS

4 Sensors 013, The modeling and simlation methodolog b FEM, hich incldes geometrical modeling, bondar conditions, meshing, material properties, process conditions and goerning eqations is schematicall presented in Figre. The eqations of motion, thermal balance, material properties and material deformation ere sed in order to calclate the stress, strain, and temperatre fields, the mold pressre distribtion and filling ratio in each step of hot imprint process. The Green-Lagrange strain-displacement Eqation (1) describes the basic relations beteen large displacements and strain: = = γ γ γ ε ε ε ε (1) Here ε, ε, ε, γ, γ, γ are linear strains in the,, direction and shear strains in the,, plane, hile,, are displacements in the,, direction. Heat transfer condctiit is described according to the formla: ( ) q T k t T T c T p = ) ( ) ( ρ () here k is thermal condctiit, ρ is densit, c p is heat capacit, T is temperatre, q is rate of the heat generation. COMSOL Mltiphsics soles contact problems sing an agmented Lagrangian method. This method is a combination of the penalt and Lagrange mltiplier methods. This means a penalt method ith penetration control. The sstem is soled b iteration from the determined displacement. These displacements cased b incremental loading, are stored and sed to deform the strctre to its crrent geometr. If the gap distance beteen the slae and master bondaries at a gien eqilibrim iteration is becoming negatie (i.e., the master bondar is penetrating the slae bondar), the ser defined normal penalt factor p n is agmented ith Lagrange mltipliers for contact pressre T n : = otherise e T g if g p T T n n T g p n n n np 0 (3) here g is the gap distance ariable beteen slae and master bondar [19,0]. Generall it is impractical to se FEM to anale periodical micrometer-scale patterning of the mold, bt if the cross-sectional shape of the mold is constant in one direction, as in Figre 3, to-dimensional stress analsis is possible. Moreoer, if the pattern of the mold is reglar and

5 Sensors 013, smmetric, e can assme a to-dimensional plane strain model of a nit cell (Figre 3) b taking smmetric bondar conditions into accont. Figre 3. Cross-sectional shape of (a) the mold and sbstrate, and (b) the analed section. (a) (b) A to-dimensional (-D) FEM model of a nickel mold (lamellar profile ith period of 4 µm) and polcarbonate sbstrate ith bondar conditions is presented in Figre 4, here depth of the mold (h m ) is 100 nm, thickness of the polcarbonate (h p ) 3 mm, half idth of the land (W) 1 µm, half idth of the ridge (S) 1 µm. The mold is made of nickel allo, hich is a more rigid material than polcarbonate, so it as assmed in the FE simlation that the mold has rigid contact srfaces. Smmetric bondaries ere sed on the left and right sides of the model. A smmetric bondar indicates that displacement and temperatre gradients across the bondar are ero. Fied displacement and temperatre are applied at the bottom srface of the sbstrate. The initial temperatre of the mold and polcarbonate is 93 K. Dring the preheating step the mold s temperatre as defined as linear fnction T = f(t,t) of mold s heating temperatre T (Kelin) and mold displacements t (meters). When the temperatre reaches the maimm ale (41 K), it remains stable dring all frther steps. The air inflence beteen mold and polcarbonate is neglected. Also in order to improe the conergence of the simlation and aoid stress concentrations small radis arcs ere implemented in the A and B areas of the mold (Figre 4). The pressre is described as a linear fnction f(t) dependent on mold displacement t from 0 to m ith 10 8 m steps. Figre 4. Comptational scheme of the nickel mold hot imprint into polcarbonate. pressre smmetric in dir master W h m A B MOLD S T = f(t,t) smmetric in dir smmetric T smmetric in dir h p slae POLYCARBONATE smmetric T smmetric in dir fied in and dir T = 93 K

6 Sensors 013, The accrac and conergence of the soltion depends on the choice of mesh as ell. The mesh of the model, sing trianglar elements, is presented in Figre 5. The trianglar element is defined b si nodes, each haing three degrees of freedom: horiontal and ertical displacement, and temperatre. The Lagrange-Qadratic finite element tpe, hich is sed for -D modeling of solid strctres, as chosen. Meshes of the contact area and at the smmetrical regions are finer than in other areas ith smaller deformation. The model consists of 5,583 finite elements. It is recognied that interaction beteen strctral parts has a great inflence on the reslts for soling a mlti-field contact problem. Smoothing of contact edges proides a significant improement in conergence behaior and the refiner mesh allos one to define contact accratel. The refined mesh as sed arond arcs so that more points come onto contact at the same time. The arc of the mold s contact edge consists of 1 nodal points. The horiontal molding and PC contact edges ere discretised coarsel (mesh 3 times spare) and hae the same mesh densit. Comparisons of nmerical and theoretical reslts sho that the defined mesh ensres the necessar accrac of the soltion. Figre 5. Finite element mesh of the sstem of nickel mold and polcarbonate. Materials sed for the mold and sbstrate, and their properties are listed in Table 1. Nickel as sed as the mold, and it as assmed to be isotropic and linearl elastic, hile amorphos polcarbonate (glass transition temperatre 43 K) as sed as sbstrate. Table 1. Material properties of the mold and sbstrate. Mold Sbstrate Material Nickel Polcarbonate Densit, Kg/m ρ(t) Thermal condctiit, W/mK 90.9 K(T) Thermal epansion, 1/K Heat capacit, J/(Kg K) 445 c p (T) Elastic modls, GPa 00 E(T) Poisson s ratio 0.31 ν(t)

7 Sensors 013, Polcarbonate s densit, thermal condctiit, heat capacit, elastic modls and Poisson s ratio as fnctions of temperatre ere taken from COMSOL Mltiphsics 3.5a material librar (Figre 6). Figre 6. Thermal dependencies of polcarbonate material properties (COMSOL Mltiphsics 3.5a material librar). (a) Heat capacit (b) Yong modls (c) Thermal condctiit (d) Poisson s ratio (e) Densit In the polcarbonate dring hot imprint process large deformations are indced. The micro hot imprint process is being performed near the glass transition temperatre of polcarbonate, here it behaes like elasto-plastic material. The mold and sbstrate are restrained from moing in the direction. Therefore it is possible to se the D Plane Strain application mode, hich assmes, that the -component of the strain is ero. The total strain ector ε of polcarbonate material consists of thermal ε th and elastic ε el strain ectors so that: ε = eel ε th (4) The polcarbonate material model ith a nonlinear behaior is an elasto-plastic material, here the stress-strain relationship or the constittie eqation is: σ ( ε ε ε ) = Dε el = D p th (5) here D is the elasticit matri and the stress and the strain are both gien in colmn ector form: σ σ σ σ =, τ τ τ ε ε ε = γ γ γ ε (6)

8 Sensors 013, Thermal strain depends on the present temperatre T, the stress-free reference temperatre T ref, and the thermal epansion ector α ec : For elasto-plastic material: ε ε ε ( T T ) ε th = = α ec ref γ (7) γ γ th α α α α ec = 0 (8) 0 0 The polcarbonate as an elasto-plastic material that has ield criterion and hardening model settings. The ield criterion is interpreted as an eqialent stress σ e. When the eqialent stress is eqal to a material ield parameter σ Y the material ill deelop plastic strains. If σ e is less than σ Y, the material is elastic and the stresses ill deelop according to the elastic stress-strain relations. Eqialent stress can neer eceed the material ield since in this case plastic strains old deelop instantaneosl, thereb redcing the stress to the material ield. As a ield fnction the Von Mises fnction as chosen: σ 1 ( ) ( ) ( ) Y = σ1 σ σ σ 3 σ1 σ 3 (9) here σ 1, σ, σ 3 are principal stresses and σ Y is the eqialent stress [1]. The ield stress leel is gien b: ( ) ε ( T ) σy ( T ) = E T (10) here T present temperatre, E(T) Yong s modls fnction, ε (T) ield strain fnction. The polcarbonate E(T) fnction is chosen from the COMSOL Mltiphsics material librar, ε (T) fnction as created as a linear fnction according to reslts pblished in [,3]. Yield strain aries from 8% to 5%, hen the temperatre aries from 93 K to 417 K. The hardening model is a phenomenon here ield stress increases ith frther plastic strain. Isotropic hardening has been proposed to define the modification of the ield srface dring plastic deformation. Using this hardening model, it as assmed that the initial ield srface epands niforml ithot distortion and translation as plastic flo occrs. The isotropic tangent modls E Tiso = MPa [,3]. A Lagrangian formlation as sed for incremental general nonlinear analsis in COMSOL Mltiphsics. The model as soled sing heat transfer and the solid stress-strain application modes ith thermal contact problem beteen nickel mold and polcarbonate. This mltiphsical hot imprint model of polcarbonate incldes the heat transport, strctral mechanical stresses and strains reslting from the

9 Sensors 013, temperatre distribtion. It allos s to ealate temperatre distribtions and stresses in the polcarbonate dring the hot imprint process. 3. Reslts and Discssion As described in the preios section, the hot imprint process as diided into three steps: heating, imprinting and demolding. The initial temperatre of the mold and polcarbonate is 93 K, the same as the ambient temperatre and the imprint force is eqal to ero at the beginning of the heating step. In this step the temperatre of the mold increases p to the chosen 41 K and throgh the initial contact beteen mold ridges and polcarbonate sbstrate it as preheated (Figre 7). Dring the heating process, deformation of polcarbonate starts (Figre 8). Figre 7. Von Mises stress distribtion (color map, Pa) and temperatre fields (in Kelins) represented b lines in the polcarbonate after the heating process. Figre 8. Distribtion of total displacements (in meters) after heating process.

10 Sensors 013, Polcarbonate is elastic, de to this the polmer from the contact area moes to the empt cait of the mold. The cait of the mold is partiall filled ith heated polcarbonate. After the heating a stead ariation of temperatre in the range from 95 K (in the bottom of polcarbonate) to 413 K (in the place of contact ith the mold) as obsered; this is presented b contor lines (Figre 7). Von Mises stress in polcarbonate reaches 10.6 MPa (Figre 7). Dring the imprint step (Figre 9), the mold moes don from the initial point abot 900 nm (it corresponds 5 atm pressre) and presses the polcarbonate, at the same time the contact force beteen the mold and polcarbonate increases and plastic strains appears. Maimm Von Mises stress (33.6 MPa) is located in the contact area beteen the mold s corner and polcarbonate. The arros in Figre 9 sho that total displacement after the hot imprint process is dierted donards. In the demolding step, the hot mold (T = 41 K) is demolded and finall polcarbonate is cooled, and it sstains the form of the mold (Figre 10). Figre 9. Von Mises stress distribtion in the deformed polcarbonate (in Pa) and total displacements (in meter) represented b arros after imprint step. Figre 10. Areas of the permissible ield and profile of the eperimental microrelief represented b the red line.

11 Sensors 013, One of the most important qalitatie parameters in a hot imprint process is the filling ratio of the mold s microrelief. It is defined as a ratio of filled area and total area of microstrctre. Figre 10 represents dependence of the non-filled cait on mold displacement throgh the hole hot imprint process. As shon in Figre 11, non-filled cait decreases er slol dring the heating step. At the end of this step the filling ratio is abot 70%, and the mold s displacement 10 7 m. Then the empt cait decreases er qickl (abot seen times) to 10%, becase the top srface of the polcarbonate becomes a soft material (it reaches the glass transition temperatre). Throgh the remaining part of the imprint step, the empt cait decreases to %. After the demolding step the empt cait increases slightl and remains at abot 4%. Figre 11. The dependence of non-filling cait erss mold displacement. The elastic strain appears hen the mold displacement reaches 0.1 µm, and plastic strain appears hen the mold displacement reaches 0.9 µm and then hen the mold is demolded residal strains appear in the polcarbonate. The nmerical model as erified eperimentall. The same eperimental scheme as in the nmerical simlation as sed. A lamellar microstrctre of 4 µm period and 100 nm depth as replicated into polcarbonate (3 mm thickness) for 15 seconds at 148 C and nder 5 atm pressre. Flat embossing eperiment as performed sing a flat thermal pressre deice (designed at the Institte of Materials Science of Kanas Uniersit of Technolog, Kanas, Lithania). The original constrction ensres controlled pressre, force, temperatre and eposre time (P = 1 5 atm, T = 0 00 C, t = s). Dring the eperiment, the mold as being heated p to the reqired temperatre, before the microstrctre as imprinted onto the polmer. Dring the mechanical hot imprint process, the polmer as compressed ith a high load. The polmer is plastic, so after demolding it remains deformed. A ie of the replica, obtained b sing atomic force microscop, is presented in Figre 1.

12 Sensors 013, Figre 1. (a) 3D and (b) D ie of the replica. (a) (b) In order to compare the area of the microstrctre imprinted onto the polcarbonate ith the area of the nickel mold, data from AFM measrements ere sed. Eperimental reslts ere integrated sing Simpson s rle. Calclations sho that the area not filled ith polcarbonate is abot 10%, hereas the theoreticall empt area is 4%. Graphical comparison of the theoretical microrelief ith the eperimental one (red line) is presented in Figre 10. This shos that the difference beteen model and realit is not big and model is sitable for theoretical ealation of the material behaior dring the thermal imprint process. 4. Conclsions A mathematical non-linear model of the hot imprint process of a nickel mold into polcarbonate as created sing an elasto-plastic material model. The finite element model, implemented b COMSOL Mltiphsics softare, allos s to determine temperatre fields, displacements and stresses in each step of the hot imprint process. The filling ratio of the mold is the main parameter of the replica s qalit. Therefore dependence of the empt cait erss the mold s imprint displacement as obtained in all steps of the hot imprint process. Nmericall it as determined that after the hot imprint process the empt cait remains at abot 4%. The finite element model as erified sing eperimental inestigations. In the eperimental and nmerical reslts, lamellar form replicas ere obsered, and differences beteen eperimentall and nmericall obtained filling ratios are ithin alloable limits. In addition the aerage measred depth of the replica is abot 100 nm, the same as the calclated ale. Acknoledgments The research as spported b Lithanian State Science and Stdies Fondation and the Research Concil of Lithania. Conflicts of Interest The athors declare no conflict of interest.

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