Smulatng njecton mouldng of mcrofeatured components T. Tofteberg 1 * and E. Andreassen 1 1 SINTEF Materals and Chemstry, Oslo, Norway terje.tofteberg@sntef.no; erk.andreassen@sntef.no Numercal smulaton of njecton mouldng of a mcrofeatured component s reported n ths paper. The smulaton s dvded n two; frst a large scale smulaton of the man geometry s performed and then a mcroscale smulaton of the structural detals usng the pressure from the large scale smulaton as a boundary condton. The smulatons show that even though the polymer soldfes n less than 5 ms, there s enough tme to make a good replcaton of the mcrofeatures as has also been shown expermentally. It s also possble to solate dfferent physcal effects nfluencng the replcaton. The shear heatng n the mcrofeature s n ths case not mportant. However, the shear thnnng wthn the mcrofeatures helps replcaton sgnfcantly. The shear stress at the wall s n the lower range of where wall slp has been reported to occur for other polymer melts. Even though wall adheson s not consdered n these smulatons, t s shown that the adheson energy s comparable n sze wth the pressure-volume work requred to push the polymer nto the mcrofeature and could therefore be an mportant factor nfluencng replcaton. Introducton Mcrofeatured components are commonly used as optcal storage meda (CD, DVD Blu-ray Dsc), and they are meetng new challenges n the world of mcrofludcs [1]. When patternng a polymer surface on the mcroscale the physcal propertes can be changed dramatcally, e.g. to make superhydrophobc surfaces [2]. The smulatons performed n ths work relates to an expermental study n whch a dffractve optcal element (DOE) was njecton moulded [3]. Ths DOE s an essental part of a low-cost nfrared spectrometer, whch e.g. can be used to dentfy dfferent types of polymers. In Fgure 1 the shape of the DOE s shown on the macro and mcroscale. The gratng shown n part b) of the fgure spans an area of 10 mm x 10 mm whch means ~3000 perods. One of the materals used n the experments was Topas 5013S-04, a Cyclc Olefn Copolymer (COC), and the smulatons presented here wll consder ths materal. Prevous attempts on smulatng mcrofeatures The specfc problem related to the smulaton of mcrofeatured parts s the large sze scale ratos between the overall geometry and the mcrofeatures. In our case the heght of the gratng s less than 0.1 % of the thckness of the part that we are mouldng. Tryng to smulate both sze scales n one smulaton can lead to unphyscal results [4, 5]. Another approach whch s smlar to the one employed n ths work, s to frst perform a large scale smulaton consderng only the overall geometry. Then the pressure and temperature feld from ths smulaton s used as a boundary condton for a local smulaton of the mcrofeature [6-8]. Fgure 1 a) Drawng of the njecton moulded part wth dmensons [mm]. The mcrofeatures are located n the central square area and a pressure sensor located n the cavty opposte to the mcrofeatures s ndcated wth a dark grey crcle. b) The topography of the mcrofeatures measured on an njecton moulded DOE usng AFM. Expermental The smulatons were performed wth ANSYS CFX 11.0 [9] and Moldflow MPI 6.1 [10]. CFX s a fnte volume program developed as a general flud mechancs solver whle Moldflow s the most wdely used njecton mouldng software. We have used Moldflow smulatons for smulatng the fllng of the man geometry and used data from these smulatons as boundary condtons for local smulatons around the mcrofeatures. Snce Moldflow smulatons are routnely done n both ndustry and academa we wll not go nto the computatonal detals nvolved, but rather consder the novel mcroscale smulaton. On the mcroscale we have been solvng a multphase problem where we have consdered both the flow of ar and of a polymer melt wthn a N mould. The effect of the mould s only taken nto account as a boundary condton. The multphase problem s solved by usng the so called nhomogeneous model n CFX, meanng that the two phases have two separate flow felds. These felds are calculated by solvng two sets of Naver-Stokes equatons. The two flow felds are ndependent except for n the nterphase regon where they nteract va nterphase transfer terms. Proceedngs of the Polymer Processng Socety 24th Annual Meetng ~ PPS-24 ~ June 15-19, 2008 Salerno (Italy)
Governng equatons The equaton of contnuty s wrtten for each phase as ( α ρ ) t ( α ρ ) = 0 + u, (1) where u s the velocty vector, t the tme, the dvergence operator, α the volume fracton and ρ the densty. The subscrpt descrbes the phase and can be ether polymer melt or ar. The melt s treated as a generalzed Newtonan flud and the momentum balance s wrtten ( α ρ u ) t α p + + u ( ( α ρu )) = T ( u + ( u ) ) ( α η ) + M j ( 2) smulaton s actually of an nfnte seres of such peaks. It has prevously been shown that placng the nlet drectly at the mcrofeature (where the nterphase s located n Fgure 2) neglects mportant contrbutons to the pressure loss and temperature feld [4]. Because of ths we also nclude the lower part of the geometry. The unstructured mesh s seen n the same fgure. where p s the pressure and η the vscosty. M j s the nterphase transfer term whch obeys Newton s thrd law by havng M j = -M j and s expressed as M j = C α u u ( u u ) D ρα, (3) j j j where C D s a constant nterphase transfer coeffcent. As can be seen, f α s ether zero or unty the transfer term vanshes. The densty ρ s the total densty taken as a lnear nterpolaton between the two phases,.e. ρ=α ρ + α j ρ j There are now 9 unknowns, the two volume fractons, the sx velocty components and the pressure. The equatons requred are two tmes Equaton (1) (one for each phase), sx tmes Equaton ( 2) (two phases, three components), and the fact that the volume fractons add to unty. In addton we wll ntroduce a new varable, the temperature. The temperature feld s shared by both fluds and n the energy equaton we nclude conductve and convectve heat transfer and a term descrbng vscous dsspaton. ( ρc T ) p t 2 ( κ T ) + ηγ& + u ( ρc T ) p = (4) T s the temperature, c p s the heat capacty, κ s the heat conductvty and γ& s the shear rate. All varables and parameters, ncludng the velocty feld n Equaton (4) are taken as a lnear nterpolaton usng the same procedure as descrbed for the densty n Equaton (3). Computatonal doman and mesh The computatonal doman for the mcroscale smulaton s shown n Fgure 2. It represents one perod of the dffractve gratng, 600 nm hgh and 3 µm long. By usng perodc boundary condtons the Fgure 2 The computatonal doman for the mcroscale smulaton representng one perod of the dffractve gratng shown n Fgure 1b. The mesh s unform and unstructured. The ntal state of the volume fracton varable s shown wth red ndcatng polymer and blue ar. Boundary condtons At the walls we have appled no-slp condtons for the polymer and free slp for the ar. A heat transfer coeffcent s defned to be h = 5000 W/m 2 K at the wall. At the top of the computatonal doman we have a zero pressure outlet where ar s allowed to escape from the smulaton. There are two perodc boundares, meanng that any flud leavng on the rght sde wll reappear on the left, thus mmckng that there s a repeatng gratng consstng of an nfnte number of copes of the geometry n Fgure 2. An the nlet we prescrbe a tme dependent pressure obtaned from Moldflow smulatons of the entre geometry shown n Fgure 1a. In the expermental work descrbed n ref. [3] the cavty pressure was measured and as can be seen n Fgure 3, there s very good Proceedngs of the Polymer Processng Socety 24th Annual Meetng ~ PPS-24 ~ June 15-19, 2008 Salerno (Italy)
agreement between smulated and measured cavty pressure. The temperature of the melt enterng through the nlet s set so that the temperature gradent s constant. η η ( T & γ ) o ( T ) ( T ) ( T ) ηo = n η & o γ 1+ τ A1( T D2) = D1exp A2 + ( T D2), 1 (5) The parameters used n ths equaton were taken from the Moldflow database. All physcal propertes for both fluds, except for the melt vscosty are set to a constant value. Values are gven n Table 1. In order to mprove convergence, the vscosty of ar was ncreased. It s however stll so low that the flow of the ar does not nfluence the flow of the polymer. Fgure 3 Measured (symbols) and smulated (lnes) cavty pressure vs. tme. Mould temperature 110 C. The locaton of the pressure sensor s ndcated n Fgure 1a. Intal condtons The mcroscopc smulaton starts as the flow front hts the mcrofeature. Snce the mcrofeature s so small compared to the thckness of the part, the flow front s essentally parallel wth the mould wall t hts and the ntal shape of the flow front used n the smulatons can be seen n Fgure 2. The ntal temperature s set to be the same as the melt temperature n nozzle of the njecton mouldng machne. Ths ntal temperature s based on Moldflow smulatons of the flow front temperature as can be seen n Fgure 4, where t can be seen that the temperature of the flow front s almost unform n space. Table 1 Physcal constants for the materals used n the smulatons. The sx frst parameters are Cross-WLF parameters for equaton (5). COC Ar A1 [-] 37.16 0 A2 [K] 51.6 0 D1 [Pa s] 4.884 10 15 0.1 D2 [Pa s] 343.15 0 τ* [Pa] 3433 0 n [-] 0.4617 0 ρ [kg/m 3 ] 1049.3 1 κ [W/mK] 0.163 0.026 c p [J/kgK] 1800 1004 2D/3D smulaton Usng CFX, all smulatons are performed n 3D. In ths case we only need a 2D smulaton. Ths s done by makng a mesh whch s one element thck n the out of plane drecton. Symmetry boundary condtons are appled on the two faces parallel to the computatonal doman. Results and Dscusson Fgure 4 The smulated temperature at the flow front for an njecton molded DOE wth njecton velocty 20 cm 3 /s, mould temperature 60 C and nlet melt temperature 270 C. The mcrofeatured area s shown wth a dotted lne. Materal propertes The vscosty of the polymer s descrbed usng a sx parameter Cross-WLF model whch s the same model used n the Moldflow smulatons. Fgure 5 shows the flow front as a functon of tme and processng parameters. It can be seen that the speed of the flow front gradually decreases before comng to a stop because the melt has been cooled by the mould wall. When ncreasng the njecton velocty (gong from left to rght n the fgure) the degree of fllng ncreases, as when ncreasng the mould temperature (gong from top to bottom n the fgure). It was also found expermentally that the degree of replcaton changed from ~20% at the settngs n the upper left corner to ~100% at the settngs n the lower rght corner. [3] The results presented next all relate to the smulaton marked T110 v100 n Fgure 5 (mddle-rght), whch s one of the smulatons wth the hghest njecton velocty and where we expect to have the hghest shear rates. Proceedngs of the Polymer Processng Socety 24th Annual Meetng ~ PPS-24 ~ June 15-19, 2008 Salerno (Italy)
Fgure 5 The smulated flow front as a functon of tme and processng settngs. The thck black lne s the mould wall and the contours are the flow front separated by 0.1 ms. T refers to the mould temperature [ C] and v to the njecton velocty [cm 3 /s]. The x-axs s the horzontal dstance [µm] and the y-axs the vertcal [nm]. For all smulatons the flow front was horzontal y=0 at t=0 and the lower contour represents the flow front at t = 0.6 ms. Contours wth the same colour ndcate the same tme value. Proceedngs of the Polymer Processng Socety 24th Annual Meetng ~ PPS-24 ~ June 15-19, 2008 Salerno (Italy)
Temperature dstrbuton The rate of coolng for the polymer melt as t hts the mould can be seen n Fgure 6. It can be seen here that the melt s cooled very rapdly (2-3 ms) from the melt temperature of 270 C and down to a temperature around 160 C where the vscosty ncreases to such hgh levels that the polymer stops behavng lke a lqud and the flow front stops as seen n Fgure 5. When the vscosty rses above a certan level we are no longer capable of obtanng convergence and the smulaton s aborted. It can also be seen n Fgure 6, that the dfference between the maxmum and mnmum temperature wthn the computatonal doman s small. Ths s because the mcrofeature (and the computatonal doman) s so small that any thermal gradent that may exst wll mmedately vansh due to thermal conducton. Also note that the temperature frst decreases relatvely slowly before the rate ncreases and decreases agan. Ths s because the heat flow s proportonal to the temperature dfference between the melt and the mould and to the contact area. At the start of the smulaton the contact area s small leadng to a relatvely small coolng rate. Then the contact area ncreases and wth t also the coolng rate before, near the end of the smulaton, the temperature dfference between the mould and the melt s gettng smaller and the coolng rate decreases. Fgure 6 Smulated temperature and vscosty n the computatonal doman for an njecton moulded DOE wth settngs: T mold = 110 C, v nj = 100 cm 3 /s and h = 5000 W/m 2 K. The sold lnes show the average value and the dotted lnes show the max and mn value at each tme step. Shear rates The polymer melt s hghly shear thnnng. Is shear thnnng n the mcrofeature an mportant factor for the replcaton of mcrofeatures? The shear rates for the same part as s shown n Fgure 6 (hgh njecton speed) are shown n Fgure 7. The hghest shear rate s ~2000 s -1 at ths tme step. Accordng to the Cross- WLF model used, ths s enough to reduce the vscosty by more than a factor 10 relatve to the zero shear rate vscosty. Hence, the shear thnnng effect seems to be mportant, also wthn the mcrofeatures. It can also be noted that the maxmum shear stress at the wall observed durng the smulatons was ~0.1 MPa whch n the same order of magntude as the values where wall slp has been reported to occur for lnear polyethylene melts [11,12]. No data have been found dscussng the crtcal shear stress for wall slp wth COC melts. Fgure 7 Smulated shear rates for the njecton moulded part from Fgure 6 at t = 0.8 ms. It can be seen that the shear rates are well above 100 s -1, approachng 2000 s -1 near the wall at the flow front. Shear heatng It s of mportance to know f the local shear heatng n the mcrofeature s a determnng factor nfluencng replcaton. The local rate of shear heatng can be 2 wrtten asη & γ. If we ntegrate ths varable n space, we obtan the total effect [W] of the shear heatng wthn the computatonal doman. The results are shown n Fgure 8 for the same smulaton as n Fgure 6. The total effect ncreases at the begnnng of the smulaton as the pressure ncreases, but when the polymer melt soldfes the shear rate drops and the shear heatng falls wth t. Integraton n tme gves the total energy produced by shear heatng: t = 2 E shear η & γ dvdt 0 V The curve n Fgure 8 was ntegrated up to 2.5 ms, and the total energy due to shear heatng was found to be 43 kj/m 3. Dvdng ths energy by ρc p gves an estmated temperature ncrease of 0.02 C. Ths shows that the shear heatng wthn the mcrofeature n ths case s of mnute mportance and could have been removed from Equaton (4), wthout much nfluence on the results. Proceedngs of the Polymer Processng Socety 24th Annual Meetng ~ PPS-24 ~ June 15-19, 2008 Salerno (Italy)
The smulatons show that even though the temperature of the polymer melt decreases to below an effectve noflow temperature n 2-3 ms, there s stll enough tme to fll the mcrometer szed structural detals. It s also possble to solate physcal phenomena nfluencng the replcaton qualty. It s observed that the shear heatng n the mcrofeatures n ths case can be neglected, that the shear thnnng s mportant for the fllng of the mcrofeatures and that the flow regme s smlar to regmes where wall slp has been observed to occur for polyethylene melts. Fgure 8 The smulated local shear rate ntegrated over the entre computatonal doman V, dvded by the volume of the mcrofeature V µ [W/m 3 ]. Same smulaton settngs as n Fgure 6 Adheson effects The total work that forced the polymer melt nto the mcrofeature s the same as the work dsspated as heat n the smulaton. It has been argued that for submcrometer structures, where the surface area s large compared to the volume, the adheson force between the wall and the melt may be an mportant factor nfluencng replcaton [13]. The rato between the total energy requred to force the polymer nto the mcrofeature and the mould surface area of the mcrofeature s n ths case 0.012 J/m 2. It has been reported that the adheson energy between a steel wall and polyethylene (PE) s 0.02 0.04 J/m 2 dependng on the grade of PE. We would expect that the adheson energy between the mould wall (Nckel n ths case) and the polymer melt (a cyclc olefn copolymer) s of the same order of magntude as ths, but we do not have specfc measurements relatng to these two materals. It s stll lkely that the adheson energy between the mould wall and the polymer plays an mportant role n the replcaton of mcrometer szed features. Smulaton tme The smulatons performed n ths work took approxmately 48 hours on one core of an Intel Core 2 Duo T7600 2.33GHz processor whle the author had other thngs to do on the second core. The smulatons also ndcate that the adheson energy between the mould wall and the polymer s mportant when replcatng mcrometer szed or smaller features. References 1. G. M. Whtesdes Nature 2006, 442, 368. 2. E. Puuklanen; T. Raslanen; M. Suvanto; T. A. Pakkanen Langmur 2007, 23, 7263. 3. T. Tofteberg; H. Amédro; E. Andreassen Polym. Eng. Sc. 2008, Submtted. 4. T. Tofteberg; E. Andreassen n PPS Europe/Afrca Regonal Meetng, Gothenburg, 2007. 5. L. Yu; L. J. Lee; K. W. Koellng Polym. Eng. Sc. 2004, 44, 1866. 6. D. Yao; B. Km J. Mcromech. Mcroeng. 2002, 12, 604. 7. S. W. Km; L. S. Turng Polym. Eng. Sc. 2006, 46, 1263. 8. T. Erksson; H. K. Rasmussen J. Non-Newton. Flud. 2005, 127, 191. 9. ANSYS CFX-11.0, ANSYS Europe Ltd., 2007. 10. Moldflow Plastcs Insght (MPI) 6.1, Moldflow Corporaton, 2007. 11. M. M. Denn Annual Revew of Flud Mechancs 2001, 33, 265. 12. S. H. Anastasads; S. G. Hatzkrakos Journal of Rheology 1998, 42, 795. 13. H. Pranov; H. K. Rasmussen; N. B. Larsen; N. Gadegaard Polym. Eng. Sc. 2006, 46, 160. Conclusons We have demonstrated a novel approach to smulate njecton mouldng of mcrofeatured components. Wth ths method t s possble to get a reasonable estmate for the replcaton qualty of mcrofeatures. The model correctly accounts for the postve correlaton observed expermentally between njecton speed and degree of replcaton and between mould temperature and degree of replcaton. Proceedngs of the Polymer Processng Socety 24th Annual Meetng ~ PPS-24 ~ June 15-19, 2008 Salerno (Italy)