Modeling of Through-Mask Electrochemical Micromachining

Transcription

1 A publcaton of 85 CHEMICAL ENGINEERING TRANSACTIONS VOL Guest Edtors: Smonetta Palmas Mchele Masca Annalsa Vacca Copyrght 014 AIDIC Servz S.r.l. ISBN ; ISSN The Italan Assocaton of Chemcal Engneerng DOI: /CET Modelng of Through-Mask Electrochemcal Mcromachnng Alexey D. Davydov* a Tatyana B. Kabanova a Vladmr M. Volgn ab a Frumkn Insttute of Physcal Chemstry and Electrochemstry Russan Academy of Scences Lennsk pr. 31 Moscow Russa b Tula State Unversty pr. Lenna 9 Tula Russa davydov@elchem.ac.ru The through-mask electrochemcal machnng of features of mcron dmensons was studed theoretcally. The Laplace equatons for the electrc feld potental and the equaton of workpece surface evoluton were used as the mathematcal model of the process. The problem was solved numercally usng the methods of fnte and boundary elements and also the Level Set method. By usng the numercal experments the effect of parameters whch characterze the mask geometry and the process condtons on the ntal dstrbuton of current densty over the workpece surface and the varaton of current dstrbuton n the course of etchng was studed. In partcular the dependences of dmensonless average current densty on a fracton of unprotected areas were obtaned at varous values of mask thckness and unprotected areas (rectangular grooves or crcles). It s shown that the hgher s the unprotected area densty the mask thckness and unprotected area wdth the hgher s the ntal average current densty. It s shown that the ntal nonunformty of average current densty changes n the course of machnng leadng to a change n a rato between the anodc dssoluton rates of unprotected areas of dfferent wdths. In the ntal perod of treatment the smaller s the wdth of uncovered area the hgher s the anodc dssoluton rate. Then n the course of machnng the anodc dssoluton rate on the narrow unprotected areas steeply decreases and can become lower than that on the wder areas. As a result the depth of unprotected areas of dfferent szes wll be dfferent. The result of modelng enables one to predct the fnal depth of the features on the workpece surface. 1. Introducton The through-mask electrochemcal machnng s one of effectve methods of mcrofabrcaton (Datta 1998) whch s used to fabrcate precson nozzles (Datta 1995) regular relfs (Volgn and Davydov 004) complex patterned substrates (Raffelstetter and Mollay 010) mcro-dmples arrays (Qan et al. 010) mcro probe arrays (Km et al. 011) etc. Ths method enables one to machne not only bulk metal (Kern et al. 007) but also thn metal flms (West et al. 199). Varous combnatons of template anodc dssoluton and cathodc electrodeposton (Herraz-Cardona et al. 013) can be used for producton of hghly porous metal layers and complex herarchcal patterns. The shape and dmensons of a workpece whch s fabrcated by the through-mask electrochemcal machnng depend on a large number of parameters: the mask thckness shape and dmensons of unprotected areas (West et al. 199) mask wall angle (L et. al 011) the machnng mode and condtons of mass transfer (Alkre and Delgann 1988) etc. Therefore the process of electrochemcal machnng s frequently studed by the theoretcal methods (Davydov et al. 004). The theoretcal study of the process s commonly performed usng the mathematcal models whch nvolve the Laplace equaton for the electrc feld potental (Shenoy et al. 1996) or concentraton of dssolvng metal ons (Madore et al. 1999). The regulartes of electrochemcal shapng of ndvdual elements have been much studed (L et al. 010). The results of these studes enabled one to estmate the effect of hydrodynamcs of electrolyte soluton flow; to predct the formaton of metal slands on the substrate and develop the methods whch provde the absence of the slands; to take nto account the effect of the nclne of sde mask walls on the geometry of machned surface etc. Frequently a mask has the blanks of dfferent shapes and dmensons. Please cte ths artcle as: Davydov A. Kabanova T. Volgn V. 014 Modelng of through-mask electrochemcal mcromachnng Chemcal Engneerng Transactons DOI: /CET

2 86 In ths case t s mportant to take nto consderaton a mutual effect of mask elements (Raffelstetter and Mollay 010). By now the regulartes of electrochemcal machnng of mcro features usng the masks wth the elements of dfferent shapes and szes have not been studed adequately. The am of ths work s to study theoretcally the regulartes of mask electrochemcal machnng on the bass of recent achevements n the feld of modelng of electrochemcal machnng (Hnduja and Kuneda 013).. Statement of problem In the general case the mask can have the blanks of dfferent shapes (for example grooves crcles etc.) and szes and they are nonunformly arranged over the workpece surface (Fgure 1). A dependence of the current densty and consequently the rate of metal anodc dssoluton on the protected surface area leads to the formaton of elements of dfferent depth on the workpece surface areas wth dfferent geometrc parameters. For nstance n zone I where a fracton of unprotected surface area s smaller than n zone II the resultng relef elements wll be deeper than that n zone II (Fgure 1). In addton due to the screenng effect the depth of the elements wll dffer also wthn each zone. For nstance n zone I the element of relef wth a larger fracton of unprotected area (4) wll be deeper than that wth smaller unprotected area (3). At equal characterstc szes of mask elements of dfferent geometrc shapes (b D where b s the wdth of the groove-shaped unprotected area (5) and D s the dameter of the round unprotected area (6)) the machnng wll yeld the elements of dfferent depth (Fgure 1 zone III (5) and (6)): due to a larger contrbuton of the edge effect the axsymmetrc element wll be deeper than the straght-lne groove. Fgure 1: A scheme of electrochemcal machnng usng the mask wth the blanks of dfferent shapes and szes: (1) a workpece (substrate); () a mask As a rule several elements wth varous shapes and szes should be formed on the workpece surface. Due to the overcuttng and edge effects the szes of elements and the rates of ther formaton wll be dfferent. In order to predct the workpece surface geometry and also to determne the shape and szes of mask elements whch s requred to produce the prescrbed workpece surface geometry t s advantageous to apply the methods of mathematc modelng. 3. Mathematcal model and method of numercal soluton The electrochemcal machnng wth partal nsulaton of workpece surface s smulated by the model of shapng whch gnores the concentraton changes n the soluton (Davydov et al. 004). Ths approxmaton s acceptable n the cases of suffcently ntense pumpng or strrng of electrolyte soluton. The dstrbuton of electrc potental over the nterelectrode gap s calculated by usng the Laplace equaton.

3 For convenence of soluton and analyss of the results the mathematcal model s gven n the dmensonless form. The nterelectrode gap (S) was taken as a unt length and the appled voltage (U) was taken as a unt electrc potental. 87 X x y a ϕ S ηε V χu Ya Φ I τ t S S U χu S (1) where X Y a are the dmensonless coordnates; Φ s the dmensonless potental; I s the dmensonless current densty; τ s the dmensonless tme; η s the current effcency; ε s the volumetrc electrochemcal equvalent; χ s the conductvty of electrolyte soluton; a s the subscrpt whch characterzes the workpece surface. Thus obtaned system of dmensonless equatons s as follows: ( grad ) 0 dv Φ () V Ya Φ τ n Ya 1+ X (3) where n s a vector normal to the workpece surface. To solve the system of Eqs. () and (3) the boundary and ntal condtons should be prescrbed. For the machnng scheme under consderaton when the polarzaton of electrodes can be gnored the boundary condtons for the dmensonless potental wll be as follows: Φ Φ n on the anode (workpece) on the cathode ( tool - electrode) on the nsulator The ntal condton s that at the ntal nstant of tme the workpece surface s a plane passng through the orgn of coordnates: Y ( 0) 0 (5) a The boundary value problem for Eqs. () (5) s a problem wth movng boundary. Then the equatons whch descrbe the transport processes and the moton of computatonal regon boundary should be calculated smultaneously. Ths presents a consderable dffculty. The numercal soluton s frequently smplfed by usng the quas-steady state approxmaton. Wthn ths approxmaton the entre tme of machnng was dvded nto a number of tme steps. For each tme step: 1) the dstrbuton of electrc potental was calculated (at the electrode geometry correspondng to the begnnng of the step); ) a new shape of the workpece surface was determned (at the dstrbuton of the current densty correspondng to the begnnng of the step). At each tme step the boundary-value problem for the Laplace Eq. () wth the boundary condtons (4) was solved numercally by the method of boundary elements. The boundary-value problem was reduced to the equvalent boundary ntegral equaton. As a result of numercal soluton of Eq. () the dstrbuton of the current densty over the workpece surface was determned. It was used to solve Eq. (3) and determne the geometry of workpece surface for the next tme step. 4. Results of modelng and dscusson At the frst stage the ntal stage of electrochemcal machnng wth partal nsulaton of workpece surface was smulated.e. the dstrbuton of dmensonless current densty (anodc dssoluton rate) over the workpece surface was calculated. The calculatons were performed at varous values of machnng parameters such as the blank wdth mask thckness a fracton of actve workpece surface area.e. a rato of unprotected area to the total surface area of the workpece. Based on these results the general regulartes for a mask wth actve areas of dentcal shapes and szes can be formulated as follows: An (4)

4 88 average current densty monotoncally decreases wth decreasng dstance between the mask elements (ncreasng actve surface area) and ncreasng mask thckness. An average current densty for the axsymmetrc (crcle) unprotected areas s hgher than that for the lnear (band strp) unprotected areas. In the majorty of cases of practcal mportance the masks have actve areas of dfferent szes. In ths case the rates of formaton of dfferent elements wll dffer; as a result the elements of dfferent depth wll form on the workpece surface. A dfference between the element depths can be estmated by usng the above regulartes for the unform mask. However t s dffcult to determne an actual actve surface area. Fgure : A fragment of zone of electrochemcal machnng and (b) computatonal regon At the second stage the electrochemcal machnng usng the masks wth actve areas of two dfferent szes was performed. Fgure (a) gves a fragment of zone of electrochemcal machnng wth (1) the anode () the cathode and (3) the mask. The nterelectrode gap (4) s flled wth the electrolyte soluton. The mask contans the blanks (5) (6) of dfferent wdth. The planes of symmetry that bound the computatonal regon are denoted by 7. Fgure b gves the computatonal regon. The computatonal nodes (Fgure b) are shown wth crcles; the arrows show the drecton of numberng the computatonal nodes on the outer boundary (8) (corresponds to Fgure a lnes 1 and 7) and nner (mask/electrolyte) boundary (9). For convenence of modelng t was assumed that there s a small gap between the ntal workpece surface and the mask. The presence of the gap of the order of to 0.1 had no consderable effect on the results of modelng; however t enabled us to smplfy sgnfcantly the algorthm of calculatons due to the presence of two dfferent surfaces (the workpece surface and the mask surface). To provde the requred accuracy of smulaton at the mnmum amount of computaton a nonunform mesh of boundary elements was used. To mantan the accuracy n the course of surface evoluton the adaptve remeshng was realzed. As t was found by the test calculatons a tme step of and shorter provded the stablty of numercal smulaton. In addton a Level Set method was used for dscretzaton of spatal dervatve n equaton (3):

5 89 n n n n n 1 Φ Y 1 + a + Ya Ya Y n n a 1 + Δ 1+ max 0 + mn a Ya 0 n X + 1 X X X 1 Y τ (6) where s the number of computatonal node on the workpece surface; n s the number of tme step; and Δ τ s the tme step. Fgure 3 gves the results of modelng for a mask of dmensonless thckness (a) and (c) h 0. 1; (b) and (d) h 0. 01; the dmensonless szes of blanks are b 0. 1 (on the left) and b 1 (on the rght); (a) and (b) a fracton of actve surface area s 7.5 %; (c) and (d) a fracton of actve surface area s 55 %. Dashed lnes show the workpece surface at the tme when the elements on dfferent unprotected areas reach equal depth. Fgure 3: Workpece surface evoluton n the electrochemcal machnng usng a mask wth actve areas of 0.1 (the left parts of the fgures) and 1 (the rght parts of the fgures) From Fgure 3 t s seen that n the ntal perod of machnng a hgher rate of anodc dssoluton s observed on smaller actve areas. As a result of machnng the actve workpece surface area ncreases. At a constant current ths leads to a decrease n the current densty and consequently the anodc dssoluton rate. To the frst approxmaton assume that the anodc dssoluton s sotropc and the current through a blank s constant. Then dya b dτ b + πy a I av (7) where I av s dmensonless average current densty. From (7) t follows that the anodc dssoluton rate decreases faster on the smaller unprotected areas.

6 90 Thus n the course of machnng a rato between the anodc dssoluton rates n dfferent blanks permanently vares. Under certan condtons the anodc dssoluton rate on the larger unprotected areas becomes hgher than that on the smaller unprotected areas. At a shorter tme of machnng the elements wth narrower unprotected areas wll be deeper whereas at a longer tme the elements wth wder unprotected areas wll be deeper. The tme (or depth) when equal depth of dfferent elements s reached depends on the mask thckness and a fracton of actve surface area (Fgure 3). 5. Conclusons A scheme of numercal modelng of through-mask electrochemcal machnng whch enables one to predct the shape and szes of workpece surface s developed. The developed mathematc model enables one to determne the tme dependence of unprotected area depth and to determne the treatment tme whch s requred to obtan equal depth for the areas of dfferent szes. Acknowledgements Ths work was supported by the Russan Foundaton for Basc Research project no and the Mnstry of Educaton and Scence of the Russan Federaton project no of the Basc Part of the State Program. References Alkre R. Delgann H The Role of mass transport on ansotropc electrochemcal pattern etchng Journal of The Electrochemcal Socety Datta M Fabrcaton of an array of precson nozzles by through-mask electrochemcal mcromachnng Journal of The Electrochemcal Socety Datta M Mcrofabrcaton by electrochemcal metal removal IBM Journal of Research and Development 4(5) Davydov A.D. Volgn V.M. Lyubmov V.V. 004 Electrochemcal machnng of metals: Fundamentals of electrochemcal shapng Russan Journal of Electrochemstry Herraz-Cardona I. Gonzalez-Buch C. Ortega E. Garca-Anton J. Perez-Herranz V. 013 Energy effcency mprovement of alkalne water electrolyss by usng 3D N cathodes fabrcated va a doubletemplate electrochemcal process Chemcal Engneerng Transactons Hnduja S. Kuneda M. 013 Modellng of ECM and EDM processes CIRP Annals - Manufacturng Technology 6() Kern P. Veh J. Mchler J. 007 New developments n through-mask electrochemcal mcromachnng of ttanum Journal of Mcromechancs and Mcroengneerng Km Y. Youn S. Cho Y.-H. Park H. Chang B.G. Oh Y.S. 011 A mult-step electrochemcal etchng process for a three-dmensonal mcro probe array Journal of Mcromechancs and Mcroengneerng 1 Art. No L W. Quanda W. Xuqng H. Yucheng D. Bngheng L. 010 Fnte element smulaton and expermental study on the through-mask electrochemcal mcromachnng (EMM) process Internatonal Journal of Advanced Manufacturng Technology L D. Zhu D. L H. Lu J.-G. 011 Effects of mask wall angle on matrx-hole shape changes durng electrochemcal machnng by mask Journal of Central South Unversty of Technology Madore C. Potrowsk O. Landolt D Through-mask electrochemcal mcromachnng of ttanum Journal of The Electrochemcal Socety Qan Sh. Zhu D. Qu N. L H. Yan D. 010 Generatng mcro-dmples array on the hard chrome-coated surface by modfed through mask electrochemcal mcromachnng Internatonal Journal of Advanced Manufacturng Technology Raffelstetter P. Mollay B. 010 On the modelng of shape evoluton n through-mask electrochemcal mcromachnng of complex patterned substrates Electrochmca Acta Shenoy R.V. Datta M. Romankw L.T Investgaton of sland formaton durng through-mask electrochemcal mcromachnng Journal of The Electrochemcal Socety Volgn V.M. Davydov A.D. 004 Modelng of multstage electrochemcal shapng Journal of Materals Processng Technology 149(1-3) West A.C. Madore C. Matlosz M. Landolt D. 199 Shape changes durng through-mask electrochemcal mcromachnng of thn metal flms Journal of The Electrochemcal Socety