Morphology of composite films: a computer study. Stanislav Novák

Size: px

Start display at page:

Download "Morphology of composite films: a computer study. Stanislav Novák"

Derek Lambert
6 years ago
Views:

1 Supeficies y acío 9, 48-5, Diciembe 1999 Mophology of composite films: a compute stuy Stanislav Novák Depatment of Physics, J.E.Pukynì Univesity, Èeské mláeže 8, Ústí na Labem, Czech Republic Ruolf Hach Depatment of Physics, J.E.Pukynì Univesity, Èeské mláeže 8, Ústí na Labem, Czech Republic an Depatment of Electonics an acuum Physics, Chales Univesity, Holešovièkách, Paha 8, Czech Republic Miloš Sobotka Depatment of Electonics an acuum Physics, Chales Univesity, Holešovièkách, Paha 8, Czech Republic Fou moels, all with spheical objects, wee pepae to stuy the possibility of at least patial econstuction of 3D stuctues fom thei D sections. The citeion to juge whethe the stuctue consists of constant size objects is given. The eos cause by the influence of limite numbes of objects an of igitization ae iscusse. It is illustate the usage of the integal tansfom metho to unfol the stuctues. The esults of the film analysis ae pesente. 1. Intouction Mophological analysis of composite metal/ielectic thin films is impotant both fo the chaacteization of the films themselves an fo the analysis of thei popeties. Theoetical appoach to the poblem often leas to nealy invincible ifficulties. The main goal is to eive some spatial chaacteistics of the film such as the aveage numbe of objects pe unit volume N, the istibution function of the chaacteistic imensions of objects D, o the mean chaacteistic imension. We usually ae able to fin out some statistical paametes fom plana sections of the films o fom thei pojections. ssuming futhe spheical objects an knowing the mean numbe of section cicles pe unit aea N an the istibution function of iametes of section cicles D it is possible to solve the impotant poblem, fist solve by Wicksell 0. This leas to bel integal equation ( x) x (1) x whee an is ensity function of the istibution function D an D, espectively. Its solution can be witen [] as ( x) 1 x () π x x x fo 0. In the special case of the constant sphee iametes, we fin out that the equation (1) leas to fo 0. (3) If we have the section then the mean an the vaiance of the cicle iametes ae π (4) 4 σ ( 3 3π ). (5) 48 We can use these equations to test whethe the stuctue has the constant sphee iamete. compute moel is moe convenient fo mophological analysis of moe complex systems. The islans in composite films with low metal volume faction can be nealy spheical an anomly istibute [3]. In ou pape we eal with such a kin of composite films, islans of which have iffeent iamete an ae anomly istibute in a polyme matix.. Moels We pepae fou iffeent moels of composite films to analyse the influence of vaious conitions an phenomena on the unfoling pocess. ll basic moels have the main egion pixels (x, y, an z iections) an 100 pixels wie boe fo a compensation of bounay effects. The numbe of geneate objects is always the same objects in the inne pat (except fo the iscussion of eos). They ae anomly geneate, not touching each othe in feet, the minimum istance between objects is anothe paamete of ou moels. Fom the chosen filling facto (0.) of the film an the numbe of objects, we coul calculate the sphee iametes. Once we have geneate the composite stuctue we make a anom section paallel to the (x, y)-plane. In the next fou figues, we can always see the pojection of iffeent 48

Supeficies y acío 9, 48-5, Diciembe 1999 FIG. 1. Moel M1 with constant aii of sphees, R 16.84. stuctues geneate by ou moels, on the left sie, an the anom section of each stuctue, on the ight sie.

shows the moel M whee the aii of sphee objects ae unifomly istibute in the inteval <R 1, R>. In Fig.

4 - moel M4. Hee ae one pat of sphees geneate with a mean iamete R 1 an secon pat of sphees with anothe mean iamete R. 3.

2 Supeficies y acío 9, 48-5, Diciembe 1999 FIG. 1. Moel M1 with constant aii of sphees, R stuctues geneate by ou moels, on the left sie, an the anom section of each stuctue, on the ight sie. The pixel is the basic length unit eveywhee. Fig. 1 shows the moel M1 of the film with constant object aii R, whil Fig. shows the moel M whee the aii of sphee objects ae unifomly istibute in the inteval <R 1, R>. In Fig. 3 we can see the moel M3 of the film with Gaussian (nomal) istibution of the iametes of objects, an the supeposition of two Gaussian cuves is goun fo the geneating of object iametes in the Fig. 4 - moel M4. Hee ae one pat of sphees geneate with a mean iamete R 1 an secon pat of sphees with anothe mean iamete R. 3. Moments of the object istibution When we take into account a anom section of a composite film with a constant iamete D of sphees, we can see a set of cicles with iamete in the inteval <0, D>. If we consie these cicle iametes as anom vaiables then thei mean an the vaiance have to be given by Equations (4) an (5), espectively. s was peviously note we can use these fomulae to test whethe the object iametes ae actually constant. In this case it is also possible to eive the size of the sphee objects. The conition to be the sphee objects constant, is: σ ( ) 3 3 π π (6) FIG. 3. Moel M3 with Gaussian istibution of sphee aii with paametes: mean aius R 15.8, stana eviation σ 0.5 get the next values of the atio vaiance to mean squae: , , an In accoance with the given moels, we see that we can exclue the moels M to M4 fom ou futhe thinking because of the fomula (6). Howeve, the value fo moel M1 coespons to the citeion (6) not exactly. It leas to consieation about the pecision of the esults taken fom expeimental analysis of the plana sections of the composite films. The possible uncetainty has the two easons: fluctuations cause by the limite numbe of objects in the sample image igitization eo of the sample The next infomation, that the analysis of the section of composite stuctue yiels, is the stuctue-filling facto. It is equal to the coveage etemine fom the section, in the case of unifomly istibute objects. That fact hols fo any size an fom istibution of objects, as was showe aleay by a Fench geologist Delesse in 1847 [4]. Nevetheless, in this case the esults ae influence by the both above-mentione possible uncetainties. In the next two sections, we iscuss the eos of analysis of composite film sections base in all cases on the moel M1. Influence of limite numbe of objects In left Fig. 1 is shown the stuctue consisting of 1000 objects in active egion. The plana section (in the pplying the same analysis to the stuctues in ou fou moels M1 to M4 with the paametes pesente above we FIG.. Moel M with unifom istibution of sphee aii in the inteval <R 1, R >, R , R 3.0. FIG. 4. Moel M4 whee the istibution of the sphee aii has esulte fom the supeposition of two Gaussian istibutions with the paametes: mean aius R , stana eviation ó 1 0.5, elative numbe of the sphees 50%; mean aius R 0.0, stana eviation ó 0.10, elative numbes of the sphees 50%. 49

3 Supeficies y acío 9, 48-5, Diciembe 1999 ight) contains about 300 cicles an we got fo the atio (6) the value fom statistical analysis of thei iametes. Repeating the compute expeiments we get a set of values accoing Gaussian istibution N(ì, ó) with the mean ì compae to Eq. (6) - an with stana eviation ó In eal cases, we usually have no possibility to epeat sectioning of the stuctue an we have to o ou analysis fom one section. Then is suitable to apply the well-known 'thee sigma ule'. It says it is extemely impobable to have one paticula measue value iffeing moe than thee sigma fom the exact value, uing measuement of any physical value with Gaussian istibution of magnitues. In ou situation, the value of sigma will epen on the numbe of objects in the section. Table 1 illustates how many objects we have to take into consieation to get the esult with esie pecision. We can conclue that the fomula (6) is accepte with the uncetainty given in the Table 1 fo the stuctues fille by ientical sphee objects. nothe esults confim the possibility to estimate the stuctue-filling facto fom the coveage in a chosen section. Influence of igitization eo The pevious esults wee obtaine fo ieal case when each cicle iamete woul be eive exactly, e.g. by analogue measuements on micophotogaph at a big magnification. In fact, an obvious analysis is one with cetain magnitue of each pixel on the igitize micophotogaph. It bings the othe eo to the esults, which is supeimpose to the pevious uncetainty. In oe to analyze it we use the ata fom the moel M1. We have igitize this section aea with vaious magnitues of pixels, now. The esults of the statistical analysis we see in Table. The fist column gives how many pixels we can istinguish on the section aea. In the secon column is numbe of pixels fo the biggest cicle iamete D MX an thi one gives elative eo of the obtaine esult ( ) σ. The both types of the eos ae supeimpose in eal cases when we analyze existing plana sections of composite films. Compaing both the Tables 1 an we see that the influence of the limite numbe of objects is mostly moe impotant. Table. Statistical eo cause by igitization. ea of section [pixels] Diamete D MX [pixels] Eo of igitization [%] ,000 1, ,000, ,000 4, ,000 10, Size istibution of objects When the stuctues stuie consist of constant size objects, the analysis base on the statistical moments is sufficient. Howeve, in many cases the stuctues ae fome by objects - often again spheical - with cetain size istibution. The goal of the analysis is then to eive this size istibution. s pespective seems to be, the metho base on the integal tansfom [5]. It is vey ifficult to etemine the istibution of object sizes expeimentally. Nevetheless, the objects in the composite laye have spheical shape in many cases. In these cases the metho base on the so-calle Cho Length Distibution of Dak Segments (CHLD ak) [5] is applicable an seems to be vey pespective. This metho was peviously use fo the econstuction of the iscontinuous cicula metal islan aii (D poblem) fom so-calle ak chos [5]. Dak chos ae segments on the anomly oiente o paallel lines (in the anisotopic case) istibute ove the igitize Table 1. Reache pecision of the esults fo iffeent numbe of objects. Numbe of objects in the section 3 σ [%] , , FIG. 5. Constuction of the CHLD ak. Only segments that fully intesect an object ae taken into account (white fagments) 50

4 Supeficies y acío 9, 48-5, Diciembe 1999 D micophotogaph of the thin film intesecting the islans (Fig. 5). CHLD ak metho uses the statistics (nomalise histogams of lengths) of these chos as input ata an pouces istibution of the iametes on the output. If P(D) is the istibution of the cicula object iametes an χ (l) is the istibution of ak chos, a simple integal equation connects these two functions: l ( l) P( D) 1 1 D D χ (7) D In thee imensions, instea of lines we shall have anomly oiente o paallel planes, an cho lengths coespon to aeas of the cicles that oiginate fom intesection of the planes with objects. The integal equation emains vali. In fact, this equation is classifie as the oltea integal equation of the fist kin an this equation can be easily numeically solve by back-substitution. Its iscete equivalent is uppe tiangula matix. Because of the statistic natue of the input ata an because of the thei finiteness they seem to be noisy. Duing the solution some egulaization must take place to avoi hoible oscillations of the finite solution [6]. Binomial filteing of both input an output ata is suitable metho that can accomplish this [7]. Unfotunately any egulaization metho causes the loss of the infomation. In this case iscete peaks (objects with the same aii) ae wiene into peaks with finite half-with (see Fig. 6). The input ata fo the econstuction obtaine fom the plana section of moel M1 ae seen in Fig. 7. FIG. 6. Raius istibution in moel M1 (Fig. 1), actual (histogam) an econstucte (cuve). FIG.7. Size istibution of the cicle aeas in the plana section of the moel M1 (Fig. 1). The metho econstucts object sizes vey well with two exceptions: Due to the egulaization it cannot coectly econstuct istibution of object sizes if the objects ae of the same imension. Nevetheless the position of the maximum of the econstucte cuve coespons with the position of the iscete peak (see Fig. 6). Small imensions ae econstucte baly. Thee ae seveal easons fo it. The eo uing igitisation pocess manifests oneself mainly at small object sizes. nothe eo occus uing numeical solution. The back-substitution metho computes fist lage sizes an these esults ae use fo the computation of the smalle ones. ny eo is popagate such way an istibution at small sizes is affecte much moe (see Figs. 8 an 9) The use of this metho can help us to fin out the size istibution of objects in space. In the Figues 6, 8, an 9 ae epicte both the actual size istibutions by means of histogams an the size istibutions econstucte by the integal tansfom metho fom one section of the stuctue (continuous cuve). These istibutions ae taken fom ou vaious moels with constant aii of sphees (Fig. 6) o with Gaussian istibutions of sphee aii (Fig. 8 an 9). It is evient that this metho only appoximates the ight solution. Nevetheless, it povies valuable infomation about size istibution of objects although the pimay infomation was not complete. It is pobably impossible to gain it in othe way. 5. Discussion Ou moels ae well useable to stuy the unfoling poblems. Moeove, it is easy to tansfom them to stuy 51

5 Supeficies y acío 9, 48-5, Diciembe 1999 FIG. 8. Raius istibution in moel M3 (Fig. 3), actual (histogam) an econstucte (cuve).stuctues with othe foms of objects. We can conclue that the fomula (6) is well acceptable to fin out the stuctues with constant sphees but it is pobably ha to say it fo othe foms of objects. Even in case of sphees, the consieations about possible eos ae impotant. The uncetainty, cause by limite numbe of objects o by igitisation, was shown. Both the types of eos ae supeimpose in eal cases. We can say that the influence of the limite numbe of objects is mostly impotant fo the possible eo The metho of integal tansfom helps to econstuct the size istibution of objects in 3D stuctues. We ae convince that the metho is wiely useable not only fo spheical types of objects an that can be aapte fo the etemination of spatial istibution of objects, too. We ae woking on this poblem to be able both to obtain the spatial istibution of objects fom aea occuence in plana sections of composite films an to eive the paametes of composite films fo othe foms of objects. FIG. 9. Raius istibution in moel M4 (Fig. 4), actual (histogam) an econstucte (cuve). Refeences [1] S. D. Wicksell, Biometika 17, 84(195), 18, 15(196). [] D. Stoyan, W. S. Kenall, an J. Mecke, Stochastic Geomety an Its pplications (kaemie-elag, Belin, 1987). [3] H Bieeman an L Matinù, Plasma Deposition, Teatment an Etching of Polymes (caemic Pess, New Yok, 1990). [4] M.. Delesse, C. R. ca. Sci. (Pais) 5, 544(1847). [5] R. Hach an M. Sobotka, Inten. J. Electonics 69, 55(1990). [6] S. Chistiensen, in: Fomulae an Methos in Expeimental Data Evaluation, ol. 3, Es. R. K. Bock, K. Bos, S. Bant, J. Myheim, an M. Regle (Euopean Physical Society, CERN 1984) pat Q. [7] S. K Mita an J. F. Kaise, Hanbook fo Digital signal pocessing, (John Wiley & Sons, New Yok, 1993). cknowlegments This wok was suppote by The Ministy of Eucation of Czech Republic (Poject LB984), by the Euopean Commission une Contact INCO- COPERNICUS No. ERB IC 15 CT , an by Gant gency of Chales Univesity (gants GUK-50/1997 an GUK-179/1999). 5

PH126 Exam I Solutions

PH126 Exam I Solutions PH6 Exam I Solutions q Q Q q. Fou positively chage boies, two with chage Q an two with chage q, ae connecte by fou unstetchable stings of equal length. In the absence of extenal foces they assume the equilibium