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MECH466-Lecture-5

O pwith stthteon onmemration of gu w 9a win obtain fr

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Unformatted text preview: oteworthy when strongly polar molecules or elements within materials are in contact. © N. Dechev, University of Victoria 33 The Effective Contact Area Between Two Surfaces A rough surface is generally modeled as a rough plane with a mean height and a standard deviation using a Gaussian A physical model to predict stiction in MEMS surfaces, we are interested in the distance probability density distribution of heights, as shown below: function of the surfaces, because this function is going to give us the amount of surface at a speciﬁc distance, and hence A t h t si l mode redict ic i c in ﬁgu S its inﬂuence on the poyalcaenerglyt.o pWith stthteon onMEMration of ﬁgu w 9a win obtain, fr the t istance ht bab ribu i e fu surfaces,re e , re e terested inom dhe heigprodistility tdonsitynctions, this d tt e ce aces, bility f this o n han z . F g t e G function ofishansurfprobabecauseunctifun,ctiob (is)goinortohgive aussian [Image from us theisaributnon,f tsurfcalculation eiciﬁc disltlance, and vience urfaces d t mou i t o he ace at a sp s as fo ows. Ha h ng s Merlijn-van-Spengen,et-al] its inﬂuendistoibutionotala (z) and hb (ith givecoby guration of with ce rn the t s h energy. W z) the n nﬁ za Surface a ﬁgure 9, we obtain, from the height distribution functions, 1 za )2 ¯ this distance probability function, hab ((z .−For the Gaussian ¯ ha (z, za ) = √ exp − z) and distribution, the calculation πs as follows2σa2 aving surfaces .H σa 2 i with distributions ha (z) and hb (z) given by 2 za Figure 8 Gaace Modeling of the equilibrium.Surfussiaan distribubetweenhetwo surfaces in distance tion of surface ights. contact. Determination of equilibrium distance is not trivial,b(and= √ exp −− ()z 2σz2b ) , a 1 (z z − ¯ ¯ h z, zb ) 1 ¯ b and ¯ p h (z, z ) = √ σb ex2π − Figure 8 Gaus n distribution o su ace 2σ good reference is W. Merlijn.SvansiaSpengen,f etrfal.heights. σ 2π urface b the distribution of the distance between the surfaces 7) (h a a 2 a a a zb 0 Surface b Surf ace a zb 0 1 √ σb 2π ¯¯ hab (z, za + zb ) = (z − zb )2 ¯ , 2 1 2σb exp − exp − ab (z) (z − (za + zb ))2 ¯¯ 2 2 the distribution of the distance σetween the surfaces2haσ(z+ iσb ba ) s 2π b 2 + σ 2 ¯¯ hab (z, za + zb ) = a 1 b exp − 2 from 2case a2in wbhich special π σ + σ (z − (za + zb ))2 ¯¯ . 2 2 σa2 + σb is . (8) [Image For the za + zb = d0 , the equilibrium ¯ ¯ M tance, we will call the distan diserlijn-van-Spengen,et-al] ce distribution function hd0 (z). (8) S l rf a o f s Figure 9. Modeuing ce aurface roughnesses and equilibrium Of course, the Gaussian approximation is rather crude, and distance. For tbecopecialecasesin ul hich n athe zborces 0 ,ettheeequtilibriurfaces are he s mes l ss u ef w whe z + ¯f = d b w en he su m ¯ distavery we willbecause dif thece eviatioution fm the n hd0distribution nce, weak, call the o stan d 34 distrib ns fro unctioreal (z). di u anc s. I e ing k surf t e o disn ib es a n, we ibri m Figstre 9.eMod...
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