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
MerlijnvanSpengen,etal]
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
diserlijnvanSpengen,etal] 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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 Fall '07
 Podhorodeski
 Force, Moment Of Inertia, Stress, Surface tension, Van der Waals force, N. Dechev

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