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A mathematical model for the cleansing of silicon substrates by fluid immersion
A Mathematical Model for the Cleansing of Silicon Substrates by Fluid Immersion S. B. G. O ' B R I E N 1 AND B. H. A. A. V A N D E N B R U L E Philips Research Laboratories, P.O. Box 80000, 5600 JA Eindhoven, The Netherlands
Received September 27, 1990; accepted December 10, 1990 The importance of cleanliness for the efficientmanufacture of integrated circuits is well known in the microprocessorindustry. The efficiencyof traditional cleansingmethods is known to decreasedramatically for dirt particles smaller than about 1 ~m in radius. Experimentallyit has been shown that the passage of a liquid-gas phase boundary along a substrate can be exploited to effectparticle removal. A model is derived here which explainsthe success of this process, illustrates the dependency of the method on the speed of immersion, and further showsthat it is theoreticallyindependent of particle size. ©1991Academic Press, Inc.
1. INTRODUCTION A dirt particle adhering to an integrated circuit (IC) decreases the stability and reliability of the IC. Furthermore if the process used in the manufacture of the IC is inherently "dirty," then the percentage of chips which function acceptably is also decreased. For rand o m access memories ( R A M s ) , it is generally accepted that particles whose size is about 1020% of the m i n i m u m pattern size of the IC are critical. For ICs of current interest, e.g., the 1-megabit static R A M and the 4-megabit dynamic RAM, the m i n i m u m pattern size is 0.7 ~zm. Particles of the order of 0.1 tzm are thus critical. It is generally felt that the minim u m pattern size will decrease for m a n y years to come and, thus, so will the critical particle size. The microprocessor industry has traditionally placed an emphasis on prevention of particle deposition by carrying out processes in "clean" environments with the philosophy that it is easier to prevent than to remove, but substrate pollution invariably takes place. Dirt particles occur in two general categories: in-
organic, e.g., metal oxides, and organic, e.g., bacteria. Existing cleansing methods fall into four categories (1): water jet scrubbing, mechanical scrubbing, ultrasonic cleaning, and megasonic cleaning. These are generally used in combination, but for submicron particles their success has been rather limited, though megasonic cleaning has been successful down to 0.3 # m (2). Below this level none of the existing processes are satisfactory. The existence of such a lower limit m a y at first sight seem puzzling but a brief look at the physics clarifies the matter. If we assume that there is no chemical bonding between a dirt particle (assumed spherical for simplicity) and a substrate as in Fig. 1, then adhesion is primarily caused by van der Waals forces for particles smaller than about 10 u m in radius. For this idealized case the adhesion force FA can be represented by
l To whom correspondenceshould be addressed at current address: OCIAM, Mathematical Institute, 2h-2g St. Giles, Oxford OX 1 3LB, England.
AR
FA = 6H2 ;
H,~R
[1]
where FA is the L o n d o n - v a n der Waals force, A is the H a m a k e r constant, R is the particle radius, and H is the gap between the particle and the substrate. During cleansing a force must be exerted on the particle which opposes
210 0021-9797/91 $3.00 Copyright © 1991 by Academic Press, Inc. All rights of reproduction in any form reserved.
Journal of Colloid and Interface Science, Vol. 144, No. 1, June 1991
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/
/ / FIG. 1. Spherical particle adhering to substrate during passage of phase boundary.
the adhesion force. Using the methods elucidated above, the removal force is generally exerted on the surface area of the sphere or on its volume. It will thus be proportional to the second or third power of the particle radius R. All other factors remaining the same, if we reduce the particle size then the forces of adhesion as in Eq. [ 1] will eventually dominate the removal forces as the former are linear in R. A cleansing technique which just succeeds for R = 1 izm will fail when R = 0.1 ~m. Once the problem is explained in this fashion, a theoretical solution seems obvious: simply invent a method which exerts a removal force linear in R. This is much more difficult than it seems but an ingenious solution was devised in (3) and extensively tested. The crux of the idea can be seen by referring to Fig. 1. The substrate to be cleaned is immersed in water and as the dirt particle passes through the liquid/gas phase boundary, the surface tension forces which originate at the contact line around the sphere can oppose the adhesion forces given favorable wetting conditions (contact angles). The contact angle for any triplet of solid, liquid, and gas is a constant for that combination. One way of changing it is to chemically treat the solid surface (socalled silylation) thus changing its affinity for liquid molecules. In Fig. 1, the substrate is not well wetted, the contact angle a being about 70 °. (A solid is said to be well wetted if the contact angle, measured through the liquid, is
small.) Similarly the particle is poorly wetted, the result being a favorable removal force as denoted by Ft. The crucial factor is that the capillary forces can be shown to be linear in R so the method is essentially independent of particle size. In this paper we shall summarize the experimental work performed. A theoretical model will then be developed to explain the phenomenon. Furthermore, as the experiments indicate, we shall see that the process is velocity dependent; i.e., successful cleaning only occurs if the immersion takes place at a low enough velocity. This fact will also be theoretically explained and some suggestions will be made for improving the process. 2. E X P E R I M E N T A L W O R K
Precise details of the experiments are to be found in (3). In summary a number of silicon substrates were contaminated with TiO2 (rutile), c~-Fe203 (hematite), and SiO2 (amorphous silica). Where necessary the contaminated substrates were silylated to favorably change the contact angles. The substrates were then passed slowly through a water interface (see Fig. 2). Before and after immersion the particles were sized and counted using an
Water
FIG. 2. Immersion of water. Journal of Colloid and Interface Science, Vol. 144,No. 1, June 1991
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electron microscope. In general from 70 to 97% of the particles was removed, given favorable wetting conditions as explained in the next section. The method was equally successful for particles as small as 0.1 #m provided the immersion velocity was in the range from 1 ~zm/s to about 1 cm/s. Almost no particles were removed for velocities of the order of several centimeters per second or higher. A typical result is shown in Fig. 3 for hematite particles. Immersion was stopped when the substrate was half submerged so only the left half has been cleaned. 3. STATICS
In the last section it was experimentally indicated that colloidal particles can be removed from a (silicon) substrate by the passage of a liquid-gas phase boundary. The most important parameters governing this process are the wetting properties of the substrate and the particles which determine contact angle values. For clarity we direct attention to the particular case of silica particles. The aim of the
analysis which follows is to produce at least a qualitative explanation of the phenomenon whereby no cleansing effect is observed when the phase boundary passes at a speed of about 10 cm/s, while high efficiencies are observed at lower speeds. Experiment suggests the existence of a critical velocity above which the process no longer works. As it might be expected that increased submersion velocity would increase the viscous drag and hence improve the cleansing efficiency, an understanding of the mechanism of removal is important if the process is to be optimized. It is thus important to identify the most significant forces acting on a particle. We begin with a purely static analysis of the forces on a particle floating in a horizontal liquid interface. The forces acting on the floating particle ( H = ~ ) in Fig. 4 are the gravity forces, the hydrostatic force of the liquid, and the surface tension force which acts along the circle of contact (the solid-liquid-gas contact line) and whose direction is completely determined by the position of the interface on the particle (angle ~)
FIG. 3. Dark field photograph of silicon substrate before and after cleansing. Journal of Colloid and Interface Science, Vol. 144,No. 1, June 1991
CLEANSING ~,.
O F SILICON S U B S T R A T E S
gas
of F~ is obtained by differentiating Eq. [4], giving the result 0 = 0/2. The component of this force in the horizontal direction is Fv cos a and in general we can expect a particle to come away from the wall if (F.~)maxCOS o~ > F A ,
F1G. 4. Removal of particle from horizontalsubstrate.
and the contact angle of the fluid with the particle (angle 0). The body force is (4/3)TrR3opg, where R is the particle radius, op is the particle density, and g is the gravitational acceleration. The hydrostatic force is obtained by integrating the pressure over the submerged part of the body and is given by /'rr/2- 4~
Fn = 2zrR2og J-~/2
[h + R (cos 4~
- sin 13)] sin/3 cos/3d/3
[2]
which on integration reduces to
FH = -- 7rR2ogh sin20 71"
+ -~ R3pg(cos3d? - 3 cos 4~ - 2),
[3]
p being the density of the liquid and h being the submergence as shown in Fig. 4, which is roughly of order R as we shall see in the next section. The surface tension forces are given by F~ = 2~rR3~ sin 0 sin(0 - 0),
213
[4]
with 3' being the surface tension of a liquidgas interface. For particle radii less than 10 #m the surface tension forces dominate and are consequently the only forces considered in this paper. Referring now to Fig. 1 where a silicon substrate is being lowered vertically into water, we note that for an acute substrate/liquid contact angle a there is a nonzero component of the resultant surface tension force acting normally away from the substrate. This force is opposed by the van der Waals adhesion force Fa given in Eq. [1]. The maximum value
[5]
a being the contact angle between the fluid and the substrate. This is then a necessary condition for removal but as experiment shows it is not sufficient. A full understanding of the process must lie in the dynamics of the problem which we investigate in the next section. It is possible, however, to determine when removal is impossible by consideration of Eq. [5]. For a substrate with substrate/liquid contact angle a = 20 ° and particle/liquid contact angle 0 = 0 °, it follows easily from Eq. [ 5 ] that removal cannot take place. If instead a = 0 = 76 ° then the removal forces are theoretically large enough to effect removal. This has been experimentally verified (3) as long as the immersion velocity was small enough. In the former case almost no cleansing took place regardless of speed of immersion. It is also relevant to point out at this stage that the ingenuity of the method lies in the fact that F~ like FA is proportional to R, unlike previous cleansing methods which exerted forces which are O ( R 2) or O(R3). Thus for very small R the Fv forces remain large enough to oppose the FA forces. 4. D Y N A M I C S
In general the contact angle at the contact line of a solid, liquid, and gas in equilibrium is a constant depending on the particular triplet considered. Variation of this angle can occur in the case of a moving contact line, experimental evidence indicating that the dynamic value can differ from the static angle. Furthermore contact angle hysteresis leads to a difference between advancing and receding angles. For the purposes of this paper we shall in general assume that the contact angle remains constant at its static value and ignore any hysteresis. This will be further discussed Journal of Colloid and Interface Science, Vol. 144, No. 1, June 1991
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later. The most direct way of performing a dynamic analysis would be to consider the situation in Fig. 1 as the fluid meniscus comes in contact with a particle, gradually envelops, and finally immerses it. As a result of the small dimensions involved, the Reynolds number Re is also small and the relevant equations of motion are the Stokes creeping flow equations Vp
=
]~VZu,
where u is the fluid velocity and p is the pressure. In addition we have the continuity equation for incompressible flow. A strategy for dealing with the problem is the solution of the above equation with the intention of finding the stress tensor. This problem would be extremely difficult to solve in view of its threedimensional nature and the existence of a moving free boundary. Furthermore the existence of a moving contact line gives rise to circumstances wherein it becomes necessary to enforce a slip boundary condition in the neighborhood of the contact line. Otherwise a stress singularity results (4). It could only be undertaken numerically, of course, and as the aim is to understand the process rather than to churn out numerical results, we choose instead to make simplifying assumptions in the hope of obtaining analytical results. The static analysis of the last section neglected viscous forces arising in the fluid and, as these are extremely important in Stokes flow, it is probable that the solution to the problem lies there. We might expect the viscous drag in the vertical direction to play a role, so we use the well-known Stokes law, which gives an expression for the drag on a sphere in a free stream, as an estimate of the vertical drag on the particle. It is given by F = 6rr~RV,
where ts is the fluid viscosity and V is the immersion velocity. On comparison of these forces with the surface tension forces we find that an immersion velocity of the order ofsevral meters per second is necessary for an approximate balance. Furthermore, higher veJournal of ColloM and Interface Science, Vol. 144, No. 1, June 1991
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locities would yield even higher drag forces which should then improve the cleansing efficiency. In any case the net effect of these drag forces could only be to exert a couple on the particle moving it along the substrate rather than away from it. As experimentally indicated in (3), the particles are translated away from the substrate and not along it. Because a fluid immersion velocity of several meters per second was not even vaguely approached experimentally, we make the assumption that the vertical drag plays no significant role. Thus as a further and extremely important assumption we will consider the situation in Fig. 4 as our model situation instead of that in Fig. 1. Thus wall effects are removed, the fluid level rising as a horizontal surface everywhere except in the neighborhood of a particle where capillary effects bend the meniscus. For analysis purposes, this has the further simplifying effect of giving rise to an axisymmetric fluid meniscus whose shape, at least in the static case, is easier to find. From a practical point of view, this setup has a number of advantages which are later discussed, the problem being that it is experimentally more difficult to achieve. We consider then Fig. 4 as the fluid level rises. If at a particular point the resultant surface tension force (which in this case is wholly opposing the adhesion force) is greater than the adhesion force, the particle will tend to move away from the wall. Then, as in lubrication theory, we expect the generation of very large viscous forces which offer great resistance to the further separation of sphere and substrate. Unlike the previously considered drag forces, these do act in the direction of motion, or rather they tend always to oppose motion. However, there is a vital difference here compared with our previous static model. It is no longer sufficient for the surface tension forces merely to be greater than the adhesion forces: this no longer guarantees removal. The viscous forces, which will presumably vary monotonically with the velocity of the particle, retard its removal. Meanwhile the fluid meniscus is passing over the particle, and complete removal requires that the latter have moved far
CLEANSING OF SILICON SUBSTRATES
enough away from the substrate to prevent it readhering before the surface tension forces on the particle drop to zero (i.e., before the particle passes wholly through the liquid/ phase boundary). In other words we require the particle to attain sufficient separation before the surface tension forces die away. As the particle starts from rest, we have
fo (Fr - FA -- F~)dt = my(t),
[6]
where F , are the viscous forces, and it becomes clear that the time for which the surface tension forces operate, and hence the velocity of fluid immersion, is crucial to ensure that the impulse acting on the particle is large enough. The model which we now propose is one-dimensional, in the direction of the resultant surface tension force. To formulate it precisely we must derive expressions for the viscous force on a spherical particle near a wall, and the surface tension force as a function of time (or immersion velocity). At this stage we can already hypothesize that anything which increases the area under the force-time curve (impulse) expressed in Eq. [6] will enhance the removal of a particle, e.g., increase in surface tension 3", decrease in fluid viscosity ~, and increase in time for which surface tension forces act. The further development of our model will verify these factors. We first derive an expression for the surface tension forces. 5. SURFACE TENSION FORCES AS A FUNCTION OF THE TIME
The basic form of the surface tension forces is given by Eq. [4] with q~ = q~(t). We assume that the meniscus rises smoothly on the particle with no sticking or sudden jumping and consider the situation occurring in Fig. 4 as the fluid level rises. The velocity at which the undisturbed meniscus rises is constant but will not be the same as the rate of rise of the circle of contact on the sphere. The former is given by dho~/dt = V, where ho~ is the height of the water meniscus far from the particle (at ov ),
215
i.e., the globally observed height. The difference between the two heights arises due to the effect of viscous and capillary forces on the fluid-free surface. The capillary number, Ca = isV/3", is the dimensionless quantity giving the ratio between the viscous and the capillary forces. For the parameter values of interest in the present paper (# = 10 -3 kg m -1 S - 1 , V 10 -1 m s -1, 3' = 7.26 X 10 -2 N m - l ) , Ca is small so we neglect the effect of the viscous forces in the formation of the free surface. We thus view the dynamic formation of the meniscus as a series of quasi-steady state problems. This avoids all the problems associated with a dynamic contact line; the shape of the meniscus is instead described by the wellknown Laplace capillary equation. Solution of this will yield a relationship between the undisturbed fluid meniscus height and the position of the contact line on the sphere as indicated by the angle q~ (see Fig. 4). We recall that the meniscus is axisymmetric in shape, the governing differential equation being the corresponding form of the Laplace equation written in cylindrical polar coordinates (r, z):
dZz/dr z 3" [1 + (dz/dr)2] 3/2
+
1
dz/dr
r [1 + (dz/dr) 2] 1/2
}
-pgz=O.
[7]
The zero on the right-hand side arises from a choice of axes such that z = 0 when r = ~ . We first nondimensionalize using as length scale a = (gp/3,)1/2 which has a value of approximately 0.368 × 103 m -1 for pure water at 20°C. If we scale as follows
y=az,
x=ar,
~=aR,
[8]
the problem in dimensionless form now becomes
[1 +
d2 y / dx 2 (dy/dx)2] 3/2 dy / dx + x[1 + (dy/dx)2] 1/2 - y = 0
[9]
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with the boundary conditions dy/dx
= - cot(0 + qs0) atx= dy/dx---~O
Ecosq~0
asx---~ oo.
[10] [11]
The first boundary condition reflects the fact that the contact angle condition must be satisfied at the sphere surface, while the second arises because the fluid is infinitely flat at ~ . The condition that y --~ 0 as x --~ oo is contained intrinsically in the equation. This equation can be solved numerically, this being rather laborious and complicated by the fact that it is a two-point boundary condition with one boundary at ~ . Previously tabulated results (5) were of no avail for the parameter ranges of interest here, so numerical solutions were obtained using finite elements. These were used to check the analytic solutions obtained in this paper, and also to solve the more interesting problem where the presence of a vertical wall removes the axisymmetry from the problem as in Fig. 1. These solutions will be reported in a later paper. In the present paper we require the position of the circle of contact on the sphere as a function of the time, t, for all values of t, essentially. Numerically we would have to solve the problem for m a n y different values of 4~ to obtain reasonable accuracy. A closed form solution is very desirable and noting the fact that the problem contains a small parameter, e (_-_10-3) for submicron particles, a perturbation solution is indicated. As E is so small, we expect good results even from a one- or two-term series. The zero-order solution will be a completely flat undisturbed meniscus. We formulate the problem in the following way: we assume that the undisturbed water level is given by an expression y = Ke + nE In E (if we work to this order in e) where y --~ 0 as e --~ 0 as it should; i.e., in the absence of the sphere the meniscus is completely flat. The inclusion of the EIn e terms is not a priori obvious but is necessary for the successful matching of the inner and outer solutions to be presently discussed. It is then necessary to modify the left-hand side of Eq. [9] by adding Journal of Colloid and Interface Science, Vol. 144, No. 1, June 1991
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~E In ~ + K~to it. Furthermore, we assume that the position of the circle of contact is known relative to the sphere as expressed in the angle q~ (or ~b0), and the a b s o l u t e position of the sphere is also known. As part of our solution we will have to estimate ~ and K: for this we need an extra boundary condition. Note that the y = 0 axis now passes through the center of the sphere so we know the height of the point of contact above the axis. This is the required extra condition. F r o m a physical point of view it appears that the solution far from the sphere will differ somewhat from that in its immediate vicinity. Far away the meniscus is nearly flat, while nearby we expect a local rise in order to satisfy the contact angle condition. If we recall van Dyke's maxim, that a perturbation problem is likely to be singular if the perturbation parameter is the ratio of two lengths, as here, then we must expect nonuniform behavior. We seek an outer solution in the form yO = ~yl(x) + ~ In c,
[12]
which is equivalent to assuming y and d y / d x small and neglecting higher order terms. On substituting [12 ] into [9 ], we obtained the modified Bessel equation of the second kind of zero order with a nonzero right-hand side for Yl, xy~ + y'l - x y l = Kx,
[ 13 ]
with the solution yl = B K o ( x ) - K,
[14]
where K o ( x ) is the modified Bessel function of second kind of zero order, satisfying the outer boundary condition [ 11 ]. The constant B will be determined later from the matching with the inner solution. For the inner solution we scale the variables as follows: x = e~', y = ~Y. We apply two scaled inner boundary conditions (b.c.'s), Y(~" = cos q~0) = sin q~0 dY
- - (~" = cos 4~o) -- - c o t ( 0 + 4'o). d~-
[15] [16]
CLEANSING OF SILICON SUBSTRATES
217
The first condition fixes the origin at the center of the sphere, and the second is the contact angle condition. The inner equation to lowest order can be written as
seen. However, after position 2, the only possible means of moving the contact line higher on the sphere is to lower the meniscus to position 3 at infinity. These solutions are not realizable during immersion, and the meniscus d2y l dY l fdy]3 flicks clear. If we consider the physical example :0. of this paper where 0 = oe = 75 °, then as the This is a Bernoulli equation in d Y / d £ yielding circle of contact moves upward from the bottom of the sphere, the surface tension forces the following solution: do not begin to oppose the adhesion forces ¥ = 6 ln(~'+ ~ f ~ - 62) + w. [18] until the contact line has passed above the From b.c.'s [10] and [11] we obtain 6 = cos(0 center line and ~b0 = (rr/2 - 0). At this point + q~o)COS 4~o and o~ -- sin q~o - 6 In(cos 4~0 the surface tension forces act horizontally, and + Vcos24~o - 62). Matching of the inner and hereafter have a positive component in the outer series can now take place yielding the upward direction until ~b0= 55 ° in the present case when the contact circle moves free. Note relations also that zero time for our model occurs when fl=-6, ~=-261n2+6"yc-~o, r/=6, the adhesion forces just equal the surface ten[19] sion forces. Figure 5 gives a representation of the surface where 3% is Euler's constant. As we are only tension forces versus the time for fast and slow interested in the relationship between the immersion. In reality this graph will be trunheight of the meniscus far from the sphere and cated at each end: at the left-hand end because the position of the circle of contact, there is removal cannot begin to take place until the no need to derive a composite series. A comforces are large enough, at the right-hand side parison of our solutions with previous nubecause of the phenomena discussed in the merical results (5) gives about 0.1% error for last paragraph. e = 0.01. For a particle of radius 0.1 tim, e - 3 × 10-5, so our two-term series should be ex6. T H E V I S C O U S F O R C E S tremely accurate. If we translate our axes so As a particle starts to move away from the that the meniscus at infinity is given by y = 0, substrate (see Fig. 4) we expect the occurrence then the relationship between the physical of strong viscous forces opposing the separaheight, h (see Fig. 4), and the angle ~bo,defintion. In order to approximate this flow we note ing the position of the circle of contact, is that the so-called "squeeze flow" between the h = ( 1 / a ) E r [ l n E + In(cos (k0
a-y+7-
-+Ttdrj
+Vcos2~bo-62 ) - l n 4 + 7 c ] .
[17]
[20]
The solutions obtained here can be used to investigate the behavior of the fluid meniscus as it rises from the middle to the top of the sphere (see Fig. 4). It can thus be observed that the contact line first rises gradually, but at a particular point (when d [ a h + ~ sin 40]/ d4~0 = 0), equilibrium ceases to become possible and the meniscus flicks clear of the sphere. Considering Fig. 6, as the meniscus at infinity rises from position I to 2, then the corresponding positions on the sphere can be
0,50 - -
slow
---
fast
0,40
/,
~ 0.30
x
/
0.20
0.10
0.00 0,00
1.57 t(s)
FIG. 5. Comparison of impulse delivered by F~ for slow
and fast immersion. Journal of Colloid and Interface Science, Vol. 144, No. 1, June 1991
218
O'BRIEN AND VAN DEN BRULE
2
= S ( r ) , where S ( r ) = H + r 2 / 2 R for r ~ R represents the separation between the substrate and the sphere, we obtain Vr = 1 / 2 # d p / d r [ z 2 - S ( r ) z ] .
FIG. 6. Position of circle of contact on sphere as fluid rises (contact angle 90 ° ).
Applying global continuity to the whole region fromz= 0toz= S(r) andr=Otor= r yields ~S(r)
2rrr Ja
sphere and the substrate may be modeled using the lubrication approximation, as the layer of fluid separating the two bodies is so thin. In the derivation which follows we use a redefined set of cylindrical polar coordinates whose origin is the intersection of a perpendicular from the sphere center to the substrate and the substrate itself. Distance from the substrate is denoted by z, the radial coordinate by r. We thus assume that Vr >> V~ and derivatives in the r direction are much smaller than those in the z direction. We seek a quasi-steady state solution; i.e., at any time the radial flow problem will be treated as a steady state hydrodynamic problem. The continuity equation is given by 10(rVr)
r
O~
OVz
+ ~ z = 0.
[21]
Using the following scales, assuming terms in Eq. [ 21 ] are of equal order, and noting that V is the velocity with which the sphere moves away from the wall, we have R* = r / R ; Uz = V z / V ;
[22]
and, for H / R ~ 1, the orders of magnitude of the individual stress components, r~r = rzz = too = O ( t ~ V / H ) , rr~ = O ( ~ R V ] H 2) allow the equation of motion (Navier-Stokes) to be simplified to dp 02V, dr - # Or2
[23]
withp = p ( r ) . Now using the no slip boundary conditions along the substrate and on the sphere surface, i.e., Vr = 0 on z = 0 and on z Journalof Colloidand InterfaceScience,Vol. 144,No. 1, June 1991
[251
Vrdz = -~rr2V.
Hence we obtain [26]
d p / d r = 6 R l s V / S 3.
We now obtain an expression for the pressure by straightforward integration and by now further integrating the stress over the sphere surface, we find F , = ~r(- 6taRV)
H/R + 1
+ H/~--R + 2 ( H / R
]
+ 0.5) 2 '
[27]
which for H ~ R reduces to F _~ - 6~r#VR2/ H . It is interesting to note that the above problem has been solved exactly (6) and for the dimensions under consideration (R = 0.3 #m, H ~ 1 nm) the above approximation yields a solution with 0.5% error. The advantage of using the approximate rather than the exact solution is that the former is far simpler in form and will allow us to obtain analytical solutions of our equation of motion. 7. EQUATION OF MOTION: SOLUTIONS
Z= z/H; Ur = H V r / R V
[24]
Referring to Fig. 4 we consider a force balance on a sphere moving away from the substrate, the sphere being considered a particle. Equilibrium occurs as result of a balancing of the inertial, surface tension, viscous, and adhesive (van der Waals) forces on the particle. Then, considering Eqs. [1], [4], and [27] we can formulate an initial value problem rnSd = 2~rR'y sin ~b sin(0 - q~) AR 6X 2
67r#R 2 -x
[281
X '
where m is the mass of the particle, the x di-
CLEANSING
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SILICON
rection points vertically upward in Fig. 4 in a one-dimensional Cartesian system, and x represents the smallest separation between sphere and substrate as a function of the time. q~ is also known as a function of the time. The boundary conditions are x ( 0 ) = H,
2(0) = 0,
[29]
where the dot is used to signify differentiation with respect to time. Using the following scales, we nondimensionalize
x* = x / H ,
t* = t v / R
[30]
219
SUBSTRATES
it is clear that the mass of the particle is tiny so we do not expect corresponding inertial terms to be important. Our solutions thus assume a quasi-steady state, the viscous forces playing an important damping role. We ignore the boundary condition involving the velocity and apply, in dimensionless form, x*(0) = 0.
[321
The solution to this equation is given by x*2(t*) = exp{ 2 [ / ( t * ) - 1(0)]}
whence we obtain
KSi* = G( t* )
),
X ~2
2?exp[2l(t*)]fo'*exp{-2l(O)}dO,
1 dx* X ~
dr*
'
[31] where
where
G( t* ) = sin[ 4~(R t * / V ) ]sin[ O - eh(rt*/V ) ], K-
2°pV2H 33,
X=
3/aV
6 '
A H23` 12~r '
- 3Ca, 3` where op is the density of the particle and Ca is the capillary number. This problem can be considered as a singular perturbation initial value problem as it is clear that the coefficient K is extremely small, much smaller than the other coefficients. However, to lowest order it turns out that the outer solution obtained by solving Eq. [ 31], ignoring the inertia terms and using only the first of the b.c.'s [29 ], gives rise to a solution which also satisfies the second b.c. Thus the problem does not display boundary layer behavior. No sudden changes in solution behavior occur because the surface tension forces are originally in equilibrium with the adhesion forces at time zero. Only in the case where a particle was suddenly immersed to a certain depth would genuine singular behavior occur. We thus choose to neglect the first term and essentially use only the outer solution of the problem using the initial condition on the displacement to determine the integration constant. Physically this is also perfectly reasonable as the first term represents the acceleration terms. In the physical problem
l(t) * =
f"
G(r)dr.
[33]
These solutions were checked using Gears' method for stiff ordinary differential systems (including the acceleration terms). It is gratifying to note that [33 ] is in agreement with what we already predicted in Section 4 and Fig. 5. The important factor in Eq. [33] is the 2/)t which identifies the dimensionless group 3`/#V, the reciprocal of the capillary number Ca which has arisen naturally in the course of the analysis. To optimize the process it is necessary to keep Ca as small as possible. Allowing Ca --~ ov lets the displacement x* --~ 1, its initial value, while allowing Ca --~ 0 lets x* --~ oe. This can be done by decreasing the velocity of immersion, increasing the surface tension, or decreasing fluid viscosity. The dimensionless form of Eq. [ 33 ] somewhat masks the fact that it is qualitatively represented by Fig. 5, the different time scaling being contained in the capillary number. If we instead solve the equation without nondimensionalizing we obtain as solution
x2(t) = H2exp
F~(r)dr
2A exp[ 27 f t F , ( r ) d r ] 36~-ttR [3~ d × f 0 ' e x p [ - 3-~2Y
('r,(v)dv]dr,
[341
Journal of Colloid and Interface Science, Vol. 144, No. 1, June 199 l
220
O'BRIEN AND VAN DEN BRULE
where Fo(t) = sin O(t)sin[0 - q~(t)]. In this instance the relevance of Fig. 5 can be seen more easily. Decreasing the velocity of immersion increases the area under the surface tension-time graph and maximizes the distance. The dimensionless solution has the merit of pointing out the relevant dimensionless groups. Application of the results attained here to the physical problem (Fig. 1 ), which makes the rather crude assumption that the meniscus is axisymmetric, in order to attain an estimate for the length of time for which the fluid meniscus remains in contact with the sphere (Fig. 1; 0 = a = 75 ° , R = 0.3/~m, # = 1 0 - 3 k g m -1 s - l , A = 1.5 × 10-19j, O = 103 kg m-3), predicts a cutoff velocity of the order of 25 cm/s. Figure 7 gives an idea of the trends involved. Above this value the process should no longer work, although experiment suggests that it is somewhat lower in the 10-15 cm/s range. It is worth noting that for the idealized problem of Fig. 4 the estimated critical velocity is significantly higher. It must be emphasized that, for the simple model chosen here, quantitative results can really only give order-ofmagnitude estimates. There are a number of factors in the problem which produce this uncertainty, not the least being the initial distance between the particle and the substrate ( H ) which we have taken to be 1 n m and the value of the Hamaker constant A. The former is known to be of the order of 1 nm, but a recent article ( 1 ) suggests that a more realistic value would be 0.6 nm. Using this latter value brings
-5 -6 -7 -8 -9
-10 -3.20
<____ r -2.96
i
__
-2,72
~ -2.48
L -2.24
-2.00
log(Ca)
FIG. 7. Terminal position of particle as function of immersion speed. Journal of Colloid and Interface Science, Vol, 144, No. 1, June 1991
theory much closer to the experimental values. Uncertainty also creeps in when the position of the dynamic meniscus is approximated by the static solution. It is worth noting that, as the capillary number Ca represents a comparison of the viscous and inertial forces, small values of Ca justify the static solution. However, it is rather unfortunate that the approximation becomes worse at precisely the point we are most interested in: when the removal method fails. A further possible source of error is estimating when a particle is actually free from the substrate. As the nature of the van der Waals forces is known to change for a separation of 10 -8 m, we arbitrarily assumed a particle to be free when it reached this distance, which is 10 times the initial separation. We also assume that the surface tension forces do not change over this distance. An alternative criterion for determining when a particle is free would be that the Stokes' forces are of the same order as the adhesion forces. 8. DISCUSSION
The approximate model derived here captures the essential features of the experimental process, i.e., the significance of the viscous forces and the velocity-dependent nature of the mechanism. One dynamic factor missing from the model is the variation of contact anNe with substrate velocity for the simple reason that no values of the dynamic contact angle for the case considered in this paper are known, be it theoretical or experimental. Reference to experimental values for capillaries (7) suggests that the contact angle does not vary much for the range of velocities considered here. However, this is far from certain, but until such time as more is known about the dynamic contact angle, the current model cannot be improved. The experimental method on which this paper is based might, however, serve as a possible means of answering an old question concerning the contact angle. It is known that the apparent or dynamic angle changes with substrate speed, but it is generally believed that the actual contact angle remains at the static value (8) while a
CLEANSING OF SILICON SUBSTRATES sudden change in phase boundary orientation at a distance of about 100 nm from the contact line gives rise to the apparent change. As such length scales are too small to allow direct measurement, an indirect method incorporating the experimental method of this paper could be used. The basic idea is to set up a series of test experiments with particles of decreasing radius and look for significant changes in the efficiency of the method when the critical length scale is reached, if it exists. The analysis in this paper identifies the capillary n u m b e r as the critical dimensionless parameter and indicates that the critical velocity of immersion can be increased by decreasing the ratio of u / ' r for the cleaning fluid. It was initially thought that increasing the viscosity of the fluid would improve the process, but this is clearly not the case as the model indicates. A further rather obvious i m p r o v e m e n t is also indicated: the experimental process as shown in Fig. 1 has the disadvantage that part of the removal power of the surface tension forces is lost due to the effect of the inclination a. The theoretical setup in Fig. 5 removes this problem completely and results in a m u c h
221
higher theoretical critical velocity. In practice this arrangement would be difficult to obtain when dealing with submicron particles. However, a more favorable arrangement than Fig. 1 could be obtained by submerging the substrates at an angle, thus striving for a setup between the extremes of Figs. 1 and 4. If the length of a substrate is given by L, then the time taken to clean one substrate would be of the order of R~ V instead of L~ V.
REFERENCES 1. Kim, S., and Lawrence, C. J., Chem. Eng. Sci. 43, 991 (1988). 2. Shwartzman,S., Mayer, A., and Kern, W., RCA Rev. 46, 81 (1981). 3. Leenaars, A. F. M., in "Particles on Surfaces: Detection, Adhesion and Removal" (K. L. Mittal, Ed.), p. 361. Plenum, New York, 1988. 4. Huh, C., and Scriven, L. E., J. Colloid Interface Sci. 35, 85 ( 1971 ). 5. Huh, C., and Scriven, L. E., Z Colloid Interface Sci. 30, 323 (1969). 6. Brenner, H., Chem. Eng. Sci. 16, 242 (1961). 7. Hoffman, R. L., aT. Colloid Interface Sci. 50, 228 (1975). 8. Dussan E. B., V, Annu. Rev. F[uid Mech. 11, 371 (1979).
Journal of Colloid and Interface Science, Vol. 144, No. 1, June 1991
Received September 27, 1990; accepted December 10, 1990 The importance of cleanliness for the efficientmanufacture of integrated circuits is well known in the microprocessorindustry. The efficiencyof traditional cleansingmethods is known to decreasedramatically for dirt particles smaller than about 1 ~m in radius. Experimentallyit has been shown that the passage of a liquid-gas phase boundary along a substrate can be exploited to effectparticle removal. A model is derived here which explainsthe success of this process, illustrates the dependency of the method on the speed of immersion, and further showsthat it is theoreticallyindependent of particle size. ©1991Academic Press, Inc.
1. INTRODUCTION A dirt particle adhering to an integrated circuit (IC) decreases the stability and reliability of the IC. Furthermore if the process used in the manufacture of the IC is inherently "dirty," then the percentage of chips which function acceptably is also decreased. For rand o m access memories ( R A M s ) , it is generally accepted that particles whose size is about 1020% of the m i n i m u m pattern size of the IC are critical. For ICs of current interest, e.g., the 1-megabit static R A M and the 4-megabit dynamic RAM, the m i n i m u m pattern size is 0.7 ~zm. Particles of the order of 0.1 tzm are thus critical. It is generally felt that the minim u m pattern size will decrease for m a n y years to come and, thus, so will the critical particle size. The microprocessor industry has traditionally placed an emphasis on prevention of particle deposition by carrying out processes in "clean" environments with the philosophy that it is easier to prevent than to remove, but substrate pollution invariably takes place. Dirt particles occur in two general categories: in-
organic, e.g., metal oxides, and organic, e.g., bacteria. Existing cleansing methods fall into four categories (1): water jet scrubbing, mechanical scrubbing, ultrasonic cleaning, and megasonic cleaning. These are generally used in combination, but for submicron particles their success has been rather limited, though megasonic cleaning has been successful down to 0.3 # m (2). Below this level none of the existing processes are satisfactory. The existence of such a lower limit m a y at first sight seem puzzling but a brief look at the physics clarifies the matter. If we assume that there is no chemical bonding between a dirt particle (assumed spherical for simplicity) and a substrate as in Fig. 1, then adhesion is primarily caused by van der Waals forces for particles smaller than about 10 u m in radius. For this idealized case the adhesion force FA can be represented by
l To whom correspondenceshould be addressed at current address: OCIAM, Mathematical Institute, 2h-2g St. Giles, Oxford OX 1 3LB, England.
AR
FA = 6H2 ;
H,~R
[1]
where FA is the L o n d o n - v a n der Waals force, A is the H a m a k e r constant, R is the particle radius, and H is the gap between the particle and the substrate. During cleansing a force must be exerted on the particle which opposes
210 0021-9797/91 $3.00 Copyright © 1991 by Academic Press, Inc. All rights of reproduction in any form reserved.
Journal of Colloid and Interface Science, Vol. 144, No. 1, June 1991
211
CLEANSING OF SILICON SUBSTRATES
/
/ / FIG. 1. Spherical particle adhering to substrate during passage of phase boundary.
the adhesion force. Using the methods elucidated above, the removal force is generally exerted on the surface area of the sphere or on its volume. It will thus be proportional to the second or third power of the particle radius R. All other factors remaining the same, if we reduce the particle size then the forces of adhesion as in Eq. [ 1] will eventually dominate the removal forces as the former are linear in R. A cleansing technique which just succeeds for R = 1 izm will fail when R = 0.1 ~m. Once the problem is explained in this fashion, a theoretical solution seems obvious: simply invent a method which exerts a removal force linear in R. This is much more difficult than it seems but an ingenious solution was devised in (3) and extensively tested. The crux of the idea can be seen by referring to Fig. 1. The substrate to be cleaned is immersed in water and as the dirt particle passes through the liquid/gas phase boundary, the surface tension forces which originate at the contact line around the sphere can oppose the adhesion forces given favorable wetting conditions (contact angles). The contact angle for any triplet of solid, liquid, and gas is a constant for that combination. One way of changing it is to chemically treat the solid surface (socalled silylation) thus changing its affinity for liquid molecules. In Fig. 1, the substrate is not well wetted, the contact angle a being about 70 °. (A solid is said to be well wetted if the contact angle, measured through the liquid, is
small.) Similarly the particle is poorly wetted, the result being a favorable removal force as denoted by Ft. The crucial factor is that the capillary forces can be shown to be linear in R so the method is essentially independent of particle size. In this paper we shall summarize the experimental work performed. A theoretical model will then be developed to explain the phenomenon. Furthermore, as the experiments indicate, we shall see that the process is velocity dependent; i.e., successful cleaning only occurs if the immersion takes place at a low enough velocity. This fact will also be theoretically explained and some suggestions will be made for improving the process. 2. E X P E R I M E N T A L W O R K
Precise details of the experiments are to be found in (3). In summary a number of silicon substrates were contaminated with TiO2 (rutile), c~-Fe203 (hematite), and SiO2 (amorphous silica). Where necessary the contaminated substrates were silylated to favorably change the contact angles. The substrates were then passed slowly through a water interface (see Fig. 2). Before and after immersion the particles were sized and counted using an
Water
FIG. 2. Immersion of water. Journal of Colloid and Interface Science, Vol. 144,No. 1, June 1991
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O'BRIEN AND VAN DEN BRULE
electron microscope. In general from 70 to 97% of the particles was removed, given favorable wetting conditions as explained in the next section. The method was equally successful for particles as small as 0.1 #m provided the immersion velocity was in the range from 1 ~zm/s to about 1 cm/s. Almost no particles were removed for velocities of the order of several centimeters per second or higher. A typical result is shown in Fig. 3 for hematite particles. Immersion was stopped when the substrate was half submerged so only the left half has been cleaned. 3. STATICS
In the last section it was experimentally indicated that colloidal particles can be removed from a (silicon) substrate by the passage of a liquid-gas phase boundary. The most important parameters governing this process are the wetting properties of the substrate and the particles which determine contact angle values. For clarity we direct attention to the particular case of silica particles. The aim of the
analysis which follows is to produce at least a qualitative explanation of the phenomenon whereby no cleansing effect is observed when the phase boundary passes at a speed of about 10 cm/s, while high efficiencies are observed at lower speeds. Experiment suggests the existence of a critical velocity above which the process no longer works. As it might be expected that increased submersion velocity would increase the viscous drag and hence improve the cleansing efficiency, an understanding of the mechanism of removal is important if the process is to be optimized. It is thus important to identify the most significant forces acting on a particle. We begin with a purely static analysis of the forces on a particle floating in a horizontal liquid interface. The forces acting on the floating particle ( H = ~ ) in Fig. 4 are the gravity forces, the hydrostatic force of the liquid, and the surface tension force which acts along the circle of contact (the solid-liquid-gas contact line) and whose direction is completely determined by the position of the interface on the particle (angle ~)
FIG. 3. Dark field photograph of silicon substrate before and after cleansing. Journal of Colloid and Interface Science, Vol. 144,No. 1, June 1991
CLEANSING ~,.
O F SILICON S U B S T R A T E S
gas
of F~ is obtained by differentiating Eq. [4], giving the result 0 = 0/2. The component of this force in the horizontal direction is Fv cos a and in general we can expect a particle to come away from the wall if (F.~)maxCOS o~ > F A ,
F1G. 4. Removal of particle from horizontalsubstrate.
and the contact angle of the fluid with the particle (angle 0). The body force is (4/3)TrR3opg, where R is the particle radius, op is the particle density, and g is the gravitational acceleration. The hydrostatic force is obtained by integrating the pressure over the submerged part of the body and is given by /'rr/2- 4~
Fn = 2zrR2og J-~/2
[h + R (cos 4~
- sin 13)] sin/3 cos/3d/3
[2]
which on integration reduces to
FH = -- 7rR2ogh sin20 71"
+ -~ R3pg(cos3d? - 3 cos 4~ - 2),
[3]
p being the density of the liquid and h being the submergence as shown in Fig. 4, which is roughly of order R as we shall see in the next section. The surface tension forces are given by F~ = 2~rR3~ sin 0 sin(0 - 0),
213
[4]
with 3' being the surface tension of a liquidgas interface. For particle radii less than 10 #m the surface tension forces dominate and are consequently the only forces considered in this paper. Referring now to Fig. 1 where a silicon substrate is being lowered vertically into water, we note that for an acute substrate/liquid contact angle a there is a nonzero component of the resultant surface tension force acting normally away from the substrate. This force is opposed by the van der Waals adhesion force Fa given in Eq. [1]. The maximum value
[5]
a being the contact angle between the fluid and the substrate. This is then a necessary condition for removal but as experiment shows it is not sufficient. A full understanding of the process must lie in the dynamics of the problem which we investigate in the next section. It is possible, however, to determine when removal is impossible by consideration of Eq. [5]. For a substrate with substrate/liquid contact angle a = 20 ° and particle/liquid contact angle 0 = 0 °, it follows easily from Eq. [ 5 ] that removal cannot take place. If instead a = 0 = 76 ° then the removal forces are theoretically large enough to effect removal. This has been experimentally verified (3) as long as the immersion velocity was small enough. In the former case almost no cleansing took place regardless of speed of immersion. It is also relevant to point out at this stage that the ingenuity of the method lies in the fact that F~ like FA is proportional to R, unlike previous cleansing methods which exerted forces which are O ( R 2) or O(R3). Thus for very small R the Fv forces remain large enough to oppose the FA forces. 4. D Y N A M I C S
In general the contact angle at the contact line of a solid, liquid, and gas in equilibrium is a constant depending on the particular triplet considered. Variation of this angle can occur in the case of a moving contact line, experimental evidence indicating that the dynamic value can differ from the static angle. Furthermore contact angle hysteresis leads to a difference between advancing and receding angles. For the purposes of this paper we shall in general assume that the contact angle remains constant at its static value and ignore any hysteresis. This will be further discussed Journal of Colloid and Interface Science, Vol. 144, No. 1, June 1991
214
O'BRIEN
AND
VAN
later. The most direct way of performing a dynamic analysis would be to consider the situation in Fig. 1 as the fluid meniscus comes in contact with a particle, gradually envelops, and finally immerses it. As a result of the small dimensions involved, the Reynolds number Re is also small and the relevant equations of motion are the Stokes creeping flow equations Vp
=
]~VZu,
where u is the fluid velocity and p is the pressure. In addition we have the continuity equation for incompressible flow. A strategy for dealing with the problem is the solution of the above equation with the intention of finding the stress tensor. This problem would be extremely difficult to solve in view of its threedimensional nature and the existence of a moving free boundary. Furthermore the existence of a moving contact line gives rise to circumstances wherein it becomes necessary to enforce a slip boundary condition in the neighborhood of the contact line. Otherwise a stress singularity results (4). It could only be undertaken numerically, of course, and as the aim is to understand the process rather than to churn out numerical results, we choose instead to make simplifying assumptions in the hope of obtaining analytical results. The static analysis of the last section neglected viscous forces arising in the fluid and, as these are extremely important in Stokes flow, it is probable that the solution to the problem lies there. We might expect the viscous drag in the vertical direction to play a role, so we use the well-known Stokes law, which gives an expression for the drag on a sphere in a free stream, as an estimate of the vertical drag on the particle. It is given by F = 6rr~RV,
where ts is the fluid viscosity and V is the immersion velocity. On comparison of these forces with the surface tension forces we find that an immersion velocity of the order ofsevral meters per second is necessary for an approximate balance. Furthermore, higher veJournal of ColloM and Interface Science, Vol. 144, No. 1, June 1991
DEN
BRULE
locities would yield even higher drag forces which should then improve the cleansing efficiency. In any case the net effect of these drag forces could only be to exert a couple on the particle moving it along the substrate rather than away from it. As experimentally indicated in (3), the particles are translated away from the substrate and not along it. Because a fluid immersion velocity of several meters per second was not even vaguely approached experimentally, we make the assumption that the vertical drag plays no significant role. Thus as a further and extremely important assumption we will consider the situation in Fig. 4 as our model situation instead of that in Fig. 1. Thus wall effects are removed, the fluid level rising as a horizontal surface everywhere except in the neighborhood of a particle where capillary effects bend the meniscus. For analysis purposes, this has the further simplifying effect of giving rise to an axisymmetric fluid meniscus whose shape, at least in the static case, is easier to find. From a practical point of view, this setup has a number of advantages which are later discussed, the problem being that it is experimentally more difficult to achieve. We consider then Fig. 4 as the fluid level rises. If at a particular point the resultant surface tension force (which in this case is wholly opposing the adhesion force) is greater than the adhesion force, the particle will tend to move away from the wall. Then, as in lubrication theory, we expect the generation of very large viscous forces which offer great resistance to the further separation of sphere and substrate. Unlike the previously considered drag forces, these do act in the direction of motion, or rather they tend always to oppose motion. However, there is a vital difference here compared with our previous static model. It is no longer sufficient for the surface tension forces merely to be greater than the adhesion forces: this no longer guarantees removal. The viscous forces, which will presumably vary monotonically with the velocity of the particle, retard its removal. Meanwhile the fluid meniscus is passing over the particle, and complete removal requires that the latter have moved far
CLEANSING OF SILICON SUBSTRATES
enough away from the substrate to prevent it readhering before the surface tension forces on the particle drop to zero (i.e., before the particle passes wholly through the liquid/ phase boundary). In other words we require the particle to attain sufficient separation before the surface tension forces die away. As the particle starts from rest, we have
fo (Fr - FA -- F~)dt = my(t),
[6]
where F , are the viscous forces, and it becomes clear that the time for which the surface tension forces operate, and hence the velocity of fluid immersion, is crucial to ensure that the impulse acting on the particle is large enough. The model which we now propose is one-dimensional, in the direction of the resultant surface tension force. To formulate it precisely we must derive expressions for the viscous force on a spherical particle near a wall, and the surface tension force as a function of time (or immersion velocity). At this stage we can already hypothesize that anything which increases the area under the force-time curve (impulse) expressed in Eq. [6] will enhance the removal of a particle, e.g., increase in surface tension 3", decrease in fluid viscosity ~, and increase in time for which surface tension forces act. The further development of our model will verify these factors. We first derive an expression for the surface tension forces. 5. SURFACE TENSION FORCES AS A FUNCTION OF THE TIME
The basic form of the surface tension forces is given by Eq. [4] with q~ = q~(t). We assume that the meniscus rises smoothly on the particle with no sticking or sudden jumping and consider the situation occurring in Fig. 4 as the fluid level rises. The velocity at which the undisturbed meniscus rises is constant but will not be the same as the rate of rise of the circle of contact on the sphere. The former is given by dho~/dt = V, where ho~ is the height of the water meniscus far from the particle (at ov ),
215
i.e., the globally observed height. The difference between the two heights arises due to the effect of viscous and capillary forces on the fluid-free surface. The capillary number, Ca = isV/3", is the dimensionless quantity giving the ratio between the viscous and the capillary forces. For the parameter values of interest in the present paper (# = 10 -3 kg m -1 S - 1 , V 10 -1 m s -1, 3' = 7.26 X 10 -2 N m - l ) , Ca is small so we neglect the effect of the viscous forces in the formation of the free surface. We thus view the dynamic formation of the meniscus as a series of quasi-steady state problems. This avoids all the problems associated with a dynamic contact line; the shape of the meniscus is instead described by the wellknown Laplace capillary equation. Solution of this will yield a relationship between the undisturbed fluid meniscus height and the position of the contact line on the sphere as indicated by the angle q~ (see Fig. 4). We recall that the meniscus is axisymmetric in shape, the governing differential equation being the corresponding form of the Laplace equation written in cylindrical polar coordinates (r, z):
dZz/dr z 3" [1 + (dz/dr)2] 3/2
+
1
dz/dr
r [1 + (dz/dr) 2] 1/2
}
-pgz=O.
[7]
The zero on the right-hand side arises from a choice of axes such that z = 0 when r = ~ . We first nondimensionalize using as length scale a = (gp/3,)1/2 which has a value of approximately 0.368 × 103 m -1 for pure water at 20°C. If we scale as follows
y=az,
x=ar,
~=aR,
[8]
the problem in dimensionless form now becomes
[1 +
d2 y / dx 2 (dy/dx)2] 3/2 dy / dx + x[1 + (dy/dx)2] 1/2 - y = 0
[9]
Journal of Colloid and Interface Science, Vol. 144, No. 1, June 1991
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O'BRIEN
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with the boundary conditions dy/dx
= - cot(0 + qs0) atx= dy/dx---~O
Ecosq~0
asx---~ oo.
[10] [11]
The first boundary condition reflects the fact that the contact angle condition must be satisfied at the sphere surface, while the second arises because the fluid is infinitely flat at ~ . The condition that y --~ 0 as x --~ oo is contained intrinsically in the equation. This equation can be solved numerically, this being rather laborious and complicated by the fact that it is a two-point boundary condition with one boundary at ~ . Previously tabulated results (5) were of no avail for the parameter ranges of interest here, so numerical solutions were obtained using finite elements. These were used to check the analytic solutions obtained in this paper, and also to solve the more interesting problem where the presence of a vertical wall removes the axisymmetry from the problem as in Fig. 1. These solutions will be reported in a later paper. In the present paper we require the position of the circle of contact on the sphere as a function of the time, t, for all values of t, essentially. Numerically we would have to solve the problem for m a n y different values of 4~ to obtain reasonable accuracy. A closed form solution is very desirable and noting the fact that the problem contains a small parameter, e (_-_10-3) for submicron particles, a perturbation solution is indicated. As E is so small, we expect good results even from a one- or two-term series. The zero-order solution will be a completely flat undisturbed meniscus. We formulate the problem in the following way: we assume that the undisturbed water level is given by an expression y = Ke + nE In E (if we work to this order in e) where y --~ 0 as e --~ 0 as it should; i.e., in the absence of the sphere the meniscus is completely flat. The inclusion of the EIn e terms is not a priori obvious but is necessary for the successful matching of the inner and outer solutions to be presently discussed. It is then necessary to modify the left-hand side of Eq. [9] by adding Journal of Colloid and Interface Science, Vol. 144, No. 1, June 1991
BRULE
~E In ~ + K~to it. Furthermore, we assume that the position of the circle of contact is known relative to the sphere as expressed in the angle q~ (or ~b0), and the a b s o l u t e position of the sphere is also known. As part of our solution we will have to estimate ~ and K: for this we need an extra boundary condition. Note that the y = 0 axis now passes through the center of the sphere so we know the height of the point of contact above the axis. This is the required extra condition. F r o m a physical point of view it appears that the solution far from the sphere will differ somewhat from that in its immediate vicinity. Far away the meniscus is nearly flat, while nearby we expect a local rise in order to satisfy the contact angle condition. If we recall van Dyke's maxim, that a perturbation problem is likely to be singular if the perturbation parameter is the ratio of two lengths, as here, then we must expect nonuniform behavior. We seek an outer solution in the form yO = ~yl(x) + ~ In c,
[12]
which is equivalent to assuming y and d y / d x small and neglecting higher order terms. On substituting [12 ] into [9 ], we obtained the modified Bessel equation of the second kind of zero order with a nonzero right-hand side for Yl, xy~ + y'l - x y l = Kx,
[ 13 ]
with the solution yl = B K o ( x ) - K,
[14]
where K o ( x ) is the modified Bessel function of second kind of zero order, satisfying the outer boundary condition [ 11 ]. The constant B will be determined later from the matching with the inner solution. For the inner solution we scale the variables as follows: x = e~', y = ~Y. We apply two scaled inner boundary conditions (b.c.'s), Y(~" = cos q~0) = sin q~0 dY
- - (~" = cos 4~o) -- - c o t ( 0 + 4'o). d~-
[15] [16]
CLEANSING OF SILICON SUBSTRATES
217
The first condition fixes the origin at the center of the sphere, and the second is the contact angle condition. The inner equation to lowest order can be written as
seen. However, after position 2, the only possible means of moving the contact line higher on the sphere is to lower the meniscus to position 3 at infinity. These solutions are not realizable during immersion, and the meniscus d2y l dY l fdy]3 flicks clear. If we consider the physical example :0. of this paper where 0 = oe = 75 °, then as the This is a Bernoulli equation in d Y / d £ yielding circle of contact moves upward from the bottom of the sphere, the surface tension forces the following solution: do not begin to oppose the adhesion forces ¥ = 6 ln(~'+ ~ f ~ - 62) + w. [18] until the contact line has passed above the From b.c.'s [10] and [11] we obtain 6 = cos(0 center line and ~b0 = (rr/2 - 0). At this point + q~o)COS 4~o and o~ -- sin q~o - 6 In(cos 4~0 the surface tension forces act horizontally, and + Vcos24~o - 62). Matching of the inner and hereafter have a positive component in the outer series can now take place yielding the upward direction until ~b0= 55 ° in the present case when the contact circle moves free. Note relations also that zero time for our model occurs when fl=-6, ~=-261n2+6"yc-~o, r/=6, the adhesion forces just equal the surface ten[19] sion forces. Figure 5 gives a representation of the surface where 3% is Euler's constant. As we are only tension forces versus the time for fast and slow interested in the relationship between the immersion. In reality this graph will be trunheight of the meniscus far from the sphere and cated at each end: at the left-hand end because the position of the circle of contact, there is removal cannot begin to take place until the no need to derive a composite series. A comforces are large enough, at the right-hand side parison of our solutions with previous nubecause of the phenomena discussed in the merical results (5) gives about 0.1% error for last paragraph. e = 0.01. For a particle of radius 0.1 tim, e - 3 × 10-5, so our two-term series should be ex6. T H E V I S C O U S F O R C E S tremely accurate. If we translate our axes so As a particle starts to move away from the that the meniscus at infinity is given by y = 0, substrate (see Fig. 4) we expect the occurrence then the relationship between the physical of strong viscous forces opposing the separaheight, h (see Fig. 4), and the angle ~bo,defintion. In order to approximate this flow we note ing the position of the circle of contact, is that the so-called "squeeze flow" between the h = ( 1 / a ) E r [ l n E + In(cos (k0
a-y+7-
-+Ttdrj
+Vcos2~bo-62 ) - l n 4 + 7 c ] .
[17]
[20]
The solutions obtained here can be used to investigate the behavior of the fluid meniscus as it rises from the middle to the top of the sphere (see Fig. 4). It can thus be observed that the contact line first rises gradually, but at a particular point (when d [ a h + ~ sin 40]/ d4~0 = 0), equilibrium ceases to become possible and the meniscus flicks clear of the sphere. Considering Fig. 6, as the meniscus at infinity rises from position I to 2, then the corresponding positions on the sphere can be
0,50 - -
slow
---
fast
0,40
/,
~ 0.30
x
/
0.20
0.10
0.00 0,00
1.57 t(s)
FIG. 5. Comparison of impulse delivered by F~ for slow
and fast immersion. Journal of Colloid and Interface Science, Vol. 144, No. 1, June 1991
218
O'BRIEN AND VAN DEN BRULE
2
= S ( r ) , where S ( r ) = H + r 2 / 2 R for r ~ R represents the separation between the substrate and the sphere, we obtain Vr = 1 / 2 # d p / d r [ z 2 - S ( r ) z ] .
FIG. 6. Position of circle of contact on sphere as fluid rises (contact angle 90 ° ).
Applying global continuity to the whole region fromz= 0toz= S(r) andr=Otor= r yields ~S(r)
2rrr Ja
sphere and the substrate may be modeled using the lubrication approximation, as the layer of fluid separating the two bodies is so thin. In the derivation which follows we use a redefined set of cylindrical polar coordinates whose origin is the intersection of a perpendicular from the sphere center to the substrate and the substrate itself. Distance from the substrate is denoted by z, the radial coordinate by r. We thus assume that Vr >> V~ and derivatives in the r direction are much smaller than those in the z direction. We seek a quasi-steady state solution; i.e., at any time the radial flow problem will be treated as a steady state hydrodynamic problem. The continuity equation is given by 10(rVr)
r
O~
OVz
+ ~ z = 0.
[21]
Using the following scales, assuming terms in Eq. [ 21 ] are of equal order, and noting that V is the velocity with which the sphere moves away from the wall, we have R* = r / R ; Uz = V z / V ;
[22]
and, for H / R ~ 1, the orders of magnitude of the individual stress components, r~r = rzz = too = O ( t ~ V / H ) , rr~ = O ( ~ R V ] H 2) allow the equation of motion (Navier-Stokes) to be simplified to dp 02V, dr - # Or2
[23]
withp = p ( r ) . Now using the no slip boundary conditions along the substrate and on the sphere surface, i.e., Vr = 0 on z = 0 and on z Journalof Colloidand InterfaceScience,Vol. 144,No. 1, June 1991
[251
Vrdz = -~rr2V.
Hence we obtain [26]
d p / d r = 6 R l s V / S 3.
We now obtain an expression for the pressure by straightforward integration and by now further integrating the stress over the sphere surface, we find F , = ~r(- 6taRV)
H/R + 1
+ H/~--R + 2 ( H / R
]
+ 0.5) 2 '
[27]
which for H ~ R reduces to F _~ - 6~r#VR2/ H . It is interesting to note that the above problem has been solved exactly (6) and for the dimensions under consideration (R = 0.3 #m, H ~ 1 nm) the above approximation yields a solution with 0.5% error. The advantage of using the approximate rather than the exact solution is that the former is far simpler in form and will allow us to obtain analytical solutions of our equation of motion. 7. EQUATION OF MOTION: SOLUTIONS
Z= z/H; Ur = H V r / R V
[24]
Referring to Fig. 4 we consider a force balance on a sphere moving away from the substrate, the sphere being considered a particle. Equilibrium occurs as result of a balancing of the inertial, surface tension, viscous, and adhesive (van der Waals) forces on the particle. Then, considering Eqs. [1], [4], and [27] we can formulate an initial value problem rnSd = 2~rR'y sin ~b sin(0 - q~) AR 6X 2
67r#R 2 -x
[281
X '
where m is the mass of the particle, the x di-
CLEANSING
OF
SILICON
rection points vertically upward in Fig. 4 in a one-dimensional Cartesian system, and x represents the smallest separation between sphere and substrate as a function of the time. q~ is also known as a function of the time. The boundary conditions are x ( 0 ) = H,
2(0) = 0,
[29]
where the dot is used to signify differentiation with respect to time. Using the following scales, we nondimensionalize
x* = x / H ,
t* = t v / R
[30]
219
SUBSTRATES
it is clear that the mass of the particle is tiny so we do not expect corresponding inertial terms to be important. Our solutions thus assume a quasi-steady state, the viscous forces playing an important damping role. We ignore the boundary condition involving the velocity and apply, in dimensionless form, x*(0) = 0.
[321
The solution to this equation is given by x*2(t*) = exp{ 2 [ / ( t * ) - 1(0)]}
whence we obtain
KSi* = G( t* )
),
X ~2
2?exp[2l(t*)]fo'*exp{-2l(O)}dO,
1 dx* X ~
dr*
'
[31] where
where
G( t* ) = sin[ 4~(R t * / V ) ]sin[ O - eh(rt*/V ) ], K-
2°pV2H 33,
X=
3/aV
6 '
A H23` 12~r '
- 3Ca, 3` where op is the density of the particle and Ca is the capillary number. This problem can be considered as a singular perturbation initial value problem as it is clear that the coefficient K is extremely small, much smaller than the other coefficients. However, to lowest order it turns out that the outer solution obtained by solving Eq. [ 31], ignoring the inertia terms and using only the first of the b.c.'s [29 ], gives rise to a solution which also satisfies the second b.c. Thus the problem does not display boundary layer behavior. No sudden changes in solution behavior occur because the surface tension forces are originally in equilibrium with the adhesion forces at time zero. Only in the case where a particle was suddenly immersed to a certain depth would genuine singular behavior occur. We thus choose to neglect the first term and essentially use only the outer solution of the problem using the initial condition on the displacement to determine the integration constant. Physically this is also perfectly reasonable as the first term represents the acceleration terms. In the physical problem
l(t) * =
f"
G(r)dr.
[33]
These solutions were checked using Gears' method for stiff ordinary differential systems (including the acceleration terms). It is gratifying to note that [33 ] is in agreement with what we already predicted in Section 4 and Fig. 5. The important factor in Eq. [33] is the 2/)t which identifies the dimensionless group 3`/#V, the reciprocal of the capillary number Ca which has arisen naturally in the course of the analysis. To optimize the process it is necessary to keep Ca as small as possible. Allowing Ca --~ ov lets the displacement x* --~ 1, its initial value, while allowing Ca --~ 0 lets x* --~ oe. This can be done by decreasing the velocity of immersion, increasing the surface tension, or decreasing fluid viscosity. The dimensionless form of Eq. [ 33 ] somewhat masks the fact that it is qualitatively represented by Fig. 5, the different time scaling being contained in the capillary number. If we instead solve the equation without nondimensionalizing we obtain as solution
x2(t) = H2exp
F~(r)dr
2A exp[ 27 f t F , ( r ) d r ] 36~-ttR [3~ d × f 0 ' e x p [ - 3-~2Y
('r,(v)dv]dr,
[341
Journal of Colloid and Interface Science, Vol. 144, No. 1, June 199 l
220
O'BRIEN AND VAN DEN BRULE
where Fo(t) = sin O(t)sin[0 - q~(t)]. In this instance the relevance of Fig. 5 can be seen more easily. Decreasing the velocity of immersion increases the area under the surface tension-time graph and maximizes the distance. The dimensionless solution has the merit of pointing out the relevant dimensionless groups. Application of the results attained here to the physical problem (Fig. 1 ), which makes the rather crude assumption that the meniscus is axisymmetric, in order to attain an estimate for the length of time for which the fluid meniscus remains in contact with the sphere (Fig. 1; 0 = a = 75 ° , R = 0.3/~m, # = 1 0 - 3 k g m -1 s - l , A = 1.5 × 10-19j, O = 103 kg m-3), predicts a cutoff velocity of the order of 25 cm/s. Figure 7 gives an idea of the trends involved. Above this value the process should no longer work, although experiment suggests that it is somewhat lower in the 10-15 cm/s range. It is worth noting that for the idealized problem of Fig. 4 the estimated critical velocity is significantly higher. It must be emphasized that, for the simple model chosen here, quantitative results can really only give order-ofmagnitude estimates. There are a number of factors in the problem which produce this uncertainty, not the least being the initial distance between the particle and the substrate ( H ) which we have taken to be 1 n m and the value of the Hamaker constant A. The former is known to be of the order of 1 nm, but a recent article ( 1 ) suggests that a more realistic value would be 0.6 nm. Using this latter value brings
-5 -6 -7 -8 -9
-10 -3.20
<____ r -2.96
i
__
-2,72
~ -2.48
L -2.24
-2.00
log(Ca)
FIG. 7. Terminal position of particle as function of immersion speed. Journal of Colloid and Interface Science, Vol, 144, No. 1, June 1991
theory much closer to the experimental values. Uncertainty also creeps in when the position of the dynamic meniscus is approximated by the static solution. It is worth noting that, as the capillary number Ca represents a comparison of the viscous and inertial forces, small values of Ca justify the static solution. However, it is rather unfortunate that the approximation becomes worse at precisely the point we are most interested in: when the removal method fails. A further possible source of error is estimating when a particle is actually free from the substrate. As the nature of the van der Waals forces is known to change for a separation of 10 -8 m, we arbitrarily assumed a particle to be free when it reached this distance, which is 10 times the initial separation. We also assume that the surface tension forces do not change over this distance. An alternative criterion for determining when a particle is free would be that the Stokes' forces are of the same order as the adhesion forces. 8. DISCUSSION
The approximate model derived here captures the essential features of the experimental process, i.e., the significance of the viscous forces and the velocity-dependent nature of the mechanism. One dynamic factor missing from the model is the variation of contact anNe with substrate velocity for the simple reason that no values of the dynamic contact angle for the case considered in this paper are known, be it theoretical or experimental. Reference to experimental values for capillaries (7) suggests that the contact angle does not vary much for the range of velocities considered here. However, this is far from certain, but until such time as more is known about the dynamic contact angle, the current model cannot be improved. The experimental method on which this paper is based might, however, serve as a possible means of answering an old question concerning the contact angle. It is known that the apparent or dynamic angle changes with substrate speed, but it is generally believed that the actual contact angle remains at the static value (8) while a
CLEANSING OF SILICON SUBSTRATES sudden change in phase boundary orientation at a distance of about 100 nm from the contact line gives rise to the apparent change. As such length scales are too small to allow direct measurement, an indirect method incorporating the experimental method of this paper could be used. The basic idea is to set up a series of test experiments with particles of decreasing radius and look for significant changes in the efficiency of the method when the critical length scale is reached, if it exists. The analysis in this paper identifies the capillary n u m b e r as the critical dimensionless parameter and indicates that the critical velocity of immersion can be increased by decreasing the ratio of u / ' r for the cleaning fluid. It was initially thought that increasing the viscosity of the fluid would improve the process, but this is clearly not the case as the model indicates. A further rather obvious i m p r o v e m e n t is also indicated: the experimental process as shown in Fig. 1 has the disadvantage that part of the removal power of the surface tension forces is lost due to the effect of the inclination a. The theoretical setup in Fig. 5 removes this problem completely and results in a m u c h
221
higher theoretical critical velocity. In practice this arrangement would be difficult to obtain when dealing with submicron particles. However, a more favorable arrangement than Fig. 1 could be obtained by submerging the substrates at an angle, thus striving for a setup between the extremes of Figs. 1 and 4. If the length of a substrate is given by L, then the time taken to clean one substrate would be of the order of R~ V instead of L~ V.
REFERENCES 1. Kim, S., and Lawrence, C. J., Chem. Eng. Sci. 43, 991 (1988). 2. Shwartzman,S., Mayer, A., and Kern, W., RCA Rev. 46, 81 (1981). 3. Leenaars, A. F. M., in "Particles on Surfaces: Detection, Adhesion and Removal" (K. L. Mittal, Ed.), p. 361. Plenum, New York, 1988. 4. Huh, C., and Scriven, L. E., J. Colloid Interface Sci. 35, 85 ( 1971 ). 5. Huh, C., and Scriven, L. E., Z Colloid Interface Sci. 30, 323 (1969). 6. Brenner, H., Chem. Eng. Sci. 16, 242 (1961). 7. Hoffman, R. L., aT. Colloid Interface Sci. 50, 228 (1975). 8. Dussan E. B., V, Annu. Rev. F[uid Mech. 11, 371 (1979).
Journal of Colloid and Interface Science, Vol. 144, No. 1, June 1991