Chapter 25
Stripe Patterns Formed in Particle Films: Cause and Remedy 1,3
1,4
2
Downloaded by STANFORD UNIV GREEN LIBR on October 11, 2012 | http://pubs.acs.org Publication Date: October 15, 1996 | doi: 10.1021/bk-1996-0648.ch025
Eiki Adachi , Antony S. Dimitrov , and Kuniaki Nagayama 1
Nagayama Protein Array Project, ERATO, JRDC 5-9-1 Tokodai, Tsukuba, Ibaraki 300-26, Japan Department of Life Sciences, Graduate School of Arts and Sciences, University of Tokyo, Komaba, Meguro-ku, Tokyo 153, Japan
2
When a droplet of suspension of particles dries on a glass surface, particles collect near the edge of the droplet (contact line) and often leave a striped pattern of particles as water evaporates. During drying processes, the motion of the droplet contact line resembles stick-slip motion and it shrinks towards the center of the droplet with an oscillatory motion. To explain the oscillatory motion and the mechanism of the stripe formation, we formulated a mathematical model that includes a friction force which the contact line feels when particles flow from the inside of the droplet to the droplet edge. As a result of competition between this friction force and surface tensions at the contact line, the droplet oscillates as it dries and generates a striped film composed of particles.
When a suspension droplet dries on a solid substrate, we often observe striped patterns of particles, which remain on the substrate after evaporation of the water. Natural and artificial stripe patterns are shown in Figure 1 for examples. Natural strips, on a car body and on a cup wall, are composed by dusts and coffee powders, respectively. The stripe of dusts was found after a rainy day on a waxed car body, which means droplets were formed on it. The coffee powder stripe was found on a Monday morning, which was left on a desk on the last Friday night. These kind of patterns, of course, can be obtained by using artificial particles such as polystyrene particles. They imply existence of common principle for striping. When we examine the drying process, we observe that the droplet contact line shrinks towards the center of the droplet with an oscillatory motion (1), causing generation of a particle-array film at the contact line. The shrinking motion of the droplet can be broadly classified as a "stick-slip" motion (2), but the motion we 3
Current address: National Institute for Physiological Sciences, Okazaki, Aichi 444, Japan Current address: L'Oreal Tsukuba Center, 5-5 Tokodai, Tsukuba 300-26, Japan
4
0097-6156/96/0648-0418$15.00/0 © 1996 American Chemical Society In Film Formation in Waterborne Coatings; Provder, T., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1996.
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25. ADACHI ET AL.
Stripe Patterns Formed in Particle Films
419
Figure 1. The stripes formed in nature. (A) the stripe of dusts on a car body are some time found after a rainy day when the car body is waxed, which means droplets can be formed on the body. (B) the stripe of coffee powders on a cup wall is formed when one leaves his cup with coffee on a desk for several night. We can find other stripes, for examples, in a kitchen, in a bath room, and in a toilet, where water is there. (C) when a suspension droplet including mono-dispersed particles dries on a glass plate, regular stripes can be obtained.
In Film Formation in Waterborne Coatings; Provder, T., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1996.
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420
FILM FORMATION IN WATERBORNE COATINGS
observed is different from the stick-slip motion that has been observed by other researchers (3-5). In general, an object undergoing stick-slip motion periodically converts its kinetic and internal energy to thermal energy. The periodic energy dispersion is caused by coupling of the friction at the contact surface with the object motion. In addition to the energy dispersion, the motion of the suspension droplet accompanies discharges of suspended particles from the inside to the boundary of the droplet (6). The particles assemble to form particle-array films (7,8) at the droplet contact line, which is caused by particle flow induced by the evaporation of water from the film surface near the receding contact line. We call this process, convective self-assembly of particles, which characteristically occurs in the thin liquid layer of particle suspensions (9). Since the particle flow is viscous, the flow affects the stick-slip motion of the droplet as a source of friction at the droplet contact line. On the other hand, the shrinking motion affects the self-assembly of particles. This coupled system must show a pattern or structure in the final particle-array films. Model of Droplet Edge for Formulation of Striping To investigate the pattern resulting from these competing processes, we derived a mathematical model for the motion of the contact line of the shrinking droplet by introducing the friction of the contact line into the equations of motion, and setting this friction proportional to the particle flow at the contact line. In the model we assume a circular, sessile suspension droplet containing spherical particles on a solid substrate. We further assume the droplet edge is composed of a monolayer of particles that wets the surface with the suspension (see Figure 2). We define the contact line as the intersection of the extrapolated meniscus with the plane of the wetting-film surface. To derive a relation between the particle flow and the motion of the droplet contact line, we first assume the droplet shrinks towards its center. We further assume that a water flow (J ) induced by evaporation (J ) from the array and the film surface, coupled with viscous drag on the particles, carries the particles from inside of the droplet to the boundary of the particle arrays. Therefore, the total particle motion can be described as a flow, J . Although Je causes energy transfer, we simply consider J is the cause of inducing of J and Je Since the suspension flow (a mixture of J and Jw) is viscous, a friction force, s, acts on the wetting-film surface close to the contact line so as to prevent shrinkage of the droplet. The surface tension of the wetting film (y ) also tends to prevent droplet shrinkage, but the surface tension of the droplet (y ) tends to cause droplet shrinkage. As a result, y competes y with and s at the droplet contact line. w
e
P
e
w
P
f
L
L
f
Motion of Contact Line and Stripe Number Depending on Particle Volume Fraction Let h, h, and ν be the shear viscosity, the wetting-film thickness, and the velocity of the suspension flow at the film surface, respectively. Since a velocity of the suspension flow, w, is written as u=zv/h (at z=0, w=0; z=A, w=v) at low ν (laminar flow) (10), σ is written as
In Film Formation in Waterborne Coatings; Provder, T., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1996.
25. ADACHI ET AL.
421
Stripe Patterns Formed in Particle Films
(1) When the suspension is treated as a continuous fluid, ν is defined from the averaged momentum of the water and particles as h
1 zv 1 -^dzp— = -pv = m J + m J P
P
w
(2)
w
where /wp and /w are the masses of the particle and the water molecules, respectively. Downloaded by STANFORD UNIV GREEN LIBR on October 11, 2012 | http://pubs.acs.org Publication Date: October 15, 1996 | doi: 10.1021/bk-1996-0648.ch025
w
In eq 2, ρ is the average density of the suspension, defined as Ρ =ΡΦ + Ρ
1
ΡΛ ~Φ)
where φ is the particle volume fraction in the wetting film and p and p are the mass density of a particle (m /V \ and the mass density of a water molecule ( w / F ) , respectively (F and F are the volumes of a particle and of a water molecule, respectively). The flows, J and J , are defined as p
?
P
w
?
w
w
w
?
w
Vw
Vρ
Where, v and v are the velocities of particle and water molecules, respectively. From eq 2, ν is expressed by J . Therefore, eq 1 takes the form P
w
?
_2η
σ
βφ + 41-φ) h βφ[φ + ε(1-φ)]
where, β = v / v and ε = Pw/pp This equation relates the particle flow, J , to the motion of the contact line through an equation of motion of the droplet. When p*, R, and θ are a effective mass density of the contact line (adjustable parameter), the droplet radius, and the effective contact angle of the droplet, respectively, the equation of motion is written by using eq 3 P
w
P
2
*d R 2
s\ 2η
βφ+ε(1-φ)
3
dt h βφ[φ+ε(1-φ)] where / is time. The first and second terms on the right side of eq 4 are the wetting film surface tension and the horizontal projection of the droplet surface tension, respectively. The third term on the right side of eq 4 represents the friction force. Since the solutions that represent the motion of the droplet contact line, J = J (/) and R = R(t), are necessary, we need another relation between J and R. This relation can be obtained by considering two conservation laws for the particle number and for the total volume of the particle and water molecules during the array formation and film wetting. If we assume that the wetting film thickness at the contact line is equal to the particle diameter, the particle number conservation law can be written as P
P
In Film Formation in Waterborne Coatings; Provder, T., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1996.
P
422
FILM FORMATION IN WATERBORNE COATINGS
2π\%άηηφ (Γ)
= 2πγP^dtrh
ρ
J
(5)
P
where R = R(t = 0). The left side represents the total particle number in the particle array and in the wetting film. The right side represents the particle number transferred to the particle array and the wetting film from the droplet across the contact line from time 0 to J. In eq 5, φ is the volume fraction of the particles in the particle array and in the wetting film. The total volume of particles and water molecules, which move from inside the suspension droplet to the wetting film, should be balanced with the volume of water evaporating from the particle array and the wetting-film surface and with the volume of the new wetting film formed at the contact line. This conservation law can be expressed as 0
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Ρ
2KRh(V Jp+VwJw) P
e
w
R
(6)
.dR
R 0
2nJ V \ drr[l^ (r)]+2nR-^h p
at Combining eqs 5 and 6 then yields d(RJ ) p
R— + V RJ dt P
hV []+(0-l)
dt
F
P
h_d_( J y dt\ w
dR dt.
R
(7)
By combining eqs 4 and 7, we derive a non-linear differential equation for the motion of the contact line as 3
dR 3
dt
dR
2
^d R
2
+ 2 λ ^+
2
2
(λ +μΊ^Γ-ν (λ +μ ) dt
dt
5
(8)
J_dR R dt
dt
dt' JeVw
where
Υ_(r -r cose)h f
L
βφ
A =
j
2η
2
P'h
and ε is assumed to be one for simplicity. Note that the fifth term on the left side of eq 8 is nonlinear. When this term is sufficiently small (i.e., R is large compared with dR/dt), eq 8 simplifies to a third-order linear differential equation. If the contact angle, Θ, and the volume fraction, φ, are assumed to be constant during the formation of stripes, an oscillatory solution of eq 8 is obtained as a function of / as
xt
R = R(t) = ^ + V \t - ^-[e s
ύη(μ + 2a) - sin 2a] j
In Film Formation in Waterborne Coatings; Provder, T., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1996.
(9)
25. ADACHI ET AL.
Stripe Patterns Formed in Particle Films
423
Where
λ
-ι
a = - tan —
μ
Downloaded by STANFORD UNIV GREEN LIBR on October 11, 2012 | http://pubs.acs.org Publication Date: October 15, 1996 | doi: 10.1021/bk-1996-0648.ch025
The corresponding boundary conditions for eq 9 are
The number of stripes per unit length, N , along the droplet radius is estimated by the particle density in a particle-array film. Since the particle density is proportional to J (t)/V(i) [V(t)= d/?(/)/d/], an area becomes distinguishable from other areas when V(t) tends to zero with finite J (t). The area is then recognized as a dense stripe. Therefore, the number of real roots of the equation, V(t) = 0, is equivalent to the number of stripes. It is clear from eq 9 that the real roots exist only in a limited range, 0
