Vertical Composition Profile During Hydrodynamic-Evaporative Film Thinning:
The Physics of Spin Casting Dilute Solutions
Stefan Karpitschka,∗ Constans M. Weber, and Hans Riegler
We analyze the evolution of the vertical composition profile during hydrodynamic-evaporative
film thinning as it typically occurs during spin casting mixtures of non-volatile solutes and volatile
solvents. We assume that the solvent dominates the hydrodynamic-evaporative film thinning. The
internal spatio-temporal evolution of the composition is analyzed with a diffusive-advective approach. The analysis provides transparent physical insights into the influence of the experimental
conditions on the evolution of the internal composition. We present power laws that link the process
control parameters to the composition evolution, process duration, and final solute coverage. The
analysis reveals a characteristic Sherwood Number as fundamental process parameter. It identifies
for which stages of the process our analysis is quantitatively relevant and discloses the dominance
of either diffusion or evaporation. The analysis is valid for dilute solutions e.g., for the deposition
of solute (sub)monolayers. But it is also relevant for the deposition of thicker (polymer) films.
PACS numbers: 68.03.Fg, 47.57.eb, 47.85mb, 82.70.-y
Many studies have analyzed this process experimentally [2–45] and theoretically [1, 46–161]. Most of
them focus on the radial (in)stability of the film [2–
5, 7, 10, 12, 13, 16, 18, 20, 26, 29, 30, 32, 40–44, 46,
51, 52, 54, 55, 60, 61, 70, 71, 73–75, 78–80, 83–85, 91–
94, 97, 99, 100, 103, 106, 109–117, 119–121, 124, 126–
136, 139–145, 147–151, 153, 154, 157, 158, 161], or systems with complicated rheologies such as polymer solutions [2–4, 6, 7, 9–11, 13, 17, 21, 26–28, 30, 31, 33–35, 37–
39, 43, 45, 50–52, 58, 65, 76, 81, 89, 98, 105, 122, 125, 155,
156] (the main application of spin casting is depositing
polymer films) or other non-newtonian liquids [40, 44, 46,
49, 51, 54, 78, 94, 95, 102, 103, 142, 143, 161]. Rather few
studies [48, 57, 65–68, 76, 78–80, 86, 146, 155, 156] aim at
elucidating the general, fundamental physics of this process. Due to different approximations/approaches there
is no agreement on the final amount of solute deposition [65, 86, 156, 162]. Very few studies explicitly investigate the evolution of the internal film composition. Often
they assume specific cases and/or approximations such as
solute ”boundary layers” [65, 76, 78], or ”split mechanism
models” [86]. Some analysis is solely based on numerics [66–68] or combining numerics with analytical considerations assuming complicated nonlinear viscosity and
diffusivity behavior [156]. In any case, all those studies
early phase
(spin-off dominated)
film
Introduction.—Excess amounts of liquid deposited on a
spinning, planar, wettable substrate form a thinning film
of uniform thickness h as the liquid flows outward [1].
Evaporation of volatile film components adds to the
film thinning (Fig. 1). Nonvolatile components continuously enrich, eventually exceed saturation, and precipitate. The general, fundamental physics questions in this
hydraulic-evaporative (spin cast) process are: How thins
the film with time? How evolves the film composition
during thinning? How much nonvolatile solute is finally
deposited?
height h
arXiv:1205.3295v3 [physics.flu-dyn] 10 Jan 2014
Max-Planck-Institut für Kolloid- und Grenzflächenforschung, Potsdam-Golm, Germany
(Dated: January 13, 2014)
z
late phase
(evaporation dominated)
solvent evaporation (rate E)
flow field
spinning support
solute enrichment
r
angular frequency w
2
spin-off coefficient K µ w
FIG. 1: Schematics of spin-casting with processes dominating
early and late stages.
use vertical ”initial” Peclet numbers based on inappropriate (see [86]) ”initial” film heights [174]. Thus their
validity ranges are ill-defined and the insight into the
general physics of this interesting hydraulic-evaporative
process is limited.
Increasingly spin casting is used for fundamental nucleation and growth studies [163, 164] and for the deposition of structured (sub)monolayers (particle arrays,
“evaporation-induced self-assembly” [164–172]). This
means dilute solutions, which behave rather ideally during most of the spin cast process. This is the motivation
and basis of our analysis.
In this report we focus on the evolution of the
vertical composition profile during the hydrodynamicevaporative film thinning. To reveal its general aspects
we neglect any complicated, non-linear solution behavior. Thus, for the first time palpable, universal relations
between the process parameters and the compositional
evolution are presented. The key process parameter, its
characteristic Sherwood number Shtr is introduced. It is
demonstrated that the analysis is indeed quantitatively
valid for low solute concentrations but also yields valuable insights into the behavior of higher concentrated so-
2
10
Hydrodynamic film thinning (E = 0, dashed) dominates ≈30% of the spin cast time; ≈70% is evaporationdominated (K = 0, dashed). All this agrees well with
experimental results (see inset in Fig. 2 and supplement [176]).
II. Solute concentration evolution — During spin casting, non-volatile solute enriches at the free surface (where
the solvent evaporates) and migrates into the film via
diffusion. The spatio-temporal evolution of the solute
concentration c is described by
ttr ÷tsc
1
only evaporatio
0.01
Reflectometry
Fit of Eq. (5)
4
htr = 3.08 ± 0.01 µm
tsc = 1.20 ± 0.01 s
2
0
0.0
0.001
0.0
Toluene, 3000 rpm, 20°C
6
h [µm]
h÷htr
8
0.1
only spinning
n
0.2
0.4
0.2
0.6
0.8
1.0
1.2 t [s]
0.4
0.6
0.8
1.0
∂t c = D ∂z2 c + K z 2 (3h − z) ∂z c,
t/ t sc
FIG. 2: Universal thinning curve h/htr = f (t/tsc ) (dashed
lines: thinning due to spinning/evaporation only). Inset:
Measured thinning curve in physical units for toluene and
its fit according to the theory (Eq. 5).
lutions including polymeric solutes.
I. Hydrodynamic-evaporative film thinning — The
thinning of a Newtonian, volatile liquid film of thickness
h on a rotating support is described by [48]:
dh/dt = −2K h3 − E ,
(1)
with spin-off coefficient K = ω 2 /(3ν), ω = rotational
speed, ν = kinematic viscosity and E = evaporation rate.
The fundamental form of Eq. 1,
dξ/dτ = −ξ 3 − 1 ,
(2)
is obtained by rescaling ξ = h/htr and τ = t/t∗sc i.e., by
the system inherent “natural” scales (see [146]):
htr = (E/2K) /3 ,
(3)
t∗sc = (2E 2 K)− /3 .
(4)
1
1
htr is the “transition height” where evaporative and hydrodynamic thinning are equal. t∗sc is the ”reduced process duration” (Eq. 6).
The inverse of Eq. 2 can be integrated [146] [175]:
)
√ (
1 − 2ξ
3
1
1 − ξ + ξ2
τ (ξ) =
π + 2 arctan √
, (5)
+ √ log
6
(1 + ξ)2
3
3
with integration constants so that τ = 0 for ξ → ∞.
The total spin cast time (from h → ∞ to h = 0) is
tsc = (2π/33/2 )(2E 2 K)− /3 = (2π/33/2 ) · t∗sc .
1
(6)
Fig. 2 shows the universal thinning curve (the inverse
of Eq. 5) on scaled axes. The transition time, ttr , at
which htr is reached, is universal for all spin cast processes described by Eq. 1:
√
ttr = 2π − 3 log 4 /(4π) · tsc ≈ 0.309 · tsc . (7)
with boundary conditions
D ∂z cz=h = E cz=h ,
∂z cz=0 = 0.
(8)
(9)
The advective term in Eq. 8 is derived from the radial
velocity field u(r, z) of a Newtonian fluid rotating with
its solid support (no slip), with a free surface at the top
(no stress) [1]:
u(r, z) = 3K r z (h − z/2) .
(10)
r and z are the radial respectively vertical coordinates.
The radial volumetric flux φ dz is
φ dz = 2π r u(r, z) dz = 6π K r2 z (h − z/2) dz.
(11)
With the continuity equation this yields the thinninginduced vertical motion of the horizontal stream lines:
Z z
∂r φ dz ′ = −K z 2 (3h − z) . (12)
dZ/dt = −1/(2π r)
0
In order to solve Eq. 8, h(t) is required but not known
explicitly, only its inverse, t(h) (Eq. 5). Since h(t) respectively τ (ξ) are bijective, the time variable can be changed
1
to ξ: ∂t c = dτ /dt · dξ/dτ · ∂ξ c = −(2E 2 K) /3 (ξ 3 + 1) ∂ξ c.
Rescaling y = z/h (y ∈ [0, 1], from substrate to surface)
avoids the moving boundary. This leads to:
∂y2 c
(ξ y)2 (3 − y) y
∂ξ c = −
∂y c,
−
−
Shtr ξ 2 (ξ 3 + 1)
2(ξ 3 + 1)
ξ
(13)
∂y cy=1 = Shtr ξ cy=1 ,
∂y cy=0 = 0,
(14)
where the Sherwood number, Shtr , parameterizes the ratio of evaporative to diffusive mass transport on the characteristic length scale of the system, htr :
Shtr = E htr /D = E
4/3
(2K)− /3 /D.
1
(15)
By scaling c with the initial solute concentration c0 the
initial condition is cξ→∞ = 1. Thus the system is
parametrized completely by Shtr . Eq. 13 can be solved
numerically with ξ as independent variable. Eq. 5 then
provides τ as function of ξ i.e., finally, c(t, z).
3
1.2
20
2.0
Sh = 4.00
Sh = 1.00
Sh = 0.25
Time
/
c c0
3.0
0.4
0.6
0.8
100
1.0
z/h
FIG. 3: Solute concentration profiles (resulting from Eq. 13)
for three different Sherwood numbers Shtr at three different
moments respectively heights h/htr .
c c0
N A c0 htr
III. General aspects of the concentration evolution —
Based on the solution of Eq. 13 the impact of Shtr (i.e.,
of the individual system parameters K, E, and D) is
now analyzed. Fig. 3 shows profiles of c during film thinning for Shtr larger and smaller than 1 (i.e., convection
dominating over diffusion and vice versa) and for film
heights larger and smaller than htr , respectively. It reveals the competition between evaporative enrichment,
spin-off, and diffusive dilution. In general, larger Shtr
means more pronounced gradients in c. If h ≫ htr , solute enrichment occurs only locally near the free surface
and c ≈ c0 near the substrate. If h ≪ htr , c also increases
near the substrate. Shtr = 1 and h = htr mark the transition: The solute gradient just reaches the film/substrate
interface and c just begins to increase globally.
Fig. 4 shows as function of h/htr : A) The total solute
A
20
10
5
2
1
20.0
10.0
5.0
2.0
1.0
Time
Total Solute Amount
Surface
c c0
Sh 10- 3
Sh 2 102
B
Sh
Sh
Sh
0.25
1.00
4.00
C
Sh
Sh
Sh
1.00
0.25
1.00
4.00
cmax
hmax
0.5
1.0
5.0
10.0
0.1
1
10
100 Sh
FIG. 5: Normalized final solute coverage, N (h → 0)/(Ac0 htr )
and c/c0 at the film surface for h = htr as function Shtr .
Rh
amount N per unit area A (N/A = 0 c dz); B) c/c0
at the surface respectively substrate/film interface; C)
The difference between the surface and the substrate/film
interface concentrations, ∆c/c0 i.e., the relative enrichment. Both, h = htr and Shtr = 1 mark transitions
between distinctly different behaviors. I) For h ≫ htr
spin-off dominates film thinning. Therefore, globally
c/c0 remains approximately constant and N/A decreases.
Nevertheless, there is a surficial solute enrichment due
to evaporation. But this ∆c/c0 only becomes substantial for Shtr > 1. It has a maximum at h = hpeak
(hpeak ≈ htr for Shtr ≈ 1). II) For h ≪ htr , evaporation
dominates. Solvent loss leads to increasing c/c0 while
N/A and ∆c/c0 remain constant (because spin-off becomes negligible). Remarkably, for h < htr , N/(Ac0 htr )
remains approximately constant and independent from
Shtr whereas ∆c/c0 increases with increasing Shtr .
Fig. 5 shows c(z = h = htr )/c0 , the concentration at
the free surface for h = htr , and N (h → 0)/(Ac0 htr ),
the rescaled final coverage. Both are plotted as function of Shtr . For Shtr ≪ 1 diffusion dominates and
c is mostly homogeneous (Fig. 3). In this case c0 Etsc
is the total amount of solute that is not spun off with
the solvent. Distributing this into a film with htr yields
c/c0 = Etsc /htr = 2π/33/2 ≈ 1.2 (Fig. 5). The final
solute coverage for Shtr < 1 is (in agreement with [88])
1
0.10
0.1
0.01
Γ = N (h → 0)/A ≈ c0 htr ≈ 0.8 c0 (K/E)− /3 .
Substrate
5.00
0.9
0.8
1.0
0.2
1.0
5
1
0.001
4.
0.0
10
2
1
1.5
1.1
c0 htr
c z h htr c0
h/htr = 0.25
NAh 0
5.0
h htr
FIG. 4: For various Sherwood numbers Shtr are shown as
function of the reduced film thickness h/htr : A) Total solute
amount N per per unit area A (scaled by c0 htr ); B) Concentrations c at the surface respectively film/substrate interface (scaled by c0 ); C) ∆c, the difference between the surface
and the substrate/film interface concentrations, respectively
(scaled by c0 ).
(16)
For Shtr ≫ 1 it is only ≈ 6% lower.
IV. Relative surficial enrichment maximum — Fig. 6
shows ∆cpeak /c0 and its spatio-temporal properties as
function of Shtr . Symbols denote the results from the
numerical analysis. The solid lines show power laws for
Shtr < 1 which are rationalized by analyzing the underlying processes.
Panel A) shows hpeak rescaled by htr i.e., the film thickness at which ∆c = ∆cpeak as function of Shtr . ∆cpeak
emerges from the competition between evaporative enrichment, spin-off, and diffusional equilibration. Evaporation and spin-off dominate at opposite ranges of h.
For large h, surficial spin-off efficiently suppresses enrichment: ∆cpeak requires 2Kh3peak < E (from Eq. 1).
However, for small h, diffusional equilibration is fast, so
4
10.00
hmax htr
A
1.00
0.10
tmax tsc
B
1.00
0.10
0.01
cmax C0
100
C
1
0.01
10
4
0.001
0.01
0.1
1
10
Sh
100
FIG. 6: ∆cpeak /c0 as a function
of Shtr and related
spatio
temporal properties (hpeak =h∆c
and tpeak =t∆c
). The
peak
peak
symbols indicate numerical results, the lines the underlying
power laws for Shtr < 1.
∆cpeak requires D/hpeak < E. Optimizing both conditions simultaneously (E is linear and protagonistic in
both cases, and therefore cancels out) reveals the same
power law as numerics (which supplies the prefactor):
hpeak ≈ (D/K) /4 ≈ 1.2 htr Shtr − /4 .
1
1
(17)
Panel B) shows the time tpeak rescaled by tsc i.e., the
time at which ∆c = ∆cpeak . Before reaching hpeak , thinning is dominated by spin-off. Hence tpeak can be estimated by inserting hpeak into h = (2K t)−1/2 (the solution
to Eq. 1 with E = 0):
tpeak /tsc ≈ 0.31 E /3 K − /6 D− /2 ≈ 0.35
2
1
1
p
Shtr .
(18)
Panel C presents ∆cpeak /c0 , reflecting the balance between evaporative
enrichment and diffusive equilibration
3
i.e., Shtr h
= E hpeak /D = Shtr /4 . With Eq. 17 this
peak
means:
∆cpeak /c0 ≈ 0.46 E K − /4 D− /4 ≈ 0.55 Shtr
1
3
3/4
.
(19)
Discussion and Conclusions — Section I introduces
the system-specific fundamental length and time scales
(htr , t∗sc ) for the spin casting process of an ideal Newtonian volatile liquid. These reduce the general spin cast
equation (Eq. 1) to its fundamental form [146] and lead
to a universal film thinning behavior (Eq. 5). The total
spin coating time, tsc is calculated (Eq. 6) as function of
the process parameters (E, K, D). For any combination
of these parameters, in the first 30% of the process time,
thinning is governed by hydrodynamics. During 70% of
the time evaporative thinning dominates until complete
drying.
Sections II through IV analyze the spin cast process of
a mixture of a volatile solvent and a non-volatile solute
assuming constant process parameters E, K, and D.
It is found that the spatio-temporal evolution of the
solute concentration within the thinning film is universally characterized by a Sherwood number, Shtr , scaled to
the system-inherent fundamental length, htr (it reflects
the competition between evaporative solute enrichment
and diffusional dilution at htr ). For Shtr < 1 (diffusion dominates) the spatio-temporal occurrence of the
relative surficial enrichment maximum is related to the
process parameters via universal power laws (Eqs. 17,
18, and 19, see Fig. 6). These findings are rationalized
semiquantitatively with the underlying physics. At last,
the final solute coverage Γ is calculated from the process
parameters (Eq. 16).
To examine the relevance of our analysis for real cases,
where E, K, and D are not necessarily always constant
(e.g. depending on the solute concentration), we assume
a typical solvent (e.g. toluene) with molar mass MS ≈
0.1 kg/mol, density ρS ≈ 103 kg/m3 , ν = 6 · 10−7 m−2 s−1 ,
and E = 10−6 m/s [18]. We consider two examples: A) A
typical polymer solution [162]; B) A nanoparticle solution
for submonolayer deposition.
Case A): Polymer with MP = 100 kg/mol, ρP =
103 kgm−3 , and c0,P = 10 kgm−3 (i.e., mole fractions
x0,P = 10−6 and x0,S ≈ 1), DP = 10−11 m2 s−1 [173] and
K = 1011 m−2 s−1 (≈ 4000 rpm). This yields: htr ≈ 1.7 ·
10−6 m (Eq. 3), Shtr ≈ 0.17 (Eq. 15), cP (htr )/c0,P ≈ 1.2
(Figs. 3 and 5) and Γp ≈ 17 ∗ 10−6 kgm−2 (Eq. 16) i.e.,
a final film thickness of ≈ 17 nm. This is in reasonable
agreement with experimental results [162].
Case B): Spheres with radius rNP = 36 nm. DNP =
10−11 m2 s−1 (estimation: Stokes-Einstein) as in case A)
and thus htr and Shtr are identical to case A). A 50%
monolayer coverage means ΓNP ≈ 1014 m−2 i.e., c0,NP ≈
5 · 10−5 mol/m3 (x0,NP ≈ 5 · 10−9 , x0,S ≈ 1) with Eq. 16.
In both cases the initial solute mole fraction is small.
Therefore film thinning to htr will follow Eq. 1 with K
and E remaining approximately constant because thinning is hydrodynamically dominated and the solute concentration barely changes. Hence Γ can be calculated
with Eq. 16, which is not affected by changes in K and
E occurring after reaching htr .
For both examples Shtr < 1. Therefore our analysis regarding hpeak , tpeak , and ∆cpeak is relevant even
quantitatively because all three maxima occur at h >
hpeak i.e., the (surficial) solute concentration has not yet
increased substantially above c0 (Figs. 5 and 6). Accordingly, also Eqs. 17, 18 and 19 are applicable.
Of course, evaporative film thinning at h < hpeak
will eventually increase the solute concentration and
thus change E and K. However, in particular for
”evaporation-structured” submonolayer particle array
deposition, non-ideality will only become relevant for
h ≪ htr , when h is already in the range of rNP , because
5
typically x0,NP ≪ 1. For low solubilities our analysis
reveals quantitatively whether solute aggregation occurs
first at the top surface (if evaporative solute enrichment
dominates) or globally homogeneous (or heterogeneously
at the substrate) if diffusive equilibration is more efficient
at the corresponding film thickness. Last not least, our
analysis predicts a quantifiable behavior based on measurable parameters of the initial solution. In particular
the predicted Γ and h(t) [18] are easily measurable. Thus
not only the validity of the approach can be evaluated.
It is possible to specifically address the influence of individual parameters of the multi-parameter spin casting
process. For instance, parameters not related to mixing
properties (e.g., temperature changes induced by evaporation) can be probed, because a sufficient reduction of
c0 will per definition render the solution behaving approximately “ideally”. All this proposes our approach
as a future basis for incorporating specific (non)-linear
properties of specific systems. Due to the predominantly
analytic approach such a refined approach still will reveal
general physical insights.
We thank Andreas Vetter and John Berg for scientific discussions, Helmuth Möhwald for scientific advice
and general support. SK was funded by DFG Grant
RI529/16-1, CMW by IGRTG 1524.
Electronic address: [email protected]
[1] A. G. Emslie, F. T. Bonner, and L. G. Peck, J. Appl.
Phys. 29, 858 (1958).
[2] F. Givens and W. Daughton, J. Electrochem. Soc. 126,
269 (1979).
[3] W. Daughton and F. Givens, J. Electrochem. Soc. 129,
173 (1982).
[4] L. White, J. Electrochem. Soc. 130, 1543 (1983).
[5] B. Chen, Polym. Eng. Sci. 23, 399 (1983).
[6] A. Weill and E. Dechenaux, Polym. Eng. Sci. 28, 945
(1988).
[7] W. McConnell, J. Appl. Phys. 64, 2232 (1988).
[8] L. Strong and S. Middleman, AICHE J. 35, 1753 (1989).
[9] K. Skrobis, D. Denton, and A. Skrobis, Polym. Eng. Sci.
30, 193 (1990).
[10] L. Spangler, J. Torkelson, and J. Royal, Polym. Eng.
Sci. 30, 644 (1990).
[11] D. Bornside, C. Macosko, and L. Scriven, J. Electrochem. Soc. 138, 317 (1991).
[12] L. Peurrung and D. Graves, J. Electrochem. Soc. 138,
2115 (1991).
[13] J. Britten and I. Thomas, J. Appl. Phys. 71, 972 (1992).
[14] M. Hershcovitz and I. Klein, Microelectron. Reliab. 33,
869 (1993).
[15] F. Horowitz, E. Yeatman, E. Dawnay, and A. Fardad,
J. Phys. III 3, 2059 (1993).
[16] L. Peurrung and D. Graves, IEEE Trans. Semicond.
Manuf. 6, 72 (1993).
[17] C. Extrand, Polym. Eng. Sci. 34, 390 (1994).
[18] A. Oztekin, D. Bornside, R. Brown, and P. Seidel, J.
Appl. Phys. 77, 2297 (1995).
∗
[19] R. Vanhardeveld, P. Gunter, L. Vanijzendoorn,
W. Wieldraaijer, E. Kuipers, and J. Niemantsverdriet,
Appl. Surf. Sci. 84, 339 (1995).
[20] D. Birnie, B. Zelinski, and D. Perry, Opt. Eng. 34, 1782
(1995).
[21] J. Gu, M. Bullwinkel, and G. Campbell, J. Appl. Polym.
Sci. 57, 717 (1995).
[22] J. Gu, M. Bullwinkel, and G. Campbell, Polym. Eng.
Sci. 36, 1019 (1996).
[23] D. Birnie and M. Manley, Phys. Fluids 9, 870 (1997).
[24] D. Birnie, J. Non-Cryst. Solids 218, 174 (1997).
[25] F. Horowitz, A. Michels, and E. Yeatman, J. Sol-Gel
Sci. Technol. 13, 707 (1998).
[26] S. Gupta and R. Gupta, Ind. Eng. Chem. Res. 37, 2223
(1998).
[27] D. Hall, P. Underhill, and J. Torkelson, Polym. Eng.
Sci. 38, 2039 (1998).
[28] D. Haas and J. Quijada, in Sol-Gel Optics V, edited
by Dunn, BS and Pope, EJA and Schmidt, HK and
Yamane, M (2000), vol. 3943 of Proceedings of the Society of Photo-Optical Instrumentation Engineers (SPIE),
pp. 280–284.
[29] D. Birnie, J. Mater. Res. 16, 1145 (2001).
[30] K. Strawhecker, S. Kumar, J. Douglas, and A. Karim,
Macromolecules 34, 4669 (2001).
[31] C. Walsh and E. Franses, Thin Solid Films 429, 71
(2003).
[32] J. Burns, C. Ramshaw, and R. Jachuck, Chem. Eng.
Sci. 58, 2245 (2003).
[33] J. Kim, P. Ho, C. Murphy, and R. Friend, Macromolecules 37, 2861 (2004).
[34] P. Jukes, S. Heriot, J. Sharp, and R. Jones, Macromolecules 38, 2030 (2005).
[35] C. Chang, C. Pai, W. Chen, and S. Jenekhe, Thin Solid
Films 479, 254 (2005).
[36] D. Birnie, S. Hau, D. Kamber, and D. Kaz, J. Mater.
Sci.-Mater. Electron. 16, 715 (2005).
[37] K. Cheung, R. Grover, Y. Wang, C. Gurkovich,
G. Wang, and J. Scheinbeim, Appl. Phys. Lett. 87
(2005).
[38] P. Yimsiri and M. Mackley, Chem. Eng. Sci. 61, 3496
(2006).
[39] E. Mohajerani, F. Farajollahi, R. Mahzoon, and
S. Baghery, J. Optoelectron. Adv. Mater. 9, 3901
(2007).
[40] N. H. Parmar and M. S. Tirumkudulu, Phys. Rev. E 80
(2009).
[41] N. H. Parmar, M. S. Tirumkudulu, and E. J. Hinch,
Phys. Fluids 21 (2009).
[42] D. P. Birnie, III, D. E. Haas, and C. M. Hernandez,
Opt. Lasers Eng. 48, 533 (2010).
[43] P. Mokarian-Tabari, M. Geoghegan, J. R. Howse, S. Y.
Heriot, R. L. Thompson, and R. A. L. Jones, Eur. Phys.
J. E 33, 283 (2010).
[44] K. E. Holloway, H. Tabuteau, and J. R. de Bruyn,
Rheol. Acta 49, 245 (2010).
[45] D. T. W. Toolan and J. R. Howse, J. Mater. Chem. C
1, 603 (2013).
[46] A. Acrivos, M. Shan, and E. Petersen, J. Appl. Phys.
31, 963 (1960).
[47] B. Washo, IBM J. Res. Dev. 21, 190 (1977).
[48] D. Meyerhofer, J. Appl. Phys. 49, 3993 (1978).
[49] S. Matsumoto, Y. Takashima, T. Kamlya, A. Kayano,
and Y. Ohta, Ind. Eng. Chem. Fundam. 21, 198 (1982).
6
[50] S. Jenekhe, Polym. Eng. Sci. 23, 830 (1983).
[51] S. Jenekhe and S. Schuldt, Ind. Eng. Chem. Fundam.
23, 432 (1984).
[52] W. W. Flack, D. S. Soong, A. T. Bell, and D. W. Hess,
J. Appl. Phys. 56, 1199 (1984).
[53] P. Sukanek, J. Imag. Technol. 11, 184 (1985).
[54] S. Jenekhe and S. Schuldt, Chem. Eng. Commun. 33,
135 (1985).
[55] B. Higgins, Phys. Fluids 29, 3522 (1986).
[56] J. Kaplon, Z. Kawala, and A. Skoczylas, Chem. Eng.
Sci. 41, 519 (1986).
[57] D. Bornside, C. Macosko, and L. Scriven, J. Imag.
Techn. 13, 122 (1987).
[58] S. Shimoji, Japan. J. Appl. Phys. 26, L905 (1987).
[59] M. Yanagisawa, J. Appl. Phys. 61, 1034 (1987).
[60] Y. Tu, J. Colloid Interface Sci. 116, 237 (1987).
[61] L. Stillwagon, R. Larson, and G. Taylor, J. Electrochem.
Soc. 134, 2030 (1987).
[62] S. Middleman, J. Appl. Phys. 62, 2530 (1987).
[63] T. Rehg and B. Higgins, Phys. Fluids 31, 1360 (1988).
[64] T. Papanastasiou, A. Alexandrou, and W. Graebel, J.
Rheol. 32, 485 (1988).
[65] C. Lawrece, Phys. Fluids 31, 2786 (1988).
[66] D. Bornside, C. Macosko, and L. Scriven, J. Appl. Phys.
66, 5185 (1989).
[67] T. Ohara, Y. Matsumoto, and H. Ohashi, Phys. Fluids
A 1, 1949 (1989).
[68] S. Shimoji, J. Appl. Phys. 66, 2712 (1989).
[69] Y. Matsumoto, T. Ohara, I. Teruya, and H. Ohashi,
JSME Int. J. 32, 52 (1989).
[70] J. Hwang and F. Ma, J. Appl. Phys. 66, 388 (1989).
[71] F. Ma and J. Hwang, J. Appl. Phys. 66, 5026 (1989).
[72] B. Dandapat and P. Ray, Int. J. Non-Linear Mech. 25,
569 (1990).
[73] F. Ma and J. Hwang, J. Appl. Phys. 68, 1265 (1990).
[74] L. Stillwagon and R. Larson, Phys. Fluids A 2, 1937
(1990).
[75] J. Hwang and F. Ma, Mech. Res. Commun. 17, 423
(1990).
[76] C. Lawrence, Phys. Fluids A 2, 453 (1990).
[77] Y. Tu and R. Drake, J. Coll. Interf. Sci 135, 562 (1990).
[78] C. Lawrence and W. Zhou, J. Non Newtonian Fluid
Mech. 39, 137 (1991).
[79] B. Reisfeld, S. Bankoff, and S. Davis, J. Appl. Phys. 70,
5258 (1991).
[80] B. Reisfeld, S. Bankoff, and S. Davis, J. Appl. Phys. 70,
5267 (1991).
[81] P. Sukanek, J. Electrochem. Soc. 138, 1712 (1991).
[82] C. Wang, L. Watson, and K. Alexander, IMA J. Appl.
Math. 46, 201 (1991).
[83] S. Kim, J. Kim, and F. Ma, J. Appl. Phys. 69, 2593
(1991).
[84] A. Potanin, Chem. Eng. Sci. 47, 1871 (1992).
[85] L. Stillwagon and R. Larson, Phys. Fluids A 4, 895
(1992).
[86] R. Yonkoski and D. Soane, J. Appl. Phys. 72, 725
(1992).
[87] T. Rehg and B. Higgins, AIChE J. 38, 489 (1992).
[88] D. Bornside, R. Brown, P. Ackmann, J. Frank,
A. Tryba, and F. Geyling, J. Appl. Phys. 73, 585 (1993).
[89] W. Levinson, A. Arnold, and O. Dehodgins, Pol. Eng.
Sci. 33, 980 (1993).
[90] B. Dandapat and P. Ray, Int. J. Non-Linear Mech. 28,
489 (1993).
[91] J. Kim, S. Kim, and F. Ma, J. Appl. Phys. 73, 422
(1993).
[92] M. Forcada and C. Mate, J. Colloid Interface Sci. 160,
218 (1993).
[93] F. Ma, Probab. Eng. Eng. Mech. 9, 39 (1994).
[94] M. Spaid and G. Homsy, J. Non-Newton. Fluid Mech.
55, 249 (1994).
[95] A. Borkar, J. Tsamopoulos, and R. Gupta, Phys. Fluids
6, 3539 (1994).
[96] B. Dandapat and P. Ray, J. Phys. D-Appl. Phys. 27,
2041 (1994).
[97] J. Gu, M. D. Bullwinkel, and G. A. Campbell, J. Electrochem. Soc. 142, 907 (1995).
[98] M. da Souza, K. Leaver, and M. Eskiyerli, Comput.
Mat. Sci. 4, 233 (1995).
[99] C.-T. Wnag and S.-C. Yen, Chem. Eng. Sci. 50, 989
(1995).
[100] S.-C. Yen and C.-T. Wnag, J. Chin. I. Ch. E. 26, 157
(1995).
[101] R. van Hardeveld, P. G. L. van IJzendoorn, W. Wieldraaijer, E. Kuipers, and J. Niemantsverdriet, Appl. Surf.
Sci. 84, 339 (1995).
[102] S. Burgess and S. Wilson, Phys. Fluids 8, 2291 (1996).
[103] J. Tsamopoulos, M. Chen, and A. Borkar, Rheol. Acta
35, 597 (1996).
[104] T. Okuzono, K. Ozawa, and M. Doi, Phys. Rev. Lett.
97, 136103 (2006).
[105] P. Sukanek, J. Electrochem. Soc. 144, 3959 (1997).
[106] E. Momoniat and D. Mason, Int. J. Non-Linear Mech.
33, 1069 (1998).
[107] B. Dandapat and P. Ray, Z. Angew. Math. Mech. 78,
635 (1998).
[108] B. Dandapat and G. Layek, J. Phys. D-Appl. Phys. 32,
2483 (1999).
[109] I. McKinley, S. Wilson, and B. Duffy, Phys. Fluids 11,
30 (1999).
[110] P. Wu and F. Chou, J. Electrochem. Soc. 146, 3819
(1999).
[111] S. Wilson, R. Hunt, and B. Duffy, J. Fluid Mech. 413,
65 (2000).
[112] A. Kitamura, Phys. Fluids 12, 2141 (2000).
[113] F. Chou and P. Wu, J. Electrochem. Soc. 147, 699
(2000).
[114] B. Dandapat, Phys. Fluids 13, 1860 (2001).
[115] R. Usha and R. Ravindran, Int. J. Non-Linear Mech.
36, 147 (2001).
[116] R. Usha and T. Gotz, Acta Mech. 147, 137 (2001).
[117] T. Myers and J. Charpin, Int. J. Non-Linear Mech. 36,
629 (2001).
[118] R. Usha and B. Uma, Z. Angew. Math. Phys. 52, 793
(2001).
[119] A. Kitamura, Phys. Fluids 13, 2788 (2001).
[120] R. Usha and B. Uma, Z. Angew. Math. Mech. 82, 211
(2002).
[121] A. Kitamura, E. Hasegawa, and M. Yoshizawa, Fluid
Dyn. Res. 30, 107 (2002).
[122] P. de Gennes, Eur. Phys. J. E 7, 31 (2002).
[123] D. Haas and D. B. III, J. Mat. Sci. 37, 2109 (2002).
[124] S.-K. Kim, J.-Y. Yoo, and H.-K. Oh, J. Vac. Sci. Technol. B 20, 2206 (2002).
[125] D. W. Schubert and T. Dinkel, Mat. Res. Innovat. 7,
314 (2003).
[126] G. Sisoev, O. Matar, and C. Lawrence, J. Chem. Technol. Biotechnol. 78, 151 (2003).
7
[127] G. Sisoev, O. Matar, and C. Lawrence, J. Fluid Mech.
495, 385 (2003).
[128] B. Dandapat, P. Daripa, and P. Ray, J. Appl. Phys. 94,
4144 (2003).
[129] R. Usha and R. Ravindran, Int. J. Non-Linear Mech.
39, 153 (2004).
[130] L. Schwartz and R. Roy, Phys. Fluids 16, 569 (2004).
[131] O. Matar, G. Sisoev, and C. Lawrence, Phys. Fluids 16,
1532 (2004).
[132] B. Dandapat, B. Santra, and A. Kitamura, Phys. Fluids
17 (2005).
[133] R. Usha, R. Ravindran, and B. Uma, Acta Mech. 179,
25 (2005).
[134] G. Sisoev, O. Matar, and C. Lawrence, Chem. Eng. Sci.
60, 2051 (2005).
[135] E. Momoniat, T. Myers, and S. Abelman, Int. J. NonLinear Mech. 40, 523 (2005).
[136] L. Wu, Phys. Rev. E 72 (2005).
[137] R. Usha, R. Ravindran, and B. Uma, Fluid Dyn. Res.
37, 154 (2005).
[138] R. Usha, R. Ravindran, and B. Uma, Phys. Fluids 17
(2005).
[139] T. Myers and M. Lombe, Chem. Eng. Process. 45, 90
(2006).
[140] L. Wu, Phys. Fluids 18 (2006).
[141] O. K. Matar, G. M. Sisoev, and C. J. Lawrence, Can.
J. Chem. Eng. 84, 625 (2006).
[142] H. Tabuteau, J. C. Baudez, X. Chateau, and P. Coussot,
Rheol. Acta 46, 341 (2007).
[143] J. P. F. Charpin, M. Lombe, and T. G. Myers, Phys.
Rev. E 76, 016312 (2007).
[144] L. Wu, Sens. Actuator A-Phys. 134, 140 (2007).
[145] K. E. Holloway, P. Habdas, N. Semsarillar, K. Burfitt,
and J. R. de Bruyn, Phys. Rev. E 75 (2007).
[146] V. Cregan and S. O’Brien, J. Colloid Interface Sci. 314,
324 (2007).
[147] Y. Zhao and J. S. Marshall, Phys. Fluids 20 (2008).
[148] O. K. Matar, G. M. Sisoev, and C. J. Lawrence, Chem.
Eng. Sci. 63, 2225 (2008).
[149] C.-K. Chen and M.-C. Lin, Math. Prob. Eng. 2009,
948672 (2009).
[150] S. Mukhopadhyay and R. P. Behringer, J. Phys.Condes. Matter 21 (2009).
[151] C.-K. Chen and D.-Y. Lai, Math. Probl. Eng. (2010).
[152] C.-K. Chen and D.-Y. Lai, Math. Prob. Eng. 2010,
987981 (2010).
[153] J.-Y. Jung, Y. Kang, and J. Koo, Int. J. Heat Mass
Transf. 53, 1712 (2010).
[154] A. McIntyre and L. Brush, J. Fluid Mech. 647, 265
(2010).
[155] P. Temple-Boyer, L. Mazenq, J. Doucet, V. Conedera,
B. Torbiero, and J. Launay, Microelectr. Eng. 87, 163
(2010).
[156] A. Muench, C. P. Please, and B. Wagner, Phys. Fluids
23, 102101 (2011).
[157] N. Modhien and E. Momoniat, Appl. Math. Model. 35,
1264 (2011).
[158] C. K. Chen, M. C. Lin, and C. I. Chen, J. Mech. 27, 95
(2011).
[159] B. S. Dandapat and S. K. Singh, Int. J. Non-Linear
Mech. 46, 272 (2011).
[160] B. S. Dandapat and S. K. Singh, Commun. Nonlinear
Sci. Numer. Simul. 17, 2854 (2012).
[161] M. C. Lin and C. K. Chen, Appl. Math. Model. 36, 2536
(2012).
[162] K. Norrman, A. Ghanbari-Siahkali, and N. B. Larsen,
Annu. Rep. Prog. Chem. C 101, 174 (2005), in particular references 25 through 37 cited therein.
[163] H. Riegler and R. Köhler, Nat. Phys. 3, 890 (2007).
[164] J. K. Berg, C. M. Weber, and H. Riegler, Phys. Rev.
Lett. 105, 076103 (2010).
[165] M. A. Brookshier, C. C. Chusuei, and D. W. Goodman,
Langmuir 15, 2043 (1999).
[166] E. Rabani, D. R. Reichman, P. L. Geissler, and L. E.
Brus, Nature 426, 271 (2003).
[167] T. P. Bigioni, X.-M. Lin, T. T. Nguyen, E. I. Corwin,
T. A. Witten, and H. M. Jaeger, Nat. Mater. 5, 265
(2006).
[168] T. Hanrath, J. J. Choi, and D.-M. Smilgies, ACS Nano
3, 2975 (2009).
[169] A. T. Heitsch, R. N. Patel, B. W. Goodfellow, D.-M.
Smilgies, and B. Korgel, J. Phys. Chem. C 114, 1616
(2010).
[170] E. Klecha, D. Ingert, and M. P. Pileni, J. Phys. Chem.
Lett. 1, 14427 (2010).
[171] A. C. Johnston-Peck, J. Wang, and J. B. Tracy, Langmuir 27, 5040 (2011).
[172] A. G. Marin, H. Gelderblom, D. Lohse, and J. H. Snoeijer, Phys. Rev. Lett. 107, 085502 (2011).
[173] J. Rauch and W. Kohler, Phys. Rev. Lett. 88, 185901
(2002).
[174] except for [86, 146], which reveal very little about the
compositional evolution.
[175] The expression Eq. (43) presented in [79] is not correct!
[176] See supplemental material at http://link.aps.org/
supplemental/... for experimental thinning curves, a
step-by-step derivation of Eq. 13, and details on the numerical methods.