Influence of Nanoscale Geometry on the Dynamics of
Wicking into a Rough Surface
SUPPLEMENTARY MATERIAL A
Assuming that the film of fluid imbibing the roughness is only as thick as the height
of the nanostructures such that all of the surfaces except for the top of the nanostructures are
immersed, it can be shown, according to Bico et al’s analysis,1 that wicking into a rough,
chemically homogeneous surface is only energetically favorable when
cos   cos  c 
1  s
r  s
(S1)
where θ is the contact angle the liquid makes with a flat surface of the substrate material, θc is
the critical angle (0° ≤ θc ≤ 90°), s is the fraction of the unit cell area that is made up by the
top of the nanostructures and r refers to the roughness of the surface which is computed as
the ratio of the actual surface area to projected area.
When this condition is satisfied and wicking occurs, the energy gained by the fluidsolid system, E, will result in a pressure, ΔP, which causes the wicking through the
nanostructures. The relationship between E and ΔP is given as
dE  PdV
where
and
(S2)

  cos 

 1
h  cos  c

(S3)
dV  1   s hdz
(S4)
P 
γ refers to the surface tension of the imbibing fluid, h refers to the height of the
nanostructures, z refers to the displacement of the wicking front and V refers to the volume of
fluid. Note that equations (S2) - (S4) refer to wicking in a unit width of the sample over a
small distance of dz.
Because of this driving pressure, ΔP, the fluid will invade the spaces between the
nanostructures with a mean velocity of Umean (= dz/dt) which is related to ΔP in the following
way:
U mean 
Ph 2
3z
(S5)
μ refers to the viscosity of the fluid and β refers to the drag enhancement factor that accounts
for the impedance to the capillary flow caused by the nanostructures. Note that when β = 1,
equation (S5) essentially describes Poiseuille’s flow on a flat, smooth plane.
Combining equations (S3) and (S5) together and integrating with respect to time, the
displacement-time relationship of the wicking process can be obtained as
 2h cos   cos  c
z  
cos  c
 3
1
1 1
 2 12
 t  D 2 t 2

(S6)
In Bico et al’s analysis, β is a value that has to be empirically determined for each surface of
interest. This can be done by obtaining D1/2 from the gradient of the graph of z vs t1/2 and
substituting the relevant liquid (γ, μ), material (cosθ) and geometric parameters (h, cosθc) to
compute β.
SUPPLEMENTARY MATERIAL B
Figure S1: Schematic diagram illustrating the process flow for the fabrication of nanofins. Si
– silicon; PR – photoresist; Au – gold.
P-type (100) Si substrates of area 1 x 1 cm2 were first subjected to standard RCA-I
and RCA-II cleaning steps. The native oxide layer was subsequently removed with 1 min of
immersion in HF. 300nm of positive photoresist (Ultra-i-123) was then spin-coated onto the
Si substrates and cured at 90°C for 90s. To obtain a hexagonal pattern of photoresist nanofins,
the photoresist was exposed to laser interference lithography (LIL) with a HeCd laser source
(λ = 325nm) and Lloyd’s-mirror set-up. Two exposures were performed with the sample
tilted at an in-plane angle between 0° to 90° with respect to the initial position for the second
exposure. After post-exposure baking at 110°C for 90s and development using Microposit
MF CD-26, photoresist nanofins were left on the Si surface. An oxygen plasma treatment
(power = 200W, pressure = 0.2 Torr, oxygen flow rate = 20 sccm, etching time = 30s to 45s)
was then employed to reduce the size of the photoresist nanofins.
Next, 30nm of Au was thermally evaporated onto the Si substrates at a chamber
pressure of 10-6 Torr. The photoresist nanofins were then dissolved with acetone in an
ultrasonic bath, leaving a gold film with nanofin holes on the Si surface. Metal assisted
chemical etching was then carried out by immersing the substrates in a solution of H2O, HF
and H2O2 at room temperature. The height of the nanofins can be controlled by the length of
etching time which varies from 5min to 30min. The concentrations of HF and H2O2 were 4.6
M and 0.44 M, respectively. A standard commercial Au etchant was then used to remove the
Au film and the samples were dried with a nitrogen gun after rinsing in deionized water.
Figure S1 shows a schematic diagram describing the Si nanofin fabrication process. The SEM
pictures of the nanofins are given in Fig. S2 and the roughness data for each sample are given
in Table S1.
To measure the displacement of the wicking front with time, 1μl of silicone oil (ρ =
760 kg/m3, γ = 3.399 ×10-2 N/m, µ = 3.94×10-2 Pas, θ = 18°) was deposited at one end of the
sample surface and the entire wicking process was recorded at 1000 frames per second with a
high speed camera. Measurements and analysis were carried out using Photron FASTCAM
Viewer.
Figure S2: SEM pictures of nanofin samples A - K used for this study. Insets show top view
of nanofins. Each scale bar represents 2μm.
Table S1: Roughness data of nanofins used in this study.
r
A
5.55
B
3.94
C
4.12
D
4.05
E
6.46
F
5.34
G
5.05
H
3.83
I
2.63
J
7.94
K
6.30
s
0.24
0.13
0.11
0.14
0.17
0.09
0.13
0.15
0.15
0.22
0.18
SUPPLEMENTARY MATERIAL C
Starting with the schematic diagram of the nanofins shown in Fig. 1b in the main text,
we first note that the length of the rounded ends of the nanofins, p', which is not necessarily
equivalent to m/2, is much smaller than p for our samples such that the length of the nanofin
can be taken as p + p' ≈ p. As a result, we have
s 
pm
( p  q)( m  n)
r
and
2 h p  m 
1
 p  q m  n
(S7)
(S8)
Next, consider wicking in the z (normal) direction. Since the energy lost to form drag
in a unit cell of nanofin, ΔEform, is equal to knphΔP (Eq. (3) in the main text), we can obtain
the energy loss due to form drag per unit width of the sample, dEform, as


dz
  1   s  fPhdz
dE form  E form 
 ( p  q)( m  n) 
where f 
(S9)
kpn
. f represents the fraction of fluid that is stagnant, which is
(1   s )( p  q )( m  n)
also equivalent to the fraction of energy lost to form drag. The remaining energy, when
averaged over the entire fluid volume, dV, will then give the reduced pressure driving the
capillary flow, ΔP', as
P' 
dE  dE form
dV
 P1  f 
(S10)
Equation (S10) can be thought of as describing a situation in which all of the fluid in a
unit width of the sample, dV, is flowing with reduced pressure of ΔP' due to form drag
instead of the previously described situation where a reduced volume of fluid, dV – dVform is
flowing with the original driving pressure, ΔP, which is not reduced by form drag. Note that
dVform = [Vformdz]/[(p+q)(m+n)]. The transition from the second case to the first by means of
equations (S9) and (S10) is required because equation (S6), to which β is applied to obtain
the kinematics of wicking, is derived based on the scenario in the first case1. Nevertheless, it
should be noted that dE - dEform is the same in both cases i.e. ΔP'dV= ΔP (dV – dVform).
To account for viscous losses in z (normal) direction, a unit cell of nanofin is modeled
as an open channel of height, h, and length, m + n, which contains (1 - s)(p + q)(m + n)h kpnh amount of actively flowing fluid. Note that the volume of fluid, kpnh, is stagnant due to
form drag. The width of the channel, wn, is therefore
wn  1  f 1   s  p  q 
(S11)
and the mean velocity of flow, taking into account form drag, can be given by2
h 2 wn
P' h 2 wn
P




1

f
z 3 4h 2  wn 2
z 3 4h 2  wn 2
2
U mean

2



(S12)
Finally, by comparing equation (S12) with equation (S5), we obtain
 1
1 f
 (normal )  

 4h 2
 2  1


 wn

(S13)
For wicking in z (parallel) direction, there is no form drag, as discussed above, and
thus, f = 0. Consequently, β (parallel) can be expressed as
  parallel  
4h 2
wp
2
1
(S14)
where wp = (1 - s)(m + n) for capillary flow in the z (parallel) direction. Note that a unit cell
of nanofin is modeled as an open channel of height, h, and length, p + q, which contains (1 -
s)(p + q)(m + n)h amount of actively flowing fluid for wicking in z (parallel) direction.
As a preliminary check on the validity of equations (S13) and (S14), let us consider
the effect of varying the nanofin dimensions on β. At one extreme, in the absence of
nanostructures (p  0, p’  0, m  0, q  ∞, n  ∞), β (normal) = β (parallel) = 1 and the
only obstruction to the capillary flow is viscous dissipation provided by the basal plane which
is intuitively obvious. On the other hand, if the nanostructures are so much larger than the
gaps between them (q  0, n  0, p  ∞, m  ∞), β (normal) = β (parallel)  ∞ as there is
essentially no room for wicking to take place. Thus, it can be seen that equations (S13) and
(S14) give reasonable predictions even in limiting cases.
REFERENCES
1. Bico, J., Tordeux, C. & Quéré, D. Rough wetting. Europhysics Letters (EPL) 55, 214–220
(2001).
2. Mai, T. T. et al. Dynamics of Wicking in Silicon Nanopillars Fabricated with Interference
Lithography and Metal-Assisted Chemical Etching. Langmuir 28, 11465–11471 (2012).