SUPPLEMENTARY INFORMATIONS
Optical Detection and Sizing of Single Nano-Particles
Using Continuous Wetting Films
Yves Hennequin,1 Cédric P. Allier,1* Euan McLeod,3,4 Onur Mudanyali,3,4 Daniel Migliozzi,2 Aydogan
Ozcan3,4,5,6 and Jean-Marc Dinten1
1
CEA, LETI, MINATEC, 17 rue des martyrs, 38054 Grenoble cedex 9, France
2
École Polytechnique (member of ParisTech), 91128 Palaiseau, France
3
Electrical Engineering Department, University of California, Los Angeles, California 90095, USA
4
Bioengineering Department, University of California, Los Angeles, California 90095, USA
5
California NanoSystems Institute, University of California, Los Angeles, California 90095, USA
6
Department of Surgery, David Geffen School of Medicine, University of California, Los Angeles,
California 90095, USA
* To whom correspondence should be addressed.
1. SHAPE OF THE LIQUID MICROLENSES
The shape of the liquid polymer film around a particle is described according to.1-4 We reproduce the
calculation of Kralchevsky and Nagayama2 and Fiegel3 to which we add an explicit account of the van
der Waals forces and the subsequent disjoining pressure in the film.
Linearization of the Young-Laplace equation.
The meniscus shape is obtained by solving of the Laplace equation:
𝛻. (
𝛻𝜁
√1+|𝛻𝜁|2
)=
𝑃𝑜𝑢𝑡 −𝑃𝑖𝑛
𝛾
Eq. S1
where 𝜁(𝑟) is the local height of the interface above the film thickness ℎ0 far from the particle (Fig.
S1a). ∆𝑃 = 𝑃𝑜𝑢𝑡 − 𝑃𝑖𝑛 is the difference in pressure outside and inside the film and 𝛾 is the surface
tension. The pressure inside the liquid meniscus 𝑃𝑖𝑛 includes the hydrostatic pressure 𝜌𝑔𝜁 and a term
for the disjoining pressure 𝛱(ℎ0 + 𝜁). The latter can arise from the long-range van der Waals forces,
steric repulsion or electric double layers acting between the substrate-film and film-air interfaces. The
disjoining pressure is typically negligible for micron-thick film but it becomes the leading term for thin
films2 ℎ0 ~ 100 nm. Considering only the van der Waals forces, we make an estimation of this term in
the following section. Mechanical equilibrium far from the particle implies that 𝑃𝑜𝑢𝑡 (ℎ0 ) = 𝑃𝑖𝑛 (ℎ0 ) +
𝜕𝛱
𝛱(ℎ0 ) with 𝛱 ′ (ℎ0 ) = ( 𝜕ℎ )
ℎ0
<0
and a first-order Taylor expansion of the disjoining pressure
gives2 𝛱(ℎ0 + 𝜁) = 𝛱(ℎ0 ) + 𝜁𝛱′. Solutions of Eq.S1 are not trivial, but tractable in the case where the
film thickness varies only slowly over its lateral length 𝐿 such that(ℎ0 + 𝜁)/𝐿 ≪ 1 (small-gradient
approximation). This condition is true if the lateral extent of the meniscus is much greater than the
particle size. In such case, it is possible to re-write Eq. S1 into the linearized form:2
𝛻 2 𝜁 = 𝐿−2 𝜁
Eq. S2
with
𝐿−2 = (𝜌𝑔 − 𝛱 ′ )/𝛾
Eq. S3
Estimation of the lateral extent L of the microlenses.
For thick films the disjoining pressure is negligible and 𝐿 corresponds to the capillary length of the
liquid film 𝐿 = √𝛾/𝜌𝑔, typically of millimeter order. For a thin film of thickness ℎ0 , the disjoining
pressure resulting from the van der Waals forces is written 𝛱(ℎ) = −𝐴123 /6𝜋ℎ3 where 𝐴123 is the
Hamaker constant for materials1,
2
interacting through a slab of material.3,
5
From Eq. S3, we then
obtain the lateral extent 𝐿 of the meniscus for a film of thickness ℎ0 :
𝐿−2 =
𝜌𝑔
𝛾
𝐴
123
− 2𝜋ℎ
4𝛾
Eq. S4
0
The Hamaker constant is estimated from the DLP theory5,6 assuming that all the materials have the
same absorption frequency in the UV range5 (𝜈𝑈𝑉 = 3 1015 Hz):
3
𝐴123 = 4 𝑘𝐵 𝑇𝐷0 (𝜀1 , 𝜀2 , 𝜀3 ) +
3ℎ𝜈𝑈𝑉
8√2
𝐷1 (𝑛1 , 𝑛2 , 𝑛3 )
(𝜀 −𝜀 )(𝜀 −𝜀 )
𝐷0 (𝜀1 , 𝜀2 , 𝜀3 ) = (𝜀1 +𝜀3 )(𝜀2 +𝜀3 )
1
𝐷1 (𝑛1 , 𝑛2 , 𝑛3 ) =
3
2
3
(𝑛12 −𝑛32 )(𝑛22 −𝑛32 )
√(𝑛12 +𝑛32 )(𝑛22 +𝑛32 ).(√(𝑛12 +𝑛32 )+√(𝑛22 +𝑛32 ))
Eq. S5
Eq. S6
Eq. S7
Supplemental Figure S1: (a) Schematic representation of a particle in the polymer film spread on a flat
substrate. 𝑅 is the radius, ℎ0 is the thickness of the film far away from the particle, 𝑧𝑐 is the height of the
spherical cap not covered by the liquid film, 𝜁(𝑟) is the height of the liquid meniscus, such that lim 𝜁(𝑟) = 0.
𝑟→∞
(b) Lateral extent L of the meniscus as a function of the thickness of the film ℎ0 calculated according to equation
S4-S7.
with 𝜀𝑖 and 𝑛𝑖 the dielectric constant and the refractive index of material (i), respectively. 𝑘𝐵 is
2
Boltzman's constant. Taking 𝜀𝑔𝑙𝑎𝑠𝑠 = 4.7 and 𝑛𝑔𝑙𝑎𝑠𝑠 = 1.5 for the glass and 𝜀𝑓𝑖𝑙𝑚 = 1.5 and 𝑛𝑓𝑖𝑙𝑚
=
2.136 for the PEG film (7) the Hamaker constant for the system is 𝐴123 = −6.3 10−21 𝐽. We thus find
that the disjoining pressure term dominates for film of thicknesses below 400 nm. Figure S1b shows
the dependence of the lateral extent 𝐿 on the film thickness for these parameters. For instance, for ℎ0 =
100 nm we find 𝐿 = 63 µm. The value of 𝐿 is somewhat sensitive to the precise values of the refractive
indices with relative variations of ~ 20% for ∆𝑛 = 10−2. In any case 𝐿 is always much larger than the
particle diameter and the small-gradient approximation is justified.
Solutions of the Young-Laplace equation
The locus of contact between the film and the particle forms a rim of radius 𝑟𝑐 = 𝑅𝑠𝑖𝑛𝛹𝑐 defined by its
opening angle 𝛹𝑐 (figure S1a). The boundary conditions for 𝜁(𝑟) are therefore
𝑟→∞
lim 𝜁(𝑟) = 0
Eq. S8
𝜁(𝑟𝑐 ) = 𝑅(1 + 𝑐𝑜𝑠𝛹𝑐 ) − ℎ0
Eq. S9
and
Additionally,
𝜕𝜁(𝑟)
𝜕𝑟
(𝑟𝑐 ) = −tan(𝛹𝑐 − 𝜃)
Eq. S10
where 𝜃 is the contact angle of the liquid film on the particle. In the wetting case, the film meets the
particle smoothly and 𝜃 = 0°.
The only axisymmetric solution of Eq. S2 satisfying the above boundary conditions is:1-4
𝜁(𝑟) = −𝐿 tan𝛹𝑐
𝑟
𝐿
𝑟
𝐾1 ( 𝑐 )
𝐿
𝐾0 ( )
,
Eq. S11
where 𝐾0 and 𝐾1 are the modified Bessel functions of order 0 and 1, respectively. In the case where
𝑟 ≪ 𝐿 the Bessel functions can be replaced by their asymptotic limits for small arguments, this leads to
the approximation:3,4
𝑟
𝜁(𝑟) = −2𝑧𝑐 [𝑙𝑛 (2𝐿) + 𝛾𝑒 ]
Eq. S12
where 𝑧𝑐 is the height of the spherical cap not covered by the liquid film and 𝛾𝑒 is the EulerMascheroni constant giving the logarithmic dependence of 𝜁(𝑟) described in the main text (see Eq. 1).
Having the dependence of the length 𝐿 on the film thickness ℎ0 , it is possible to use the boundary
condition at the particle (Eq. S9) to find the unique solution for a given particle size, film thickness and
wetting conditions by solving for 𝛹𝑐 numerically.
2. DESCRIPTION OF MOVIE M1.AVI
This movie shows in real-time the evaporation of a droplet with 200nm particles as seen in a mixed
bright-field and fluorescence microscopy (×10 NA = 0.3). The pinned contact line of the droplet on the
substrate can be seen at the top. Due to the higher evaporative flux near the contact line, the particles
are advected by an outward flow and become eventually trapped in the polymer. The movie starts at a
second stage when the evaporation front moves radially inwards and spreads a uniform polymer film
with particulate inclusions over the substrate.
3. DESCRIPTION OF MOVIE M2.AVI
This movie shows in real-time a close up on the moving evaporative front of a droplet with 200 nm
particles as seen in bright-field microscopy (×5 NA = 0.15 - the effective magnification of the optical
system is 2.5). The particles are literally revealed by the polymer film.
4. CONTRAST ENHANCEMENT USING CONTINUOUS WETTING FILM
Supplemental Figure S2. Evaporated droplet containing 200 nm particles and a volatile low-molecular
weight ethylene glycol (EG). (A) As soon as the water is evaporated, the particles can be clearly
identified due to the microlens effect. (B) Minutes later the EG evaporates and the particles cannot be
seen anymore.
5. LENSFREE IMAGING OF 100 NM PARTICLES UNDER CONTINUOUS WETTING FILM
Supplemental Figure S3 (a) Lensfree holographic imaging (cropped from a larger FOV of 28 mm2) of an
evaporated drop with continuous polymer thin film and 100 nm beads. The white rectangle in (a) is expanded in
(b); scale bar is 50µm. The SNR of individual 100 nm is about 5. (c) Holographic reconstruction of the pixel
super-resolved lensfree hologram shown in (b). The SNR of an individual 100 nm particle is about 10. For
comparison purposes, transmission and fluorescence microscopy images (×5 objective, NA=0.15) are also
shown in (d) and (e), respectively.
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