LH AN JP WW DV LABEL:7 [7]

# Wetting on Silicone Surfaces

Lukas Hauer<EMAIL_ADDRESS>Institute for Biology, Humboldt-
Universität zu Berlin, 10115 Berlin, Germany  Physics at Interfaces, Max
Planck Institute for Polymer Research, Ackermannweg 10, 55128 Mainz, Germany
Abhinav Naga  Department of Physics, Durham University, DH1 3LE, United
Kingdom  Rodrique G. M. Badr  Institut für Physik, Johannes Gutenberg-
Universität Mainz, Staudingerweg 7-9, 55099 Mainz, Germany  Jonathan T. Pham
Department of Chemical and Environmental Engineering, University of
Cincinnati, Cincinnati, 45221 OH, USA  William S. Y. Wong  Department of
Applied Physics, School of Science, Aalto University, 02150 Espoo, Finland
Doris Vollmer  Physics at Interfaces, Max Planck Institute for Polymer
Research, Ackermannweg 10, 55128 Mainz, Germany

###### Abstract

Silicone is frequently used as a model system to investigate and tune wetting
on soft materials. Silicone is biocompatible and shows excellent thermal,
chemical, and UV stability. Moreover, the mechanical properties of the surface
can be easily varied by several orders of magnitude in a controlled manner.
Polydimethylsiloxane (PDMS) is a popular choice for coating applications such
as lubrication, self-cleaning, and drag reduction, facilitated by low surface
energy. Aiming to understand the underlying interactions and forces, motivated
numerous and detailed investigations of the static and dynamic wetting
behavior of drops on PDMS-based surfaces. Here, we recognize the three most
prevalent PDMS surface variants, namely liquid-infused (SLIPS/LIS),
elastomeric, and liquid-like (SOCAL) surfaces. To understand, optimize, and
tune the wetting properties of these PDMS surfaces, we review and compare
their similarities and differences by discussing (i) the chemical and
molecular structure, and (ii) the static and dynamic wetting behavior. We also
provide (iii) an overview of methods and techniques to characterize PDMS-based
surfaces and their wetting behavior. The static and dynamic wetting ridge is
given particular attention, as it dominates energy dissipation, adhesion, and
friction of sliding drops and influences the durability of the surfaces. We
also discuss special features such as cloaking and wetting-induced phase
separation. Key challenges and opportunities of these three surface variants
are outlined.

PDMS, LIS, SLIPs, elastomers, gels, SOCAL, brushes, liquid-like, self-
assembled monolayer, soft wetting, wetting ridge, cloaking, elastocapillarity,
viscoelasticity

††preprint: APS/123-QED

##  I Introduction

Silicones are abundant materials with global production margins exceeding a
million tonnes annually. As a surface coating, the material is widely used in
numerous industries, ranging from the medical and energy sectors to the
personal care and automotive/aerospace industries. The low surface energy of
silicones promotes low friction and adhesion with contacting solids and
liquids, providing desirable properties for self-cleaning, lubrication, anti-
icing/-biofouling, drag-reduction, and enhanced heat and mass transfer [1, 2,
3, 4, 5, 6, 7, 8, 9, 10]. The most common silicone is polydimethylsiloxane
(PDMS). PDMS is biocompatible, water repellent, and flexible while being
chemically/thermally stable [11, 12, 13, 14, 15, 16, 17, 18, 19]. The
mechanical properties of PDMS can be tuned over many orders of magnitude and
the synthesis is safe and mostly straightforward when using commercially
available two-part systems, requiring no advanced training. Thus, PDMS is
readily used in labs as a model system for soft materials.
Due to the chemical and mechanical properties, PDMS is rigorously used as a
surface coating: The softness and flexibility of PDMS yield excellent
lubrication properties [20, 21, 22] which can be utilized for joint
lubrication [23] or drag reduction [24, 25]. The low (lateral) adhesion is
exploited for applications in anti-icing [26, 8, 27, 28], anti-marine fouling
[29], and anti-microbial formation [30, 31, 32, 33, 34]. Heat and mass
transfer can be optimized as PDMS surfaces provide high nucleation rates while
maintaining drop mobility [35, 36, 37]. PDMS is also used to interface
biological systems, e.g., to control the growth and the size of cells [38,
39].
PDMS-based surfaces come in different forms with various material and wetting
properties [40]. Variations in polymer chain configuration give a broad design
space for PDMS-based surfaces. The chain-chain interactions (cohesion and
friction) vary with the tunable chain length or with the addition of covalent
cross-links. The adhesive interactions with the underlying substrate can be
physical (e.g., capillary forces) or chemical (e.g., covalent bonding).

To provide optimal surface functionality, understanding the wetting
interactions in a given application is essential. Misinterpretation of these
interactions can lead to misuse, malfunction, and deterioration of the
surface. However, such understanding is challenging as the interactions are
complex with strong coupling between the different components, and require
consideration on many levels. During wetting, various forms of mass, momentum,
and energy exchange take place on length scales ranging from the molecular
structure of the polymer ($\mathrm{\AA}$) to the macroscopic drop
($\mathrm{cm}$) [41, 42, 43]. The associated timescales vary from mere
picoseconds up to several days. The liquid drop can mix to some extent with
the underlying polymer [44, 45, 46]. The sub-components that make up the
polymer matrix of solid PDMS can undergo chemical reactions or phase changes.
This alters the molecular structure of either or both the surface and the
liquid [47, 41]. Liquid PDMS chains (e.g., oil) can engulf and cloak the drop
[48, 49, 50]. The cloak changes the drop affinity to the surface and therefore
affects the wettability. Interactions between the drop and surface are
particularly high near the so-called ”three-phase-contact line” [51, 52]. At
this location, the drop meets with the surface and the surrounding medium (a
third fluid such as air). Here, the surface tension of the sessile drop exerts
pulling stress on the surface. On many rigid surfaces such as metals or glass,
the pulling stress is minor compared to the internal material stresses. PDMS,
however, is soft, even when it comes as a rubber. Hence, the surface tension-
induced pulling stress deforms the surfaces to an annular ”wetting ridge” [53,
54, 55, 56]. On very soft surfaces the wetting ridge grows large enough that
it becomes visible by the bare eye.

This review aims to present, discuss, and compare the physicochemical
properties of variations of PDMS surfaces during static and dynamic wetting.
We first introduce the chemical and structural features of PDMS and PDMS-based
surfaces. We focus on the three most frequently used PDMS-based surfaces: i)
liquid infused surfaces  (cf. lubricant infused surface - LIS [57]/slippery
lubricant infused porous surface - SLIPS [58]), where liquid PDMS oil
impregnates a porous substrate structure, Fig. 1a. ii) elastomeric surfaces
(cf. silicone gel surface), where PDMS chains are cross-linked to give
structural integrity, Fig. 1b. iii) slippery omniphobic covalently attached
liquid-like surfaces (cf. SOCAL, liquid-like surfaces [59, 60, 61, 62]), where
PDMS chains form a nanometric thin surface layer, Fig. 1c.
After establishing the structural differences in chain-chain and chain-surface
organization on the different surfaces, we discuss their characteristic
wetting behavior. We consider static and dynamic wetting - with a particular
focus on the wetting ridge. This feature is ubiquitous amongst PDMS-based
surfaces and the most pivotal element for wetting, drop mobility, and surface
durability. Contact friction mostly dissipates in the ridge. Therefore it is
critical for lubrication and drop sliding. However, it also accumulates
surface material, and contacting objects may entrain surface material in the
wetting ridge [63]. Eventually, this can cause surface deterioration and
malfunction. We discuss secondary effects such as PDMS-drop cloaking which
influences the wetting behavior and surface durability.
We emphasize the importance of the surface-associated wetting properties as
they are directly related to lubrication and durability: These are the two
main concepts for optimized surface functionality. This perspective will help
to gain a generalized, fundamental understanding of these most important PDMS-
based surface variants, which helps to create guidelines for optimized and
long-lasting PDMS surfaces.

Figure 1:  PDMS surface with sessile water drop. Scale bars serve as order of
magnitude guide. Depending on their molecular chain configuration and
substrate arrangement, PDMS surfaces can be a) liquid-infused, b) elastomeric,
or c) molecularly attached (SOCAL) to a rigid substrate. The thickness of the
surface varies from mm and $\mu$m (LIS, elastomeric) to nm (in particular
SOCAL). Top row shows shadowgraphs of sessile drops ($10~{}\mathrm{\mu l}$) on
each PDMS surface type, each exhibiting $\theta>90^{\circ}$. a) Low molecular
weight chains with free ends (green points) behave as a Newtonian liquid,
obtaining viscosities between $1~{}\mathrm{mPa~{}s}\leq\eta\leq
10~{}\mathrm{kPa~{}s}$. Rough/porous solid substrate texture provides support
for oil lubrication and increases surface retention. b) Crosslinks (red
points) create a network mesh with elasticity. Free chains (PDMS oil) can
reside inside the network and rearrange freely. The degree of crosslinking and
the amount of free chains inside the network determine the softness and
lubrication of elastomer surfaces. c) PDMS chains are covalently bonded to
anchor sites (gray points). The free ends remain flexible and maintain liquid-
like surface properties on SOCAL surfaces. Few chains are grafted on both
chain ends to the surface. Illustration and shadowgraphs adapted from [64].

##  II PDMS-Based Surface - Structure and Properties

PDMS-based surfaces come in various configurations with associated material
properties and wetting behaviors. We consider three common types of PDMS-based
surfaces, i) PDMS oil-based surfaces (e.g., liquid-infused surfaces), Fig. 1a,
ii) PDMS elastomeric surfaces (e.g., crosslinked polymers), Fig. 1b, and iii)
PDMS SOCAL surfaces (e.g., liquid-like polymer chains), Fig. 1c. Each PDMS-
based surface has the same monomeric building block, and consequently, some
intrinsic properties are shared among the surface types.
The monomer of PDMS has an inorganic siloxane backbone and two methyl side
chains, Fig. 2a center. The silicon-based backbone sets it apart from alkyl
(hydrocarbon) or perfluoroalkyl (fluorocarbon) chemistries, hence the
classification of siloxanes (silicon-oxygen bond).
The siloxane backbone is more flexible than a carbon backbone. This
flexibility is attributed to the large $\mathrm{Si-O-Si}$ bond angle of
$143^{\circ}-150^{\circ}$ [65, 66, 62] which separates the pendant dimethyl
groups. In addition, the alternating oxygen atoms do not allow side groups,
resulting in low rotational energy around the backbone
($3.3~{}\mathrm{kJ/mol}$) - e.g., lower than what is noted between carbon-
carbon bonds with methylene $\mathrm{CH_{2}}$ configurations
($13.8~{}\mathrm{kJ/mol}$) that typically occur in comparable organic
polymers, such as polyethylene glycol [67]. As a result, PDMS chains are
highly flexible, and readily re-orientate to a minimal energy state, while
possessing a low glass transition temperature
($\approx-150^{\circ}~{}\mathrm{C}$ [68]).
The organic content in PDMS is relatively low compared to other silicones due
to the small organic methyl group ($\mathrm{CH_{3}}$, is the smallest organic
group). The apolarity of the methyl group yields weak interactions with polar
solvents such as water and alcohols [69, 70]. This is reflected in the low
solubility which, in the case of water, is only $\approx 770~{}\mathrm{ppm}$
[71, 72, 73]. Vice versa, PDMS is even insoluble in water [29]. The poor
interaction strength between PDMS and water, however, does not hamper the
diffusive mobility of $\mathrm{H_{2}O}$ molecules in PDMS ($\approx 2\times
10^{-9}~{}\mathrm{m^{2}/s}$ [74, 73]) and is comparable with many other
liquid-liquid diffusivities.
The weak interaction strength with many solvents [38] results in low PDMS
surface energy. Consequently, adhesion between contacting liquids and PDMS is
relatively weak. Water repels from PDMS surfaces as reflected in the contact
angles ranging between $90^{\circ}-110^{\circ}$ [75, 76, 77, 78, 79, 80]. The
flexibility of PDMS chains maintains slippage and thus provides mobility for
contacting drops [81, 82, 83, 84]. Depending on the surface type, aqueous and
organic drops can slide off when tilting the surface by a few degrees.
However, on some PDMS surfaces drops remain pinned even when tilting the
surface by $90^{\circ}$.
PDMS chains can be adjusted in length (i.e., molecular weight), cross-linkage,
and chemical/physical surface attachments (”grafting”). Such variations shape
distinctive surface types and features including viscosity, elasticity, film
thickness, and durability.

Figure 2: Chemical structure of polydimethylsiloxane. a) PDMS can be
oligomeric ($n<100$) with low viscosity ($1-5~{}\mathrm{mPa~{}s}$) or assemble
into long chains with high viscosities (up to $\eta=10~{}\mathrm{kPa~{}s}$).
The silicon-oxygen backbone provides chain flexibility while the methyl side
chains yield water repellency. The end groups can be inert (trimethylsilyl) or
reactive (hydroxyl, vinyl). b) Hydrosilylation reaction between linear vynil-
terminated PDMS chains (top) and cross-linking agents with methylhydrosiloxane
monomers create branches and networked elastomeric polymers. Here, the cross-
linking agent is a polymer chain with $x$ dimethylsiloxane and $y$
methylhydrosiloxane monomers that are randomly arranged. The network branches
are randomly distributed on the cross-linker chain and terminal on the PDMS
base chain. c) Hydroxyl-terminated PDMS chains can be grafted on surface
hydroxyl groups that naturally occur on metals/metalloid oxides, or can be
functionalized with hydrogen plasma.

###  II.1 Liquid-Infused Surfaces

The design of liquid-infused surfaces is to some extent nature-inspired:
carnivorous plants such as the Nepenthes pitcher plant utilize a combination
of leaf texture and lubrication (the plant uses water as a lubricant) to trap
insects such as ants [85]. Liquid-infused surfaces comprise a micro- or
nanostructured texture, infiltrated with liquid PDMS oil, also termed silicone
oil. Typically, silicone oils are PDMS chains terminated with trimethyl
groups, Fig 2a. The end groups are nonreactive; therefore PDMS remains liquid
with a linear chain topology [13], Fig. 2a, left. The oil viscosity is tunable
via the number of monomer units in the chain and ranges from
$1~{}\mathrm{mPa~{}s}$ to $10~{}\mathrm{kPa~{}s}$ [86]. Short PDMS chains
(”oligomers” [87]) with $n<100$ monomer units are typically very mobile with
low viscosities of $\eta\leq 50~{}\mathrm{mPa~{}s}$. Due to the low surface
tension of PDMS oils, almost all surfaces are ”siliconephilic”, enabling rapid
spreading of oil into the porous, micro-/nanostructure textures [88, 89, 90].
Micro- and/or nanostructures are added to the surface to enable stronger oil
retention by capillary forces [91]. These surfaces are summarized under the
surface class named ”SLIPS” (slippery liquid-infused porous surface) or ”LIS”
(lubricant-infused surface) [58, 57]. The oil provides lubrication, resulting
in static friction forces orders of magnitude lower than dry friction. Most
liquid drops slide off when tilting the surface by merely a few degrees.
However, the infused oil can also be taken along, inducing degradation of the
coating.
Various strategies for oil replenishment [92], such as microfluidic setups
[93] or lubricant reservoirs [94, 95] have been implemented to enhance the
lifetime of the surface.

###  II.2 Elastomeric Surfaces

Crosslinked PDMS coated on substrates forms elastomeric surfaces, Fig. 1b.
Vinyl- or hydroxyl-terminated PDMS chains (Fig. 2a, right) are reactive and
widely used in cross-linking reactions. In Sylgard 184 (a commercial PDMS
elastomer), these reactive PDMS chains are mixed with a platinum-catalyzed
methylhydrogensiloxane, $\mathrm{(HCH_{3})SiO}$, cross-linker. In a
hydrosilylation reaction, these shorter siloxane-based cross-linker chains
($n\approx 10$) undergo an addition reaction with the vinyl groups on the PDMS
chain ends [96], Fig. 2b. Other derivatives of the synthesis exist. For
example, the vinyl group can become incorporated in the siloxane backbone or
other cross-linking agents may be utilized to realize addition and
condensation crosslinking-reactions [97, 98, 99, 100].
The mixture of activated PDMS chains and cross-linker is coated on a surface
by e.g., spin-coating, dip-coating, or blade-coating [101, 102, 103], before
the reaction proceeds. Crosslinking reactions (independent of their
formulation) will eventually arrive at the ”gel point”. The gel point is
achieved when a percolated bulk network forms and the material loses fluidity
[104]. Physically, this is an equilibrium state where the original PDMS liquid
mixture becomes a soft solid (i.e., a gel). Three specific changes to
physicochemical properties arise. i) Thermal rheology; as with other thermoset
polymers, once a PDMS elastomer reaches the gel point, it can no longer reflow
upon heating (unlike thermoplastics). This occurs above the glass transition
temperature (or even melting point) of an equivalent linear (identical
$\mathrm{M_{w}}$, uncrosslinked) PDMS chain. ii) Solvation; prior to gelation,
the PDMS liquid mixture is readily soluble in certain solvents (e.g., hexane
or toluene [69]). However, post-gelation, cross-linked PDMS will experience
swelling instead. The cross-links preserve the integrity of the elastomeric
polymer network. iii) Mechanical; passing the gel point induces an abrupt
increase of the viscosity and the emergence of elasticity [105]. Depending on
the cross-linker to oligomer ratio, the material obtains a stiffness between
$1~{}\mathrm{kPa}$ and $10~{}\mathrm{MPa}$ [106, 107, 79, 108, 109].
Even under optimal stoichiometric conditions, not all reactive chains will be
crosslinked to the network. At least 3$\%$ of the chains will remain free
within the network [69]. These free chains alter the material properties. They
can accumulate at the coating surface towards air and in the wetting
ridge[110] where they separate from the network [111, 112]. This has
consequences for static and dynamic wetting [113, 44, 114].

###  II.3 SOCAL Surfaces

The free chain ends of PDMS can be covalently anchored to a surface, creating
a nanometer-thin coating [115, 116, 117, 118, 119], Fig. 1c. Most chains are
anchored at one end while the other remains free. Few chains can be grafted on
both ends, forming chain loops [120, 121]. The free-moving end is almost as
flexible as liquid oligomers or oils and the surface interface is comparably
slippery. Therefore, these surfaces are often called ”liquid-like” or ”SOCAL”
(slippery, omniphobic, covalently attached, liquid) surfaces.
Surface anchoring is done either by a ”grafting-from” or a ”grafting-to”
reaction. ”Grafting-from” implies that chain polymerization is initiated at
the anchor site on the surface and proceeds by adding PDMS monomer units. In a
”grafting-to” reaction, an already polymerized chain binds to the surface
[122, 115, 116]. ”Grafting-from” can be attained by e.g., highly reactive
chloro-terminated siloxanes (monomers or oligomers). Long PDMS chains
($n>1000$) will experience ”grafting-to” anchoring. In both instances, they
are reported to form covalent bonds and create molecularly thin layers [123].
Surface anchor sites are typically provided by hydroxyl groups, Fig. 2c.
Anchor sites can be created using chemical treatment, e.g., oxygen plasma
exposure or alkali activation. Many metals/metalloid oxides (e.g.,
$\mathrm{SiO_{2}}$, $\mathrm{TiO_{2}}$, $\mathrm{Al_{2}O_{3}}$,
$\mathrm{NiO}$) naturally form hydroxyl groups under ambient water vapor
exposure [124].
Grafted PDMS chains exist in a stretched or a collapsed state, depending on
the affinity to the surrounding solvent and the chain density [125]. Polar
liquids (i.e., water) collapse chains as hydrophobic aggregation occurs to
minimize surface energy. Organic liquids can stretch out chains if they are
preferably solvated under similar chemical affinity. Anchored chains swell
when exposed to PDMS oil. These solvent-dependent chain conformations lead to
unique wetting behaviors: While both advancing $\theta_{a}$ and receding
angles $\theta_{r}$ follow trends expected by surface tension variations
(i.e., high for water, low for organic liquid), contact angle hysteresis
$\Delta\theta=\theta_{a}-\theta_{r}$ is low for both liquids [126, 127]. This
is attributed to the nanometric smoothness of such surfaces and the
flexibility exhibited by grafted chains. At the molecular level, grafted
chains rotate almost freely, reducing contact friction [128] Thus, contact
angle hysteresis $\Delta\theta$ remains relatively low. Notably,
$\Delta\theta$ depends on the grafting density and grafted chain length (i.e.,
the molecular weight) [124]. For example, $\Delta\theta$ was measured to be
lower on SOCAL surfaces with $6~{}\mathrm{kDa}$ chains, compared to both
shorter ($0.77~{}\mathrm{kDa}$) and longer ($117~{}\mathrm{kDa}$) chains
[115].

##  III Soft Wetting Mechanisms

###  III.1 Static Wetting Ridge

All surface types share the formation of an annular wetting ridge around the
three-phase contact line [129, 130, 56, 131]. The wetting ridge is defined as
the region that rises above the original unperturbed level of the surface. On
liquid-infused surfaces, the wetting ridge consists of PDMS oil only, whereas
on elastomers, it may consist of a mixture of crosslinked and liquid PDMS,
depending on the swelling ratio. On SOCAL surfaces, the wetting ridge will
encompass the local stretching of the grafted polymer chains. As PDMS chains
are organized differently on the liquid-infused, elastomer, and SOCAL
surfaces, the material forces in the wetting ridge differ, as well. Thus, each
surface type forms a characteristic wetting ridge with a specific geometry.
The wetting ridge significantly contributes to the overall drop morphology and
the wetting dynamics.

Figure 3: Different manifestations of the wetting ridge on different PDMS
surfaces. a) Liquid wetting ridge on a liquid-infused surface. Neumann balance
at the three-phase contact line (box) and the apparent contact angle
$\theta_{app}$. The right half illustrates the starved limit and the pseudo-
Young’s balance. Macroscopically, the vector center of the surface tensions
aligns with the baseline of the substrate. The image on the right shows a
liquid wetting ridge, taken with a fluorescence confocal microscope, adapted
from [92]. b) Elastic wetting ridge on an elastomeric surface. Bulk elasticity
creates a compliance force $f_{el}\propto\mu\delta_{e,\mathrm{I}}$, resulting
in wetting ridges smaller than the liquid ones. The right half shows material
dimpling around the wetting ridge when the surface coating is very thin. The
right image shows an elastic wetting ridge, taken with an x-ray microscope on
elastomeric PDMS ($3~{}\mathrm{kPa}$), adapted from [132]. c) Molecular
wetting ridge on a SOCAL surface. The entropic force of the PDMS chain $f_{m}$
limits the surface tension-induced chain stretch. Right half shows surface
swelling when liquid-PDMS affinity/solubility is high. The right image shows a
molecular wetting ridge, computed with coarse-grained molecular dynamics
simulation, adapted from [133].

####  III.1.1 Liquid

In addition to the hydrophobic nature of PDMS oil, the impregnated oil acts as
a lubricant. The film lubrication reduces the (static) friction of contacting
objects (both solids and liquids) on the surface immensely. Low friction
typically leads to excellent self-cleaning properties [57]. Furthermore,
lubricant allows surfaces to ”self-heal” from abrasion damages by the
replenishment and re-flow of oil. However, once the PDMS oil is partially
depleted from the surface, contacting drops are no longer lubricated and lose
their mobility [134]. Maintaining the PDMS oil on the surface is, hence, the
most important priority for the design of liquid-infused surfaces [91, 135,
28].
Oil depletion is already triggered by contacting water drops that induce the
formation of annular wetting ridges, Fig. 3a. Wetting ridges formed on
lubricant-infused surfaces can be visible to the naked eye, Fig. 1a. Still,
the characteristic sizes are usually in the micro-domain (ca.
$200~{}\mathrm{\mu m}$), and optical magnification helps to reveal more
detail, Fig. 3a right.
For drops smaller than the capillary length, gravity can be neglected and
capillary interactions between the lubricant and the water drop solely govern
the morphology of the wetting ridge. In contrast to wetting on inert rigid
surfaces, a single angle, such as the Young angle [136], is insufficient for
this wetting configuration [137]. On soft surfaces, the surface material (the
lubricant) adapts to the wetting drop as the interfacial tension of the drop
pulls on the surface. The surface becomes curved in the vicinity of the
wetting ridge. In equilibrium, the (liquid) wetting ridge is in a so-called
”Neumann configuration”, which implies a balance of the three fluid
interfacial tensions [$\vec{\gamma}_{w}+\vec{\gamma}_{o}+\vec{\gamma}_{ow}=0,$
subscripts correspond oil-ambient, water-ambient, and oil-water], Fig. 3a left
box. This tension balance yields a closed triangle, characterized by the
associated three Neumann angles $\vartheta_{1}$, $\vartheta_{2}$, and
$\vartheta_{3}$ [138]. The angles are related to the interfacial tensions as

$\frac{\gamma_{w}}{\sin{\vartheta_{1}}}=\frac{\gamma_{o}}{\sin{\vartheta_{2}}}=\frac{\gamma_{ow}}{\sin{\vartheta_{3}}}.$
(1)

For water and PDMS oil, $\gamma_{w}>\gamma_{o}+\gamma_{ow}$, hence no Neumann
configuration exists, and we expect the oil to fully spread around the drop
(cf. Sec. Free Chain Cloaking Layer). In other situations, the height of the
wetting ridge (or the location of the three-phase contact line) keeps growing
until the lubricant pressure in the wetting ridge matches the lubricant
pressure in the solid textures. In equilibrium, each of the fluid interfaces
assumes a minimal energy surface [139] where the mean curvature is constant
[140, 141, 142]. The oil interface deforms significantly in the surroundings
of the wetting ridge but straightens, afar. The ”catenoid” surface
approximates all these properties. This is possible because the surface
curvature in the two principal normal directions has equal magnitude but
opposite signs, such that they perfectly cancel out.
While the classical Young equation/angle fails for lubricant-infused surfaces,
a similar angle can be recovered in the limit when the wetting ridge is small
compared to the size of the drop, Fig. 3a center. This limit typically
prevails in the ”starved limit” when the infused PDMS oil is scarce. The
resulting ”pseudo Young angle” $\theta_{app}^{S}$ [143] is

$\cos{\theta_{app}^{S}}=\frac{\Upsilon_{s}^{\mathrm{eff}}-\Upsilon_{sl}^{\mathrm{eff}}}{\gamma_{w}}.$
(2)

The effective surface tensions of the solid have contributions from both,
texture ($\Upsilon_{s}$ and $\Upsilon_{sl}$) and PDMS oil ($\gamma_{o}$ and
$\gamma_{ow}$), i.e.,
$\Upsilon_{s}^{\mathrm{eff}}=\Upsilon_{s}\varphi+\gamma_{o}(1-\varphi)$ and
$\Upsilon_{sl}^{\mathrm{eff}}=\Upsilon_{sl}\varphi+\gamma_{ow}(1-\varphi)$,
where $\varphi$ is the texture fraction of the solid surface.
Moreover, in the starved limit, the shape of the wetting ridge is not only
defined by capillarity. Disjoining pressures must also be considered when the
oil film becomes very thin ($\lessapprox 100~{}\mathrm{nm}$) because the oil-
ambient interface starts to interact with the oil-solid interface. Such
interactions are characteristic of very thin films and are summarised in the
”disjoining pressure” framework [144]. The interactions take place on a
molecular level and can stem from a combination of van der Waals interactions,
electrostatic attraction or repulsion, and steric forces [145]. For static
PDMS oil films, the interactions are usually purely dispersive, and the
disjoining pressure is

$\Pi(h)=-\frac{A_{H}}{6\pi h^{3}},$ (3)

where $h$ is the film thickness and $A_{H}$ is the Hamaker constant. $A_{H}$
depends on the involved materials and can be approximated with the Lifshitz
theory [146]. In general, the disjoining pressure can be either attractive
(unstable film) or repulsive (stable film). For PDMS oil on a solid-textured
substrate $\mathcal{O}(A_{H})\approx-10^{-19}~{}\mathrm{J}$ (cf. [147] for a
comprehensive material list). In case of severe film starvation, the
disjoining pressure needs to be considered within the total free energy [148].
A consequence is that the oil interface assumes shapes, different from the
catenoid [140].
When the surface holds sufficient PDMS oil, the contact angle defined in Eq.
(2) breaks down because the assumption of a small wetting ridge no longer
holds. The interface of the ridge now contributes to the total free energy. It
becomes challenging to define a baseline from which to measure the (pseudo)
angle, as the wetting ridge conceals the footprint of the drop. In practice,
apparent contact angles $\theta_{app}$ are utilized, which are defined as the
angle of the drop at the Neumann point to the horizontal plane, Fig. 3a left.
The size of the large ridge $\delta_{l}$ and the size of the drop $r$ are
related to the difference between $\theta_{app}$ and $\theta_{app}^{S}$ [149]
per

$\delta_{l}=r\left(\cos{\theta_{app}}-\cos{\theta_{app}^{S}}\right).$ (4)

####  III.1.2 Elastic

Wetting ridges also occur on elastomeric surfaces [150]. However, the
mechanisms that govern the ”elastic wetting ridge” are more complex than those
governing liquid ridges. Capillary forces around the perimeter of a wetting
drop induce mechanical stress in the PDMS material $(\bm{\sigma})$.
Consequently, the material around the three-phase contact line deforms to a
wetting ridge, similar to the liquid wetting ridge. In addition to the surface
tension of the PDMS surface, the elasticity of the network bulk contributes to
the ridge geometry. Depending on the bulk stiffness, wetting ridges on
elastomers can grow up to tens of microns. With the steady improvement of
optical techniques, elastic wetting ridges have been resolved with an
increasing level of detail. Such modern state-of-the-art techniques are e.g.,
shadowgraphy, interferometry, and laser scanning confocal microscopy (cf.
Methods and Techniques, [53, 132, 151, 152], and Fig. 3.)
The shape of the elastic wetting ridge can be found by balancing the bulk
elasticity and the surface stress of the network to the imposed surface
tension of the water drop at the three-phase contact line. The surface tension
of the water pulls on the surface, acting like a point/line load on the PDMS
material, reading

$\vec{\gamma}_{w}=\gamma_{w}\left(\cos{\theta}\vec{t}+\sin{\theta}\vec{n}\right)\delta\left\\{x-x_{\mathrm{cl}}\right\\}.$
(5)

Here, $\vec{n}$ and $\vec{t}$ denote the surface normal and tangential vectors
respectively. $\delta\left\\{x-x_{\mathrm{cl}}\right\\}$ denotes the Dirac
delta function and $x_{\mathrm{cl}}$ localizes the horizontal position of the
three-phase contact line. The surface stress balance is then

$\bm{\sigma}\vec{n}-\Upsilon_{\mathrm{e}}\left(\nabla\cdot\vec{n}\right)\vec{n}=\vec{\gamma}_{w}.$
(6)

The spring-load system is a simple analogy to this wetting ridge where the
resulting displacement - i.e., the height of the wetting ridge
$\delta_{e,\mathrm{I}}$ [153, 154] \- scales with the drop surface tension
pulling on the surface per

$\delta_{e,\mathrm{I}}\propto\frac{\gamma_{w}}{\mu}\sin{\theta}.$ (7)

The compliance of the wetting ridge is characterized by $\mu^{-1}$, where
$\mu$ is the shear modulus of PDMS. PDMS elastomers have a hydrophobic contact
angle $\theta$ between $90^{\circ}-110^{\circ}$ [75, 76, 77, 78, 79, 80], Fig
1b. Here, the solid surface tension $\Upsilon_{\mathrm{e}}$ is considered
symmetric around $x_{\mathrm{cl}}$. This, however, is not fulfilled as PDMS-
air and PDMS-water interfaces have differing surface tension
($\Upsilon_{el}\approx 40~{}\mathrm{mN/m}$ and $\Upsilon_{e}\approx
20~{}\mathrm{mN/m}$). To match the asymmetric surface tension around the
contact line, Eq. (6) can be adjusted with Heaviside functions.
Solutions of this system of equations (and variations) show that the
competition between the bulk elasticity $(\mu)$ and the capillary point load
$(\gamma_{w}\sin{\theta})$ produces shapes that are comparable with
experimental measurements [132].
Considering only these two components, however, produces a singularity at the
contact point of the load. The surface tension of the network material
$\Upsilon_{e}$ regularizes this singularity by opposing the deformation with
plastic stresses [137, 155]. The length scale at which the surface tension-
regularization emerges is

$\delta_{e,\mathrm{II}}\propto\frac{\Upsilon_{e}}{\mu}.$ (8)

When the wetting drop induces an elastic wetting ridge where
$\mathcal{O}(\delta_{e,\mathrm{I}})$ is below
$\mathcal{O}(\delta_{e,\mathrm{II}})$ the Neumann configuration [Eq. (1)] is
recovered. The ratio of the two elastic length scales
$\left[\alpha=\delta_{e,\mathrm{I}}/\delta_{e,\mathrm{II}}\right]$ demarcates
the two limits in which the stress can be treated:

$\alpha\propto\frac{\gamma_{w}}{\Upsilon_{e}}\sin{\theta}\begin{cases}\ll
1&\mathrm{Neumann}\text{-}\mathrm{limit,}\\\ \gg
1&\mathrm{bulkelasticity}\text{-}\mathrm{limit}.\end{cases}$ (9)

Straining solid (crystalline) surfaces can alter the molecular surface
structure, yielding a local dependency of the surface energy on the strain
[156]. Consequently, the surface energy and the surface tension are not equal
[$\gamma_{e}\neq\Upsilon_{e}$]. The surface energy is a function of the
surface strain $\gamma_{e}(\epsilon_{\Sigma})$, and the relation to the
surface tension is

$\Upsilon_{\mathrm{e}}=\frac{\mathrm{d}}{\mathrm{d}\epsilon_{\Sigma}}\left[(1+\epsilon_{\Sigma})\gamma_{e}(\epsilon_{\Sigma})\right].$
(10)

Eq. (10) is known as the Shuttleworth equation [157, 158]. While originally
introduced in 1950, it obtained recent momentum in the wetting community. The
Shuttleworth effect was utilized to show anisotropic wetting behavior on
unidirectionally stretched PDMS surfaces [159, 160, 161] and differing
dependencies of $\Upsilon_{\mathrm{e}}$ and $\gamma_{e}$ on the cross-link
density [162]. Other studies found contradictory behavior and the absence of a
$\gamma_{e}(\epsilon_{\Sigma})$ dependency [163]. Their observations were
rationalized by only partially stretched elastomers (i.e., below the max.
chain length) or oil-swollen elastomers where mobile chains at the surface
enable the relaxation of strain-induced local molecular surface
inhomogeneities [164, 165, 113]. In that sense, the PDMS elastomer surfaces
resemble rather a liquid interface than a solid one. However, surface
modifications, e.g. oxidation of surface molecules, can create a thin ”surface
skin” that is chemically and mechanically different from the bulk PDMS [166,
167]. Upon stretching, the oxidized skin breaks, and non-oxidized PDMS
molecules replenish the gap. Such surface-modified PDMS elastomers exhibit a
strain-dependent surface energy/tension per Eq. (10).
Further experimental findings indicate that the surface tension might not only
depend on strain [per Eq. (10)] but on the strain rate, which could be
interpreted as a form of complex surface rheology [168].

####  III.1.3 Molecular

A wetting ridge on a SOCAL surface is tiny but possible: chains may stretch
around the contact line, or ungrafted chains migrate freely to the contact
line. Because of the nanometric thickness of the SOCAL surface, direct
experimental observations of ”molecular wetting ridges” have not been
demonstrated. However, theoretical calculations - both on a molecular level
and in a mean-field framework - predict their formation [169, 170, 171, 133].
The molecular wetting ridge is, again, a result of the interplay of the
capillary forces induced by the wetting drop and the material response of the
surface. However, the motivation of the linear elasticity framework from a
continuum mechanical point of view is less obvious as the length scales are
much smaller. Deformations occur (at most) over the length of a chain
(collapsed to stretched) in contrast to elastomers that undergo deformations
far exceeding the lengthscales of the spacing between crosslinking nodes.
Further, the chain configuration introduces isotropy as the chains are not
laterally reticulated. Therefore, the ridge deformation on SOCAL surfaces is
usually considered in a thermodynamic framework of the individual polymer
chains. The restoring force on the molecular chain level originates from
entropy [59]. Stretched chains have a reduced conformational state. Thus, the
entropy is low. This state is undesirable for the grafted chain. It strives to
increase the possible conformations and maximize its entropy by recovering the
unstretched state [172, 173]. Notably, the thermodynamic consideration for a
swollen chain recovers a linear relation between the deformation and the
restoring force per chain

$f_{m}=k_{B}T\frac{H_{B}}{nb^{2}},$ (11)

where $k_{B}$ is the Boltzmann constant, $T$ is the temperature, $H_{B}$ is
the thickness of the grafted chain layer, $n$ is the number of monomers in a
chain, and $b$ is the size of a monomer. With this, the height of the
molecular ridge is expected to scale as

$\delta_{m}\propto\frac{\gamma_{w}nb^{2}}{\sqrt{\sigma}k_{B}T}\sin{\theta},$
(12)

where $\sigma$ is the grafting density for the SOCAL surface ($\sigma\approx
0.1~{}\text{chains/nm}^{2}$ [121]).

###  III.2 Free PDMS Chains and Cloaking

####  III.2.1 Phase-Separated Wetting Ridge

In Sylgard 184 PDMS elastomers synthesized under the standard recommended
crosslinking procedure (10:1 base to crosslinker), around $5~{}\mathrm{\%wt}$
of the chains remain free. Free chains can be considered as a solvent that
swells the crosslinked network [174, 69, 175]. The volume ratio between the
swollen network and the dry network defines the swelling ratio
$Q=V_{\mathrm{swollen}}/V_{\mathrm{dry}}$. Depending on the degree of
crosslinking, PDMS elastomers display different swelling ratios in
equilibrium. For example, a gel prepared with Sylgard 184 with a 60:1 mixing
ratio can be swollen up to $Q\approx 16$ with low molecular weight
(oligomeric) PDMS chains [112].
Synersis - i.e., the extraction of free chains from the network onto the
surface [176, 33, 177, 178] \- can cause lubrication [113]. This essentially
makes (swollen) PDMS elastomers similar to liquid-infused surfaces on which
contaminants (such as water) hardly stick [29].
Synersis is typically induced by global effects such as temperature variations
[179] or the progression of the crosslink reaction [33], affecting the entire
surface of the PDMS elastomer. Wetting drops, however, trigger the
accumulation of free chains locally around the three-phase contact line,
caused by the surface tension of the drop [180, 44]. On highly swollen PDMS
elastomers, this can lead to the phase separation of free chains at the tip of
the wetting ridge [111, 112]. The extent of phase-separation (measured by the
size of the separated tip, $h_{\mathrm{oil}}$) is proportional to the swelling
ratio $Q$, i.e., the number of free chains in the gel. On PDMS-based SOCAL
surfaces, such kinds of phase-separated tips have also been observed in
molecular dynamics simulations [133]. Due to the small length scales of the
surface, direct experimental visualization is still lacking.

####  III.2.2 Cloaking Layer

The surface tension of a wetting drop yields the formation of a wetting ridge
and the accumulation of free chains (PDMS oil). A sharp three-phase contact
line at the tip of the ridge implies a (Neumann) balance [Eq. (1)] of the
competing surface tensions of the water ($\gamma_{w}$), the oil
($\gamma_{o}$), and water/oil ($\gamma_{ow}$). The Neumann balance is
fulfilled with a negative ”spreading coefficient”:

$S=\gamma_{w}-\gamma_{ow}-\gamma_{o}\leq 0.$ (13)

Generally, $S\leq 0$ indicates the presence of a contact line (partial
wetting), and $S>0$ expresses the absence of a contact line in equilibrium
(complete wetting). As all three interfaces are liquid, their surface tensions
are easily accessible with direct measurements (cf. Shadowgraphy in Methods
and Techniques), and their values are listed in Tab. 1. Notably, for these
values, the spreading coefficient is positive $(7-16~{}\mathrm{mN/m})$, and
thus, PDMS oil tends to spread on top of the water drop [48]. The PDMS engulfs
the drop with a thin layer, Fig. 4a,b. This behavior is called ”cloaking”, and
significantly changes the wetting configuration [181]. Due to the cloak, the
surface energy of the drop cap changes, affecting the total free energy and
consequently the wetting affinity of the drop and the contact angle,
respectively. A cloaking layer of PDMS oil makes the Neumann configuration
impossible. Instead, the effective surface tension of the drop changes.

Table 1: Interfacial tension between water-air, PDMS oil-air, and PDMS oil-water interface at room temperature. Differences in the values can stem from differing PDMS chain lengths ($\eta\approx 5-300~{}\mathrm{mPa~{}s}$) and measurement errors. Interface |  | Surface Tension $[\mathrm{mN/m}]$
---|---|---
water-air | $\gamma_{w}$ |  $72$ (self-measured)
PDMS oil-water | $\gamma_{ow}$ |  $40$ [182], $38$ [183]
PDMS oil-air | $\gamma_{o}$ |  $18-22$ [183], $21$ [182], $25$ [184]

Naively, the ”cloaked” surface tension $\gamma_{c}$ of the drop effectively
sums up from two interfaces (water-oil and oil-air) that replaces the
uncloaked surface (water-air), i.e., $\gamma_{c}=\gamma_{ow}+\gamma_{o}$.
However, the concepts of surface tension and surface energy consider shared
surfaces of two bulk phases. Surface tension and energy result from excess
energy that decays at some length scale towards the bulk phases. The length
(or the vicinity) in which the excess energy decays is also reflected in the
stress isotropy, being isotropic in the bulk phases, and anisotropic near the
interface. Following this reasoning, $\gamma_{c}$ is only the sum of
$\gamma_{ow}$ and $\gamma_{o}$ when the cloaking layer is thick enough and the
surface tension has ”bulk properties”. However, the layer can be substantially
thinner, especially upon cloak formation. In this case, the two interfaces
(water-oil and oil-ambient) of the thin cloak layer start to interact with
each other via the disjoining pressure $\Pi(h_{c})$ yielding stabilizing
stress contributions which depend on the thickness of the cloak $h_{c}$. The
stabilizing disjoining pressure contribution adds to the surface tension of
the cloak with [185, 186, 187]

$\gamma_{c}\left(h_{c}\right)=\gamma_{ow}+\gamma_{o}+\Pi(h_{c})h_{c}+\int_{h_{c}}^{\infty}\Pi(h)dh.$
(14)

Note, that for $h_{c}\to\infty$, the disjoining pressure vanishes and
$\gamma_{c}=\gamma_{ow}+\gamma_{o}$ is recovered. In the opposite limit, when
$h_{c}\to 0$, $\gamma_{c}\to\gamma_{w}$ such that the surface tension of the
uncloaked drop is recovered.
The formation of the cloak is a transient process, associated with time scales
of the oils rooting in viscous or diffusive oil transport from the surface to
the drop cap. Hence, the temporal evolution of the cloak formation can be
measured: experimentally, this has been illustrated with pendant drop
measurements, where drops hang on swollen PDMS elastomer surfaces [180, 188,
189]. In these experiments, the surface tension of the hanging drop
($38~{}\mathrm{\mu l}$) changed from the characteristic
$\gamma_{wa}=72~{}\mathrm{mN/m}$ to $\gamma_{c}\approx 64~{}\mathrm{mN/m}$
after more than $\approx 3$ minutes, Fig. 4c. Besides oil parameters (such as
viscosity), the drop size influences the cloak formation time [180]. The
coupling in Eq. (14) between $\gamma_{c}$ and $h_{c}$ indicates that one is a
proxy for the other. Hence, the temporal evolution of $\gamma_{c}$ may be
linked to the gradually increasing cloak thickness. Note, that
$\gamma_{wo}+\gamma_{o}\approx 64~{}\mathrm{mN/m}$, Tab. 1, indicating that
the cloak reached a bulk-like thickness for $t\to t_{\infty}\approx
3~{}\mathrm{min}$.

Figure 4:  Sessile drop on lubricated surface, cloaked by PDMS oil. a) Three-
dimensional view of PDMS oil on the drop, obtained by laser scanning confocal
microscopy. b) Side view of the angular averaged 3D stack. Images adapted from
[133] c) Temporal evolution of pendant drop on PDMS elastomer with swollen
with $200~{}\mathrm{mPa~{}s}$ (red) and $350~{}\mathrm{mPa~{}s}$ (green) PDMS
oil. The shaded region shows standard deviation of experimental repeats. Data
adapted from [189].

###  III.3 Dynamic Wetting Ridge

####  III.3.1 Liquid

Sliding drops experience friction, stemming from shear dissipation. On rigid
surfaces, the source of dissipation is either found at the three-phase contact
line (for small speeds) or at the footprint of the drop where a hydrodynamic
boundary layer forms (at higher speeds). On PDMS surfaces, the main source of
dissipation, however, can be localized in the PDMS layer instead of the drop.
This is the case when the lubricant is more viscous than the drop [190]. The
wetting ridge at the three-phase contact line moves with the drop and the
accumulated material in the ridge is constantly reorganized, Fig. 5a. During
reorganization, the material undergoes shear and strain that dissipates energy
due to the viscous/viscoelastic nature of the PDMS material. Note, that shear
is not only evoked in the ridge but on all parts of the PDMS surface that are
in contact with the drop (i.e., the drop footprint); however, the dissipation
in the wetting ridge is the dominant source of sliding friction. Hence, to
quantify the sliding friction, it is sufficient to only focus on the wetting
ridge.
In general, the dissipation
$P_{\mathrm{diss}}=\int_{V_{\mathrm{ridge}}}\bm{\sigma}:\dot{\bm{\epsilon}}dV$,
depends on the volume of the wetting ridge $V_{\mathrm{ridge}}$, the strain
rate $\dot{\bm{\epsilon}}$ (i.e., the rate of reorganization), and the
material stress $\bm{\sigma}$. The stress in the wetting ridge follows from
material laws, and thus, changes with the fabric of the different surface
types. PDMS oils are Newtonian fluids that show a linear relationship between
stress and shear rate, $\bm{\sigma}=\eta\dot{\bm{\epsilon}}$. If the oil is
very viscous, the drop mobility is significantly hampered [190, 191]. Still,
lubrication enables higher drop mobility on lubricated compared to non-
lubricated surfaces which tend to be rougher and have a much larger number of
surface defects that slow down the drop [24]. The shear stress in the liquid
wetting ridge scales as $\sigma\propto\eta v/h_{r}$. The dissipation rate is
found by integrating the viscous stress over the ridge area. Furthermore, we
must account for the variation of dynamic contact angle of the wetting ridge
with speed, Fig 5b,c), which leads to, $P_{\mathrm{diss}}\propto v\beta\phi
r\left(\eta v/\theta_{o}\right)$. $\phi$ accounts for the texture density
which is important as only texture contacts along the perimeter add to the
overall friction. On parts where the ridge contacts oil, it is assumed that no
significant shear builds up and can, therefore, be neglected.
$\beta=\ln{(h_{r}/\epsilon)}$ and $\epsilon$ account for cut-off length scale
(taken to be $\approx 100~{}\mathrm{nm}$) [192]. $\theta_{o}$ is the opening
angle, formed between the horizontal plane and the wetting ridge. This angle
changes with the speed according to Tanner’s law
$\theta_{o}\approx\left(\beta\eta v/\gamma\right)^{1/3}$ [193]. Including
Tanner’s law into the expression for the dissipative power yields
$P_{\mathrm{diss}}\propto v\phi r\left(\beta\eta v/\gamma\right)^{2/3}$. The
ratio between the viscous stress and the surface force in the liquid wetting
ridge are condensed in the capillary number $\mathrm{Ca}=\eta v/\gamma$.
Hence, the dissipative scaling can be written compactly as

$P_{\mathrm{diss}}^{\mathrm{vis,I}}\propto\phi
vr\left(\beta\mathrm{Ca}\right)^{2/3}.$ (15)

Despite its simplicity, this relation has been remarkably successful at
capturing the scaling between dissipation and capillary number in experiments
performed with various surface geometries and lubricant viscosities by
independent groups, Fig. 6a.
On flat lubricated surfaces, the ridge meniscus behaves analogously to a
Landau Levich Derjaguin film: the viscous stress elevates the film in the
vicinity $l_{\mathrm{dyn}}$ behind the advancing and the receding wetting
ridge. The elevated height scales as $h_{\mathrm{dyn}}\propto
r\mathrm{Ca}^{2/3}$, and the vicinity length as $l_{\mathrm{dyn}}\propto
r\mathrm{Ca}^{1/3}.$ Depending on whether considering the advancing or
receding ridge, $\mathrm{Ca}$ is formed with $\gamma_{ow}$ and $\gamma_{o}$,
respectively. As many oil surfaces utilize a porous structure to maintain
better retention of the oil film, $h_{\mathrm{dyn}}$ only forms when this
height exceeds the height of the porous structure $h_{\mathrm{p}}$ [90].
Combining the dynamic $h_{\mathrm{dyn}}$ scaling with this inequality,
produces the critical capillary number
$\mathrm{Ca}^{*}=(h_{\mathrm{p}}/r)^{3/2}$ that needs to be exceeded to form
$h_{\mathrm{dyn}}$ [191]. The associated (secondary) dissipation rate within
this domain scales (non-linearly in $v$) as

$P_{\mathrm{diss}}^{\mathrm{vis,II}}\propto\gamma vr\mathrm{Ca}^{2/3}.$ (16)

Figure 5:  Dynamic geometries of liquid wetting ridge during drop sliding. The
oil film underneath the drop and behind the drop, both, resemble a Landau-
Levich film. The dynamic film thickness a) underneath the drop front
$h_{\mathrm{dyn,f}}$, and b) behind the drop rear end $h_{\mathrm{dyn,b}}$,
both display the scaling $\propto\mathrm{Ca}^{2/3}$, characteristic for
Landau-Levich films. Data in b) and c) adapted from [50] and insets in b)
adapted from [190].

Since the scaling between power and the capillary number is the same for both
dissipation in the Landau Levich film and at the foot of the wetting ridge,
most experiments have the same scaling regardless of the dissipation
mechanism. While this makes the scaling law remarkably universal, it also
makes it difficult to directly deduce which of the two mechanisms is dominant.
At high capillary numbers (Ca $\gtrsim 10^{-2}$), a different scaling law was
observed between friction on velocity, with a lower exponent on the velocity.
In this regime, drops begin to move more rapidly. The mechanism for this
regime is still unknown. A tentative explanation that has been proposed is
that the wetting ridge shape does not have time to fully develop in this
regime [194].

Figure 6:  Dissipation-speed profile ($P_{\mathrm{diss}}/r\propto v^{k}$) on
a) liquid-infused silicone oil (green) $\eta=100~{}\mathrm{mPa~{}s}$ and
perfluorinated oil (red) $\eta=30~{}\mathrm{mPa~{}s}$ ($k\approx 1.66$), b)
$\mu=250~{}\mathrm{kPa}$ elastomer ($k\approx 1.23$), and c)
$6~{}\mathrm{kDa}$ chain SOCAL surfaces ($k\approx 1.25$). a) Bright colors
correspond to cases where $\mathrm{Ca}^{*}<\mathrm{Ca}$ and dark colors to
cases where $\mathrm{Ca}^{*}>\mathrm{Ca}$, with
$\mathrm{Ca}^{*}=(h_{\mathrm{p}}/r)^{3/2}$.b) The colors correspond to various
coating thickness: green - $1100~{}\mathrm{nm}$, black berry -
$350~{}\mathrm{nm}$, red - $90~{}\mathrm{nm}$, yellow - $55~{}\mathrm{nm}$. c)
Data points were measured on SOCAL surface, with chains: pink - grafted from,
blue - grafted to substrate (c.f. PDMS based surfaces - structure and
properties). Data reproduced from a) red [195] green [190, 191], b) [196], and
c) green [197] rest [64].

####  III.3.2 Elastic

On elastomeric surfaces, the wetting ridge dissipation is more complex [198,
199]. Dissipation in elastomers is non-Newtonian and depends on the
(oscillatory) strain rate $\omega$. The crosslinks add (visco)elasticity to
the material fabric and the resulting stress is well represented with the
Chasset-Thirion model, i.e.,
$\sigma\propto\epsilon\mu\left(\tau\omega\right)^{m}$ [200]. The moving
wetting ridge oscillates with $\omega\propto v/\delta_{\mathrm{e,I}}$. $\tau$
is the rheological relaxation time and $m<1$ yields characteristic shear-
thinning of PDMS [201, 202]. These two values vary depending on the degree of
crosslinking and the number of free chains in the network [203]. Softer
elastomers with more free chains usually have both smaller $\tau$ and smaller
$m$. The strain in elastic ridges scales with the ratio of the two
elastocapillary lengths
$\epsilon\propto\delta_{\mathrm{e,I}}/\delta_{\mathrm{e,II}}$ and the strain
rate with $\dot{\epsilon}\propto\epsilon\omega$. Together with the volumetric
scaling $V\propto\delta_{\mathrm{e,I}}^{2}$, the dissipative work in the
viscoelastic wetting ridge scales as

$P_{\mathrm{diss}}^{\mathrm{vel}}\propto\gamma_{w}vr\alpha^{2}\mathrm{Ca}_{e}^{m},$
(17)

Fig. 6b. Here, we introduce the ”viscoelastic” capillary number
$\mathrm{Ca}_{e}=\tau v/\delta_{e,I}$. The thickness of the surface coating
affects the shape of the wetting ridge when the length scales of both become
comparable [204, 205]. Surface material that accumulates in the elastic
wetting ridge cannot be replenished. Thus, the surface dimples around the
wetting ridge, in lack of surface material, Fig. 3b. Very thin coating
surfaces also limit the size of the wetting ridge as the material is confined
(or attached) to an underlying surface. The ridge geometry is in addition to
$\delta_{e,\mathrm{I}}$ and $\delta_{e,\mathrm{II}}$ governed by the thin film
height. The friction that builds up in the wetting ridge is consequently
reduced, as the total dissipation volume, $V_{\mathrm{ridge}}$ decreases
[196].
Wetting ridge relaxations that are different from expected viscoelastic
materials were observed on PDMS elastomers having free oligomer chains [206].
The reorganization of the free chains inside the elastomer follows poroelastic
behavior, which is typically slower than viscoelastic dissipation [207].
Wetting ridges that move slowly dissipate energy in a different mode, as the
material fabric can change significantly inside the ridge. As discussed
earlier, a phase of pure liquid can form at the tip of the ridge where the
dissipative picture resembles the liquid-infused case. The amount of phase
separation scales inversely with the moving speed of the ridge [114]. Thus, a
complex back coupling between speed and dissipation is expected. While concise
models for dissipation on swollen PDMS elastomers are missing, first clues
suggest that the separation lowers dissipation: the accumulated oligomeric
free chains have low viscosity and lubricate the drop and effectively reduce
contact friction [113, 44].

####  III.3.3 Molecular

In SOCALs, a mobile drop sliding over the surface reorganizes the streched
chains in the ridge constantly. While low, the contact angle hysteresis is not
zero. This indicates a non-symmetric process: the differing capillary force at
the advancing and receding side leads to a non-zero net force as the drop
moves. The force that builds up during sliding was measured with a drop
friction force instrument [64]. This force scales with the drop speed
approximately with $P_{\mathrm{diss}}\propto v^{5/4}$, Fig 6c. While
preliminary, these measurements have some important implications: as the
variation in synthesis does not appear to influence the effective friction
force, therefore, the grafting density and chain length may not play a
significant role. However, rigorous investigations are still lacking and
decisive conclusions should not be drawn at this stage.

##  IV Coating Comparison

Figure 7: Comparison of PDMS-based surface structures and wetting mechanisms
on the three variants. Illustrations adapted from [64].

In this review, we discussed wetting behaviors on PDMS-based surfaces. We
discuss how this relatively simple polymer can create a wide design space for
surfaces, via polymer-chain configurations, polymerization, and surface
anchoring. The complexity in surface manufacturing varies for different
surface variants: Creating liquid-infused surfaces without a micro-/nanometric
texture is straightforward. It can be achieved by simply spreading PDMS oil on
a surface. The process is equally facile for PDMS elastomeric surfaces, but
only if no special requirements regarding the coating thickness exist. The
”grafting-from” technique for SOCAL surfaces requires a more elaborate process
such as ”chemical-vapor-deposition” (CVD), requiring controlled atmosphere
conditions, and sometimes toxic chemicals (e.g. chlorosilanes or
alkoxysilanes).
The various surface designs open up a wide variety of concepts and mechanisms.
Today, they are increasingly utilized and tuned for a plethora of
applications. While similar at first glance, wetting mechanisms such as the
wetting ridge develop differently on each PDMS-based surface variant. In the
context of a specific application, this brings advantages and disadvantages.
For example, self-cleaning are increasingly essential for preserving the
energy efficiency of solar panels, bio-contamination reduction of medical
equipment, or repelling stains and dirt on outdoor equipment and textiles. For
this, all three PDMS-based surface variants are conceptually suitable as they
strive to provide drop mobility (low static friction), a feature that is
essential for self-cleaning.
Liquid-infused surfaces have freely flowing PDMS oils providing excellent
lubrication properties and high drop mobility. An important avenue for future
research is to understand how particulate contaminants adheres to the
lubricant layer and whether they enter the solid crevices. However, one
limitation of liquid-infused surfaces is that lubricant is depleted by the
(comparatively) large wetting ridge when they are exposed to harsh
environments. Thus, to make these surfaces viable for technological
applications, further work is needed to systematically understand the
mechanisms of lubricant depletion. Elastomer surfaces, on the other hand, have
good material retention and hence mechanical durability, provided by the
crosslink network. This advantage, however, brings along higher dissipation
margins and declined drop mobility. SOCAL surfaces seem to overcome both
challenges, as retention is high due to covalent surface anchors, while
dissipation remains low, although not as low as liquid-infused surfaces. This
is attributed to a minuscule ridge and the inherent flexibility of molecularly
anchored chains that behave in a liquid-like manner. Particulate matter hardly
sticks and can be removed easily. However, once the surface experiences
critical damage, the nano-thickness of the SOCAL surface can be permanently
abraded away. Bulky liquid-infused and elastomer surfaces provide plenty of
surface material that - in case of surface damage - can self-heal and increase
the surface lifetime.
We currently observe a transition in soft wetting research, shifting from a
fundamental understanding perspective to a practical application focus,
encompassing fields such as energy and biomedical science. Still, to utilize
soft wetting effectively in any application, fundamental understanding is
critical, as illustrated in the self-cleaning example, above. This review
serves to provide a series of guidelines that link the current state-of-the-
art soft wetting knowledge - in particular on silicone surfaces - to
mechanistic design features desirable for a myriad of applications.

##  V Methods and Techniques

###  V.1 Optical Characterization

####  V.1.1 Shadowgraphy

The Shadowgraphy method captures shadows of macroscopic objects (e.g., drops
of $>1~{}\mathrm{\mu l}$) with high-resolution optical cameras and back-
illumination. Shadowgraphs are typically well-contrasted and thus suitable for
contour profile extractions of static/dynamic drops using image processing.
For example, shadowgraphs are the basis of goniometry, i.e., one of the most
common and simplest methods to measure surface wettability. Here, the wetting
contact angle is extracted from the drop contour and recovered by various
fitting techniques ( e.g. Young-Laplace equation, spherical/elliptical fit, or
tangent fit [208, 209]). Goniometry was utilized to demonstrate surface
adaptation and dissipation on PDMS gels [44, 210]. The overall shape of the
drop contour is governed by the surface tension. If the drop becomes larger
than the capillary length, gravity distorts the contour shape from a perfect
(hemi)sphere (i.e., oblate sessile drops and prolate pendant drops). When the
density of the drop and the prevailing gravity is known, the surface tension
can be measured from shadowgraphs (cf. pendant drop measurement). With
optically-assisted magnification, microscopic features such as wetting ridges
become visually detectable [160]. In place of (vis)light, x-rays were used for
high-definition imaging of static wetting ridges [132].

####  V.1.2 Confocal Microscopy

Confocal microscopy is a non-invasive optical technique that allows us to
image samples across both horizontal and vertical planes. Three-dimensional
images can also be reconstructed by stacking the two-dimensional
horizontal/vertical images. Confocal microscopy has been used to study the
static and dynamic properties of drops on liquid-infused surfaces and soft
elastomeric surfaces [152, 211, 112, 114]. Fundamentally, confocal microscopy
is an extension of fluorescence microscopy where light that is out of focus is
filtered out using a pinhole placed at the conjugated point of the optical
path. Laser scanning confocal microscopes scan across a sample point by point
and combine the scans to produce 3D images with a horizontal resolution of
around 200 nm and a vertical resolution of around 1 $\mu$m. The relatively
slow point-by-point scan can be accelerated with a spinning archimedean disk.
When using confocal microscopy, different fluorescent dyes must be added to
the different phases (e.g. drop and coating) being imaged. The use of dyes
enables the detection of interfaces between phases of similar refractive
index. Typically, inverted confocal microscopes (objective lens below
specimen) are used to image drops because this configuration makes it possible
to image the contact region between the drop and the surface - which is the
region that governs the wetting properties - with minimal optical artifacts.
The primary limitation of this technique is that it requires transparent
substrates, such as glass.

####  V.1.3 Interferometry

Interferometric microscopy enables in-plane depth measurements with nano-to-
microscopic resolution. In wetting, this has been utilized to directly
visualize nanometric features such as contact angles/lines [212, 213], drop
profiles [214], wetting ridges [53, 215], and thin films [216, 217, 218, 195,
219]. Coherent light of a “diagnostic beam” travels through the specimen and
is brought into superposition with a reference beam with identical coherency.
The emerging interference pattern gives the optical path (geometric depth
times refractive index) difference between both light beams. The optical path
length of the reference beam is calibrated. Different ways of beam calibration
brought up various flavors of interference techniques such as Reflection
Interference Contrast Microscopy, Digital Holographic Microscopy, or Michelson
interferometry, each varying in practical complexity and precision. With
information on the refractive index of the specimen, the geometrical depth of
the specimen can be determined with a precision of $\leq 10\mathrm{nm}$.

####  V.1.4 Ellipsometry

Ellipsometry is a non-destructive/non-invasive optical technique that allows
the characterization of thin films with sub-Angstrom resolution. The fast
scanning speed ($<1$ min for single-wavelength, $<1$ s for multi-wavelength)
enables in situ measurements of dynamic films. The technique was used to
investigate e.g., precursor spreading of PDMS oil on glass [220, 221, 222] or
wetting on SOCAL surfaces [223, 224]. Ellipsometry measures changes in the
polarization state of an incident light beam upon interaction with the
surface. In ellipsometric measurements, linear polarized light is directed to
a film. Depending on the thickness and material properties of the film, the
amplitude and phase of the reflected light change due to light-film
interactions (absorption/scatter). The light-film interactions are different
for the p and s components of the light (parallel and lateral to the plane of
the incident), resulting in an altered, elliptical polarization state of the
reflected light. The ratio of s/p amplitudes before and after reflection, and
the phase shift contains information on the film thickness and optical
constants (e.g., refractive index). Ellipsometers either use a single
wavelength or multiple wavelengths. In the former case, the angle of incidence
is varied and the layer thickness and refractive index are fitted. When
working with multiple wavelengths, usually one angle of incidence is used and
the film thickness and refractive index are fitted as a function of
wavelength. PDMS films on oxidized surfaces (i.e., glass or silicon oxide) are
prone to measurement errors as both have almost identical refractive indices.

Figure 8:  Different imaging techniques are suitable for different length
scales. Most light-based techniques such as (Vis)shadowgraphy or confocal
microscopy are bound by the diffraction limit. Interferometry and ellipsometry
overcome this limit by exploiting wave superposition and phase shifts,
bringing down the resolution limit to sub-nano.

###  V.2 Mechanical Characterization

####  V.2.1 Classical Rheology

Rheology describes the stress-strain/shear behavior and the mechanical
properties of soft materials such as PDMS. From a wetting perspective, these
mechanical properties provide information that can be related to the dynamics
and dissipation in the wetting ridge and around the contact line. The term
rheology is generic, which includes various geometries and experimental setups
to measure dynamic mechanical properties. The most common for silicone-based
materials - particularly related to wetting - is parallel-plate shear
rheology. In these experiments, silicone-based elastomers, swollen networks,
or melts are loaded between two plates. The top plate is typically rotated in
a sinusoidal fashion while recording both the stress and the strains. The
lower plate can be connected to a Peltier plate, or the entire system can be
enclosed into an environmental chamber if temperature or humidity control is
needed. Typically, an amplitude strain sweep is conducted at a constant
frequency to determine the linear region. A constant strain is then chosen,
and a frequency sweep is then conducted to quantify the mechanical properties
of the material as a function of frequency (i.e., rate dependence) in the
linear regime. Common values that are extracted from the measurement as a
function of frequency are the storage $(G^{\prime})$ and loss modulus
$(G^{\prime\prime})$. For more liquid-based PDMS (i.e. oils), viscosities can
also be measured through a similar method. The storage modulus describes the
tendency of the material to store energy (solid-like character) whereas the
loss modulus is the tendency to dissipate energy (liquid-like character).
Their ratio, $G^{\prime\prime}/G^{\prime}=\tan\delta$ can be associated with
damping.

####  V.2.2 Indenter

The use of indentation and contact mechanics is useful for characterizing
surface and near-surface properties. This can include properties like adhesion
and friction, lubricant layer properties, and the mechanical properties of a
substrate. Such experiments can be conducted across various size scales, from
macro- [225, 226, 227] to micro- [228, 229] to nanoscales [230]. The
experimental setup generally consists of a spherical or cone shaped probe,
which is attached to a force measurement tool (e.g. a deflecting cantilever or
a load cell). For the nanoscale, an AFM tip is usually employed. In a typical
experiment, the probe approaches the surface at a designated rate until it
comes into contact, at which point the probe and surface can jump into
contact. The probe continues to indent into the surface until reaching a pre-
defined distance or force. The probe is then either retracted from the surface
directly or held stationary for some time and then retracted. During this
entire process, the force and distance are recorded, allowing for
quantification of mechanical properties during the approaching process,
relaxation of the material during the holding time, and adhesion during the
retraction process. Combining indentation with contact mechanics models can
enable the quantification of mechanical properties while integrating force-
distance curves during retraction determines the work required to separate two
surfaces. By sliding laterally, friction measurements can be conducted with a
similar type of experimental setup. From the wetting perspective, indentation
can be used as a tool to characterize mechanical characteristics, which can
then be related to the wetting ridge formation. For example, measuring forces
during the holding time can provide a route to separate the viscoelastic and
poroelastic relaxation timescales associated with wetting ridges [231].
Moreover, particle indentation methods can also be connected to fluid
separation and surface stress. The indentation process can also provide
information for pure liquid surfaces like capillary forces [232], detection of
nanobubbles [233] or quantification of lubricant layer thicknesses [234] for
SLIPS.

####  V.2.3 Force Microscopy

Forces that build up during drop-sliding on PDMS films can be measured with
cantilever set-ups [197]. Such ”force microscopes” were used to measure
dissipation on PDMS surfaces of all types. The cantilever is vertically placed
over the PDMS surface. Glass capillaries, metal blades, and rods with ring
ends have been successfully utilized as cantilevers [197, 235, 236, 196, 237].
The cantilever top is fixed while the lower end hangs freely several
micrometers above the surface. Displacements of the lower end - which can be
tracked with e.g. shadowgraphy - relate to acting forces and can be extracted
with the cantilever spring constant. The spring constant is calibrated by
measuring the natural frequency or by applying well-defined loads. To measure
dynamic wetting forces on PDMS surfaces, a sessile drop is pushed against the
cantilever by moving the surface horizontally. The major drawback of this
technique is the invasiveness of the capillary to the drop. It is still not
completely understood how the cantilever itself influences the measured
friction forces.

Figure 9:  Force microscopic measurement conducted on a confocal microscope.
Here, $10~{}\mathrm{\mu l}$ water drops slide on on SOCAL surfaces. a) Atto-
dyed, steady-state drop footprints, and faintly the probing metal blade. b)
Kymograph of the blade deflection upon drop contact, measured with the
reflection channel of an $xt$-scan. c) Calibration curve to determine the
spring constant of the blade, here approx. $0.21~{}\mathrm{N/m}$.

###  V.3 Computational Methods

####  V.3.1 Molecular Dynamics (MD) Simulations

Molecular dynamics (MD) simulations are particle-based methods that resolve
the evolution of the particle positions and momenta by solving Newton’s second
law of motion [238]. Due to the molecular resolution, MD is utilized to obtain
explicit insight into the polymer behavior of PDMS chains during wetting. In
particular, MD is used to investigate the nanoscopic wetting ridges of SOCAL
surfaces, and the conformation/dynamics of PDMS chains, or to calculate
relevant thermodynamic quantities such as surface tensions or micro-rheology.
The particles can represent atoms but also larger, coarse-grained structures
(e.g. molecules), which affects the interaction force fields between
particles. Interaction forces may be atomistic (quantum mechanical) or
effective (e.g. Lennard-Jones potential) such that they reproduce the most
important physical properties of the system of interest. By virtue of their
resolution, MD simulations are computationally expensive since one needs to
solve the equation of motion for every particle. Consequently, MD setups are
usually microscopic in time and space resolution.

####  V.3.2 Lattice Boltzmann Method (LBM)

LBM is a computational method typically applied to solve fluid mechanics
problems [239, 240, 241]. In contrast to microscopic methods (e.g. MD) that
compute the motion of individual (or clusters) of fluid particles, LBM is a
mesoscopic method based on kinetic theory. The central quantity computed in
LBM is the probability distribution function, which gives the number of fluid
particles moving at a given velocity at a given position. From the
distribution function, we can obtain detailed information that is difficult to
obtain experimentally, such as the velocity field, pressure distribution, and
viscous dissipation in the fluid. LBM has been applied to study drop dynamics
on flat and textured surfaces, including liquid-infused surfaces [242]. One of
the key advantages of LBM is that coarse-grained molecular interactions (for
example, between liquids and solid surfaces) and complex solid geometries can
be implemented efficiently. Since LBM focuses on the collective behavior of
fluid particles rather than on individual fluid particles, it is less
computationally intensive than microscopic simulations, which means larger
drops can be simulated relative to the surface features (height of pillars in
the case of liquid-infused surfaces).

Figure 10:  Typical time and length scale domains of wetting problems, treated
with different computational methods. Density functional theory (DFT) is
utilized in the sub-nano domain. The next larger domain sizes are treated with
atomistic and coarse-grained molecular dynamics (MD) simulation. The Lattice
Boltzmann method (LBM) utilizes kinetic theory and is mesoscopic, linking the
microscopic and the macroscopic world. Mean field theory is utilized for large
molecule ensembles and macroscopic problems.

####  V.3.3 Mean Field Models

Mean field approaches regard the collective behavior of fluids and solids and
are an alternative to MD and LBM, particularly for macroscopic systems. Mean
field models were utilized e.g., for static and dynamic wetting on PDMS gels
[210, 243, 170, 244, 245], characterization of wetting ridges [246, 247, 248,
249], water sorption in PDMS [250], or to model cloaking [251]. Mean field
models exploit that individual fluctuations of atoms, molecules, or coarse-
grained particles diminish when averaged over sufficiently large ensembles. As
a result, energy and momentum are not properties of single beads, but (weakly)
smooth mean fields which are time and space-dependent. The variety of mean-
field approaches is large: Momentum-based approaches applied to fluids result
in the well-known Navier-Stokes equations and their many sub-frameworks
(lubrication theory, Stokes equation, potential theory) [252, 253, 254].
Solids are frequently described with linear elasticity theory. Complex
materials require rheological models (e.g. Maxwell/Kelvin-Voigt bodies).
Consideration of the (free) energy field gives rise to the Allen-Cahn/Cahn-
Hilliard equations [255, 256], gradient dynamics [257], or the heat equation
[258]. The spatiotemporal distribution of the mean-field quantities is
expressed with partial differential equations (PDE). Finding closed-form
solutions for these PDEs is the exception as the dependent variables (e.g.,
velocity and temperature) can be coupled and the system geometry is often
complex. Hence, numerical tools, such as the Crank-Nichelson method and the
finite element method are utilized to solve the time and space components,
respectively [259, 260, 261]. Multiple phases and their shared interfaces are
handled with explicit (interface tracking, e.g., arbitrary Lagrangian-Euler
[262]) or implicit (interface capturing, e.g. volume of fluid [263, 264])
methods. Commercial and open-source software libraries simplify the practical
application of such tools. Mean field models assume material properties (e.g.,
densities, viscosities, compliances, mobilities, surface tensions, etc.) a
priori, which can be a drawback for various PDMS wetting-related questions
(e.g. adaptation or cloaking). Also, the individual molecular behavior of PDMS
molecules is inaccessible.

##  VI Acknowledgments

We wish to thank H.-J. Butt, Z. Cai, S. Karpitschka, F. Schmid, and P. Stephan
for stimulating and helpful discussions. L.H. thanks T. Boeddeker and K.
Sporbeck for feedback on the manuscript. This research was motivated by the
DFG Priority Programme 2171. Financial support came from the DFG Emmy Noether
Programme No. 460056461 (L.H.), the EPSRC National Fellowship in Fluid
Dynamics Grant No. EP-X028410-1 (A.N.), the European Union’s HORIZON research
and innovation program under the Marie Sklodowska-Curie grant agreement No.
101062409 and the Academy of Finland grant agreement No. 13347247 (W.S.Y.W),
the US National Science Foundation through CBET grant 2326933 (J.T.P.), and
the Max Planck Center for Complex Fluid Dynamics - University of Twente
(D.V.).

##  VII Author Declarations

The authors have no conflicts to disclose.

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