Encyclopedia of Polymeric Nanomaterials

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| Editors: Shiro Kobayashi, Klaus Müllen

Controlling Wetting Properties of Polymers

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DOI: https://doi.org/10.1007/978-3-642-36199-9_369-1
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Keywords

Surface Tension Contact Angle Contact Line Finite Layer Lifshitz Theory 
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Synonyms

Dynamics and stability of thin films; Intermolecular forces

Definition

Wetting properties of a liquid comprise static and dynamic aspects. In the static case, the contact angle, spreading coefficient, and effective interface potentials are relevant. For dynamics, properties like the wetting or dewetting velocity, influenced by viscosity and viscoelasticity as well as by the hydrodynamic boundary conditions between the liquid and its confining media are important.

Introduction

Equilibrium and dynamical wetting properties of liquid systems with nanoscopic dimensions on solid substrates are largely determined by the intermolecular interactions [1, 2] between all of a system’s constituent parts. Thus, the control of wetting properties relies on a thorough understanding of the microscopic mechanisms at the root. These interactions can be grouped as short ranged and long ranged. Observed macroscopic or collective phenomena result from a balance between these interactions at disparate length scales. In equilibrium [3, 4, 5, 6], one example of a macroscopic property is the surface or interfacial tension, γ ij , between a condensed phase, “i” and either another condensed or a vapor phase, “j.” For i a liquid and j a vapor, it is often convenient to suppress the indices, and γ is therefore representative of the liquid–vapor surface tension; this convention is adopted here. A second example of an equilibrium wetting property is the contact angle, θ, at the three phase contact line. In the dynamic case [3, 4, 7, 8, 9], one may consider among many other examples (i) the speed with which such a contact line moves. The contact line motion is a result of unbalanced interfacial tensions or other external perturbations, and is damped by friction in the bulk fluid and friction with the boundaries. One may also consider (ii) how the ripples on a surface smooth out due to surface tension. Similarly, with respect to layers thin in comparison to the range of the intermolecular interactions, it is interesting to note (iii) how quickly a stable mode relaxes or how quickly an unstable mode grows. In all of these equilibrium and dynamic cases, intermolecular interactions, comprising short-ranged repulsive and long-ranged attractive interactions, play an important role in the observed phenomena.

The intermolecular interactions to be described are, first, steric interactions arising from the Born repulsions between the bound electron clouds. These repulsive interactions dominate at the smallest scales, playing an important role in e.g., the determination of the contact angle. Per unit area, these repulsive interaction potentials are here termed ϕ r. The second set of interactions are the van der Waals forces. These forces are not only ubiquitous, they also enjoy an important place in the history of modern physics as aptly described by Parsegian [2]: Planck exploited the idea that the cavity walls of a black body influence the spectrum of allowed wavelengths, and used this idea to begin quantum theory with a prediction of the temperature dependence of the black body spectrum and energy density; Casimir then determined the force acting between these walls by considering the derivation of the energy density with respect to the distance; Lifshitz generalized the distance-dependent interaction to arbitrary media, i.e., a theory not restricted to vacuum. These van der Waals forces occur between all atoms and molecules, independent of the charge, polarization or polarizability, and size of the objects. Van der Waals interactions between atoms or molecules are always attractive. Though depending on the balance between different condensed phases, interfaces may repel one another. Per unit area, the van der Waals potential is denoted ϕ vdW. The total effective interface potential is then built up from the sum of all interactions
$$ \phi ={\phi}_{\mathrm{r}}+{\phi}_{\mathrm{vdW}}+\cdots, $$
(1)
where other terms such as unbound charge interactions or hydrophobic interactions may be present. For these other contributions, the comprehensive text by Isrealachvili may be consulted [1]. Thus, at long length scales the attractive forces dominate while, at smaller scales, the repulsive forces are dominant, resulting in a balance at typical length scales of one to tens of nanometres.

In the following, the van der Waals forces that are responsible for many surface and interfacial phenomena will be described in more detail. This potential is then generalized to systems with several layers. After describing the layered systems, wetting phenomena in the context of this effective interface potential are described.

Intermolecular Interactions

The short-ranged interaction between particles in the effective interface potential is a steric one arising from the Born repulsion between the electron clouds of atoms. This interaction is often modeled empirically using an inverse power law
$$ {W}_{2,\mathrm{rep}}\sim \frac{1}{r^M}, $$
(2)
with an exponent M = 12 and with r the distance between the two particles; other forms can be used [3]. Using the inverse power form, and assuming linear additivity, the repulsive interaction term for two semi-infinite continuous media interacting across a gap with thickness h can be determined. Upon integrating Eq. 2, the result is
$$ {\phi}_{\mathrm{r}}(h)=\frac{C}{h^{M-4}}=\frac{C}{h^8}, $$
(3)
where C is a positive constant, ensuring a repulsive short-range interaction.
Attractive van der Waals forces have their origin in the fluctuating permanent or induced dipole moments of all atoms and molecules. Dipoles produce an electric field that induces similarly oriented moments in surrounding charge pairs. Since the interaction between like-oriented dipoles is attractive, and because all atoms and molecules contain at least an instantaneous dipole moment, this van der Waals force is always present. These forces can be broken up into three parts, depending on the permanency of the involved dipoles. If both atoms and molecules are polarized, then one speaks of the Keesom interaction energy, which describes the angle-averaged interaction between two permanent dipoles. If only one molecule is permanently polarized, then the interaction energy is referred to as the Debye interaction describing dipole-induced dipole interactions. Lastly, even if the average dipole moment of a molecule is zero, there exists an instantaneous dipole moment which arises from the instantaneous distribution of charges within the molecule. The resulting interaction is called the London dispersion interaction. In all these cases, the scaling with distance of the interaction is the same, and the van der Waals interaction energy between two isolated atoms separated by a distance r in vacuum is given by
$$ {W}_{2,\mathrm{v}\mathrm{d}\mathrm{W}}=-\frac{C_{\mathrm{vdW}}}{r^N}. $$
(4)
For nonretarded dipole interactions (that is, r small enough such that the finite speed of light need not be considered), the exponent is N = 6 < M (cf. Eq. 2). C vdW is a constant depending on the temperature, T, and the spectrum of absorption frequencies, V i , of the atoms. This constant contains contributions from the three dipole interactions described above [1, 2].

At first sight, the interaction energy in Eq. 4 appears as rather short range. However, proceeding with a linear additivity assumption as was first done by Hamaker in 1937, it is easily shown that for macroscopic bodies the interaction can be much stronger.

For two spheres of radius R separated by a minimum distance D, the potential of Eq. 4 can be integrated to obtain an interaction potential in the limit where separations are small compared to the size of the spheres, \( D \ll R \). The result is
$$ {W}_{\mathrm{ss}}(D)=-\frac{AR}{6D}, $$
(5)
where, A is called the Hamaker coefficient. When the spheres are composed of dielectric materials and separated by a vacuum (or air), typical values for A are of order 10–20 to 10–19 J ≈ 100 kT at room temperature with k Boltzmann’s constant.
A second geometry of significant practical interest is that of a flat thin film of thickness h separating a solid substrate and air. In this case, the van der Waals potential, ϕ vdW(h), which is the analogue to W ss except that it describes instead the interaction energy per unit area, is given by
$$ {\phi}_{\mathrm{vdW}}(h)=-\frac{A}{12\pi {h}^2}. $$
(6)
This interaction is similar to the one for two spheres in that it also contains a Hamaker coefficient, A, and in that it varies as an inverse power of the separation between the interfaces which is smaller than the exponent for the molecule-molecule interaction. The same procedure for determining the van der Waals interaction potential between extended bodies can be investigated for a zoology of geometries, and many examples can be found in Israelachvili [1] and Parsegian [2], along with the limitations to the used additivity assumption. The important point is that although the interaction between two atoms or molecules is of rather short range, the effect of all collective interactions between atoms in a macroscopic body gives rise to a much stronger interaction.

It must be stressed that the simple additivity assumption – which can be used to predict Eq. 5 – is, strictly speaking, invalid. The problem with this assumption is that the van der Waals interactions between two given atoms are influenced by neighboring atoms, meaning that simple additivity cannot be assumed. However, as described in Israelachvili [1] and Parsegian [2], the predictions for the geometric dependence of the forces obtained using the additivity assumption are similar to those obtained using the more correct continuum theories developed by Dzyaloshinskii [10] and co-workers (the Lifshitz continuum theory) when: the permittivities of the media are not too different; and when retardation effects are not important (see [2], sections “PR.1” and “L2.D.3,” for a detailed discussion of the limitations of the additivity result).

The Lifshitz theory is based on quantum electrodynamics in continuous media, such that the nonadditivity of interactions is completely avoided. Therefore, the Hamaker coefficients predicted using an additivity assumption are not recovered. Under the aforementioned limitations, this is the major difference between the predictions of the Hamaker and Lifshitz theories.

Within the more accurate and general Lifshitz theory, a useful approximation [1, 4, 11, 12] for the Hamaker coefficient can be made. To make the estimate, the frequency dependence of the permittivities, ϵ(v), should be known. In practice, these functions are tabulated for some common substances, or suitable models can be used. If the three media are dielectric (e.g., polymer), and if the dominant electronic absorption frequency, v e , for the three media is almost the same, a simple closed form expression for the Hamaker coefficient can be derived (if a metal medium is involved, a dielectric function, ϵ(v), which is different from the one used to predict Eq. 7 must be used. See [1, 2]). For dielectric bodies 1 and 3 interacting across dielectric medium 2, as shown in Fig. 1a, the Hamaker coefficient is given by
Fig. 1

Schematic description of three relevant thin film geometries for the consideration of van der Waals interactions. (a) A three layer system, showing semi-infinite bodies 1 and 3 interacting through medium 2, which can e.g., be a thin film. (b) A four layer system, showing semi-infinite bodies 1 and 4 interacting through media 2 and 3; typical experiments with such a system are polymer films on oxidized silicon wafers, in air. (c) A five layer system, for instance: air/photoresist/compatibilizer/SiO2/Si

$$ \begin{array}{ll}{A}_{1/2-3/2}\hfill & =\frac{3}{4}kT\left(\frac{\upepsilon_1-{\upepsilon}_2}{\upepsilon_1+{\upepsilon}_2}\right)\left(\frac{\upepsilon_3-{\upepsilon}_2}{\upepsilon_1+{\upepsilon}_2}\right)\hfill \\ {}\hfill & +\frac{3\uppi \hslash {v}_e}{4\sqrt{2}}\frac{\left({n}_1^2-{n}_2^2\right)\left({n}_3^2-{n}_2^2\right)}{{\left({n}_1^2-{n}_2^2\right)}^{1/2}{\left({n}_3^2+{n}_2^2\right)}^{1/2}\left[{\left({n}_1^2+{n}_2^2\right)}^{1/2}+{\left({n}_3^2+{n}_2^2\right)}^{1/2}\right]}.\hfill \end{array} $$
(7)
The Hamaker coefficient is now defined with the subscripts i/j–k/j, indicating that the bodies made up of materials i and k interact across medium j; this notation is useful for describing multilayer samples, as is done below. In Eq. 7, ϵ i and n i are the dielectric permittivities and indices of refraction (in the visible) of medium i, ℏ is Planck’s reduced constant, and v e is a dominant electronic absorption frequency in the UV, typically around 3 × 1015s−1.
For a thin film on a solid surface, the film is medium 2, lying between the gas phase (medium 1) and the substrate (medium 3). This system is described by Eq. 1 along with Eqs. 3, 6, and 7, completing the total effective interface potential for two extended planar bodies 1 and 3 interacting across a medium 2 of thickness h:
$$ \phi (h)={\phi}_r+{\phi}_{\mathrm{vdW}}=\frac{C}{h^8}-\frac{A_{1/2-3/2}}{12\uppi {h}^2}. $$
(8)

For applications involving polymer films on, e.g., Si substrates with various surface treatments in air, the first term in Eq. 7 does not contribute more than a few percent of the total Hamaker coefficient, thus A is only weakly temperature dependent in these cases. For these systems, since the denominator in the second term is strictly positive, the relative sizes of the indices of refraction give the sign of A, determining whether the interaction is attractive or repulsive. Given the form of Eqs. 5 and 6, it is seen that a positive Hamaker coefficient gives rise to an attractive interaction, while in the opposite case of negative A, a repulsive interaction is expected. Thus, if the index of refraction of the intervening medium, n 2, satisfies n 1 < n 2 < n 3 (similarly, n 3 < n 2 < n 1), we expect a repulsive interaction between the interfaces (i.e., a stable thin film); in the case where n 2 is either greater than or less than both of n 1 and n 3, an attractive interaction is expected (i.e., a thin film drains to become thinner).

Multilayered Samples

In many cases of practical interest, the effective interface potential described in Eq. 8 should be augmented to include the possibility of more than a single layer between two semi-infinite layers. As mentioned above, a silicon wafer may have a SiO2 layer of some thickness. Typically, the oxide layer is 1 or 2 nm thick for “native” oxide layers, or some hundreds of nanometres for thermally grown layers. Si wafers are useful in part because they are amenable to many surface treatments, particularly with regard to polymers. In addition to the oxide layer, an applied self-assembled monolayer (SAM) [11, 13] or an amorphous polymer film, such as a fluoropolymer layer may be added to the wafer. In such cases, the substrate to be used in the experiment of interest already has two layers in addition to the two semi-inifinite ones (practically, layers thicker than ~100 nm can be treated as semi-infinite). The last of these mentioned layers is typically used to endow the topmost layer with a lower surface energy.

For a single additional layer with thickness d as shown in Fig. 1b, the total van der Waals interaction can be determined using an additivity assumption [12] and it is written as
$$ \begin{array}{ll}{\phi}_{\mathrm{vdW}}(h)=-\frac{1}{12\uppi}\left[\frac{A_{1/2-3/2}}{h^2}+\frac{A_{1/2-4/2}-{A}_{1/2-3/2}}{{\left(h+d\right)}^2}\right],\hfill & \left(\mathrm{two}\;\mathrm{layer}\right)\hfill \end{array}, $$
(9)
which accounts for the two limits \( d \ll h \) and \( d \gg h \). Parsegian [2] has outlined the procedure for determining the van der Waals interactions for an arbitrary number of layers, although the procedure does not cover the full range of d/h. It must also be emphasized that in Eq. 9 the dependence is only on h – additional terms are necessary when d is also allowed to vary as when media 2 and 3 are fluid (a case which is beyond the scope of this contribution).
As a final example of a multilayer system, we quote the formula for a system with three finite layers and bounded by two semi-infinite media [2, 12], as shown in Fig. 1c. The typical example of a system with two finite layers bounded by two semi-infinite media, as described in Fig. 1b and Eq. 9, is a polymer film covering an oxidized silicon wafer. However, it may be that a compatibilizing layer such as a SAM is needed. SAMs are popular since they are relatively easy to prepare and give reproducible surface properties that may be widely varied; this is also the case for spin coated compatibilizing layers on SiO2/Si substrates. In such cases, a SiO2 layer with thickness d3 is covered with a compatibilizer of thickness d 4 on top of which a polymer layer of thickness h may be placed. For this system, the h-dependent effective interface potential is given by
$$ \begin{array}{ll}{\phi}_{\mathrm{vdW}}(h)\hfill & =-\frac{1}{12\uppi}\left[\frac{A_{1/2-3/2}}{h^2}+\frac{A_{1/2-4/2}-{A}_{1/2-3/2}}{{\left(h+{d}_3\right)}^2}\right.\hfill \\ {}\hfill & \begin{array}{ll}\kern1em +\left.\frac{A_{1/2-5/2}-{A}_{1/2-4/2}}{{\left(h+{d}_3+{d}_4\right)}^2}\right],\hfill & \left(\mathrm{three}\;\mathrm{layer}\right)\hfill \end{array}.\hfill \end{array} $$
(10)
Looking at the various forms for the effective interface potentials described (see Eqs. 6, 9 and 10), it can be seen that a considerable amount of control on the wetting properties of a thin polymer film may be achieved: the topmost layer of the involved interfaces dominates the size of the contact angle, while the lower layers greatly influence the stability and form of the liquid–vapor interface. These ideas are described in detail below.

Equilibrium Wetting Properties

In the context of an effective interface potential, many wetting properties of liquids and solids can be described. A first example is that of the surface tension, γ, of a liquid–air interface, defined as the work required to augment the surface area of a condensed phase by one unit. The work required to separate a symmetric one layer system, as shown in Fig. 1a with medium 1 the same as medium 3 and described by Eq. 8, from contact to large distances, h → ∞, represents twice the surface tension since two surfaces are created. Thus, the surface tension can be estimated using
$$ \upgamma =\frac{1}{2}\left(\underset{h\to \infty }{ \lim}\phi (h)-\phi \left({D}_0\right)\right), $$
(11)
where D 0 is a characteristic separation distance for two surfaces in ideal contact. Using the approximation \( \phi \approx {\phi}_{\mathrm{vdW}} \) [1], the surface or interfacial tension is then
$$ \upgamma \approx \frac{A_{1/2-1/2}}{24\uppi {D}_0}. $$
(12)
Using a symmetric version of Eq. 7, the surface tension of polystyrene, for example, can be estimated. With n 1 = n 3 = 1.59, ϵ1 = ϵ3 = 2.6ϵ0, with ϵ0 the permittivity of vacuum, and n 2 = ϵ2/ϵ0 = 1, the Hamaker coefficient can be estimated as \( {A}_{\mathrm{air}/PS-\mathrm{air}/PS}=7\times {10}^{-20}\mathrm{J} \). Using this coefficient, and using D 0 = 0.165 nm [1], the surface tension of PS is predicted to be 0.035 J/m2, in good agreement with tabulated values. Computing the surface tensions of other liquids [1] in this way provides similar agreement to within about 20 %, provided the assumptions made in deriving Eq. 7 are satisfied. It is noted that when hydrogen bonding interactions are present, this scheme shows significant deviations from experimentally measured surface tensions [1, 14].

An important system property in wetting dynamics is the so-called spreading coefficient, \( S={\upgamma}_{\mathrm{SV}}-\left({\upgamma}_{\mathrm{SL}}+\upgamma \right) \) where the subscripts S, L, V denote the solid, liquid, and vapor phases with their associated interfacial tensions. S describes the energy difference between a dry substrate (γsv) and a wet substrate (γSL + γ). Therefore, if S > 0, one expects a liquid to spread to cover the substrate. In the opposite case, S < 0, partial wetting is observed and in equilibrium the liquid forms the so-called Young angle, θ Y, with the solid. See Fig. 3 for a schematic description of possible wetting states arising from various S. Having demonstrated that it is possible to access surface tensions through the effective interface potential, it is then also possible to predict the spreading coefficient if one knows the associated optical properties described in Eq. 7.

With knowledge of the van der Waals effective interface potential and the surface tension, it is also possible to make statements concerning the stability of nanoscopic polymer layers. Depending on the application, one might wish to obtain a stable coating on top of a surface, or one may wish that the surface remains dry. For the simple three layer system described by Eq. 8, the stability of the layer is determined by the sign of the Hamaker coefficient: a positive coefficient leads to a spinodally unstable regime for heights \( h\gtrsim {\left(C/A\right)}^{1/\left(M-N\right)} \), where the numerical coefficient has been omitted. In the spinodal regime, films spontaneously break up due to the attraction between the interfaces, “1/2−3/2.” Using the thin film lubrication approximation, and by balancing surface tension with the van der Waals interface potential, a by now standard linear stability analysis [3, 4, 8] shows that the mode which grows the fastest is the one with wavenumber, q = q max, given by
$$ {q}_{\max }={\left(\frac{-8\uppi \upgamma}{\phi_{\mathrm{vdW}}^{{\prime\prime} }(h)}\right)}^{1/2}, $$
(13)
where the prime denotes differentiation with respect to the film height, h. Combining Eqs. 6, 12 and 13, it is seen that the fastest growing wavelength is λmax ≈ h 2/D 0. For D 0 ≈ 0.2 nm and h ≈ 5 nm, spinodal wavelengths of some hundreds of nanometres may be expected, which are indeed seen in experiments [15]. In a spinodally unstable film, the distribution of holes will be intrinsically correlated with the preferred length scale λmax.
When the Hamaker coefficient for a system with one finite layer separated by two semi-infinite media is instead negative, the film is stable to both spinodal dewetting and nucleated dewetting. However, if the effective interface potential arises from a system with two (or more) finite layers with two bounding semi-infinite media, such as is described by Eq. 9 or Eq. 10 and shown in Fig. 2, it is possible for films to either be spinodally unstable or to be metastable (that is, unstable to nucleated dewetting). In such cases, the stability of the film is dependent on its thickness, as shown in Fig. 2, and in sweeping the oxide layer thickness it is possible to observe a wetting transition. In Fig. 2, the effective interface potential for the air/PS/SiO2/Si system shows regions for which: (i) the potential is concave down corresponding to a spinodally unstable film (e.g., h ≈ 3 nm); and (ii) the potential is concave up corresponding to a metastable film (e.g., h > 5 nm). See Fig. 2b, c for examples of dewetting films for which spinodal and nucleated dewetting has been observed. In the case of a metastable film, an external stimulus of sufficient strength, such as a dust particle, is necessary in order to overcome the potential barrier leading to the global potential minimum. For \( {\phi}^{{\prime\prime}}\approx 0 \), thermal fluctuations can induce dewetting.
Fig. 2

(a) Effective interface potentials, ϕ(h), for thin films of polystyrene on variously coated Si wafers. From top to bottom at h = 4 nm, the lines correspond to Eqs. 8, 9, 10 and 8. The “×” on the red curve marks ϕ(h min) as in Eq. 15. The minimum for the grey dashed curve is at ϕ ≈ −10.8 mJ/m2, which accounts for the larger contact angle of PS on OTS as compared to that on SiO2 (cf. Eq. 15). OTS refers to a self-assembled monolayer of octadecyltrichlorosilane. (b, c) 50 × 50 μm2 atomic force microscopy topography images of 2 kg/mol polystyrene films dewetted from d = 2.4 nm thick SiO2 covered Si substrates at 55 °C. Undisturbed film thicknesses were (b) h = 3.5 nm and (c) h = 6.6 nm; in both cases the height scale is 20 nm

In the late stages of the breakup of nanoscopic polymer films, a collection of droplets is left behind. These droplets may equilibrate with a film of thickness h min, see Fig. 3b, which is determined by the interplay between the short-range repulsion (C) and the long-range attraction (A1/2−3/2). In equilibrium, the tension in the film must be equal to that in the droplet with the Young contact angle, θ = θ Y, given by
$$ \upgamma \cos {\theta}_{\mathrm{Y}}={\upgamma}_{\mathrm{SV}}-{\upgamma}_{\mathrm{SL}}. $$
(14)
Following [3], the tension in the flat film is given by \( \upgamma +{\upgamma}_{SL}+\phi \left({h}_{\min}\right) \), while at the edge of the droplet we have a tension given by \( {\upgamma}_{SL}+\upgamma \cos {\theta}_{\mathrm{Y}} \). Setting these two tensions equal and recognizing that \( \phi \left({h}_{\min}\right)<0 \), the connection between the macroscopic contact angle and the microscopic effective interface potential can be made:
$$ -\phi \left({h}_{\min}\right)=\upgamma \left(1- \cos {\theta}_{\mathrm{Y}}\right). $$
(15)
Referring to Fig. 2, it is seen that the effective interface potential for the OTS covered Si substrate has a much deeper minimum than the effective interface potentials for the substrates on which the polymer layer is exposed to SiO2. Indeed, SAM substrates are often used for their low energy surface properties, a fact which is reflected in the contact angles observed, and elucidated by Eq. 15. In practice, the relationship between the effective interface potential minimum and the contact angle can be used to reconstruct the effective interface potentials [15].
Fig. 3

Different wetting states depending on the spreading parameter, S. (a) For S > 0 complete wetting occurs. (b–d) For S < 0, partial or nonwetting states will be observed. The inset in (b) shows the smooth transition, the form of which is dictated by the effective interface potential, between either the substrate or the equilibrium film with thickness h min and the macroscopic spherical cap. The schematic shape shown here corresponds to a potential with a form similar to the one marked with an “×” in Fig. 2a, or, Eq. 8 with a positive Hamaker coefficient

An important point to be made is that the contact angle is indeed a macroscopic quantity. At length scales for which the van der Waals interaction is relevant, the fine structure of the contact line is not a kink with angle θ Y. At these scales, the interfaces are necessarily curved, due to the fact that the film should smoothly transition between its flat equilibrium value, h min, and a spherical cap with contact angle θ Y, see the inset of Fig. 3b. At such scales, it is useful to work with an energy functional that contains both the van der Waals interaction as well as a contribution for the surface tension. With these two contributions, the energy functional takes the form
$$ \mathcal{G}(h)={\displaystyle \iint \upgamma \sqrt{1+{\left(\nabla h\right)}^2}+\phi (h)\kern0.5em dx\kern0.5em dy,} $$
(16)
where the term associated with the square root is the differential area of the fluid–air interface, which in this model is no longer flat. Without going into the details [3, 4, 5, 6], this energy functional can be used to determine the liquid–air interface shape that connects the undisturbed film to the asymptotic spherical cap shape. At large distances from the contact line, the extremum shape is given by a spherical cap with a modified contact angle, θ L. This angle can be related to the Young angle and the effective interface potential through
$$ \cos {\theta}_{\mathrm{L}}= \cos {\theta}_{\mathrm{Y}}-\frac{\tau }{\upgamma R}, $$
(17)
where R is the radius of the solid–liquid interface underneath the dewetted drop and τ is referred to as the line tension, its strength and sign depending directly on the effective interface potential [4, 6]. For polymeric liquids, droplets with radii \( R\lesssim 1\upmu \mathrm{m} \) can show significant deviations away from the Young angle [4].

The form of Eq. 16 for the energy is powerful, and amenable to many alterations accounting for the effects of gravity (usually only relevant for scales larger than the capillary length \( {\kappa}^{-1}=\left(\rho g/\upgamma \right) \), where ρ and g are the fluid density and standard gravity, a case躜that is rare for nanoscopic systems), imposed electric fields (i.e., electrowetting), and roughness (e.g., Cassie-Baxter and Wenzel models), among other examples. Details and examples may be found in de Gennes et al. [3], Blossey [4], Bormashenko [5] and Dietrich et al. [6] and elsewhere.

Nonequilibrium Processes

The process of transitioning between a nonequilibrium to an equilibrium situation entails a dissipation of energy. The amount of available energy is determined by the initial and final states. The dissipation rate is determined by the balance of damping and driving forces. The most important driving forces for nanoscopic systems are usually those already described above, with their origins in the intermolecular interactions. The damping occurs in the bulk with viscous or viscoelastic processes. Additionally, dissipation associated with fluid in contact with the confining interfaces of the liquid [4, 7, 8, 9] occurs. In particular, when considering nanoscopic systems, this solid/liquid dissipation may become important. Dissipation at the boundary entails a flow of the liquid along the boundary, which is known as slip [4, 16, 17].

In modelling dynamic processes, energy dissipation (either in the bulk or at the surfaces) can be balanced with the loss of excess energy, as described for example by Eq. 16. Thus, understanding dynamic processes requires a detailed knowledge of the interaction energies present in the system of interest. For polymeric nanomaterials, these energies are often dominated by the van der Waals intermolecular interactions, which give rise to the wetting properties, described herein.

Related Entries

References and Further Reading

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Authors and Affiliations

  1. 1.Experimental PhysicsSaarland UniversitySaarbrückenGermany