Nonequilibrium configurations of swelling polymer brush layers induced by spreading drops of weakly volatile oil

Polymer brush layers consist of densely spaced polymer chains that are covalently attached at one end to a solid substrate. In dry state and in poor solvents, they form dense collapsed polymer layers on the substrate. Upon exposure to a good solvent, they swell. The degree of swelling is controlled by the balance of the osmotic pressure of the solvent and the elastic stretching of the polymer chains^1,2 and varies under the influence of many external stimuli, such as temperature, pH value, solvent composition, electric fields, and illumination. This responsiveness can result in strong variations of many physical properties, including adhesion and fouling, friction and lubrication, mass transport, and release with a wide variety of possible applications, as described in various review articles including Refs. 3–6. While most applications involve polymer brushes completely immersed in a solvent, recent years have seen an increasing interest in the wetting of polymer brushes and other soft materials, i.e., situations where responsive soft substrates are simultaneously exposed to the liquid solvent and to an ambient gas that is more or less saturated by solvent vapor^6–9 or to a second immiscible liquid.¹⁰ In particular, in dynamic situations where a drop of solvent is initially deposited onto a dry brush layer in a dry ambient atmosphere, this gives rise to a coupling between the spreading dynamics of the liquid, the evolution of the solvent vapor (for volatile liquid), and the swelling of the substrate with all the concurrent changes of its physical properties including the equilibrium contact angle. This specific responsiveness of polymer brush layers has been denoted as adaptive wetting.⁷ Equilibrium properties of adaptive wetting systems, including also polyelectrolyte layers,¹¹ have been studied for quite some time and led to two persistent puzzles: Schroeder’s paradox that adaptive wetting layers exposed to fully saturated solvent vapor are usually less swollen than upon immersion into bulk liquid and the fact that even good solvents often display partial wetting on brush layers, despite the—by definition—strong affinity between polymer and a good solvent.¹² One additional challenge of adaptive wetting systems is that they often display multiple and very long relaxation times. This can make it difficult to judge whether “true equilibrium” is actually established in a given experimental situation. For instance, exposing polymer brushes to solvents of variable composition can lock in metastable molecular configurations that affect the wetting properties for months, as recently reported in the work of Schubotz et al.⁶ using a combination of contact angle measurements and sum frequency generation spectroscopy.

The competition of different time scales becomes particularly evident in dynamic wetting situations when the intrinsic relaxation time scales interfere with the time scale of contact line motion that may be due to an externally imposed rate of change of the drop volume or arise from the intrinsic hydrodynamic spreading or evaporative retraction of the drop. In the work of Butt et al.⁷, recently the very general qualitative consequences of an intrinsically exponential contact angle relaxation process for the phenomenology of dynamic wetting experiments including, for instance, the appearance of contact angle hysteresis if the displacement rate of the contact line across the substrate is comparable to the relaxation time of the substrate (wettability) adaptation was pointed out. To understand these processes for a specific system, it is essential to identify the actual relaxation processes involved in wettability adaptation and contact line motion. The spreading of drops on polymer brushes includes solvent transport by hydrodynamic drop spreading and solvent sorption by the brush layer. In the case of nonvolatile solvents, the latter can only take place by sorption at the solid–liquid interface followed by imbibition of the solvent within the polymer brush layer. This process has been pictured either as a diffusive process of individual molecules¹³ or as a hydrodynamic imbibition process like the imbibition of fluid into porous media.^14,15 The latter gives rise to a liquid front that propagates with $x(t)∝(t)$ according to the classical Washburn law.¹⁶ For volatile liquids, solvent evaporation, diffusion in the vapor phase, and subsequent condensation into the brush layer provide an additional pathway that can affect the coupled dynamics of drop spreading and swelling of the adaptive substrate.^17,18 For inert solid substrates, the effect of evaporation and condensation on drop spreading has been studied extensively, see, e.g., Refs. 19–24. In this case, the competition between the divergence of both evaporation rate and viscous stress near the contact line leads to a complex scenario that results, for instance, in finite receding contact angles even for completely wetting liquids.^25,26 For water-soluble polymers, solvent uptake by condensation from the vapor phase can lead to a substantial reduction of the equilibrium contact angle and thereby facilitate drop spreading, as demonstrated by Lequeux and co-workers.^17,18 Those authors also described the dynamics of the wetting process using an analytical model of the evaporation/condensation and imbibition process. For adaptive polymer brushes, the effect of vapor condensation might be even more important given the strong driving force arising for solvent sorption as initially dry brushes swell. At this stage, however, the role of evaporation and condensation on the dynamic wetting of adaptive substrates remains underexplored and poorly understood. This applies to the experimental perspective as well as to the one of modeling. For the latter, particular challenges arise from the need to incorporate multiple phases (liquid, vapor, dry polymer, swollen polymer) and their various transition and transport pathways. The resulting multi-scale aspects couple processes strongly localized near the three-phase contact line to the macroscopic dynamics of the bulk of the drop and the brush and vapor far away from the contact line. Furthermore, note that intricacies of contact line modeling are not limited to the wetting of polymer substrates but are related to fundamental questions in the physics of wetting.^27–31 Similarly, the modeling of evaporation and condensation is related to fundamental questions of phase change dynamics, in particular, to the distinction of mass transfer across the interface limited by the actual phase change and by the diffusive transport of the vapor within the gas surrounding the drop.^24,30,32,33 For a recent review see the introduction of Ref. 34. Of the wide range of approaches to the modeling of related dynamic phenomena, in particular, molecular dynamics simulations^35–37 and mesoscopic hydrodynamic models^13,38 have been applied to the wetting of polymer brushes.

In the present work, we study the spreading dynamics of drops of an oil, hexadecane (HD), with a low but finite vapor pressure and contact angle on a oleophilic polymer brush layer of poly(lauryl methacrylate) (PLMA)^39,40 and the resulting inhomogeneous swelling dynamics of the adaptive substrate formed by the brush layer. Using video imaging and microscopic interferometry, we quantify the macroscopic spreading dynamics and demonstrate the emergence of a halo of partially swollen brush layer ahead of the moving contact line in the later stages of the spreading process (Fig. 1). This halo can reach extensions of several hundred micrometers on a time scale of several hours and can assume different long-living i.e., quasi-stationary, nonequilibrium configurations depending on the containment of the evaporating solvent vapor. A gradient dynamics model for the evolution of three independent fields is developed and numerically solved. It reproduces the temporal evolution of the halo and provides insight into the relative importance of competing transport mechanisms through the vapor and within the brush layer.

Silicon wafers (100.0 $±$ 0.5 mm diameter and 525 $±$ 25 µm thickness, boron-doped with (100) orientation, 510 $Ω$ cm, Okmetic) were cut into 2 × 2 cm² pieces for characterization and synthesis. Lauryl methacrylate (LMA, 96%, CAS 142-90-5), copper(II) chloride (⁠$CuCl2$⁠, 97%, CAS 7447-39-4), $α$-bromoisobutyryl bromide (BiBB, 98%, CAS 20769-85-1), N,N,N′,N″,N″-pentamethyldiethylenetriamine (PMDETA, 99%, CAS 3030-47-5), triethylamine (TEA, 99%, CAS 121-44-8), (3-aminopropyl) triethoxysilane (APTES, 99%, CAS 919-30-2), ascorbic acid (⁠$>$99%, CAS 50-81-7), sulfuric acid (⁠$H2SO4$⁠, 98%, CAS 7664-93-9), hydrogen peroxide (⁠$H2O2$⁠, 30%, CAS 7722-84-1) were purchased from Sigma-Aldrich, toluene (99.8%, CAS 108-88-3) was purchased from Alfa Aesar, and n-hexadecane (99%, CAS 544-76-3) was purchased from Acros Organics and used as received without purification. Ultrapure water (resistivity 18.2 M$Ω$ cm) was obtained from a Millipore Synergy UV system.

The oxidized Si wafers were functionalized with bottle brushes of poly(lauryl methacrylate) (PLMA), i.e., a polymer with a polymethacrylate backbone functionalized with fully saturated lauryl side chains that provide a oleophilic character. Surface functionalization was conducted in a grafting-from approach employing the surface-initiated activators regenerated by electron transfer atom transfer radical polymerization (SI-ARGET-ATRP). This method requires little (typically ppm) metal catalyst and provides better oxygen tolerance compared to conventional ATRP methods.^41,42 Three pre-functionalization steps (surface hydroxylation, silanization, and initiator coupling) were performed following standard procedures as described in the literature⁴³ before starting the actual polymer brush synthesis. The specific SI-ARGET-ATRP recipe was adapted from Ref. 44 with minor adjustments to the reactant ratios. Ascorbic acid (AA) (40 mg, 227 µmol) and ethanol (3.5 ml) were mixed in a glass vial (10 ml, 2 cm diameter). CuCl₂ (28 mg, 210 µmol) and PMDETA (100 µl, 480 µmol) were mixed in ethanol (10 ml). A volume of 0.5 ml Cu catalyst solution was added to the glass vial containing AA. Monomer (4 ml, 13.65 mmol) was added to the vial, and the mixture was stirred. The initiator-modified substrate was inserted into the reaction solution, and the glass vial was sealed with a screw-top lid. Reaction solutions were not degassed, and glass vials contained $∼$4 cm³ volume of ambient air. After 3 h of reaction time, the substrates were rinsed with toluene, water, and ethanol and dried with a nitrogen stream.

The dry thickness of the polymer brushes $Hdry$ was measured to range between 180 and 220 nm using a Spectroscopic Ellipsometer (SE) with Nanofilm-EP3 SE (ACCURION GmbH, Göttingen, Germany) at angles of incidence of 60°, 65°, and 70° in a spectral range of 400–995 nm. Optical images of the spreading drops were recorded using an upright microscope (Nikon Eclipse, L150) with 10× objective with a working distance of 5 mm under a color camera (Basler a2A5328 - 15ucBAS). The macroscopic spreading behavior was quantified by imaging under white light illumination. The samples are placed in the center of a cylindrically symmetric cell with a diameter of 3.5 cm. In “closed configuration”, this cell is covered with a microscope slide ∼0.4 mm above the substrate. In “open configuration”, the sidewalls of the cell and the microscope objective above the sample define the far-field boundary conditions for the evaporation and provide decent protection from ambient air currents in the laboratory, together with macroscopic shields placed around the microscope. Quantitative information about local swelling profiles was obtained using interferometric imaging under monochromatic illumination with a narrow band green filter (⁠$λ=532(10)nm$ Thorlabs, FL05532-1). More detailed information about the analysis steps are provided in the supplementary material, Fig. S1.

The theoretical description of the system is based on the framework of gradient dynamics as employed in the mesoscopic hydrodynamic modeling of complex wetting.^45–47 In particular, we extend an earlier model by Thiele and Hartmann¹³ for a nonvolatile liquid on a polymer brush. The system is described employing an underlying free energy functional $F[h,ζ,ρvap]$ that depends on three independent fields: the thickness of the oil layer $h(r⃗,t)$⁠, the excess brush thickness due to the local degree of swelling $ζ(r⃗,t)$⁠, and the local vapor density $ρvap(r⃗,t)$ (Fig. 1). Here, $r⃗=(x,y)$ and $t$ are the substrate coordinates and time, respectively. While the model is presented in the general form below, in all the numerical calculations, we only consider radially symmetric geometries. Moreover, we assume that the extension of the experimental chamber in the vertical direction is small as compared to its horizontal dimensions such that the vapor quickly equilibrates in the vertical direction and $ρvap$ can be considered to only depend on $r⃗$ and $t$⁠. A detailed assessment of this approach can be found in Ref. 34.

Variation of the free energy with respect to $h$⁠, $ζ$⁠, and $ρ$ yields the corresponding three chemical potentials. Taking the conservation of the number of molecules of the fluid across all phases into account, the time evolution of each field at any position can be written as the sum of a conserved flux driven by gradients of the corresponding chemical potential and nonconserved fluxes $Ji$ arising from the transfer of hexadecane between the different fields due to evaporation $(i=ev)$ and imbibition $(i=im)$⁠.

Here, $Jim$⁠, $Jev$⁠, and $Jev′$ are nonconserved fluxes that describe the transfer of oil between the three fields: transfer by imbibition from the bulk liquid into the polymer layer, transfer by evaporation/condensation between bulk liquid and vapor phase, and transfer by evaporation/condensation between the partly saturated brush layer and the vapor phase. Note that from now on, we only consider radially symmetric geometries and employ $r$ as radial coordinate. A detailed description of the model, derivations of the relevant equations, and the values of all parameters are provided in the  Appendix.

Oil drops are deposited onto the polymer brush substrate to spread under two different conditions. In the open configuration, the samples are mounted in a sample cell open to the ambient air. In the closed configuration, we close the sample cell within seconds of depositing the drop by placing a microscope cover slip to contain any vapor of evaporated liquid. In both situations, top view video images allow us to extract the drop radius $R$ as a function of time. For both configurations, $R$ initially increases algebraically with time as $tν$ and an exponent of $ν≈1/6$ (Fig. 2). Contact angles $θ$ extracted from droplet height profiles (Fig. 3) using interferometry images recorded with the same conditions show that $θ$ decreases algebraically with an exponent $μ≈−1/2$⁠. As expected, the values of $ν$ and $μ$ are consistent with the elementary geometric relation $r∝θ−3$ for spherical caps of fixed volume for $θ≪1$ as valid at short times. After ∼15 to 20 min, the spreading process saturates, and the macroscopic drop shape approaches a nearly stationary state for both open and closed configurations. For the open configuration, the contact angle keeps decreasing long after the radius has saturated (open blue symbols in the left panel of Fig. 2(a). We attribute this continued decrease to a combination of gradual drop evaporation and a small contact angle hysteresis of ∼0.5^◦.

The numerical values of $ν$ and $μ$ deviate from the classical exponents $νT=1/10$⁠, and $μT=−3/10$ given by Tanner’s law that describes the spreading of nonvolatile Newtonian liquids on solid substrates with a perfect no-slip boundary condition.⁴⁸ Qualitatively, this is not surprising. The interface between the swollen brush and the bulk drop is rather diffuse, and displays dilute, flexible polymer chains that are easily deformed by the strong viscous stresses close to the contact line. Both, the diffuseness and the possibility of local shear thinning or slip, will apparently lead to an effective hydrodynamic boundary condition that alleviates the stress divergence and thereby promotes faster spreading than in Tanner’s law.³⁰ Moreover, local evaporation and condensation also affect fluid transport.¹⁹ 

At first glance, one might also be surprised that the two different forms of vapor containment lead to the same type of macroscopic spreading behavior regarding drop radius and contact angle. This arises from the fact that the brush layer is initially dry in both cases. A significant difference in the spreading behavior can only be expected once the system has time to experience the difference in the boundary conditions for the vapor. At the very least, molecules in the vapor must have had enough time to diffuse to the edge of the experimental cell. For a cell diameter of a few centimeters, this is the case after a characteristic diffusion time $Tdiff=L2/Dvap$⁠, which amounts to about ten seconds for a vapor diffusion coefficient D_vap = 10⁻⁵ m²/s for hexadecane in air.

To illustrate that the swelling state of the brush layer does indeed affect the spreading behavior, we performed spreading tests on brush layers that were pre-equilibrated in saturated HD vapor inside the closed chamber for up to three weeks. This leads to homogeneous pre-swelling of the brush layer by a factor of $≈2$ compared to the dry thickness. The chamber is then quickly opened to deposit an HD drop and immediately closed again. The subsequent spreading of the drop results in a slower algebraic drop spreading with an exponent of $νsat≈1/8$ (Fig. S2 in the supplementary material). Pre-swelling thus clearly affects the macroscopic spreading dynamics in our system, similar to earlier reports for polyelectrolyte layers.¹¹ 

Of primary concern in the present work is, however, not the macroscopic spreading behavior of the drop but the effect of drop spreading on the swelling of the polymer brush layer. Immediately after deposition, the drop quickly spreads across the dry polymer brush layer (see the video in the supplementary material). After only a few tens of seconds, a colorful halo emerges, indicating that a zone of partly swollen polymer brush layer appears ahead of the moving contact line. While the initial development of the halo is independent of the vapor containment, its subsequent behavior at long times is very different: In the open configuration, the halo initially extends its width $W$ but then saturates after 15–20 min, right panel of Fig. 2(a). In contrast, in the closed configuration, $W$ grows indefinitely, right panel of Fig. 2(b). Then, at a very late stage, its outermost edge becomes somewhat “wavy”, rendering its exact width difficult to determine. We attribute this waviness to the presence of very small heterogeneities of the surface energy that give rise to very large excursions of the contact line position as the contact angle approaches zero. [Even in the absence of this waviness, the exact position of the outer edge of the halo is not very well defined. While the edge is usually easily detectable by the eye, the onset of the increase in film thickness is in fact rather gradual, as seen below in Figs. 5 and 6. As an operational definition of the outer edge, we use an intensity variation of 10% compared to the region far from the drop (see Fig. S3 in the supplementary material)]. The difference between the two configurations becomes very clear from magnified images of the contact line region. They are given in Figs. 4(a) and 4(b) and very clearly show how the halo assumes a stationary state in the open configuration while it continues to widen in the closed one.

The same behavior is seen in the brush swelling ratio profiles $α(r̃,t)=h(r̃,t)/Hdry$ (Fig. 5) that we extract from the analysis of the monochromatic interferometry images. Note that, here, $r̃=r−R$ is the radial distance to the contact line. In the open configuration, these profiles converge onto a universal curve for t ≥ 1 h with a maximal swelling ratio of nearly 5 close to the contact line at $r̃=0$⁠. Far away from the contact line, the film remains in its dry state with $α=1$ at all times. In contrast, in the closed configuration, the profile does not converge but continues to evolve even on our maximal experimental time scale of 24 h. While the maximum of swelling ratio close to the contact line remains nearly constant at a value of about 4—only slightly smaller than in the open configuration—the brush layer continues to swell across the entire sample. Even far away from the contact line, after 24 h, the swelling ratio reaches values up to 2. The comparison between the open and the closed configuration thus clearly proves that vapor phase transport is crucial for the spreading-induced swelling of the brush layer, despite the very low vapor pressure of HD.

To explicitly demonstrate the simultaneous contributions of liquid imbibition and vapor phase transport, we perform additional experiments with a substrate purposefully broken into two pieces. Within the chamber, the two parts of the substrate are then placed next to each other, separated by a small gap as indicated by the black dashed lines in Fig. 4(c). A drop is deposited onto the left piece, the cell is closed, and the spreading process is observed. As the drop spreads, as expected, a halo develops close to the contact line. After a few hours in the closed cell, the brush layer also starts to swell on the right piece. Yet, comparing the color variation far away from the contact line on the two separated pieces, it becomes clear that the brush layer on the left-hand piece swells more quickly than the one on the right-hand piece. From this observation, we conclude that the brush swells faster if it is simultaneously fed by both direct liquid imbibition and condensation from the vapor phase. In contrast, the right-hand piece still shows significant but slower swelling as it is only fed via oil condensation from the vapor phase. This experiment thus unambiguously demonstrates that in the present system, both transport mechanisms operate in parallel and that they are both of appreciable importance. It remains intriguing, though, that the brush layer in the open configuration assumes a stationary state with a pronounced gradient in swelling ratio once the macroscopic spreading process has saturated. Such gradients in a stationary state are incompatible with thermodynamic equilibrium and can only exist in the presence of persistent gradients in chemical potential. Despite their longevity, the observed brush profiles must therefore reflect an ongoing nonequilibrium process in the system.

To reach a detailed understanding of the dominant transport processes and of the origin of the nonequilibrium stationary state characterized by steady profiles, we perform numerical calculations of the combined drop spreading and brush swelling process using the gradient dynamics model described in Sec. II D. In all simulations, the drops are placed at $t=0$ on an initially dry sample in a chamber with a dry atmosphere. (For numerical reasons, we actually chose small but finite initial oil saturations of 4% and 10% for the brush layer and the atmosphere, respectively, rather than numerically ill-defined completely dry initial conditions.) The open configuration is implemented by imposing a constant vapor saturation of 10% along the right edge of the simulation box, while for the closed configurations, a no-flux condition is used [see Fig. 1(b)]. Within a fraction of the first second, oil quickly penetrates and completely saturates the brush layer directly underneath the drop [indicated by the saturated orange in the left column of Figs. 6(a) and 6(b)]. At the same time, the oil evaporates from the drop surface and quickly generates an almost saturated vapor phase directly above the drop [blue shading of the gas layer in the left column of Figs. 6(a) and 6(b)]. Diffusion subsequently allows the oil molecules to spread out in the radial direction both in the vapor phase and within the brush layer, as visualized by the softening gradient of the vapor saturation profiles in the top panels as well as of the brush saturation profiles in the bottom panels of Figs. 6(a) and 6(b). The solid lines in the latter panels correspond directly to the thickness profiles of the brush layers. Note that the brush model predicts the existence of a wetting ridge, as shown in the work of Greve et al.³⁸ The wetting ridge is too small to be visible in Fig. 6 due to our choice of parameters, namely, the strength of the brush potential.

A further observation in Fig. 6 is that after 1 h (middle column), the open and the closed configurations show almost identical vapor saturation and brush swelling profiles. Only at a later stage (see right column), the vapor saturation becomes nearly uniform in the closed configuration while an almost linear vapor saturation profile develops in the open configuration. This key difference between the two configurations arises from the different boundary conditions imposed on the vapor concentration profile on the right-hand boundary. The different vapor saturation profiles are accompanied by different brush swelling profiles: In the open configuration, the profile after 24 h is much closer to the one after 1 h than in the closed configuration.

These results are summarized in the right column of Fig. 5, which provides a direct comparison with the experimental profiles in the left column that we have discussed above. The model reproduces all salient features of the experimental observations, namely, the (near) stationary character of the profiles in the open configuration and the gradual evolution along with a continuous swelling far away from the contact line for the closed configuration. Note that the absolute swelling ratios slightly differ between experiment and simulations, likely because the assumption of a fully collapsed brush in the dry limit $ζ→0$ of the model is idealized. Moreover, the decay of the stationary halo profile to a constant height in the open configuration [Fig. 5(a)] is slower in the model than in the experiment. This is a consequence of the implementation of the experimental open-to-ambient-air situation via lateral boundary conditions far away from the drop in our modeling approach.

Achieving the (semi)quantitative agreement shown in these graphs, including the absolute time scales, requires careful adjustment of several parameters in the model. The most important parameter to be fixed turns out to be the ratio between the vapor diffusion coefficient of HD, here assumed as $Dvap10−5m2/s$⁠, and the (also diffusive) oil transport coefficient within the brush layer, $Dbrush$⁠. Good agreement of the profiles is only achieved if the diffusion in the brush is chosen substantially smaller than $Dvap$⁠. The numerical results shown here correspond to $Dbrush=10−10m2/s$⁠. This value is one order of magnitude lower than the self-diffusion coefficient of liquid HD at ambient conditions.⁴⁹ At the same time, it is higher than the reported values for the diffusion coefficient of HD in bulk polymers. (A reference value for HD in bulk polypropylene is $10−13m2/s$⁠.) However, the latter is expected to depend strongly on the actual polymer and also on the density and free volume in the material. For a brush layer, this should be related to the grafting density. Our method might thus provide a new and unique method to estimate solvent transport coefficients within a swelling polymer brush layer. Such information should be of interest whenever one considers the response time of polymer brushes to external stimuli, e.g., in sensing applications. There are, however, a few caveats. First of all, the value provided here should be considered an averaged “effective” diffusion coefficient within the limitations of our model. The model neglects possible variations of $Dbrush$ with the degree of solvent saturation in the brush between the two limiting cases indicated above. Moreover, the absolute value of $Dbrush$ is expected to depend also on the transfer coefficients that relate the fluxes $Jim$⁠, $Jev$⁠, and $Jev′$ to the differences between the chemical potentials of the oil in the adjacent phases. The values assumed for these quantities (see the  Appendix) are subject to a substantial uncertainty that can have an impact on the absolute value of $Dbrush$⁠. To minimize the influence of this uncertainty, here, we assume that both diffusive processes are slower than the actual phase change, i.e., we consider a diffusion-limited case. A more detailed analysis of the absolute values would require a more extensive set of experiments to further constrain the numerical parameters.

Notwithstanding these limitations, several additional conclusions can be extracted from the numerical simulations: The consequences of the faster transport in the vapor phase can be seen in Fig. 7. There, we show the local brush swelling rate (blue lines) and the contribution due to evaporation from the brush layer into the vapor phase (green lines) for the simulations corresponding to the snapshots in Fig. 6. The faster diffusion in the vapor phase leads to a quickly increasing vapor saturation in the vicinity of the contact line, while the underlying brush layer is still dry. In consequence, the brush layer acts as a sink and swells by absorbing oil from the vapor phase. This corresponds to initially negative values of the brush evaporation rate close to the contact line (green) accompanied by the positive total brush swelling rate (blue). At later times (t = 1 h), the situation has reversed: The brush layer is now fairly swollen close to the contact line. The brush layer is efficiently fed with oil by imbibition within the polymer layer. In consequence, the brush saturation exceeds the local vapor saturation and the flux from the brush into vapor becomes positive, indicating net evaporation close to the contact line. Farther away from the contact line, the original situation prevails: The vapor saturation is higher than the brush saturation and brush swelling is dominated by oil condensation from the vapor. At very late stages (t = 24 h), clear differences in the fluxes appear between the open and the closed configuration. As one might expect, for the open configuration the low vapor saturation far away from the contact line leads to continuous evaporation of oil from the brush layer. This explains the existence of the nonequilibrium stationary state related to steady swelling profiles: They result from the balance between continuous evaporation and continuous influx of oil by imbibition within the brush layer. This continuous flux stabilizes the prevailing gradients in brush layer thickness characterizing the stationary state. A simplified version of a similar mechanism was in fact already proposed in the work of Seker et al.¹⁵ to explain imbibition of volatile fluids into a porous medium that is surrounded by a dry atmosphere. For the closed configuration in our experiments, far away from the contact line, net condensation dominates even at very large times as the vapor approaches full saturation more quickly than the brush layer. The fact that after 24 h the brush layer still displays a substantial thickness gradient despite the high saturation is due to the fact that the vapor phase is still not completely saturated at the right-hand side of our simulation box. Given the fact that the equilibrium adsorption isotherm of our system is very steep upon approaching complete saturation, even a minor undersaturation of 5–10% still leads to a substantial reduction of the brush layer thickness.

Finally, it is worthwhile to comment on the observed very long relaxation times and the fact that even in the closed configuration, the system still evolves after 24 h. At first glance, this may seem surprising given the fact that the characteristic time scale for vapor diffusion in the system is 10 s. Because of the combination of the low absolute vapor pressure of HD and the high sorption capacity of the brush layer, the transient states in our system display a substantially larger lifetime. While the diffusion time is indeed of the order of a few seconds, transporting the equivalent of a film of a few hundred nanometers height of liquid HD as required to saturate the brush layer takes much longer: A simple estimate yields an equilibration time for the system of $Teq=ρliqLΔζ/Dvapρvap≈2.4×105s$⁠, which is of the order of days. This is consistent with the observation that after 24 h the brush layer is still far from being homogeneously swollen. From this expression, we see that equilibration should accelerate with increasing vapor pressure, as intuitively plausible. Preliminary experiments with drops of tetradecane and dodecane with vapor pressures at room temperature of 1.55 and 18 Pa, respectively, instead of 0.2 Pa for hexadecane confirm this expectation (data not shown). For water drops with a vapor pressure of 2300 Pa spreading on swellable responsive surface coatings, including polymer brush and polyelectrolyte layers, the influence of vapor phase transport should be even more important.

In summary, we have demonstrated that the spreading of drops of volatile hexadecane on oleophilic polymer brush layers of PLMA is accompanied by the formation of a halo of partly swollen brushes. Swelling kinetics and the extent of the halo are controlled by the balance of two competing transport mechanisms: direct imbibition of oil from the drop through the polymer brush layer and vapor phase transport in combination with evaporation and condensation at the brush–vapor interface. Numerical simulations with a mesoscopic hydrodynamic model based on a gradient dynamics framework reproduce the experimentally observed time-dependent swelling profiles for slowly evaporating drops in both an open atmosphere and in a closed cell. We consider it a success that the complexity of this time-dependent multiphase phenomenon can be reproduced using such a conceptually simple and self-consistent model that allows us to identify the roles of each physical process. Matching the numerical results to the experimental data provides a method to estimate the hitherto unknown diffusion coefficient of the solvent within the polymer brush layer, which for the present system is found to be ∼100 000 times lower than the diffusion coefficient in vapor. The combination of this small diffusion coefficient and the low vapor pressure explains the very long relaxation times of more than 24 h. We anticipate that vapor phase transport should play an important role in many dynamic wetting phenomena on swellable polymer materials and coatings, in particular for aqueous drops with their characteristic high vapor pressure. Our experiments also suggest that the strong gradients in the local swelling of such responsive systems can be achieved by regulating the local vapor saturation in a controlled manner. This may be of interest to sensing applications, e.g., in combination with fluorescence-based detection of the swelling state.¹⁰ 

The supplementary material includes; information about the interferometry analysis and additional figures for the characterization of macroscopic drop spreading on a pre-saturated PLMA brush layer and the halo width determination.

S.d.B., U.T., and S.H. acknowledge support by the Deutsche Forschungsgemeinschaft (DFG) within SPP 2171 by Grant Nos. BE8029/1-2, TH781/12-1, and TH781/12-2.

The authors have no conflicts to disclose.

Özlem Kap: Data curation (equal); Formal analysis (equal); Investigation (lead); Methodology (lead); Software (equal); Visualization (lead); Writing – original draft (equal); Writing – review & editing (equal). Simon Hartmann: Data curation (equal); Formal analysis (lead); Investigation (equal); Methodology (equal); Software (equal); Visualization (lead); Writing – original draft (equal); Writing – review & editing (equal). Harmen Hoek: Data curation (equal); Formal analysis (equal); Investigation (supporting); Methodology (equal); Software (equal); Visualization (supporting); Writing – original draft (supporting). Sissi de Beer: Conceptualization (equal); Methodology (equal); Project administration (equal); Supervision (equal); Writing – review & editing (equal). Igor Siretanu: Conceptualization (equal); Funding acquisition (supporting); Resources (equal); Supervision (equal). Uwe Thiele: Conceptualization (equal); Funding acquisition (lead); Project administration (lead); Resources (equal); Supervision (lead); Validation (equal); Writing – original draft (supporting); Writing – review & editing (equal). Frieder Mugele: Conceptualization (lead); Funding acquisition (lead); Methodology (equal); Project administration (lead); Resources (lead); Supervision (lead); Validation (equal); Writing – original draft (equal); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.

In more complex systems, the dynamics of the drop/film profile couples to other dynamic quantities characterizing the system. Then, the model can be extended to a gradient dynamics of multiple coupled order parameter fields, e.g., effective solvent and solute height profiles for films/drops of mixtures of simple liquids^45,56 or drop profile and surfactant concentration profiles in and on the liquid for films/drops covered by a soluble surfactant.^46,47 The gradient dynamics form makes modeling such complex systems conceptually simple as it automatically takes into account that the various competing dynamical processes are all driven by the same energy. Thermodynamic consistency is ensured by a few simple rules for the mobilities.⁴⁷ A potentially large number of parameters is a natural consequence of the various energetic contribution and transport processes. The gradient dynamics form makes modeling such complex systems conceptually simple as it automatically takes into account that the various competing dynamical processes are all driven by the same energy. Thermodynamic consistency is ensured by a few simple rules for the mobilities.⁴⁷ A potentially large number of parameters is a natural consequence of the various energetic contributions and transport processes.

For a thermodynamically sensible description in the gradient dynamics framework (A2), we first transform all three order parameter fields to particle numbers per area, i.e., the per area number of liquid molecules in the drop $ψ1(x,t)=ρliqh(x,t)$⁠, within the brush $ψ2(x,t)=ρliqζ(x,t)$⁠, and in the ambient air $ψ3(x,t)=ρvap(x,t)[d−h(x,t)−ζ(x,t)]$⁠. Here, we have introduced the vapor particle density $ρvap(x,t)$ and the constant liquid particle density $ρliq$⁠. Conveniently, all variations of the free energy functional with respect to the particle number densities $ψa$ then correspond to effective chemical potentials $μa=δF/δψa$⁠.

Note that the given description of the transport processes includes some unwanted side effects. In particular, it allows for evaporation (condensation) of liquid from (to) the brush in areas that are covered by the drop. This can be fixed by modulating the respective transfer coefficient $Mev′=Mev′(h)$ with a smooth step function such that it is close to zero when the drop profile height $h(x,t)$ is larger than a small threshold value and otherwise constant. As our model incorporates a thin liquid adsorption layer to avoid the contact line singularity,²⁷ we choose the threshold height slightly larger than the equilibrium adsorption layer height. Similarly, we modulate the two transfer coefficients $Mev$⁠, $Mim$ in order to suppress any imbibition or evaporation of liquid from the film when the profile height is smaller than the threshold value. This is necessary mostly for two reasons: First, if the adsorption layer was coupled to the vapor or to the brush, the height of the adsorbed film would increase slightly such that the pressures in film, vapor, and brush balance. While this effect may be very subtle, it can take up a substantial amount of liquid across a large domain, effectively draining the drop as the adsorption layer adapts to changes in the atmosphere or brush state. Second, gradients in the brush or vapor pressures would also evoke a gradient in the film pressure, hence causing a liquid flux through the adsorption layer. In this way, the model would bypass the “slow” diffusive transport processes by rapidly transferring liquid away from the droplet via the adsorption layer, where it then evaporates or absorbs. As an alternative to the modulation of the transfer coefficients described above, both effects could also be suppressed by employing an ultrathin adsorption layer height, which would on the other hand inhibit contact line motion.

Having established a simple dynamical framework, next we discuss the underlying free energy functional $F[ψ1,ψ2,ψ3]$ that determines all the chemical potentials $μi=δF/δψi$ driving the dynamics.

In the following, we perform time simulations of these equations using the finite-element element method implemented in the C++ library oomph-lib.⁶⁵ Moreover, we make use of polar coordinates and perform all simulations for a radially symmetric geometry, effectively reducing the spatially two-dimensional cartesian domain to a one-dimensional radial domain.

The model parameters used to generate the simulation data in Figs. 5 and 6 are given in Table I. The parameters are chosen such that they closely match the experiments.

The scale of the brush energy [Eq. (A11)] is dominated by its prefactor $HdrykBT/ℓK3$ while the term in brackets is roughly of the magnitude $−0.8$ for a saturated brush. For the employed parameters, the brush therefore contributes to the per area free energy with $fbrush≈−1.3J/m2$⁠, i.e., the brush energy is much larger than the interface energies. Hence, we conclude that the intake of liquid into the brush is strongly driven by the brush potential rather than by the capillary energy of the drop.