ΔW, necessary to separate the different phases to an infinite distance in vacuum.n, formed at the interface.n requires a microscopic description of the interface, approximate macroscopic methods are adopted in practice. For example, the Good–Girifalco–Fowkes equation, one of the most known among the combining relations present in the literature, AB (≡ −1/2 ΔW), as a function of dispersive (van der Waals) and polar components; the latter term is further parametrized in terms of the electron-donor and the electron-acceptor contributions. For each phase, three parameters are determined by an experimental procedure in which contact angles of three (or more) reference liquids are measured. These parameters are then used to solve a (overdetermined) system of equations, leading to results too often depending on the chosen triad of liquids.combining methods provide a thickness-independent value of γ AB . The outcome of these approachesΔW—is observed upon reduction of the polymer film thickness below 5–10-fold the macromolecular size, R g , that is ≲30–300 nm (see The performance of multiphase systems, as nanocomposites and hybrid nanomaterials, depends on the work of adhesion,, necessary to separate the different phases to an infinite distance in vacuum. (1) This quantity, whose estimation is a major requirement in several engineering applications, is proportional to the number of contacts,, formed at the interface. (2) Because the determination ofrequires a microscopic description of the interface, approximate macroscopic methods are adopted in practice. For example, the Good–Girifalco–Fowkes equation, one of the most known among the combining relations present in the literature, (3−5) describes the interfacial free energy between two immiscible materials A and B, γ(≡ −1/2), as a function of dispersive (van der Waals) and polar components; the latter term is further parametrized in terms of the electron-donor and the electron-acceptor contributions. For each phase, three parameters are determined by an experimental procedure in which contact angles of three (or more) reference liquids are measured. These parameters are then used to solve a (overdetermined) system of equations, leading to results too often depending on the chosen triad of liquids. (4,6) In the case of thin polymer films (7,8) deposited onto solid substrates (9−13) —systems widely employed in a large number of everyday life applications, e.g., as antifouling coatings, (14) dielectric (15) and conducting media, (12) lubricating layers, (16) etc.— (3−5,17) provide a thickness-independent value of γ. The outcome of these approaches (14,18,19) commonly used at both the academic and industrial level, hence, implies that the interaction strength between two materials placed in intimate contact is not affected by confinement at the nanoscale level. In neat disagreement with this assumption, in this Article, by measuring the equilibrium adsorbed amount of thin polymer films deposited onto a solid substrate, we prove that the number of contacts between polymer and substrate, a quantity directly proportional to the work of adhesion, is actually correlated to sample geometry. A substantial drop in adsorbed amount—and in turn in—is observed upon reduction of the polymer film thickness below 5–10-fold the macromolecular size,, that is ≲30–300 nm (see Figure 1 ), or when the adsorbing coating is capped by another polymer layer (see Figure 2 ). We explain these effects considering how interfacial interactions are tamed by nanoconfinement.

Figure 1 Figure 1. Thickness dependence of the equilibrium adsorbed amount. (A – D) Equilibrium adsorbed amount (Γ∞) normalized to the value measured in bulk samples [Γ bulk ∞ = Γ∞(L > 7R g )] as a function of the spin-coated film thickness (L) normalized to the macromolecular size (R g ), for PtBS74, where 74 is the molecular weight in kg/mol (A, red diamonds), PMMA320 (B, orange triangles), P4MS72 (C, green squares), and PS (D, circles) of molecular weight 49, 99, 325, 488, 560, 1000, and 1460 kg/mol (ordered respectively to the molecular weight from light blue to dark purple, see Table S1). The black dashed lines mark the bulk value of the equilibrium adsorbed amount. (E) Sketch of a substrate/polymer monolayer system of spin-coated thickness L and adsorbed layer thickness h ads . (F) Kinetics of adsorption for PS1000 at 433 K, the thickness of the adsorbed layer is plotted as a function of annealing time for spincoated monolayer samples of varying thickness L (130 nm blue circles, 235 nm green circles, and 325 nm red circles). Data superimpose in the linear growth regime, indicating a constant value of the adsorption rate. After the crossover point, marked by the arrows, the adsorption kinetics transitions from linear to logarithmic growth, and the effect of nanoconfinement becomes relevant: the equilibrium adsorbed amount, Γ∞, decreases with L. The inset shows the same data set on a linear time scale.

Figure 2 Figure 2. Correlation between equilibrium adsorbed amount and effective Hamaker constant. (A–C) Equilibrium adsorbed thickness for single-layer systems (see sketch in Figure 1E) of PtBS74 (A, red diamonds), PMMA350 (B, orange triangles), and P4MS72 (C, green squares) as a function of the effective Hamaker constant (A eff ) acting at the polymer/substrate interface. (D) Equilibrium adsorbed amount normalized to bulk value as a function of the ratio A eff (L)/A eff (L 0 = 7R g ) for thin films of PS of different molecular weight (same symbols as in Figure 1D). We remark that data sets of Γ∞/Γ bulk ∞ vs A eff (L/L 0 ) for different molecular weights do not collapse in a master plot, because A eff (L) is not a linear function of L. Moreover, while for data sets A–C discriminating a constant value of the adsorbed amount at more negative Hamaker constants is hindered by experimental errors, for data in panel D a regime where the adsorbed amount is independent of A eff is clearly visible. (E) Bilayer systems (see sketch in the top right corner of the same panel): normalized adsorbed amount for layers PS1000 (320 nm pink triangles, 140 nm orange triangles) capped by thin layers of PMMA. In these experiments A eff is controlled by changing the PMMA layer thickness. For example, the Hamaker constant of the polymer bilayer system air/PMMA(150 nm)/PS(320 nm)/SiO 2 (2 nm)/Si equals that of the monolayer system air/PS(103 nm)/SiO 2 (2 nm)/Si. The adsorption kinetics for these bilayer systems are presented in Figure S12. Data for (uncapped) single layer of PS of the same molecular weight (purple circles) have been added for comparison.

L (corresponding to the range 2–30 R g ) onto Si substrates, covered by a native oxide layer (≈2 nm). Samples are annealed over time under isothermal conditions above the glass transition temperature to favor adsorption of chains onto the supporting substrate; In our experiments, we spin-coat ultrathin polymer films of different film thickness(corresponding to the range 2–30 (20) ) onto Si substrates, covered by a native oxide layer (≈2 nm). Samples are annealed over time under isothermal conditions above the glass transition temperature to favor adsorption of chains onto the supporting substrate; (21) the nonadsorbed material is washed away in a good solvent, following Guiselin’s experiment. (22) The adsorbed layer thickness, an operational parameter directly proportional to the number of chains adsorbed per unit area, Γ, is measured via ellipsometry and atomic force microscopy (Materials and Methods section in the Supporting Information (SI)). For a given polymer and a given molecular weight, Γ is strictly proportional to the number of directly adsorbed monomers, and thus to the number of contacts between macromolecules and substrate (see eqs S1–S6 ).

∞, the equilibrium adsorbed amount corresponding to the value reached after prolonged annealing, is observed as the film thickness decreases. This quantity is directly related to the depth of the interfacial potential (see the 1 Figure F shows the adsorption kinetics for films of polystyrene (PS) of three different spin-coated thicknesses. At short times, the growth of the adsorbed layer proceeds irrespectively of the film thickness, which implies the lack of nanoconfinement effects on the monomer adsorption rate. (23) At later times, a pronounced reduction in Γ, the equilibrium adsorbed amount corresponding to the value reached after prolonged annealing, is observed as the film thickness decreases. This quantity is directly related to the depth of the interfacial potential (see the SI for additional details on the kinetics of adsorption). Nanoconfinement, therefore, mitigates the strength of interfacial interactions, without affecting the mechanisms of adsorption (see Figure 1 F, and Figures S1, S2, S10, and S12 ). This observation is in line with our previous work, (23,24) showing that adsorption originates from the perpetuated repetition of fluctuations in the density of chains located in the layer in direct contact with the substrate. The monomer adsorption rate is, hence, not affected by the global dynamics of the whole film nor by the peculiar relaxation mode active at the free surface. (25)

∞. We discarded those components of the interfacial potential acting only on short-range, e.g., electronic and polar interactions, since those vanish at distances from the interface 1–2 orders of magnitude smaller than those observed here.A eff (see the supplementary text in the ∞ upon reduction of L could be ascribed to the thickness dependence of A eff . To understand the origin of the trends in Figure 1 , we searched for a material property connected to the polymer/substrate potential and having a similar thickness dependence as Γ. We discarded those components of the interfacial potential acting only on short-range, e.g., electronic and polar interactions, since those vanish at distances from the interface 1–2 orders of magnitude smaller than those observed here. (2) We have noticed that a change in polymer thickness yields a variation in the effective Hamaker constant of the system, (26) (see the supplementary text in the SI ). This parameter, indicating the strength of the long-ranged interactions at the polymer/substrate interface, is directly proportional to the depth of the interfacial potential (27,28) and, consequently, to the equilibrium number of molecules in direct contact with the adsorbing medium after long annealing time. (29) We have, hence, tested if the changes in Γupon reduction ofcould be ascribed to the thickness dependence of

h∞ (and therefore Γ∞, see 2 /Si). In line with our hypothesis, smaller adsorbed amounts are measured in thinner films, which are characterized by less negative values of the Hamaker constant (see the A eff ). In our analysis, we considered unretarded Hamaker constants; we verified that our findings are not affected by retardation effects—arising from the London component of van der Waals interactions—which would require a nonzero time to cover the layer thickness (see Our results confirm a linear correlation between the equilibrium thickness of the absorbed layer(and therefore Γ, see eq S6 ) and the effective Hamaker constant (see Figure 2 A–D) of single-layer systems (i.e., air/polymer/SiO/Si). In line with our hypothesis, smaller adsorbed amounts are measured in thinner films, which are characterized by less negative values of the Hamaker constant (see the SI for details on the determination of). In our analysis, we considered unretarded Hamaker constants; we verified that our findings are not affected by retardation effects—arising from the London component of van der Waals interactions—which would require a nonzero time to cover the layer thickness (see Figures S7–S9 ).

∞(L) could be ascribed to finite size effects or any other unknown or uncontrolled parameter which could arise from sample preparation, we designed a second type of experiment where we modified the effective Hamaker constant without altering the thickness of the adsorbing layer. This condition was achieved by preparing bilayer polymer systems 2 /Si) in which the thickness of PS, L, is kept constant, while the thickness of the capping PMMA layer is varied from 0.1- to 1-fold L. Importantly, a reduction in A eff can be obtained upon either increasing the thickness of the PMMA layer in bilayer systems or upon reduction of the thickness of the PS layers in the monolayer system (see 2 decreases with increasing the thickness of the PMMA layer (see A eff . To discard that the trend in Γ) could be ascribed to finite size effects or any other unknown or uncontrolled parameter which could arise from sample preparation, we designed a second type of experiment where we modified the effective Hamaker constant without altering the thickness of the adsorbing layer. This condition was achieved by preparing bilayer polymer systems (30−32) (i.e., air/PMMA/PS/SiO/Si) in which the thickness of PS,, is kept constant, while the thickness of the capping PMMA layer is varied from 0.1- to 1-fold. Importantly, a reduction incan be obtained upon either increasing the thickness of the PMMA layer in bilayer systems or upon reduction of the thickness of the PS layers in the monolayer system (see Figure S6 ). Results (see Figure 2 E) show that the number of chains of PS adsorbed onto SiOdecreases with increasing the thickness of the PMMA layer (see Figure S5 ), as expected for less negative values of

∞ and A eff in bilayer samples of constant PS film thickness proves that sample preparation and the presence of free surfaces in contact with air are not responsible for the observed trend in the monolayer systems. By changing the thickness of the capping PMMA layer, while keeping constant that of PS, we were able to modify Γ∞ in an identical manner as when reducing the thickness of PS in monolayers (see 2 interface depends on the value of A eff only and not on the thickness of the single spin-coated layers. The correlation between Γandin bilayer samples of constant PS film thickness proves that sample preparation and the presence of free surfaces in contact with air are not responsible for the observed trend in the monolayer systems. By changing the thickness of the capping PMMA layer, while keeping constant that of PS, we were able to modify Γin an identical manner as when reducing the thickness of PS in monolayers (see Figure S12 ). Importantly, in Figure 2 E data of different PMMA/PS bilayers overlap with those of single layers of PS. This observation corroborates that the number of contacts at the PS/SiOinterface depends on the value ofonly and not on the thickness of the single spin-coated layers.

N, covering a broad range of molecular weights (≈50–1500 kg/mol). A linear relation between Γ∞ and A eff was verified for all samples, regardless of chain length (see N–1/2 (∼R g –1), all data collapse in a master plot where the normalized adsorbed amount is presented as a function of L/R g , see Finally, we have extended our investigation to thin films of monodisperse PS samples of different polymerization degree,, covering a broad range of molecular weights (≈50–1500 kg/mol). A linear relation between Γandwas verified for all samples, regardless of chain length (see Figure S11 ). We remark that, while the thickness dependence of the Hamaker constant does not depend on the molecular weight, the number of chains necessary to cover a surface of unit area does. Because the number of contacts a polymer chain makes with a flat substrate scales as(∼), all data collapse in a master plot where the normalized adsorbed amount is presented as a function of, see Figure 1 D.

∞(L) is reached at thicknesses L > L 0 ∼ 7R g . In our reasoning L 0 indicates the thickness at which the number of polymer/substrate contacts reaches a given value, ensuring bulk work of adhesion. Remarkably, the onset of confinement effects on the adsorbed amount (∼7R g ) is in line with the length scale within which properties of the air/polymer surface are affected by the substrate.in silico experiments. Such a plot reveals that a size-independent value of Γ) is reached at thicknesses∼ 7. In our reasoningindicates the thickness at which the number of polymer/substrate contacts reaches a given value, ensuring bulk work of adhesion. Remarkably, the onset of confinement effects on the adsorbed amount (∼7) is in line with the length scale within which properties of the air/polymer surface are affected by the substrate. (33) While it seems unrealistic to rationalize such unexpectedly large molecular length scale via mechanisms involving adsorbed chains having at least one monomer in the polymer/surface layer, (34) our results indicate that confinement effects are directly ascribable to changes in work of adhesion, in agreement with the outcome of recentexperiments. (35)

In view of the results in Figure 2 , we conclude that in thin polymer coatings the equilibrium adsorbed amount deviates from bulk value, following the modification in the strength of the interaction potential induced by confinement in nanoscopic geometries. We remark that the trend observed for the impact of nanoconfinement on the kinetics of adsorption, see Figure 1 F, is shared by other nonequilibrium phenomena, e.g., physical aging, (36) where the kinetics of equilibration (in our case the monomer adsorption rate) are not affected by a change in thermodynamic driving force (the depth of the interfacial potential).