Experimental Study and Modeling of Boundary Lubricant

Article pubs.acs.org/Macromolecules

Experimental Study and Modeling of Boundary Lubricant Polyelectrolyte Films Anne-Sophie Bouchet,†,‡ Colette Cazeneuve,§ Nawel Baghdadli,§ Gustavo S. Luengo,*,§ and Carlos Drummond*,†,‡ †

CNRS, Centre de Recherche Paul Pascal (CRPP), Avenue Albert Schweitzer, F-33600 Pessac, France Université Bordeaux 1, CRPP, F-33600 Pessac, France § L’Oréal Research and Innovation, Aulnay sous Bois, France ‡

ABSTRACT: Strongly adsorbed polyelectrolytes are central in the conditioning properties of many personal care products. Whereas the mechanism of polymer adsorption is rather well understood, less is known about the actual mechanism of polyelectrolyte lubrication. We investigated the adsorption on mica and the lubricant properties of a strong polyelectrolyte, poly(diallyldimethylammonium chloride), polyDADMAC, in aqueous solutions of different salt concentrations. We found that the adsorption of the polymer was enhanced and the morphology of the adsorbed layer modified by increasing salt concentration. The lubricant properties of the adsorbed polyelectrolyte layer were good at low compressions but rapidly deteriorated at larger applied pressure. A complex velocity dependence of the friction was observed, with a maximum value at intermediate velocities and hysteresis in an acceleration/ deceleration cycle. A progressive increase in separation between the rubbing surfaces with velocity was also observed. We developed a model that describes the complex behavior of friction observed, taking into account the hysteretic elastic deformation of the polymer layer under shear and the dilatency due to the elastohydrodynamic effects resulting from its low elastic modulus. Our results can help to explain the significant lasting of the lubricating properties when polyDADMAC is used in conditioners, where polymer-lubricated hair-hair contacts exhibit low shear forces.

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INTRODUCTION Polyelectrolytes (PE) have a large number of applications as surface modifiers. For this reason, many experimental and theoretical studies of PE adsorption have been reported. While most of theoretical studies have addressed the thermodynamic equilibrium properties of PE, this issue may be out of reach in experimental studies because of the long relaxation times involved. These effects are more important at large polymer concentrations, where PE may assume a metastable condition, with history-dependent properties. The type of surface is an important factor in the adsorption process. Apart from obvious parameters such as surface energy or charge density, it is known that other aspects like surface topography can play an important role. As an example, the accumulation of polymers on hair is observed close to the outer scales where protein may be more exposed.1 A more detailed description of the hair’s structural characteristics can be found elsewhere.2 Simple PE systems based on homopolyelectrolytes have been used in cosmetic products since the 1970s, in particular in shampoos and hair conditioners. They are formulated in the presence of other species as cationic and anionic surfactants, lipophilic compounds (i.e., fatty alcohols), and silicones. Good conditioning (lubrication) properties are observed in terms of hair combing and manageability. Whereas much work has been dedicated to the thermodynamics of polymer stability and © XXXX American Chemical Society

precipitation (i.e., formation of coacervates with surfactants), less is known about the mechanism whereby these homopolymers have a lubricating effect on negatively charged surfaces (such as hair) or how the particular structure conditions the behavior under shear, much different as compared to dense end-grafted nonadsorbing polyelectrolytes (polymer brushes).3 Remarkable lubrication properties have been reported for polymer brushes.3 This performance has been attributed to the small interlayer penetration between the opposing polymer layers and to the effect of hydration layers surrounding the charged macromolecules. However, when the pressure applied between the surfaces is increased or the solvency conditions are worsened, a clear deterioration of lubrication is observed.4,5 This results from the higher energy dissipation occurring between the strained interpenetrating polymer chains. For polymer brushes in contact, friction arises because of polymer segments being dragged through the interpenetration zone.6 The scenario is somehow different for adsorbing polymers: intersurface bridging attraction or shear-induced expulsion of polymer chains can also occur which can increase adhesion and friction.7 Received: January 23, 2015 Revised: March 6, 2015

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DOI: 10.1021/acs.macromol.5b00151 Macromolecules XXXX, XXX, XXX−XXX

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the crystal, Δf can be related to the adsorbed mass per unit area (Γ ≡ Δm/A) by the Sauerbrey equation19

Some experimental studies have addressed a number of natural and biomimetic lubricant systems based on charged biopolymers.8−12 On the other hand, there are only few reports on tribology of adsorbed homopolyelectrolytes. Studies with weakly charged adsorbing polyelectrolytes have shown only a limited improvement of lubrication properties at large adsorbing densities, although the resistance to wear is notoriously improved.5,13 Rutland et al. studied adsorbed layers of polycations of different charge density; poor lubrication properties were observed.14 Similar results were reported by Kampf and co-workers in a study of interaction between chitosan-coated mica surfaces. 11 The influence of the preparation path on the behavior of adsorbed PE layers of quaternized poly(4-vinylpyridine) under shear was verified by Ruths and co-workers.15 In this work we explored the adsorption and lubricant properties of a cationic polyelectrolyte from solutions at concentrations larger than those usually investigated in the literature, but commonly used in many applications. We investigated the adsorption and the behavior under compression and shear of adsorbing polyelectrolyte poly(diallyldimethylammonium chloride), polyDADMAC, a fully water-soluble strong cationic polyelectrolyte which has been extensively investigated in solution, owing to its widespread use in the industry of cosmetics.

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Δf =

− 2f 2 Δm Aρq c

(1)

where ρq and c denote the density and speed of sound in the quartz, and A is the surface area of the resonator. If the properties of the bulk liquid are different before and after adsorption, the influence of change in solvent density and/or viscosity must also be considered. If the adlayer is not thin and rigidly attached, more complicated models considering the influence of viscoelastic properties of the adsorbed material must be used. One common practice is to describe the adsorbed film as a homogeneous viscoelastic Voigt-like film with a complex shear modulus G*= G′ + iG″ in contact with a viscous solvent (G′ and G″ are the storage and loss modulus of the film). G′ and the film viscosity (η ≡ G″/ω, ω being the angular frequency 2πf for a given harmonic) may be frequency-dependent. A complete description of this and other models have been reported.17,19 We used 5.0 MHz quartz resonators with silica-coated gold electrodes. The sensors were rinsed twice with Milli-Q water and ethanol, then irradiated with ultraviolet light for 15 min, and rinsed again with ethanol and blow-dried with nitrogen gas prior to use. At the beginning of each experiment, the resonator was placed in the cell and immersed in Milli-Q water adjusted to salt concentration of the polymer solution to be studied for at least 30 min, until a stable baseline was obtained. This enabled the system to thermally equilibrate and the silica surface charge to equilibrate at the measurement conditions. Solutions of the polymer at concentration Cp 0.5% w/w in water were studied. We investigated the effect of background salt concentration Cs (NaNO3) on polymer adsorption. To analyze the measured data, Δf and ΔDis were fitted for the odd harmonics (n = 3 to n = 13) using the QTM software, written by Johannsmann.20 Atomic Force Microscopy (AFM). The structure of the layers adsorbed at the solid/solution interface was examined using a NanoScope IIIa Multimode atomic force microscope (Digital Instruments), in contact mode. Triangular silicon nitride cantilevers (200 μm long) with silicon tip probes were used (Bruker). They were irradiated with ultraviolet light for 15 min prior to use. The polymer solutions were hold in a fluid cell and sealed by a silicone O-ring. Both were rinsed with ethanol and distilled water and then dried using filtered dry nitrogen gas. The solid substrate used was muscovite mica (METAFIX, France) cleaved using adhesive tape immediately before use. Images presented show height data captured in contact mode.21 The repulsion between the polymer layer adsorbed on the substrate and the polymer coated SiO2 AFM tip allows imaging using steric or electrical double layer forces. When the force while scanning (determined by the deflection set point) was small enough, the morphology of the adsorbed layer was determined without the tip physically contacting the sample: the tip−surface force is set below the value necessary to push through the adsorbed layer (soft-contact mode21,22). Working in these conditions can be difficult: at larger deflection values, the tip physically contacts (and passes through) the polymer layer and a different AFM image is obtained, as discussed below. Typical imaging scan rates were varied between 0.5 and 2 Hz, and proportional and integral gains between 0.5 and 5 were used. Variation of scan frequency, scan angle, and gains had no effect on the observed morphology. All images are unmodified except for flattening along scan lines. Curves of force versus tip−substrate separation were also measured. We considered the tip to be in contact with the surface when the voltage vs displacement response varied steeply in an approach− retraction cycle: we used this data to calibrate the response of the photodiode for each force curve, as described in the literature.23 We did not attempt to calculate the actual interaction force from the deflection data because the tip geometry and the precise spring constant of the cantilever were not known.

EXPERIMENTAL SECTION

Materials. PolyDADMAC (Merquat 100, Polyquaternium 6) with nominal molecular weight of 1.5 × 105 g/mol was obtained from Lubrizol (Rouen, France) as a 40% w/w solution in water. The amount of unreacted monomer (diallyldimethylammonium chloride) in solution was determined to be 2.7% by 1H NMR (JEOL ECS 400 MHz). No evidence of the presence of added surfactants was detected. No salt was added during the preparation of the material. 0.5% w/w polymer solutions (30 mM in DADMAC monomers) were prepared by further dilution in water (resistivity of 18 MΩ·cm: Millipore), after adjusting salt concentration by adding sodium nitrate (NaNO3, Aldrich, France). Solutions were gently stirred longer than 24 h before use. The hydrodynamic radius of the polymer can be estimated to be 45 nm at 1 mM NaNO3 and 40 nm at 10 mM NaNO3 from the data by Liu et al.16 Polymer Adsorption. We investigated the adsorption of PolyDADMAC on mica and silica and the interaction forces between coated surfaces before and after rinsing with salt solutions. As mentioned before, in this work we focused on conditions of relatively large polymer concentrations during adsorption, Cp, which have been less studied in the past. We investigated two salt concentrations, Cs, 1 and 10 mM of NaNO3, which will be designed as high and low Cs. For atomic force microscopy and quartz crystal microbalance experiments, we removed unadsorbed polymer by extensive rinsing with an aqueous solution of similar or different Cs after 30 min of adsorption. Longer adsorption times (typically overnight) were used for the surface forces apparatus tests to investigate the properties of the adsorbed polymer before and after rinsing. Methods. Quartz Crystal Microbalance with Dissipation Monitoring (QCMD). Polymer adsorption on silica was measured on a commercial quartz crystal microbalance with dissipation monitoring (QCMD E1, Q-Sense). The principles of the technique have been extensively described in the literature.17 The resonance frequency f of an AT-cut quartz resonator is measured; a change in the effective mass of the resonator due to material adsorption produces a variation of its resonance frequency, Δf. The damping of crystal oscillation is also measured and used to calculate the “dissipation factor”, Dis, which is the inverse of the quality factor of the resonance peak.18 The measured Δf and ΔDis can be related to the properties of the material adsorbed on the quartz crystal by using adequate models.17 If the adsorbed mass is evenly distributed, rigidly attached and low compared to the mass of B


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Figure 1. PolyDADMAC adsorption. Resonance frequency and dissipation measured by QCMD. Polymer solution injection: Cp = 0.5 mg/mL at t = 400 s. Rinsing: (a, b) Cs = 1 mM, t = 1800 s (c, d) Cs = 10 mM, t = 2200 s. The quartz crystal was in contact with water for a time period longer than 30 min before injection to ensure thermal stability. Data for harmonics n = 3 to n = 13 are shown. Cs was the same before and after rinsing in both cases. Surface Forces Apparatus (SFA). A surface forces apparatus (SFA) modified for nanotribological studies was used to measure the interaction between polymer-coated mica surfaces.24 In this technique two back-silvered molecularly smooth mica surfaces are glued to cylindrically curved silica lenses using Epikote 1004 (Shell) and placed in cross cylinder configuration. With this technique it is possible to study well-controlled contacts, owing to the molecular smoothness of mica. Multiple beam interferometry (MBI) was used to measure the separation D between the surfaces with subnanometric resolution and the radii of curvature of the surfaces, R.25 The separation between surfaces was controlled with an accuracy of a fraction of a nanometer by using a piezoelectric nanopositioner (Physik Instruments). The force−distance profile between the surfaces was determined by changing the position of the spring attached to the lower surface and measuring the actual variation in the separation between the surfaces. Any difference between these two quantities indicates the deflection of the spring as a consequence of interaction force between the surfaces. If a normal force, L, is applied to the surfaces in contact, the cylinders are elastically flattened at the point of contact and a circular region of contact is formed. The load-induced contact radius and the area of contact, A, can be directly measured by MBI. For friction experiments, we induced a lateral relative motion between the surfaces by voltage-driven bimorph strips attached to the lower surface. Shearing cycles were carried out by moving this surface at constant velocity, V, over a certain distance, then the driving direction was reversed.26 The upper surface was attached to a vertical double cantilever spring whose deflection was monitored using strain gauges (SurForce Inc.).The friction force, Ff, between the surfaces can be calculated from the experimentally measured spring force, Fspring, with an accuracy of ±5 μN.27 The shear stress σ can then be calculated as σ = Ff/A.

polymer adsorption are presented in Figure 1. As can be observed, the adsorption of polymer changes with Cs. Larger salt concentrations resulted in larger frequency and dissipation shifts upon polymer injection. The small changes in Δf upon rinsing suggest that little polymer desorption takes place in the time scale of the experiment. To describe the data, the adsorbed polymer may be modeled as a homogeneous viscoelastic layer with constant G*. In addition, the viscosity of the bulk fluid must be included. It is difficult to have an independent measure of this quantity at the high shear rates explored in the QCMD experiments; this is particularly worrisome for the case of polymer solutions, which often show frequency-dependent viscosity (non-Newtonian behavior). The scenario is somehow simpler after rinsing, when the irreversibly adsorbed polymer is in contact with salt solutions. For this reason, we did not attempt a quantitative fit of the results before rinsing. The results of fit after rinsing are presented in Table 1. Values Table 1. Thickness T and Viscoelastic Properties of the Adsorbed Polymer Layers

a

Cs (mM)

Ta (nm)

G′ (kPa)

G″ (kPa)

1 10

17 4

13 26

88 106

Assuming the density of the adsorbed layer is 1 mg/mL.

extracted from viscoelastic modeling of the data indicate that a thicker polymer film was adsorbed at low Cs values. Nevertheless, it is important to emphasize that QCMD measures the hydrated thickness of the film, which includes an important fraction of water. In contact with water, the adsorbed polyelectrolyte is more or less swollen, depending on

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RESULTS Adsorption. Quartz Crystal Microbalance. Typical data of resonance frequency and dissipation shift measured during C


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Macromolecules the experimental conditions and in particular of Cs. In good solvent conditions for the polymer, the hydrated thickness is substantially larger than the dry (collapsed) thickness of the polymer layer. Both the adsorbed polymer and associated water perturb the oscillation of the quartz crystal and are included in the mass measured by QCMD. We will discuss further this point in relation to AFM and SFA results. Atomic Force Microscope: Morphology of the Adsorbed Layer and Normal Interaction Forces. The morphology of the adsorbed polymer layers in contact with water was determined by AFM in contact mode. Typical results are presented in Figure 2. Several points deserve to be highlighted. In general,

swollen in contact with water, as corroborated by the rough aspect observed. It is informative to consider the long-range tip−substrate interaction force, shown in Figure 3. Each curve corresponds to

Figure 3. Tip−surface interaction force for layers of PolyDADMAC adsorbed on mica, after adsorption and rinsing. Cp 0.5%. Cs during adsorption 1 mM (circles) and 10 mM (squares). Cs of rinsing 1 mM (open symbols) and 10 mM (full symbols). Lines are exponentialdecay fits of the data at separations larger than 5 nm.

an average of at least 15 approach−retraction cycles. In general, a long-range tip−surface repulsive interaction is observed, but its extension varies with Cs during adsorption and after rinsing. The lines included in the figure correspond to fits to a pure exponential decay of the force measured at separations larger than 5 nm. For the data obtained after rinsing at Cs 1 mM NaNO3 (open symbols), the fitted decay lengths closely match the Debye length expected at 1 mM (9.6 nm), which suggests that the repulsive force is dominated by tip−surface electrostatic interaction. This is true regardless of Cs during adsorption. On the contrary, after immersing the adsorbed layers in higher Cs solutions (10 mM NaNO3, full symbols), the extension of the interaction depends upon salt concentration during adsorption. In the case of low adsorption Cs, the fitted decay length matches the Debye length for Cs of 10 mM (3.0 nm). On the contrary, a much longer decay length was observed when the polymer layer was adsorbed from a solution of larger Cs, as can be observed in Figure 3. In the latter case, the force is not just the result of electrostatic interaction. Steric forces due to the adsorbed polymer must be considered. Thus, it is clear that increasing Cs favors the adsorption of the polyelectrolyte, and a thicker coating is obtained. Enhanced adsorption due to the reduced repulsion between PE molecules (screening-enhanced adsorption) has been often reported.29 This has important implications for lubricant properties of the coating layer, as discussed in the following section. It is also apparent that the relatively large polymer concentration studied here favors a nonflat conformation of PE molecules on the mica surfaces; a similar observation has been reported several times in the past.7,28 In summary, the longer range of the electrosteric forces measured after the adsorption at large Cs (Figure 3) indicates that increasing salt concentration favors the adsorption of the polymer by reducing intermolecular electrostatic repulsion, which is in agreement with the SFA results described below; in addition, the larger size of the “asperities” observed on the height profiles after rinsing at small Cs (Figure 2) indicates

Figure 2. PolyDADMAC adsorption. AFM height micrographs in contact mode and typical height profiles. Cs during adsorption 10 mM and rinsing (a) Cs 1 mM and (b) Cs 10 mM. Dashed lines indicate the position where the height profiles were measured. The scale bars correspond to 200 nm.

we observed a complete coverage of the mica surfaces by the polymer, which is expected for the large polymer concentrations investigated. Nevertheless, the observed morphology was modified by scanning conditions. When a very small cantilever deformation (small tip−layer interaction force) was imposed during imaging, the roughness of the layer was significantly larger, and its morphology was more heterogeneous; features 1−2 nm taller than the background can be observed uniformly distributed on the adlayer. A similar observation has been reported by other groups in the past.28,7 Nevertheless, from our experimental results we cannot elucidate if this asperities correspond to brushlike zones or are entangled networks. This operating condition (soft contact) was rather unstable, in particular when the adsorption was carried out at a low Cs. A small increase in the applied force (intentional or due to thermal drifts) resulted in smoother images. The observed morphology appears smoother when the rastering tip penetrates into the adsorbed polymer layer (the upper section in Figure 2a and the lower part in Figure 2b). Before the penetration, a rougher profile is observed after rinsing at low Cs, even though the amount of polymer adsorbed is unchanged (cf. Figures 2a and 2b): the apparent height of the adsorbed layer depends upon Cs, in agreement with QCMD results. The rougher aspect of the polymer layer at low Cs is due to a larger degree of hydration of the polymer. Thus, the questions of smoothness and extension of the adsorbed layer must then be handled with care: the hydrated polymers are D


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Figure 4. Normal force profiles between two polymer-coated mica surfaces measured by SFA. The conditions of adsorption and rinsing were (a) Cs = 1 mM and (b, c) 10 mM. (a) and (b) show data for approach (full symbols) and separation (open symbols) of the surfaces. Data before (circles) and after (squares) rinsing are shown. (c) Effect of shear: the long-range repulsive force is progressively decreased after the surfaces are sheared under pressure. The different symbols correspond to different approach profiles measured at different times of shear: before shear (full circles), after 1 h of shear (open circles), and after completing the sliding diagram at different loads (squares).

Figure 5. Typical friction traces measured between PE coated surfaces: (a−c) Cs = 1 mM; (d−f) Cs = 10 mM. (a) V = 18 nm/s, L = 1.58 mN. (b) V = 1197 nm/s, L = 1.58 mN. (c) V = 7184 nm/s, L = 2.64 mN. (d) V = 1197 nm/s, L = 1.05 mN. (e) V = 2395 nm/s, L = 1.05 mN. (f) V = 7184 nm/s, L = 1.58 mN.

if the measured force is fitted using a simple exponential decay, the decay length obtained exceeds what could be anticipated for the ionic strength of the solution. These results indicate that the repulsive force originates from the steric interaction due to the forced overlap of the polymer layer of the opposite surfaces. After rinsing, the threshold distance of the observed repulsive interaction is about twice the gyration radius of the polymer studied. By removing unabsorbed polymer from the device, the ionic strength was reduced from about 15 mM (considering only the counterions of the polyelectrolyte) to 1 mM. The thickness reduction of adsorbed polyelectrolyte layer with increasing ionic strength has been theoretically predicted30 and experimentally observed previously.31 A scaling law H ∼ N/ (sns)1/3 has been proposed for the brush height H, where N denotes the number of monomer units in the polymer chains, s the area per chain, and ns the concentration of salt. This scaling correctly describes the increment in uncompressed polymer thickness observed: reducing ionic strength by a factor of 15 generates an increment by a factor 2.5 in the height of polymer layer (defined as half the separation at which the interaction forces are first detectable).

enhanced swelling of the adsorbed polymer at lower salt concentrations, in agreement with the QCMD results. Surface Forces Apparatus: Normal Forces. Interaction forces between identically polymer-coated mica surfaces were measured using a SFA. Typical results for F/R measured between surfaces coated at different Cs are presented in Figure 4. Because of the larger size of interacting surfaces, a greater accuracy in F/R can be obtained as compared to AFM curves. In addition, MBI allows the actual surface separation to be precisely measured, which is not possible with conventional AFM force measurements. The results for Cs 1 mM are presented in Figure 4a. A longrange repulsive interaction with little hysteresis in an approach−retraction cycle is observed. The “hard wall thickness” (surface separation measured when F/R = 100 mN/m) was 2.7 nm. It was unchanged after removing unadsorbed polymer from the device and the adsorbed layer being exposed to a 1 mM solution of NaNO3. On the contrary, a substantial increase in the repulsive force range was observed after rinsing, as seen in the figure. In all cases, the shape of the force profile deviates from an exponential decay. Nevertheless, E


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Figure 6. Velocity dependence of the shear stress between two PE-coated surfaces adsorbed from a solution Cs = 1 mM, before (a) and after (b) rinsing with a solution Cs = 1 mM. (a) L = 0.51 mN (circles), 1.58 mN (squares), and 2.64 mN (triangles). The data were measured increasing (full symbols) and decreasing (open symbols) V. (b) L = 0.52 mN (crossed squares), 1.04 mN (triangles), 1.58 mN (squares), and 2.64 mN (circles). The difference between the acceleration and deceleration branches is evident.

Figure 7. (a) Velocity and (b) load dependence of the friction force between two PE-coated surfaces, adsorbed from a solution Cs = 10 mM after rinsing with a salt solution of the same concentration. (a) L = 1.58 mN. The two data sets were measured under identical conditions but separated by a few hours of shear (first circles and then squares) to illustrate the effect of progressive polymer removal. The surfaces were separated and at rest for 30 min between the two sets. (b) V = 598 nm/s. (c) Comparison of normal (full circles) and friction force (open squares) for the same conditions of adsorption and rinsing.

on V in a nontrivial manner, the dynamics of the system can be more complicated. As can be observed in the figure, the characteristics of the friction traces vary with experimental conditions. At low velocities, Ff reaches the steady-state value smoothly: the movement of the upper surface is progressively decelerated and a plateau in the friction signal can be clearly identified. On the contrary, a more complicated scenario occurs at larger speeds: a force spike is observed in the friction trace before the stationary state is achieved. The velocity reversal spike was less noticeable (or hardly visible) in the case of adsorption at larger Cs. However, it became more evident after extensive shear, which was accompanied by polymer removal, as described before. Commonly, the appearance of a spike in the friction signal is identified with the static friction force, and it is called stiction spike. However, as observed in the low V friction traces, Ff continuously decreases at low speeds. This apparent contradiction can be understood if the entire sliding diagram (Ff or σ vs V) is investigated. Typical sliding diagrams are presented in Figures 6 and 7a. To build these diagrams, V was progressively increased until reaching its maximum value and then decreased; Ff is the stationary force value measured at each velocity after few back and forth cycles. A few aspects may be highlighted: • In general, larger Ff values were observed in the accelerating branch of the diagrams, and smaller values of Ff were measured when V was being reduced. In addition, a progressive increase in surface separation D (dilatency, ΔD) with increasing V was observed (Figure 8). This change was almost entirely irreversible in the time scale of the experiment: only a small reduction of D was observed in the deceleration branch of the sliding diagram, and the initial value of D was typically not recovered upon decreasing V or after stopping the

As can be observed in Figure 4b, a long-range repulsive interaction between the coated surfaces is also observed in the case of adsorption at Cs =10 mM. In addition, a thicker “hard wall” of ∼6 nm is observed before and after rinsing. In this case, noticeable hysteresis is observed in an approach−retraction cycle, particularly before rinsing and/or shearing (circles). This hysteresis was more important for larger speeds of displacement of the surfaces during the cycle. The increased hard wall thickness and the hysteresis in approach−retraction cycles strongly suggest enhanced polymer adsorption at larger Cs due to reduction of interchain electrostatic repulsion, in agreement with AFM results. On the contrary, after rinsing the extension of electrostatic forces is more important for low Cs, in agreement with AFM and QCMD results. As observed in the past for similar systems, the extension of interaction force changed with the history of the samples.7 The repulsion appeared at shorter surface separations after successive approach−retraction cycles or upon shear of the coating layers, as illustrated in Figure 4c. This was accompanied by a progressive reduction of the hard wall thickness. However, no significant adhesion between the coated surfaces was ever observed, indicating that bridging, if present, is outweigh by the osmotic repulsion due to the adsorbed polymer layer. Surface Forces Apparatus: Shear. An extensive study of Ff as a function of V and L was performed for the different adsorption Cs described in the previous sections. Typical friction traces, measured under different operation conditions, are presented in Figure 5. The alternate nature of the data is due to the applied reciprocating motion. When there is a finite Ff between the surfaces in contact, the displacement of the lower surface drags the upper surface, deforming the vertical spring until the elastic spring force matches Ff at the imposed V. At this point, sliding is initiated. Nevertheless, as Ff may depend F


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described by many authors.33 However, it is expected that Ff will increase again at larger V (beyond the range of V explored in this work) as a consequence of the viscous drag of the fluid around the surfaces, stabilizing the motion.

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DISCUSSION It has often been reported that polyelectrolytes act as good lubricants. As illustrated by our results, this is true as long as little interpenetration between the opposite layers is forced by the applied load; lubricant properties worsen rapidly as confinement increases. To understand the friction phenomena, it is necessary to describe how energy is being dissipated. Polymer bridging has often been advanced as a possible reason for friction between surfaces coated by an adsorbing polymer.7 However, the absence of adhesion and the reduction of Ff at low V argue against the importance of this mechanism in the system investigated in this work. It is probably important in the case of thin PE layers adsorbed at low Cp, but it appears to be less significant in the case studied here. A second pathway for energy dissipation which must be considered is the irreversible loss of elastic energy due to hysteretic cycles of deformation in the polymer layer. It has been shown that this mechanism can determine the boundary lubrication properties of structured polymer coatings.34 As evidenced by AFM micrographs, the adsorbed polymer films present nanometric asperities which are confined at the large compressions studied in this work. Similar results were reported by Block and Helm in a study on an adsorbing polyelectrolyte.28 Thus, it seems reasonable to consider the polymer lubricant layer as having protrusions that are confined when the surfaces are brought in contact under load. As the system is continuously evolving, we do not conceive these protrusions as static surface features; nevertheless, a number of dynamically evolving asperities will be present on each lubricant layer. We believe these asperities are deformed under shear (if the strain rate is larger than the characteristic relaxation of the confined asperity), snapping off after a certain maximum deformation is exceeded and releasing the stored elastic energy. The continuously increasing value of Ff vs V at low V can be reasonably explained by the competition between thermal and mechanical mechanisms of snapping off. However, this mechanism fails to completely describe the different regimes observed in the sliding diagrams; in particular, it does not predict the friction decreasing regimes at high V. Quantitatively, the process of energy dissipation can be described by modifying a model developed to explain the complex sliding curves observed with weakly adhering boundary lubricated surfaces.35,36 In that model it was assumed that the contact area could be depicted as composed of N independent junctions, each of average area δA. Individual junctions undergo successive process of formation and rupture at rest or under shear. Two processes are responsible for junction breaking. The junctions either detach spontaneously (a process characterized in the absence of strain by a characteristic time τ0) or when they are deformed above a given threshold, l*. In that model, it was assumed that junction deformation reduced the energy barrier controlling the rupture process, as proposed by Schallamach to describe the friction of rubber. The characteristic lifetime of a stressed junction thus becomes τ0 exp(−γf Elas/kBT), where γ is a constant with the dimension of length and f Elas= δAGVt/d is the force applied to the junction (G is the shear modulus, d the thickness of the junction, and t the time after bonding of the junction). Another characteristic time must be considered: the mean time to

Figure 8. Variation of surface separation D during a friction experiment. L = 1.04 mN. V was first increased and then decreased in discrete steps during the experiment from 11 nm/s to a maximum value of 1796 nm/s and down to 3 nm/s. The arrow signals the time when the decreasing V was set back to the initial value in the decelerating branch.

motion. Interestingly, the observed dilatency ΔD was rather independent of applied L. • The measured shear stress σ increased with applied L, which implies that Ff is not proportional to the measured area of contact A. Increasing L produces a progressive reduction in D, increasing the confinement and the extension of the intimate contact between the opposite polymer layers. The typical dependence of Ff on L is presented in Figure 7b. Ff increases almost linearly with the applied load, except for very small loads when it falls below the detection limit of our setup. It is only in this regime of low compression (L < 0.3 mN) that exceptional lubrication properties are observed, probably due to the absence of interlayer interpenetration. It is apparent that no adhesive contribution occurs: Ff vanishes at small loads. It is also evident that lubricant properties of the adsorbed PE layer quickly worsen as L increases. The region of exceptional lubrication is limited to L < 0.3 mN. As can be observed in Figure 7c, significant normal forces are observed at much larger separations than friction forces. However, under substantial compression, Ff rapidly increases. Similar results have been previously reported.32 • A nonmonotonic σ vs V (or Ff vs V) dependence is observed. σ is small at low velocities, and it increases with increasing V until a broad maximum is reached at a speed Vm. Further increase in V gives rise to a gradual reduction in Ff. Similar trends for the sliding curve were reported in a previous study of PE friction by Ruths et al.15 (in their study only the Vincreasing and the plateau regimes were observed). It is then clear why a friction spike is observed at large V: to accelerate the surfaces to a given V (from rest or upon velocity reversal), it is necessary to progressively increase their relative speed. If V is larger than Vm, the force opposing the movement will momentarily exceed steady-state value of Ff, and a spike appears in the friction trace. This effect was less marked at high Cs during adsorption due to the thicker and more viscous polymer layer adsorbed in that case. However, the stiction spike observed gets increasingly important as the polymer layer gets thinner since the difference between the maxima of the stress at intermediate velocities and the stress at larger V becomes more important (cf. Figure 7a). The negative slope on the sliding diagram may compromise the stability of the movement of the surfaces, as previously G


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Figure 9. Velocity dependence of shear stress between two PE-coated surfaces, adsorbed from a solution Cs = 1 mM. (a) L = 1.04 mN; (b) L = 2.64 mN. The lines are the best fit obtained with the nonadhesive friction model, as described in the text. The fitting parameters are G = 17 MPa, d* = 1.5 nm, τ0 = 0.35 s, and l* = 0.8 nm in (a) and 1.2 nm in (b). (c) Shear stress normalized by its maximum value for L = 0.52 mN (full circles), 1.58 mN (open circles), and 2.64 mN (squares). Vm increases with applied L.

reactivate a junction, τ, which is assumed to be independent of the shear rate. In this adhesive model, the dissipation of energy comes from the elastic energy stored during the deformation of the junctions in the adhesive state and then irreversibly lost after the rupture process. As extensively described in previous works, this model gives the following expressions (2) for the elastic component of the friction (FElas) and the mean lifetime of a junction, ⟨t⟩b FElas =

tb kT N B ⟨t ⟩b + τ γα

∫1

eα

not true in the nonadhesive case studied here. In the absence of wear, the characteristic distance between two asperities, d* determined by the adsorption conditions of the polyelectrolyte, Cp and Csis constant. Thus, the probability of a collision between asperities on the opposite surfaces will be proportional to V. The junction reactivation time, τ, then is proportional to d*/V, vanishing at large V. In the absence of motion τ will diverge, given that there is no adhesion between the asperities. This modification in the model has some consequences on FElas. The most relevant is that the regime FElas ∼ V−1 at high velocities disappears since the processes of mechanical rupture and asperity collision (junction reformation) are both accelerated at the same pace. Thus, ϕ and FElas do not vanish at large V as was the case for adhesive junctions which explains the plateau observed at large V. On the contrary, unlike what is observed for the adhesive case, ϕ vanishes at low V, as the collision frequency vanishes (τ → ∞) in the limit V → 0. The model presented above correctly describes our results at low and intermediate V (as well as those reported by Ruths et al.15 in an earlier study on friction of adsorbed PE). As there are a large number of fitting variables (γδA, d*, l*, τ0, and G), it is possible to find different sets of parameters that correctly describe the data. However, the number of possible solutions can be reduced if few sliding curves are adjusted simultaneously. An example is presented in Figure 9. The same values of typical asperity distance d*, relaxation time τ0, and shear modulus G were used for two different conditions of L. The two data sets are reasonably described by varying the critical asperity deformation before forced rupture, l*, which increases at larger L. Interestingly, the fitted value for G is about an order of magnitude larger than the (macroscopic) value obtained from the sliding curve or what can be estimated from the normal force profile. This is reasonable since the modulus of the asperity must be larger than the modulus of the film, which includes an important proportion of water. More severe tests of the model described will require a better independent mechanical and morphological characterization of the adsorbed polymer layer. It is also noteworthy that the characteristic velocity at which the measured stress reaches the V-independent regime seems to be independent of the applied load (Figure 9). If polymer bridging were at the origin of energy dissipation, this velocity would be related to the characteristic time of bridge reactivation, which is likely to be load dependent.35 On the contrary, in the model proposed here, the reactivation time τ only depends on d* and V and is independent of L, in agreement with the experimental results. Studies at different polymer coverage will help to better characterize the influence

⎡ t ⎤ dη ln(η) exp⎢− b (η − 1)⎥ η ατ ⎣ ⎦ 0

⎞ ⎛ t t2 ⟨t ⟩b = tb exp⎜− b (e α − 1)⎟ + 2b ⎠ α τ0 ⎝ ατ0

∫1

eα

⎡ t ⎤ dη ln(η) exp⎢− b (η − 1)⎥ ⎣ ατ0 ⎦

(2)

with α = γδAGl*/dkBT. The fraction of junctions in the bounded state, ϕ, is a function of V and is given by eq 3: ϕ(V ) =

⟨t ⟩b ⟨t ⟩b + τ

(3)

The model predicts four friction regimes. At very low sliding velocity, V ≪ l*/τ0, the junctions are always broken by thermal fluctuations and the friction is proportional to the sliding velocity (FElas ∼ AGVτ0/d). At higher driving velocity, when V becomes comparable to l*/τ0, there is a competition between the thermal and mechanical process of junction rupture, giving in a first approximation a logarithmic velocity dependence for the friction.35,37,38 It will be given by FElas ∼ A(kBT/γδA) ln(V/ V0). When l*/τ ≫V ≫ l*/τ0, the junctions are mainly elastically unbounded. Friction is almost velocity-independent35,38,39 and given by FElas ≅ 1/2AGl*/d. Finally, at even higher velocity, as V becomes comparable to or larger than l*/τ, the number of junctions in the bonded state decreases since each junction now spends relatively more and more time in the unbounded state and the friction decreases as the inverse of sliding velocity,35,38,39 FElas ∼ V−1. This model can be applied to the problem discussed in this work if the process of elastic deformation and snapping-off of colliding asperities on the opposite surfaces is identified with the bonding and breaking of an adhesive junction. Thus, we face the same competition between thermal and mechanical processes of rupture as described above. However, an important difference between the two cases must be introduced. In the adhesive case, it was considered that the characteristic time of junction reformation, τ, was independent of V. This is obviously H


Article

Macromolecules

separated and brought back into contact show that the lubricant layers have not been permanently modified. The observed dilation due to the softness of the boundary lubricants has a significant influence in the behavior under shear of soft surfaces. This effect must be considered in the field of biolubrication, which typically involves the shear of soft substrates (cartilages, mucous membranes, hair, etc.). Our results can explain the significant lasting of the lubricating properties when this kind of polymer is used in conditioners, where symmetric polymer containing surfaces contacts exhibit low shear forces.

of bridging on friction between surfaces coated by adsorbing polyelectrolyte. The described model fails to predict the reduction of Ff at larger V values. One possible hypothesis is that at large shear rates deformed asperities have not enough time to completely recover after snap-off before colliding with a new asperity in the opposite surface; the rate of dissipation of elastic energy (and Ff) is then reduced at large V. The appearance of a negative dFf /dV can then be identified with a characteristic relaxation time of the confined polymer. However, two facts argue against this explanation. First, the shear rate γ̇c, of the order of few hundred s−1 when negative dFf /dV is first observed, appears to be too low compared with reasonable relaxation times of the confined polymer. Second, γ̇c increases with increasing applied load, as can be seen in Figure 9c. This seems counterintuitive: it is reasonable to expect that molecular relaxations would be slowed down by increasing the applied pressure. We have already discussed how the viscous forces due to the shear of the confined liquid layer stabilize the dynamics of the system at large speeds. However, the shear stress is not the only nonzero component of the stress tensor. The normal component to the force of hydrodynamic origin (lift force, FL) also acquires more importance at higher speeds. Given the symmetry of our contact, no lift force is to be expected if the surfaces and polymer coating are not deformable. However, the polymer layer is relatively soft. The symmetry breaking due to its deformation under stress leads to a normal component of the stress tensor, as extensively discussed by Skotheim and Mahadevan.40 This explains the dilatency ΔD observed at large speeds; the reduced confinement is responsible for the Ff reduction at large V. The observed ΔD depends on the mechanical properties of the adsorbed layer and the stiffness of the measuring device. We can use the results of ref 33 to estimate ΔD. The lift force can be estimated as FL = [μ2V2/(2G + λ)](HlR2/h03), where μ denotes the viscosity of the fluid, G the shear modulus of the film, λ the first Lamé coefficient, Hl the thickness of the soft adsorbed layer, and h0 the thickness of the liquid film separating the surfaces. The factor 2G + λ in the denominator can be expressed as 2G + λ = E[(1 − ν)/((1 + υ)(1 − 2υ))], where E denotes the Young modulus of the adsorbed layer (E ∼ 3 MPa, calculated from the measured normal forces under large compression) and ν the Poisson’s ratio. Using that expression for FL (and the spring constant of our experimental setup) and assuming ν ≈ 0.25 (as predicted for an array of independent Flory coils in good solvent41), we obtain ΔD = 0.5 nm at V = 10−5 μm/s, if a film of water of h0 = 0.5 nm is considered (the measured dilatency; cf. Figure 8); as the calculated ΔD matches the value assumed for h0, the outcome appears self-consistent. This result must be taken just as an indication: it is not evident that the expression based on continuum mechanics and hydrodynamic considerations will be valid at the molecular level investigated in this work. However, this result strongly supports the idea of a hydrodynamic origin of the observed dilatency. At larger applied L (smaller initial D), a more important force is required to achieve similar D values after lift (ΔD is almost L independent). This explains why γ̇c increases with L. It appears that modifications of the confined polymer layer occur after dilation. The observed hysteresis in Ff in an acceleration/deceleration cycle is due to the failure of D to recover its initial value at lower V. Even though this change seems irreversible while the polymer layers are kept under confinement (cf. Figure 8), the fact that reproducible results are obtained for normal and frictional forces after the surfaces are

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CONCLUSIONS We have studied the friction between surfaces coated with selfassembled polyelectrolyte layers. We found that the thickness and the roughness of the adsorbed polymer layer are modified by the ionic strength of the solution. In the conditions of large polymer concentration investigated, no evidence of polymer bridging was observed. We are able to describe the boundary lubricant properties of the adsorbed polyelectrolyte by using a simple model that considers the energy loss due to hysteretic cycles of deformation in the adsorbed layer. Because of its generality, this model should be of use to describe friction in many different systems, particularly in the absence of adhesion between the rubbing surfaces. At larger velocities, dilatency is observed due to the lift hydrodynamic force. This effect may have significant influence on understanding soft surfaces under shear, e.g., in biolubrication.

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AUTHOR INFORMATION

Corresponding Authors

*E-mail [email protected] (C.D.). *E-mail [email protected] (G.S.L.). Notes

The authors declare no competing financial interest.

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REFERENCES

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Experimental Study and Modeling of Boundary Lubricant

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