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Adsorption of stearic acid at the iron oxide/ oil interface - theory, experiments and modeling Aditya Jaishankar, Arben Jusufi, Jessica L. Vreeland, Shane P. Deighton, Joseph R Pellettiere, and Alan M. Schilowitz Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.8b03132 • Publication Date (Web): 09 Jan 2019 Downloaded from http://pubs.acs.org on January 11, 2019

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Adsorption of stearic acid at the iron oxide/oil interface - theory, experiments and modeling Aditya Jaishankar,∗ Arben Jusufi,∗ Jessica L. Vreeland, Shane Deighton, Joseph Pellettiere, and Alan M. Schilowitz∗ ExxonMobil Research and Engineering 1545 Route 22 East, Annandale, NJ - 08801, USA E-mail: [email protected]; [email protected]; [email protected]

∗ To

whom correspondence should be addressed

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Abstract Improved friction performance is an important objective of equipment manufacturers for meeting improved energy efficiency demands. The addition of friction reducing additives, or friction modifiers (FMs), to lubricants are a key part of the strategy. The performance of these additives is related to their surface activity and their ability to form adsorbed layers on the metal surface. However, the extent of surface coverage (mass per unit area) required for effective friction reduction is currently unknown. In this paper,we show that full coverage is not necessary for significant friction reduction. We first highlight various features of surface adsorption that can influence the surface coverage, packing and free energy of adsorption of organic FMs on iron oxide surfaces. Using stearic acid in heptane and hexadecane as model lubricant formulations, we employ a combination of experiments and molecular dynamics (MD) simulations to show how dimerization of acid molecules in the bulk solvent, and crystallographic orientation of the surface modifies surface adsorption. In addition, we show that the solvent can strongly influence adsorption kinetics; MD simulations reveal that hexadecane tends to align on the surface, increasing the energy barrier for the adsorption of stearic acid to the surface. Furthermore, we present a combined approach using MD and molecular thermodynamic theory to calculate adsorption isotherms for stearic acid on iron oxide surfaces, which agrees well with experimental data obtained with a Quartz Crystal Microbalance (QCM). Our results suggest that, while friction of systems lubricated with organic FMs decreases with increasing coverage, complete coverage of the surface is neither practically achievable nor necessary for effective friction reduction for the systems and conditions studied here.

Introduction Friction reduction is a key component of improving energy efficiency and decreasing carbon dioxide emissions in machines, engines and other equipment. Many metal-to-metal contacts inside operating equipment experience boundary lubrication conditions where the surfaces are not separated by a hydrodynamic film, and instead encounter very thin adsorbed or reacted tribofilms which are thinner than the surface roughness(1). In general, friction can be controlled by material 2


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selection, surface finish and lubricants. However, under boundary lubrication conditions, friction modifier (FM) molecules are required to decrease friction. Typical organic FMs such as stearic acid perform by adsorbing to the metal surface and forming a protective barrier between the contacting surfaces, preventing asperity contact (2, 3). In contrast, molybdenum containing friction modifiers form reacted tribofilms which are produced by mechanochemically induced reactions occurring at the contact. In addition to FMs, lubricants contain complex blends of other performance additives to impart important attributes to the lubricant. Many of these additives — such as antiwears, detergents, and dispersants — are surface active and compete for lubricated surfaces. The activity of most organic FMs is dependent on their ability to compete with other additives and adsorb on the surface to form a friction reducing film. Therefore, combining additives with the correct balance of surface activity is critical. To do this successfully, we measure free energies of adsorption, ∆Gads , of various types of molecules, which quantifies the affinity of a molecule for a surface, and determines the outcome of competitive adsorption processes(4, 5). Measurements and DFT calculations show that in the absence of solvent, stearic acid is capable of forming chemisorbed, crystalline adsorbed layers on hematite (Fe2 O3 ).(6, 7). However, relatively little work has been published on the fundamental adsorption properties such as the free energy of adsorption, and rate of adsorption of lubricant additives on various metals and coatings in hydrocarbon solvents(8–12). Beltzer and Jahanmir, in a series of articles, showed that the friction benefit of organic friction modifiers like stearic acid is directly related to surface coverage (13, 14). They demonstrated that the adsorption isotherm could be obtained by measuring friction as a function of concentration. It was assumed that a dense crystalline monolayer with complete coverage of the steel is needed to fully reduce friction. They assumed that friction is directly proportional to surface coverage. Consequently there is no indication from their work of how much coverage is needed to actually reduce friction. Prediction of friction reduction is also complicated by other factors such as solvent effects (15), contact pressure (16), unsaturated bonds (17), and sliding speed (18).

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In this paper, we clarify some important factors that can influence the adsorption of nonreacting friction modifiers. The properties of the adsorbed layer can in turn affect friction in the contact. We chose stearic acid in heptane and hexadecane as our model system because it is able to form adsorbed monolayers without undergoing tribochemical reactions. We first discuss various aspects and features of the adsorption of stearic acid on iron oxide surfaces. We provide a combined approach of Molecular Dynamics (MD) simulations and a molecular-thermodynamic theory (MTT) to predict adsorption isotherms.The MD/MTT approach has been recently developed for surfactants at the water/vapor interface,(19) and is employed here for the first time for FMs at an oil/solid interface. Using Fourier transform infrared (FTIR) measurements and MD simulations, we discuss the influence of dimerization of stearic acid in the bulk, and how to accurately calculate free energies of adsorption in the presence of dimerization. Next, we show that complete coverage of a surface by stearic acid is not necessary for friction reduction. We use the random sequential adsorption (RSA) model to demonstrate that complete coverage is also not achievable from a practical viewpoint because of packing considerations. The solvent can also influence the adsorption of stearic acid, and we show using experiments and MD simulations how the alignment of solvent molecules near the surface can lead to strong adsorption kinetic effects. Finally, we show that crystallographic orientation of the surface influences the properties of the adsorbed film.

Experimental Solution preparation Stearic acid (Grade I, ≥98.5%) was obtained from Sigma-Aldrich, and heptane and hexadecane (HPLC and spectrophotometry grade) were obtained from J. T. Baker. A commercially available, lubricant basestock grade sample of PAO2 was used. All materials were used without further purification. Solutions of stearic acid in heptane were prepared by carefully weighing the desired quantity of acid and solvent, followed by vigorous stirring over many hours at room temperature. At the end of the stirring, the solutions were visibly clear with no suspended particles or turbidity. 4


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Solutions were stored in amber bottles and covered with aluminum foil to minimize photocatalytic degradation of the carboxylic acids.

Friction measurements Friction measurements were performed using a high frequency reciprocating rig (HFRR) purchased from PCS Instruments (London, UK) Because the default geometry of the experiment is a ball reciprocating on a flat disc, the HFRR was modified to use a piston ring reciprocating on a cylinder liner to simulate the more realistic conditions of an engine. Both the piston cylinder liner as well as the piston ring were cast iron. The outer radius of the ring and the inner radius of the cylinder liner were matched to be at 94 mm. A 11 mm x 15 mm section of the cylinder liner was rubbed against a 9 mm arc section of the ring (1 mm thickness) An 800 g load was applied and the piston ring reciprocated at 10 Hz with a 2 mm stroke length for 2 hours (Hertzian contact pressure of approximately 1 MPa). This Hertzian pressure was calculated by assuming a cylinder on flat geometry, which approximates our set up closely(20). We calculate the non-dimensional Hersey number to be 2 × 10−8 , which is well within the regime for Boundary lubrication(21). The temperature was held constant at 50◦ C.

QCM experiments QCM experiments were performed with a commercially-available instrument from Biolin Scientific (QSense QCM-D, E4 flow cells). Hydrocarbon-resistant Kalrez o-rings and seals were used in the flow cells. The injection flow-rate was maintained at 150 uL/min using a peristaltic pump (Cole-Palmer). Solvent-resistant PVC solva tubing (Cole Palmer) was chosen for the peristaltic pump. Teflon inlet tubing and fittings were used as far as possible. The quartz sensors (Biolin Scientific) were PVD coated with iron oxide (Fe3 O4 ). We chose Fe3 O4 (which is a mixture of FeO and Fe2 O3 ) because it represents a more realistic surface; there often is a thin layer of corrosion products on iron and steel surfaces, which consist of a mixture of iron oxides (22). We performed experiments on all three solvents mentioned above, but we only present data on heptane 5


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and hexadecane for brevity (the data obtained with PAO2 as the solvent was nearly identical to that obtained for solutions in hexadecane). The rms surface roughness measured by AFM as reported by the manufacturer is 1.4 nm. Prior to experiments, the sensors were thoroughly cleaned in ethanol, toluene and heptane, and the flow modules were oxygen plasma cleaned for 30 seconds to remove any residual organic matter. During measurements, the pure solvent was first injected for 18 - 20 hours to determine the solvent contribution from the total frequency shift. Subsequently, a series of stearic acid in solvent solutions, each of different concentration, were injected for 2 hours each, and the total frequency shift measured. The adsorbed mass was calculated by removing the solvent contribution to the total frequency shift and the Sauerbrey equation using the method outlined in a previous publication (23). The experiments were done in a flow-loop; the volume of solution looped through the flow cells was large enough to ensure that any stearic acid lost from the solution arising from surface adsorption made a negligible change in bulk concentration.

FTIR spectroscopy Fourier transform infrared (FTIR) spectroscopy measurements were performed using a Thermo Scientific Nicolet 6700 FTIR instrument in transmission mode. Data was averaged over 64 scans at a resolution of 2 cm−1 . A KBr transmission cell (International Crystal Laboratories) with a path length of 1 cm was used. Attenuation screens were placed in the path of the beam to prevent saturation of the detector. The IR spectrum of the solvent (heptane or hexadecane, depending on the experiment) was first collected and used as the background spectrum for all subsequent solutions containing stearic acid (averaged over 128 scans). In this manner, we were able to isolate the peaks at 1713 cm−1 and 1766 cm−1 , which correspond to the hydrogen-bonded cyclic dimer and monomer forms of stearic acid in nonpolar solvents, respectively(24).

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Powder adsorption experiments Iron oxide powder (Fe3 O4 , particle diameter ≤ 5 µm) was purchased from Sigma-Aldrich (product no. 310069). The specific surface area of the powder was independently measured using the BET adsorption isotherm technique(25) and was found to be 8.79 m2 /g. We first measured a calibration curve for the concentration of stearic acid in heptane by measuring the IR absorbance of the carbonyl group in dimer form (1713 cm−1 ) as a function of concentration. Next, a weighed quantity of iron oxide powder was added to a series of solutions containing varying concentrations of stearic acid in heptane. The powder-containing solutions were then gently stirred for 24 hours, following which all the iron oxide powder was extracted using a rare-earth magnet to leave behind stearic acid solutions of reduced concentration. The final concentration of stearic acid in these solutions was measured by comparing the IR absorbance against the calibration curve. The surface coverage of stearic acid on the iron oxide powder can then be determined by calculating the difference in its concentration before and after adsorption. Surface coverage measurements obtained from this technique need no corrections because adsorbed mass is directly measured from the decrease in the bulk concentration of stearic acid.

Modeling methods We used MD simulations to calculate the free energy of adsorption of a stearic acid molecule on b Fe3 O4 surface, ∆Gm ads , and the dimerization free energy of stearic acid in the bulk, ∆Gd . Using

Lammps MD,(26) we employed the following procedure for the adsorption calculations: A single stearic acid molecule dissolved in 1,400 heptane or 900 hexadecane molecules is sandwiched between two solid surfaces positioned at the top and bottom of a simulation box with fixed lateral dimensions. We performed energy minimization with a subsequent 10 ps NV T simulation (N: number of particles, T : temperature, V : volume) with a 0.1 fs time step and a temperature of 100 K. After this relaxation another NV T simulation was carried out in which the temperature was ramped up to the target value of 298.15 K over 0.5 ns at 1 fs time step. In order to achieve the correct liquid density between the solid layers, we then applied an atmospheric load on the top 7


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layer for 2 ns. The top layer serves here as a piston which compresses the liquid to its equilibrium density which corresponds to an equilibrium liquid thickness of about 10 nm in heptane and 12.9 nm in hexadecane. The final bulk density of heptane was (708 ± 23) g/l and for hexadecane (794 ± 14) g/l. These values are 5.0 % and 3.1 %, respectively, above experimental values for these alkanes and in line with a previous simulation work.(27) In all these stages the boundary atoms of the top and bottom layer (outermost layers of atoms) were allowed to move only laterally while the rest of the crystal atoms could move in any direction. The only exception concerns the top layer atoms during the piston stage when all top layer atoms were allowed to move in all directions due to the applied load. For the liquid molecules, a Nosé-Hoover thermostat was applied with a time relaxation constant of 0.1 ps.(28) The top and bottom solid layers were thermostated with the Langevin method, using a 0.1 ps time constant.(29) Long-range electrostatic interactions were calculated with the slab implementation of the PPPM method at a relative force accuracy level of 10−4 .(30) All bonds involving hydrogen atoms were constrained using the SHAKE algorithm.(31) After the piston run (2 ns) we prepared configurations for calculations of free energy of adsorption. At this stage, the overall box dimensions were fixed at (5.8×5.7×12.9) nm3 in heptane and (5.8×5.7×14.5) nm3 in hexadecane. Periodic boundary conditions were imposed only laterally. For the free energy of adsorption calculations we utilzed a similar Umbrella sampling approach as in a previous work where the free energy of adsorption was calculated for aqueous surfactants at the liquid/vapor interface.(19) For this study, we placed a single stearic acid molecule sufficiently far away from the bottom surface to ensure bulk conditions. The distance depends on the extent of surface effects on the solvent molecules; for heptane 3 nm above the surface is sufficient whereas for hexadecane surface 5-6 nm is required. The profile reaches bulk behavior at these distances. This is based on the free energy profile of stearic acid as a function of distance z from the solid surface, as discussed below. We then pulled the stearic acid molecule at (2 nm/ns) towards the surface (in the z-direction which denotes the surface normal) until contact. During the pulling stage configurations were stored every 25 ps. These atomic position snapshots served as initial configurations in subsequent umbrella sampling runs, in which the acid group (COOH) was held

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at a given distance z through a harmonic umbrella potential (32). The harmonic force constant was set to 600 kcal/(mol nm2 ) for z > 1 nm and to 900 kcal/(mol nm2 ) for smaller separations in which solvation or acid-surface effects become strongly pronounced. For the same reasons we used bin sizes of 0.1 nm except for the smaller separation range (z ≤ 1 nm) where the bin sizes were reduced to 0.05 nm. Unless otherwise noted, typical umbrella runs lasted for at least 15 ns in which the first 2 ns of those runs were used for further equilibration, whereas the last 13 ns were used for the umbrella sampling analysis. Here we set the time step to 2 fs using the multiple time step method,(33) where bonded and non-bonded interactions were calculated every 1 fs and 2 fs, respectively. Free energy profiles were calculated using the weighted histogram analysis method (WHAM) (34, 35). For the calculation of the free energy of dimerization we placed two stearic acid molecules in a bulk heptane (or hexadecane) system with cubic box dimensions, and we used a similar simulation protocol as in the slab system described above. We only replaced the piston stage by an isobaric MD run of 2 ns in which the equilibrium density of heptane (or hexadecane) was reached. The time constants for the Nosé-Hoover thermostat and barostat were set to 0.1 ps and 1 ps, respectively. Thereafter, the box dimensions were fixed to be a cube of side 6.1 nm, and the two stearic acid molecules were positioned at a distance of 2.5 nm away from each other. Stearic acid molecule coordinates are defined by the center-of-mass positions of the acid (COOH) entities due to their ability to form hydrogen bonds, which is the primary driver for stearic acid dimerization. The two acid groups were slowly (2 nm/ns) pulled towards each other (in the x-direction) to contact separation. During this process the y and z-directions were constrained so that the interaction coordinate is solely defined by the x-component. During this pulling stage configurations were stored every 25 ps. These atomic position snapshots served as initial configurations in subsequent umbrella sampling runs, in which the two acid groups were held at a given distance x through a harmonic umbrella potential (32). The harmonic force constant and bin size were set to 600 kcal/(mol nm2 ) and 0.1 nm for x >0.8 nm, respectively. In order to capture any solvation or hydrogen bonding effects in the free energy profile near the contact range, the force constant was increased to 900

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kcal/(mol nm2 ) and the umbrella bin size reduced to 0.05 nm. The MD time step was set to 2 fs and each umbrella run lasted 10 ns in heptane and 30 ns in hexadecane. The first 2 ns of those runs were used for equilibration and the remaining time were used for the umbrella sampling analysis. As in the free energy of adsorption calculations WHAM was used to compute the free energy of dimerization(34, 35). In all MD calculations we employed the all atom L-OPLS forcefield for the liquid system(36) that has been further refined with respect to the acid atoms (27, 37). The van-der-Waals forcefield parameters for the atom types comprising the solid top layer of the system were set to σ = 0.3 nm and ε = 0.2 kcal/mol. Interatomic interactions among solid layer atoms were turned off since they were positionally restrained. However, cross interaction parameters of solid atoms with liquid atoms were determined through the Lorentz-Berthelot mixing rule using the corresponding Lennard-Jones parameters. The top layer atoms do not carry any electrostatic charges since this layer only serves as a boundary piston wall. In contrast, the bottom Fe3 O4 layer atoms carry electrostatic charges, in addition to van-der-Waals interactions with Lennard-Jones parameters taken from Ewen et al. (37). All partial charges of Fe atoms were calculated based on the charge equilibration method (38), using the ‘QEq’ option in the forcite tool of Materials Studio (39).

Results and Discussion Dimerization Stearic and other carboxylic acids can dimerize in hydrocarbon solvents through hydrogen bonds(40– 42). A solution of carboxylic acid in a hydrocarbon solvent can hence be considered as a two component mixture of monomers HA and dimers (HA)2 ; these compete with each other for surface adsorption. To determine the thermodynamics of adsorption and consequently the distribution of monomers and dimers adsorbed on the surface, we need to determine the free energy of dimerization of stearic acid in hydrocarbon solvents. These thermodynamics dictate the overall free energy of adsorption onto iron oxide surfaces of stearic acid solutions in hydrocarbon solvents. We first 10


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use MD to determine the equilibirum constant of dimer dissociation, Kd , and independently verify the value with experiments. The free energy profile, W (r), when two stearic acid molecules approach each other is shown in Figure 1. Two dimensions (y and z) were fixed during the Umbrella sampling runs; the interaction coordinate r is along x. The profile deviates from its bulk value (set to zero) at 1.3 nm and becomes strongly negative as the two acid groups approach each other. The two minima indicate two regions of hydrogen bonding between the acid groups: The first minimum at around 0.4 nm indicates a single hydrogen bond. A second hydrogen bond is established at a closer distance of 0.3 nm that is deeper and narrow. A steep repulsion occurs upon decreasing the separation further. As a first approximation one could relate the equilibrium constant Kd to these free energy minima. However, because there are at least two preferred states, we integrate the entire well region:(43, 44)

∆Gdim = − kT ln ∆Gdim = − kT ln

4πR3 3vs

R r1 −[W (r)+2kT ln(r/r ))/kT ] 2 ! 1 r dr r0 e , RR

−W (r)/kT r 2 dr r1 e ! R r1 −W (r)/kT dr 4πR3 r12 r0 e , RR −W (r)/kT r 2 dr 3vs e r1

(1) (2)

with r0 = 0.26 nm and r1 = 0.82 nm defining the boundary range for dimerization in heptane, i.e. the lower and upper limit of the free energy well. Eq. (1) and Eq. (2) were normalized to the molecular volume of the solvent, vs , which was calculated based on its liquid density at 298 K: vs = 0.24 nm3 for heptane and 0.49 nm3 for hexadecane.(45) R = 2.5 nm is the maximum distance range of the free energy calculations. Note that the free energy, W (r), in the numerator of Eq. (1) is corrected by an entropic term since we imposed constraints in the potential of mean force (PMF) calculations in all three dimensions.(46) Integration of this entropic term leads to Eq. (2). We simplify the equations further due to the large distance ranges, R. In the non-bounded range, r > r1 , Figure 1 shows that W (r) ≈ 0, particularly in the heptane case. In the hexadecane case, the small barrier in the profile at r ≈ 0.7 nm turns out to have negligible effects on the final results.

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Hence, setting W (r) = 0 for r > r1 we obtain 4πr12 ∆Gdim = −kT ln vs [1 − ( rR1 )3 ]

Z r1

−W (r)/kT

e r0

4πr12 dr ≈ −kT ln vs

Z r1

−W (r)/kT

e

dr .

(3)

r0

In the last approximation on the right-hand side of Eq. (3) we used the fact that (r1 /R)3 1 in both solvent cases. Eq. (3) accounts for all Boltzmann-weighted free energy values for dimerization, i.e. for all possible density of states within the distance range defined by the free energy well shown in Figure 1. Numerical integration of the MD calculated free energy profile in the well regime using Eq. (3) yields a value of ∆Gdim = −24.5 kJ/mol in heptane. In hexadecane we obtain a value of ∆Gdim = −22.9 kJ/mol using r1 = 0.56 nm in Eq. (3). With a difference of less than 10% it is not conclusive whether this is a true solvent effect in the dimerization free energy.

Figure 1: The free energy of dimerization, W (r) as a function of separation r between the acid groups of the two stearic acid molecules in heptane (black solid line) and in hexadecane (blue dashed line), both at 298 K calculated through MD. The two minima at 0.3 nm and at 0.4 nm indicate a double and a single hydrogen bonding dimer state, respectively. The upper bound of the free energy well is r1 = 0.83 nm in the heptane case and r1 = 0.56 nm in the hexadecane case. To experimentally verify the value of the free energy of dimerization obtained from MD simulations, we performed FTIR measurements on varying concentrations of stearic acid in heptane. 12


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In the inset to Figure 2, we show a typical FTIR spectrum of a 0.1 wt.% solution of stearic acid in heptane. The two peaks at 1766 cm−1 and 1713 cm−1 correspond to IR absorbance peaks of

Figure 2: IR Absorbance of various concentrations of stearic acid in heptane at ν = 1713 cm−1 . The red dashed line corresponds to a fit of Eq. (7) to the absorbance data, from which we obtain the equilibrium constant for stearic acid dissociation Kd = 2.4 × 10−5 . This corresponds to a dimerization free energy ∆Gd = −26.4 kJ/mol. This result is in reasonable agreement with the value of ∆Gd obtained from MD simulations (-24.5 kJ/mol). stearic acid in monomer and cyclic dimer forms respectively (24). We use the relative intensities of the absorbance peaks to calculate the equilibrium concentrations of monomer and cyclic dimer in solution, and consequently the dissociation constant of stearic acid in heptane. The equilibrium reaction for the dissociation of a stearic acid dimer can be written as kd

−− * (HA)2 ) − − 2 HA.

(4)

k−d

in which kd and k−d are the rate constants for the dissociation and association reactions respectively. The equilibrium dissociation constant Kd is given by Kd = kd /k−d . Using chemical equilibrium calculations, we can show that the concentration of dimers cd can be written in terms of the

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total acid concentration ct as s

! 8ct 1+ −1 , Kd

cd = 4ct − Kd

(5)

where cd and ct are expressed in units of mole fraction, and the concentration of monomer is given by

cm = ct − 2cd .

(6)

Note that at low total concentrations, we can use a binomial expansion approximation in Equation Eq. (5), to show that cd ≈ 0 and hence most stearic acid molecules exist as monomers in solution. Similarly, appropriate expansions show that at high concentrations, most stearic acid molecules exist as dimers. The free energy of adsorption is especially sensitive to the surface coverage at low total concentrations. The IR absorbance I = εd cd l, where εd is the molar attenuation coefficient of dimers and l is the path length of the absorbance cell, can be written using Eq. (5) as " I = εd

125ρs Ms

s 4ct − Kd

8ct 1+ −1 Kd

!!# l.

(7)

Here, cd is expressed in units of molarity, which introduces the numerical pre-factor as well as the density of the solvent ρs and the molecular weight of the solvent Ms . In Figure 2, we show the fit of Eq. (7) to absorbance data for the dimer peak (1713 cm−1 ) obtained for various concentrations of stearic acid in heptane. Here, the fitting parameters are εd , and Kd . The obtained value of Kd = 2.4 × 10−5 yields a value of free energy of dimerization ∆Gdim = −RT ln(Kd−1 ) = −26.4 kJ/mol. This is in good agreement with the value obtained from molecular dynamics simulations (∆Gdim = −24.5 kJ/mol). The experimental value of ∆Gdim for stearic acid in hexadecane yields an identical number, within the error bounds of the measurements. These numbers are also in agreement with those published in the literature over many

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decades, which range between -28 kJ/mol and -22 kJ/mol for various combinations of solvents and surfaces (8–11, 47–49). This value of ∆Gdim indicates that dimerization of carboxylic acids in hydrocarbon solvents is strongly thermodynamically favorable, and most stearic acid molecules in solution exist as dimers at high total concentrations of stearic acid. Therefore, for a complete picture of adsorption of stearic acid on surfaces, it is critical to understand the contribution of monomers and dimers of stearic acid to the total surface coverage. d To determine the free energy of adsorption of the monomer, ∆Gm ads , and dimer, ∆Gads , individ-

ually, we consider the thermodynamic cycle in Figure 3, where we conceptualize two stearic acid monomers going through multiple thermodynamic steps in a cyclic manner. The steps are: (1) two

Step 1

Step 4

Step 2

Step 3

Figure 3: Schematic figure showing the gedanken thermodynamic cycle used to determine the free energy of adsorption of stearic acid dimers. Because free energies are state variables, we can use a closed cycle to determine the relationship between the free energies of adsorption of the monomer and dimer. We use the following steps: (1) Adsorption of two stearic acid monomers, (2) Dimerization of stearic acid on the iron oxide surface, (3) Desorption of the stearic acid dimer, (4) Dissociation of dimer in the bulk into two monomers. This closed cycle has zero net change in free energy and hence yields a relationship between the adsorption energy of monomeric and dimeric stearic acid, given in Eq. (8). monomers adsorbing on the surface, (2) monomers dimerizing on the surface, (3) desorption of dimer from surface and (4) dissociation of dimer into two monomers. Because of the closed loop

15


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Page 16 of 42

nature of this cycle, the free energies of these steps are related through the equation

s d b 2∆Gm ads + ∆Gd − ∆Gads − ∆Gd = 0

(8)

s d b in which ∆Gm ads , ∆Gd , ∆Gads and ∆Gd are the free energies of (1) adsorption of a monomer, (2)

dimerization of stearic acid on the surface, (3) adsorption of dimer onto the surface, and (4) dissociation of dimer into two monomers in the bulk. From molecular dynamics simulations, we find that ∆Gsd is negligibly small. This is likely due to the fact that the formation of new bonds between stearic acid pre-adsorbed on the surface comes at the expense of breaking existing hydrogen bonds between stearic acid and the surface, and these compensate each other. We thereb fore ignore this term. Eq. (8) then reduces to ∆Gdads = −(2∆Gm ads + ∆Gd ). Using the value of

∆Gbd = −∆Gdim = 26.4 kJ/mol determined from MD simulations and rewriting free energies in terms of equilibrium constants using the relationship ∆Gi = −RT ln Ki , where R is the gas constant, we find that the equilibrium constants for adsorption of monomer and dimer, Km and Kd respectively, are related through 26400 Km2 . Kd = exp − RT

(9)

We model the adsorption of monomeric and dimeric stearic acid onto our surfaces using a multi-component Langmuir adsorption isotherm (50), where the adsorbed mass on the surface m is given by

m = M∞ ∑ i

Ki ci 1 + ∑i Ki ci

(10)

where the subscript i represents either a monomer quantity or dimer quantity, Ki is the adsorption equilibrium constant of species i and ci is the concentration of species i expressed in units of mole fraction. Eq. (9) and Eq. (10) in conjunction with Eq. (5) and Eq. (6) can then be used to fit an experimentally measured adsorption isotherm to determine the mass at saturation, M∞ , the

16


Page 17 of 42

individual equilibrium constants Km and Kd , and subsequently the free energies of adsorption. In Figure 4, we show adsorption isotherms measured using the QCM for stearic acid in heptane (a) and hexadecane (b). While calculating the masses, the measured frequency shifts were corrected for solvent replacement effects (i.e. adsorbing stearic acid molecules replace solvent molecules initially adsorbed on the surface) according to the method outlined in a previous publication (23). The symbols are the measured data and the lines are fits to Eq. (10). The values 6 0

2

2

]

]

1 2 0 1 0 0

A d s o rb e d m a s s [n g /c m


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

8 0 6 0 4 0 2 0

5 0 4 0 3 0 2 0 1 0 0

0 0 .0

4 .0 x 1 0

-4

8 .0 x 1 0

-4

1 .2 x 1 0

-3

0 .0 0

7 .5 0 x 1 0

-4

1 .5 0 x 1 0

-3

2 .2 5 x 1 0

-3

T o ta l m o le fr a c tio n [- ]


(a) Stearic acid in heptane

(b) Stearic acid in Hexadecane

Figure 4: Adsorption isotherms of stearic acid in (a) heptane and (b) hexadecane. Although the free energy of adsorption ∆G differs only slightly between solvents, the final surface coverage of stearic acid in heptane is twice that in hexadecane. We attribute this difference to solvent-mediated differences in adsorption kinetics, as discussed below. d of ∆Gm ads and ∆Gads for each of the solvents calculated from the determined equilibrium constants

is shown in Table 1. We also show the values for free energies calculated from MD simulations; these results are discussed later in the paper. These data show that the free energy of adsorption Table 1: Free energies of adsorption for monomeric and dimeric stearic acid in heptane and hexadecane. Values determined by MD are discussed later in the paper. Solvent Heptane Hexadecane

d m ∆Gm ads [kJ/mol] ∆Gads [kJ/mol] ∆Gads [kJ/mol] (MD) -26.1 -23.6 -26.4 -26.4 -24.3 -27.6

∆Gdads [kJ/mol] (MD) -28.2 -32.3

of monomers and dimers are not different by a significant amount. Using these values in Eq. (10), along with Eq. (5) and Eq. (6) allows us to calculate the individual surface coverages of monomeric 17


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and dimeric stearic acid.

Molecular thermodynamic theory Recent molecular dynamics simulations have focused on understanding the structure of adsorbed stearic acid films on friction (51, 52). In MD, the control parameter is the surface coverage, Γ (units: molecules/area), and friction performance is then calculated as a function of Γ. However, in principle, under equilibrium conditions the control parameter is the bulk concentration of stearic acid, and it is difficult to compare MD results with experiments. The missing link is the adsorption isotherm that needs to be measured or predicted through molecular modeling. In practice, the actual surface coverage can be affected by other factors such as competitive adsorption, kinetics, squeeze out under high-pressure and high-shear conditions. From the perspective of MD, predicting the adsorption isotherm requires simultaneous sampling of bulk and interface partitioning. Using atomistic models this is computationally infeasible since exchanges between adsorbed molecules at the interface and the bulk can be extremely rare events for molecules with strong surface affinities of several kT . We provide a pathway that helps overcome this modeling challenge by combining the MD approach with a molecularthermodynamic theory (MTT) framework. Molecular-thermodynamic theory (MTT) is an established method for calculating adsorption isotherms of aqueous surface active compounds on fluid/fluid interfaces (53–55) and on fluid/solid interfaces (56). Recently an MD-MTT approach was developed to predict surface tension isotherms for aqueous surfactant solutions.(19) A similar approach is used here for adsorption isotherm prediction for the solid-liquid interface of a hydrocarbon based solution containing stearic acid as the surface active compound. First, the chemical potential of the solutes in the bulk and at the interface, µ b and µ σ , respectively, are identified. In the dilute bulk concentration range, µib = µib,0 + kT ln Xi + kT ln

18

Λ3i , vs


(11)

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Langmuir

where we distinguish between stearic acid monomers (i = m) and dimers (i = d). The bulk partitioning of the two entities is determined by its equilibrium constant Kd , calculated in the previous section. In Eq. (11) µib,0 denotes the chemical potential at infinite dilution, Λi the thermal de Broglie wavelength of species i, and vs being the molecular volume of the solvent. The last term in Eq. (11) occurs due to the conversion from solute number density into mole fraction and is usually absorbed in µib,0 . For the MD-MTT approach this term is, however, important. As surface coverage increases, the molecules begin to interact with each other, and non-ideal contributions to the chemical potential µ σ become relevant on the surface. These excess terms are predominantly of repulsive nature in the case of stearic acid. In contrast to aqueous surfactants adsorbed on solid interfaces, we may ignore micelle formation or self-aggregation on surfaces due to solvent phobicity. Since stearic acid is highly soluble in hydrocarbons they do not micellize in solution or at the interface so that homogeneous adsorption is the primary mechanism. There is a weak attraction between stearic acid molecules on the surface, as they may form a single hydrogen bond (interface dimerization)(23). However, the occurrence of such bonds requires a certain orientation of the molecules with respect to each other. As the surface concentration is increased these specific orientations cannot be preserved due to chain packing effects. Hence, we ignore specific attractive pair interactions on the surface and assume homogeneous adsorption as the bulk concentration is enhanced. This is very similar to homogeneous adsorption of nonionic surfactants at the oil/water interface where the interactions are predominantly of repulsive nature.(57) Since adsorbed molecules prefer a specific distance from the solid surface (at around 0.3 nm),(52) the adorbed stearic acid molecules are mainly located on a plane above the Fe3 O4 layer. We therefore employ a 2D-hard disk model for the equation of state of adsorbed molecules to account for their steric interactions. We use the same expression for this repulsive contribution in the chemical potential as in (53) µiσ = µiσ ,0 + kT ln(Γi a/(1 − Γi a)) + kT Γi a

19

Λ3i 3 − 2Γi a + kT ln + µiσ ,def , 2 (1 − Γi a) aL


(12)

Langmuir 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 20 of 42

where Γi denotes the surface concentration of monomers (i = m) or dimers (i = d), and a denotes the effective hard-disk area of stearic acid. Note that Γd = 2Γm , however, µmσ ,0 6= µdσ ,0 . This inequality leads to different strengths of adsorption for dimers and monomers, as discussed in the previous section. The second to last term in Eq. (12) is needed due to the conversion of number density in the entropic part (2nd term) into the packing fraction Γi a. The conversion term also contains the thickness of the interface region, L, which is identified from the free energy profile, see below. The last term in Eq. (12), µiσ ,def is not part of the conventional 2D-adsorption isotherm equation at oil/water interface. As the concentration is increased, stearic acid molecules adsorbed on solid surfaces have to specifically orient themselves (mainly normal or slightly tilted to the surface) due to chain packing effects, and an additional chain deformation chemical potential contribution has to be introduced: µiσ ,def = fdef Γ2i .

(13)

This contribution has been evaluated previously for surfactant adsorption on solid surfaces forming a lamella structure, similar to our situation here. The entropic penalty of freely rotating chain molecules in the bulk and to the constraints imposed by a lamellar configuration of oriented linear chains on surfaces has been analytically estimated.(56) The prefactor fdef in Eq. (13) for linear chains is given by fdef =

π2 Nc Lc4 , 8

(14)

where Lc = 0.46 nm is the size of a hydrocarbon segment and Nc = (nc + 1)/3.6 is the number of such segments of a hydrocarbon chain with hydrocarbon number nc .(56) For stearic acid the number of hydrocarbons (methylene plus methyl) groups is nc = 17. At thermodynamic equilibrium µib = µiσ and the adsorption isotherm is calculated using Eq. (11)Eq. (12), which yields: ∆Giads Γi a 3 − 2Γi a π2 ln Xi = + ln + Γi a + Nc Lc4 Γ2i kT 1 − Γi a (1 − Γi a)2 8

(15)

vs with ∆Giads ≡ µiσ ,0 − µib,0 + kT ln aL is the free energy of adsorption at infinite dilution. Note that

20


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Langmuir

the thermal wavelength Λi has been cancelled out above. Eq. (15) is the main equation that relates surface coverage to bulk concentration. It contains the following parameters for each adsorbing species: the free energy of adsorption (∆Giads ), the hard disk area (a), as well as the hydrocarbon segment parameters Nc and Lc estimated above. The effective headgroup area has been determined previously with a value of a = 0.17 nm2 (23). The missing parameter in Eq. (15) is the free energy of adsorption, ∆Giads . It is obtained from potential of mean force calculations in MD. Figure 5 shows the MD calculated free energy of transferring a stearic acid molecule from bulk to surface contact at around z0 = 0.6 nm. The range of the interface region in the liquid layer is defined as L = z2 − z1 withz2 marking the range of deviation of the PMF from its bulk value and z1 being the upper bound of the free energy well. For heptane z1 = 1.0 nm and z2 = 2.0 nm, whereas for hexadecane z1 = 0.9 nm and z2 = 4.0 nm. The reason for the oscillations in the free energy profile for hexadecane seen in Figure 5 compared to heptane are explained in the section on solvent effects. As for the well depth of the PMF, a value of -32.5 kJ/mol is calculated from the free energy difference at the minimum (z = 0.65 nm) and the bulk value (set to zero). However, for a very narrow free energy well, other density of states as defined by neighboring free energy values around the minimum become important as well. Small fluctuations lead to significant free energy changes for such narrow wells. Instead of using ∆Gm ads as the minimum of the free energy profile, we integrate over the well region, i.e. between z0 = 0.6 nm and z1 (values given above). The difference between the chemical potential of stearic acid in the interface region and in the bulk region, both at infinite dilution, is calculated from the transfer free energy as a function of the distance, z:

∆Gm ads

Z z1 vs 1 exp(−Wads (z)/kT )dz + kT ln , = −kT ln z −z z aL 1 0 0 Zz 1 aL = −kT ln exp(−Wads (z)/kT )dz vs (z1 − z0 ) z0

(16) (17)

where Wads (z) is the transfer free energy for a single stearic acid molecule obtained through MD calculated PMF. According to Eq. (17) we obtain a value of ∆Gm ads = −26.4 kJ/mol in heptane

21


Langmuir 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

and ∆Gm ads = −27.6 kJ/mol in hexadecane. The adsorption free energy for a dimer is calculated through Eq. (8), again with ∆Gsd = 0 and ∆Gbd determined through MD, as described in the previous section. We obtain ∆Gdads = −28.2 kJ/mol in heptane and ∆Gdads = −32.3 kJ/mol in hexadecane. Note that these values suggest physisorption as the main adsorption mechanism for stearic acid in heptane/hexadecane on Fe3 O4 . The calculated values for ∆Gm ads are within the range of our own experimentally derived values, see Table 1, consistent with previous experimental results reported in literature.(8–11, 13, 14, 16, 17) In our QCM experiments we also observed that adsorbed stearic acid material could easily be washed off using heptane, suggesting relatively weak adsorption compared to what could be expected for chemisorbed material. Interestingly, recent density functional theory (DFT) calculations of hexanoic acid showed that it chemisorbs on hematite (Fe2 O3 ).(7) The calculated adsorption energy is in the order of -230 kJ/mol, which is much lower than our values in Table 1. However, it should be noted that those DFT calculations were done in vacuum at 0 K on hematite. Presence of solvent, surface chemistry, and thermal conditions could all affect the adsorption mechanism of being chemisorption or physisorption. More studies are needed to resolve the role of these factors for the adsorption free energy.

Surface coverage and random sequential adsorption Using Eq. (15) we calculated the total adsorbed surface concentration (mass per area), m = (Γm + Γd )/(Mw NA ), with Mw being the molar mass of stearic acid and NA being Avogadro’s number. The MD/MTT prediction of the mass adsorbed is within the range of data obtained from QCM and powder experiments, as shown in Figure 6. The discrepancy between the results obtained from QCM and powder experiments may be explained due to differences in crystallographic orientation of the surface material, which is discussed in more detail in section on crystallographic orientation. Also shown as a dashed line in Figure 6 is the upper limit for adsorption based on the random sequential adsorption (RSA) model(58). This model predicts that, because adsorbed molecules arrive at the surface randomly, achieving perfect, efficient packing requires cooperative surface rearrangements of multiple molecules. Therefore, despite the theoretical maximum packing frac22


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Langmuir

Figure 5: Free energy as a function of distance z between the position of the center of mass of the acid group and the Fe3 O4 surface for stearic acid in heptane (black solid line) and in hexadecane (blue dashed line). Note that the free energy gain between the bulk value at large separations and at its minimum (z = 0.65 nm) is not affected by the solvent. tion of hard disks on a surface being approximately 0.9, at surface concentrations near jamming the surface packing fraction reaches a barrier at 0.547. Although RSA is developed for adsorbed disks on a 2D-surface (equilibrium constant Kads 1) it is striking how close the predicted surface density Γmax = 0.547/a = 3.2 nm−2 = 152 ng/cm2 agrees with the experimental data using the calculated hard disk area of a = 0.17 nm2 .(23) The surface affinity of stearic acid is relatively strong (calculated Kads = 4 × 104 1) and one can ignore desorption so that the RSA model prediction becomes applicable.(6, 7) The MD/MTT prediction together with the RSA estimation of maximum coverage imply that although high packing densities of stearic acid is thermodynamically favorable, it is difficult to achieve in practice (59). Full coverage is therefore limited by jamming (surface packing of 0.55) and not by the maximum packing of 2D-hard disks (0.91 packing fraction). Earlier work(60) reported that arachidic acid (C20) forms submonolayer coverage on aluminum oxide. This is in general agreement with our QCM results and RSA arguments. However, more recent work showed that, “under the appropriate conditions” organic acids can adsorb 23


Langmuir 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Figure 6: Adsorption isotherm of stearic acid in heptane on Fe3 O4 . The adsorbed mass is plotted as a function of bulk mole fraction of stearic acid in its monomeric form Xm . Experimental isotherms determined by 2 hour QCM experiments (blue squares) and by powder experiments (black circles) are also shown. The line is the prediction using MD/MTT, Eq. (15). The horizontal dashed line denotes the upper adsorption limit based on the random sequential adsorption (RSA) model. The discrepancy between the isotherms obtained from the QCM and iron oxide powder is explained in the section on crystallographic orientation. into highly ordered, close packed, crystalline layers on aluminun oxide surfaces but also noted that formation of such a close packed layer is highly variable.(59) It was noted that “the slowest and most unpredictable part of the assembly occurs during the incorporation of the last 20-25% of the monolayer”. The discrepancy was explained as due to an extremely slow kinetic effect related to the dynamic nature of the assembly process. It is interesting that an obstacle was observed for incorporating the last 20-25% into the monolayer. We suggest that this obstacle is analogous to our jamming barrier. Until this critical surface concentration is reached which, based on RSA is 0.547, adsorption of stearic acid readily occurs and friction is reduced. However, the formation of a close-packed crystalline, thermodynamically preferred layer is dependent on surface morphology, solvent, impurities and time. Even under appropriate conditions full coverage can take a week or more to occur(59). We also note the discrepancy in Figure 6 between the isotherm obtained by

24


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Langmuir

the QCM vs. powder adsorption methods, and this is explained in the section on crystallographic orientation. Nevertheless, the highest observed coverage on magnetite is lower compared to a recently reported adsorption study of stearic acid dissolved in dodecane in contact with hematite (α−Fe2 O3 (0001) surface).(17) In that study maximum coverage of 3.6 molecules/nm2 was measured which is higher than the RSA limit of 3.2 molecules/nm2 . We attribute the higher coverage to chemisorption of stearic acid on hematite, as a recent DFT study found for this surface material.(7) RSA is strictly limited to physisorption which is the case in our study. This conclusion is based on our own experiments and calculations of a relatively weak free energy of adsorption (> -30 kJ/mol), and supporting values found in the literature (9–11). Moreover we observed in the QCM that adsorbed stearic acid material could be easily washed off the surface using heptane. To show that complete coverage is not required for effective friction reduction as assumed by Jahanmir and Beltzer(13), in Figure 7a we show the results of friction measurements performed with the HFRR. As seen in Figure 7, the friction coefficient decreases with increasing bulk concentration. Because we have measured adsorption isotherms, we can re-plot the friction data as a function of absolute surface coverage, which is shown in Figure 7b. We observe friction reduction at relatively low surface coverages of ∼ 35 ng/cm2 , which is far from the coverage required for completely saturating the surface with stearic acid molecules (152 ng cm−2 ×(0.91/0.547) = 252 ng cm−2 ). The red dashed line in Figure 7a represents an adsorption isotherm calculated purely from the friction data. Because we have shown that the friction coefficient depends on surface coverage (Figure 7b), we assume a linear relationship between the fractional coverage θ and the friction coefficient f and we have θ=

f0 − f f0 − f∞

(18)

where f0 is the friction coefficient of the pure basestock, i.e., in the absence of stearic acid, and f∞ represents the plateau value of friction, i.e., where the friction does not decrease further upon increasing the concentration of stearic acid. Note that although the fractional coverage θ can approach unity as defined above, θ = 1 does not represent complete coverage of the surface, with 25


0 .1 6

0 .1 6

0 .1 5

0 .1 5

F r ic tio n c o e ff. [- ]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

F r ic tio n c o e ffic ie n t [- ]

Langmuir

0 .1 4 0 .1 3 0 .1 2

0 .1 4 0 .1 3 0 .1 2 0 .1 1

0 .1 1 0 .1 0

Page 26 of 42

0 .0

5 .0 x 1 0

-4

1 .0 x 1 0

-3

1 .5 x 1 0

-3

2 .0 x 1 0

0 .1 0

-3

0

5

1 0

1 5

2 0

2 5

3 0

3 5

S u rfa c e c o v e ra g e [n g /c m


(a)

4 0

4 5 2

5 0

]

(b)

Figure 7: (a) Friction coefficient as a function of total stearic acid concentration in the polyalphaolefin basestock PAO6, measured using the modified HFRR ring-on-liner setup. Friction coefficient drops with increasing stearic acid concentration, in agreement with the results of Jahanmir and Beltzer(13, 14). The red dashed line is the fit of the Langmuir-like isotherm obtained from the θ values found using Eq. (18). From this fit, we obtain the free energy of adsorption to be ∆G = −26.4 kJ/mol. (b) Friction coefficient from ring-on-liner experiments as a function of surface coverage of stearic acid. (13, 14). The surface coverage at minimum friction is far from the coverage at saturation. every possible adsorption site occupied by an acid molecule. As we describe above, by our definition θ = 1 is achieved when the surface coverage reached the jammed packing ratio of 0.547 predicted by the RSA model. We can use the value of θ so obtained in Eq. (10) (with i = 1) to find Keq and consequently, ∆Gads = −26.4 kJ/mol. This value is very close to the value obtained from QCM experiments as well as MD simulations. There are some assumptions involved in the construction of Figure 7. First, in this work we only consider surface coverage effects on friction, and we ignore any tribochemical processes that might be occurring. The situation is likely to be different for friction modifiers that form reacted tribofilms (such as MoDTC and MoDTP that react to form MoS2 layers), and the amount of triboproduct formed is likely to be concentration dependent.(61) The friction measurements were done on cast iron, which might behave differently compared to the Fe3 O4 QCM sensors. However there is likely to be a corrosion nanolayer of Fe3 O4 on the surface of the cast iron, and the choice for Fe3 O4 QCM sensors was motivated to mitigate this difference. Furthermore, the solvent used in the 26


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Langmuir

QCM experiments was PAO2; we do not show the data in this paper but the adsorption isotherms were identical to those obtained for hexadecane. The surface coverages shown in Figure 7 were obtained from QCM measurements on PAO2 (refer Figure 4 for the hexadecane data). The similarity in data is to be expected because of the minimal difference between hexadecane (hydrocarbon with 16 carbon atoms) and PAO2 (20 carbon atoms) as a solvent. However, the solvent used in the friction measurements was PAO6. The latter has a higher molecular weight. In some cases, the solvent can influence surface packing, as we discuss in more detail below.

Solvent effects A comparison of Figure 4(a) and (b) reveals that the plateaus of the adsorption isotherms, M∞ , are 54 ng/cm2 and 127 ng/cm2 for stearic acid in hexadecane and heptane, respectively. However, as Table 1 shows, the free energy of adsorption of stearic acid in hexadecane and heptane is very similar; the solvent influences the surface coverage of stearic acid without influencing ∆Gads significantly. Bain et al. (62, 63) have discussed various aspects of the influence of solvent on the selfassembled monolayer formation of thiols on gold surfaces. For example, solvent can inter-mingle with the adsorbed layer, especially when there is strong geometric matching between the solvent and the tails of the adsorbate, such as linear long-chain adsorbates in hexadecane. Allara and Nuzzo showed that for n-alkanoic acids adsorbing onto an aluminum surface, equilibrium surface coverage is not achieved even after allowing for an adsorption time of more than two weeks (59). Bain et al. also suggest that for the adsorption of long-chain alkane thiols on gold surfaces, the adsorption initially proceeds quickly, but the packing of molecules is disordered (62). This makes achieving complete coverage a slow process, because the replacement of solvent by adsorbate happens over long timescales. Moreover cooperative molecular rearrangements are required for the consolidation of adsorption sites so that the surface packing is efficient enough to allow for complete coverage. We have quantified the consequence in the previous section using the RSA model. 27


Langmuir 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

To understand the origin of the solvent effects observed in our experiments, we performed MD simulations to compute free energy of adsorption of stearic acid in heptane and in hexadecane. Details of the procedure are explained in the experimental section. There are two main differences between the heptane and hexadecane cases. The system size of the latter is slightly larger: the thickness of the equilibrium liquid layer is 12.9 nm vs 10.2 nm in heptane. The other important difference is the simulation time during umbrella sampling due to longer relaxation times of hexadecane molecules compared to heptane. We only achieved convergence of the free energy depth at contact by simulating for at least 30 ns for each umbrella bin in hexadecane. The resulting free energy profile as a function of distance z of the acid group from the bottom Fe3 O4 surface is shown in Figure 5 for stearic acid in heptane and in hexadecane. The location of the minimum at contact (z = 0.65 nm) as well as the depth of that minimum are solvent independent, as expected. This is a verification that we ran the hexadecane simulations long enough. Shorter simulations, e.g. 15 ns, were not sufficient to reach this required agreement in free energy gain between bulk and surface contact. The solvent effects are pronounced in two ways: first, Figure 5 shows that the influence of the surface reached much farther into the liquid in hexadecane (up to 4 nm away from the bottom surface compared to 2.0 nm in heptane, as defined by the distance from the surface where the free energy reaches its bulk value of 0). By imposing a larger liquid slab we ensure bulk conditions in the middle of the slab. Second, it appears that there are some small barriers in heptane between 0.5 nm - 2 nm. Those barriers are of the order of 3.5 − 4 nm above the surface. In the heptane case solvent layering is less extended (z < 2 − 2.5 nm) and the peaks are not as pronounced as in the hexadecane case. Adsorption of a solute on the surface requires break-

Figure 8: Left figure: carbon density profiles for heptane and hexadecane as a function of the distance from the surface z. Right figure: MD snapshot showing layering of hexadecane solvent molecules (green chains) near a Fe3 O4 surface (iron: purple spheres; oxygen: red spheres). A stearic acid molecule on the surface is represented by a ball-stick model, in which the carbon atoms of the methylene and methyl groups are shown asblue-colored spheres (hydrogen atoms not shown for clarity). ing up those structures and removing them from the surface. In addition, the geometric similarity between stearic acid and hexadecane can also lead to significant intermingling of adsorbate and solvent, leading to slower adsorption kinetics (4, 62). As we see in Figure 8, the hydrocarbon tail of stearic acid is perfectly aligned along the layered hexadecane molecule and can be kinetically trapped in this position. Note, however, that the global minimum reached at z = 0.65 nm is caused by specific interaction of the headgroup of stearic acid with the polar surface, and not by solvent effects.

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Langmuir

Crystallographic orientation Iron oxide exists in various phases, each of which have unique crystallographic orientations (66). For example, in magnetite coatings that are sputtered onto substrates, the amount of α − Fe2 O3 formed on the surface of magnetite is influenced by the temperature, the saturation pressure of oxygen and the crystallographic orientation of the underlying magnetite (67). Different crystallographic orientations also have different densities of defects. These defects increase the energy of surfaces and can modify the free energy of adsorption and surface coverage. (68, 69). To highlight the difference the surface can play in surface coverage and free energy of adsorption, we compared adsorption data for stearic acid on iron oxide surfaces obtained from the QCM to adsorption on iron oxide powder. The QCM sensors were sputter coated in a PVD process and hence the crystallographic orientation of the coating is likely to be very different from the random, polycrystalline nature of the iron oxide powder. In Figure 9, we show adsorption isotherms measured with powder adsorption experiments for stearic acid in (a) heptane and (b) hexadecane. For 1 7 5

9 6 h o u rs

1 5 0

1 5 0

9 6 h o u rs

2

2

]

]

1 7 5

1 2 5



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1 0 0

2 h o u rs

7 5 5 0 2 5 0

1 2 5 1 0 0

2 h o u rs

7 5 5 0 2 5 0

0 .0

1 .0 x 1 0

-4

2 .0 x 1 0

-4

3 .0 x 1 0

-4

0 .0

1 .0 x 1 0

T o ta l c o n c . [m o le fr a c .]

-4

2 .0 x 1 0

-4

3 .0 x 1 0

-4

4 .0 x 1 0

-4

5 .0 x 1 0

-4

T o ta l c o n c . [m o le fr a c .]

(a)

(b)

Figure 9: Adsorption isotherms of stearic acid measured using powder adsorption experiments in (a) heptane and (b) hexadecane. Square symbols are measurements after allowing for a 2 hour adsorption time, while circular symbols are after allowing for 24 hours of adsorption. In both solvents, we observe kinetic effects, and the adsorbed mass is lower after 2 hours of adsorption compared to 24 hours. The mass adsorbed as measured by powder experiments is higher than the mass measured from QCM experiments (see Figure Figure 6). This difference arises in part from differences in crystallographic orientation of Fe3 O4 between the QCM sensors and the powder.

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each solvent, we performed experiments with powder soak times of 2 hours (squares) and 96 hours (circles). In both solvents, the mass of stearic acid adsorbed after two hours is lower than that adsorbed after 96 hours, indicating the presence of slow adsorption kinetics. After 96 hours, the mass of stearic acid adsorbed in the two solvents is similar. When these data are compared to the coverage values obtained from QCM measurements, we find that the coverages obtained for stearic acid in hexadecane do not agree with each other, although the agreement for heptane is good. There are also differences between the free energies of adsorption ∆Gads when comparing the QCM and powder soaking experiments. In Table 2 we present ∆Gads values for stearic acid adsorbing onto iron oxide coated QCM sensors and iron oxide powder in heptane and hexadecane. The affinity for stearic acid is higher for the iron oxide powder compared to the coated QCM sensors. Table 2: Free energies of adsorption for monomeric stearic acid molecules on Fe3 O4 in heptane and in hexadecane as measured by the experimental and modeling methods described in this paper. The MD values correspond to adsorption on Fe3 O4 (111). Solvent

Heptane Hexadecane

∆G [kJ/mol] Powder (96 hrs)

QCM

MD

-33.2 -29.1

-26.1 -26.4

-26.4 -27.6

To understand the effect of crystallographic orientation and surface chemistry on the free energy of adsorption, we performed MD simulations of three different Fe3 O4 surfaces. Figure 10 shows MD results of free energy of adsorption profiles for the Fe3 O4 (111) and Fe3 O4 (001) surface orientations. As can be seen crystallographic orientation has a significant impact on the free energy of adsorption. The adsorption energy difference between the two surfaces is around 10 kJ/mol comparing the values at the PMF minima. The strength of adsorption becomes even stronger when the surface is fully terminated with OH groups (hydroxyl termination), which is not uncommon when metal surfaces are exposed to humidity (70). Comparing the adsorption of stearic acid on plain Fe3 O4 (001) with hydroxyl-terminated Fe3 O4 (001), the adsorption strength is enhanced by roughly 5 kJ/mol. We have not validated test results experimentally; however, because we have not changed the forcefield the modeling approach is self-consistent. We conclude 31


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Figure 10: Free energy profiles as a function of the distance z between stearic acid in heptane and the solid surface calculated using MD. Profiles are shown for three different surfaces characterized by the orientation of the Fe3 O4 crystal: Fe3 O4 (111) (black solid line), Fe3 O4 (001) (red dashed line), and hydroxyl terminated Fe3 O4 (001) (blue dotted line). that surface effects likely play an important role in adsorption of molecules. In the case of Fe3 O4 the most dominant surface orientation is in the (111) direction. However, in many experimental samples there might be a variety of surface orientations and termination. As mentioned above, the orientation of Fe3 O4 sensors used in QCM may vary from surface orientations of Fe3 O4 powder. We reasonably assume that Fe3 O4 powder exhibits a distribution of crystallo/graphic orientations, and a measured ∆Gads is an average value. In contrast, a QCM sensor is likely dominated by a single or a few crystallographic orientation of Fe3 O4 . If the sensor surfaces are dominated by the most prominent orientation of Fe3 O4 (111) then this may explain the differences in the free energy of adsorption values shown in Table 2. Wood et al. (17) also attribute differences between planar surface adsorption (as measured by Polarized Neutron Reflectometry) and powder adsorption experiments to crystallographic plane orientation. Here we quantify the influence the orientation of the exposed crystal plane can have on adsorption. However, we recognize that many other factors might be at play leading to the observed adsorption differences, such as surface defects(71) and 32


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surface roughness (72).

Conclusion In this paper, we have described various features of adsorption of stearic acid onto iron oxide surfaces, highlighting phenomena especially pertinent to friction reduction applications. Most importantly, we show that stearic acid achieves a minimum friction plateau without fully covering the iron oxide substrate. This conclusion is based on the fact that the isotherm does not reach full coverage under our experimental conditions. We do not see any evidence for a crystalline adsorbed layer of stearic acid, and yet we see a reduction in friction. Here, we discuss reasons for the absence of a crystalline adsorbed layer of stearic acid, for example, slow adsorption kinetics and solvent effects. We found that in agreement with jamming theory, minimum friction occurs at a surface coverage of 0.55 in terms of packing fraction which is below the theoretical crystalline packing limit. We find that stearic acid monomers and dimers exhibit equal affinity for an iron oxide surface; moreover the combination of the MTT with MD calculations is able to predict adsorption isotherms for this system. We find that the specific crystallographic orientation of the iron oxide surface and extent of hydroxylation can change the affinity of stearic acid for the surface. When comparing heptane and hexadecane, hexadecane molecules align along the surface, significantly slowing down the kinetics of adsorption of stearic acid. We propose that geometric matching between solvent and additive, as well as tendency of solvent molecules to align along the surface might be an important parameter in determining lubricant additive function and performance. However, solvents can also strongly influence adsorption through solubility effects(73); this discussion is outside the scope of this article. Moreover, the simulations and experiments presented here were performed under static conditions in the absence of shear. The structure and alignment of an adsorbed monolayer can be strongly influenced by shear (51). These and other effects will be discussed in future publications.

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Table of Contents Graphic

Hexadecane

Stearic acid in Heptane

Aligned Stearic acid

Stearic acid in Hexadecane

Iron oxide

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oil interface - theory

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