Non-linear adsorption modeling of fatty esters and oleic estolide esters via boundary lubrication coefficient of friction measurements

Wear 262 (2007) 536–544

Non-linear adsorption modeling of fatty esters and oleic estolide esters via boundary lubrication coefficient of friction measurements夽 Todd L. Kurth a,∗ , Jeffrey A. Byars a , Steven C. Cermak b , Brajendra K. Sharma c , Girma Biresaw a a

Cereal Products and Food Science Research, National Center for Agricultural Utilization Research, Agricultural Research Service, United States Department of Agriculture, 1815 N. University Street, Peoria, IL 61604, USA b New Crops and Processing Technology, National Center for Agricultural Utilization Research, Agricultural Research Service, United States Department of Agriculture, 1815 N. University Street, Peoria, IL 61604, USA c Department of Chemical Engineering, Pennsylvania State University, University Park, PA 16802, USA Received 23 March 2005; received in revised form 16 June 2006; accepted 28 June 2006 Available online 15 September 2006

Abstract The frictional behaviors of a variety of fatty esters (methyl oleate (MO), methyl palmitate (MP), methyl laurate (ML), and 2-ethylhexyl oleate (EHO)) and oleic estolide esters (methyl oleic estolide ester (ME) and 2-ethylhexyl oleic estolide ester (EHE)) as additives in hexadecane have been examined in a boundary lubrication test regime using steel contacts. Critical additive concentrations were defined and used to perform novel and simple Langmuir analyses that provide an order of adsorption energies: EHE ≥ ME > EHO > MP > MO ≥ ML. Application of Langmuir, Temkin, and Frumkin–Fowler–Guggenheim (FFG) adsorption models via non-linear fitting demonstrates the necessary inclusion of cooperative effects in the applied model. Fits of the steady-state coefficient of friction (COF)-concentration data for EHE, ME, and EHO indicate slight cooperative adsorption. MO, MP, and ML data require larger attractive interaction terms (α ≤ −2.3) to be adequately fit. Primary adsorption energies calculated via a general adsorption model are necessarily decreased while total adsorption energies correlate well with values obtained via critical concentration analyses. To account for multiple surface-site coverage a multiple-site model was defined. The intuitive assumption of multiple-site coverage of more massive components suggests deceptively increased calculated adsorption energies for typically applied models (e.g. FFG, Langmuir). Published by Elsevier B.V. Keywords: Adsorption; Estolide; FFG; Friction; Langmuir; Temkin

1. Introduction As part of our effort to develop new bio-based lubricants, members of our lab have sought to characterize a wide range of bio-based materials [1,2]. Biresaw et al. previously examined a series of natural oils and methyl ester control systems as lubricant additives in a boundary lubrication test regime (Ballon-Disk). Critical to their performance in hexadecane is their ability to adsorb to a steel surface. To characterize such sys-

夽 Names are necessary to report factually on available data; however, the USDA neither guarantees nor warrants the standard of the product, and the use of the name by the USDA implies no approval of the product to the exclusion of others that may also be suitable. ∗ Corresponding author at: Degussa Goldschmidt Corporation, 900 South Palm Street, P.O. Box 1018, Janesville, WI 53547-1018, USA. Tel.: +1 608 314 3058; fax: +1 608 752 3186. E-mail address: [email protected] (T.L. Kurth).

0043-1648/$ – see front matter. Published by Elsevier B.V. doi:10.1016/j.wear.2006.06.020

tems, Jahanmir and Beltzer previously noted the ability to obtain coefficient of friction (COF)-derived adsorption isotherms [3]. Subsequently, adsorption energies may be obtained via fits to a variety of adsorption models (e.g. Langmuir, Temkin, FFG). In our previous work we noted the non-Langmuirian behavior of methyl fatty ester control systems (methyl stearate, methyl oleate (MO), methyl palmitate (MP), methyl laurate (ML)) and applied non-linear fitting and a general adsorption model [4]. In this paper we extend that work to include more complex estolide esters (methyl oleic estolide ester (ME), 2-ethylhexyl oleic estolide ester (EHE)) and an additional ester control system (2-ethylhexyl oleate (EHO)). Estolide esters display good pour points, viscosities, and oxidative stability and are thus suitable choices for bio-based lubricants [5,6]. Additionally, these oligomeric systems provide more complex surface interactions for which the theoretical models may be tested and developed. For each of the additives discussed here a cooperative adsorption model is required to adequately fit the data. The analyses of

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the oligomeric systems indicate the necessary consideration of multiple-site coverage and/or adsorption due to mass or chemical functionality of each additive. Although commonly applied to data obtained for twocomponent solutions exposed to complex surfaces, the Langmuir adsorption isotherm was derived for the adsorption of ideal gases to a uniform surface [7]. This may be understood in the context of a common first-order equilibrium reaction: ka

A + B C

(1)

kb

A is the concentration of unoccupied surface sites (adsorbent), B the concentration of adsorpt (un-bound additive) and C is the concentration of adsorbate (bound additive/occupied sites). The equilibrium constant K (ka /kb ) may be defined by both the equilibrium equation and the Arrhenius equation, respectively: C = e−Eads /RT AB

(2)

Fig. 1. Example of general adsorption isotherms, arbitrarily E = −2 kcal/mol. α is varied from −3 to 3 to simulate the dependence of the model on the cooperative interaction parameter.

Eads is the adsorption energy, R the molar gas constant, and T is the temperature. Adsorbate surface coverage, θ, is defined by the ratio of occupied sites to total number of sites: C/A0 , where A0 is the initial (total) concentration of sites. Noting that the equilibrium concentration of sites is A0 − C, the Langmuir equation may be solved for θ:

(α < 0). As previously noted by Brunauer et al. [10], the Langmuir and the non-linearized Temkin and FFG models are just special cases of the same model if the cooperative interaction term is allowed to be zero, positive, or negative, respectively:

K=

KB θ= 1 + KB

(3)

Rowe and Jahanmir and Beltzer utilized this model to define the relationship between wear and additive concentration and between COF and additive concentration, respectively [3,8]. Jahanmir and Beltzer utilized the linear relationship of surface coverage and COF to allow the direct determination of θ from friction measurements: f = fb (1 − θ) + fa ,

θ=

f − fb fa − f b

(4)

f is the measured COF and fa and fb are the coefficients of friction of the additive and base lubricants, respectively. These θ and the Langmuir model allow the simple determination of K and thus Eads via linear plots of Eq. (5): 1 1 1 = +1 θ KB

(5)

Jahanmir and Beltzer also noted the importance of additive lateral (cooperative) interactions effects on adsorption unaccounted for in the Langmuir model. To incorporate such effects those authors utilized a linearized Temkin model, which assumes by its application only repulsive lateral interactions, α > 0, and a primary adsorption energy, E [3,9,10]: Eads = E + αθ

(6)

A model treating attractive interactions, termed Frumkin– Fowler–Guggenheim (FFG), has also been defined and applied to adsorption data [11]. It has the same fundamental form as the Temkin equation but assumes attractive adsorbate interactions

K = e(−E−αθ)/RT

(7)

This model that does not assume a sign or magnitude of the lateral interaction term, α, may be considered a general adsorption model. In Fig. 1, isotherms of this general model demonstrating the effect of α are shown. Unlike the seminal Langmuir model, the general model may not be rearranged to a linear form (e.g. Eq. (5)) without simplifying assumptions [10]. The necessary linearizing assumptions constrain the fitting procedure to independent regions of surface coverage. i.e. θ ≤ 0.2, 0.2 ≤ θ ≤ 0.8 and 1 ≥ θ ≥ 0.8 are fit by different equations that are not equivalent to the general adsorption model. Due to its simplicity, the “mid-coverage” equation is utilized almost exclusively for linearized Temkin analyses [10]: θ=

RT E ln B − α α

(8)

The application of this linearized Temkin model has been demonstrated by Jahanmir and Beltzer and Biresaw et al. [1–3]. The slopes and y-intercepts of θ versus ln B plots provided the values of α and E and thus the total adsorption energies via Eq. (6) [1–3]. Unfortunately, the focus of previously published studies was the attainment of Eads and the lateral interaction terms were not presented nor discussed [1–3]. The multi-variable fits using the linearized Temkin model will certainly provide improved correlation relative to analogous Langmuir fits. Thus superior correlations of linearized Temkin fits are not necessarily indicative of the model’s success. Further, large amplitude changes that occur over relatively small concentration ranges are not readily observable via such analyses. The use of now ubiquitous iterative non-linear fitting computational software allows an absolute examination of the adsorption isotherms independent

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of previously applied linearizing assumptions and their resulting limitations. Our initial work demonstrated the need for non-linear methods of analysis and the required inclusion of cooperative interactions to adequately model COF-derived adsorption data for the simple ester control systems [4]. The work presented here extends the application of the general adsorption model and also demonstrates the need for consideration of multiple-site coverage/adsorption when treating systems of divergent molecular mass and/or chemical functionality. 2. Experimental 2.1. Materials The lubricant additives were: methyl laurate, ML, Sigma (St. Louis, MO), methyl palmitate, MP, and methyl oleate, MO, from Aldrich Chemical Co. (Milwaukee, WI). These were obtained commercially at greater than 99% purity and used as supplied. MO was also synthesized for comparison. The base lubricant was HPLC grade hexadecane also from Aldrich. The lubricant formulations comprised 0.0007–0.6 M of each additive in hexadecane. Oleic acid (90%) and boron trifluoride diethyl etherate were purchased from Aldrich. Perchloric acid (70%), methanol, and 2-ethylhexyl alcohol were purchased from Fisher Scientific Co. (Fairlawn, NJ). Potassium hydroxide was obtained from J. T. Baker Chemical Co. (Phillipsburg, NJ). Sodium hydrogenphosphate was obtained from EM Science (Gibbstown, NJ). Ethanol was purchased from AAPER Alcohol and Chemical Company (Shelbyville, KY). Filter paper was obtained from Whatman (Clifton, NJ).

molecular weight of 830.1 g/mol (estolide number 1.54) while the methyl oleic estolide ester (ME) had an average molecular weight of 1015.6 g/mol (estolide number 2.55). 2.3. General ester synthesis Oleic acid (100.0 g, 354.0 mmol) and either a 0.5 M BF3 /methanol or 0.5 M BF3 /2-ethylhexyl alcohol solution (5.0 equiv., 1.77 mol) were combined together followed by magnetic stirring at 60 ◦ C. Esterification reactions were complete >99% in 3–4 h as monitored by HPLC. The completed reactions were transferred to a separatory funnel and were washed with saturated NaCl (2× 50 mL). The pH of the organic layer was adjusted to 5.3–6.0 with the aid of pH 5 buffer (NaH2 PO4 , 519 g in 4 L H2 O, 2× 50 mL). The esters were dried over sodium sulfate and filtered through a Buchner funnel with Whatman #1 filter paper. All reactions were concentrated in vacuo, then K¨ugelrohr-distilled at 100–120 ◦ C at 0.1–0.5 mmHg to remove any excess alcohol. Methyl oleate (MO) and 2-ethylhexyl oleate (EHO) esters were produced in greater than 93% yield with acid values 0.87 mg KOH/g or lower. 2.4. Test specimen Balls and disks were obtained commercially (Falex Corporation, Sugar Grove, IL). The specifications of the balls were: 52100 steel; 12.7 mm (0.5 in.) diameter; 64–66 Rc hardness; extreme polish. The specifications of the disks were: 1018 steel; 25.4 mm (1 in.) OD; 15–25 Rc hardness; 0.36–0.46 ␮m (14–18 ␮in.) roughness. Prior to use, the balls and disks were thoroughly decontaminated by consecutive sonications in fresh HPLC grade methylene chloride (Aldrich) and HPLC grade hexanes (Aldrich).

2.2. General estolide ester synthesis 2.5. Friction test method An acid-catalyzed condensation reaction was conducted without solvent in a 500 mL baffled and jacketed reactor with a three-neck reaction kettle cover that had been pre-treated with an acidic wash. The reaction solution was mixed with an overhead stir motor using a glass shaft and a Teflon blade. Oleic acid (100.0 g, 354.0 mmol) was heated to 60 ◦ C under house vacuum. Once the desired temperature of 60 ± 0.1 ◦ C was reached, perchloric acid (0.05 equiv., 26.5 mmol, 2.3 mL) was added and the flask was placed under vacuum and stirred. After 24 h, corresponding alcohol (0.86 equiv., 457.6 mmol) was added to the vessel, vacuum was restored, and the mixture was stirred for an additional 3 h. The completed reactions were quenched by the addition of KOH (22.3 mmol, 1.25 g, 1.2 equiv. based on HClO4 ) in 90% ethanol/water (10 mL) solution. The solution was allowed to cool with stirring for 30 min. The material was filtered through a Buchner funnel with Whatman #1 filter paper. The organic layer was dried over sodium sulfate and filtered. The reaction was concentrated in vacuo, then K¨ugelrohrdistilled at 100–110 ◦ C at 0.1–0.5 mmHg to remove any excess alcohol followed by K¨ugelrohr-distillation at 180–190 ◦ C at 0.1–0.5 mmHg to remove any remaining un-reacted fatty esters. The 2-ethylhexyl oleic estolide ester (EHE) had an average

Friction was measured under point contact conditions using a Ball-on-Disk configuration via a Falex Friction & Wear Test Machine, Model Multi-Specimen (Falex Corporation, Sugar Grove, IL). In the Ball-on-Disk configuration, a ball, in contact with a stationary cylindrical disk, moves in contact around the disk at a specified speed. The resistance to the motion of the ball, i.e. the frictional force (torque), is measured by a load cell connected to the disk. The COF is obtained by dividing the friction force by the normal force pressing the ball against the disk. The ball was fixed on the upper specimen holder with a point contact radius of 11.9 mm (0.468 in.). The disk was fixed on the lower specimen holder that is enclosed in a lubricant reservoir. The reservoir was filled with 50 mL of lubricant solution to completely submerse the ball and disk assembly. The disk assembly was then raised and made to contact the ball. The upper specimen was allowed to rotate at 5 rpm (6.22 mm/s). As 5 rpm was reached, load was applied at 50 lb/s to reach 400 lb (1.78 kN). The temperature of the specimen and lubricant throughout each test was 25 ± 2 ◦ C. The friction measurements were carried out for 30 min. Repeat trials of additive samples deviated less than 5% from the initial measurement. The COF of pure hexadecane

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was measured for each additive preparation studied. The average value obtained over 15 tests was 0.44 ± 0.01. Repeat runs of used samples produced identical data as fresh samples indicating a lack of chemical reaction. 2.6. Scar width measurements After each test, scar widths were measured using a Leica StereoZoom 6 (Leica Microsystems, Bannokburn, IL) with digital micrometer attached to the specimen holder (Mitutoyo, Model 164-162, Japan). Four points along the track were measured to obtain an average value. 2.7. Model fitting procedure Eqs. (3) and (7) may be combined and solved for additive concentration (B) in terms of surface coverage (θ), adsorption energy (E), and cooperative interaction energy (α) [10]: B=

θ (1 − θ) e(−E−αθ)/RT

(9)

This equation is equivalent to the Langmuir, Temkin, or FFG model depending on the necessary α (see Fig. 1). Application of this general model without constraint allows the data to determine the necessary sign and magnitude of the lateral interaction term. θ versus concentration data were fit via the optimization of E and α using an iterative BFGS quasi-Newton or Nedler-Mead simplex optimization procedure and represent best fits using Matlab (version 7, Mathworks, Natick, MA). Linear fits (Eqs. (5) and (8)) and repeat non-linear fits of Eq. (9) (constrained and unconstrained α) were carried out using the iterative least squares fitting as implemented in Origin (version 7.5, OriginLab, Northampton, MA). Inverse concentrations were used to provide similar weighting of each data point.

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in wear rate would correspond directly with the COF-time variation during the wear-in region. However, it has been shown that the equivalent scar measurements and equivalent time frames require that the average wear rates are equivalent for all the samples, despite the local differences in the wear-in regions. The equivalent average wear rates allow our analysis and therefore, wear does not affect the analysis of our results and the applicability of our model. 3.2. COF-time spectra COF versus time spectra were generated for each of the methyl esters studied via Ball-on-Disk, e.g. Figs. 2 and 3. The MO data are representative of those for MP and ML. Shown in Figs. 2 and 3 are spectra for MO and ME solutions, respectively, of varying concentration in hexadecane. It is clear that within each spectrum two regions may be defined: a transient dynamic region and a steady-state region. The dynamic region is characterized by significant autocorrelation in COF. The COFvariability is attributed to the high initial contact pressure (high load, small contact area). At high pressures the wear and deformation of the softer 1018 steel disk material occurs. As the test progresses, the contact surface area of the ball and disk increases to a point at which the pressure is no longer sufficient to cause macroscopic wear and deformation of the test specimen. In the steady-state region, however, nascent surface is maintained with each cycle. Given adequate time (15 min), debris

3. Results 3.1. Effect of additive concentration on wear scar The scars on the ball and disk specimen were examined after each 30 min test. The harder ball specimen showed minimal wear and were not examined extensively. The disks showed a scar width of 3.05 ± 0.03 mm, for all disks, independent of the additive chemistry or concentration in hexadecane. Apparently the disk scar widths are a function of the metallurgy of the specimen and test conditions. The consistency of the disk wear scar indicates equivalent test parameters (e.g. pressure, contact surface area) in the steady-state region. Analogous to ASTM method 5183 for a 4-ball COF test [12], these similar contact areas for each COF test are required for quantitative COF comparisons. As evidenced by the COF data, the wearing in process is significantly different for different concentrations and lubricant additives and it would be useful to have the in situ measured wear with respect to test time and lubricant. We attempted this previously in our lab and were unable to obtain useful and consistent real-time wear data with our equipment. The variations

Fig. 2. Methyl oleate (MO) COF-time spectra, shown, are representative of methyl palmitate (MP) and methyl laurate (ML): (a) 0.0063 M; (b) 0.025 M; (b) 0.1 M; (d) 0.5 M in hexadecane.

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Fig. 3. Methyl oleic estolide ester (ME) (in hexadecane) COF-time spectra, shown, are representative of 2-ethylhexyl oleic estolide ester (EHE) and 2ethylhexyl oleate (EHO).

is removed from the wear track and the steady-state condition is obtained. 3.3. COF-concentration dependence The COF-time spectrum for 0.00625 M MO in hexadecane, Fig. 2a, is characteristic of the lower concentrations and pure hexadecane measured. Fig. 2c and d (0.1 and 0.5 M, respectively) are characteristic of high concentration data. The spectral responses at low and high concentrations are subtle and continuous [1,2]. In each, the wearing-in process occurs, leading to steady-state COF that diminishes with increasing additive concentration. Fig. 2b (0.025 M) is a COF-time spectrum characteristic of a “critical” concentration in which the transient wearing-in process is clearly observed with the return to low steady-state COF values. Stick-slip behavior in the steady-state is also observed for critical concentrations. Similar behavior and the presence of such critical points of lubricant failure, at low surface coverages, have been noted previously [8,13–16]. Characteristic critical concentration features were not observed for EHE, ME, and EH. In each of the COF-time spectra shown in Figs. 2 and 3, the average COF values of the steady-state regions (15–30 min) are indicated by horizontal solid black lines. The concentration dependent COF behaviors are observable via plots of these average steady-state COF values versus additive concentration (Fig. 4). One may observe the correlation of the critical region COF-time spectra, e.g. Fig. 2b, with sharp transitions in the

Fig. 4. COF-concentration profiles for 2-ethylhexyl oleic estolide ester (EHE), methyl oleic estolide ester (ME), 2-ethylhexyl oleate (EHO), methyl palmitate (MP), methyl oleate (MO), and methyl laurate (ML). Dashed lines indicate critical concentrations, Bc .

COF-concentration plots for MO, MP, and to a lesser extent ML (Fig. 4d–f, respectively). Fig. 4a–c shows the average steady-state COF versus concentration data for EHE, ME, and EHO, respectively. These data show that the transition from high to low COF values is not relatively sharp. The presence of these gradual transitions (typical of Langmuir behavior) clearly correlates with the COF-time spectral characteristics. The transition (critical) concentrations exhibit wear-in with no return to low COF values and no stickslip behavior. 4. Discussion 4.1. Langmuir critical concentration analyses Critical concentrations, Bc , may be defined as concentrations of additive sufficient to lower the COF to half its original value (e.g. the inflection point of a sigmoidal fit). These are depicted in Fig. 4 by vertical dashed lines, from which one may quantify the trend in critical concentration: EHE < ME < EHO < MP < MO ≤ ML (Table 1). Frewing and others reported that such critical regions are defined by the surface coverage (adsorbate concentration) at which sufficient asperity separation occurs [8,13–16]. In previous work the Langmuir model was used to define critical temperatures (at high concentration) for additive lubricant failure in the boundary

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Table 1 Friction derived adsorption data Additive

EHE ME EHO MP MO ML

Ec a,b

Bc (M)

0.0013 0.0016 0.0053 0.009 0.02 0.023

−3.9 −3.8 −3.1 −2.8 −2.3 −2.2

Langmuir fitsc

“Mid-range” linearized Temkin fits

General model fitsd

Eads

α

E

Eads a

α

E

Eads a

−3.9 −3.8 −3.1 −0.72 −1.2 −0.60

1.8 1.6 3.6 0.20 1.5 0.45

−4.1 −4.6 −5.3 −2.9 −1.5 −2.4

−3.2 −3.8 −3.5 −2.8 −0.75 −2.2

−1.2 −0.90 −0.86 −3 −3 −2.3

−3.4 −3.4 −2.7 −1.3e −1.0e −1.1

−4.0 −3.9 −3.1 −2.8 −2.5 −2.2

Adsorption parameters are in kcal/mol at 298 K. a θ = 0.5. b Eq. (10). c Eq. (5). d Eq. (9). e Maximal values.

lubrication regime [8,13–16]. This is directly analogous to our work in that temperature and/or concentration may be varied to affect the steady-state adsorption equilibrium that determines surface coverage. We define an equation relating critical concentration (at room temperature) to adsorption energy Ec (Eads at Bc ): RT ln

θc = −Ec (1 − θc )Bc

are inherently incapable of treating complex behavior. However, critical concentration calculated adsorption energies correlate well with the linear Langmuir values when sufficiently Langmuirian isotherms are observed, Table 1.

(10)

Using θ c = 0.5, T = 298 K, and R = 1.987 × 10−3 kcal/mol, Ec is obtained from our COF isotherms: EHE −3.9 kcal/mol, ME −3.8 kcal/mol, EHO −3.1 kcal/mol, MP −2.8 kcal/mol, MO −2.3 kcal/mol, and ML −2.2 kcal/mol (Table 1). Necessarily, these results correlate with those obtained via non-linear Langmuir fits of the data. However, the simplicity of this calculation is desirable. 4.2. Langmuir and linearized Temkin fits The steady-state COF-concentration plots in Fig. 4, display clear differences in transition verticality (magnitude of transition slope). EHE, ME, and EHO qualitatively appear to exhibit Langmuirian behavior, while MP, ML, and MO indicate varying degrees of transition verticality typical of cooperative adsorption isotherms [11]. Langmuir (Eq. (5)) and linearized Temkin (Eq. (8)) fits for each compound studied were performed. Superior correlations were necessarily obtained for each fit with the multi-variable linearized Temkin model. The ME analyses are representative, Fig. 5. A subtle but clear deviation from linearity is observable for each Langmuir fit and the Temkin fits rely solely on a few data points in the critical region. The linearized Temkin fits do not yield a set of consistent adsorption parameters for the similarly structured esters studied here (Table 1). Additionally, repulsive interactions are required by the application of the linear fitting of the Temkin model irrespective of reality. Non-linear representations of the linearly derived adsorption parameters (utilizing Eq. (9)) demonstrate further the inadequacy of the linear fitting via the Langmuir and linearized Temkin models (e.g. Fig. 5). Not only are the linear fits highly dependent on the variation of a fraction of the total data points but also they

Fig. 5. Adsorption model fits for methyl oleic estolide ester (ME). (a) Non-linear general model fit (solid line) and non-linear representations of fits obtained via the linearized Temkin model (dotted line) and the Langmuir model (dashed line) that are shown in 5b and 5c, respectively. (b) Linearized Temkin fit. (c) Linear Langmuir fit.

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a more negative total adsorption energy (Table 1). The critical concentration calculated adsorption energies values agree remarkably well with each of the general model total adsorption energies except MO. A lower presumed Eads for MO of −2.3 kcal/mol would require a decrease in primary adsorption energy; this implies a relatively lower propensity for adsorption of MO than that found via complex iterative least squares fitting of the vertical transition. It is readily apparent that the nonlinear analyses provide consistent adsorption parameters (α, E, and Eads ) that correlate well with the qualitative features of the raw data. The consideration of attractive cooperative interaction terms provides increased resolution allowing the differentiation of the ester systems. The observed differences may be attributed directly to the divergent geometries and intermolecular interactions of the fatty alkyl chains. 4.4. Cooperative effects

Fig. 6. General cooperative model fits for (a) 2-ethylhexyl oleic estolide ester (EHE); (b) 2-ethylhexyl oleate (EHO); (c) methyl palmitate (MP); (d) methyl oleate (MO); (e) methyl laurate (ML), utilizing α and E values from Table 1.

4.3. Non-linear fitting of adsorption isotherms The calculation of Ec for all additives studied and the nonlinear fitting of the adsorption isotherms for each of the estolide esters studied is straightforward. Non-linear fitting of the essentially vertical transitions found for MP, MO, and ML is more difficult. Due to the backward bending function for α ≤ −2.5 (see Fig. 1), θ values are nearly concentration independent other than at the critical concentration. The ML data allow the simplest non-linear analysis in that α = −2.3 provides a best fit solution of sufficient transition verticality (Fig. 6e, Table 1). The more vertical transitions of the MO and MP data sets require α ≤ −2.5 to be adequately fit (Fig. 6c and d). The transition verticality is so severe that data points having intermediate θ values were not obtained. α = −3 provides maximal E values of −1.0 and −1.3 kcal/mol, for MO and MP, respectively. Fortuitously, the similar adsorption parameters of MO and ML allow the direct observation of the inter-relationship of the calculated primary adsorption and cooperative interaction energies. In Fig. 6 it is clear that MO and ML have essentially the same critical concentrations (Table 1). However, MO exhibits relatively low and high surface coverage, θ, before and after the critical concentration, respectively. The non-linear general model correctly accounts for these subtle variations and assigns MO a lower propensity for adsorption and concomitantly

Several authors have proposed that cooperative adsorption isotherms are indicative of the reorganization of previously adsorbed molecules/layers [17,18]. With the nascent steel surface it is likely that isolated adsorpts (MO, MP, ML) have low energy conformations parallel to the metal surface. As subsequent molecules adsorb to the surface the less energetically favorable parallel interactions are displaced. While the low concentration equilibrium is determined by solvated and adsorbed parallel systems, the higher concentration equilibrium conditions are determined by solvated and perpendicularly aligned adsorpts. For this reason a backward bending function as in the FFG model is expected (e.g. α < −3, Fig. 1). An alternate description of the non-physical vertical region of the FFG model is a two-phase equilibrium: “Clusters of adsorbed molecules coexist on the surface with single adsorbed molecules, similar to a van der Waals equation of state” [11]. In Table 1 one may observe that the α terms utilized for the fitting of MP, ML, and MO data indicate that lateral interaction energies dominate the calculated total adsorption energies. For clusters of interacting adsorpts (ν > 2, α = νQ) the molecular lateral interaction energies, Q, may be relatively small compared to the adsorption energies E or Eads . The diminished lateral interaction terms of the EHE, ME, and EHO are indicative of the more complex surface interactions of these systems, not necessarily that cooperative interactions are not absent and insignificant. It is likely that these systems intermolecularly interact and exist in a continuum of geometries relative to the surface. 4.5. Multiple-site coverage Irrespective of the model or method of fitting chosen, the estolide esters exhibit improved lubricity at relatively low concentrations (more negative adsorption energies, e.g. Fig. 5). It is not certain that this behavior may be attributed solely to increased propensity for the estolide systems to adsorb to the steel surface via a single adsorptive interaction. It is likely that the more massive and multiple ester oligomers cover and/or adsorb to more that one site per adsorbate. Multi-site

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coverage/adsorption is not considered by the previously applied models; however, such multi-site behavior may be accounted for by an additional parameter n to the initial equilibrium equation (Eq. (1)): ka

nA + BC kb

(11)

n is the number of sites covered by one adsorbate. Considering only a single adsorptive interaction per adsorbate resulting in multiple-site coverage, the equilibrium equation may be simply defined: K0 =

C 1 = e−E0 /RT = K nAB n

(12)

K0 is the true equilibrium constant and E0 is the corresponding adsorption energy. This equation provides qualitative insight into the variation of Eads that may be accounted for by multiple-site coverage by a single adsorbate. For adsorbates that cover more than one site, inconsideration of n will result in deceptively more negative calculated adsorption energies. Adsorbates that cover and adsorb to multiple sites are expected to exhibit yet more negative adsorption energies than predicted via this simple model. Prior to our non-linear analyses, a clear dependence of the calculated adsorption energies, Ec , on the molecular weight was evident, Fig. 7. Solving Eq. (12) for Eads (Ec may also be used) demonstrates a theoretical basis for a similar dependence on n: Eads = −RT ln(n) + E0

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expectation that mass per covered site would increase with MW for simple surface coverage. A fit of the data to Eq. (13) is clearly insufficient but predicts reasonable E0 = −2.61 kcal/mol, Fig. 7. This suggests that while multiple-site coverage may account for some Ec dependence on MW it cannot account for it entirely. To obtain improved fits to the data an exponential term was introduced: Ec = −RT ln(nγ ) + E0

(14)

The exponential term, γ, is simply a correction factor that provides insight into the dependence of n on MW; without γ the slope of the Ec versus n plot (Fig. 7) is not variable. Obtained γ are entirely independent of the normalization; i.e. assumed coverage of ML has no effect on the obtained slope. A fit obtained via optimization of both parameters, γ = 1.81 and E0 = −2.27, indicates that the trend resulting in the large γ parameter is apparently not due to the dominance of any particular point, Fig. 7. This suggests that the mechanism leading to the MW dependence of Ec is present in each system studied, not just the oligomeric estolides. Increasing dependence of Ec on MW (γ > 1) correlates with multiple adsorptive interactions (not accounted for in our model) which would result in such exponential terms. Thus, the data seem to indicate that multiple-site coverage and multiple adsorptive interactions are required to account for the observed MW dependence. 5. Conclusion

(13)

Assuming that each ML adsorbate covers one site and has one adsorptive interaction (E0 = Eads ) and that the larger esters will cover proportionately more sites, the MW values may be normalized to the molecular weight of ML (i.e. MW/214.34 = n). This provides rational estimates of n for each additive despite the

A general adsorption model was applied to boundary COF derived adsorption isotherms for estolide esters and simple ester control systems in hexadecane solution with steel surfaces. It was found that non-linear methods of analysis yield more consistent and reasonable adsorption parameters than those obtained from previously applied linear methods. The non-linearly obtained adsorption parameters correlate well with those calculated via a simple critical concentration analyses. A multiple-site model was defined and considered. It is apparent that while multi-site coverage is a contributing factor to the relatively superior lubricity at low additive concentrations the increased propensity of the estolide esters to adsorb to a steel surface is likely due to multiple adsorptive interactions per adsorbate. Acknowledgements The authors gratefully acknowledge the contributions of Armand Loffredo and Megan Goers. We also thank Sevim Erhan and the Food and Industrial Oil Research Unit (NCAUR) for use of their laboratory and equipment. References

Fig. 7. Ec vs. MW of each additive studied, E0 = −2.61, γ = 1 (solid line) and fit of Eq. (14), E0 = −2.27, γ = 1.81 (dashed line).

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