A Theoretical Explanation of the Lowering of Frictional Forces with Layer Height of Long Chain Polar Lubricants BY M. J. SUTCLIFFE AND A. CAMERON Dept. of Mechanical Engineering, Imperial College, London, S. W.7. Received 23rd June, 1972 Assuming the anisotropic elastic constants of stearic acid as a function of layer height to be as determined by Akhmatov and Panova, the shear strengths of mono layers and multi layers are calculated. A computer programme describing the slip of a monolayer of an orthorhombic fatty acid crystal system about the methyl end groups is adapted to represent layer height by the use of a range of interaction potentials. Although the computed shear strengths are too high in magnitude compared with experimental results, as expected from strength calculations of an ideal crystal system, the correct trend of behaviour between multilayers and monolayers is shown.1. INTRODUCTION Various theories of one phase multimolecular film formation exist, physical and chemically based. Godrey considers a single chemisorbed layer as a basis, while Drauglis and Allen have put forward an ordered liquid model which, through a local short range force, produces long range order. Akhmatov cites long polar molecules having true elasticity of shape and places great importance on the elastic and mechanical properties of long chain hydrocarbon lubricants being decreasing func- tions of the layer thickness. has deter- mined elasticity constants of stearic acid from monolayers to 100 monolayers. In the literature, Denisov and Toporov,6 have shown multilayers to display a lower shear strength than monolayers.The purpose of this paper is to try and test some of the theories available in terms of a computer programme already in use to describe monolayers, assuming experi- mental constants available in the literature. As the programme describes shear between methyl planes of polar molecules in an orthorhombic crystal structure, the paper is most applicable to Akhmatov’s quasi solid theory and not to liquid ordered models, which require some order of randomness of the molecules. As in all inter- molecular theories which consider ideal and perfect systems, the actual magnitudes of derived forces are usually far too high compared with experimental results, but it is hoped to show a trend of behaviour between mono and multilayers.Using the “ layer stack ” method, Panova 2. FRICTIONAL MODEL The frictional model on which the theory is based is as follows. Firstly an orthorhombic crystal structure of long chain hydrocarbons is assumed, according to Muller and Smith.* When one surface is moved relative to the other, shear is assumed to take place between the terminal methyl groups. When multi- layers are considered, slip is assumed to take place at the mid-point between the two solid phases. Secondly, the assumption is made that all the energy needed for movement is dissipated and none is recovered ; clearly if some energy were recovered 26M. J . SUTCLIFFE AND A . CAMERON 27 Z AXIS t FIG. 1.-Arrangement of molecules in hydro- carbon crystal as determined by Miiller, where 0 represents methyl groups, 0 = methylene groups, a = 7.426 A, b = 4.956A (repeat dis- tance in z direction), d = 3.09 8, (gap distance), e = 0.94 A, (gap distance), k = 1.203 A, y setting angle, 8 = tetrahedral angle, - - - = top chains, - = bottom chains.I 2 . 1 1 ib $ INTERACTION ENERGY /CHAIN FIG. 2.-Compression diagram of interaction energy /chain against distance compressed ' GAP DISTANCE explaining the mode of energy dissipation. BETWEEN PLANES28 REDUCTION OF FRICTION the observed frictional force would be lower by that amount. Thirdly, during shear the terminal methyl groups are assumed to be capable of internal rotations. They are allowed to rotate about the C-C bond in the xy plane and to tilt in the z direction, which, when used simultaneously, gives a good model of pure axial rotation about the C-C bond.The potential barrier of the methyl group is taken as 3 kcal/mol for a 120" rotation about the C-C a ~ i s . ~ - l l For a deformation of 20°, if a sinusoidal potential curve is ass~med,~ the energy dissipated is of the order of 0.5 kcal/mol (the maximum potential is assumed to occur at 60"). The mode of energy dissipation used was first suggested in the discussion of Cameron's paper l2 by van Battun and Broeder, who pointed out that when the chains were moved under a load, the vertical distance between chains may change. The frictional work is then made up of two terms-see fig. 2. The load N is proportional to the gradient of the compression curve at El and gap distance d l , at point A .At some other point B the system is in the same state of compression, i.e., comparable gradients at energy E2 and gap distance d2. (1) The energy profile of the interaction forces The two terms then are AE = (El -EZ), (2) the work against the normal load N Ad where 2 Ad = (d2 - d l ) , as both planes move by opposite and equal reaction. The total energy dissipated is E = AE+N*Ad; the force is E AE = x=x+T- where X is the distance moved in the shearing action, i.e., the two compressed states at D(0) and D(X). Therefore the coefficient of friction p is F AE Ad p = - =-+- N N X 3. PROGRAMME A computer programme was developed which allowed the distances between the X-ray scattering centres of a long chain molecule at the origin of a given set of axes and all the other chains within a radius of 25 A to be determined under all conditions of movement (translation, rotation and deformation of the terminal methyl group).These were used to calculate the attractive potentials. There was a sub-routine to calculate all relevant H-H repulsive distances under all conditions of shear encountered. All distances were resolved along three orthogonal axes in terms of the tetrahedral angle 8 between the C-C chain bonds and CH2 groups, the setting angle y of the orthorhombic crystal system chosen. Provision was provided through the angle B for the chains to be oriented to the vertical and the terminal methyl group was allowed to rotate about the CC axis and to be deformed in the z axis. The periodicity of the interaction potentials of the source chain over the unit cell was shown and the setting angle of the system was calculated at 44" in excellent agreement with Smith's experimental results.*M. J . SUTCLIFFE AND A . CAMERON 29 4. P 0 TEN TI ALS AND COMPRESS I BI LI TY DIAGRAMS A Slater-Kirkwood attractive potential was used to describe the contribution from the C-C and C-H pairs to the dispersive van der Waal forces. The coeffi- cients used were 22.6 and 8.68 kcal/mol for CC and CH contributions respectively. The total interaction energy of the HH pairs were described by a Buckingham " 6-exp " potential of the form described by Coulson and Haig l 3 as U(R,) = - A/R: + B exp( - CR,) with R1 the distance in atomic units. The H-H potentials used are described in table 1. In addition, one Lennard- Jones potential (l/r12) was used with coefficient 1.35 x lo4 kcal/mol.Hirschfelder 2o has used an empirical curve fitting procedure to produce U(R,) = ART6+0.818 Rf exp( - 2R,). TABLE 1 Barton l4 3.397 13.207 2.434 Hendrickson 3.573 15.93 2.434 Hendrickson-Bartell l6 3.573 21.66 2.497 Bartell l7 3.573 10.51 2.160 Muller 4.496 100.9 2.645 Slater l9 3.573 16.8 2.43 t y p e A B C By assuming that a situation analogous to Hooke's law exists between the com- pressive deformation and external pressure, relationships connecting all parameters of energy in boundary layers can be derived, i.e., AA = AZ/Z,,= -PIEb (3) where AA = compressive strain ; AZ = change of distance (I- lo) ; lo = equilibrium distance ; Eb = elasticity modulus ; p = external pressure .'.(z-zo)/zo = -p/Eb. RSC HFE L DEH 8.0 - E - 2 6.0 a a .* .I 4.0 5 2 8 2.0 \ ;? a, a ._ u O O 4 4 6 .I - 2.0 E QU I L I BRI U M '0'51 TlON 3.09 EEN STROMS 0 FIG. 3.-Compression diagram for scattering centre interaction potentials.30 REDUCTION OF FRICTION Usingp = -a4/aZ, i.e., force is the negative gradient at any point on an (energy/cm2, distance) graph, it can be shown that where & is the interaction energy/cm2 at the equilibrium position lo, and by using eqn (3) it follows by introducing A , the area of cross section of the chain to give interaction energylchain, a, that 1 EbA (@--@o) = - -(Z-Z0)? 2 lo (5) Fig. 3 shows computed compression graphs using the potentials in table 1. For each potential, <Do is different; all the potentials except Hirschfelder’s display a potential minimum consistent with the crystal model.As lo and A are the same for each case, the only factor contributing to the different gradients will be the elasticity Eb. 5. ELASTICITY AS A FUNCTION OF LAYER HEIGHT Akhmatov considers that long chain polar molecules under bounda1.y lubrication conditions assume the properties of a “ quasi solid ”, or crystalline solid which display gradients of physical constants in the z direction. Hardy 21 attributes the anomalous compressive strain FIG. 4.-Compression diagram of normal stress against strain of stearic acid, as a function of layer height, from Panova’s results using layer stack method. nature to the influence of the field of the solid phase, and other authors to structural changes caused indirectly by the solid.Panova has managed to measure the elasticity of multilayers ranging from 100 layers to a monolayer using a “ layer stack method ”. The graphical relationship between these layers to each other is shown in fig. 4 for stearic acid. AkhmatovM. J . SUTCLIFFE AND A . CAMERON 31 cites these results and gives the absolute value of Young’s modulus of 100 molecular layers as 0.25 x lo6 kg/cm2 and of a monomolecular layer as 3.47 x lo6 k g / ~ m ~ . ~ Using these values and fig. 4, a graph of elasticity (log scale) against monolayers can be drawn as in fig. 5 ; from this graph the elasticity of any number of layers can be extrapolated. ELASTICITY IN X AND Y DIRECTIONS (105kg/cm2) OF ORTHORHOMBIC CELL. no. of monolayers FIG. 5.-Graph of elasticity (log scale) against number of monolayers from Panova’s results.It can also be seen that Akhmatov associates the value of the elasticity of the methyl gap in a monomolecular layer with the elasticity of the chain.” Cameron and Sutcliffe 22 showed that the gapcompressibilityshould be of the order of the values in the x and y direction of the unit cell; for the purposes of this paper Akhmatov’s assumption will be used for reasons given in the next section. 6. RELATIONSHIP BETWEEN LAYER HEIGHT AND POTENTIAL FUNCTIONS Using eqn (5), the theoretical compression curve formula, in conjunction with the information in fig. 5 ; the elasticity as a function of layer height and elasticity can be drawn, fig. 6, which shows a compression graph as a function of layer height and elasticity. By comparison with fig.3, the actual potential profiles obtained, it can be seen that a Barton potential function can be used to describe the effect across the methyl slip plane midway between two solid phases of five molecular layers of lubri- cant and Miiller and Lennard-Jones potentials used to represent three layers.32 REDUCTION OF FRICTION \@ No OF LAYERS 0 E = 981Nlm‘ ( lo6 kglcm’) EQUlLl BRIUM POSl T I ON 309 8 3 0 24 *‘ 6- E= 98.1 Nlrn’ \ .CI .U 4 = l . O kcal lmol - 2 L FIG. 6.-Compression graph of stearic acid as a function of layer height and elasticity. If the gap compressibilities were taken to be the same as the xy plane, the Slater and Bartell potentials describe a monolayer which implies that the softest potential available could only describe three layers at the most.Using Akhmatov’s assumption of the gap compressibility, Bartell’s potential describes three layers which meant that another suitable potential had to be developed to describe the slip plane character- istics of a monolayer. - 0 5 6 - Q) - 2 1 .5 FIG. 7.-Compression curves of fitted potential, representing a monolayer, as a function of ROT/ ROCK ratio and distance along the x-axis. This potential was developed using a polynomial fit employing a least squares method. A curve to the fourth power provided an adequate approximation of the exponential type 4(Rl) = Co+C,exp(-R,)+C2exp(-2Rl)+. . . +C,exp(-nR,) with the coefficients C, = 2.38 x lo4 ; C1 = -7.19 x lo5 ; C2 = 8.08 x lo6 ; C3 = -4.01 x 107 ; c4 = 7.39 x 107.This is shown in fig. 7 compressed at the equilibrium position D(O.0).M . J . SUTCLIFFE AND A . CAMERON 33 Thus every suitable potential can be used to represent the properties of the slip plane under compression, differing layer thicknesses represented by a different potential, i.e., Barton 5M Hendrickson-Bartell 5M Slater 3-5M Bartell 3M Muller 21 3M Lennard-Jones 3M Polyfit potential 1M where M = no. of monolayers 7. EFFECT OF LOAD AND METHOD OF CALCULATING RESULTS A consequence of having gradients of physical constants in the z direction is that a multilayer will shear about its weakest point, which will be the methyl plane midway between the two solid phases. This implies that when a load is applied to the multi- layer, the slip plane methyl gap compresses less than the others surrounding it.Again, an anisotropic law is taken to govern the gap compression as a function of height ; because the probable structural changes under compression are unknown, O I - \ D (2.4) D (2.4) U .- 4 2 0 9 2.0 I 2- - 2 / DISTANCE BETWEEN PLANES IN ANGSTROMS - FIG. 8.-Compression diagram for Barton's scattering centre potential (- - -) representing five layers and Miiller's scattering centre potential (--) representing three layers as a function of ROTIROCK ratio and distance along the x-axis. the model must cope with the anisotropy within its parameters-the mutual approach of CH3 groups. Akhmatov quotes an inverse law between layer thickness and yield strength implying shear across the slip plane, thus it seems reasonable to apply an inverse proportion law governing CH3-CH3 approach.All the results were cal- culated using these principles and any excessive decrease across the gaps is assumed to be taken up by the chain compressions. The basic model of shear across the unit cell was taken from the ideas expressed by van Battum and Broeder explained in the "friction model " (Section 2). 2-B34 REDUCTION OF FRICTION To do this, compression diagrams of all the potentials were needed as a function of internal rotation at the equilibrium point and at the maximum potential profile on the x-axis-which occurs at a point D = 2.4A from the origin. Fig. 7 and 8 show examples of these diagrams, viz the polyfit monomolecular potential ; Muller's 3M potential representation and Barton's 5M potential representation.Friction forces were calculated using the amount the gap between the planes lifts the load (the load is proportional to equal gradients on the two curves), the remaining interaction potential profile AE, and the energy dissipation required to rotate or deform the terminal groups. Programmes were run to obtain compression diagrams at all potential maximum profiles throughout the unit cell. It was found that values of AE were 60-80 % of AE, x-axis values-but Ad values were approximately constant. Pro- grammes were also run to give results for a 3 and 5 molecular layer at an angle of 20" to the vertical. The results were either interpreted as " kinetic friction "-taken as the statistical average of energy dissipated/cm of the shear path-(which was found to be the only way x-axis results could be combined with other orientations) and " static friction " of the x-axis which is taken as the maximum frictional force of the shear path, i.e., the force to initiate motion.8. RESULTS TABLE Z-RIGID MOLECULES-STATIC-X-AXIS potential layers FO PST Barton 5M 2.083 0.5 Hendrickson 5M 1.596 0.5 Slater 3-5M 0.965 0.49 Bartell 3M 1.74 0.4 Muller 3M 1.74 0.47 Lennard-Jones 3M 0.675 0.48 Where M represents no. of layers, Fo = intercept on specific force axis at zero normal stress kg/cm2 x lo3. N.B. 1 . 0 ~ lo3 kg/cm2 N 0.981 N/m2 (the graphs are presented in both units). Rotation = ROT/ROCK; ST = " static " coefficient of friction-taken as the slope on the (specific force, stress) graphs ; p~ = " kinetic coefficient " of friction-taken as the slope on the (specific force, stress) graphs.TABLE 3 .-ROTATION ~/O-KINETIC-X-AXIS Barton 5M 0.393 0.094 Hendrickson 5M 0.393 0.094 Slater 3-5M 0.180 0.10 Bartell 3M 0.326 0.08 MuIIer 3M 0.328 0.085 Lennard-Jones 3M 0.128 0.09 potential layers Fo PST TABLE 4 . 4 / 2 0 STATIC-X-aXIS potential Barton Hendrickson Slater Bartell Muller Lennard- Jones Polyfit layers 5M 5M 3-5M 3M 3M 3M 1M FO 2.22 2.22 2.22 2.02 2.7 2.02 2.41 PST 0.25 0.25 0.3 0.15 0.325 0.215 0.23M. J . SUTCLIFFE AND A . CAMERON 0) 2 2 o 2 0 3 O a 6 35 1.0 N / m 2 1 - EN€ RGY DISSIPATED KINETIC CASE -0.5 Nlm' TABLE 5.4120 KINETIC-X-AXIS 0.4 0.2 potential Barton Hendrickson Slater Bartell Muller Len nard-Jones Polyfit - MONOLAYER TILTED AT Zoo I ia - 5.0NIm2 TO THE VERTICAL 11.0 N l m 2 layers 5M 5M 3-5M 3M 3M 3M 1M FO 0.43 0.43 0.43 0.394 0.55 0.394 0.45 PK 0.047 0.047 0.055 0.03 0.04 0.04 0.044 Table 5 can be summarized as follows-by taking the average of the layer number and the average 70 % of Fo for a 360" average layers FO 70 % Fo 5M 0.43 0.305 3M 0.446 0.310 1M 0.45 0.3 15 See fig.9-the graphs have been extrapolated to enable the relationship to be seen Programmes were run to obtain compressive data on the x-axis with the chains more clearly. inclined at 20" to the vertical, compare table 6 and see fig. 10 for these data. 1 ENERGY DISSIPATED OVER 1.4 1 ON X-AXIS, STATIC CASE 5 .0N/m2 normal stress kg/cmZ x lo3 FIG. 9.-Graph of specific force against normal stress as a function of layer height, rotation 0/20, over the x-axis.TABLE 6 layers Fo PK 5M 0.058 3M 0.078 all in the range 1M 0.098 0.03-0.05REDUCTION OF FRICTION - FIG. 10.-Graph of specific force against normal stress as a function of layer height, rotation 0/20, and shift of chains to the vertical, over the x-axis. 9. DISCUSSION The coefficients of friction are of the correct order for fatty acid lubricated systems. The magnitude of the shear forces are generally too high by an order of magnitude, i.e., Godrey gives several values for the shear stress of stearic acid as a function of pressures ranging from 25 kg/cm2 at 250 kg/cm2 of pressure to 140 kg/cm2 at pres- sures of 1000 kg/cm2 and 440 kg/cm2 of pressure. Attempts have been made to calculate strength properties of solid materials from a knowledge of intermolecular forces, e.g.by De Boer 23 and Kemball 24 but in all cases the theoretical values are far greater than experimental values. The discrepancy is attributed to dislocations in the crystalline It is suggested that dislocations would affect the magnitude of the forces involved but not the coefficient of friction, as was found in the case of comparing x-axis values to other orientations of the crystal where AE was 60-80 % of x-axis values at zero load but Ad was constant. The correct relationship between layer height and shear forces has been shown ; dislocations should affect them all equally. If the layers are tilted 20" to the vertical the shear forces are less. D. Godrey, Symposium of Properties of Surfaces, A.S.T.M. Special Publication, 1962, NO. 340, 109. E.Drauglis and C. M. Allen, Wear, 1969,14, 363. A. S. Akhmatov, Molecular Physics of Boundary Friction (Israel Programme for Scientific Translation, 1966). Chapter 7. Panova, see ref. (3), p. 255-264. P. V. Denisov, Thesis (Moscow, 1954). Investigation of the Phenomena of Adhesion in Flat Steel Surfaces. Y . P. Toporov, Research in Surface Forces. Vol. 21 ; (Consultants Bureau, New York, 1966), p. 312. ' A. Muller, Proc. Roy. SOC. A, 1927, 114, 542. A. E. Smith, J. Chem. Phys., 1953, 21,2229. K. S . Pitzer, Disc. Faraday SOC., 1951, 10, oo00. lo J. Oosterhoff, Disc. Faraday Soc., 1951, 10, oo00. l 1 S. Mizushima, Structure of Molecules and Internal Rotations (Academic Press, New York, l 2 A. Cameron, Amer. SOC. Lubr. Engineers, 1960, 2, 195. l3 C. A. Coulson and Haig, Tetrahedron, 1963,19, 527. l4 D. H. R. Barton, J. Chem. SOC., 1948, 340. l6 J. B. Hendrickson and L. S. Bartell, J. Amer. Chem. Soc., 1961,83,4537. 1954. T. B. Hendrickson, J. Amer. Chem. Soc., 1961, 83,4537.M. J . SUTCLIFFE AND A . CAMERON 37 L. S. Bartell, J. Chem. Phys., 1960, 32, 831. l 8 A. Muller, Proc. Roy. SOC. A, 1936, 154, 624. l9 A. Muller, Proc. Roy. SOC. A, 1941, 178, 227. 2o J. 0. Hirschfelder, J. Chem. Phys., 1950, 00, 130. 21 W. B. Hardy, Proc. Roy. SOC. A, 1925,108,000. 22 M. J. SutclXe and A. Cameron, to be published. 23 J. H. de Boer, Trans. Faraday SOC., 1936, 22, 10. 24 C. Kemball, Adhesion and Adhesive-Fundunzentuls and Practice (Society of Chemical Industry, 2 5 C. A. Wert and R. M. Thomson, Physics of Solids (McGraw Hill, N.Y. 1969), p. 108-113. London), p. 69.