Adhesive Force due to a Thin Liquid Film between Two Smooth Surfaces (Wringing Mechanism of Gage Blocks)

Journal of JSEM, Vol.14, Special Issue (014) s36-s41 Copyright C 014 JSEM Adhesive Force due to a Thin Liquid Film between Two Smooth Surfaces (Wringing Mechanism of Gage Blocks) Kenji KATOH 1 and Tatsuro WAKIMOTO 1 1 Department of Mechanical Engineering, Osaka City University, Osaka 558-8585, Japan (Received 9 January 014; received in revised form 8 April 014; accepted 7 June 014) Abstract It is well known as the wringing phenomenon of gage blocks that a strong adhesive force appears between two smooth surfaces when a thin liquid film is applied on the interface. A theoretical and experimental study is conducted to discuss the wringing mechanism. The liquid film may be separated into a large number of small puddles when two surfaces are rubbed together. One can expect a large adhesive force due to the resultant of surface tension acting on the circumference of each puddle. Numerical results for oil distribution between two surfaces show that the total perimeter length of liquid film reaches 10500 m on the surface of 3cm, which leads to 1.0 10 6 (N/m ) for the adhesive force. The adhesive forces measured experimentally roughly agree with the theoretical value. A simple model is proposed to explain the adhesive force dependence on the velocity of pulling apart two surfaces. Key words Surface Tension, Gage Blocks, Wringing Phenomenon, Surface Roughness, Viscosity 1. Introduction It is well known that a strong adhesive force appears between two smooth solid surfaces when a thin liquid film is applied on the interface. This phenomenon is usually applied to the wringing of gage blocks used in precise measurement of linear dimension [1]. The roughness of gage blocks is typically order of 0.01μm and it is very important for gage blocks to be wrung without length uncertainty. In the actual system, the adhesive force reaches several times as large as atmospheric pressure when two surfaces are rubbed together to spread the oil uniformly as a thin film between them. Many authors have been trying to explain the wringing mechanism from the intermolecular force or the viscous force of the oil film [- 9]. Based on the order of roughness and of oil film thickness, the intermolecular force should not be related to the wringing mechanism. Mainly the influences of surface tension or viscous force have been considered. However, the wringing of gage blocks has not been treated since 1980 s, although the mechanism why such a large adhesive force is generated is still an open problem The authors had proposed a theoretical model for the wringing mechanism based on the surface tension acting on the oil film distributed on the contact surface [10]. In this report, the surface characteristics of gage blocks are observed by a STM (Scanning Tunneling Microscope). Then the state of solid contact and the distribution of oil film are numerically simulated between two surfaces having statistically equivalent roughness to the observed gage block surfaces. Using the simulated results, the adhesive force is estimated based on the model proposed in the preceding report. The adhesive forces are measured experimentally for several kinds of liquids with different viscosities, and the validity of the theoretical model is examined through the comparison with the measured results. In addition, the effect of viscous force on the wringing mechanism is discussed from a simple theoretical consideration.. Theoretical Model of Adhesive Force due to Surface Tension In this section, a brief explanation is given to describe the theoretical model for the adhesive force in the preceding report [10]. The surface area of actual gage block is about 3cm and its roughness and flatness are less than 0.1μm. When two gage blocks are adhered to each other, the resulted linear length should exactly be the sum of two gage blocks with the deviation less than the roughness stated above [, 3]. The number of contact points and the real contact area between two gage blocks should be much larger than those observed in contacts between usual surfaces. Figures 1(a) and 1(b) schematically show the states of contacts between usual rough surfaces and between gage blocks, respectively. The vacant spaces surrounded by metal contacts may be fulfilled with a liquid (a) Usual surfaces (b) Gage blocks Fig. 1 State of contact between two surfaces -s36-

Journal of JSEM, Vol.14, Special Issue (014) such as a thin oil film when gage blocks are wrung as shown in Fig. 1(b). When the upper surface is pulled apart from the lower surface, the gas-liquid surface tension γ appears and acts as a drag on the circumference of the oil film as shown in Fig. 1(a) or 1(b), and following adhesive force should be generated as the total drag. F L (1) where L indicates the total perimeter length around oil films. The drag F in Eq. (1) should be trivial for usual contact between rough surfaces, because the surface tension γ is at most the order of 0.03 (N/m) and the length L does not become so large. On the other hand, however, the oil films between two gage blocks may be separated into great number of small puddles bounded by many contact points, as shown in Fig. 1(b). Now let us imagine that the surface roughness is simply downsized with keeping geometric similarity. The number of oil puddles increases with minus second power of the roughness size, and the total length around the perimeters increases with minus first power. Therefore the length L in Eq. (1) may be quite large and the drag F cannot be neglected. In the preceding report, the length L is estimated from a simple consideration based on the characteristics of gage block surface observed by a SEM [10]. The result showed that the length L may be order of 5000m on the gage block surface of 3cm and the adhesive force estimated from Eq. (1) reaches F~5 10 5 (N/m ). This is about five times as large as the atmospheric pressure and is the same order as experienced in the practical system. 3. Distribution of Oil Films and Correct Estimation of Adhesive Force The estimations of L and F in the preceding report were not rigorous, because they were calculated based on a rough estimation for the statistics of roughness on the surface. In this report, in order to discuss the validity of the theoretical model of Eq. (1), first the surface of gage blocks is precisely observed by using a STM. Then the state of contact is numerically simulated between two surfaces with statistically equivalent surface characteristics to gage blocks. From the calculation of the total perimeter length L, the adhesive force in Eq. (1) is estimated precisely. 3.1 Characteristics of gage block surface Figure show examples of measured roughness curves on gage blocks obtained by STM. Figures (a) and (b) correspond to the roughness curves measured in longitudinal (x) and in transverse (y) direction on the rectangular surface, respectively. Although the statistical characteristics are almost similar to each other, a slight directional dependence can be observed. The standard deviation calculated from Fig. is about 5nm and it was confirmed that the probability distribution of roughness heights can be approximated by usual Gaussian profile. 3. Estimation of total perimeter length and adhesive force The roughness curves like Fig. were measured at 10 different positions on the gage block surface and the power spectrum was analyzed. Figures 3(a) and (b) show examples of measured power spectrum in x and y direction, (a) x-direction (a) x-direction 1 (b) y-direction Fig. Roughness curve on gage blocks surface (b) y-direction Fig. 3 Power spectrum of surface roughness on gage blocks -s37-

K. KATOH and T. WAKIMOTO (a) h = 0 Fig. 5 Ratio of contact area between two gage blocks surfaces Fig. 6 Total perimeter length of oil film on gage blocks (b) h = 5 nm Fig. 4 State of contact between two gage blocks respectively. The horizontal and longitudinal axes in the figures indicate the wave number and the power spectrum, respectively. Although the distributions of power spectrum are almost similar in both directions, the spectrum of x direction is slightly smaller than that of y direction in low wave number region, while vice versa in high wave number region. Some directional dependence may appear since the surface of gage blocks is finished by rapping. Giving random phase shift to each wave number component of averaged two-dimensional power spectrum, the gage block surface with the same statistical characteristics is simulated from the inverse Fourier transform. Then the state of contact is numerically simulated between two surfaces to detect the oil film distribution. Figure 4 shows an example of simulated contact state within surface area (1 1μm). The protrusions overlapped between two surfaces may be distorted elastically and the contacted areas are depicted by black parts as shown in Fig. 4. The white parts in the figures correspond to cavities in which oil may be fulfilled. As shown in Fig. 4(a), the oil film is distributed to many puddles with complicated configuration like fractals. When two gage blocks are pulled apart, gas-liquid interface appears and surface tension acts as the drag on the perimeter of each puddle of white parts. Figures 4(a) and (b) show the results for different average distance h between two surfaces. Figure 4(a) corresponds to h=0 when the averaged heights of two rough surfaces as shown in Fig. coincide with each other and the length at the wringing is strictly equal to the sum of two gage blocks. Figure 4(b) shows the result for h=5nm. As shown in the figures, the oil film distribution is dramatically changed even when the distance is slightly increased. Figure 5 shows the calculated result for the ratio of contact area to the total surface area, p (0 p 1). The ratio p is about 0.5 at h=0 and monotonously decreases with increasing h. As suggested in the past literature [8], the ratio of contact area is quite large as compared with (0.001~0.1) for the contact between usual surfaces. Figure 6 shows the perimeter length of oil films L varying with h. The length L can be calculated from the sum of the perimeters around the white parts as shown in Fig. 4. As expected, L becomes maximum for h=0 at which the contact area is about 0.5 as shown in Fig. 5. The perimeter length gradually decreases in proportion to -1 power of h and becomes one third as large as the maximum at h=10nm. As shown in Fig. 6, the perimeter length L reaches about 3.5 10 7 m/m at h=0, corresponding to L 10500m on gage blocks of 3cm area, which is comparable to the rough estimation in the preceding report [10]. The adhesive force of Eq. (1) is then estimated as F=10.5 10 5 (N/m ) (i.e., about 10 times as large as the atmospheric pressure) for -s38-

Journal of JSEM, Vol.14, Special Issue (014) usual oils having γ 0.03 (N/m). It is noted that all spaces of white area shown in Fig. 4 may not be necessarily fulfilled with oil and the oil film may be distributed heterogeneously over the gage block surface. Hence the adhesive force in the actual system should be somewhat smaller than that estimated above. 3.3 Effect of viscosity In the above discussion, we have considered the effect of surface tension on the wringing force when two gage blocks are pulled apart. On the other hand, Bowden and Tabor [5] proposed the adhesive force due to viscosity F µ (N/m ) expressed as the following relation. F 3 µ d 1 µ = 3 µ d V = () 8 t 8 3 H H where d and H indicate the equivalent diameter of contact surface and the oil film thickness, respectively. μ is the viscosity of oil. t and V in the above equation are the time necessary to separate the two surfaces and the velocity of pulling apart two gage blocks, respectively. The theoretical model of Bowden and Tabor assumed that the oil film of uniform thickness H is distributed on the circular surface with diameter d. When two gage blocks are pulled apart, the oil flows toward the center of the circular surface and the pressure gradient is generated in the oil film to counterbalance the viscous drag. The adhesive force represented by Eq. () can be estimated from the integration of negative pressure on the wall surface by using the approximate solution of wedge flow equation. Some authors expected a quite large adhesive force from Eq. () such as several times as great as the atmospheric pressure for d=cm circular surface equivalent to gage block surface (i.e., 3cm ), if the film thickness is assumed as H=0~50nm, referring to the measured results [, 3]. However, it is apparent that the adhesive force of Eq. () must not exceed the atmospheric pressure because the model is originally based on the negative pressure on the wall. In other words, it is not meaningful to imagine the oil flow larger than that generated by the maximum pressure difference (i.e., atmospheric pressure). If the oil is distributed homogeneously over the gage blocks surface as a very thin film, the pressure gradient could not drive the flow against large viscous drag. In the actual system, however, the oil film may be separated into many small puddles as shown in Fig. 4. In such a case, a large pressure gradient may be possible in each puddle to drive the thin film. Here the order of viscous drag is estimated in a simple manner when the oil flows in each small puddle. Assuming i as the number of oil puddles (thickness H) per unit area and (1 p) as the ratio of occupied area by all puddles, the equivalent diameter of a 4 ( 1 p) small puddle can be estimated as d~. The total πi viscous drag can be obtained from the sum of Eq. () over whole puddles as: 3 μd 1 3 μ( 1 p) 1 3 μ( 1 p ) V F µ ~ i = = (3) 8 t it 3 H π H πih Fig. 7 Number of oil puddles on gage block surface As is clear from the above equation, the viscous drag should be negligible when the number i becomes quite large. Figure 7 shows the calculated results of the number of oil puddles (i.e., number of white parts shown in Fig. 4). As shown in the figure, the number of oil puddles per unit area reaches about 8 10 13 when the average distance between two gage blocks surfaces is zero. Now let us estimate the viscous drag from Eq. (3). We assume here that the average oil film thickness is about H=0nm as stated before, p=0.5 from Fig. 5, and t=1s for the typical time to be separated. Then the viscous drag is estimated as F µ 1.3(N/m ) from Eq. (3) for the machine oil having viscosity μ =0.17 (Pa s). Even if one considers the separating velocity as V=10mm/min referring to the experimental results shown in section 4, the viscous drag calculated from the right hand side in Eq. (3) is F µ 1.1 10 4 (N/m ), which is still quite small compared with the atmospheric pressure or the drag of surface tension. Other than Eq. (3), we may consider the viscous stress produced from the vertical flow of velocity V, when two surfaces are pulled apart. The viscous force per unit area can be estimated simply from the viscous stress tensor as: V Fµ~ µ ( 1 p) (4) H Inserting each value stated above into Eq. (4), F µ ~1.4 10 3 (N/m ) is obtained. This is again quite small compared with the atmospheric pressure. From these results, the adhesive force due to viscosity should not exceed the atmospheric pressure and we can neglect the effect of viscosity on the wringing phenomenon. 4. Experimental Results and Discussion The wringing force between two gage blocks were actually measured by a usual tensile testing machine for three kinds of oils, in order to examine the validity of the theoretical model in the preceding section. The velocity of separating two gage blocks was widely changed from 0.05 to 1mm/min. Table 1 shows the physical properties for Table 1 Physical properties of test liquids (0 C) Surface tension (N/m) Viscosity (Pa s) Spindle oil 3.04 10-1.90 10 - Machine oil 3.4 10-1.68 10-1 Chainsaw oil 3. 10-3.03 10-1 -s39-

K. KATOH and T. WAKIMOTO test oils used in this experiment. As shown in the table, the surface tension is almost similar to each other, while the viscosity is varied more than 1000% among three kinds of oils. The measurement of adhesive force was repeated more than 0 times for each experimental condition and the average was calculated. Figure 8 shows the measured adhesive forces F (N/m ) for three kinds of oils at 0 C. The horizontal axis indicates the pulling velocity V. When the gage blocks are adhered, only a minute amount of oil is necessary. Since the adhesive force could be sensibly dependent on the oil film distribution, it may scatter greatly at each measurement. The standard deviation of measured adhesive force is 1.1 10 5 (N/m ) (16% of average adhesive force) in this experiment. As shown in Fig. 8, the adhesive forces are not so different between three kinds of oils, in spite of large variance of viscosity. The influence of viscosity on the adhesive force can be neglected as expected in the preceding section. Although some literatures suggested that the adhesive force varies dependent on the viscosity of test liquids, the wettability between oil and solid surface should be more important. The contact angles of three kinds of oils on the gage blocks surface were measured by the contact angle meter (Kyowa Kaimen Kagaku Co. Ltd., CA-A type). The receding contact angles are 0 for all oils and the advancing contact angles range from 15 to 5. The oils used in this experiment can wet well the surface of gage blocks and the oil film may be distributed uniformly on the surface. Hence the surface tension has the most important effect on the wringing mechanism of gage blocks, if usual oils are used as in this experiment. The solid curve drawn in Fig. 8 indicates the calculated result of Eq. (1) in which the surface tension averaged between three kinds of oils is used. The theoretical result is somewhat larger than the experimental results. This is because the calculated adhesive force is obtained from the total perimeter length of all oil puddles. As stated in section 3, all puddles are not necessarily filled with the oils and the total surface tension should be smaller than that expected in the theoretical curve shown in the figure. The experimental results shown in Fig. 8 increase with V and finally approach a critical value. This tendency suggests that the oil film distribution, i.e., the configuration of liquid surface as shown in Fig. 1, may change dependent on the velocity to separate two gage blocks. Figures 9 and 10 show some models of oil behavior in a puddle when Fig. 8 Measured adhesive force between gage blocks (a) Plain view (b) Vertical view Fig. 9 Oil behavior in space between gage block surfaces (low velocity of pulling apart) (a) Plain view (b) Vertical view Fig. 10 Oil behavior in space between gage blocks surfaces (high velocity of pulling apart) gage blocks are pulled apart. The plain and vertical views are drawn in each figure. Figure 9 corresponds to the case when the velocity V is not large. As shown in the figure, the oil can follow the expansion of the puddle by the retreat of solid contact as shown by arrows in the plain and vertical view. Then the liquid surface appears in the middle of the oil film as shown in the vertical view. On the other hand, when the velocity is quite large as shown in Fig. 10, the oil cannot follow the quick expansion of the puddle. In this case, the liquid surface appears close to the contact area as shown in the figure. If one compares the perimeter length between Fig. 9 and Fig. 10, the perimeter of Fig. 10 -s40-

Journal of JSEM, Vol.14, Special Issue (014) is apparently larger. The model of Fig. 10 corresponds to the theoretical model drawn by the solid line in Fig. 8. Now let us estimate the adhesive force for the oil film shown in Fig. 9. The profile of perimeter shown in the vertical view is assumed as a circle having the same area as the original oil puddle shown in the plain view. The average surface area was calculated for all the oil puddles as shown by white parts in Fig. 4 and the equivalent diameter was obtained. The perimeter length is calculated as πd of the circle (d: diameter). The adhesive force is calculated from the total perimeter length multiplied by surface tension and the result is drawn by the broken line in Fig. 8. As shown in the figure, the experimental results in this paper distribute between the solid line corresponding to the model of Fig. 10 and the broken line. As the velocity V becomes large, the oil in a small puddle still can follow the expansion, while the oil in a large puddle cannot follow the movement. Hence the number of puddles in which the oil behaves like Fig. 10 increases with V and the adhesive force may approach the solid line as shown in Fig. 8. The above model is rather simple and the oil would behave more complicatedly in the actual system. The actual behavior of oil between two surfaces should be observed experimentally to ensure the validity of the theoretical model proposed here. However, the proposed model can explain the mechanism how such large adhesive force appears and the actual behavior of adhesive force dependent on the velocity of pulling apart, which cannot be explained by other models discussed in the literatures. 5. Conclusion The total perimeter length of oil film L on the interface between two gage blocks was estimated from the numerical simulation for contact problem between two surfaces with equivalent statistical characteristics. When two gage blocks are adhered, the oil film is separated into large number of small puddles. The numerical simulation shows that the length may reach L=10500m on 3cm surface. The product of surface tension and L can explain the adhesive force experienced in the actual system. The viscous force was estimated from the fluid dynamic model for the oil behavior in each small puddle. The result suggested that the influence of viscosity on the wringing phenomenon can be negligible. Actually almost the same adhesive force is observed experimentally for three kinds of oils among which the viscosity varies more than 1000%. The measured adhesive force increases with the separating velocity and approaches a constant value. A rather simple theoretical model was proposed based on the behavior of oil film in the puddle. As the velocity increases, the gas-liquid interface is more likely to be generated near the solid contacts since the liquid in the puddle cannot follow the movement. In such a case, the perimeter configuration becomes more complicated and the adhesive increases due to large gas-liquid interface length. The model can explain the tendency observed experimentally. Nomenclature d diameter of oil puddle [m] F adhesive force per unit area [N/m ] H oil film thickness [m] h distance between gage blocks [m] i number of oil puddles L total perimeter length [m] p ratio of solid contact area t time necessary to pull apart gage blocks [s] V velocity of pulling apart gage blocks [mm/s] density [kg/m 3 ] γ surface tension [N/m] μ viscosity [Pa s] References [1] Titov, A., Malinovsky, I., Belaidi, H., Franca, R. S. and Massone, C. A.: Precise Interferometric Length and Phase-change Measurement of Gauge Blocks Based on Reproducible Wringing, Applied Optics, 39-4 (000), 56-538. [] Tsugami, K.: Block Gauges (in Japanese), The Nikkan Kogyo Shimbun, Ltd. (196), 31-40. [3] Tsumura, K. and Fujii, K.: Cohesive Force of Block Gauges: Study on Wringing Force of Block Gauges (Report 1) (in Japanese), Journal of the Japan Society of Precision Engineering,37-7 (1971), 509-515. [4] Fujii, K. and Tsumura, K.: Estimation of Wringing of Block Gages (in Japanese), Journal of Japan Society of Lubrication Engineers, 17-8 (197), 51-57. [5] Bowden, E.P. and Tabor, D.: Friction and Lubrication of Solids, Clarendon Press Oxford (1954), 304-306. [6] Budgett, H. M.: The Adherence of Flat Surfaces, Proc. Royal Soc. London (Ser. A), 86 (191), 5-35. [7] Rolt, F. H. and Barrell, H.: Contact of Flat Surfaces, Proc. Royal Soc. London (Ser. A), 116 (197), 401-45. [8] Kato, T. and Sakurai, Y.: Measurement of Real Contact Area of Wrung Gauge Block and Effect of the Area on Wringing Force (in Japanese), Journal of the Japan Society of Precision Engineering, 43-8 (1977),905-908. [9] Kato, T.: Effect of the Surface Integrity on the Wringing Force (in Japanese), Journal of the Japan Society of Precision Engineering, 48-3 (198), 394-398. [10] Katoh, K. and Tsutsumi, S.: Consideration of Wringing mechanism of Gage Blocks (in Japanese), Transactions of the Japan Society of mechanical Engineers, Series C, 58-549 (199),1634-1639. -s41-