Journal of Undergraduate Research
Volume 5, Issue 8 - May 2004

Measurement and Modeling of Sliding Friction on Soft Curved Surfaces

Alison Rennie

ABSTRACT

The need to accurately measure and model the sliding friction of hydrogels is motivated by the belief that ocular discomfort of contact lens wearers is directly related to the coefficient of friction between the eyelid and the hydrogel. Ways to measure the friction coefficient of hydrogels and interpret the results are the goals of this research.

BACKGROUND

Soft contact lenses are made from hydrogels, comprised of polymer matrices that contain 30-70% water[1]. The material properties of hydrogels make them extremely sensitive to water content and humidity. Any pressure on the material surface surfaces squeezes the hydrogel and reduces its water content[2]. Therefore, friction testing of hydrogels must be done under conditions that closely resemble those of the eye.

The human eye has multiple modes of operation, but blinking is considered most relevant because the moving eyelid exerts the greatest normal force on the lens. Normal force and contact pressure created by the eyelid during blinking are about 20 mN and 3.5-4.0 kPa (36-41 g/cm2), respectively. Blinking speed is an average of 12 cm/s[3, 4]. A previous study[5] considered the pressure field developed by fluid in the hydrogel expelling in the radial direction, but this investigation considers the only resistance to hydrogel penetration to be the elasticity of the fully hydrated hydrogel. Precision sensors and a stable environment are needed to make the necessary friction measurements at these delicate conditions. This technology can be found in a commercially available reciprocating sliding tribometer[6], and preliminary results show that differences between brands of commercially available contact lenses can be detected using this instrumentation.

SETUP

In the tribometer setup (Figure 1), a contact lens is removed from its packaging and placed on a rigid hemispherical mold [1] without any modification. This is done as quickly as possible to limit the amount of surface evaporation before the test begins. The hydrogel is then used as a counterface against a 1-mm radius glass ball. Tangential and normal forces produce cantilever displacement that is measured by optical sensors. The tribometer sensors report both a positive and a negative friction coefficient from consecutive reciprocating sliding passes, which are opposite in direction. A pass is defined as half of one full cycle, in which the moving stage starts at one end and goes to the other. In a full cycle the stage would return to its start position.

Figure 1. Tribometer setup

Figure 1. Tribometer setup

Because of the large difference in hydrogel and ball radii, the hydrogel-to-ball contact can be modeled as either a sphere-on-flat contact or sphere-on-sphere contact without significant pressure deviation (Figure 2). To solve contact geometry for pressure, normal force was integrated over a partial sphere and resulted in simple equations for contact pressure. The contact pressures obtained with the 1-mm ball were higher (by 1 order of magnitude) than those of the eye, but consistent results could not be obtained with lower contact pressures.

Figure 2. Contact pressure calculations for flat and curved hydrogel models

Figure 2. Contact pressure calculations for flat and curved hydrogel models

THEORY

Preliminary tests were run with the following three variables: sliding speed, contact pressure (normal load with constant area), and test duration. From the Coulomb friction model,

Ff = μFn,

where Ff is the friction force and Fn is the normal force. Results of these preliminary tests showed that friction could not be calculated by dividing the tangential force by the normal force because friction increased as normal load increased. Another contribution to the normal force could be a meniscus force, Fm, pulling the glass ball to the hydrogel. The Coulomb friction model then becomes

Ff = μ(Fn + Fm) = μFn + μFm.

This equation takes the form of a line, y=mx+b. The coefficient of friction is now the slope of a best-fit line through a series of data points collected under varying normal load tests (Figure 3). Further, the y-intercept is a product of the meniscus force (Fm) and the friction coefficient. Thus both the friction coefficient and the meniscus force can be calculated from the expression for the least squares regression line. This was found to be a reliable method of data interpretation for reproducing coefficients of friction of as low as 0.0310 ± 0.002.

Figure 3. Friction force vs. applied normal force for difference sliding speeds

Figure 3. Friction force vs. applied normal force for difference sliding speeds

TESTING

Experimentation followed this procedure:

All experiments were run using this protocol. If any step took longer than expected or any defects in the lens were noticed, the lens was discarded and a new lens used. A test matrix was run varying sliding speed from 63-6300 µm/s and normal load from 10-50 mN, with reciprocation distance and test duration being held constant.

RESULTS

The raw data from the tribometer shows that the friction increases over time (Figure 4). There is a time interval at the start of the test during which the friction coefficient remains constant before it increases, usually about 30 seconds.

Figure 4. Raw plot from the tribometer data.

Figure 4. Raw plot from the tribometer data.

Friction coefficient values of 0.018-0.035 were obtained, with increasing friction values as the sliding speed increases (Figure 3). Data was used only from the steady time interval at the start of the test. From equation (2), meniscus forces were found to be on the order of 20 mN.

A Linear Variable Differential Transformer (LVDT) was added to the reciprocating stage to gain positional data. The graph shows many unique characteristics (Figure 5). First, the top and bottom of the plot are curved outward, indicating higher friction toward the middle of the path. Second, the coefficient of friction is higher at the beginning of a path than at the end. Third, the friction is not consistent between consecutive passes (forward and reverse). Fourth, the outward curve suggests that static friction may not always be higher than kinetic friction on a squeezed hydrogel.

Figure 5. Plot of data from the LVDT as friction coefficient vs. lens position

Figure 5. Plot of data from the LVDT as friction coefficient vs. lens position

DATA INTERPRETATION AND DISCUSSION

From Figure 4, the coefficient of friction increases over time, but is steady for a predictable interval at the start of the test (t < 30s). The only condition that changed during the test was the water content of the lens due to surface evaporation, so the overall increase in friction was assumed to be due to relative surface dryness. This assumption was validated by running extra-dry argon gas over a sample at regular intervals during a friction test. A tube was held 1cm away with gas flowing parallel to the reciprocation path at approximately 3.1LPM. The results show that friction increases during the intervals where the argon gas was running over the sample and decreases when the argon gas stream is removed (Figure 6).

Figure 6. Plot of friction coefficient vs. time for intervals where argon gas was blown over the surface.

Figure 6. Plot of friction coefficient vs. time for intervals where argon gas was blown over the surface.

Figure 5 provides information about measuring the friction of a curved hydrogel under the constraints of periodic reciprocation. First, kinetic friction is not constant along a single-direction path. A possible explanation for the outward curve characteristic is that the lens is curved, so the friction should increase as the cantilever head passes over the peak point on the hydrogel where the normal force is greatest. An explanation for the overall decrease in friction over one pass is that at the beginning of the pass, the glass ball has just passed over that point in the previous pass (moving the opposite direction), but the other end of the reciprocation path has had more time to rehydrate to a condition that produces less friction.

To test the hypothesis that portions of the lens have higher friction due to some solution having been squeezed out, sliding speed was varied to examine the friction at specific points on the path. If sliding speed is increased, more solution will be squeezed away, causing higher friction. With a higher sliding speed, there was a greater difference in friction between the beginnings and ends of the paths (Figure 7). A corollary test was run with constant time but varied reciprocation distance and velocity. This test validated the significance of time between passes over a single point because constant time produced relatively constant slopes, while varied times produced varied friction coefficient slopes. The greater slopes indicate a higher friction coefficient at higher sliding speeds.

Figure 7. Plot of changes in friction coefficient for different sliding speeds

Figure 7. Plot of changes in friction coefficient for different sliding speeds

Friction values are not consistent between consecutive paths; however, they are consistent in the previously described shape. The y-axis shift is assumed to be due to the inconsistency of the stiffness of the cantilever that responds to the tangential forces.

Static friction is not always higher than kinetic friction for squeezed hydrogels. This is shown by the outward curve in the data plot—the coefficient of friction can increase after overcoming static friction rather than decreasing. With an incompressible, dry material, there should be obvious sharp corners where static friction turns to kinetic friction.

CONCLUSIONS

Soft materials with high water content provide unique challenges to determining friction coefficient, not only because of their fragile material properties, but because of the intricate applications in which they are used.

A commercially available tribometer was chosen to measure hydrogel friction coefficient because of its ability to apply and measure very small forces. The curved surface of the contact lens presented many problems because of contact geometry, surface dehydration, and interpretation of experimental results. Tests were run varying sliding speed and normal load, making sure that contact lens was held on the mold simply by the solution. Humidity was carefully monitored to assure consistent surface drying, and experimental results were interpreted from plots of friction coefficient vs. time and friction coefficient vs. position.

A conclusion from Figure 4 is that friction increases over time after a steady interval at the start of the test. This was assumed to be due to surface evaporation and was validated by running an inert gas over the surface and looking at the friction of a single point on the reciprocation path.

Adding an LVDT provided more insight into experimental results (Figure 5). First, the shape of the friction graph on one pass (half cycle) can be explained as follows: the outward curve in the middle is due to the cantilever ball passing over the peak height of the hydrogel, and the higher friction at the start of the pass can be explained by some solution having been squeezed out at that point. Second, friction is not consistent between the opposite paths in one cycle due to cantilever stiffness inconsistency. Third, static friction is not always higher than kinetic friction when conditions change over the course of the testing path.

The implications of these results emphasize the importance of hydration of soft contact lenses and suggest that the pressures caused by blinking can contribute to the dehydration of a contact lens and ocular discomfort. The effects of lower contact pressure, humidity control, and real time lens rehydration are the planned additions to this work.

ACKNOWLEDGEMENTS

This work was supported in part by a grant from Vistakon. Ms. Rennie was awarded a University Scholars Program grant for summer 2003. This research was completed with the assistance of Pamela L. Dickrell and was supervised by Dr. W.G. Sawyer in the Tribology Lab at the University of Florida.

REFERENCES

  1. Martin, D.K. and B.A. Holden, Forces Developed beneath Hydrogel Contact-Lenses Due to Squeeze Pressure. Physics in Medicine and Biology, 1986. 31(6): p. 635-649.
  2. McCutchen, C.W., The Frictional Properties of Animal Joints. Wear, 1962. 5(1): p. 1-17.
  3. Nairn, J.A. and T. Jiang, Measurement of the Friction and Lubricity Properties of Contact Lenses. ANTEC '95, 1995: p. 3384-3388.
  4. Hung, G., F. Hsu, and L. Stark, Dynamics of Human Eyeblink. American Journal of Optometry and Physiological Optics, 1977. 54(10): p. 678-690.
  5. Pascovici, M.D. and T. Cicone, Squeeze-film of unconformal, compliant and layered contacts. Tribology International, 2003. 36(11): p. 791-799.
  6. Scherge, M. and S.N. Gorb, Biological micro- and nanotribology: nature's solutions. Nanoscience and technology. 2001, Berlin ; New York: Springer. xiii, 304.

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