Behaviour of complex fluids between highly deformable surfaces: isoviscous elastohydrodynamic lubrication

Post-doctoral Marie Curie Fellow: Juan de Vicente (jvicente@ugr.es)

Supervisors : Dr. Jason R. Stokes (Unilever) and Prof. Hugh A. Spikes (Imperial College) FRC, Unilever R&D Colworth, Colworth House, Sharnbrook, MK44 1LQ, UK

Abstract Because of its occurrence in many biotribological contacts, there is growing interest in the lubrication regime known as “isoviscous elastohydrodynamic lubrication (EHL)” or “soft-EHL”. In particular, we are interested in understanding the behaviour of film forming and friction properties of complex fluids in soft contacts, more precisely with respect to the formulation of personal care products and the behaviour of foodstuffs in the tongue-palate contact.

“Soft-EHL” lubrication occurs in lubricated contacts between non-conforming bodies when the contact pressure is sufficiently high to cause elastic deformation of one or both of the surfaces, but is not high enough to produce a significant increase of lubricant viscosity in the contact inlet and thereby influence fluid entrainment. In this work, a commercial tribometer (PCS Instruments) has been adapted to measure friction in the soft-EHL lubrication regime; a simulated point contact has been designed using elastomer surfaces. Couette friction is measured for a wide range of entrainment speeds using Newtonian fluids. At slow speeds friction is found to be constant. At intermediate speeds friction decreases because fluid starts to be entrained and surfaces separate. At high enough speeds full film lubrication is reached; in this situation a thick film of lubricant separates the surfaces and friction increases due to viscous dissipation. Finally, a Newtonian “master curve” is generated when plotting the friction coefficient as a function of the product entrainment speed × viscosity. A full numerical friction model has also been developed by simultaneously solving the 2D-Reynolds equation and the elasticity equation of the surfaces. A good correlation is found with the experimental Newtonian results in the full-film lubrication regime. Complex multiphase fluids (Non-Newtonian) are then investigated and Couette friction results compared to Newtonian predictions. In particular, the lubricating properties of a range of fluids are tested as follows: xanthan and guar gum polymer solutions, sunflower oil-in-water/glycerol emulsions, and Carbopol based yield stress fluids. Interestingly, the

friction vs. speed × viscosity curve fits the Newtonian master curve to a reasonable degree in the mixed and hydrodynamic regime. This suggests that the viscosity of the fluid in the film dominates in these regimes.

1 BACKGROUND EHL film thickness is dependent on the elastic properties of the surfaces but not on the pressure-viscosity coefficient of the lubricant. In practical terms, this type of lubrication occurs either when one of the containing surfaces has low elastic modulus, such as rubber or human tissue, or, for stiffer materials, where the lubricant has a very low pressure-viscosity coefficient and is typically water. This type of lubrication was first analysed systematically for line contact by Dowson and Higginson (1) and Dowson and Swales (2) in the 1960s and for elliptical contact by Hamrock and Dowson in the late 1970s (3) but became more clearly identified with the development of lubrication regime maps in the 1970s, which demarcated between the four hydrodynamic regimes of rigid-isoviscous, rigid-piezoviscous, elastic-isoviscous and elastic-piezoviscous lubrication (4)(5). Initially, the main application of elastic-isoviscous lubrication was to engineering systems such as seals, windscreen wipers and wet tyres, all of which consist of an elastomer rolling or sliding on a stiff surface. More recently, however, there has been increasing interest in biotribological systems, many of which also involve the lubricated contact of highly elastic bodies (6-10). Typical examples are synovial joints, human skin lubricated by creams and oils and the tongue-palate contact during the oral processing of food. There have been two main strands of previous research on soft-EHL, numerical modelling and experiment. The main thrust of modelling work has been to simultaneously solve the Reynolds and elasticity equations in order to identify the range of conditions over which elastic-isoviscous analysis is valid and to develop regression equations for predicting minimum and central film thickness. A number of equations have been developed to predict film thickness in the isoviscous-elastic regime,

both in line (1)(11) and in elliptical contact (3)(12)-(14). Probably the most commonly-used for elliptical contact are still those due to Hamrock and Dowson (3), which for circular contact reduce to:

22.064.03.3

'

!== WUH

R

h

c

c [1]

21.065.08.2

'

!== WUH

R

h

m

m [2]

with

'RE

UU

!=

" and

2'RE

WW

!= [3]

Here, hc and hm are the central and minimum film thickness respectively. U is the entrainment, W is the applied load, η is the lubricant dynamic viscosity, R’ is the reduced radius in the entrainment direction and E’ is the reduced elastic modulus. The latter two terms are defined by:

21

'/1/1/1

xxRRR +=

[4] ( ) ( )

2

2

21

2

1/1/1/2 EEE !! "+"=#

respectively, where Rx1, Rx2, E1, E2, ν1, ν2 denote the radii in the entrainment direction, the Young’s moduli and the Poisson’s ratios of the two contacting bodies. As well as film thickness measurements there has also been a considerable amount of work to measure the friction properties of soft contacts. The bulk of this was carried out in the 1960s and 1970s and has been well-documented by Roberts (15). Of particular interest was the influence of adhesion and substrate viscoelastic hysteresis on friction. It was suggested that friction increases at high loss tangent of the elastomer and with increasing extent of deformation (expressed as the ratio of contact pressure to elastic modulus) (16)(17). Most work has focussed on mixed lubrication, where friction decreases with entrainment speed, but some studies have also reported friction measurements in full film lubrication (15)(18).

2 EXPERIMENTAL

2.1 MTM rig Friction measurements were carried out using a mini-traction machine (MTM), shown schematically in figure 1. In this apparatus a ball is loaded and rotated against the flat surface of a rotating disc immersed in lubricant at a controlled temperature. Both bodies are independently-driven to achieve any desired sliding-rolling speed combination and the ball shaft is angled to minimise spin in the contact. Friction is measured by a load cell attached to the ball motor and, in the MTM, its value is generally determined by taking a pair of measurements at

the same slide-roll ratio but with the ball moving respectively faster and slower than the disc. These two values are then differenced.

Figure 1: MTM friction device. Adaptation to measure in

soft contacts. In the current study a soft contact (E’ = 10.9 MPa) was obtained by using a stainless steel ball (AISI 440; radius R = 9.5 mm) against a silicone elastomer disc (NDA Engineering Equipment Limited, Kempston, UK). The discs were 46 mm diameter and 4.5 mm thick, cut from elastomer sheets, and were clamped on top of a supporting, stainless steel disc. The root-mean-square roughness, Rq of the steel ball used in this study was 10 nm, while the roughness of elastomer discs was 800 ± 100 nm. All tests were carried out at an applied load of W = 3.0 N, a temperature of 35ºC and a slide-roll ratio of SRR = 0.50, where this is defined as the ratio of the absolute value of sliding speed,

DBuu ! to the entrainment speed,

(uB+uD)/2 and uB and uD are the surface speeds of the ball and disc respectively. Couette friction was measured over an entrainment speed range of 0.004 to 1.2 m/s.

2.2 Materials In the current study a range of different fluids are tested: i.- Newtonian fluids (from 7.54×10-4 to 3.12 Pas) were obtained by dissolving various concentrations of corn syrup (CS) in distilled water. ii.- Two different biopolymers were also investigated: xanthan gum and guar gum. These polymers were chosen due to their different conformation and shear-thinning behaviour in aqueous solution. iii.- Oil-in-water emulsions were prepared by adding the solution of glycerol in water to sunflower oil and homogenising. In order to keep the system as simple as possible, no surfactant was added to the formulation. iv.- Finally, “yield stress” fluids consisting of polyacrylic acid microgel suspensions (2-4 microns diameter) were prepared upon neutralization of Carbopol C934 in aqueous solution.

Elastomer disc

Lubricant

heaters

Steel ball

3 FRICTION MODEL Full numerical solution of the soft-EHL problem was carried out in order to calculate friction coefficient within a soft-EHL contact and compare this with experiment (6). This involved simultaneously solving the isoviscous 2-D Reynolds equation and the elasticity equation, Eqs. 5 and 6, respectively, using the forward iterative approach described by Hamrock and Dowson (3),

dx

dhU

y

ph

yx

ph

x!12

33=""#

$%%&

'

(

(

(

(+"#

$%&

'

(

(

(

( [5]

'')','(

'

2),( dydx

r

yxp

Eyx

A

!!="

# [6a]

dxdyyxpWA

!!= ),( [6b]

where p is the pressure. ω is the total elastic deflection and r is the distance between point (x,y) and point (x’,y’). The solution approach was a standard finite difference one with a constant spacing 128×128 grid and Reynolds cavitation boundary condition. Non-dimensional units were used throughout; '/ Rxx = , '/ Ryy = ,

'/ Rhh = , '/ RsS = , '/ R!! = , '/ Epp = ,

''/ REUU != , 2''/ REWW = , 2

''/ REFF = .

Solutions were found over a range of values of U = 1.5×10-7 to 4×10-5 (corresponding, for the values of E’ and R’ in the experimental work to Uη = 0.015 to 4 Pa.m) and W = 0.00075 to 0.0245 (corresponding, for the values of E’ and R’ in the experimental work to W = 0.75 to 24 N) Friction was calculated by integrating the shear stress over the lubricant-filled region of the contact;

!!=A

dxdyF " [7]

The shear stress is given by;

x

ph

h

u

z

u s

z !

!"±=#

$

%&'

(

!

!=

= 20

))* [8]

where the first term is due to Couette flow and the second, pressure gradient-dependent term, results from Poiseuille flow. Substituting the sliding speed this gives;

x

ph

h

USRR

!

!"±=2

. #$ [9]

Regression equations were obtained to predict friction coefficient over the range of U and W studied. For Poiseuille friction coefficient, speed and load dependence both fitted closely a power law expression of the same

form as those used for film thickness prediction. Eq. 10 shows the best-fit expression.

70.065.046.1

!= WUPoiseuille

µ [10] Couette friction coefficient could not be fitted to such a simple expression and was found to depend less strongly on load and speed at high load and low speed than predicted from a simple power law expression based on results at low load and high speed. This is believed to be because Couette friction originates from two different regions of the contact, the central, “Hertzian” zone and the periphery, each of which has a different form of load and speed dependence. Based on this, Eq. 11 was found to give good fit over the whole speed and load range studied;

( )11.036.076.071.096.08.3

!! += WUWUSRRCouette

µ [11]

4 RESULTS AND DISCUSSION Figure 2 shows plots of measured log10(Couette friction coefficient) versus log10(entrainment speed) for all of the Newtonian fluids tested.

101

102

103

10-2

10-1

Water 50% CS 72% CS 86% CS 92% CS 93% CS 95% CS

Frictio

n c

oe

ffic

ien

t,

µ

Entrainment speed, U (mm/s)

Figure 2: Couette friction coefficient versus

log(entrainment speed) for a range of different viscosity Newtonian fluids.

At first sight the various tests show little correlation. If, however, the same friction coefficient results are plotted versus the product of viscosity and entrainment speed, (Uη) as shown in figure 3, it can be seen that they fall on a single “Stribeck curve.” It should be noted that the friction results span a Uη range of more than six orders of magnitude. From figure 3, it appears that full film lubrication is reached at approximately Uη = 0.02 Pa.m and that friction coefficient rises linearly (on a logarithmic scale) above this threshold. Figure 3 also shows the predicted Couette friction coefficient. It can be seen that theory predicts similar values to those measured experimentally.

When comparing predictions of friction coefficient with experiment it is important to appreciate that the friction measured by the MTM is only the Couette friction. Recently, a novel experimental approach has been developed to measure Poiseuille friction coefficient using the MTM device (7).

10-6

10-5

10-4

10-3

10-2

10-1

100

10-2

10-1

Theory, Eq. (11)

Frictio

n c

oe

ffic

ien

t,

µ

Entrainment speed * viscosity, U!, (N/m)

Figure 3: Friction coefficient versus log(entrainment

speed×viscosity) for a range of different viscosity fluids. Note how the data collapses onto one master Stribeck

curve

4.1 Polymer solutions As observed in figure 4, Stribeck friction curves corresponding to polymer solutions have been found to collapse in the mixed regime by multiplying the entrainment speed by a K factor for each concentration. This factor represents the average effective dynamic viscosity of the polymer solution over the shear rate range present in the contact inlet (9).

10-6

10-5

10-4

10-3

10-2

10-2

10-1

10-6

10-5

10-4

10-3

10-2

10-2

10-1

10-6

10-5

10-4

10-3

10-2

10-2

10-1

10-6

10-5

10-4

10-3

10-2

10-2

10-1

10-6

10-5

10-4

10-3

10-2

10-2

10-1

10-6

10-5

10-4

10-3

10-2

10-2

10-1

XG (wt %) Water 0.005 0.02 0.07 0.2

Entrainment speed x Scaling factor, UK (N/m)

GG (wt %) Water 0.05 0.2 0.4 0.6

Friction c

oeffic

ient,

µ

Figure 4: Friction coefficient versus log(entrainment

speed×K) for xanthan and guar biopolymer solutions.

4.2 Emulsions In figure 5 emulsion friction results are mapped onto the Newtonian master curve by multiplying the entrainment speed by two alternative viscosities. In the case of large viscosity ratios (p ≥ 5.8), the viscosity of the oil is used (the dispersed phase) while at low viscosity ratios (p ≤ 1.3), the viscosity of the polar phase is used. This has the effect of mapping the friction results of all of the emulsions tested on the same Newtonian master curve. It suggests that for low p values the polar phase or bulk emulsion is controlling the lubricant film thickness and filling the contact, while at high p values the oil phase is dominant (10).

Figure 5: Master Stribeck curve for emulsions using the viscosity of sunflower oil and the polar phase as needed to

force collapse. Solid line represents the Newtonian master curve. Viscosities used to force collapse are shown

in brackets (Pa s).

4.3 Microgel suspensions In figure 6, Stribeck curves for Carbopol suspensions are shown. At low U values unneutralized (UN) Carbopol suspensions have similar friction coefficients to those of water. At high speeds mixed-lubrication is observed and UN suspensions have lower friction than water, presumably due to the slightly higher viscosity caused by the presence of microgels. Stribeck curve for neutralized (N) Carbopol passes through a maximum between the boundary and mixed regimes. Lubrication properties in boundary and mixed regimes for neutralized Carbopol may be related to the confinement of swollen microgel particles in the interfacial film rather than bulk rheological properties (8).

Fric

tion

coef

ficie

nt, µ

10-2

10-1

p = 47, (0.035) Pa s) p = 10.9 (0.035) )Pa s) p = 5.8 (0.035) Pa s) p = 1.3 (0.0267) Pa s) p = 0.7 (0.0478) Pa s) p = 0.2 (0.173) Pa s) p = 0.09 (0.38) Pa s) Newtonian master curve

Entrainment speed × viscosity, Uη (N/m) 10-6

10-5

10-4

10-3

10-2

10-1

100

101

10-6

10-5

10-4

10-3

10-2

10-2

10-1

Water 0.1 wt % UN 0.15 wt % UN 0.2 wt % UN 0.5 wt % UN 0.1 wt % N 0.15 wt % N 0.2 wt % N 0.5 wt % N Master curve

Friction coeffic

ient,

µ

Entrainment speed x viscosity, U! (N/m)

Figure 6: Friction coefficient for unneutralized (UN) and neutralized (N) Carbopol solutions.

5 CONCLUDING REMARKS A tribometer has been developed to measure friction in a simulated soft-EHL contact. When using Newtonian lubricants, a transition from boundary lubrication to full-film isoviscous-elastic lubrication is observed. A numerical solution of the point contact, isoviscous-elastic problem has been obtained for comparison with experimental results. This shows reasonably close agreement with measured values for Couette friction. Newtonian and complex multiphase fluids are then tested and friction results compared to the model. A master Stribeck curve was obtained using Newtonian fluids. This master curve was later used to study lubricating properties of xanthan and guar biopolymer solutions. It was found that, in spite of their conformation being very different, these biopolymer solutions behaved effectively as Newtonian fluids with an apparent constant viscosity corresponding to the one measured at high shear rates. On the other hand, Carbopol microgels particles mainly operated in the boundary regime and showed an increase in friction with increasing entrainment speed suggesting that some kind of confinement of the swollen microgels is taking place. The nature and properties of the lubricant film depend strongly on the viscosity ratio, p, of the phases in a surfactant-free emulsion. At p values of 5.8 and above, (i.e. the dispersed oil is at least 5.8. times as viscous as the dispersion medium), the oil phase properties control the lubricant film formation and friction. But at lower p values, lubricating film formation is dominated by the aqueous phase. This difference probably arises from the non-deformable nature of the high viscosity ratio droplets, which means that these are entrained and confined in the contact inlet, thereby wetting the surfaces and coalescing to form a pool of viscous oil. Similarly, at low viscosity

ratio, the more viscous aqueous phase forms a pool of liquid in the contact region which cannot be displaced by the less viscous and deformable oil droplets. These results suggest that it may be possible to control which fluid is confined between shearing surfaces by manipulating the viscosity ratio of the two phases.

6 ACKNOWLEDGEMENTS A Post-doctoral Marie Curie Fellowship, from the E.U. awarded to J. de Vicente is acknowledged and we are also grateful to Unilever Research for permission to publish these results.

7 REFERENCES [1]Dowson, D. and Higginson, G.R., Elastohydrodynamic

Lubrication. SI Edition, Pergamon Press, 1977. [2]Dowson, D. and Swales, P.D., An elastohydrodynamic

approach to the problem of the reciprocating seal.” Proc. 3rd Intern. Conf. Fluid Sealing, Paper F3, BHRA, (1967).

[3]Hamrock, B.J. and Dowson, D., “Elastohydrodynamic lubrication of elliptical contacts for materials of low elastic modulus 1 – fully flooded conjunction.” Trans. ASME, J. Lubr. Techn. 100, pp. 236-245, (1978).

[4]Hamrock, B.J. and Dowson, D., “Minimum film thickness in elliptical contacts for different regimes of fluid film lubrication.” Proc 5th Leeds-Lyon Symp. on Tribology, Elastohydrodynamics and Related Topics, MEP, Bury St Edmunds, Suffolk, pp. 22-27, (1979).

[5]Esfahanian, M., and Hamrock, B. J. “Fluid-film lubrication regimes revisited.” Trib. Trans, 34, 628-632, (1991).

[6]J. de Vicente, J.R. Stokes, and H.A. Spikes, "The frictional properties of Newtonian biological fluids in rolling-sliding soft-EHL contact", submitted to trib. Let., (2005).

[7]J. de Vicente, J.R. Stokes, and H.A. Spikes, "Rolling and sliding friction in compliant, lubricated contact", submitted to Proc. I. Mech. Eng. E, (2005).

[8]J. de Vicente, J.R. Stokes, and H.A. Spikes, "Soft lubrication of model hydrocolloids", accepted for publication in Food Hydrocolloids, (2005).

[9]J. de Vicente, J.R. Stokes, and H.A. Spikes, "Lubrication properties of non-absorbing polymer solutions in soft elastohydrodynamic (EHD) contacts", Tribology International, 38(5), 515-526, (2005).

[10]J. de Vicente, J.R. Stokes, and H.A. Spikes, "Viscosity ratio effect in the emulsion lubrication of soft EHL contact ", submitted to Trans. ASME, (2005).

[11]Herrebrugh, K., “Solving the incompressible isothermal problem in EHL through an integral equation.” Trans. ASME J. Lubr. Tech. 90, pp. 262-270, (1968).

[12]Hooke, C.J. and O’Donoghue, J.P.,

“Elastohydrodynamic lubrication of soft, highly deformed contacts.” J. Mech. Eng. Sci. 14, pp. 34-48, (1972).

[13]Biswas, S. and Snidle, R.W., “Elastohydrodynamic lubrication of spherical surfaces of low elastic modulus.” Trans. ASME J. Lubr. Techn. 98, pp. 524-529, (1976).

[14]Jamison, W.E., Lee, C.C. and Kauzlarich, J.J., “Elasticity effects on the lubrication of point contacts.” ASLE Trans. 21, pp. 299-306, (1979).

[15]Roberts, A.D., “Studies of lubricated rubber friction,

Part 1: Coupled optical observations to friction measurement.” Trib. Intern. 10, pp. 115-122, (1977).

[16]Moore, D.F., “The elastohydrodynamic transition speed for spheres sliding on lubricated rubber.” Wear 35, pp. 159-170, (1975).

[17]D F Moore, The Friction and Lubrication of Elastomers. Pergamon Press, 1972.

[18]McClune, C.R. and Tabor, D. “An interferometric study of lubricated rotary face seals.” Trib. Intern. 11, pp. 219-227, (1978).