Study of Non-Newtonian Fluids’ Load-Carrying Capacity for Polyoxyethylene Oxide Water-Based Lubricants

Abstract

Water-based lubricants have become increasingly prevalent across various fields due to their accessibility, cooling properties, and environmentally friendly characteristics. This study investigated the non-Newtonian properties of polyoxyethylene oxide (PEO) aqueous solutions. The rheological behaviors of 1%, 2%, and 3% PEO aqueous solutions were assessed using a flat plate rheometer. Shear strain responses were comprehensively analyzed, resulting in the derivation of the corresponding power law functions. The total loads of 1%, 2%, and 3% PEO aqueous solutions can be obtained by the numerical integration of Reynolds equations. Results indicate that at high shear strain rates, load-carrying capacity increased; however, the rate of increase gradually diminished as the shear strain rate rose. In practical applications, shear stress is subject to fluctuations; negative viscosity occurs resulting in reduced hydrodynamic pressure and potential lubrication failure. Full viscosity and incremental viscosity are introduced, with the latter being identified as a crucial factor that provides a more direct characterization of the relationship between shear stress and shear strain rate. This factor significantly influences the load-bearing capacity of the lubrication film in non-Newtonian fluids.

1. Introduction

Lubricants play a crucial role in reducing the friction in mechanical operations. In the past 20 years, water-based lubricants have been widely used in metal processing processes, such as cutting, grinding, drawing, and cold rolling, as well as in the field of hydraulic transmission. These lubricants have the advantages of wide availability, low cost, abundant resources, non-flammability, good cooling properties, easy cleaning, and environmental friendliness [1]. However, water has the disadvantages of low viscosity, poor film-forming ability, metal corrosion, high melting point, and low boiling point. Therefore, it is difficult for water to be used alone as a lubricating medium. To improve the lubricating performance of water, various additives are usually added, such as thickeners, extreme pressure additives, dispersants, rust inhibitors, and oil agents [2].

Currently, many research studies have been carried out on water-based lubricants. First, in terms of application, water-based lubricants can greatly enhance the performance of silicon carbide ceramic sliding bearings, as was found in a research study [3], and the cooling performance of water-based lubricants was more than 50% higher than that of lubricating oil when they were used in ceramic bearing spindle prototypes [4]. Water-soluble groups can be introduced on the surface of nano-titanium dioxide, and their anti-wear and friction-reducing properties were experimentally verified in Wang’s research [5]. Phillip et al. studied the effect of ionic liquid water solutions on the friction and wear performance of silicon nitride tribo-pairs, especially during the run-in period [6]. Zhang [7] studied the sliding behavior of silica Ball-Shale contacts in polyacrylamide aqueous solutions. Second, to enhance the properties of water-based lubricants, researchers have developed various novel additives and modified materials, such as water-soluble nanoparticles and polymers, to improve their anti-wear and friction-reducing characteristics. Water-soluble synthetic copper nanoparticles, which can be used as water-based lubricant additives, were obtained by coating copper nanoparticles with the methyl polyethylene glycol ester of xanthan gum [8]. In addition, water-soluble nano-silica additives can be prepared by modifying the surface of nano-silica particles, and how the particle size of water-based lubricant additives affected their performance was studied in Wang’s research [9]. A novel fullerene-acrylamide copolymer, obtained through free radical polymerization, can improve the load-carrying capacity of water-based lubricants [10]. Duan et al. [11] enhanced the anti-wear performance of water-based thickeners with polymer additives. Wang et al. [12] synthesized an ion liquid with good water solubility, which significantly improved the tribological performance of water-based lubricants as an additive. Xu et al. [13] improved the thermal tribological properties of steel/ultra-high molecular weight polyethylene contact in water-based lubrication using microgels. Shetty [14] developed a simulation compound of polyethylene glycol/fat thickener in water. Zhang et al. [15] used different forms of CuO nanostructures as additives in water-based lubricants to improve their tribological performance. Finally, studies on the friction behavior and lubricating properties of lubricants provide a better understanding of their performance in practical applications, thereby promoting the technological advancement and application of water-based lubricants. Liu et al. [16] studied the lubrication properties of various polymer aqueous solutions using a homemade water-based lubricant film thickness-measuring instrument and a UMT wear-testing machine. Some advantages of using comb-shaped poly(ethylene glycol) methyl ether acrylate as a water-based lubricant were found by Jia et al. [17] based on studying performances of the comb-shaped poly(ethylene glycol) methyl ether acrylate, and its friction behavior was discussed. Chen et al. [18] tested and studied the friction behavior of rough polydimethylsiloxane surfaces in hydrophobic polymer aqueous solutions. Fang et al. [19] studied the effect of polyethylene glycol hydrolysis on the lubricating properties of styrene-ethylene-butene-styrene block copolymers.

Research on water-based lubricants focuses more on their performance; however, more in-depth studies and basic experimental analyses on frictional mechanisms are needed. Few studies address the non-Newtonian lubricating characteristics of water-based lubricants. This study was based on this premise; different concentrations (1%, 2%, and 3%) of water-based lubricants were prepared using polyethylene oxide (PEO, referring to polyethylene oxide polymers with a relative molecular mass exceeding 25,000) as a thickener. PEO is known for its excellent water solubility and low chemical toxicity. Additionally, it displays distinct non-Newtonian fluid characteristics. The rheological properties of different concentrations of PEO aqueous solutions were studied through rheometer experiments. Based on experimental results, changes in stress, viscosity, and load-carrying capacity with increasing shear strain rate were analyzed, and the impact of the non-Newtonian properties of PEO aqueous solutions on lubricating performance is discussed herein. Finally, changes in load-carrying capacity resulting from the shear-induced thinning of PEO aqueous solutions are explained by introducing full viscosity and incremental viscosity. This study of the lubrication mechanisms and tribological properties of water-based lubricants will deepen our understanding of them and promote the development of water-based lubricants, which is of great significance.

2. Experimental Study and Theoretical Analysis Model

2.1. Experimental Study of PEO Aqueous Solution Lubricant Rheology

Main materials: ultra-pure water, produced by Jiangsu Shuyang Xishimeng Trading Co., Ltd., Shuyang, China and polyoxyethylene (PEO), produced by Shanghai IKka Biotechnology, Shanghai, China with a relative molecular mass of 600,000.

Main instruments: DF-101S heat-collecting magnetic stirrer and HAKKERheo Win Mars40 rheometer.

Before the experiment, PEO was added to ultra-pure water to prepare PEO aqueous solutions with different concentrations (1%, 2%, and 3%) at a temperature of 40 °C; these were fully stirred and dissolved by DF-101S magnetic stirrers, and homogeneous liquids were obtained, which did not precipitate for 24 h at rest. In the experiment, PEO aqueous solutions were measured using a HAAKE rheometer at a temperature of 50 °C to obtain their rheological properties. Constitutive curves of shear strain rate-shear stress are shown in Figure 1.

The constitutive curves of 1%, 2%, and 3% PEO aqueous solutions demonstrate significant pseudoplasticity, with increasingly pronounced behavior as the concentration rises. Within the shear strain rate range of 0 to 1000/s the curves are relatively smooth; however, at higher shear rates (i.e., greater than 1000/s), data dispersion becomes more pronounced. As the shear strain rate increases, the shear stress grows slowly and tends to stabilize at a constant value, or may even show a decreasing trend. This phenomenon can be attributed in part to the ejection of some liquid during the experiment when the shear strain rate surpassed 1000/s, resulting in greater variability in these data.

Shear strain rate-shear stress curves were fitted with the Ostwald model to obtain the rheological constitutive equation; the expression of the Ostwald model is as follows:

$$\tau =\eta {\dot{\gamma }}^m$$

(1)

where $\dot{\gamma }$ is the shear strain rate; τ is shear stress; m is the rheological index; and η is the viscosity coefficient.

The constitutive equations of solutions with different concentrations are listed in

Table 1. Constitutive equation expressions of liquids with different proportional concentrations fitted by power functions.

	Pure Water	1% PEO	2% PEO	3% PEO
Fitted curve (power function)	$\mathrm{\tau }=0.001\dot{\gamma }$	$\tau =0.2249{\dot{\gamma }}^{0.7021}$	$\mathrm{\tau }=1.7827{\dot{\gamma }}^{0.5701}$	$\mathrm{\tau }=4.7088{\dot{\gamma }}^{0.5321}$

. Pure water is treated as a Newtonian fluid with a reference viscosity of 0.001 mPa·s. The table indicates that as concentration increases, the viscosity coefficient η also rises, while the rheological index m shows a decreasing trend, indicating that shear-thinning non-Newtonian behavior becomes more pronounced. It is crucial to note that, in dimensional analysis, the viscosity coefficient of a non-Newtonian fluid should not equal viscosity.

Figure 2 illustrates viscosity-shear strain rate variation curves for 1%, 2%, and 3% PEO aqueous solutions, while the viscosity of pure water is represented as a horizontal line at 1 mPa·s. Viscosity displays a logarithmic linear relationship with the shear strain rate, indicating pseudoplastic behavior. Notably, these logarithmic lines are nearly parallel, reflecting their similar logarithmic slopes. As observed in the constitutive equation, the data dispersion increases at high shear rates (i.e., greater than 1000/s).

2.2. Analysis Model and the Constitutive Equation

The fluid lubrication analysis model is shown in Figure 3, and the following conditions should be assumed: the interface boundary conditions for the steady-state problem are that the lower surface is a sliding surface, i.e., at z = 0, u = U, and the upper surface is fixed, i.e., at z = h, u = 0. In this case, the corresponding power-law fluid constitutive equation, the Reynolds equation, can be written as follows:

$${\displaystyle \frac{\partial }{\partial x}}\left[{\displaystyle \frac{m\rho }{(4m+2)}}{\left({\displaystyle \frac{h^{2m+1}}{\eta }}{\displaystyle \frac{\partial p}{\partial x}}\right)}^{1/m}\right]+{\displaystyle \frac{\partial }{\partial y}}\left[{\displaystyle \frac{m\rho }{(4m+2)}}{\left({\displaystyle \frac{h^{2m+1}}{\eta }}{\displaystyle \frac{\partial p}{\partial y}}\right)}^{1/m}\right]=U{\displaystyle \frac{\partial (\rho h)}{\partial x}}$$

(2)

where η represents the viscosity coefficient, ρ denotes the density, and m is the rheological index. Particularly, when m = 1, Equation (2) transforms into the Reynolds equation for a Newtonian fluid. In this case, it represents viscosity in the conventional sense.

2.3. Numerical Analysis

Assume that the density ρ of the solution does not vary with pressure, and the nondimensionalized equation of Equation (2) is as follows:

$${\displaystyle \frac{\partial }{\partial X}}\left({\epsilon }_X{\displaystyle \frac{\partial P}{\partial X}}\right)+{\displaystyle \frac{\partial }{\partial Y}}\left({\epsilon }_Y{\displaystyle \frac{\partial P}{\partial Y}}\right)={\displaystyle \frac{\partial H}{\partial X}}$$

(3)

where ${\epsilon }_X=k_0{H}^2{\left({\displaystyle \frac{H}{{\eta }^{\mathrm{*}}}}\right)}^{{\displaystyle \frac{1}{m}}}{\left({\displaystyle \frac{\partial P}{\partial X}}\right)}^{{\displaystyle \frac{1}{n}}-1}$, ${\epsilon }_Y=k_0{H}^2{\left({\displaystyle \frac{H}{{\eta }^{\mathrm{*}}}}\right)}^{{\displaystyle \frac{1}{m}}}{\left({\displaystyle \frac{\partial P}{\partial Y}}\right)}^{{\displaystyle \frac{1}{n}}-1}$, and $k_0={\displaystyle \frac{m{h}_0}{U\left(4m+2\right)}}{\left({\displaystyle \frac{H_0{P}_H}{{\eta }_0{L}_x}}\right)}^{{\displaystyle \frac{1}{m}}}$. The definitions of each parameter in the equation can be found in reference [20,21].

The nondimensional Equation (3) contains differential terms and needs to be discretized. After applying the forward difference and central difference methods and further rearranging, the $P_{i,j }$ can be obtained as follows:

$$P_{i,j}={\displaystyle \frac{r_{i,j}-∆X(H_{i,j}-H_{i-1,j})}{{\epsilon }_0}}$$

(4)

where, $r_{i,j}={\epsilon }_{i-1/2,j}{P}_{i-1,j}+{\epsilon }_{i+1/2,j}{P}_{i+1,j}+{\epsilon }_{i,j-1/2}{P}_{i,j-1}+{\epsilon }_{i,j+1/2}{P}_{i,j+1}$, ${\epsilon }_0={\epsilon }_{i+1/2,j}+{\epsilon }_{i-1/2,j}+{\epsilon }_{i,j+1/2}+{\epsilon }_{i,j-1/2}$, ${\epsilon }_{i\pm 1/2,j}={\displaystyle \frac{1}{2}}\left[{\left({\epsilon }_X\right)}_{i,j}+{\left({\epsilon }_X\right)}_{i\pm 1,j}\right]$, ${\epsilon }_{i,j\pm 1/2}={\displaystyle \frac{1}{2}}\left[{\left({\epsilon }_X\right)}_{i,j}+{\left({\epsilon }_X\right)}_{i,j\pm 1}\right]$, and $∆X=X_i-X_{i-1}$.

The solution domain is divided into 121 × 121 grids, an initial pressure distribution is given, and a new pressure distribution is solved by substituting Equation (4). The new pressure distribution is iterated repeatedly until the pressure difference obtained from the previous two times is less than 1 × 10⁻⁵.

3. Results

3.1. Pressure

Assuming the pressure (P) at both the inlet and outlet was 0, Equation (1) was plotted for different values of m = 0.7021, 0.5701, and 0.5321, with corresponding η of 0.2249, 1.7827, and 4.7088, respectively. The minimum film thickness (h_min) was 1 × 10⁻⁶ m, and the maximum film thickness (h_max) was 1.21 × 10⁻⁶ m. The sliding velocity (U) was 1 m/s, and the length of the slider (L) was 0.05 m. The pressure distribution was calculated assuming a constant density ρ, as illustrated in Figure 4. In this figure, x represents the dimensionless horizontal coordinate, while P denotes the dimensional vertical coordinate. The graph clearly shows that pressure values increased as the non-Newtonian behavior intensified (with smaller values of m).

3.2. Load-Carrying Capacity

By performing pressure integration, the resultant load can be calculated. The influence of h_min on the load-capacity is shown in Figure 5, where the values for m are 1, 0.7021, and 0.5321, while the viscosity coefficients correspond to 0.001, 0.2249, and 4.7088, respectively. In (a), the load-capacity of the film is presented for h_min ranging from 0.1 mm to 0.5 mm, while (b) shows the load-capacity for h_min ranging from 1 µm to 3.8 µm. Figure 5 illustrates that the load-carrying capacity of pure water exceeds that of the 1% PEO solution when the minimum film thickness is less than h_min = 2 × 10⁻⁴ m, despite pure water having a lower viscosity coefficient. Furthermore, the load-carrying capacity of the 3% PEO solution is markedly superior to that of pure water, as shown in Figure 5a. When h_min is less than 2 × 10⁻⁶ m, the load-carrying capacity of pure water surpasses that of the 3% PEO solution, even though the latter possesses a higher viscosity coefficient, as shown in Figure 5b.

Numerical simulation findings indicate that the load-carrying capacity of water, despite its low viscosity characteristic as a Newtonian fluid, can exceed or even significantly outstrip that of a non-Newtonian fluid with a high viscosity coefficient when the lubricant film is sufficiently thin. This observation suggests that, in the context of bearing design, the lubrication efficacy of non-Newtonian fluids exhibiting high macroscopic viscosity does not surpass that of Newtonian fluids under conditions of extremely reduced film thickness. Furthermore, this could precipitate an earlier onset of mixed lubrication (a lubrication condition where both a liquid film and solid-to-solid contact occur simultaneously), potentially leading to increased surface wear on frictional components.

Based on the analysis derived from the constitutive equations of pure water and various non-Newtonian fluids presented in Table 1, it was observed that as the shear strain rate increased, the shear stress of the non-Newtonian fluids ultimately decreased to a level lower than that of the reference solvent (water), as illustrated in Figure 6. Nonetheless, this premise was valid only when pure water was treated as a Newtonian fluid. In reality, due to the limited shear strain rate applied in the experimental setup, an adequately high shear strain rate would result in pure water exhibiting shear-thinning characteristics.

4. Discussion

To discuss the impact of actual viscosity variations on load-bearing capacity, we introduce total viscosity and incremental viscosity as follows:

Total viscosity (${\mathrm{\eta }}_0$) is defined by Equation (5):

$${\mathrm{\eta }}_0={\displaystyle \frac{\mathrm{\tau }}{\dot{\mathrm{\gamma }}}}={\displaystyle \frac{\mathrm{f}\left(\dot{\mathrm{\gamma }}\right)}{\dot{\mathrm{\gamma }}}}$$

(5)

where τ and $\dot{\gamma }$ represent shear stress and the shear strain rate, respectively.

The viscosity obtained from a conventional rheometer is total viscosity. This differ from incremental viscosity (${{\mathrm{\eta }}_0}^{\prime }$), which is defined as the derivative of the shear stress to the shear strain rate (Equation (6)). Further processing of experimental results is required.

$${{\mathrm{\eta }}_0}^{\prime }={\displaystyle \frac{\mathrm{d}\mathrm{\tau }}{\mathrm{d}\dot{\mathrm{\gamma }}}}={\displaystyle \frac{\mathrm{d}\mathrm{f}\left(\dot{\mathrm{\gamma }}\right)}{\mathrm{d}\dot{\mathrm{\gamma }}}}={\displaystyle \frac{∆\mathrm{f}\left(\dot{\mathrm{\gamma }}\right)}{∆\dot{\mathrm{\gamma }}}}$$

(6)

Figure 7’s incremental viscosities were obtained by differentiating fitting functions provided in Table 1. The 1%, 2%, and 3% PEO aqueous solutions demonstrated a shear-thinning phenomenon, characterized by a pronounced reduction in viscosity at lower shear strain rates. Subsequently, as the shear strain rate increased, the viscosity diminished more gradually. At sufficiently high shear strain rates, the viscosities converged toward a line with a negative slope. The incremental viscosity ${{\eta }_0}^{\prime }$ for 1%, 2%, and 3% PEO water solutions was lower than the ${\eta }_0$. This discrepancy arose primarily because the fitted curves represent average values, which exhibit reduced variability relative to the total viscosity. Given that the exponent of the power function is greater than zero, the derivative of the ${{\mathrm{\eta }}_0}^{\prime }$ forms a decreasing line, yet its value remains positive throughout.

In practical applications, shear stress is subject to fluctuations, particularly pronounced in the latter segments of the curve. Employing numerical methods to compute the ${{\mathrm{\eta }}_0}^{\prime }$ based on the second reduction equation of Equation (6) can lead to significant oscillations in the curve, as shown in Figure 8, with ${{\eta }_0}^{\prime }$ potentially attaining negative values multiple times.

In both theoretical and practical contexts, negative viscosity is a viable phenomenon. As the shear strain rate increases, the shear stress of the liquid decreases rather than increases. As shear stress is a primary factor affecting hydrodynamic pressure, a reduction in shear stress results in a decrease in hydrodynamic pressure. Given that positive viscosity is already quite small under actual working conditions, internal fluctuations within the lubricant can easily cause incremental viscosity to become negative. Consequently, shear stress no longer increases; instead, it begins to decrease, a situation commonly referred to as lubrication failure.

Expressing total viscosity without derivation proves more convenient. When the film thickness is particularly small or the shear strain rate is high, distortions may arise in describing the relationship between the current shear stress and shear strain. Incremental viscosity provides a more direct characterization of the relationship between shear stress and shear strain rate under current conditions, and it can elucidate certain special phenomena and their underlying causes, such as negative viscosity and lubrication failure.

5. Conclusions

PEO water solutions exhibit pronounced non-Newtonian characteristics, and their shear-thinning behavior becomes more prominent at higher concentrations.

When using PEO water solutions as lubricants, shear thinning could cause non-Newtonian behavior. Significant non-Newtonian effects can lead to a noticeable reduction in the lubrication carrying capacity when the lubricant film is sufficiently thin.

Incremental viscosity not total viscosity provides a more direct characterization of the relationship between shear stress and shear strain rate, and determines the carrying capacity of non-Newtonian lubricating film in fluid hydrodynamic lubrication.