Some Considerations to the Energy Dissipation of Frictionally Stressed Lubricating Greases

Abstract

The introduction of mechanical energy during a friction process stimulates the system to eliminate this disturbance and find ways for energy dissipation. There are two principal situations: the system is either near equilibrium or far from equilibrium. Near equilibrium, it can be expected that the disturbance will be damped after a certain time, and the system will settle in a stationary state at a level where it began. However, the situation could be entirely different when the system is far from equilibrium. After a phase of instability and crossing a critical parameter, there is a probability of a change in the order level. This means that a new structure will be formed. This paper describes some aspects of the criteria that lead a friction process inside the grease film to instability and examines the influence of different dependencies. In this publication, the dependencies are extended to verify the stability criterion. Finally, the rest phase of a thixotropic experiment is examined from the perspective of potential instability and, thus, the possibility of self-organizing processes occurring.

1. Introduction

If we look at a system in which a continuous friction process takes place, we see a system in a state of non-equilibrium. The process can often be in a stationary non-equilibrium. A well-known observation is the change from a strongly time-dependent process to a time-independent process. This applies to friction processes with and without the presence of a lubricant. Friction processes, e.g., in a material area such as we find in lubricants, are no exception. If, in addition to an energy exchange, we also see an exchange of substances across a defined system boundary, the investigation is carried out on an open system. This is also typical for tribological systems. In an open system, we investigate the possibility of organizing the energy dissipation process from within the system. This self-organized dissipation can lead to the formation of structures, so-called dissipative structures. The object of investigation is the volume of lubricating grease stressed by friction. The subject of investigation in this work is the volume of lubricating grease subjected to friction. In addition to presenting some previous studies that indicate the possibility of instability and thus self-organization processes in grease films, the investigations are conducted with extended dependencies. A stability criterion is examined in this context. Finally, this type of stability analysis is applied to the resting phase of previously friction-stressed lubricating grease (thixotropic effect) to gain insights into the nature of ongoing processes in the unloaded grease.

2. Lubricating Greases

Lubricating greases are extremely visco-elastic materials. They usually consist of a base oil and a solid, the so-called thickener. Base oils are often semi-synthetic or fully synthetic. However, bio-oils such as castor oil, rapeseed oil or sunflower oil and the like are also being used more and more frequently [1]. In general, the structure of a lubricating grease is understood to mean the geometry, arrangement and distribution of the solid. The distribution of agglomerates in a base oil requires more of a macroscopic view. The geometry and formation of a network, on the other hand, is an observation on the microscale (Figure 1 and Figure 2).

Under frictional stress, the main mechanism is the shearing of grease volume [2,3]. This stress leads to a structural change, which in turn reduces the friction in the lubricating grease [4]. This can typically be investigated in rheometer tests. If the stressed volume is left at rest for a longer period of time after shearing, there is a structure formation during this period—a thixotropic effect. Both phenomena, the structural change caused by a frictional load and the structure formation after a frictional process, are driven by the effort to reach a stable process position. This can also occur when there are no effective attractions (equilibrium) and the system organizes energy dissipation on its own.

3. Checking the Stability of the System

3.1. The Investigations to Date

It is impossible to say whether a structure formation, i.e., a self-organized process, is triggered in a system subject to friction. One way to check the probability of this is to examine the stability of the process. A prerequisite for the formation of new structures far from equilibrium is the passage of instability. The search for conditions for triggering an instability provides an opportunity to analyze a probable structure formation.

An investigation into the system-inherent organization of energy dissipation can use the stability criterion of Prigogine [5]. This means that for systems far from equilibrium, the following holds

$${\displaystyle \frac{\partial }{\partial t}}\left({\delta }^{2}S\right)=\delta X·\delta J>0$$

(1)

This goes back to Einstein, who suggested using $\left({\delta }^{2}S\right)$ for the study of fluctuation (quoted in [6]) The term $\left({\delta }^{2}S\right)$ characterizes the stability of a system. Close to equilibrium, the process is stable and acts as an attractor. Far from equilibrium, the conditions may be different and a disturbance does not disappear, but it is amplified in a critical process situation. The time derivative of this function is associated with entropy production. In a non-equilibrium situation, however, it is not the thermodynamic forces and flows in a stationary state that are investigated, but their disturbances (see Equation (1)).

There are a number of studies that deal with stability investigations of very different tribological friction pairings, e.g., [7,8,9].

If the system of the grease volume element stressed by friction is considered, criteria can also be observed that indicate possible instability. The thermodynamic force $X_h$, which causes a heat flow, and the heat flow $J_h$ are investigated. With [10]

$$J_h=-\lambda gradT$$

(2)

$$X_h={\displaystyle \frac{gradT}{T^2}}$$

(3)

and the proposal [11]

$$J_h=(\tau ·\dot{\gamma }·V),$$

(4)

we can write an equation for the production of entropy via disturbances, as follows:

$${\displaystyle \frac{\partial }{\partial t}}\left({\delta }^{2}S\right)=\delta \left({\displaystyle \frac{(\tau ·\dot{\gamma }·V)}{-\lambda ·T}}\right)\delta (\tau ·\dot{\gamma }·V)$$

(5)

A condition was found for the possible occurrence of an instability (more details in [11])

$${\displaystyle \frac{\partial \tau }{\partial ϵ}}{\displaystyle \frac{\partial \lambda }{\partial ϵ}}>0$$

(6)

with $ϵ$ being an introduced parameter that describes the distance to the equilibrium. The condition (Equation (6)) shows that instability is possible for the general case of friction in the lubricating grease film. A more detailed analysis with a breakdown into different mechanisms [11] also shows the possible occurrence of instability. Interestingly, the probability increases with increasing solids’ content (thickening). For example, it appears to be a necessary condition that

$${\displaystyle \frac{\partial {\gamma }_{critic}}{\partial ϵ}}<0\mathrm{and} {\displaystyle \frac{\partial F_f}{\partial ϵ}}>0$$

(7)

with ${\gamma }_{critic}$ being the critical deformation of the lubricant volume at the transition to the dominant plastic deformation range. ${\gamma }_{critic}$ becomes smaller with increasing solids’ content as observed for Li-soap- and PU-grease samples. $F_f$ presents the fragmentation rate.

Another hint about the possibility of checking conditions that indicate a probable structure formation comes from Prigogine [5] and Klamecki [12]. They indicate that cyclic energy dissipation behavior can be associated with dissipative structures. The extension of the model of [12] shows

$${\displaystyle \frac{d{E}_i}{dt}}=z{E}_k{E}_i+a_i{E}_0-b{E}_i^m+c_i{E}_i{E}_j{E}_k$$

(8)

with the following explanation:

• Mechanical dissipation (fragmentation) $E_i$ always runs in parallel with thermal dissipation $E_k$;
• The energy dissipation rate is influenced by the amount of energy that is applied to the system $E_0=E_k+E_i+E_j$;
• There is an influence of energy dissipated up until now;
• The energy dissipation rate changes relative with the energy dissipated in the mechanism i in relation to the energy dissipated in j and k (mechanical dissipation – coagulation).

In a numerical test (Geogebra, University of Vienna), conditions were found that indicate cyclic energy dissipation (Figure 3).

There is also an indication of the role of solids’ content in the possible structure formation when investigating the intensification of a friction process, leading to increased grease wear, thus reducing friction.

$$E_f=E_{f0}+E_{f\dot{\gamma }}-E_{fw}$$

(9)

with $E_{f\dot{\gamma }}$ being the frictional energy rate due to an increase in deformation and $E_{fw}$ being the reduction caused by wear. This then leads to (see more details in [13])

$${\displaystyle \frac{1}{2}}{\displaystyle \frac{\partial }{\partial t}}\left({\delta }^{2}S\right)=-{\displaystyle \frac{1}{T2}}\left[{(\mathrm{factor}}_1){\left({\displaystyle \frac{\partial \rho }{\partial ϵ}}\right)}^2+{\displaystyle \frac{1}{{\lambda }^2}}{(\mathrm{factor}}_2){\displaystyle \frac{\partial \rho }{\partial ϵ}}{\displaystyle \frac{\partial \lambda }{\partial ϵ}}\right]{\left(\delta ϵ\right)}^2$$

(10)

and we obtain the condition

$${\displaystyle \frac{\partial \rho }{\partial ϵ}}{\displaystyle \frac{\partial \lambda }{\partial ϵ}}>0,$$

(11)

with the right-hand side becoming negative. This shows that there exist conditions where increasing the solid fraction (thickening) facilitates the possibility of self-organization.

3.2. Investigation with Extended Dependencies

I would like to look again specifically at the influence of solids’ content and expand the dependencies. Now, we have $\tau =\tau (\psi ,\dot{\gamma })$ and $\lambda =\lambda \left(\psi \right)$, where $\psi$ represents the solids’ content, $\dot{\gamma }$ the shear rate and $\lambda$ the thermal conductivity.

Thus, we have

$${\displaystyle \frac{1}{2}}{\displaystyle \frac{\partial }{\partial t}}\left({\delta }^{2}S\right)={\left({\displaystyle \frac{V}{T}}\right)}^2\left[{\displaystyle \frac{{\dot{\gamma }}^3}{{\lambda }^2}}{\displaystyle \frac{\partial \tau }{\partial \psi }}\left(\lambda {\displaystyle \frac{\partial \tau }{\partial \psi }}-\tau {\displaystyle \frac{\partial \lambda }{\partial \psi }}\right)\right]{\left(\delta \psi \right)}^2+{\displaystyle \frac{1}{\lambda }}{\left({\displaystyle \frac{\partial \tau }{\partial \dot{\gamma }}}\dot{\gamma }+\tau \right)}^2{\left(\delta \dot{\gamma }\right)}^2$$

(12)

Only the terms that make a relevant contribution are displayed [14].

Moreover, we find that

$${\displaystyle \frac{\partial \lambda }{\partial \psi }}>0,$$

(13)

allowing us to obtain a probability for the formation of a structure. The possible influence of increased solids’ content on the probability of structure formation is also confirmed here.

4. Thixotropic Effects from the Perspective of the Observations Made

A thixotropy study [15] of a grease model is now considered, and the effect of structure formation in the so-called resting phase is investigated. Classical experiments according to Czarny [16,17] and the more recent investigations byPaszkowski and Olsztynska-Janus [18] observe time-dependent structure formation after intensive shearing. The latter study, in [19], shows which mechanisms and which effects are to be expected in the friction process (shearing process) in the lubricating grease film. Notable results of their investigations are outlined as follows:

• Hydrogen bonds (binding OH groups) between lithium soap fibers are destroyed when grease is sheared;
• The observations indicate that the hydrocarbon chains of individual fibers become joined during the flow of grease;
• After the shear process, investigations have shown that hydrogen bonds destroyed during shearing form again. In addition, the number of COOH groups was found to decrease, which indicates a change in the chain structure of floccules.

This test specification is based on the proposal by [18]. However, it serves here as proof that a structural change takes place during the resting phase. The different end and start points of Phase 1 and Phase 3 are not the focus of this investigation. The behavior of the storage modulus curve (G^′) indicates a high probability of structure formation. It can be assumed that with longer observation times of the resting phase, a temporally constant course will show up.

Let us look for the driving forces behind the observed mechanisms in the resting phase. I suggest a description on different scales:

• Macroscale: Continuum material characterized by macroscopic properties, such as storage modulus $G^{\prime }$, apparent viscosity ${\eta }^{*}$, etc., corresponding to instantaneous network structures.
• Mesoscale: Determined by network shape, i.e., arrangement and geometry. Structure formation increases the degree of order.
• Nanoscale: Characterized by the interaction of free chain ends. Energy is bound and the energetic state of the system is minimized. Entropy production can lead to instability.

We can write, in general, terms for the entropy production as

$${\displaystyle \frac{d{S}_i}{dt}}=X_B·J_B$$

(14)

For the observation of the process in the rest phase according to Figure 4, it should be written as

$${\displaystyle \frac{d{S}_i}{dt}}={\displaystyle \frac{X_B·({\phi }_b·\Gamma ·(1-N_0)N)}{T}}$$

(15)

It is assumed that the entropy production under consideration is influenced by the following parameters:

-

Thermodynamix force $X_B$;

-

Structure formation velocity ${\phi }_B$;

-

Content of solids $\Gamma$;

-

Portion of new interactions $(1-N_0)$;

-

Number of free ends N;

-

Portion of free ends without interactions $N_0$;

-

Temperature T;

-

Distance to equilibrium $ϵ$.

The disruption of a stationary state is now being investigated, as follows:

$${\displaystyle \frac{\partial }{\partial t}}\left({\delta }^{2}S\right)=\delta X_B·\delta J_B=\delta X_B·{\displaystyle \frac{\Gamma ·N}{T}}\delta ({\phi }_B·(1-N_0)N).$$

(16)

This leads to

$${\displaystyle \frac{\partial }{\partial t}}\left({\delta }^{2}S\right)=\delta X_B·\delta J_B={\displaystyle \frac{\partial X_B}{\partial \epsilon }}·{\displaystyle \frac{\Gamma ·N}{T}}\left((1-N_0){\displaystyle \frac{\partial {\phi }_B}{\partial \epsilon }}-{\phi }_B{\displaystyle \frac{\partial N_0}{\partial \epsilon }}\right){\left(\delta \epsilon \right)}^2$$

(17)

With the assumption that at a deviation from equilibrium thermodynamic forces is seen [20]:

$${\displaystyle \frac{\partial X_B}{\partial ϵ}}>0.$$

(18)

Thus, it can be expected that

$${\displaystyle \frac{\partial N_0}{\partial ϵ}}<0$$

(19)

and

$${\displaystyle \frac{\partial \phi }{\partial ϵ}}<0$$

(20)

The second condition shows us a possibility that the ongoing process exhibits instability and that self-organization is possible. In other words, instability on the nanoscale leads to structure formation on the mesoscale.

5. Results

In many cases, tribological systems are open systems that, like most processes, exhibit temporal dynamics leading to a stable non-equilibrium. Disturbances that occur close to a stable process situation are often eliminated, and the system moves to equilibrium or a stationary state. This stable state acts as an attractor on the system. If disturbances take effect far from the equilibrium, the consequences can be quite different. Under certain critical conditions, the system checks the energy dissipation process and may find ways to achieve stability by forming a structure. There are various examples that can be given from the point of view of tribology.

In the studies presented here, the object of observation is a volume of lubricating grease subjected to frictional stress. The question of whether structure formation caused by energy dissipation is possible in this very heterogeneous material is investigated. To this end, stability investigations that have already been carried out and more recent extensions are presented. The extraordinarily interesting phenomenon of the thixotropic behavior of lubricating greases is examined against the background of the approach taken here. In the interpretation, an attempt is made to provide information on the driving forces that cause a structural change in the resting phase.