Phenomenological studies on magnetic and mechanical remanence effects in magnetorheological fluids

While magnetorheological fluids (MRF) are already used in devices available on the market, the complex response of these fluids to magnetic and mechanical actuation is still under evaluation and thus full-fledged models suitable for device development are not available, yet. However, there are many investigations pertaining several microphysical aspects that make the response of MRF so complex. In particular, the microstructure determining the rheological properties of the MRF can be manipulated by magnetic fields leading to phenomena such as cluster formation and locally ordered alignment of the embedded ferromagnetic particles (as briefly reviewed in the next sections). This is partly due to the generally complex behavior of ferromagnetic materials when it comes to magnetization in external fields, which is associated with nonlinear responses, hysteresis phenomena and thus particularly memory effects (remanence). In MRFs these phenomena are connected to complex mechanical effects associated with the above-mentioned phenomena, such that, e.g. also mechanical memory effects are encountered. As a consequence, demagnetization routines, which are common to achieve well-defined initial states prior to the characterization of a ferromagnetic material, have to be supplemented by defined mechanical actuation to resolve memory effects ('shear history') related to microstructures. In this paper such a routine has been devised experimentally and it is shown to what extent reproducible measurements for the characterization of the MRFs can be obtained in that manner. However, while these investigations stimulate hypotheses on the underlying microphysics (which is also topic of earlier works by other authors), we do not present a full-fledged model describing an MRF subjected to magneto-mechanical actuation. Thus, in particular, while the used measurement apparatus itself represents a possible device configuration (a rotating shaft subjected to a controllable counter-torque), the design of such devices is also not in the focus of this work but is rather one of the next steps in our research.

Magnetorheological (MR) fluid consists of small, spherical, ferromagnetic particles dispersed in a carrier fluid such as hydrocarbon oil and additives that affect the properties of the MR fluid as sedimentation, abrasiveness, rheology, or in-use-thickening [1]. The particles are mostly made of carbonyl iron and have a particle size range of 1–10 µm in diameter. In the absence of a magnetic field, the viscosity of the MR fluid is very low. When the ferromagnetic particles are exposed to a magnetic field, they align in a chain-like structure along the magnetic field lines, affecting the rheological behavior of the MR fluid and increasing the viscosity, resulting in higher shear stress. The magnetorheological effect occurs when a magnetic field is applied creating a yield stress that causes the MR fluid to behave like a semi-solid [2]. When using MR fluids with a high concentration of iron particles, not only the formation of simple single-row chains is observed, but also the formation of more complex structures, such as thicker chains and aggregations. Within this process, the chains tend to aggregate with each other and form longer and thicker columnar structures [3–7].

Jolly et al [4] and Kubìk et al [5] identified at least two stages for the alignment of the particles, which initially form single chain-like structures and then migrate to longer and thicker chain structures. The MR fluid can be used in the so-called 'valve mode' [8, 9], particularly in dampers, and in the 'shear mode' [10–14], used in clutches, brakes or other dissipative devices where two surfaces move against each other [15–20].

The technology based on MR fluids has been known for a long time [2] and much research is being done to improve many aspects and solve different problems. For example, there are many investigations to improve the particle-based simulation and calculation to predict the behavior of an MR device [3, 6, 10, 21] or to adapt the MR formulation to solve settling and abrasion problems [1]. Research is also being carried out to optimize designs for fast force response [4, 22, 23] or to characterize the torque response [24]. In addition, experimental work has been carried out to determine sufficiently good macroscopic and microscopic models of MR fluids [3, 8, 21, 25–27] or to determine the effect of particle size, distribution or volume fraction on the MR fluid behavior [18, 28]. Despite this research effort, the technology is used in only a few commercially available products.

Many models are used to describe the relationship between the magnetic field, the shear rate, and the resulting shear stress. In case of MRFs, the Bingham model,

$$\begin{equation}\tau \left( H \right) = \,{\tau _0}\left( H \right) + \eta \dot \gamma \end{equation} \tag{ 1 }$$

is a particularly useful model to describe the non-Newtonian behavior, where ${\tau}$ is the shear stress caused from the MR fluid as a function of the yield stress ${\tau}$ ₀ (depending on the applied magnetic field strength H), the dynamic viscosity η and the shear rate ${\dot \gamma}$. It should be noted that the Bingham model (1) does not include terms to describe transient behavior, remanence, or temporal effects, which, however, is often desired.

Zubieta et al [29] modelled a commercial MR fluid with the Bingham model (1) and pointed out that the calculated and measured data recorded with an Anton Paar rotary rheometer showed inconsistencies at the lower shear rates. To improve the model, the Bingham model was extended by adding a third fitting parameter to form the familiar Herschel–Bulkley model [29, 30]. This model gave better agreement with the calculated and measured data, especially at low shear rates. Other authors [5, 24, 30, 31] find similar non-linear effects in the measurement data and try to describe these effects with non-linear models.

Pei and Peng [22] mentioned a shear history effect as an external influence, where the pre-shear conditions affect the responses of the MR fluid. Rankin et al [32] and Shan et al [33] discuss the influence of shear history on the behavior of the MR fluid in different forms. Rankin et al [32] studied the behavior of carbonyl iron particles in viscoplastic media as a function of increasing and decreasing magnetic fields evaluated in strain sweeps. In addition to the field dependence, the MR fluid also showed a history dependence, which was explained as a field-dependent evolution of the suspension microstructure. The first magnetic field sweep resulted in a smaller storage modulus for the increasing magnetic field than for the decreasing magnetic field. The following sweep showed approximately the same values as for the first decreasing magnetic fields.

Shan et al [33] investigated the effect of shear history experimentally with shear ramp tests and found similar results: the shear stress at low shear rates was lower in the very first shear rate ramp-up than in the subsequent shear rate ramp-down process. The complex evolution of MR fluid structures may influence subsequent responses.

Another non-linear influence on the behavior of MR fluids is the transient response to the step change in current respectively the magnetic field. Jolly et al [4] introduced two time-constants in different orders of magnitude of microstructure formation. Kubìk et al [5] explained three different time dependencies: the hydrodynamic response time, the particle structure development response time, and the rheological response time.

Also, in other literature it can be seen in the measurement data that the step responses do not become stationary immediately but show a certain temporal evolution on a larger time-scale. For example, in the data from Senkal and Curicak [14], the resulting torque is not constant after 0.8 s and continues to increase. In the case of Güth et al [12], when measuring the transient behavior of the torque, the torque continues to increase even though the current is already constant. An increase in torque over time despite constant current can be seen also in the measurement data of Böse et al [13].

It is known that there are non-linear effects the description of which has been attempted with various models or simulations [6]. Moreover, there is a shear history effect, and the transient behavior of MR fluids is associated with at least two constants [4]. A certain time-dependent change in the shear stress can also be seen in various measurement data despite the application of a constant magnetic field.

As with pure magnetic remanence and shear rate, the behavior of MR fluids is more complex; applying the same magnetic cycle and shear rate to MR fluids does not mean that the MR fluid returns to the same state. We found this shear history in our measurements as well, and it goes beyond the simple hysteresis effect.

Some of the authors work in a company, where MR technology is used for rotary haptic feedback devices, which can adapt the mechanical resistance when rotating, to realize different end stops or varying difficulty of rotation. One of the quality requirements for series production was a 100% end-of-line (EOL) control. Strange temporal effects occur here, and the question is how to get a grip on the device and make reproducible measurements. We developed an appropriate measurement routine to investigate and control the influence of the effects. Part of this routine is also a generalized demagnetization routine that removes not only the magnetic remanence but also the mechanical remanence, the so-called 'magneto-mechanical remanence effect'.

Experience has shown that such shear history and remanence effects occur in our devices, as described in the literature, and need to be investigated further. Our aim here is to understand the severity of these effects and how they can be managed through a controlled routine. The resulting measurement guideline aims to provide a procedure for a realistic and comparable characterization of MRFs, combining all effects (time dependence, shear history, non-linearity) from the literature into one routine.

The MRF used in these studies is MRF-140BC sourced from LORD Corporation (USA). It consists of synthetic oil (hydrocarbon-based) and 40 vol% iron particles, which corresponds to 86 wt%, which is frequently applied and representative for many other commonly used MRFs.

2.1.1. Redispersing the MRF.

2.1.2. Analyzing the MRF.

2.1.3. Particle size.

After preparation, a qualitative and quantitative analysis of the iron particles in the fluid was conducted. First, in order to provide a qualitative impression of the MRF's microstructure at the microscopic level, we provide a scanning electron microscope (SEM) illustrating typical particle shapes (figure 1). The SEM image was taken with a JEOL SEM, type JSM-7100F. The used MR fluid (MRF-140BC) is a commercially available standard product subject to the manufacturers' specifications that has been used continuously in our recent research and development activities. We note that this picture was taken in 2019 during earlier work with this particular MR fluid and it merely serves to provide a general impression about the microstructure. In particular, no data used in the present investigation has been derived from this SEM image. Nonetheless the samples investigated in this work can be expected to show similar characteristics. In the SEM picture, the majority of iron particles are spherical particles with partially attached so-called satellites in size ranges of up to 6 µm. The particles seem to have an average diameter in the order of of 2–3 µm.

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Second, to characterize the specific samples investigated in this work, the particle size distribution was assessed by laser diffraction synchronized with image analysis using a Microtrac Sync instrument. The instrument is equipped to measure both particle size and shape by laser diffraction according to ISO 13320:2020 and dynamic image analysis according to ISO 13322–2. It operates using static light diffraction over a measurement range of 0.0215 µm–2000 µm. The result of this analysis is shown in figure 2. Here, the d50 value (median) of the particle diameter is 2.68 µm. This matches qualitatively with the result from the SEM image (figure 1).

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The design of the experimental setup (figure 3) is similar to that of a parallel plate magneto-rheometer which characterizes the MRF in shear mode. With our setup, magnetic field strengths of over 1000 000 A m⁻¹ and shear rates up to 10 000 s⁻¹ can be achieved.

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The experimental setup consists of three rotationally symmetrical main components: a stator (green), a MRF chamber (light red) and a rotor (yellow). The stator comprises a chamber containing the MR fluid to be tested. Inside this chamber there is a rotating disc, that is part of the rotor (yellow), which, in turn, is driven from above by a servo motor.

The magnetic flux is generated in the stator by two coils (Coil 1 and Coil 2), which are electrically wired such that their respective magnetic fluxes are oriented in the same direction. The coil current is supplied by a custom-developed power supply controller providing a voltage with pulse width modulation operating at a frequency of 100 kHz utilizing a bridge circuit. The total magnetic flux runs along the red dashed line shown in figure 3. The light green and dark yellow components are made of Vacoflux 50, a material with high magnetic permeability and saturation magnetization. In this setup, magnetic flux densities of more than 2.3 T can be achieved. All other components, like the sealing system, are made of nonferromagnetic materials.

Inside the MRF chamber there are two so-called interaction areas (or working gaps, see figure 3). Here the MRF is exposed to the adjustable magnetic field, where the shear effect of the MRF and thus the associated force is generated. The geometry of the coils is chosen to produce an approximately uniform field distribution across these working gaps. One working gap is located at the top of the disc and the other at the bottom. Both are penetrated by the same magnetic field and experience a similar shear rate as the gap sizes are the same and the radial velocity varies identically. The gap extends from a radius of r_i = 10 mm to r_o = 20 mm with a center at r = 15 mm. The disc extends further outwards than the working area to reduce spurious fringe effects and to support the development of a continuous flow profile.

The stator is mounted on a torque transducer, a piezoelectric reaction torque sensor with a measuring range from −25 to 25 Nm (Kistler 9349A), which is regularly zeroed with a second torque transducer to avoid offset errors. The piezoelectric sensor is read out by a Kistler charge amplifier LV5074 and used to measure the torque generated by the setup. The setup is attached to a floating bearing to minimize concentricity problems.

Upon application of a torque T, shear stresses ${\tau}$ are exerted onto the fluid, which are given by [10]

$$\begin{equation}\frac{T}{2} = \frac{{4\pi }}{3}\tau \left( H \right)\left( {r_o^3 - r_i^3} \right)\end{equation} \tag{ 2 }$$

where r_i is the inner radius, r_o is the outer radius and T is the total torque. As the experimental setup has two interaction areas (top and bottom), each gap contributes only half of the total torque.

The magnetic field inside the magnetic circuit and especially in the interaction area cannot be measured by any method without affecting the resulting field strength and thereby the fluidic properties. To compensate for this missing measured variable in the best possible way, the field distribution is calculated by finite elements method (FEM). For the MRF material properties, the magnetization B(H) from the manufacturer's data sheets is used. Figure 4 shows the resulting field strength (left) and the magnetic flux density (right) of the FEM calculation for a coil current of 1 A. The field strengths are highest in the working gap and the main part of the magnetic circuit remains below the saturation limit of about 2.2 T, therefore little stray flux is generated in the rest of the MRF chamber, and no saturation effects are to be expected.

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In figure 5 the relationship between coil current and magnetic field strength, evaluated in the upper working gap, can be seen. According to our simulation (using material data from the data sheet), saturation only sets in at over 900 000 A m⁻¹ while our measurements are performed at field strengths up to 535 100 A m⁻¹.

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Mixing is performed as described in section 2.1 and the measurements are carried out using the experimental setup (figure 3). The measurements take place at room temperature (23 °C). The disc is rotated at a constant angular velocity of ω = 180° s⁻¹ for all measurements to avoid shear rate dependent influences, which we do not want to consider in this work. With a gap width of h = 500 µm and a mean diameter r = 15 mm of the interaction area this results in an average shear rate of ${\dot \gamma}$ ₀= 94.2 s⁻¹.

To characterize the MRF of interest, five different magnetic field strengths are selected to investigate the impact on the magnetic field on the MRFs properties. As the measurement data is scattered and random outliers should not be included in the interpretation, a large amount of measurement data were collected and statistically analyzed to obtain representative statements. As known from the literature [1, 8], temperature and ageing of the MR fluid play a significant role in the MR effect. To spread the influence of these evenly over all field strengths and to keep the relative deviation small, the field strengths are run in ascending order from the lowest field strength H₁ = 61.4 kA m⁻¹, referred to as stage 1 (S₁), to the highest field strength H₅ = 535.1 kA m⁻¹, called stage 5 (S₅). A series of measurements from S₁ to S₅ is referred to as an epoch. These epochs are repeated 25 times for statistical data collection without changing the MR fluid. After every five epochs from S₁ to S₅, a pause is made to store the data.

In the measurement, a servomotor drove the rotor until the shear stress ${\tau}$ reached a steady state, which depends on the magnetic field applied. The current to the field coils is applied during the entire measurement state starting at an angle Ψ_i = 0° and ending Ψ_end corresponding to the above mentioned steady state. After a measurement, the magnetic circuit had to be demagnetized to avoid effects of remanence. To that end, a generalized demagnetization routine was developed, that removes not only the magnetic remanence but also the 'mechanical remanence' (shear history effect), where we refer to the combined effect as "magneto-mechanical remanence effect".

Extensive material characterization experiments have shown that the shear history effect is significant in our experiments. For this reason, an attempt was made to put the system into an initial state where the history should have little influence. This was largely achieved by a general demagnetization routine that reduces the magnetomechanical remanence by magnetic demagnetization with exponentially decreasing magnetization amplitude and by mechanical motion. Bansevičius et al [34] report experiments in which pure magnetic demagnetization is not sufficient to return the particle structures to a desired random arrangement, but rather an additional mechanical excitation is required, which is consistent with the experience made in our experiments.

However, even with the generalized demagnetization routine discussed below, a small remanent effect remains such that the 'demagnetized' MRF appears slightly more viscous for vanishing magnetic fields compared to the 'virgin' material.

We thus aimed at identifying an effective demagnetization routine returning the system to the exhibiting the same friction or viscosity as in the initial ('virgin') demagnetized initial state, in which the arrangement and the MR fluid does not exhibit any remanence. In order to evaluate the demagnetization routine we identified and used in our subsequent measurements, the magnetic circuit (including the MRF) was magnetized using an 1 A coil current, which corresponds to a magnetic flux density of B = 1.60 T, where in parallel to the magnetization, seven revolutions at a reference angular velocity of 180° s⁻¹ were performed, which corresponds to an average shear rate of ${\dot \gamma}$ ₀ = 94.2 s⁻¹. After this, the driving motor and the magnetization current were switched off. Using this routine, the required shear stress ${\tau}$ after seven revolutions was always the same such that we consider this as a well-defined reference state for the magnetized medium.

To evaluate to what degree the MRF returned to the initial ('virgin') state, we measured the shear stress ${\tau}$ required to maintain an angular velocity of ω= 180° s⁻¹ (${\dot \gamma}$ ₀= 94.2 s⁻¹) without any magnetic field — we refer to this shear stress as 'idle shear stress' in the following.

In the simplest case (case 1), no demagnetization routine is performed at all. Alternatively (case 2), a standard demagnetization procedure is performed, however without any rotation present. To that end, an alternating current with a frequency of 10 Hz and an initial amplitude of 1 A and an exponential decay (using a time constant of about 2 s and switching of the current after 5 s) was applied to the coils. Finally, as case 3, a generalized demagnetization routine using the same decaying current, but incorporating a continuous rotation of the system at a rate of ω= 180° s⁻¹ (equivalent to ${\dot \gamma}$ ₀ = 94.2 s⁻¹) for a duration of five complete revolutions (1800°) was evaluated.

The effect of the various demagnetization tests are shown in figure 6, where ${\tau}$ ₀ denotes the idle shear stress for the virgin MRF, i.e. an MRF that has neither magnetically nor mechanically been excited. Due to system-related concentricity issues, which caused minor variations of ${\tau}$ ₀ during one rotation, an average value over time was determined, which is ${\overline \tau}$ ₀ = 0.4 kPa. For the three cases considered, after full magnetization and subsequent application of the routine as outlined above, the associated ide shear stresses, ${\tau}$ ₁ to ${\tau}$ ₃, have been determined. For each routine, the test was performed four times to demonstrate the robustness of the measurement data and to be able to calculate a time and ensemble average of each to eliminate concentricity problems.

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The characteristics of ${\tau}$ ₁ (red) is associated to the case neither demagnetization nor mechanical actuation was performed, resulting in a high idle shear stress of ${\overline \tau}$ ₁ = 5.8 kPa, because the residual magnetic field still affects the particle structures. For the second case, i.e. ${\tau}$ ₂ (yellow), magnetic demagnetization was performed at ω = 0° s⁻¹, resulting in ${\overline \tau}$ ₂ = 1.6 kPa, which is obviously significantly lower than ${\tau}$ _1, because no residual field influences the particle structures and therefore the viscosity is lower. Finally, magnetic and mechanical excitation (ω = 180° s⁻¹) was performed in the third case associated with ${\tau}$ ₃ (blue) resulting in an even lower idle shear stress of ${\overline \tau}$ ₃ = 1.3 kPa, since not only the magnetic but also the mechanical remanence is reduced.

To test the difference in shear stresses between ${\overline \tau}$ ₂ and ${\overline \tau}$ ₃ for statistical significance, an unpaired t-test was performed, giving a p-value of 0.0013 and a t-value of 5.158. Using conventional criteria, this difference is considered highly statistically significant at p < 0.01.

Demagnetizing in motion supposedly breaks any microstructural chains mechanically and demagnetizes the entire magnetic circuit (incl. MRF) efficiently. In addition, the mechanical rotation induces a flow that apparently brings all particles (including those in the 'dead space', i.e. the volume in the MRF chamber, which is not penetrated by the magnetic field) into the area of the working gap, thus applying the demagnetizing process to these particles as well. One could expect that demagnetizing while standing and then rotating should have the same effect on the chains. However, this is not exactly the case, as can be seen in the comparison of curves ${\tau}$ ₂ and ${\tau}$ ₃, since in the second case the rotation associated with the measurement of the idle shear stress corresponds to mechanical actuation albeit after demagnetization. In addition, demagnetizing in motion results in a more time efficient measurement routine.

Note however, that the mean value of the initial 'virgin' shear stress before the first test cycles ${\tau}$ ₀ is lower than ${\tau}$ ₃, so demagnetizing in motion comes closest, albeit not reaching the virgin state completely. We could not find any routine or other influences that reached the base shear stress ${\tau}$ ₃, only fresh, unloaded MRF yielded the same low shear stress. The difference between ${\overline \tau}$ ₀ and ${\overline \tau}$ ₃ is therefore Δτ = 0.9 kPa and is sufficiently small in view of our typical measurement range (up to 125 kPa). The remaining difference Δ${\tau}$ is an indication of an irreversible activation of the MR fluid, which is described in the next section.

In many attempts to characterize the MR fluid and to find a generalized demagnetization routine, we came across an interesting behavior: During the very first epoch, i.e. the first measurement series with successively increasing field strengths ('stages'), for each particular field strength, smaller shear stress values than in all the following epochs were obtained and these values thus could not be reproduced in later epochs. Figures 7(a)–(e), shows a continuous series of measurements, i.e. epochs 1–25 each successively involving five different field strengths H₁ to H₅ (stages), with the first epoch marked in red and the second to fifth epochs marked in black. It is important to note that between the individual stages (and thus also between epochs), the system was demagnetized using the general demagnetization routine from 3.1, which means that this effect exists independent from remanent behavior that is eliminated by our general demagnetization routine.

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As can be seen in figure 7, the shear stresses during very first epoch is clearly weaker than the following ones for all field strengths. The shear stress evaluated at 1500° is shown for each stage and epoch in figure 7(f). In this plot it is easy to see that the first epoch is significantly lower and then the shear stress increases epoch by epoch.

At low field strengths (stage 1) a steady-state level is reached more quickly, while at higher field strengths more repetitions are required to reach a steady shear stress. The closer the field strengths are to the maximum magnetization (stage 3–5), the longer it takes to reach a steady state. At the second epoch in (a) the curve already resembles the following repetitions, which can be seen by the fact that the black curves are aligned with the blue ones. At the higher field strengths (b) to (e), not only the red but also the black curves (epochs 2–5) tend to lie below the colored curves representing the later epochs.

3.2.1. Possible explanations.

A closer look at figure 7, particularly at subfigure (c) to (e), reveals that the values of the shear stress at 1500° are not stationary and continue to increase with further rotation. It can also be seen from the measured data in much of the literature that the shear stresses are not immediately stationary. On the contrary, it takes a long time to reach a steady state. We wanted to investigate this further and find a steady state shear stress at which the shear stress stops increasing. To do this we had to increase the angle of rotation Ψ_end.

Figure 8 shows the course ${\tau}$(t,Ψ) of a representative shear stress measurement out of stage S₅ at a constant angular velocity of ω = 180° s⁻¹ (${\dot \gamma}$ ₀= 94.2 s⁻¹) and a constant current of 1 A, corresponding to H₅. The shear stress initially jumps to a high value of τ_b = 62 kPa, the 'bend point' (at t ≈ 0 s). Then, contrary to expectation, the gradient is relatively high and changes to a lower gradient at t ≈ 1 s. This lower gradient further decreases and reaches a steady state value of τ= 125 kPa at t ≈ 40 s, which is twice the value of τ_b . This behavior is called long-term non-stationary behavior and is, to the best of our knowledge, not described in the literature.

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To statistically validate this unexpected behavior, we use repeated measurements (25 epochs) for each field strength (stage) as reported above, see also the description in 2.3. As already discussed, before each measurement, the generalized demagnetization routine from 3.1 was applied. For the following statistics, the first 5 epochs were excluded from the evaluation to eliminate the initial irreversible MR fluid activation described in 3.2. Figure 9 shows the corresponding shear stress curves ${\tau}$ as a function of the rotation angle Ψ, where the epochs 6–25 were averaged and plotted as mean and standard deviation. Furthermore, representative slopes at the end of the stages are plotted.

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The angle of rotation Ψ_end has been adjusted to enable approaching a steady state but kept as short as possible to avoid unnecessary temperature rise and hence ageing of the MRF. For H₃ to H₅, the steady-state value has not yet been fully reached, but the end condition of a gradient of less than 1.7 Pa/° has been achieved.

The shear stress values increase with increasing magnetic field strength. This supports the expectation from the relevant literature [13] and the manufacturer's data sheet [37]. The time taken for the shear stress ${\tau}$ to reach a steady state depends on the field strength and particularly increases with increasing field strength. At the highest field strength measured, H₅, the system takes more than 7200° (≈40 s) to reach a steady state. At the lowest field strength, H₁, the shear stress is already stationary shortly after the applied current jump. For H₂ to H₄ the times to reach steady state are between those for H₁ and H₅.

However, the shear stresses are not scalable and the shapes of the characteristics themselves are not self-similar. The longer time required at high magnetic field strengths could be attributed to the formation of non-equilibrium particle structures. Two hypotheses for this phenomenon are briefly discussed in the appendix—Hypothesis 2, Hypothesis 3.

A potential considered cause is the problem of 'in-use thickening' [1]. In-use thickening occurs after a certain energy input and causes the MRF to thicken and partially lose its desired properties. To counteract this effect in our experimental setup, the geometry of our cell features a large dead volume in the edge regions. Carlson [1] gives a formula for calculating the so-called lifetime dissipated energy (LDE). In order to perform a worst-case estimation, we consider a maximum current of 1 A at a typical angular velocity of ω = 180° s⁻¹ (${\dot \gamma}$ ₀= 94.2 s⁻¹) which results in about τ= 120 kPa. The power is therefore calculated to P = 22 W. Next, we assume that this power is applied for a total duration of 25 times 420 s (i.e. the duration of one epoch), which is a maximum use time that occurs in our measurements. The LDE is calculated as to 38 500 J cm⁻³. According to the manufacturer, any LDE values below about 10⁷ J cm⁻³ do not induce significant aging of the MRF. Therefore, in-use thickening is ruled out as a reason for this strange behavior at the first measurement.

The experimental setup has two coils whose windings feature an ohmic wire resistance of R = 8.8 Ω. The coils have an inductance of L = 0.2...1 H, depending on the experimental composition and filling. Thus, the coils with the winding resistances form a resistor-inductor circuit with a time constant ${\tau}$ _t, which yields an exponential settling behavior when a supply voltage U₀ is switched on as a step. If the values R = 8.8 Ω and L = 1 H provided above are used for an approximate calculation of the time constant, ${\tau}$ _t = L/R = 0.11 s results. After five ${\tau}$ _t the coil current should reach 99% of U₀/R, so at latest after 0.55 s, 99% of the magnetic field should be established. This, in turn, would be reached at H₅ after 100° rotation.

Since a fast increase in shear stress is necessary for crisp feedback in haptic applications, the magnetic field and therefore the current must increase quickly. That is why a specially developed power supply with current regulation is used. This current control ensures that the current reaches the set value of 1 A within 20 µs, which could be confirmed by current measurements. At ω = 180° s⁻¹ this means a rotation of approx. 3.6° until the current rise is finished.

This means that neither with nor without current controller the current rise time is responsible for the long field strength dependent slopes, which last up to more than 7,000° at 1 A.

In the following, the green curve (H₅) in figure 9 is used to discuss the shape of the curve. There are supposedly three underlying phases with three different time constants and phenomena [4, 5].

The first phase is the initial shear stress jump to the bend point (see figure 8). Initially, there are many free, unbound particles homogeneously distributed in the MRF with a concentration of 40 vol%. If the current jump is switched on, the magnetic field forms in the gap within 20 µs, which is achieved by the current controller, as explained above. Due to the magnetic forces, chains of particles are formed [3, 26, 28] quickly and abruptly that span across the gap along the field lines, thereby forming a connection of magnetically well conducting material between the two surfaces. This is fast because the particles only align themselves according to the magnetic field and do not have to perform any significant translational movements. Here, the so-called chain formation rate is high.

Due to the presence of chains, the magnetic resistance (or reluctance) of the magnetic circuit decreases locally, and the field distribution changes. The first chains formed are quickly saturated and the stray field ensures that further chains are formed until the stray field becomes low, at which point the process moves into the second phase.

The second phase of the curve covers the area from the bend point up to about 300°. Here, a medium slope with a longer time constant can be seen. There are fewer free unbound particles in the MRF, and the magnetic stray field is influenced by the initial chains and therefore lower. New chains form with a medium chain formation rate. The particles have to perform movements. This happens until the concentration of free particles, or the magnetic stray fields have fallen below a certain threshold, which still needs to be investigated in more detail. The magnetic resistance decreases, and the overall magnetic flux increases as the coil current remains constant.

The third phase has a low chain formation rate. Here, there are hardly any free particles left in the MRF or the magnetic stray field is already strongly reduced by the already existing chains, which are not magnetically saturated. Further chains are formed only slowly. Particles which are washed up from outside the working gap by the flow are integrated into chains. At this stage, the simple chains already present are linked together to form more massive and complex structures, including thicker chains and aggregates. These developments allow greater forces to be absorbed within the structures [3–7]. Here the magnetic resistance decreases further and the magnetic flux increases, this causes this formation effect to reinforce itself until equilibrium is reached at some point. This is when the steady state is reached.

Figure 10 shows the results of the FEM calculation for the magnetic field strength in the interaction areas.

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At H₁ (figure 10(a)), the stray field is mainly concentrated in the actual working gap. At low flux, proportionally fewer MRF chains are required to transport the flux across the working gap with acceptable losses. The steady state is quickly reached because after the first initial chains are formed, the stray field is no longer high and no new chains can form.

At H₅ (figure 10(b)) the flow is much larger. More chains are required for the flux to cross the gap. Many chains are formed, but the chains quickly become saturated due to the high flux. Even outside the interaction area, a stray field can still be seen. The lack of a steady state, or the fact that it takes so long at high magnetic field strengths, may be due to the formation of non-equilibrium particle structures.

For H₁ the scattering range is smaller than for H₅. This results in a larger virtual interaction area for H₅. Particles magnetically attracted by the field or carried by the fluid flow remain in the working gap and form new chains that contribute to the shear stress.

In short, a larger current means more magnetic flux, which results in a higher magnetic field. This forms more iron chains, which in turn decreases the magnetic resistance. This results in a higher flux. Overall, this increases the number of chains and the shear stress. With higher coil currents, this process takes longer due to the chain formation rate until an equilibrium is reached.

By measuring coil current and voltage during the measurements, it is also evident that the inductance L of the experimental setup increases. This supports the assumption that iron accumulates in the gap. This will be investigated in more detail in continued research.

In addition, by opening the experimental setup and inspecting the MRF chamber, particle accumulation within the working gap became visible. This observation is currently limited to a mere qualitative observation since it can only be made after disassembling the setup and it would therefore require the development of a methodology for quantitative assessment in a reproducible manner. This is particularly challenging as it is hardly possible to 'recycle' the entire MRF after disassembly (as part of the liquid gets lost) and use it again in a new measurement trial (after suitable homogenization). In any case, it is believed that the accumulation of particles within the working gap contributes to an increased iron concentration, which subsequently leads to the formation of complex structures and aggregation and therefore is a major contribution to the aforementioned irreversible MR fluid activation.

Finally, we note that the hypotheses cannot be simply verified or proven experimentally, at least not with reasonable efforts. Therefore, complex simulations, developing a method to detect changes in concentration and dedicated experiments have to be developed and performed to confirm or disconfirm these hypotheses, which, however, goes far beyond the focus of this paper.