## Abstract

We use a recent experimental technique to measure in situ shear and normal stresses during magnetorheological finishing (MRF) of a borosilicate glass over a range of magnetic fields. At low fields shear stresses increase with magnetic field, but become field-independent at higher magnetic fields. Micromechanical models of formation of magnetic particle chains suggest a complex behavior of magnetorheological (MR) fluids that combines fluid- and solid-like responses. We discuss the hypothesis that, at higher fields, slip occurs between magnetic particle chains and the immersed glass part, while the normal stress is governed by the MRF ribbon elasticity.

©2010 Optical Society of America

## 1. Introduction

Magnetorheological finishing (MRF) is a sub-aperture polishing process for fabrication of precision optical surfaces. It has been applied to polishing various types, shapes, and sizes of materials without introducing subsurface damage. The MRF process is based on a magnetorheological fluid that consists of magnetic carbonyl iron (CI) particles, non-magnetic polishing abrasives, and water or other non-aqueous carrier fluids and stabilizers. The MR fluid, in the form of a convected ribbon, stiffens in the presence of a magnetic field to form a stable, sub-aperture polisher. Optical parts are moved through the polishing zone to polish the surface.

In a recent article we have demonstrated an in situ method that allows the direct measurement of drag and normal forces during MRF [1]. Measurement of the contact area with optical metrology allows the extraction of the shear and normal stresses, as well as of the average and maximum material removal rates [1,2]. The technique has been applied to soft and harder glasses, as well as to optical ceramics. We have also studied the dependence of the shear and normal stress and material removal rate on process parameters such as abrasive nanodiamond concentration, penetration depth, magnetic field strength, and the relative velocity between the part and the rotating MR fluid ribbon [3]. We refer to these publications for an extensive discussion of this method and comparison with previous work in the literature [1–3]. For the first time, this approach provides a direct evaluation of the interaction between the MR fluid and the part, and thus focuses on the fundamentals of the MRF removal process.

The effects of magnetic field on MRF has attracted some attention. For example, Schinaerl et al. [4] reported on the measurement of the normal forces as a function of the magnetic field, but not the resulting contact (normal and shear) stresses. The in situ measurement of stresses provides additional insights. The work by Miao et al. [2,3] reported the interesting observation that the measured shear stress and material removal rates were practically independent of the applied magnetic field leading to a magnetic induction in the range 2.2 to 2.7 kG (in the absence of a MR fluid but with the magnets under steady operational currents.) From the industrial point of view this is an important observation, demonstrating that the MRF process performance metrics are independent of the magnetic field when the field is sufficiently high to produce a stiff ribbon. On the other hand, it is well known that the yield stress of MR fluids depends quadratically on the applied field.

The goal of this report is to put together the framework to understand how the intensity of the magnetic field in MRF affects measurable performance metrics, such as material removal rate and surface shear stress. A further goal is to identify the micromechanical response of the MR fluid as a function of the applied magnetic field. Sections 2 and 3 present the experimental details and measurements, while section 4 discusses two models of magnetorheological fluids and the applicability of these models to MRF.

## 2. Experimental details

#### 2.1 Borosilicate glass

Four borosilicate glass (BK7) parts, thin flats from Schott Glass, were used for the spotting experiment. All of the substrates were round disks with a diameter of 50 mm, a thickness of 2 mm, a surface flatness less than 1 μm [5], and a root-mean-square surface roughness of less than 2 nm [6]. The mechanical properties for BK7 glass are as follows: Young’s modulus, 81 GPa, Vickers hardness, 6 GPa, fracture toughness 0.8 MPa∙m^{1/2}.

#### 2.2 Magnetorheological finishing spot taking

The spot-taking machine (STM) was used as the research platform to take spots on BK7 glass [3,7]. The standard aqueous MR fluid (made by us in our MRF group), containing carbonyl iron (CI), nanodiamond abrasives, deionized water and stabilizers, was used as the finishing slurry. The MR fluid contains 45 vol.% of CI particles, which has an average size of 4 μm. Screening experiments were conducted to obtain the range of magnet currents where a constant ribbon height could be obtained across the whole magnet current range by adjusting pump rate while wheel speed was kept constant. The screening experiment showed that when the magnet current varied from 5 A to 22.5 A, a constant ribbon height could be achieved by adjusting pump rate.

The settings applied in this experiments were as follows: the mixing rate 1000 rpm, the ribbon height 1.7 mm, penetration depth 0.3 mm, wheel speed 150 rpm, with magnet current varied from 5 A to 22.5 A at an increment of 2.5 A. Pump speed was adjusted to keep the ribbon height constant at 1.7 mm. The ribbon height is measured and preset by the numerical control system associated with the STM with an in-line sensor at an accuracy of 0.1 mm. It is kept the same for various magnetic field strengths by adjusting the pump rate. The viscosity was held at 45 cP. The spots were placed at 8 mm away from the center of the part. Parameters were not quite the same as those used in previous work [3]. These changes (higher ribbon height and lower wheel speed) were dictated by the requirement for operating the STM over a larger range of magnetic fields.

One spot at each magnet current setting was placed on a BK7 part on day 1, day 2 and day 3, respectively. Therefore, in total three spots were taken at each magnet current setting. The manipulation of magnet current was randomized to minimize any systematic errors in the measurement. We found that, as expected, the magnetic field strength, measured with a Gauss meter at a location of 0.3 mm above the apex of the wheel [2], is linearly proportional to the magnet current. A higher magnet current setting results in a stronger magnetic field strength. The interpolation of the magnetic field strength to low magnet current shows a nonzero intercept at zero magnet current, implying that there might be remanent magnetization on the wheel. The magnet currents of 5-22.5 A correspond to a magnetic induction of 0.98-3.08 kG, respectively (magnetic fields in the range 78-245 kA/m in SI units).

#### 2.3 Characterization

A Zygo Mark IV laser interferometer was used to measure the deepest depth of penetration (ddp), volume, and area of the spots taken at various magnet currents. As discussed in Section 2.1, the parts were flats with surface flatness less than 1 μm. Initial surface figure of the part was measured before taking each spot. Surface figure was re-measured after spotting. Subtracting initial surface figure from post surface figure produces a reliable and accurate influence function of the spot, eliminating the initial surface figure error. Drag force (*F _{d}*) and normal force (

*F*) were recorded using a dual load cell, see ref [3]. for details. Peak removal rate (PRR) was calculated by dividing the ddp by spotting time. Volumetric removal rate (VRR) was obtained by dividing the spot volume by spotting time. The experimental data are reported as an average of three measurements taken on day 1, day 2 and day 3.

_{n}## 3. Experimental results

The MRF spotting experiments were conducted within a three-day period after the MR fluid was loaded into the STM machine. Table 1
gives the experimental data including the depth of deepest penetration (ddp), spot area (*A _{s}*), spot volume, drag force (

*F*), normal force (

_{d}*F*), shear stress (

_{n}*τ*), pressure (

*p*), the measured ratio of drag force over normal force (

*F*/

_{d}*F*), peak removal rate (PRR, defined by the ratio of ddp to spot time) and volumetric removal rate (VRR) for spots taken on BK7, for various magnet current settings. The data are given as an average of three spot measurements. At very low magnet current, the MR fluid ribbon is very soft, resulting in very low removal rate and a shallow spot. If the spot is too shallow, it is hard to identify by the interferometer in the measurement of influence function. 2s or 3s spot produces a reasonably deep spot. However, at high magnet current, the ribbon is stiff, which tends to result in a higher removal rate and deeper spot. This will lead to drop out on the interferometer figure map. Therefore, spot time is reduced to obtain a reasonably deep spot. The adjustment of spot time does not affect F

_{n}_{d}, F

_{n}, PRR or VRR as discussed in [2].

Both drag force and normal force remain almost constant at various magnet currents. This is similar to what was found in our previous work [3] where normal force and drag force remained constant or decreased slightly with magnet current. The difference in the magnitudes of forces measured here and in prior work [3] is attributed to the different wheel speed used in the two experiments. The wheel speed in the present work is 150 rpm, 50 rpm lower than that in prior work [1]. In addition, the ribbon height and fluid viscosity are also different in these two experiments, which may contribute to the difference in measured forces.

Shear stress, drag force divided by spot area, is shown against magnet current in Fig. 1 . Increasing magnet current leads to an increase in shear stress when magnet current is in the range of 5-15 A, presumably because the ribbon stiffens. However, when the magnet current increases to a certain level, above 15 A (2 kG), increasing the magnet current has no impact on the stiffness of MR fluid ribbon. Similar effects are observed in the normal stress, see Fig. 2 .

In Table 1, our data for shear stress, pressure, PRR and VRR shows a consistent trend that all these quantities increase with magnet current (or magnetic field strength) before it reaches a critical point, 12.5 A, where they start to level out. The influence of magnet current on MRF removal is complex since varying magnet current affects the width and stiffness of the ribbon, and also the interaction metric between the ribbon and the part surface. This paper contributes to the understanding of this missed aspect.

Figure 3 shows the peak removal rate (PRR) and volumetric removal rate (VRR) as a function of magnet current. Like shear stress, both peak removal rate and volumetric removal rate increase with magnet current when it is below 12.5 A (1.88 kG). When the magnet current is below 12.5 A (1.88 kG), the ribbon is soft. Increasing the magnet current stiffens the MR fluid ribbon, therefore, resulting in an increasing peak/volumetric removal rate. However, additional increases in magnet current do not lead to an increase in either peak removal rate or volumetric removal rate. This can be explained by a trade-off effect as we discussed in prior work [1]. The increase in magnet current, on one hand, stiffens the MR fluid ribbon, and hence increases removal rate. On the other hand, increasing magnet current narrows the MR fluid ribbon and reduces the spot size, which causes a decrease in removal rate.

## 4. A model of magnetorheological fluids under MRF conditions

We now turn to a discussion of the dependence of the measured shear and normal stresses on the magnetic field. We first start by identifying the magnetic response of the CI particles.

Jang et al. [8] have measured the magnetization curve for CI particles used in an industrial magnetorheological fluid. This fluid is water-based, and contains CI particles with mean diameter 1.1 µm and range of diameters 0.5-2.2 µm. Apart from size, these CI particles are not very different from the CI particles used in our STM platform. The calculated magnetic susceptibility χ = M/H is shown in Fig. 4 . The saturation value of the polarization is Js = 2.05 Tesla. The low-field susceptibility is similar to that measured by Gorodkin et al. [9]

Based on these data, we conclude that in our experiments the magnetic induction is 0.98 to 3.08 kG, i.e. the magnetic field H in SI units is 78 to 245 kA/m, the polarization (µ_{0} M) is in the range 0.85 to 1.75 T, and the magnetic susceptibility is in the range 8.6 to 5.7. We now turn to the prediction of the yield strength of the MR fluids.

A way of modeling MR fluids is via a Bingham stress-strain-rate law described by a yield stress τ_{Y} and a post yield viscosity η_{p} (for plastic viscosity.) The question we address in this section is the dependence of the yield stress τ_{Y} on the magnetic field H. We have used two models to predict the mechanical response of MRF fluids. The model by Jolly et al. [10] is analytical, while the model by Ginder and Davis [11] is based on a finite element computation.

Jolly et al. [10] proposed a model of the yield strength of a magnetic fluid, based on the extension and eventual breaking of magnetic particle chains. Yield is assumed to occur when the force-displacement relation of a chain of magnetic particles reaches a maximum value. In this model, the saturation in the magnetization of the CI particles is explicitly accounted for: near the area where two CI particles touch, the magnetization is well into the saturated region, whereas closer to the particle center the magnetization is still dependent on the magnetic field H. The predicted yield stress τ_{Y} (units Pa) is

_{p}is the solid CI particle polarization, µ

_{1}is the surrounding fluid relative permittivity ( = 1 for MRF applications), and h is a dimensionless particle-to-particle distance (measured with respect to the particle diameter). For the case of touching chains it is given by

The particle polarization J_{p} is given by

In this expression, α is a parameter that gives the relative size of the saturated region to the unsaturated region of a particle, and is found from the fact that the flux is continuous across the saturated and unsaturated portions of the CI particle. For our data, we found that the parameter α varied from 1 at low magnetic fields (i.e. no magnetic saturation within the CI particles) to about 0.65 for a magnetic field of 245 kA/m.

The important point in this model is shown in Eq. (3). The particle magnetic polarization J_{p} is driven by two contributions. The first contribution depends on the magnetic induction B, and the second on the particle saturated polarization J_{s}.

Jolly et al. [10] also show that, for strains lower than at yield, the MR fluid has an elastic modulus because, before yielding, the fluid behaves like a non-linear elastic solid. Jolly et al. compute a shear modulus G, which we convert to a Young’s modulus E = 3G assuming incompressibility. The result is

The elastic modulus can be converted to a normal stress by multiplying by the average compressive strain caused when the glass part is depressed into the MR fluid ribbon. In our experiments, the part is depressed by 0.3 mm and the total fluid height is 1.7 mm, i.e. the average strain is (2/3)(0.3 mm/1.7 mm). The factor of (2/3) in the average strain arises due to the circular ribbon shape and the flat shape of the work. Therefore the normal stress is

Finally, we can compute an apparent friction coefficient, i.e. the ratio of shear stress to normal stress.

Ginder and Davis [11] have adopted a model similar to that of Jolly et al. [10] However, Ginder and Davis compute the magnetic field details by using finite element methods, therefore do not need to make any special assumptions about the regions that are saturated or unsaturated. In their model, they use a saturation magnetization µ_{0} Ms = 2 Tesla and a low field relative permeability of 1000 (representative of pure iron), and compute the yield stress τ_{Y} and shear modulus G.

The results by Ginder and Davis are cast in terms of the average magnetic induction B_{ave} (Tesla). Their results are also for CI volume fractions ϕ = 0.2 and 0.5. We have calculated the average induction B_{ave} in our data from

Ginder and Davis [11] also predict a shear modulus G, which we first convert to Young’s modulus, and then convert to an average normal stress based on a compressive strain of (0.3 mm / 1.7 mm), i.e. governed by the amount of the glass depression into the MRF ribbon as compared to the ribbon height, as indicated in Eq. (5).

Figures 5(a)
–5(c) show the dependence of the fluid yield stress τ_{Y}, fluid normal pressure σ_{n}, and their ratio on the applied magnetic field H, as well as our measurements.

From Fig. 5(a) we infer that the predicted shear stress is within 15-20% of our measurements at low fields (H < 150 kA/m), but not so at higher fields. At higher fields the measured shear stress is independent of magnetic field, but the models predict an increasing yield stress. We conclude that at lower magnetic fields the shear stress at the glass/fluid interface is controlled by the fluid yield stress, i.e. the fluid is yielding.

Figure 5(b) shows that the measured fluid normal stress is well correlated with the model predictions that reflect the amount of depression of the part (0.3 mm) into the 1.7 mm high ribbon. We conclude that the measured fluid normal stress is governed by the elasticity of the MR fluid and the amount of depression of the part into the ribbon height.

The ratio of the shear and normal stresses, shown in Fig. 5(c), combines these two effects, i.e. the fluid yield stress and the normal pressure governed by the fluid elasticity. The model predictions overestimate by about 30% our measured ratio of shear to normal stress. We observe that the Jolly model, because of its inherent assumptions in the computation of the local magnetic field, predicts a constant ratio of shear to normal stresses.

## 5. Discussion

The comparison of our measured shear and normal stresses in the previous section showed that at lower magnetic fields the measured shear stress is very close to the fluid yield stress, i.e. the fluid yields within the contact region. However, at higher fields, the measured shear stress is independent of the magnetic field. On the other hand, the normal stress is entirely dependent on the fluid elasticity and the average strain due to the immersion of the part into the MRF ribbon.

It is important to observe here that the two models by Ginder and Davis [11] and Jolly et al. [10] predict that, at high magnetic fields, the fluid yield stress saturates, i.e. reaches a value that is independent of the magnetic field, and only determined by the saturated magnetization of the particles. However, for a CI volume fraction of ϕ = 0.45, the Ginder and Davis model predicts that the saturated yield stress is about 180 kPa, which is much higher than the measured field-independent shear stress of about 55 kPa. We conclude, therefore, that our measured field-independent shear stress is not due to the particle magnetic saturation. The difference between the saturated model value (180 kPa) and the measured value (55 kPa) is so large that it is unlikely that magnetic saturation is the explanation of our measurements at larger fields.

Another hypothesis may be offered that also explains the observation of field-independent shear stress at the glass/ribbon interface at larger magnetic fields. This explanation cannot be captured by the magnetic particle models of MRF chains, because these models do not include the interaction of the magnetic MRF ribbon with the glass surface. It is possible that, if slip is allowed to occur at the ribbon/glass interface, then the shear stress at that interface will remain constant and governed by the mechanical interaction between the MR fluid and the glass surface. This hypothesis will explain also the fact that the field-independent shear stress depends on the material that contacts the MRF ribbon [1].

Based on this hypothesis, we may also infer a comprehensive model of how the applied magnetic field affects the interaction between the magnetic ribbon and the glass surface. At lower magnetic fields, the yield stress of the magnetic particle chains is relatively lower than the shear stresses induced by the MRF flow conditions, the magnetic chains continually break up and reform, and hence the measured shear stress is governed by the inherent yield stress of the fluid. At higher magnetic fields, the magnetic particle chains are stronger and do not yield under the action of the imposed hydrodynamic shear stresses. Now the MR slurry behaves like a solid, and the imposed rate of deformation can be accommodated only by the continual slipping at the interface of the MRF ribbon and the glass part.

On the other hand, the normal stress at the fluid/glass interface is controlled by the strain due to the penetration of the part into the MRF ribbon and by the elasticity of the MRF ribbon. We conclude that both the fluid-like yielding and solid-like elastic properties of the MRF ribbon are important in determining the shear and normal stresses that develop under MRF processing conditions.

There are important aspects of the CI particle micromechanical interactions that our approach does not include. For example, Gorodkin et al. [9] have shown that the CI particle susceptibility at low fields depends on the CI particle size, and this is a feature that is currently under investigation. Similarly, the magnetization curves of various types of CI particles over the full range of magnetic fields from the initial linear response to full saturation is an important feature [8], but we expect that the saturation magnetization is weakly dependent on particle size. Given these observations, a full micromechanical model of CI particle chain formation must include size effects as well as the initial and full saturation responses

## 6. Conclusions

We have demonstrated that our in situ method of measuring the shear and normal stresses during MRF processing reveals the contribution of the magnetic field on the micromechanical interaction between the MRF ribbon and the glass part. Our data for shear stress, pressure, PRR and VRR show a consistent trend that all these quantities increase with magnet current (or magnetic field strength) before it reaches a critical point, 12.5 A, where they start to level out. The influence of magnet current on MRF removal is complex since varying magnet current affects the width and stiffness of the ribbon, and also the interaction between the ribbon and the part surface. This paper contributes to the understanding of these effects.

Ribbon height is maintained constant at various magnetic field strengths. A high magnet field narrows the fluid ribbon and makes the ribbon stiffer, resulting in a much smaller polish zone. These effects may explain an almost unchanged VRR, Fd, and Fn.

Over the range of magnetic fields we tested, we have observed both fluid-like and solid-like behavior of the MR fluid. Our data are consistent with a view of MRF processing that depends on the applied magnetic field. For lower fields, the MR fluid consists of magnetic particle chains that continually yield under the action of the flow induced shear stresses. As a result, the measured shear stress at the ribbon/glass interface is controlled by the fluid yield stress, and the fluid MR fluid indeed behaves like a fluid. For higher magnetic fields, the magnetic particle chains are too strong to yield under the action of the hydrodynamic shear stresses, and the measured shear stress is independent of the magnetic field. We have offered the hypothesis that, at higher magnetic fields, the MR fluid behaves like a solid and now slip occurs between the MRF ribbon and the glass interface.

Our comparisons of data and existing models of magnetic particle chain mechanical properties also show that our measured normal stress is governed by the elasticity of the MRF ribbon and the amount of penetration of the glass into the ribbon. These conclusions point to a more complex behavior of MR fluids that, under magnetic fields, exhibit both fluid- and solid-like responses. This work, therefore, provides a fundamental understanding of how MR fluid behaves under varying magnetic field during MRF.

## Acknowledgements

This work was supported by the U.S. Department of Energy Office of Inertial Confinement Fusion under Cooperative Agreement No. DE-FC52-08NA28302, the University of Rochester, and the New York State Energy Research and Development Authority. The support of DOE does not constitute an endorsement by DOE of the views expressed in this article.

## References and links

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**5. **Zygo Mark IVxp interferometer, Zygo Corp., CT. This instrument is a four inch HeNe Fizeau interferometer with a wavelength of 632.8 nm. Peak-to-valley (pv) for surface flatness and depth of deepest penetration (ddp) of the spot was measured in microns. The spot is expected to be less than 0.2 μm deep for achieving a good measurement, and spotting time was adjusted to stay below this upper limit.

**6. **Zygo New View 5000 noncontacting white light interferometer, Zygo Corp., CT. The surface roughness data were obtained under the following conditions: 20 × Mirau; high FDA Res.; 20 μm bipolar scan length; Min/Mod: 5%, unfiltered.

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