Research on chain-model transition identification of magnetic dipole theory for magneto-rheological fluid

Abstract

Chain formation model is very useful to characterize the magneto-rheological phenomenon and prepare good magneto-rheological fluids. The single-chain model is common to explain the process of chain formation for ferromagnetic particles under magnetic field. With the increment of magnetic field and ferromagnetic particle content, the chain will transit from the single chain to multi-chains. However, there are few literatures involved in this phenomenon. This study investigates the effect of magnetic field and ferromagnetic particles content on the transition. The static yield stresses at different magnetic fields were measured under quasi-static mode for different magneto-rheological fluid samples. The results show that the transition of chain model can be identified on two parameters including the amplitude of static yield angle, ${\mu }_y$ , and the inclined angles distribution, $\sigma$ . The single-chain model is only effective under low magnetic field and ferromagnetic particles below content of 30% volume. With the increment of magnetic field and ferromagnetic particles content, the transition from single chain to multiple chains will be observed, which validates that there is a transition from the single chain to multi-chains.

Introduction

Magneto-rheological fluid (MRF) is a kind of smart material.^1,2 MRF possesses the salient characteristics such as rapid response, reversible, and low power requirement, which has some primary engineering applications.^3–5 To depict the mechanical rheological property of MRF, there are two categories of models: continuous models^6,7 and discrete models^8,9 in general. Comparing with the former ones, the latter models are based on the basic physical law, and thus, it is more general. ¹⁰ Within the discrete models, the chain mechanism of ferromagnetic particles based on the magnetic dipole theory is a relatively effective method to depict the micro-macro mechanism of magneto-rheological (MR) material. Great efforts have been made in the analysis and prediction of the dynamic behavior of MRF with magnetic dipole theory. For example, Bossis and Lemaire ¹¹ developed a dipolar-sphere model based on the calculation of forces between two particles without considering the influence of nonlinear magnetization. Ginder and Davis ¹² applied the Frolich–Kennelly equation to depict the nonlinear magnetization when further to investigate the model. Li et al.^13,14 performed the numerical simulation of the chain-formulation process of MRF. Lemaire et al. ¹⁵ studied the effect of Brownian forces on the internal structure of the aggregates. To investigate the angle distribution of inclining chains in MRF, Peng and Li ¹⁶ proposed a micro-macro model. Guo et al. ¹⁷ proposed a yield shear stress model with an exponential distribution of angles. The mentioned above belongs to the single-chain model which is simple and effective to understand the dynamic rheological behavior. However, the single-chain model neglects the interaction among the adjacent chains. In fact, the formed chain structures are changing dependent on magnetic field and ferromagnetic particles content.^18–20 Xu et al. ²¹ proposed a kind of double-chain model and the results were more accurate than single-chain model while characterizing an MR damper. Tao ²² discovered that the built chain structures had a body-centered tetragonal lattice structure in some cases. It can be found that both the single-chain model and the complex multi-chain models can effectively describe the micro-dynamics of ferromagnetic particles in the presence of magnetic field in some cases. It is natural to hypothesize that there might be a transition from the single chain to multi-chains. However, there are few researches about this.

Consequently, the purpose of this study is to figure out the transition mechanism of micro-dynamics of ferromagnetic particles under magnetic field and its impact factors through the experimental method for MRF. To implement the purpose, the static yield shear stress of MRFs was first analytically derived on the basis of the magnetic dipole theory. A kind of quasi-static method was then used to determine general inclined angles of built chains with MCR 301 magnetorheometer. The impact factors of transition mechanism of micro-dynamics of ferromagnetic particles from single to multi-chains are analyzed. At last, the distribution of the inclined angles of the chains was applied to find out the transition point from single chain to multi-chains and predict yield stress. The study consists of four sections. The following section will address the research background and some theoretic foundation. Experiments and the corresponding results and discussions are performed in section “Experiments.” The conclusion is made in section “Conclusion.”

Background

The model used in this study derives from the research of Peng and Li. ¹⁶ The uniformly sized spherical particles disperse in carrier oil acting as dipolar particles which aggregate to single chains aligning along the local magnetic field, H. Each particle is assumed as a sphere with same radius R and owns a magnetic moment m under corresponding field with magnetic susceptibility $\chi$ . The induced shear force T(θ) of an intact single chain, which attaches two plates by their two ends, with N particles inclining with θ, shown in Figure 1, is

$$\begin{array}{cc}\multicolumn{1}{c}{T(\theta )}& =\sum _{k=1}^{N-1}\frac{3{\mu }_0{m}^2}{4\pi r_k^4}(5{\cos }^2\theta -1)\sin \theta \hfill \\ \multicolumn{1}{c}{}& =\sum _{k=1}^{N-1}\frac{4\pi {\mu }_0{R}^6{\chi }^2{H}^2}{3k^4{(2R+\delta )}^4}(5{\cos }^2\theta -1){\cos }^4\theta \sin \theta \hfill \\ \multicolumn{1}{c}{}& =\frac{4\pi A{\mu }_0{R}^6{\chi }^2{H}^2}{3{(2R+\delta )}^4}(5{\cos }^2\theta -1){\cos }^4\theta \sin \theta \hfill \end{array}$$

(1)

where $A=\sum _{k=1}^{N-1}1/k^4\approx 1.08$ , ${\mu }_0=4\pi \times {10}^{-7}\mathrm{kg}\mathrm{m}{\mathrm{s}}^{-2}{A}^{-2}$ , $m=(4/3)\pi R^3\chi H$ , and δ is the space between two neighboring particles when applying a vertical magnetic field without shearing, and $r_k$ is the center distance between the first and the kth particle in a single chain.

Figure 1.

Stretch of formed and deformed single chain.

Concerning that not all chains incline with the same angle, when subjected to magnetic field and shearing, normal distribution of θ is introduced. Therefore, one obtained the shear stress $\tau \prime$ from chains with particle volume fraction φ as follows

$$\begin{array}{c}\tau \prime =\underset{-\pi /2}{\overset{\pi /2}{\int }}T(\theta )•P•D(\theta )d\theta \hfill \\ =\frac{4\pi {\mu }_0\mathrm{PA}{R}^6{\chi }^2{H}^2}{3{(2R+\delta )}^4}\hfill \\ \times \underset{-\pi /2}{\overset{\pi /2}{\int }}(5{\cos }^2\theta -1){\cos }^4\theta \sin \theta \frac{1}{\sqrt{2\pi }\sigma }{e}^{-\frac{{(\theta -\mu )}^2}{2{\sigma }^2}}\mathrm{d}\theta \hfill \\ =\frac{{\mu }_{0}A{R}^3{\chi }^2{H}^2\phi }{{(2R+\delta )}^3}\hfill \\ \times \underset{-\pi /2}{\overset{\pi /2}{\int }}(5{\cos }^2\theta -1){\cos }^4\theta \sin \theta \frac{1}{\sqrt{2\pi }\sigma }{e}^{-\frac{{(\theta -\mu )}^2}{2{\sigma }^2}}\mathrm{d}\theta \hfill \end{array}$$

(2)

where µ is the mean and σ is the standard deviation which depict the general inclination and distribution of inclined angles of all chains, respectively, and P accounts for the number of chains in per unit area with $P=3\phi (2R+\delta )/(4\pi R^3)$ , and $D(\theta )=1/(\sqrt{2\pi }\sigma )e^{-{(\theta -\mu )}^2/2{\sigma }^2}$ .

In order to apply equation (2) to different MRFs, Song et al. ²³ and Tian et al. ²⁴ added shear stress $\tau ″$ produced by carrier medium when calculating MRFs’ total shear stress $\tau$ , that is

$$\tau =\tau \prime +\tau ″$$

(3)

For determination of magnetic susceptibility $\chi$ of particles, Frohlisch–Kennelly equation ¹² is adopted

$$\chi (H)=\frac{{\chi }_0{M}_s}{{\chi }_{0}H+M_S}$$

(4)

to calculate $\chi$ under a certain local magnetic field, H, where $M_S$ is saturation intensity of magnetization and ${\chi }_0$ is initial magnetic susceptibility. For pure iron, ${\mu }_0{M}_s=2.1\mathrm{T}$ and ${\chi }_0$ could be set as 1000 reasonably. ¹²

In above equations, local magnetic field H is a critical factor. However, in practical measurement result, external magnetic field $H_0$ could be got more easily than H. The relation between $H_0$ and H is given by^15,25

$$H=\frac{H_0}{{\mu }_S(H)}$$

(5)

where ${\mu }_S(H)$ accounts for the average magnetic permeability of the suspension and could be estimated by the Maxwell-Garnet theory ²⁶

$$\frac{{\mu }_S-1}{{\mu }_S+2}=\phi \frac{\nu -1}{\nu +2}$$

(6)

$$\nu =1+\chi$$

(7)

where $\nu$ is the internal permeability of the iron particles. Combining equations (5) and (6), H could be confirmed approximately.

There are inevitably various dimensional and shaped particles, which result in the difficulty about the confirmation of δ. According to equation (2), let

$$c=\frac{\xi (R,\delta )}{{\xi }_0}=8\times {\left(\frac{R}{2R+\delta }\right)}^3$$

(8)

where $\xi (R,\delta )=(R/(2R+\delta ){)}^3$ , ${\xi }_0=\xi (R,0)=1/8$ . It is easily known that c could be close to 1 with a big R and small δ, for example, when R = 5 µm and δ = 0.01 µm, c = 97%. Therefore, for simplifying the simulation, δ = 0 is assumed when applying relatively big particles.

Experiments

Materials and measurements

The base carrier medium are lubricating grease which are made from dimethyl silicone oil (350cst, Dow Corning (Zhangjiagang) Silicone Co., Ltd, China) and lithium soap thickener (powder, Hin Leong Trading (Pte) Ltd, China). The base matrix and carbonyl iron (average radius R = 5 µm, the particle density is about 7.86 g/cm³, Beijing Xing Rong Yuan Technology Co., Ltd, China) are mixed. Two MRFs with 13 wt% thickener in the base greases are prepared. The produce process of the first group contains two thickening processes in 150°C and 190°C while the second one does not. This distinction results in different base carrier medium for the two group samples. According to the particle volume fraction, the first group samples are named after MR1-05, MR1-10, MR1-15, MR1-20, MR1-25, and MR1-30, while the second group ones after MR2-05, MR2-10, MR2-15, MR2-20, MR2-25, and MR2-30, respectively.

The MRFs are sheared in parallel-plate geometry with a diameter of 20 mm and a gap of 1 mm under 25°C on a commercial rheometer (MCR 301, Anton Parr, Austria) with a magneto-controllable accessory MRD 180. The tests are all in the quasi-static shear mode in which shear rate is constant and very low under different magnetic field. Initially, sample is homogenized at 50 s⁻¹ for 100 s. Then, desired magnetic field is applied without shearing for 120 s. At last, a very low shear rate is applied and the stress–time data are recorded at the same time.

Results and discussion

Yield angle ${\mu }_y$ and single-chain model area

Figure 2 shows the stress–time relationship of MR1-05 under various external magnetic field. This study is performed at a constant shear rate of 0.01 s⁻¹. The results indicate that there are three regimes: pre-yield, in-yield, and post-yield regimes. In the pre-yield, shown in the inset of Figure 2, shear stresses have a linear relationship with the time, while shear stresses are nearly unvarying in the post-yield regime. This is in accordance with the former report. ²⁷ The in-yield regime is a transitory regime between pre-yield and post-yield regimes. This regime is important for the chains from the complete chains to the broken. To describe this complex dynamic behavior easily, an imaginary yield point is introduced. The imaginary yield point is the intersection of two lines, which is shown in Figure 3. The two lines are fitted, respectively, by the pre-yield regime and post-yield regime. X-coordinate value $t_y$ of the point is used to obtain yield angle ${\mu }_y$ by equation

$${\mu }_y=\arctan \left(\frac{t_y}{\stackrel{·}{\gamma }}\right)=\arctan \left(\frac{t_y}{0.01}\right)$$

(9)

where the Y-coordinate value ${\tau }_y$ accounts for static yield shear stress. The whole values of ${\mu }_y$ are listed in Table 1.

Figure 2.

Stress–time curve at constant shear rate of 0.01/s with increasing external magnetic field of MR1-05. The inset figure shows the pre-yield regime with its fitted straight line, in-yield regime, and post-yield regime.

Figure 3.

Yield point in the shear–time curve.

Table 1.

Yield angles of MR1 and MR2.

H0 (kA/m)	µy
	MR1-05	MR1-10	MR1-15	MR1-20	MR1-25	MR1-30
10.07	0.306	0.326	0.288	0.318	0.293	0.187
23.3	0.211	0.187	0.193	0.194	0.165	0.113
30.9	0.196	0.16	0.16	0.164	0.151	0.094
42.6	0.162	0.154	0.159	0.155	0.15	0.089
51.43	0.155	0.154	0.152	0.156	0.152	0.093
61.69	0.158	0.159	0.164	0.156	0.159	0.086
76.77	0.16	0.158	0.153	0.158	0.161	0.091
91.84	0.163	0.159	0.164	0.159	0.196	0.1
108.4	0.166	0.153	0.194	0.191	0.191	0.103
125.3	0.16	0.152	0.193	0.191	0.207	0.104
140.8	0.156	0.192	0.208	0.206	0.209	0.102
157.6	0.164	0.201	0.198	0.202	0.206	Unknown
174.6	0.156	0.197	0.203	0.205	0.208	Unknown
H 0 (kA/m)	µy
	MR2-05	MR2-10	MR2-15	MR2-20	MR2-25	MR2-30
10.07	0.267	0.263	0.33	0.429	0.261	0.161
23.3	0.156	0.143	0.178	0.248	0.157	0.095
30.9	0.13	0.127	0.134	0.191	0.127	0.082
42.6	0.128	0.126	0.129	0.139	0.126	0.078
51.43	0.125	0.127	0.126	0.138	0.126	0.071
61.69	0.13	0.126	0.131	0.132	0.125	0.079
76.77	0.125	0.133	0.128	0.132	0.13	0.082
91.84	0.132	0.132	0.136	0.128	0.132	0.089
108.4	0.125	0.135	0.136	0.13	0.128	0.089
125.3	0.133	0.134	0.157	0.156	0.157	0.092
140.8	0.13	0.155	0.16	0.156	0.152	0.092
157.6	0.128	0.165	0.159	0.16	0.161	Unknown
174.6	0.135	0.158	0.16	0.166	0.162	Unknown

Figure 4 shows the variation of ${\mu }_y$ with $H_0$ increasing. After an initial reduction, there are two constant regimes for MRFs which contain particle content less than 30 vol%. The first-constant regimes become narrower gradually with particle content increasing. Especially for the 30 vol% samples, the constant regimes of ${\mu }_y$ are about 0.09 different from the others, and when $H_0$ rises up to a certain, ${\mu }_y$ cannot be gotten any more. This is attributed to a stress fluctuation phenomenon, shown in Figure 5. This fluctuation might be due to the interferences among the high concentrated chains. To clearly explain the phenomenon, more experiments and theoretical analysis are needed. This interesting phenomenon will be studied further in the future.

Figure 4.

Variation of ${\mu }_y$ with $H_0$ of MR1 and MR2.

Figure 5.

Stress–time curve at constant shear rate 0.01/s of (a) MR1-30 and (b) MR2-30.

In the beginning, samples exclusive of the 30 vol% are discussed. In the initial-reduction regime, the magnetic field is so low that it is hard to build the intact chain model. Thus, the value of ${\mu }_y$ is not constant and presents downtrend with $H_0$ in the initial-reduction regime. It means that with the chains becoming more and more complete, the yield angle will decrease. When $H_0$ exceeds about 30.9 kA/m, ${\mu }_y$ enters into the first-constant regime. If every chain is independent of each other, ¹⁶ it could be thought that chains with the same model will have the same yield angles when under the same magnetization and carrier medium. In macroscopic view, when under a magnetization, if MRFs have the same ${\mu }_y$ , they should have a same chain model. Specific to the first-constant regime, first the corresponding local magnetic field H is closed to induce the saturation magnetization seen from Figure 6. Then, there is a nearly same yield angle ${\mu }_y$ . Thus, there should be a same chain model in the first-constant regime. Furthermore, the corresponding model to this regime could be the simplest chain model, that is, single-chain model. Continuing to increase $H_0$ , ${\mu }_y$ will enter into the second-constant regime. In this regime, the chain model will transmit to multi-chain model because of the variation of ${\mu }_y$ , 0.16 to 0.2 for MR1 and 0.13 to 0.16 for MR2, respectively. If not, ${\mu }_y$ could not have transformed. Analogically for the MRFs with 30 vol% particle content, they are taken for holding a complex chain model on account of the different ${\mu }_y$ , about 0.09.

Figure 6.

The nonlinear M-H relation of the pure iron.

Thus, the single-chain model area is found based on the value of ${\mu }_y$ . It is at low external magnetic field and below particle content of 30 vol% and becomes narrower and narrower with particle content increasing. However, what kinds of complex chain models are hard to confirm beyond the single-chain model area.

Angle distribution σ

Referring to equation (3), static yield stress ${\tau }_y$ might be expressed as follows

$${\tau }_y={\tau }_y^{\prime }+{\tau }_y^{″}$$

(10)

where ${\tau }_y^{\prime }$ and ${\tau }_y^{″}$ are the yield stresses from particle chains and base carrier, respectively. Figure 7 shows relationship between shear ratio ${\tau }_y^{″}/{\tau }_y$ and H₀ of MR1-05 and MR2-05. The result suggests that, for MR2-05, the values of ${\tau }_y^{″}/{\tau }_y$ are all less than 3%. Thus, ${\tau }_y^{″}$ are small enough to be negligible. For MR1-05, when $H_0$ is up to 30.9 kA/m and above, the values of ${\tau }_y^{″}/{\tau }_y$ are less than 10%. The base carrier also affects the stress tinily. With particle content increasing, the value of ${\tau }_y^{″}/{\tau }_y$ will decrease further. In other words, the effect of ${\tau }_y^{″}$ could decrease further. Thus, the static yield stress ${\tau }_y$ can be equal to ${\tau }_y^{\prime }$ roughly in the first-constant regime of ${\mu }_y$ .

Figure 7.

The relation of ${\tau }_y^{″}/{\tau }_y$ and H₀ of MR1-05 and MR2-05.

However, checking the regime of MR1-05 below 30.9 kA/m in Figure 7, the value of ${\tau }_y^{″}/{\tau }_y$ could be large, especially 77% at 10.01 kA/m. The reason is that the structure of the carrier grease of MR1 is more complex than that of MR2 after thickening process for MR1. That could result in difficult motion of particles. However, from another point of view, this difficulty will be helpful to block the break of the built chains. As a result, the yield stress could increase, which is indicated by the yield angle of MR1, 0.2, bigger than that of MR2, 0.16. In conclusion, the influence of the carrier medium on the performance of MRF is mainly on the inclined angles and distribution of the chains.

Focusing on first-constant regime of ${\mu }_y$ , corresponding $\sigma$ by fitting experimental data are listed in Table 2, when setting the uniform ${\mu }_y$ 0.16 and 0.2 for MR1 and MR2, respectively. The results show that $\sigma$ has a downtrend with external magnetic field increasing. The smaller value of σ indicates a more concentrated distribution of the chains. The following equation is used to model $\sigma$ with respect to $H_0$

$${\sigma }_i\left(H_0\right)=a_i{H}_0^{m_i}+b_i$$

(11)

where a_i, m_i, and b_i are constants, and i = 1 or 2 represents MR1 and MR2, respectively. Thus, the $\sigma$ model of MR1-05 can take the form of

$${\sigma }_1\left(H_0,5\%\right)=10.4615{H_0}^{-0.3961}-0.9088$$

(12)

while model of MR2-05 is

$${\sigma }_2\left(H_0,5\%\right)=10.7176{H_0}^{-0.3632}-1.1751$$

(13)

Table 2.

The σ of MR1 and MR2.

H0 (kA/m)	σ
	MR1-05	MR1-10	MR1-15	MR1-20	MR1-25
10.07	–	–	–	–	–
23.3	–	–	–	–	1.965
30.9	–	1.59	1.5	1.613	1.712
42.6	1.45	1.32	1.23	1.32	1.298
51.43	1.29	1.164	1.08	1.164	1.147
61.69	1.15	1.02	0.94	1.013	0.992
76.77	0.97	0.85	0.784	0.843	0.823
91.84	0.83	0.723	0.645	0.709	–
108.4	0.716	0.624	–	–	–
125.3	0.631	0.555	–	–	–
140.8	0.564	–	–	–	–
157.6	0.501	–	–	–	–
174.6	0.453	–	–	–	–
H0 (kA/m)	σ
	MR2-05	MR2-10	MR2-15	MR2-20	MR2-25
10.07	–	–	–	–	–
23.3	–	–	–	–	–
30.9	1.9	1.665	1.619	1.678	1.565
42.6	1.573	1.361	1.316	1.303	1.316
51.43	1.398	1.2	1.157	1.195	1.147
61.69	1.233	1.036	1.005	1.043	0.988
76.77	1.041	0.867	0.838	0.867	0.827
91.84	0.882	0.729	0.71	0.733	0.693
108.4	0.769	0.628	0.609	0.626	0.597
125.3	0.676	0.552	–	–	–
140.8	0.602	–	–	–	–
157.6	0.544	–	–	–	–
174.6	0.47	–	–	–	–

Figure 8 shows the relationship of ${\epsilon }_i(\phi )=({\sigma }_i(H_0,\phi )-b_i)/({\sigma }_i(H_0,5\%)-b_i)$ and $H_0$ , where $\phi$ accounts for volume fraction of particles. It indicates that ${\epsilon }_i(\phi )$ is almost constant and independent of $H_0$ , that is, other samples could have similar models of

$${\sigma }_1(H_0,\phi )=a_1(\phi ){H_0}^{-0.3961}-0.9088$$

(14)

and

$${\sigma }_2(H_0,\phi )=a_2(\phi ){H_0}^{-0.3632}-1.1751$$

(15)

Figure 8.

The $\epsilon -H_0$ curve of MR1 and MR2.

The value of $a_{\mathrm{i}}(\phi )$ listed in Table 3 shows that with $\phi$ increasing, $a_i$ decreases initially and results in decreasing $\sigma$ . It indicates that the distribution of chains becomes more and more concentrated. The downtrend of $a_i$ does not continue, when $\phi$ is up to 20%. The possible reason is that, in the small $\phi$ region, more particles mean more complete chains, thereby resulting in a more concentrated distribution. When $\phi$ is up to 20%, most chains are formed, and even a small amount of chains begin to aggregate to another complex model, thus $\sigma$ does not decrease any more.

Table 3.

The values of a of MR1 and MR2.

φ (%)	a 1	a 2
5	10.4615	10.7176
10	9.8204	9.8954
15	9.4217	9.7389
20	9.7405	9.8434
25	9.8715	9.6591

Hither, based on the above analysis and experiments, theoretical values of τ₀ in the first-constant regime of ${\mu }_y$ are got, shown in Figure 9. It can be seen that the proposed models provide a good agreement with experimental data.

Figure 9.

Comparison of the static yield stress of (a) MR1 and (b) MR2 from experimental results and equation (2).

Conclusion

The chain mechanism of ferromagnetic particles based on the magnetic dipole theory can depict the performance of MR material very well. In this study, the model is applied to identify the transition from single-chain model to multi-chain model. Some analysis and experiments are investigated. Some conclusions can be made in the following:

• It is possible to identify different chain models of MRF according to the value of yield angle. The existed area of the single-chain model can be found, which is at low external magnetic field and below 30 vol% composition of ferromagnetic particles.

• The carrier medium has significant effect on the mechanical properties of MRFs which have different mean value and deviation of the chain distribution.

• With ferromagnetic particle content and magnetic field increasing, the whole inclined-angle distributions will be more and more concentrated. Meanwhile, they possess a series of similar mathematic models.