Predicting High-Concentration Aggregation in Magnetic Colloidal Suspensions Using Tunnel Theory

Abstract

Accurate prediction of aggregation in suspensions is crucial for diverse engineering applications. This paper develops a sequential theoretical strategy, based on tunnel theory, to predict the aggregation configuration in magnetic compound fluids (MCF) by evaluating their volume concentration C_v. We formulated the viscosity η, resistance R, and capacitance C resulting from aggregation as functions of C_v. This involved a theoretical procedure using tunnel theory, refined using experimental data, including vertical force F_v arising from the concentration gradient, as well as electrical conductivity σ and permittivity ε. The theoretical formulation for η was further refined by considering hypothetical aggregation configurations, specifically non-uniform particle distribution and agglomerations approximated as spheroids with axis ratio κ, along with experimental data on shear flow. For R and C, the formulations were refined using experimental data for σ and ε, together with the relationship between C_v and the applied magnetic field H_v derived from tunnel theory and F_v. This sequential theoretical analysis yielded final formulations for η, R, and C as functions of H_v and initial volume concentration C_v,o. Specifically, η was expressed as a function of κ and C_v,o for the shear and stress–shear strain γ’ relationship under conditions of H_v < 200 mT, 11 < C_v,o < 30 vol.%, and γ’ < 300 1/s. R and C were determined under conditions of H_v < 150 mT and 11 < C_v,o < 30 vol.%. These findings pave the way for novel theoretical predictions of C_v, R, and C based solely on H_v data, a capability crucial for designing diverse materials.

1. Introduction

Suspensions of nano-particles exhibit diverse behaviors in numerous engineering applications. These applications, reviewed in detail by [1], span various fields: energy devices (e.g., heat exchangers and photovoltaic systems for heating and cooling), automotive components (e.g., engines, clutches for enhanced efficiency, and cooling), and manufacturing processes (e.g., grinding, turning, milling, and lubricating). Such systems utilize particle suspensions to reduce energy consumption, emissions, and air pollutants from fossil fuel combustion, as well as to minimize waste production and raw material usage. Beyond these roles, suspensions also function as composite materials, exhibiting properties like electro-strictivity or piezo-resistivity. This allows their application in flexible electrodes, flow batteries, or capacitors [2,3], conductive inks [4], and various sensors. Nano-particles are particularly effective in a wide range of suspensions, owing to the diversity of available materials, including pure metal (Au, Ag, Cu, and Fe), metal oxides (CuO, SiO₂, Al₂O₃, TiO₂, ZnO, Fe₃O₄, Fe₂O₃, and MgO), carbides (SiC and TiC), and carbon-based forms (diamond, graphite, fibers, and nanotubes) and particle shapes (spheres and cylinders) [1]. For example, suspensions of 10 nm Fe₃O₄ in solvents like water or kerosene are known as magnetic fluids (MFs) or ferrofluids [5]. MFs have been investigated since their invention by Dr. Pappell at NASA in 1963, initially to transport fuel in liquid rockets efficiently. This research led to many discoveries across diverse physical and engineering fields, ultimately establishing the discipline of ferrohydrodynamics.

For these applications, suspension performance is evaluated by properties such as thermal conductivity, viscosity η, electrical conductivity σ, permittivity ε, and specific heat capacity [1]. Particle aggregation significantly affects these suspension properties, necessitating investigations at high particle volume concentrations C_v. However, current research often focuses on comparatively small C_v values. For instance, theoretical and experimental studies typically limit C_v to <0.12 for thermal conductivity and η, <0.05 for σ, and <0.1 for ε and specific heat capacity [1,6]. Specifically, theoretical analysis rarely extends beyond the hypothesis of small C_v and weak magnetic fields. This limitation stems from the difficulties in simulating suspension performance and formulating empirical equations, both theoretically and experimentally, at not only high C_v or high volume fraction ϕ but also the following non-uniform distributions of particles.

Experimental observations at high C_v reveal non-uniform particle distribution upon application of a magnetic field H_v for current colloidal suspension, as shown in Figure 1a. This non-uniformity results from particle aggregation induced by the magnetic field. Despite the complex behavior of these non-uniform distributions, theoretical and experimental investigations into their formation remain limited to a few studies. These include experimental observations of non-uniform distributions for Fe₃O₄ nano-particles in suspensions [7] and powdery magnetic particles in air [8,9], as well as theoretical analyses simulating interparticle force interactions [10,11]. Regarding chain-like particle aggregation, studies include experimental observations using various techniques such as scanning electron microscopy (SEM), transmission electron microscopy (TEM), light scattering, and X-ray scattering [12], alongside theoretical simulations employing Monte Carlo (MC) methods and Brownian dynamics [5,13]. Research on related properties has investigated the viscosity of Fe₃O₄ nano-fluids [14]; experimental measurements of σ in MFs [15], cylindrical fillers [16], and polyions [17]; σ simulated via percolation theory [18,19]; and experimental measurements of ε in MFs [5,20]. However, studies demonstrating C_v–H_v, η –H_v, σ–H_v, and ε–H_v relationships are notably scarce.

The magnetic compound fluid (MCF), which Shimada has developed as another suspension of nano-particles, as well as MF, brings about high C_v distribution in the region of the applied non-uniform magnetic field in the case of a strong magnetic field or high fluid concentration. Regarding MCF, MF, and MCF rubber liquid, which is a mixed MCF in a rubber liquid, σ has been investigated experimentally in earlier work [21]. MCF consists of 10 nm oleic acid-coated Fe₃O₄ spheres (derived from MF) and 1 μm carbonyl iron (Fe) spheres; it is easily produced by mixing MF and Fe powder. The aggregates formed by Fe₃O₄ and Fe, with Fe₃O₄ acting as a conjunction among the Fe particles, are long and thick, as shown in Figure 1b. The aggregation is approximated by a spheroid. The innumerable spheroids formulated the non-uniform distribution according to the magnetic field strength. The aggregation provides the phenomenon of dense particles that the particles cannot move easily by a shear flow and Brownian motion, which has been elucidated by earlier work [21], as shown in Appendix A.1. The numerous aggregates of MCF particles under a magnetic field lead to spike structures, as shown in Figure 2. These aggregates provide a characteristic established for needle-like structures [22]. The resulting spikes are more rigid than those in MF. In contrast, MF forms short, thin, single-chain-like clusters with fluid spikes. Magnetorheological fluid (MRF) forms condensed aggregates, producing spikes that are more rigid than MF and resemble a sierra; however, these aggregates form thick, long chains only at more dilute concentrations. MRF is a magnetically responsive suspension containing millimeter-sized metal particles. Due to this condensation, MRF aggregates cannot be approximated by simple shapes like spheroids at high concentrations. Consequently, MCF and MRF are high-concentration suspensions, behaving as illiquid suspensions, whereas MF behaves as a fluid. Therefore, MCF and MRF under a magnetic field do not present the liquidity of a fluid.

The present research focused on MCF due to the high concentration and non-uniform aggregate distribution, which induces a configuration where the distance between the particles is very small, and the formation of the aggregated particles cannot be broken up easily. Therefore, the multi-barriers in the ordinary quantum problem can be applied to the configuration consisting of Fe₃O₄ and Fe, as shown in Figure 1b, which exhibits the potential energy V(x) on the barrier among the particles in Appendix A.2. As for MCF rubber liquid and MCF rubber material, Shimada theoretically explained the electrical properties (resistance R and capacitance C) of rubber–magnetic particle composites using tunnel theory [23,24]. This theory might also predict the properties such as η, σ, ε, etc. in high-C_v suspensions for MCF, which is the object of the present study. This strategy would advance the theoretical investigation of these properties of colloidal suspension with not only high C_v but also non-uniform distributions of particles by tunneling theory.

MFs are non-conductive or exhibit negligible conductivity, whose mechanism likely involves ionic transport between particles during collisions (e.g., Brownian motion and shear flow). Interparticle distances in MFs remain long, even under a magnetic field. In contrast, high-concentration MCFs develop non-uniform particle aggregation under a magnetic field. This aggregation shortens interparticle distances, enabling electron transmission via the tunnel effect. Similarly, under shear flow, high-C_v MCFs form dense, non-uniform distributions, further reducing interparticle distances and enabling electron tunneling.

Building on this understanding, the present study theoretically predicts MCF behavior, specifically for high-concentration fluids with non-uniformly distributed, spheroid Fe₃O₄ and Fe particles. Tunnel theory results were experimentally modified using η, σ, and ε to refine predictions for MCFs. The final formulations (C_v–H_v, η–H_v, R–H_v, and C–H_v) effectively predict C_v, η, R, and C from H_v across diverse engineering applications. These convenient relations facilitate novel material design. This tunneling approach offers a novel theoretical analysis for suspension properties in shear hydraulic flow.

2. Theoretical Analysis

2.1. Theoretical Paradigm

Figure 3 outlines the theoretical procedure, which is composed of Phases 1–5 encompassing theoretical analysis, experimentation, and verification.

Phase 1 derives C_v from theoretical analysis and experiment 1. It begins with theoretical results (a1 in Figure 3) for the electrical properties (R and C) of a rubber–magnetic particle composite, based on tunnel theory from previous studies [23,24]. The C_v*–H relation (d1) is then obtained by combining these theoretical R–Δ results (a1) with experimental σ–H data (b1).

Phase 2 modifies C_v using experiment 2. The C_v*–H_v relation (d1) is refined using experimental vertical force data (F_v–H_v, b3), yielding a modified C_v*–H_v formula (d2).

Phase 3 integrates shear flow experiments (b) and verification (c). Experimental τ–γ ’ data (b10), obtained from shear flow in a cone-rotating rheometer, modifies the theoretical spheroid axis ratio κ (c1). This modified κ then verifies the τ–γ ’ relation (c2), leading to the comprehensive C_v*–H formula (d2) under conditions of C_v,o and γ ’.

Phase 4 derives electrical properties from experiment (b) and verifications (c2 and c3). Experimental σ and ε data establish σ–H_v (b4) and ε–H_v (b5) relations, which in turn yield R–H_v (b6) and C–H_v (b7). From these, and incorporating experimental vertical force F_v–H_v data (b3), the experimental R–F_v (b8) and C–F_v (b9) relations are obtained. On the other hand, combining experimental F_v–H_v data (b3) with the theoretical C_v*–H_v formula (d2) establishes the theoretical F_v–C_v relation (d3). This theoretical F_v–C_v relation (d3) then modifies the experimental R–H_v (b6) and C–H_v (b7) relations, using the C_v*–H_v theoretical formula (d2), to refine the theoretical R–F_v (b8) and C–F_v (b9) relations.

The final objective, of predicting empirical equations for C_v*, R, and C as functions of H_v, is delineated by the bold-lined squares (d2, c3, c4) in Figure 3.

2.2. Phase 1

Phase 1 corresponds in Figure 3 as shown in Figure 4. Solving the ordinary quantum problem for multi-barriers (Appendix A.1) yields the dimensionless resistance R^* to compressive strain cs, as presented in Figure A3 [24] (Appendix A.3) using L_o’ = 10⁻⁶ m. Figure 5a, along with its approximate expression (a1), is obtained by modifying Figure A3 with the relation l/2r = 1 − cs = 1 − Δ/L’_o. The representative resistance R_∞ represents the maximum resistance at magnetic field saturation, a phenomenon demonstrated in “b. experiment 1”. Similarly, based on particle location in a unit volume (Appendix A.4), the relation between compression Δ and dimensionless volume concentration C_v^* can be obtained by using l/2r’ =1 − Δ/L’_o, as shown in Figure 5b (a2), which was elucidated previously [25]. From “b. experiment 1”, the relationship between σ and magnetic field strength H (Appendix A.5) is shown in Figure 5c (b1), also a previously established result [21]. Electrical conductivity σ_∞ represents the electrical conductivity at the maximum magnetic field. This σ–H relation is then used to modify the relation R–H (b2).

Figure 5d is derived by combining the theoretical R–Δ results (a1) with experimental R–H data (b2). Specifically, by setting a value for R_x, Δ_x is determined from Figure 5a and H_x from Figure 5c, thereby establishing the Δ_x–H_x relation. Next, Δ_x is replaced by C_v,x* from Figure 5b (a2). Finally, the C_v*–H relation, along with its approximate expression (d1), is obtained and presented in Figure 5e. The ultimate result of C_v,x* is the tendency that it enhances as H increases and saturates after that. It depends on the tendency of the relation between σ and H, as shown in Figure 5c. As a result, the relationship of C_v,x* depends on the one of σ.

2.3. Phase 2

Phase 2 corresponds to Figure 3, as shown in Figure 6. Particles concentrate within the magnetic field region, driven by the magnetic field gradient. Applied magnetic fields are generally non-uniform due to their symmetrical configuration. For nano-particles, however, magnetic pressure generated by ferrohydrodynamics is independent of particle concentration. The force resulting from particle concentration is given by Equation (1) [10].

$$\overrightarrow{f_c}={\displaystyle \frac{1}{2{\mu }_0}}∆{\chi }_m\overrightarrow{\nabla }c{V}_p{\left|\overrightarrow{B}\right|}^2$$

(1)

The concentration gradient force $\overrightarrow{f_c}$ is a function of the concentration gradient of the ferrofluid $\overrightarrow{\nabla }c$. This theorem states that particle aggregation in a magnetic field induces a non-uniform particle distribution, which creates a concentration gradient force. Therefore, measuring the force generated on the magnetic fluid in a magnetic field allows evaluation of the C_v distribution. The relationship between F_v and H_v was experimentally determined, as shown in the following experimental section. At the maximum vertical magnetic field of 141 mT (H_v,mean), the experimental data for F_v (Figure 7a; b3 in Figure 6) coincided with the approximate equation for C_v* (Figure 5e; d1 in Figure 6). This yielded the consummate formula shown in Equation (2) (d2), which has a 1% uncertainty. One example of this coincidence is shown in Figure 7b. The distinctive tendency of the ultimate results is as follows. By the effect that the attractive force of the magnetic field induces the particles into the magnetic field region, C_v provides the distribution as shown in Figure 7b. As the smaller magnetic field distribution and C_v,o, C_v at the magnetic field region are smaller than C_v,o and the rest of the particles remain outside the magnetic field.

$$\begin{array}{c}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{C}_V^{*}=5.713\times {10}^{-2}{e}^{0.1133C_{V,0}} \left(133<H_v<200 mT\right)\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}(-1.104\times {10}^{-6}{H}_v^2+4.168\times {10}^{-4}{H}_v+2.122\times {10}^{-2})e^{0.1133C_{V,0 }}\left(48<H_v<133 mT\right)\hfill \\ \left(2.374\times {10}^{-4}{H}_v+2.768\times {10}^{-2}\right)e^{0.1133C_{V,0 }}\left(H_v<48 mT\right)\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}at C_{v,o}<30 vol.\%\hfill \end{array}$$

(2)

2.4. Phase 3

Phase 3 corresponds to Figure 3, as shown in Figure 8. The modified C_v*–H_v formula (Equation (2); d2 in Figure 8) is verified for its application to the viscosity η +Δη, as measured by a cone-rotating rheometer (Equation (3)). Here, Δη is the enhanced viscosity due to a magnetic field. Aggregated magnetic particles form chain-like configurations under a magnetic field, which can be approximated as spheroids. The rotational viscosity (α) for these particles is shown in Equation (5), where g(κ) (Equation (8) [26,27]) is the rotary diffusion constant for spheroids. These spheroidal particles are distributed non-uniformly due to aggregation, as shown in Figure A6b (Appendix A.6). Function f(H_v) is obtained when a magnetic field is applied to a rotating cone, as described by Equation (9). This function is derived from the theory of spheroids in a magnetic field, as presented in Equation (A12) (Appendix A.6 [28]).

$$\tau =(\eta +\mathsf{\Delta }\eta ){\gamma }^{\prime }$$

(3)

$$\mathsf{\Delta }\eta ={\displaystyle \frac{\alpha \eta }{2}}f(H_v)$$

(4)

$$\alpha =4\varphi {\eta }_{s}g\left(\kappa \right)\left(0<\kappa <1\right)$$

(5)

$$\alpha =6\varphi {\eta }_s(\kappa =1)$$

(6)

$$\varphi =C_v/100$$

(7)

$$g\left(\kappa \right)={\displaystyle \frac{1-{\kappa }^4}{\left(2-{\kappa }^2\right)\frac{{\kappa }^2}{\sqrt{1-{\kappa }^2}}\ln \left(\frac{1+\sqrt{1-{\kappa }^2}}{\kappa }\right)-{\kappa }^2}}$$

(8)

$$f\left(H_v\right)=\lambda \left({\displaystyle \frac{L_1}{2\xi }}-{\displaystyle \frac{L_2}{2{\xi }^2}}-{\displaystyle \frac{L_3}{2\xi }}{H_v^{*}}^2\right)-{\displaystyle \frac{1}{2}}{\displaystyle \frac{\xi -tanh\xi }{\xi +tanh\xi }}(\lambda \ne 1)$$

(9)

$$\lambda ={\displaystyle \frac{1-{\mathsf{\kappa }}^2}{1+{\mathsf{\kappa }}^2}}$$

(10)

$$\xi ={\displaystyle \frac{{\mu }_o{m}_B{H}_v}{k_B{T}_B}}, L_2=1+{\displaystyle \frac{3}{{\xi }^2}}-{\displaystyle \frac{3}{\xi }}coth\xi ,L_3=-\left({\displaystyle \frac{6}{\xi }}+{\displaystyle \frac{15}{{\xi }^3}}\right)+\left(1+{\displaystyle \frac{15}{{\xi }^2}}\right)coth\xi , {\displaystyle \frac{L_1^2\xi }{\xi -L_1}}={\displaystyle \frac{\xi -tanh\xi }{\xi +tanh\xi }}$$

(11)

κ is modified using experimental viscosity data from the cone-rotating rheometer (b10 in Figure 8) as shown in Equation (12) and exemplified in Figure 9a (c1). Then, the τ–γ ’ relation is derived from Equations (3) and (4) and compared with the experimental data (Figure 9b; c2 in Figure 8). For this derivation, H_v in Equation (2) is replaced by H_v,mean as in Equation (7) (Figure 9c). Additionally, H_v in Equation (4) is calculated by averaging H_v values across each radial distance to determine Δη. The quantitative discrepancy between theory and experiment diminishes with increasing C_v,o, H_v,mean, and γ ’. Thus, the present theory can predict the τ–γ ’ relation. The tendency, as shown in Figure 9b, is that the discrepancy between the experiment and theory on τ becomes larger at extremely large or small γ ’.

$$\begin{array}{l}\hspace{1em}\hspace{1em}\kappa =\left(8\times {10}^{-5}{e}^{8.3\times {10}^{-3}{H}_{v,mean}}\right)exp\left(8.99\times {10}^{-2}{e}^{-1.2\times {10}^{-2}}{\times C}_{v,o}\right)\\ \times exp\left\{\left(6.9\times {10}^{-3}{e}^{-1.6\times {10}^{-2}{H}_{v,mean}}\right)exp\left[\left(4\times {10}^{-4}{H}_{v,mean}-3.84\times {10}^{-2}\right){\times C}_{v,o}\right]\times {\gamma }^{\prime }\right\} (11\\ <C_{v,o}<30 vol.\%, H_{v,mean}<120 mT, {\gamma }^{\prime }<300 1/s)\end{array}$$

(12)

We confirm that vertical viscosity exerts a greater effect than horizontal viscosity on spheroid formation in the cone-type rheometer, with spheroids aligning vertically along the axial direction. As for horizontal viscosity, the radial viscosity is obtained by parameters α, f (derived from Equation (A12) in Appendix A.6 by replacing H_v with H_r), and g, as shown in Equations (12)–(14) through Equations (3) and (4). For horizontal viscosity calculations, ξ and H_v are replaced by the radial magnetic field strength H_r. Δη is then compared between H_v and H_r (Figure 9d). A larger major axis length of the spheroid correlates with a larger Δη; however, Δη under H_v is larger than under H_r. Figure 9d shows an example of the comparison between experimental and theoretical results (Equations (5) and (13)). The pale green area indicates quantitative agreement between the experimental and theoretical results. Consequently, vertical viscosity predominates when assuming a spheroid. Therefore, only vertical parameters, such as the magnetic field H_v and vertical force F_v, need to be considered. If aggregation is postulated as a prolate spheroid, both the axis ratio κ and the viscosity enhancement Δη can be predicted within the delineated range shown in Figure 9d. This theoretical convenience allows for effective estimation of chain formation, as demonstrated by the aggregation.

$$\alpha =8\varphi {\eta }_{s}g\left(\kappa \right)\left(0<\kappa <1\right)$$

(13)

$$g\left(\kappa \right)={\displaystyle \frac{1-{\kappa }^4}{\left(2{\kappa }^2-1\right)\frac{{\kappa }^2}{\sqrt{1-{\kappa }^2}}\ln \left(\frac{1+\sqrt{1-{\kappa }^2}}{\kappa }\right)+1}}$$

(14)

$$f\left(H_r\right)=\lambda \left({\displaystyle \frac{L_1}{\xi }}-{\displaystyle \frac{L_2}{2{\xi }^2}}\right)+{\displaystyle \frac{{H_r^{*}}^2-1}{2}}{\displaystyle \frac{\xi -tanh\xi }{\xi +tanh\xi }}(\lambda \ne 1)$$

(15)

2.5. Phase 4

Phase 4 corresponds to Figure 3, as shown in Figure 10. The experimental section details the measurement of σ and ε data, which define the relations σ–H_v (b4 in Figure 10) and ε–H_v (b5). These relationships then yield approximate equations for R–H_v (b6) and C–H_v (b7) at each radial distance for a 141-mT H_v,mean, as respectively shown in Figure 11a,b. Utilizing the experimental vertical force data H_v–F_v (Figure 7a; b3 in Figure 10), these relations further derive R–F_v (b8) and C–F_v (b9), shown in Figure 11c,d.

On the other hand, combining the experimental vertical force data F_v–H_v (Figure 7a; b3 in Figure 10) with Equation (2) for C_v*–H_v (d2) yields the F_v–C_v relation (Figure 11e; d3 in Figure 10). Applying this relation F_v–C_v to the previously derived R–F_v (b8) and C–F_v (b9) relations, we obtain the modified final relations for R–H_v (d4) and C–H_v (d5). These are shown in Figure 11f,g, and Equations (16) and (17), and verified against the experimental data of R and C. The experimental tendency that R linearly decreases and C slightly increases is qualitatively explained by the theory as shown in Figure 11f,g. The quantitative coincidence is smaller for R than for C.

$$R=-7.231\times {10}^2{e^{-0.107C_{V,0}}H}_v+1.466\times {10}^5{e}^{-0.02C_{V,0}} \left(H_v<150 mT, 11<C_{v,o}<30 vol.\% \right)$$

(16)

$$\begin{gathered} C=\left(-3\times {10}^{-16}{C}_{v,o}+9\times {10}^{-15}\right)H_v^2+\left(3\times {10}^{-14}{C}_{v,o}-8\times {10}^{-13}\right)H_v+9\times {10}^{-12}{e}^{0.1043C_{V,0}} \\ \left(H_v<150 \mathrm{m}\mathrm{T}, 11<C_{v,o}<30 \mathrm{v}\mathrm{o}\mathrm{l}.\%\right) \end{gathered}$$

(17)

3. Experiment

3.1. Material

A few MCFs with different volume concentrations were prepared from two components: oleic acid–coated Fe₃O₄ particles in MF (40 wt% Fe₃O₄ in water-based solvent, W40, Ichinen Chemicals Co., Ltd., Tokyo, Japan) and spherical Fe particles (HQ by Yamaishi Co., Ltd., Noda, Japan). The compositions are shown in

Table 1. Composition of liquid specimen containing magnetic particles.

vol. %	HQ [g]	MF [g]	Water [g]
11.5	40	20	-
19.6	46	20	-
29.6	20	20	10
32.2	23	20	-
saturation magnetization [mT]	175	27
particle [m × 10−6] (shape)	1.2 (sphere)	0.01 (sphere coated oleic acid)

. The volume concentrations in the table comprise both Fe₃O₄ and Fe with reference level [22]. The measurement range was 11 vol.% < C_v,o < 33 vol.%.

MCF was produced by mixing HQ, MF, and water with an ultrasonic stirrer (UR-20P, Tomy Seiko, Co., Ltd., Tokyo, Japan).

3.2. Measurement

The MCF was set on a flat substrate, and a magnetic field was applied using a permanent magnet, as shown in Figure 12a. Multiple spikes formed in the MCF due to the aggregation of particles aligned along the magnetic field. The permanent magnet was a neodymium type (Niroku Seisakusho Co., Ltd., Kobe, Japan) with dimensions of 10 × 15 mm and a thickness of 5 mm. Its surface magnetic field intensity was approximately 300 mT. The coordinate system is shown in Figure A6a (Appendix A.6). The magnetic field was measured with a tesla meter (TM-701, Kanetec Co., Ltd., Nagano, Japan) by putting the probe at an angle toward the direction of the following magnetic field: the vertical magnetic field was measured as the strength perpendicular to the magnet surface, and the horizontal (or radial) magnetic field as the one parallel to the magnet surface. The measurement range was less than 150 mT.

The probe for measuring F_v was a ϕ3-mm acrylic resin cylinder brought into contact with the MCF spike, as shown in Figure 12b. The electrical properties σ and ε were measured using two copper plates inserted into the MCF spike (Figure 12b,d). The probes were moved by a compression-testing machine (SL-6002; IMADA-SS Co., Ltd., Toyohashi, Japan). H_v was measured by a load cell installed within the compression-testing machine, and σ and ε were measured using an inductance–capacitance–resistance (LCR) meter (IM3536; Hioki Co., Ltd., Ueda, Japan), with the measurement area shown in Figure 12c.

τ–γ ’ was measured using a rotational rheometer with a cone, as shown in Figure 12e. The MCF was placed between the cone and the base under the same magnetic field as used in the measurements of H_v, σ, and ε. The cone had a surface angle of 4°. Under motor-driven (BXM460-GFH2, Oriental Motor Co., Ltd., Tokyo, Japan) rotation of the cone, the torque applied to the cone was measured by a torque detector (SS-501, Ono Sokki Co., Ltd., Yokohama, Japan). The measurement range was 50 1/s < γ ’ < 500 1/s.

The repetition times in all measurements were more than 5 times.

4. Results and Discussion

4.1. Vertical Force

The measured vertical pressure as a function of magnetic field strength, which comprises vertical (H_v, as shown in Figure 13a) and horizontal (H_r) magnetic field components, is shown in Figure 13b. The figure also shows H_v and H_r. The vertical direction is along the axial direction z, and the horizontal one is along the radial direction r. H_v is obtained from the vertical pressure, as shown in Figure 13a. The measurement error of the vertical pressure was less than 5%. F_v is also obtained from the vertical pressure. As shown in Figure 12a, the MCF spike becomes slanted at larger radial distances because the particles aggregate along the outer edge of the radius. This phenomenon causes F_v to become larger along that edge.

The parameter F_v is important in machining techniques; for instance, processing pressure must be controlled when processing or polishing objects such as metal or lenses with a machine tool. In particular, magnetic abrasive finishing (MAF) has recently garnered attention [29]. This is because a polishing liquid, in which the abrasive material is compounded in a magnetically responsive liquid such as MF, MCF, or MRF, can be controlled by a magnetic field. Especially, MCF has been more successful in MAF than MF and MRF [30]. At the target polished area, the polishing liquid can be concentrated by the magnetic field, thereby increasing the polishing efficiency. In the prediction and estimation of polishing efficiency, F_v is a key parameter because the polishing amount is evaluated using factors including F_v, as defined by the well-known Preston equation [30]. Therefore, the measurement or numerical prediction of F_v is a common challenge in the field of MAF. The obtained F_v data support the advancement of engineering applications such as MAF using MCF, which has emerged as a prominent research focus.

4.2. Electrical Properties

Figure 14 shows the measured electrical properties σ and ε as a function of the vertical magnetic field strength H_v. R and C are obtained from σ and ε, as shown in Figure 11a,b. The measurement errors of σ and ε were less than 5%. The theoretically analyzed R (c3 in Figure 3) is obtained by combining C_v from phase 1 (d1) with the experimental R (b8), as shown in Figure 11f. Similarly, C (c4) is derived by combining C_v from phase 1 (d1) with the experimental C (b9), as shown in Figure 11g. Quantitatively, R is more readily explained than C. Qualitatively, as H_v increases, R decreases, indicating enhanced electrical conductivity due to a larger C_v. Concurrently, C increases with H_v, reflecting enhanced capacitance due to a larger C_v. This behavior is attributed to increased aggregation with higher H_v. These quantitative tendencies are attenuated at larger C_v,o values, as a high initial C_v,o limits further aggregation.

In the field of MAF, both magnetic and electric fields enhance polishing efficiency. For instance, novel MSF research has explored the integration of magnetic and electric fields in MCFs [29]. Electropolishing, on the other hand, utilizes electrically responsive fluids such as electro-rheological fluid (ERF) to control polishing efficiency with an electric field [31]. ERF comprises dielectric particles that, similar to those in MF, MCF, and MRF, aggregate into chain-like or distributed structures under an electric field. Both MCF and ERF exhibit a small ε, which induces an electrical body force, thereby creating F_v. Critical technical information regarding σ and ε is therefore needed. However, data on these parameters, as presented in Figure 14, have been scarce. Such information could advance engineering techniques for enhancing polishing efficiency and improving predictive evaluations. Our study’s σ and ε findings will thus promote progress across various engineering applications, including MAF.

Moreover, given the rare application of tunnel theory to electrical properties, our theoretical strategy, incorporating this theory, offers a significant breakthrough.

4.3. Shear Flow

Figure 9b compares experimental τ–γ ’ (c2 in Figure 3) with theoretical predictions based on κ (Equation (12); c1) and C_v (Equation (2); d2). The measurement error of shear stress was less than 10%. This comparison reveals that τ–γ ’ can be accurately predicted by Equations (3) and (4) within the range of H_v < 200 mT, 11 < C_v,o < 30 vol.%, and γ’< 300 1/s. However, the predictive accuracy decreases with smaller C_v,o and H_v.

The theoretical and numerical evaluation and prediction of η in colloidal suspensions has long been a persistent challenge, especially in rheology. Knowledge of η is crucial for the advancement of engineering applications, such as the design of chemical processes and polymer materials. Attractive colloidal systems are widely utilized in everyday applications, such as foods, cosmetics, and pharmaceuticals, as well as in agricultural engineering [32]. Thermal conductivity and heat transfer performance are critical research areas relevant to η, as particle aggregation significantly influences suspension properties [33]. However, due to the elusive nature of the complex morphological structures of particle aggregates (e.g., chains, spheroids, or non-uniform distributions, especially those involving magnetic particles), empirical, theoretical, or numerical understanding of the η–C_v relation considering these configurations remains limited. Many novel and conventional models for nano-particle aggregation under shear rheology have been proposed [5,34,35,36]. However, these current models, such as the Einstein and Krieger–Dougherty models, are primarily formulated for dilute suspensions [37] and thus fail to explain the η–C_v relation in high-concentration particle systems. Aggregation arises from particle–particle interactions driven by forces like van der Waals forces and Brownian motion. Such aggregation leads to high η, non-Newtonian effects, impaired flow behavior, and reduced thermal performance. Therefore, current models require refinement, modification, and experimental validation. Consequently, the η–C_v relation, considering aggregate configurations as demonstrated by Equation (4) in the present study, is invaluable. Moreover, similar to its application in electrical properties, the present theoretical strategy, applying tunnel theory to η, represents a significant breakthrough. This is because the tunnel theory paradigm has not been extensively applied to advance the understanding of shear flow and hydraulic phenomena until recently.

5. Conclusions

Controlling the aggregation of magnetic particles in colloidal suspensions using external magnetic fields enables the production of diverse materials, devices, and systems. To design such engineering applications, accurately predicting the aggregation configuration is critical. The present theoretical strategy, based on tunnel theory, evaluates aggregation using C_v. By focusing on MCF and incorporating experimental data to reconcile quantitative discrepancies, C_v was derived as a function of H_v and C_v,o (Equation (2)) under the conditions H_v < 200 mT and C_v,o < 30 vol.%. Similarly, η was determined as a function of κ (derived from shear flow experiments), H_v, C_v,o, and γ’ (Equation (12)), for H_v < 200 mT, 11 < C_v,o < 30 vol.%, and γ’ < 300 1/s. R and C were also obtained as functions of H_v and C_v,o (Equations (16) and (17)) for H_v < 150 mT and 11 < C_v,o < 30 vol.%. These empirical equations represent a crucial breakthrough for emerging technological trends.

Unlike existing theoretical and experimental research, this strategy addresses high particle volume concentrations, opening new avenues for theoretically predicting C_v, R, and C solely from H_v data. Although particle aggregation is known to form non-uniform, chain-like structures, often simplified as spheroids, constructing an empirical equation from such complex models proves challenging. As a prerequisite for designing diverse materials, a robust theoretical paradigm is essential. Thus, the present theoretical strategy is critical, and its application of tunnel theory to shear flow and electrical properties is particularly novel and valuable.