Magnetoviscosity of semidilute ferrofluids and the role of dipolar interactions: Comparison of molecular simulations and dynamical mean-field theory

Extensive molecular simulations on a model ferrofluid are performed in order to study magnetoviscous and viscoelastic phenomena in semidilute ferrofluids. Simulation results of the nonequilibrium magnetization, shear viscosity, and normal stress differences are presented. Rotational and configurational contributions to the shear viscosity are analyzed and their influence on the magnetoviscous effect is discussed. The simplified model of noninteracting magnetic dipoles describes the nonequilibrium magnetization and the rotational viscosity, but does not account for configurational viscosity contributions and normal stress differences. Improved mean-field models that overcome these limitations show good agreement with the simulation results for weak dipolar interactions where the models should apply. Comparisons to simulation results for various interaction strengths allow us to determine the range of validity of the mean-field models.

Figure 1

Schematic plot of the flow geometry together with the orientation of the magnetic field considered in the present study.

Figure 2

Normalized equilibrium magnetization as a function of the Langevin parameter $h$ for different volume fractions $\varphi$ and interaction strengths $\lambda$. Circles, squares, diamonds, and triangles correspond to $(\varphi ,\lambda )=(0.157,4)$, (0.209,2), (0.0393,4), and (0.0785,2), respectively. Big solid symbols are the result of the present simulations using the reaction field method, while small open symbols denote the results employing an Ewald summation technique [15]. Solid lines connecting the data [15] are a guide to the eye.

Figure 3

Nonequilibrium magnetization $M_x∕M_{\mathrm{sat}}$ as a function of the inverse of the number of particles $N$. The magnetic field is oriented in the gradient direction of the shear flow; the Langevin parameter is chosen as $h=1.0$. Squares and diamonds correspond to reaction field cavity radii of $r_{\mathrm{RF}}=3.0$ and 3.5, respectively, while black circles correspond to $r_{\mathrm{RF}}=2.5$. The volume fraction and dipolar interaction parameter are chosen as $\varphi =0.05$ and $\lambda =2.0$, respectively. The reduced shear rate is chosen as $\dot{\gamma }*=0.1$.

Figure 4

Nonequilibrium magnetization $M_x∕M_{\mathrm{sat}}$ as a function of the Langevin parameter $h$. The magnetic field is oriented in the gradient direction of the shear flow. Circles and squares correspond to volume fractions $\varphi =0.05$ and 0.1, respectively. Solid, open, and shaded symbols correspond to different values of $\lambda$, resulting in Langevin susceptibilities of ${\chi }_{\mathrm{L}}=0.2$, 0.4, and 0.8, respectively. The reduced shear rate is chosen as $\dot{\gamma }*=0.1$. The solid line denotes the result for noninteracting magnetic dipoles, dashed, dotted, and dash-dotted lines correspond to the dynamical mean-field model, for ${\chi }_{\mathrm{L}}=0.2$, 0.4 and 0.8, respectively. See Sec. 3 for a summary of both models.

Figure 5

Nonequilibrium magnetization perpendicular, $M_x∕M_{\mathrm{sat}}$, and parallel, $M_y∕M_{\mathrm{sat}}$, to the magnetic field as a function of the reduced shear rate $\dot{\gamma }*$. The magnetic field is oriented in the gradient direction of the shear flow with $h=2$. The volume fraction and dipolar interaction parameter are chosen as $\varphi =0.05$ and $\lambda =1.0$, respectively. The solid line denotes the result of the NI model, dashed lines those of the DMF model. Straight gray lines are the result for the low shear rate limit.

Figure 6

Reduced rotational viscosity ${\eta }_{\mathrm{rot}}^{*}$ as a function of the Langevin parameter $h$. The same conditions and the same symbols as in Fig. 4 are chosen, in particular, $\dot{\gamma }*=0.1$. The solid and dashed lines denote results for the NI and DMF models, respectively.

Figure 7

Different contributions to the reduced shear viscosity ${\eta }_{yx}^{*}$ as a function of the Langevin parameter $h$. Circles and squares correspond to $\varphi =0.05$, $\lambda =1.0$ and $\varphi =0.1$, $\lambda =0.5$, respectively. Solid and gray symbols show the rotational and configurational contributions, respectively. The solid lines are the predictions of the NI model, dashed lines the additional prediction of the configurational contribution by the DMF model. The same flow conditions as before are used, with a reduced shear rate of $\dot{\gamma }*=0.1$.

Figure 8

The relative viscosity change $\Delta \eta ∕{\eta }_0-1$ is shown as a function of the Langevin parameter $h$. The same conditions and the same symbols as in Fig. 4 are used. The solid lines correspond to the NI model, dashed lines to the DMF model ($\varphi =0.05$, $\lambda =2.0$ and $\varphi =0.1$, $\lambda =1.0$) prediction.

Figure 9

Reduced shear viscosity ${\eta }_{yx}^{*}$ as a function of the volume fraction $\varphi$. Open symbols correspond to $h=0$, while solid black symbols show the result for $h=20$. Circles, squares, and diamonds correspond to $\lambda =0.5$, 1.0, and 2.0, respectively. The reduced shear rate is chosen as $\dot{\gamma }*=0.1$. The solid line denotes the NI model result [4], dashed $(\lambda =0.5)$ and dash-dotted $(\lambda =2.0)$ lines the DMF model, each for $h\to \infty$.

Figure 10

The relative viscosity change $\Delta \eta ∕{\eta }_0-1$ is shown as a function of the volume fraction $\varphi$. Circles, squares, and diamonds correspond to $\lambda =0.5$, 1.0, and 2.0, respectively. The solid line corresponds to the maximum viscosity increase predicted by the NI model, dashed $(\lambda =0.5)$ and dash-dotted $(\lambda =2.0)$ lines the maximum viscosity increase predicted by the DMF model.

Figure 11

Rotational viscosity ${\eta }_{\mathrm{rot}}^{*}$ as a function of shear rate $\dot{\gamma }*$. Circles, squares, and diamonds correspond to $h=1.0$, 2.0, and 4.0, respectively. The volume fraction $\varphi =0.05$ and dipolar interaction strength $\lambda =1.0$ have been chosen. The solid line corresponds to NI model prediction in the effective field approximation.

Figure 12

Reduced normal stress differences $N_1^{*},N_2^{*}$ as functions of the Langevin parameter $h$. The same flow conditions and parameters as well as the same symbols as in Fig. 4 are used. For better visibility, only $\varphi =0.1$ and $\lambda =0.5$ (open symbols) and $\lambda =1.0$ (gray symbols) are shown. The dashed lines are a fit based on the DMF model (see Sec. 3).

Figure 13

Ratio of normal stress differences $-N_2∕N_1$ as a function of the Langevin parameter $h$. The same flow conditions and parameters as well as the same symbols as in Fig. 4 are used. The dashed line is the prediction of the DMF model.

Figure 14

The first normal stress difference $N_1^{*}$ as a function of the volume fraction $\varphi$. Circles, squares, and diamonds correspond to $\lambda =0.5$, 1.0, and 2.0, respectively. The reduced shear rate is chosen as $\dot{\gamma }*=0.1$. A strong magnetic field $h=20$ is applied in the gradient direction of the flow. The dashed lines are the result of a quadratic fit.