Magnetic field tuning of mechanical properties of ultrasoft PDMS-based magnetorheological elastomers for biological applications

PDMS-based MREs were fabricated using Sylgard^TM 527 elastomer (Dow Corning) and CIP (spherical, 3.9–5.2 μm mean diameter). Sylgard^TM 527 was first mixed in equal weights of part A and part B and then thoroughly mixed with CIP (Chemical Store Inc.) at CIP volume fractions of $\Phi$ = 0%, 9%, 17%, and 23%, and then poured to a thickness of ∼5 mm into the culture dishes (35 mm in diameter). The samples used for shear rheology were fabricated by pouring the same mixture into 35 mm dishes containing a Teflon mold to decrease the diameter to 20 mm in order to fit the rheometry measurement requirement. The mixtures were degassed for 5 min to remove air bubbles introduced during the preparation process, placed on a hot plate at 60 °C for 4 h, and then left at room temperature for 24 h to ensure fully cross-linking of the polymers.

Figure 1 shows the experimental set up, consisting of an electromagnet (from GMW), soft iron core and a neodymium iron boride (NdFeB) magnet (from CMS Magnets Inc.) for applying magnetic field during rheology, indentation and interferometry studies. The GMW electromagnet was placed on the measurement stage of the instrument. The iron core, 31.75 mm in diameter and 19.05 mm in height, was used to separate the samples from the NdFeB magnet, magnify the field strength and improve the field uniformity at the sample. The cylindrical N45 NdFeB magnet was 31.75 mm in diameter and 6.35 mm in height. The overall magnetic field applied at the sample is the vector sum of the fields from the electromagnet and the NdFeB magnet, which were also magnified by the iron core. For example, to achieve zero magnetic field at the sample, the current through the electromagnet was set so the magnetic field generated by the electromagnet cancelled that from the permanent magnet. For all tests, the magnetic field strength was set from 0 to 95.5 kA m⁻¹ with steps of 15.9 kA m⁻¹ and measured by a Lakeshore®-410 gaussmeter. The electromagnet was cooled by an industrial chiller (CW-3000) and powered by a Volteq power supply. To reduce vibrational noise in the measurements, the chiller was shut off as the test proceeded and was turned back on in between tests. Heat generated by the electromagnet increased the sample temperature by <2 °C during the measurements.

Zoom In Zoom Out Reset image size

Magnetic field-dependent oscillatory rheology was carried out with a Kinexus lab+ rheometer (Malvern Instruments) in strain-controlled mode fitted with the magnetic field setup described above. The shear storage (G') and loss (G'') moduli were measured as a function of the frequency, strain, and magnetic field for elastomers of three different CIP volume fractions ($\Phi = 9,{ }17,{ }23{{\% }}$). First, a frequency sweep from f = 10 to 0.01 Hz at $\gamma = 1{{\% }}$ was carried out, followed by shear strain sweeps from $\gamma = 2 - 20{{\% }}$ at f = 1 Hz at six different fields from H = 0–95.5 kA m⁻¹. The MREs were naturally sticky, and slip was thus avoided.

Indentations on MREs were performed on a custom micro-indenter similar to that described by Rennie et al [25] and Schulze et al [26]. To measure the MRE response to compressive loads we brought a spherical indenter into contact of the MRE surface up to a target load of 5 mN at 50 µm s⁻¹ and then immediately retract the indenter at the same rate. All experiments were conducted with the MRE inside the annulus of the electromagnet (figure 1). Ten indentations were made at each magnetic field strength (0–95.5 in 15.9 kA m⁻¹ steps). This protocol was followed for each MRE sample.

Indentation force and displacement were measured by a capacitance probe (Capacitec) and optical, linear encoder (Renishaw). The capacitance probe has a resolution of 25 µm V⁻¹ and measures the deflection of a calibrated titanium cantilever. The normal stiffness of the cantilever is, k_normal = 1385.9 N m⁻¹. A spherical, 2 mm radius, ruby probe was attached to a 5 mm long cylindrical aluminum rod and was used for all indentations. The probes and encoder were wired into a data acquisition device (National Instruments, USB-6343) which communicated with a custom programmed code (Matlab). The code used Matlab's Appdesigner software as well as µManager Java libraries to control the instrument.

Magnetic field-dependent surface roughness measurements were conducted on a scanning white light interferometer (Zygo, NT6100). These measurements were made using the same MRE and EM configuration as the rheology and indentation. Surface scans were acquired at each magnetic field strength interval between 0 and 95.5 kA m⁻¹ with a 20× objective set at 0.5× optical zoom (10× magnification) over a ∼500 × ∼700 µm rectangle. A planar shift was applied to the data and average (Ra) and root-mean-squared (Rq) roughness was recorded at each field strength.

Figure 2 shows the bulk rheological response measured by the rheometer as a function of magnetic field strength, frequency and strain. For all magnetic field strengths and a broad range of frequencies, the ultrasoft PDMS-based MREs behave as a chemically cross-linked gel. The shear storage modulus measured in the linear viscoelastic regime (${\gamma _{\text{o}}} = 2{{\% }}$), increased quadratically with increasing magnetic field strength and CIP volume fraction up to ∼41× for $\Phi = 23{{\% }}$ (figure 2(A)), which is an order of magnitude larger than the increase in shear modulus $\left( {{{{G_{\left( H \right)}^{\prime}}}/ {{G_{\left( {H = 0} \right)}^{\prime}}}}} \right)$ reported for isotropic rubber-like MREs subjected to the same range of magnetic field strengths [1, 2]. There is a slight frequency dependence in the viscoelastic moduli over the frequency range of about three orders of magnitude (figure 2(B)). Shear strain amplitude sweeps from $\gamma = 2\% - 20{{\% }}$ reveal a weakening of the shear storage and loss moduli. (figure 2(C)). For the highest CIP volume fraction and largest magnetic field strength, both moduli decrease by ∼1/3 (G' ∼ 100→30kPa, G'' ∼ 13→4kPa). While collagen and other biopolymer networks have been shown to be strain-stiffening [27], a less often reported deformation mechanism is strain-weakening [28]. Ultrasoft PDMS-based MREs weaken as the strain increases and this effect is stronger at higher magnetic fields.

Zoom In Zoom Out Reset image size

Figure 3 shows the compressive indentation measurements, as described in section 2.4, for $\Phi = 0\% ,{ }9\% ,{ }17\% {\text{ and }}23\%$ at various magnetic field strengths. Figures 3(A) and (B) compare the indentation force F as a function of the compressive displacement for $\Phi = 0$ and $\Phi = 9{{\% }}$. For $\Phi = 0$, the force vs indentation depth curve and measured elastic modulus were unaffected by the strength of the magnetic field (figure 3(A)), as expected. However, for MREs containing $\Phi = 9{{\% }}$, the slope of the force vs indentation curve increases monotonically with increasing magnetic field strength (figure 3(B)). The elastic moduli are calculated by fitting the unloading portion of the indentation force vs depth curves with the JKR adhesive contact model [29]

$$\begin{equation}\delta = \,\frac{1}{R}{\left[ {\frac{{3R}}{{4E}}(F + 3\Delta \gamma \pi R + \sqrt {6\Delta \gamma \pi RF + {{\left( {3\Delta \gamma \pi R} \right)}^2}} } \right]^{\frac{2}{3}}}.\end{equation} \tag{ 1 }$$

Zoom In Zoom Out Reset image size

where R is the radius of the indenter while the elastic modulus, E, and work of adhesion, Δγ, are the fit parameters. The depth of indentation into the material, δ, is taken as the difference between the stage displacement, z, and the cantilever deflection, δ_c, where $\delta = z - {\delta _{\text{c}}}$. Since this contact model accounts for adhesion, the unloading portion of the curve, which is most affected by adhesion, was used for the fit. Figure 3(C) shows the obtained elastic moduli as a function of the magnetic field strength for $\Phi = 0\% ,{ }9\% ,{ }17\% {\text{ and }}23\%$. The elastic modulus increased quadratically with increasing magnetic field strength and CIP volume fraction up to ∼11× increase for $\Phi = 23\%$, which is an order of magnitude larger than the increase in elastic modulus $\left( {{{{{E_{\left( H \right)}}}}/ {{{E_{\left( {H = 0} \right)}}}}}} \right)$ reported for rubber-like MREs subjected to the same range of magnetic field strengths [1, 2].

Figure 4 shows the results of the surface roughness measurements with white light interferometry. The root-mean-square roughness, R_q, was used as a quantitative measure of the roughness across the entire surface. Surface profiles for $\Phi = 9\% ,{ }17\% ,{ }23{{\% }}$ at 95.5 kA m⁻¹ (figure 4(A)) show an increase in R_q at both large and small length scales. Figure 4(B) shows the surface profiles for $\Phi = 23{{\% }}$ at various magnetic field strengths. R_q was found to increase monotonically with magnetic field strength and $\Phi$, as shown in figure 4(C). Interestingly, R_q also increased quadratically with increasing magnetic field strengths up to ∼7× increase for $\Phi = 23\%$.

Zoom In Zoom Out Reset image size

We used a 2D power spectral density (C(q), PSD) analysis to study the evolution of the surface topography with magnetic field strength. We follow the methods of Dash et al [30] and Jacobs et al [31] to obtain the 2D-PSD of the measured surfaces given by equation (2),

$$\begin{equation}C\left( q \right) = \frac{1}{{{L^2}}}{\left| {{{\mathop \iint \nolimits }}\frac{{{d^2}r}}{{2\pi }}{e^{iq \cdot r}}\langle z\left( r \right)\rangle } \right|^2},{ }r = \left( {x,y} \right)\end{equation} \tag{ 2 }$$

where L [2] is the scanned area of length L, q is the spatial frequency, and z is the height. Figure 4(D) gives the PSD curves for the $\Phi = 23{{\% }}$ sample for all magnetic field strength values. We consider two parameters: (a) the plateau height at low wavelengths, which is related to the average height of the rough surfaces; and (b) the slope of the tail of the PSD curve which gives an idea of the fractal dimension of the roughness. For increasing magnetic field strength, the plateau height increases monotonically while maintaining a similar slope as q increases.

Finally, we examine the Hurst exponent (h) to determine the degree of self-affinity of each surface. The Hurst exponent was obtained by fitting a power law to the tail of the PSD to equation (3) [32, 33]:

$$\begin{equation}C\left( q \right) \propto A{q^{ - 1 - 2h}}.\end{equation} \tag{ 3 }$$

The PSDs for the $\Phi = 23{{\% }}$ sample (figure 4(D)) have nearly constant h (h = 0.97 ± 0.01) which signifies that no particular spatial wavelength is amplified by applied magnetic field or particle inclusion. Combining this knowledge with the fact that the amplitude of R_q increases with magnetic field, we have determined that the topography of the surface remains similar only increasing in magnitude with magnetic field strength.

The inclusion of CIP increases the zero-field shear modulus and RMS surface roughness of the MREs approximately by the square of the CIP volume fraction, as shown in figures 5(A) and (B) respectively. The dependences of the zero-field (H = 0) shear storage moduli ($G^{^{\prime}}_0$) and RMS surface roughness $\left( {{R_{q,{\text{H}} = 0}}} \right){ }$on the CIP volume fraction ($\Phi$) can be fit by the following equations,

$$\begin{equation}G_0^{\prime} = 35000\left( {{\text{Pa}}} \right){\Phi ^2} + 760\left( {{\text{Pa}}} \right)\end{equation} \tag{ 4 }$$

$$\begin{equation}{ }{R_{q,{\text{H}} = 0}} = 4100{ }\left( {{\text{nm}}} \right){\Phi ^2} + 12\left( {{\text{nm}}} \right).\end{equation} \tag{ 5 }$$

Zoom In Zoom Out Reset image size

In the magnetic field regime (H < ∼100 kA m⁻¹ ), the shear storage modulus increases quadratically with respect to the magnetic field strength, which is consistent with a recent theory that has been compared to data for PDMS elastomers loaded with CIPs [24]. While analytic models [34, 35] have shown success in predicting the magnetic field-dependent increase in shear storage modulus for rubber-like MREs, they fail to predict the large increase observed in ultrasoft PDMS-based MREs. Here, we propose a single fit parameter equation for the shear storage modulus as a function of CIP volume fraction ($\Phi$) and magnetic field strength (H):

$$\begin{equation}G^{^{\prime}}_{\left( {\Phi ,{\text{ H}}} \right)} = 760\left( {{\text{Pa}}} \right) + 35000\left( {{\text{Pa}}} \right){\Phi ^2} + \alpha {\Phi ^2}{\mu _0}{H^2}\end{equation} \tag{ 6 }$$

where $G^{^{\prime}}_{\left( {\text{H}} \right)}$ is the shear storage modulus of the MRE, $\alpha$ is a fit parameter, $\Phi$ is the CIP volume fraction, ${\mu _0}$ is the vacuum permeability, and H is the magnetic field strength. The fit parameter $\alpha = 134\,$ (dimensionless) was determined by averaging $\alpha = \frac{{G^{^{\prime}}_{\left( {\text{H}} \right)} - {G^{^{\prime}}_0}}}{{{\Phi ^2}{\mu _0}{H^{\,2}}}}$ for each magnetic field strength and volume fraction. The proposed equation fits the experimental data well (figure 6(A)) but begins to deviate at higher CIP volume fractions. Equation (6) can be extended to the elastic modulus by approximating the MRE as a perfectly elastic material that conserves volume such that $E = 3G$ resulting in [36]:

$$\begin{equation}{E_{\left( {\Phi ,H} \right)}} = 2280\left( {{\text{Pa}}} \right) + 10500\left( {{\text{Pa}}} \right){\Phi ^2} + 3\alpha {\Phi ^2}{\mu _0}{H^{\,2}}\end{equation} \tag{ 7 }$$

Zoom In Zoom Out Reset image size

The equation agrees well with the experimental data as shown in figure 6(B). Equations (4), (6) and (7) provide researchers with a method for tuning the moduli of ultrasoft PDMS-based MREs to a wide range of desired biological systems.

Similarly, the magnetic field-dependent RMS surface roughness, which originates from the magnetic interactions between the magnetic particles [37], can also be fit by:

$$\begin{equation}{ }{R_{q\left( {\Phi ,{ }H} \right)}} = 12\left( {{\text{nm}}} \right) + 4100\left( {{\text{nm}}} \right){\Phi ^2} + \beta {\Phi ^2}{H^{\,2}}\end{equation} \tag{ 8 }$$

where $\beta = 4100{ }\left( {{\text{nm}} \cdot \frac{{{m^2}}}{{{A^2}}}} \right)$ is the fit parameter. Interestingly, the fit also agrees well with the experimental data as shown in figure 6(C), which suggests similarities in the underlying mechanisms driving the magnetic field-dependent change in moduli (volumetric) and surface roughness (interfacial).