### Detection of bead transport on the MMA—phase-locked and phase-slipping regimes

Figure 2 presents the opti-MMA signal from a region in which SPM beads were dispersed as the frequency of the external rotating magnetic field, *ω*, was varied from 0 to 100 Hz. Five distinct regimes of behavior were identified that are characteristic of the *ω* and the number of the SPM beads. In regions (*i*) and (*vi*) an external magnetic field was not applied to the MMA and the photodetector signal was characterized by a background signal that was set by the number of beads in the optical sensing area and optical properties of the system. In region (*ii*) a low-frequency of rotation of the magnetic field, i.e., *ω* < 19 Hz, was applied to the MMA producing a periodic photodetector signal. In this range of *ω* the beads moved between the micro-mirror and silicon substrate in a phase-locked mode that was synchronized with *ω*, as illustrated in Fig. 1b. Optical imagines revealed that the minima of the photodetector signal (*V*_{min} ~ 785 mV) corresponded to all the beads positioned on-top of the micromagnets, *P1* in Fig. 1b, covering the highly reflective chrome regions that acted as micro-mirrors. The peak signal (*V*_{max} ~ 815 mV) corresponded to all the beads positioned on the silicon substrate, i.e., between adjacent micro-mirrors, *P2* as shown Fig. 1b. In this position the beads cover the less reflective silicon substrate increasing the total reflected light intensity.

In region (*iii*) the beads motion started to decouple from the external magnetic field at frequency equal to and higher than the critical frequency, *ω*_{c}. At this frequency, optical microscopy revealed that the beads started to become immobilize on the edge of micro-mirror and oscillated with an amplitude, *ΔA*, as shown in Fig. 1c. The amplitude of the photodetector signal was observed to decrease in region (*iii*), which was a result of the fact that *ΔA* decreased as *ω* increased^{22}. As the *ω* increased further it reached the immobilization frequency, *ω*_{i}, at which all the beads were in a phase-slipping mode and thus immobilized of the MMA. The critical and immobilization frequency of the 2.8 µm SPM beads were determined to be 19 ± 2 Hz and 40 ± 2 Hz, respectively. Region (iv) corresponded to the signal as the frequency was increased from 20 to 39 Hz. In this region, *ω*_{c} ≤ *ω* < *ω*_{i} and the amplitude of the signal continued decreasing as the oscillation frequency increased. The decreased *ΔA* resulted from an increasing portion of the beads entering a phase-slipping regime. In region (v), *ω* increased from 40 to 100 Hz where *ω* ≥ *ω*_{i}. In this region, all the beads were in the phase-slipping regime having a minimum oscillation motion at the edge of micro-magnets. This produced an average signal of ~ 800 mV approximately halfway between *V*_{min} and *V*_{max} with a peak-to-peak frequency ~ 5 mV. This resulted from the fact that SPM beads were trapped in a position between the silicon and micro-mirror, as shown in Fig. 1c.

### Sensitivity of the opti-MMA system: optical configuration and external frequency of rotation of the magnetic field

The magnitude of the photodetector signal was studied for four optical configurations, i.e., FOV1, 2, 3 and 4, as a function of the number of the beads in the sensing area. Figure 3 presents the normalized output voltage, *V*_{min}*/V*_{max}, for a known quantity of SPM beads for the four optical configurations at a frequency of 1 Hz, which is in the phase-locked transport regime (*ω* < *ω*_{c}). The relationship between nominalized voltage and the quantity of SPM was similar for all four FOVs, i.e., *V*_{min}*/V*_{max} decreased in a linear fashion with the number of SPM beads in the FOV. The sensitivity, *S*, of the optical detection systems was defined as

$$S=left|{(V}_{min}/{V}_{max})/Nright|,$$

(1)

where N is the number of SPM beads in the field of view, was found to be dependent on the optical configuration used to detect the beads. The sensitivity for FOV1, FOV2, FOV3 and FOV4 was 0.1307, 0.011, 0.0007, and 5 × 10^{–5} per bead, respectively. These results suggest that the sensitivity of the detection system was higher for optical systems with a smaller field of view. For instance, a single 2.8 µm bead could be easily detected in FOV1 while the limit of detection was two beads in FOV2, four beads in FOV3, and thirty-two beads in FOV4. Although the sensitivity of detection in larger fields of view was low, they had a wider area of detection and thus were capable of higher throughput, e.g., the numbers of detectable SPM beads in FOV4 varied from 32 to more than 800.

Figure 4 presents the dynamic response of the opti-MMA as a function of *ω* for specific densities of SPM beads in the four different FOVs. For each optical system the *V*_{min}*/V*_{max} was constant at low frequencies for a specific quantity of beads until the critical frequency was reached. This is consistent with the model of SPM beads moving in the phase-locked mode at low frequencies that was described above. The *ω*_{c} was found to be 19 ± 2 Hz for all bead and optical configurations except for 3 beads in FOV1. This exception can be attributed to the propensity of the beads to form clusters on a single micromagnet under these conditions. The *V*_{min}*/V*_{max} increased for *ω* higher than critical frequency until it reaches a constant value that was determined by the total number of beads in the FOV. It was also observed that the more beads that were in the FOV the larger the *V*_{min}*/V*_{max} change in the frequency range *ω*_{c}

< *ω* < *ω*_{i}. Taken together these results demonstrated that the magnitude of the photodetector signal at a specific frequency could be used to determine the number of beads in the FOV for a specific optical detection configuration.

### Theoretical analysis of the opti-MMA signal

A model has been developed to describe the signal of the opti-MMA detection system for a defined number of SPM beads as a function of the mode of transport. This model is based on describing the change in the photodetector signal as the SPM beads moved from the reflective micromagnets to the supporting silicon substrate, and a *V*_{min} and *V*_{max} are determined as the beads move in a synchronous manner under the external magnetic in either the phase-locked, *ω* < *ω*_{c} , or phase-slipping, *ω* ≥ *ω*_{i}, regime. In the frequency regime between these two limits, i.e., *ω*_{c} ≤ *ω* < *ω*_{i}*,* the motion of the beads was influenced by variations in their physical properties, i.e., size and magnetization, and the optical signal was a function of *ω*.

The photodetector signal was determined by the optical properties of the system. In the case where there are no beads on the surface of the MMA the photodetector signal was *V*_{o} = *R*_{s}* G Ω P*_{o}, where *R*_{s} is the responsively of photodetector, *G* is the transimpedance gain, *Ω* is a scale factor and *P*_{o} is the input light power to photodetector. The photodetector power was proportional to the reflected light intensity, *P*_{o} = *I*_{o}*R*_{o}, where *I*_{o} is the laser intensity at the surface of the MMA and *R*_{o} is the effective reflectance of the MMA. When SPM beads are on MMA their strong absorption and scatter behavior must be considered. If the reflectance of the MMA were uniform the opti-MMA signal without bead motion would be

$$Vleft( t right) = V_{o} – R_{s} GOmega I_{o} (R_{o} – R_{p} )NA_{p}$$

(2)

where *R*_{p} is the *effective* reflectance of the *N* beads of area *A*_{p}. The optical adsorption of the SPM beads will be determined by their physical properties, and we anticipate it will be proportional to their loading with iron oxide nanoparticles and follow a second order scale law with radius. The optical scattering of the SPM beads lies in the Mie scattering regime and thus also follow a second order scale law with radius,

In the phase-locked transport mode, the motion of the beads was described by a frequency dependent overlap parameter, *ζ(t)*, which was defined as the overlap of the beads with the micro-mirrors at different time points. In the case in which all beads moved in a synchronized phase-locked manner *ζ(t)* = *cos(ωt/2).* Thus, the value of *ζ(t)* is equal to 1 and 0 at *P1* (*θ*_{xz} = 0˚) and *P2* (*θ*_{xz} = 180˚), respectively, as shown in Fig. 1b. The output voltage of the photodetector for *N* beads on the MMA may expressed as

$$Vleft(tright)={V}_{o}-{R}_{s}GvarOmega N{A}_{p}{I}_{o}left[zeta left(tright){R}_{m}^{^{prime}}-(1-zeta left(tright)){R}_{Si}^{^{prime}}right]$$

(3)

where ({R}_{m}^{^{prime}}) and ({R}_{Si}^{^{prime}}) are the effective reflectivity of the micro-mirrors and silicon substrate, respectively, in the presence of the beads, i.e., ({R}_{Si}^{^{prime}})= *R*_{Si}–*R*_{p}. The minimum and maximum output voltages can be expressed in their simplest forms to be

$${V}_{min}={V}_{o}-{R}_{s}GOmega N{A}_{p}{R}_{m}^{{prime}}{I}_{o}, ;text{at} ;;theta_{xz}=180^{circ}, ;text{and}$$

(4a)

$${V}_{max}={V}_{o}-{R}_{s}GOmega N{A}_{p}{R}_{Si}^{{prime}}{I}_{o}, ; text{at} ;;theta_{xz}=0^{circ}.$$

(4b)

Figure 3 present the result of the calculations of *V*_{min}*/V*_{max} that were carried-out based Eq. (4). The empirical and materials parameters used for these calculations are listed in Table S1 (Supporting information). There is reasonably good agreement between the theoretical sensitivity and experimental observations for all four FOVs, i.e., *R*^{2} values are presented in the Fig. 3 caption. However, the model produced *V*_{min}*/V*_{max} values that were consistently larger than the experimental results for FOVs 2–4. Figure 1d indicates that the light reflected from the MMA is not uniform. This suggests the position of the beads must be considered and would produce higher theoretical *V*_{min}*/V*_{max} values.

At higher external field frequencies, *ω* > *ω*_{c}, the motion of beads was not fully synchronized with external field. In this regime the frequency, the photodetector signal was determined by *ΔA* and the population of beads in the phase-slipping regime. Based on the analysis of the speed of a SPM bead in the NLM system we defined.

$$A=C((omega -sqrt{{omega }^{2}-{omega }_{c}^{2}})/{omega }_{c}),$$

where *C* was a constant obtained empirically^{21}. This behavior can be defined using a second overlap parameter

$$psi (t)=((omega -sqrt{{omega }^{2}-{omega }_{c}^{2}})/{omega }_{c})cos(omega t/2)$$

for the fraction of beads in phase-slipping mode. The population of phase-locked and phase-slipping beads at certain oscillation frequencies could be determined using a cumulative distribution function which had been determined to have the form

$$Nleft(omega right)=left{begin{array}{ll}N; & quad omega le {omega }_{c}\ frac{N}{2}erfc left(frac{left(frac{omega }{2pi }right)-mu }{2sqrt{sigma }}right); & quad { omega }_{c} < omega le {omega }_{i}\ 0; & quad omega > {omega }_{i}end{array}right.$$

(5)

where *µ* is the mean and *σ* is the standard deviation obtained experimentally. Figure 5 presents the population of beads moving in a phase-locked and phase-slipping regime as a function of frequency.

Thus, the photodetector signal for *ω* > *ω*_{c} can be expressed in its simplest form in terms of *N(ω), ζ(t)* and (psi left(tright))

$$begin{aligned} Vleft( t right) & = {V_o} – {R_s}GvarOmega Nleft( omega right)left( {{A_p} – zeta left( t right)R_m^prime {I_o} – {A_p}left( {1 – zeta left( t right)} right)R_{Si}^prime {I_o}} right)\ & quad – {R_s}GvarOmega left( {N – Nleft( omega right)} right)left( {{A_p}psi left( t right)R_m^prime {I_o} – {A_p}left( {1 – psi left( t right)} right)R_{Si}^prime {I_o}} right) end{aligned}$$

(6)

Calculations of *V*_{min}*/V*_{max} were carried-out as a function of *ω* based Eq. (6) and the results are presented as lines in Fig. 4. The materials parameters and constants used for calculation were listed in Table S1. The theoretical sensitivity of the opti-MMA system in the transport regime where the beads no longer are phase-locked and was in reasonable agreement with the experimental observations for all four FOVs.

Although this theoretical model produced a reasonable description of the response of the opti-MMA system, several assumptions have been made in its derivation. First, the incident laser light intensity, *I*_{o}, in FOV was constant and homogenous. As we have already seen, variation in the laser intensity as a function of time and position will lead to systematic overestimation of the signal and increased noise. This can be corrected for a specific optical system. Second, we have assumed that the SPM beads adsorb and scatter light uniformly from the areas that they occupy on the MMA. This assumption had limited impact on the results presented in this study due to the strong optical activity of the beads, but obviously will need to be modified for smaller beads that may be bound to large analytes, such as, mammalian or bacterial cells. Third, we have assumed that we can accurately describe the SPM bead motion using Eq. (4), which is a function of the external magnet field. The primary sources of error in this assumption are the interactions of the beads with the MAA surface and each other that lead to transport behavior that is not consistent with Eq. (5). Nonspecific adhesion of the beads with the surface was observed for 1–5% of beads and these beads did not move under the external field and could not be detected in the *V*_{min}*/V*_{max} values. The aggregation of beads also led to a significant change in transport behavior, i.e., the critical immobilization frequencies decreased significantly for aggregates^{23}. The change in transport behavior resulting from bead aggregation can be observed in the optical signal at higher densities of beads in Fig. 4c for 90 beads on the MMA.