## 2 Material and Methods  
### The Underlying Theory  
The modified Helmholtz resonator (HR) proposed in this paper is shown in Figure 1a. It consists of a cavity and a spiral neck with inlet and outlet. There are several well-known 2D spirals including Archimedes spirals, hyperbolic spirals, Theodorus spirals, Fibonacci spirals, Cornu spirals, etc. [14]. Among them, the neck of the proposed HR in this paper is designed along the Archimedes spiral, which is the trajectory of a point moving with constant velocity on a rotating straight line with constant angular velocity in the horizontal plane. Archimedes spirals are common in the natural world and widely used in practical applications such as the flat coil spring and the planar spiral coil [15,16]. There are some other spiral structures modified from the Archimedes spiral, such as the rectangular, hexagonal and octagonal spirals usually investigated as planar spiral inductors [17], which have discontinuities in their curvatures that may yield non-uniformity of velocity or acceleration of fluid moving inside. For the purposes of this paper, we needed a uniform cross-section, a smooth change of curvature as well as a long length of spiral defined in a 2D plane simultaneously. Considering the engineering restrictions mentioned above, the Archimedes spiral was selected in this paper. This kind of structure can be easily fabricated using 3D printing or etching techniques, which are common in the field of sensors. For the biosensor application, the HR was filled with two kinds of fluid, i.e., the fixed fluid and the test solution. This HR can be equivalent to an HR with a straight neck with sectional area \(S_{n}\), as shown in Figure 1b. The length of the neck filled with fixed fluid and test solution is marked as \(L_{f}\) and \(L_{s}\), respectively. For a small HR, a small volume usually corresponds to a large resonant frequency, but the frequency can be lowered by making the neck length much longer. For a small HR, the well-known frequency formula \(\omega_{\rm HR}=c_{0}\sqrt{S_{n}/(LV)}\) obtained using the classical mass-spring system was justified by [PERSON] [18]. The proposed HR can be considered as a lumped mechanical system consisting of mass and spring, as shown in Figure 1b, to calculate the resonant frequency. The mass (\(m\)) of the equivalent mass-spring system can be obtained as  
\[m=m_{f}+m_{s}=\rho_{f}S_{n}L^{\prime}{}_{f}+\rho_{s}S_{n}L^{\prime}{}_{s} \tag{1}\]  
where \(\rho\) is the density and \(L^{\prime}\) is the equivalent length. The subscripts \(f\) and \(s\) indicate the fixed fluid and test solution, respectively. \(L^{\prime}{}_{f}\) and \(L^{\prime}{}_{s}\) are usually longer than \(L_{f}\) and \(L_{s}\) because of the radiation mass loadings. They can be computed as \(L^{\prime}{}_{f}=L_{f}+\alpha\) and \(L^{\prime}{}_{s}=L_{s}+\beta\), where \(\alpha\) and \(\beta\) are the end correction lengths determined as \(\alpha~{}=~{}0.36\sqrt{S_{n}}\), \(\beta~{}=~{}0.48~{}\sqrt{S_{n}}\)[19]. In a polar system \((r,\theta)\), the Archimedes spiral can be expressed as \(r=a\theta\) and the arc length of the Archimedes spiral can be obtained by  
\[L_{\rm as}=\frac{1}{2}a\Big{(}\theta\sqrt{1+\theta^{2}}+\ln\Big{(}\theta+\sqrt {1+\theta^{2}}\Big{)}\Big{)} \tag{2}\]  
where \(a\) is the distance between the starting point and the origin of the polar coordinate system, and \(\theta\) is the rotation radian of the spiral. The stiffness (\(k\)) of the equivalent mass-spring system can be obtained by  
\[k=\frac{PS_{n}}{\Delta L}~{}=~{}\frac{\rho_{s}c_{s}{}^{2}S_{n}^{2}}{V} \tag{3}\]  
Then, the resonant frequency can be calculated using  
\[f_{hr}=\frac{1}{2\pi}\sqrt{\frac{k}{m}}=\frac{c_{s}}{2\pi}\sqrt{\frac{\rho_{ s}S_{n}}{\big{(}\rho_{f}L^{\prime}{}_{f}+\rho_{s}L^{\prime}{}_{s}\big{)}V}} \tag{4}\]  
Thus, the HR functions as a muffler when it is linked to a duct as a side-branch (Figure 2), and the maximal noise attenuation will occur at the resonant frequency of the HR.