Microwave measurement of giant unilamellar vesicles in aqueous solution

A microwave technique is demonstrated to measure floating giant unilamellar vesicle (GUV) membranes in a 25 μm wide and 18.8 μm high microfluidic channel. The measurement is conducted at 2.7 and 7.9 GHz, at which a split-ring resonator (SRR) operates at odd modes. A 500 nm wide and 100 μm long SRR split gap is used to scan GUVs that are slightly larger than 25 μm in diameter. The smaller fluidic channel induces flattened GUV membrane sections, which make close contact with the SRR gap surface. The used GUVs are synthesized with POPC (16:0–18:1 PC 1-palmitoyl-2-oleoyl-sn-glycero-3-phosphocholine), SM (16:0 Egg Sphingomyelin) and cholesterol at different molecular compositions. It is shown that SM and POPC bilayers have different dielectric permittivity values, which also change with measurement frequencies. The obtained membrane permittivity values, e.g. 73.64-j6.13 for POPC at 2.7 GHz, are more than 10 times larger than previously reported results. The discrepancy is likely due to the measurement of dielectric polarization parallel with, other than perpendicular to, the membrane surface. POPC and SM-rich GUV surface sections are also clearly identified. Further work is needed to verify the obtained large permittivity values and enable accurate analysis of membrane composition.


Analytical model of the SRR and complex permittivity extraction of GUV membranes
To analyze, design, and extract the permittivity of MUT locating at the split, an appropriate mathematic model of the proposed SRR is required. Some equivalent circuitbased models have been proposed to meet the demand [1]- [3]. However, these existing models have to fit and adjust lumped-element parameters to agree with the measured Sparameters. Moreover, these fitted parameters are not always unique, which brings a great uncertainty for permittivity extraction. To address this issue, a novel analytical model is proposed to obtain S-parameters of the SRR based on necessary structural dimensions and dielectric information.
The proposed SRR can be modeled as shown in Fig. S1, which is composed of a coupled line and a gaped ML, discussed in Section 1 and 2, respectively. Then, using the even-odd mode analysis technique [4], they are combined together for the final 2-port Sparameters in Section 3. For the well-matched MLs on the left and right ends, they only shift the phase of the combined S-parameters. So the total phase of the SRR will be revised using the measurement result as reference.

S-parameters of Asymmetric Coupled Line
For a four-port asymmetric coupled line made up by two adjacent MLs a and b with a spacing s and arbitrary widths W 1 andW 2 in Fig. S2, its S-parameters can be expressed as (1) where S 11 =S 22 , S 21 =S 12 , S 33 =S 44 , S 43 =S 34 , S 31 =S 13 =S 42 =S 24 , S 41 =S 14 =S 32 =S 23 , because of the symmetry and reciprocity [5]. , the elements V i and I i (i=1, 3) represent the voltage and current at the port i of the network. And the impedance matrix is 11 , we have the following static capacitances to ground per unit length for each conductor (a and b): (1) Differential Mode (π) where the self-capacitance coefficients c 11 , c 33 are solved using c p +2c f from eqs. (3)-(4) in [7], the mutual-capacitance coefficients c 31 for two adjacent asymmetric lines is solved using C 0 ACPS• ε eff ACPS from eqs. (10) and (39) in [8].
To order to obtain the same modal impedances for two conductors, in the above equations, C aπ =C bπ and C ac =C bc should be met. Then we have It is worth noting that mn=-1, which will be used in some later calculations. For a special case when the conductor a and b in Fig. S2 are the same, m=-1, n=1. Then, the mode impedances are written as The reflection and transmission coefficients of each mode in terms of corresponding mode impedances and electrical lengths are expressed as For the asymmetric a and b (as a result, we use π and c instead of odd and even denoting modes), these equations (21)-(23) have verified by V. K. Tripathi et al [9].  One can easily prove that (28) and (29) The reflection and transmission coefficients for each mode are the same on each of the conductor a and b since z aπ =z bπ and z ac =z bc . Now we obtain Similarly, to apply an overall 1 V voltage at Port 3, the distribution of voltages at two conductors shown in Fig. S4 can be solved: for Port 1 (34) for Port 3 (2) Common Mode (c) Similarly, S 13 , S 33 , S 43 , S 23 can be also solved:  [12]. These models do not consider a top dielectric layer above the metal, e.g., the microfluidic channel filled with a solution. In this section, a π-network of capacitors [13] shown in Fig. S5 will be used to model this area for an effective and simple quantification. The following will introduce how to solve C end and C gap in Fig. S5, respectively.
Although some fitting equations are reported [13], these equations are generally not applicable to the multiple layer system of substrate/metal layer/GUV bottom surface/solution inside the GUV/GUV top surface/PDMS/air shown in Fig. S6. So, some more practical and applicable to the multiple layer system for C gap calculation needs to be established. For C end , however, most of electric fields emitted from edge only penetrate substrate material and terminate at the back conductor, rather than another conductor, so the fitting equation of C end is applicable. According to [13], where s is the gap width, ε sub and h are the dielectric constant and thickness of the substrate, respectively, and W is the width of the ML, which is different from the length w and perpendicular to the cross section in Fig. S6. Chen et al's work [17], the total capacitance per unit length between two adjacent metals shown in Fig. S6 are written by and (48) where x=air, sub, PDMS, and solution, K is incomplete elliptic integral of the first kind with variables k x and k x '. It is worth to be mentioned that in Fig. 1(b) the microfluidic channel does not cover the whole metal layer but only ~100 μm long on each side taking the split as the center. So w=100 μm is selected for the calculations of eqs. (43)-(48).
From Fig. 1(b), the microfluidic channel is only 25 μm wide (⊗ direction in Fig. S6), so most of the split is covered by PDMS, not solution. Similarly, the capacitance C wall per unit length for the cross section without the microfluidic channel is expressed as Consequently, C gap can be obtained as a shunt connection using C channel and C wall in parallel: where w channel =25 µm is the width of microfluidic channel, and w wall =75 µm is the width of electrodes covered by PDMS.

Combinations of Coupled Line and Gaped ML
The S-parameters of the symmetric SRR can be solved by analyzing its odd and even modes. Under odd and even mode excitations, it becomes to Fig. S7 (a) and (b) [4]. From  The following will take the odd mode as an example to demsonstrate how to solve S 11 o . The solving method of S 11 e is similar to S 11 o . It is analyzed by the following signal flow [18], where Γ sc =-1 for short-circuit. To write S 11 o =b 1 /a 1 directly is still difficult, so an additional step is necessary, i.e., to match Port 3 (let Γ o =0) and then solve the 2-port network [S 11 ' S 13 '; S 31 ' S 33 '] first, as shown in Fig. S9.
For S 11 ', the signal flow from Node 1 to 3 may be 1→3, 1→2→4→3, and 1→6→8→3.    The rest of the work is to solve the reflection coefficient Γ o and Γ e in Fig. S7 (a) and (b). Taking Γ o for example, it is S 11 of the 1-port network in Fig. S11, which is decomposed from Fig. S7(a). The 2-port S-parameters [S 11 " S 12 "; S 21 " S 22 "] of the transmission line TL1 and TL2 in series in Fig. S11 can be easily solved. The calculations of conductor loss and 45 o mitered bend can be found in [19], [20]. Then, S 11 of the network, i.e., Γ o , is expressed as eq.
(57). The similar process is also used to obtain Γ e .
For the SRR loading 0.1 M glucose-water solution in Fig. 1(b), the calculated |S 21 | from 1 GHz to 12 GHz using the proposed analytical model is shown in Fig. S12. The frequency-dependent complex permittivity of the solution [21] is used to calculate eq. (43).
To analyze the respective resonant frequencies of odd-mode and even-mode, |S 11 o | for oddmode and |S 11 e | for even-mode are also demonstrated, respectively, in Fig. S13 (a) and (b).
We can see that 1 st and 2 nd resonant frequencies of the odd-mode, i.e., 2.68 GHz and 7.99 GHz, correspond to 1 st and 3 rd resonant frequencies of the complete SRR, i.e., 2.54 GHz and 7.54 GHz as shown in Fig. S12, whereas 1 st and 2 nd resonant frequencies of the even-mode, i.e., 5.08 GHz and 10.99 GHz, correspond to 2 nd and 4 th resonant frequencies of the complete SRR, i.e., 4.77 GHz and 9.90 GHz. The deviations are caused since only half of full length is calculated for |S 11 o | and |S 11 e |, which is a requirement of the even-odd mode analysis technique [4]. It is also observed that the values at these frequencies for the combined |S 21 | are different from the single |S 11 o | and |S 11 e |, due to the combination calculation in eq. (52). The dimension (not included in Fig. 1c) of the proposed SRR in the calculation is shown in Fig. S14.

Complex Permittivity Extraction of GUV Membrane
In Fig. S12, 1 st frequency locates at 2.54 GHz with a |S 21 | of ~-20.17 dB, much lower than the measured ~-4.51 dB in Fig. 1(d), implying that the proposed coupled-line model has a stronger coupling between two adjacent MLs. The same thing also affects other resonant frequencies. The reason is that the calculation of the mutual-capacitance coefficients c 31 for two adjacent asymmetric lines uses the coplanar stripline (CPS)-based conformal mapping [7], but it is unlike two coplanar MLs of the SRR. For the CPS, almost all electric field streamlines emitted from an electrode terminate at another one; for two adjacent MLs of the SRR, a lot of electric field streamlines emitted from an electrode terminate at the back conductor [7]. So the calculated c 31 is larger than what it should be.
Moreover, the proposed model is derived based on the assumption of TEM propagation, which has to be used to meet the prerequisite of eq. (6.2), but it is only strictly satisfied in a homogeneous medium. Based on this assumption, the field distribution at any transverse plane perpendicular to z-direction in Fig. S2 can be treated as a linear combination of the proposed two fundamental TEM-modes, i.e., π Mode and c Mode in Section 1. Then the derivations in Section 1 can be further performed for a relatively simple and intuitive analysis of the SRR.
To extract the permittivity of the measured GUV membranes using the proposed model, some additional revisions are required. According to the above discussion, c 31 is the  Fig. S15(a). But, 2 nd resonant frequency shifts to 8.1 GHz with a |S 21 | of -1.53 dB, therefore. So the above adjustment is more suitable for permittivity extraction at 2.7 GHz, and c 31 and the length of the coupled line need to be adjusted again when extracting permittivity at 7.9 GHz. Figure S15( respectively, for good agreement at 2.7 GHz and 7.9 GHz, are compared with the measured data together in Table S1. The relative errors at 2.7 GHz and 7.9 GHz from Table S1 are 1.00% and 1.90%, respectively, suggesting that the proposed model can obtain an accurate permittivity of MUT. It is worth noting that the above comparison is made for a wide permittivity range (from the solution to the air). In fact, for the minute permittivity change without and with a GUV membrane, the proposed model is expected to obtain a lower relative error.   Until this step, however, the additional phase shift induced by the left and right MLs in Fig. S1 is not taken into account. In Fig. S15 Fig. 3 can be transformed to corresponding complex permittivities in Fig. 8. The same procedures are also done at 7.9 GHz. In the transformation, the width of the flattened GUV membrane (along W direction in Fig. S6) w GUV is selected as 25 μm. But, according to the discussion in 5 th paragraph of the main body, the possible w GUV value ranges from 20.8 μm to 25 μm. For other possible choices of w GUV , the extracted complex permittivities may demonstrate a slight deviation. Figure S16 shows a complete calculation flow graph for GUV membrane complex permittivity using the proposed model.