Drag force of polyethyleneglycol in flow measured by a scanning probe microscope

We propose a method to measure the drag force due to synthetic polymers in flowing fluids by using a scanning probe microscope (SPM). Methoxy polyethyleneglycol thiol (mPEG-SH) was attached to the cantilever probe of the SPM, which was further immersed in flows of glycerol and polyethyleneglycol (PEG) solutions. The mPEG-SH-bonded cantilever detects the extra force due to polymer-polymer and polymer-fluids interaction in flowing fluids. The conformation of the mPEG-SH polymer bonded to the probe of the cantilever was predicted as having a stem and ellipsoidal-flower shape, and the drag force due to the deformed mPEG-SH was calculated. The forces detected by experiments using the SPM and the forces obtained by model calculations were compared and found to be reasonably close.

Figure 1

(a) Whole image of the experimental setup. (b) Flow channel attached on the sample stage of the SPM. (c) Schematic of the flow channel connected to two syringes by tubes. The size of the small stainless-steel piece in the channel is indicated. (d) An example of the experimental data detected by the SPM. When the sample solution was at rest, the measured force was close to 0. When the sample solution was flowing, a cantilever attached to the SPM detected the drag forces.

Figure 2

(a) Top view of the flow channel where the microcell was immersed. (b) The probe cartridge holding the microcell. (c) The probe head of the SPM where the probe cartridge was mounted. (d) The lateral view of the sample stage. The cantilever probe was positioned in the middle of the channel. (e) A close-up image and an illustration of the cantilever. The cantilever was immersed in the flow at an approximate angle $\theta ={15}^{\circ }$.

Figure 3

Schematic illustration of the experimental procedure. A cantilever with the original gold-coated probe was positioned in flowing solutions of (a) glycerol and (b) PEG. The cantilever was further immersed in a mPEG-SH solution (c) to attach thiol-end to the gold-coated probe (d). The mPEG-SH-coated cantilever probe was further immersed in (e) glycerol and (f) PEG solutions.

Figure 4

Schematic illustration of the cross-sectional area of the flow channel. The stainless-steel piece helped to stabilize the flow, producing uniform flow fields at the inlet.

Figure 5

Local velocity profiles measured at the positions P1, P2, and P3 in 22 wt% and 35 wt% glycerol solutions. The flow rate was 4 ml/min.

Figure 6

Normalized velocity profile for each glycerol solution.

Figure 7

An example of the change in the resonant frequency ΔFreq during a QCM experiment.

Figure 8

Forces detected by the naked probe cantilever and by the polymer-bonded cantilever, in flowing solutions at several velocities. The viscosities of the 8 wt% glycerol solution and of the 1 wt% PEG solution are the same.

Figure 9

$C_{\mathrm{d}}$-Re plot for the naked cantilever probe in glycerol and PEG solutions.

Figure 10

Comparison between the $C_{\mathrm{d}}$ values of a naked cantilever probe, and a polymer-bonded cantilever probe (a) in glycerol solutions and (b) in PEG solutions.

Figure 11

The $\mathrm{\Delta }{C}_{\mathrm{d}}$ in each solution plotted as a function of their viscosities.

Figure 12

(a) Schematic of the cantilever, tilted with an angle $\theta ={15}^{\circ }$ to the flow. (b) Schematic of the mPEG-SH attached to the gold-coated probe of the cantilever from the lateral view. (c) Upside-down image of the cantilever to illustrate the V-shaped probe, where mPEG-SH molecules attach to its both lateral sides. (d) Close-up figure of the probe to indicate $l_{\mathrm{i}}$ and $\mathrm{\Delta }l$ along the edge. (e) Close-up figure of the probe from the downstream view showing the apex angle.

Figure 13

(a) Force balance between the stem part and the ellipsoidal-flower part. (b) Close-up figure of the polymer, focusing on the number of monomers.

Figure 14

Comparison of Δ$F$ on the left and the right side of Eq. (12). The former was obtained experimentally (Exp.), and the latter was obtained by calculations (Calc.). Δ$F$ is the difference between the force measured by the polymer-bonded probe cantilever and the force measured by a naked probe cantilever. To eliminate the measurement noise, the force measured by each cantilever was fitted by least squares prior to the calculation of Δ$F$. Experiments were done with multiple cantilevers, therefore the experimental value is indicated with the error bar.