Janine K.
Nunes
*^{a},
Jiang
Li
^{b},
Ian M.
Griffiths
^{c},
Bhargav
Rallabandi
^{d},
Jia
Man
^{e} and
Howard A.
Stone
^{a}
^{a}Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, USA. E-mail: nunes@princeton.edu
^{b}School of Mechanical Engineering, University of Science and Technology Beijing, Beijing 100083, China
^{c}Mathematical Institute, University of Oxford, Oxford, OX2 6GG, UK
^{d}Department of Mechanical Engineering, University of California, Riverside, California 92521, USA
^{e}School of Mechanical Engineering, Key Laboratory of High Efficiency and Clean Mechanical Manufacture (Ministry of Education), Shandong University, Jinan 250061, China

Received
18th October 2020
, Accepted 21st December 2020

First published on 13th January 2021

The dynamics of the wrapping of a charged flexible microfiber around an oppositely charged curved particle immersed in a viscous fluid is investigated. We observe that the wrapping behavior varies with the radius and Young's modulus of the fiber, the radius of the particle, and the ionic strength of the surrounding solution. We find that wrapping is primarily a function of the favorable interaction energy due to electrostatics and the unfavorable deformation energy needed to conform the fiber to the curvature of the particle. We perform an energy balance to predict the critical particle radius for wrapping, finding reasonably good agreement with experimental observations. In addition, we use mathematical modeling and observations of the deflected shape of the free end of the fiber during wrapping to extract a measurement of the Young's modulus of the fiber. We evaluate the accuracy and potential limitations of this in situ measurement when compared to independent mechanical tests.

For polyelectrolyte systems, such as DNA–protein systems, the formation of stable wrapped complexes due to electrostatic interactions is determined by numerous parameters. The most important of these parameters include the length of the polymer, its linear charge distribution, its persistence length, the size, charge and curvature of the complexing object, and salt concentration.^{1} While the resulting thermodynamically stable conformations of these complexes have been investigated, the dynamics of this wrapping process can be challenging to study at such small dimensions. Inspired by macromolecular wrapping behavior, we describe a model experimental system to study the dynamics of spontaneous electrostatic wrapping of a slender high-aspect-ratio object around a curved oppositely charged solid object at micrometer to millimeter length scales, and use both analytical and numerical analyses to extract useful properties of the system.

The dynamics of flexible fibers moving in a viscous fluid is an important aspect of the rich class of problems involving fluid–structure interactions, and is essential for understanding, for example, microscopic biological motion involving slender flexible filaments (e.g., cilia, flagella), the development of micro-actuators, and many industrial practices involving flexible fiber suspensions.^{7,8} In the self-driven wrapping motion of a fiber around an oppositely charged curved particle, the microfiber sweeps through the fluid with decreasing length, and it adopts a unique deflected shape during wrapping. We model this behavior to enable an in situ “measurement” of the Young's modulus of the fiber, which we compare with independent mechanical testing.

Fiber designation | a (μm) |
Q
_{1} (mL h^{−1}) |
Q
_{2} (mL h^{−1}) |
---|---|---|---|

Soft | 17.5 | 0.16 | 1.8 |

Soft | 20 | 0.16 | 1.2 |

Soft | 22.5 | 0.16 | 1.0 |

Soft | 27.5 | 0.16 | 0.8 |

Rigid | 20 | 0.1 | 0.75 |

Ultraviolet (UV) light was used to initiate the cross-linking reaction in the flowing oligomer solution. The UV light was supplied by a fluorescence light source (120 W mercury short arc lamp) on a Leica DMI4000B inverted microscope via the 20× magnification objective lens; spot size = 1.4 mm. No optical filters were used to modify the broad spectrum of light supplied from the lamp. For fiber fabrication, we measured the UV light intensity to be 35 ± 5 mW cm^{−2} using a UVA/B digital light meter (spectrum range 290–370 nm), based on a sensor area = 0.8 cm^{2}. The oligomer solution was exposed to 200 ms pulses of UV light.

The fibers were collected in DI water and washed 3 times in 1 wt% aqueous solution of Tween 80 non-ionic surfactant, followed by at least 2 washes in 0.1 wt% Tween 80 solution. To limit excessive adhesion to all surfaces, both fibers and particles were stored in a 0.1 wt% aqueous solution of Tween 80. After cleaning, a dye adsorption assay (see ESI†) was used to estimate the amount of bound carboxyl groups in the fibers. Lastly, to ensure that the entire length of the fiber was in the field of view during the wrapping measurements, the fibers were manually cut immediately before experiments. The fiber lengths were 3–5 mm, and the fiber radii a ranged from 17.5–27.5 μm. Polymerization conditions were selected to minimize variations in the Young's modulus among the fibers with different radii. However, the fiber modulus is not constant in this study, and increases slightly with increasing a.

A probe with a spherical indentation tip was used to deform the samples of the fibers and the particles immersed in water. The radius of the spherical tip of the probe was 39.5 μm, and the spring constant of the cantilever was 3.56 N m^{−1}. Though the spherical tip is of comparable size to the fibers, the contact area is expected to be small with a small indentation depth, so we utilize the traditional Hertzian model; this approach is consistent with measurement of spindle-shaped cells where reasonable values were obtained for the reported elastic modulus.^{16} A single force curve was generated per particle or fiber for at least 4 different particles or fibers in the batch, and an average Young's modulus, E, per batch was determined. The Young's moduli of the soft and rigid fibers (a = 20 μm) were measured to be 82 ± 5 kPa and 210 ± 30 kPa, respectively. The modulus of the disk-shaped particles was found to be 2500 ± 900 kPa.

During the first revolution of fiber wrapping, we record the length of fiber adhering to the particle (from one free end), and we find that the length varies linearly with time (Fig. 1c). Moreover, within the limits of experimental error, this constant wrapping rate is independent of the initial length of the free end of the fiber for the range of lengths tested (1–3 mm) (Fig. S2, ESI†). This result suggests that viscous drag over the length of the fiber plays a negligible role in the dynamics of this wrapping process, and so the wrapping rate is determined by a dissipative mechanism close to the contact point.

Since the fiber wraps with a constant speed, we measure the wrapping rate (in mm s^{−1}) to study the effects of the radius of the “soft” fiber a and the particle radius r, which is shown in Fig. 2. The wrapping rate can also be presented in terms of an angular frequency, which is shown in ESI† (Fig. S3). For constant a, we observe a roughly linear increase in the wrapping speed with increasing r, for 125 μm ≤ r ≤ 1 mm, as shown in Fig. 2a. From the trend, we expect a positive horizontal intercept (corresponding to a wrapping rate = 0 mm s^{−1}), indicating that there is a critical radius r_{c} below which the fiber cannot wrap; in this case, r_{c} ≲ 125 μm. In Section 4.2, we develop a theory to predict r_{c} for different particle, fiber and solution properties.

In contrast, we observe an inverse relationship between the wrapping speed and a when r is constant (Fig. 2b). The trends in wrapping rate as a function of fiber and particle radii indicate that the fiber wraps if the gain in interaction energy outweighs the energetic cost of bending—the difference is dissipated due to motion through the fluid. Viscous dissipation occurs both as a result of having the free end of the fiber sweep through the fluid and by the squeezing out of fluid through the small gap as the surfaces make contact. The bending energy per unit length scales as Ea^{4}/r^{2}, where E is the Young's modulus of the fiber, thus we expect that, as the radius of the particle increases (decreasing curvature), less energy is required to deform the fiber, which may lead to an increase in the rate of wrapping (Fig. 2a). Similarly, we expect that the wrapping rate may decrease if the radius of the fiber increases, i.e., the bending rigidity of the fiber increases, and so more energy is required to bend a larger radius fiber around a particle of a given radius (Fig. 2b).

For the results shown in Fig. 3b, we consider the behavior of the soft and rigid fibers. We observe that the rigid fiber can only wrap around the largest particle in our study, r = 1 mm. Otherwise, the trend in wrapping rate due to salt concentration for the rigid fiber is the same as the soft fiber, except that the transition to the no-wrapping regime occurs at a lower concentration of NaCl (0.1 mM). Though the Young's modulus of the rigid fiber is approximately 2.6 times larger than that of the soft fiber (based on nanoindentation measurements; refer to Section 2.3), the wrapping rate is about one order of magnitude smaller than that of the soft fiber (dashed lines in Fig. 3b). This result suggests a significant effect of fiber Young's modulus on the dynamics of the wrapping rate. We explore this dependence in more detail in the subsequent section by deriving a model to quantify the Young's modulus of the fiber from the deflected shape in situ during wrapping for more convenient characterization and a theory to predict the critical radius for wrapping.

Our experiments demonstrate the role of salt in screening the electrostatic interaction between the fiber and particle, inhibiting wrapping above one critical concentration, and inhibiting adhesion (without wrapping) above a second higher critical concentration. However, we note that if salt is added after the fiber has wrapped around the particle (initially in a salt-free medium), there is no observable effect of added salt concentration. Moreover, we note that an adhered fiber does not unwrap from the particle for any of the conditions tested in this study.

4.1.1 Model assumptions and geometrical setup.
Prior to wrapping, the fiber is in a prestressed configuration that may not be straight (see Fig. S4, ESI†). The fiber initially makes contact with the particle at a single point. The fiber subsequently wraps around the particle, with the part of the fiber lying above the contact point wrapping in a clockwise fashion while the part that lies below the contact point wraps in an anti-clockwise fashion (Fig. 1).

We define a coordinate system (x,z) such that the origin lies at the particle center (see Fig. 4 for a schematic). We model only the part of the wrapping fiber above the contact point, which wraps in a clockwise fashion. We model the fiber shape with angle from the x-axis, θ, and arclength, s, coordinates, where s ∈ [0,L] is the arclength of the fiber measured from the first contact point (s = 0) to the end (s = L) (Fig. 4). We suppose that immediately before contact, in the absence of any motion, the fiber's initial state is described by θ_{p}(s) with θ_{p}(0) = 0. This information is extracted from the experiment (see ESI† for details). We set t = 0 to be the time at which the fiber first makes contact with the particle, of radius r, and without loss of generality assume this is at the far-left point of the particle. The initial fiber configuration is thus θ(s,0) = π/2 + θ_{p}(s). We suppose that the fiber subsequently wraps around the particle in a clockwise fashion and, based on the experimental observations, we assume it wraps with constant and known angular frequency ω. At time t, the fiber will have thus wrapped an angle α_{w} = ωt. At this time, the part of the fiber described by 0 ≤ s ≤ s_{w}(t) = rα_{w}(t) is wrapped while the part of the fiber described by s_{w}(t) ≤ s ≤ L is free; the total length of the detached fiber is L_{d}(t) = L − s_{w}.

Fig. 4 Schematic illustrating the wrapping motion of the fiber. The fiber is parameterized using angle-arclength coordinates θ(s,t). |

The x and z coordinates of the fiber at time t are then given by

(1) |

(2) |

The fiber is observed to detach from the particle tangentially to the surface. In the experiments, the deflection of the fiber is typically small compared with the deflection of the prestressed state. As a result, it is convenient to work with the deflections from the prestressed configuration. Letting ψ denote the free-fiber configuration relative to the deflection from the prestressed state, we may write the fiber configuration as

(3) |

4.1.2 Model equations.
A cylindrical fiber of length L_{d} and radius a moving in a fluid of viscosity μ experiences a force per unit length due to viscous resistance^{17}

where n and t denote the unit normal and tangent to the fiber (Fig. 4); v_{t} and v_{n} are the velocity components of the fiber in the tangential and normal directions to the fiber surface, and are given by

The fiber will also experience a resistance due to friction with the base of the Petri dish. This resistance scales with a/d where d is the spacing between the fiber and the base. In our experiments, d is not measured, but appears to vary during wrapping and between different experimental trials. Here, a/d is assumed to be small and so we neglect this effect, however we note that it may lead to a small correction in the prediction of the Young's modulus of the fiber described in Sections 4.1.3 and 4.1.4.

where T and F are the internal normal and tangential forces on the fiber respectively. The anticlockwise bending moment on the fiber, M, is assumed to be proportional to the deviation of the curvature from the prestressed state and is thus given by

where we recall that E is the fiber Young's modulus, while I = πa^{4}/4 is the second moment of area of the fiber's cross-section.

These express, respectively, zero internal tension, shear and bending moment at the end, and the tangential departure of the fiber from the particle at the detach point.

(4) |

(5a) |

(5b) |

A force balance in the directions parallel and perpendicular to the fiber and a balance of moments on the fiber yields, respectively,

(6a) |

(6b) |

(6c) |

(7) |

At t = 0, the fiber makes first contact with the particle. Prior to this, the fiber is stationary, so no external forces act and thus f_{t} = f_{n} = 0, and the fiber will be in its prestressed configuration, θ(s,0) = π/2 + θ_{p}(s). The initial x and z coordinates are determined by (1) and (2), and F = T = M = 0 from (6).

At time t > 0, the fiber described by 0 ≤ s ≤ s_{w} wraps onto the particle and its location is thus known trivially. For s_{w}(t) ≤ s ≤ L, we must solve for the fiber shape using (3), (6) and (7). The system is solved subject to the following boundary conditions

(8) |

4.1.3 Model reduction.
While we may solve the model presented in its full form, we make two observations that simplify the model. First, the initial prestressed state is typically close to straight (θ_{p} ≪ 1), and second, the deformation of the fiber from this prestressed state is weak (ψ ≪ 1). Exploiting these features we find that

which is effectively a local rigid-body rotation of an elastic, but nearly straight, fiber. Substituting this into (6) leads to the reduced system

where ξ = (s − s_{w})/L_{d} and

subject to the boundary conditions

We emphasize that, while the right-hand side of (13) contains explicit time dependence, β(t) varies in such a way as to ensure that the Young's modulus E remains constant throughout the experiment, as is expected since this is a material property. We will verify this in Section 4.1.4.

Thus the deflection of the tip relative to the horizontal is

where D is the dimensional deflection of the fiber tip from the prestressed state. Eqn (16) provides a simple method for estimating the Young's modulus from the maximum fiber deflection from the centerline.

v_{t} ≈ 0, v_{n} ≈ −(s − s_{w})ω, | (9) |

(10) |

(11) |

The system behavior is thus characterized entirely by the dimensionless parameter β(t), given by (11). We note that the time dependence of the parameter β arises due to the change in the free-fiber length L_{d} as it wraps during the experiment and that all other parameters in (11) remain constant during the experiment. This equation can be rearranged to provide a prediction for the Young's modulus in terms of physical parameters

(13) |

The final deflection of the profile from the prestressed state is

(14) |

(15) |

We may use this expression as a simple way of determining the Young's modulus when the deflection from the prestressed state is not too large and the prestressed state is close to flat. In dimensional terms, this result may be rearranged to give a prediction for the Young's modulus,

(16) |

While the above strategy provides an estimate of the Young's modulus, we may obtain a more accurate estimate from determining the parameter β(t) from (11) if we are able to fit the entire experimental fiber profile to the model eqn (10). Then, (13) provides the Young's modulus for the fiber. In the following section we shall show how in practice both of these strategies are helpful to obtain a prediction for the Young's modulus, depending on the experimental setup and in each case we compare this with measurements made using nanoindentation.

4.1.4 Comparison with experiments.
First, we consider the experiment in which a soft fiber wraps around a particle of radius 0.25 mm. We perform the algorithm to predict the Young's modulus outlined in ESI,† for a snapshot taken from an experiment conducted in 0.1 mM NaCl. The rotation rate is found from experiments to be ω = 0.037 s^{−1}.

In Fig. 5, we show the experimental fiber deflection after accommodating for the prestressed state and the prediction from the theory (14) and observe a good agreement in the shape. For completeness, we also compare this with the full theory in which we make no assumption on the prestressed fiber shape and the subsequent deflection (see ESI† for details). We observe almost no discernible difference between this prediction and the full numerical system, which confirms the validity of the simplified theory to describe the fiber dynamics.

Fig. 5 Comparison of the prediction of the deflection of a soft fiber wrapping around a 0.25 mm radius particle in 0.1 mM NaCl for the fiber configuration given by the solution to the reduced model for small-deflection prestressed states (14) (red dashed) and the experimental data (green solid). The match between the data and the theory is excellent. We also show in blue dot dashed the prestressed state and in black the solution to the full model presented in the Model equations section (see ESI† for details). The difference between the two theories is barely discernible, supporting the use of the simplified model (14). |

We use three independent snapshots and match the experimental profiles with the theoretical model and obtain a prediction of E = 71 ± 6 kPa, where the error here corresponds to one standard deviation (see Fig. S6, ESI† as an example). The smallness in the error and the good fit to the experimental shape provides confidence in the reliability of this method. This also emphasizes the fact that in (13), although both β and L_{d} vary with time, the predicted Young's modulus remains constant. Nano-indentation measurements give a Young's modulus of this fiber of 82 ± 5 kPa, which is not too dissimilar to the prediction.

We now attempt a similar approach for a rigid fiber wrapping around a particle of radius 1 mm in 0.1 mM NaCl. In this case, the rotation rate of the fiber is more than five times slower than in the soft fiber case (ω = 0.0067 s^{−1}). Coupled with the fact that this fiber is more rigid, we observe only marginal deflection of the fiber, which is at the level of the pixel resolution of the images (see Fig. S7, ESI†). Indeed, for an experiment conducted on the rigid fiber, we observe a deflection of a single pixel from the prestressed state due to the viscous drag forces. As a result, it is unwise to match the shape of the fiber as we did in the case of the soft fiber. In this case, we instead use (16), to give an estimate of the Young's modulus. However, when determining the error bounds in this case we know that the deflection is at least 0.5 pixels and less than 1.5 pixels. Thus we obtain an estimate for the Young's modulus of 134 ± 67 kPa. For comparison, the Young's modulus of the rigid fiber was measured to be 210 ± 30 kPa by nanoindentation. While we acknowledge the reduced accuracy within which we can predict the Young's modulus in this case, we observe that this predicts a higher Young's modulus than that of the soft fiber, and so we are able to distinguish between the two fibers.

(17) |

This favorable energetic contribution is offset by the bending energy which, per wrapped length, is EI(1/r − 1/r_{p})^{2}/2, where 1/r_{p} = −∂θ_{p}/∂s is the signed curvature of the fiber in its initial (prestressed) configuration.^{20} In our experiments, the curvature of the initial state is typically small relative to that of the particle and will be neglected below. The net energy cost per unit time to wrap the fiber around the particle at a speed ṡ_{w} = ωr is then

(18) |

(19) |

We estimate σ as the charge corresponding to the first few molecular-thickness layers at the fiber surface. Denoting the bulk carboxylate ion density by n_{e} and the radius of a carboxylate ion by a_{i} = 0.35 nm (based on the size of the formate anion^{21}), the outermost k layers of the fiber surface carry a charge ke(2a_{i})n_{e}. For the 20 μm radius soft particle, we measure n_{e} ≈ 0.017 ions per nm^{3} (see ESI†), yielding that the top one to two “layers” of ions carry a charge in the range σ ≈ 1.8–3.6 mC m^{−2}. We note that this range of surface charge corresponds to zeta potentials σλ_{D}/ε in the range 25–50 mV at a 1 mM NaCl concentration, which is typical of similar charged hydrogels.^{22,23} For simplicity, we use the mean value within the range of surface charge estimated above, σ = 2.7 mC m^{−2}. We note that σ is expected to remain relatively independent of concentration since the fibers and particles are seeded with bulk charges at a fixed density.

We now estimate r_{c} for a soft fiber (E = 82 kPa) with radius a = 20 μm wrapping in pure (deionized) water (corresponding to Fig. 2a). We use n_{∞} = 10^{−7} M (the concentration of ions in deionized water) to calculate λ_{D} and, with the value of σ estimated above, find r_{c} ≈ 0.2 mm, consistent with the experimental observation for the initiation of wrapping. Alternatively, the expression (19) can be inverted to obtain a prediction for the radius of the thickest fiber, a_{c}, that will wrap a particle of radius r, given by

(20) |

To predict ω one needs to balance the energy gained from wrapping with the rate of dissipation. While the mechanism of dissipation remains unclear, we make some estimates of viscous dissipation here. The energy dissipated due to the drag on the fiber is estimated as [see (4) and (9)] and is approximately μω^{2}L_{d}^{3}/log(L_{d}/a). Very close to the contact point, there is an additional lubrication force associated with squeezing the fluid between the fiber and the particle. This force per unit length of the fiber scales as f_{lub} ∼ −μa^{3/2}h^{−3/2}v_{n}, where h is the separation distance between the surfaces.^{24} Assuming a weakly curved fiber consistent with (9), we estimate h ∝ x^{2}/r and obtain an energy dissipation rate , where h_{m} is a molecular cutoff length.

For the experimentally measured parameters we find that both of these sources of dissipation are of similar magnitude, but both are smaller than the rate of energy gained from wrapping. However, the experimental observation that the wrapping rate is independent of the detached length L_{d} (see ESI†) indicates that the actual dissipation mechanism in the experiments must be local to the region around the contact point. Such a mechanism may involve an interplay of electrostatic, elastic and viscous forces on length scales much smaller than the fiber radius.^{25} For example, viscous dissipation is likely amplified by a local flattening of the fiber cross-section by the lubrication pressure^{26} or by electrostatic or adhesive forces. Energy may also be dissipated within the hydrogel matrix of the fiber as the contact point moves along it.^{27} Quantifying these sources of dissipation requires a detailed analysis of the near-contact dynamics and is an interesting line of future inquiry.

In addition, we present a model to describe the shape of a fiber as it wraps around a particle due to an attractive force between the fiber and particle. This model provides a convenient and useful approach to estimating the modulus of the fiber in situ. The full model comprising integro-differential equations can describe arbitrarily large deflections of the fiber from its original prestressed state. By making the assumption that the deflections are not too large simplifies the model to a set of coupled partial differential equations. Making the further assumption that the initial prestressed state is close to straight enables us to write down the profile shape explicitly. The shape of the fiber is characterized by a single dimensionless parameter β, which corresponds to a balance between the viscous forces exerted on the fiber and the resistance to deformations through the bending stiffness. Matching the theoretical model to the experimental images provides a prediction of β, which in turn gives a prediction for the Young's modulus.

For soft fibers that deflect by a measurable amount we are able to extract a prediction for the Young's modulus that does not differ significantly from snapshot to snapshot. For stiffer fibers, or for experiments in which the wrapping rate is slower, the deflections are more difficult to measure accurately. In this case the model is still able to make a prediction, but with necessarily larger associated errors. This model is applicable in a wide range of other scenarios in which a fiber is deflected due to an external viscous flow, and may be useful in probing the material properties in scenarios in which other invasive methods are difficult.

- R. G. Winkler and A. G. Cherstvy, in Strong and Weak Polyelectrolyte Adsorption onto Oppositely Charged Curved Surfaces, ed. M. Müller, Springer Berlin Heidelberg, Berlin, Heidelberg, 2014, pp. 1–56 Search PubMed.
- X. Yi, X. Shi and H. Gao, Phys. Rev. Lett., 2011, 107, 098101 CrossRef.
- P. Guo, D. Liu, K. Subramanyam, B. Wang, J. Yang, J. Huang, D. T. Auguste and M. A. Moses, Nat. Commun., 2018, 9, 1–9 CrossRef.
- B. Roman and J. Bico, J. Phys.: Condens. Matter, 2010, 22, 493101 CrossRef CAS.
- R. D. Schulman, A. Porat, K. Charlesworth, A. Fortais, T. Salez, E. Raphaël and K. Dalnoki-Veress, Soft Matter, 2017, 13, 720–724 RSC.
- C. Py, P. Reverdy, L. Doppler, J. Bico, B. Roman and C. N. Baroud, Phys. Rev. Lett., 2007, 98, 156103 CrossRef.
- A. Lindner and M. Shelley, Fluid-Structure Interactions in Low-Reynolds-Number Flows, The Royal Society of Chemistry, 2016, pp. 168–192 Search PubMed.
- P. Katsamba and E. Lauga, Phys. Rev. E, 2019, 99, 053107 CrossRef CAS.
- J. K. Nunes, K. Sadlej, J. I. Tam and H. A. Stone, Lab Chip, 2012, 12, 2301–2304 RSC.
- A. Perazzo, J. K. Nunes, S. Guido and H. A. Stone, Proc. Natl. Acad. Sci. U. S. A., 2017, 114, E8557–E8564 CrossRef CAS.
- M. Slutzky, H. A. Stone and J. K. Nunes, Soft Matter, 2019, 15, 9553–9564 RSC.
- D. Qin, Y. Xia and G. M. Whitesides, Nat. Protoc., 2010, 5, 491–502 CrossRef CAS.
- J. C. Love, D. B. Wolfe, H. O. Jacobs and G. M. Whitesides, Langmuir, 2001, 17, 6005–6012 CrossRef CAS.
- D. Dendukuri, P. Panda, R. Haghgooie, J. M. Kim, T. A. Hatton and P. S. Doyle, Macromolecules, 2008, 41, 8547–8556 CrossRef CAS.
- L. Cheng, X. Xia, L. Scriven and W. Gerberich, Mech. Mater., 2005, 37, 213–226 CrossRef.
- S.-A. Xie, T. Zhang, J. Wang, F. Zhao, Y.-P. Zhang, W.-J. Yao, S. S. Hur, Y.-T. Yeh, W. Pang, L.-S. Zheng, Y.-B. Fan, W. Kong, X. Wang, J.-J. Chiu and J. Zhou, Biomaterials, 2018, 155, 203–216 CrossRef CAS.
- H. A. Stone and C. Duprat, Fluid-Structure Interactions in Low-Reynolds-Number Flows, The Royal Society of Chemistry, 2016, pp. 25–77 Search PubMed.
- J. N. Israelachvili, Intermolecular and Surface Forces, Academic Press, San Diego, 3rd edn, 2011, pp. 253–289 Search PubMed.
- H. Ohshima, Biophysical Chemistry of Biointerfaces, John Wiley & Sons, 2011 Search PubMed.
- B. Audoly and Y. Pomeau, Elasticity and geometry: from hair curls to the non-linear response of shells, Oxford University Press, 2010 Search PubMed.
- N. A. Lange and J. G. Speight, Lange's Handbook of Chemistry, McGraw-Hill Education, New York, 16th edn, 2005 Search PubMed.
- S. Schmidt, H. Wang, D. Pussak, S. Mosca and L. Hartmann, Beilstein J. Org. Chem., 2015, 11, 720–729 CrossRef CAS.
- A. H. Faraji, J. J. Cui, Y. Guy, L. Li, C. A. Gavigan, T. G. Strein and S. G. Weber, Langmuir, 2011, 27, 13635–13642 CrossRef CAS.
- D. Jeffrey and Y. Onishi, Q. J. Mech. Appl. Math., 1981, 34, 129–137 CrossRef.
- T. Tang, C.-Y. Hui and A. Jagota, J. Appl. Phys., 2006, 99, 054906 CrossRef.
- R. H. Davis, J.-M. Serayssol and E. Hinch, J. Fluid Mech., 1986, 163, 479–497 CrossRef.
- B. Andreotti and J. H. Snoeijer, Annu. Rev. Fluid Mech., 2020, 52, 285–308 CrossRef.

## Footnote |

† Electronic supplementary information (ESI) available. See DOI: 10.1039/d0sm01857k |

This journal is © The Royal Society of Chemistry 2021 |