Universal behavior of hydrogels confined to narrow capillaries

Flow of soft matter objects through one-dimensional environments is important in industrial, biological and biomedical systems. Establishing the underlying principles of the behavior of soft matter in confinement can shed light on its performance in many man-made and biological systems. Here, we report an experimental and theoretical study of translocation of micrometer-size hydrogels (microgels) through microfluidic channels with a diameter smaller than an unperturbed microgel size. For microgels with different dimensions and mechanical properties, under a range of applied pressures, we established the universal principles of microgel entrance and passage through microchannels with different geometries, as well as the reduction in microgel volume in confinement. We also show a non-monotonic change in the flow rate of liquid through the constrained microgel, governed by its progressive confinement. The experimental results were in agreement with the theory developed for non-linear biaxial deformation of unentangled polymer gels. Our work has implications for a broad range of phenomena, including occlusion of blood vessels by thrombi and needle-assisted hydrogel injection in tissue engineering.

Pressure difference dependence of Qgap, Qgel and their sum Q fitted to experimental data (circles) in the leakage-dominated regime and the following linear response regime with the best fitting parameters g = 0.38 μm and ΔPc/E0 = 1.9. Contributions to the total flow rate (black) from the flow through the leaking gaps (blue) and the microgel (red) are shown separately. Note that the leakage flow is insignificant in the higher pressure difference regime ΔP > E0 shown in Fig. 3 (main text). Figure S6. Schematics of microgel confinement. Axial coordinates x0 and x are used for the undeformed and deformed states of the microgel; x0 = 0 corresponds to the center of the unperturbed microgel and x = 0 corresponds to the entrance of the constriction. We assume that a diskshaped thin slice in undeformed state at position x0 with respect to its center deforms homogeneously without bending into a disk-shaped thin slice at x(x0). Front and back microgel tip positions are xf and xb, respectively, while the front and back boundary positions of the deformed gel are x'f and x'b (axial boundaries of microgel portion in contact with the microchannel walls). We neglect the deformation of the back and front caps (dark shaded regions) assuming λ⊥ = λ|| = 1 inside the caps (xb ≤ x ≤ x'b and x'f ≤ x ≤ xf). With these boundary conditions, we numerically solve the differential force balance equations (55) and (56). Figure S7. Numerical results for the deformation ratio profiles, normalized concentration profiles, volume change factor, normalized flow rate, and normalized front tip position of the microgel in the constriction. Geometrical parameters are α = 30°, dc = 38 μm, D0 = 103 μm in (a-d) and α = 15°, dc = 42 μm, D0 = 94 μm in (e). Qualitatively similar results are obtained for different sets of geometrical parameters. (a) Deformation ratios (ratios of deformed to undeformed lengths) in axial λ|| (upper solid curves) and radial λ⊥ (lower dotted curves) directions as functions of axial coordinate x along the deformed microgel normalized by the constriction diameter dc. Different colors correspond to different microgel locations: microgel completely localized in the tapered region (x'f < 0, red lines); microgel at the point of entry into the constriction (x'f = 0, green lines); partially constricted before translocation (x'f > 0, ΔP < ΔPmax, blue lines), and microgel at the translocation instability point (ΔP = ΔPmax, grey lines). The vertical lines at x = 0 mark the constriction entrance in (a) and (b). (b) Concentration profile along the confined microgels. Colors correspond to the same microgel locations along tapered and constriction zones as in (a). Note that for microgel partially in constriction (blue and grey lines) the maximum concentration is near the boundary between tapered zone and constriction. (c) Dependence of the relative change in volume λ⊥ 2 λ||, on the radial deformation ratio λ⊥. Logarithmic axes. Colors correspond to the same microgel locations along tapered and constriction zones as in (a). Black dashed line representing the asymptotic dependence λ⊥ 2 λ|| ~ λ⊥ 4/3 (obtained from equation (63)) predicted for the case of no external axial stress is in reasonable agreement with the numerical solutions in the presence of external axial stress due to flow through the microgel (colored lines). (d) Dependence of the flux Q (normalized by E0/R0, see equation (73)) through the microgel on the pressure difference ΔP across it (normalized by E0). Solid black curve shows numerical calculation described in Section 4, including flow through the caps. Red line shows asymptotic linear dependence Q ~ ΔP for low pressure differences. Blue line shows asymptotic scaling Q ~ ΔP -2.0 for high pressure differences. Colored circles correspond to the same microgel locations as in (a)-(c). (e) Normalized front tip position of the microgel in the constriction xf/dc as a function of normalized pressure difference ΔP/E0. Black curve is the result of the numerical calculations for α = 15°, dc = 42 μm, D0 = 94 μm. Markers show the experimental results (see Fig. 4b of the main text) for these geometrical parameters. The red line with a slope 2.3 (see equation (59)) is in good agreement with numerical and experimental results over a wide range of ΔP/E0 for the strongly constricted microgel. This scaling relation (equations (2) and (6) in the main text) is used in the theoretical analysis of experiments (equations (3)(4)(5) in the main text).

Supplementary
The resistance R is obtained by integration over cylindrical rings of radius r, thickness dr and length L(r).

Experimental setup
The experimental setup for microgel studies is described elsewhere 1 . Briefly, a MF device was interfaced with upstream and downstream water reservoirs. Microgels were introduced into the MF device from a syringe attached to the upstream reservoir using a three-way valve. The pressure difference, ΔP, applied along the microchannel was controlled by varying the difference in heights of the two reservoirs. The motion of the microgel was recorded with a high-speed camera (Canon EX-F1) and analyzed by a code written in MATLAB.

Supplementary Note 2. Determination of the flow rate of water through the microgel trapped at the entrance to the constriction
We employed fluorescence recovery after photobleaching (FRAP) in combination with confocal laser scanning microscopy (CLSM) to measure the rate, Q, of water flow through a microgel trapped at the entrance to the constriction, as a function of applied pressure difference ∆P.
Before introducing a microgel into the MF channel, a 200 μm depth scan was performed at increments of 5 μm in the direction perpendicular to the imaging plane of the microchannel (z-direction).
By using 3D reconstruction of the z-stack of CLSM images, we ensured that the microchannel had a close-to-circular cross-section (Supplementary Fig. S1a and b). FRAP experiments were performed at constant position, z = 0 μm, that is in, the middle plane of the circular microchannel (see Supplementary   Fig. S1a), in order to avoid the attenuation of intensity that occurs in CLSM experiments when zcoordinate is varied. With the assumption of axisymmetry (explained in the derivation following the present section), the measurement of fluorescence intensity in the middle plane is sufficient to describe the fluorophore concentration throughout the geometry of the circular microchannel.
After introducing a microgel with an unperturbed diameter D0 = 104 μm into a microchannel (dc = 38 μm, α = 30°), a pressure difference not exceeding ΔPmax was applied to the MF system to confine the microgel in the tapered zone of the microchannel. An intense (100% power of 5.4 mW), 25 s laser pulse photobleached a rectangular (250 μm × 150 μm) region in the microchannel-at-large using the NIS Element software. To monitor the recovery of fluorescence, the program set the instrument to the attenuated beam (10% power) and a series of 20 images was recorded with 5 s intervals between image capturing. Photoemission intensity data, I(x,r), were collected from the images (selected images shown in Supplementary Fig. S2a), where x is the coordinate in the flow direction measured from the left edge of the image, and r is the radial coordinate measured from the axis of the channel. The reference photoemission intensity, I0(x,r), was obtained from the image taken with the attenuated beam prior to the photobleaching event.
To determine the average velocity, U, of the fluid flowing through the microgel from the CLSM data, the normalized first axial moment M1/M0 of the cross-section-averaged fluorophore concentration, defined as was calculated from the CLSM images taken at time intervals of 5 s. In equation (1), I(x,r) is the fluorescence intensity distribution; x and r are the axial and radial positions coordinates; and l and d are the length and diameter of the microchannel in the image (see Supplementary Fig. S2a). The double integrals in equation (1) were calculated in MATLAB using the trapezoidal rule 2 . As shown in the next section, when the fluorescence intensity profile after bleaching is nearly a square pulse (that is, the 2D projection of the bleached region is nearly a rectangle), and the fluorophore distribution is axisymmetric, M1/M0 can be approximated by the following function of time The derivation of the equation (2) is given below. In equation (2), X0 is the intercept of M1/M0 at t = 0.
Since the radial dependence of fluorescence intensity can be measured either from the top half, or the bottom half of the CLSM image (the distribution is symmetric), the average value of M1/M0 (shown with black squares in Supplementary Fig. S2b) from the two halves was fit to equation (2); this is shown in ± 14 μm 3 /s, where d/2 is the radius of the microchannel, measured from the CLSM images to be 60 μm.
The above procedure was implemented for a range of pressure differences varying from the value required to block the microchannel, up to ∆Pmax. For each pressure difference, the FRAP experiment was repeated three times. Between repetitive FRAP measurements, a 5 min time interval was allowed for the system to reach a steady state. The variation in the average flow rate of the liquid through the microgel as a function of pressure difference is shown in Supplementary Fig. S3. For pressure difference below 2000 Pa, the flow rate of water was high, due to its leakage through the gaps between microchannel walls and the microgel. The estimation of this effect is given in Supplementary Note 4.

Supplementary Note 3. Interpretation of photobleaching data
In the derivation of equation (2), we followed the work of Aris 3 . Consider the pressure-driven flow of a Newtonian fluid through a circular tube. The flow occurs in the x direction, and r and θ are the crosssectional coordinates in the polar form. The governing equation for any disturbance, Δρ, in the unbleached fluorophore concentration distribution is the time-dependent convective diffusion equation Here, Df is the diffusion coefficient of the fluorophore, x is the direction of flow, and u is the velocity in the flow direction where U is the average velocity through the tube. The operator 2 ∇ in equation (3) is the gradient in the cross-sectional coordinates r and θ . Note that Δρ is a disturbance in the concentration of the fluorophore.
For example, if the initial concentration of the fluorophore is ρ0 and the concentration at a later time is ρ, then the disturbance, Δρ, in the concentration is defined as Δρ = ρ -ρ0. In the experiment, the disturbance is introduced in the form of a rectangular region by photobleaching, and is, therefore, non-zero only in that region. As shown in the derivation that begins in the next paragraph, the axial velocity of the center of mass of the concentration disturbance is equal to the average velocity of the fluid. This result is straightforward to understand, when diffusion is absent, but is valid even when there is diffusion-induced radial and axial smearing of the concentration, which can be explained as follows. Since the initial fluorescence intensity distribution produced by photobleaching is rectangular, it is symmetric about the center of mass at t = 0. As time proceeds, convection advances to the two edges of the rectangle to the right via the parabolic velocity profile, thereby establishing concentration gradients in the radial direction (see Fig. S3a). At the left edge, there is radial diffusion of the fluorophore from the center towards the walls, along with axial diffusion in the flow direction. Conversely, at the right edge, the fluorophore diffuses radially from the walls to the center, and axially -in the direction opposite to the flow. The key is that the diffusional distortions produced at the two edges complement each other exactly, their axial moments sum to zero, and do not contribute to the overall axial motion of the center of mass of the distribution. This is demonstrated more rigorously in the derivation below.
Multiplying equation (3) by x n , and integrating in x from 0 to l yields ( ) Here, Cn is the n th axial moment of Δρ, defined as For example, C0, the zeroth axial moment of Δρ, is In deriving equation (5), we have assumed that the concentration disturbance at the two ends of the tube, x = 0 and x = l, is negligible. This is a valid assumption provided that the photobleached region far from the edges of the tube throughout the duration of the experiment. This condition was maintained in FRAP experiments. By integrating equation (5) where Mn is the cross-section-integrated value of Cn, where dA is an elemental cross-sectional area. Examination of equation (5) for n = 0 gives the governing equation for C0 (equation (7)) 2 0 with the boundary condition The function Γ(r, θ ) is obtained by axially integrating the initial concentration distribution Δρ(x, r, θ, 0).
The above problem has the following eigenfunction expansion solution The functions φ j and µ j are the eigenfunctions and eigenvalues, respectively, of the following linear problem with the boundary condition The constants Bj are derived from the initial condition for C0 in equation (14) using biorthogonality condition, Note that Γ is the axial integral of the initial concentration distribution (see equation (14)). Since ∬ Γ = 0 , the constant B0 = zero. In addition, for the special case of a square pulse in the initial concentration profile that is imposed in the experiment, Bj = 0 for all 0 j ≠ , as well, implying that Bj = 0 for all j. Note that for a square pulse, Γ is independent of the cross-sectional coordinates. The null eigenfunction of the operator in equation (16) (i.e. µ 0 = 0) is φ 0 = 1, and satisfies the following biorthogonality Equation (20), combined with equation (19), gives Bj = 0 for 0 j ≠ .
Since Bj = 0 for j, the solution for C0 reduces from equation (15) to: (21) Expressing equation (8) Thus, by fitting the temporal variation in the ratio M1/M0 to the linear equation (15), we obtain U as the slope. For a square pulse at t = 0, the spatial distribution of the fluorophore concentration is axisymmetric at every instant, which enables the description of the cross-sectional variations of the fluorophore concentration with only the radial coordinate, r. The elemental area, dA, for the double integral is 2πr dr.
With this simplification, the definition of M1/M0 given in equation (1) can be used.
In practice, the assumptions of axisymmetry and square pulse are not exact. To be more accurate, all the eigenmodes in the solution of C0 in equation (15) (except the j = 0 mode) have to be considered, which would result in the following expression for M1/M0: A linear regression of equation (26) will provide the value of U. However, the calculation of M1/M0 would require the fluorescence intensities throughout the cross-section to be measured as a function of r and θ, which is not possible in our experimental setup due to the time required for a complete cross-sectional confocal scan, and the dependence of the intensity on the z-position.

Supplementary Note 4. Estimation of the effect of leakage of water through the gaps between the microgel and microchannel walls
At small pressure difference ∆P applied to the system, leakage of water may occur due to the nonconformal contact between the microgel trapped at the entrance to the constriction and microchannel walls. If the hydraulic resistance to flow through these gaps is smaller than that for the flow through the gel, the total water flow will be dominated by the leakage through the gaps, thus biasing the results. In this Supplementary Note, we estimate the effect of applied pressure difference, ∆P, on the leakage flow and show that significant leakage occurs only for ∆P < E0.
Since the geometry of the non-conformal contact between the microgel and the wall is not known a priori, our estimation was based on the assumed gap geometry. First, we verified that a small mismatch between the spherical microgel shape and the elliptical channel cross-section cannot explain the experimentally observed pressure difference required to close the gaps. Thus we concluded that the mismatch between the channel geometry and the microgel shape should have different characteristics.
We considered the gaps between the microgel and the channel wall to be due to surface roughness ( Supplementary Fig. S4) and estimated the effective spacing g of surface irregularities on microchannel wall, as well as the pressure difference, ΔPc, required to close the gaps between the microgel and the channel surface.
From Hertzian theory of an elastic contact between two spheres 4 , we estimate the diameter, a, of the contact between the spheres and applied force, ℱ, as functions of strain (h): a ≈ (Dc h) 1/2 and ℱ ≈ E* (Dc h 3 ) 1/2 .
For the geometry shown in Supplementary Figure S4c, L is the axial length of region of contacts, and LD0 is proportional to the area of contact spots. Assuming the number of contacts n ≈ LD0/g 2 , the total force imposed on the microchannel wall is nℱ ≈ LD0ℱ /g 2 . This force is supported by the pressure difference ΔP. Since the radial wall pressure is proportional to ΔP, the total contact force is nℱ ≈ LD0ℱ/g 2 ≈ ΔP LD0, and thus, the force acting on a single contact surface is ℱ ≈ ΔP g 2 . Substitution of the force ℱ from equation (27) with Dc ≈ g and E* ≈ E0, gives The gaps are closed when h ≈ a ≈ g. Thus, from equation (28), the pressure difference ΔPc required to close the leaking gaps is , which agrees with experimental results (see Supplementary Fig. S3a). For ΔP > ΔPc, water flows only through the microgel without leakage.
The flow rate of water through the gaps is predicted as Qgap = ΔP/Rgap, where is the hydraulic resistance of the gaps 5 . The prefactor in square brackets is due to water leakage occurring simultaneously through D0/g gaps. The axial length of the contact region can be determined from the Hertzian theory for an elastic contact between two cylinders as 4 L ≈ (E*ℱ) 1/2 , where the force on walls is, Combining equations (27-31), we obtain which is plotted in Supplementary Fig. S5, as blue curve.  Fig. 3b and Supplementary Fig. S5). Hence, the total flow rate is given by Q = Qgap + Qgel.
We estimated the size, g, of gaps and the pressure, ΔPc, required to close the gaps by fitting the theoretical prediction for Q(ΔP) to the experimental Q-ΔP data in the leakage dominated regime and the following linear response regime. Such a fit resulted in the best fitting parameters g = 0.38 ± 0.02 μm and ΔPc/E0 = 1.9 ± 0.2, which agreed with experimental data (Supplementary Fig. S5). We note that the reported surface roughness of PDMS surface is in the submicrometer range 6 .
We therefore conclude that at pressure differences ΔP < E0, a large number of small-roughness elements with submicrometer size can provide leakage of water comparable to the flow within the gel.
The size of imperfections (on the order g ≈ 0.1 μm), the value of ΔP required for closing the gaps (on the order ΔPc ≈ E0), and the decrease in flow rate Q with increasing ΔP in the leakage-dominated regime (equation (32) and Supplementary Fig. S5), all agree with the experimental results. Furthermore, we estimate the hydraulic resistance of gaps as Rgap ≈ 0.1 -1 Pa⋅s/μm 3 from the left-most three data points in Supplementary Fig. S3a, while the hydraulic resistance of microgel R0 = 1 Pa·s /μm 3 is observed for ΔP > E0 in the linear response regime (equation (75) and red lines in Fig. 3b and Supplementary Fig. S5).
Hence, by comparing the two flow resistances, we conclude that for ΔP > E0 the flow of water occurs primarily through the microgel. We should also note that with a cubic (g -h) 3 scaling, the leakage flow is highly sensitive to the height of the gaps. Thus, once the gaps are smaller than a critical height (corresponding to equal leakage and microgel flow rates), the leakage flow rapidly diminishes.

Supplementary Note 5. Theoretical predictions of the behavior of microgels in confinement 1. Deformation free energy density
Consider a microgel prepared in a good solvent at the monomer number density ci that is swollen to an equilibrium concentration c0 and volume V0. The equilibrium swelling corresponds to the deformation factors λ0 = (ci /c0) 1/3 in all three dimensions of the microgel. In the microchannel, the microgel is locally deformed by additional factors λx = λ|| in the axial direction and λy = λz = λ⊥ in the transverse directions to a new number density c of monomers The free energy of the microgel is the sum of elastic and osmotic components 7 . We use the predictions of scaling theory for these two contributions to the microgel free energy. The osmotic pressure π of the microgel in a good solvent increases proportionally to the 9/4 power of its concentration 8,9 π ≈ ( / 3 )( 3 ) 9/4 , We assume that individual network chains between crosslinks deform affinely, that is, proportionally to the dimensions of the entire microgel. In this case, the dimensions of the chain in the deformed microgel  (35) and (37)) as 10 where the sum of squares of the deformation factors in equation (37) is ∑ λ 2 = 2λ ⊥ 2 + λ ∥ 2 . Note that the free energy per unit volume of the fully swollen microgel fc0/N normalized by the modulus of the fully swollen gel, is universal and depends only on the dimensionless deformation factors This is the reason why normalizing pressure difference ∆P by the modulus E0 of the swollen microgels generated universal dependences in Figs. 4b and 5b. Note that we have chosen the numerical coefficient in equation (42) to be 11/50, to assure that E0 is the Young's modulus.
Below we present the scaling relations between (i) the position of the microgel and the applied pressure difference, (ii) the reduction in microgel volume and the degree of microgel confinement, and (iii) the translocation pressure difference and the degree of confinement.

Deformation of a spherical microgel
In the steady state the size and shape of the spherical microgel under compression are determined by the condition of force balance, and below we calculate forces acting on the microgel. The walls of the microchannel impose a non-uniform deformation on the spherical microgel. Below, we describe the calculation of the deformation of a spherical microgel, neglecting the bending effects and the deformation of the caps (Supplementary Fig. S6). Such an approach is more applicable for the microgel confinement in a microchannel with small entrance angles α.
The total free energy of deformation, F, is an integral over the whole volume of the undeformed where x0 denotes the axial coordinate in the undeformed microgel state (with the origin at the center of the microgel) and 0 ( 0 ) = 4 ( 0 2 − 4 0 2 ) is the circular cross-sectional area of the undeformed microgel at x0 (Supplementary Fig. S6).
Consider infinitesimal disk-shaped slices of the undeformed microgel. We assume that when the microgel is confined in the microchannel, each disk of diameter We minimize the total free energy via standard variational method 4 . In addition to the large deformation described in equation (44), we define an infinitesimal deformation by x → xu = x + u(x). This variation in the axial coordinates of the disks, δx = xu -x = u(x), yields the variations of the deformation and the variation in total free energy Substituting λ⊥u and λ||u from equation (45) into this δF, expanding it in series up to first order in u and u', and using integration by parts, we obtain , and defining local stress in j-direction we simplify equation (47) as This extra free energy is stored in the microgel due to the conversion of the work δW performed on it. Here, ℱ int ( ) is the body force (force in x-direction per unit volume) acting on disks.
Assumptions of the "undeformed caps" and "no bending of disks" imply zero stress at the back and front boundaries of the deformed part of the microgel Note that σxx(x'b\f) = 0 requires λ⊥(x'b\f) = λ||(x'b\f) = 1 at the front and back boundaries of the deformed section of the microgel. Inside the deformed part of the microgel ℱ int ( ) = − ′ ( ) results in where ′ ( ) is the water pressure gradient. For a steady state, these conditions correspond to the force balance at the back and front boundaries of the deformed part of the microgel (equation (51)) and inside the deformed section of the microgel (equation (52)), respectively.

Pressure gradient along the microgel
In order to relate the water pressure gradient P '(x) = dP/dx to the deformation along the microgel, we modeled each elementary vertical slice of a microgel (a disk of diameter D(x) and thickness dx) at the axial position x by a densely packed set of parallel "pipes" with a diameter ξ(x) (the correlation length), which scales with polymer concentration as ξ = ξ0 (c/c0) -3/4 ; thus, ξ = ξ0 (λ⊥ 2 λ||) 3/4 (see equation (33)). The pressure difference dP across the pipe is dP = q dRξ, where dRξ is the hydraulic resistance of each pipe and q is the volumetric flow rate of water through the pipe (e.g., in units of μm 3 /s). Each disk of diameter D(x) and length dx accommodates m ≈ (D/ξ) 2 pipes, and the total resistivity of a disk containing m-pipes acting in parallel is dR = dRξ/m. The flow rate through the disk is Q = mq.
The flow of liquid inside the pipes is assumed to be laminar and described by the Poiseuille law, which gives the resistance of a pipe dRξ = (128η dx)/(π ξ 4 ), where η is the viscosity of the liquid 5 . Thus the resistivity of a disk at position x is and the pressure gradient at position x is dP/dx = (128 η Q)/(π D 2 (x) ξ 2 (x)). Note that Eq. (53) can be treated as Darcy's law for the flow through a porous medium with non-uniform permeability along the microgel as κ = ξ²/32, since Darcy's law predicts 5 dR = (4ηdx)/(πD 2 κ). By substituting the correlation length ξ = ξ0 (λ⊥ 2 λ||) 3/4 for good solvent conditions, we obtain , which must be balanced by the body force inside the deformed microgel, as was given in equation (52).
By using the geometry of the setup and the free energy density of the microgel (given in equation (42)) in the calculation of the stress components σjj, we obtain ∥ ′ ( ) where λ||' = d λ||/dx, and we have used an abbreviation for the characteristic flow rate (in units of length), ℓ 0 = (51200 η Q)/(11π ξ0 2 E0). Note that ℓ 0 ~ Q appears in λ||'/λ|| in the above equation since flow deforms the microgel in the axial direction. In addition to equation (55), the definition of λ⊥ given in equation (44), leads to where λ⊥' = d λ⊥/dx. In equations (55) and (56), we used a function Here, the positive and negative signs are for the two sides of the unperturbed microgel, x0 < 0 and x0 > 0 respectively, as the unperturbed diameter of a disk increases or decreases with x0 , depending on which side of the gel it is.

Results of numerical simulations
The coupled differential equations (55)  In order to determine the position xc, where the function G(x) (introduced in equations (55) and (56)) changes its sign, we also performed an axial length integration From the geometry defined in Supplementary Fig. S6, the center of the undeformed microgel (x0 = 0) corresponds to ( c ) = 1 2 � 0 2 − f 2 .
In Supplementary Fig. S7, we illustrate the numerical results of our theoretical approach For the highest increase in polymer concentration of c/c₀ ≈ 3.3, the largest reduction in permeability by the factor of 5 (or the smallest ratio κ/κ0 ≈ 0.2) was obtained. In Supplementary Fig. S7c, we plot the local volume change factor (λ⊥ 2 λ||) as a function of local radial deformation factor (λ⊥) using the data in Supplementary Fig. S7a. The relation λ⊥ 2 λ|| ~ λ⊥ 4/3 (obtained for no external axial stress from equation (63), dashed line) is in good agreement with the numerical data. In Supplementary Fig. S7d, we plot the calculated flow rate as a function of pressure difference and compare the numerical data with scaling predictions.
Finally, Supplementary Fig. S7e shows the numerical result for the normalized front tip position of the microgel in the constriction, xf/dc > 0, as a function of normalized pressure difference, ΔP/E0, compared to experimental data. Fitting a function in the power-law form, xf/dc ≈ (ΔP/E0) a , to the numerical data leads to a = 2.3 ± 0.1 for a broad range of pressure differences. The universal scaling is also in agreement with the experimental results for the strongly constricted microgels. In the next section, we discuss scaling relations for volume reduction, V/V0, and rescaled translocation pressure difference ΔPmax/E0.

Scaling predictions for a deformed microgel
In the Sections 2-4 above, we considered spherical microgels. Here, in order to obtain universal scaling predictions, we consider a microgel with a cylindrical undeformed shape and an equilibrium swollen diameter D0 with its symmetry axis oriented in x-direction.
The size dependence of translocation pressure difference, i.e., the power-law ΔPmax ≈ (D0/dc) 14/3 , is universal with an exponent 14/3, independent of the shape of the tapering zone. However, the prefactor (amplitude) of this power-law, K(α), is not universal and depends on the geometry of the microchannel and the microgel. To this end, we consider simulation results to obtain the amplitude K(α) and compare them with experimental data.
In Supplementary Fig. S8a, we plot the simulation results for rescaled translocation pressure difference, ΔPmax/E0, as a function of degree of confinement, D0/dc, for different tapering angles, α. We observe (i) persistent power-law behavior ΔPmax ≈ (D0/dc) 14/3 , and (ii) saturation of the angle-dependent amplitude for large angles. This saturation is evident in Supplementary Fig. S8b, in which we plot the calculated in equation (72) is inversely proportional to the thickness of the caps 0 − ( 0 /2 − δ). At small deformations (δ ≪ 0 ) the flow resistance is equal to that of the unperturbed microgel R ≈ R0 = (16/π) 2 η/(ξ0 2 D0) for δ ≪ 0 (73) and is therefore constant independent of compression δ and pressure difference ΔP. This implies that most of the fluid flows through the whole microgel rather than through the small volume in the vicinity of the compressed microgel section of size ( 0 /2 − δ). The high resistance of longer pores (with length ~ 0 ≫ ( 0 /2 − δ)) that pass through the central section of the microgel is compensated by the smaller number of such pipes. Thus we conclude that at small pressure difference, Δ ≪ 0 , the flux Q of water through the microgel is linearly proportional to the pressure difference in good agreement with numerical calculations (red line with unit slope in Supplementary Fig. S7d) and experiments (red line with unit slope in Fig. 3b).
Balance of elastic and osmotic stresses at undeformed equilibrium state leads to 9 E0 ≈ 2kT/ξ0 3 . For the Young's modulus E0 = 2570 Pa of the microgel used in the flow experiments, and kT ≈ 4.11·10 -3 Pa·μm 3 at room temperature, we estimate the pore size in undeformed microgel to be ξ0 ≈ 15 nm.
Substituting this value into the equation (73) for the resistance R0 of the undeformed gel of diameter D0 ≈ 100 μm with viscosity η ≈ 0.9·10 -3 Pa·s for water at room temperature, we obtain R0 ≈ 1 Pa·s/ μm 3 . This estimate is in excellent agreement with the experimental results -the best linear fit to the data (red line in