Geometry-induced friction at a soft interface

Significance

In solving a jigsaw puzzle, we put together pieces having shapes that fit into each other. In nature, too, objects come in different shapes, with some that are compatible with each other and others that are not. If the objects are made of soft material, two incompatible pieces can also be made to fit together. What effect does this aspect of geometry have on the dynamical behavior of the system? We show through experiments and a theoretical model that soft, incompatible surfaces moving relative to each other, experience a significantly different frictional interaction than that experienced by geometrically compatible surfaces of the same materials. The mechanism can have important implications for the mobility of soft objects like biological cells on curved surfaces.

Abstract

Soft and biological matter come in a variety of shapes and geometries. When soft surfaces that do not fit into each other due to a mismatch in Gaussian curvatures form an interface, beautiful geometry-induced patterns are known to emerge. In this paper, we study the effect of geometry on the dynamical response of soft surfaces moving relative to each other. Using a simple experimental scheme, we measure friction between a highly bendable thin polymer sheet and a hydrogel substrate. At this soft and low-friction interface, we find a strong dependence of friction on the relative geometry of the two surfaces—a flat sheet experiences significantly larger friction on a spherical substrate than on flat or cylindrical substrate. We show that the stress developed in the sheet due to its geometrically incompatible confinement is responsible for the enhanced friction. This mechanism also leads to a transition in the nature of friction as the sheet radius is increased beyond a critical value. Our finding reveals a hitherto unnoticed mechanism based on an interplay between geometry and elasticity that may influence friction significantly in soft, biological, and nanoscale systems. In particular, it provokes us to reexamine our understanding of phenomena such as the curvature dependence of biological cell mobility.

The coming together of soft objects with incompatible geometries often leads to a rich glossary of patterns and phenomena, for example, the plethora of phases seen in bent-core liquid crystals (1), size-selection in the self-assembly of systems with incompatible building blocks (2, 3) and the emergence of beautiful wrinkle patterns when thin sheets are confined to substrates with incompatible geometries (4–7). In the last case above, the incompatibility lies in the Gaussian curvature mismatch between the thin sheet and the substrate and is a consequence of the Gauss’s Theorema Egregium (8). This problem has been studied in various settings, including a flat sheet on spherical liquid drop (6), a flat sheet on a spherical solid substrate (9–11), and a thin spherical shell on a flat liquid surface (7). The ground state obtained in these problems usually involves a nontrivial stress distribution arising due to geometrical incompatibility. While many recent papers have studied the effect of such geometrical incompatibility-induced stress distribution on static wrinkle patterns, its effect on the dynamics of such systems remains relatively unexplored.

In this paper, we study the effect of geometrical incompatibility on friction at a soft interface subject to small relative velocity ($v\sim$ 10 nm/s). Friction is an intrinsic feature associated with relative motion between two bodies. Given its ubiquity and importance, friction has been studied since very early times, yet many fundamental questions remain open (12, 13). Particularly, recent advances in nano- and biotribology have brought many surprises, for example, the recent observation of the dependence of friction on relative geometry and commensurability of two-dimensional layered atomic materials sliding relative to each other (14–16).

The interface studied in this paper consists of a thin flat elastic sheet placed on a low-friction hydrogel substrate having different geometrical shapes, viz. flat, cylindrical, and spherical. We find that the frictional force at this interface shows a strong geometry dependence and is significantly larger for the geometrically incompatible configuration of a flat sheet-spherical substrate than the other two geometrically compatible configurations: flat sheet-flat substrate and flat sheet-cylindrical substrate. We also observe a dependence of friction on the sheet size and see a transition in the friction behavior at an intermediate value of sheet radius for the flat sheet-spherical substrate system. We show that these effects are a consequence of the coupling of the stress developed in the sheet due to its geometrically incompatible confinement with the curvature of the interface, resulting in an increased normal force.

The experimental setup is shown schematically in Fig. 1A. Thin flat sheets of polystyrene (thickness $h=$ 50 nm to 800 nm) cut into circular discs of radius $W_0$ are placed on the surface of swollen hydrogel substrates. The hydrogel substrate loses water through evaporation in the ambient maintained at a temperature of 24 °C, and shrinks in size, generating relative motion at the thin sheet–hydrogel interface. We monitor the relative change between the sheet and substrate sizes and use it to obtain a measure of friction similar to the method of measuring the viscosity of a liquid from the terminal velocity of a sedimenting sphere.

Fig. 1.

Geometry-dependent tribometry. (A) Schematic of the experimental setup. (B) A typical image of the spherical hydrogel substrate obtained with bright field transmission mode imaging. The image is analyzed to obtain the substrate radius $R$. (C) Image of the sheet in the same system as in (B) captured by fluorescence imaging, which is used to obtain the projected radius $w$ of the sheet. (D) The variation of the normalized substrate radius with time $\tilde{R}(t)$. (E) The variation of the normalized sheet radius with time $\tilde{W}(t)$. (F) $\tilde{W}$ plotted as a function of $\tilde{R}$. A linear fit to the initial part of the function $\tilde{W}(\tilde{R}$) (red dashed line) is used to obtain the slope $m$. The limits $m$$=$ 0 (green dotted line) and $m$$=$ 1 (blue dotted line) represent no friction and large friction, respectively. The value of $m$ is a measure of the strength of friction.

We image the substrate and the sheet separately using two different imaging schemes implemented in sequence: a brightfield transmission mode imaging for the substrate and fluorescence imaging for the sheet which is tagged with the fluorescent dye Nile red. Typical images of the substrate (in this case, a sphere) and the sheet obtained for the same system are shown in Fig. 1 B and C, respectively. Pairs of such images, recorded at equal intervals of time ($\mathrm{\Delta }t\sim$ 10 min), are analyzed to obtain the substrate radius ($R$) and the sheet radius ($w$). The projected radius of the sheet measured from the image ($w$) is used to obtain the real radius of the sheet measured along the curved surface of the substrate: $W=R\mathrm{s}i{n}^{-1}(w/R)$.

We calculate the normalized substrate and sheet radii at any instant of time $t$ as $\tilde{R}(t)=R(t)/R_0$ and $\tilde{W}(t)=W(t)/W_0$, where $R_0$ and $W_0$ are their initial radii, respectively. We observe that $\tilde{R}(t)$ usually decreases at a constant rate (Fig. 1D where $\dot{\tilde{R}}$$-$0.011 ${\mathrm{h}}^{-1}$), as expected for evaporation-driven water loss proportional to the surface area. A decrease in $\tilde{R}$ is accompanied by a corresponding decrease in $\tilde{W}$ (Fig. 1E), due to the force of friction that opposes any relative motion between the sheet and the substrate. If the sheet–substrate interface was friction-less, the radius of the sheet would not change at all with time, and there would be large relative motion at the interface. On the other hand, in the limit of large friction, there would be no relative motion between the sheet and the substrate, and the radius of the sheet would change by the same factor as the substrate. On the $\tilde{R}-\tilde{W}$ plot (Fig. 1F), these two limiting cases correspond to straight lines with slopes $m=$ 0 and 1, respectively. The above discussion suggests that $m$ can be used as a measure of the strength of the frictional interaction. We show in SI Appendix that the velocity dependence of friction for hydrogels (17–20) allows us to write friction in terms of $m$ and that for small values of $m$, friction is proportional to $m$. Our setup therefore gives us a way to determine the small force of friction between the sheet and the substrate, thus working like a sensitive tribometer.

The $\tilde{W}(t)$ and $\tilde{R}(t)$ data presented in Fig. 1 D and E respectively, when plotted as $\tilde{W}(\tilde{R})$ lies between the two straight lines corresponding to $m$$=$ 0 and 1. When considered over a large range of $\tilde{R}$, $\tilde{W}(\tilde{R})$ is logarithmic, a behavior reminiscent of the logarithmic relaxation observed in experiments on crumpling of thin sheets (21, 22) (see SI Appendix for details). For the present discussion, we focus on the initial slope:

$$m=\underset{\tilde{R}\to 1}{lim}\frac{\mathrm{d}\tilde{W}}{\mathrm{d}\tilde{R}},$$ [1]

which we obtain by considering a small initial part $1>\tilde{R}\gtrsim$ 0.9 of the $\tilde{W}(\tilde{R})$ curve and fitting a straight line to it.

As with most materials, friction in hydrogels is proportional to the normal force; however, with a very small coefficient of friction ($\sim$0.01) (18, 23). Gravitational force, which is the dominant contributor to the normal force for macroscopic objects, has a very small value for the thin sheets ($\sim \rho g{W}^{2}h$). However, for the data in Fig. 1F, we obtain a finite value of $m$$=$ 0.15. When we repeat the experiment by placing sheets of different initial radii $W_0$, on spherical substrates of initial radius $R_0\approx$ 6 mm, we observe that the value of $m$ increases monotonically with $W_0/R_0$ as shown in Fig. 2A. We note that in the flat sheet-spherical substrate geometry, the strength of geometrical confinement depends on the parameter $W_0/R_0$. For a small value of $W_0/R_0$, the sheet sees almost a flat substrate, while for a value of $W_0/R_0\sim$ 1, the sheet feels the curvature of the substrate strongly. Therefore, the dependence of $m$ on $W_0/R_0$ indicates that the force of friction is being affected by the substrate geometry.

Fig. 2.

Effect of geometrical incompatibility on friction. (A) $m(W_0/R_0)$ for flat sheets ($h$$=$ 302 nm) on spherical substrates ($R_0\sim$ 6 mm). The red curve is a fit to Eq. 3. (B) $m(W_0/R_0)$ for flat sheets ($h$$=$ 302 nm, 376 nm) on flat substrates. (C) Variation of the slopes measured along the axial direction $m_{\mathrm{a}}$ (blue) and the curved direction $m_{\mathrm{c}}$ (red) with the mean curvature $1/(2R_0)$ for flat sheets ($h$$=$ 376 nm, $W_0$$=$ 2.6 mm, and 1.7 mm) on cylindrical substrates. We observe larger values of $m$ in (A) than in (B) or (C). (B and C) are geometrically compatible configurations, as the Gaussian curvatures of the sheet and the substrate match, and therefore ${\sigma }_{\mathrm{geom}}=0$. On the other hand, (A) is a geometrically incompatible configuration with ${\sigma }_{\mathrm{geom}}\ne 0$ and a corresponding pressure $P\sim {\sigma }_{\mathrm{geom}}.{\kappa }_{\mathrm{mean}}$.

In order to further verify the role of geometry, we measure $m$ for a flat sheet on a flat hydrogel substrate. Fig. 2B shows $m(W_0/R_0)$ for $h$$=$ 302 nm and 376 nm, with $R_0\approx$ 6 mm. We see that a flat sheet “slips” on a flat substrate, with $m<$ 0.1. The larger value of $m$ for the flat sheet-spherical substrate than for the flat sheet-flat substrate configuration confirms the role of curvature. However, a flat surface has neither Gaussian curvature (${\kappa }_{\mathrm{Gauss}}$$=$ 0) nor mean curvature (${\kappa }_{\mathrm{mean}}$$=$ 0), while a sphere has both: ${\kappa }_{\mathrm{Gauss}}$ = ${(1/R)}^2$ and ${\kappa }_{\mathrm{mean}}$ = $1/R$. With an aim to untangle the effects of the mean and Gaussian curvatures, we perform experiments on cylindrical substrates having ${\kappa }_{\mathrm{mean}}$ $=$ 1/(2$R$) and ${\kappa }_{\mathrm{Gauss}}$$=$ 0.

In our experiments on cylindrical substrates of radius $R$ and axial length $M$, we measure the sheet radii along both the axial direction ($W_{\mathrm{a}}$) and the curved direction ($W_{\mathrm{c}}$) and normalize them with their initial values, respectively: $\widetilde{W_{\mathrm{a}}}=W_{\mathrm{a}}/(W_{\mathrm{a}}(t=0))$, and, $\widetilde{W_{\mathrm{c}}}=W_{\mathrm{c}}/(W_{\mathrm{c}}(t=0))$. We then determine the slopes of the $\widetilde{W_{\mathrm{a}}}(\tilde{M})$ and $\widetilde{W_{\mathrm{c}}}(\tilde{R})$ curves to obtain $m_{\mathrm{a}}$ and $m_{\mathrm{c}}$, respectively. Fig. 2C shows the variation of $m_{\mathrm{a}}$ and $m_{\mathrm{c}}$ with the initial mean curvature of the cylindrical substrate 1/(2$R_0$). We observe that both $m_{\mathrm{a}}$ and $m_{\mathrm{c}}$ have small values ($<$0.1), with no clear trend of variation with increasing mean curvature. We, therefore, conclude that the substrate’s Gaussian curvature plays an important role in the mechanism responsible for the larger value of $m$ in flat sheet-spherical substrate configuration.

The adhesion of a thin sheet to a substrate with a mismatched Gaussian curvature comes with an unavoidable geometry-dependent stress in the sheet (11). For example, deforming a flat sheet of radius $W_0$ into a sphere of radius $R_0$ results in a strain ${ϵ}_{\mathrm{geom}}\sim {(W_0/R_0)}^2$ and a corresponding in-plane stress ${\sigma }_{\mathrm{geom}}\sim Y{(W_0/R_0)}^2$ (24), where $Y=Eh$ is the sheet’s stretching modulus. An in-plane stress ${\sigma }_{\mathrm{geom}}$ causes a normal force per unit area $P\sim {\sigma }_{\mathrm{geom}}{\kappa }_{\mathrm{mean}}$, where ${\kappa }_{\mathrm{mean}}$ is the mean curvature of the interface (SI Appendix). As the force of friction is proportional to the normal force, we get $m\propto P\sim {\sigma }_{\mathrm{geom}}{\kappa }_{\mathrm{mean}}$ (25). Both ${\sigma }_{\mathrm{geom}}$ and ${\kappa }_{\mathrm{mean}}$ have nonzero values in the case of a sphere and hence a nonzero value of $P$. On the other hand, ${\sigma }_{\mathrm{geom}}=0$ for a cylinder, and both ${\sigma }_{\mathrm{geom}}=0$ and ${\kappa }_{\mathrm{mean}}=0$ for a flat substrate, leading to $P=$ 0 in both these cases. This model explains the observation in Fig. 2 and suggests a mechanism through which geometrical incompatibility can affect the force of friction at a soft interface.

We further establish the validity of our model by applying it to understand the dependence of $m$ on $W_0/R_0$ for the flat sheet-spherical substrate system (Fig. 2A). The in-plane stress induced in a flat sheet when it adheres to a spherical substrate has been calculated by solving the Föppl-von Kármán equations, for example in ref. 9. It is found that for small sheets with $W_0/R_0<\zeta$, where $\zeta$ represents the threshold value of $W_0/R_0$, the stress is tensile everywhere in the sheet. However, for sheets with $W_0/R_0>\zeta$, while the stress remains tensile near the center ($0<r<L$), a compressive zone develops near the edge of the sheet ($L<r<W$). Further, the compressive stress can relax by out-of-plane buckling, e.g., through the formation of wrinkles, etc., which costs negligible bending energy for very thin sheets. We therefore assume that the outer zone $L<r<W$ becomes almost stress-free. As shown in ref. 9 the length $L$ scales with the sheet size as $\frac{L}{R}\sim {(\frac{W}{R})}^{1/3}$. We can, therefore, write the following expression for strain in the sheet:

$$\begin{array}{c}\hfill {ϵ}_{\mathrm{geom}}\sim \left\{\begin{array}{cc}{(\frac{W_0}{R_0})}^2\hfill & \frac{W_0}{R_0}\le \zeta \hfill \\ {(\frac{L}{R_0})}^2\sim {(\frac{W_0}{R_0})}^{2/3}\hfill & \frac{W_0}{R_0}>\zeta \hfill \end{array}\right.,\end{array}$$ [2]

Corresponding to the strain ${ϵ}_{\mathrm{geom}}$, there would be a stress ${\sigma }_{\mathrm{geom}}=Y{ϵ}_{\mathrm{geom}}$ and a corresponding pressure $P\sim \frac{{\sigma }_{\mathrm{geom}}}{R}$. Further, since $m\propto P$, we expect the following functional form for the variation of $m$ with the sheet size for the same value of mean curvature:

$$\begin{array}{c}\hfill m=\left\{\begin{array}{cc}a{(\frac{W_0}{R_0})}^2\hfill & \frac{W_0}{R_0}\le \zeta \hfill \\ a{\zeta }^{4/3}{(\frac{W_0}{R_0})}^{2/3}\hfill & \frac{W_0}{R_0}>\zeta \hfill \end{array}\right.,\end{array}$$ [3]

Here, $a$ is a constant that may depend on the elastic, frictional, and geometrical properties of the interface. The coefficient of the term corresponding to $\frac{W_0}{R_0}>\zeta$ has been obtained by assuming that $m$ is continuous across $\frac{W_0}{R_0}=\zeta$. According to Eq. 3$m$ has a cusp at $W_0/R_0=\zeta$, signifying that the frictional behavior undergoes a transition at the sheet radius above which the compressive zone appears near the edge of the sheet. We fit Eq. 3 to our $m(W_0/R_0)$ data with $\zeta$ and $a$ as free parameters. The function fits the data well (red line in Fig. 2A). It may be useful to note here that we have made some simplifying assumptions in the above model. For example, there may be a nontrivial and inhomogeneous residual stress present in the outer compressive zone $L<r<W$ as opposed to it being completely stress-free as assumed. However, we find that this simple model captures the essential features of the observed frictional response of the system’s behavior reasonably well and helps us illustrate the principal underlying mechanism.

We repeat the experiment with sheets of different thicknesses: h = 52 nm, 302 nm, 472 nm, and 787 nm and fit Eq. 3 to the $m(W_0/R_0)$ data for each value of $h$. Values of the fitting parameters $a$ and $\zeta$, so obtained, are plotted as functions of $h$ in Fig. 3 B and C. Since $m(W_0/R_0)$ has a cusp at $W_0/R_0=\zeta$, where $m=a{\zeta }^2$, we scale $W_0/R_0$ with $\zeta$ and $m$ with $a{\zeta }^2$ and plot the data for various thicknesses together in Fig. 3A. The data for different thicknesses collapse onto a single curve. The scaled version of Eq. 3:

$$\begin{array}{c}\hfill \frac{m}{a{\zeta }^2}=\left\{\begin{array}{cc}{(\frac{W_0/R_0}{\zeta })}^2\hfill & \frac{W_0/R_0}{\zeta }\le 1\hfill \\ {(\frac{W_0/R_0}{\zeta })}^{2/3}\hfill & \frac{W_0/R_0}{\zeta }>1\hfill \end{array}\right.,\end{array}$$ [4]

Fig. 3.

Friction phase diagram. (A) Variation of $m/(a{\zeta }^2)$ with $\frac{W_0/R_0}{\zeta }$ for flat sheets ($h$$=$ 52 nm, 302 nm, 472 nm, and 787 nm) on spherical substrates. All the data collapse onto a single curve when plotted in terms of scaled variables. The red curve is a plot of Eq. 4. (B and C) The variation of the parameters $a$ and $\zeta$, respectively, with $h$, obtained when Eq. 3 is fitted to the $m(W_0/R_0)$ data for each individual sheet thickness. The dashed curves are fits to the power-laws $y=\mathrm{A}{x}^{\alpha }$. (D) Friction phase diagram showing $m$ (color coded) as a function of sheet size $W_0/R_0$ and sheet thickness $h$. The values of $m$ are obtained from Eq. 3 using $a(h)$ and $\zeta (h)$ as obtained above by fitting.

is plotted as a red curve in Fig. 3A and fits the data well.

Both $a$ and $\zeta$ show dependence on $h$ (Fig. 3 B and C), which we model by power-law functions $y=\mathrm{A}{x}^{\alpha }$ and find by fitting that $\zeta \sim$ 0.027$h^{0.48}$ and $a\sim$ 8.3$h^{-0.41}$, shown as dashed curves in the respective graphs. We find that $\zeta$ increases with $h$. Recall that $W/R=\zeta$ marks a transition from a phase where the whole sheet is under tensile stress to a phase where a compressive zone develops in a part of the sheet. This transition is a generic feature of the geometrically incompatible confinement of a thin flat sheet to a spherical substrate and has been studied previously in the context of the onset of patterns in the form of wrinkles, crumples, folds, etc. in the sheet (4–6, 9–11). The system’s energy in the problem is composed of multiple elastic and interfacial energy contributions, and the solution depends on the hierarchy of the magnitudes of these different energy contributions. For example, in a recent paper Box et al. (10) have studied this problem for highly bendable sheets and have shown that $\zeta \sim {(\frac{\gamma }{\mathit{Eh}})}^{1/2}$. In their analysis, Box et al. (10) have considered the substrate to be extremely rigid and completely undeformable. In contrast, in our case, although we find that placing the sheet on the hydrogel substrate does not induce any measurable global deformation of the substrate (SI Appendix), we do observe wrinkle and crumple-like structures that suggest an involvement of local deformation of the substrate. Hohlfeld et al. (9) and Davidovitch et al. (11) studied this problem taking into account the deformation energy of the substrate, and showed that $\zeta \sim {(\frac{K_{\text{sub}}h}{E})}^{1/4}$, where $K_{\text{sub}}$ represents the substrate stiffness. It may, however, be noted that the presence of friction and dynamics in our case brings in additional subtleties that have not been considered before, to the best of our knowledge.

Since $m$ depends both on $h$ and $W_0/R_0$, in order to obtain a comprehensive picture we make a 2D color plot, as shown in Fig. 3D, by computing $m$ using Eq. 3 and the power-law fits for $a(h)$ and $\zeta (h)$ obtained above. We find, as also noted before, that $m$ increases monotonically with $W_0/R_0$. On the other hand, $m$ shows a nonmonotonic dependence on sheet thickness: It is small for both very small values of $h$ as well as for large values of $h$, reaching a maximum for an intermediate value, e.g., for $W_0/R_0\sim$ 1, $m$ is maximum around $h\sim$ 1.5 $\mathrm{\mu }m$. The small value of $m$ for very thin sheets is due to their low stretching modulus $Y$, which results in smaller ${\sigma }_{\mathrm{geom}}$. On the other hand, for very thick sheets, the surface energy may be insufficient to cause the elastic deformation required to conform the sheet to the incompatible geometry of the substrate. This observation highlights that although the effect of the frictional mechanism studied here may be negligible for rigid systems, it may play an important role in a large class of soft systems.

Friction is ubiquitous in any dynamical phenomena involving relative motion at an interface and plays a particularly important role in soft systems, where it shows certain unique features not seen typically in rigid systems (26). This paper presents a simple experimental technique to study friction at a thin sheet–hydrogel interface. Our study reveals a strong dependence of friction on geometry. In particular, friction is significantly modified when the thin sheet and the substrate have incompatible geometries, viz., a mismatch in Gaussian curvatures. A thin sheet under geometrically incompatible confinement has become a prototypical system to study the emergence of rich patterns of wrinkles, crumples, and folds due to the unavoidable and often nontrivial stress distribution caused by confinement. The present work demonstrates that such a stress distribution can have a significant effect on the frictional response of the system as well. The insight obtained from the present study can have important implications for our understanding of friction in many biological, nanoscale, or other soft systems. For example, it motivates us to reexamine our understanding of phenomena such as the curvature dependence of the migration of cells called “curvotaxis” (27) and the dependence of friction on roughness at soft interfaces (26).

Methods

Preparation of Thin Sheets.

Thin sheets are prepared by spin coating a dilute solution of polystyrene (average $M_{\mathrm{w}}$$=$ 192 K, Sigma-Aldrich) and a small amount ($\sim$0.002%) of fluorescent dye Nile-red (Sigma-Aldrich, 72485) in toluene (anhydrous 99.8%, Sigma-Aldrich, 244511). The solution is thoroughly mixed and filtered before spin coating on clean microscope glass slides. Sheets of different thicknesses are obtained by varying the concentration of polystyrene between 1 wt % and 9 wt %. Circular discs of radius $W_0$$=$ 0.75 mm to 5 mm are cut near the center of the slide, where the thickness is uniform, and first floated on the surface of a deionized water bath and later transferred to the swollen hydrogel substrate.

Preparation of Hydrogel Substrates.

Commercially available polyacrylamide particles (diameter $\sim$ 2 mm in the dry state) are soaked in deionized water to obtain the swollen hydrogel spheres of diameter $\sim$ 12 mm, which are used as the spherical substrates. The swollen hydrogel spheres have Young’s modulus of $114$KPa, as measured using Universal Testing Machine (UTM), and quite a smooth surface (optical micrograph in SI Appendix). Flat substrates are obtained by cutting the swollen hydrogel spheres using a sharp, flat blade. Cylindrical substrates are obtained by cutting the swollen hydrogel spheres with a sharp blade bent into a cylindrical shape. Information regarding the characterization of the geometry and roughness of the cylindrical surfaces so obtained is provided in SI Appendix. We notice that the surfaces obtained by cutting are rougher than the surfaces of the original hydrogel spheres.

Imaging Setup.

The imaging setup is shown schematically in Fig. 1A. We image the sheet and the substrate separately using two different imaging schemes. While a brightfield illumination in the transmission mode is used to image the hydrogel substrate, a fluorescence-based technique is used to image the sheet. A green laser ($\lambda$$=$ 532 nm) is used to excite fluorescence in the Nile red dye present in the polystyrene sheet, which is captured using the camera after filtering out the green light using a low pass filter. The two different imaging schemes are implemented in sequence, one after the other. Pairs of such images are recorded at equal intervals of time ($\mathrm{\Delta }t\sim$ 10 min).

Determination of the Sheet and Substrate Dimensions.

We determine the radii of the sheet $W(t)$ and the hydrogel substrate $R(t)$ from the fluorescence and bright field images, respectively, by using image analysis codes written in Python. The size of the substrate at any instant of time is determined from the bright-field image, for example, as shown in Fig. 2B. The red channel of the image is thresholded, and a connected component analysis is run to identify the substrate in the image. An ellipse-fitting algorithm is then used to determine the dimension of the substrate in the case of spherical and flat surfaces. In order to maintain uniformity in comparison, we consistently use the major axis length as a measure of $R$. [It may be noted here that as the shape of the substrate remains unchanged even as its size decreases (SI Appendix), using either the major or the minor axis as a measure of $R$ gives the same value of $\tilde{R}=R/R_0$ at any instant of time as long as the same axis is also used to determine $R_0$.] In the case of a cylindrical substrate, the dimension of the best-fit bounding rectangle is used instead.

The size of the sheet is determined from the red channel of the fluorescent image in a two-step process. In the first step, a rough estimate of the sheet center is made by thresholding the image, running a connected component analysis, and finding the best-fit ellipse. In the second stage, we obtain a refined measure of the sheet center and radius by analyzing its radial profile. Using the center obtained in the first step as the origin we calculate the radial intensity profile $⟨I⟩(r)=\frac{1}{2\pi }{\int }_0^{2\pi }I(r,\theta )d\theta$, where $r$ is the distance from the center and $\theta$ is the polar angle. We then convolve $⟨I⟩(r)$ with the derivative of a Gaussian to find a smooth derivative $I^{\prime }(r)$ of the function $⟨I⟩(r)$. In the convolution, we note the value ($I_{\max }^{\prime }$) and position of the peak ($R$). The process is now repeated by taking different center positions in a small neighborhood of the center position estimated in the first step, and we look at the variation in $I_{\max }^{\prime }$ with the center position. We use the center and the radius corresponding to the peak value of $I_{\max }^{\prime }$ to determine the correct center and radius of the sheet with subpixel resolution. While the above method is used for spherical and flat substrates, for cylindrical substrates where the response along the curved and axial directions may be different, we measure lengths along the two directions by fitting an ellipse to the sheet image using the connected component analysis.

Supplementary Material

Appendix 01 (PDF)

Movie S1.

Flat circular polystyrene sheet (bright red) of thickness h = 472 nm and initial radius W₀ = 2.65 mm placed on a spherical hydrogel substrate (dark color) of radius R₀ = 5.77 mm which shrinks at a rate d$\tilde{R}$/dt = -0.012 h⁻¹ due to evaporation. The sheet is compressed as the hydrogel shrinks due to friction at the interface, resulting in a slope of m = 0.15 in the $\tilde{W}\left(\tilde{R}\right)$ curve. Playback speed: 2946 × real-time.

Movie S2.

Flat circular polystyrene sheet (bright red) of thickness h = 376 nm and initial radius W₀ = 1.85 mm placed on a flat hydrogel substrate (dark color) of radius R₀ = 6.22 mm which shrinks at a rate d$\tilde{R}$/dt = -0.017 h⁻¹. The flat sheet slips on the flat substrate, and the $\tilde{W}\left(\tilde{R}\right)$ curve has a slope m = 0.05. Playback speed: 3154 × real-time.

Movie S3.

Flat circular polystyrene sheet (bright red) of thickness h = 376 nm and initial radius W₀ = 3.66 mm placed on a cylindrical hydrogel substrate (dark color) of radius R₀ = 5.33 mm which shrinks at a rate of d$\tilde{R}$/dt = -0.022 h⁻¹. We obtain slopes m_c = 0.05 and m_a = 0.027 along the curved and axial directions of the cylinder, respectively. Playback speed: 2893 × real-time.

Data, Materials, and Software Availability

All study data are included in the article and/or supporting information.