Fabrication of slender elastic shells by the coating of curved surfaces

Abstract

Various manufacturing techniques exist to produce double-curvature shells, including injection, rotational and blow molding, as well as dip coating. However, these industrial processes are typically geared for mass production and are not directly applicable to laboratory research settings, where adaptable, inexpensive and predictable prototyping tools are desirable. Here, we study the rapid fabrication of hemispherical elastic shells by coating a curved surface with a polymer solution that yields a nearly uniform shell, upon polymerization of the resulting thin film. We experimentally characterize how the curing of the polymer affects its drainage dynamics and eventually selects the shell thickness. The coating process is then rationalized through a theoretical analysis that predicts the final thickness, in quantitative agreement with experiments and numerical simulations of the lubrication flow field. This robust fabrication framework should be invaluable for future studies on the mechanics of thin elastic shells and their intrinsic geometric nonlinearities.

ARTICLE

Received 1 Oct 2015 | Accepted 26 Feb 2016 | Published 4 Apr 2016

Various manufacturing techniques exist to produce double-curvature shells, including injection, rotational and blow molding, as well as dip coating. However, these industrial processes are typically geared for mass production and are not directly applicable to laboratory research settings, where adaptable, inexpensive and predictable prototyping tools are desirable. Here, we study the rapid fabrication of hemispherical elastic shells by coating a curved surface with a polymer solution that yields a nearly uniform shell, upon polymerization of the resulting thin lm. We experimentally characterize how the curing of the polymer affects its drainage dynamics and eventually selects the shell thickness. The coating process is then rationalized through a theoretical analysis that predicts the nal thickness, in quantitative agreement with experiments and numerical simulations of the lubrication ow eld. This robust fabrication framework should be invaluable for future studies on the mechanics of thin elastic shells and their intrinsic geometric nonlinearities.

DOI: 10.1038/ncomms11155 OPEN

Fabrication of slender elastic shells by the coating of curved surfaces

A. Lee1, P.-T. Brun2, J. Marthelot3, G. Balestra4, F. Gallaire4 & P.M. Reis1,3

1 Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA. 2 Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA. 3 Department of Civil and Environmental Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA. 4 Laboratory of Fluid Mechanics and Instabilities, EPFL, CH1015 Lausanne, Switzerland. Correspondence and requests for materials should be addressed to P.M.R. (email: mailto:[email protected]

Web End [email protected] ).

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Hollow chocolate eggs, rabbits and bonbons have been fabricated since the 1600s by pouring molten chocolate into a mold and draining the excess. Solidication upon

cooling ceases the ow and results in a solid shell of nearly constant thickness1. Beyond chocolatiers, the polymer industry abounds with needs to fabricate thin shell structures, and a plethora of manufacturing processes have been developed for this purpose, including: injection2, rotational1 and blow molding3, as well as dip coating4. Common to all of the above techniques are limitations in the thickness of the shells (e.g., B0.5 mm for injection molding) and its uniformity (typically B20% for rotational molding5), as well as a striking lack of predictive theoretical models due to the multi-physics complexity of the processes. Rotational molding, for example, involves coating the inner surface of a hollow mold with a polymer melt, which is then rotated biaxially while applying a decreasing heating prole until a solid shell is formed1. As another example, injection molding, is geared for mass-production manufacturing and requires costly precision-machined molds that are inexible to variations in the geometry of the part2. In these processes, the optimization of the control parameters is largely tuned empirically, with compromises on versatility, predictability and reproducibility5. As such, these techniques are not directly applicable to laboratory settings, where adaptable, inexpensive and predictable rapid-prototyping tools are more desirable. This is particularly the case for the fabrication of thin, smooth and exible three dimensional structures.

For at and cylindrical surfaces, a variety of thin lm coating techniques are well established6. A signicant advantage for these geometries is that, when compared to their double-curved counterparts, they are more amenable to theoretical modeling to predict how the nal lm thickness depends on the control parameters7,8. In these cases, the ow driven by viscous stresses and held by capillary forces is frozen as the media cools, cures or dries, yielding a defect-free and uniform nish. As a result, these robust coating techniques have matured to be ubiquitous in industry. To generate (ultra-) thin sheets, spin-coating is now widespread (e.g., in microuidics) to attain constant and tunable lm thicknesses9,10. Similarly, spin-casting exploits centrifugal forces on a rotating cylindrical surface to evenly distribute a polymer solution and fabricate nearly constant thickness shells in a highly controllable manner11. This technique was instrumental in identifying the role of imperfections on the critical buckling conditions of cylindrical shells in the 1960s (refs 12,13). For double-curved surfaces, there is a need for simple and versatile fabrication methods that are analogous to the coating of bers, plates and cylinders and able to yield uniform, controllable and predictable results.

Here, we introduce a simple and robust mechanism to fabricate hemispherical thin elastic shells by the coating, drainage and subsequent curing of polymer solutions on curved molds. Our process is analogous to spin-coating (itself not applicable on curved surfaces), albeit with a gravity-driven ow in lieu of centrifugal forces. Through a systematic series of experiments using elastomers, we show that drainage can lead to coatings that are frozen in time as the polymer cures, thereby leading to a nearly uniform thin elastic shell. A theoretical analysis of the underlying lubrication ow during drainage, which includes the evolution of the rheological properties of the polymer as it cures, is able to accurately predict the nal thickness of the shell as a function of the material properties of the polymer and the geometry of the substrate. Importantly, the nal shell thickness is found to be independent of the initial conditions such as the height of pouring and the volume of poured uid, as well as the initial thickness prole. Moreover, we nd that the shell thickness can be tuned over one order of magnitude by changing the

waiting time between the preparation of the polymer solution and the moment of pouring onto the mold. Our analysis demonstrates that the robustness and exibility of this mechanism are inherent consequences of the loss of memory in the ow eld. Our approach provides a fast, robust and predictable mechanism to fabricate thin shells with exibility in their material and geometric properties by tuning the control parameters.

ResultsElastic shells of uniform thickness from viscous coating. In Fig. 1a, we present a series of photographs that illustrate our coating process. A silicone-based liquid polymer solution is poured onto a rigid sphere (mold), drains under the effect of gravity and eventually covers the surface. We used both vinylpolysiloxane (VPS) and polydimethylsiloxane (PDMS), at different mixing and curing conditions (see Methods for details), to achieve a variety of rheological properties. With time, cross-linking of the polymer lm that emerges from the drainage process yields a thin elastic shell that can be readily peeled from the mold. The nal thickness of these elastic shells hf is found to be uniform (to within 6.6% (VPS) and 8.7% (PDMS) over the hemisphere).

The above procedure was repeated with molds in a range of radii (1rR[mm]r375, see Fig. 1b), and we found that hfBR1/2, as shown in Fig. 1c. This result is robust and independent of either the details of the polymer or the curing temperature. The square-root dependence of hf on R can be rationalized by balancing the characteristic curing time, tc, of the polymer

a

b

E

A B

C D

c

103

102

101

Final thickness, h f(m)

B C

D E

VPS-32

VPS-8

VPS-22

Equation (1)

PDMS at 20 C PDMS at 35 C PDMS at 40 C

A

101 102 103

( 0R/ g c)1/2 (m)

Figure 1 | Coating process and resulting thickness of elastic shells. (a) Liquid VPS-22 (see Methods) is poured onto a sphere (R 38 mm),

then drains under gravity and eventually cures to produce an elastic shell (see Supplementary Movie 1). The time interval between each frame is 2 s. (b) Similar procedure to that of (a) for spheres in a range of radii, 1rR[mm]r375. (c) Thickness of the elastic shells, hf, as a function of

m0R= rgtc

p

, for various polymer solutions (VPS and PDMS) and temperatures (for PDMS). See Methods for details. The solid line corresponds to equation (1). Inset: An elastic shell is cut along a meridian for the thickness measurements. Scale bars, 10 mm.

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NATURE COMMUNICATIONS | DOI: 10.1038/ncomms11155 ARTICLE

solution and the characteristic drainage time, m0R= rgh2f

,

that is

obtained when balancing the viscous stresses and gravity in the lubrication layer, such that

hf

m0R

rgtc

a

z

g

b

e[p10]

s

R r

R r

h( , t)

u( , t)

; 1

where m0 is a characteristic viscosity of the polymer (e.g., its initial value), r its density, and g is the acceleration of gravity. The fact that all the data in Fig. 1c collapses onto a master curve (irrespective of the polymer and curing temperature, over a wide range of R) supports this scaling analysis. Below, we shall develop a theoretical description that more formally recovers this scaling, both analytically and numerically.

The dynamics of coating. We proceed by experimentally characterizing the coating dynamics that, upon curing of the polymer, results in a thin elastic shell (Fig. 1a). As a representative case, we focus on VPS Elite 32 (hereafter referred to as VPS-32, see Methods) poured onto a sphere with R 38 mm. A broader

exploration with other silicone-based polymers is provided in the Supplementary Table 1, as well as Supplementary Figs 1 and 5, nonetheless, yielding similar results.

A schematic diagram of our system is presented in Fig. 2a, for a hemispherical mold, aligned such that gravity is parallel to the axis that connects its center to the pole; g g ez. Both the local

thickness, h(f, t), and the free surface velocity, u(f, t), of the draining lm are assumed axisymmetric and vary in both time, t, and space (i.e., zenith angle, f). At the pole (f 0), the uid

velocity vanishes due to symmetry. Elsewhere, the velocity is predominantly in the longitudinal direction, ef. This is supported by the representative velocity eld shown in Fig. 2b, obtained through PIV (see Methods), at t 60 s in a 1 1 cm2 region of

the lm located at f 60. Moreover, the instantaneous local

velocity is found to increase with f (Fig. 2c).

In Fig. 2d, we plot the time-series of the free surface velocity at the specic location u(f 60, t); the ow progressively slows

down and eventually comes to a halt in nite time. This leaves a coating of the nal thickness, hf, on the mold. The velocity prole and its arrest are found to correlate directly to the change in the viscosity, m, as the polymer cures (Fig. 2d), which was determined through the rheometry at the appropriate shear rate (see Methods). Note that the initial drainage and subsequent curing regimes are separated by the characteristic curing time, tc, which is signicantly larger than the initial drainage time, tdm0R= rgh2i

,

y

x

c

250

Equation (4)

Experiment

Velocity, u(m s1 )

200

150

100

Zenith angle, ()

d

102

103

Viscosity, (Pa s)

50 20 30 40 50 60 70

102

Viscosity Velocity

Equation (5) Equation (4)

Numerics

Velocity, u (m s1 )

101

101

t ~ d d << t << c

101 102 103

c

Time, t (s)

Figure 2 | Spatial and temporal variation of the ow velocity.(a) Schematic diagram of the coating problem; h(f, t) is the thickness of the viscous lm and u(f, t) is the ow velocity during drainage. (bd) All data is for VPS-32 at 20 C. (b) Instantaneous velocity eld at t 60 s in a

1 1 cm2 region of the lm located at f 60 of a sphere (R 38 mm),

obtained through PIV. (c) Dependence of the instantaneous local velocity (at t 60 s) on f. (d) Time variation of the velocity, u(f 60, t)

orange circles, and the viscosity, m(t) blue triangles, of the polymer. The characteristic curing time, tc, separates the drainage and curing regimes for both u(f, t) and m(t). The dash-dot line is the best t for the viscosity: equation (5) with m0 7.10.2 Pa s, a 5.30.7,

b (2.060.09) 10 3, and tc574 11 s. The solid and dashed lines

are the predictions from our model for the velocity eld using equation (4) and direct numerical simulations, respectively.

where hi is the initial average coating thickness.

For example, in the representative case above for VPS-32, we nd td5:9 s 0:01 tc (using hi 2 mm, R 38 mm, r 1,160

kg m 3, m0 7.1 Pa s and tc574 s). A direct consequence of

this separation of timescales is that there is loss of memory in the process, such that hf should be independent of hi. This prediction will be thoroughly examined in the Discussion, below. Returning to the time evolution of u and m (Fig. 2d), at early times totd

there are some disturbances due to initial conditions and we do not attempt to describe this regime. During intermediates times td t tc

, m is approximately constant, and the velocity is

set by viscous drainage with uB1/t (ref. 14). For t4tc, as the curing of the polymer accelerates, m increases sharply with time, and consequently, the ow velocity slows down dramatically.

The separation of the drainage and curing timescales can be leveraged to further tune the nal thickness of the shell. Since hf is

dictated by the interplay between the drainage timescale and polymerization timescale, tc, the nal thickness can be increased by accelerating the curing process. One strategy to achieve this would be to alter tc by modifying the kinetics of cross-linking (e.g., through additives or temperature), which would also modify

the viscosity of the thin lm or the elastic modulus of the nal shell. An alternative is to shift the origin of the process by waiting for a time, tw, between the preparation of the polymer and the instant when the mixture is poured onto the mold. This waiting procedure offers an additional lever in tuning the properties of the fabricated shells.

Having presented our robust and versatile mechanism to fabricate thin elastic shells by the coating and subsequent curing of a polymer lm, we proceed by rationalizing this process through a theoretical framework that is able to predict hf.

Nonlinear drainage ow solution. It is well known that the thickness at the pole of a thin viscous lm draining on a spherical

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surface is given by h0 t hi 1 4t=3td

1=2 (refs 14,15).

We now seek to generalize this solution to account for the temporal and spatial variation (in f) of the lm. We shall rst consider a Newtonian uid and, once the nonlinear drainage ow solution is obtained, then include curing effects (i.e., the time dependence of m(t) shown in Fig. 2d).

Starting with the lubrication equations14 that describe our axisymmetric ow on a hemispherical substrate and performing a nonlinear expansion of the form h f; t

h0 t

f2h2 t

(see

Supplementary Note 4) yields

h f; t

h0 t

1

f2

10 1 c 1

!

" #

5=2

4 3

t td

Rt 1

f2

10

2

; 2

where c is a numerical factor that depends on the initial condition (e.g., c 1 if the initial thickness prole is uniform). In the

limit of t td, equation (2) simplies to

h f; t

sinf: 4

This prediction for the variation of u on both f (at xed t) and t (at xed f) is in agreement with the experimental velocity proles shown in Fig. 2c,d for td t tc (i.e., in the regime

after the initial drainage, when the polymer viscosity is approximately constant, and prior to curing). In particular, u is found to be almost linear in f as the cubic term of the Taylor expansion is f3/30 in lieu of the conventional f3/6 of the sine. Strikingly, the velocity eld in this regime is independent of both gravity and viscosity and is solely set by the geometry of the problem, so that no material parameters enter the prediction.

Including the effects of curing into the ow solution. The curing of the polymer has not yet been taken into account in our model, which, as is, yields a vanishing coating thickness since equation (3) states that hB1/t1/2. To do so, the above framework is modied by considering a time-varying viscosity18 using a piecewise function of the form

m t

m0 exp bt

; if t tc; m1ta; if t4tc;

s

3m0R

4rgt

: 3

The memory loss of the ow mentioned earlier arising from the separation of the drainage and curing timescales is well captured by this description given that hi is absent from equation (3).

Moreover, there is a weak dependence on f (8.7% s.d.); a general result that has also been observed in the thinning of an air bubble formed in a uid bath16, as well as in the thin air layer that supports a drop bouncing on a uid interface17.

As an indirect validation of equation (3), we substitute it into the free surface velocity equation describing the parabolic ow prole on a sphere, u rgh2sinf/(2m0)14, to obtain

u f; t

3 8

1

f2

10

a

103

Equation (6)

Experiment

Numerics

1

Final thickness, h f(m)

Final thickness, h f(m)

10 100 101 102 103

2

102

400

h f(m)

300

Outside Underside

200

100

0 45 0 45

5

Zenith angle, ()

1

with m1 m0 exp btc

b

90

t ac chosen to ensure continuity at tc and

where b and a are tting parameters. Equation (5) is tted to the experimental data and found to accurately describe the viscosity evolution (Fig. 2d). Combining this description for the viscosity with the lubrication equations yields a complete model for our system (the full details are provided in Supplementary Note 5), including an expression for the nal thickness,

hf

3m0R

4rg

Radius, R (mm)

60 0 20 40 60 80 100

95

Experiment

Equation (6) Order 4 expansion

85

s

80

75

1 K

6

that is consistent with equation (3) but with K k e btc

=b

1

f2

10

70

65

tce btc= a 1

Figure 3 | Inuence of the geometry on the nal shell thickness.(a) Comparison between theory, numerics and experiments for the dependence of hf on R, for the representative case of VPS-32. The results are consistent with the power law hfBR1/2 and in agreement with equations(1) and (6). Inset: Final thickness of shells obtained by pouring VPS-32 on the outside or the underside of a hemisphere with R 25 mm. The error

bars of the data correspond to the standard deviation of three thickness measurements performed at three different locations of the shell near the apex (f 0). (b) Final thickness of a shell fabricated by pouring PDMS on

the outside of a hemisphere with R 38 mm compared to equation (6);

solid line. The dashed line is the prediction obtained by rening the expansion to the next order, O 4

, which adds

(instead of K t),

where k 1 when there is no delay between the preparation

and the coating with the polymer solution.In Fig. 3a, we compare experimental results (circles) for hf of

the shells fabricated with VPS-32 to the prediction (solid line) from equation (6) and nd good agreement between the two. It is important to note that our model has no adjustable parameters; all numerical coefcients (a, b and tc) are independently determined once and for all from the viscosity prole and then used in the theory. Note that the proles obtained when coating either the outside or the underside of complementary spherical molds are nearly identical (Fig. 3a, inset).

In Fig. 3b, we test the shell thickness prole, hf(f), and nd

that the experimental results (circles) are in excellent agreement with equation (6).

Our theoretical framework is now further validated through numerical simulations (see Methods, as well as Supplementary Figs 6 and 7, for details). The fully nonlinear governing equations (see equation (7) in Methods) were integrated directly, with the appropriate initial conditions and the desired viscosity prole,

Zenith angle, ()

4800 f4 to the terms in the parentheses of equation (6); see Supplementary Note 4 for details. The error bars of the data for hf (y-axis) correspond to the standard deviation on multiple measurements. The error bars of the data for f (x-axis)

correspond to the size of the angular range used to bin the data.

41

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b

a c

Normalized film thickness, h( ,t)/h i

1.4

300

Pouring height, H (cm)

4 5 6 7 8 9 10 11

Equation (2) Numerics

Final thickness, h f(m)

Normalized film thickness, h( ,t)/h i

1.5

1.2

250

1

0

5

10

20

30

200

1

0.8

t /

150

0.6

2

0.4

100

0.5

h h h

Equation (6)

0.2

50

h h

0 0 10 20 30 40 50 60 70 80 90

0 0 1 2 3 4 5 6 7

0 0 10 20 30 40 50 60 70 80 90

Zenith angle, ()

Pouring volume, V (ml)

Zenith angle, ()

Figure 4 | Insensitivity of the coating mechanism to the initial conditions. (a) Time-series of the normalized lm thickness, h(f,t)/hi, with a constant viscosity, at t=td0; 2; 5; 10; 20, and 30. Dashed lines correspond to the numerically computed evolution of an initially sinusoidal thickness

prole of the form h(f,t 0)/hi 1 0.375 cos (10f) (see Methods and Supplementary Fig. 7). Solid lines are the theoretical prediction, equation (2),

with uniform initial thickness prole, h(f,t 0)/hi 1. The various parameters are: average initial lm thickness hi 0.2 mm, sphere radius R 20 mm, and

material properties for VPS-32. (b) Shell thickness, hf, obtained for different pouring conditions: pouring height, H, and volume, V, poured onto a spherical mold with R 20 mm using VPS-32. (c) Four different initial conditions used in the numerics converge to the same nal thickness, hf, and agree well with

equation (6) (black dashed line). The simulation parameters are: hi 2 mm, R 38 mm, hi,1/hi 1, hi,2/hi 1 0.475 cos(2f), hi,3/hi 1 0.475 cos(2f),

hi,4/hi 1 0.475 cos(10f), all with the material properties of VPS-32.

Normalized waiting time, w/ c

either constant or time-varying according to equation (5). The results from these numerical simulations are in agreement with both the experimental data and the theoretical predictions for the surface velocity over time and nal shell thickness (dashed lines in Fig. 2d and Fig. 3a, respectively). In particular, we have computed the time evolution for a coating lm that has an initial sinusoidal thickness prole (dashed line in Fig. 4a), which rapidly converges to the analytically derived equation (2), plotted in Fig. 4a as solid lines for different times.

DiscussionOur above results establish the basis for the rapid and robust coating process to fabricate spherical elastic shells of nearly uniform thickness, and with radii spanning over two orders of magnitudes (1rR[mm]r375). As the radius of the sphere is decreased below Ro10 mm, the agreement between our model and the experiments deteriorates due to the inuence of the meniscus that connects the ow on the hemisphere to the puddle that forms as the uid drains. This effect is not accounted for in our model, but we expect it to be negligible when R c, where

c

g= rg

p

is the capillary length with surface tension, g, that prescribes the relative magnitude of capillary and gravitational forces. Since c 1:3 mm and c 1:4 mm for VPS and PDMS,

respectively, the deviations of the theory from the data for small R are consistent with the onset of these surface tension effects (see Figs 1c and 3a).

Our model, e.g., equation (6), uses the physical parameters for the rheology of the polymer. PDMS was found to behave as a Newtonian uid for small shear rates (but the viscosity varies with time; see Fig. 5b) whereas VPS exhibited shear thinning (see Supplementary Figs 3 and 4). An estimate of the relevant shear rate is therefore needed. We used the value _

g0:1 s 1, assuming

a uniform shear rate across the sample. In reality, _

g varies from zero at the apex to its peak value at the equator (and also depends on R). However, our choice for _

g is representative of the applied shear rates over the hemisphere and leads to good agreement between theory and experiments for most values of R and f. We have analyzed the sensitivity of the predictions for hf with respect to _

g and found that it is small (see Supplementary Note 1).

The variation of the thickness from the pole to the equator of the hemispherical shells was found to be, at most, 6.6% (VPS) and

a

12

VPS-32, experiment

PDMS, experiment

VPS-32, equation (6) with k=exp(- ) PDMS, equation (6) with k=exp(- )

Normalized final thickness, h f/h f,0

10

8

6

4

2

0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

b

100

100

90

80

80

=0

=20 min

=10 min

=30 min

70

Viscosity, (Pa s)

Viscosity, (Pa s)

60

60

40

50

20

40

0 10 20 30 40 50 60

30

Time, t (min)

20

10

0 10 20 30 40 50 60 70

Time, t + w (min)

Figure 5 | Varying the shell thickness by delaying pouring. (a) The shell thickness (normalized by its value hf,0 when tw 0) can be tuned by

delaying the pouring time by tw from the moment of preparation of the polymer solution. Results for both VPS-32 and PDMS are shown.

(b) Viscosity of PDMS versus the sum of the measuring time, t, and the effective waiting time, dtw (d 2.020.02 from tting all the curves to the

master curve obtained for tw 0). Inset: Viscosity as measured after

holding the mixture for a time tw in a quiescent state prior to testing

in the rheometer.

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8.7% (PDMS) from the experiments, 8.7% in the theoretical model, and 8.4% in the numerical analysis (using s.d.). For PDMS, the thickness prole follows equation (6) without any adjustable parameters. This agreement validates our model for the case of time varying viscosities (Fig. 3b). For VPS, shear thinning effects lead to an increase of the thickness at the apex (see Fig. 3a); where the viscosity is largest. These effects are not captured by our model, yet they do not prevent the shell from being uniform within 6.6% variations. We argue that these same effects are the source of the difference between the numerics and the measured free surface velocity for large times (see Fig. 2c).

When the polymer is poured on the underside of a mold, curvature can suppress the Rayleigh-Taylor instability and thereby prevent the formation of dripping droplets15. Therefore, the uniformity bounds of the shell that were just stated are ensured, as long as the modied Bond number B rgRhi/g

(which characterizes the relative importance of gravity and surface tension, g) remains smaller than the critical value, Bco8

(ref. 15). On the other hand, when the outside of a mold is used, ngering instabilities can occur at the advancing front of the ow, but this can be precluded by pouring a sufciently large volume of liquid over the surface14. A critical volume can be derived14 and we estimated it be of the order of 1 ml for an hemisphere of radius R 20 mm, in agreement with what was observed experimentally

for the sensitivity analysis in Fig. 4b. Under these conditions, pouring on the underside or the outside of complementary molds yields identical shells of the same uniform thickness, hf. Note that during this process an E90% of the volume drains out of the hemisphere. Even if this technique is an excellent rapid-prototyping method, it may not be suitable for large scale industrial applications. Similar limitations are found for spin-coating.

The physical principles that underlay the dynamics of the coating process are rationalized by our analytical model above, to which the separation between the initial drainage and curing timescales is key. Drainage occurs signicantly faster than the polymer curing, such that the memory of the ow vanishes before it is arrested by cross-linking to yield the nal elastic shell. Consequently, geometry prevails, and the curvature of the mold together with the rheology of the polymer set both the dynamics of the ow and the nal thickness of the shell (hfBR1/2). The robustness of this mechanism and its insensitivity to the initial conditions are now corroborated by both experiments and simulations. We measured the thickness of shells obtained for different values of the height from which the polymer is poured onto the mold (4rH[cm]r10), as well as the volume poured(0.9rV[ml]r6.3), and nd that hf is constant to within5.6% across these various conditions (Fig. 4b). Furthermore, simulations that were initiated with four signicantly different initial uid distributionsuniform, sinusoidal, as well as tapered proles towards the pole and the equatorall converge to the same nal shell thickness, which agrees well with the prediction from equation (6), as shown in Fig. 4c.

Since the nal shell thickness is directly connected to the curing time, hf can be continuously tuned by waiting a time tw between the preparation of the polymer mixture and the instant when it is poured onto the mold. In Fig. 5a, hf is plotted versus tw, for the representative experiments with both VPS-32 and PDMS. We nd that hf can be increased by as much as 60% for VPS-32 and elevenfold for PDMS. Substituting e btw for k in equation (6) allows for a direct comparison to the experimental result, with favorable agreement in the case of VPS-32 (solid curve in Fig. 5a). For the PDMS, however, an additional adjustment to our framework is required since we found that its rheology differs if the curing occurred in a quiescent state (e.g., when waiting in bulk for tw before pouring) versus when sheared (e.g., during coating). Rheometry measurements were performed where the

values of tw were systematically varied (inset of Fig. 5b). If the time axis for each of these tests is shifted by dtw (the constant factor d 2.020.02 was determined by tting), all of the data

collapses onto the master curve obtained for tw 0. We have

therefore concluded empirically that PDMS cures d times faster when quiescent compared to under shear, but we have not been able to nd this specic result in the literature. We speculate that the shifting factor required for collapse will likely depend on the shear rate and the specics of the polymer. With this additional information at hand, substituting e bdtw for k in equation (6)

accounts for the effective waiting time, and yields a prediction for hf (dashed line in Fig. 5a) that is in agreement with the experimental data for PDMS. Our model is therefore able to accurately capture the elevenfold continuous variation of the shell thickness obtained when pouring partially cured polymer solutions. As explained in the Supplementary Note 2, we did not have to consider d for VPS-32 because the entire bulk of this more viscous polymer solution is experiencing sustained shear while it was sequentially poured onto a series of identical molds. It is important to note that our theoretical description is only applicable if twotc.

In summary, we show that coating hemispherical molds with a polymer solution yields thin uniform shells whose thickness can be accurately predicted. Moreover, the nal shell thickness can be tuned by modifying the time between polymer preparation and the moment of pouring. The resulting shells are a realization of the drainage dynamics, driven by gravity, slowed down by viscous stresses and eventually arrested by the curing of the polymer. The robustness and exibility of this mechanism are inherent consequences of the loss of memory in the ow eld. The generality of this framework should open the door for future studies to fabricate slender solid structures in a variety of other geometries. A particularly interesting case outside the scope of the current study is the coating of ellipsoidal molds19, with two distinct principal curvatures, where the difference between the pouring direction and the orientation of the surface could also play a role. Furthermore, our fabrication technique could be important in the ongoing revival of the mechanics of thin elastic shells, in particularly since it enables fully elastic structures that can reversibly explore strong geometric nonlinearities in their post-buckling regime2029.

Methods

Experiments. Curing of the PDMS (Sylgard 184, Dow Corning) was performed in a convection oven at 20, 35 and 40 C. The base and curing agent were mixed in a 10:1 weight ratio using a centrifugal mixer for 30 s at 2,000 r.p.m. (clockwise), and then for 30 s at 2,200 r.p.m. (counterclockwise). We sped up the curing process using a cure accelerator (36559 Cure Accelerator, Dow Corning) that was mixed to the PDMS elastomer in the weight proportion 5:1 (PDMS-base:Cure-accelerator). VPS (Elite Double 8, 22 and 32, Zhermack, referenced throughout the text as VPS-8, VPS-22 and VPS-32, respectively) was mixed at room temperature (20 C) with a base/cure ratio 1:1 in weight for 10 s at 2,000 r.p.m. (clockwise), and then 10 s at 2,200 r.p.m. (counterclockwise).

The various polymer solutions (VPS and PDMS) were characterized with a rheometer (AR-G2, TA Instruments) as a function of time and at a constant temperature. The shear rate was xed at _

g u=h 0:1 s 1, consistently with the

characteristic drainage velocity and lm thickness (see Supplementary Note 1 and Supplementary Fig. 2). The data for the measured viscosity was then tted with the piecewise model m t tc

m0exp bt

and m t4tc

m1ta (see Fig. 2d and

Supplementary Fig. 1).

The velocity eld of the draining polymer was measured using an open-source package for particle imaging velocimetry (PIVlab30). A powder spray (Sparkler, Body Shop) was sputtered onto the surface of the ow and imaged using a digital microscope camera (Discovery VMS-004, Veho).

Upon curing, the nal thickness, hf, of the hemispherical elastic shells was measured with an optical microscope after cutting the shell along a meridian using a scalpel (insets of Figs 1c and 3a).

Model. For the analytical description of the lubrication ow, we consider a hemisphere of radius R, initially coated with a uid of initial average thickness

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NATURE COMMUNICATIONS | DOI: 10.1038/ncomms11155 ARTICLE

hi (h(f, 0) may vary spatially). Taking advantage of the azimuthal symmetry, our model is derived in a zenith coordinate system (see schematic in Fig. 2a). By assuming hi R and considering mass conservation, the ow velocity, u (u, v),

can be regarded to be essentially one-dimensional and predominantly tangential to the surface of the sphere in the f direction (the velocity normal to the interface is v uh=R u). Under low Reynolds number conditions, the lubrication

equations31 for this ow yield the following nonlinear partial differential equation (see derivation in Supplementary Note 3):

ht

13mRsinf h3sinf gR hfff 2hf hffcotf

hfcsc2f

7

where the subscripts denote differentiation with respect to time and zenith angle,i.e., q/qt and q/qf, respectively. Further assuming that the thickness of the uid lm varies slowly along f and that the effects of surface tension are negligible (the latter is valid except close to the moving front), equation (7) can be simplied to14

ht

rg3mRsinf h3sin2f

rg h cosfR sinf io

f;

0: 8

Under the above conditions, the ow is primarily governed by viscous forces and the component of gravity along the ow such that the drainage time, td,is the relevant time scale of the problem. At the pole (f 0), the thickness

varies according to the well established drainage law, h0 t hi 1 43 t=td

1=2

f

(refs 14,15), which we generalize further in our problem, in the context of time-varying viscosities.

Numerical simulations. A numerical procedure was developed to solve equation (7). The zenith angle is discretized uniformly, and we exploit the periodic domain and employ the Fourier spectral method32 to compute spatial derivatives with a high degree of accuracy. The effect of numerical diffusion is minimized by performing the time integration with the second-order Crank-Nicolson MATLAB routine ode23t.m. The computational time to derive a solution for a set of geometric and physical parameters is of the order of a few minutes.

To verify the numerics, we compare the numerical solution for an initially uniform lm with the analytical solution obtained in the limit Ehi=R ! 0 and

f 0, namely

0;~t

1 4~t=3

1=2, when using hi and td for

nondimensionalization. Supplementary Fig. 6 shows good agreement between the analytical and the numerical solutions, using N 256 discretization points.

To complement the comparison between the numerics and the 2nd-order asymptotic solution shown in Fig. 4a, we have obtained results with both methods for sinusoidal initial thickness proles of the form f; 0

1 Acosf with

A 0:5; 0; 0:5

f g. These results are plotted in Supplementary Fig. 7 and conrm

that our asymptotic solution is indeed able to predict the correct dynamics for moderate non-uniform lm distributions.

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Acknowledgements

This work was supported by the National Science Foundation, CAREER CMMI-1351449 (A.L. and P.M.R.), and by the European Research Council, ERC-2011-StG, PE8 SIMCOMICS (G.B. and F.G.)

Author contributions

P.M.R. and F.G. conceived project. A.L., P.-T.B. and J.M. performed experiments and analyzed data. G.B. and F.G. performed simulations and analyzed data. F.G., G.B. andP.-T.B. developed theoretical model. F.G. and P.M.R. supervised the research. A.L.,P.-T.B., J.M., G.B., F.G. and P.M.R. wrote paper.

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How to cite this article: Lee, A. et al. Fabrication of slender elastic shells by the coating of curved surfaces. Nat. Commun. 7:11155 doi: 10.1038/ncomms11155 (2016).

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