nl3c01058_si

Tunable Mechanical Response of Self-Assembled

Nanoparticle Superlattices

Somayajulu Dhulipala,

†

Daryl W. Yee,

‡

Ziran Zhou,

Rachel Sun,

†

Jos ́e

Andrade,

Robert J. Macfarlane,

‡

and Carlos M. Portela

∗

,

†

Department of Mechanical Engineering, Massachusetts Institute of Technology,

Cambridge, Massachusetts, USA - 02139

‡

Department of Material Science and Engineering, Massachusetts Institute of Technology,

Cambridge, Massachusetts, USA - 02139

Department of Mechanical and Civil Engineering, California Institute of Technology,

Pasadena, California, USA - 91125

E-mail: cportela@mit.edu

Phone: +1 617 715 2680

Materials

Thymine (Thy), 11-bromo-1-undecanol, 2-bromoisobutyryl bromide, potassium carbonate,

trimethylamine, styrene, N,N,N’,N”,N’”-Pentamethyldiethylenetriamine (PMDETA), trisodium

citrate dihydrate, citric acid monohydrate, and ethylenediaminetetraacetic acid tetrasodium

salt dihydrate were purchased from Acros. Hydrochloric acid (HCl), melamine, and copper

bromide were purchased from Sigma Aldrich. 2- Aminoethanethiol was purchased from TCI

America. Hydrogen tetrachloroaurate(III) trihydrate was purchased from Beantown Chem-

ical. Hexamethyldisilazane was purchased from EMD Millipore. General solvents and basic

alumnina were purchased from Fisher Scientific. All chemicals, including solvents, were used

without further purification, except styrene, which was passed through a short column of

basic alumina to remove the inhibitor prior to polymerization. Note that some solvents were

purchased as an anhydrous grade, as noted in the experimental details.

Instrumentation

Centrifugation was performed with an Eppendorf 5424 centrifuge. UV-vis spectra and mea-

surements were obtained on a Cary-60 spectrophotometer. Thermal annealing was per-

formed with a Techne Gradient Thermal Cycler. Gel permeation chromatography (GPC)

characterizations were performed on an Agilent Technologies GPC system, with THF as the

eluent at 1.0 mL/min, and monodisperse polystyrene as the standard. Transmission elec-

tron microscopy (TEM) was performed with an FEI Tecnai G2 Spirit TWIN transmission

electron microscope. Small-angle X-ray scattering (SAXS) measurements were performed at

the 12ID-B station at the Advanced Photon Source (APS) at Argonne National Laboratory.

The samples were probed using 13.3 keV X-rays with an exposure time of 1.0 second. The

sample-to-detector distance was calibrated with a silver behenate standard. The beam was

collimated using two sets of slits, and a pinhole was used. Scattered radiation was detected

using a Pilatus 2M detector. Scanning electron microscopy (SEM) and focused ion beam

(FIB) milling were performed with a FEI Helios Nanolab 600 Dual Beam System. Imaging

was done at an accelerating voltage of 5 kV and current of 86 pA. FIB milling of the WB

NCTs were done at an accelerating voltage of 30 kV and current of 93 pA and done in Clean-

ing Cross Section Mode. FIB milling of the polymer film was done in progressive stages at

varying doses. Nanomechanical Compressions were performed using an Alemnis Standard

Assembly (ASA) with a 20

m flat punch diamond tip (from Synton-MDP) at a strain-rate

of 10

−

In-situ

nanomechanical compressions were done with the ASA platform while

imaging with a Zeiss GeminiSEM at 5kV accelerating voltage. Spectroscopic analysis of

samples was done using an Oxford Instruments Ultim Max Energy Dispersive Spectroscopy

(EDS) Probe. Data were collected and processed using the AztecLive Software.

Material Synthesis and Characterization

We have previously reported the synthesis of the thymine (Thy) initiator for the atom transfer

radical polymerization (ATRP) of the Thy-terminated polystyrene (PS). The procedures are

briefly summarized below:

1,2

5.0 g thymine (40 mmol), 1.0 g 11-bromo-1-undecanol (4 mmol),

1.1 g potassium carbonate (8 mmol) and 200 mL of dimethyl sulfoxide (DMSO) were added

to a 500 mL round bottom flask. The mixture was stirred and bubbled with nitrogen for

2 minutes, and then sealed with a septum and stirred for 48 hours. The resulting mixture

was poured into water and chloroform for extraction. The organic phase was collected

and washed with brine (3x), dried with sodium sulfate, followed by solvent removal under

reduced pressure. The resulting white solid was recrystallized in ethyl acetate/hexanes to

afford

(Fig. S1). Using very gentle heat, 592 mg

(2.0 mmol) was dissolved in 150 mL

of dichloromethane, and 0.39 mL of trimethylamine (2.8 mmol) was added to the solution.

The mixture was cooled to 0

◦

C, and a solution of 552 mg 2-bromoisobutyryl bromide (2.4

mmol) in dichloromethane ( 10 mL) was added dropwise under a nitrogen atmosphere. The

reaction mixture was allowed to warm to room temperature and stirred for 14 hours, and

then washed with sodium carbonate (2x) and brine (2x). The organic phase was dried with

sodium sulfate, followed by solvent removal under reduced pressure. The resulting mixture

was purified by flash chromatography (dichloromethane/ethyl acetate 8:3), to obtain the

final Thy-initiator product. All characterization matches previous literature reports.

Thy-PS-SH Synthesis

Thy-initiator (1 eq), PMDETA (1 eq), styrene (200 eq), and anhydrous anisole (30 v% to

styrene) were added to an oven-dried 100 mL Schlenk flask with a stir bar. The reaction

flask was then sealed with a rubber septum and 3 freeze-pump-thaw cycles were performed

before filling the flask with nitrogen. The septum was removed to add CuBr (1 eq) under

positive nitrogen flow, and the flask was quickly resealed, purged, refilled with nitrogen, and

returned to room temperature. The reaction mixture was stirred at 110

◦

C and quenched

Figure S1: Synthetic scheme for the Thy initiator

with liquid nitrogen at 3 hours to obtain the polymer of the desired molecular weight (14

kDa). The reaction mixture was then added dropwise to methanol and the precipitated

polymer was collected, redissolved in dichloromethane, and precipitated in cold methanol. 3

precipitations were done, ending with a polymer with a chain end bromine.

The polymer was subsequently dissolved in N,N-dimethylformamide (DMF, 10 mg/mL)

with a small amount of triethylamine (40 eq). The solution was bubbled with nitrogen for

5 minutes followed by the addition of 2-aminoethanethiol (15 eq). The reaction flask was

sealed and the solution stirred for 60 hours, followed by precipitation in cold methanol. The

precipitate was redissolved in dichloromethane and reprecipitated in cold methanol 3 more

times, for a total of 4 cycles. The resulting white powder was dried under reduced pressure

to give the thiol-terminated Thy-PS-SH polymer. GPC results indicated that the polymer

had a molecular weight of 14.06 kDa with a dispersity of 1.06.

Gold Nanoparticle Synthesis

Spherical gold nanoparticles (AuNPs) were synthesized using a modification of the Turkevich

method to produce low dispersity, citrate-stabilized AuNPs.3-5 All glassware and stir bars

were cleaned with aqua regia before use.

A sodium citrate/citric acid buffer was prepared by first adding 485.3 mg of trisodium

citrate dihydrate and 115.6 mg of citric acid monohydrate to 800 mL of Milli-Q purified

water in a 1 L flask. The buffer solution was then heated with stirring until boiling and left

to boil for 15 minutes.

A stock solution of ethylenediaminetetraacetic acid tetrasodium salt dihydrate (EDTA

salt) was then prepared by adding 60.8 mg EDTA salt to 10 mL of Milli-Q purified water. 1

mL of the EDTA solution was then added to the boiling solution and vigorously stirred for

10 minutes.

Gold precursor solution was prepared by dissolving 159.0 mg HAuCl4 in 9 mL of Milli-Q

purified water. After 10 minutes, 3 mL of gold precursor solution was swiftly injected into

the boiling, stirring solution. The solution turned from clear to dark red 30 seconds after

the addition of the gold precursor. The reaction was boiled for 30 minutes before being

allowed to gradually cool to 90

◦

C. After 30 minutes for temperature equilibration at 90

◦

another 3 mL of gold precursor was swiftly injected, followed 30 minutes later by the final 3

mL gold precursor addition. After 30 minutes, 400 mL of the reaction solution was removed,

followed by the addition of 485.3 mg sodium citrate in 400 mL of Milli-Q purified water.

The solution was then equilibrated at 90

◦

C for another 30 minutes.

A new gold precursor solution was prepared by dissolving 157.6 mg HAuCl4 in 9 mL of

Milli-Q purified water. Once the reaction solution was finished equilibrating at 90

◦

C, an

additional 3 rounds of swiftly injecting 3 mL of gold precursor solution into the reaction

mixture was performed, with 30 minutes between injections. After the final injection, the

reaction was kept at 90

◦

C for another 30 minutes before cooling to room temperature to

obtain the AuNPs.

To prepare samples for TEM, 1 mg of thiol-terminated polystyrene was dissolved in 1 mL

of acetone and then added rapidly to 1 mL of the AuNP solution with vigorously stirring.

The resulting suspension was allowed to stir for 15 minutes. Dark red/pink precipitates were

observed almost immediately upon addition of the polymer solution. The functionalized

nanoparticles were isolated and purified via 2 rounds of centrifugation, first in DMF, and

then toluene. Particles were then dropcast from toluene onto formvar coated TEM grids.

Analysis was performed in ImageJ. AuNPs used had a measured diameter of 30.3 nm and a

dispersity of 8.5 (Fig. S2).

Figure S2: TEM images of the AuNPs used in this work had a measured diameter of 30.3

nm and a dispersity of 8.5 %.

Thy-NCT Synthesis

In a 40 mL glass vial, 14 mg of Thy-PS-SH was dissolved in 14 mL of acetone to make a clear

polymer solution. The polymer solution was then added rapidly to 14 mL of the citrate-

capped AuNPs and the resulting suspension was allowed to stir for 15 minutes. Dark red/pink

precipitates were observed almost immediately upon the addition of the polymer solution.

The functionalized nanoparticles were recovered by centrifugation at 6k RPM for 3 minutes,

and purified and concentrated by three more cycles of centrifugation: Cycle 1 - Redispersed

in 10 mL of DMF, centrifuged at 5.5k RPM for 45 minutes. Cycle 2 – Redispersed in 1

mL of DMF, centrifuged at 5.5k RPM for 45 minutes. Cycle 3 – Redispersed in 1 mL of

toluene, centrifuged at 5.5k RPM for 45 minutes. After the third cycle, the purified NCTs

were dispersed in 300

L of toluene. The optical density (OD) of the solution, as measured

by UV-Vis spectrometry, was OD 176 at 535 nm.

Thy-Melamine NCT Winterbottom Assembly

A Si chip was used as the substrate for Winterbottom growth and cleaned with acetone

and isopropanol prior to insertion into a 0.5 mL centrifuge tube. The Si chip was inserted

at a slight angle such that the target growth surface is pointed away from the opening of

the centrifuge tube. Positioning the substrate in this way minimized the amount of non-

Winterbottom NCT assemblies that was deposited on the target growth surface.

A fresh melamine stock solution was prepared prior to assembly by dissolving 25 mg of

melamine in a solution of 1 mL DMSO and 1 mL DMF. A 495

M melamine solution in

toluene was then prepared by mixing 5

L of the melamine stock solution with 995

L of

toluene for immediate use. Thy-Melamine NCT Winterbottom assemblies were formed in

the 0.5 mL centrifuge tube with the Si chip by adding 290

L of the prepared Thy-NCTs

(OD 176 at 535 nm), 60

L of toluene, and 60

L of the freshly prepared 495

μ

M melamine

solution. To crystallize the NCT Winterbottoms, the centrifuge tube was placed in the

thermal cycler, heated to 65

◦

C, and then cooled to 25

◦

C at a rate of 0.1

◦

C/2 minutes.

NCT Solvent Exchange and Drying

Post thermal cycling, a black pellet was seen at the bottom of the centrifuge tube and the

supernatant of the solution was a very faint pink. The Si chip was observed to have a blue

shimmer to it. The Si chip was transferred to another 0.5 mL centrifuge tube that was filled

with 200

μ

L of toluene. The transfer of the Si chip from the original centrifuge tube to the

new tube has to be done rapidly; if the substrate dries out, the resultant NCT Winterbottom

assemblies will lose their crystallinity and become amorphous.

A non-solvent for the NCTs, n-decane, was gradually added without mixing (slowly added

onto the walls of the tube) to increase the vol% of n-decane in solution by 10% every 25

minutes. After reaching 70 vol% n-decane, the substrate was left to stand in the solution

for 45 minutes. The substrate was then rapidly transferred to another 0.5 mL centrifuge

tube containing 250

L of 80 vol% n-decane / 20 vol% toluene and left to stand for another

25 minutes. 250

L of n-decane was then added to increase the vol% of n-decane in

solution to 90 vol% and the substrate left to stand in the solution for another 25 minutes.

The substrate was then rapidly transferred to a new 0.5 mL centrifuge tube containing 250

L of n-decane and left to stand in it for 25 minutes. To complete the drying process, the

substrate was then removed from the centrifuge tube and left to dry on a glass petri dish.

The dried substrate was observed to have a faint gold shimmer. Care must be taken to

minimize damage to the target growth surface during the transfer of the substrate from one

centrifuge tube to another.

Preparation of Thy-PS-SH films

To fabricate the Thy-PS-SH films, a polymer solution was first prepared by dissolving 14

kDa Thy-PS-SH in anisole (20 mg/mL). 20

L of the solution was then drop-cast onto a Si

chip and left to dry.

SEM Images of the NCT Winterbottom Assemblies

Figure S3: SEM images of the NCT Winterbottom assemblies. a) Top-down image of the

substrate showing that the Winterbottom assemblies were found everywhere on the target

growth surface, b) Winterbottom assemblies imaged at 52

◦

, c) top-down image of an indi-

vidual Winterbottom assembly, and d) another Winterbottom assembly imaged at 52

◦

Preparation of NCT Winterbottoms for Compression

Given the density of WBs on the substrate surface and the compression flat punch tip

diameter of 20

m, it was necessary to use the FIB to isolate candidate WB structures prior

to compression, i.e. the FIB was used to mill away all NPSLs that were within a 40

diameter of the center of the target WB. The milling was done in Cleaning Cross Section

Mode. To minimize redeposition of the unwanted structures on the target structure, care

was taken to ensure that the direction of milling started from the edge of the unwanted

structure that was furthest from the target structure and ended at the edge that was closest

to the target structure.

SAXS of NCT Winterbottom Assemblies

Small angle X-ray scattering (SAXS) measurements were performed at the 12ID-B station

at the Advanced Photon Source (APS) at Argonne National Laboratory. The samples were

probed using 13.3 keV X-rays in transmission mode with an exposure time of 1.0 second.

1-dimensional SAXS data was obtained via radial averaging of the 2-dimensional scattering

pattern. Data was then transformed into profiles of scattering intensity as a function of

scattering vector

. Face-centered cubic (FCC) ordering were confirmed, with structural

parameters:

FCC

√

(S1)

FCC

√

(S2)

Figure S4: SAXS data of the dried NCT Winterbottom assemblies. Indexing of the first

three peaks (Table S1) indicate that the NCT assemblies have FCC symmetry.

Table S1: Peak positions and structural parameters for the SAXS data in Fig. S4. As the

peaks in the NCT assemblies appeared as faint shoulders in the

peaks, they were difficult

to measure accurately and were thus omitted in the peak position calculations.

Sample

Peak Position

d (nm) a (nm) Structure

NCT

Winterbot-

tom Assemblies

0.022

1.64

1.92

34.21

48.34

FCC

Ideal FCC

1.15

1.63

1.91

FFT Analysis of lattice order

To measure the interparticle spacing, we obtained the pixels-to-length scaling using the scale

bar on the original SEM image. Since the resulting FFT had the same pixel dimensions as the

original SEM image, we measured the distance of peak intensity values in the FFT image in

pixels. This was then converted to nanometers using the pixels-to-length scaling. We obtain

an average interparticle spacing of 34.6 nm. We then calculated the standard deviation of

these particles to be 5 nm, which ensured random particle seeding in the DEM simulations.

By sectioning the region of interest on the image into 300 by 300 pixel squares, we computed

the FFT of each individual section. We then took the mean and standard deviation across all

these square FFT sections. The top seven to nine peak intensity locations were then identified

and the standard deviation values were extracted for those locations. We then obtained a

final average standard deviation by averaging over these standard deviation values at the

peak locations.

To measure the widening and flattening of the FFT plots in Fig. 2e,f, and g we define

a metric called sharpness (or yellowness according to the color scale). The sharpness was

measured on a scale of (background) 1.245 to 29.65 (sharpest). Two different sharpness

limits—one at 21 and one at 22—were used to find the widening and flattening of the

FFT plot. The pixel on the MATLAB figure that was farthest out (both horizontally and

vertically) was measured for each sharpness limit. Further, the

- and

-coordinates of

all 4 bounding-box vertices were noted. The overall dimension of the figure itself was also

measured. To get the bounding box height, the

-coordinate of the top point of the bounding

Figure S5: (a) Change in size of Bounding box of FFT with increasing strain. (b) Variation

of normalized bounding box height and width with strain. (c) Variation of bounding box

aspect ratio with strain.

box was subtracted from the bottom point of the bounding box and the difference was

normalized by the overall figure

-pixel dimension. The same procedure was applied for the

bounding box width, except using the

-coordinates of the sides of the bounding box. We

generated three different metrics: bounding box height, bounding box width, and bounding

box aspect ratio. The variation of the bounding boxes with strain is shown in Fig. S5a. The

normalized height and width are plotted in Fig S5b and the Aspect ratio is shown in Fig.

S5c. Since sharpness limit matters for each dimension, we obtain a linear trend for bounding

box height with a sharpness limit of 22, but in the case of bounding box width, we get a

linear trend with a sharpness limit of 21.

Geometric Analysis of Pillars

Figure S6: (a) angled view of tapered pillar with important geometric features annotated.

(b) top view of pillar with top and bottom area colored. (c) Variation of taper angles of the

pillars with surface area-to-volume ratio. (d) Variation of aspect ratio with surface area-to-

volume ratio. (e) Height of pillars for structures of different surface area-to-volume ratio.

An SEM image of a FIBed pillar is shown in Figure S6a with a taper angle (

) of

≈

In order to account for the taper, the surface area and volume of the pillars need to be

calculated using a modified formulation using the top and bottom areas and perimeters of

the pillar. Figure S6b shows how the area and perimeter of the top (

) and bottom

(

) are measured for a pillar. Once these are obtained we assume that the perimeter of

the pillar varies linearly while the area varies quadratically with height from top to bottom.

This is a reasonable assumption for a linear taper. In such a case, the area and perimeter

at any arbitrary height

from the top of the pillar are given by

(

) =

1

−

0

2

and

(

) =

1

−

0

respectively. Here

and

are the area and perimeter of the top

surface of the pillar respectively, and

and

are the area and perimeter of the bottom

surface respectively. The surface area is calculated as

(

)

sec

(S3)

Here,

is the taper angle. Observing from Figure S6d that the taper angle is

, the value

of sec

≈

1. Therefore, this expression can be simplified to give

(S4)

The volume is given by

(

)

dh.

(S5)

Thus, the surface area-to-volume ratio of the tapered pillars can be calculated by taking a

ratio of Eqns. S4 and S5.

Aspect ratio of pillars.

the aspect ratio for the pillar is defined as

√

(

0

1

)

and

its variation with the surface area-to-volume ratio is shown in Figure S6d. It can be seen

that the aspect ratio varies from

∼

25 for the winter bottoms (WBs) to about

∼

5 for

the highest surface area-to-volume pillars.

The variation of the height of the pillar with surface area-to-volume ratio is shown in Figure

S6e. The mean height across all samples is 1.96

μ

m and the standard deviation is 0.398

μ

Mechanical Analysis of Pillars

Figure S7: (a) True stress-true strain and engineering stress-engineering strain plot for a

winterbottom structure. (b) Variation of relative error in stress between engineering and

true values with the engineering stress. (c) variation of effective stiffness of pillar with

aspect ratio for 1% applied strain obtained through FEA. (d) Variation of relative error of

stiffness with aspect ratio obtained through FEA.

Effective stress for tapered pillar.

The effective stress for the tapered pillars is

calculated by dividing the load by the average of the top and bottom cross-sectional areas

of the tapered pillar. For taper angles of 5

, it has been shown that using this metric for a

stress estimate provides a less than 5% error in effective measured properties.

The effective

stress is given by

eff

(S6)

where

is the total load on the pillar. The distribution of taper angles for the pillars of

various SA/V is shown in Figure S6b.

Influence of aspect ratio on the obtained mechanical properties.

For validating

that there is not a large change in cross-section (at which point the contributions from the

substrate and indenter due to the low aspect ratio become relevant) we can look at the true

stress-true strain plots for the low aspect ratio structures. Figure S7a shows the true stress-

true strain and engineering stress-engineering strain plot for a Winterbottom where—up to

yield—the two plots are nearly equivalent. This is captured further in Figure S7b where

the relative error between the engineering and true stress is

10% up to yield thereby

enabling us to extract relevant information about stiffness and yield from these structures.

Additionally, we performed FEA simulations of circular pillars with different aspect ratios

to obtain their stiffness and compare with the theoretical material stiffness. We impose

fixed boundary conditions at the bottom of the pillar to mimic the substrate, and a roller

boundary condition on the top to mimic the flat-punch indenter tip. The assumption of

a roller boundary condition can be validated from the post-compression SEM images of a

Winterbottom structure as shown in Fig 2g-i, where we see crack formation and propagation

on the top surface indicating possibility of relative motion of particles on the horizontal

plane (i.e., that fixed boundary conditions would present an over-constraint). The sides of

the pillar are assumed to be free. The constituent material is assumed to be linear elastic

with a Young’s modulus of 1.6 GPa and a Poisson’s ratio of 0.34 corresponding to that of

polystyrene at room temperature.

The results from the simulation are shown in Fig. S7c,d

where for very low aspect ratios there is a 35% deviation from the expected value. In our

case, however, the lowest aspect ratios start at 0.33 for the Winterbottom which corresponds

10% deviation from the expected value. Further, this simulation serves only as an

upper limit to the measured modulus since the assumption of a fixed boundary condition

may be may be more restrictive than the actual contact between the substrate and the

Winterbottom structure. In order to account for the error in our measured stiffness, we

introduce an aspect ratio correction factor that we obtained through FEA, that we multiply

to the stiffness and strength values for samples of various aspect ratios. The values for this

correction factor ranges from 0.95 (for the lowest-aspect ratio structures) to 1 (for structures

with aspect ratio greater than 1).

Compression of polystyrene pillars

Figure S8: (a) SEM image of a polystyrene pillar (b) Variation of equivalent Young’s modulus

with Surface Area/Volume Ratio. (c) Variation of Strength with Surface Area/Volume Ratio.

To verify the effect of stiffening of the polymer, polystyrene pillars (Fig. S8a) of SA/V

ratios ranging from 1

13 1/

μ

m to 3

73 1/

μ

m were prepared via FIB-milling from the cast

Thy-PS-SH film. The pillars were prepared via a multi-stage FIB process: in a typical

process, a square pillar of length 12

μ

m was first prepared (30 keV, 6.5 nA). A second cut

was then done using circle mode from outside-in to a diameter of 10

μ

m (30 keV, 2.8 nA).

A final cut was then made using circle mode from outside-in to the desired pillar diameter

(30 keV, 280 pA). Attempts to mill these pillars using lower currents resulted in significant

drift and consequently, pillars of poor quality. These pillars were compressed quasi-statically

and their modulus and strength were measured. The trends for Modulus to SA/V ratio

and Strength to SA/V ratio are shown in Fig. S8b,c. We observe a trend similar to those

seen in the NPSLs which indicates the stiffening is primarily attributed to the polymer. We

also note that the stiffness (2–11 GPa) and strength (100–500 MPa) values obtained for the

polystyrene pillars are comparable to those obtained from the compression of the NSPLs for

the given Surface Area/Volume ratio. Further, these compressed pillars were analyzed using

EDS to determine if there was any Gallium-ion (Ga) implantation that could explain the

stiffening of the outer layer. The EDS data is shown in Fig. S9. The presence of Ga ions

is shown in Fig. S9b in yellow, and are present in the regions exposed to the FIB. They

are primarily seen in the regions where the base substrate is exposed. The base substrate is

Silicon and is shown in pink in Fig. S9d. The polystyrene pillars are comprised of carbon

atoms which are shown in red in Fig. S9c. There is an overlap between the Ga ions and

the carbon atoms (polystyrene pillars) at the walls of the pillar. We believe this Ga-ion

implantation is responsible for the strengthening of the polystyrene pillars as well as the

NSPLs.

Figure S9: (a) SEM image of a compressed polystyrene pillar (Scale bar = 10

m). EDS

images of (b) Gallium atoms (c) Carbon atoms (d) Silicon atoms.

Core-Shell Model

Figure S10: (a)Schematic of composite spring. (b) Predicted effective stiffness with and

without a stiffened top layer. (c) Predicted effective stiffness considering the simplified

formulation and the unsimplified formulation. (d)Variation of mean squared error with

surface layer thickness for the unsimplified stiffening model, supporting the assumption of

FIB effects spanning one particle-layer only.

Here, we provide the derivation of the core-shell stiffness of the NPSL pillars. A schematic

of this model is shown in Fig. 5a. The system can be modeled as a composite spring as shown

in Fig. S10a. We derive the theoretical model assuming that the top layer is exposed to the

Ga ions although it is not fully clear from the EDS that this is indeed the case. However,

we have also derived the model for the case where we ignore the stiffening at the top.

Furthermore, we note that the consideration of this layer in the model provides a variation

in the effective stiffness of less than 2%. Let

be the Young’s modulus of the core which

is assumed to be different from the Young’s modulus of the surface,

. For the composite

spring, the effective stiffness is given by

eff

total

/ε

total

(S7)

where

total

πr

2

is the total stress experienced by the composite spring and

total

∆

is the total strain experienced by the composite spring. In the case of a composite spring,

the forces are equal across springs in series and summed over springs in parallel while the

displacements are summed in series and equal across springs in parallel. Therefore, for the

given network the total force on the spring is given by

total

s,c

s,t

where

s,t

the force on the top layer of surface particles,

is the force on the core, and

s,c

is the force

on the surface particles on the walls of the pillar. Similarly, the total displacement of the

composite spring is given by ∆

total

= ∆

s,t

+ ∆

and ∆

= ∆

s,c

. These displacements

are related to the strains as

total

∆

total

s,t

∆

s,t

, and

s,c

∆

−

. Similarly, the

forces are related to the stresses as

total

s,t

πr

(

πr

−

πrt

πt

), and

s,c

πrt

−

πt

). Further, the surface stresses are related to the surface strains as

while the core stress is related to the core strain as

Eε

. In terms of strain,

the total elongation is given by

∆

total

s,t

+ (

−

)

(S8)

The total force can be written in terms of stress as

total

s,c

(

πr

−

πrt

πt

s,c

πrt

−

πt

) =

s,t

πr

. Using this relation, the strains can be derived to be

s,t

total

πr

2

and

total

(

πr

2

−

πrt

πt

2

)

πrt

−

πt

2

. This equation can be substituted in Eq. S8 to obtain

∆

total

πr

(

−

)

total

(

πr

−

πrt

πt

)

+ (2

πrt

−

πt

)

(S9)

Substituting this into Eq. S7 yields

eff

((

πr

−

πrt

πt

)

+ (2

πrt

−

πt

)

(

πr

−

πrt

πt

)

+ (2

πrt

−

πt

)

+ (

−

)

(S10)

The surface area to volume ratio is given by

and the ratio of the surface

layer thickness to the height of the pillar is can be defined as

t/h

and

can be used

to substitute for

and

in Eq. S10 to yield

eff

+ (

−

)

−

αt

+ 1

1 +

−

αt

+ 1

(S11)

Additionally, by ignoring the stiffening of the top layer, this equation simplifies to

+ (

−

)

−

αt

+ 1

(S12)

The two theoretical models whose effective stiffness are presented in equations S11 and S12

are compared in figure S10b and the relative error between the two models is calculated to

be less than 2% indicating that the consideration of the layer does not significantly affect

the analysis. Furthermore, under the assumption that

αt <<

1, Eq. S11 can be

simplified to

eff

+ (

−

) (

αt

−

)

−

+ 1

(S13)

It is seen from figure S10c that the simplified and unsimplified forms of the model start to

diverge at an

αt

value of greater than 10. This indicates that although the effective stiffness

varies quasi-linearly at low surface area-to-volume ratios it starts to saturate at higher values

Obtaining the values of

and

t

. We use a generalized reduced gradient nonlinear

solver to minimize the means squared error(MSE) between the theoretical predictions and

experimental values and solve for

and

to get a best fit with the experimental data.

Figure S10d shows the variation of the mean squared error with the surface layer thickness.

The optimal value for

and

are 40.166 GPa and 41.1 nm respectively. Since this value

is very close to the inter-particle spacing of 36.4 nm we assume the value of

to be 36.4 nm

and calculate the corresponding

to be 44.77 GPa.