Bhushan B. Nanotribology and Nanomechanics: An Introduction

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14 Mechanical Properties of Nanostructures 767

Table 14.3. Summary of physical properties of PPMA and PMMA materials

Structure T

(

◦

C) Water Thermal Thermal Elastic modulus (E) (GPa) Hardness (H)

absorption expansion Conduc. (measured)

after coeﬃcient (K

−1

) (0–50

◦

C) (MPa)

24h (%) (W/mK)

Bulk Nano- Nano- Beam

(literature) indentation indentation bending

(literature) (measured)

PMMA 104–106

0.3–0.4

2–3×10

−4

(< T

)

0.193

3.1–3.3

4.43

5.0 2.0 410

0.1–0.3

6×10

−4

(> T

)

PPMA

35–43

– – – – – 3.7 0.7 190

[80,81]

[79]

[95]

[96]

[97]

[98,99]

[100]

768 Bharat Bhushan

Hardness (GPa)

2.0

1.5

1.0

0.5

0.0

Elastic modulus (GPa)

Contact depth (nm)

0 500100 200 300 400 0 500100 200 300 400

Displacement (nm)

Time (s)

100

0 500100 200 300 400

Mean stress (GPa)

2.0

1.5

1.0

0.5

0.0

Contact stiffness (N/mm)

Time (s)

600 0 500100 200 300 400 600

PMMA beam

PPMA beam

PMMA beamPPMA beam

Fig. 14.21. (a) Hardness and

elastic modulus of 2.9 µ m

thick PPMA and 3.4µmthick

PMMA beams as a function

of contact depth, (b) inden-

tation displacement, mean

stress and contact stiﬀness as

a function of time for 2.9 µm

thick PPMA and 3.4µmthick

PMMA beams

values obtained at contact depth of 100nm from ﬁve indents. The H (0.41GPa) and

E(5.0GPa) of the PMMA beams are both higher than H (0.19 GPa) and E (3.7GPa)

of the PPMA beams. Table 14.3 compares the present data with the published data.

We note that the E of PMMA is comparable to the published data. It should be

14 Mechanical Properties of Nanostructures 769

300 nm PPMA film

0.20 0.4 0.6 0.8

Normal load (mN)

1.0

–400

–800

Coefficient of friction

Friction

During scratch

After scratch

Before scratch

Scratch depth (nm)

Region A

Region B

Region C

Region A

Region B

Region C

0.20 0.4 0.6 0.8

Normal load (mN)

1.0

–400

–800

Coefficient of friction

Scratch depth (nm)

300 nm PMMA film

Fig. 14.21. (continued)

and coeﬃcient of friction as

a function of increasing nor-

mal load and SEM images of

three regions over scratches:

at the beginning of the scratch

(indicated by A on the scratch

depth proﬁle), and at the point

of initiation of damage (in-

dicated by B on the scratch

depth proﬁle) and at the end

of scratch (indicated by C on

the scratch depth proﬁle) for

PPMA and PMMA thin ﬁlms

(after [65])

noted that the H and E were also measured on 3µm PPMA and 3 µm PMMA thin

ﬁlms/SU8-25,and the results werevery close tothe valuesobtained from thebeams.

Indentation Creep

In the creep tests, the changes in indentation displacement, mean stress (hardness)

and contact stiﬀness were monitored [65]. Figure 14.21b shows the indentation dis-

placement, mean stress and contact stiﬀness as a function of time for PPMA and

770 Bharat Bhushan

PMMA beams [65]. The indentation depth corresponding to 30 µN is about 60 nm

for PPMA and about 30nm for PMMA. The indentation depths of both polymer

beams increase with time. The PPMA beam exhibits a faster increase in indentation

depth (from about 60nm to about 90 nm) than the PMMA beam (from about 30nm

to about 50nm). This indicates that a higher hardness is associated with a higher

creep resistance. In contrast with indentation displacement, the mean stresses of

both polymer beams decrease with time, indicating that stress relaxation occurred

during the hold segment. The creep tests were also conducted on the polymer thin

ﬁlms and the results were similar to the polymer beams.

Scratch Resistance

The polymer ﬁlms were annealed at 95

◦

C and 175

◦

C to PPMA and PMMA re-

spectively to simulate the thermal processing used in the micromolding process

for polymer beam fabrication. The annealing treatment seemed to eﬀectively simu-

late the beam fabrication process, because the H, E and creep measurements show

that the PPMA and PMMA thin ﬁlms have similar mechanical properties with

PPMA and PMMA beams. In order to evaluate the scratch resistance and adhe-

sionof PPMAand PPMA beams/SU8-25substrate,nanoscratchmeasurementswere

needed. Since polymer beamswere too narrow(width 5 µm) to perform nanoscratch

tests, the nanoscratch experiments were conducted on 300nm PPMA and PMMA

thin ﬁlms/SU8-25, assuming that they have similar scratch resistance and adhesion

with PPMA and PMMA beams.

Figure 14.21c shows that the scratch depth proﬁles, coeﬃcient of friction, and

SEM images of three regions over scratches: at the beginning of the scratch (indi-

cated by A on the scratch depth proﬁle), at the point of initiation of damage (indi-

cated by B on the scratch depth proﬁle), and towards the end of the scratch (indi-

cated by C on the scratch depth proﬁle) for PPMA and PMMA thin ﬁlms [65]. The

scratch results of polymer thin ﬁlms are very diﬀerent from metallic and ceramic

thin ﬁlms [82]. Firstly, the coeﬃcientof frictionvalueis very high,evengreater than

one, and the coeﬃcient of friction proﬁle jumps up and down frequently. Secondly,

after the polymer thin ﬁlms were damaged, the scratch depth proﬁle also jumps

up and down considerably, indicating the scratch tip moved up and down during

scratching. The SEM images of PPMA show that during scratching, the materials

were plowed and accumulated in front of the tip, instead of being pushed aside,

because the polymers were so soft. When the plowed materials reached certain

amount, the tip was almost stuck, so the tip jumped up instead of ramping down.

This may explain the oscillation in the scratch depth proﬁle. At region B., the co-

eﬃcient of friction increased and the in situ scratch depth also changed abruptly,

indicating that the ﬁlm was delaminated. The critical load of PPMA and PMMA are

about the same, around 0.22mN.

14.3.5 Bending Tests of Polymeric Microbeams Using a Nanoindenter

As mentioned earlier, to avoid tip penetration into polymer beams, the tip was dip-

coated with PMMA. Figure 14.22 shows the load–displacement curves of PPMA

14 Mechanical Properties of Nanostructures 771

Load (μN)

Displacement (nm)

0 150

50 100

3.5 μm thick, 25 μm long PPMA beam

E = 0.7 r 0.1 GPa

3.5 μm thick, 30 μm long PMMA beam

E = 2.0 r 0.1 GPa

Fig. 14.22. Bending results

on PPMA and PMMA beams

with PMMA coated diamond

conical tip (after [65])

and PMMA beamsmeasured with the PMMA coated tip [65] loading curve was lin-

ear andthe unloadingcurve doesnotexhibitmuch plastic deformation,whichmeans

that the bending tests were performed mostly in the elastic region of the samples

and the elastic modulus can be calculated from the loading curve. Any slight degree

of deformation of the tip coating during the bending tests was assumed negligible

compared to beam bending. Based on Fig. 14.22, the calculated elastic modulus for

PPMA and PMMA beams are 0.7 ±0.1GPa and2.0 ±0.1GPa, respectively. The

standard deviation was calculated based on ﬁve measurements. Calculated elastic

moduli of PPMA and PMMA beams from bending tests are lower than the values

obtained from indentation.

Eﬀect of Soaking and Temperature

The primary operating environment for polymer BioMEMS is an aqueous solution

(e.g., cell culture medium). Absorption of water by the polymer matrix has the po-

tential to change the properties of the material. In order to determine if immersion

772 Bharat Bhushan

in water has a signiﬁcant eﬀect on material behavior,tests need to be run on samples

that have beensubjected to an aqueousenvironmentfor a considerabletime. In addi-

tion, in the case of employingBioMEMS for in vivo or elevated temperaturein vitro

applications, the eﬀect of human body temperature (37.5

◦

C) on polymer properties

may be important. The T

of PPMA (35–43

◦

C) is aroundthe body temperature and

the T

of PMMA (104–109

◦

C) is above body temperature. It is necessary to study

how the mechanical properties of PPMA and PMMA are aﬀected when operating at

body temperature as compared to room temperature.

To simulate the aqueous working environment of the polymer BioMEMS, the

bending tests were also performed on soaked samples in addition to the dry samples

and the results were compared.To make the soaked samples, the PMMA and PPMA

beams were soaked in DI water for 36 hours before the bending tests. In addition,

the nanoindentation tests were conducted in a range of temperature (22–37.5

◦

C) to

study the eﬀect of temperature on nanomechanical properties of polymer beams. In

order to heat the sample, the polymer beam sample was placed on a heating stage,

which can increase the temperature up to 200

◦

Fig. 14.23. (a) Soaking eﬀect on bending of PMMA beam, (b) temperature eﬀect on bending

of PPMA beam, and (c) soaking and temperature eﬀects on H and E of PPMA and PMMA

beams [65]

14 Mechanical Properties of Nanostructures 773

Figure 14.23a shows the eﬀect of soaking on elastic modulus of PPMA beams.

After soaking, the calculated elastic modulus (0.5±0.1GPa)ofPPMAwaslower

than the elastic modulus (0.7±0.1GPa) obtained at dry conditions [65]. But for

PMMA, after soaking, the calculated elastic modulus does not change (ﬁgure not

shown). The more open structure of PPMA versus PMMA might allow more water

to penetrate into the matrix, thus having a greater eﬀect on the modulus than with

PMMA (closed, hydrophobic structure). The open structure of PPMA is due to the

larger side chain compared to the side chain of PMMA.

Figure 14.23b shows the eﬀect of temperature on load–displacement curves of

PPMA beams. As the temperature increased from room temperature (22

◦

C) to hu-

man body temperature (37.5

◦

C), the slope of the loading curve decreased [65]. The

of PMMA is 35–43

◦

C, so it is understandable that the elastic modulus, thus

the slope of the loading curve, would decrease when the temperature increased be-

yond 35

◦

C. However, since the T

of PMMA is about 100

◦

C, at 37.5

◦

C, the load–

displacement curve of PMMA beam was similar to 22

◦

C (ﬁgure not shown).

The soaking and temperature eﬀects on hardness and elastic modulus of PPMA

andPMMA beams werealsostudied andthe resultsare shownin Figure14.23c[65].

It can be seen that for PPMA beam, the hardness and elastic modulus decreased af-

ter 36 h soaking in DI water, and also decreased at 37.5

◦

C, which is consistent

with the bending test results. For PMMA beam, the hardness decreased after soak-

ing, and also decreased slightly at 37.5

◦

C. However, the elastic modulus of PMMA

beam did not change after soaking or heating up to 37.5

◦

C, which is in agreement

with the bending results. The lower elastic modulus of PPMA compared to PMMA

makes it an attractive option for certain applications, such as cantilevers. However,

the changes in the properties of PPMA as a function of aqueous environment and

temperature would require careful calibration at operating conditions when imple-

menting PPMA microstructures in a biological setting.

14.4 Finite Element Analysis of Nanostructures with Roughness

and Scratches

Micro/nanostructures have some surface topography and local scratches dependent

upon the manufacturing process. Surface roughness and local scratches may com-

promise the reliability of the devices and their eﬀect needs to be studied. Finite ele-

ment modeling is used to perform parametric analysis to study the eﬀect of surface

roughness and scratches in diﬀerent well deﬁned forms on tensile stresses which

are responsible for crack propagation [71,72]. The analysis has been carried out on

trapezoidal beams supported at the bottom whose data (on Si and SiO

nanobeams)

have been presented earlier.

The ﬁnite element analysis has been carried out by using the static analysis

of ANSYS 5.7 which calculates the deﬂections and stresses produced by the ap-

plied loading. The type of element selected for the present study was SOLID95

type which allows the use of diﬀerent shapes without much loss of accuracy. This

774 Bharat Bhushan

Semicircular asperities Grooved asperities

Longitudinal direction

R = 25 nm L = 25 nm

Fig. 14.24. (a) Plots showing

the geometries of modeled

roughness–semicircular and

grooved asperities along the

nanobeam length with de-

ﬁned geometrical parameters

element is 3-D with 20 nodes,each node havingthree degreesof freedom which im-

plies translation in the x, y and z directions. The nanobeam cross section is divided

into six elements each along the width and the thickness and forty elements along

the length. SOLID95 has plasticity, creep, stress stiﬀening, large deﬂection and large

strain capabilities. The large displacementanalysis is used for large loads. The mesh

is kept ﬁner near the asperities and the scratches in order to take into account varia-

tion in the bending stresses. The beam materials studied are made of single-crystal

silicon (110) and SiO

ﬁlms whose data have been presented earlier. Based on bend-

ing experiments presented earlier, the beam materials can be assumed to be linearly

elastic isotropic materials. Young’s modulus of elasticity (E) and Poisson’s ratio (ν)

for Si and SiO

are 169 GPa [91] and 0.28 [87], and 73 GPa [92] and 0.17 [101],

respectively. A sample nanobeam of silicon was chosen for performing most of the

analysis as silicon is the most widely used MEMS/NEMS material. The cross sec-

14 Mechanical Properties of Nanostructures 775

Fig. 14.24. (continued) (b) Schematic showing semicircular asperities and scratches in the

transverse direction followed by the illustration of the mesh created on the beam with ﬁne

mesh near the asperities and the scratches. Also shown are the semicircular asperities and

scratches at diﬀerent pitch values

tion of the fabricated beams used in the experiment is trapezoidal and supported

at the bottom so nanobeams with trapezoidal cross section is modeled, Fig. 14.10.

The following dimensions are used w

= 200nm, w

= 370nm, t = 255nm, and

P = 6 µm. In the boundary conditions, the displacements are constrained in all di-

776 Bharat Bhushan

rections on the bottom surface for 1µm from each end. A point load applied at the

center of the beam is simulated with the load being applied at three closely located

central nodes on the beam used. It has been observed from the experimental results

that the Si nanobeam breaks at around 80µN. Therefore, in this analysis, a nominal

load of 70µN is selected. At this load, deformations are large, and large displace-

ment option is used.

To study the eﬀect of surface roughness and scratches on the maximum bending

stresses the following cases were studied. First the semicircular and grooved asper-

ities in the longitudinaldirection with deﬁned geometrical parameters are analyzed,

Fig. 14.24a. Next semicircular asperities and scratches placed along the transverse

direction at a distance c from the end and separated by pitch p from each other are

analyzed, Fig. 14.24b. Lastly the beam material is assumed to be either purely elas-

tic, elastic–plastic, or elastic–perfectly plastic. In the following, we begin with the

stress distribution in smooth nanobeams followed by the eﬀect of surface roughness

in the longitudinal and transverse directions and scratches in the transverse direc-

tion.

14.4.1 Stress Distribution in a Smooth Nanobeam

Figure 14.25 shows the stress and vertical displacement contours for a nanobeam

supported at the bottom and loaded at the center, [71,72]. As expected, the maxi-

mum tensile stress occurs at the ends while the maximum compressive stress occurs

under the load at the center. Stress contours obtained at a section of the beam from

the front and side are also shown. In the beam cross section, the stresses remain con-

stant at a given vertical distance from one side to another and change with a change

in vertical location. This can be explaineddue to the fact that the bendingmoment is

constant at a particular cross section so the stress is only dependent on the distance

from the neutral axis. However, in cross section A-A the high tensile and compres-

sive stresses are localized near the end of the beam at top and bottom, respectively,

whereas the lower values are spread out away from the ends. High value of tensile

stresses occurs near the ends because of high bending moment.

14.4.2 Eﬀect of Roughness in the Longitudinal Direction

The roughness in the form of semicircular and grooved asperities in the longitu-

dinal direction on the maximum bending stresses are analyzed [72]. The radius R

and depth L are kept ﬁxed at 25 nm while the number of asperities is varied and

their eﬀect is observed on the maximum bending stresses. Figure 14.26 shows the

variation of maximum bending stresses as a function of asperity shape and the num-

ber of asperities. The maximum bending stresses increase as the asperity number

increases for both semicircular and grooved asperities. This can be attributed to

the fact that as asperity number increases, the moment of inertia decreases for that

cross section. Also the distance from the neutral axis increases because neutral axis

shifts downwards. Both these factors lead to the increase in the maximum bending