robotics.sciencemag.org/cgi/content/full/5/41/eaaz4239/DC1

Supplementary Materials for

Actuation of untethered pneumatic artificial muscles and soft robots using

magnetically induced liquid-to-gas phase transitions

Seyed M. Mirvakili*, Douglas Sim, Ian W. Hunter, Robert Langer

*Corresponding author. Email: seyed@mit.edu

Published 15 April 2020, Sci. Robot. 5, eaaz4239 (2020)

DOI: 10.1126/scirobotics.aaz4239

The PDF file includes:

Working mechanism Modeling Energy analysis Fig. S1. The hardware schematic for magnetically induced thermal soft robotic grippers. Fig. S2. The circuit diagram for MITPAM. Fig. S3. Measured data and simulation results for magnetic field strength inside the coil. Fig. S4. The specs for the molds used to fabricate the soft robotic grippers. Fig. S5. Illustration of the response of various types of magnetic particles under magnetic field. Fig. S6. Characterization of the MNPs based on their thermal response to various magnetic field strengths. Table S1. Heat parameters of the fluids used in this work. References (38–41)

Other Supplementary Material for this manuscript includes the following: (available at robotics.sciencemag.org/cgi/content/full/5/41/eaaz4239/DC1)

Movie S1 (.mp4 format). Magnetothermal actuation of a MITPAM demonstrating 20% strain under a 2-kg load excited at an input power of 900 W. Movie S2 (.mp4 format). MITPAM robotic arms under no load. Movie S3 (.mp4 format). MITPAM robotic arms under a 250-g load. Movie S4 (.mp4 format). MITPAM robotic arms under a 500-g load. Movie S5 (.mp4 format). Lifting an egg by magnetothermal actuation of a soft gripper. Movie S6 (.mp4 format). Lifting a ball by magnetothermal actuation of a soft gripper. Movie S7 (.mp4 format). Controlled magnetothermal actuation of a soft gripper.

Additional Experimental Information

1 2

3

5

6

7

9

4

10

1: Li-ion batteries (3000 mAh).2: Mini induction heater unit.3: Induction coil (dinner: 17 mm).4: Ferromagnetic rod (iron nail).5: Glass syringe (2 mL).

6: Electronic switch control (power switch).7: Heat sink.8: Copper plate sealing the syringe.9: Blunt tip dispensing needle (gauge 14).10: Soft gripper.

8

A B

(A) All the physical components except the Arduino board that are required to actuate the softgripper. (B) Circuit diagram of the electronic components used for actuation. In this diagram,the micro-controller unit is powereddirectlyby the pair of 4V Li-ion batteries inseries.Themicro-controller used in this scenario was an Arduino Uno with an input voltage range of 7 – 12 V.

A

...X6

VDC VDCB Component Value

R1,2

R3,4

D1,2

D3,4

T1,2

C1-6

L1,2

L3

470Ω (5W)10kΩ

UF400712V 1WIRFP250

330nF 600VAC 50kHz100μH

3-6 turns coil from copper pipe

R1

R2

R3

R4

D1

D2

D3

D4

T1

T2

C1 C6

L1

L2

L3

(A) The coil used for induction heating andmeasuring the magnetic field characteristics along the coil axis. The images are taken beforecoating the coils with paint to prevent shorting of the coil during the experiment. (B) The circuitschematic that is used to generate the high frequency alternating magnetic field. Magnetic nano-particles can be modeled as the secondary winding of a “transformer” with the primary windingbeing the induction coil.

Fig. S1. The hardware schematic for magnetically induced thermal soft robotic grippers.

Fig. S2. The circuit diagram for MITPAM.

-200 -150 -100 -50 0 50 100 150 2000

2

4

6

8

10

0 50 100 150 200 250 300 350 4000

2

4

6

8

10

12

38.4 W

283.8 W

H(k

A/m

)

Distance (mm)

L/2−L/2

H (

kA/m

)

Power (W)

y

x

a b

R

dw

A B CF = axb

a = 0.5354

b = 0.5

(A) Illustration and specifications of the coil that was used for deriving Eq. 4 and 5. (B) Magneticfield along the coil axis. The model (Eq. 5) is fitted to the data to estimate the field inside thecoil .(C) The magnetic field strength in the center of the coil as a function of the input power.

Small Medium Large

l 43 85.6 122

w 6 11.4 16

t 3.15 6.3 9

m 1.5 2.5 3.5

n 5.7 11.3 16

tw 0.64 0.9 1.3

tch 0.9 2.2 3

lch1 39.4 79.75 113.5

lch2 37.75 76.2 109

d1 4 8.5 12

d2 3.76 7.8 11

d3 9.27 18.4 26.75

ln 2.3 4.3 7

tn 1 1.3 2

wn 1.4 3 4

A Bl

w

t

m

n

tw

lch1

tch

lch2

d1

d2

d3

tn ln

wn

(A) Top and sideviews of the structure of the molds used to make both models of the soft grippers. The channeland nodes for the actuator fluid passage can be seen in the center of the mold’s cavity. (B) Thedimension for different elements of the molds for small, medium, and large grippers. All units arein millimeters (mm).

Working Mechanism

Upon exposure to an alternating magnetic field, metals (with a grain size of greater than 1 µm)

generate heat due to formation of an eddy current. This induced current in the metallic piece

generates a Joule heating effect.

The effective heating power (per mass) due to an eddy current for a polydispersion system

Fig. S3. Measured data and simulation results for magnetic field strength inside the coil.

Fig. S4. The specs for the molds used to fabricate the soft robotic grippers.

with grain diameter mean square of 〈d2〉 equals (38):

〈Pe〉 =(πµofH)2

20ρeρm

⟨d2⟩, (1)

where ρe is the electrical resistivity of the metallic particles, ρm is the volumetric mass density

of the sample, f is the magnetic oscillation frequency, H is the magnetic field strength, µo is the

vacuum magnetic permeability (µo = 4π × 10−7 H/m). The mean square of the grain diameter

is 〈d2〉 = d2o exp(2β2) where do and β are parameters of the lognormal function. In this form of

induction of heating, the sample can be treated as anRL circuit where theL represents inductance

of the secondary winding of a transformer with the primary winding being the induction heating

coil, and R represents the Joule heating effect.

For magnetic nanoparticles such as ferrimagnetic materials (e.g., Fe3O4) dispersed in a liquid,

Brownian-Neel relaxation (for single domain particles such as superparamagnetic nanoparticles)

and hysteresis losses (for multi-magnetic domains) are the dominant heating mechanisms (Fig.

S5). In order to achieve heat generation by the magnetic nanoparticles, the period of magnetic

field oscillation should be shorter than the Brownian relaxation time (τB), Neel relaxation time

(τN ), and the overall effective relaxation time which is τ = (1/τB +1/τN)−1, if both mechanisms

are desired. In Brownian relaxation, the nanoparticles rotate to align with the applied magnetic

field, however, in Neel relaxation the magnetic moment inside the particle aligns itself with the

applied magnetic field.

For multi-domain magnetic particles, the domains are aligned in different directions. How-

ever, across the magnetic domain walls magnetization direction gradually aligns with the mag-

netization of the neighboring domain (Fig. S5D). In multi-domain magnetic particles, when

exposed to an oscillating magnetic field, the domain walls jump over the voids and imperfections

(known as Barkhausen jumps which causes the Barkhausen noise) and generate the hysteresis

heating. Heating power density (a.k.a., Specific Absorption Rate (SAR) and Specific Loss Power

(SLP)) in hysteresis heating is proportional to the area of the hysteresis in the magnetization (M )

vs. magnetic field (H) curve, and the frequency (f ) as the following equation suggests:

Ph =fµo

ρm

∮MdH. (2)

It is observed that particles that exhibit ferromagnetic behavior (i.e., hysteresis), at low mag-

netic fields (below 5 kA/m or 63 Oe) the Ph scales with H3 (39). This third-order power law is

in distinction with the second-order power law for the power scaling with magnetic field in eddy

current induction heating mechanism. The magnitude of the generated heat due to hysteresis is

proportional to the frequency (∝ f ) while for eddy current it is proportional to the square of the

frequency (∝ f 2). We chose f ≈ 150 kHz which gives us enough heat for exciting the pneumatic

actuator and is easy to generate with high power MOSFETs in a compact circuit.

Considering the size of the nanoparticles we used in this work (i.e., 200 nm – 300 nm), we

hypothesized that hysteresis loss is the dominant heating mechanism. To test this hypothesis, we

examined the behavior of heating power as a function of the magnetic field. The sample was

prepared by mixing 1.134 g of the MNP with 11.25 mL silicone oil. For the experiment, 0.6 mL

of the resulting solution was transferred to a 1 mL vial, and the sample was then placed inside

a bigger vial. The gap between the two vials was filled with a thermally insulating material

(aerogel). The coil temperature was kept at 17◦C during the experiment by running constant

temperature water through the coil (Fig. S6A). The magnetic field was varied from 7.38 kA/m to

17.45 kA/m at frequency of 148 kHz. The power balance equation can be written as:

P = CdT

dt+ L

(T (t)− To

), (3)

where To is the ambient temperature,C is the heat capacity (J/K), and L is the heat loss coefficient

(W/K). Eq. 3 can be solved analytically in form of:

T (t) = To + ∆T∞(1− exp(−t/τ)

), (4)

where ∆T∞ = P/L and τ = C/L. The measured profiles (from 11 s to 160 s) were fitted with

an exponential function in form of:

T (t) = T∞(1− exp(−t/τ)

), (5)

B B

Single Domain Magnetic Particles Brownian Relaxation Néel Relaxation

B

Multi-domain Magnetic Particles

A B C

D FEMulti-domain Ferrimagnetic Particles

(A) Single domain magnetic particles such as superparamagnetic nano-particles whichare typically 5 nm to 10 nm in size. (B) Under the magnetic field, the magnetic nanoparticlesphysically rotate to facilitate the Brownian relaxation. (C) In Neel relaxation, the magnetic mo-ment of the nanoparticles is rapidly aligned within the domain under an external magnetic field.(D) Multi-domain magnetic nanoparticles which are usually larger than 100 nm in size. The insetshows how the magnetic moment transforms from one domain to another. (E) Multi-domain fer-rimagnetic particles such as Fe3O4. The red and blue circles represent the tetrahedral (occupiedby Fe3+) and octahedral (occupied by both Fe3+ and Fe2+) sub-lattices in the crystal structure,respectively. (F) Application of a magnetic field aligns the magnetic domains inside the nano-material.

where T∞ is the temperature difference between the vial and the ambient at steady state and τ is

the heating time constant (Fig. S6B). The rate of increase in temperature right after the excitation

can be expressed as: (dT

dt

)t=0

=T∞τ. (6)

The (dT/dt)t=0 for each temperature profile is plotted as a function of the excited magnetic field

and fitted with (H/a)n (Fig. S6C). From the fit, n was found to be 4.63 with a = 22.3. The value

of n, which is greater than 2, suggests that hysteresis loss is the dominant heating mechanism.

Fig. S5. Illustration of the response of various types of magnetic particles under magnetic field.

A B C

Magne�c Par�cles

Silicone Oil

Op�cal Fiber

GaAs Cube

Interrogator

Thermal Insultator

(A) Illustration of the apparatus that was used to characterize the magnetic nano-particles (MNPs). (B) Temperature profiles of the MNPs in silicone oil excited at different magneticfield intensities. (C) Initial rate of temperature increase as a function of excited magnetic fieldintensity. Square dots and the dashed-line represent the measured data and fitted model, respec-tively.

Modeling

Output force of a pressure-driven cylindrical actuator, such as McKibben artificial muscle, is

related to the contraction strain (ε), the differential pressure between the ambient and pressure

inside the confined bladder (P ), the initial bias angle of the braiding (θo), and the initial radius of

the muscle (ro) (40) as the following equation suggests:

F (P, ε) = (πr2o)P [a(1− ε)2 − b], (7)

where a = 3/ tan2(θo) and b = 1/ sin2(θo). This model is developed under the assumption of

full transmission of the pressure inside the bladder to the external braiding without considering

the stiffness of the muscle and geometry variations at both ends of the muscle. At zero strain we

can find the blocking force to be Fblock = (πr2o)P [a − b] and at zero force the maximum strain

εmax is εmax = 1 −√b/a. To account for elasticity of the muscle, the term P can be replaced

by P − Pe where Pe is the pressure needed to deform the bladder elastically. The effect of the

geometry variations at both ends of the muscle can also be included in the model by multiplying

the strain with a correction factor k. In this work, we used the base model to develop our model.

Fig. S6. Characterization of the MNPs based on their thermal response to various magnetic field strengths.

From the braiding geometry, we can find the change in volume within the braided sleeve to be:

V (ε) = Vo

[b(1− ε)− a

3(1− ε)3

], (8)

where Vo is the initial volume of the braided sleeve. From Maxwell’s relations isothermal com-

pressibility (κ) can be derived to be (41):

κ = − 1

V

(∂V

∂P

)T, (9)

which is the fractional change in volume of a system with pressure at constant temperature and

can be expressed in terms of the thermal expansion coefficient (α) and thermal pressure coeffi-

cient (γ) as:

κ =α

γ, (10)

where α is defined as the fractional change in the volume of a system with temperature at constant

pressure and can be written as:

α =1

V

(∂V

∂T

)P, (11)

and γ is defined as the fractional change in the pressure of a system with temperature at constant

volume and can be written as:

γ =(∂P

∂T

)V. (12)

Both α and γ can be determined experimentally. Assuming that κ is independent of P and V at

low temperature and pressure ranges, we can solve Eq. 9 and combine it with Eq. 8 to rewrite

Eq. 7 as:

F (T, ε) = (πr2o)[γ(T − To)−

1

κln(b(1− ε)− a

3(1− ε)3

)][a(1− ε)2 − b

], (13)

where To is the temperature at P = Po and V = Vo.

Energy Analysis

The core working principle of our proposed actuation mechanism is based on liquid-to-gas phase

transition of a fluid via induction heating. A more complete version of Eq. 3 which includes the

phase transition heat as well (assuming not all of the liquid evaporates) is as follows:

Qind = Qw +Qv +Ql (14)

where Qind is the heat generated by induction heating, Qw is the heat required to increase the

temperature of the system to the boiling point of the liquid, Qv is the heat of vaporization, and

Ql is the heat loss. Latex exhibits poor thermal conductivity and we can assume the heat loss to

be negligible for the sake of analysis. Therefore, the heat into the system can be estimated from

the following equation:

Qind = m[C(Tb − To) +Hv

], (15)

where m is the mass of the liquid, C is the heat capacity of the liquid, Tb is the boiling temper-

ature, To is the temperature of the actuator before excitation, and Hv is the heat of vaporization

(Table S1).

Parameter C(J/kg·K) Tb (◦C) Hv(kJ/kg)Water 4200 100 2257

Engineered Fluid 1183 61 112

From equation 15, the input power to the system, and the excitation time we can find the efficiency

to be < 1% which is very similar to other thermal actuator technologies.

Table S1. Heat parameters of the fluids used in this work.