1

Supplementary Information for:

Influence of structural relaxation on thermomechanical and shape memory performances

of amorphous polymers

1, 2Ming Lei,

2,3Kai Yu,

1Haibao Lu*,

3H. Jerry Qi*,

1 State Key Laboratory of Science and Technology on Advanced Composites in Special Environments,

Harbin Institute of Technology, Harbin 150080, P.R. China

2 Department of Mechanical Engineering, University of Colorado, Denver, CO 80202, USA

3 The George W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta,

GA 30332, USA

Corresponding authors: HL: luhb@hit.edu.cn; HJQ: qih@me.gatech.edu

Files include:

S1. Experimental results.

S2. 1-D transient heat transfer modification.

S3. Numerical algorithm.

S4. Analysis of Heat Transfer.

S5. Modeling errors of shape memory behaviors.

S6. Modeling shape memory behaviors considering volume relaxation.

S7. Modeling shape memory behaviors based on TTSP in equilibrium state.

S8. Setting- and measuring- temperature deviation.

S9. Predicting the complex shape recovery process.

S10. Demonstrating the two-step shape recovery process.

2

S1. Experimental results.

S1.1 Thermal expansion tests

The coefficient of thermal expansion (CTE) was measured by a DMA testing machine (Model

Q800, TA Instruments, New Castle, DE, USA) in the uniaxial tension mode. The temperature in the

chamber was set to 90 oC for 10 mins to reach thermal equilibrium. To measure the change in the

sample’s length, a small force of 0.001N was applied and maintained during the entire experiment. The

temperature was decreased from 90 oC to 20

oC at a rate of 0.5

oC/min. After holding at 20

oC for 10 mins,

the temperature was then increased to 90 oC at the same rate. This thermal cycle was repeated three times

until the experimental curves became stable; only the data from the last cooling trace were reported in Fig.

S1.

Figure S1. Measured and fitting quasi-equilibrium thermal strain kT from 90

oC to 20

oC,

0 T 90k

L L

.

S1.2. DMA tests at multiple temperature rates

DMA tests at multiple temperature rates were also performed to further examine their effects on

the hysteresis loops. Fig. S2 shows the experimental results. In Fig. S2(a), the SMP sample was first

stabilized at 90 oC for 10 mins, then cooled to -30

oC with two cooling rates (2

oC/min and 1

oC/min)

being alternatively applied to a 20 oC temperature range. After reaching -30

oC, the sample was held for

10 mins and then reheated to 90 oC with the same temperature rates as those in the cooling trace. In a

parallel test as shown in Fig.S2(b), the temperature rates are set to be 5 oC/min and 1

oC/min alternatively.

3

In multiple-temperature-rate DMA tests, since different temperature rates were used in the

cooling and heating steps, the gap between the curves tends to be wide and narrow alternatively, which is

most obvious for a drastic change of temperature rates in Fig. S2(b).

(a)

(b)

Figure S2. Measured storage modulus and tan𝛿 : (a) at the multiple rates of 1 oC/min and 2

oC/min; (b)

at the multiple rates of 1 oC/min and 5

oC/min.

S1.3. DMA tests of different isothermal holding time

As shown in Fig. 1(a), a holding step following the cooling trace will lead to an increase of

storage modulus. In this section, we examine the influence of holding times on the thermomechanical

behaviors of SMPs. In Fig. S2, two samples were first stabilized at 90 oC for 10 mins, and then cooled to

20 oC at 2

oC/min. After reaching 20

oC, they were respectively held for 1min and 30 mins, followed by

reheating to 90 oC at 2

oC/min.

4

Figure S3. Measured storage modulus and tan𝛿 at 2 oC/min holding at 20

oC respectively for 1min and

30 mins.

It is seen that during stabilization at 20 oC, the storage modulus of the SMP sample increases and

tan𝛿 decreases. The change is more obvious for a longer holding time. During reheating, the difference

due to different holding times is obvious at the beginning, but is eliminated above ~35 oC. This

phenomenon indicates that the variation of dynamic modulus during the holding step is a time dependent

process, and its influence will be eliminated in the reheating process, which is similar to the rejuvenation

after physical aging [1, 2]. The experimental results show that a long isothermal holding time generated a

glass with higher storage modulus and lower loss tangent.

S1.4. DMA tests at multiple temperature rates

To study the influence of isothermal holding temperatures, two groups of DMA tests were

conducted where the heating and cooling rates were all set to be 2 oC/min. In method I, as shown in Fig.

S4(a) and (b), five samples were cooled from 90 oC to 40, 20, 0, -20, -50

oC respectively, followed by

holding for 30 mins, and finally reheating to 90 oC. In method II, as shown in Fig. S4(c) and (d), one SMP

sample was held for 10 mins at, respectively, 40, 20, 0, -20 and -50 oC during both heating and cooling

traces. The inset view in Fig. S4(c) plots the temperature evolution of method II.

5

(a)

(b)

(c)

(d)

Figure S4. Measured dynamic modulus at 2 oC/min: (a) storage modulus and (b) tan𝛿 with holding

method I for 30 mins; (c) storage modulus and (d) tan𝛿 with holding method II for 10 mins, and for

comparison the dashed line with holding method I at -50 oC for 30 mins.

In the low temperature range (-50 oC ~ 0

oC), the storage modulus obtained with holding method

II is slightly higher than the reference obtained by holding method I, as shown in Fig. S4(c). This

phenomenon indicates that structural relaxation plays an important role in the hysteresis loops. The

structural relaxation rate is lower with the negative deviations from the glass transition temperature [3].

The isothermal holding at relatively high temperature (40 o

C, 20 o

C, 0 o

C and -20 oC) introduces more

structural relaxation and generates higher storage modulus, which cannot be achieved by holding at low

temperature (-50 oC) for a long time.

6

S1.5. Heating rate dependent shape recovery behaviors

Fig. S5(a) shows the strain versus time in the entire shape memory cycles in three tests when the

cooling rate in the fixing step and the heating rate in the recovery step are set to be, respectively, 1 oC/min,

2 oC/min and 5

oC/min. Fig. S5(b) shows the strain versus temperature in the recovery step. As shown in

Fig. S5, all the SMP samples can eventually recover their original shape at Tr. A higher heating rate will

shift the recovery curve to a higher temperature. Such an observation is consistent with the previous study

by Yu and Qi [4]. The small strain overshooting at the onset of the shape recovery process is due to the

thermal expansion [5].

(a) (b)

Figure S5. Shape memory cycles, respectively, at the ramp rate of 1 o

C/min, 2 oC/min and 5

oC/min:

(a) Strain versus time. (b) Strain versus temperature.

S1.6. Heating rate dependent shape recovery behaviors

Here, multiple heating rates were also applied in the recovery step. In the fixing step, all the SMP

samples were cooled at 2 oC/min. In the recovery step, 2

oC/min was first used to heat samples from 𝑇𝑙 to

50 oC. Then 1

oC/min and 5

oC/min were alternatively used to heat the samples from 50 to 70

oC at an

interval of 5 oC. In another parallel test, 5

oC/min and 1

oC/min were alternatively used. Eventually, after

the temperature reached 70 oC, 2

oC/min was used to heat samples to 𝑇𝑟. Fig. S6(a) shows the strain

evolution in the entire shape memory cycle, and the inset view plots the temperature evolution in the

multiple-rate range. Since the temperature of the sample at 5 o

C/min - 1 o

C/min multiple rate is always

7

higher than that of the sample at 1 oC/min - 5

oC/min multiple rate, its recovery speed is higher. Fig. S6(b)

shows the strain evolution in the recovery step as a function of temperature. Five intersections are

observed, which correspond to the five switches of temperature rates.

(a) (b)

Figure S6. Shape memory cycles in two multiple-temperature rates of 1 o

C/min - 5 o

C/min and 5

oC/min - 1

oC/min: (a) Strain versus time. (b) Strain versus temperature.

S2. 1-D transient heat transfer modification

During the tests in which temperature is changing continuously, the heat transfer introduces

another response hysteresis in addition to the structural relaxation. Inside the DMA machine, the output

temperature is measured from the air, which has a deviation compared with temperature in the neutral

layer of samples.

For the thin film samples with the thickness far smaller than the other two dimensions, it is

enough accurate to treat the heat transfer problem as 1-D along the thickness. The temperature

distribution and boundary conditions are symmetric along the neutral layer of samples. The 1-D transient

heat transfer revision can be simplified as:

2

2

k kT T

t c z

, (S1)

the natural boundary condition I on the surface is:

0

0

( )kInf k z

z

Th T T

z

, (S2)

and the symmetric boundary condition II is:

8

2

0k

lz

T

z

, (S3)

where the z is the coordinate along the thickness, thermal conductivity, c heat capacity, density,

l sample thickness, and h the polymer-air convection heat transfer coefficient.

S3. Numerical algorithm

The multi-branch thermoviscoelastic model is employed to represent different relaxation modes

in the polymer system, as shown in Fig. S7 [6, 7]. Table S1 summarized the newly developed constitutive

model.

Table S1. 1D out-of-equilibrium thermoviscoelastic model.

Strain-stress relation:

Ther

j kdT , ( ,j r g ),

i i i

M e n , 1

neq i

M M

i

,

i i i

e eE , i i i

n n .

TTSP in the out-of-equilibrium state:

i i i

ref , T T( , x) (T ,1 x)

k f

i i i

k fT ,

1

2

(T T )log[ (T, x )]

(T T )

MT T T

M

Cx

C

, sT T ,

for Tk , kT T ,

kTx x ,

for f

i

T ,

i

fT T , 1ifT

x x

. 1 1

ln[ (T, x )] ( )cT T Tref

b g

AFx

k T T , sT T ,

Structural relaxation:

T TT

i

f ki

f i

R

,

9

0 1[1 ]

1

refk g

i i

R T T

e

.

The coupled evolution governing function of the modified TTSP model considering structural

relaxation, Eqs. 6, 8, 9, 11 and 20, can be solved by implicit iterative. And from Eq. 11, the temperature

changing rate 𝑞 can be introduced. Thus in the 𝑝 + 1 step, the fictive temperature of ith branch can be

represented as a weighted average of 𝑇𝑓𝜏𝑖 (𝑝)

and 𝑇𝑘(𝑝+1)

:

( 1) ( 1) ( )

( 1)

( 1)

p i p i p

k R fi p

f i p

R

t T TT

t

, (S4)

where the ∆𝑡 is the time increase in each step, 𝜏𝑅𝜏𝑖 (𝑝+1)

= 𝜏𝑖(𝑝+1)

+ 𝜇(𝑝+1), and 𝜏𝑖(𝑝+1)

is:

( 1) ( 1) ( 1)

T Tk f

i p i p i p

ref . (S5)

Eeq

E1

E2

E3

EN

τ1

τ2

τ3

τN

Thermal Expansion

Component

Equilibrium

Branch

Nonequilibrium

Branches

Figure S7. 1D rheological representation of the multi-branch model.

According to Eq. 9, ( 1)

Tk

p is:

( 1)( 1) 1

T ( 1)

2

(T T )log( )

(T T )k

pp k M

p

k M

Cx

C

,

( 1)p

k sT T , (S6a)

10

( 1)

T ( 1)

1 1ln( ) ( )

k

p c

p ref

b k g

AFx

k T T

,

( 1)p

k sT T . (S6b)

According to Eq. 9, ( 1)

Tf

i p

is:

( 1)

1( 1)

T ( 1)

2

(T T )log( ) (1 )

(T T )f

i p

f Mi p

i p

f M

Cx

C

,

( 1)i p

f sT T

, (S7a)

( 1)

T ( 1)

1 1ln( ) (1 ) ( )

Tf

i p c

i p ref

b f g

AFx

k T

,

( 1)i p

f sT T

. (S7b)

𝑇𝑠 is the intersection point of Williams-Landel-Ferry (WLF) equation and Arrhenius equation [8].

Considering that both 𝜏𝑅𝜏𝑖 (𝑝+1)

and ∆𝑡 are positive, the value range of 𝑇𝑓𝜏𝑖 (𝑝+1)

is 𝑇𝑓𝜏𝑖 (𝑝+1)

∈

[min(𝑇𝑓𝜏𝑖 (𝑝)

, 𝑇𝑘(𝑝+1)

) ,max(𝑇𝑓𝜏𝑖 (𝑝)

, 𝑇𝑘(𝑝+1)

)]. 𝑇𝑓𝜏𝑖 (𝑝+1)

can be solved from governing function Eq. S4. If

𝑇𝑓𝜏𝑖 (𝑝+1)

= 𝑇𝑘(𝑝+1)

, the modified TTSP in the out-of-equilibrium state is returned to the one in equilibrium

state.

For the modeling of DMA tests, storage modulus, loss modulus and loss tangent can be solved by

Eq. 5.

For the simulation of shape memory cycles by the 1-D multi-branch model as shown in Fig. S7,

the governing equation has been derived [6, 7]. The governing functions, respectively, in the strain-

controlled shape fixing process and free shape recovery process are:

strain-controlled shape fixing process:

( 1)

( 1) ( )

( 1)( )

i pi p i p

e ei pt

, (S8a)

stress-free shape recovery process:

( 1) ( 1) ( 1) ( 1)( 1) ( 1) ( ) ( )

1 1

( 1)i p j j p i p j j pN N

i p i p i p j p

n n n neqj jeq

E E

t E t t E t

. (S8b)

11

The total strain ( 1)p is the summation of thermal strain ( 1)

k

p

T and mechanical strain ( 1)p

M :

( 1) ( ) ( 1) ( ) ( 1)(T T )

k

p p Ther p p p

T j k k M , ,j r g . (S9)

Tk(1) Tk(2) Tk(3) Tk(M)Tk(M-1)

e1 e2 e3 eM-1 eM

Symmetry

z=l/2

Convection

z=0

TInf

Figure S8. 1D heat transfer model.

For the heat transfer modification, the thermal distribution satisfies a symmetric condition along

the thickness [9]. For simplicity as shown in Fig. S8, we introduce a heat transfer model with thickness of

𝑙

2 and discretize it into 𝑀 thermal elements with 𝑀 points. 𝑙 is the thickness of the thin film sample. The

thicknesses of the 1st and M

th element are

∆𝑙

2 (∆𝑙 =

𝑙

2(𝑀−1)), and for 2

nd to 𝑀− 1th

element thicknesses are

∆𝑙. Thus, the discretization form of Eq. S1 ~ S3 is:

( ) ( 1) ( 1) ( 1)

( ) ( ) ( 1) ( 1)(2 1)p p p p

k n k n k n k nT f T f T f T

, 2 1n M , (S10a)

Boundary Condition I: ( ) ( 1) ( 1) ( 1)

(1) (1) (2)(2 2 1) 2 2p p p p

k k Inf kT g f T g T f T , (S10b)

Boundary Condition II: ( ) ( 1) ( 1)

( ) ( ) ( 1)(2 1) 2p p p

k M k M k MT f T f T

, (S10c)

where 𝑓 and 𝑔 are heat transfer parameters, 𝑓 =𝜆∙∆𝑡

𝜌∙𝑐∙∆𝑙2 and 𝑔 =

ℎ∙∆𝑡

𝜌∙𝑐∙∆𝑙. By the heat transfer modification,

the temperature in the neutral layer is set as the sample temperature ( 1) ( 1)

( )

p p

k k MT T (or

( 1) ( 1)

2

p plk k z

T T

).

12

Figure S9. Measured dynamic modulus at 0.5 oC/min.

S4. Analysis of Heat Transfer

According to the Eq. S10, we use a theoretical model of heat transfer with 20 elements (𝑀=20),

and the parameters are set as Table S2 [10]. The value rang of polymer-air interfacial thermal

conductivity ℎ usually is 1~100 corresponding to free convection or forced convection. Considering the

working condition of DMA machine, ℎ is set as 10.

Table S2. Material parameters of heat transfer

Material parameters Values

𝜆 (Wm-1

K-1

) coefficient of thermal conductivity 0.15

𝜌 (Kgm-3

) polymer density 1050

ℎ (Wm-2

K-1

) polymer-air convection heat transfer coefficients 10

𝑐 (JK-1

Kg-1

) heat capacity 640

The results of heat transfer in the heating trace are shown in Fig. S10, in which the air

temperature 𝑇𝐼𝑛𝑓 is set as the temperature measured by the DMA testing machine. From Fig. S10(b), it is

clear that the polymer-air interface contributes to the main temperature hysteresis in heat transfer, and the

13

temperature is approximately uniform inside the sample. Thus, it is reasonable to set temperature in the

neutral layer as the sample temperature ( 1) ( 1)

( )

p p

k k MT T in the modeling of dynamic modulus and shape

recovery behaviors.

In Fig. S10, the maximum temperature hysteresis due to heat transfer is only about 2 oC at 5

oC/min. Compared with Fig. 1, the thermal analysis indicates that the temperature hysteresis due to heat

transfer makes a small contribution to the hysteresis phenomenon in DMA tests. Thus, the main reason of

hysteresis in DMA tests could be attributed to the structural relaxation.

(a)

(b)

Figure S10. (a) Temperature variation in the heating trace at 5 oC/min. (b) Temperature distribution at

𝑇𝐼𝑛𝑓 = 50 oC in the heating trace, respectively, at heating rate of 1

oC/min, 2

oC/min and 5

oC/min.

S5. Modeling errors of shape memory behaviors.

Fig. S11 shows the modeling error of shape memory behaviors at 5 oC/min. The light blue area

plots the range of 10 experimental curves, and the blue line plots the most probable experimental curve.

The maximum error between modeling curve and the most probable experimental curve is about 5.3 oC

(~1 min). It can be seen that the model has a good agreement with the experimental results.

14

Figure S11. The modeling error of shape memory behaviors at 5 oC/min.

S6. Modeling shape memory behaviors with thermal expansion

The thermal strain in polymers is also found to be influenced by temperature rates, and this

process is called volume relaxation. Here, in order to discuss the influence of volume relaxation on shape

memory behaviors, we also incorporate the structural relaxation of volume in our model.

Structure relaxation has widely been employed to describe the volume relaxation [11-14]. In

order to describe the volume relaxation, we can introduce the fictive temperature of volume relaxation fvT ,

It has been found that different thermomechanical properties of a given polymer have different structural

relaxation states [15, 16]. For example, Ritland [16] has found that the fictive temperatures of density and

index of refraction for a given glass are different. Thus, without loss of generality we assume that fvT is

different from fT .

Considering the volume relaxation, volume strain 0(T ,T , t)k k induced by temperature jump

from 0Tk to Tk could be divided into two parts, equilibrium volume strain

0(T ,T )eq

k k and out-of-

equilibrium volume strain 0(T ,T , t)neq

k k [14]:

15

0 00

0 0 0

(T , t) V(T , ) (T , ) V(T , ) (T , t) (T , )(T ,T , t)

V(T , ) V(T , ) V(T , )

k k k k k kk k

k k k

V V V V

, (S11a)

00

0

(T , ) V(T , )(T ,T )

V(T , )

eq k kk k

k

V

, (S11b)

0

0

(T , ) V(T , )(T ,T , t)

V(T , )

neq k kk k

k

V t

. (S11c)

fvT corresponds to the fictive structure-equilibrium volume, 0 0(T ,T , t) (T ,T , )neq neq

k k fv k .

Considering the materials is isotropic, the volume strain is three times as linear thermal strain, and we

have 3k

eq eq

T and 3k

neq neq

T :

0(T ,T ,T ) 3 3( )k k k

k eq neq

k fv T T T , (S12a)

0 0(T ,T ) (T T )

k

eq Ther

T k k r k k , (S12b)

(T ,T ) (T T )k

neq

T k fv fv k , (S12c)

in which k

eq

T is the equilibrium linear thermal strain, k

neq

T is the out-of-equilibrium linear thermal strain,

and is Ther Ther

r g .

If the volume relaxation satisfies a logarithmic relaxation process with a relaxation time Rv ,

Tool’s equations can be introduced:

d 1

( )fv

fv k

Rv

TT T

dt . (S13)

Nguyen et al. [14] deduced the expression of neq by substituting Eq. S12 into Eq. S13:

d 1

3neq

neq k

Rv

dT

dt dt

. (S14)

In order to smooth the volume relaxation, Kovacs et al. [12] introduced a discrete spectrum with

33 branchs, each branch satisfied the evolution equation of Eq. S14, and the total thermal strain satisfied

an addictive decomposition scheme:

16

33

1

neq neq

i

i

, (S15a)

d 1

3neq

neqi ki ii

Rv

dTg

dt dt

, (S15b)

0.15

16ig for 1 16i , (S15c)

0.85

17ig for 17 33i , (S15d)

where i

Rv is the structure relaxation time of volume, and weighing factor ig weighs the contribution

from the ith branch. Thus, the 1

st ~ 16

th branches totally weigh the short-time contribution as 0.15, and 17

th

~ 33th branches totally weigh the long-time contribution as 0.85. Kovacs et al. [12] argued that

i

Rv is a

function of kT and neq :

0

0 exp[ (T T )] exp[ ]3

neqi i RvRv Rv k k

x

(S16)

where the temperature parameter defines the temperature dependence, and the weight factor Rvx

defines the contribution from structure to 0

i i

Rv Rv .

In the following, we also use 1-D linear thermal strain relaxation to discuss the influence of

volume relaxation. Considering Eq. S12, Eq. S15 and Eq. S16, we have governing functions Eq. S17. By

choosing 1

109

i

Rvi

Rv

, there are only 6 parameters to determine. Ther

r and Ther

g are shown in Table 2,

and the rest of parameters are shown in Table S3.

33

1

neq neq

i

i

, (S17a)

1

neqneqi ki ii

Rv

d dTg

dt dt

, (S17b)

17

0 exp[ (T T )] exp[ ]

neqi i ref RvRv Rv k v

x

. (S17c)

Table S3. Material parameters of volume relaxation

Material parameters Values

𝜃 (K-1

) temperature parameter 0.25

𝑥𝑅𝑣 (Kgm-3

) weight factor of structural relaxation on volume 0.85

𝜏0𝑅𝑣17 (s) 17

th reference structural relaxation time of volume 5

𝑇𝑣𝑟𝑒𝑓

(oC) reference temperature of volume relaxation 60

More details about the shape memory behaviors considering volume relaxation has been

discussed by Castro et al. [17]. The strain versus temperature during the recovery process considering

volume relaxation is shown in Fig. S12. The volume relaxation is quick above glass transition

temperature, but sluggish under glass transition temperature [14, 17]. Therefore, we can get a better

prediction at the onset of shape recovery by considering the volume relaxation under glass transition

temperature. However, comparing with the mechanical deformation of shape memory polymers, the

thermal strain is very small. Thus, the contribution of volume relaxation to the predictions is small.

(a)

18

(b)

(c)

Figure S12. Shape memory cycles considering volume relaxation: (a) 1 oC/min, (b) 2

oC/min, and (c) 5

oC/min.

S7. Modeling shape memory behaviors based on TTSP in equilibrium state.

In order to evaluate the impact of structural relaxation on the shape memory cycles, we also use

the equilibrium TTSP to model the shape memory cycles. Here we assume that the structure is in

equilibrium state 𝑇𝑘 = 𝑇𝑓𝜏𝑖 in all the temperature changing rates, and also take the heat transfer into

account. Thus Eq. 8 is reduced into WLF law and Arrhenius-type equation, and Narayanaswamy mixing

parameter x , reference nonaffine relaxation time 0 and temperature attenuation factor lose their

19

functions. Detailed procedure can be found in previous literatures [7]. The parameters were still set

according to Table 2 and S1.

(a)

(b)

Figure S13. Modeling shape memory cycles based on the TTSP in equilibrium state, respectively, at

the ramp rate of 1 oC/min, 2

oC/min and 5

oC/min: (a) Strain versus time. (b) Strain versus temperature.

Fig. S13 exhibits modeling results of shape recovery behaviors based on the TTSP in equilibrium

state. Clearly, the modeling curves move to a low temperature (short time) compared with experimental

results. The reason is that this equilibrium model does not consider the structural relaxation of 𝜏𝑖 under

temperature loads.

S8. Setting- and measuring- temperature deviation.

(a)

(b)

Figure S14. Comparison of experimental/modeling setting- and experimental measuring- temperature

20

history in shape memory cycles: (a) Entire temperature history. (b) In the range of multiple

temperature rates.

Fig. S14 exhibits the experimental/modeling setting and experimental measuring temperature

history in shape memory cycles. We set the temperature history exactly the same in experiments and

simulation. However, the temperature history measured by DMA testing machine has a little deviation

compared with the setting. Obviously, the temperature rate switch is a gradual change in experimental

measurement, but a sudden change in experimental/modeling setting.

S9. Predicting the complex shape recovery process.

In this part, we show some theoretical predictions of the complex shape recovery process with two steps.

a. Different temperature rates

Here, three parallel tests at different rates were modeled. In the first shape recovery step, the

samples were first heated from 𝑇𝑙 to 45 oC at 1

oC/min, 2

oC/min, and 5

oC/min, then were held for 5 mins,

at last were cooled to 𝑇𝑙 at the same rate. In the second step, after holding at 𝑇𝑙 for 5 mins, the samples

were heated from 𝑇𝑙 to 𝑇𝑟 at the same rate, and finally were held isothermal for 15 mins. The results are

shown in Fig. S15.

It can be seen that a higher temperature rate will generate less relaxation in the first shape

recovery step, and correspondingly the second temporary shape is closer to the deformed shape.

(a)

21

(b)

(c)

Figure S15. Theoretical predictions of two-step shape recovery process at different temperature rate 1

oC/min, 2

oC/min and 5

oC/min: (a) Temperature history in the entire shape memory cycles. (b) Strain

versus time in the entire shape memory cycles. (c) Strain versus temperature in shape recovery process.

b. Different holding times

Here, three parallel tests holding for different times were modeled. In the first shape recovery step,

the samples were first heated from 𝑇𝑙 to 50 oC at 2

oC/min, then were held respectively for 1 min, 5 mins

and 10 mins, at last were cooled to 𝑇𝑙 at 2 o

C/min. In the second step, after holding at 𝑇𝑙 for 5 mins, the

samples were heated from 𝑇𝑙 to 𝑇𝑟 at 2 oC/min, and finally were held isothermal for 15 mins. The results

are shown in Fig. S16.

22

It can be seen that a shorter holding time will generate less relaxation in the first shape recovery

step, and correspondingly the second temporary shape is closer to the deformed shape.

(a)

(b)

23

(c)

Figure S16. Theoretical predictions of two-step shape recovery process with different holding times, 1

min, 5 mins and 10 mins: (a) Temperature history in the entire shape memory cycles. (b) Strain versus

time in the entire shape memory cycles. (c) Strain versus temperature in shape recovery process.

Comparing with Fig. S15 & S16, the second temporary shape in the two-step free shape recovery

process is determined by temperature rates and holding times. Different combinations of temperature rates

and holding times may generate similar relaxation. For a given partial relaxation, the temperature history

is not unique. Our previous work showed that the complex shape recovery process can be unified by the

reduced time [7]. However, the inverse design of temperature history in the complex free shape recovery

process is still an open question until now.

S10. Demonstrating the two-step shape recovery process.

In this study, most experiments were performed in the DMA chamber. So we could visually

observe the experimental process. To demonstrate the two-step shape recovery process, we designed an

experiment in the water bath.

As shown in Fig. S17, the original strip samples (50mm*2mm*2mm) were bent into 90 o in water

bath at 70 oC, held isothermal for 5 mins, and then cooled down to room temperature. By this shape fixed

process, the first temporary shape was obtained as shown in Fig. S17(b). Then, we used two different

thermal histories to activate the samples: the one-step actuation and the two-step actuation, respectively.

In the one-step actuation, the sample was held in water bath at 65 oC (~Tg). After 40 s, the bent

sample fully recovered its strip shape as shown in Fig. S17(c).

The two-step actuation contains two kinds of thermal histories as discussed in the Section 4.3.3.

Here, we first put the sample into water bath at 45 oC (~Tg-20

oC). After 2 mins, the sample partially

relaxed to the second temporary shape as shown in Fig. S17(d). The second temporary shape is stable. If

now it was cooled down to room temperature, it could bear mechanical loads with the second temporary

24

shape. Or if increasing the temperature of water bath to 65 oC, the sample would fully recover its long

strip shape after 40 s as shown in Fig. S17(d).

Figure S17. Visually demonstrate the two-step shape recovery process. (a) Original samples

(50mm*2mm*2mm). (b) Shape fixed samples. (c) The traditional one step shape recovery process, at

65 oC after 40 s. (d) and (e) are the two-step shape recovery process: (d) the second temporary shape

after holding in 45 oC for 3 mins; (e) then holding in 65

oC for 40 s to get the fully recovered sample.

25

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