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Polymer 53 (2012) 2453e2464

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Polymer

journal homepage: www.elsevier .com/locate/polymer

Effect of physical aging on the shape-memory behavior of amorphous networks

Jinwoo Choi a, Alicia M. Ortega b, Rui Xiao a, Christopher M. Yakacki c, Thao D. Nguyen a,*

aDepartment of Mechanical Engineering, The Johns Hopkins University, Baltimore, MD 21218, USAbDepartment of Mechanical Engineering, The University of Colorado at Boulder, CO 80310, USAcDepartment of Mechanical Engineering, The University of Colorado Denver, Denver, CO 80217, USA

a r t i c l e i n f o

Article history:Received 19 January 2012Received in revised form27 March 2012Accepted 31 March 2012Available online 10 April 2012

Keywords:Shape-memory polymersPhysical agingViscoelasticity

* Corresponding author.E-mail address: vicky.nguyen@jhu.edu (T.D. Nguye

0032-3861/$ e see front matter � 2012 Elsevier Ltd.doi:10.1016/j.polymer.2012.03.066

a b s t r a c t

This paper presents an experimental and modeling study of the effects of physical aging on the shape-memory performance of (meth)acrylate-based networks composed of tert-butyl acrylate (tBA) cross-linked by various concentrations of poly(ethylene glycol dimethacrylate) (PEGDMA). The experimentsmeasured the unconstrained recovery response of samples stored at 20 �C (Tg � 36 �C) for zero to 180days and evaluated the effects of storage on the strain fixity, activation temperature, and initial recoveryrate. A thermoviscoelastic model recently developed for amorphous networks near the Tg was applied tostudy the influence of structural and viscoelastic relaxation and the aging time and temperature on therecovery response. Results showed that the activation temperature and the initial recovery rate increasedwith the aging time, producing a sharper initial recovery response. The thermoviscoelastic model pre-dicted that the magnitude of these effects depended on the aging temperature. There was an optimumaging temperature that maximized the initial recovery rate. These results suggest that physical aging canbe manipulated to accelerate the recovery performance of shape-memory polymer devices.

� 2012 Elsevier Ltd. All rights reserved.

1. Introduction

Shape-memory behavior in polymers offer the capability ofstoring a programmed (deformed) shape indefinitely and recov-ering the original shape in response to a stimulus. Amorphousshape-memory polymers (SMPs) are a subset of the wider class ofthermosets and can be programmed to achieve large, recoverable,shape changes by exploiting the mechanisms of the glass transi-tion. All thermosets exhibit some degree of shape-memorybehavior, but SMPs are distinguished from conventional ther-mosets in their ability to be programmed, stored, and deployedover a useful and often narrow range of temperatures. A typicalshape-memory programming process involves heating thematerial above the glass transition temperature (Tg), deforming toa desired shape, then cooling to below the Tg to fix in thetemporary shape. For thermosets, the permanent shape is definedby the equilibrium configuration of the crosslinked network, andshape storage and recovery occur through dramatic alterations inthe molecular mobility with temperature across the glass tran-sition. Above Tg, polymer chains are highly mobile, and thenetwork responds quickly to an applied deformation by

n).

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rearranging to a new equilibrium configuration. The deformedconfiguration is entropically less favorable and the network willrecover quickly to the undeformed equilibrium configuration ifunloaded above the Tg. The molecular mobility decreases withtemperature and more time is needed for the network to relax toan equilibrium configuration in response to an applied deforma-tion and temperature. Cooling below Tg drives the networkfurther away from equilibrium, which kinetically prevents thechains from large-scale motions and allows the material to storeindefinitely a programmed shape.

While below Tg, the polymer network is in a nonequilibriumstate and continues to relax slowly towards equilibrium. Storing anSMP device for an extended period of time can lead to viscoelasticrecovery and premature loss of the programmed shape; conversely,the network structure can continue to densify, causing the chains tobecome more immobile. This structural relaxation process, alsoknown as physical aging, and its effects on the volumetric responseto temperature changes have been studied extensively for glassforming materials [27]. In isothermal recovery experiments, whichmeasure the time-dependent volume change of a material to a stepchange in temperature, structural relaxation is characterized bya nonlinear, non-exponential time-dependent response and anasymmetric response with respect to heating and cooling [12].Samples equilibrated at a lower initial temperature then rapidlyheated to the test temperature show a delayed but faster relaxation

J. Choi et al. / Polymer 53 (2012) 2453e24642454

response than materials equilibrated at a higher temperature thenrapidly cooled to the same test temperature.

Structural and viscoelastic relaxation during extended aging canhave similar effects on the unconstrained recovery response ofa programmed amorphous shape-memory polymer. These effectsmay be significant for materials designed for biomedical applica-tions because they have an onset glass transition temperature near40 �C for deployment, close to body temperature, but are stored atroom temperature near 20 �C. The proximity of the storagetemperature to the glass transition temperature may allow signif-icant stress and structural relaxation to occur. The former mayproduce an appreciable loss of shape fixity while the latter maydramatically alter the shape recovery response. Recent studies haveexamined the effect of hydrolytic aging on the thermomechanicalproperties and the shape fixity and recovery performance ofamorphous and semicrystalline shape-memory polymers[24,25,29,38]. Few studies have examined the effects of physicalaging on the shape-memory performance. Tobushi et al. [34]measured the unconstrained recovery response of a polyurethaneSMP foam aged for 2e180 days at Tg � 60 �C. They found nosignificant strain recovery during aging at this low temperature forall time periods up to 180 days. In contrast [17], showed that thephysical aging of polyurethane SMP at Tlow ¼ Tg � 30 �C for 72 dayscaused the divitrification point to shift to higher temperatures indifferential scanning calorimetry measurements. This resultsuggests that aging could shift the activation of strain recovery tohigher temperatures.

The primary emphasis of most SMP studies centers on theprogramming and recovery stages of the shape-memory behavior.For example, studies have shown that deformation temperatureduring programming can influence the activation temperatureduring recovery [6,8,37], and that multi-step deformation at twoprogramming temperatures can yield a triple-shape-recoveryresponse [15]; however, the storage stage of the shape-memorypolymers is often overlooked in these studies. It is unclear if therelaxation mechanisms of physical aging during storage wouldsignificantly affect the relationships between programming andrecovery.

The objective of this study was to evaluate the effect of physicalaging on the unconstrained recovery response of two (meth)acry-late-based networks composed of tert-butyl acrylate (tBA) cross-linked by poly(ethylene glycol dimethacrylate) (PEGDMA). Thenetworks were tailored to display equal glass transition tempera-tures (Tg ¼ 56 �C) measured by the peak of the tan delta, whileindependently varying the crosslink density of the networks. Thestress relaxation and structural relaxation behavior of the materialswere characterized using dynamic thermal and frequency scantests and a newly developed isothermal recovery experiment. Theeffect of aging up to 180 days at Tg � 36 �C on the strain fixityperformance and the unconstrained recovery was measured, anda thermoviscoelastic model previously developed for amorphousSMP was applied to investigate the effect of the aging temperatureson the recovery response [22]. The model integrates nonlinearstructural relaxation into a finite-deformation viscoelasticityframework to describe the glass transition behaviors underlyingthe shape-memory response of amorphous networks. Specifically,the nonlinear AdameGibbs model was used to describetemperature-dependence and structure-dependence of the stressand structural relaxation times, and multiple nonlinear discreteprocesses were used to model the broad distributions of stress andstructural relaxation times. A method was developed to determinethe model parameters for the relaxation spectrums from the stressand structural relaxation function measured using dynamicmechanical and isothermal recovery experiments. Nguyen et al.[22] showed that the model successfully predicted the activation

temperature and recovery rate of the tBA-co-PEGDMA networks.Here, we apply the model to investigate the effects of structuralrelaxation and viscoelasticity on physical aging and the recoveryresponse after aging.

2. Method

2.1. Materials and sample preparation

Amixture of 42.9 wt% poly(ethylene glycol) dimethacrylatewithMn ¼ 550 (PEGDMA550) and 57.1 wt% poly(ethylene glycol)dimethacrylate with Mn ¼ 330 (PEGDMA330) was used. Whenpolymerized, the mixture produced an equal Tg compared tohomopolymerized tert-butyl acrylate (tBA). This mixture was usedas the PEGDMA crosslinking agent in this study. Comonomersolutions were prepared by mixing 10 and 40 wt% PEGDMAcrosslinking agent with remainder tBA to create networks of equalTg and varying crosslinking density once polymerized. The photo-initiator, 2,2-dimethoxy-2-phenylacetophenone, was added to thecomonomer solution at a concentration of 1 wt% of the totalcomonomer weight and mixed manually until fully dissolved. Allmaterials used in the comonomer-photoinitiator mixture werepurchased from Aldrich and used as received. Glass slides werecoated with Rain-X (SOPUS Products) to provide a non-reactingrelease agent. The comonomer-photoinitiator mixture was injec-ted between two coated glass slides, separated by 1 mm spacersand secured by binder clips. Polymerization was accomplished byexposing the entire configuration to UV light (Blak-Ray, ModelB100AP) for 10 min (intensity w8 mW/cm2). The degree ofconversion was measured using near-infrared (NIR) spectroscopy(Nicolet 750 Magna FTIR Spectrometer). Samples were scannedpre- and post-polymerization and conversions were based on the]CeH absorption peak located at z6165 nm. All polymer samplesshowed conversion levels of 96e97% [23]. All samples were cutfrom bulk slide samples with a laser cutter and polished with500e600 grit sandpaper prior to testing to remove edge effectscaused by the laser cutting.

2.2. Experimental methods

2.2.1. Thermomechanical characterizationThree sets of experiments were performed to characterize the

thermomechanical properties of the acrylate networks. Thetimeetemperature superposition tests measured the tempera-ture dependence of viscoelastic behavior and the frequencydependence of the stress relaxation function; isothermalrecovery tests measured the structural relaxation function; andconstant cooling-rate tests measured the coefficients of thermalexpansions (CTE) of the rubbery and glassy materials. All exper-iments were performed on thin film specimens under uniaxialtension using a TA Q800 Dynamic Mechanical Analyzer. The filmsmeasured 1.0 mm thick, 5.0 mm wide, and 25.0 mm long. Unlessotherwise specified, the gage lengths between the grips wereapproximately 10 mm.

The CTE tests were performed with the DMA set to zero-forcemode. Samples were equilibrated at 120 �C then cooled to 0 �C at1 �C/min. The tests measured the thermal contraction, defined asthe change in length over the original length, as a function oftemperature. The rubbery linear CTE was calculated from the slopeof a line fitted to the 80e100 �C region of the strain-temperaturecurve then multiplied by 3 to give the rubbery volumetric CTE, ar.Similarly, the glassy linear CTE was calculated from the slope ofa line fitted to the 0�e20 �C region and multiplied by 3 to give theglassy volumetric CTE, ag. The intersection of the two lines definedthe reference glass transition temperature Tref

g . This Trefg

Þ

J. Choi et al. / Polymer 53 (2012) 2453e2464 2455

corresponded to the onset glass transition temperature previouslydetermined from dynamic thermal scan measurements of thestorage modulus [22].

The timeetemperature superposition (TTS) tests were per-formed to measure the temperature dependence of the stressrelaxation time and stress relaxation spectrum. The TTS principledescribes the interrelation between the time and temperaturedependence observed for the mechanical behavior of thermo-rheologically simple materials. It postulates that the frequency andtemperature dependence of the relaxation function can beexpressed as,

Gðlog u; TÞ ¼ Gðlog uþ log aTðTÞ; T0Þ; (1)

where T0 is a reference temperature, and aT (T) is the temperature-dependent shift factor. We used the WilliamseLandeleFerry (WLF)equation to model the temperature dependence of the shift factor[7]; Chapter 11),

log aT ¼ �C01ðT � T0Þ

C02 þ T � T0

; (2)

where C01 and C0

2 are the WLF constants for the reference temper-ature T0.

For the TTS tests, the DMA was programmed to run a dynamicthermal scan in multifrequency mode. The test temperature wasincreased from 0 �C to 120 �C at 5 �C increments. The samples wereannealed for 5 min at each test temperature then dynamicallystrained by 0.2% at 0.32 Hz, 1.0 Hz, 3.2 Hz, 10.0 Hz, and 31.6 Hz. Thefrequency dependence of the storage modulus was measured ateach test temperature, then shifted in logarithmic frequency toa reference temperature T0 ¼ 75 �C according to Eq. (1) to deter-mine the shift factor aT (T) and to form the master curve for thefrequency dependence of the storage modulus at T0. This was laterused to determine the viscoelastic relaxation spectrum. Thetemperature dependence of the shift factor was fitted to the WLFEq. (2) to calculate the WLF constants C0

1 and C02 taking T0 as the

reference temperature. These were transformed to the WLFconstants for Tref

g as the reference temperature using the followingrelations,

Cg1 ¼ C0

1C02

C02 þ Tref

g � T0; Cg

2 ¼ C02 þ Trefg � T0: (3)

The isothermal recovery test was developed to measure thestructural relaxation spectrum. The test measured thetemperature-dependent and time-dependent thermal contrac-tion ε (t, T) of an un-programmed specimen to a rapid temper-ature change DT. From this, we calculated the nonequilibriumvolumetric deformation as dðt; TÞ ¼ 3ðεðt; TÞ � εNÞ. When theexperiments were performed at high temperatures T[Tref

g ,thermal contraction occurred too quickly to measure with suffi-cient time resolution. Near Trefg , thermal contraction occurred tooslowly to measure completely in a single experiment. Previousstudies showed that the structural relaxation functionrðt; TÞ ¼ �dðt; TÞ=ðDaDTÞ, where Da ¼ ar � ag , measured atdifferent temperatures were superposable when plotted asa function of reduced time [2,14]:

rðlog t; TÞ ¼ rðlog uðT ; dðt; TÞÞÞ; u ¼Zt0

dtaðT; dðt; TÞÞ; (4)

The nonequilibrium volume deformation dðt; TÞ represents thenonequilibrium structure, and aðT ; dÞ is the temperature-dependent and structure-dependent shift factor. We used the

nonlinear AdameGibbs model [1,10,21,27] to describe thetemperature-dependence and structure-dependence of the shiftfactor,

aðT; dÞ ¼ exp

"� Cg

1log e

Cg2

�T � Tf ðdÞ

�þ T

�Tf ðdÞ � Tref

g

�T�Cg2 þ Tf ðdÞ � Trefg

�!#

;

Tf ðdÞ ¼ T þ d

Da;ð5

where Cg1 and Cg

2 are the WLF constant determined from the TTStests. The shift factor a (T,d) reduces to theWLF shift factor aT (T) forstructural equilibrium, where d ¼ 0.

For the isothermal recovery tests, the DMAwas set to zero-forcemode to measure the thermal contraction in response to a rapidchange in temperature. The gage length was increased to 21 mm tominimize the effects of heat conduction from the grips. Sampleswere equilibrated at 70 �C for 30 min, cooled to either Trefg þ 8+C orTrefg þ 6+C at 1 �C/min and equilibrated for 60min. The temperaturewas further decreased by DT ¼ 4 �C at 6 �C/min, and the thermalcontractionwas measured for 60 min. This procedure was repeatedfor successive temperature decreases of DT until either Tref

g � 8+Cor Tref

g � 10+C was reached. In the following developments, weassumed the cooling rate of each temperature step was sufficientlyfast such that structural relaxation during the temperature rampcan be ignored and d (t,Tn) represents the nonequilibrium volu-metric deformation resulting from n instantaneous temperaturejumps. The departure from equilibrium for the nth temperaturedecrease was calculated from the thermal expansion as,dðt; TnÞ ¼ 3εðtÞ �Pn

i¼1 arDTi, where ε (t) was the thermalcontraction measured from the beginning of the test at 70 �C. Theresponse function for the reduced time interval corresponding totþn�1 � t � t�n was approximated as,

rðuðtÞ � un�1Þ ¼ 1DaDTn

�dðtÞ � d

�t�n�1;Tn�1

��; tþn�1 � t � t�n ;

(6)

where un�1 corresponds to the time t�n�1 at the end of the previoustemperature step and u0 ¼ 0. This assumed that the non-equilibrated strain at the end of the previous hold period (i.e.,non zero values of dðt�n�1;Tn�1Þ) did not evolve significantly furthertowards equilibrium during the remainder of the test. This wasa reasonable assumption because the relaxation time increasedwith decreasing temperature and d according to Eq. (5). For eachtemperature step, we calculated the reduced time un (Tn,dn) andused it to shift the recovery response to form a master curve for thetime-dependence of the structural relaxation function r (u) at Trefgaccording to Eq. (6). This was later used to determine the structuralrelaxation spectrum.

2.2.2. Aging experimentsSample geometry for unconstrained recovery experiments was

a flat dog-bone with a gauge cross-sectional area of 3 mm � 1 mmand a gauge length of approximately 9 mm. Tests were performedusing a mechanical tester (Model 5567, Instron, Norwood, MA). Athermal chamber (Model 3119-506-A2B3, Instron, Norwood, MA)was used to control the test temperature using liquid nitrogen andheating coils under forced convection. Although humidity was notcontrolled in the environmental chamber, there were no problemswith condensation during cooling. Strain was measured both byphysical measurement of gauge marks placed on the sample (priorto and after elongation, extended storage, and recovery) and viaa video extensometer (Model 2663-821, Instron, Norwood, MA)during elongation and recovery.

J. Choi et al. / Polymer 53 (2012) 2453e24642456

Unconstrained recovery tests consisted of five steps:

1. The sample was equilibrated to a deformation temperatureTd ¼ 56 �C (Tg) and deformed to predetermined strain levels(dependent on the network composition) at an extension rateof 5mm/min. The strain levels were 95% and 32% for the 10wt%and 40 wt% crosslinked networks, respectively. The extensionrate corresponded to strain rates of 0.34 � 0.13%/s and0.37 � 0.11%/s, respectively.

2. The temperature of the sample was decreased from Td to 20 �C(Tg� 36 �C) as quickly as possible, while the extensionwas heldconstant.

3. The sample was unloaded after a 10 min isothermal hold at20 �C to ensure thermal equilibrium.

4. The sample was then removed from the Instron thermalchamber and placed unconstrained in an incubator (Model2005, VWR, Radnor, PA) held at 20 �� 0.1 �C for a pre-determined average storage time of 0, 1, 14, 91 � 1, or 183 � 1days. Samples were sealed in polyethylene Ziploc� bagsthroughout storage. Humidity was not controlled within theincubator.

5. Lastly, the sample was placed back into the Instron thermalchamber to measure free-strain recovery. The sample washeated at a rate of 2 �C/min and the strain was monitoreduntil the recovery strain reached a constant value. Data wascollected from a minimum of two samples to (n � 2), withthe exception of the 40 wt% crosslinking agent compositionat 1 day (n ¼ 1).

2.3. Constitutive model

We presented a detailed development of a three-dimensionalthermoviscoelastic model for amorphous SMPs in [22]. The impor-tant features of the model are summarized here for the case ofuniaxial tension to particularize to the aging experiments. We alsodeveloped methods to determine the parameters of the stress andstructural relaxation spectrum from the master curve of thefrequency-dependence of the storage modulus and time-dependence of the structural relaxation function. These are pre-sented in detail following the model development.

We define a coordinate system e1, e2, and e3 to align with thedimensions of a uniaxial film specimen. For uniaxial tension froman applied stretch l2 along the e2 direction, the Cauchy stress tensorhas only one nonzero component s2, which is the the normal stressin the e2 direction. The deformation gradient has three nonzerocomponents,

½F� ¼24 l1 0 0

0 l2 00 0 l3

35; (7)

where l1 ¼ l3 are stretches along the e1 and e3 directions caused byPoisson’s contraction. The deformation gradient is decomposedmultiplicatively into thermal and mechanical components toincorporate the effects of thermal contraction. For the triaxialdeformation state in Eq. (7), this translates to li ¼ Q1=3

T lMi, fori ¼ 1..0.3, where is the volumetric thermal deformation. To modelstructural relaxation, we assumed that the thermal deformationcan be decomposed additively into an instantaneous deformationd0 ¼ ag (T � Ti) and P time-evolving deformation processes,

QT ¼ 1þ agðT � TiÞ þXPk

dk; (8)

where ag is the glassy coefficient of thermal expansion (CTE) and Tiis the initial temperature at time t ¼ 0. The temperature, T (t), candepend on time to describe a general temperature history. The sumd ¼ PP

kdk is the departure from the instantaneous volume changeratio. Each structural relaxation process evolves to equilibriumaccording to the nonlinear rate equation proposed by the KAHRmodel of [14];

vdkvt

¼ � 1

sgRkaðT ; dÞ

ðdk � DakðT � TiÞÞ; dkð0; TÞ ¼ 0; (9)

where sgRkare the characteristic structural relaxation times at Tref

gand Dak are the partial CTEs. The sum of the partial CTEs give thedifference between the rubbery and glassy CTE,

PPkak ¼ ar � ag .

These parameters characterize the structural relaxation spectrum.The function aðT; dÞ is the AdameGibbs equation for thetemperature-dependent and structure-dependent shift factorexpressed in terms of d,

aðT ;dÞ ¼ exp

"� Cg

1log e

Cg2

�T�Tf

�þT�Tf �Tref

g

�T�Cg2þTf �Tref

g

�!#

;

Tf ¼ Tiþd

ar�ag:

(10)

The variable Tf is the fictive temperature introduced by [35]; andTrefg is the glass transition temperature measured in constantcooling-rate tests at a reference cooling rate. The fictive tempera-ture depends on the temperature through d, and Tf (T) signifies theequilibrium temperature at which a quenched structure at T wasfrozen. At equilibrium, Tf ¼ T and Eq. (10) becomes theWilliameLandeleFerry (WLF) equation for the temperaturedependence of the relaxation time [7]. During cooling, Tf decreasesto Tref

g and Eq. (10) becomes the Arrhenius equation.Stress relaxation is modeled by N parallel discrete relaxation

processes. The deformation of each relaxation process is describedby the multiplicative decomposition of the deformation gradientinto elastic and viscous parts. For the triaxial deformation state inEq. (7), this yields,

lMi ¼ leiklvik ; for k ¼ 1.N; (11)

where i ¼ 1..3 represents the three coordinate directions. These arefurther split into volumetric and deviatoric (distortional) compo-nents to model the different time-dependence of the volumetricand distortional stress response,

QM ¼ lM1lM2lM3; lMi ¼ Q�1=3M lMi;

Qek ¼ le1k

le2kle3k

; leik ¼ Qe�1=3

k leik :(12)

We assumed that the stress response can be split additively intoan equilibrium distortional part seq, for the high-temperaturerubbery response, N time-dependent nonequilibrium distortionalparts sneqk , and a time-independent volumetric part p. We appliedthe ArrudaeBoyce network model [3] with Langevin chain statis-tics to describe the compliant equilibrium response and a Neo-Hookean model to describe the stiff nonequilibrium response.The stress relations for uniaxial tension can be summarized asfollows,

doublehyphen;22:8pcsi ¼ seqi þPNksneqik

þ p;

doublehyphen;22:7pcseqi ¼ 1Jmeq�l2Mi

� x2M

�; sneqik

¼ 1Jmneqk

�le

2

ik � xe2

k

�; p ¼ 1

JkðQM � 1Þ;

meq ¼ mNTT0

lL

xML�1

�xMlL

�; x2M ¼ 1

3

�l2M1 þ l2M2 þ l2M3

�; xe

2

k ¼ 13

�le

2

1kþ le

2

2kþ le

2

3k

�; (13)

J. Choi et al. / Polymer 53 (2012) 2453e2464 2457

where J ¼ QMQT is the total volume change ratio. The parameter kis the bulk modulus, mneqk are the nonequilibrium shear moduli, mNand lL are the chain modulus and limiting stretch of the rubberynetwork, and LðxÞ ¼ cothx� 1=x is the Langevin function. Therubbery shear modulus at Tlow is given by mr ¼ mNlLL

�1ð1=lLÞ andthe glassy modulus equals mg ¼ mr þ

PNi m

neqk . Finally, we applied

a first order rate equation to model the time evolution of theviscous deformation, which for uniaxial tension can be written as,

_lvik

lvik¼ 1

2hgSkaðT ; dÞsneqik

; lvikð0Þ ¼ 1; (14)

where hgSk are the characteristic viscosities of the relaxationprocesses at Trefg . The characteristic stress relaxation times at Trefgare defined as sgSk ¼ hgSk=m

neqk , and these along with mneqk charac-

terize the viscoelastic relaxation spectrum.

2.4. Parameter determination

A semi-analytical procedure developed by [9] was applied todetermine the parameters mneqk and sSk of the viscoelastic relaxationspectrum. First, the thermoviscoelastic model was linearized forthe case of small-strains and structural equilibrium (for hightemperature conditions), then a corresponding fractional-derivative viscoelastic model was developed to fit the TTS data atT0. The storage and loss moduli for the linearized discrete model atT ¼ T0 and fractional model can be written as,

G0discðuÞ ¼ mr þ

PNk

mneqk u2s02

Sk

1þ u2s02

Sk

; G0fracðuÞ ¼ mr þ

Dm�ðuxÞ2aþðuxÞacos

�ap2

��1þ ðuxÞ2aþðuxÞacos

�ap2

� ;

G00discðuÞ ¼ PN

k

mneqk us0Sk1þ u2s02

Sk

; G00fracðuÞ ¼

Dm�ðuxÞasin

�ap2

��1þ ðuxÞ2aþðuxÞacos

�ap2

�;(15)

where Dm ¼ PNk m

neqk ¼ mg � mr and s0Sk ¼ sgSkaT ðT0Þ are the relax-

ation times of the discrete model at T0. The fraction derivativemodel describes a continuous relaxation function using twoparameters, the characteristic relaxation time x and fractional order0 � a � 1 that describes the frequency range of the relaxationfunction. The two parameters were fitted to the master curve of thestorage modulus measured by the TTS test, then sgSk and mneqk ofthe discrete model were determined from x and a by equating thecumulative relaxation functions of the linearized discrete andfractional models. This strategy is significantly more efficient thandirectly fitting the 2N parameters of the discrete model to TTS data.

The viscoelastic relaxation spectra hdiscðnÞ and hfracðnÞ infrequency space n ¼ 1=sS were calculated for the linearizeddiscrete and fractional models from their respective complexmoduli GðiuÞ ¼ G0ðuÞ þ iG00ðuÞ by taking the inverse Stieltjes

transform of GðiuÞ=iu [9]. The cumulative spectrum HðnÞ ¼ R n0 hðnÞ

can be expressed analytically as,

HdiscðnÞ ¼ PNkmneqk hn� ni;

HfracðnÞ ¼ Dmap

�arctan

�ðnxÞaþcosðapÞsinðapÞ

�� p

�12� a

��;

(16)

where the brackets < > represent the step function. To determinemneqk , we first defined a power-law distribution for the relaxationfrequencies nk ¼ 1=s0Sk ,

nk ¼ nmin

�nmax

nmin

�k�1N�1

; (17)

The parameters mneqk were evaluated as,

mneq1 ¼ 12

�Hfracðn1Þ þ Hfracðn2Þ

�;

mneqk ¼ 12

�Hfracðnkþ1Þ � Hfracðnk�1Þ

�; 1 < k < N � 1;

mneqN ¼ dm� PN�1

kmneqk ;

(18)

such that Hdisc form a stair-case approximation of Hfrac. Theparameters of the discretemodel were used to calculate the storagemodulus in Eq. (15) and the results were compared to the experi-

mental data for the master curve to validate the procedure. For alltBA-co-PEGDMA, a good fit to the master curve was obtained forN ¼ 20 processes spanning nmin ¼ 1=s and nmax ¼ 109=s.

We developed a similar strategy to determine the parameters ofthe discrete relaxation spectrum sgRk

and Dak. We first fit thestructural relaxation function r (u) measured in the isothermalrecovery tests to the Kohlrausch-Williams-Watts model relaxationmodel,

rKWW ðuÞ ¼ exp�� ðu=cÞb

�; (19)

The KWW model describes a continuous relaxation functionusing two parameters, a characteristic relaxation time c and0 � b � 1, which corresponds to the time-range of the relaxtionfunction. The relaxation spectrum for the continuous KWW model

Table 1Model parameters for tBA-co-PEGDMA networks with different wt% crosslinkingcontent.

Parameter 10 wt% 40 wt% Physical Significance

Trefg (�C) 36 40 Tg from qcool ¼ 1 �C/min

ar (10�4/�C) 6.9 6.0 Rubbery CTEag (10�4/�C) 2.34 2.7 Glassy CTEc (s) 900 200 KWW structural relaxation

time at Trefgb 0.38 0.35 KWW exponentx (10�6 s) 1.0 3.0 Stress relaxation time of fraction

model at 75�Ca 0.70 0.62 Fractional orderCg1 (�C) 13.76 16.21 First WLF constant

Cg2 (�C) 32.46 49.50 Second WLF constant

mr (MPa) 0.533 3.43 Rubbery shear modulusk (MPa) 1666.67 1666.67 Bulk modulusmg (MPa) 555.56 555.56 Glassy shear modulus

J. Choi et al. / Polymer 53 (2012) 2453e24642458

can be calculated using an inverse Laplace transform of rKWW ðuÞ.An analytic solution can be obtained for b ¼ 0.5, but in general, therelaxation spectrum can be expressed as an infinite series [16],

hKWW ðsÞ ¼ � c

ps2XNk¼0

ð�1Þkk!

sinðpbkÞGðbkþ 1Þ�sc

�bkþ1; (20)

where GðÞ is the Gamma function. To determine Dak, we firstdefined a power-law distribution for the structural relaxation timessk ¼ sgRk

,

sk ¼ smin

�smax

smin

�k�1P�1

; (21)

Then Dak were evaluated from the cumulative relaxation spec-trum HKWW ðsÞ ¼ R s

0 hKWW ðsÞ as,

Da1 ¼ Da2

ðHKWWðs1Þ þ HKWWðs2ÞÞ;

Dak ¼ Da2

ðHKWWðskþ1Þ � HKWWðsk�1ÞÞ; 1 < k < P � 1;

DaP ¼ Da� PP�1

kDak:

(22)

For all tBA-co-PEGDMA, a good fit to the structural relaxationfunctionwas obtained for P ¼ 20 processes spanning smin ¼ 10�5c

and smax ¼ 200c.

100 102 104 106 108

100

101

102

103

Frequency (Hz)

Stor

age

Mod

ulus

(MPa

)

40wt%

10wt%

a

Fig. 1. Experimental data and model fit for (a) the master curve of the frequency responsecurve for the time-dependence of the reduced structural relaxation function for a referenc

3. Results

The results of the thermomechanical characterization experi-ments are summarized in Table 1 for the tBA-co-PEGDMA networkswith 10 wt% and 40 wt% crosslink density. Fig. 1(a) shows thefrequency dependence of the storage modulus at the referencetemperature T0 for the 10 wt% and 40 wt% networks comparing theresults of the TTS test andfits for the fractional derivativemodel in Eq.(15). The fractional derivativemodel providedagoodfit to thedataupto the onset of the glassy plateau at high frequencies, where thefrequency shift looses accuracy and theWLF relation begins to breakdown. Fig. 1(b) plots the reduced structural relaxation functionmeasured in the isothermal recovery tests and the fit for the KWWrelaxation function in Eq. (19). The thermal contraction measured athigher temperatures (larger reduced times) exhibited a noticeableovershoot that caused the reduced relaxation function to appearnoisy. This strain overshoot coincided with a temperature overshootat the end of the temperature ramp during rapid cooling. The resultsshowed that the tBA-co-PEGDMA networks exhibited a broadstructural relaxation spectrum, with b between 0.3 and 0.4, andviscoelastic spectrum, with a between 0.6 and 0.7. Appendix A listsparameters for the discrete structural relaxation spectrum, sgRk

andDak, determined from x and a, and the viscoelastic relaxation spec-trum sgSk andmneqk determined fromc and b for thedifferent networks.

The parameters in Table 1 were applied for finite element simu-lations of the aging experiments and the unconstrained recoveryresponse of the aged samples. The finite element simulationsmodeled a strip under uniaxial tension, and applied nominally thesame temperature and loadinghistory used in the agingexperiments.To model the shape-memory programming, an engineering strain of95%and32%wasapplied to the10wt%and40wt%materials at strain-rates of 0.34%/s and 0.36%/s at a constant temperature T¼ Ti¼ 56 �C.The temperature was decreased to Tg� 36 �C (20 �C) at 7 �C/min and5 �C/min for the two materials while maintaining the applied strain.The stresswas unloadedwithin 2min, and the temperaturewas heldconstant at Tg� 36 �C for 0,1, 14, 90, and 180 days tomodel aging. Tomodel recovery, the temperaturewas increased to 100 �C at 2 �C/min.Results for the unconstrained recovery experiments and simulationswere compared in terms of the initial, mid, and final recovery rates,and the activation temperature, Ta. The three recovery rates werecalculated from the slope of the strain recovery curve in the range of80%e50%, 60%e40%, and 50%e20% of the programmed deformation.The activation temperature was defined as the intersection of linesfitted to the initial recovery region and thermal expansion region(T < 10 �C) during heating.

10 wt%

40 wt%

100 102 1040

0.2

0.4

0.6

0.8

1

Reduced Time (s)

Stru

ctur

al R

elax

atio

n Fu

nctio

n

b

of the storage modulus for a reference temperature of Trefg ¼ 75 �C, and (b) the master

e temperature Trefg measured by the isothermal recovery test.

20 30 40 50 60 700

20

40

60

80

100

Temperature (°C)

Stra

in (%

)

ExperimentModel

a

20 30 40 50 60 700

20

40

60

80

100

Temperature (°C)

Stra

in (%

)

ExperimentModel

b

20 30 40 50 60 700

20

40

60

80

100

Temperature (°C)

Stra

in (%

)

ExperimentModel

c

20 30 40 50 60 700

20

40

60

80

100

Temperature (°C)

Stra

in (%

)ExperimentModel

d

Fig. 2. Strain recovery of 10 wt% tBA-PEGDMA of different aging times, (a) 0 days (b) 14 days (c) 91 days and (d) 180 days, comparing experimental data and model prediction.

J. Choi et al. / Polymer 53 (2012) 2453e2464 2459

Figs. 2 and 3 plot the unconstrained recovery response for the10 wt% and 40 wt% materials for different aging times, comparingexperiments and simulations. The specimens did not exhibitsignificant strain recovery during storage at Tg � 36 �C. The 10 wt%specimens recovered less than 5% of the programmed strain after

20 30 40 50 60 700

5

10

15

20

25

30

35

Temperature (°C)

Stra

in (%

)

ExperimentModel

a b

20 30 40 50 60 700

5

10

15

20

25

30

35

Temperature (°C)

Stra

in (%

)

ExperimentModel

c d

Fig. 3. Strain recovery of 40 wt% tBA-PEGDMA of different aging times, (a) 0 days (b) 14 d

180 days. The 40 wt% showed a larger variation in strain recovery,between 5% and 20% of the programmed strain, but the variationdid not correspond to the storage time. Aging produced a dramaticeffect on the initial strain recovery response [23]. Figs. 4 and 5 plotthe activation temperature, Ta, and initial recovery rate for different

20 30 40 50 60 700

5

10

15

20

25

30

35

Temperature (°C)

Stra

in (%

)

ExperimentModel

20 30 40 50 60 700

5

10

15

20

25

30

35

Temperature (°C)

Stra

in (%

)

ExperimentModel

ays (c) 90 days and (d) 180 days, comparing experimental data and model prediction.

100 101 10240

44

48

52

56

Aging Time (days)

Activ

atio

n Te

mpe

ratu

re(o C

) ExperimentModel

a

101

102

103

Initi

al re

cove

ry ra

te (%

/min

) ExperimentModel

100 101 102

Aging Time (days)

b

Fig. 4. The (a) activation temperature and (b) initial recovery rate for different aging times for the 10 wt% material comparing experimental data and model prediction.

J. Choi et al. / Polymer 53 (2012) 2453e24642460

aging times for the 10 wt% and 40 wt% specimens comparingexperiments and simulations. For the 10 wt% specimens, theexperiments measured a 7 �C increase in the activation tempera-ture and a five fold increase in the initial recovery rate for aging for180 days at Tg� 36 �C. A nearly 3 fold increase in the initial recoveryrate was measured for the 40 wt% specimens. For the 10 wt%specimens, the simulations accurately predicted the initial recoveryrate and activation temperature for aging after 0 days and 14 days,but calculated a significantly higher recovery rate and activationtemperature after 90 and 180 days. The model more accuratelycaptured of the recovery rate for the 40 wt% specimens for all agingtimes, but it consistently produced an activation temperature thatwas 5 �C higher than observed in experiments.

A parametric study evaluated the effect of structural andviscoelastic relaxation on the unconstrained recovery responseafter aging. The characteristic relaxation times x and c were variedby a factor of 10 and a and b were varied by �0.2 from the valuesmeasured for the 10 wt% network. Fig. 6 compare these recoveryresponse after aging for 14 days at Tg � 36 �C for the differentcombinations of structural relaxation and viscoelastic parameters.The results showed that structural relaxation controlled the acti-vation of the recovery response while viscoelasticity stronglyinfluenced the response after activation. A larger characteristicstructural relaxation time, c, caused the material to relax moreslowly towards equilibrium, producing a more mobile structure atthe end of the cooling period. This led to a larger strain recoveryduring storage. The effect also allowed the material to evolvefurther towards equilibrium during storage and resulted in a higheractivation temperature and faster initial recovery rate. A narrowerdistribution of structural relaxation times (larger b) had a similareffect, producing a higher activation temperature and faster initialrecovery rate. Both a larger characteristic stress relaxation time, x,

100 102

40

45

50

Aging Period (days)

ExperimentModel

Activ

atio

n Te

mpe

ratu

re(o C

)

a

Fig. 5. The (a) activation temperature and (b) initial recovery rate for different aging t

and smaller a significantly decreased the recovery rate, but neitherparameters had an effect on the activation temperature. Theseresults showed that structural relaxation solely determine theactivation of strain recovery after aging.

Fig. 7 compares the recovery response obtained from thesimulations for the 10 wt% material after storage for 14 days atdifferent temperatures ranging from Tg e 56 �C to Tg � 26 �C(0 �� 30 �C). In general, strain recovery during storage increasedwith the storage temperature, and the initial recovery rate andactivation temperature increased with the storage time. Storage atthe lowest temperature, Tg � 56 �C, had little effect on the shape-fixity and -recovery response even after 91 days. At this tempera-ture, the molecular mobility was too low for appreciable relaxationto occur. At the highest temperature, Tg � 26 �C, the material failedto adequately vitrify and storage time had little effect on therecovery response. The higher molecular mobility also led tosubstantial viscoelastic strain recovery during storage. For largerstorage times, 14 days and longer, there was a optimal temperature,Tg � 36 �C, at which the initial recovery rate and activationtemperature were maximum. This important result suggests thatphysical aging can be used to improve the recovery performance ofan otherwise sluggish shape-memory material.

4. Discussion and conclusions

This study evaluated the effect of aging on the shape-recoveryresponse of a family of tBA-co-PEGDMA amorphous networks withdifferent crosslink densities. We measured the unconstrainedrecovery response for samples after different aging times rangingfrom 0 to 180 days at Tg � 36 �C then applied a large-deformationthermoviscoelastic model, previously developed for amorphousnetworks, to study how viscoelasticity and structural relaxation

100 102100

101

102

Aging Period (days)

Initi

al re

cove

ry ra

te (%

/min

) ExperimentModel

b

imes for the 40 wt% material comparing experimental data and model prediction.

20 30 40 50 60 70

10

30

50

70

90

Temperature (oC)

Stra

in (%

)

χ=9000 sχ=90 s

a

20 30 40 50 60 70

10

30

50

70

90

Temperature (oC)

Stra

in (%

)

ξ=10-5 sξ=10-7 s

b

20 30 40 50 60 70

10

30

50

70

90

Temperature (oC)

Stra

in (%

)

β=0.58β=0.18

c

20 30 40 50 60 70

10

30

50

70

90

Temperature (oC)

Stra

in (%

)α=0.9α=0.5

d

Fig. 6. Effect of (a) structural relaxation time c, (b) viscoelastic relaxation time x, (c) breadth of structural relaxation function a, and (d) breadth of viscoelastic relaxation functionafter aging 14 days.

J. Choi et al. / Polymer 53 (2012) 2453e2464 2461

influence the shape-memory response after aging. The modelemployed a discrete spectrum of nonlinear structural relaxationprocesses and stress relaxation processes, and used the AdameGibbsmodel to describe the temperature-dependence and structure-dependence of the stress and structural relaxation times. Theparameters for the stress relaxation spectrumwere determined fromstandard dynamic mechanical analysis at small strains and underconditions of structural equilibrium, while parameters for thestructural relaxation spectrum were obtained from a new-ly developed isothermal recovery test that measured the time-dependent thermal contraction response to successive temperaturedecreases near the onset of the glass transition. The model wasapplied to simulate the effect of physical aging on the recovery

0oC

20oC30oC

20 30 40 50 60 70

10

30

50

70

90

Temperature (oC)

Stra

in (%

)

10oC

Fig. 7. Effect of annealing temperature on strain recovery after aging for 14 days.

behavior of the 10 wt% and 40 wt% tBA-co-PEGDMA networksdeformed to 95% and 32% strain. Both experimental and modelingresults showed that aging delayed the activation of shape recoveryandproduced an accelerated recovery responsewith negligible effecton the strain-fixity performance. A parameter study confirmed thesingular role of structural relaxation in determining the activationtemperature and initial recovery rate after aging. Viscoelasticityprimarily affected the final recovery rate and had little influence onthe initial recovery behavior.

The model accurately predicted the recovery rate and activationtemperature after aging for 0 days and 14 days for the 10 wt%experiments and the recovery rate for all aging times for the 40wt%specimens. However, it predicted a significantly faster recovery rateand higher activation temperature after 90 and 180 days thanobserved in experiments for the 10 wt% specimens and it consis-tently produced an activation temperature thatwas 5 �C higher thanobserved in experiments for the 40 wt% specimens. Several factorslikely contributed to the differences between experiments andmodeling. For the 10wt%material, the discrepancies at longer agingtimes suggest that improved characterization of the structuralrelaxation function is needed for longer times. The current methoduses timeetemperature superposition to determine the structuralrelaxation function at longer times by measuring the isothermalrecovery response at higher temperatures. Recovery occurs morerapidly at higher temperatures and significant recovery may haveoccurred during the finite cooling period between the temperaturestep of the recovery test, which were neglected in determining thestructural relaxation function. The large applied tensile strain mayhave induced nonlinear viscoelastic behavior or a nonlinearcouplingbetweendeformation and the rateof physical aging [12,19].Struik [33] and coworkers have shown that large deformations canshift the viscoelastic spectrum to shorter times and dramaticallydecrease the rate of aging, though the details of these nonlinear

Table 3Activation temperature for different aging times and temperatures.

�C Tg � 56 �C Tg � 46 �C Tg � 36 �C Tg � 26 �C

0 day 41.8 41.8 41.9 42.31 day 41.9 41.9 42.4 43.614 days 42.0 43.5 46.5 45.691 days 44.1 49.1 52.0 46.8

J. Choi et al. / Polymer 53 (2012) 2453e24642462

interactions remain unknown. The AdameGibbs and KAHR modelsemployed here provide a good description of structural relaxationnear equilibrium, but are inadequate for describing the phenomenafar from equilibrium. Consequently, a single set of parameters maynot be able to describe a wide range of thermal histories (see forexample [11]. For the 40 wt% specimens, we suspect that batch-to-batch variation in the material properties provided an additionalsource of error in themodel prediction. Different batches ofmaterialswere used for thermomechanical characterization to determine themodel parameters in Table 1 and for measuring the shape recoveryresponseafteraging. In this study, the recovery responseof the40%wtspecimensexhibitedmorevariability than the10wt%specimens [23].For example, the activation temperature after aging 7 daysmeasuredfor 4 different samples varied by 2�C, which is comparable to theobserved differences between experiments and modeling. Wedemonstrated in a previous study [22] that the model accuratelypredicted the strain recovery response of the 40wt%materialwithnoaging when the same batch of materials was used for thermo-mechanical characterization and recovery measurements

Themodeling study also showed that the aging temperature hada dramatic but non-monotonic effect on the recovery response.Therewas an optimal storage temperature below Tg thatmaximizedthe recovery rate. At the optimal temperature, the chain mobilitywas low enough to inhibit viscoelastic strain recovery and preventloss of programmed shape, but still high enough to permit structuralrelaxation to achieve faster recovery rates. Aging for as little as 14days at the optimum temperature nearly tripled the initial recoveryrate of the 10 wt% material without noticeable loss of programmedstrain during aging. This study introduces a new perspective on thedesign and application of SMP devices. SMP materials have beenproposed primarily for use as sensors and actuators, and they havegained increased attention over the past decade for their potentialapplications in biomedical devices. Thematerials have the potentialto promoteminimally invasive surgery and adapt to patient specificcharacteristics [5,30,39]. Nevertheless, SMPs have yet to experiencea major breakthrough into the biomedical industry because of thetremendousdesign challengesassociatedwith exploiting the shape-memory effect in vivo. One of the major challenges is activatingshape recovery within a clinically relevant time frame. For example,an orthopedic suture anchor used in rotator cuff repairs takes onaverage 225e380 s to implant [13], while cardiovascular stents aredeployed near instantaneously via balloon expansion or pseudoe-lasticity [31,32]. SMP medical devices need to be implanted andactivatedwithin these short time frames to gain clinical acceptance.

Shape-memory behavior in polymers is a highly time-dependent and temperature-dependent process that posesunique dilemmas for designing medical devices. Efforts to increaseshape-recovery rates have examined laser heating [4,18,28] andmagnetic heating [20,26,36,40] to rapidly heat a device to andabove body temperature. These methods are limited by themaximum allowable temperature of 60 �C to prevent cell and tissuedamage and they pose a tremendous regulatory approval burden.Other methods have been proposed that take advantage of theshape-memory programming conditions to lower the activationtemperature, which has been correlated to the temperature atdeformation [6,8,37]; however, lowering the activation tempera-ture may cause premature shape-recovery during storage.

Table 2Initial recovery rate for different aging times and temperatures.

% min�1 Tg � 56 �C Tg � 46 �C Tg � 36 �C Tg � 26 �C

0 day 17.89 17.84 17.79 18.031 day 17.88 17.97 19.35 22.2414 days 18.19 24.15 50.51 29.5991 days 27.78 119.47 340.13 30.50

The results of this study suggest anovel, counterintuitivemethodfor achieving increased recovery rates by the simple act of storage atan optimal temperature and time. Figs. 2e5 demonstrate shape-recovery evolving towards a more ideal response as aging occurs,where the activation temperature increases and the onset of shaperecovery becomes more abrupt. The physically-aged responseresults in increased recovery rates and is better suited to resistpremature activation. Application of the thermoviscoelastic modelsuggests there is an optimal aging temperature, Tage, thatmaximizesthe initial recovery rate and activation temperature, while pre-venting significant loss of programmed strain during recovery. Inthis study, Tage ¼ Tg � 36 �C, but the optimal aging temperature islikely dependent on the chemistry and morphology of the polymernetwork, and is a function of the aging time. Higher temperaturesare needed for shorter aging times to achieve the same relaxed state,which is supported by the model predictions in Tables 2 and 3.

In summary, this study measured the effects of aging on theunconstrained recovery response of two tBA-co-PEGDMA amor-phous networks with different crosslink densities for differentaging times, 0e180 days, at Tg � 36 �C. Aging delayed the activationof shape recovery and produced a faster initial recovery responsewith negligible effect on the strain fixity performance. A thermo-viscoelastic model, recently developed for amorphous networksnear the Tg, was applied to study the effect of aging temperatureand time on the activation temperature and initial recovery rate.The aging temperature had a dramatic but non-monotonic effect onthe recovery response. There was an optimal aging temperaturebelow Tg that maximized the initial recovery rate and activationtemperature. At the optimal temperature, the chain mobility waslow enough to inhibit viscoelastic strain recovery and prevent lossof programmed shape, but still high enough to permit structuralrelaxation to achieve faster recovery rates. Overall, this newlyproposed method of increased shape-recovery performance asa function of physical aging could potentially help solve someinherent challenges in the design SMP biomedical devices.

Acknowledgments

This work was funded in part by the National Science Foundation(CMMI-0758390) and the Laboratory Directed Research and Devel-opment program at Sandia National Laboratories. Sandia is a multi-program laboratory operated by Sandia Corporation, a LockheedMartin Company, for the United States Department of Energy undercontract DE-ACO4-94AL85000. A. M. Ortega would like to acknowl-edge the guidance of Professors Ken Gall and Alan Greenberg inexperimental design and analysis and financial support by GrantNumber F31AR053466 from the National Institute of Arthritis andMusculoskeletal and Skin Diseases as well as the National ScienceFoundation Alliances for Graduate Education and the Professoriategrant at the University of Colorado (NSF HRD-0639653).

Appendix A. Relaxation spectra

Table A.4 through A.7 list the parameters of the discrete stressand structural relaxation spectrum for the 10 wt% and 40 wt%specimens. The spectra were calculated semi-analytically asdescribed in Section 2.4.

Table A.4Discrete structural relaxation spectrum for 10 wt% specimens.

k sr (s) Dak=Da� 10�6

1 0.00900 4.532 0.0218 1.893 0.0528 2.644 0.128 3.685 0.310 5.126 0.751 7.117 1.82 9.838 4.41 13.59 10.7 18.510 25.9 25.011 62.6 33.112 152 42.813 368 52.914 891 61.115 2157 63.216 5226 54.717 12661 36.118 30672 15.719 74303 3.7820 180000 0.803

Table A.7Discrete stress relaxation spectrum for 40 wt% specimens.

k ss (s) Dmk=Dm

1 0.000781 0.8742 0.00233 0.8473 0.00692 1.674 0.0206 3.315 0.0613 6.616 0.18 13.37 0.54 27.38 1.62 55.59 4.81 10010 14.3 12711 42.6 10212 127 57.513 378 28.314 1124 13.815 3345 6.8516 9955 3.4317 29628 1.7318 88185 0.87719 262469 0.445

J. Choi et al. / Polymer 53 (2012) 2453e2464 2463

Table A.5Discrete stress relaxation spectrum for 10 wt% specimens.

k ss (s) Dmk=Dm

1 0.00484 0.7062 0.0144 0.8133 0.0429 1.764 0.128 3.845 0.380 8.566 1.13 19.87 3.37 48.68 10.0 1129 29.8 15710 88.8 11511 264 50.312 786 20.513 2341 8.8114 6968 3.9515 20742 1.8116 61735 0.83617 183747 0.38818 546896 0.18119 1627758 0.084120 4844778 0.0734

Table A.6Discrete structural relaxation spectrum for 40 wt% specimens.

k sr (s) Dak=Da� 10�6

1 0.0002 2.202 0.0006 1.103 0.0018 1.634 0.0056 2.405 0.0171 3.526 0.0520 5.147 0.158 7.498 0.485 10.829 1.46 15.4710 4.44 21.7511 13.5 29.7812 41.1 39.0313 125 47.6214 379 51.5615 1154 45.8516 3510 29.7917 10672 12.0218 32450 2.3619 98666 0.1420 300000 0.34

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