Modeling the solvent-induced shape-memory behavior of glassy polymers

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    its heated rubbery state. Cooling the material below the glasstransition temperature Tg reducepolymer chains, xing the deformthermodynamic state. The temindenitely until the material isamorphous SMPs can experiencxity below Tg when stored forwater or in a humid environmsolvents, such as water, can bmemory eect of amorphous nalso be harnessed to athermallyalternative to temperature-drivenbiomedical applications, such aswhere the controlled deliverychallenge in a surgical environm

    The rst experimental stud

    higher levels of humidity. Lu et al.11 described the solvent-

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    recovery examined the eect ourethane SMP.5,6 The authors ogradually with the absorption ofin the Young's modulus and yiel

    Department of Mechanical Engineering, Jo

    21218, USA. E-mail: vicky.nguyen@jhu.edu

    This journal is The Royal Society ofs the molecular mobility of theed shape in a nonequilibrium

    porary shape should be xedheated above the Tg. However,e a signicant loss in shapea prolonged period of time inent.5 While the absorption ofe detrimental to the shape-etworks, the phenomena canactivate shape recovery. Thisshape recovery is attractive forimplantable medical devices,

    of heat poses an intractableent.ies of solvent-driven shape

    driven shape recovery behavior of a styrene-based SMPimmersed in toluene, while Du and Zhang12 investigated theresponse of a poly(vinyl alcohol) SMP in good versus poorsolvents. Smith et al.13,14 studied the relationship between theglass transition temperature, water absorption, and uniaxialtensile stress response for a family of (meth)acrylate networks.Their experimental results showed an initial rapid decrease inthe elastic modulus upon immersion in water. The toughnessalso initially increased, then abruptly decreased aer a pro-longed immersion time.

    The absorption of solvent precipitates shape recovery inamorphous polymers by depressing the glass transitiontemperature below the storage temperature. This also causesthe material to become more compliant (i.e., more rubbery)compared to the dry polymer at the same temperature. TheModeling the solvof glassy polymers

    Rui Xiao and Thao D. Nguy

    We present a constitutive model fo

    and shape-memory behavior of a

    depresses the glass transition tem

    In a shape-memory application, th

    the time-dependence and tem

    constitutive model for nite elem

    the time-dependent eect of di

    meth(acrylate) copolymer network

    experiments of specimens in air an

    1. Introduction

    Thermally activated shape-memory polymers (SMPs) havegarnered signicant attention in recent years for their potentialbroad range of applications and tailorable properties.1 Severaldierent mechanisms can be employed to achieve the shapememory eect in polymers,24 but the two most common onesare themelt transition of semicrystalline polymers and the glasstransition of amorphous polymers. For amorphous polymers,the temporary shape is obtained by deforming the material in

    Cite this: DOI: 10.1039/c3sm51210j

    Received 30th April 2013Accepted 27th August 2013

    DOI: 10.1039/c3sm51210j

    www.rsc.org/softmatter

    PAPERf water absorption on poly-bserved that the Tg decreasedwater, resulting in a reductiond strength, as well as a loss of

    hns Hopkins University, Baltimore, MD,

    Chemistry 2013nt-induced shape-memory behavior

    n*

    the eect of solvent absorption on the thermomechanical properties

    orphous polymers. The absorption of low concentrations of solvent

    erature by increasing the molecular mobility of the polymer chains.

    s can lead to premature shape recovery and signicant alterations to

    erature-dependence of shape recovery. We implemented the

    nt analysis and developed a computational model that considered

    usion to study the solvent-induced shape-memory behavior of a

    The model was validated by comparing to isothermal shape recovery

    water at dierent temperatures.

    the programmed shape. For large programmed tensile defor-mation, this phenomena can lead to buckling instabilities, asdemonstrated by Zhao et al.7 and Wang et al.8 for PMMA inethanol. Huang9 also demonstrated the potential biomedicalapplications of solvent-driven shape recovery. Chen et al.10

    reported a novel shape memory polyurethane containing pyri-dine moieties with excellent moisture absorption propertiesand measured how the temperature and humidity inuencedthe shape recovery performance. They found that a fasterrecovery rate could be achieved with higher temperature and

    View Article OnlineView Journaleect of low solvent concentration on the glass transition ofamorphous networks has been modeled as an increase in themolecular mobility caused either by an increase in freevolume1517 or congurational entropy of the polymersolventsystem. The pioneering congurational entropy theory ofDimarzio and Gibbs18 used statistical mechanics to derive animplicit expression for the glass transition temperature of a

    Soft Matter

  • relaxation time depends inversely on the congurationalentropy of the polymer network. Cooling the polymer below the

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    View Article OnlineTg progressively reduces the congurational entropy and causesthe relaxation time to increase exponentially preventing thematerial from relaxing to equilibrium.

    In this work, we extend the thermoviscoelastic model of Xiaoet al.22 by modifying the nonlinear AdamGibbs model toincorporate the eect of low solvent concentrations on thetemperature-dependent relaxation behavior of amorphouspolymers. We implemented the model for nite element anal-ysis and developed a computational model to study solvent-driven shape recovery of a (meth)acrylate network. Thecomputational model also described the diusion of solventinto the polymer matrix to more accurately describe the time-dependence of shape recovery. The model parameters wereobtained from dynamic mechanical analysis and diusionexperiments. The model was validated by comparing toisothermal free recovery experiments of specimens in air andwater at dierent temperatures.

    2. Methods2.1 Materials and specimen preparation

    Methyl acrylate (MA), methyl methacrylate (MMA), poly-(ethylene glycol) dimethacrylate (PEGDMA), with typicalmolecular weight Mn 550, and photoinitiator 2,2-dimethoxy-2-phenylacetophenone (DMPA) were purchased from Sigma-Aldrich and used in their as-received condition. The MAMMAPEGDMA solution wasmixed in a 5 : 4 : 1 mass ratio, and DMPAwas added to the comonomer solution at a concentration of 0.5wt% of the total comonomer weight. The polymer solution waseither injected between two glass slides to make tensile testspecimens or a thin-walled tube mold to make hollow cylin-drical specimens as described in Yakacki et al.28 The specimenswere placed in a UV oven (Model CL-1000L Ultraviolet Cross-linker) for 15 minutes to polymerize. The specimens were thenannealed in an incubator at 80 C for 1 hour to achieve fullpolymerization.polymersolvent system as a function of the concentration, andsize and exibility of the solvent molecules. Chow19 extendedthe work of Dimarzio and Gibbs18 by deriving an explicitexpression for Tg that showed good agreement with experi-mental data. Here, we adopt the congurational entropyapproach to develop a constitutive model for the eect of lowsolvent concentration on the thermomechanical properties andshape-memory performance of amorphous polymers. Previ-ously, we developed thermoviscoelastic constitutive models forthe glass transition behavior of amorphous polymers, thatincluded the time-dependent mechanisms of stress relaxation,structural relaxation, and stress-activated yielding and viscousow below Tg.2022 The models were applied successfully todescribe the eects of temperature, mechanical loading,22,23 andphysical aging24 on the shape-memory response. The thermo-viscoelastic models used the nonlinear AdamGibbs model2527

    to describe the temperature-dependence and structure-depen-dence of the relaxation time. In the AdamGibbs model theSoft Matter2.2 Experimental

    In previous works, we performed three sets of experiments tocharacterize the thermomechanical properties of amorphouspolymers in their dry state. These were timetemperaturesuperposition tests to measure the temperature-dependentviscoelastic properties, isothermal volume recovery tests tomeasure the structural relaxation properties, and isothermaluniaxial compression tests to measure the temperature-depen-dent and rate-dependent yield and post-yield behavior. Themethods for these tests were described in detail in Nguyenet al.,20,21 Choi et al.,24 Xiao et al.22 Here we describe additionalexperiments developed to measure the eect of water absorp-tion on the mechanical properties and shape recovery behaviorof the polymer network.

    2.2.1 Diusion tests. Rectangular specimens with dimen-sion 20.0 20.0 0.95 mm3 were used to measure the diu-sion coecient of water into the polymer network. Eachspecimen was weighed before testing. The specimens were thenimmersed in de-ionized water and placed in an incubator ateither 25 C or 30 C. The specimens were removed periodicallyand weighed using a high resolution digital balance with 104

    gram resolution. The specimens were returned immediately tothe water bath and placed into the incubator aer measure-ment. The measurements were performed on 4 specimens foreach temperature.

    2.2.2 Frequency sweep tests. The frequency sweep testswere performed on dry and saturated tension lm specimensusing a TA Q800 Dynamic Mechanical Analyzer (DMA) inmultifrequency mode. Dry polymer lm specimens were heatedfrom 20 C to 80 C in 5 C increments. The specimens wereannealed at each test temperature for 5 minutes to allow forcomplete heat conduction through the sample, then subjected toa 0.2% dynamic strain at 0.3 Hz, 1.0 Hz, 3.0 Hz, 10.0 Hz, and 30.0Hz to measure the storage modulus and tan delta. The samefrequency sweep was also performed on saturated samples atroom temperature tomeasure the eect of solvent absorption onthe storage modulus and tan delta. Saturated specimens wereprepared by immersing the dry tension lm specimens in de-ionized water and placed in an incubator at 25 C for 48 hours.This immersion time was sucient to achieve equilibrium inprior diusion tests described in Section 2.2.1. The saturatedspecimens were not subjected to measurements at highertemperatures to prevent signicant evaporation.

    2.2.3 Isothermal uniaxial tension tests. Isothermal uniaxialtension tests were performed on dry and saturated specimensusing a custom-built micro-tensile setup described in Joshiet al.29 to measure the eect of water absorption on the tensileproperties. The experiments used dog-bone shape specimenswith a 6.0 2.0 1.0 mm3 gage section cut from the lm spec-imen using a digital control mill (Fig. 1). Digital image correla-tion (DIC)wasused tomeasure the local strain in the gage sectionas described in Joshi et al.29 The specimens were pulled to 40%engineering strain at a strain rate of 0.002 per second.

    2.2.4 Isothermal recovery tests. Isothermal shape recoveryexperiments were performed to compare the response of dryspecimens in air and specimens immersed in water. TheThis journal is The Royal Society of Chemistry 2013

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    View Article Onlineexperiments used two dierent specimen geometries: thin lmspecimens 20.0 5.0 0.75 mm3 in size, and tube specimenswith length 15.0 mm, outer radius 10.0 mm, and thickness 0.8mm. The lm specimens were equilibrated at 60 C and thenstretched to 25% engineering strain in 100 seconds using the TAQ800 DMA. The deformed specimens were cooled down to 20 Cat 5 Cmin1 and annealed for ve minutes before unloading tonominally zero force (0.001 N). For recovery in air, the lmspecimens were heated in the DMA, set to zero force mode,to the test temperature at 10 C min1 then held constant for1 hour. The displacements of the grips were measured undernominally zero force (0.001 N) and used to calculate the shapexity ratio dened as,

    Rt Lt L0Lmax L0 ; (1)

    where L0, Lt, and Lmax are the initial length, current length attime t, and the maximum length at the end of the programmedstep. The shape recovery experiments in air were performed atfour test temperatures: 41 C, 44 C, 47 C and 50 C. Forrecovery in water, the specimens were immersed in de-ionizedwater and placed in an incubator at the test temperature. Thetwo test temperatures were 25 C and 30 C. The momentaryshape was recorded using a digital camera and the length Lt wasmeasured using the GNU Image Manipulation Program.

    Similarly, the tube specimens were equilibrated at 60 C in anincubator. The top andbottomof the tubewere pinched togetherusing tweezers to create a temporary shape (Fig. 2). Thedeformedspecimen was removed from the oven to room temperature to xthe programmed shape. To recover the permanent tube shape,the programmed specimen was immersed in de-ionized water ateither 25 C or 30 C. The specimen was imaged using a digitalcameraduring the recovery process to track thedistance betweenthe top and bottom of pinched section (Fig. 2). The shape xityratio was dened as,

    Rt Ht H0Hmin H0 ; (2)

    Fig. 1 Dog-bone shaped tensile test specimen.where H0, Ht, and Hmin are the initial height, current height attime t, and the height at the end of the programmed stepbetween the top and bottom of the pinched section (Fig. 2).

    2.3 Constitutive model

    We present in this section an extension of the constitutivemodel developed by Xiao et al.22 to incorporate the eect of

    http://www.gimp.org/.

    This journal is The Royal Society of Chemistry 2013solvent absorption on the thermomechanical behavior ofamorphous polymers spanning the glass transition. We rstreview the key features of the nonlinear thermoviscoelasticmodel described previously in detail in Nguyen et al.20,21 andXiao et al.,22 then focus on developing a formulation for thedependence of the stress and structural relaxation times on thesolvent concentration.

    To model the swelling, time-dependent thermal, andmechanical deformation, we assume that the deformationgradient, which maps material lines in the reference congu-ration to spatial lines in the deformed conguration, can bedecomposed multiplicatively into isotropic swelling andthermal components, FS JS1/31 and FT JT1/31, and amechanical component, FM. The mechanical component isdecomposed further into N1 parallel elastic and viscouscomponents:

    F (JSJT)1/3FM, FM FekFvk, for k 1.N1. (3)

    We assume molecular incompressibility, such that thediusion of solvent molecules into the polymer network causesthe following volume change,

    Js 1rp

    rsf; (4)

    where rp, rs is the density of the polymer and solvent and f is

    the mass ratio between solvent and polymer; thusrp

    rsf is the

    volumetric ratio between the solvent and polymer.Thermal deformation is assumed to undergo structural

    relaxation with an instantaneous deformation given by theglassy coecient of thermal expansion (CTE), ag, and N2 time-dependent partial deformations dneqk ,

    JTT ; dneq expagT Ti

    exp

    XN2dneqk

    !; (5)

    Fig. 2 The geometry of the tube specimen showing both the undeformed anddeformed shape.where Ti is the initial temperature at time t 0. The totaldeparture from the instantaneous response is given by

    dneq PN2kdneqk . We assume that d

    neq is related to the ctive

    temperature Tf of the nonequilibrium structure as,

    Tf 1Da

    dneq Ti, where Da ar ag is the dierence betweenrubbery and glassy CTE. The ctive temperature was originallyproposed by Tool30 to represent the temperature at which thenonequilibrium structure would be in equilibrium. We assume

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    View Article Onlinethe following rst-order nonlinear kinetic relation originallyproposed by Kovacs et al.31 for the partial deformations,

    _dneq

    k 1

    sRk

    dneqk DakT Ti

    ; (6)

    where sRk is the structural relaxation time dependent on thetemperature, ctive temperature, and solvent concentration.

    The parameter Dak is the partial CTE withXN2k

    Dak Da.

    Polymers exhibit dierent time-dependent response tovolumetric and isochoric (volume preserving) deformation.32

    Consequently, the mechanical deformation gradient and itselastic component are split into volumetric and isochoriccomponents, FM JM1/3FM and Fek Je

    1=3k F

    ek. This allows the

    stress response to be represented by an equilibrium distortionalpart seq represented by the ArrudaBoyce33 model, N1nonequilibrium distortional parts sneqk represented by the Neo-Hookean model, and a time-independent volumetric part p,

    s 1J

    mN

    3

    Tf

    T0

    lL

    leffL 1

    leff

    lL

    bM 1

    3IM11

    |{z}

    seq

    XN1k

    1

    Jmneqk

    b ek

    1

    3I e1k1

    |{z}

    sneq

    12J

    kJM

    2 1|{z}p

    1; (7)

    where bM FMFTM and bek FekFeTk are the le Cauchy-Greendeformation tensor of the mechanical deformation and itselastic component. The latter is an internal variable describingthe departure from the equilibrium mechanical response. Thevariables IM1 I:bM and Ie1k I:bek are the rst invariant of the

    le Cauchy-Green deformation tensors, while leff 13IM1

    rdenotes the eective chain stretch of the ArrudaBoyce33

    network model. The following nonlinear evolution equation34 isadopted for internal viscoelastic variable bek,

    12L vb

    ekb

    e1k

    1

    2ySksneqk ; (8)

    where ySk is the shear viscosity dependent of the temperature,ctive temperature, solvent concentration, and ow stress

    s 12sneq : sneq

    r. The shear relaxation time is dened from the

    shear viscosity as, sSk ySk/mneqk .2.3.1 Relaxation time of a polymersolvent system. Previ-

    ously, we applied the AdamGibbs25 model to describe thetemperature dependence of the structural and stress relaxationtimes. The central hypothesis of the AdamGibbs model is thatnetwork rearrangements in response to decreasing tempera-tures require the cooperative motion of progressively largernumber of molecular groups. By relating the number of coop-eratively rearranging groups to the congurational entropy Sc,Adam and Gibbs25 developed a formulation for the structuralrelaxation time that depends inversely on Sc,

    sR A exp

    B

    TSc

    ; (9)Soft Matterwhere A and B are a scaling parameters. For a system in ther-modynamic equilibrium, we assume that the congurationalentropy of the polymersolvent system depends on thetemperature and number of solvent molecules, Sc(N,T). For adilute system, we further assume that the presence of solventmolecules in low numbers contributes a small increment to thecongurational entropy of the pure polymer system thatdepends only on the number of solvent molecules,

    Sc(N,T) Sc(0,T) + DSc(N). (10)

    This assumption was used by Chow19 to develop a statisticalmechanics model for the eect of dilute solvent concentrationon the glass transition temperature. The model was appliedsuccessfully to predict the decrease in the glass transitiontemperature of polystyrene with absorption of a variety oforganic solvent molecules. We adopt the approach of Chow19

    here to develop an extension of the AdamGibbs model for theeect of the dilute solvent concentration on the relaxationtimes.

    The entropy of a system in equilibrium can be determinedfrom the congurational partition function Q as,

    Sc k ln Q kTvln Q

    vT

    p

    ; (11)

    where k is the Boltzmann constant. Substituting eqn (11) intoeqn (10) gives,

    DSc k lnQN;TQ0;T

    kT v

    vT

    ln

    QN;TQ0;T

    p

    : (12)

    The BragWilliams theory35 is used to evaluate the ratio ofpartition functions in eqn (12), for the possible arrangements ofsmall solvent molecules in the polymer lattice. The partitionfunction Q(N,T) represents N solvent molecules randomlydistributed among N + L lattice sites, where L is the number ofempty lattice sites, while the partition function Q(0,T) repre-sents the state before mixing (N 0). The ratio of partitionfunctions Q(N,T)/Q(0,T) can be written as,19

    QN;TQ0;T

    N L!N!L!

    exp EkT

    : (13)

    where E is the change in interaction energy caused by mixing.Dening the lattice coordination number as z and 3NN, 3LL and3NL as the energies of each NN, LL, and NL pair, E can be esti-mated as follows,

    E z2N L

    N2

    N L2 3NN 2NL

    N L2 3NL L2

    N L2 3LL!

    z2N3NN z

    2L3LL

    zNLN L 3NN 3LL 23NL: (14)

    The second term in the above equation is the interactionenergy aer mixing while the other two terms are the interac-tion energy before mixing.This journal is The Royal Society of Chemistry 2013

  • polymersolvent system to be expressed relative to the drysystem at the reference temperature as follows,

    g ref!2 3

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    View Article OnlineSubstituting eqn (13) and (14) into eqn (12) and applyingStirling's approximation, ln(N!) N ln N N, gives thefollowing

    DSc z[q ln(q) + (1 q)ln(1 q)], (15)

    where z k(N + L) and q NN L is the solvent number fraction.

    Note that while we assumed that the entropy change DS istemperature independent in eqn (10), the result is achieved byusing the BraggWilliams theory. The number of solventmolecules N and the number of the total lattice sites N + L aredened for the polymersolvent system as follows,

    N msNAMs

    ; N L zmpNAMp

    ; (16)

    wherems,mp are the mass andMs,Mp are the molar mass of thesolvent and polymer respectively, and NA is Avogadro's number.This allows z and the number fraction q to be expressed in termsof the following form as,

    z zmpRMp

    ; q MpzMs

    f; (17)

    where R is the gas constant.For a nonequilibrium polymersolvent system, we assume

    that the congurational entropy near equilibrium can be simi-larly decomposed as in eqn (10) into a contribution for thenonequilibrium pure polymer system dependent on the ctivetemperature Tf and an increment dependent on the number ofsolvent molecules,

    Sc(N,Tf) Sc(0,Tf) + DSc(N). (18)

    Furthermore, we assume that DSc has the same dependenceon the solvent concentration as eqn (15). The nonlinear AdamGibbs model is applied for Sc(0, Tf),2527

    Sc0;Tf

    C 1T2

    1Tf

    ; (19)

    where C is a scaling parameter and T2 is the Kauzmanntemperature for an ideal glass.

    Combining eqn (9), (10), (15) and (19), gives the followingexplicit function for the dependence of the structural relaxationtime for each relaxation process on the temperature, nonequi-librium structure, and solvent concentration,

    sRkT ;Tf ; q

    Ak exp

    B=z

    T

    C=z

    1

    T2 1Tf

    q lnq 1 qln1 q

    0BB@

    1CCA

    AkaT ;Tf ; q

    : (20)

    The function a(T,Tf,q) is dened as the shi factor. For thepurpose of parameter identication, we dene the relaxationtime for the dry polymer system (q 0) at an equilibriumreference temperature as, sgRk sRk T refg ; Trefg ; 0, where Trefg isthe glass transition temperature measured for a referencecooling rate. This allows the structural relaxation time of theThis journal is The Royal Society of Chemistry 2013sRkT ;Tf ; 0

    sgRkexp Cg1log eC2 T Tf T Tf Tg

    T

    C

    g2 Tf T refg

    4 5;(22)

    where Cg1 and Cg2 are the WLF constants dened at the reference

    glass transition temperature.We assume that the stress relaxation time has the same

    dependence on temperature, structure, and solvent concentra-tion. To represent the stress-activated viscoplastic owbehavior, we further assume that the stress relaxation time alsodepends on the ow stress according to a modied Eyringmodel.36 In the modied Eyring model, the Arrhenius temper-ature dependence is replaced by the AdamGibbs dependenceon temperature, structure, and solvent concentration ineqn (23),

    sSkT ;Tf ; q; s

    sgSk aT ;Tf ; q

    a

    T refg ;T

    refg ; 0

    QST

    s

    sy

    sinh

    QS

    T

    s

    sy

    1; (23)

    where sgSk is the stress relaxation time at Trefg . The parameterQs is

    a scaling parameter, sy is the yield strength and the ow stress s.Lastly, to incorporate the strain soening phenomena in theglassy state, the phenomenological evolution equation devel-oped by Boyce et al.37 is used for yield strength,

    _sy 1syT

    QSsinh

    QS

    T

    s

    sy

    aT refg ;T refg ; 0aT ;Tf ; q

    1 sysyss

    sy; syt 0 sy0 ;

    (24)

    where sy0 and syss are the initial and steady-state yield strength.

    2.4 Finite element model

    The constitutive model was implemented in an open-sourcenite element program Tahoe (Sandia National Laboratories)and applied to simulate the isothermal recovery behavior ofspecimens in air and water to validate the model. We modeledboth the rectangular and tube specimens described in Section2.2.4. Fig. 3 shows the nite element model of the rectangulartension specimen. Only one eighth of the specimen was simu-lated because of symmetry. The mesh was discretized usingtrilinear hexahedral elements, and a mesh convergence study

    http://sourceforge.net/projects/tahoe/.sRkT ;Tf ; q

    sgRk aT ;Tf ; q

    a

    T refg ;T

    refg ; 0

    ;

    a

    T refg ;T

    refg ; 0

    exp B=z

    T refg C=z

    1

    T2 1T refg

    !0BBBB@

    1CCCCA:

    (21)

    For the dry polymer system, the expression for the relaxationtime can be written as,20Soft Matter

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    View Article Onlinewas performed to determine the mesh density. The niteelement model solved both the Fickian diusion problem forthe time-dependent spatial distribution of the solventconcentration,

    c f1 f (25)

    and the mechanics problem for the deformation and stressresponse of the specimens. An iterative staggered scheme wasapplied to couple the two problems. The diusion problem wassolved to calculate the solvent concentration cn+1 at time n + 1.This was used to solve the mechanics problem for thedisplacement eld, and the process was repeated until thesolution for both converged. The displacement boundaryconditions for the mechanics problem during the programmingstage at T 60 C were set as follows,

    ux(x 0, y, z) 0, uy(x, y 0, z) 0, uz(x, y, z 0)

    Fig. 3 Finite element model of rectangular specimen. 0, uz(x, y, z 10) u(t), (26)

    where the applied displacement u(t) resulted in an engineeringstrain of 25% aer 100 seconds. The remaining boundarieswere le traction free. The temperature was decreased to either25 C or 30 C at a rate of 5 C min1 and held constant for anadditional 5 minutes. The strip was unloaded by applying aboundary condition at the top surface z 10mm that decreasedthe traction to zero at a constant rate over a period of 20seconds. To simulate shape recovery in water, the diusionproblem was solved with the initial condition c(x, y, z, t# tr) 0,where tr is the time at the beginning of the recovery step. Theboundary conditions for the solvent concentration were,

    c(x 0.375, y, z, t $ tr) cN, c(x, y 2.5, z, t $ tr) cN, c(x, y, z 10, t $ tr) cN,(27)

    where cN is the equilibrium concentration at the recoverytemperature. Zero ux was specied for the remainingboundaries.

    Soft MatterFor the tube geometry, we modeled the 2D plane strainproblem rather than the full 3D problem to reduce thecomputational time. Fig. 4 shows the nite element model forthe tube geometry. The mesh was discretized by bilinearquadrilateral elements. A signicantly higher mesh density waschosen where there would be large bending stresses. A meshconvergence study was performed to determine the meshdensity. The bottom point of the tube, x 0, y 0, was xed in xand y to remove rigid body motions. A spherical indenter withradius r 0.5 mm with a modulus 1000 times greater the glassymodulus of the SMPs was used to deform the tube at T 60 Cat a velocity of 0.18 mm s1 for 100 seconds. The temperaturewas decreased from the initial 60 C to the recovery temperatureat a rate of 5 C per minutes, under the constraint of thespherical indenter. The temperature was held constant for5 minutes before removing the constraint. To simulate recoveryin water, the diusion problem was solved with solventconcentration at the inner and outer surfaces set at cN. Theinitial concentration was set to zero to represent the drypolymer.

    3. Parameter determination

    The parameters and relaxation spectra, obtained from ther-momechanical tests previously described in (ref. 2022 and 24)are listed in Table 1. The discrete stress and structural relaxa-tion spectra are given in Tables 2 and 3. The following describesmethods developed to measure the diusion coecient and

    Fig. 4 Finite element model of tube specimen.parameters for the shi factor of the polymersolvent system ineqn (21).

    3.1 Relaxation times

    The density of the polymer and water for this study were 1.12 gcm3 and 1.0 g cm3. For the polymersolvent system, thefollowing additional parameters were determined for therelaxation time: the molar mass for the solvent and polymer Msand Mp as well as the coordination number z in eqn (17), theKauzmann temperature T2, and the scaling parameters B/z andC/z in eqn (20). The MAMMAPEGDMA polymer system wascomposed of two monomers, methyl acrylate and methylmethacrylate, with molar mass 86 g mol1 and 100 g mol1

    respectively. Since the ratio between the two monomers was

    This journal is The Royal Society of Chemistry 2013

  • Physical signicance

    Rubbery coecientGlassy coecient

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    View Article OnlineTable 1 Parameters of the constitutive model for the polymersolvent system

    Parameter Values

    ar (104/C) 6.33

    ag (104/C) 1.8

    65 : 4, the molar mass of the copolymer was approximated asMp 92 g mol1. We neglected the crosslinker contribution,considering the polymer was lightly cross-linked. We assumedz 1 for the coordination number. The remaining threeparameters, B/z, C/z and T2, were obtained from dynamic

    x (10 s) 7.0 Stress relaxation time at Tref (75 C)a 0.66 Stress relaxation spectrum breadthc (s) 80 Structural relaxation time at T Trefgb 0.3 Structural relaxation spectrum breadthT2 (C) 1.7 Kauzmann temperatureB/z 5395 Scaling parameter related with activation energyC/z 1377 Scaling parameter related with congurational heat capacitymN (MPa) 0.96 Shear modulus of equilibrium networklL 4 Limiting chain stretch of equilibrium networkmneq (MPa) 443.3 Non-equilibrium shear modulusk (MPa) 1666.7 Bulk modulusQS/sy0 (

    K MPa1) 110 Activation parameter for viscous owsyss/sy0 0.43 Ratio of steady-state to initial yield strengthsy (MPa) 5000 Characteristic yield timec1N (%) 2.28c2N (%) 2.36D1 (mm

    2 s1) 3.94 106D2 (mm

    2 s1) 5.41 106

    Table 2 Discrete structural relaxation spectrum of dry MAMMAPEGDMAdetermined from the parameters c and b in Table 1 as described in Xiao et al.22

    k sgRk sDakDa

    1 0.0008 0.03052 0.0052 0.02423 0.0335 0.04124 0.2171 0.06855 1.4060 0.10926 9.1036 0.16117 58.944 0.20498 381.65 0.19829 2471.1 0.120210 16 000 0.0418

    Table 3 Discrete stress relaxation spectrum of dry MAMMAPEGDMA deter-mined from the parameters x and a in Table 1 as described in Xiao et al.22

    k sgSk (s)DmkDm

    1 0.0111 0.00352 0.0668 0.00813 0.4009 0.02784 2.4055 0.10165 14.433 0.29126 86.598 0.33927 519.59 0.16298 3117.5 0.04479 18705 0.012810 112 231 0.0055

    This journal is The Royal Society of Chemistry 2013frequency sweep tests described in Section 2.2.2. We rstapplied the frequency tests at multiple temperatures toconstruct the master curve of the storage modulus and deter-mined the WLF constants Cg1 and C

    g2 at T

    refg for the dry polymer

    as described in ref. 21, 22 and 32. The Kauzmann temperature

    and the ratio of the two scaling parametersB=zC=z

    are related to

    the WLF constant Cg1 and Cg2 as,

    B

    CT2 C

    g1C

    g2

    log e; T2 T refg Cg2 : (28)

    To calculate B/z and C/z, we performed a dynamic frequencysweep test at room temperature Tr to measure the storage

    Equilibrium water concentration at 25 CEquilibrium water concentration at 30 CWater diusion coecient at 25 CWater diusion coecient at 30 Cmodulus of the saturated polymer. We then shied the storagemodulus of the saturated material at Tr to the master curve ofdry polymer at a reference temperature Tref 75 C to obtain thefrequency shi D as shown in Fig. 5. From eqn (20), D is relatedto the parameters as follows,

    Fig. 5 Dynamic frequency sweep tests.

    Soft Matter

  • where qN is the equilibrium solvent number ratio, which isrelated with equilibrium mass ratio through eqn (17). Thismethod assumes that the saturated specimen is in structuralequilibrium at Tr. This is reasonable because the storagemodulus measured for the saturated system is within the glasstransition region (Fig. 5). We also observed in experiments thatthe storage modulus did not evolve with aging time for thesaturated sample. Combing eqn (28) and (29) yields theparameters B/z and C/z.

    in de-ionized water at 25 C and 30 C. The solvent weightfraction at any given time t is dened as:

    ct Wt W0WN ; (31)

    where W(0),W(t), andW(N) refer to the initial weight of the dryspecimen, wet specimen at time t, and the saturated specimen.The analytical solution for the weight fraction of the specimenat time t can be obtained as follows assuming small solventconcentration,

    solvent concentration and diusion coecient at 30 C are

    D aTr;Tr; qNaTref ;Tref ; 0

    exp B=zTr

    C=z

    h 1T2

    1Tr

    qNlnqN 1 qNln1 qN

    B=zTref

    C=z

    h 1T2

    1Tref

    0BB@

    1CCA; (29)

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    View Article Online3.2 Solvent diusion

    We applied Fick's second law to model the time-dependentdiusion process of the polymersolvent system. While Fick'slaw is not valid for systems far below Tg, where solvent uptakeoccurs by case II diusion,3,3840 our system is suciently nearTg that Fick's law can be used to accurately describe the time-dependent results of the diusion tests detailed in Section2.2.1. We solved the diusion equation for the solventconcentration in the rectangular specimen using the boundarycondition c cN at the surface and initial condition c 0 at t 0. The results for the solvent concentration can be written as,

    cx; y; z; t cN cNXNl;m;n0

    64 expDal2 bm2 gn2talbmgnL1L2L3

    cos alx cosbmy cosgnz

    al 2l 1 pL1

    ; bm 2m 1p

    L2; gn 2n 1

    p

    L3;

    (30)

    where D is the diusion coecient and cN is the equilibriumsolvent concentration. To experimentally determine the diu-sion coecient and cN for the MAMMAPEGDMAmaterial, wemeasured the solvent weight fraction of four samples immersedFig. 6 The experimental and simulation results for water diusion tests.

    Soft Matterlarger than at 25 C, which was also observed in Chen et al.41 Thediusion parameters are listed in Table 1.

    4. Results and discussion4.1 Isothermal stress response

    Fig. 7 compares the experimental data and simulation for theisothermal, uniaxial tension stressstrain response of dry andsaturated specimens at room temperature. The results showedthe dry specimen exhibited a typical hard glassy response with ayield point, post-yield soening, and viscoplastic ow. Asexpected, the saturated sample showed a compliant rubberyresponse without a denite yield point. The diusion of solventct

    L1=2L1=2

    L2=2L2=2

    L3=2L3=2

    cx; y; z; tdxdydzL1L2L3

    cN cNXNl;m;n0

    83 expDal2 bm2 gn2tal2bm

    2gn2L1

    2L22L3

    2(32)

    We obtained the equilibrium concentration cN and thediusion coecient D by tting eqn (32) to the experimentaldata as shown in Fig. 6. It was found that the equilibrium

    Fig. 7 The stressstrain response for dry and saturated specimens at roomtemperature.

    This journal is The Royal Society of Chemistry 2013

  • molecules into the polymer matrix reduced the glass transitiontemperature from above to below the room temperatureresulting in the dramatic soening of the stress response. Thesimulation results agreed well with the experimental data;though the model slightly underestimated the stress responseof the saturated specimen. This may be caused by the loss ofhydration during the experiment.

    4.2 Isothermal recovery

    Fig. 8 plots the experimental data and model prediction for theshape memory cycle, including the programming andisothermal recovery stage, of dry uniaxial tension specimens.Shape recovery was measured as a decrease in the shape xityratio, dened in eqn (1). The results showed the recovery rateincreased with increased recovery temperatures. At 41 C lessthan 30% deformation was recovered aer one hour, while at50 C full recoverywas achieved aer 15minutes. For thedryMAMMAPEGDMA material, shape-memory programmed speci-mens showed negligible recovery aer months storage at roomtemperature. However, the programmed specimens experienceda signicant loss of shapexity aer it was immersed inwater forseveral hours. Fig. 9 plots the shape xity ratio for the isothermalrecovery of specimens immersed in water at 25 C and 30 C. The

    specimens recovered to their original length aer 8 hours at25 C and 4 hours at 30 C. The model prediction showed goodagreement with experimental results at 30 C but predicted alonger recovery time, 11 hours rather than 8, for 25 C.

    For tube specimens, Fig. 10 compares the experimentallymeasured and predicted specimen shape at dierent timepoints during recovery in water at 30 C. Fig. 11 plots the shapexity ratio dened in eqn (2) for both temperatures. The modelprediction and experimental measurements showed excellentagreement.

    5. Conclusions

    Thermally activated shape-memory polymers have been exten-sively investigated both in experimental and modeling studies.However, the challenge of heating the SMP device to an activa-tion temperature restricts the use of these smart materials inbiomedical applications. The solvent-driven shape-memoryeect is a possible solution to this limitation. The absorption ofsolvent in low concentration lowers the glass transitiontemperature allowing for athermal shape deployment. Thisprocess also changes the mechanical properties by dramaticallyreducing the stiness and strength. For devices designed to bemechanical structures upon deployment, solvent absorptionmay signicantly compromise the long-term load-bearingcapability of the device. In this work, we developed a constitutivemodel to describe the eect of solvent on the thermomechanical

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    View Article OnlineFig. 8 The experimental and simulation results for shape memory programmingand isothermal, unconstrained recovery of tension specimens in air.

    Fig. 9 The experimental and simulation results for isothermal, unconstrainedrecovery of tension specimens in water.This journal is The Royal Society of Chemistry 2013Fig. 11 The experimental and simulation results for isothermal shape recoveryof tube specimens in water.Fig. 10 The momentary shape of tube samples in both experiments and simu-lation in 30 C water bath.Soft Matter

  • properties of amorphous networks. The solvent increases thechain mobility by increasing the congurational entropy. Theresult is adecrease in the relaxation timeandTg.Weextended the

    future, we will investigate polymer systems capable of large

    program at Sandia National Laboratories. Sandia is a multi-

    9 W. Huang, C. Song, Y. Fu, C. Wang, Y. Zhao, H. Purnawali,H. Lu, C. Tang, Z. Ding and J. Zhang, Adv. Drug DeliveryRev., 2013, 65, 515535.

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    View Article Onlineprogram laboratory operated by Sandia Corporation, aLockheedMartin Company, for the United States Department ofEnergy under contract DE-ACO4-94AL85000.

    References

    1 M.Behl, J. Zotzmann andA. Lendlein, Shape-memory polymersand shape-changing polymers, Springer, 2010, pp. 140.

    2 A. Lendlein and S. Kelch, Angew. Chem., Int. Ed., 2002, 41,20342057.

    3 W. Huang, Y. Zhao, C. Wang, Z. Ding, H. Purnawali, C. Tangand J. Zhang, J. Polym. Res., 2012, 19, 134.

    4 C. Wang, W. Huang, Z. Ding, Y. Zhao and H. Purnawali,Compos. Sci. Technol., 2012, 72, 11781182.

    5 B. Yang, W. Huang, C. Li, C. Lee and L. Li, Smart Mater.Struct., 2003, 13, 191.

    6 B. Yang, W. Huang, C. Li and L. Li, Polymer, 2006, 47, 13481356.

    7 Y. Zhao, C. Chun Wang, W. Min Huang and H. Purnawali,Appl. Phys. Lett., 2011, 99, 131911131911.

    8 C. C. Wang, Y. Zhao, H. Purnawali, W. M. Huang and L. Sun,React. Funct. Polym., 2012, 72, 757764.solvent absorption and rapid diusion. This will require thedevelopment of a fully coupled thermodynamic theory for themechanical and diusion processes.

    Acknowledgements

    R. Xiao and T.D. Nguyen gratefully acknowledge the fundingsupport from the National Science Foundation (CMMI-0758390)and the Laboratory Directed Research and DevelopmentAdamGibbsmodel to describe this physical process. Themodelwas implemented for nite element analysis and computationalmodelsweredeveloped to study the isothermal recovery behaviorof specimens in air and water. The models also considered thediusion process to accurately represent the real recoveryprocess. The model was able to predict quantitatively thedramatic soening of the stress response of the saturatedmaterial and the time-dependent solvent-driven shape recovery.The results show that the constitutive model developed here iscapable of describing the solvent-driven shape-memory eect. Inbiomedical applications, the SMP devices may be designed toassume a complex temporary shape and shape recovery path.The present work can assist this design process by predicting theeect of the shape-memory programming process and deploy-ment conditions (e.g. temperature, solvent) on the shaperecovery process. The polymer investigated in this work onlyabsorbs low concentrations of solvent and the solvent-drivenrecovery time of the polymer is on themagnitude of hours. In theSoft Matter10 S. Chen, J. Hu, C. Yuen and L. Chan, Polymer, 2009, 50, 44244428.

    11 H. Lu, Y. Liu, J. Leng and S. Du, Eur. Polym. J., 2010, 46, 19081914.

    12 H. Du and J. Zhang, So Matter, 2010, 6, 33703376.13 K. Smith, S. Parks, M. Hyjek, S. Downey and K. Gall, Polymer,

    2009, 50, 51125123.14 K. Smith, P. Trusty, B. Wan and K. Gall, Acta Biomater., 2011,

    7, 558567.15 F. Kelley and F. Bueche, J. Polym. Sci., 1961, 50, 549556.16 R. Simha and R. Boyer, J. Chem. Phys., 1962, 37, 10031007.17 G.U.Losi andW.G.Knauss,Polym.Eng.Sci., 1992,32, 542557.18 E. Dimarzio and J. Gibbs, J. Polym. Sci., Part A: Gen. Pap.,

    1963, 1, 14171428.19 T. Chow, Macromolecules, 1980, 13, 362364.20 T. D. Nguyen, H. J. Qi, F. Castro and K. N. Long, J. Mech. Phys.

    Solids, 2008, 56, 27922814.21 T. D. Nguyen, C. M. Yakacki, P. D. Brahmbhatt and

    M. L. Chambers, Adv. Mater., 2010, 22, 34113423.22 R. Xiao, J. Choi, N. Lakhera, C. M. Yakacki, C. P. Frick and

    T. D. Nguyen, J. Mech. Phys. Solids, 2013, 61, 16121635.23 X. Chen and T. D. Nguyen, Mech. Mater., 2011, 43, 127138.24 J. Choi, A. Ortega, R. Xiao, C. Yakacki and T. Nguyen,

    Polymer, 2012, 53, 24532464.25 G. Adam and J. H. Gibbs, J. Chem. Phys., 1965, 43, 139146.26 G. W. Scherer, J. Am. Ceram. Soc., 1984, 67, 504511.27 I. M. Hodge, Macromolecules, 1987, 20, 28972908.28 C. M. Yakacki, R. Shandas, C. Lanning, B. Rech, A. Eckstein

    and K. Gall, Biomaterials, 2007, 28, 22552263.29 S. Joshi, C. Eberl, B. Cao, K. Ramesh and K. Hemker, Exp.

    Mech., 2009, 49, 207218.30 A. Q. Tool, J. Am. Ceram. Soc., 1946, 29, 240253.31 A. J. Kovacs, J. J. Aklonis, J. M. Hutchinson and A. R. Ramos,

    J. Polym. Sci., 1979, 17, 10971162.32 J. D. Ferry, Viscoelastic Properties of Polymers, John Wiley and

    Sons, New York, NY, 1980.33 E. M. Arruda and M. C. Boyce, J. Mech. Phys. Solids, 1993, 41,

    389412.34 S. Reese and S. Govindjee, Int. J. Solids Struct., 1998, 35,

    34553482.35 W. Bragg and E. J. Williams, Proc. R. Soc. London, Ser. A, 1934,

    145, 699730.36 H. Eyring, J. Chem. Phys., 1936, 4, 283291.37 M. C. Boyce, G. G. Weber and D. M. Parks, J. Mech. Phys.

    Solids, 1989, 37, 647665.38 K.Chou,S.LeeandC.Han,Polym.Eng.Sci., 2004,40, 10041014.39 N. Thomas and A. Windle, Polymer, 1982, 23, 529542.40 P. Vijalapura and S. Govindjee, J. Polym. Sci., Part B: Polym.

    Phys., 2003, 41, 20912108.41 S. Chen, J. Hu and H. Zhuo, J. Mater. Sci., 2011, 46, 6581

    6588.This journal is The Royal Society of Chemistry 2013

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