Nonlinear Bending Mechanics of HygroscopicLiquid Crystal Polymer Networks

M. R. HaysFlorida Center for Advanced

Aero-Propulsion–FCAAPDepartment of Mechanical Engineering

Florida A&M/Florida State UniversityTallahassee, FL 32310-6046

Email: mrh04@fsu.edu

H. WangFlorida Center for Advanced

Aero-Propulsion–FCAAPDepartment of Mechanical Engineering

Florida A&M/Florida State UniversityTallahassee, FL 32310-6046Email: hongbo@eng.fsu.edu

W. S. Oates∗

Florida Center for Advanced Aero-Propulsion-FCAAPDepartment of Mechanical Engineering

Florida A&M/Florida State UniversityTallahassee, FL 32310-6046

A chemically responsive liquid crystal polymer network isexperimentally characterized and compared to a nonlinearconstitutive model and integrated into a finite element shellmodel. The constitutive model and large deformation shellmodel are used to understand water vapor induced bend-ing. This class of materials is hygroscopic and can exhibitlarge bending as water vapor is absorbed into one side ofthe liquid crystal network (LCN) film. This gives rise to de-flection away from the water vapor source which providesunique sensing and actuation characteristics for chemicaland biomedical applications. The constitutive behavior ismodeled by coupling chemical absorption with nonlinearcontinuum mechanics to predict how water vapor absorptionaffects bending deformation. In order to correlate the modelwith experiments, a micro-Newton measuring device was de-signed and tested to quantify bending forces generated bythe LCN. Forces that range between 1-8 µN were measuredas a function of the distance between the water vapor sourceand the LCN. The experiments and model comparisons pro-vide important insight into linear and nonlinear chemicallyinduced bending for a number of applications such as mi-crofluidic chemical and biological sensors.

1 INTRODUCTIONGlassy and soft elastomer liquid crystal networks

(LCNs) provide a number of fascinating material character-istics for solid state sensing and actuation. The synthesisof these materials was pioneered by Finkelmann and oth-ers [1–3]; however, the fundamental principles governing the

∗Corresponding author.

nonlinear, field-coupled mechanics are still not well under-stood. These active materials are of great interest for inte-gration into micro-electro-mechanical systems (MEMS) dueto their intrinsic field-coupled material behavior for use inbiomedical sensing, microfluidics, and small scale roboticmanipulators [4]. In such areas, polymers have receivedrecognition due to their inherent advantages over conven-tional silicon materials used in MEMS. They are especiallyrelevant in actuator and sensing environments due to theirstructure and color response to external stimuli, ease of im-plementation, and bio-compatibility.

Nematic phase liquid crystals are characterized by rigidrod shape molecules with length scales on the order of 10A.When these materials are synthesized within a polymer net-work, monodomain or polydomain liquid crystal structuresform within the polymer network. Monodomain liquid crys-tal structures depend on a number of processing factors suchas rubbed glass slides that are used to form thin liquid crys-tal network films via capillary action as the material flowsbetween the glass slides. The nematic ordering and mechan-ical characteristics also strongly depend on the liquid crys-tal density, polymer cross linking density, and temperatureat cross linking [5]. The liquid crystals provide new func-tionality within the polymer network which can be consid-ered as a molecular motor that does work on the polymernetwork. Much of the work in this area is focused on identi-fying methodologies to directly stimulate the liquid crystalssuch that an efficient transfer of energy occurs between theliquid crystals and the polymer network to create novel ac-tuators and sensors. Various stimuli have been consideredincluding heat, light, electrostatic fields, and chemicals[3].

The liquid crystal network considered here is a hygro-scopic polymer first developed by Harris et al. [6]. TheLCN initially consists of both covalent and secondary bonds.Humidity-controlled deformation occurs after the networkisconverted to a salt. This provides a hygroscopic materialwhich begins to swell under humid conditions as water infil-trates the material and interacts with the liquid crystals thatare attached to the polymer network. Different deformationmay also occur if the pH or polarity of the solvent in the en-vironment changes [7]. The LCN material deforms due toasymmetric water vapor absorption which can occur on theorder of seconds. The liquid crystal salt units are believedto expand perpendicular to the director that is pre-alignedinthe liquid crystalline phase. It will be shown that this pre-ferred expansion depends on the transverse isotropic elas-tic properties of the film which depend on the alignment ofthe liquid crystals. Order-disorder liquid crystal evolutioncan also play a role in the deformation process which is alsobriefly discussed. It will be shown that if only one side of theLCN is exposed to moisture, asymmetric swelling causes thepolymer to bend away from the water source as illustrated inFigure 1.

In the following sections, the bending force measure-ments of the LCN film is presented and compared with anonlinear continuum model. The constitutive model includesnonlinear mechanics, diffusion of water vapor, and a chem-ical potential associated with interactions between the saltunits and water vapor absorption. The constitutive modelis incorporated into an orthotropic nonlinear shell finite el-ement model and correlated with the experimental results.Discussion and concluding remarks are given in the final sec-tion.

Fig. 1. Illustration of the bending mechanism caused by asymmetric

swelling of the hygroscopic liquid crystal network.

2 EXPERIMENTAL SETUPThe synthesis of the hygroscopic liquid crystal network

is based on the materials and a synthesis procedure describedelsewhere [6]. Capillary forces were used to form a mon-odomain film between rubbed glass slides. The final poly-

mer network film thickness was controlled using spacers be-tween the glass slides which resulted in a film thickness of26 µm. All specimens were stored in a desiccant containeruntil tested. The specimen length and width used in all ex-periments were 8.8×2.6 mm2 unless otherwise noted.

Prior to conducting bending force measurements, thefilm was placed over water and the direction of maximumbending was identified experimentally. The specimens werecut in a direction such that the long axis of the cantilever filmexhibited the largest amount of bending. Based on the watervapor bending characterization presented in [6], the long axisof the film is designated to be perpendicular to the director.The bending forces were tested using the following set-upand procedure for this material and structure configuration.

2.1 BENDING MODE MEASUREMENTSA custom designed mechanical measuring device was

developed to measure micro-Newton bending forces gener-ated by the liquid crystal network. A small diameter stainlesssteel wire (McMaster-Carr P/N: 9882K11, measured diame-ter: 120µm) was placed in contact with the polymer networkand viewed under an optical microscope. The deflection ofthe wire was measured using an Olympus BX60 microscopewith a QImaging Go-3 camera to quantify the force gener-ated based on elastic properties and bending mechanics ofthe wire.

In order to ensure sufficient precision of the measure-ments, the elastic modulus of the stainless steel wire wasmeasured using both tensile test and harmonic resonancetests. The wire was tensile loaded on a MTS 1 kN testingmachine equipped with a 5 N load cell. An average elasticmodulus of 24.28 GPa with a standard deviation of 0.82 GPawas measured using five different specimens. The mechan-ical behavior was found to be linear elastic up to at least3000µε. This modulus measurement was compared to res-onance tests. In these test, the wire was clamped on oneside and the free side was placed under a microscope to op-tically measure small magnitudes of free vibration. Once thewire was deflected and released, a high speed camera (10,000fps) was used to measure the damped resonance frequencyof the wire. Using vibration dynamics and taking into ac-count damping, the elastic modulus was calculated from thedamped natural frequency of the wire. The elastic modu-lus of the stainless steel wire was found to be 24.30 GPawith a standard deviation of 0.29 GPa for 6 different mea-surements on the same wire. The wire used in the reso-nance test was used in all force measurements. Given theelastic modulus and geometry of the wire, the force requiredfor a given displacement was determined using classic bend-ing mechanics by assuming a uniform circular cross-sectiongeometry. Given the maximum force measured in the ex-periments (∼ 8 µN), the force measurement uncertainty wasapproximately±93 nN based on the dynamic measurementstandard deviation.

The magnitude of force generated by the liquid crys-tal network in the presence of water vapor was measured byplacing the actuator in line with the calibrated stainless steel

Fig. 2. The experimental set-up that illustrates the technique used

to indirectly measure forces using the stainless steel wire using an

optical microscopy set-up.

wire as shown in Figure 2. The film was fully constrainedon one end and in contact with the stainless steel wire on theother end. A fully saturated moisture sheet consisting of acotton swab on the end of a rod was attached to a translat-ing stage to control the amount of water vapor exposure asshown in Figure 2. When the stage was moved, the saturatedcotton swab moved in the direction of the liquid crystal net-work. The film would then bend in a direction away fromthe water vapor source as the distance between the water va-por source and film was reduced. The deflection of the wirewas measured optically under the microscope. The distancebetween the water vapor source and the film was quantifiedusing calibrated measurement software with the digital cam-era that was mounted on the microscope. This distance wastaken from a reference position of the film prior to bending.The bending forces are plotted in Figure 3 which show in-creases in forces from 1µN to 8µN as the water vapor sourceapproaches the film. It is noted that this is not the blockingforce, but the force induced against a “spring-like” load cre-ated by the wire in contact with the film. It will be shownthrough model comparisons that this force is within 1% ofthe blocked force.

2.2 LCN Modulus CharacterizationThe elastic modulus of the LCN was also measured and

compared with data in the literature. Due to the delicate na-ture of the films, the modulus was measured using the stain-less steel wire and optical microscopy set-up. The stainlesssteel wire was placed on a translating stage that moved per-pendicular to the longitudinal axis of the specimen. The filmwas again constrained on one end and was placed in contactwith the steel wire on the opposing end as shown in Figure 2.The base of the steel wire was displaced a prescribed amountand magnitudes of deflection of both the liquid crystal net-work and the wire were recorded while in contact. Using

0 0.5 1 1.5 2 2.5 30

1

2

3

4

5

6

7

8

X (mm)

For

ce (µ

N)

X

LCE

wire

H2O

Water vaporsource

Fig. 3. Experimental results of bending forces generated by the

LCN film that is measured with the stainless steel wire.

classical bending, the elastic modulus of the LCN was deter-mined [8]. The elastic modulus of the cantilever was found tobe 122.46 MPa with a standard deviation of 5.71 MPa for fivedifferent specimens. Results presented in [9] have shown thatthe elastic modulus is orthotropic where the modulus perpen-dicular to the liquid crystal director was 1.6 GPa and 2.4 GPaparallel to the director. These values were obtained in a drystate. In a water swollen state, the moduli were 0.16 GPaand 0.3 GPa perpendicular and parallel to the director, re-spectively.

Since the modulus measured in bending was signifi-cantly lower than dynamic mechanical analysis (DMA) datareported in the literature [9], the assumption of classic beambending was assessed by determining the modulus as a func-tion of the point of wire contact along the long axis of theLCN film. A similar analysis is reported in [10] whichshowed significant reductions in modulus measurements inpolymer microcantilevers in comparison to bulk tensile tests.The stainless steel wire was placed in contact with the LCNfilm at different points along the long axis and the stiffnesswas computed again based on classic beam theory. The mod-ulus was found to be approximately constant based on thesemeasurements as shown in Figure 4; however, the nominalmodulus was only 6 MPa. It is important to note that thesemodulus measurements were conducted on a different spec-imen that was synthesized several months after the originalexperiments were conducted. A reduction of the moduluson the order of one magnitude was measured relative to theoriginal measurements conducted during the time of the wa-ter absorption force measurements. Due to the reduction inmodulus, images were taken under a load for the smallest andlargest normalized distances and were compared with classicbeam displacement solutions. The results illustrated goodcomparisons with beam theory for both extreme cases; seethe Appendix for comparisons of data and classic beam dis-placement solutions. Based on the uncertainty in modulus,

0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

2

4

6

8

10

Normalized Distance from Clamp (x/L)

Ela

stic

Mod

ulus

(M

Pa)

Fig. 4. Variations in the elastic modulus based on classic beam the-

ory of a cantilever under a point load. Multiple measurements were

taken at each normalized distance x/L.

the DMA modulus measurement given in [9] was used in thefinite element shell model. The effect of changes in elasticmodulus on changes in concentration and bending deforma-tion will also be assessed using the model.

It was previously noted in these experiments that thefilms were cut in the direction which produced the largestbending along the long axis of the cantilever specimens. Thisis assumed to be the direction perpendicular to the directorwhich would coincide with the measurement of a lower elas-tic modulus relative to the modulus parallel to the director.In the following model comparison, it is assumed that theelastic modulus parallel to the director (perpendicular tothecantilever film’s long axis) is 1.5 times as large as the mea-sured modulus.

3 Nonlinear Mechanics ModelingThe bending behavior of the liquid crystal network is

modeled by coupling diffusion of water vapor with nonlin-ear continuum mechanics. To this end, an energy functionis introduced that includes a chemical potential energy func-tion and a mechanical energy function of the host polymernetwork. This is coupled with mass diffusion to describe therate of water vapor absorption into the material. The chemi-cal potential energy function is used to describe the chemicalaffinity for water molecules to absorb into the film via in-teraction with the salt units. The concentration of the watermolecules is introduced as an internal state variable to de-scribe this interaction. Since changes in bending were mea-sured as a function of position of the water vapor source, dif-fusion of water vapor in the air surrounding the film is alsomodeled using diffusion relations described in [11]. The dif-ferences in water vapor diffusion in air and within the liquidcrystal network are described in the following subsections.

It has been previously reported that the salt units withinthe liquid crystal network prefer to expand perpendicular tothe director during water absorption. This is believed to bedue to the transverse isotropic elastic properties which are

characterized by larger stiffness parallel to the director. Us-ing nonlinear mechanics, it is shown that a scalar order pa-rameter induces hydrostatic stresses. This is expected sincea scalar order parameter has no preferred orientation. Sincethe polymer network is transverse isotropic due to the mon-odomain liquid crystal order, the strain induced by the hydro-static stress will lead to more bending along the compliantdirection (i.e., perpendicular to the director) which is similarto experimental observations.

It is important to note that during water absorption,order-disorder liquid crystal evolution could directly lead toanisotropic deformation. The proposed model focuses onthe lowest-order scalar order parameter contribution to de-formation in terms of concentration. Vector order parametereffects could be introduced into the model to explicitly in-clude the liquid crystal director (ni) using Landau-de Gennesfree energy relations. This typically requires introducingan eight order polynomial inni or fourth order polynomialin Qi j = Q/2(3nin j − δi j ); see [3, 12] for details. Thesehigher order effects require careful treatment of the directororder reduction to ensureQi j remains traceless during order-disorder processes. This will be described in a future analy-sis [13].

3.1 Chemical Potential Coupled Nonlinear MechanicsThe chemical free energy is written as a function of the

water vapor concentration

ψc(cw) =g2(cw− c0)

2+h4(cw− c0)

4 (1)

per current volume wherecw is the concentration of water inmoles per volume in the liquid crystal network andc0 is a ref-erence concentration at room temperature, pressure, and rel-ative humidity. Whereas the number of moles is often usedin formulating a chemical potential [14, 15], the concentra-tion is used as the internal state variable to obtain stress asa function of water vapor. The amount of water concentra-tion will be quantified using relative humidity and ideal gasrelations in Section 3.3. The phenomenological parametersg andh denote the magnitude of the linear and third orderchemical potential driving forces, respectively.

The chemical potential in the spatial configuration is de-fined to be work conjugate to the concentration of water va-por

µ=∂ψc

∂cw(2)

which follows the classic description [15]. More details onthe differences of these field variables in the spatial and refer-ence configuration is described in [16] and in Section 3.2. Itwill be shown that a linear chemical potential is sufficient topredict small bending deformation while third order effectsare important in predicting larger bending deformation.

Nonlinear coupling between the deformation gradientand the water concentration is introduced within this energyfunction using the conservation of mass relation betweenthe reference and spatial configurations,cw = J−1cw, whereJ= det(FiK ) is the Jacobian andFiK is the deformation gradi-ent [17]. The concentration in the reference configuration isdenoted bycw. This requires that the free energy in the refer-ence frame be coupled to the deformation gradient accordingto

ψc =gJ2(J−1cw− c0)

2+hJ4(J−1cw− c0)

4 (3)

per reference volume, noting thatψc = Jψc [18].The stress associated with the change in water vapor

concentration is obtained from the conventional definitionof

nominal stress,siK =∂ψ

∂FiKwhereψ is the total free energy of

the solid material [18]. In this case, the water vapor coupled

stress is defined bysciK =

∂ψc

∂FiK. The functional form of this

stress tensor is

sciK = −

g2

J−1HiK (c2w− J2c2

0)−h4

HiK (3J−3c4w

−8J−2c3wc0+6J−1c2

wc20− Jc4

0).

(4)

The Cauchy stress is often useful in giving concise con-stitutive relations and in illustrating stress symmetry. Thisstress can be obtained from the nominal stress using

σci j = J−1FjK

∂ψc

∂FiK(5)

as described in [17] for the general case. This gives

σci j =−

(g2

(c2

w− c20

)+

h4

(3c4

w−8c3wc0+6c2

wc20− c4

0

))δi j

(6)for the concentration coupled Cauchy stress which illus-trates that changes in water concentration lead to hydrostaticstresses. In the experiments, the amount of bending in thefilm was limited to the small strain case; therefore, the con-centration coupled Cauchy stress will be introduced into thebending moment calculations for nonlinear shell modeling.While the strain is small, large rotation may be observed incertain cases which motivates the need for using nonlinearplates or shells. This is discussed in Section 3.4.

3.2 Elastic Coupling and Diffusion BehaviorThe chemical potential energy given by (3) is combined

with elastic energy to describe the mechanical behavior of

the host polymer network. A linear elastic energy functionis introduced to model the mechanical behavior; however,the theory can accommodate any mechanical energy such ashyperelastic neo-Hookean, Ogden, Mooney-Rivlin, etc.; see[19] for an example on soft liquid crystal elastomers using aneo-Hookean model. The elastic energy used here is

ψM =cIJKL

2EIJEKL (7)

wherecIJKL is the fourth order elastic tensor in the referencedescription andEIJ is the Green strain tensor [17]. This willbe reduced to transverse isotropic elastic properties whenim-plemented in the finite element shell model such that a largerstiffness is implemented in the orientation of the director.This gives a reasonable estimate of these liquid crystal net-works in the glassy state prior to significant water absorption.As water penetrates the material, reductions in modulus maylead to incompressible elastomer behavior. The uncertaintyin gradients of modulus and changes in the Poisson ratio to-wards an incompressible state (i.e.,ν ⇒ 0.5) will be assessedusing the model.

The total free energy is defined by the sum of the chem-ical potential and mechanical free energy as denoted byψ = ψc + ψM. The change in this free energy with respectto deformation and concentration provides stress and chemi-cal driving forces; respectfully. When combined with kineticenergy, the variation in deformation leads to the linear mo-mentum equations which will be implemented in terms ofa finite element shell model in the following section. Thechemical potential driving force is introduced into the con-servation of mass which results in a diffusion equation. Thisis used to model water vapor concentration gradients that oc-cur through the thickness of the LCN film. The concentrationgradient is introduced into the coupled Cauchy stress givenby (6) to calculate the induced bending moment from wa-ter vapor absorption. This moment will be applied to thenonlinear shell model to simulate bending from water vaporconcentration gradients. In addition, diffusion of water vaporthrough air will be analyzed to quantify changes in bendingforces as the water vapor source moves closer to the liquidcrystal network.

The evolution of water vapor concentration is governedby

˙cw =−JI ,I in Ω0 (8)

whereΩ0 is the reference volume andJI is the mass fluxin the reference configuration. This flux can be related tothe flux in the spatial configuration using the transforma-tion, Ji = J−1FiK JK [16]. It is also important to point outthat the diffusion in the spatial frame is proportional to therelative velocity between the solid and the diffusing chemi-cal [11, 16]. The mass flux is assumed to be proportional tothe gradient of the chemical potential

JI =−AIJµ,J (9)

where the chemical potential in the reference configuration

is µ=∂ψ∂cw

. The mobility tensor is denoted byAIJ [20].

A diffusion equation is obtained by substituting (9) into(8). In the numerical model, the mobility tensor is assumedto be isotropic and independent of concentrationAIJ = DδIJ .A correlation between water vapor in the air and inside thematerial is determined based on the experiments. This re-quires determining steady state diffusion within the LCNfilm to quantify bending. It also requires determining steady-state diffusion in air between the water source and the filmsurface. Since steady-state diffusion is of primary interestfor experimental comparisons, the value ofD is arbitrary inthe model.

Using the above diffusion relations, the final form of thetime-dependent conservation of mass in the reference config-uration is

˙cw = Dµ,II . (10)

This relation is used to predict: i) the change in water con-centration on the surface of the film with respect to the loca-tion of the water vapor source in the air and ii) the diffusionof water into the LCN film through the material thickness.In regards to (i), this is necessary to relate changes in bend-ing as the water source approaches the film surface based onbending force changes illustrated in Figure 3. This will alsobe important in modeling the case of large film rotation sincethe concentration may vary along the length of the LCN film.

The diffusion equation in (10) is first used to correlatesteady state concentration of water vapor in air from a ref-erence distance between the water source and the LCN filmsurface. These values are then used to correlate the amountof water present on the film’s surface and the penetration intothe material thickness as a function of the distance from thewater vapor source. Steady state values are used to comparewith static deformation measurements given in the previoussection.

3.3 Water Vapor ConcentrationAs outlined in [11], the flux in air is attributed to a com-

bination of diffusion and flux from the total molar bulk flow.A one dimensional solution is used since the water vaporsource was a flat, damp cotton swab that was larger than thefilm. In this case, a close form solution for the water vaporconcentration is given by

(1− cw

c

1− cmaxc

)=

(1− co

c

1− cmaxc

) XX0

(11)

wherec is the total concentration of the air and water vapor,cmax is the maximum water vapor concentration at the surfaceof the moisture source, andc0 is the water vapor concentra-tion at ambient conditions. The reference distance betweenthe film surface and the moisture source is defined byX whileX0 is the distance to ambient conditions.

For the relative humidity conditions in the experiments,this relation results in a linear decay of water vapor concen-tration from the reference distanceX0 where the humidityinduced bending is zero. This reference position is restrictedto X > 3 mm (see Figure 3) and defined to beX =X0 = 4 mmin the model since the bending force does not go completelyto zero atX = 3 mm. The boundary conditions for the diffu-sion problem include a fixed concentration of water vapor atX = 0 and the concentration of water vapor in the air is set toambient conditions atX = X0 = 4 mm. The size of the wa-ter vapor source is large relative to the size of the LCN filmsuch that variations along the length and width of the filmare negligible for small film rotations. For large rotations,the concentration variation along the length may contributeto changes in bending. These changes will be compared tothe ideal model that assumes constant concentration alongthe film length.

Whereas the concentration of water vapor in air atsteady-state that is based on (11) is nonlinear in general, thisrelation simplifies to a linear decay over the experimentalconditions tested. Therefore, the concentration is written as

cw(X) = a(X0−X)+ c0 for 0≤ X ≤ X0 (12)

where the parametera is obtained from (11) and depends onthe ambient and saturated air conditions at room tempera-ture, atmospheric pressure, and relative humidity. The am-bient concentrationc0 is obtained at room temperature andpressure for the nominal relative humidity measured duringthe experiments which was measured to be 35%.

The water concentration in air is obtained using the fol-lowing method. Based on the ideal gas law and a volume ofgas above the LCN film, the ambient concentration for a 35%relative humidity isc0 = 3.97×10−7 mol/m3. The volumeof gas used in this calculation is based on dimensions corre-sponding to the film geometry and maximum distance fromthe water vapor source (8.8× 2.6× 3) mm3 where the lastdimension corresponds to the distance from the film surfaceto the water vapor source:X = 3 mm. It is assumed that thevolume of gas near the film surface becomes fully saturatedwhenX = 0.25 mm. Since the water concentration scaleslinearly with the relative humidity, the maximum concentra-tion is cmax= 1.13×10−6 mol/m3 for a relative humidity of100%; see the Appendix for details on these calculations.

Inside the LCN film, the water vapor concentration de-creases through the thickness toc0 on the back side of thefilm. In the case whereg 6= 0 andh = 0 in (1), the steady-state solution is trivially satisfied by a linear decay. It willbe shown that including the fourth order term in (1) is neces-sary to fit the bending force data to experimental results for

larger bending forces. This results in a nonlinear diffusionequation that is solved numerically using the pdepe solver inMatlab. The steady-state solution is then fit to a polynomialequation and used to obtain stresses and moments within theliquid crystal network. The polynomial fits are denoted by

cw = f (X,Z)+ c0 (13)

where the functionf (X,Z) is linear if g 6= 0,h= 0 and non-linear if g= 0,h 6= 0 as illustrated in Figures 5 and 6; respec-tively. In these simulations, the time dependent water vaporconcentration has been normalized as well as the material co-ordinate through the film thickness. The polynomial fits aretaken at steady state. The coordinateZ is within the LCNfilm thickness andX corresponds to the distance betweenthe water vapor source and the film surface in the referenceconfiguration. These coordinates are parallel and thereforedirectly related; however, they are distinguished here to em-phasize the change in location of the water vapor source. Fur-thermore, the range ofX is on the order of millimeters whilethe range ofZ is on the order of 10µm; therefore, the co-ordinateX is practically constant through the film thicknessfor a fixed position of the water vapor source. The values forthe polynomial fits for steady state diffusion are given in theAppendix.

The relation in (13) is substituted into the Cauchy stressrelation (6) to quantify bending moments and bending straininduced by the water concentration gradients in the liquidcrystal network film. The moments in theX andY direc-tions are obtained based on the Cauchy stresses for imple-mentation in the finite element model. For the case of largerotation, cw 6= cw will be implemented by tracking the filmsurface during quasi-static bending and updating the watervapor induced bending moment along the surface of the film.

The water vapor induced stress couple is

0 0.2 0.4 0.6 0.8 1−0.2

0

0.2

0.4

0.6

0.8

1

1.2

Z/Z0

c(Z

)

Linear Fit

Fig. 5. Time dependent solutions of diffusion of water vapor into the

liquid crystal network. Each line corresponds to a time step starting

from zero initial conditions at t = 0. Results are plotted for a linear

chemical potential where g 6= 0 and h= 0. The linear fit is given at

steady-state.

0 0.2 0.4 0.6 0.8 1−0.2

0

0.2

0.4

0.6

0.8

1

1.2

Z/Z0

c(Z

)

Polynomial Fit

Fig. 6. Time dependent solutions of diffusion of water vapor into the

liquid crystal network. Results are for a nonlinear chemical potential

where g= 0 and h 6= 0. Again, the polynomial fit is given at steady-

state.

mc =1

j

∫ t/2

−t/2σσσcgα jξdξ (14)

wherej is the mid-surface Jacobian,gα is the reciprocal con-vected basis for the shell surface,j is the determinant of thetangent map of deformation, andξ is the coordinate throughthe film’s thicknesst; see [21] for more details. Note thatdue to the hydrostatic concentration stressσσσc, the shear stresscouplemc

xy is zero and the normal stress couples are equal,mc

xx = mcyy.

The concentration dependent stress couple in (14) is cor-related with internal bending strains to facilitate its integra-tion into FEAP. As discussed in [22], the stress couple is

work conjugate to the bending strain according tomi j =∂ψ∂κi j

whereψ is the internal energy andκi j is the bending strainrelative to the mid-surface. For the case of concentration in-duced bending, the strain induced by the concentration is de-termined from

σi j = ci jkl (εkl − εckl) (15)

whereεckl is the concentration induced strain in the deformed

state. The bending strain can be obtained through correla-tions with the concentration induced stress in (6). This isgiven by

mci j =−

∫ t/2

−t/2ci jkl εc

klξdξ =−

∫ t/2

−t/2ci jkl κc

klξ2dξ (16)

where the last relation on the right is defined in terms ofthe bending strain induced by concentration gradients and

is equivalent to the actual strainεckl only in the integral sense.

This is because the additional effect of uniform or membranestrain does not contribute to bending. This allows the bend-ing strain induced by concentration to be directly introducedinto FEAP. For simplicity, the small deformation case hasbeen considered wherej = j ≃ 1 and the reciprocal con-vected basis is aligned with the global Cartesian coordinatesystem for a flat film.

For orthotropic elastic materials, the bending strain in-duced by the concentration gradient is equivalent in both di-rections and given by

κc =12(1−ν)mc

yy

t3Ex=

12(1−ν)mcxx

t3Ey(17)

where the moment is numerically calculated using (14), (6),and the concentration gradient through the thickness. Theorthotropic moduli are denoted byEx andEy.

These relations are implemented in the following sectionusing the nonlinear finite element shell model to accommo-date the bending moment induced by the water vapor concen-tration and the internal mechanical moment induced by theelasticity of the polymer network as well as external forcesfrom the stainless steel wire in contact with the film. Sincethese films may undergo large rotation, the large deformationshell model is used to correlate the measured bending forceswith the internal moment induced by the changes in watervapor concentration.

3.4 Nonlinear Bending MechanicsThe bending force measurements due to water vapor are

compared with the induced moment in (14) using the finiteelement analysis program FEAP [23]. The shell model con-tained within FEAP is based on an exact three dimensionalgeometrical description of thin shells that can undergo largedeformation. A series of papers on the theory can be foundin [21,22,24,25]. The model utilizes classical shell theory asdescribed by a one-director Cosserat surface which leads toan efficient numerical algorithm that can handle extremelylarge deformation. The approach mitigates shear or mem-brane locking such that larger deformation problems can besimulated.

A flat plate geometry is implemented to describe the liq-uid crystal network film geometry. The boundary conditionsinclude the following set of constraints. Similar to the ex-periments, one end of the film was fully clamped. On theopposing end, the center node along this free edge was con-strained to the experimental displacement in the presence ofthe stainless steel wire. Recall that the bending force wasmeasured based on deflection of a wire of known geometryand stiffness. Model correlation to this measurement wasobtained by applying the concentration dependent bendingstrain in (17) uniformly over the entire surface of the modelto predict the effect of water vapor induced bending. Thedisplacements were small when the film was constrained bythe wire; therefore, changes in concentration along the length

of the shell surface due to displacements were neglected inthese simulations. In the case of free bending displacement,concentration dependence on bending displacements is eval-uated. The moment was increased until the reaction forceat the node constrained to a fixed displacement matched theforces from the experiments. This was done for each po-sition of the water vapor source. Through this process, thewater vapor induced moment was calculated at each experi-ment point as illustrated in Figure 7. These calculations uti-lize (14)-(17) to describe the concentration induced bendingstrain for each relative humidity condition.

The elastic modulus along the long axis of the film wasbased on data given in [6]. Since the long axis of the filmwas oriented in the direction of largest bending, this direc-tion is assumed to coincide with the direction perpendicularto the liquid crystal director. Based on modulus measure-ments given in [7], the elastic modulus parallel to the directoris nominally 1.5 times larger in the direction parallel to thedirector. This proportionality was applied to the model. ThePoisson ratio was assumed to be 0.35 since these films areglassy in the “dry” state and unlikely to be incompressiblein this state. The constant modulus assumption and Pois-son ratio introduces some uncertainty into the moment pre-dictions since it is known that the elastic behavior changessignificantly between the extremes of a dry and completelywater swollen state. This uncertainty is reduced by imple-menting an effective modulus that is a function of the waterconcentration. This “dry” state modulus was assumed to bethe modulus in ambient humidity conditions. The modulusthen varied linearly with the water concentration. An effec-tive modulus was obtained by integrating the modulus overthe film thickness and then normalizing it by the thickness.The effective modulus perpendicular to the director variedfrom 1.6 GPa in a dry state whereX = 3 mm to 360 MPafor the largest bending state whereX = 0.25 mm. This cal-culation used the nonlinear distribution of the water concen-tration since this distribution and coupled stresses provideda better model fit over a larger range of the experimentalconditions. The uncertainty in the modulus is discussed andcompared with different modeling techniques in subsequentparagraphs.

Since the calculation of the moment is based directlyon experiments, it is not ideal for general model predic-tions for variations in concentration near the film surface.A stronger understanding of the material behavior is ob-tained by implementing a material parameter fit using theparameters in (1) and (13). Improved model correlationis obtained using the steady-state diffusion solution withh = 1.35× 1026 Nm9/mol4 and g = 0 in comparison withg = 9.0× 1013 Nm3/mol2 andh = 0 as shown in Figure 7.These model fits illustrate that a linear chemical potentialwhereh = 0 is sufficient to predict bending at lower con-centrations, while higher order nonlinear effects (h 6= 0) arerequired for higher water vapor concentrations.

The model fit was then used to predict the free bendingof the LCN film. Using the effective modulus, maximumfree displacement simulations are plotted in Figure 8. In thisplot, a comparison of bending as a function of water vapor

0 0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

1

X (mm)

Ben

ding

str

ain,

%ε

FEA/Model Fit

Model h > 0, g = 0

Model g > 0, h=0

nonlinear regime linear regime

Fig. 7. Model predictions based on the moment calculated using

the bending force data and the FEAP nonlinear shell model. These

calculations are compared with the general stress couple model us-

ing (14) by varying the chemical potential parameters g and h. The

dashed curve uses g = 9.0× 1013 Nm3/mol2 and h = 0 and the

solid curve uses h = 1.35× 1026 Nm9/mol4 and g = 0. The lin-

ear and nonlinear regimes refer to the form of the chemical potential

driving force.

concentration variations during large bending deformation isanalyzed. The shell displacement is predicted by compar-ing the two cases: 1) constant concentration induced bendingstrain that is independent of the mid-surface position of theshell and 2) concentration induced bending strain as a func-tion of the film surface during large rotation. The concen-tration induced bending strain is incremented during quasi-static loading to avoid transient waves in the solution. It isshown that changes in concentration along the film lengthduring large rotation has a significant effect on the free bend-ing prediction. This is expected since the concentration in-duced bending strain is highly sensitive to positions betweenX = 0.25 mm andX = 1.0 mm; see Figure 7 in comparisonwith Figure 8.

In Figure 9, free displacement predictions are plotted fortwo different positions of the water vapor source that weremeasured experimentally. Again, the case of position depen-dent concentration relative to the film surface (“true” solu-tion) is plotted and compared with the case where changesin concentration as a function of large bending rotation areneglected (“ideal” solution). The results demonstrate highsensitivity to water vapor as significant bending rotationsbe-gin to occur when the water vapor source approaches the filmsurface. The comparison between the “true” and “ideal” so-lutions illustrate significant deviations in bending displace-ment for small values ofX since the sensitivity of water va-por increases asX decreases as previously illustrated in Fig-ure 7. Also note that under these free displacements, onlyminor in-plane displacement coupling was predicted. Thus,adecoupled linear plate model could be used in this operatingregime. However, larger displacements have been observed

experimentally and will be illustrated through a chemical po-tential parameter sensitivity analysis.

The uncertainty associated with the free displacementmodel predictions is quantified by implementing elasticproperties that vary through the thickness. Again the elas-tic tensor was assumed to be proportional to the water con-centration by varying the modulus perpendicular to the di-rector from 1.6 GPa in a dry state to 122 MPa in a saturatedstate. The modulus was increased by a factor of 1.5 parallelto the director. A two dimensional domain was implementedin Comsol 3.5 to quantify variations through the thickness.A smaller film length was simulated to resolve concentra-tion and stress variations through the thickness. Pure bend-ing was modeled by constraining one bottom corner as a pinand the opposing bottom corner as a roller. Based on thenonlinear absorption model whereg= 0 andh> 0, the cur-vature was compared for a thickness varying modulus versusthe effective modulus model. The curvature was found to becomparable for water vapor concentrations forX ≥ 1.5 mm.WhenX ≤ 1.0 mm, significant errors on the order of 30%occurred. This error is expected to be an upper bound sincethe modulus reported in the literature compares a “dry” stateto a specimen that was fully submersed in water. Smallerchanges in stiffness are expected when the material is onlyexposed to water vapor in the air. The effect of the Poissonratio (ν) was also checked. The difference in deflection in theshell model usingν = 0.35 versusν = 0.5 was negligible.

It was previously noted that the forces measured and il-lustrated in Figure 3 are not the blocked forces. These mea-surements are predicted to be relatively close to the blockedforces. By fixing the displacement at the end of the shellmodel to be zero, the blocked forces were computed usingthe nonlinear shell model and water vapor induced moments.These forces were less than 1% different than the measuredforces. This becomes apparent when considering the dis-

0 2 4 6 8

−10

12

3

0

0.2

0.4

0.6

0.8

1

y (mm)x (mm)

Dis

pla

cem

en

t (m

m)

z

Position dependentconcentration model

Position independentconcentration model

Fig. 8. Bending deformation for the maximum bending force case

where X = 0.25 mm. The results are given for the free displace-

ment case (no external edge load) for the case where concentration

is independent or dependent on the bending deformation of the film.

0 2 4 6 80

0.2

0.4

0.6

0.8

1

Length (mm)

Dis

plac

emen

t (m

m)

X=1mm (ideal)

X=1mm (true)

X=0.25mm (ideal)

X=0.25mm (true)

Fig. 9. The cross section view of the shell model illustrating in-

creases in bending with no external load. The displacements cor-

respond to two experimental water vapor locations (X = 1 mm and

X =0.25 mm) previously illustrated in Figure 7. The “true” solutions

correspond to position dependent concentration while the “ideal” so-

lution neglects position dependent concentration.

placements and bending forces of the film measured opti-cally. These displacements in the presence of the stainlesssteel wire were on the order of 1 to 10µm which are aboutthree orders of magnitude smaller than the free displacementmodel predictions.

Lastly note that these materials fatigue as they are con-tinuously exposed to water vapor. Larger free bending wasobserved prior to conducting quantitative bending force mea-surements. Similar larger bending has been reported in theliterature where rotation greater than 90 has been observed[6]. Based on these observations, the sensitivity of the chem-ical free energy parameterh was conducted using the nonlin-ear shell model. As illustrated in Figure 10, large bending isobtained by increasing the fourth order parameterh by fac-tors ranging from 2 to 8. The angle of the film at the end was24, 50, and 100 for h parameters that increased by factorsranging from 2, 4, and 8; respectively. It is also importantto note that these larger bending angles required assumingconstant concentration over the length of the film.

4 CONCLUDING REMARKSThe bending forces of a hygroscopic liquid crystal net-

work film were measured and compared to a finite deforma-tion constitutive model that was integrated into a nonlinearshell model. The results show micro Newton forces can begenerated by a film with geometry 8.8× 2.6× 0.026 mm3.Due to the small forces induced by the material, a custom-made stainless steel wire was characterized using two differ-ent methods and then used as a sensor to quantify bendingforces using an optical microscope. It was shown that thechemical potential driving force for bending changes fromlinear to nonlinear as a function of the location of the water

0 2 4 6 8 −1 0 1 2 30

1

2

3

4

5

x (mm)y (mm)

Dis

plac

emen

t (m

m)

z

Fig. 10. The changes in free displacement bending was modeled by

varying the chemical potential parameter h by factors of 2, 4, and 8.

Bending monotonically increases as this parameter increases. The

result is compared to the original case (smallest bending) previously

plotted in Figure 8.

vapor source.The experiments were correlated with a nonlinear con-

tinuum mechanics model that was coupled to a chemical po-tential energy function. It was necessary to include a higherorder fourth order chemical free energy function to predictbending forces when the water vapor source approached thefilm surface. A linear chemical potential was sufficient topredict bending as this distance increased. These bendingforces were predicted by writing the energy function in thespatial domain and correlating the free energy with energy inthe reference configuration. Such coupling gives hydrostaticdeformation from changes in water concentration without in-troducing explicit phenomenological coupling coefficients.By coupling this water concentration induced stresses with“mechanically” induced stresses from elasticity of the poly-mer network, bending deformation was predicted. Differentamounts of bending deformation were predicted along thelong and short axis of the film due to the orthotropic elasticproperties. This required computing steady state concentra-tion variations through the material thickness with the loca-tion of an external water vapor source.

The constitutive model and finite element model utilizedconcentration variations to determine hydrostatic stresses.Since these materials include liquid crystals which promotewater absorption, additional underlying liquid crystal order-disorder processes and changes in charge density may oc-cur as other chemicals in the air diffuse into the material.However in the present case, coupling with water vapor ab-sorption was found to be sufficient to predict bending overa relatively wide range water vapor concentrations. In othercases, differences in pH and polarity of the solution can af-fect the amount of bending [7]. This would require chemicaldriving forces that are a function of the pH or the introduc-tion of electrostatics and charge density variations. More-over, liquid crystal networks that have an induced chirality

either from cholesteric liquid crystals or from rotation ofanematic phase through the film thickness would require moreadvanced modeling techniques that include liquid crystal di-rector mechanics coupled to the polymer network.

It should also be noted that these films degraded af-ter repeated exposure to a humid environment and after be-ing stored in a desiccant container for several months. Theconcentration dependent bending results presented here wereconducted over approximately an hour to ensure repeatableresults. Several days and weeks later, the materials degradedin performance until no water vapor induced deformation oc-curred, although the materials were stored in a desiccant con-tainer. The nematic liquid crystal phase used in this composi-tion may play a role in this irreversible behavior. Cholestericcompositions have also been tested where it has been sug-gested that improved reliability can be achieved [6]. The in-teractions between the liquid crystal phases, director order-disorder absorption process, and polymer network mechan-ics should be tested and modeled in further detail to under-stand this behavior.

It was also shown that the elastic modulus measurementwas reduced by one to two orders of magnitude relative todynamic mechanical analysis measurements [9]. Specimenor synthesis variability may be responsible for these vari-ations in stiffness. However, these moduli measurementswere conducted using bending mechanics which has beenshown to predict lower moduli relative to uniaxial tensiontests [10]. Despite this discrepancy, the mechanical bendingexperiments resulted in a relatively constant elastic modu-lus prediction based on classic beam theory as the point loadapproached the clamped edge of the film. This suggest rea-sonably ideal clamped boundary conditions which were fur-ther confirmed by comparing bending deformation near theclamp with classic beam theory; see the Appendix.

In conclusion, the model fits illustrate relatively strongsensitivity to water vapor exposure and reasonable correla-tion with bending force data. This provides a highly sensi-tive chemical sensor that is potentially ideal for MEMS mi-crofluidics, chemical sensing, and biomedical applications.Since liquid crystals can be functionalized for sensitivity to abroad number of chemical constituents such as biological en-zymes, this provides an exciting avenue for developing novelbiomedical sensors; however, improved reliability must beachieved for these applications.

AppendixWater Vapor Concentration

The water vapor concentration is determined using theideal gas law and water vapor table values. The mass of thegas is given by

mgas= MWgasPVRT

(18)

whereMWgas is the molecular weight of the gas,P is pres-sure,V is volume,R is the gas constant, andT is temperature.

The molecular weight of the gas is based on air and water va-por according to

MWgas= yairMWair + yH2OMWH2O (19)

where the molecular weight of air isMWair = 29 g air/moleair andMWH2O = 18.016 g H2O/mole H2O. The mole frac-tion of each component is defined byyi .

The mole fractions are determined based on saturatedpressure and relative humidity. At room temperature and at-mospheric pressure (T = 25o C,P = 101.325 Pa), the rel-ative humidity is defined byhr = PH2O/P∗

H2O whereP∗H2O is

the saturated pressure. At room temperature and atmosphericpressure,PH2O = 2.8111 Pa. The mole fractions can be com-puted based on the ratio of pressures,yi = Pi/P thereforeyH2O = 2.8111/101.325= 2.774×10−2 if hr = 100%. Themole fraction of air is thusyair = 1− yH2O = 0.97226.

Using (18) and (19), the mass of the gas ismgas =8.05×10−11 kg gas. The number of moles of water is foundfrom n100%

H2O = yH2Ongas= 7.785× 10−14 mol wherengas=mgas/MWgas. Given the volume of gas above the liquid crys-tal LCN film, V0 = 2.6× 8.8× 3 mm3, cmax= n100%

H2O /V0 =

1.13×10−6 mol/m3. The concentration scales with the rela-tive humidity; therefore by applyinghr = 35%, the concen-tration under ambient conditions isc0 = 3.97×10−7mol/m3.

Polynomial FitThe parameters used in fitting the steady state diffusion

relation in (13) are based on the following equations. For thelinear fit the equation of a line is trivially satisfied by

cw = 2.5×10−4(X−X0)(1−Y)+ c0 (20)

where a normalized coordinateY has been introduced. Thiscoordinate is related to the material thickness coordinatebyY = 1

2(2t Z+1) wheret is the LCN film thickness and the ori-

gin of Z is in the center of the film’s thickness. For the non-linear steady state diffusion problem, the water vapor con-centration is

cw = 2.5×10−4(X−X0)(a1Y9+a2Y8+a3Y7+a4Y6+ . . .

a5Y5+a6Y4+a7Y3+a8Y2+a9Y+a10)+ c0(21)

Film Modulus AnalysisThe modulus measurement were based on optical mi-

croscopy comparisons of bending deformation versus classicbeam theory. Images of the bending deformation were di-rectly compared with beam displacement predictions usinglinear elastic theory of beams [8]. For a cantilever with aconcentrated loadP at a variable point along the length ofthe beamx, the displacement versus force relation is

Table 1. Polynomial coefficients used in (21).

Name Valuea1 −1.27×103

a2 5.34×103

a3 9.37×103

a4 8.88×103

a5 −4.92×103

a6 1.61×103

a7 −2.98×102

a8 2.81a9 −1.42a10 1.01

w(x) =Px2

6EI(3a− x) (22)

for the region between the clamped edge and the concen-trated load (0< x< a). The modulus and moment of inertiaareE andI , respectively. Plots of this solution in comparisonwith microscopy measurements are given in Figures 11 and12. The images correspond to the modulus from Figure 4for the normalized distances of 0.35 and 1. The comparisonsof deformation illustrate good predictions using classic beamtheory.

x (µm)

y (µ

m)

500 1000 1500 2000 2500 3000

500

1000

1500

2000

Fig. 11. Optical microscopy comparison of the film bending in com-

parison with classic beam theory. The film is clamped on the left and

the wire point load is shown on the right for the case of the smallest

normalized point load location of 0.35 from Figure 4.

AcknowledgmentsThis material is based upon work supported by the

Florida Center for Advanced Aero Propulsion (FCAAP)

x (µm)

y (µ

m)

500 1000 1500 2000 2500 3000

500

1000

1500

2000

Fig. 12. Similar comparison of beam deformation microscopy mea-

surement with classic beam theory as illustrated in Figure 11. The

location of the point load was along the cantilever tip (i.e., normalized

location of 1 in Figure 4.

and an AFOSR grant FA9550-09-1-0353. W. Oates andM. Hays also appreciate a Summer Faculty Fellowship Pro-gram through ASEE and Eglin AFRL. The authors also ap-preciate materials and synthesis help from Paul Luchette andPeter Palffy-Muhoray at the Liquid Crystal Institute at KentState University.

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