viscoelastic and poroelastic mechanical characterization of hydrated gels

Viscoelastic and poroelastic mechanical characterization of hydrated gels

Matteo Galli, Kerstyn S.C. Comley, Tamaryn A.V. Shean, and Michelle L. Oyena)

Engineering Department, Cambridge University, Cambridge CB2 1PZ, United Kingdom

(Received 10 August 2008; accepted 29 October 2008)

Measurement of the mechanical behavior of hydrated gels is challenging due to arelatively small elastic modulus and dominant time-dependence compared withtraditional engineering materials. Here polyacrylamide gel materials are examined usingdifferent techniques (indentation, unconfined compression, dynamic mechanical analysis)at different length-scales and considering both viscoelastic and poroelastic mechanicalframeworks. Elastic modulus values were similar for nanoindentation andmicroindentation, but both indentation techniques overestimated elastic modulus valuescompared to homogeneous loading techniques. Hydraulic and intrinsic permeabilityvalues from microindentation tests, deconvoluted using a poroelastic finite elementmodel, were consistent with literature values for gels of the same composition. Althoughelastic modulus values were comparable for viscoelastic and poroelastic analyses, time-dependent behavior was length-scale dependent, supporting the use of a poroelastic,instead of a viscoelastic, framework for future studies of gel mechanical behavior underindentation.

I. INTRODUCTION

Nanoindentation has become a common techniquefor characterizing the mechanical properties of a widerange of materials, including materials with time-de-pendent mechanical behavior such as polymers1 andbiological materials.2 Until recently, the majority ofnanoindentation studies concerning biological tissueswere on stiff, hard mineralized tissues, and most sam-ples were dehydrated for testing. An increasing inter-est in compliant, soft, and hydrated materials hasdriven the development of new techniques for expand-ing nanoindentation to new experimentally challengingmaterials sets. Both hydrated soft tissues and hydratedgel materials have been the subjects of recent nanoin-dentation investigations.

Time-dependent mechanical behavior is a criticalaspect of the mechanical response in many compliantand hydrated materials. Early nanoindentation studiesemphasized time-independent, elastic-plastic materialresponses but many recent studies have incorporatedviscoelastic3–7 or viscous-elastic-plastic8,9 material mod-els. Linearly viscoelastic material characterization isnow well developed for nanoindentation studies on

glassy and rubbery polymers, and such analysis has beenused (with caveats) for characterizing time-dependentnanoindentation creep responses in hydrated bone10,11

and indentation load-relaxation responses in soft bio-logical tissues.12,13 A similar analysis based on linearviscoelasticity has also been used recently for nanoin-dentation load-relaxation in hydrated gel materials.14

Several open questions remain in the viscoelastic litera-ture that are relevant in developing novel mechanical char-acterization paradigms for hydrated compliant materials.One question concerns the utility of time-domain3–14

versus frequency domain15–17 nanoindentation measure-ments for best-practice mechanical characterization oftime-dependent materials. A second question concernsthe appropriateness of viscoelastic material models forgels or hydrated tissues, as the behavior of a fluid-saturated porous solid is more appropriately described byporoelasticity18,19 than viscoelasticity.20 A recent studyexamined poroelastic data analysis for bone and demon-strated some promise for the identification of hydraulicpermeability values from nanoindentation data.21

The current study was undertaken to examine thetime-dependent mechanical response of hydrated gelmaterials, as a model for hydrated soft tissues, via spher-ical indentation experiments. Both nanoindentation andmicroindentation load-relaxation studies were performedon gels of two different nominal thicknesses. Data wereexamined in both viscoelastic and poroelastic mechani-cal frameworks. Results were compared with two sets ofcompression tests on the same gels but emphasizingfrequency-domain measurements and including dynamic

a)Address all correspondence to this author.e-mail: [email protected] author was an editor of this focus issue during the reviewand decision stage. For the JMR policy on review andpublication of manuscripts authored by editors, please refer tohttp://www.mrs.org/jmr_policyDOI: 10.1557/JMR.2009.0129

J. Mater. Res., Vol. 24, No. 3, Mar 2009 © 2009 Materials Research Society 973

mechanical analysis (DMA). Time-domain indentationrelaxation results were converted to storage modulus forcomparison with the compression results.

II. MATERIALS AND EXPERIMENTAL METHODS

A. Materials

Polyacrylamide gels were synthesized with 20%acrylamide following established recipes. Briefly, thegels were created by the addition of initiators [ammo-nium persulfate and tetramethylethylenediamine(TEMED)] to a 30% (w/v) acrylamide stock solutionwith a 19:1 acrylamide to bis-acrylamide ratio (SevernBiotech Ltd., Kidderminster, Worcs, UK). Gels werecast into two thicknesses (l), nominally 2 mm and 5mm thick, and left 24 h prior to being removed fromoriginal casting plates and submerged in distilled water.Gels were kept hydrated by storage in distilled waterwith water changes approximately every 48 h until thetime of testing.

Dehydration profiles were established by weighing thesamples as they air dried; no change of mass was foundin the first hour after removal from the bath, and thustests undertaken in nonsubmerged conditions were con-ducted no more than 60 min after removal of the gelsfrom the storage bath. Air-dehydrated gels were dehy-drated in ambient conditions at least 24 h prior to testing.Ethanol-dehydrated gels were soaked in ethanol bathsfor 24 h prior to testing.

B. Indentation tests

Nanoindentation tests were performed in displace-ment-control on a UBI nanoindenter (Hysitron, Inc.,Minneapolis, MN) using the “manual method” for sur-face determination and test engagement.14 A sphericalalumina indenter tip with a 400 mm radius was used forall tests. A ramp-hold displacement-time profile wasused with a 5 or 10 s ramp time and a 20–30 s hold timefor each test. Peak displacements for nanoindentationranged from 1 to 3.5 mm.

Microindentation tests were conducted on an Instron5544 (Canton, MA) under similar ramp-hold conditionswith a 10 s ramp time and a 300 s hold time, and peakdisplacements from 100 to 500 mm in thin gels and100 mm to 1 mm in thick gels. The indenter tip wasstainless steel and had a 1.5 mm radius; a 5 N load cellwas used for data collection.

Indentation data for both nanoindentation and micro-indentation were analyzed with an approximate solutionfor spherical indentation load-relaxation based onelastic-viscoelastic correspondence as presented previ-ously.12 Briefly, load-time data were fit (nonlinear curvefit wizard, Origin 8, OriginLab, Northampton, MA) to anexponential decay function of the form:

PðtÞ ¼ B1 þ BD expð�t=t1Þ ; ð1Þwhich was derived assuming the extensional rela-xation function for a standard linear solid (Fig. 1) wherethe single time-constant exponential decay can bewritten

EðtÞ ¼ E1 þ ED expð�t=tÞ ; ð2Þand where the time constant is related to the viscosityand dashpot-associated modulus by t = Z/E1. The para-meters E1 and ED were simply calculated from thefitting parameters B1 and BD (assuming n = 0.5) by12:

E1 ¼ B1h3=2maxð16

ffiffiffiR

p=9Þ

; ð3Þ

and

ED ¼ BD

ðRCFÞh3=2maxð16ffiffiffiR

p=9Þ

; ð4Þ

where

RCF ¼ ttR½expðtR= tÞ � 1� : ð5Þ

E1 is the equilibrium modulus while the instanta-neous modulus is given by the sum E1 + ED; the elasticfraction is fE = E(1)/E(0) = E1/(E1 + ED).

C. Unconfined compression tests

A circular cylinder of the thick (nominally 5 mm) gelwith a diameter of 18 mm was compressed betweenplastic platens using a screw driven tensile test machine(Instron 5544, Canton, MA) fitted with a 5 N load cell.An oscillatory sawtooth displacement profile with a peakto peak displacement of 0.3 mm was used to compressthe sample at a range of frequencies from 0.01 Hz to8 Hz. A preload of 3 N was applied to ensure that thesample was held in compression throughout the durationof the test. The raw force, F(t), and displacement, u(t),time series data were converted into the frequencydomain, F(o) and u(o), using a fast Fourier transform.The complex modulus of the sample was calculated as:

FIG. 1. Schematic illustration of the standard linear solid viscoelastic

model used throughout.

M. Galli et al.: Viscoelastic and poroelastic mechanical characterization of hydrated gels

J. Mater. Res., Vol. 24, No. 3, Mar 2009974

E� ¼ FðoÞ�luðoÞ�A ; ð6Þ

where l is the sample height and A is the surface areaof the sample; the real part of E* corresponds to thestorage modulus E0 and the imaginary part to the lossmodulus E00.

D. DMA tests

Dynamic Mechanical Analysis (DMA) in compres-sion was undertaken on circular cylindrical samples ofdiameter 18 mm and thickness nominally 5 mm using aDMA machine (Q800, TA Instruments, Newcastle, DE).The samples were oscillated up to a peak displacementof 5 mm over a frequency range of 1–200 Hz. Followingthe findings of Yamashita,22 a pilot study was under-taken to determine an appropriate static preload of8.5 N, which gave stable measurements of the complexmodulus. During the main compression testing it wasnoted that at each given frequency a settling time ofapproximately 1 min was required to achieve a stablemeasurement of the compression storage modulus, E0and loss modulus, E00.

E. Comparisons for time and frequency domains

Comparisons were made between indentation and un-confined compression tests by converting the indentationcreep function data to storage modulus (E0) and losstangent (tan d) via20:

E oð Þ ¼ E1 þ ED

o2t2

1þ o2t2; ð7aÞ

tan d oð Þ ¼ E00ðoÞE0ðoÞ ; ð7bÞ

where

E oð Þ ¼ ED

ot1þ o2t2

: ð7cÞ

The frequency-based functions were computed usingthe average values of E1, ED, and t for the microin-dentation and nanoindentation tests on similar (thick)samples as were used in the homogeneous compressiontests.

III. MODELING AND ANALYSIS

An axisymmetric finite element (FE) model of theindentation experiment was developed using ABAQUS(Version 6.7, SIMULIA, Providence, RI). The indenterwas modeled as an analytical rigid surface and thecontact between the indenter as the specimen surfacewas imposed to be frictionless (Fig. 2). The specimen

was assumed to be linear poroleastic and saturated andwas discretized using eight-node elements with biqua-dratic displacement interpolation, bilinear pore pressureinterpolation, and reduced integration (the mesh con-sisted of 65,830 nodes and 21,735 elements). It wasassumed that the specimen lies on an impervious stiffsubstrate and that water can diffuse freely across theentire layer surface, including the contact region.

The mechanical behavior of a linear poroelasticmaterial saturated with an incompressible fluid dependsthe elastic properties (in the present case drained elasticmodulus, E and Poisson’s ratio, n were considered)and on the hydraulic permeability k.19 The intrinsicpermeability k = ZWk, where ZW is the viscosity ofwater. (For further background on poroelasticity thereader is referred to Refs. 18, 19, and 21.) To assessthe values of these three material parameters from theresults of indentation tests an identification algorithmwas utilized.

Given the output of the experiment in terms of indenter time-load data points (t1, P1), . . . , (tm, Pm) and themodel of the experiment M(x, t), whose response isgoverned by a set of unknown parameters x, it ispossible to compute the residuals for any choice of theparameter values:

fi xð Þ ¼ Pi �Mðx; tiÞPave

; i ¼ 1; . . . ;m ; ð8Þ

where Pave is the average experimental value. In thepresent case M(x, t) is the finite element model, and theunknown parameters x are E, n, and k.

For a least squares fit, the objective of the identifica-tion procedure is then finding the set of parameters xwhich minimizes

FIG. 2. Schematic illustration of the finite element model geometry

and boundary conditions for establishing hydraulic permeability from

indentation tests. The indentation depth is h and the poroelastic layer

thickness is l.


J. Mater. Res., Vol. 24, No. 3, Mar 2009 975

F xð Þ ¼ 1

2

Xm

i¼1

ð fiðxÞÞ2 : ð9Þ

Therefore the identification problem becomesthe problem of the minimization of the objective func-tion F(x) with respect to the unknown parameters x. Inthe present work the minimization problem was solvedby using MATLAB (The MathWorks, Natick, MA)optimization toolbox and in particular the nonlinearleast-squares routine, whose successful use in similarproblems is reported in the literature.23–25

Identification was carried out for the hold segment ofeight microindentation tests, five on thin gels and threeon thick gels. No maximum value of F(x) was imposedas a convergence criterion and the identification loopwas iterated until the procedure showed no improvementin the minimization of F(x). Typically convergence wasachieved in 4–5 iterations.

IV. RESULTS

Representative load-time (P-t) relaxation curves areshown in Fig. 3 for (a) nanoindentation of a hydratedgel, (b) nanoindentation of dehydrated (ethanol-soaked)gel, and (c) microindentation of hydrated gel. On thistimescale, the microindentation results are nearly elastic,while there is some relaxation present in the nanoinden-tation data on the same timescale. Interestingly, thedehydrated sample [Fig. 3(b)], although orders of mag-nitude stiffer than the hydrated sample tested under thesame conditions [Fig. 3(a)], exhibited more viscous dis-sipation than the hydrated samples.

Numerical data derived from viscoelastic analysis ofthe full set of nanoindenation and microindentation testsare shown in Table I. As was evident in the raw load-time responses (Fig. 3), dehydration increased the elasticmodulus by three orders of magnitude regardless of de-hydration method—air or immersion in ethanol. Theelastic fraction ( fE) decrease was greater in the ethanol-soaked gels compared with the air-dried gels.Overall the calculated storage modulus values from

indentation were greater than those measured in two dif-ferent modes of homogeneous compression tests, uncon-fined compression and DMA compression [Fig. 4(a)].Loss tangent values [Fig. 4(b)] were comparable forboth compression tests, and similar to the peakcalculated for nanoindentation. In the low frequencyrange for this experiment, the loss tangent from micro-indentation also started to approach the compressivevalues. It is clear that the standard linear solid model(Fig. 1) is too simple a description of the dynamicbehavior for these gel materials, given the sharp peak ata single frequency value.A representative fit to experimental microindentation

data from the poroelastic FE model is shown in Fig. 5.Summary data from eight poroelastic FE fits to thickand thin gel microindentation tests are shown inTable II. The drained elastic modulus values are inagreement with the overall average equilibrium modu-lus, but are smaller than the individual values from theviscoelastic analysis of the same data (data notshown); this is likely because the finite layer thicknesswas accounted for in the FE model but not in theviscoelastic analysis. This is particularly an issue withthe thin gels, for which the indentation depth h wasmore than 10% of the layer thickness, l. There is alsoa trend toward decreasing modulus values with in-creasing indentation depth in the thin gel poroelasticresults, although this effect is relatively small (<20%).Drained Poisson’s ratio (nd) values for the materialincreased with increasing indentation depth, and bothhydraulic and intrinsic permeability values decreasedwith increased indentation depth.

V. DISCUSSION

Highly hydrated biological tissue and polymer gelmaterials are extremely compliant, and mechanical char-acterization of low-modulus materials holds severalchallenges. Indentation is an ideal mechanism for char-acterization of compliant materials, as there is no need totry to “grip” the sample as in a tensile test. However,new approaches are required for data analysis when itcomes to hydrated time-dependent materials, as the elas-tic-plastic contact models26,27 typically used for nanoin-dentation data analysis are completely inappropriate inthis context.

FIG. 3. Experimental indentation load-time (P-t) relaxation data

for (a) nanoindentation on hydrated gel (b) nanoindentation on

a dehydrated gel soaked in ethanol and (c) microindentation on

hydrated gel.



Under displacement control, at a fixed displacementthe load relaxes in a time-dependent material. There aredistinct relaxation mechanisms in viscoelastic and por-oelastic mechanical problems. Historically a selectionhas been made by researchers a priori as to which mech-anism is likely, and little effort has gone into distinguish-ing experimentally which model is more appropriate in agiven context. The choice is typically made based on

knowledge of the material, for example a viscoelasticsolid polymer or a poroelastic hydrated soft tissue. Herethe two approaches were compared explicitly, demon-strating that there is a clear difference and distinctionthat can be experimentally detected with sufficientlycareful experiments.

The length-scales probed here by nanoindentation andmicroindentation tests were significantly different. Therewas a two order of magnitude difference in the indenta-tion depth and an order of magnitude difference in theindenter radius. As such, given the fact that the time-constant for poroelasticity, t ¼ L2=Gk ;18,21 containsan experimental length-scale (L), it should be expectedthat the time constants observed in nanoindentation andmicroindentation were substantially different here. Forlarger experimental timescales (microindentation) therewas little relaxation (i.e., a larger elastic fraction, fE) inan experimental time-frame of tens of seconds whencompared with more obvious time-dependence observedin the same time-frame in nanoindentation (Fig. 3).There is no length-scale intrinsic to viscoelasticity— theappearance of different experimental time-constants fornanoindentation and microindentation is consistent withthe existence of a length-scale associated with poroelas-ticity.

It is also possible to identify differences between theshapes of the relaxation curves, given the fact that themechanisms of relaxation are physically different. Wefurther propose to use this curve shape as an additionalpiece of evidence to establish the active time-dependentmechanism active in any given experiment. This curve-shape difference illustrated in Fig. 6, in which a best fitfrom an FE spherical contact simulation on acrylamidegel is shown for both a poroelastic and a viscoelastic(standard linear solid) model [Fig. 6(a)]. The differencebetween the fits is subtle, but plotting the residuals[Fig. 6(b)] clearly demonstrates that the poroelastic fitis superior to the viscoelastic fit in this case: the visco-elastic fit both overshoots and undershoots at differentpoints on the experimental curve. There are thus severalindependent pieces of evidence to suggest a poroelastic,

TABLE I. Mechanical properties of poly(acrylamide) gels measured via microindentation and nanoindentation load-relaxation tests.

Number of

tests, nEquilibrium modulus,

E1 (MPa)

Dashpot-associated

modulus, ED (MPa)

Elastic fraction, fE =

E(1)/E(0) = E1/(E1 + ED) Viscosity, Z (MPa-s)

Thick hydrated gels (l = 4.4 � 0.7 mm)

Microindentation 6 0.37 � 0.11 0.04 � 0.03 0.92 � 0.04 6.7 � 4.9

Nanoindentation 10 0.37 � 0.11 0.08 � 0.02 0.81 � 0.05 0.48 � 0.35

Thin hydrated gels (l = 1.8 � 0.2 mm)

Microindentation 8 0.50 � 0.06 0.07 � 0.03 0.88 � 0.04 10.5 � 3.6

Nanoindentation 11 0.35 � 0.04 0.07 � 0.01 0.82 � 0.01 0.26 � 0.04

Thin dehydrated gels (nanoindentation)

Ethanol-dehydrated 3 145.9 � 51.9 83.4 � 50.3 0.65 � 0.05 846.5 � 662.7

Air-dehydrated 10 423.9 � 255.7 100.0 � 29.0 0.76 � 0.01 688.5 � 336.0

FIG. 4. Comparison between dynamic compression tests and the

average results from micro- and nanoindentation tests converted to

frequency-based data, illustrating (a) storage modulus, E0, and (b) loss

tangent, tand.



instead of viscoelastic, mechanism operating in theseindentation tests. Given the emphasis on poroelasticflow, it is envisioned that time-domain, and notfrequency-domain, measurements will be most suitablefor nanoindentation characterization of hydrated gelsand tissues.

In the current study, experimental measurements ofthe elastic modulus of hydrated gel materials were con-sistent between microindentation and nanoindentation,although numerical values of the obtained elastic modu-lus were larger for indentation than for homogeneouscompression testing. One potential experimental factorin overstimation of elastic modulus values is the inden-tation depth relative to the gel layer thickness. It hasbeen shown with FE models that poroelastic layers showan apparent stiffening effect when the layer thickness issmall,28 an effect similar to that for an elastic multilay-ered system. The thin gel results for microindentation

(Table I) seem to be particularly afflicted with this, sincecomparison of the viscoelastic analysis—which assumesbulk half-space behavior—and the poroelastic analysisfor a subset of the same gels (Table II) removes theoverestimation relative to all other indentation data. Thecompressive values were numerically in good agreementwith values of E = 0.22 MPa reported previously29 forsimilar 20% acrylamide gels. Therefore the indentationvalues appear to be still slightly too large, even in thickgels or when the finite layer thickness has beenaccounted for. A similar effect has been observed previ-ously in non-gel polymeric systems, in which nanoin-dentation results consistently overpredict the elasticmodulus compared with homogeneous loading.7 It isunclear why this is the case but further investigationsare clearly warranted.Gel intrinsic permeability values, obtained from inverse

FE analysis of microindentation test data, ranged from 5.6� 10�20 m2 to 1.2 � 10�19 m2. These values are in directagreement with a value of 1 � 10�19 m2 previously pub-lished30 for polyacrylamide gels of the same composition(20%) but using macroscopic homogeneous loadingexperiments. This result is encouraging, as it implies thatindentation tests can be set up for high-throughput poroe-lastic characterization, especially when modern nanoin-denters with automated x-y stage motion is available.However, each FE simulation for parameter optimizationin the current study lasted for approximately 24 h.Recently, a suggestion has been made,21 based on previousanalyses of the poroelastic spherical indentation prob-lem,31,32 that a “master curve” approach could be adoptedfor fast identification of permeability from experimental

FIG. 5. (a) Comparison between experimental and FE-identified load-

time (P-t) curve for a 200 mm indentation depth and (b) the

corresponding residuals, computed according to Eq. (8). At the end

of the identification the two relaxation (P-t) curves almost coincide,

with the residuals not exceeding �0.002 at any time point.

TABLE II. Permeability results for hydrated gels tested by micro-

indentation, based on a poroelastic finite element model.

Indentation

depth, hmax

Drained

elastic

modulus,

E (MPa)

Drained

Poisson’s

ratio, nd

Darcy

(hydraulic)

permeability,

k (m4 N-1 s-1)

Intrinsic

permeability,

k (m2)

Thin gels

100 mm 0.365 0.13 1.24 � 10�16 1.10 � 10�19

200 mm 0.329 0.17 1.30 � 10�16 1.15 � 10�19

300 mm 0.332 0.17 1.16 � 10�16 1.04 � 10�19

400 mm 0.301 0.18 1.11 � 10�16 9.89 � 10�20

500 mm 0.298 0.20 8.48 � 10�17 7.55 � 10�20

Thick gels

100 mm 0.323 0.14 1.09 � 10�16 9.73 � 10�20

200 mm 0.365 0.14 9.16 � 10�17 8.15 � 10�20

500 mm 0.356 0.18 6.25 � 10�17 5.56 � 10�20

FIG. 6. (a) Comparison between experimental and FE-identified por-

oelastic and viscoelastic load-time (P-t) curves for a thin gel specimen

and a 100 mm indentation depth, and (b) the corresponding residuals,

computed according to Eq. (8). In the poroelastic case the two P-t curves almost coincide with the residuals not exceeding �0.002 at

any time point, while the quality of the viscoelastic fit is much lower

with a maximum value of the residuals as large as 0.015.



indentation data. The original suggestion and earlier ana-lyses21,31,32 were all based on step-load creep experiments,but there is no physical reason that a similar approachcould not be adopted for displacement-controlled relaxa-tion tests under more typical experimental conditions. Therapid identification of permeability from indentation testswould allow for “permeability mapping” of biological tis-sues or inhomogeneous materials, similar in concept tothe “modulus map” approach already being used forelastic characterization of inhomogeneous materials andtissues.33

ACKNOWLEDGMENTS

M. Galli was supported by Grant No. PBELB-120953from the Swiss National Science Foundation. Theauthors thank Z. Suo and X. Zhao of Harvard University,for helpful discussions related to the interpretation ofthese data.

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