microrheological studies offer insights into polysaccharide gels
TRANSCRIPT
A
mgFcc
uose©
K
1
matelieitas
umt
0d
J. Non-Newtonian Fluid Mech. 149 (2008) 63–70
Microrheological studies offer insights into polysaccharide gels
M.A.K. Williams a,b,∗, R.R. Vincent a,b, D.N. Pinder a, Y. Hemar a,b
a Institute of Fundamental Sciences, Massey University, New Zealandb MacDiarmid Institute for Advanced Materials and Nanotechnology, New Zealand
Received 29 October 2006; received in revised form 19 March 2007; accepted 19 May 2007
bstract
Microrheological measurements have been carried out on a series of model systems of increasing structural and temporal complexity usingultiple particle tracking (MPT). Purely viscous media, entangled polymeric solutions, and subsequently a biologically relevant polysaccharide
el have all been studied. In addition, the gelled sample has been mechanically disrupted in order to induce a transient heterogeneous microstructure.or the polysaccharide sample MPT results are analysed in order to yield both one- and two-point microrheological measurements and these areompared with those obtained by additional experiments carried out using diffusing wave spectroscopy (DWS) and those obtained using aonventional rheometer.
Results from glycerol/water mixtures and polyethylene oxide solutions agree with expectations, providing confidence both in the techniquessed and the analysis employed. Moreover, a calcium-induced gel of the polysaccharide pectin yields similar results to those published recently
n actin, with the elastic modulus being frequency independent at low frequency, and both elastic and viscous moduli exhibiting a high frequencycaling with ω3/4. Furthermore, preliminary results reported here indicate that these techniques are ideally suited to the study of the temporalvolution of the spatial distribution of mechanical properties in such samples, for example during gelation or digestion.2007 Published by Elsevier B.V.
io[tetaptot
eca
eywords: Microrheology; MTP; Polysaccharide gel; Pectin; Heterogeneity
. Introduction
It is well known that a plethora of biologically relevant softaterials exhibit heterogeneous hierarchical architectures, that
rise naturally from the propensity of constituent biopolymerso self-assemble and aggregate. Furthermore, owing to the pres-nce of such structures, biopolymer solutions typically exhibitarge deviations from Newtonian behaviour. Thus, understand-ng how to probe the viscoelastic properties of such systems isxpected to have wide relevance, both from fundamental andndustrial view points. Microrheology, an extension of rheologyhat is concerned with how materials store and dissipate energys a function of length scale when exposed to stresses, is ideallyuited to this task.
Since its conception microrheology has continually offered
nprecedented new ways to examine the structures of softaterials [1]. Advances in passive microrheology, where thehermally activated motion of tracer particles is monitored, have
∗ Corresponding author.E-mail address: [email protected] (M.A.K. Williams).
aiombdu
377-0257/$ – see front matter © 2007 Published by Elsevier B.V.oi:10.1016/j.jnnfm.2007.05.006
ncluded improvements and innovations in the measurementf the mean square displacement (MSD) of embedded tracers2–5], and in the interpretation of these MSDs. Once obtained,he MSD data can be transformed to reveal the viscoelastic prop-rties of the material. Several routes exist in order to performhis transformation, all based on equivalent representations of
generalised Stokes–Einstein (GSER) equation. This inverseroblem can be tackled by direct Fourier transformation, Laplaceransformation and analytic continuation, or numerical meth-ds based on assuming a power law form of the MSD. Suchransforms are well documented [1,6].
Initially microrheology might be considered as a techniquenabling measurements of rheological properties, akin to thosearried out on a conventional mechanical rheometer, on minutemounts of sample. Indeed, this simple idea is particularlyppealing in the biophysical arena and to researchers interestedn biopolymeric structure–function relationships, where it isften the case that only small amounts of material with controlled
olecular architectures are available. However, driven largelyy the development of DWS [2,7,8] that allowed small particleisplacements to be measured that previously would have beenndetectable, microrheology’s ability to extend the frequency
6 wtoni
ratmabe
sfttttwpaaditiwic[tsptboamp
esglttioaw
ltptemtwis
toeompbfttdwtmowbtffs
bavoisaaagastapoh
2
2
A
iSsMmt
4 M.A.K. Williams et al. / J. Non-Ne
ange of rheological characterisation was immediately appreci-ted. In contrast to conventional rheometry, the small mass ofhe tracer particles ensures that inertial affects can be ignored inicrorheolgical investigations up to frequencies of some 106 Hz,
llowing the rich dynamics of soft condensed matter systems toe probed over a hitherto impossible frequency range of someight orders of magnitude.
The development of real-space particle tracking, first usingingle [9,10], and later multiple [11–14] beads, as a mechanismor extracting the MSD afforded the opportunity to examinehe variation of mechanical properties in the spatial as well asemporal dimension. While other, scattering based, techniquesypically average the system’s behaviour over several millime-res, as many as a 100 individual fluorescent beads can be trackedithin areas of some tens of microns squared. By moving theosition of the observation window and taking ensemble aver-ges of the MSD the degree of spatial heterogeneity existingt this scale can easily be ascertained, as has been elegantlyemonstrated in living cells [15,16]. However, structures exist-ng in biopolymer gels, biomaterials and tissues are well knowno exist on length scales substantially smaller than this, so thatn order to obtain the maximum information single particlesould ideally be tracked. Furthermore, when a tracer particle
s introduced into a sample it is possible that there is a surfacehemistry dependent depletion zone induced around the particle17]. Both these effects, the existence of naturally existing struc-ures that vary the local mechanical properties of soft materialsampled by tracers, for example pore-like structures [13], andarticle induced pockets of different local concentration meanhat for systems of this type the rheological properties sampledy the motion of tracers are not likely to reflect the behaviourf the bulk mechanical properties. However, rather than beingdisadvantage, such cases simply highlight the fact that havingeasurements of a sample’s rheology at different length scales
rovides additional information regarding its internal structure.While tracking single particles is a worthwhile goal, the
xtended observation time required in order to ensure that thetatistics are good enough to achieve ergodicity means that, ineneral, the MSD of each individual track in MTP is not equiva-ent to the ensemble averaged MSD. One approach that has beenaken in order to circumvent this problem is the construction ofhe van Hove correlation functions from multiple particle track-ng data giving, at a fixed time, the distribution among pathsf the mean square distance travelled [13]. A test can then bepplied in order to determine if this distribution is consistentith the statistical expectation.Considerations of such possible heterogeneity and how the
ocal behaviour is related to the bulk rheological properties led tohe development of two-point microrheology [11,18,19]. Two-oint microrheology is based on cross-correlating the motion ofwo tracer particles in order to examine the rheological prop-rties on the length scale of their separation. For homogeneousaterials it can be shown that scaling these contributions back to
he tracer size successfully yields data that compare favourablyith that measured using one-point microrheology. However,
f there are extra contributions to the particle’s motion at thecale of its size, for example motion in a more compliant deple-
3oS
an Fluid Mech. 149 (2008) 63–70
ion layer, or relatively unconstrained motion in a pore, then thene and two-point microrheology are not expected to map ontoach other. Furthermore, owing to the extended scale of the rhe-logical measurement of two-point data it is expected to agreeore closely with bulk rheological measurements, and it is inde-
endent of the size or shape of the particle and of the couplingetween the tracer and the medium. In cases where observed dif-erences between one and two-point microrheology arise fromhe presence of a depletion layer, an expression has been derivedhat is able to map the two datasets onto one another, with theepletion zone thickness as an extractable parameter [19]. Thisas successfully demonstrated for solutions of DNA [14]. In
erms of systems that are indeed believed to be locally inho-ogenous per se the methodology has been successfully tested
n biopolymeric solutions of guar, where the two-point methodas shown to extract rheological behaviour close to the bulkehaviour measured by a conventional rheometer, in contrast tohe one-point data [18]. A study using worm-like micelles alsoound reasonable agreement between all techniques over a largerequency range [20], while in contrast, it is apparent that actinolutions have a scale-dependent rheology [18].
We are particularly interested in biopolymers that assem-le into networks and in attempting to understand the spatialrrangement and the dynamics of such systems over rele-ant length and time scales. While a considerable amountf microrheological work has been carried out on actin, anmportant proteinaceous component of the cytoskeleton, weought another model system with a view to investigating thepplicability of the polymer physics models being applied toctin in describing polysaccharide gels. We selected pectin,n important gelling polysaccharide that has a crucial role inoverning the properties of plant cell walls [21]. En route, welso present results from purely viscous and flexible polymerystems (glycerol/water mixtures and polyethylene oxide solu-ions, respectively) in order to validate our experimental set-upnd analysis techniques, and finally we describe the results ofreliminary experiments designed to investigate the potentialf using these techniques for studying the evolution of spatialeterogeneity in polysaccharide systems.
. Materials and methods
.1. Materials
Glyercol (99.5%) was purchased from AJAX Chemicals,uburn, NSW, Australia, and used without further purification.PEO (Mw ∼ 900 kDa) was purchased from ACROS Organ-
cs (New Jersey, USA) and used without further purification.tock solutions of 6.2 and 10% (w/w) were prepared by dis-olving the PEO powder in Milli-Q water (water purified with a
illi-Q apparatus, Millipore Corp., Bedford, MA, with a mini-um resistivity of 18.2 M�) and stirring gently for 24 h at room
emperature.
Pectin (Mw ∼ 150 kDa), extracted from citrus peel, with a5% degree of methyl-esterification and a galacturonic contentf 90%, was kindly supplied by CP Kelco ApS, DK 4623 Lillekensved, Denmark. A stock solution of 3% (w/w) was prepared
wtonia
boNrw
k8
t
s(f
2
2
s
2
pwaPbrts
2
c1cstttwtCwqbmtGtsltard
mo
2
sTsalsccttagss
2
bi
2
tmmstcaoc
aooaai
G
wfTGdt
M.A.K. Williams et al. / J. Non-Ne
y dissolving the pectin powder in Milli-Q water and stirringvernight. The pH was adjusted to 5.6 with a solution of 1 MaOH, in order to ensure that the unmethylated galacturonic
esidues were fully charged (pKa ∼ 3.5). The final concentrationas 2.94% (w/w).CaCO3 powder with a mean particle diameter of 1 �m was
indly provided by Provencale s.a., Avenue Frederic Mistral,3172 Brignoles Cedex, France.
Glucono-δ-lactone (GDL) was purchased from Fisher Scien-ific, Bishop Meadow Rd, Loughbourough LE11 5RG, UK.
Latex particles of diameters 202 and 465 nm (2.62% (v/w)tock solutions) and fluorescent particles of diameter 541 nmFluoresbrite plain YG, 2.64% stock solution) were purchasedrom Polyscience Inc. (Warrington, PA).
.2. Sample preparation
.2.1. Glycerol/water mixturesGlycerol/water mixtures of the required ratio were prepared
imply by mixing with Milli-Q water.
.2.2. PEO solutionsSamples of different PEO concentrations and 1% (v/w) latex
articles (DWS) or 0.02% (v/w) fluorescent particles (MPT)ere prepared by mixing appropriate amounts of stock solutions
nd Milli-Q water and were stirred overnight. Three differentEO concentrations (2.2, 4, and 6% (w/w)) were studied, alletween 10 and 45 times the overlap concentration [22], cor-esponding to the semi-dilute regime. At these concentrations,he polymer solution mesh size [22] is the order of nm, which isignificantly smaller than the latex particle diameter.
.2.3. Pectin gelsIonotropic pectin gels were obtained by slowly releasing
alcium ions into a 1% (w/w) pectin solution [23] containing% (v/w) of latex beads (DWS) or 0.02% (v/w) of fluores-ent beads (MPT). The appropriate amounts of pectin and beadtock solutions were mixed and stirred, and immediately prioro loading into appropriate test cells a salt solution was added,he final mixing of which achieves the final desired concentra-ion of all components of the system. This aqueous salt solutionas composed of CaCO3 and GDL. The GDL hydrolyses with
ime, releasing protons that solubilise the Ca ions from theaCO3. These components were introduced as powders intoater, quickly mixed, and added to the pectin/bead solution asuickly as possible, in order to avoid significant calcium releaseefore mixing with pectin. The quantity of CaCO3 added deter-ines the R value, R = 2×[Ca2+]/[COO−] which can be varied
o tune the elasticity of the material, and a stoichiometric ratio ofDL [GDL] = 2×[Ca2+] is used in order to maintain the pH of
he solution [23]. After the addition of the salts the final preparedolution was stirred for a few minutes, and the samples wereoaded on the appropriate test cell and left overnight. After 24 h
he mechanical properties of the resulting gel did not undergony further significant evolution (as checked by conventionalheometry at 1 Hz). Precautions were taken in order to avoid theehydration of the sample during the gelling phase. Availablem
G
n Fluid Mech. 149 (2008) 63–70 65
icroscopy evidence suggests that the mesh size in such gels isf the order of 100 nm [24].
.2.4. Biphasic gelsMechanically heterogeneous biphasic samples were made by
uperposing two layers of pectin gel with two different R values.he bottom half of a DWS cell was loaded with an R = 0.4 gellingolution, and subsequently once the gel had stopped evolving,n R = 0.2 gelling solution was introduced directly on top andeft overnight. The interface formed was not macroscopicallytraight owing to a meniscus in the cell, but was marked on theell before addition of the second layer. For the MPT samples, aavity microscope slide was filled with an R = 0.4 gelling solu-ion, and left overnight with a cover slip in place. Subsequently,he cover slip was removed and half of the gel removed using
scalpel. The resulting space was then filled with an R = 0.2elling solution. The cover slip was then reintroduced and theample left overnight. Thus, model macroscopically biphasicamples were designed.
.2.5. Mechanically disrupted heterogeneous systemsPre-formed pectin gels (R = 0.2) were introduced into a sonic
ath for 2 min in order to introduce large scale heterogeneitiesnto the sample.
.3. Microrheology
As described in the introduction the aim of microrheology iso extract the rheological properties of soft materials from the
otion of probe particles immersed in the material [1]. Passiveicrorheology, contrary to active microrheology, is the simple
tudy of the particle’s thermal motion. It is a non-destructiveechnique and recovers the linear response of the material, inontrast to active microrheology (far from this distinction beingdisadvantage the potential complementarity of the techniquesffers the extraction of more information, akin to performingonventional small and large deformation experiments [25]).
For a viscoelastic fluid, the MSD of a probe particle will varys a local power law 〈�r2(τ)〉 ∼ τα with 0 ≤ α ≤ 1 dependingn the nature of the medium, and τ is the observation time. Inrder to link the measured MSD to the viscoelastic properties,generalised Stokes–Einstein relation (GSER) [6] can be used
s described in the introduction, with the caveats that the fluids incompressible and the boundary non-slip:
ˆ (s) = kBT
πas〈�r2(s)〉 (1)
ith G(s) is the shear modulus in the Laplace space, s the Laplacerequency and 〈�r2(s)〉 is the Laplace transform of the MSD.he complex shear modulus G*(ω) is the Fourier transform of(t), with ω the Fourier frequency. A numerical method foretermining the elastic (G′) and viscous (G′′) shear moduli fromhe MSD has been detailed by Mason [6]. The storage and loss
oduli are given as a function of the frequency ω by:
(ω) = kBT
π〈�r2(1/ω)〉Γ [1 + α(ω)](2)
6 wtoni
G
G
〈α
Wdp
2
dc4leTmTPrMl
fc1Ii
g
)〉] +
3)
√
wdtil
podndt[taoe
2
m
iwfCd(fb
〈wl
cc
D
wojps
D
wGM
〈
2
cGwrrc
3
c
6 M.A.K. Williams et al. / J. Non-Ne
′(ω) = G(ω) cos [πa(ω)/2] (3)
′′(ω) = G(ω) sin [πa (ω) /2] (4)
�r2/(1/ω)〉 is the MSD at time τ = 1/ω,(ω) = |∂ ln〈�r2(τ)〉/∂ ln τ|τ=1/ω and Γ is the gamma function.e used this method and an extension in which second order
erivatives are taken in order to better characterise the localower laws [6,22].
.4. DWS experimental arrangement
The DWS apparatus used in this study has been fullyescribed previously [26]. The samples were contained in glassells of width 10 mm, height 50 mm, and path length L ofmm, and were illuminated with a 35 mW HeNe Melles–Griot
aser operating at wavelength λ = 633 nm. The laser beam wasxpanded to approximately 8 mm on the surface of the cell.he transmitted scattered light was detected using a single-ode optical fibre (P1-3224-PC-5, Thorlabs Inc., Germany).he optical fibre was connected to a Hamamatsu HC120-08MT photomultiplier tube module, and the intensity autocor-elation functions of the scattered light were obtained using a
alvern 7132 correlator. Tests were run for 20 min to ensureow noise intensity autocorrelation functions.
For an expanded beam mode, the field autocorrelationunction, g1(t) is obtained from the measured intensity auto-orrelation function g2(t) using the Siegert equation g2(t) =+ βg2
1(t), where β is a constant depending on the instrument.n the transmission geometry, the field autocorrelation functions given by [27]:
1(t) =[(L/l∗ + 4/3)/(z0/l∗ + 2/3)]{sinh[(z0/l∗)
√k2
0〈�r2(τ
(1 + (8t/3τ)) sinh[(L/l∗)√
k20〈�r2(τ)〉] + (4/
here l∗ is the scattering mean free path, z0 the penetrationepth (assumed here to equal l∗), k0 = 2π/λ, and L the samplehickness (4 mm here). l∗ is obtained by performing an exper-ment on a water sample using 1% of latex beads, and fitting∗ using the accepted viscosity. Subsequently l∗ for future sam-les is obtained by scaling the value obtained for water, basedn the change in transmitted intensity when the sample is intro-uced, compared to the water experiment. It is known that foron-absorbing slabs of thickness L, the transmitted intensity isirectly proportional to (l∗/L) (1 + 4l∗/3L), so that by measuringhe change in transmittance, the change in l∗ can be calculated27]. Once l∗ is determined, the MSD can be obtained fromhe experimental correlation functions by inverting Eq. (1) withzero-crossing routine. That is, for each time point, the value
f 〈�r2(τ)〉 that is required in order for Eq. (1) to match thexperimentally measured g1(t) is determined.
.5. MPT experimental set-up
The Brownian motion of fluorescent polystyrene beads wasonitored with an Olympus OH2 microscope, using a 100 × oil
8ava
an Fluid Mech. 149 (2008) 63–70
(2/3)
√k2
0〈�r2(τ)〉 cosh[z0/l∗√
k20〈�r2 (τ)〉]}
k20〈�r2(τ)〉cosh
[(L/l∗)
√k2
0〈�r2(τ)〉] (5)
mmersion lens. Forty-five frames per second were recordedith a UP800 CCD camera (UNIQVISION, USA), typically
or 20 s, and digitalized with a PCDIG L frame grabber (Dalsaoreco, CA). The motions of approximately 40 particles wereetermined using the tracking software in Image-Pro PlusMedia-Cybernetics, USA), in order to obtain the MSD as aunction of observation time. The one-point MSD was obtainedy averaging over all the particles and all the initial times:
�r2(τ)〉OPMR = 〈[rα(t + τ) − rα(t)]2〉t,α (6)
here rα(t) is the position of the particle α at time t, and τ, theag time.
In the two-point correlation analysis, a correlated diffusionoefficient was calculated by averaging over all the particleombinations and all the initial times:
rr(R, τ) = 〈�riR(t, τ) · �r
jR(t, τ)〉t,i�=j (7)
here �riR(t, τ) is the displacement during the observation time
f particle i along the vector connecting the two particles i and, and R is the distance between the two particles. For an incom-ressible fluid, with R > a, where a is the bead radius, it can behown that [18]:
rr(R, s) = kBT
2πRsG(s)(8)
ith Drr(R, s) and G(s) the Laplace transforms of Drr(R, s) and
(s). Finally by drawing a parallel with the GSER a two-pointSD can be derived:
�r2(τ)〉TPMR = 2R
aDrr(R, τ) (9)
.6. Bulk rheology
Rheological measurements were performed on a stress-ontrolled rheometer (Parr Physica UDS 300; Physica, Stuttgart,ermany) using the cone-plate geometry. The angle of the coneas 4◦ and its diameter 40 mm. The sample was loaded into the
heometer and a solvent trap was fitted to avoid water evapo-ation. All the measurements were performed in duplicate at aonstant temperature of 20 ◦C.
. Results and discussion
Fig. 1 shows the MSDs obtained from our microrheologi-al investigations of a series of glycerol/water mixtures of 60,
0, 90 and 98% (w/w), plotted as a function of time on log-rithmic axes. In accordance with our expectations for purelyiscous materials the data show linear dependences of the log-rithm of the MSDs on logarithmic time, with slopes close to
M.A.K. Williams et al. / J. Non-Newtonian Fluid Mech. 149 (2008) 63–70 67
Fig. 1. The MSDs obtained from our microrheological investigations of a seriesopt
1tooetTavpcl
sabts
fl(tstoptletttt
spat
Fig. 2. (a). The MSD obtained for a 4% (w/w) solution of PEO, prepared asdescribed in the experimental section, plotted against time on logarithmic axes.The solid line shows the fit to a local power law. (b) The viscoelastic prop-em
Frwaepstb
�
waptu
f glycerol/water mixtures of 60 (©), 80 (�), 90 (♦) and 98 (�) % (w/w),lotted against time on logarithmic axes. The solid lines are predicted based onhe accepted viscosity values of these solutions.
, over the whole frequency range examined. The size of theracer particles used in the different experiments were differentwing to the availability of labelled particles and the desire tobserve significant displacements over short times in the DWSxperiments. Therefore, in order to directly compare the datasetshe measured MSDs have been scaled to a single particle size.he success of this procedure is evident in the figure and goodgreement can be seen between the techniques as reported pre-iously [9]. Although the DWS experiments proved difficult toerform at the highest concentrations, simply owing to the diffi-ulty of including sufficient tracer particles to achieve reasonable∗ values, our particle tracking data are particularly robust andhow no tendency towards anomalous sub-diffusive behaviours has been reported previously [1]. In addition, the predictedehaviour based on the accepted viscosity values of these solu-ions are shown on the figure and reasonable agreement can beeen.
Next attention was turned towards a viscoelastic solution ofexible polymers. Fig. 2(a) shows the MSD obtained for a 4%w/w) solution of PEO, prepared as described in the experimen-al section. Once again the DWS and particle tracking data werecaled to account for differences in bead size, and found therebyo produce a consistent dataset, in good agreement with previ-us studies [9]. The MSD can be seen to be well described by aower law behaviour as found previously [22,28], with a long-ime viscous behaviour, characterised by a linear region in theogarithm of the MSD versus logarithmic time plot with a gradi-nt close to 1, and viscoelastic behaviour predominant at shorterimes. Fig. 2(b) shows the viscoelastic properties derived fromhe MSDs compared with measurements made using a conven-ional rheometer, and indeed within experimental uncertaintieshere is good agreement between the methodologies.
Encouraged by the results from what might be considered as
tandard systems an ionotropic gel of the biologically importantolysaccharide pectin was studied. The gel (R = 0.4) was formeds described in the experimental section. The results obtained forhe MSDs by DWS and multiple particle tracking are shown infpc(
rties derived from the MSDs shown in the figure (solid line), compared witheasurements made using a conventional rheometer, G′(©) and G′′(�).
ig. 3(a). There is a striking resemblance between the one-pointesults, that again are consistent with each other, and previousork reported on actin solutions [10,29–31] and cross-linked
ctin networks [32,33]. Motivated by this obvious similarity wendeavoured to fit our one-point MSD to a theoretical expressionreviously used in actin work [29]. The theoretical MSD for amall bead that couples to single filament eigenmodes that arehemselves coupled to collective modes of an overdamped elasticackground is predicted to have the form:
x2(t) = A{1 − (π/t)a erf(ta)/2} + B{1 − btbΓ [−b, t]} (10)
here a = 1/2, b = 3/4, t is in units of τc, erf is the error functionnd Γ is the incomplete gamma function. The fit of the one-oint DWS and MPT data, to this functional form is shown inhe figure. Given the good agreement in the time domain it isnsurprising to find that the viscoelastic properties we obtain
rom the transformation of this data, also closely resemble thatreviously reported, in particular giving both elastic and vis-ous moduli (G′and G′′) scaling with ω3/4 at high frequenciesFig. 3(b)). Furthermore, the switch to a frequency independent
68 M.A.K. Williams et al. / J. Non-Newtoni
Fig. 3. (a) The MSDs obtained from experiments using DWS, and multipleparticle tracking, on an ionotropic gel of the polysaccharide pectin formed asdescribed in the experimental section, plotted against time on logarithmic axes.The solid line is a fit to a theoretical model, described in the text. (b) Theviscoelastic properties obtained from the transformation of this data. The solidguideline shows that the one-point microrheology results, G′ (filled symbols)and G′′ (open symbols), scale with ω3/4 at high frequencies. The filled and opensd
et
ogflroncwwpcw
i
fesimmHrtaaiafi
ttepsetwteopdt
npprafter sonification of the gel we clearly observed, simply fromthe bead movement, that some parts of the material were con-siderably less elastic than before the treatment. Furthermore,there was a significant spatial variation to the bead’s movements
quares show the two-point result for G′ and G′′, respectively and the solid andashed lines the data obtained by conventional bulk rheometry.
lastic modulus at low frequencies also looks extremely similaro data obtained on cross-linked actin filaments [32,33].
These data provide strong evidence that, at least in this regimef cross-link density, the viscoelastic properties of the pectinel can be modelled as a lightly cross-linked network of semi-exible filaments, with scaling power of 3/4 arising from theedistribution of bending modes of individual filaments inducedn compression or stretching [34–36]. Although pectin gels areot traditionally considered to be particularly filamentous inharacter the individual polymers are themselves semi-flexible,ith persistence lengths of around 15 sugar residues comparedith a contour length of some 200–500 [37]. Furthermore, theirolyelectrolytic nature and the postulated increased stiffness ofalcium-cross-linked dimers, the elementary units of the net-
ork, makes them interesting model systems.While the latest advances using fast cameras, laser traps andnterferometers are able to measure two-point data over a large
Fs(
an Fluid Mech. 149 (2008) 63–70
requency range our frequency range is currently limited. Nev-rtheless, it is safe to say that the two-point data in this systemhows some differences to the one-point, that are indeed sim-lar to that previously reported [31]. The two-point data is, inagnitude at least, slightly closer to the results of measure-ents carried out using conventional bulk rheology, also shown.owever, in general all techniques agree reasonably within the
estricted frequency range available, with variations in the elas-ic moduli similar to those found in other gelled systems [38]nd typical of sample to sample variation. Future work will aimt controlling the equivalent filament length in our system, tonvestigate whether the observed enhanced viscoelastic relax-tion at intermediate frequencies scales with the square of thelament length, as found in actin systems [31].
Additionally we sought to carry out preliminary investiga-ions into the possibility of using these techniques for assessinghe evolution of large scale heterogeneities in mechanical prop-rties that may occur for example during gelation or breakdownrocesses. Firstly, experiments were carried out on macro-copically biphasic systems, generated as described in thexperimental section. These results confirmed that, as expected,he DWS data obtained when the beam spanned both phasesas in fact representative of some average of the properties of
he two individual phases. Using multiple particle tracking how-ver, simply by focussing on two well separated areas either sidef the interface, the mechanical properties of the two individualhases were easily distinguished and were, at least at reasonableistances from the interface, the nominal values obtained whenhe phases were measured independently.
Secondly, we took a pre-formed pectin gel and disrupted theetwork using sonification. Initially we estimated, simply byerforming repeat measurements of the MSD as a function ofosition on the quiescent gel, that the spatial variation of theecorded MSD was on the order of 5%. However, immediately
ig. 4. Spatially resolved MSDs, plotted against time on logarithmic axes, mea-ured at different locations in a pectin gel (R = 0.4), (i) prior to sonification andii) immediately after sonification of the gel.
M.A.K. Williams et al. / J. Non-Newtonia
Fmf
nTFwtaspit
4
svcmtpdoaiwdartoe
A
hA
PwS
R
[
[
[
[
[
[
[
[
[
[
[
[
ig. 5. The time evolution of the MSD, plotted against time on logarithmic axes,easured on a pectin gel (R = 0.4) following relaxation toward the initial state
ollowing sonification.
ow, far in excess of the 5% observed before the gel disruption.he MSDs shown in Fig. 4 suffice to demonstrate this point.inally, we observed that at a location in the gel where the MSDas significantly enhanced immediately following sonification,
he mechanical properties evolved as a function of time, so thatfter 1 h the properties more closely approximated that of thetarting material (Fig. 5). While these measurements are of areliminary nature they reveal the potential to measure interest-ng re-healing dynamics of disrupted networks, including howhe spatial heterogeneity of the network evolves.
. Conclusion
Results from glycerol/water mixtures and polyethylene oxideolutions were found to agree with expectations for purelyiscous and flexible polymer solutions respectively, givingonfidence in the techniques used. Moreover, the subsequenticrorheological characterisation of a calcium-induced gel of
he polysaccharide pectin, yields similar results to those obtainedreviously for actin, with a low frequency, frequency indepen-ent elastic modulus (G′ > G′′) and a high frequency scalingf both G′ and G′′ with ω3/4. For this sample MPT results arenalysed in order to yield both one- and two-point microrheolog-cal measurements and these are shown to compare favourablyith those obtained by additional experiments carried out usingiffusing wave spectroscopy (DWS) and those obtained using
conventional rheometer. Furthermore, preliminary resultseported here indicate that these techniques are ideally suitedo the study of the temporal evolution of the spatial distributionf mechanical properties in such polysaccharide samples, forxample during gelation or digestion.
cknowledgements
The authors gratefully acknowledge Adrian Kitson for hiselp with Mathematica, and the MacDiarmid Institute fordvanced Materials and Nanotechnology for the funding of a
[
[
n Fluid Mech. 149 (2008) 63–70 69
hD studentship (RRV). The CCD camera used in this workas purchased with assistance from the Institute of Fundamentalcience Graduate Research Fund (IFSGRF).
eferences
[1] T.A. Waigh, Microrheology of complex fluids, Rep. Prog. Phys. 68 (2005)685–742.
[2] T.G. Mason, D.A. Weitz, Optical measurements of the linear viscoelasticmoduli of complex fluids, Phys. Rev. Lett. 74 (1995) 1250–1253.
[3] J.C. Crocker, D.G. Grier, Methods of digital video microscopy for colloidalstudies, J. Colloid Interf. Sci. 179 (1996) 298–310.
[4] F. Scheffold, S.E. Skipetrov, S. Romer, P. Schurtenburger, Diffusing-wavespectroscopy of non-ergodic media, Phys. Rev. E 63 (2001) 061404.
[5] A. Papagiannopoulos, T.A. Waigh, A. Fluerasu, C. Fernyhough, A. Mad-sen, Microrheology of polymeric solutions using X-ray photon correlationspectroscopy, J. Phys.: Condens. Matter 17 (2005) L279–L285.
[6] T.G. Mason, Estimating the viscoelastic moduli of complex fluids usingthe generalised Stokes Einstein equation, Rheol. Acta 39 (2000) 371–378.
[7] D.J. Pine, D.A. Weitz, P.M. Chaikin, E. Herbolzheimer, Diffusing wavespectroscopy, Phys. Rev. Lett. 60 (1988) 1134–1137.
[8] T.G. Mason, H. Gang, D.A. Weitz, Diffusing-wave spectroscopy measure-ments of viscoelasticity of complex fluids, J. Opt. Soc. Am. A 14 (1) (1997)139–149.
[9] T.G. Mason, K. Ganesan, J.H. vanZanten, D. Wirtz, S.C. Kuo, Particletracking microrheology of complex fluids, Phys. Rev. Lett. 79 (1997)3282–3285.
10] F. Gittes, B. Schnurr, P.D. Olmsted, F.C. MacKintosh, C.F. Schmidt, Micro-scopic viscoelasticity: shear moduli of soft materials determined fromthermal fluctuations, Phys. Rev. Lett. 79 (1997) 3286–3289.
11] T.G. Mason, A. Dhople, D. Wirtz, Concentrated DNA rheology andmicrorheology, MRS Proc. Stat. Mech. Phys. Biol. 463 (1997) 153–158.
12] J. Apgar, Y. Tseng, E. Fedorov, M.B. Herwig, S.C. Almo, D. Wirtz, Multi-ple particle tracking measurements of heterogeneities in solutions of actinfilaments and actin bundles, Biophys. J. 79 (2000) 1095–1106.
13] M.T. Valentine, P.D. Kaplan, D. Thota, J.C. Crocker, T. Gisler, R.K.Prud’homme, M. Beck, D.A. Weitz, Investigating the microenvironmentsof inhomogeneous soft materials with multiple particle tracking, Phys. Rev.E 64 (2001) 061506.
14] D.T. Chen, E.R. Weeks, J.C. Crocker, M.F. Islam, R. Verma, J. Gruber,A.J. Levine, T.C. Lubensky, A.G. Yodh, Rheological microscopy: localmechanical properties from microrheology, Phys. Rev. Lett. 90 (10) (2003)108301.
15] S. Yamada, D. Wirtz, S.C. Kuo, Mechanics of living cells measured bylaser tracking microrheology, Biophys. J. 78 (2000) 1736–1747.
16] Y. Tseng, J.S.H. Lee, T.P. Kole, I. Jiang, D. Wirtz, Micro-organization andvisco-elasticity of the interphase nucleaus revealed by particle nanotrack-ing, J. Cell Sci. 117 (2004) 2159–2167.
17] M.T. Valentine, Z.E. Perlman, M.L. Gardel, J.H. Shin, P. Matsudaira,Colloid surface chemistry critically affects multiple particle tracking mea-surements of biomaterials, Biophys. J. 86 (2004) 4004–4014.
18] J.C. Crocker, M.T. Valentine, E.R. Weeks, T. Gisler, P.D. Kaplan, A.G.Yodh, D.A. Weitz, Two-point microrheology of inhomogeneous soft mate-rials, Phys. Rev. Lett. 85 (4) (2000) 888.
19] A.J. Levine, T.C. Lubensky, Two-point microheology and the electrostaticanalogy, Phys. Rev. E 65 (2001) 011501.
20] M. Buchanan, M. Atakhorrami, J.-F. Palierne, C.F. Schmidt, Comparingmacrorheology and one and two-point microrheology in wormlike micellesolutions, Macromolecules 38 (2005) 8840–8844.
21] W.G.T. Willats, P. Knox, J.D. Mikkelson, Pectin: new insights into andold polymer are starting to gel, Trends Food Sci. Technol. 17 (3) (2006)97–104.
22] B.R. Dasgupta, S.-Y. Tee, J.C. Crocker, B.J. Frisken, D.A. Weitz, Microrhe-ology of polyethylene oxide using diffusing wave spectroscopy and singlescattering, Phys. Rev. E 65 (2002) 051505.
23] A. Strom, M.A.K. Williams, Calcium release in the absence and presenceof an ion-binding polymer, J. Phys. Chem. 107 (40) (2003) 10995–10999.
7 wtoni
[
[
[
[
[
[
[
[
[
[
[
[
[
[mations of pectin polysaccharides: determination of chain characteristics
0 M.A.K. Williams et al. / J. Non-Ne
24] C. Lofgren, S. Guillotin, A.-M. Hermansson, Microstructure and kineticrheological behaviour of amidated and nonamidated LM pectin gels,Biomacromolecules 7 (2006) 114–121.
25] T.M. Squires, J.F. Brady, A simple paradigm for active and nonlinearmicrorheology, Phys. Fluids 17 (2005) 073101.
26] Y. Hemar, D.N. Pinder, DWS Microrheology of a linear polysaccharide,Biomacromolecules 7 (2006) 674–676.
27] D.A. Weitz, D.J. Pine, in: W. Brown (Ed.), Dynamic Light Scattering,Oxford University Press, Oxford, 1992.
28] J.H. vanZanten, S. Amin, A.A. Abdala, Brownian motion of colloidalspheres in aqueous PEO solutions, Macromolecules 37 (2004) 3874.
29] J. Xu, V. Viasnoff, D. Wirtz, Compliance of actin filament networks mea-sured by particle-tracking microrheology and diffusing wave spectroscopy,Rheol. Acta 37 (1998) 387–398.
30] J. Xu, A. Palmer, D. Wirtz, Rheology and microrheology of semi-flexible
polymer solutions: actin filament networks, Macromolecules 31 (1998)6486–6492.31] J. Lui, M.L. Gardel, K. Kroy, E. Frey, B.D. Hoffman, J.C. Crocker, A.R.Bausch, D.A. Weitz, Microrheology probes length scale dependent rheol-ogy, Phys. Rev. Lett. 96 (2006) 118104.
[
an Fluid Mech. 149 (2008) 63–70
32] A. Palmer, J.Y. Xu, D. Wirtz, High frequency viscoelasticity of cross-linkedactin filament networks measured by diffusing wave spectroscopy, Rheol.Acta 37 (1998) 97–106.
33] G.H. Koenderink, M. Atakhorrami, F.C. MacKintosh, C.F. Schmidt, Highfrequency stress relaxation in semiflexible polymer solutions and networks,Phys. Rev. Lett. 96 (2006) 138307.
34] F. Gittes, F.C. MacKintosh, Dynamic shear modulus of a semiflexiblepolymer network, Phys. Rev. E 58 (1998) R1241.
35] D.C. Morse, Viscoelasticity of tightly entangled solutions of semiflexiblepolymers, Phys. Rev. E 58 (1998) R1237.
36] D.C. Morse, Viscoelasticity of concentrated isotropic solutions ofsemiflexible polymers. 2. Linear response, Macromolecules 31 (1998)7044.
37] S. Cros, C. Garnier, M.A.V. Axelos, A. Imberty, S. Perez, Solution confor-
by small angle neutron scattering, viscometry and molecular modelling,Biopolymers 39 (1996) 339–352.
38] B.R. Dasgupta, D.A. Weitz, Microrheology of cross-linked polyacrylamidenetworks, Phys. Rev. E 71 (2005) 021504.