Heterogeneous and anisotropic dynamics of a 2D gel

D. Orsi,1 L. Cristofolini,1, ∗ G. Baldi,1, 2 and A. Madsen3, 4

1Physics Department, Parma University, I-43124, Parma, Italy2CNR-IMEM Institute, I-43124 Parma, Italy

3European Synchrotron Radiation Facility, B.P. 220, F-38043 Grenoble, France4European X-Ray Free Electron Laser, Albert-Einstein-Ring 19, D-22761 Hamburg, Germany

(Dated: July 30, 2018)

We report X-ray Photon Correlation Spectroscopy (XPCS) results on bi-dimensional (2D) gelsformed by a Langmuir monolayer of gold nanoparticles. The system allows an experimental deter-mination of the fourth order time correlation function, which is compared to the usual second ordercorrelation function and to the mechanical response measured on macroscopic scale. The observeddynamics is anisotropic, heterogeneous and super-diffusive on the nanoscale. Different time scales,associated with fast heterogeneous dynamics inside 2D cages and slower motion of larger parts ofthe film, can be identified from the correlation functions. The results are discussed in view of otherexperimental results and models of three-dimensional gel dynamics.

PACS numbers: 64.70.pv, 68.18.-g, 83.80.Kn, 83.85.Hf

Many of the diverse properties of soft materials origi-nate from their complex structures and dynamics charac-terized by multiple length and time scales. A wide varietyof technologies, from paints to food science and from oilrecovery to personal care products, depend crucially onunderstanding the dynamics of complex interfacial sys-tems. Gels and their dynamics have been widely studiedin 3D bulk configurations [1–3], however, dimensionalityis known to be quantitatively important in arrested sys-tems: e.g. hard spheres caging occurs in 3D at a volumefraction Φ3D

RCP ' 60−64%, while in 2D the random closepacking is attained at an area fraction Φ2D

RCP ' 82% [4].

The effect of dimensionality on the glass transitionprocess in frictionless colloids has been investigated innumerical simulations [5], within the mode couplingapproach [6], and in experiments on granular materi-als [7, 8]. Heterogeneous dynamics of micron sized beadsconfined in 2D and interacting through complex inter-particle potentials, either attractive or repulsive, haverecently been studied by means of video microscopy andparticle tracking techniques [9, 10]. Both simulations andtheory suggest that the behavior is not significantly dif-ferent in 2D and in 3D. This is surprising as dimensional-ity usually plays a decisive role in phase transitions. How-ever, this point still lacks a definite experimental confir-mation and, as a matter of fact, a recent comparison be-tween surface and bulk dynamics in a repulsive colloidalsuspension evidenced heterogeneous ballistic motion par-allel to the surface while the dynamics in the directionperpendicular to the surface was much slower and appar-ently diffusive [11].

Here we report the first study by X-ray Photon Corre-lation Spectroscopy (XPCS) [12] of interfacial dynamicsin a strictly two-dimensional system formed by a Lang-muir monolayer of weakly attractive nano-particles at theair/water interface. We compare the microscopic dy-namics probed with XPCS and the mechanical moduli

measured by Interfacial Shear Rheometry (ISR, see refs.[13, 14]) on the same samples. Typically, dynamical het-erogeneity is identified by calculating the variance χ ofthe two-times correlation function C(t1, t2). Here, weprovide the first experimental determination – by XPCS– of the fourth order correlation function g(4)(t, τ) [15–17]giving direct information about the lifetime of dynamicalheterogeneity upon approaching dynamical arrest.

Gold nanoparticles of 70A diameter, stabilized bydodecanethiol coating, were produced by Ruggeri andcoworkers (Pisa University) following the literature [18],and thoroughly characterized by dynamic light scatter-ing and electron microscopy, both in transmission andin scanning (SEM) geometry. At the air/water interfacethe particles possess a weakly attractive interaction po-tential due to hydrophobic interactions. Uniform Lang-muir monolayers are prepared through successive slowcompression/expansion cycles at constant temperature,following the method given in ref. [19]. The fraction (Φ)of surface covered by the film is usually deduced fromthe dispersed aliquot. The determination of Φ is cru-cial [20], therefore in this work it was double checked byin-situ null-ellipsometry and by SEM imaging of mono-layers transferred onto a solid substrate. The three in-dependent determinations of Φ coincide, as reported in[21], within an errorbar of roughly ±4%. A typical SEMimage is shown in the top panel of figure 1. It evidencesstructural heterogeneity, with clusters and holes of dif-ferent sizes ranging from a few tens of nm to severalmicrons. The distribution of hole sizes, shown in the in-set, is strongly asymmetrical and compatible with a Levydistribution [22] characterized by an algebraic tail withexponent −3. No crystalline phase was detected neitherby X-ray diffraction nor by SEM. The measurements andsample preparation was always performed following thesame protocol in order to eliminate uncertainties due todifferent states of sample aging.

arX

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t] 3

Feb

201

2

2

1 µm

0 0.2 0.4 0.610

−3

10−2

10−1

Diameter l (µm)

N (

%)

FIG. 1: (color online) Top: a typical SEM image of a mono-layer with Φ = 50% transferred onto a solid substrate. Inset:distribution of holes on a log scale, and its best fit with theLevy distribution (continuous line). Bottom: sketch of thesetup used for XPCS experiment on a Langmuir monolayer.

The mechanical properties were characterized on themacro-scale by ISR [13]. As commonly found in many gelsystems the loss factor tan δ = G”/G′ is constant over themeasured concentration range, while the complex shearmodulus G∗ rapidly increases with increasing concentra-

tion, and can be modeled as a power law G∗ ∝(

Φ−Φ0

Φ0

)z

[21, 23, 24]. In the present case Φ0 = 0.27(1), close to theconcentration at which the gel forms, and the exponentis z = 0.65(1), which is smaller than 2.4 observed in pro-tein gels [25] and 4.0 found in carbon black gels [26]. Thedifference is probably due to the reduced dimensionalityof the present system.

XPCS experiments were performed using partially co-herent X-rays at the Troika beamline ID10A of the Eu-ropean Synchrotron Radiation Facility. A single bounceSi(111) crystal monochromator is used to select 8.06 keVradiation and the transversely coherent beam is definedby slit blades with carefully polished cylindrical edges.A set of guard slits placed just upstream of the sam-ple blocks parasitic scattering due to diffraction fromthe beam-defining slits. A MAXIPIX detector [27] isused to collect speckle images close to the specular re-flected beam; expressed as a function of q, the range from0.002A−1 to 0.03A−1 was covered. Focusing on the slowdynamics, speckle patterns were collected at time inter-vals of 0.01−0.2s. The measurements were performed atconstant temperature (18◦C) and increasing surface pres-sure, ranging from 10 mN/m up to 30 mN/m, by meansof the apparatus sketched in figure 1. The signals fromindividual detector pixels were grouped in the analysis inorder to provide a q-sensitive map of the dynamics andincrease the signal-to-noise ratio.

10−2

100

102

0

0.2

0.4

0.6

0.8

1

1.2

time t (s)

(g(2

) − A

) / β

a)

q// = 0.004 Å−1

q// = 0.009 Å−1

q// = 0.015 Å−1

10−3

10−2

10−1

100

101

102

q// (Å−1)

Rel

axat

ion

time,

τ (

s)

b)

Φ = 35%Φ = 69%Φ = 77%Φ = 88%

G’ (m

N/m

)

10−1

100

101

30 40 50 60 70 80 9010

−2

10−1

100

101

102

τ (

s)

Concentration, Φ (%)

c)

FIG. 2: (color online) a: scaled correlation functions g(2) forselected q|| values (lines are fit to the data) from a Langmuirfilm at Φ = 69%. b: relaxation times τ for different surfaceconcentrations Φ. Continuous lines are fits to the power lawdiscussed in the text. c: relaxation times τ versus surfaceconcentration Φ at q = 0.015A−1 (squares, left scale) andshear modulus G′ measured by ISR (circles, right scale). Thecontinuous line is the power law fit described in the text.

Intensity auto correlation functions are calculated in

the usual way: g(2) (q, t) = 〈I(q,t1)I(q,t1+t)〉〈I(q,t1)〉2 and are re-

ported in figure 2a for Φ = 69% and q⊥ = 0.005A−1.They can be modeled as

g(2) (q, t) = A+ βe−2(t/τ)γ (1)

where A is the baseline, β is the contrast, τ the relaxationtime and γ the compression coefficient, determining theshape of the relaxation. The shape can be taken as asignature of the interactions present in the system: e.g.one expects γ = 1 for Brownian motion while γ = 1.5is often found in colloidal gels [28]. We find γ = 1.5(1)for all the concentrations studied here. Figure 2b showsthe q dependence of the relaxation times τ obtained fromfits of eq. (1) to the correlation functions taken at var-ious concentrations Φ. Different kinds of dynamics canusually be distinguished by the relation between τ andq: in the present case, at all concentrations, a power lawτ ≈ q−n is found with exponent n ' 0.8(1) (continuouslines in the figure). A similar behavior has been found ina variety of arrested systems [1], e.g. in oil nanoemulsion[30], in aqueous colloidal polystyrene gels [31], and re-cently also by us in molecular layers of polymers [32]. Infigure 2c the dependence of τ on Φ is shown, measured atconstant momentum transfer (q = 0.015 A−1), and com-pared with the 2D storage shear modulus G′ measured byISR on the same film: they share roughly the same powerlaw dependence on Φ represented by the continuous line.

3

The relation between microscopic fluctuations (τ) anda macroscopic response function (G′) was predictedwithin the Bouchaud-Pitard model [29]. This model as-sumes the dynamics to be governed by micro-collapses ofthe gel, resulting in the formation of force dipoles, whichaffect the surrounding particles by inducing a strain field.With the further assumptions that the collapses occurrandomly in space and time, the stochastic equations ofmotion for the strain field can be solved analytically. Inthe slow-collapse regime, the resulting intermediate scat-tering function has a compressed shape with γ = 1.5,while the relaxation time is proportional to the macro-scopic elastic modulus of the gel and inversely propor-tional to the momentum q. Our findings confirm all themodel predictions.

Figure 3a displays the q-map of the relaxation timesτ obtained by fitting g(2) with eq. (1). The dynamics isclearly hallmarked by the anisotropic dependence of τ onthe exchanged momentum, where τ depends only on thesurface parallel component q|| (horizontal direction in themap) and not on the vertical component q⊥. This pictureholds for all the concentrations studied, proving that thenanoparticle dynamics is confined within the air/waterinterfacial plane.

The correlation functions characterize dynamics withina given time window, whose short-time limit is mainlydetermined by the amount of scattered intensity and bythe detector speed. Any faster dynamics will reducethe contrast of the correlation function to less than theSiegert factor (∼ 0.2 in the present setup). Informationon fast motion, if present, can be obtained by studyingthe missing contrast and its q-dependence. In this casethe contrast β follows a pseudo Debye-Waller law [33–

35]: β = β0 exp(− q

2a2

3

), indicating that at least parts

of the missing contrast is due to a faster rattling motioncharacterized by the localization length a. The fits aredisplayed in figure 3b. The inset shows the variation ofa: with growing Φ it decreases to reach a limiting valueof 50 A, comparable to the mean particle radius. Theextrapolation of β to q|| = 0 never reaches the Siegertfactor, suggesting that several fast dynamical modes con-tribute to reducing the contrast. For instance, more mo-bile particles with mean square displacements larger than1/qmin ' 250A[34] could be present.

With several populations of particles moving distinctly,the presence of dynamical heterogeneity in the fast dy-namics is implied and can be further characterized byhigher order correlation functions. In figure 3c we reportthe normalized two times correlation function C(t1, t2) =I(t1) ∗ I(t2) [36] measured at q|| = 0.009A−1, T = 18◦Cand Φ = 69%. Its variance χ, calculated as in ref. [37], isshown in figure 3d: it features a broad maximum aroundτc = 0.5 − 1s, which is an evidence of dynamical het-erogeneity. Thanks to the good quality of the data it ispossible to bring the analysis one step further and calcu-

q|| (10−3 Å−1)

q ⊥ (

10−3

Å−1

)

a)

5 10 15 20

4

6

8

10

12

14

0 1 2 3 4 5 610

−2

10−1

Con

tras

t, β

q||2 (10−4 Å−2)

Φ = 35%Φ = 69%Φ = 77%Φ = 88%

0 50 1000

100

200

Φ (%)

a (Å

)

b)

t1 (s)

t 2 (

s)

c)

10 20 30 40 50

10

20

30

40

50

τ10

−210

−110

010

1

0

0.2

0.4

0.6

0.8

1

1.2

χ/β

2 (

10

−4)

τ (s)

d)

FIG. 3: (color online) a: q-map of relaxation times τ mea-sured at Φ = 69% along different directions in reciprocalspace: q|| and q⊥ are respectively parallel and perpendicularto the water surface. Solid lines are isochrones. b: contrastβ as a function of squared momentum q2||, for different valuesof Φ. Dashed lines are Debye-Waller fits. Inset: localizationlength at different values of Φ. The continuous line is a guideto the eye. c: two times correlation function for Φ = 69%,T = 18◦C and q|| = 0.009A−1. The lighter stripe represents

the range over which g(4) is calculated, as described in thetext. The arrow indicates the time τ of equation (2). d:normalized variance χ of the two-times correlation functionreported in panel c; the continuous line is a guide to the eye.

late the four times correlation function g(4) defined as:

g(4) (t, τ) = 〈C (t1, t1 + τ) · C (t1 + t, t1 + t+ τ)〉t1 =

〈I (t1) · I ( t1 + τ) · I (t1 + t) · I ( t1 + t+ τ)〉t1 (2)

The first lag time τ = |t1− t2|, indicated by the lighterstripe in figure 3c, identifies the sub-diagonal of matrixC along which g(4) is calculated. The second lag time,t, corresponds to the separation between two instants ofthe selected sub-diagonal. Averages are performed overinitial time t1. We choose τ = τc because the varianceanalysis suggests that on this time scale the heteroge-neous dynamics is most pronounced.

Figure 4a reports g(4) , for q|| = 0.009A−1 and τ = τc,and compares it with the usual 2nd order correlationfunction g(2). Both functions display a full decay dur-ing the timescales accessible in the experiment but thefunctional form of g(4) appears to be characterized by apeak located at τ∗. This observation holds for all con-centrations studied, as shown in 4b. We take τ∗ as ameasure of the timescale of dynamical heterogeneity andtrack its dependence on q as shown in 4d. In accordancewith the previous discussion concerning the ”missing con-

4

10−1

100

101

102

103

0

1

2

3

x 10−4

time t (s)

g(4

) /β2

1

1.02

1.04

1.06

1.08

g(2

) (t)

g(2)

(t)g

(4)(t, τ

C)

a)

10−2

10−1

100

101

102

103

0

0.5

1

1.5

2

2.5

3

3.5x 10

−4

g(4

) / β

2

time t (s)

Φ=69%

Φ=88%

b)

0 0.005 0.01 0.015 0.020

0.5

1

1.5

2

2.5

x 10−4

q|| (Å−1)

χ*

Φ=69%Φ=88%

c)

10−3

10−2

10−2

10−1

100

101

102

q|| (Å−1)

τ, τ

* (s

)

τ* − Φ=69%τ* − Φ=88%τ − Φ=69%τ − Φ=88%

d)

FIG. 4: (color online) a: comparison between g(4) and g(2)

both measured at Φ = 88% and q|| = 0.009A−1. Lines are

guides to the eye. b: normalized g(4) for different values of Φand q|| = 0.009A−1. Lines are guides to the eye. c: Intensity

χ∗ of the peak of g(4) as a function of q||, calculated for Φ =69% and Φ = 88%; dashed lines represent their averages. d:q||-dependence of characteristic times τ∗(empty symbols) andof τ (filled symbols) for the values of Φ indicated. They allfollow an approximate ∼ q−1

|| scaling law (lines).

trast” it is found that τ∗ < τ for all q and for all con-centrations studied. In addition the q-dependencies of τ∗

and τ appear to be similar (∼ q−1|| ) even if the two time

scales originate from different dynamics. Although akinobservations have been made before, see e.g. Ref. [15],a deeper understanding of this ”scaling behavior” is stillmissing. No q⊥ dependence of the variance χ or of τ∗

was observed, indicating that all the observed dynam-ics (fast and slow) and the heterogeneous behavior aresurface phenomena of 2D nature.

For the highest concentration g(4) has a larger ampli-tude (χ∗) as observed in 4b, which in turn appears atslower times. In the analogy between temperature andpacking fraction, this behavior is in accordance with pre-dictions of numerical simulations [38]. A similar behav-ior was also found in the variance χ measured by XPCS[16] and photon correlation imaging [39] in 3D systems.Here, the decreased mobility is followed by an increasein the range of spatial and temporal correlations betweenparticles, in agreement with the general assumption thatdynamical arrest is related to the development of hetero-geneities.

Within the statistical uncertainty, no clear q-dependence could be detected in χ∗, as shown in figure 4c.This suggests that a hierarchy of different length scales is

involved in the heterogeneous behavior, and possibly thefast, non-localized population could be modeled as Levyflights [40, 41], similarly to other systems presenting dy-namical heterogeneity [15, 32].

In conclusion, we present the first XPCS experimenton a Langmuir monolayer (2D gel) displaying anisotropicand heterogeneous dynamical behavior. The data qualityallows a rare experimental characterization of the dynam-ics by calculating the fourth order temporal correlationfunction g(4). The anisotropic dynamics occurs on dif-ferent time – and length – scales: fast collective hetero-geneous rearrangements characterized by τ∗ are respon-sible for the peak in g(4), but are too fast to be directlyseen in g(2), where they only appear as reductions of thecontrast. Part of the missing contrast can be accountedfor using a Debye-Waller model to describe fast rattlingon the nano scale but the data also indicate that fastdiffusive-like motion is present. g(2) tracks a much slowerergodicity restoring dynamics characterized by the relax-ation time τ . Both fast and slow motions are anisotropicand confined to the surface plane (2D) and have similardependencies on q, and on the coverage fraction Φ. Fi-nally, we note that the relaxation time scales with Φ sim-ilarly to the 2D mechanical response measured by ISR.This, together with the non-Brownian character of theergodicity restoring dynamics, is in good agreement withthe model proposed by Bouchaud and Pitard [29].

The ESRF is acknowledged for provision of beamtime, O. Konovalov (ESRF) for access to the Langmuirthrough, G. Ruggeri and A. Pucci (Pisa) for preparingthe nanoparticles, F. Pincella (Parma) for help with theexperiments, R. Leheny (John Hopkins) and P. Cicuta(Cambridge) for useful comments and discussions.

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