Soft Condensed Matter. Biannual Report 2003/2004

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University of Konstanz

SOFT CONDENSED MATTER PHYSICS Physics Department

Biannual Report 2003

2004

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Cover (from left):

Three-particle distribution function of colloids in the solid (left) and the liquid phase (right) (B3);

Vortex landscape in a high temperature superconductor (D1);

Microscope image of a 2D system of magnetic colloidal particles (C2);

Charged membrane with counter-ions (E2).

1

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Preface

This is the fourth report on research activities in soft condensed matter at the Physics Department of the University of Konstanz, which comprises the years 2003 and 2004. As previous reports it is intended to inform experts in this research area by surveying the experimental and theoretical work at Konstanz. During the last two years, Prof. R. Klein has retired and Prof. M. Fuchs has become Professor for Theoretical physics. One of our colleagues has left: Dr. H.H.

von Gr¨unberg accepted a position as Professor at the University of Graz. The group leaders express their gratitude to all scientific collaborators, the technical and administrative staff for their support. This report has been edited on behalf of the soft-condensed matter groups by

U. Gasser C. Aegerter G. Maret

Konstanz, July 2005.

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A Polymers

1

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A1 Structural and conformational dynamics of polymer melts

Th. Vettorel^∗, M. Aichele^∗,S.–H. Chong^†, J. Baschnagel^∗, and M. Fuchs

While vitrification of simple liquids is possible upon rapid cooling only, glasses can readily be produced in polymer melts, because internal disorder and fluctuations of the macromolecules prevent ordering. Polymer glasses thus are ubiquitous and are widely used technologically.

The characteristic feature of such macromolecular systems is the large number of internal (’conformational’) degrees of freedom a single polymer possesses. Because of chain–connectivity, conformational rearrangements are slow already at high temperatures and cause characteristic anomalies in single chain transport or relaxation processes. Their role during vitrification is not yet well understood, where they may influence transport and viscoelastic properties, which change strongly close to the glass transition. While phenomenological models, mostly work- ing with ’effective field’–ansatzes are widely used, little is known about how to handle the intra– and inter–molecular correlations on similar footings.

An important challenge for first-principles approaches in polymer science thus is to derive the well-known models of chain transport and relaxation in polymer melts [1]. Recently, we proposed an atomistic theory of the structural and conformational dynamics of a bead-spring model for an unentangled polymer melt [2]. Presently, we perform quantitative comparisons between simulation and first-principles calculations. On the one hand, our results explain the onset of the viscous slowing-down, ultimately leading to kinetic arrest into an amorphous solid (the glass transition). On the other hand, we find and explain characteristic deviations from Rouse behavior.

We explored the statics of a bead-spring model for a nonentangled, glass-forming polymer melt via various structure factors, involving not only the monomers, but also the center of mass (CM). We found that the conformation of the chains and the CM-CM structure factor, which is well described by a recently proposed approximation, remain essentially unchanged on cooling toward the critical glass temperatureTc. Spatial correlations between monomers on different chains, however, depend on temperature, albeit smoothly. This implies that the glassy behavior of our model cannot result from static intrachain or CM-CM correlations. It must be related to interchain correlations at the monomer level.

Our approach is based on the mode-coupling theory (MCT) of simple liquids and the site formalism. A key assumption of our theory, which renders it tractable even if the chain lengthNis large, is to replace the site-specific surroundings of a monomer (‘site’) by an averaged one

0 5 10 15 20

q 0.5

1.0

0.0 1.0

f q c

0.2 0.4 0.8

0.0 0.6

0 2 4 6 8 q q q*

Sq q*

Sq

6 7 8 9 q

0.0 0.5 1.0

Sq C q

C

q C 0 2 4

Figure 1: Glass-form factorsf_q^c(upper panel) and rescaledα- relaxation timesτq/τq^∗ ofφq(t)(lower panel) versusq. The circles represent the result from the simulation atT = 0.47, the solid line that from MCT. The dashed line in the upper panel denotes the extrapolatedSq(multiplied by 0.1) atT_c^MCT≈0.277.

The upper inset depicts the extrapolatedSq atT_c^MCT (dashed line), and the simulatedSqatT = 0.47(solid line),0.48(dotted line), and1(long-dashed line) near the first peakq^∗= 6.9. The lower inset showsS_q^C, the static structure factor of the chain’s center of mass, at all simulated temperatures.S_q^Cexhibits a weak maximum atqC= 3.4.

(equivalent-site approximation), while keeping the full site-dependence in a chain. For coherent structural dynamics, this leads to a set of closed (scalar) equations for density correlators

Fq(t) = (1/nN)hρ^tot_~_q (t)^∗ρ^tot_~_q (0)i, where

ρ^tot_~_q =

n

X

i=1 N

X

a=1

exp(i~q·~r_i^a)

denote the total monomer-density fluctuations for wave vector ~q. Here, ~r_i^a refers to the position of the ath monomer in theith chain,nis the total number of chains, andq = |~q|. In the following, we will deal with cor- relatorsφq(t) = Fq(t)/Sq, normalized by the total melt structure factorSq =Fq(0). Concerning the dynamics of a single (or tagged) polymer (labeleds), we obtain matrix equations for

F_q^ab(t) =hexp{−i~q·[~r_s^a(t)−~r_s^b(0)]}i

for which the matrix structure has to be retained to prop- erly account for the chain connectivity. Mean-square dis-

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A1. STRUCTURAL AND CONFORMATIONAL DYNAMICS OF POLYMER MELTS 3

-5 -4 -3 -2 -1 0 1 1.0

φ _q(t)

log ₁₀(t/τ )q*

q = 4.0

q = 6.9 q = 12.8

0.2 0.4 0.8

0.0 0.6

1.0

0.2 0.4 0.8

0.0 0.6

Figure 2:φq(t)as a function oft/τq^∗ forq = 4.0(left scale), 6.9, and 12.8 (right scale). τq^∗ is theα-relaxation time atq^∗. Circles refer to the simulation results atT = 0.47, solid lines to the MCTα-master curves, and dashed lines to the MCT curves at the distance parameterǫ^MCT=−0.022.

placements (MSD) can be expressed in terms ofF_q^ab(t)for q → 0. For the statics our approximate theory has been verified for wavelengths around the average monomer separation by simulation [3]. We recently extended the test by performing quantitative comparisons of the theory for collective and single-chain dynamics with molecular- dynamics (MD) simulations.

In the simulations, we study a bead-spring model of linear chains, each containing N = 10 monomers [1].

All monomers interact via a Lennard-Jones (LJ) potential which is truncated at twice the position of the minimum and shifted to zero. We will use LJ units in the following (ǫLJ/kB = 1,σLJ = 1, monomer massm = 1). In addition, successive monomers in a chain interact via a FENE potential; the superposition of the LJ- and FENE potentials imposes chain-connectivity. To test our theory we analyzed time series obtained from isobaric MD simulations at pressurep= 1. We determined the total melt structure factorSq and the site-resolved intrachain structure factorsw^ab_q = F_q^ab(0)[3]. These quantities are the only input that the theory requires to predict the dynamics of the model. Predicted and simulated dynamics may then be compared. Such a fully quantitative comparison is novel and, even for simple fluids, has only been attempted recently.

Figure 1 shows the glass form factorsf_q^c describing the frozen–in structure of the polymer glass at the transition temperature. The local structure exhibits the largest ar- rested fraction, and concomitantly also the slowest relaxation time in the melt state, close to arrest (fig. 1, lower panel). Figure 2 looks at the full dynamics of the coherent density fluctuations. Local length scales connected with the segmental ’cage effect’ are seen to enslave other

modes, and are well described by theory.

Simulations and theory yield a subdiffusive, Rouse-like behavior of the segmental MSD close toTc. Clearly, this polymer-specific feature is also present in the simulation data at highT. However, asT increases, the cage effect loses in importance, and it is thus not clear to what ex- tent the theory can still be applied. To test this, we analyze the MSDs atT = 1, which is more than twiceT_c^MD. Here, the theory utilizesSq directly from the simulation (no extrapolation is needed). Figure 3 indicates that, be- yond the short-time regime, the agreement between theory and simulation is very good forgM(t). In particular, we findgM(t)∼t^0.63with the same exponent.

-4 -3 -2 -1 0 1 2

log Dt

10 log g(t)10

2

-4 -3 -2 -1 0 3

1

0 5 10 15

q 0 1 2 3

q q*

M

C t

0.63

~

Figure 3: Double logarithmic plot ofgM(t)(label M) andgC(t) (label C) versusDtatT = 1. The inset shows the ratioτq/τq^∗

of theα-relaxation times ofφq(t)as a function ofq. Circles and solid lines refer to the simulation and MCT results, respectively.

The dotted line denotes the power law∼t^0.63.

Our results suggest that the onset of slow relaxation in a glass-forming polymer melt can be described in terms of monomer-caging supplemented by chain connectivity.

Furthermore, a unified atomistic description of glassy arrest and of conformational fluctuations that (asymptoti- cally) follow the ‘Rouse model’ emerges from the theory.

∗ Physique des Milieux Disperses, Institut Charles Sadron, 67083 Strasbourg, France

† Institute for Molecular Science, Okazaki 444-8585, Japan

[1] K. Binder, J. Baschnagel, and W. Paul, Prog. Polym. Sci.

28, 115 (2003).

[2] S.-H. Chong and M. Fuchs, Phys. Rev. Lett. 88, 185702 (2002).

[3] M. Aichele, S.-H. Chong, J. Baschnagel, and M. Fuchs, Phys. Rev. E 69, 061801 (2004).

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A2 Soft particle model for block copolymers

F. Eurich, A. Karatchentsev, J. Baschnagel (U. Stras- bourg), W. Dieterich, and P. Maass (TU Ilmenau) Phase separation processes in polymer melts with incompatible components play a prominent role in attempts to produce technologically interesting micro- or nanoscale structures through selforganization. Of particular interest are the kinetics and morphologies of phases in thin film geometries. Thin films with chemically prepatterned con- fining walls can be used to generate tailored structures through spontaneous demixing processes [1]. Clearly, such processes occur on semi-microscopic or even macro- scopic length- and time scales, orders of magnitude larger than those related to the motion of individual monomers.

From the theoretical point of view this poses the problem to eliminate internal degrees of freedom of polymer chains, which largely become irrelevant in such processes, and to seek for a coarse-grained description of the phase kinetics in terms of a small number of collective degrees of freedom.

A promising approach is to represent a polymer chain simply by one “soft particle”, an idea that is based on the fact that polymers in the melt strongly interpenetrate. This soft-particle picture should be valid for length and time scales similar to or larger than the radius of gyrationRand the diffusion timeτD ≃ R²/DCM, respectively, where DCM denotes the center-of-mass diffusion constant. For homopolymers, Murat and Kremer have proposed a soft ellipsoid model, where ellipsoids can change their position, shape and orientation [2].

In our previous work we used a soft ellipsoid model with Gaussian chains as a microscopic input. This model was shown to reproduce well-known scaling relations for polymer melts at equilibrium [3]. Moreover, for two- component melts, phase separation processes both in the bulk and in confined geometries could be described suc- cessfully, including effects of patterned walls which peri- odically favor one of the two mixture components [4].

Recently we have extended the soft-particle model to studies of block copolymers. In these systems phase separation proceeds under the constriction of a chemical link between incompatible polymer blocks [5,6]. Depending on the relative amounts of different blocks, a variety of ordered microphases can emerge on length scales that are tunable by the respective degrees of polymerization. Be- cause of these unique features block copolymers offer a promising tool for the fabrication of new materials with some prescribed mesoscopic structure.

Following the basic ideas in [7] we represent each block in an (AB)-diblock copolymer molecule by one soft sphere

Figure 1: Structure factorSBB(q)ofB-monomers for asymmetric diblock copolymer chains with numbers of monomersNA= 70,NB = 30 and Flory-Huggins parameterχ= 0.65. Super- structure peaks are characteristic of the cylindrical phase.

with radius of gyration RX(X = A or B). The molecules’ orientation is given by the vector ~rAB =

~rA−~rBconnecting the two centers-of-mass of the spheres.

Its magnituderAB = |~rAB|will reflect the stretching of the molecule under the AB-repulsion. The kinetics of our model is driven by a free energy that depends on the centers of mass of all molecules and their internal degrees of freedom, specified by the parameters ~rAB, RA and RB. By kinetic Monte Carlo simulation we have shown that this Gaussian disphere model reproduces known bulk properties of diblock copolymer melts. First of all, it accounts for important features of the phase diagram such as the appearance of lamellar and cylindrical phases. Fig. 1 displays the structure factor of B-monomers in asymmetric chains in the strong segregation region. The superstruc- ture peaks indicate the prevalence of a triangular lattice formed by the cylinders. Vizualization of real space structures confirm this picture.

By a careful control of finite size effects we found that for equal amounts ofA andB-components the lamellar periodicityλ^∗ of the lamellar phase shows a dependence on the Flory-Huggins parameterχand chain lengthN as λ^∗≃χ^qN^pwithq≃0.2andp≃0.8, see Fig. 2. This ob- servation can be compared with recent, more microscopic simulations [8] that also yield exponentsqnotably larger than the predictionq = 1/6 of the Fredrickson-Helfand theory [6]. Note that our studies ofλ^∗cover a larger range inN than earlier studies.

Another notable feature of this model, which distinguishes it from mean field-kinetic theories for block copolymer melts, is its capability to describe diffusion properties and the orientational dynamics of individual molecules in addition to the fluctuations inA- andB-monomer densities.

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A2. SOFT PARTICLE MODEL FOR BLOCK COPOLYMERS 5

Figure 2: Double-logarithmic plot of lamellar spacingλ^∗versus Nforχ= 0.45 (∆)andχ = 0.95(N). The two sets of data can be represented asλ^∗ ∼N^pwithp= 0.77andp= 0.81, respectively.

Finally we obtained results for confined systems, a sub- ject of considerable current interest [9]. A pattern in a thin film geometry, formed spontaneously out of the disordered phase is depicted in Fig. 3, which displays do- mains with different orientations. There the slab thickness Lz has been chosen somewhat less than the periodicity λ^∗of the lamellar phase in the bulk, and walls were chosen neutral. If walls have a weak preference to one of the components(B), then with increasingLzpatterns can switch between perpendicular orientation (like in Fig. 3) and parallel orientation, as a result of the competition between the two length scales Lz andλ^∗. These findings agree with earlier investigations [10]. A systematic study of these issues, including effects of structured surfaces, is in progress [11].

[1] M. B¨oltau, S. Walheim, J. Mlynek, G. Krausch, and U.

Steiner, Nature 391, 877 (1998)

[2] M. Murat, and K. Kremer, J. Chem. Phys. 108, 4340 (1998).

[3] F. Eurich, and P. Maass, J. Chem. Phys. 114, 7655 (2001).

[4] F. Eurich, P. Maass, and J. Baschnagel, J. Chem. Phys.

117, 4564 (2002).

[5] M. W. Matsen, J. Phys.: Condens. Matter 14, R21 (2002).

[6] L. Leibler, Macromolecules 13, 1602 (1980); H. Fredrick- son, and E. Helfand, J. Chem. Phys. 87 , 697 (1987).

[7] F. Eurich “Coarse-Grained Models for the Kinetics of Polymeric systems”, PhD thesis, University of Konstanz

Figure 3: Illustration of the microphase structure of symmetric chains (χ = 0.95;N = 100)in a slab geometry (Lz = 19).

Shown are isosurfaces of the A-monomer density ρA(~r) = ρ^Atot/2withρ^Atotthe averageA-density.

(dissertation.de Berlin, 2002).

[8] A. J. Schultz, C. K. Hall, and J. Genzer, J. Chem. Phys. 7, 10329 (2002).

[9] J.-U. Sommer, A. Hoffmann, and A. Blumen, J. Chem.

Phys. 111, 3728 (1999).

[10] M. Kikuchi, and K. Binder, J. Chem. Phys. 101, 3367 (1994).

[11] F. Eurich, A. Karatchentsev, J. Baschnagel, W. Dieterich, and P. Maass, in preparation

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A3 Stochastic modeling of charge transport in polymer electrolytes

O. Dürr, A. Karatchentsev, B. M. Schulz (Universität Halle), M. Schulz (Universität Ulm), W. Dieterich, and A.

Nitzan (University of Tel Aviv)

Polyethers are a prototype class of chain molecules ca- pable of forming polymer-salt complexes with significant ionic conductivities. Materials of this type offer widespread applications as electrolytes in rechargeable batteries. More fundamental research, however, is needed to understand the mechanisms of salt dissociation and ion transport, and in turn to optimize their electrochemical properties required in applications.

The aim of this project is to improve our theoretical understanding of the mutual influence of polymer network dynamics and ion diffusion and thereby to contribute to the general area of molecular diffusion through polymer networks. Specifically we focussed on the following sub- jects.

i) Semi-microscopic lattice models for ion and chain dif- fusion

We studied a lattice model for polymer electrolytes where chain molecules carry specific binding sites for cations.

The diffusion constants of cations D+ and of lattice chains DP display Vogel-Tammann-Fulcher (VTF) behavior with a common VTF-temperature. As the ion con- centrationxincreases, chain segmental motions and ion diffusion get reduced through crosslink formation, and the corresponding VTF-temperature rises in accordance with an enhanced glass transition temperature. It turns out that VTF-temperatures for smallxallow an interpretation in terms of DiMarzio’s concept [1] of a vanishing configu- rational entropyScas a criterion for an ideal transition to an amorphous state. Another remarkable feature of this model is a “fractional Stokes-Einstein” law betweenD+

and the viscosityη∼D⁻¹_P [2].

ii) Dynamic percolation theory for ion diffusion in poly- mer networks

Dynamic percolation theory (DPT) provides a description of random walks in a disordered environment that under- goes temporal renewals. To utilize this theory for ion diffusion in a polymer network, we assumed that the statistics of renewals in an associated DPT-model can be derived from density-fluctuations of the network. As shown previously, DP-theory implemented in this way could account quantitatively for the concentration-dependent diffusion constant in an athermal model of chains and ions [3].

The “fluctuation site-bond” Monte Carlo algorithm, developed recently [4], allows one to test the DP-concept under more general conditions, in particular for systems

Figure 1: Tracer correlation factorsf(c)from full simulations for different chain lengthsr. f(c)is defined throughD(c) = D0(1−c)f(c), whereD0 is the diffusion constant for infinite dilution.

with high densitiesc. Fig. 1 shows tracer correlation factors obtained from full simulations. These were carried out using that algorithm for athermal chains of lengthr together with a small amount of point particles. In distinc- tion to [3], the present algorithm yields correlation factors which always fall below the hard-core lattice gas(r= 1) result and appear to decrease monotonously with increas- ingr. For our longest chains withr= 10the correlation factor can become of the order of10⁻¹, indicating strong dispersion in the ionic diffusivityD(ω). Indeed, a power- law type dispersion forD(ω)is obtained in the transient regime between short-time and long-time diffusion. Pre- liminary results forf(c)from DP-theory appear to follow the full simulations in a satisfactory manner but simulta- neously require significantly less computational effort.

[1] E. A. DiMarzio, J. Res. Nat. Bur. Stand.-A, Phys. Chem.

68A, 611 (1964)

[2] O. D¨urr, W. Dieterich, and A. Nitzan, J. Chem. Phys. 121, 12732 (2004)

[3] O. D¨urr, T. Volz, and W. Dieterich, J. Chem. Phys. 117, 441 (2002)

[4] B. M. Schulz, and M. Schulz, New Journal of Physics 7 (2005), in press

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A4. MIXTURES OF NANOPARTICLES AND POLYMERS 7

A4 Mixtures of nanoparticles and polymers

M. Rawiso^∗, and M. Fuchs

Since the work by S. Asakura and F. Oosawa in the 1950s, it is known that the addition of non-adsorbing small polymers with radius of gyrationRg to a suspension of colloidal particles of much larger radiusR(R≫Rg) causes phase separation to occur [1]. This behavior is of technological importance and results from the so–called ’depletion interactions’. Obviously, the centers of the polymer coils are excluded from a shell with widthRgaround each colloid. The overlap of the excluded, or depleted, shells associated with two colloids thus increases the volume accessible to the polymers, and hence their ideal en- tropy. The free energy is therefore lowered when the colloids come together (hence an attractive interaction acts between the particles). The parameter that determines the phase diagram isξ = Rg/R. ξ ≪ 1corresponds to the

’colloid limit’ which has been extensively studied and for which phase diagrams, effective interactions and structural correlations are rather well understood. The opposite case ξ ≫1, which is referred to as ’protein limit’ because of its relevance to protein crystallization or separation, has not been clarified. It has been addressed mainly theoretically, while experimental or simulation studies have not yet reached large enoughξ.

Thus it is of interest to study the phase diagrams and structural correlations in suspensions of mixed model nanoparticles and long polymer chains. We use small angle scattering methods, and liquid state theory approaches, in order to study novel nanoparticles made of fullerene C60 molecules coated with low molecular weight polystyrene (PS) chains (6 PS chains are grafted on each C60 molecule) [2]. They are mixed with high molecular weight PS chains, and dissolved in various PS good solvents.

Figure 1 shows neutron scattering intensities at low particle concentrations for varying polymer concentrations from the dilute to the semidilute regime. The size of the polymers exceeds the particle size by a factor around Rg/R ≈ 12. The signals result from the particle (partial) structure factors and the particle form factors, because scattering takes place on the whole star–shaped nanoparti- cle. A polymer–induced attractive interaction between the (soft repulsive) particles is evident since an aggregation process occurs as the polymer concentration c/c∗ is in- creased; the low–angle scattering intensity grows possibly approaching phase demixing at higher particle concentrations. Further small angle scattering experiments are however needed to be more specific about the dispersion state of the nanoparticles as well as about the average conformation of the long polymer chains.

Figure 1: Neutron scattering intensities of the fullerene–based particles (cF = 1%in weight fraction corresponding to a packing fractionφc ≈ 0.13) obtained for different (nonadsorbing) PS polymer concentrations inbetweenc= 0andc= 8c∗as la- beled; herec∗is the overlap concentration of the PS chains. The size ratio corresponds toRg/R≈12.The signal is determined by the particle form factors and the partial structure factors of the C60–based soft particles.

Data on the polymer correlations would be desirable to compare with theoretically obtained polymer segment density profiles around the nanoparticles, which are studied in our liquid state approach. While at finite particle concentrations, polymer conformations could as far only be handled as Gaussian, it is well known that self–

avoidance leads to different scaling behaviors in the segment profile very close to a hard particle. First computa- tions [3] indicate that the novel polymeric closure approxi- mations, together with the important employed concept of

’thermodynamic consistency’, qualitatively correctly describe the intermolecular segment correlations, if the in- tramolecular ones are given. Further studies at finite concentrations are required, yet, as are comparisons with experimental data.

∗ Physique des Milieux Disperses, Institut Charles Sadron, 67083 Strasbourg, France

[1] W. C. K. Poon, J. Phys.: Condens. Matter 14, R859 (2002);

M. Fuchs and K. S. Schweizer, J. Phys.: Condens. Matter 12, R239 (2002).

[2] F. Audouin, S. Nunige, R. Nuffer and C. Mathis Synth. Met.

121, 1149, 1153 (2001).

[3] Y.-L. Chen, K. S. Schweizer, and M. Fuchs, J. Chem. Phys.

118, 3880 (2003); M. Schmidt and M. Fuchs, J. Phys. Chem.

117, 6308 (2002).

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A5 Tomographic imaging of single tethered polymers using confo- cal microscopy

R. Lehner, J. Koota, G. Maret, and T. Gisler

Polymers tethered to solid substrates have widespread technical applications, such as the stabilization of colloidal suspensions against coagulation, the protection of biosensors against unspecific binding, and for lubrication and adhesion. The presence of an impenetrable substrate profoundly affects the conformation and segment distribution of a polymer chain withNsegments that is attached to it by one of its ends at a tether surface densityσ. When the interaction between monomers and substrate is repulsive, scaling theory for isolated end-tethered polymers predicts a depletion zone near the surface characterized by a slowly increasing segment distributionΦ(z)∼z^ζ, whereζis related to the critical (Flory) exponentνbyζ= (1−ν)/ν which in a good solvent (ν = 0.588) takes the valueζ= 0.7(P. G. de Gennes, Macromolecules 13, 1069 (1980)).

At larger distances, the influence of the wall should become weaker, and the segment distribution can be ex- pected to approach a maximum value given by the average segment density within the polymer layer. The slow increase of the segment distribution close to the substrate, together with its fast decay for larger distances predicted by renormalization-group (RG) calculations and computer simulations, suggest that the segment distribution of an end-tethered polymer in the limit of isolated chains is strongly asymmetric, resembling the shape of a mushroom.

Although this ’mushroom’ conformation of an isolated polymer, end-grafted to a substrate, represents the sim- plest situation of a broken symmetry for polymer statistics, detailed experimental tests of the theoretical predictions forΦ(z) in the low-density limit are lacking. We have measured the segment distributions of end-grafted polymers with uniform chain length in a good solvent, using confocal fluorescence microscopy on single DNA molecules labelled with a fluorescent intercalation dye characterized by a high affinity to DNA and a low background fluorescence in the unbound state [1].

Experiments were carried out with double-stranded DNA molecules whose one end had been functionalized with bi- otin. This functionality was then used to tether the DNA by its end to the glass substrate coated with streptavidin, providing a strong binding for the end segment, while the free segments were repelled from the substrate by electrostatic repulsion. A confocal microscope was used to record 2-dimensional fluorescence images of single DNA molecules at different heightszfrom the substrate. By re- moving background and camera noise from the raw flu-

orescence images we determined the intensity-weighted segment distributionS(z).

-2 -1 0 1 2 3 4

10^-2 10^-1 10⁰

normalizedS(z)

scaled distance from surface z/R

g

Figure 1: Intensity-weighted segment distributionS(z)of end- tethered DNA as a function of the reduced distancez/Rgfrom the substrate. Symbols: data for DNA with20^′040bp (triangles), 38^′416bp (squares),48^′502bp (circles),97^′004bp (diamonds) and145^′507bp (diamonds). Scaling factorsRgwere determined from a fit of Eq. (A.1) to the data (see text). The full line is the scaling prediction Eq. (A.1) forΦ(z/R˜ g), convolved with the experimental resolution functionKz(z).

Fitting the convolution product of the instrumental point spread function and the interpolation expression

Φ(z) = 1.786σN Rg

z Rg

−0.3

(A.1)

×

erfc z

2Rg

−erfc z

Rg

proposed by Kreer et al. (T. Kreer et al., J. Chem. Phys.

120, 4012 (2004)) to the intensity-weighted segment dis- tributions S(z) allows to determine the radius of gyration Rg. For YOYO-1-labelledλ-DNA with a contour length L = 19.8µm we obtainRg = 0.84µm. Mea- surements of the segment distributionΦ(z) for different contour lengths L should, according to the scaling prediction Eq. A.1 be described by a single function depending only onz/Rg for excluded-volume behavior. In order to vary Rg we have thus measured Φ(z) for DNA molecules with different contour lengths8.2µm ≤ L ≤ 59.4µmprepared either by enzymatic digestion and elec- trophoretic separation (yielding uniform fragments ofλ- DNA) or by allowing linearized λ-DNA to anneal into concatemers in the presence of T4 ligase. All the other parameters influencing the chain dimension, such as ionic strength, buffer composition and YOYO-1 labelling density, were kept constant.

Extracting the radius of gyration Rg from the measured S(z) of DNA with different contour lengths and rescaling thez-axis byRg, we find that the segment density profiles measured for different chain lengths do indeed

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A5. TOMOGRAPHIC IMAGING OF SINGLE TETHERED POLYMERS USING CONFOCAL MICROSCOPY 9 collapse onto a master curve (see Fig. 1), providing quanti-

tative evidence that the profiles are governed by excluded- volume chain statistics.

0 10 20 30 40 50 60

0.4 0.6 0.8 1.0 1.2 1.4 1.6

radius of gyration Rg [µm]

contour length L [µm]

Figure 2: Radius of gyrationRgdetermined from the measured intensity-weighted segment distributionsS(z), as a function of the contour lengthLof DNA labelled with YOYO-1. The line is a non-linear least-squares fit of a power lawRg =BL^νto the data, with the best-fit exponentν= 0.58. The contour lengthL was calculated from the number of basepairs,N0, and the basepair risea= 3.4A˚ by the relationL = 1.2N0a, the prefactor 1.2 taking into account the increase of the basepair rise upon YOYO-1 binding.

Fig. 2 shows that the radius of gyration Rg follows a power-law scalingL^0.58, in very good agreement with the critical exponentν = 0.588 for excluded-volume interactions. This provides clear evidence that the excluded- volume interactions are indeed governing the structure of isolated DNA chains in water at a pH and salinity compat- ible with biological function.

The scaling ofΦ(z)suggests that the end-segment distribution functionΦe(z)for end-tethered chains in a good solvent should be governed by the same scaling parameter, namely the radius of gyrationRg. We have measured the intensity-weighted end-segment distributionSe(z), using end-tetheredλ-DNA to whose second, free end an avidin- coated fluorescent bead with diameter100 nmhad been attached (see Fig. 3). The size of this marker is compara- ble to the persistence lengthp≈50 nm, so we can expect that the segment distribution is not perturbed by its presence. In order to distinguish the fluorescence signal from the bead from the one of the inner chain segments, the fluorescence of the bead was measured atλ= 605 nmwhere the fluorescence of DNA-bound YOYO-1 is very weak.

Using the ideal-chain expression Φe(z) ∝ exp

"

−

z−z0

4Rg,e

2#

(A.2)

−exp

"

−

z+z0

4Rg,e

2#

-2 -1 0 1 2 3 4 5 -2 -1 0 1 2 3 4 5

10^-3 10^-2 10^-1 10⁰

b)

distance from surface z [µm]

intensity-weighted end-segment distributionSe(z) a)

Figure 3: Intensity-weighted end-segment distributionSe(z)of aλ-DNA mushroom, as a function of the distancez from the substrate. Error bars represent the standard deviation over 3 molecules. (a) DNA labelled with 0.2 YOYO-1/bp. (b) Un- labelled DNA. Lines: Eq. (A.2) convolved with the experimental resolution functionKz(z) for YOYO-1-labelled DNA (Rg= 0.77µm) and unlabelled DNA (Rg = 0.66µm).

to determine the radius of gyration Rg,e from the measured end-segment profiles Se(z), results in a valueRg,e = 0.77µmfor λ-DNA with 0.2 YOYO-1/bp dye loading which is in very good agreement with the valueRg = 0.84µmobtained from the analysis ofS(z).

In contrast, the analysis of Se(z) from unlabelled λ- DNA yields a reduced value Rg,e = 0.66µm. This value is in excellent agreement with the radius of gyra- tionRg,f = 0.7µm obtained from static light scattering on freeλ-DNA. While independent measurements ofRg

of YOYO-1-labelled DNA are not available, our results provide strong indications that the radius of gyration of tethered excluded-volume chains is identical with the one of free chains.

The equilibrium segment distributions measured in this work provide the first direct experimental test of theoretical predictions for the conformation of end-tethered polymers in a good solvent. The combination of confocal fluorescence microscopy with its ability to provide 3-dimensional, high-resolution images, and long-chain, monodisperse DNA molecules that can be labelled selec- tively could open up new avenues for the study of adhesion and friction phenomena, such as wall-slip and interfacial rheology.

[1] R. Lehner, J. Koota, G. Maret, and T. Gisler (submitted).

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B Colloidal liquids

11

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B1 Flow curves and shear thinning of dense colloidal dispersions

O. Henrich, F. Varnik^∗, J. Crassous^†, Matthias Ballauff^† and M. Fuchs

A multiplicity of systems with biological, chemical or technological relevance, and a vast amount of things of our everyday life, are soft matter. But despite their over- whelming presence, soft materials still pose a lot of unan- swered questions to scientists. Taking particle suspensions as an example, one can say that a profound understanding of their rheological properties is still missing. Due to the non-linear nature of their flow behaviour at higher concentrations, colloidal suspensions show a variety of phenomena. Rheology is a classical field in colloidal physics, and a large number of experimental studies, mostly carried out on hard spheres, have established the basic facts: If the particle concentrations are not too high, a first Newtonian region is observed if the shear rateγ˙ is small. The reg- istered viscosity can be significantly higher than that of the solvent. At higher shear rates the pertubation of the microstructure in the suspension by the convecting forces can no longer be restored by the Brownian motion of the particles. This results in a tremendous reduction of the measured viscosity, until a second Newtonian plateau is reached. This phenomenon is usually referred to as shear- thinning. At even higher shear rates hydrodynamic interactions between the particles dominate and often induce a reverse effect, the so called shear-thickening. Highly concentrated suspensions behave like amorphous solids that elastically withstand small but finite stresses.

For the theoretical description of shear-thinning the familiar framework of linear response theory cannot be used.

The effect of the external driving on the microstructure and the dynamics of the suspension has to be understood.

To address this, a first-principles approach to the steady state properties of sheared dense colloidal dispersions has been developed [1,2]. It settles on the description of mode coupling theory (MCT) for the slow structural relaxation in concentrated colloidal dispersions and extends the theoretical framework to systems far from equilibrium. Fun- damental starting point for the non-linear response formalism is the Smoluchowski-equation:

Ω^{( ˙}^γ)=D0 N

X

i=1

∂

∂r_i· ∂

∂r_i − F_i kBT

−γ˙ ∂

∂xi

yi

(B.1)

∂

∂tΨ(ri, t) = Ω^{( ˙}^γ)Ψ(ri, t) (B.2) It covers the time-evolution of the N-body phase space probability distribution of a large number of particles in a shear field.D0is the diffusion coefficent and the forces Fi arise from particle interactions. Because of the additional shear-dependent term∝ γ˙ in (B.1), which cannot

be expressed by a potential force, the system is driven far away from equilibrium. The new aspects, entering the MCT equations of motion compared to quiescent systems, are time dependent coupling of modes, negative mode- coupling coefficients and the spatial anisotropy.

An important rheological quantity is the transverse stress

r−r’ t>0 κ

t=0

r t

Figure 1: Schematic diagram of the advection of density fluctuations by steady shear in plane Couette-flow. The shear tensor κ= ˙γxˆˆyis idempotent under this condition. Then a density fluc- tuationρqwith wavevectorqat timet= 0is related to a fluctu- ationρq(t)with time dependent wavevectorq(t) =q(1 +κt)at later timest >0. The corresponding wavelength thus decreases.

Therefore, with proceeding time, smaller and smaller Brownian movements of the particles can erase the fluctuations. This leads to the decay of internal stresses.

10^-4

γ .

10^-2 10^-2

10^-1 10⁰

σ

simulation theory

0 0.2 0.4

T

-0.04 -0.02 0 0.02 0.04

ε

sep. parameter used for fits:

σ⁺ glass

supercooled state lin. resp.

Figure 2: Flow curves: shear stress versus shear rate in theory (solid lines) and simulation (symbols) for various temperatures ranging from the supercooled fluid state to the glassy domain. In the former the shear thinning behaviour crosses over to the linear response regime at lowγ. On approaching˙ Tc, the validity of the linear response regime shifts to considerably lower shear rates. BeyondTcthe occurence of a dynamic yield stress is predicted. The inset shows the relation between the MCT separation parameter and the temperature in the simulation (from [3]).

as stationary average of the microscopic stress fluctuations: σ=hσxyi^{( ˙}^γ). It can be related to an exact Green- Kubo expression, containing the stress auto-correlation function with the shear-depending dynamics ofΩ^{( ˙}^γ), averaged in the quiescent system. The basic physical mech- anism causing shear-thinning is thought to be connected to the shear-induced advection of wavevectors and their

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B1. FLOW CURVES AND SHEAR THINNING OF DENSE COLLOIDAL DISPERSIONS 13 decorrelation [3], depicted in Figure 1.

Beginning with novel schematic mode-coupling models that already contain some of the above mentioned shear- induced aspects, it was possible to gain a theoretical understanding of universal features among shear-thinning liquids and yielding solids. We found an excellent quantitative agreement between model calculations and both simulations and experiments. Figure 2 shows simulation results for the transverse shear stress as a function of the shear rate for different temperatures in binary LJ-mixtures [4]. The discontinous transition from a shear-thinning fluid to a yielding, shear-molten solid is displayed. The flow curves exhibit a so called yield stress - the stress that needs to be exceeded, to end up in a stationary state of flow: σ⁺(ǫ) = limγ→0˙ σ(ǫ,γ). Figure 3 gives the yield˙ stresses for different temperatures in the glassy regime.

Asymptotic expansions of the correlation function of density fluctuations close to the glass transition explain the characteristic square root dependence of the yield stress that is predicted by MCT and seen in the simulations. The

0.2 0.25 0.3 0.35 0.4

Temperature 0.1

0.2 0.3 0.4 0.5

dynamic yield stress, σ+ From flow curves

MCT fit using T_c=0.4

σ_c=σ⁺(T-->T_c=0.4)

Figure 3: Temperature dependence of the dynamic yield stress σ⁺(T)obtained via fits to the low-γ˙ branch of the above flow curve. BelowTc = 0.4in the vincinity of the transition point the yield stress shows the typical square root dependence in MCT (from [4]).

experimental studies of the non-linear rheology in colloidal dispersions were performed by Prof. M. Ballauff and his coworkers. They introduced a model system, con- sisting of a thermosensitive suspension of latex particles (Figure 4). With an improved schematic model that accounts for the correct change of the shear modulus for temperatures belowTc [5], the experimental data can be matched (Figure 5). Hydrodynamic interactions, which were omitted in the present approach and in the simulations, were often considered as being crucial for the oc- currence of shear-thinning. The analysis shows that hydrodynamic interactions lead to an appreciable but non- singular slowing-down of the inherent dynamics and can be included adequately via a correct rescaling of the data.

∗ Max-Planck-Institut f¨ur Eisenforschung, Max-Planck-

Figure 4: Picture of a dilute aqueous suspension of thermosensitive core-shell latex particles, used throughout the rheology experiments in the group of Prof. Ballauff. The particles provide a model system and allow to change the effective volume fraction in situ by simply changing the temperature, which swells and collapses the particle size. In this way highly concentrated suspensions can be generated directly in the rheometer.

-4 -2 0

log₁₀Pe₀ -2

0

log10σ ¹⁰

oC 12ôC 15ôC 17ôC 18ôC 19ôC 20ôC 21ôC 22ôC 23ôC 24ôC 25ôC

Figure 5: Flow curves close to the glassy arrest: the stationary transverse stressσis depicted as a function of the shear rateγ˙for different temperatures between10^◦Cand25^◦C. The solid lines display fits to the experimental data with a newly developped schematic model.

Str. 1, D-40237 D¨usseldorf, Germany

†Lehrstuhl Physikalische Chemie I,

Universit¨at Bayreuth, D-95440 Bayreuth, Germany [1] M. Fuchs, M.E. Cates, Phys. Rev. Lett. 89, 248304 (2002).

[2] M. Fuchs, M.E. Cates, J. Phys.: Cond. M. 17, 1681 (2005).

[3] O. Henrich, F. Varnik, M. Fuchs, submitted to J. Phys.:

Cond. M.,(2005)

[4] F. Varnik, Non linear rheology and dynamic yielding, paper in preparation.

[5] M. Fuchs, M. Ballauff, J. Chem. Phys. 122, 094707 (2004).

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B2 One-Bead Microrheology with Rotating Particles

M. Schmiedeberg, H. Stark

In recent years the experimental method of microrheology emerged as a powerful tool to monitor the me- chanical properties of viscoelastic soft materials especially in biological systems including cells. The main idea is to disperse micron-sized beads into the material and monitor their motion either as reponse to external forces (active method) or due to Brownian fluctuations (passive method). Whereas in the first method, the frequency-dependent response function or compliance is measured directly, it is inferred in the second method from the particle’s positional fluctuations using the fluctuation- dissipation theorem and the Kramers-Kronig relations.

In a recent article [1], we addressed the particles’ rotational degree of freedom and laid the theoretical basis to use it in microrheology. The main theoretical problem of microrheology is how the particle compliance measured in experiments relates to the viscoelastic properties of the material quantified in its complex shear modulus. We show in Ref. [1], based on a model-viscoelastic medium, the so-called two-fluid system, that the compliance in the rotational case obeys a generalized Stokes law which is only limited by an upper inertial crossover frequency in contrast to the translational motion where the validity is restricted to a frequency window. The lower frequency is due to the fact that for small enough frequencies of the translating sphere the elastic compoment, say a polymer network, experiences compressional deforma- tions whereas the viscous fluid only allows shear flow.

Only for sufficiently high frequencies is the friction between elastic and viscous component large enough that they move together. They can then be desribed by a complex shear modulusG(ω) =µ−iωη, whereηis the viscosity andµ denotes Lam´e’s elastic constant associated with shear. Hence, the compliance for translational motion of a spherical particle assumes the simple generalized Stokes relationα(ω) = [6πG(ω)a]⁻¹. One advantage of rotating particles is that they create a pure shear field in the elastic network for all frequencies and therefore ex- tend the frequency range for which the generalized Stokes law is valid. This is, however, only true for stick boundary conditions for both the viscous and the elastic component at the particle surface.

We consider a rotating particle with an oscillating angu- lar displacement Φ(t) = Φ(ω)e^−iωt where the direc- tion of the vectorΦ(ω)characterizes the axis of rotation.

In a pure incompressible Newtonian fluid, the oscillating velocity field is described by the Helmholtz equation (∇²+k²)v(r, ω) = 0with wave numberk=p

iωρ/η.

For stick boundary conditions, the solution is a vortex

field with penetration depth δ = Im(k). The external torqueT(ω) = α⁻¹(ω)Φ(ω)to drive the oscillating particle obeys the familiar Stokes lawα⁻¹/−iω = 8πa³η only whenω <2η/(ρa²), i.e, whenδ > a.

We now study the equivalent problem for a model viscoelastic medium represented by a two-fluid model:

0 = µ∇²u+ (λ+µ)∇(∇ ·u) + Γ(v−∂u

∂t) ρ∂

∂tv = −∇p+η∇²v−Γ(v−∂u

∂t) , divv= 0. Here an incompressible Newtonian fluid with shear vis- cosityηis coupled to an elastic medium with Lam´e con- stantsλ, µvia a conventional friction term. By dimensional analysis, the friction coefficientΓ =η/ξ²contains a characteristic lengthξwhich is on the order of the mesh size in, e.g., an actin network. We neglect the mass density of the elastic medium to the one of the Newtonian fluid right from the beginning. The characteristic parameters of the theory are the reduced mesh sizeξ/a,ωe =µ/ηand ω0= 2η/(ρa²). The two frequencies quantify the fluid inertia, shear elasticity, and shear viscosity. Typical numbers for an actin solution areωe = 10³Hzandω0 = 10⁵Hz based ona= 3µm,η= 0.01Pandµ= 1 N/m². For rotating particles the oscillating fieldsu(r, ω)v(r, ω) now obey a vector Helmholtz equation in analogy to the pure Newtonian fluid, however withk²replaced by a ma- trixK² that depends on ω annd the parametersωe, ω0, andξ/a. We have solved this equation and ultimately determined the complianceα(ω)assuming that elastic and viscous components obey stick boundary conditions at the particle surface (case 1) or that the elastic network is not attached at all to the particle (case 2).

Case 1: In Figs. 1 and 2 we plot the real and imagi- nary parts of the inverse complianceα⁻¹(ω), relative to their results for a pure elastic and viscous system, as a function of ω/ωe and ω0/ωe; the mesh size is ξ/a = 0.1. For low frequencies they both exhibit constant values which correspond to the generalized Stokes relation α⁻¹ = 8πa³G(ω)withG(ω) = µ−iωη. This result is valid as long as inertial effects of the fluid are negligible. Note that unlike the translational motion, the validity of the Stokes relation extends to ω → 0; there is no lower crossover frequency. The reason is clearly that a rotating sphere only creates pure shear fields for both dynamic variablesuandv. This indicates that elastic network and viscous fluid are strongly coupled to each other and move together. Therefore, their dynamics is described by the equation of linear elasticity with the complex shear modulusG(ω)(reminiscent to a Voigt element) or alter- natively by the Navier-Stokes equation withηreplaced by η−µ/(iω). This allows an analytic formula that fits the graphs in Figs. 1 and 2 well. It especially accounts for the deviations from the Stokes law due to inertial effects at higher frequencies. In a pure Newtonian fluid, inertia

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B2. ONE-BEAD MICRORHEOLOGY WITH ROTATING PARTICLES 15

0.001 0.01 0.1 1

10

0.01 1 0.1

10 1000 100

-1 0

1 2

0.001 0.01 0.1 1

10

!

0

!

e

!

!e Re(

1

)

8a 3

inertial

Stokes

Figure 1: Case1: Real part of the inverse complianceα⁻¹, normalized to8πa³µas a function ofω/ωeandω0/ωe; the parameter isξ/a= 0.1.

0.001 0.01 0.1 1

10

0.1 0.01 10 1

1000 100

1 1.2

1.4

0.001 0.01 0.1 1

10

!

0

!

e

!

e

Im(

1

=!)

8a 3

inertial

Stokes

Figure 2: Case 1: Imaginary part of the inverse complianceα⁻¹, normalized toω8πa³ηas a function ofω/ωe andω0/ωe; the parameter isξ/a= 0.1.

becomes noticable around the frequencyω0; however for the strongly coupled viscoelastic system, the onset of inertial effects is not so clear. We therefore determined the appropriate crossover frequencies empirically by requir- ing that the compliance α(ω) deviates from the Stokes law by 10%. As it is already obvious from the graphs in Figs. 1 and 2, we find that the crossover frequencies exhibit different behavior for the real and imaginary part of α⁻¹(ω). For the real part, it scales as√ω0ωe∝µ/(ρa²) whereas for the imaginary part it behaves asω₀^0.77ω_e^0.23 forω0/ωe<1and passes over toω0forω0/ωe>1. This crossover is clearly seen in Fig. 2.

Case 2: We now address the case where the elastic net- work is not coupled to the particle surface, it only reacts to shear flow via friction drag from the fluid. In Fig. 3, we plot the real part of the inverse compliance as a function of reduced frequencyω/ωe and mesh sizeξ/a; the additional parameter is set toω0/ωe = 100. For small frequencies, the real part is close to zero, in contrast to

0.001 0.01

0.1 1

10

0.001 0.01 0.1 1 10

-1 0 1

0.001 0.01

0.1 1

10 perme−

ation

a

!

e

Re(

1

)

8a 3

Stokes inertial

Figure 3: Case2: Real part of the inverse complianceα⁻¹, normalized to8πa³µas a function ofω/ωeandξ/a; the parameter isω0/ωe= 100.

case 1, where it assumes the reference Stokes value of 8πa³µ, as already discussed. The friction between the two components is sufficiently small so the fluid permeates the elastic network without deforming it noticeably. Then for increasing frequency, an edge occurs andRe(α⁻¹)enters the region where the Stokes law is valid. Note, however, that this region is considerably reduced compared to case 1. Correspondingly, in the region of permeation, i.e., relative motion of viscous and elastic component, the imaginary part of α⁻¹ is strongly enhanced compared to the reference valueω8πηa³. An analytic formula valid when inertial effects are negligible shows that the onset of the Stokes regime occurs at a frequency which scales asξ/a.

This is in contrast to translational motion where it scales as(ξ/a)².

Our study of different boundary conditions clarifies that the interpretation of compliances measured in experiment needs care. So far, in the translational case, always stick- boundary conditions are assumed. Our results show that some slip of the elastic network changes the measurable compliance dramatically. This could lead to false interpre- tations. Interestingly, recent microrheology experiments with rotating particles in aqueous solutions of polymers measure a frequency-depend shear modulus G(ω) over more than three decades. Its values are consistent with the reduced Stokes regime in case 2 [2].

[1] M. Schmiedeberg and H. Stark, Europhys. Lett. 69, 629 (2005).

[2] E. Andabla-Reyes, P. D´ıaz-Leyva, and J. L. Arauz-Lara, Phys. Rev. Lett. 94, 106001 (2005).

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B3 Three-particle correlations in colloidal suspensions

C. Ruß, H.H. von Gr¨unberg

The static structure of a simple fluid is commonly described in terms of the m-body distribution functions g^(m), measuring the probability densities of finding two, three, and more particles at specified positions in space.

Three-body distribution functions in classical fluids have been theoretically investigated many times, but have never been measured directly.

In [1,2] we have presented experimental three-point correlation functions that have been computed from particle configurations measured by means of video-microscopy in two types of quasi-two-dimensional colloidal model fluids: a system of charged colloidal particles and a system of paramagnetic colloids. In the first system the particles interact via a Yukawa potential, in the second via a poten- tialΓ/r³. Varying the particle density in the charged system or the interaction strengthΓin the magnetic system, one can systematically explore how triplet correlations behave if the coupling between the particles changes. We have found for both systems very similar results: on increasing the coupling between the particles one observes the gradual formation of a crystal-like local order due to triplet correlations, even though the system is still deep inside the fluid phase. These are mainly packing effects as is evident from the close resemblance between the results for the two systems having completely different pair- interaction potentials, see Fig. (1).

To demonstrate that triplet correlations are significant not only locally, but also when integrated over the whole volume we have considered the Born-Green equation and have shown that in a strongly interacting system this equation can be satisfied only with the full triplet correlation function but not with three-body distribution functions obtained from superposing pair-correlations (Kirkwood superposition approximation). For details, see [1,2].

What are these triplet correlation functions useful for?

As an example we quote Ref.[3]. In order to prove that three-body interaction potentials are present in concentrated suspensions of charged colloids, we have inferred the shape and magnitude of the three-body potential from analyzing the higher-order correlations of particles within a concentrated colloidal suspension. In [3] we have computed correlation functions and extract from these functions the whole three-body interaction potentials among the colloids. Since from two-particle-correlations one obtains microscopic information only on the level of pair-interactions, we had to analyze pair- and, in addition, triplet-correlation functions, in order to get access to

0 1 2 3

r ρ^1/2 0

1 2 3 4

g(2) (r) , g(3) (r,r,r)1/3

g⁽³⁾_SA(r,r,r)^1/3 = g⁽²⁾(r) g⁽³⁾(r,r,r)^1/3

Γ = 4

Γ = 14

Γ = 46

a) magnetic

0 1 2 3

r ρ^1/2 0

1 2 3 4 5

g(2) (r) , g(3) (r,r,r)1/3 g⁽³⁾_SA(r,r,r)^1/3 = g⁽²⁾(r)

g⁽³⁾(r,r,r)^1/3 ρσ² = 0.0366

ρσ² = 0.169

ρσ² = 0.186

b) electrostatic

Figure 1: Comparison between triplet distribution function g⁽³⁾(r, r, r)^1/3 and the pair-distribution functiong(r)of (a) a paramagnetic colloidal fluid for differentΓ, and (b) the charge- stabilized colloidal fluid for different colloid densities as indi- cated. Circles are experimental data, lines are simulation data.

The curves for different densities (differentΓ) are shifted for clarity. Taken from [2]. Note the close resemblance which is remarkable considering that we have two systems with completely different pair-interaction potentials.

both two- and three-body interaction potentials. At the same time, the paper aimed at presenting an alternative experimental method to measure three-body forces, a method which may be seen to be complementary to the more direct one realized in [4].

[1] K. Zahn, G. Maret, C. Ruß and H.H. von Gr¨unberg, Three-Particle Correlations in Simple Liquids, Phys. Rev. Lett.

91 (11), 115502-1 (2003)

[2] C. Ruß, K. Zahn, and H.H. von Gr¨unberg, Triplet correla- tions in two-dimensional colloidal model liquids, J. Phys. Cond.

Mat. 15, S3509 (2003).

[3] C. Russ, M. Brunner, C. Bechinger and H.H. von Gr¨unberg, Three-body forces at work: three-body potentials derived from triplet correlations in colloidal suspensions, Europhys. Lett., 69, 468 (2005).

[4] M. Brunner, J. Dobnikar, H.H. von Gr¨unberg, and C.

Bechinger, Phys. Rev. Lett. 92, 078301 (2004)

Soft Condensed Matter. Biannual Report 2003/2004

University of Konstanz

SOFT CONDENSED MATTER PHYSICS Physics Department

Biannual Report 2003

2004

Contents

Preface

A Polymers

A1 Structural and conformational dynamics of polymer melts

A2 Soft particle model for block copolymers

A3 Stochastic modeling of charge transport in polymer electrolytes

A4 Mixtures of nanoparticles and polymers

A5 Tomographic imaging of single tethered polymers using confo- cal microscopy

B Colloidal liquids

B1 Flow curves and shear thinning of dense colloidal dispersions

r−r’ t>0 κ

t=0

r t

γ .

σ

B2 One-Bead Microrheology with Rotating Particles

B3 Three-particle correlations in colloidal suspensions