Colloidal Suspensions of Rodlike Nanocrystals and Magnetic Spheres

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Colloidal suspensions of rodlike nanocrystals and magnetic spheres under external magnetic stimulus: Experiment and molecular dynamics simulation Kathrin May, Alexey Eremin, Ralf Stannarius, Stavros D Peroukidis, Sabine H.L. Klapp, and Susanne Klein Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.6b00739 • Publication Date (Web): 27 Apr 2016 Downloaded from http://pubs.acs.org on April 30, 2016

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Colloidal suspensions of rodlike nanocrystals and magnetic spheres under external magnetic stimulus: Experiment and molecular dynamics simulation† Kathrin May,† Alexey Eremin,∗,† Ralf Stannarius,† Stavros D. Peroukidis,∗,‡ Sabine H. L. Klapp,‡ and Susanne Klein¶ †Otto-von-Guericke Universität Magdeburg, Universitätplatz 2, 39106 Magdeburg, Germany. ‡Institute of Theoretical Physics, Technical University Berlin, Secr. EW 7-1 Hardenbergstr. 36, D-10623 Berlin, Germany ¶HP Laboratories, Long Down Avenue, Stoke Gifford, Bristol BS34 8QZ, U.K. E-mail: [email protected]; [email protected] Phone: +49 30 314-28851 Abstract Using experiments and molecular dynamics simulations, we explore magnetic-fieldinduced phase transformations in suspensions of non-magnetic rodlike- and magnetic sphere-shaped particles. We experimentally demonstrate that an external uniform magnetic field causes the formation of small, stable clusters of magnetic particles which, in turn, induce and control the orientational order of the non-magnetic subphase. Optical birefringence was studied as a function of applied magnetic field and the volume fractions of each particle type. Steric transfer of the orientational order was investigated by

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molecular dynamic (MD) simulations whose results are in qualitative agreement with the experimental observations. By reproducing the general experimental trends the MD simulation offers a cohesive bottom-up interpretation of the physical behaviour of such systems and it can also be regarded as a guide for further experimental research.

Introduction Colloidal suspensions of anisometric particles show a variety of unique phenomena, depending on particle concentrations and external electric fields. Formation of ordered or liquid crystal (LC) phases, phase separation, electro-optical switching, and non-linear rheology 1–5 are a few examples of such phenomena. Electro-optical and magneto-optical properties of colloids are of particular practical interest, because of possible industrial applications in mechanical devices, electronics, photonics and even display applications. In the late forties, Onsager demonstrated that the translational entropy of rod-shaped colloidal particles can be increased at the expense of orientational entropy, resulting in the formation of a lyotropic nematic phase. 6 Such ordered phases have been extensively studied in various colloidal systems such as goethite suspensions, fd- and tobacco-mosaic viruses, or clays. 1–5 A combination of large optical anisotropy of the colloidal particles with a large paramagnetic susceptibility may be a promising strategy to produce materials that can be aligned, reoriented and manipulated with moderate magnetic fields. Here, we report an experimental and theoretical study of the magneto-optical behaviour of non-polar suspensions of rod-shaped pigment crystalline particles (rods) doped with ferromagnetic sphere-shaped nanoparticles. The dispersed crystallites studied here are typical components of inks. From the theoretical perspective, a thorough understanding of liquid crystalline (LC) behavior of anisometric particles has been archived using analytical theories 6–9 and computer simulations. 10–14 The LC particle prototypes mainly used are platelets, 10 spherocylinders, 11,12 ellipsoids 13,15,16 and spheroplatelets. 8,14 The simplest model for magnetic nanoparticles that is broadly used in theoretical studies comprises spheres with an embedded magnetic dipole 2


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point moment. Monodispersed dipolar spheres arrange into head–to–tail configurations and form chains under external magnetic fields 17 . Based on Monte Carlo simulations we have recently studied the self-assembly of a model hybrid system consisting of rods and dipolar soft magnetic spheres in a LC phase. 18,19 Even in the absence of an external field, the LC matrix facilitates the alignment of ferromagnetic chains either along or perpendicularly to the director of the LC phase. In the present work, we consider an even more realistic representation of the relative size of the rod and sphere species in accordance with the experimental counterpart system. The goal is to achieve a deeper understanding of the response of an otherwise isotropic mesogenic matrix material to an external magnetic field by launching extensive Molecular Dynamics (MD) simulations for a variety of magnetic particles concentrations. In previous experimental studies, we demonstrated that the suspensions made of elongated pigment particles Novoperm Carmine HF3C (Clariant) exhibit a nematic phase at a concentration as low as 17 v%. Even below the isotropic-nematic transition, the suspensions showed a remarkable optical response to an electric field 20 and a rich pattern forming behaviour. 21 In the present study, we demonstrate a strong magneto-optical response of this system when a small amount of the magnetic particles is added. Novoperm Carmine only suspensions do not show any appreciable magnetically induced birefringence in weak fields, since the colloidal pigment particles themselves have only a small anisotropy of the magnetic susceptibility. In order to enhance this response, we add a small amount (< 1v%) of magnetic nanoparticles. Unblended magnetic particles can be aligned already in weak external magnetic fields as small as a few mT. Their alignment is sterically transferred to the non-magnetic component of the suspension. Such an induced alignment was already described by Lekkerkerker et al., 9 by extending the Onsager theory 6 to account for different shapes of colloidal particles. They found that a mixture of anisometric particles with two different lengths has a phase transition at lower concentrations than each single component alone. Thus, in a two-component mixture of particles it is possible to align one

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component of the mixture by aligning the other one. A numerical study by Slyusarenko et al. 22 demonstrated this effect in a binary mixture of magnetic and non-magnetic rod-shaped particles. It has been confirmed experimentally in dispersions of rod-shaped Vanadiumpentoxide and spherical-shaped magnetic particles. 23 In another model system consisting of micrometre-sized non-magnetic composite dumbbells immersed in ferrofluid, magnetic field action resulted in the formation of chains of the dumbbells. 24 In that case, however, much larger particles did not form a stable dispersion and the formation of the chains could be observed directly by optical microscopy. Here, we investigate the magneto-optical effect in suspensions (inks) of elongated pigment particles doped with spherical magnetic particles. The paper is composed as follows: First, the experimental setup and suspensions are presented. By means of birefringence measurements we experimentally explore steric alignment transfer from the magnetic component to the non-magnetic one and develop a numerical model to describe this phenomenon, in the following section. The model, our MD simulation results and the discussion of our results for a binary mixture of rods and dipolar spheres are presented, and finally, the conclusions of this work are summarised.

Experimental Section Preparation of suspensions As a test material, C.I. Pigment Red 176, a blue shade benzimidazolone pigment was used. The structure formula is shown in Fig. 1. One of its commercially available forms is Novoperm Carmine HF3C (Clariant, Frankfurt am Main, Germany, used as received). The primary particles are rods with an average length of 230 nm ± 70 nm and an average diameter of 46 nm ± 20 nm (Fig. 2). 25 The particles were dispersed in the nonpolar solvent dodecane (Sigma-Aldrich, Hamburg, Germany, used as received) with the help of a commercially available dispersant Solsperse 4


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Figure 1: Chemical formula of a C.I. Pigment Red 176

a)

b)

Figure 2: a) Scanning electron microscopy (SEM) image of the pigment particles Novoperm Carmine; b) Polarising microscopy texture of the pigment only suspension at the particle volume fraction of 0.21. The width of the image is approx. 1 mm, and white arrows indicate the orientation of the polarisers. At this high concentration, the suspension is birefringent in the field-free state.

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11200 (Lubrizol, Brussels, Belgium, used as received). Details of the preparation of the suspensions with different concentrations are described in Ref. 21 Initially, an 30 wt% basis suspension was produced, and different concentrations below 20 wt% were obtained by stepwise dilution of this basis suspension. Samples were centrifuged at 10 000 rpm for 60 min to test the stability of the suspensions. None of the concentrations showed phase separation in particle-rich and particle-poor zones. Samples left untouched for 12 months neither showed any phase separation nor aggregation. For doping the suspensions with magnetic particles, a commercial available ferrofluid (APG 935, Ferrotec) was used. The ferrofluid contains magnetite nanoparticles with average diameter of about 10 nm suspended in hydrocarbons. The surfactant layer thickness can be estimated to be about 2 nm. 26 To obtain homogenous mixtures, three cycles of about 3 minutes each in an ultrasonic bath were performed. The dispersions were diluted with dodecane to obtain the desired particle volume fractions.

Experimental setup Optical measurements were made in sandwich glass cells (E.H.C. and Instec) of thicknesses 10 µm or 5 µm. The cell with the doped suspension was placed into the homogeneous field of an electro-magnet (Bmax = 800 mT). A cw He-Ne laser (λ = 632.8 nm) was used for observation in order to avoid the absorption by the red particles during the birefringence measurements. The laser beam propagating through the sample was modulated harmonically with a photoelastic modulator (PEM) and the amplitudes of the first (V1 ) and second (V2 ) harmonics were measured with a lock-in-amplifier. The birefringence was then determined by λ arctan ∆n = 2πd

V1 · J2 (δ0 ) V2 · J1 (δ0 )

(1)

where d is the cell thickness, J1 , J2 are Bessel functions, and δ0 = 2.405 rad is the amplitude of the modulation.

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Results and discussion Behaviour of dispersions of magnetic nanoparticles only In order to investigate the optical birefringence induced by the magnetic field in suspensions of magnetic particles only, we diluted the commercial ferrofluid with dodecane. The induced birefringence results from the chain formation of magnetic particles in a magnetic field. In Fig. 3a, the birefringence of a ferrofluid/dodecane mixture is shown for different concentrations of spherical magnetic particles cs . As expected, the birefringence increases with the field strength and the concentration of magnetic particles. At high field strengths, the birefringence reaches a saturation value ∆nmax . The birefringence (∆nmax ) at high field strengths linearly increases with the ferrofluid concentration cs (see Fig. 3b).) The length of the particle chains, viz. the average number of magnetic nanoparticles in a chain, can be estimated following the approach proposed by Xu et al. 27 They suggested that the birefringence due to the chaining of magnetic particles can be described approximately by the expression:

3 · coth (a · B) 3 ∆n = 0.5 · nr · cs · g · 1 − + a·B (a · B)2

(2)

where nr is the refractive index, cs is the concentration of magnetic particles, g = χ|| −χ⊥ is the anisotropy of the magnetic susceptibility, B is the magnetic field and a = µch /(kT ), where µch is the magnetic moment of a chain/cluster. This function was used to fit the experimental data. It yields µch ≈ 5 · 10−19 Am2 and g ≈ 0.1, independent of the concentration. Since the magnetic moment of a single magnetite particle of size 10 nm is µ1 = 2.34 · 10−19 Am2 , the mean chain length is estimated to be about 2 particles. Interestingly, the chain length estimated in that way does not depend on the concentration. A possible reason for this could be a violation of the chaining model, given by Eq. 2, at high concentrations.

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Binary mixtures of pigments with ferromagnetic particles Even though the chain lengths are small, the magnetic chains induce a noticeable alignment of the pigment particles by steric interactions. Birefringence curves of different concentrations of magnetic and pigment particles are shown in Figs. 4 and 5. The index r stands for rods (the pigment particles) and s stands for magnetic particles. In Fig. 4, the concentration of magnetic particles is constant (cs = 0.3 vol%), while the concentration cr of pigment particles is varied. In Fig. 5, cs is varied while the concentration of pigment particles is held constant (cr = 5 vol%).

Figure 4: Birefringence of mixtures with magnetic particle concentration cs = 0.3 vol% at a) low and b) high pigment particles concentration (cr ). The mixture of pigment particles and magnetic particles exhibits considerably higher

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birefringence in comparison to the pigment only suspension, where the birefringence is nearly zero, and also in comparison to pure magnetic particles (black curve in Fig. 4a). In Fig. 6 the maximum birefringence (saturation value of the curve) at the field strength of B = 640 mT is shown. At constant cs and low pigment particle concentrations the birefringence at high field strengths increases linearly, while for high cr a different behaviour is observed (Fig. 6a). This dramatic increment of birefringence can be explained with an onset of the liquid crystal phase of the pigment suspensions at high concentrations. On the other hand, the maximum birefringence at a constant cr shows a linear dependence on cs (see Fig. 6b) in the investigated concentration range. Similar behaviour was found when the concentration of the magnetic particles cs was varied at a constant concentration of the rods cr (compare Fig. 5 with Fig. 4).

Numerical estimations using the Lekkerkerker approach Numerical calculations of the birefringence were made by an iteration method developed by Herzfeld et al., 28 which was also applied by Slyusarenko et al. 22 for a mixture of nonmagnetic colloidal rods and magnetic rods. The model is based on the minimisation of the free energy of a two component system: 22

∆F µ0 = − 1 + ln c + (1 − x) ln (1 − x) + x ln x+ N kB T kB T (1 − x)σ1 + x σ2 + c [b11 (1 − x)2 ρ11 + 2 b12 x (1 − x) ρ12 + Z 2 b22 x ρ22 ] − ax cosβ f2 (Ω) dΩ

(3)

where µ0 is the chemical potential, c is the number density of the particles, x is the fraction the magnetic rods (2) of the total number of rods, (1-x) is the fraction of nonmagnetic rods (1), bij =

π 8

(Di + Dj )Li Lj , β is the angle between H and µ (i.e. the magnetic

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moment of rods 2).

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The minimisation is done using Matlab software for the iterative determination of the orientational distribution functions for both particle types. For the calculations, it is assumed that both particle types are mono-dispersed and the chains of magnetic particles are treated as magnetic rods with the lengths of two spherical particles and the widths of one. Thus, the magnetic particles are assumed to be cylinders with the size of 28 nm ×14 nm and with the magnetic moment of two spherical magnetic particles, µ2 = 4.68 · 10−19 Am2 . With the calculated distribution functions, the order parameters of both particles (i = 1, 2) can be calculated:

Si =

Z

1 fi (3 cos2 θ − 1) dΩ 2

(4)

In colloidal suspensions, the relation between the birefringence ∆n and the order parameter S is 29

∆n = ∆nsp c S

(5)

where ∆nsp is the specific birefringence. The overall birefringence of the two particle system is expressed as:

∆n = ∆nsp,r cr Sr + ∆nsp,s cs Ss .

(6)

The index r corresponds to the pigment particles and s to the magnetic particles. To compare the numerically estimated order parameter of the mixtures to the experimental birefringence, the values of the specific birefringence ∆nsp,r and ∆nsp,s are required. The latter can be extracted from the birefringence data of magnetic particle only suspensions (Fig. 3). As expected, the measured birefringence and calculated order parameters for constant concentrations are approximately proportional to each other. From that, the specific bire1

In Eq. 3, the interparticle interactions between the "dipolar" magnetic "rods" are neglected.

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fringence can be determined as ∆nsp,s = 0.0785. With this value, the birefringence curves ∆n(B) could be fitted well with Eq. 6 containing numerically estimated order parameters Sr and Ss . The specific birefringence of about ∆nsp,r = 0.7 for all concentrations was determined from the fits. The resulting birefringence curves are shown in Figures 4 and 5 as solid lines. The numerical results fit the experimental data well, especially the behavior of the maximum birefringence at constant pigment particle concentration shows goods agreement. In contrast, the maximum birefringence at high cr deviates from the numerical results. This behaviour is a consequence of the polydispersity, which leads to a decreased transition concentration to the nematic phase. 9 However, the concentration still remains relatively high with cr > 8v%. In addition, the polydispersity of particle sizes leads to smearing of the critical concentration for the transition into the nematic phase. In the numerical model, the critical concentration is much higher than the experimental value. The pigment particle concentration, at which a nematic phase is observed, is about cr = 12 v%. But already at concentrations of cr = 6 v% birefringent regions can be observed in the suspension without a magnetic field.

Molecular dynamics simulation Model Our model fluid is a binary mixture of Nr uniaxial rods and Ns magnetic spheres. The rods are represented as prolate, nonmagnetic ellipsoids, that interact via the Gay-Berne (GB) potential, using a standard parametrisation. 16 We have chosen the aspect ratio (length to width) equal to l/σ0 = 3. The reason we have used this value is that the phase behaviour of a mono-dispersed system of this aspect ratio is well studied and is considered an established reference system. 30 Furthermore, this aspect ratio is in the lowest accessible limit of the experimentally studied suspensions (see section Simulation Results and Discussion) and Ref. 21 The magnetic spheres are modelled via dipolar soft spheres (DSS), with an embedded central 14


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dipole moment µi (i = 1, . . . , Ns ). They interact via a soft repulsive potential and a dipoledipole interaction. 18,19 The DSS particles have a diameter σs . We have set σs equal to 1/4 the width of the rods, i.e. σs∗ = σs /σ0 = 0.25, which is consistent with experimental estimation of the relative sizes of the species. The reduced temperature is set T ∗ = kB T /ǫ = 1.2 (with kB being Boltzmann’s constant and ǫ is energy parameter). We have chosen this value since the mono-dispersed system exhibits both isotropic and nematic phases at this temperature by varying the number density. 30 The highest value of the reduced dipole moment we have used p is µ∗ = µ/ ǫ0 σs3 = 2.4; the corresponding coupling parameter λ = µ2 /kB T σs3 takes values less than 4.8 that characterise moderate rather than strongly coupled magnetic particles. 31–33

Long-range interactions are evaluated with the three dimensional Ewald sum. 34 Finally, the interactions between rods and spheres are described via a modified GB potential. 15,16 For further details on the interaction potential see Ref. 19 The model we have implemented for the rod-sphere mixture is generic for a range of systems in which the lengths scales of the rods and spheres are comparable. For example, the same model has been applied previously 16,18,19 for systems with 1 ≤ σs∗ ≤ 2. Only when the diameter of magnetic particles is far beyond the typical rod size (as it is the case, e.g., for micron-sized particles in LC matrices 35 ) alternative models should employed. We have examined systems for various total number densities ρ∗ = Nσ03 /V (with N = Ns + Nr ), fraction of particles xa = Na /N (where a=r,s for rods and spheres, respectively), and resulting concentrations ca = xa ua ρ∗ 100% (where ua is particles’ volume and V is the total volume of the system). Here, we have used the geometrical features of the particles to roughly estimate their "volume" since they interact via soft potentials. Note also that the fraction of DSS particles is given by xs = 1 − xr . We have performed extensive MD simulations using N particles at constant volume V and (kinetic) temperature T. 36 The equations of motion are solved using the leap-frog algorithm. 36,37 The reduced moment of inertia is set for both particles to I ∗ = I/mσ02 = 1.0 p and the time step ∆t∗ = 0.002 (where the reduced time t∗ = ǫ0 /mσ02 t and m is the mass 15


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of particles). Equilibration requires extensive runs of 5 · 105 time steps and a further 5 · 105 time steps are used for calculation of ensemble averages of quantities of interest. The orientational order of each species has been quantified by calculation of the global nematic order parameter S (a) by diagonalising the corresponding ordering matrix 38 and extractˆ r and ing the largest eigenvalues. The corresponding eigenvectors define the principal axes n E D P s ˆ s . We have also calculated the normalized magnetization defined by M = | N µ ˆ |/N n s . i=1 i

The DSS particles can self assembly into clusters. This has been quantified by using a

simple criterion to define a "bond" between two particles. Specifically, two DSS particles i and j are considered bonded if rij < rcl , where rcl is set to 1.2σs . This value is between the first maximum and first minimum of the DSS particles pair correlation function of the most systems studied. Using our criterion, the categorisation of the DSS particles into clusters includes: (i) chain clusters consisting of at least three particles (for which two particles of the clusters have one bond), (ii) ring clusters (for which all particles of the cluster have two bonds), (iii) branched clusters (for which at least one particle of the cluster has three bonds) and (iv) "free" (non-bonded) particles and pairs of DSS particles. The global order parameter S (s) of the DSS is not an appropriate quantity to characterise the orientational order of the magnetic particles in the LC matrix since the latter self-assemble into finite clusters. Therefore, we have introduced a method to quantify the orientational order of the clusters in the LC matrix. Below, we describe the steps we have implemented and the necessary calculated quantities: (i) Firstly, from the ordering matrix of each cluster (ring or chain) the extracted eigenvalues are sorted S+cl > S0cl > S-cl giving the ˆ cl ˆ cl ˆ cl corresponding eigenvectors n +, n o and n − (hence, a local principal axis frame is assigned to each cluster). (ii) Secondly, ensemble averages are evaluated using the order parameters

ˆ cl of the clusters defined as Sbcl = P2 (H · n b ) , with b = +, −, o and P2 the second Legendre

polynomial. The quantity Sbcl monitors the orientational order of the local principal axis of

ˆ cl specific type of clusters with respect to the direction of the magnetic field H. The n + local director defines the major axis of the rings (of elliptical shape) and of ’snake’ chains. The

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non-uniaxiality of the shape of the clusters is shown by the inequality S0cl and S-cl . Indeed, the non-vanishing ∆ = S0cl − S-cl parameter indicates deviation from a uniaxial shape.

Simulation results and discussion In this subsection we explore the response of GB-DSS mixtures to a uniform magnetic field given by H = Hˆz at various field strengths H ∗ = µH/kB T . The field H is coupled to the permanent dipole µi of particle i through the potential Ui = −ˆ µi · H. We consider five state points [(ρ∗ , xs ) = (0.350, 0.20)], [(ρ∗ , xs ) = (0.560, 0.50)], [(ρ∗ , xs ) = (0.778, 0.64)], [(ρ∗ , xs ) = (1.00, 0.72)], and [(ρ∗ , xs ) = (1.40, 0.80)], that correspond to the isotropic phase of a field-free GB-DSS mixture. For these state points the concentration of rods is the same (cr = 44%) whereas the concentration of DSS cs is varying (increases by increasing the concentration xs ). Initially, we have set µ∗ = 2.4. The nematic order parameter S (r) and the magnetisation hM i are shown in Fig. 7 a and Fig. 7 b, respectively as a function of the magnetic field strength. When the field is off (H ∗ = 0) the nematic order parameter and the magnetisation are essentially zero. The nematic order parameter remains essentially zero for relative low cs < 0.229% (see Fig. 7a) whereas for higher cs increase with H ∗ until saturation is reached at H ∗ = 8.3. The saturation value also depends on the volume fraction of DSS and increases with increasing cs . Evidently, the rodlike species exhibit a field induced isotropic-nematic phase transition. These findings describe qualitatively the behaviour of the experimental counterpart system ˆ r and (compare Fig. 7a with Fig. 5). We should note here that the macroscopic directors n ˆ s are on average parallel to the magnetic field H and a uniaxial nematic phase is formed. n The DSS particles are oriented along the direction of the magnetic field giving rise to a net magnetization that also saturates for small magnetic fields (see Fig. 7b).

It is interesting to note that the saturation value of the nematic order parameter increases 17


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Figure 7: (a) Nematic order parameter of the rod species S (r) and (b) magnetization hM i of the magnetic particles as a function of the external magnetic field strength for a GB-DSS mixture with µ∗ = 2.4. The results are taken for constant concentration of rods cr = 44% by varying the concentration of spheres cs .

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linearly with increasing cs as it is shown in Fig. 8a. This is also in qualitative agreement with trends observed in experiments of the birefringence dependence on the magnetic field and cs (compare Fig. 6b with Fig. 8a) (where the birefringence can be roughly considered proportional to the nematic order parameter). From the above discussion, it is apparent, that the optical properties in real colloidal suspensions as well as the orientational order in model systems depend crucially on the concentration of the magnetic particles in the solution. To further examine the dependence of the orientational order on the magnetic properties of DSS particles we have considered a system with smaller magnetic moment (i.e. of lower dipolar coupling strength). By decreasing µ∗ the saturation value of the order parameter S (r) decreases for the same cs as it is shown in Fig. 8a. This finding transferred to real suspensions means that the optical phenomena (e.g. birefringence) are expected to be less pronounced or even absent at relative small magnetic particles interactions. Therefore, not only the concentration of the magnetic particles but also the interparticle interactions influence the response of these systems to an external magnetic field.

From the experimental perspective, it is a challenge to visualise small magnetic particles of diameter 10 − 20 nm using developed tomography techniques. 39 On the other hand, MD simulations are a useful tool that offers direct visualisation of the model GB-DSS system. It is straightforward to calculate physical quantities that are related to the structure of magnetic particles in the liquid crystalline matrix as described in the subsection Model. Hence, MD provides a bottom-up physical insight into the link between macroscopic properties (e.g. order parameters and birefringence) and inter-particle correlations. Below we have performed such an analysis. The microscopic structure of DSS in various LC states is illustrated in Fig. 9. For relatively small volume fractions of magnetic particles, cs = 0.229% (at the saturation value of S (r) for H ∗ = 8.3), the mean size of the clusters that are formed by DSS is hni = 1.36. This 19


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Figure 8: (a) Saturation value of the nematic order parameter of rods S (r) as a function of cs for constant cr = 44%, and various values of dipolar moment µ∗ . (b) Mean size of the clusters hni -open squares- (irrespective of their type) and mean size of chainlike clusters hnc i -open triangles- at the saturation value of S (r) . The corresponding values of hni and hnc i when the field is off are shown by solid squares and triangles, respectively.

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means that on average each cluster consists of 1.36 particles. Furthermore, for this cs only a small amount of 23% of DSS particles form chainlike clusters. The calculated mean size of the clusters is shown in Fig. 8b. For even lower volume fractions of cs = 0.0573%, almost all DSS particles (over 97%) are "non-bonded" with hni = 1.08 (see Fig. 8b). A characteristic snapshot of this state is given in Fig. 9c. By increasing cs (at the saturation value of S (r) for H ∗ = 8.3) the mean size of the clusters hni and hnc i increases, though slightly, as it is indicated in Fig. 8b. In addition, and most importantly, the percentage of DSS particles that form chainlike clusters reaches 58% for the highest volume fraction (cs = 0.916%) we have examined. Representative snapshots that show the formation of chainlike clusters (ferromagnetic chains) are given in Fig. 9. It should be noted that we have not observed neither ringlike clusters nor branched structures. Finally, the orientational order of the chainlike clusters for cs > 0.229% (at the saturation value of the S (r) ) is S+ch > 0.8 which is considerably higher than the global order parameter of the rods. This indicates that the ferromagnetic chains are strongly aligned along the field. The ∆ parameter is nearly zero indicating that the chains are linear uniaxial objects. We should note that chain clusters are present also in the field–off states but their sizes are smaller (see Fig. 8b). Furthermore, we observe a continuous assembly and dis–assembly of chains, revealing a dynamic equilibrium between ’bonded’ and ’non–bonded’ magnetic particles. The experimental observations can be rationalised and interpreted using the above findings. The self assembly of DSS particles into linear ferromagnetic chains hinders the motion of the anisotropic particles perpendicularly to the magnetic field. Consequently, the ferromagnetic chains induce orientational order, acting as ’rodlike’ entities, along the magnetic field. What is more important for the induction of alignment to the rod species is the percentage of DSS particles that form chainlike clusters. As we have already mentioned their mean size does not change considerably with cs ; it increases only by 1.38. On the other hand, the percentage of DSS particles that form chains triples (from 23% to 58%) going from relative low volume fraction cs = 0.299% to higher cs = 0.916% (measured at the saturation

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value of S (r) ). Hence, both the clusters size and the quantity of chainlike clusters in the sample determines the induced alignment of the rod species. Therefore, the MD simulations indicate the (hitherto only suspected) microscopic mechanisms that are responsible for the macroscopic behaviour of such systems.

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(b)

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(c)

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Figure 9: Representative simulation snapshots of a GB-DSS mixture with µ∗ = 2.4 subject to external magnetic field of H ∗ = 8.3 at constant cr = 44% and various cs : (a) [(ρ∗ , xs ) = (1.40, 0.80)] with cs = 0.916% [nematic state], (b) [(ρ∗ , xs ) = (0.778, 0.64)] with cs = 0.407% [nematic state], and (c) [(ρ∗ , xs ) = (0.35, 0.20)] with cs = 0.0573% [isotropic state]. The ˆ r (green) and the direction of the magnetic field H (orange) are sketched as arrows. director n In the right column we have removed the rod species for clarity.

We now turn back to the diagram that is shown in Fig. 8a. In the system with smaller 22


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dipole moment magnitude (µ∗ = 1.8), the first maximum and minimum of the DSS particles’ pair correlation function are shifted to longer inter-particle distances. In addition, the first maximum is much less intense in comparison to the system with stronger dipolar moment of the DSS particles. Hence, the reduction of the saturation value of S (r) with µ∗ (for the same cs ) can be interpreted by the fact that both less "tight" and smaller clusters are formed. Therefore, the motion of the rodlike species perpendicularly to the field is less hindered by the clusters and the field induced orientation becomes less pronounced.

Conclusions In summary, using experiments and Molecular-Dynamics simulations we have studied the phase transformations and the corresponding self-organisation of binary mixtures of solid rodlike pigment particles and magnetic spheres subject to an external uniform magnetic field. In relatively low magnetic fields (up to B = 700 mT) suspensions of the pigment particles show very low magnetically induced birefringence. In contrast, appreciable birefringence is observed when a small volume fraction of magnetic particles is added to the system. The birefringence is, depending on the concentration of the pigment particles, about one order of magnitude higher than the birefringence of the pure ferrofluid/dodecane mixture of the same volume fraction of magnetic particles. This means, that the magnetic particles align anisometric pigment particles in magnetic field. Since single spherical particles cannot align the pigments, we suggest that the magnetic particles form anisometric structures, which is supported by the observation of the magnetically induced birefringence in pure ferrofluid. The magnetic particles either form aligned chains in the magnetic field, or existing chains align in an external field B. The exact process of chain alignment or formation needs further investigation. The Molecular Dynamics simulations have shown that the magnetic particles self-assemble into chains even in the zero–field (isotropic) states. Under an external magnetic

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field, the chains align along the direction of the field and increase their size. This process is dynamic and is accompanied by continuous assembly and disassembly of chains. Despite that the estimated 27 chain length in pure magnetic particle suspensions is estimated to be only two particles, it has a huge effect on the field-induced birefringence in the pigment dispersions. The Molecular-Dynamics simulations confirm, that even for a small number of chain-like aggregates of magnetic particles an overall alignment of the non-magnetic particles can be achieved. In simulations, the alignment of the non-magnetic particles is achieved already at low concentrations of chain-like aggregates of magnetic particles. The birefringence observed in the experiments is nearly two orders of magnitude higher than for pure clay particles. 29 In addition, the simulation results qualitatively agree well with the experimental results. Especially, the linear behaviour of the saturation birefringence in dependence of the magnetic particle concentration can be verified.

Acknowledgement Financial support from the German Science Foundation (DFG) via the priority programme SPP 1681 is gratefully acknowledged.

References (1) Fraden, S.; Maret, G.; Caspar, L. D.; Meyer, R. B. Isotropic-nematic phase transition and angular correlations in isotropic suspensions of tobacco mosaic virus. Phys. Rev. Lett. 1989, 63, 2068–2071. (2) Dogic, Z.; Fraden, S. Smectic phase in a colloidal suspension of semiflexible virus particles. Phys. Rev. Lett. 1997, 78, 2417–2420.

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(3) Dogic, Z.; Fraden, S. Cholesteric phase in virus suspensions. Langmuir 2000, 16, 7820– 7824. (4) Dogic, Z.; Philipse, A. P.; Fraden, S.; Dhont, J. K. G. Concentration-dependent sedimentation of colloidal rods. J. Chem. Phys. 2000, 113, 8368–8380. (5) Zhang, Z. X.; van Duijneveldt, J. S. Isotropic-nematic phase transition of nonaqueous suspensions of natural clay rods. J. Chem. Phys. 2006, 124, 154910. (6) Onsager, L. The effecs of shape on the interaction of colloidal particles. Annals of the New York Academy of Sciences 1949, 51, 627–659. (7) Straley, J. P. Ordered phases of a liquid of biaxial particles. Phys. Rev. A 1974, 10, 1881–1887. (8) Mulder, B. Isotropic-symmetry-breaking bifurcations in a class of liquid-crystal models. Phys. Rev. A 1989, 39, 360–370. (9) Lekkerkerker, H. N. W.; Coulon, P.; Van Der Haegen, R.; Deblieck, R. On the isotropic liquid crystal phase separation in a solution of rodlike particles of different lengths. J. Chem. Phys. 1984, 80, 3427–3433. (10) Veerman, J. A. C.; Frenkel, D. Phase behavior of disklike hard-core mesogens. Phys. Rev. A 1992, 45, 5632–5648. (11) Bolhuis, P. G.; Stroobants, A.; Frenkel, D.; Lekkerkerker, H. N. W. Numerical study of the phase behavior of rodlike colloids with attractive interactions. J. Chem. Phys. 1997, 107, 1551–1564. (12) McGrother, S.; Williamson, D.; Jackson, G. A re-examination of the phase diagram of hard spherocylinders. J. Chem. Phys. 1996, 104, 6755–6771. (13) Camp, P. J.; Allen, M. P. Phase diagram of the hard biaxial ellipsoid fluid. J. Chem. Phys. 1997, 106, 6681–6688. 25


Langmuir

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(14) Peroukidis, S. D.; Vanakaras, A. G. Phase diagram of hard board-like colloids from computer simulations. Soft Matter 2013, 9, 7419–7423. (15) Cleaver, D. J.; Care, C. M.; Allen, M. P.; Neal, M. P. Extension and generalization of the Gay-Berne potential. Phys. Rev. E 1996, 54, 559–567. (16) Antypov, D.; Cleaver, D. J. The role of attractive interactions in rod–sphere mixtures. J. Chem. Phys. 2004, 120, 10307–10316. (17) Jordanovic, J.; Jäger, S.; Klapp, S. H. L. Crossover from Normal to Anomalous Diffusion in Systems of Field-Aligned Dipolar Particles. Phys. Rev. Lett. 2011, 106, 038301. (18) Peroukidis, S. D.; Klapp, S. H. L. Spontaneous ordering of magnetic particles in liquid crystals: From chains to biaxial lamellae. Phys. Rev. E 2015, 92, 010501. (19) Peroukidis, S. D.; Lichtner, K.; Klapp, S. H. L. Tunable structures of mixtures of magnetic particles in liquid-crystalline matrices. Soft Matter 2015, 11, 5999–6008. (20) Eremin, A.; Stannarius, R.; Klein, S.; Heuer, J.; Richardson, R. M. Electro-optical Materials: Switching of Electrically Responsive, Light-Sensitive Colloidal Suspensions of Anisotropic Pigment Particles. Adv. Fun. Mater. 2011, 21, 556–564. (21) May, K.; Stannarius, R.; Klein, S.; Eremin, A. Electric-Field-Induced Phase Separation and Homogenization Dynamics in Colloidal Suspensions of Dichroic Rod-Shaped Pigment Particles. Langmuir 2014, 30, 7070–7076. (22) Slyusarenko, K.; Reshetnyak, V.; Reznikov, Y. Magnetic field control of the ordering of two-component suspension of hard rods. Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences 2013, 371, 20120250. (23) Kredentser, S.; Buluy, O.; Davidson, P.; Dozov, I.; Malynych, S.; Reshetnyak, V.; Slyusarenko, K.; Reznikov, Y. Strong orientational coupling in two-component suspensions of rod-like nanoparticles. Soft Matter 2013, 9, 5061–5066. 26


Page 26 of 29

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(24) Takahashi, H.; Nagao, D.; Watanabe, K.; Ishii, H.; Konno, M. Magnetic Field Aligned Assembly of Nonmagnetic Composite Dumbbells in Nanoparticle-Based Aqueous Ferrofluid. Langmuir 2015, 31, 5590–5595. (25) Schmidt, M. U.; Hofmann, D. W. M.; Buchsbaum, C.; Metz, H. J. Crystal Structures of Pigment Red 170 and Derivatives, as Determined by X-ray Powder Diffraction. Angew. Chem. Int. Edit. 2006, 45, 1313–1317. (26) Rosensweig, R. Ferrohydrodynamics; Cambridge Monographs on Mechanics; Cambridge University Press, 1985. (27) Xu, M.; Ridler, P. J. Linear dichroism and birefringence effects in magnetic fluids. J. Appl. Phys. 1997, 82, 326–332. (28) Herzfeld, J.; Berger, A. E.; Wingate, J. W. A highly convergent algorithm for computing the orientation distribution functions of rodlike particles. Macromolecules 1984, 17, 1718–1723. (29) van der Beek, D.; Petukhov, A. V.; Davidson, P.; Ferré, J.; Jamet, J. P.; Wensink, H. H.; Vroege, G. J.; Bras, W.; Lekkerkerker, H. N. W. Magnetic-field-induced orientational order in the isotropic phase of hard colloidal platelets. Phys. Rev. E 2006, 73, 041402. (30) Miguel, E. D.; Rull, L. F.; Chalam, M. K.; Gubbins, K. E. Liquid crystal phase diagram of the Gay-Berne fluid. Mol. Phys. 1991, 74, 405–424. (31) Sreekumari, A.; Ilg, P. Slow relaxation in structure-forming ferrofluids. Phys. Rev. E 2013, 88, 042315. (32) Klokkenburg, M.; Erné, B. H.; Meeldijk, J. D.; Wiedenmann, A.; Petukhov, A. V.; Dullens, R. P. A.; Philipse, A. P. In Situ Imaging of Field-Induced Hexagonal Columns in Magnetite Ferrofluids. Phys. Rev. Lett. 2006, 97, 185702.

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(33) Borbáth, T.; Borbáth, I.; Günther, S.; Marinica, O.; Vékás, L.; Odenbach, S. Threedimensional microstructural investigation of high magnetization nano–micro composite fluids using x-ray microcomputed tomography. Smart Mater. Struct. 2014, 23, 055018. (34) Schoen, M.; Klapp, S. H. L. Nanoconfined fluids: Soft Matter between two and three dimensions. Reviews of Computational Chemistry 2007, 24, 1–517. (35) Araki, T.; Tanaka, H. Colloidal Aggregation in a Nematic Liquid Crystal: Topological Arrest of Particles by a Single-Stroke Disclination Line. Phys. Rev. Lett. 2006, 97, 127801. (36) Allen, M. P.; Tildesley, D. J. Computer Simulation of Liquids; Oxford Science Publ; Oxford University Press, 2006. (37) Wilson, M. R.; Allen, M. P.; Warren, M. A.; Sauron, A.; Smith, W. Replicated data and domain decomposition molecular dynamics techniques for simulation of anisotropic potentials. J. Comput. Chem. 1997, 18, 478–488. (38) Camp, P. J.; Allen, M. P.; Masters, A. J. Theory and computer simulation of bent-core molecules. J. Chem. Phys. 1999, 111, 9871–9881. (39) Schümann, M.; Günther, S.; Odenbach, S. The effect of magnetic particles on pore size distribution in soft polyurethane foams. Smart Mater. Struct. 2014, 23, 075011.

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Δn

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Colloidal Suspensions of Rodlike Nanocrystals and Magnetic Spheres

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