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Memory Effects in Mixtures of Liquid Crystals and Anisotropic Nanoparticles

Written By

Marjan Krašna, Matej Cvetko, Milan Ambrožič and Samo Kralj

Submitted: March 15th, 2011 Reviewed: March 22nd, 2011 Published: August 23rd, 2011

DOI: 10.5772/22372

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1. Introduction

For years there is a substantial interest on impact of disorder on condensed matter structural properties (Imry & Ma, 1975) (Bellini, Buscaglia, & Chiccoli, 2000) (Cleaver, Kralj, Sluckin, & Allen, 1996). Pioneering studies have been carried out in magnetic materials (Imry & Ma, 1975). In such system it has been shown that even relatively weak random perturbations could give rise to substantial degree of disorder. The main reason behind this extreme susceptibility is the existence of the Goldstone mode in the continuum field describing the orienational ordering of the system. This fluctuation mode appears unavoidably due to continuous symmetry breaking nature of the phase transition via which a lower symmetry magnetic phase was reached. For example, the Imry Ma theorem (Imry & Ma, 1975), one of the pillars of the statistical mechanics of disorder, claims, that even arbitrary weak random field type disorder could destroy long range ordering of the unperturbed phase and replace it with a short range order (SRO). Note that this theorem is still disputable because some studies claim that instead of SRO a quasi long order could be established (Cleaver, Kralj, Sluckin, & Allen, 1996).

During last decades several studies on disorder have been carried out in different liquid crystal phases (LC) (Oxford University, 1996), which are typical soft matter representatives. These phases owe their softness to continuous symmetry breaking phase transitions via which these phases are reached on lowering the symmetry. In these systems disorder has been typically introduced either by confining soft materials to various porous matrices (e.g., aerogels (Bellini, Clark, & Muzny, 1992), Russian glasses (Aliev & Breganov, 1989), Vycor glass (Jin & Finotello, 2001), Control Pore Glasses (Kralj, et al., 2007) or by mixing them with different particles (Bellini, Radzihovsky, Toner, & Clark, 2001) (Hourri, Bose, & Thoen, 2001) of nm (nanoparticles) or micrometer (colloids) dimensions. It has been shown that the impact of disorder could be dominant in some measured quantities. In particular the validity of Imry-Ma theorem in LC-aerosil mixtures was proven (Bellini, Buscaglia, & Chiccoli, 2000).

In our contribution we show that binary mixtures of LC and rod-like nanoparticles (NPs) could also exhibit random field-type behavior if concentration pof NPs is in adequate regime. Consequently, such systems could be potentially exploited as memory devices. The plan of the contribution is as follows. In Sec. II we present the semi-microscopic model used to study structural properties of LCs perturbed by NPs. We express the interaction potential, simulation method and measured quantities. In Sec. III the results of our simulations are presented. We calculate percolation characteristics of systems of our interest. Then we first study examples where LC is perturbed by quenched random field-type interactions. We analyze behavior

  1. in the absence of an external ordering field B,

  2. in presence of B, and

  3. Binduced memory effects.

Afterwards we demonstrate conditions for which LC-NP mixtures effectively behave like a random field-type system.


2. Model

2.1. Interaction potential

We consider a bicomponent mixture of liquid crystals (LCs) and anisotropic nanoparticles (NPs). A lattice-spin type model (Lebwohl & Lasher, 1972) (Romano, 2002) (Bradač, Kralj, Svetec, & Zumer, 2003) is used where the lattice points form a three dimensional cubic lattice with the lattice constanta0. The number of sites equals N3, where we typically set N= 80. The NPs are randomly distributed within the lattice with probability p(For p= 1 all the sites are occupied by NPs).

Local orientation of a LC molecule or a nanoparticle at a site riis given by unit vectors siandmi, respectively. We henceforth refer to these quantities as nematic and NP spins. We take into account the head-to-tail invariance of LC molecules (De Gennes & Prost, 1993), i.e., the states ±siare equivalent. It is tempting to identify the quantity siwith the local nematic director which appears in continuum theories. We allow NPs to be ferromagnetic or ferroelectric. In these cases mipoints along the corresponding dipole orientation. Also other sources of NP anisotropy are encompassed within the model. For example, mimight simulate a local topological dipole consisting of pair defect-antidefect.

The interaction energy Wof the system is given by (Lebwohl & Lasher, 1972) (Romano, 2002) (Bradač, Kralj, Svetec, & Zumer, 2003)


The constantsJij(LC), Jij(NP), and wijdescribe pairwise coupling strengths LC-LC, NP-NP, and LC-NP, respectively. The last two terms take into account a presence of homogeneousexternal electric or magnetic fieldB=BeB, where eBis a unit vector; the B2 term acts on nematic spins while linear Bterm acts on magnetic spins.

Only first neighbor interactions are considered. ThereforeJij(LC), Jij(NP), and wijare different from zero only if iand jdenote neighbouring molecules. The Lebwohl Lasher-type term describes interaction among LC molecules, whereJij(LC)=J0. Therefore, a pair of LC molecules tend to orient either parallel or antiparallel. The coupling between neighboring NPs is determined with Jij(NP)=JNP0which enforces parallel orientation. On the contrary, neighboring LC-NP pairs tend to be aligned perpendicularly by the interaction strength wij= w< 0.

We also consider the case when the anisotropic particles act as a randomfield. For this purpose we use the interaction potential (Bellini, Buscaglia, & Chiccoli, 2000) (Romano, 2002).


The first LC-LC ordering term is already described above. In the second term the quantity wiplays the role of a local quenched field. LC molecules are occupying all the lattice sites and only a fraction pof them experiences the quenched random field. These ”occupied” sites are chosen randomly. In the cases wi= w> 0 or w< 0 the random field tends to align LC molecules alongeior perpendicular to it, respectively. The direction of the unit vector eiis chosen randomly and is distributed uniformly on the surface of a sphere.

In all subsequent work, distances are scaled with respect to a0and interaction energies are measured with respect to J(i.e., J= 1).

2.2. Simulation method

Each site is enumerated with three indices: p, q, r, where1pN, 1qN,1rN. The equilibrium director configuration is obtained by minimizing the total interaction energy with respect to all the directors by taking into account the normalization condition|npqr|2=1. The resulting potential to be mimimized readsW*=pqrWpqr*, where


and λpqrare Lagrange multipliers. We minimize the potential W*and obtain the following set of N3equations which are solved numerically. We give here the corresponding equations just for the free energy given in Eq. (2).


where the vector function gis defined as


The system of Eq. (4) is solved by relaxation method which has been proved fast and reliable. We used periodic boundary conditions for spins at the cell boundaries, for instance, the “right” neighbor of the spin with indices (N, q, r) is the spin with indices (1, q, r), and similarly in other boundaries.

2.3. Calculated parameters

In simulations we either originate from randomly distributed orientations of directors, or from homogeneously aligned samples along a symmetry breaking direction. In the latter case the directors are initially homogeneously aligned alongex. We henceforth refer to these cases as the

  1. randomand

  2. homogeneouscase, respectively.


  1. randomcase can be experimentally realized by quenching the system from the isotropic phase to the ordered phase without an external field (i.e., B= 0). This can be achieved either via a sudden decrease of temperature or sudden increase of pressure.

  2. The homogeneouscase can be realized by applying first a strong homogeneous external field Balong a symmetry breaking direction.

After a complete alignment is achieved the field is switched off.

In order to diminish the influence of statistical variations we carry out several simulations (typically Nrep ~ 10) for a given set of parameters (i.e., w, pand achosen initial condition).

From obtained configurations of directors we calculate the orientational correlation function G(r). It measures orientational correlation of directors as a function of their mutual separation r. We define it as (Cvetko, Ambrozic, & Kralj, 2009)


The brackets

denote the average over all lattice sites that are separated for a distance r. If the directors are completely correlated (i.e. homogeneously aligned along a symmetry breaking direction) it follows G(r) = 1. On the other hand G(r) = 0 reflects completely uncorrelated directors. Since each director is parallel with itself, it holds G(0) = 1. The correlation function is a decreasing function of the distance r. We performed several tests to verify the isotropic character of G(r), i.e.G(r)=G(r).

In order to obtain structural details from a calculated G(r) dependence we fit it with the ansatz (Cvetko, Ambrozic, & Kralj, 2009)


where the ξ, m, and sare adjustable parameters. In simulations distances are scaled with respect to a0 (the nearest neighbour distance). The quantity ξestimates the average domain length (the coherence length) of the system. Over this length the nematic spins are relatively well correlated. The distribution width of ξvalues is measured by m. Dominance of a single coherence length in the system is signalled by m= 1. A magnitude and system size dependence of sreveals the degree of ordering within the system. The case s= 0 indicates the short range order (SRO). A finite value of sreveals either the long range order (LRO) or quasi long range order (QLRO). To distinguish between these two cases a finite size analysis s(N) must be carried out. If s(N) saturates at a finite value the system exhibits LRO. If s(N) dependence exhibits algebraic dependence on Nthe system possesses QLRO (Cvetko, Ambrozic, & Kralj, 2009).

Note that the external ordering field (B) and NPs could introduce additional characteristic scales into the system. The relative strength of elastic and external ordering field contribution is measured by the external field extrapolation length (De Gennes & Prost, 1993)ξB~J/B. In the case of ordered LC-substrate interfaces the relative importance of surface anchoring term is measured by the surface extrapolation length (De Gennes & Prost, 1993)de~J/w. The external ordering field is expected to override the surface anchoring tendency in the limitde/ξB1. However, if LC-substrate interfaces introduce a disorder into the system, then instead of dethe so called Imry-Ma scale ξIMcharacterizes the ordering of the system. It expresses the relative importance of the elastic ordering and local surface term. It roughly holds (Imri & Ma, 1975):


where wdiswmeasures the disorder strength. Parameter din the exponent of Eq. (8) denotes the dimensionality of physical system: in our case d= 3, thusξIMwdis2.


3. Results

3.1. Percolation

One expects that systems might show qualitatively different behaviour above and below the percolation threshold p= pcof impurities. For this reason we first analyze the percolation behaviour of 3Dsystems for typical cell dimensions implemented in our simulations.

On increasing the concentration pof impuritiesa percolation threshold is reached at p= pc. This is well manifested in the P(p) dependence shown in Fig. 1, where Pstands for the probability that there exists a connected path of impuritysites between the bottom and upper (or left and right) side of the simulation cell. In the thermodynamic limit Nthe P(p) dependence displays a phase transition type of behaviour, where Pplays the role of order parameter, i.e., P(p> pc) = 1 and P(p< pc) = 0. For a finite simulation cell a pretransitional tail appears below pc, and at p~ 0.30 the P(p) steepness decreases with decreasing N. In simulations we use large enough values of N, so that finite size effects are negligible.

Figure 1.

The percolation probabilityPas a function ofpand system sizeN3. For a finite value ofNthe percolation threshold (p=pc) is defined as the point whereP= 0.5. We obtainpc~ 0.3 roughly irrespective of the system size. (∆)N= 60; (○)N= 80.

3.2. Structural properties in absence of external fields

We first consider the case where LC is perturbed by random field. Therefore LC configurations are solved by minimizing potential given by Eq. (2).

In Fig. 2 we plot typical correlation functions for the randomand homogeneousinitial conditions. One sees that in the randomcase correlations vanish for rξ(i.e., s= 0) which is characteristic for SRO. On the contrary G(r) dependencies obtained from homogeneousinitial condition yield s> 0.

Figure 2.

G(r) forp>pcandp<pcfor thehomogeneousandrandomcase,B= 0,w= 3,pc~ 0.3,N= 80. (•)p= 0.2,homogeneous; (▲)p= 0.7,homogeneous; (○)p= 0.2,random; (∆)p= 0.7,random.

More structural details as pis varied for a relatively weak anchoring (w= 3) are given in Fig. 3. By fitting simulation results with Eq. (7) we obtained ξ(p), m(p) and s(p) dependences that are shown in Fig. 3. One of the key results is that values of ξstrongly depend on the history of systems for a weak enough anchoring strength w. A typical domain size is larger if one originates from the homogeneousinitial configuration. We obtained a scaling relation between ξand p, which is again history dependent. We obtain ξp0.92±0.03for the homogeneouscase and ξp0.95±0.02for the randomcase.

Information on distribution of domain coherence lengths about their mean value ξis given in Fig. 3b where we plot m(p). For the homogeneouscase we obtain m~ 0.95, and for the randomcase m~ 1.17. A larger value of mfor the randomcase signals broader distribution of domain coherence length values in comparison with the homogeneouscase. Our simulations do not reveal any systematic changes in mas pis varied. Note that values of mare strongly scattered because structural details of G(r) are relatively weakly m-dependent.

In Fig. 3c we plot s(p). In the randomcase we obtain s= 0 for any p. Therefore, if one starts from isotropically distributed orientations ofsi, then final configurations exhibit SRO. In the homogeneouscase sgradually decreases with p, but remains finite for the chosen anchoring strength (w= 3).

Figure 3.

Structural characteristics aspis varied forB= 0 andw= 3. a)ξ(p), b)m(p), c)s(p). (▲)homogeneous, (∆)random. Lines denote the fits to power law.

For two concentrations we carried out finite size analysis, which is shown in Fig. 4. One sees that s(N) dependencies saturate at a finite value of s, which is a signature of long-range order. We carry out simulations up to values N= 140.

Figure 4.

Finite size analysiss(N) forp<pcandp>pcfor thehomogeneouscase;B= 0,w= 3, (∆)p= 0.2; (○)p= 0.7. Lines denote average values ofs.

We now examine the ξ(w) dependence. The Imry-Ma ( Imry & Ma, 1975) theorem makes a specific prediction that this obeys the universal scaling law in Eq. (8): ξw2holds for d= 3. We have analyzed results for p= 0.3, p= 0.5, and p= 0.7, using both random and homogeneous initial configurations and we fitted results with


We expect that even in the strong anchoring limit, the finite size of the simulation cells will induce a non-zero coherence length. The fit with Eq. (9) shows Imry-Ma behavior at low wonly for cases where we originate from randominitial configurations. The fitting parameters for some calculations are summarized in Table 1.

Initial conditionpγξ0ξ
r (random)0.32.11±0.3362±171.38±0.57
r (random)0.51.97±0.1937±40.35±0.32
r (random)0.72.20±0.3236±70.00±0.36
h (homogeneus)0.33.29±0.23297±600.90±0.28
h (homogeneus)0.53.29±0.13159±140.80±0.15
h (homogeneus)0.73.15±0.2699±180.50±0.22

Table 1.

Values of fitting parameters defined by Eq. (9) for representative simulation runs.

Figure 5.

ξ(w) variations for different initial configurations forN= 80. Imry-Ma theorem is obeyed only for therandominitial configuration.

3.3. External field effect

We next include external field Band still consider system described by interactional potential given by Eq. (2). A typical G(r)dependence is shown in Fig. 6 where we see the impact of B. We plot G(r) for both homogeneousand randominitial configuration in the presence of external field and without it. For B= 0 it holds ξ(hom)> ξ(ran) , where superscripts (hom) and (ran) denote correlation lengths in samples with homogeneousand randominitialconfigurations, respectively. The reasons behind this are stronger elastic frustrations in the latter case (denotation random samples). Furthermore, ξ(ran) roughly obeys the Imry-Ma scaling for low enough external fields (i.e. ξ(ran)< < ξBwhereξB~J/B), suggesting ξ(ran) ~ ξIM. The presence of Bbecomes apparent when ξB< ξIM, which is shown in Fig. 6.

In Fig. 6 we see that the presence of external field can enforce a finite value of salso in randomsamples.

In Fig. 7 we plot ξas a function of 1/Bfor both homogeneousand randomsamples. For strong enough magnetic fields one expectsξ~ξB1/B. On the other hand for a weak enough Bthe value of ξis dominantly influenced by the disorder strength. Indeed, we observe a crossover behavior in ξ(B) dependence on varying B. The crossover between two qualitatively different regimes roughly takes place at the crossover field Bc. We define it as the field below which the difference between ξ(ran) and ξ(hom) is apparent. Below Bcthe disordered regimetakes place, where ξexhibits weak dependence on B, i.e. ξ~ ξIM. Above Bcthe ordered regimeexists, whereξ~ξB1/B. Therefore, for BBcit holds ξ(ran) ~ ξ(hom) ~ ξBand in the random regimeone observes ξ(hom)> ξ(ran) ~ ξIM.

Figure 6.

The orientational correlation function as a function of separationrbetween LC molecules. Inrandom samples G(r) vanishes for large enough values ofrforB= 0 while inhomogeneus samplesit could saturate at a finite plateau (ifporware low enough). ForB> 0 a finite plateau can be observed also inrandom samples. Parameters:p= 0.3,w= 2.5.

The corresponding s(B) dependence is shown in Fig. 8. As expected smonotonously increases on increasing B, because the external field tends to increase the degree of ordering. Note that in random samples s(B= 0) = 0 and the presence of Bgives rise to s> 0.

In Fig. 9 we show m(B) dependence. For weak enough fields (B< < Bc) one typically observes m(ran)> m(hom) ≈ 1. Therefore, in random sampleswe have larger dispersion of ξvalues than in homogeneous samples. With the increasing external field both m(ran) and m(hom) asymptotically approach the value m= 1. In the latter case the distribution of ξvales is sharply centered at ξ~ ξB.

The crossover field Bcas a function of pis shown in Fig. 10. Indicated lines roughly separate ergodic (B> Bc) and nonergodic regimes (B< Bc). With increasing pone the degree of frustration within the system increases. Consequently larger values of Bare needed to erase disorders induced memory effects.

Figure 7.

Correlation lengthξas a function of 1/Bforhomogeneousandrandomsamplesfor three different concentrations of impurities. Theξ(B) dependence displays a crossover between thedisorderedandordered regime.Thedisordered regimesextends at (B>Bc), whereξ(hom)>ξ(ran). In theordered regime(B<Bc) one observesξ(ran) ~ξ(hom) ~ξB. Parameters:w= 2.5,N= 100.

Figure 8.

Thes(B) dependence forhomogeneousandrandomsamplesfor two differentp.Fors(B= 0) we obtains(ran) = 0. In thedisordered regimeit holdss(hom)>s(ran) ands(hom) ~s(ran) in theordered regime. Parameters:w= 2.5,N= 100.

Figure 9.

Them(B) dependence forhomogeneousandrandomsamplesfor two differentp. In thedisordered regimeit holdsm(ran)>m(hom) ≈ 1. In theordered regimewe obtainm(ran) ~m(hom) which asymptotically approach one on increasingB.Parameters:w= 2.5,N= 100.

Figure 10.

The crossover fieldBcon varyingp. Indicated dotted curve rougly separates ergodic (B>Bc) and nonergodic regimes (B<Bc). With increasingpone the degree of frustration within the system increases. Consequently larger values ofBare needed to erase disorders induced memory effects. The points are calculated and the dotted line serves as a guide for the eye. Parameters:w= 2.5,N= 100.

3.4. Memory effects

We further analyze how one could manipulate the domain-type ordering with external magnetic or electric ordering field. For this purpose we originate from the randominitial configuration. We then apply an external field of strength Band calculate configuration for different concentrations of impurities. Then we switch off the field and calculate again the configuration, to which we henceforth refer as the switch-off configuration. The corresponding calculated sand ξbehaviour is shown in Fig. 11 and Fig. 12. Dashed lines mark values of observables in the presence of field of strength B, while full lines mark values after the field was switched off. From Fig. 11 we see that the presence of external field develops QLRO or LRO (we have not carried time consuming finite size analysis to distinguish between the two cases). This range of ordering remained as the field was switched off, although the correlation strength is reduced. Note that above the threshold field strength the degree of ordering in the switch-off configurationis saturated, i.e., becomes independent of B.

The corresponding changes in ξare shown in Fig. 12. With increasing Bthe ξvalues for samples with different pdecrease and converge to the same value, which is equal to the external field coherence length. In the switched-off configurationthe average domain coherence length increases and again for a large enough value of Bsaturates at a fixed value.

Figure 11.

s(B) forw= 4,N= 60;randomcase. Dashed curves: configurations are calculated in the presence of external fieldB. Full curves: configurations are calculated after the field was switched off.

Figure 12.

ξ(B) forw= 4,N= 60;randomcase. Dashed curves: configurations are calculated in the presence of external fieldB. Full curves: configurations are calculated after the field was switched off.


4. Mixtures

We next consider a mixture of LCs and NPs. The presence of NPs enforces to LC a certain amount of disorder. Our expectation is that if one quenches the system from the isotropic phase the established domain pattern could be stabilized by NPs. In the following we show that there indeed exists a regime where a binary mixture behaves like LC system perturbed by a random field-type perturbation.

We calculate the LC correlation by minimizing Eq. (1). In simulations mixtures are quenched from the isotropic phase. The LC correlation function is calculated from Eq. (6) from which we extract ξand sby using Eq. (7). Typical results are shown in Fig. 13 where we plot ξ(p) and s(p). A strong presence of disorder is observed for concentrations roughly between p= 0.1 till percolation threshold. This is indicated by s(p) ~ 0, which signals presence of short range order. For p> pcthe s(p) becomes again apparently larger than zero.


5. Conclusions

We study structural properties of nematic LC phase which is perturbed by presence of anisotropic NPs. Simulations are performed at the semi-microscopic level, where orientational ordering of LCs and NPs is described by vector fields taking into account head-to-tail invariance. Such modeling approximately describes entities exhibiting cylindrical symmetry. We focused on orientational ordering of LC molecules as a function of concentration p of NPs or random sites, interaction strength w between LC molecules and perturbing agents and external ordering field strength B.

Figure 13.

Structural characteristics for the mixture. One sees that therandomfield regime extends roughly betweenp=pRF~ 0.1 andp=pc~ 0.3. a)ξ(p) and b)s(p) dependence. The other interaction constants are set to 1.

We determined percolation properties of NPs, which exhibit the percolation threshold pc ~ 0.3 in three dimensions. Then we first studied cases, where impact of NPs could be mimicked by a random field type interaction. Studies for B = 0 showed that the Imry-Ma type behavior is expected only in the case, where ensembles were quenched from the isotropic phase. In this case a short range ordering is realized. Studies in presence of an external ordering field B followed. We estimated boundaries separating ergodic and nonergodic regimes. We explored memory effects by exposing LCs to different strengths of B and then switching it off. We determined regimes where memory effects are apparent and are roughly proportional to values of B. Finally we demonstrated under which conditions structural behavior in mixtures of NPs and LCs could be mimicked by random-field type models. The findings of our investigations might be useful in order to design soft matter based memory devices in mixtures of LCs and appropriate NPs.



Matej Cvetko acknowledges support of the EU European Social Fund. Operation is performed within the Operative program for development of human resources for the period 2007-2013.


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Written By

Marjan Krašna, Matej Cvetko, Milan Ambrožič and Samo Kralj

Submitted: March 15th, 2011 Reviewed: March 22nd, 2011 Published: August 23rd, 2011