Calorimetric study of the Paranematic-to-Nematic transition of polydomain side-chain...

EPJ manuscript No.(will be inserted by the editor)

Experimental and theoretical study of the nanoparticle-driven

blue phase stabilisation

Brigita Rozic1, Vassilios Tzitzios2, Eva Karatairi2,3, Uros Tkalec1,4, George Nounesis2, Zdravko Kutnjak1, George

Cordoyiannis1,5, Riccardo Rosso6, Epifanio G. Virga6, Igor Musevic1, and Samo Kralj1,7,a

1 Condensed Matter Physics Department, Jozef Stefan Institute, Jamova cesta 39, Ljubljana 1000, Slovenia

2 Institute of Radioisotopes and Radiodiagnostic Products, National Centre for Scientific Research Demokritos, Aghia Paraskevi

15310, Greece

3 Department of Materials Science, University of Patras, 26500 Patras, Greece

4 NAMASTE Centre of Excellence, Jamova 39, Ljubljana 1000, Slovenia

5 EN FIST Centre of Excellence, Dunajska 156, Ljubljana 1000, Slovenia

6 Dipartimento di Matematica and CNISM, Universita di Pavia, via Ferrata 1, I-27100 Pavia, Italy

7 Department of Physics, University of Maribor, Koroska cesta 160, Maribor 2000, Slovenia

Received: date / Revised version: date

Abstract. High-resolution ac calorimetry and optical polarising microscopy have been employed in order

to study the effects of various types of nanoparticles on the stabilisation of liquid-crystalline blue phases.

Imposing stabilisation of blue phases, achieving a 20 K range for blue phase III, has been observed for CdSe

nanoparticles of 4.5 nm diameter and hydrophobic coating. On the contrary, the effect of hydrophilic aerosil

silica nanoparticles of 7.5 nm diameter and hydrophilic coating is very week. The presented experimental

data are accompanied by a theoretical model that accounts for the observed phenomena.

1 Introduction

The cholesteric blue phases (BPs) appear in a narrow tem-

perature range between the isotropic and the cholesteric

phase of highly-chiral thermotropic liquid crystals [1–4]

Send offprint requests to:a Present address: [email protected]

and, scarcely, in lyotropic systems [5,6]. Three distinct

and thermodynamically stable BPs have been observed

upon decreasing temperature, named as blue phase III

(BPIII), blue phase II (BPII) and blue phase I (BPI) [7,8].

Their structure, as it has been revealed by previous stud-

ies, is macroscopically amorphous for BPIII and of simple

2 Brigita Rozic et al.: Experimental and theoretical study of the nanoparticle-driven blue phase stabilisation

and body-centered cubic symmetry for BPII and BPI, re-

spectively [9–11]. The blue phases are gifted with a high

potential for applications, such as three-dimentional lasers

and fast displays [12–16]. Nevertheless, their extensive use

in the industry is hampered by the narrow temperature

range at which they usually appear. Intense research ef-

forts that include pioneer synthetic methods and confine-

ment of liquid crystals (LCs), employment of experimental

techniques and theoretical studies are in progress the last

ten years in order to achieve a wide-range stabilisation

of the blue phases and understand the underlying mecha-

nisms [17–30].

In this paper detailed experimental results are reported

on blue phase stabilisation by doping a chiral LC with dif-

ferent kinds of nanoparticles. It is demonstrated that the

effect strongly depends on the type of nanoparticles dis-

persed in the liquid crystal. Moreover, a theoretical model

is presented, accounting for the trapping of nanoparticles

in the disclination lines in order to interpret the observed

phenomena. In Sec. II the samples and experimental se-

tups are described. The theoretical background is pre-

sented in Sec. III. The experimental results are shown in

Sec. IV and discussed in Sec. V. In the last section we

summarise the main conclusions. The theoretical consid-

erations needed to qualitatively explain the measurements

are presented in a more detailed way in the Appendices.

In the Appendix I the phase separation tendency on enter-

ing an orientationally-ordered phase is briefly presented.

In the Appendix II the influence of nanoparticle’s sur-

face treatment on interaction with its surrounding is anal-

CE8 liquid crystal

CdSe nanoparticles

tri-octyl phosphine

oleyl amine

Cr-HexG*-HexJ*-HexJ*-SmC*-SmA-N*-BPI-BPII-BPIII-I

OH

OH

OH

Aerosil-300 nanoparticles

4.5 nm 7 nmOH

OHOH

OH OH

O

O

Fig. 1. The chemical structure of the liquid crystal CE8 (or

8SI*) and its phase sequence can be seen at the top part. A

simple schematic representation of the CdSe and the aerosil-

300 nanoparticles can be seen at the bottom left and right part,

respectively.

ysed. In the Appendix III the influence of nanoparticles

on BPIII stability is estimated.

2 Samples and apparatus

The liquid crystal compound 4-(2-methyl butyl) phenyl 4-

n-octylbiphenyl-4-carboxylate has been supplied by Merck

and used without any additional treatment. It is assessed

in the literature under the acronyms CE8 or 8SI*. Upon

cooling from the I phase, it exhibits three distinct, ther-

modynamically stable and well-characterised blue phases

[24]. The chemical structure of CE8 and its phase transi-

tion sequence are shown at the top of Figure 1.

Brigita Rozic et al.: Experimental and theoretical study of the nanoparticle-driven blue phase stabilisation 3

The CdSe nanoparticles (NPs) have been synthesised

at the National Centre for Scientific Research “Demokri-

tos”. The procedure was as follows: 0.5 mmol Cd(acac)2

was injected in 20 ml oleyl amine (OA), under continu-

ous magnetic stirring until a clear solution was acquired.

Elemental selenium (0.5 mgr-at.) was dissolved in 2 ml tri-

octyl phosphine. The Se-TOP solution was then added in

the Cd-OA dispersion at 120 oC and the reaction was com-

pleted in 1 hour at 250 oC in order to attain nanoparticle

nucleation and growth. All the experimental procedures

were carried out under Ar mantle. Nanoparticles precipi-

tated after cooling the reaction mixture to room temper-

ature by the addition of 40 ml ethanol. The so-produced

CdSe NPs are essentially monodisperse and have an av-

erage diameter of 4.5 nm [24]. They are highly soluble in

non-polar solvents such as hexane, toluene and chloroform.

Prior to mixing with CE8 their solution was inserted in

an ultrasonic bath in order to eliminate any possible ag-

gregation. A simple schematic representation of them is

shown at the bottom left part of Figure 1. Further details

on the preparation and the characterisation of the CdSe

NPs have been described elsewhere [31].

The hydrophilic aerosil type-300 nanoparticles are man-

ufactured by Degussa. They are spherical SiO2 particles

with a diameter 2R ∼ 7 nm, treated with hydroxyl groups

on an active surface area a = 300m2/g. Prior to mixing

with the liquid crystal, the aerosil powder was heated to

500 oC for few hours in order to destroy preformed particle

aggregates mediated by hydrogen bonding and to evapo-

rate any possible humidity absorbed by them. A simple

schematic representation of these NPs is shown at the bot-

tom right part of Figure 1.

Apart from the bulk CE8, several mixtures with CdSe

and aerosil nanoparticles have been prepared and mea-

sured. In particular, the x = 0.02, 0.07 and 0.20 concen-

trations were prepared for CE8 and CdSe nanoparticles,

where x stands for the ratio of the mass of NPs over

the mass of CE8. Moreover, a mixture of x = 0.05 was

prepared for CE8 and aerosils. This concentration corre-

sponds to the range of the so-called soft-regime, where the

nanoparticles form adaptive networks [32,33]. High-purity

toluene has been used as a solvent for the preparation of

CE8 and CdSe mixtures (due to the good solubility of

the nanoparticles in the particular solvent), whereas high-

purity acetone has been used for the CE6 and aerosil ones,

following the well-established protocol described in [34].

High-resolution ac calorimetry has been used in order

to obtain the heat capacity temperature dependence for

pure CE8 and several mixtures with nanoparticles. Apart

from the principal ac mode, our apparatus can operate

also in the so-called relaxation or non-adiabatic scanning

mode. The ac mode is sensible only to the continuous en-

thalpy changes, but the phase of the ac temperature oscil-

lations provides information on the nature of the transi-

tion, i.e. continuous (second order) or discontinuous (first

order). The relaxation mode senses both the continuous

and discontinuous (i.e. latent heat) enthalpy changes. A

detailed description of the technique can be found else-

where [35]. Sample quantities of 40mg were placed in silver

cells and mounted in the calorimeter. The heat capacity of


the cell was subtracted and the result was divided by sam-

ple mass in order to derive the net specific heat capacity

Cp of the samples.

Optical observations of the blue phases of bulk CE8

were performed using a polarising optical microscope Nikon

Eclipse E600. The sample was placed between untreated

indium tin oxide (ITO) glass plates and then inserted in

an Instec STC-200 heating stage with temperature stabil-

ity of 0.1 K. Afterwards, it was examined under crossed

polarisers by slowly decreasing the temperature from the

isotropic phase. A Nikon Coolpix E5400 digital camera

has been used for capturing images.

3 Theoretical background

3.1 Parameters describing the mixtures

We consider a nematogen - non-nematogen mixture. The

local volume concentration ϕ of the NPs is the conserved

parameter. The spatially averaged volume concentration

of NPs and LC molecules in the mixture is given by ϕ =

Nnpvnp/V and 1−ϕ, respectively. Here over-bar (..) marks

the spatial average, Nnp counts number of NPs, vnp mea-

sures the volume of a nanoparticle and V stands for the

volume of the sample.

At the mesoscopic level the LC orientational ordering

is described by the symmetric, traceless tensor [37]

Q =3∑

i=1

λi−→e i ⊗−→e i. (1)

It is expressed in terms of its eigenvalues λi and eigen-

vectors −→e i. We consider uniaxial nematic LCs, which in a

bulk equilibrium exhibit uniaxial ordering conventionally

expressed as [37]

Q = S

(−→n ⊗−→n − 1

3I

). (2)

Here −→n is the nematic director field, S is the uniaxial

orientational order parameter, and I stands for the unit

tensor. The unit vector −→n , exhibiting the head-to-tail in-

variance (i.e. ±−→n states are equivalent), points along the

local uniaxial ordering direction. The order parameter S

quantifies the extent of fluctuations along this direction.

3.2 Free energy

We express the free energy Ω of a mixture as

Ω =

∫(fm + fc + fe + fsδ(

−→r −−→r s)) d3−→r , (3)

where δ stands for the delta function, −→r s locates NP-

LC interfaces and the integral runs over the sample vol-

ume. The mixing (fm), condensation (fc) and elastic (fe)

free energy densities are expressed as

fm =kBT

vlc(1− ϕ) ln(1− ϕ) +

kBT

vnpϕ lnϕ+ χϕ(1− ϕ),

(4)

fc =A0(T − T∗)

2QijQij −

B

3QijQjkQki +

C

4(QijQij)

2,

(5)

fe =L

2Qjk,iQjk,i + 2κLεiklQijQlj,k, (6)

where a summation over the repeated indices is as-

sumed. Only the most essential and simplified terms, nec-


essary in order to explain the experimental observations,

are included.

The term fm describes the isotropic mixing for the

two components within the Flory theory [38], kB is the

Boltzmann constant, T is the absolute temperature, vlc

estimates the volume of an average LC molecule, and χ

stands for the Flory-Huggins parameter [38].

The condensation term fc enforces orientational LC

ordering below a critical temperature Tc. The quantities

A0, B, C, are material constants and T∗ describes the

supercooling temperature of a pure bulk LC component.

The elastic term fe penalises the elastic distortions

which tend to enforce a twisted orientational ordering. The

quantity L is the bare nematic elastic constant represent-

ing the LC elastic properties in a single elastic constant

approximation. κ is the chiral parameter, which is pro-

portional to the pitch of the cholesteric helix in a bulk

cholesteric phase.

The conditions at the NP-LC interface are determined

by the surface anchoring term fs. Its anisotropic part ∆fs

is conventionally expressed as ∆fs =w2

(Qij −Q

(s)ij

)(Qij −Q

(s)ij

), where w > 0 is the anchoring strength fa-

voring the nematic ordering determined by Q(s) at an in-

terface. For example, if a uniaxial orientational ordering

is enforced along the unit vector −→n e (the so called easy

axis), then

Q(s) = Ss

(−→n e ⊗−→n e −

1

3I

), (7)

where Ss describes the surface enforced degree of the

uniaxial ordering.

3.3 Mechanical force

In order to calculate a mechanical force−→F on a NP im-

mersed in orientationally-ordered LC medium we use the

expression [40,39],

−→F =

∫T (E)−→v d2r. (8)

Here T (E) is the Ericksen stress tensor and the integration

is performed over the NP surface, where a local outer unit

normal points along −→v . The Eriksen stress tensor can be

expressed in terms of the volume LC free energy density

f = fc + fe as

T (E) = If −∇Q⊙ ∂f

∂∇Q, (9)

where

(A⊙B)ij :=∑h,k

AhkiBhkj (10)

for any pair of third-rank tensors A and B.

4 Experimental Results

In this section the experimental results are presented. In

Figure 2 optical polarising microscopy images of the blue

phase structure of bulk CE8 are shown. In Figure 3 the

heat capacity temperature profiles obtained by the ac mode

along the isotropic (I), BPIII, BPII, BPI, cholesteric (N∗)

and smectic A (SmA) phases are shown for the bulk CE8

and three mixtures with CdSe NPs. When CE8 is doped

with NPs, all phase transitions are smeared and shifted to

lower temperatures. A substantial increase in the range of

BPIII as a function of NPs concentration is observed. This


Fig. 2. Images of the BPI structure of bulk CE8 placed in ITO

glasses and observed under crossed polarisers. The image A

(top part) displays a larger sample area, while image B (bottom

part) corresponds to a smaller area using three times higher

magnification. In both images the peculiar platelet structure

of BPI can be visualised.

range, being 1.2 K wide for bulk CE8, is extended to 10 K

for x = 0.02 and it approaches 20 K for x = 0.20 of NPs.

Concerning the other two blue phases, BPII already dis-

appears for x ≥ 0.02 and BPIII is rather mildly affected.

The resulting T-x phase diagram for CE8 and CdSe NPs

is displayed in Figure 4.

390 400 410 420

2

3

4

BPI-BPIIIBPIII-IN*-BPI

SmA-N*

T (K)

2

3

4 BPIII-I

BPI-BPIIIN*-BPI

(d)

(c)

Cp (J

K-1

g-1)

SmA-N*

2

3

4BPIII-I

BPI-BPIIIN*-BPISmA-N*

BPII-BPIII

(b)

2

3

4BPIII-I

N*-BPI

BPI-BPII

SmA-N*

(a)

Fig. 3. The heat-capacity temperature profiles obtained in

ac runs are presented here. From top to bottom, the results

for bulk CE8 (a), x = 0.02 (b), 0.07 (c) [24] and 0.20 (d)

concentrations of CdSe nanoparticles can be seen.

The heat capacity temperature profiles for the bulk

CE8 and the mixture x = 0.05 with aerosil NPs on the

phase transition behaviour is shown in Figure 5. Contrary

to the strong impact of CdSe NPs, observed already for

x = 0.02 mixture, a very week influence of the aerosils is

realised in this case. The phase transitions occur at tem-

peratures very close to the ones of CE8 and their shape

remains bulk-like. In addition, BPII does not disappear

for the aerosil mixture studied here, though its range is

decreased for few mK compared to bulk CE8. The con-


0.00 0.05 0.10 0.15 0.20

-20

-10

0(b)

SmA

N*BPI

BPII

BPIII

I

T-

T IBPI

II (K

)

x

0.00 0.05 0.10 0.15 0.20

-20

-10

0

x

T-T IB

PIII

(K)

I

BPIII

BPII

BPIN*

SmA

(a)

Fig. 4. The phase diagram of CE8 and its mixtures with CdSe

NPs is presented upon cooling (a) and heating (b). The solid

lines serve only as guides to the eye.

siderable difference between the effect of the two types of

nanoparticles can be depicted in Figure 6, where the re-

sulting phase diagrams are plotted in the same range of

concentrations.

The results for CE8 and CdSe mixtures are repro-

ducible on heating and cooling, as it has been confirmed

by the present measurements. A very small hysteresis ob-

served between heating and cooling runs is attributed to

the first order character of these phase transitions.

405 410 415 420

2

4

6

8

10

BPIII-I

BPII-BPIII

BPII-BPIBPI-N*

N*-SmA

Cp (J

g-1K

-1)

T (K)

(b)

2

4

6

8

10

BPII-BPIII

BPII-BPI

BPI-N*N*-SmA

BPIII-I

(a)

Fig. 5. The heat-capacity temperature profiles obtained in ac

runs are presented for bulk CE8 [24] at the top layer (a) and for

x = 0.05 concentration of aerosil nanoparticles at the bottom

layer (b).

5 Discussion

Henceforth, the possible mechanisms behind the observed

huge temperature widening of the BPIII phase are dis-

cussed. The 1st possible mechanism was originally pro-

posed by Kikuchi et al. [17] who studied a polymer-stabilised

BPI structure. They have reported that the monomers as-

semble at the disclination cores. The overall volume of

relatively costly defect cores is reduced and, consequently,

the temperature interval of the BP structure hosting the

disclination lines is increased.


0.000 0.025 0.050-15

-10

-5

0(b)

SmA

N*

BPIBPII

BPIII

I

T-

T IBPI

II (K

)

x

0.000 0.025 0.050

-15

-10

-5

0

x

T-T IB

PIII

(K)

I

BPIII

BPII

BPI

N*

SmA

(a)

Fig. 6. The phase diagrams are presented, plotted only in

the range of similar concentrations of CdSe NPs at the top

layer (a) and aerosil NPs at the bottom layer (b) for a direct

comparison. The solid lines on both layers serve only as guides

to the eye.

The proposed scenario explains in addition the NP-

induced widening of BPIII phase for appropriate parti-

cles. In the following, the essential conditions that are

necessary for the NP-induced BPIII phase stabilisation

are discussed. For the case that NPs collect at the core

of defects it is shown in Appendix III that their assem-

bling increases the temperature range ∆TBP of the sta-

bility of the BPIII phase. The main mechanism behind

this is a partial replacement of the relatively energetic

core volume of the disclinations with NPs. Therefore, the

trapped NPs strongly reduce the effective condensation

free energy costs, while the remaining free energy terms

are rather weakly affected. The estimate Eq.(33), derived

in Appendix III, yields a dependence∆TBP (x)/∆TBP (0) ∝

1/(1−xb), with b being a constant, which is in line with the

experimental observations. In the limit of large x values,

NPs tend to saturate the defect lines and the described

mechanism is not anymore effective. The NPs are expected

to be essentially non-uniformly distributed within the discli-

nation lines. Such a random character stabilises the amor-

phous BPIII-type structure and not the more ordered BPI

and BPII structures. Note that the effect of NPs on the

I-BP transition with respect to the BP-N∗ one is much

weaker because i) the penalty of introducing defects is

less energetic and ii) the size of defect cores is larger and,

consequently, the core volume replacement is less effective.

We next discuss the conditions which favour the as-

sembling of NPs at the disclination lines. For this pur-

pose one needs initially a homogeneous mixture of NPs

and LC in the isotropic phase. Therefore, the effective

Florry-Hugins parameter χeff , introduced in Appendix I

(Eq.(17)), should be below a critical value. On entering

BPIII, the average orientational ordering is established

and consequently χeff strongly increases, see Eq.(17), as

it is demonstrated in the Appendix I. In order to promote

phase separation this term should be positive. If it is in-

creased above the threshold value, the NPs tend to sepa-

rate from the LC medium. It is of interest to assemble the

expelled NPs at the disclination lines. But for this mech-


Fig. 7. The hybrid plane parallel cell of thickness h hosting

a nanoparticle. At the top plate (z = h) the radial uniaxial

orientation is enforced. On the contrary, the bottom plate (z =

0) imposes a uniaxial homeotropic orientation along the z-axis.

The free boundary condition is imposed at the cylinder wall

(r = R). The system exhibits a cylindrical symmetry.

anism to be effective, it is necessary that the NPs have

appropriate surface coating. Note that the expression for

χeff includes also the term which describes the LC wet-

ting properties at the LC-NP interfaces. This contribution

might play an important role for a strong enough surface

interaction wetting strength W . It supports a phase sep-

aration tendency if W > 0, i.e., the interface tends to

locally decrease the degree of LC ordering.

In order to estimate the effects of NP’s surface coating

on interaction with its surrounding and the consequent

NP trapping capabilities, we study theoretically a force

exhibited on NP within a plane parallel hybrid cell ex-

hibiting a cylindrical symmetry. A cylindrical-shape NP

is set at an arbitrary position z at the symmetry axis of

the cell as shown in Figure 7. At the top plate (z = h) a

strong uniaxial radial anchoring is imposed, i.e.

Q(s)rad = Seq

(−→e r ⊗−→e r −

1

3I

), (11)

where −→e r is the unit vector along the radial cylindrical

coordinate and Seq stands for the equilibrium nematic or-

der parameter (see Eq.(7)). At the bottom plate (z = 0)

a strong homeotropic anchoring is imposed, i.e.

Q(s)hom = Seq

(−→e z ⊗−→e z −

1

3I

), (12)

where −→e z is the unit vector along the z-axis. For such a

geometry the boojum topological defect in enforced at the

top plate [41]. For given boundary conditions it is char-

acterised by a finger-like structure, where the finger tip is

due to topologically reasons melted (i.e., Q = 0). The boo-

jum core structure is analysed in detail in Ref.[41]. Due

to the symmetry of the problem the mechanical force on

the immersed NP points along ±−→e z if it is different from

zero. Note that there exist at least three qualitatively dif-

ferent “attractors” for the NP : i) the melted point at the

boojum finger tip, ii) a homogeneous ordering along −→e z

at z = 0, and iii) a radial ordering at z = h. We consider

three different surface coatings of the NP. At the NP sur-

face we either strongly enforce a) Q = 0, b) Q = Q(s)rad,

c) or Q = Q(s)hom. The hight and diameter of NP is set to

be equal to the biaxial order parameter correlation length.

The position of NP is varied along the z-axis, and for each

position we calculate the equilibrium nematic configura-

tion and the corresponding mechanical force on the NP.


The mathematical details of simulation are given in Ap-

pendix II. In Figure 8 the force is plotted as a function

of z-coordinate for the three different surface treatments.

One can see that the NP, which is enforcing melting, ra-

dial or homogeneous configuration, tends to assemble at

the finger tip, top plate or bottom plate, respectively. The

main information of this study is that the NPs tend to

assemble at locations which exhibit similar local struc-

ture. Therefore, in order to to efficiently assemble NPs

at a disclination core it is essential to choose the surface

coating in such a way that the effective NPs resembles the

defect core structure.

The experimental results confirm the simulation out-

comes. The aerosils locally enforce a relatively strong ne-

matic ordering. On the other hand, the CdSe NPs are

coated with flexible chains. Therefore, the effective struc-

ture of CdSe NPs more resembles the essentially disor-

dered structure at disclination lines. In addition, the aerosil

particles tends to form a network structure, which pre-

vents the collection of particles at disclination lines. In

the experiment we probed a concentration within the so-

called soft network regime [32,33], ranging in the interval

0.01 < x < 0.10, where the aerosil nanoparticles form

adaptive networks. Therefore, the CdSe NPs are expected

to have a much stronger tendency to assemble at discli-

nations. Consequently, the reduction of the energetically

costly condensation term is less effective in aerosil-liquid

crystal mixtures.

The measured stability diagrams in the (x, T ) plane

are topologically similar to stability diagrams, where x

0.2 0.4 0.6 0.8

-0.04

-0.02

0.00

0.02

0.04

F/F o

(h-z)/h

radial homogeneous melted

Fig. 8. The force F experienced by differently surface-treated

NPs as a function of distance z within the cell shown in Figure

7. The nanoparicle enforces locally either i) melting of the LC

order (dashed line), or ii) radial uniaxial order (full line), or

iii) homogeneous ordering (dash-dotted line) along the z-axis.

Note that for the case i) the force equals zero at the boo-

jum finger tip. The characteristic sizes of the nanoparticle are

comparable to the biaxial order parameter correlation length,

which is roughly equal to h/10 in the simulation. Fo stands for

the reference force.

is replaced by chirality strength. This suggests that an

increased chirality itself could be responsible for the ob-

served phenomena. It could be that the flexible chains

covering the CdSe NPs could adopt and even amplify the

chiral tendency of the LC molecules. An increase in the

chirality widens BPIII much more than BPI phase and

changes the I-BPIII transition temperature much less than

the other transition temperatures, in agreement with the

present experimental results. However, note that the chi-

rality strength is defined as a ratio between the chiral elas-

tic and the condensation free energy term. In the disclina-


tion trapping mechanism the condensation penalty is de-

creased and consequently the effective chirality strength is

increased which explains the above mentioned similarities

in (x, T ) and (chirality strength, T ) phase diagram.

6 Conclusions

The effects of various types of nanoparticles in the stabili-

sation of BPs have been presented. A substantial widening

of BPIII is achieved reaching a range of 20 K for large con-

centrations of CdSe nanoparticles. BPII disappears and

BPI is weakly affected. Our theoretical estimates support

the assumption that the main mechanism responsible for

the BPIII stabilization is the effectively reduced condensa-

tion free energy penalty due to assembling of NPs at the

disclination lines. The effect of CdSe nanoparticles were

recently explored for another liquid crystal compound and

they have revealed very similar results [24,42], establish-

ing a robust method to stabilise BPIII. In particular, the

theoretical analysis suggests that in order to achieve NP-

driven stabilization it is advantageous to i) enforce LC-

NP phase separation on entering the LC orientationally

ordered phase and ii) to drive NPs towards disclination

cores. The latter process is pronounced if NPs effective

configuration locally resembles the core structure of the

disclinations. Note that there might be also a qualitatively

different mechanism behind the observed behavior which

our measurements can not rule out. The BP stability dia-

gram as a function of x is similar to the one obtained as a

function of chirality strength [36]. This suggests that the

presence of appropriate NPs could enhance the effective

chirality of the system which does not require assembling

of NPs at defect cores. In this case, the molecular chiral-

ity of LCs is amplified by the presence of homogeneously

distributed NPs. However, for the first mechanism (assem-

bling the NPs at the cores) the effective chirality strength

of the system is also increasing on increasing x. Namely,

the chirality strength expresses a ratio between the elastic

and the condensation free energy penalty. In case that the

NPs assembled at the disclination cores, they apparently

reduce the effective free energy condensation whereas they

only slightly effect the elastic free energy penalty, and the

effective chirality of the system is increased. This explains

the resemblance between (x,T) and (chirality strength,T)

phase diagram also if the NP assembling mechanism em-

phasised in this paper is prevailing.

7 Acknowledgments

This work was performed with the financial support of EU

(Project No. HPRN-CT-2002-00169) and Slovenian Office

of Science (Projects No. J1-9368, J1-0155, P1-0125). E.K.

acknowledges the support of the Graduate Fellowship Pro-

gram of N.C.S.R. “Demokritos”. G.C. acknowledges the

EN FIST Centre of Excellence and U.T. acknowledges the

financial support from the NAMASTE Centre of Excel-

lence.

Appendix I: Phase separation

In the appendix we estimate the influence of the on-

set of LC orientational ordering on the phase separation

of a mixture of NPs and liquid crystal. For this purpose


the LC ordering is approximated with a spatially averaged

uniaxial order parameter S and volume concentration of

nanoparticles ϕ. The average free energy density is ex-

pressed as f = fm + fc + fe + fs . The patially averaged

terms are approximately given by [43,44]

fm ∼ kBT

vlc(1− ϕ) ln(1− ϕ) +

kBT

vnpϕ lnϕ+ χϕ(1− ϕ),

(13)

fc ∼(1− ϕ

) (a0(T − T∗)S

2 − bS3+ cS

4), (14)

fe ∼(1− ϕ

)LS

2/ξ

2, (15)

fs ∼(1− ϕ

)ϕW, (16)

where a0 = 2A0/9, b = 2B/27, c = C/9. The(1− ϕ

)term in fc and fe accounts for the part of the volume not

taken up by LC. In general T∗ = T∗(ϕ). The average do-

main length ξ estimates the elastic distortions in the LC

orientational order. A simple binary modeling [45] sug-

gests T∗ = T0 − λϕ, where T0 > 0 and λ > 0 are indepen-

dent of ϕ. Such dependence reflects relatively weaker inter-

actions among the LC molecules due to the NPs, which

act as ”impurities“. Furthermore, the average contribu-

tion of the interface is proportional to(1− ϕ

)ϕ, because

this free energy term is absent in the limits ϕ = 0 and

ϕ = 1. W includes assembled interface properties which

are independent of ϕ.

With this in mind one can express the effective Flory-

Huggins [38] parameter χeff as

χeff = χ+ a0λS2+W. (17)

Appendix II : LC mediated force on an im-

mersed object

Systems that exhibit a cylindrically symmetric LC or-

dering are considered. For this purpose the LC is con-

fined within a cylindrical plane parallel cell of thickness

h, which is schematically shown in Figure 7. The posi-

tive orthonormal triad −→e r,−→e ϑ,

−→e z describes the unit

vectors of the cylindrical coordinate system. Here −→e r is a

radial unit vector, −→e z points along the z-coordinate, while

−→e ϑ = −→e z ×−→e r.

We take into account the expression for Ω (see Eq.(3)),

where we consider a pure LC in the nematic phase and

set κ = 0, ϕ = 0. We parameterise Q in terms of scalars

qm, qδ, q0 as [46]

Q = −2q0−→e ϑ ⊗−→e ϑ + (q0 + qδ)

−→e r ⊗−→e r +

(q0 − qδ)−→e z ⊗−→e z + qm(−→e r ⊗−→e z +

−→e z ⊗−→e r).

(18)

We restrict our attention to cylindrically symmetric

solutions where the parameters q’s depend only on the

coordinates r and z. It is convenient to introduce the re-

duced temperature θ = 24A0(T−T∗)CB2 = T−T∗

T∗∗−T∗, where T∗∗

is the nematic superheating temperature and the super-

heating nematic order parameter s0 = B4C . For this scal-

ing the bulk nematic order parameter equilibrium value

is given by θ ≤ TIN−T∗T∗∗−T∗

= 8/9 by Seq = s0(1 +

√1− θ

)and metastable states exist in the interval θ ∈ [0, 1]. We

further introduce the bare biaxial correlation length ξb =√4LC/B2 and the dimensionless scaled quantities Q =


Q/s0, q0 = q0/s0, qm = qm/s0, qδ = qδ/s0 Then, all

lengths are scaled with respect to h, i.e. r = r/h, z = z/h,

ξb = ξb/h. Henceforth, the tildes are skipped. Details

about the parametrisation Eq.(18) and the employed scal-

ing are given in [46].

The resulting dimensionless volume free energy densi-

ties read

fc=θ3 (q

2m+q2δ+3q20)−4q0(q

2m+q2δ−q20)+

12 (q

2m+q2δ+3q20)

2,

fe=2ξ2b

(3|∇q0|2+|∇qδ|2+|∇qm|2+ (3q0+qδ)2+q2m

r2

).

(19)

where ∇ stands for the dimensionless Nabla operator.

The minimisation of the free energy yields the follow-

ing bulk Euler-Lagrange equations for q0(r, z), qδ(r, z),

qm(r, z):

0=ξ2b

[∆q0−

3q0+qδr2

]− θ

6 q0+13 (q

2δ+q2m−3q20)−

q02 (3q20+q2δ+q2m)

(20)

0=ξ2b

[∆qδ−

3q0+qδr2

]− θ

6 qδ+2q0qδ−qδ2 (3q20+q2δ+q2m),

(21)

0=ξ2b [∆qm− qmr2]− θ

6 qm+2q0qm− qm2 (3q20+q2δ+q2m),

(22)

where ∆ = ∂2

∂r2 + 1r

∂∂r + ∂2

∂z2 is the dimensionless Laplace

operator. The equations are solved by a standard relax-

ation method for given boundary conditions.

Our interest is to calculate the force on a NP immersed

in the LC cell. For this purpose the Ericksen stress tensor

must be evaluated at the LC-NP interface and the in-

tegration Eq.(8) has to be performed. We consider only

cases where the cylindrical symmetry of the system is

preserved, i.e. we put a cylindrically symmetric NP at

the symmetry axis of the cell as shown in Figure 7. Us-

ing the parametrisation and scaling described above, the

relevant components of the dimensionless Ericksen tensor

T (E) = T (E) (4C)3

B4 are expressed as

−→e z·T (E)−→e z=

ξ2b

(3(q20,r−q20,z)+(q2δ,r−q2δ,z)+(q2m,r−q2m,z)+

q2m+(3q0+qδ)2

r2

)

+ θ6 (3q

20+q2δ+q2m)+2q0(q20−q2δ−q2m)+ 1

4 (3q20+q2δ+q2m)

2,

(23)

−→e z·T (E)−→e r=−2ξ2b(3q0,rq0,z+qδ,rqδ,z+qm,rqm,z).

(24)

The resulting non-vanishing dimensionless force F =

F (4C)3

h2B4 along the z-axis reads

F2π=

∫top

−→e z·T (E)−→e zrdr−∫bottom

−→e z·T (E)−→e zrdz+

∫wall

−→e z·T (E)−→e rrdz,

(25)

where the 1st, 2nd and 3rd integral run over the top,

bottom and lateral surface of the cylindrical NP shown


in Figure 7. At the surfaces we either enforce melting or

conditions given by Eq.(11) or Eq.(12).

Appendix III : Nanoparticle-induced tempera-

ture shift

In this appendix we estimate influence of NPs on the

isotropic-blue phase-cholesteric (I-BP -N∗) phase sequence

upon varying the temperature. The I-BP and BP-N∗ phase

transition temperatures of bulk samples are labeled as TIB

and TBN , respectively.

Originating from Eq.(3) we consider only the most es-

sential terms in a BP phase that are needed to explain

the measurements. In a analysis the structural differences

among the BP structures can be neglected. Our attention

is confined in conditions for which the topological discli-

nation lines are stable. The latter are the hallmark of BP

structures. In the cores of the disclination lines the ne-

matic order is either essentially melted (i.e. S ∼ 0) or

strongly biaxial [47]. The core average radius is roughly

given by the relevant (uniaxial or biaxial) order parameter

correlation length ξ.

In the following the free energy cost ∆Ω of a repre-

sentative lattice cell unit of volume V0 is estimated within

a BP structure hosting disclination lines of a total length

hd, in absence of NPs. The corresponding condensation

free energy penalty ∆Ωc of introducing the disclination

line inside an orientationally ordered medium is roughly

given by

∆Ωc ∼ a0(TIB − T )S2πξ2hd. (26)

This penalty opposes to formation of a BP structure

with respect to the competitive defectless chiral phase N∗.

However, the former can be stabilised by the so-called

surface elastic term in fe, characterised by the saddle

splay elastic constant K24. This elastic term can be via

the Gauss theorem [37] expressed as a surface contribu-

tion at interfaces facing the LC body. Using the nota-

tion implemented in Eq.(3) it holds [48] K24 = k24S2 ∼

2LS2. The corresponding free energy contribution∆Ω24 of

the saddle-splay term at the interface separating the line

disclination core volume and its surrounding is roughly

given by

∆Ω24 ∼ −πk24S2hd. (27)

For K24 > 0 this term favors formation of BP struc-

tures. Furthermore, the presence of disclinations is ac-

companied by director field distortions in the LC body.

The corresponding elastic penalty ∆Ωe can be roughly

expressed [23] as

∆Ωe ∼πKhd

4ln

(Rmax

ξ

), (28)

where K = kS2 ∼ 2LS2stands for an average Frank

elastic constant [37,48] and Rmax is the cutoff radius. The

distance 2Rmax roughly determines the distance between

neighboring disclination lines running in similar direction.

The stability of the BP structures with respect to N∗

below TIB is determined by the condition ∆Ωc +∆Ω24 +

∆Ωe < 0. Hence, TBN is defined by

∆Ωc +∆Ω24 +∆Ωe = 0. (29)


The corresponding temperature width ∆TBP = TIB −

TBN of the BP stability is given by

∆TBP =k24 − k

4 ln(

Rmax

ξ

)a0ξ2

. (30)

On the other hand the I-BP transition is given by the

conditions Ω ≡ Ωc + ∆Ω24 + ∆Ωe = 0 and ∂Ω∂S

= 0,

where the condensation term Ωc which drives the I-BP

transition, is given by

Ωc ∼(a0(T − T∗)S

2 − bS3+ cS

4) (

V0 − πξ2hd

). (31)

Note that below TIB it holds Ωc < 0.

We next consider the case that NPs are added to LC

medium and that they collect at the disclination lines.

The condensation free energy penalty is decreased due to

the reduced volume occupied by the energetically costly

isotropic (or strongly biaxial) phase [17]. It roughly holds

∆Ωc ∼ a0(TIB − T )S2 (

πξ2hd −Nnpvnp). (32)

where Nnp ∝ x counts number of nanoparticles within

the defect line of length hd, and vnp is the NP’s volume.

Furthermore, we assume that the changes in ∆Ω24 and

∆Ωe are relatively negligible.

From the conditions given by Eq.(29) and Eq.(30) it

follows ∆TBP (x)∆TBP (x=0) ∼

(1− Nnpvnp

πξ2hd

)−1

. Taking into account

x ∼ mnp

mlc∼ ϕ

ρnp

ρlc, we finally obtain

∆TBP (x)

∆TBP (x=0)∼(1−x

ρlcρnp

V0πξ2hd

)−1=(1−ϕ

V0πξ2hd

)−1.

(33)

Heremnp (ρnp) andmlc (ρlc) stand for mass (densities)

of NPs and LC molecules, respectively.

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