MATERIAL PROPERTIES OF MIXTURES
R. Repnik, Z. Bradač and S. Kralj
Faculty of Natural Sciences and Mathematics
University of Maribor
Koroška cesta 160
Maribor, 2000, Slovenia
ABSTRACT
Domain structure plays an important role in mechanical properties of materials. In some cases typical linear domain size can
be effectively controlled by adding a perturber (e.g. nanoparticles) to the system. To demonstrate this we study the domain
evolution of the system using the Brownian molecular dynamics and the Lebwohl-Lasher lattice model. This minimal model
well describes the main universal features of the coarsening dynamics following a sudden temperature driven phase transition
from the orientationally disordered to ordered phase. A typical corresponding condensed matter system is the isotropicnematic liquid crystal (LC) phase transition. We calculate the saturated domain size as a function of the perturber
concentration, which acts as the random anisotropy field. We further estimate how surface imprinted and frozen in domain
pattern in LC cells due to surface memory effect could affect macroscopic properties of the system.
Introduction
Recent years face increased interest in composite materials. These consist of several components, combination of which
yields desired material (i.e. mechanic, electric or optic) properties, which can not be realized in a one component system. In
this contribution we focus on a domain structure of a bicomponent mixture A+B, representing a simplest possible composite
system exhibiting complex behavior. We show how the domain structure of the component A can be tuned via the component
B.
The mixture of interest consists of the component A displaying critical behavior on varying the temperature T. At the phase
transition with decreasing temperature a continuous symmetry of the parent phase is broken. In the continuum picture the
v
v
lower temperature phase is characterized by the so called gauge field φ (r ) and the order parameter field η (r ) [1]. In the
simplest case these fields are scalars. The gauge field exhibits a Goldstone fluctuation mode, trying to recover the lost
symmetry in the symmetry broken phase. In case of imposed contradicting constraints this field typically exhibits spatial
variations on the length scale ξφ that is imposed geometrically (i.e., in absence of external fields it does not exhibit a material
dependent spatial response). The order parameter field is in the pure bulk sample different from zero only in the lower
temperature phase. In general it responds to local distortions on the length scale given by the order parameter correlation
length ξη . It attains the maximal value at the phase transition temperature Tc and monotonously decreases on increasing
T − Tc . As the B component we consider a noncritical and chemically inert material that imposes constraints to A.
If the component A is quenched into the lower temperature phase then a randomly chosen configuration of the symmetry
breaking gauge field value is established in causally disconnected parts. This choice is based on local fluctuation mediated
preferences. Consequently a domain structure appears [2,3] that is well characterized by a single domain length ξd. With time t
the domain growth gradually enters the dynamic scaling regime, where the power law ξ d ∝ t γ is obeyed. The universal
scaling coefficient γ strongly depends whether a conservation law for the order parameter exists or not. In addition it depends
on the order parameter and space dimensionality d. The gauge field evolution across a domain wall obeys the geodesic rule,
i.e., it follows the shortest possible path in the continuum field space. It is still open problem how a component B could affect
and stabilize the described domain evolution.
This phenomenon is of interest for various branches of science, even for cosmology [2,4]. Note that the statistics of the
established domain pattern in a pure bulk system A can be well described by the Kibble-Zurek mechanism [4] that was
originally introduced to explain the formation of topological defects in the early universe following the big bang. Basic
ingredients of this mechanism are symmetry breaking and causality (i.e., a finite speed with which information spreads in a
system). Generality of these principles gives rise to a broad universality of the phenomenon.
The influence of B on A can be in some cases treated as random field [5,6,7]. The pioneering work on the resulting domain
structure was done by Imry and Ma [8]. They studied the influence of non-correlated quenched disorder on systems with a
continuously broken symmetry and a bending energy density proportional to the square of the gradient of the relevant order
parameter. Their Imry-Ma argument claims that an arbitrary weak random field with one fold symmetry (i.e. the field is linearly
coupled to the conjugated order parameter) breaks the system into a domain like pattern provided the spatial dimension is less
than four. Consequently, the (quasi) long-range order of the system is replaced by a short-range order. The characteristic
−
2
domain size ξd is predicted to scale as ξ d ∝ w 4 − d , where w is a measure of the disordering strength. Note that applicability of
this argument to various condensed matter systems is still disputable [9].
In the paper we use for demonstrative purpose liquid crystals (LCs) [1] as the component A. These are materials that can
potentially form liquid crystalline phases. LC phases represent an intermediate state between ordinary liquids and crystals. On
one hand they flow like an ordinary liquid. In addition they posses some long (or quasi) range ordering in the orientational or
translational order or both of them. Note that LC phases and structures are often chosen as testing ground [2,4,9] of basic
mechanisms in physics for the following main reasons. They are typical representatives of soft matter systems [1]. They earn
this name because a relatively small amount of locally supplied energy is needed for their response on experimentally
accessible length and time scales. The samples with different LC structures are relatively easy prepared because they can be
readily shaped with confining surfaces [10] or external fields due to their liquid and soft characteristics. LC phases are
transparent for light and have electro-optic anisotropic properties. Therefore they can be studied with visible light or, vice
versa, they can be used to manipulate the light. Further, there exist a large variety of different LC phases and structures (e.g.
induced by different confining cavities) within them exhibiting a rich gallery of physical phenomena. With this respect
phenomena exhibiting universal laws are of particular interest. The chemistry of LCs is relatively well developed so that
desirable behaviour can also be tailored by chemists.
We consider only the thermotropic LCs (i.e. the phases are reached via temperature variation) and the nematic phase [1]. This
phase represents the simplest LC phase. On lowering temperature it is reached from the isotropic (ordinary liquid) phase via
st
the weakly 1 order phase transition in which the continuous orientational symmetry is broken. In the nematic phase the
molecules tend to be oriented locally parallel. Consequently in bulk samples (in which the influence of confining boundaries is
negligible) the molecules are in the thermodynamically stable state aligned on average homogeneously along a single
symmetry breaking direction. As a component B one commonly chooses nano-particles [11,12] (e.g. aerosils) or porous
confinements [13,14,15] (aerogels, Vycor glass, Russian glasses, Controlled pore glasses) characterized by submicrometer
typical length scales.
In the paper we study the domain structure of A that is influenced by B. For A we chose the nematic LC phase that is
quenched from the isotropic phase. For B we chose either nanoparticles or a confining surface exhibiting surface memory
effect. The plan of the paper is as follows. In Sec. II we introduce the model and the simulation approach. In Sec. III we first
study coarsening dynamics of the pure A system and then introduce perturbers B. In the last section we summarize the
results.
Lattice model and Brownian molecular dynamics
In our simulation we use the Lebwohl-Lasher semi-microscopic lattice model [16]. The pairwise interaction gij between a pair of
v
v v
v
rod-like LC molecules at ri and r j = ri + Δrij , is expressed as
g ij = −
( )2
J v v
li ⋅ l j
r6
.
(1)
v
The positive constant J describes interaction between the first neighbours, r = Δrij , and the orientation of the i-th and j-th
v
v
molecule is described by the unit vectors li and l j , respectively. The interaction energy of the whole sample is calculated as
the sum over all pair interactions in the sample. The molecules are placed in the hexagonal lattice with the characteristic lattice
length a0. The number of molecules in simulation ranges from 103 to106.
In
the
laboratory
cartesian
coordinate
system
(x,y,z)
the
orientation
of
molecules
is
parametrized
as
v
v
v
li = (sin ϑ cos ϕ , sin ϑ sin ϕ , cos ϑ ) . Here the angles ϑ = ϑ (ri , t ) and ϕ = ϕ ( r j , t ) represent dynamic variables of the model. The
rotational dynamics of the system is driven by the Brownian molecular dynamics [17,3]. At each time interval Δt (one sweep)
the molecular orientation at the i-th site in the local cartesian frame (x', y', z') is updated obeying the equations
ϑi ( x ') = −
ϑi
( y ')
∂g ij
DΔt
+ ϑr(,xi ') ,
∑
( x ')
k BT j ≠i ∂ϑ j
(2)
∂g ij
DΔt
=−
+ ϑr(,yi ') .
∑
( y ')
k BT j ≠i ∂ϑ j
Here kB is the Boltzman constant and D the rotational diffusion constant. The z'-axis of the local frame is directed along the
long axis of the molecule. The angles ϑi
(x ')
and ϑi
( y ')
correspond to small rotation of the i-th molecule about x' and y' axes.
The gradient of the potential for these two rotations is calculated numerically. The quantities ϑr ,i
variables obeying the Gaussian probability distribution centered at
Δϑr ,i ( x ') = Δϑr ,i ( y ') are proportional to
ϑr ,i ( x ') = ϑr ,i ( y ') =0,
( x ')
and ϑr ,i
( y ')
are stochastic
where the distribution widths
T .
The corresponding model in the continuum limit is the Landau-de Gennes (LdG) model [1] in terms of the tensor order
parameter
Q in the approximation of equal nematic elastic constants. The link between the LdG mesoscopic and our
semimicroscopic ordering is roughly given by
v v
v
Q(r ) = li ⊗ li − I / 3 ,
(3)
where <…> stands for the time averaging over relatively fast molecular fluctuations and over the neighbours of the i-th site.
The uniaxial nematic ordering, which is realised in most nonperturbed or weakly perturbed nematics, is commonly given by [1]
v v
Q = S (n ⊗ n − I / 3) .
( v )2 − 1⎞⎟⎠ / 2
v
Here S = ⎛⎜ 3 n ⋅ li
⎝
(4)
v
stands for the continuum order parameter field. The gauge unit vector field n , pointing along the
average local LC orientation, is referred to as the nematic director field. The isotropic phase is characterized by S=0, and a
v
rigidly homogeneously aligned structure along n is reflected in S=1. In case that LC is strongly elastically distorted in locally
enters in biaxial states. The extent of biaxiality is well measured with the dimensionless parameter [18] β2:
(Tr Q )
= 1− 6
Tr (Q )
3 2
β2
2 3
(5)
It ranges in the interval between β=0 and β=1. These limiting values reflect a uniaxial state and a state with maximal biaxiality,
respectively.
Simulation results
In the following we study the domain pattern of the component A, represented by the LC phase suffering the isotropic nematic
(I-N) phase transition. It is of interest how the component B affects this pattern. We begin by describing the domain coarsening
dynamics in bulk. Then we add a concentration p of impurities (component B) the influence of which on A can be mimicked by
a random anisotropy (RA) field [19,20]. The orientation of this field randomly varies from the site to site. We study how the
concentration of impurities affects the stabilized average domain length. Afterwards we consider the case in which the LC
phase is enclosed between two parallel plates, representing the component B. We carry out the I-N quench and assume that
after some time the domain pattern is frozen in at one plate due to the so called surface memory effect (SME) [10,21,22], that
is known to play important role in LC devices. The surface frozen pattern then serves as the boundary condition that affects
the domain growth within the cell.
We first follow the coarsening dynamics in the pure bulk LC. In the simulation we monitor the evolution of the spatially
averaged nematic order parameter <S> and of the average domain size ξd after the quench. We calculate local values of S as
the largest eigenvalue of the Q tensor (Eq.(3)). We estimate a typical linear size ξd of an average domain of a pattern using
two different approaches that are described in detail in Ref.[3]. In the first one ξd is determined geometrically and in the second
one solely on energy grounds. Our simulations show that both estimates of ξd coincide in the regime, where domain pattern is
clearly manifested. This indicates that the domain wall energy costs dominate the domain growth.
In Fig.1 we show the time evolution of the domain configuration in bulk LC sample, where periodic boundary conditions were
imposed. Soon after the quench the scaling regime is entered, where the domain size scales as ξ d ∝ t γ , γ ≈ 0.5 . A typical
domain pattern is depicted in the inset of the picture. In the continuum picture within each domain a rather uniform
v
configuration of the gauge field n is established. Contradicting orientational tendencies at domain walls can give rise to
topological line dislocations. The annihilation of these topological defects of different topological charge enables the domain
growth [2]. There is grown numerical evidence that understanding of the annihilation process of a pair of isolated topological
defects gives essential information on domain coarsening dynamics (i.e. from the velocity of the defect pair annihilation one
can well estimate the domain coarsening dynamics). A detail study of such annihilation process is given in [23]. Note that on
approaching the late coarsening regime the domain pattern becomes gradually less pronounced and visually the structure
becomes dominated by dislocation lines.
Fig. 1. The domain coarsening dynamics obeys well the scaling law ξ d ∝ t . In the inset we plot
3
a typical domain pattern. The number of particle in the simulation is equal to N , N=40.
0.5
We next study the influence of spatially quenched impurities on the domain pattern. For this purpose we introduce a
concentration p of randomly distributed impurities, imposing random direction in space. The coupling strength of impurities with
LC molecules is equal to the LC molecule – LC molecule coupling. In Fig. 2a we plot the domain growth dynamics for p=6%
and different system simulation cell sizes. One sees that the saturation domain length ξd(p) is independent of the cell size. In
Fig. 2b we show the ξd(p) dependence. With increasing p the values of ξd(p) monotonously decrease, where ξd(p) exceeds the
average distance between impurities.
(a)
(b)
Fig. 2. The domain growth as a function of concentration p of impurities. (a) ξd time variation for
3
3
3
3
different cell sizes and p=6%. The following cell sizes where chosen: N =24 , 32 and 50 , where
N counts number of lattice cites along a given Cartesian coordinate within a cubic simulation cell.
(b) ξd(p) dependence.
We proceed by analyzing the influence of SME and the imprinted domain pattern on macroscopic appearance of a LC cell.
The geometry of the problem is depicted in Fig. 3. The LC molecules are enclosed in a plan-parallel cell of thickness L=Nza0.
At the bottom plate (z=0) we enforce isotropic tangential anchoring (i.e. the surface favours in plane orientation within which all
orientations are equivalent). At the top plate (z=L) we set homeotropic anchoring (i.e. molecules are forced to be oriented
along the z-axis). The cell length along the x and y direction is set to Nxa0 and Nya0, respectively. At the lateral boundaries we
impose periodic boundary condition.
Fig.3. The geometry of the problem. At the bottom plate we freeze in the domain pattern when the
degree of nematic ordering is high enough (i.e., S=0.3). The imprinted domain pattern afterwards
imposes the boundary condition to the LC structure within the cell.
In the simulation we quench the system from the isotropic phase deep into the nematic phase. After the quench the average
degree of nematic ordering begins to grow and the domain pattern is formed. When the degree of ordering is high enough (in
the simulation we choose S=0.3) we impose ϑ(x,y,z=0)=0 and froze in the established domain pattern at z=0. Therefore, we
assume that at this moment the orientational fluctuations are enough suppressed, so that the molecules at the surface get
trapped by the strong short ranged surface potential. With time the average domain length within the cell tends to increase
while the surface restricts it locally with the frozen in pattern. In the simulation we monitor the outcome of this competition.
After a long enough time (i.e. 10000 sweeps for NxNyNz ∼104) the equilibrium pattern is formed characterized by a biaxial layer
of thickness ξb. The resulting equilibrium structure is shown in Figs.4,5,6. In Fig.4 we plot the LC structure at subsequent (x,y)
planes starting from the frozen in layer. One sees that the average domain length increases with increasing z. In the region,
where domains persist, the elastic distortions are rather strong and consequently the system locally enters into the biaxial
state. This is clearly presented in Fig.5 where the biaxial profile of different simulation cells are superimposed. We see that the
thickness of the surface layer displays a relatively weak dependence on L and is therefore defined by the intrinsic LC length
(the biaxial correlation length [1]). Our simulations suggest ξb≈7a0. Above it essentially uniform structure exists aligned on
average along the z-axis, as shown in Fig.6. In case of homogeneous (i.e. single domain) alignment along one direction in the
(x,y) plane, the molecules would gradually reorient towards the z-direction on increasing z. In this case the surface biaxial layer
would disappear and the optic properties of the cell would be qualitatively different.
Fig. 4 The domain structure within (x,y) planes at different values of z.
Fig. 5 The degree of biaxiality on increasing z. Different simulation runs are superimposed in
order to show, that the structure is characterized by the biaxial surface layer of thickness ξb∼7a0.
In the main figure we focus on the region close to the bottom cell plate. The numbers (Nx,Ny,Nz) in
the inset describe the size of the samples used in simulations.
2
Fig. 6 Superimposed β (z) dependence and the LC ordering in the (x,z) plane. Above ξb the
molecules are aligned on average along the z-axis.
Conclusions
We investigate theoretically a bicomponent system A+B, where we monitor the domain pattern of the system A, that is
controlled by the component B. As a demonstrative condensed matter system we choose for A the liquid crystal material
focusing on the temperature driven isotropic (ordinary liquid) to nematic (orientationally ordered liquid) phase transition. As B
we chose either randomly dispersed nano-particles or a confining surface. To investigate the phase behavior of A we use
semi-microscopic Lebwohl-Lasher model, which well describes universal features of phase transitions in which continuous
orientational symmetry is broken. We monitor the domain kinetics via Brownian molecular dynamics.
We show that following a temperature quench the component A forms discernable domain structure soon (i.e. after the so
called Zurek time [4]) after the quench. With time the established domain pattern enters the scaling regime, where the
γ
characteristic domain size obeys the scaling law ξ d ∝ t . For the Lebwohl-Lasher system we obtain γ ≈ 0.5 . The domains
grow on average in order to reduce the domain wall elastic penalties. The qrowth is enabled by annihilation of topological
defects. In pure or weakly perturbed system A the domain pattern becomes gradually less pronounced and individual line
dislocations begin to dominate the pattern. Finally (at least in perfectly pure system) the defectless monodomain equilibrium
structure is established. On adding a small concentration p of nanoparticles acting as a random anisotropy field, one could
stabilize the domain pattern at finite saturated domain lengths ξ d ( p) . The stability of such a system was discussed in detail in
[24]. We show that with increasing p the saturated domain length monotonously decreases and that ξ d ( p) exceeds the
average nearest neighbor distance among nanoparticles. We further demonstrate that the combination of quench driven
domain pattern and surface memory effect could qualitatively change the optic appearance of a LC cell. Note that imprinted
domain patterns are often observed and we believe that their origin is dynamical [22].
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