Wetting of a spherical particle by a nematic liquid crystal.
ABSTRACT We discuss how the curvature of a substrate influences wetting by a nematic liquid crystal concentrating on the surface of a spherical particle. Our investigation is based on Landau-de Gennes free energy formulated in terms of second-rank nematic order parameter Q(ij). We review the method to treat wetting transitions in curved geometries and calculate the wetting phase diagram in terms of the temperature and a surface coupling parameter. We find that the length of the prewetting line which corresponds to the boundary-layer transitions introduced by Sheng [Phys. Rev. A 26, 1610 (1982)] gradually decreases with a decrease in particle radius until it vanishes completely below a critical radius of about 100 nm. The prewetting line ends at a critical point which we study in detail. By interpreting the effect of curvature as an effective shift in temperature in Landau-de Gennes theory, we are able to formulate a good estimate for the critical temperature as a function of the inverse particle radius. It demonstrates that splay deformations around the particle significantly influence nematic wetting of curved surfaces.
- SourceAvailable from: Eugene M Terentjev[show abstract] [hide abstract]
ABSTRACT: Mixing model colloidal particles with a thermotropic nematic liquid crystal results in a soft solid with significant storage modulus (G' approximately 10(3)-10(5) Pa). The soft solid comprises a network of particle aggregates, formed by the exclusion of particles from emergent nematic domains as the mixture is cooled below the isotropic-nematic transition. The unusually high storage modulus of the colloid-liquid-crystal composites may be due to the local frustration of nematic order within the particle aggregates. The birefringent soft solid is potentially important as a switchable electro-optical material that can be readily handled and processed.Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics 07/2000; 61(6 Pt A):R6083-6.
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ABSTRACT: By means of a Landau-de Gennes mean field model, we predict the existence of a nonspontaneous surface nematic phase in a smectogenic compound in contact with a suitable solid substrate. In the bulk the system does not show any nematic phase, the latter being solely induced by the substrate-liquid crystal interaction. Depending on the strength of the surface potential, a prewetting line, terminating at a critical point, may appear. For strong enough coupling, a new surface smectic phase can be induced, accompanied by a reentrant behavior. Our analysis might explain some recent experimental results [T. Moses, Phys. Rev. E 64, 010702(R) (2001)]: to validate it we suggest possible further experimental investigations.Physical Review E 08/2002; 66(1 Pt 1):010701. · 2.31 Impact Factor
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ABSTRACT: Using a temperature controlled atomic force microscope, we have measured the temperature dependence of the force between a flat silanated glass surface and a silanated glass microsphere, immersed in the isotropic phase of the nematic liquid crystal 5CB (4'-n-pentyl 4-cyanobiphenyl). At separations of several nanometers, we observed a weak, short range attractive force of the order of 100 pN, which was increased by decreasing the temperature. The temperature dependence of the amplitude and the range of this attractive force can be described by a combination of van der Waals and a mean-field prenematic force due to the surface-induced nematic order. This is supported by ellipsometric study and allows for the determination of the surface coupling energy of 5CB on a silanated glass surface.Physical Review E 12/2001; 64(5 Pt 1):051711. · 2.31 Impact Factor
Wetting of a spherical particle by a nematic liquid crystal
Yokoyama Nano-structured Liquid Crystal Project, ERATO, Japan Science and Technology Agency,
5-9-9 Tokodai, Tsukuba 300-2635, Japan
Fachbereich Physik, Universita ¨t Konstanz, D-78457 Konstanz, Germany
Yokoyama Nano-structured Liquid Crystal Project, ERATO, Japan Science and Technology Agency,
5-9-9 Tokodai, Tsukuba 300-2635, Japan
and Nanotechnology Research Institute, AIST, 1-1-1 Umezono, Tsukuba 305-8568, Japan
?Received 10 July 2003; published 27 February 2004?
We discuss how the curvature of a substrate influences wetting by a nematic liquid crystal concentrating on
the surface of a spherical particle. Our investigation is based on Landau–de Gennes free energy formulated in
terms of second-rank nematic order parameter Qij. We review the method to treat wetting transitions in curved
geometries and calculate the wetting phase diagram in terms of the temperature and a surface coupling
parameter. We find that the length of the prewetting line which corresponds to the boundary-layer transitions
introduced by Sheng ?Phys. Rev. A 26, 1610 ?1982?? gradually decreases with a decrease in particle radius until
it vanishes completely below a critical radius of about 100 nm. The prewetting line ends at a critical point
which we study in detail. By interpreting the effect of curvature as an effective shift in temperature in
Landau–de Gennes theory, we are able to formulate a good estimate for the critical temperature as a function
of the inverse particle radius. It demonstrates that splay deformations around the particle significantly influence
nematic wetting of curved surfaces.
DOI: 10.1103/PhysRevE.69.021714 PACS number?s?: 61.30.Hn, 64.70.Md, 61.30.Cz, 82.70.Dd
The wetting of surfaces by a fluid has tremendous indus-
trial applications as exemplified by the famous Lotus effect
?1?. For three decades, wetting has been intensively studied
?2,3? and, in connection with nanostructuring of surfaces and
microfluidic, it gains further importance. In his seminal pa-
per ?4?, Cahn argued on the basis of mean-field theory that a
two-phase system moving along the coexistence line towards
its critical point exhibits complete wetting beyond the wet-
ting temperature Tw. Furthermore, a prewetting line exists
which starts on the coexistence curve at Twand ends at a
critical point located in the region of either phase 1 or 2.
Cahn’s work was extended ?5? and then wetting of curved
surfaces of cylinders and spheres was studied ?6–9?, also
within density-functional theory ?10?. The main features are
that complete wetting cannot occur on curved surfaces and
that the prewetting line vanishes with increasing curvature.
Experimental observations of wetting phenomena on spheres
and cylinders are reported in Refs. ?11? and ?12?.
Surface phenomena in liquid crystals are widely studied
?13,14? partly due to their importance in liquid crystal dis-
plays. Sheng was the first to investigate the so-called
boundary-layer transition in nematic liquid crystals close to a
planar substrate and above the isotropic–nematic phase tran-
sition ?15?. It corresponds to the prewetting transition men-
tioned above. Detailed investigations were then performed
by Poniewierski and Sluckin ?16,17? who also elaborated
upon the connection to wetting. Experiments confirm the sur-
face induced nematic order ?18,19? and recently the
boundary-layer transition was observed ?20?. Numerical
studies and density functional theory established complete
orientational wetting ?21?.
In this article we report on the effect of curvature on
nematic wetting layers. Our work is stimulated by recent
investigations of surface-induced nematic order around col-
loidal particles and their effect on the stability of colloidal
dispersions ?22–24?. So far theoretical studies have been
based on the harmonic approximation of the Landau–de
Gennes free energy functional. Here, we will employ com-
plete Landau–de Gennes theory to study orientational wet-
ting around a spherical particle. Since the conventional
method to treat wetting transitions in planar geometries
?4,15? is no longer applicable for systems with curved
boundaries, we will apply a method outlined in Refs. ?6,8?,
The results of our investigation are summarized in Fig. 1,
where we plot the prewetting line for different reduced par-
ticle radii as a function of the temperature and surface-
coupling parameter w, also called the surface-ordering field.
Let us first review the planar geometry ?15? ?that corresponds
to an infinite particle radius? and place it within the context
of wetting. At temperatures well above the reduced bulk-
transition temperature ?IN?0.125, the nematic scalar order
parameter Q¯0at the surface assumes its small ‘‘thin-layer’’
value. When traversing ?INfor w?0.037, it jumps to a value
closer to that of nematic bulk parameter Q¯b. In the treatment
by Cahn ?4?, this corresponds to a situation where the nem-
*Electronic address: firstname.lastname@example.org
PHYSICAL REVIEW E 69, 021714 ?2004?
1063-651X/2004/69?2?/021714?7?/$22.50©2004 The American Physical Society
atic phase only partially wets the surface. When w?0.037,
the surface order parameter exhibits a boundary-layer transi-
tion at the prewetting line where it jumps from its thin-layer
to the larger ‘‘thick-layer’’ value. The prewetting line ends at
a critical point (?*,w*) indicated by a plus sign. For w
?w*, the surface order parameter behaves smoothly with a
decrease in temperature. Following the analysis of Cahn ?4?,
the nematic phase completely wets the bounding surface at
?INif w is chosen larger than the so-called wetting surface-
coupling parameter wm?0.037 ?25? introduced in full anal-
ogy to the wetting temperature. To see this we note that at
?INthe thick-layer value of the surface order parameter is
always larger than the bulk value Q¯b. So between the sub-
strate and isotropic phase, we can fill in a macroscopically
thick layer of nematic phase, i.e., complete wetting. Now,
Fig. 1 clearly indicates the effect of curvature on the
boundary-layer transition. The length of the prewetting line
gradually decreases with a decrease in particle radius and the
line completely vanishes below a reduced critical radius of
R¯0*?28.9, corresponding to 100 nm for a typical nematic
mesogen. As a further effect of curvature, we observe that
complete wetting at ?INis suppressed since the surface ten-
sion at the interface between the nematic and isotropic
phases grows according to the square of the distance from
the center of the sphere. We will demonstrate in this article
that, compared to wetting with, e.g., binary fluids, elastic
distortion of the director field close to the particle surface has
a pronounced effect on the wetting properties of nematic
In the following, we review theory needed to calculate the
phase diagram of Fig. 1 ?Sec. II? and then discuss details of
our results ?Sec. III?. We finish with a conclusion ?Sec. IV?
where we suggest extensions of the present work and com-
ment on experimental verification.
II. DESCRIPTION OF THE MODEL
A. Free energy
We begin with a description of our model by writing the
free energy of the system in terms of the local orientational
order parameter of a nematic liquid crystal for which we
adopt the second-rank symmetric and traceless tensor
Q??(r) ?26?, also called the alignment tensor ?27?. The free
energy density fbulkof a nematic liquid crystal in the bulk
is given by the sum of the local and elastic energy. We write
the former in terms of Landau–de Gennes expansion ?26?
with A,B and C phenomenological coefficients. Greek indi-
ces denote Cartesian coordinates and summation over
repeated indices is implied. For simplicity, we adopt the
?(1/2)L1(??Q??)2, where L1is the elastic constant.
The phenomenological surface free energy density fsde-
scribes the capability of the bounding surfaces to induce ori-
entational order right at the surface. In this article, following
the work of Sheng ?15?, we employ the simple form fs
??WQ??????, where the phenomenological surface-
coupling parameter W, also called the surface-ordering field,
characterizes the strength of anchoring and ??is the unit
vector normal to the surface. This surface energy is a
straightforward generalization of that used by Sheng ?15?,
who discussed the boundary-layer transition of a nematic on
a flat substrate. The minus sign in front of W?0 implies that
at the surface homeotropic ordering is preferred. We adopt
this simple surface energy because one of the aims of this
article is a concise presentation of the method needed to treat
the effect of curvature on the boundary-layer transition.
Based on the present investigation, more general types of
surface free energy that contain, e.g., quadratic terms in
Q??, can be studied.
The total free energy of the system is now written as F
??Vd3rfbulk??Sd2rfs, where V is the volume occupied by
the liquid crystal and S denotes the bounding surfaces. To
simplify the discussion below, we use reduced quantities. We
rescale the orientational order parameter to Q???sQ¯??with
??L1/Cs2??27L1C/8B2, where 2?2?ndenotes the nem-
atic coherence length at the isotropic–nematic phase transi-
tion. The rescaled free energy of the system then reads
where r¯?r/?n, ?¯???n??and ??A/Cs2?27AC/8B2gives
the reduced temperature. The unit of the free energy ?f
?Cs4?64B4/729C3is proportional to the latent heat and
the reduced anchoring strength is defined as w?W?n/L1s.
In this article we consider the effect of one spherical par-
ticle on the phase transition behavior of a nematic liquid
crystal close to the particle’s surface. We place the center of
the sphere at the origin and denote the radius by R0
?R¯0?n. According to the symmetry of the system, we
choose the uniaxial order-parameter profile Q¯??(r¯)?Q¯(r¯)
3???), where eˆris the unit vector along the radial
FIG. 1. Prewetting lines for different reduced particle radii R¯0
?see numbers close to the plus signs? as a function of temperature ?
and surface-coupling parameter w. The lines start at the bulk phase
transition temperature ?IN?1/8 and end at critical points (?*,w*)
indicated by the plus sign. Surface-layer transitions do not occur
FUKUDA, STARK, AND YOKOYAMAPHYSICAL REVIEW E 69, 021714 ?2004?
direction ?that represents the nematic director? and the scalar
order parameter Q¯(r¯) only depends on r¯, the ?reduced? dis-
tance from the center of the particle. With this choice, the
free energy of Eq. ?1? is then written in terms of Q¯(r¯) as
dr ¯ r ¯2?f¯?Q¯??3?
with the bulk local energy f¯in terms of Q¯being
Note that the second term of the integrand in Eq. ?2? is as-
sociated with splay deformation of the orientational order
around the spherical particle. It is specific to a nematic liquid
crystal and therefore absent in similar investigations of wet-
ting in a binary fluid whose composition is specified by a
scalar order parameter ?9?. In addition, this second term can
be viewed as an effective shift in temperature, depending on
radial coordinate r¯.
For later use, we summarize the properties of the bulk
isotropic–nematic transition deduced from the local free en-
ergy ?3?. The bulk isotropic–nematic phase transition occurs
at reduced temperature, ?IN?1/8, and the nematic order pa-
rameter at the point of transition is Q¯IN??6/4?0.612. The
metastable nematic phase exists in the temperature range of
?IN???9/64 with a limiting order parameter of Q¯*
?3?6/16?0.459 at the superheating temperature, ??9/64.
Finally, the isotropic phase becomes unstable at the super-
cooling temperature, ??0.
B. Determination of the order-parameter profile
and phase behavior
The order-parameter profile Q¯(r¯) that minimizes the free
energy ?1? is determined by ?F/?Q¯?0, which yields the
following Euler–Lagrange equation:
together with boundary conditions
Boundary condition ?6? implies that the bulk is in the isotro-
pic state. Note that in the nematic state, the spherical sym-
metry of our system with its radial order parameter profile is
an artificial situation since it creates global splay deforma-
tion around the particle. We, therefore, restrict our discussion
to the isotropic bulk state assuming ???IN?1/8.
The boundary-layer transition is monitored by a jump of
the scalar order parameter at the surface, Q¯?r¯?R¯0?Q¯0. In the
planar case, which follows from our system for R¯0→?, the
Euler–Lagrange equation, Eq. ?4?, with the vanishing second
and third terms can be integrated at once and together with
boundary condition ?5?, the surface order parameter follows
from ?2f¯(Q¯0)?1/2?w. When multiple solutions exist, the
Q¯0??2f¯(Q¯)?1/2?w?dQ¯has to be identified. Furthermore,
a boundary-layer transition between different branches of
minima is indicated in a Maxwell construction. This path-
way, outlined by Sheng ?15? and by Cahn ?4?, is no longer
possible for curved surfaces. A first integral of Eq. ?4? no
longer exists. This is obvious from mechanical analogy with
the replacement of Q¯→x and r¯→t, where the second and
third terms of Eq. ?4? represent a time dependent potential
and a friction term, respectively ?7,9?. Instead, we follow a
method outlined in Refs. ?6? and ?8? and use it in a version
introduced in Ref. ?9?. We solve Eq. ?4? for fixed Q¯0at the
particle surface. We thus arrive at a family of orientational
profiles for which we calculate the free energy F¯(Q¯0) which
is now a function in the variable Q¯0. On the other hand, we
consider the variation ?F¯of our free energy within the fam-
ily of profiles, just introduced. Since these profiles satisfy the
bulk Euler–Lagrange equation, Eq. ?4?, the bulk term in the
variation vanishes and the surface term gives
where we replaced ?F¯by dF¯(Q¯0). The condition dF¯/dQ¯0
?0 for a minimum of the total free energy F¯(Q¯0) then
?(dQ¯/dr¯)?r¯?R¯0as a function of Q¯0, the possible surface
order parameters are the intersections with the constant w
?see Fig. 2?. Their free energies are calculated by integrating
Eq. ?7? and the absolute minimum of F¯(Q¯0) then gives the
stable surface-order parameter Q¯0.
Suppose we find two solutions, Q¯1and Q¯2, of Eq. ?5?,
then a first-order phase transition between the branches of
the two minima occurs if
This is a variant of Maxwell’s construction illustrated in Fig.
2; since the two shaded regions possess equal areas, the
boundary-layer transition takes place. Note that the third so-
lution, Q¯3, corresponds to a maximum of the free energy and
is therefore unstable. If surface constant w decreases relative
WETTING OF A SPHERICAL PARTICLE BY A . . .PHYSICAL REVIEW E 69, 021714 ?2004?
to the value in Fig. 2, thin-layer parameter Q¯1is stable,
whereas thick-layer parameter Q¯2is realized for increasing
The critical points in the phase diagram of Fig. 1 occur
when (dQ¯/dr¯)?r¯?R¯0as a function of Q¯0possesses a saddle
point ?see Figs. 3?a?–3?c??,
which determines the ‘‘critical temperature’’ ?*. By varying
surface coupling parameter w, the number of possible solu-
tions of boundary equation ?5? changes from three to one or
vice versa, so the ‘‘critical anchoring strength’’ w* for a
given particle radius R¯0is the solution of Eq. ?5? at the
In Fig. 3, we show plots of ?(dQ¯/dr¯)?r¯?R¯0as a function
of Q¯0for various particle radii R¯0. The parameter of the
curves in each plot is the temperature. In Fig. 3?a?, we
present, as a reference, the results for R¯0??, i.e., a planar
surface. Note that in this case the single curves can be deter-
mined completely analytically ?15?. At the bulk transition
temperature ?IN?0.125, the derivative ?(dQ¯/dr¯)?r¯?R¯0be-
comes zero for bulk order parameter Q¯b. The critical point
in the phase diagram of Fig. 1 is given by the superheating
temperature ?*?9/64?0.140625 and w*?9/128?0.0703.
For ???*, the curves are monotonic functions of Q¯0.
A large but finite radius of R¯0?200 is chosen in Fig. 3?b?.
Although it is similar to that in Fig. 3?a?, the derivative
?(dQ¯/dr¯)?r¯?R¯0never reaches zero for any finite R¯0. This is
partly due to the effective shift in temperature, mentioned
earlier in the discussion following Eq. ?3?.
Critical radius R¯0*, where the length of the prewetting
line shrinks to zero, is determined by the requirement that
critical condition ?9? is satisfied at ???IN. We numerically
obtain R¯0*?28.9, and the appropriate plot is presented in
Fig. 3?c?. For any R¯0?R¯0* ?see Fig. 3?d? for R¯0?10], the
curves are monotonic functions for all ???IN. Therefore, a
boundary-layer transition can no longer occur. On the basis
FIG. 2. Illustration of generalized Maxwell construction ?see
Eq. ?8??. The first-order surface-layer transition occurs between the
small thin-layer (Q¯1) and the large thick-layer (Q¯2) order param-
eters when the areas of the two shaded regions are equal.
FIG. 3. Plots of ?(dQ¯/dr¯)?r¯?R¯0vs Q¯0for various particle radii: R¯0? ?a? ? ?planar surface?, ?b? 200, ?c? 28.9 ?critical radius?, and ?d?
10. The numbers on the lines indicate ?. The curves at the critical temperature ?* are plotted as thick lines.
FUKUDA, STARK, AND YOKOYAMA PHYSICAL REVIEW E 69, 021714 ?2004?
of the plots in Fig. 3, we determined the phase diagram of
Fig. 1, which shows the prewetting lines for different particle
radii, as already discussed in Sec. I.
For typical nematic mesogens, the nematic coherence
length 2?2?nis of the order of 10 nm ?26?, so in real units
the critical radius becomes R0*?100 nm. This means that in
colloidal dispersions with particle radii larger than approxi-
mately 100 nm, the boundary-layer transition should be ob-
In Fig. 4 we illustrate the order-parameter profiles for the
reduced particle radius R¯0?200 and surface-coupling pa-
rameter w?0.06 by plotting the scalar order parameter Q¯as
a function of radial distance r¯?measured from the center of
the sphere?. At the bulk transition temperature ?IN?0.125
?see curve ?1??, the order parameter decays on a length scale
of about four times the nematic coherence length 2?2?n,
indicating that the particle is wetted by the nematic phase.
However, due to the spherical geometry, complete wetting
cannot occur since the surface tension at the interface be-
tween nematic and isotropic phases grows as r¯2. Curves ?2?
and ?3? show the respective thick-film and thin-film solutions
right at the prewetting line at reduced temperature of ?
?0.13224. The discontinuity in the surface order parameter
is clearly visible.
In the following, we discuss further details of the prewet-
ting line. From Fig. 1 we know that the possible anchoring
strengths w where a boundary-layer transition occurs are
bounded from above by the critical value w* and from below
by a value which we denote by wt. At anchoring strength
wt, the prewetting line intersects the coexistence line at the
bulk transition temperature, ?IN?0.125. Or, mathematically
speaking, for ?INcondition ?8? of Maxwell’s construction is
fulfilled at wt. In Fig. 5 we plot both w* and wtas a func-
tion of the inverse particle radius, 1/R¯0. We find that the
dependence of w* on the particle radius is rather weak and
that the width of the region bounded by the two curves
?where boundary-layer transitions occur? decreases almost
linearly with 1/R¯0. If we view the effect of curvature on this
width as expansion in terms of 1/R¯0, this means that the
linear term is far more dominant than the higher-order con-
tributions, which is reasonable for R¯0?R¯0*?28.9.
Finally, in Fig. 6, we plot the critical temperature ?* as a
function of 1/R¯0?see the plus symbols?. We notice again
that, for the planar surface (1/R¯0?0), the critical tempera-
?9/64?0.140625 coincides with the superheating
temperature of the nematic phase. Now, the decrease of ?*
with increasing curvature (1/R¯0) reflects suppression of the
boundary-layer transition due to elastic deformation in the
order parameter induced by the curved surface of the par-
ticle. To show that distortion of the director field contributes
significantly to this decrease, we present the following esti-
mate. The linear part of the Euler–Lagrange equation, Eq.
?4?, reads Q¯???6/r¯2?(2/r¯)(dQ¯/dr¯)/Q¯??(d2Q¯/dr¯2)?0.
The first and the fourth terms already appear in planar geom-
etry. The second term results from distortion of the director
field. It is, therefore, specific to the nematic problem. For
thicknesses of the wetting layer much smaller than the par-
ticle radius, we can approximate it by 6/R¯0
also appears when critical wetting is studied in systems with
a scalar order parameter. From the profiles in Fig. 3, we
observe that, at the critical temperature, dQ¯/dr¯?r¯?R¯0and Q¯0
only weakly depend on 1/R¯0. With dQ¯/dr¯?r¯?R¯0??0.07,
2. The third term
FIG. 4. Scalar order parameter Q¯as a function of radial distance
r¯?measured from the center of the sphere? for R¯0?200 and w
?0.06. Curve ?1? is at ?IN. Curves ?2? and ?3? are the respective
thick-film and thin-film profiles right at the prewetting line for ?
FIG. 5. Critical anchoring strength w* ?triangles? and anchoring
strength wt?circles?, where the prewetting line intersects the coex-
istence line at ?IN, plotted as a function of 1/R¯0. As a guide to the
eye, the lines connect the numerical data points.
FIG. 6. Critical temperature ?* ??? as a function of the inverse
particle radius 1/R¯0. The horizontal dotted line indicates the bulk
transition temperature ?IN?0.125. The solid and dashed lines are
estimates for ?* ?see the text for details?.
WETTING OF A SPHERICAL PARTICLE BY A . . .PHYSICAL REVIEW E 69, 021714 ?2004?