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Coarsening dynamics in nematic liquid crystals
PHYSICA
Physica B 178 (1992) 56-72 North-Holland
Coarsening dynamics in nematic liquid crystals B e r n a r d Yurke a, A n d r e w N. Pargellis a, Isaac Chuang b and Neil T u r o k c aAT& T Bell Laboratories, Murray Hill, NJ 07974, USA bmlT Research Laboratory for Electronics, Cambridge, MA 02139, USA cJoseph Henry Laboratories, Princeton University, Princeton, NJ 08544, USA
The dynamics of symmetry breaking phase transitions and the dynamics of defect tangles generated following the symmetry breaking are of interest in a number of fields of physics ranging from condensed matter physics to particle physics and cosmology. For the case when continuous symmetries are involved, as of yet, there has been little experimental work. Here we report on our studies of the dynamics of defects generated in a sudden quench of a uniaxial nematic liquid crystal from the isotropic phase to the nematic phase. The symmetry breaking is from 0 ( 3 ) to D ~ . In this phase quench, stringlike defects belonging to the 7r1 homotopy class, pointlike defects belonging to the 772 homotopy class, and texture defects belonging to the % homotopy class are created. Although the % defect density scales as t-l, nonscaling behavior is seen for % defects.
I. Introduction
Symmetry breaking phase transitions in which the symmetry of a system is spontaneously broken to some lower symmetry are of importance in a number of fields of physics including condensed matter physics, particle physics, and cosmology. There has been considerable interest in the dynamics that occurs when such systems are rapidly quenched from one phase to the other. Topological defects are generated in such quenches and the dynamics of the defects ("coarsening dynamics") often exhibit scaling laws. Spinodal decomposition of binary mixtures is an example of such a phase quench for which there has been considerable theoretical and experimental work [1-5]. For those systems the topological defects are the domain walls between the two phases of the binary mixture. The mechanism by which defects are generated in a phase quench is referred to in the particle physics and cosmology literature as the Kibble mechanism [6]. The general idea is that the rate at which the quench takes place determines the size of the patches over which the direction of symmetry breaking is correlated. Patches having different directions of symmetry
breaking try to merge with each other smoothly to minimize gradient energies. However, because of topological constraints, invariably there will be regions where patches with different directions of symmetry breaking will not be able to merge with each other smoothly. As a result there will be regions where the gradient energy becomes singular. These are the topological defects. Depending on the symmetries involved, the defects need not be the two-dimensional sheetlike structures called domain walls. They could be stringlike structures called disclination lines or pointlike structures called monopoles. In addition, it is possible to have a topological defect that is nonsingular except for short instances of time when the defect unwinds. These nonsingular defects are called texture defects in the literature [7]. Using the notation of homotopy theory [8-10], a systematic method for classifying topological defects, we will refer to domain walls as % defects, disclination lines as % defects, monopoles as % defects, and texture as % defects. Considerable work has been done on the dynamics of quenches involving the breaking of a discrete symmetry for which the topological defects are domain walls, such as in the spinodal
0921-4526/92/$05.00 © 1992 - Elsevier Science Publishers B.V. All rights reserved
B. Yurke et al. / Coarsening dynamics in nematic liquid crystals
decomposition of binary fluids [1-5]. The study of phase transition dynamics and the dynamics of defect tangles created in phase quenches involving the breaking of a continuous symmetry, for which the topological defects are strings, points, or texture defects, is somewhat less advanced. Theoretical work [1t-17[ and, more recently, experimental work has picked up in this area. Research has concentrated on predicting and measuring the defect density as a function of time. Studies of the coarsening dynamics of stringlike defects in two-dimensional films of nematic liquid crystals [18, 19] and pointlike defects in two-dimensional films of cholesteric liquid crystals [20] have been carried out. Here we report on our own studies [21, 22] of the coarsening dynamics of bulk (three-dimensional) samples of uniaxial nematic liquid crystals. Cosmologists have also displayed considerable interest in symmetry breaking phase transitions in which 7rl, 7r2, or % defects are generated [7, 23-25]. In an expanding universe which started out hot and dense it is likely that symmetry breaking phase transitions have taken place. For example, the most successful particle theory to date, the Weinberg-Salam theory of electromagnetic and weak interactions, indicates that at a temperature of about 100 GeV the electroweak symmetry of the universe would have been broken. The phase transition would have been first order and % defects would have been formed during the phase quench. These texture defects would be unstable to decay via a baryon number violating process. It is thus possible that the electroweak phase transition may account for the observed baryon-antibaryon asymmetry in the universe [26-29]. Models invoking topological defects have also been proposed to account for the large scale inhomogeneity of luminous matter in the universe ]7, 23-25]. Topological defects, by perturbing space-time, could generate density fluctuations which would grow to allow hydrogen gas to be gathered in regions with sufficiently high density that stars and galaxies could form. The statistics of the distribution of galaxies created by such a process would depend on how the topological defects move after the phase quench
57
[30, 31]. It is thus important to understand the coarsening dynamics of defect tangles if one is to construct models that successfully "predict", for example, the galaxy-galaxy correlation function measured by observational astronomy. Considerable work has been done on models invoking ~-~ defects, cosmic strings [23]. More recently work on cosmological models employing % defects [7, 30, 31] has become of interest. Zurek [32] has suggested that the physics of cosmic string formation could be studied by investigating the formation of vortices in liquid 4He when the liquid is subjected to a rapid pressure quench from the normal phase to the superfluid phase. This experiment still remains to be done. We have preferred to work with liquid crystals instead because of the advantages afforded by a material whose phase transition occurs close to room temperature and whose defects can be directly imaged through an optical microscope. It should be noted that the correspondence between the behavior of liquid crystal or liquid helium defects and cosmological defects is not perfect. Liquid crystal dynamics are dissipative with inertial effects being negligible. The dynamics of cosmic strings is likely to be highly dissipative only for a relatively short time after the phase transition when the temperature of the universe is still sufficiently high so that the strings have large numbers of particles from which to scatter. At late times, when the universe is cold, cosmic strings would experience little damping and would whip back and forth at velocities that are a sizeable fraction of the speed of light. Though the Hubble expansion also provides a damping term in the field equations governing the dynamics of cosmic defects, it is generally not large enough to dominate the dynamics. However, when the universe is undergoing inflationary expansion this damping term would dominate. In this case the dynamics of disclination lines in liquid crystals could also closely mimic the dynamics of cosmic strings when expressed in comoving coordinates. Although the dynamics of defects in condensed matter systems may not always correspond closely to the dynamics of the analogous cosmological defects, the study of condensed matter systems
58
B. Yurke et al. / Coarsening dynamics in nematic liquid crystals
could significantly enhance our understanding of both.
2. Defects in uniaxial nematic liquid crystals
Liquid crystals exhibit a variety of mesophases [33-35] between the liquid phase and the solid phase and thus exhibit a variety of symmetry breaking phase transitions. Here we report on studies of the phase transition from the isotropic phase (the ordinary liquid phase) to the uniaxial nematic phase. The material we chose to work with, 4-cyano-4'-pentylbiphenyl (commercially known as K15 or 5CB) consists of long rodshaped molecules. In the isotropic phase the molecules are oriented randomly and are free to diffuse among themselves. In the nematic phase the molecules line up parallel to each other, that is, the molecules give up rotational entropy in order to maintain high translational entropy. Let n denote a vector pointed along a molecule. The liquid crystal does not distinguish between n and - n , that is, the material has inversion symmetry. The symmetry of the isotropic phase is the full rotation group 0(3). In the transition to the nematic phase, this symmetry is spontaneously broken to the symmetry of a cylinder, D~ h. Because of the inversion symmetry it is only necessary, however, to consider the symmetry breaking of the SO(3) subgroup of 0 ( 3 ) to 0 ( 2 ) . The vacuum manifold, which is the manifold of all possible ground states of the nematic phase, is the projective two-sphere, that is, a sphere with antipodal points identified with each other [8, 36]. According to homotopy theory for a projective two-sphere, 7rl, ~'2, and % defects may exist in a uniaxial nematic liquid crystal. The defects that are created in greatest abundance in a rapid pressure or temperature quench from the isotropic phase to the nematic phase are 7r~ defects in which the director field winds by + 7r or - 7r as one traverses a loop around the disclination line. These defects are called type + 1/2 and type - 1/2 disclination lines respectively. Figure 1 shows the director field configuration surrounding a + 1/2 and a - 1/2 disclination line. In the case shown the director field winds by +'rr
,/
+1/2
f
,y
-1/2
X
Fig. 1. An example of a type +1/2 and type - 1 / 2 disclination line.
and --rr, respectively, around the z-axis. Other types of winding by +Tr are possible. Disclination lines with a winding of -+27r are possible, but they are not topological singularities since Z
S
z/
,y
Fig. 2. A n example of a type +1 disclination line which is escaped along the z-axis.
"Y
Fig. 3. A monopole-antimonopole pair on a type +1 disclination line.
B. Yurke et al. +1/2
Coarsening dynamics in nematic liquid crystals
59
+1/2
+1
+1 Fig. 4. A type + 1 disclination line being pulled apart into two +1/2 disclination lines.
xS Fig. 6. The director field configuration for a texture.
Fig. 5. Defect tangle produced by a thermal quench of a freestanding film of nematic liquid crystal approximately 500 p~m thick. The picture is approximately 790 p.m wide. Type 1/2 defects are visible as dark sharp strings. Diffuse type 1 defects and monopoles are also visible in this photograph.
60
B. Yurke et al.
Coarsening dynamics in nematic liquid crystals
Fig. 7. Sequence of photographs showing texture decay in a freely suspended film of nematic liquid crystal.
B. Yurke et al, / Coarsening dynamics in nematic liquid crystals
®®
61
" ? [
,!
t=O.O sec
t=O
t=8.6 sec
t=60
~ii!iiiii!iiii!!!!:!!ii!~! ~ ~iiiiii~iiiiiii!!!iiiiii!!i!i~i~ iiiiiiiiiiii!iiiiiiiiii!iiiiiii!!!iiii!i!~=
t = 1 4 . 0 sec
t=80
.....
iiiiiii~i ~¸¸
~iiil;iiiii!~ili!!!~i~! iiiiiiiiiiiiiiiiiiiiii!i!~
t=27.5 sec
t=lO0
Fig. 8. Comparison of observed texture decay (left column) with numerical simulation (right column). Each video frame is 260 ~m wide.
62
B. Yurke et al. / Coarsening dynamics in nematic liquid crystals
the singularity at the core can be removed by suitably reorienting the molecules near the core [37-39]. This is referred to as escape in the third dimension. Figure 2 shows a type + 1 disclination line having a winding of +2~r about the z-axis. The singularity at the core is avoided by having the molecules tilt upwards and become parallel to the z-axis as one approaches the core. Figure 3 shows a type + 1 disclination line in which the direction of escape is downward both at the upper and lower end of the disclination line. However, in between there is a segment along which the direction of escape is upward. Where the segments with different directions of escape meet one has a pointlike singularity, known as a ~'2 defect or monopole [38]. At the upper point singularity the molecules all point radially inward, i.e., one has a "hedgehog" monopole. The lower monopole is a hyperbolic "hedgehog" and has the opposite monopole charge. Figure 4 shows that a type +1 disclination line can be pulled apart into two + 1 / 2 disclination lines. Such junctions of two type + 1 / 2 disclination lines with a type + 1 disclination line are commonly seen in defect tangles produced by rapid phase quenches. For comparison, the photograph in fig. 5 shows a defect tangle produced by a thermal quench of a freely suspended film of K15. The crisp narrow black lines are type 1/2 defects. The more diffuse bright lines are the type 1 disclination lines. A number of terminations of type 1 disclination lines on type 1/2 disclination lines can be seen. Also some of the type 1 disclination lines appear to be pinched into a dark spot at points along their lengths. These points are monopoles. Although % defects can in principle be created, they are rarely created in phase quenches from the isotropic phase to the nematic phase. Figure 6 shows a possible texture configuration. It consists of two concentric type 1 disclination lines of opposite winding number. At large distances from the defect the director field becomes uniform, in particular, the molecules orient their long axis along the z-axis. We have found that by inserting the tip of a glass needle into a freestanding film of nematic liquid crystal,
--500 ~m thick, type 1 disclination lines can be created as the needle is dragged through the fluid. In this manner we were able to produce defect tangles that were rich in type 1 disclination lines and that thus optimize the chances of creating a % defect configuration. The photographs of fig. 7 show what we believe to be a % defect in the process of unwinding. The first photograph in the sequence has no singularity present and can be interpreted as two loops of type 1 disclination lines. This texture unwinds by first having the two loops merge at one point to form a singularity which then decays to form a monopole-antimonopole pair. In successive frames of the figure, the monopoleantimonopole pair unwind the loop and then annihilate each other. During the run in which these photographs were taken, the microscope was switched back and forth between a camera and a video tape recorder. Figure 8 shows video frames of the texture unwinding. The first frame shows the moment at which the texture becomes singular at one point. Successive frames again show the unwinding of the texture. Alongside each video frame is shown the director field energy density of a computer simulation of texture decay in which the initial configuration consisted of a type - 1 disclination line inside a +1 disclination line in a configuration similar to that of fig. 6, but the two loops were placed offcenter with respect to each other. The outer loop overtakes the inner loop at one point creating a singularity which decays into a monopoleantimonopole pair. The singularity and the monopole-antimonopole pair show up in the energy density plots as pointlike concentrations of the strain energy. We note that ~'3 defects have been reported for cholesteric liquid crystals [40].
3. Equations of motion Strictly speaking, the uniaxial nematic liquid crystal in the nematic phase is characterized by a traceless symmetric tensor of second rank. However, it is more convenient for most purposes to work with n, the vector pointing along the long
B. Yurke et al. I Coarsening dynamics in nematic liquid crystals
axis of the molecule. In terms of the n vector field the elastic free energy F has the form
63
string as having a constant line tension T and a constant mobility F. Letting r denote the local radius of curvature, the dynamics are given by
F= ½{K,(~7-n) 2 + K2(n.V x n) 2 + K3ln x V x nl 2)
(1)
and is referred to as the Frank free energy [33, 34]. The constants K1, K2, and K 3 are respectively the splay, twist, and bend elastic constants. Similar terms arise in the free energy of superfluid 3He where the orbital angular momentum vector now plays the role of n. Since the reorientation viscosity of liquid crystals is generally sufficiently strong to make inertial terms negligible, the equation of motion for a liquid crystal is given by
On Y at
_
~F
8n~'
(2)
where y is the damping constant and the functional derivative is taken under the constraint that In[ 2 = 1. The equations of motion generated in this manner are fairly complicated and difficult to deal with. One often makes the equal constant approximation in which K 1 = K 2 = K 3 = K. In this case the field equations take the simplified form Ol'l a
Y Ot- = K[V2n~ + (7n~).(Vn~)n~].
(3)
This is the damped nonlinear sigma model. It is similar to the nonlinear sigma model used by cosmologists to study the evolution of a universe populated by global defects [41]. Because of the identification of n with - n when applied to the nematic liquid crystal eq. (3) admits stringlike solutions. The string line tension is proportional to In(R/ Re) where R is the radius of the disclination line (essentially the distance from the string to the nearest neighboring string) and Rc is the core radius. Similarly, the dissipation is proportional to ln(R/Rc). Because these logarithmic dependences of the string tension and the viscous dissipation are weak functions of R, to a good approximation one can model the dynamics of a
dr F dt
T r
(4)
For a type 1/2 disclination loop of radius r, this equation, upon integrating, yields r =
•/2T T-
(to - t ) ,
(5)
where t o is the time of collapse. The radius of the loop is thus expected to decrease as the square root of the time to collapse.
4. Coarsening dynamics The defects that are generated in the greatest abundance during a temperature or pressure quench from the isotropic to the nematic phase are type 1/2 disclination lines. Monopoles and texture defects tend to be rare. Also, since the type 1 defects are escaped in the third dimension, they are much less energetic than the type 1/2 defects. The interaction of type 1/2 defects thus dominate the coarsening dynamics. A scaling argument of the Lifshitz-Slyosov type [42] can be made to determine the type 1/2 string density p (line length per unit volume) as a function of time. One assumes that there is only one characteristic length ~. The string density is then proportional to ~-2 p ~ 1/~2.
(6)
The characteristic radius of curvature is also proportional to ~, so the characteristic line tension force per unit length fT is given by
ZT
(7)
The characteristic friction force per unit length fr is given by
fr = r v ,
(8)
64
B. Yurke et al. / Coarsening dynamics in nematic liquid crystals
where v is the characteristic velocity of the string. Equating the line tension force to the friction force one thus finds that the characteristic velocity is proportional to 1/~ v ~1/~.
(9)
The elastic energy per unit volume, E, stored in the disclination lines is given by E= T~/~ 3 ~ T p .
(10)
The rate W at which energy is being dissipated by friction is given by W = - - f F V ~ / ~ 3 oc I ' p 2 '
(11)
where eqs. (8) and (9) were used to obtain the proportionality. Equating the rate of change of the elastic energy dE/dt with the rate W at which energy is being dissipated yields the differential equation dp dt
F 2 c ~ p ,
(12)
where c is a constant of proportionality. This equation has the solution o
-
T c r ( t - to) '
a crystalline solid phase to nematic phase transition at 23°C and a nematic phase to isotropic phase transition at 35°C. We measured the slope Ap/AT of the coexistence curve to be 2.47 MPa/ K, between 0.7 and 17 MPa. A schematic of the apparatus used in these studies is depicted in fig. 9. The sample cell had two sapphire windows which, for most of the runs reported, were spaced 234-+ 23 pom apart. The view port to the cell was 3 mm in diameter. To prevent the pinning of defects to the windows, the windows were treated with the homeotropic alignment material n,n-dimethyl-noctadecyl-3-aminopropyltrimethoxysilyl chloride (otherwise known as D M O A P ) , using standard procedures [45]. Pressure was applied to the liquid crystal material in the sample cell via a diaphragm separating the liquid crystal from the pressurizing fluid which was water. Pressure jumps of less than 35 ms duration were applied to the sample cell by manually opening a valve. Transmission microscopy in which the cell was illuminated from the rear with unpolarized light was employed. A 10x objective having a depth of field large compared to the separation between the windows was used so that the disclina-
ILLUMINATION
(13) DIAPHRAGM
where t o is an integration constant. At late times the string density p thus scales as t -1. Our experimental measurements of the decay of the string density with time are described in section 6.
5. Apparatus In the following sections we describe experimental data on the dynamics of defects produced in rapid pressure quenches from the isotropic phase to the nematic phase. The nematic liquid crystal, 4-cyano-4'-n-pentylbiphenyl [43, 44], used in our studies was obtained from B D H Chemicals and used without further purification. At atmospheric pressure this material has
,I, SAMPLE CELL
~
VALVE
PRESSURE GAUGE
[ ~ MICROSCOPE PUMP
E ~]VALVE
L
I FLUID RESERVOIR
Fig. 9. Schematic of pressure quench apparatus.
B. Yurke et al. / Coarsening dynamics in nematic liquid crystals
tion lines throughout the cell would all appear in focus. No polarization filters were used. The microscope images were recorded on a high speed video recorder at a rate of 200 frames per second. For the study of the decay of the type
65
1/2 string density, video frames were digitized and image processed to determine the projected line length. A description of the image processing techniques used can be found in Chuang's thesis [46].
t = l . O sec
t = l . 7 sec
t = 2 . 9 sec
t=4.8 sec
Fig. 10. A coarsening sequence of a uniaxial nematic liquid crystal defect tangle. The dark threadlike defects are type 1/2 disclination lines. The tangle is 230 ixm thick and 360/xm wide.
B. Yurke et al. / Coarsening dynamics in nematic liquid crystals
66
6. Coarsening data A nematic liquid crystal subjected to a rapid pressure quench from the isotropic phase to the nematic phase generates a dense tangle of defects. The defect t~ingle consists predominately of type 1/2 disclination lines which undergo heavily damped motion in a direction to reduce the total strain energy of the liquid crystal. Figure 10 shows video images of the coarsening of a defect tangle at 1.0, 1.7, 2.9, and 4.8s after a pressure jump Ap of 4.69 MPa from an initially isotropic state in equilibrium at approximately 35°C and 3.6 MPa. Each picture is 360 txm wide. Figure 11 shows the coarsening data obtained from video images such as those shown in fig. 10. Data for the type 1/2 string density p as a function of time for four different pressure jumps each starting in the isotropic state at approximately 35°C and 3.6 MPa are shown. The projected string length was measured. The threedimensional string density (line length per unit volume) was obtained by assuming that the defect tangle is three-dimensional with the probability distribution of orientation of each line segment being uniform over the sphere. Each data point represents an average of 10 coarsening sequences taken at a given Ap. The data closely follow lines depicting the t-1 scaling ob-
256
'._:1.. 'ai~:~.~el . . . .
~-- 128 •~
IAi~.'a.b0' 'L
....
~_ ...... ........ + + : ~ .... -~ ~ zx.,. ........... +.+ %~
64
~'.......
I ....
-
4
~>i .... ...........
4 ........
e-
~
32
" <
16 12 -
I
'9,".
%! "-..
..... ~ , ~ . ~ 2 ' ~
1.0
2.0
4.0 8.0 Time [sec]
Fig. 11. T y p e 1/2 disclination time. D a t a were accumulated j u m p s AP: + , A P = 2 . 0 0 M P a ; 2.62MPa; ~, AP = 4.69MPa. scaling of t ~.
16.0
32.0
line density as a function of for four different pressure A, A P = 2 . 2 8 M P a ; *, A P = The dashed lines depict a
tained in section 4. A scaling relationship of p = t -~ with u = 1.02 _+ 0.04 is obtained from a least squares fit to this data. Finite size effects limit the reliability of the data for late times t. In particular, since the defects are not pinned to the sapphire windows of the sample cell, the defect tangle eventually collapses to a two-dimensional tangle and the type 1 / 2 disclination line density crosses over from a scaling of t 1 to the t 1/2 scaling studied by Orihara, Ishibashi, and Nagaya [18, 19] for a two-dimensional network. To take into account the uncertainties introduced by these finite size effects we quote our measured bulk coarsening exponent with greater error u = 1.02 _+ 0.09.
7. Loop data As the defect tangle coarsens and type 1/2 lines intersect each other and interconnect (intercommute), occasionally loops are formed. An example of this is shown in fig. 12. To check the model used for the coarsening argument of section 4, the collapse of type 1/2 loops was measured as a function of time. Figure 13 shows the time dependence of the loop radius for a type 1/2 disclination loop. By parameterizing the loop collapse as r = ( t - to) s, a least squares fit to the data gave a loop collapse exponent a = 0.49 _+ 0.002 for this data set. The measurement of seven such loop collapses gave an average loop collapse exponent a = 0.50 -+ 0.03 in good agreement with eq. (5). We also recorded the time of ,birth (the time of the intercommutation event that gave rise to the closed loop) and the time of death of all the type 1/2 disclination loops of the Ap = 4.69 MPa run. We included every type 1/2 loop that died after 1.00 s regardless of the number of type 1 disclination lines attached to the loop. From this data the loop density (the number of loops per unit volume) as a function of time could be determined. One expects the loop density to scale as the inverse of the correlation volume ~:3. The loop density is thus expected to scale as t -3/2 with time. Figure 14 shows our measured loop density as a function of time. At late times the loop density appears to scale with
B. Yurke et al. / Coarsening dynamics in nematic liquid crystals
t = l . 2 sec
67
t = 2 . 3 sec
t = l . 7 sec
I _ _ t=3.0 sec
t = 4 . 0 sec
t = 4 . 6 sec
Fig. 12. Sequences of video frames showing the production and collapse of a type 1/2 loop.
the expected power law of t 3/2 indicated by the solid black line. At early times (between 1 and 2s) the loop density does not appear to scale
~
I
' ' ' ' l
'
' ' ' 1
. . . .
I'
' ' ' l
'
properly. In fact, the density increases and builds up to a maximum at 2 s. There thus appears to be nonscaling behavior in the loop density. Cumulative Loop Density
' ' ' 1 ' ' " 1
48
4..]
.~ 32
-
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
2 E 1
16 o
0.5 ,-1
8 6
0.25 0.125
0.1
0.2
0.4 1.0 2.0 Time [sec]
4.06.0
Fig. 13. Loop radius as a function of time to collapse.
0.5
1.0
2.0 4.0 Time [sec]
8
Fig. 14. Loop density as a function of time.
16
B. Yurke et al. / Coarsening dynamics in nematic liquid crystals
68
8. Monopole data Figure 15 shows a sequence of video frames depicting the birth and death of a monopole. Although other mechanisms exist for forming monopoles, the most prevalent monopole production mechanism observed after a rapid pressure quench from the isotropic to the nematic phase consists of the collapse of a type 1/2 disclination loop having two type 1 disclination lines attached to it, as seen in the sequence of video frames. The figure also shows the most c o m m o n means by which monopoles are destroyed. The monopoles follow the type 1 disclination lines on which they are attached to the nearest termination of the type 1 on a type 1/2 string. The m o n o p o l e is destroyed when it
I
reaches the termination as can be seen by the absence of the m o n o p o l e in the last frame of the sequence. Again, if the coarsening is characterized by a single length scale sc, one would expect the m o n o p o l e density to scale as ~ 3 or as t 3/2. We have recorded the birth and death time of each m o n o p o l e contained in the sequence of runs A p = 2.28 MPa, 2.62 MPa, and 4.69 MPa. Monopoles are rare. So, in order to improve our statistics, we rescaled the time axis of the Ap = 2.28 MPa and the 2.62 MPa data run so that data of these runs could be c o m p a r e d with that of the 4.69 MPa run for equal string density. The resulting composite data is depicted in fig. 16. As with the loop density, the monopole density does not scale as expected. There is a sudden onset of m o n o p o l e production at early times. The mono-
/
t=5.9 see
t=6.8 see-
t=7.1 see
t=8.3 see
t=9.1 see
t--lO.6 s e c
Fig. 15. Monopole creation and annihilation sequence.
B. Yurke et al. / Coarsening dynamics in nematic liquid crystals
69
Cumulative Monopdle Density
_=
(a)
(b)
0.5 r
0.5
I ~-i
q I I I
1.0
2.0 4.0 Time [sec]
P I I i
8.0
16.0
Fig. 16. Monopole density as a function of time.
pole density reaches a maximum and then decays away more quickly than expected. Our observations of the loop density and the monopole density show that, strictly speaking, a nematic defect tangle does not scale with time. That is, the defect tangle at later times is not simply a magnified version of the defect tangle at earlier times. It may thus be likely that whenever more than one type of topological defect is present a defect tangle does not scale. One could, of course, argue that we have not reached the scaling regime. However, the system has had more than one decade in time (35 ms to 1 s) to reach the scaling regime before data is taken. Also type 1/2 disclination lines which dominate the coarsening dynamics exhibit the scaling expected for stringlike defects. Here we offer a partial explanation for what may be responsible for this nonscaling behavior. Depicted in fig. 17 is a type 1/2 disclination line that has been contorted so as to produce a loop. In order for the loop to generate a monopole upon collapse, the loop must have the same winding number along its entire length. Hence, the two sections of the disclination line near each other repel each other with a force F R which is proportional to the inverse of the distance d between them, F R o: K / d .
(14)
This repulsive force must be overcome by the curvature force F c which pulls the two disclina-
d~!
(c) Fig. 17. Loop production.
tion lines toward each other. This force is given by Fc ~ Kln(d/Rc)/R
,
(15)
where R c is the core radius and R is the radius of curvature. The factor l n ( d / R c ) accounts for dependence of the line tension on the defect line size. The radial extent of the defect is of the order of the distance of the defect to the next nearest defect d. If one takes d ~ R ~ ~:, then by equating F R with F c one finds a critical length scale ~:c which characterizes a string density threshold for the production of type 1/2 loops capable of producing monopoles A
~c = e R c ,
(16)
where A contains the geometrical factors that make the proportionalities, eqs. (14) and (15), equalities. The geometrical factor A is expected to be of order unity. If, however, it is 10, then e A is of order 2 × 1 0 4 . An A of 10 would thus provide a mechanism of amplifying a small length scale, the core radius R e which is of the order of 3 0 A , to a length scale ~:c of 6 0 ~ m which can easily be seen with optical microscopy.
70
B. Yurke et al. / Coarsening dynamics in nematic liquid crystals
9. Event frequencies As a defect tangle coarsens a number of processes happen that can be characterized by a well-defined event in time. For example, strings can intercommute, type 1 disclination lines can be pulled apart into type 1/2 disclination lines, monopoles are created and annihilated, loops collapse, and so forth. We have constructed a classification scheme for these events. For the case of the A P = 4 . 6 9 M P a data set we have systematically recorded all events occurring between 2 s and 5 s in the 400 ixm × 312 txm field of view of the video. For times between 1 s and 2 s we recorded all the events in one quarter of the video screen. For this data set we distinguished between the 16 types of events listed in table 1. As a listing of possible types of events, this table
is incomplete, since we have seen other types of events in other runs, such as monopoleantimonopole annihilation. We also observed 19 out of a total of 485 events (3.9%) which we were not able to classify because of obscuration. Table 2 lists the event frequencies for the various types of events. In the time interval 2.0 s ~< t < 5.0 s a total of 354 events were observed. In the time interval from 1.0 s < t < 2.0 s a total of 131 events were found. Also listed in table 2 are the classification numbers for the events. These numbers are defined as follows. An T is the difference in the n u m b e r of terminations of type 1 defects on type 1/2 defects before and after the event takes place. An M is the change in the number of monopoles after the event takes place. An L is the change in the number of links. Anls and An½s are
Table 1 Defect event classifications. Code
Event pictograph
a
X-)(
e
]E-)(
3
]lq[-)(
4
. ~ - -Itlr
6
"]1"][-_
7
-]fqF-
Description Intercommutation of two ± 1/2 strings (the two initial strings are separated in space) Decay of a ± 1 across two -+1/2's into two ± 1/2's Decay of two ± l's across two - 1/2's into two ± 1/2's lntercommutation of a ± 1/2 and a ± 1 resulting in a ± 1/2 with two T-intersections Unlinking of two ± l's from a ± 1/2
11
T--][-
12
C)--
Unlinking of two -+l's from a -+1/2 resulting in a single -+1/2 and a monopole carrying -+1 Decay of a ± 1 connecting two parts of the same -+1/2 into a single -+1/2 Decay of a ± 1 connected at both ends to a straight segment of -+1/2 Decay of a monopole carrying -+1 connected at both ends to a straight segment of -+1/2 Decay of a monopole sitting on a -+1 by absorption into a -+1/2 Collapse of a -+1/2 loop (no end products)
13
(~_,
Collapse of a -+1/2 loop with a -+1 across it
8 9
3D-D ""..Z.."-
10
16
~5----~-
17
X-X
Collapse of a -+ 1/2 loop with two ± l's coming out at either end, resulting in a single -+1 Collapse of a ± 1/2 loop with two attached -+l's, resulting in a monopole carrying -+1 Collapse of a -4-1/2 loop with four attached -+l's, resulting in a four--+ 1 vertex which escapes Intercommutation of two ± 1/2's resulting in two ± 1/2's connected by a -4-1
71
B. Yurke et al. / Coarsening dynamics in nematic liquid crystals
Table 2 Defect events, their probabilities, and associated observables. Probabilities were measured from the observation of 485 events from the AP = 4.69 MPa data. Code
Event
% Prob. <2s
% Prob. >2s
An v
An M
An L
Anls
1
X- )(
31.9
23.7
2
7[-)(
13.8
8.8
-2
-1
-1
3
]I-IF---)(
0.7
1.1
-4
-1
-2
4
-auf"- -/ql-
0
0.3
+2
+1 -1
6
71-][- ~
5.8
10.2
-2
7
71-][-- ~
0.7
0.6
-2
8
-]D-~
23.9
27.1
-2
-1
0
2.3
-2
-1
1.5
0.6
-2
9
""~-..--""-
10
"~".~-~-"
11
T-]I-
1.5
2.5
12
O-"
7.8
9.3
13
(~- ~
0
1.7
-2
14
-,,~,---~
5.1
5.9
-2
15
-,,~--- ~
1.5
1.4
-2
16
"<~>-@
0
0.6
-4
17
X-X
0
0.3
+2
t h e c h a n g e in t h e n u m b e r o f t y p e 1 a n d t y p e l i n e s in t h e l o c a l v i c i n i t y o f t h e e v e n t . N o t e events 8 and 9 are not topologically distinct a r e g e n e r a l l y e a s y to d i s t i n g u i s h d e p e n d i n g whether the type 1/2 or type 1 disclination exhibits greater curvature.
1/2 that but on line
Acknowledgement I s a a c C h u a n g w o u l d l i k e to t h a n k t h e M I T V I A f o r t h e i r s u p p o r t in c o o r d i n a t i n g this p r o j e c t .
+1
An1~s
-1
-1
-1
-1 -1 - 1
-1 -1
+1
-1 -1 +1
+1
References [1] H. Furukawa, Adv. Phys. 34 (1985) 703. [2] P.C. Hohenberg and B.I. Halperin, Rev. Mod. Phys. 49 (1977) 435. [3] P.W. Voorhees, J. Stat. Phys. 38 (1985) 231. [4] J.D. Gunton, M. San Miguel and P.S. Sahni, in: Phase Transitions and Critical Phenomena, eds. C. Domb and J.L. Lebowitz (Academic Press, London, 1983) Vol. 8, p. 267. [5] A.J. Bray, Phys. Rev. Lett. 62 (1989) 2841. [6] T.W.B. Kibble, J. Phys. A 9 (1976) 1387. [7] N. Turok, Phys. Rev. Lett. 63 (1989) 2625. [8] N.D. Mermin, Rev. Mod. Phys. 51 (1979) 591.
72
B. Yurke et al. / Coarsening dynamics in nematic liquid crystals
[9] L. Michel, Rev. Mod. Phys. 52 (1980) 617. [10] M. KI6man, Points, Lines, and Walls in Liquid Crystals, Magnetic Systems, and Various Ordered Media (Wiley, Chichester, 1983). [11] Fong Liu, M. Mondello and N. Goldenfeld, Phys. Rev. Lett. 66 (1991) 3071. [12] M. Mondello and N. Goldenfeld, Phys. Rev. A 42 (1990) 5865. [13] H. Toyoki and K. Honda, Prog. Theor. Phys. 78 (1987) 237. [14] H. Toyoki, Phys. Rev. A 42 (1990) 911. [15] H. Nishimori and T. Nukii, J. Phys. Soc. Jpn. 58 (1989) 563. [16] F. de Pasquale and P. Tartaglia, Phys. Rev. B 33 (1986) 2081. [17] G.F. Mazenko and M. Zannetti, Phys. Rev. B 32 (1985) 4565. [18] H. Orihara and Y. Ishibashi, J. Phys. Soc. Jpn. 55 (1986) 2151. [19] T. Nagaya, H. Orihara and Y. Ishibashi, J. Phys. Soc. Jpn. 56 (1987) 3086. [20] T. Nagaya, H. Hotta, H. Orihara and Y. Ishibashi, J. Phys. Soc. Jpn. 60 (1991) 1572. [21] I. Chuang, R. Durrer, N. Turok and B. Yurke, Science 251 (1991) 1336. [22] I. Chuang, N. Turok and B. Yurke, Phys. Rev. Lett. 66 (1991) 2472. [23] A. Vilenkin, Phys. Rep. 121 (1985) 263. [24] C.T. Hill, D.N. Schramm and J.N. Fry, Comments Nucl. Part. Phys. 19 (1989) 25. [25] W.H. Press, B.S. Ryden and D.N. Spergel, Astrophys. J. 347 (1989) 590. [26] N.S. Manton, Phys. Rev. D 28 (1983) 2019. [27] V. Kuzmin, V.A. Rubakov and M.E. Shaposhnikov, Phys. Lett. B 155 (1985) 36.
[28] M.E. Shaposhnikov, Nucl. Phys. B 287 (1987) 757 and 299 (1988) 797. [29] N. Turok and J. Zadrozny, Phys. Rev. Lett. 65 (1990) 2331. [30] A.K. Gooding, D.N. Spergel and N. Turok, Astrophys. J. 372 (1991) L5. [31] C. Park, D.N. Spergel and N. Turok, Astrophys. J. 372 (1991) L53. [32] W.H. Zurek, Nature 317 (1985) 505. [33] G. Vertogen and W.H. de Jeu, Thermotropic Liquid Crystals (Springer, Berlin, 1988). [34] P.G. de Gennes, The Physics of Liquid Crystals (Clarendon Press, Oxford, 1974). [35] D. Demus and L. Richter, Textures of Liquid Crystals (Verlag Chemie, New York, 1978). [36] M. K16man, L. Michel and G. Toulouse, J. Phys. 38 (1977) L-195. [37] P.E. Cladis and M. Kl6man, J. Phys. 33 (1972) 591. [38] C. Williams, P. Pierafiski and P.E. Cladis, Phys. Rev. Lett. 29 (1972) 90. [39] C.E. Williams, P.E. Cladis and M. K16man, Mol. Cryst. Liq. Cryst. 21 (1973) 355. [40] Y. Bouligand, J. Phys. 35 (1974) 959. [41] N. Turok and D.N. Spergel, Phys. Rev. Lett. 66 (1991) 3093. [42] I.M. Lifshitz and V.V. Slyosov, J. Phys. Chem. Solids 19 (1961) 35. [43] P.P. Karat and N.V. Madhusudana, Mol. Cryst. Liq. Cryst. 40 (1977) 239. [44] Shin-Tson Wu and R.J. Cox, J. Appl. Phys. 64 (1988) 821. [45] F.J. Kahn, Appl. Phys. Lett. 22 (1973) 386. [46] I. Chuang, The Dynamics of Strings, Monopoles, and Texture in Nematic Liquid Crystals, M.Sc. Thesis, Massachusetts Institute of Technology (1991).
Physica B 178 (1992) 56-72 North-Holland
Coarsening dynamics in nematic liquid crystals B e r n a r d Yurke a, A n d r e w N. Pargellis a, Isaac Chuang b and Neil T u r o k c aAT& T Bell Laboratories, Murray Hill, NJ 07974, USA bmlT Research Laboratory for Electronics, Cambridge, MA 02139, USA cJoseph Henry Laboratories, Princeton University, Princeton, NJ 08544, USA
The dynamics of symmetry breaking phase transitions and the dynamics of defect tangles generated following the symmetry breaking are of interest in a number of fields of physics ranging from condensed matter physics to particle physics and cosmology. For the case when continuous symmetries are involved, as of yet, there has been little experimental work. Here we report on our studies of the dynamics of defects generated in a sudden quench of a uniaxial nematic liquid crystal from the isotropic phase to the nematic phase. The symmetry breaking is from 0 ( 3 ) to D ~ . In this phase quench, stringlike defects belonging to the 7r1 homotopy class, pointlike defects belonging to the 772 homotopy class, and texture defects belonging to the % homotopy class are created. Although the % defect density scales as t-l, nonscaling behavior is seen for % defects.
I. Introduction
Symmetry breaking phase transitions in which the symmetry of a system is spontaneously broken to some lower symmetry are of importance in a number of fields of physics including condensed matter physics, particle physics, and cosmology. There has been considerable interest in the dynamics that occurs when such systems are rapidly quenched from one phase to the other. Topological defects are generated in such quenches and the dynamics of the defects ("coarsening dynamics") often exhibit scaling laws. Spinodal decomposition of binary mixtures is an example of such a phase quench for which there has been considerable theoretical and experimental work [1-5]. For those systems the topological defects are the domain walls between the two phases of the binary mixture. The mechanism by which defects are generated in a phase quench is referred to in the particle physics and cosmology literature as the Kibble mechanism [6]. The general idea is that the rate at which the quench takes place determines the size of the patches over which the direction of symmetry breaking is correlated. Patches having different directions of symmetry
breaking try to merge with each other smoothly to minimize gradient energies. However, because of topological constraints, invariably there will be regions where patches with different directions of symmetry breaking will not be able to merge with each other smoothly. As a result there will be regions where the gradient energy becomes singular. These are the topological defects. Depending on the symmetries involved, the defects need not be the two-dimensional sheetlike structures called domain walls. They could be stringlike structures called disclination lines or pointlike structures called monopoles. In addition, it is possible to have a topological defect that is nonsingular except for short instances of time when the defect unwinds. These nonsingular defects are called texture defects in the literature [7]. Using the notation of homotopy theory [8-10], a systematic method for classifying topological defects, we will refer to domain walls as % defects, disclination lines as % defects, monopoles as % defects, and texture as % defects. Considerable work has been done on the dynamics of quenches involving the breaking of a discrete symmetry for which the topological defects are domain walls, such as in the spinodal
0921-4526/92/$05.00 © 1992 - Elsevier Science Publishers B.V. All rights reserved
B. Yurke et al. / Coarsening dynamics in nematic liquid crystals
decomposition of binary fluids [1-5]. The study of phase transition dynamics and the dynamics of defect tangles created in phase quenches involving the breaking of a continuous symmetry, for which the topological defects are strings, points, or texture defects, is somewhat less advanced. Theoretical work [1t-17[ and, more recently, experimental work has picked up in this area. Research has concentrated on predicting and measuring the defect density as a function of time. Studies of the coarsening dynamics of stringlike defects in two-dimensional films of nematic liquid crystals [18, 19] and pointlike defects in two-dimensional films of cholesteric liquid crystals [20] have been carried out. Here we report on our own studies [21, 22] of the coarsening dynamics of bulk (three-dimensional) samples of uniaxial nematic liquid crystals. Cosmologists have also displayed considerable interest in symmetry breaking phase transitions in which 7rl, 7r2, or % defects are generated [7, 23-25]. In an expanding universe which started out hot and dense it is likely that symmetry breaking phase transitions have taken place. For example, the most successful particle theory to date, the Weinberg-Salam theory of electromagnetic and weak interactions, indicates that at a temperature of about 100 GeV the electroweak symmetry of the universe would have been broken. The phase transition would have been first order and % defects would have been formed during the phase quench. These texture defects would be unstable to decay via a baryon number violating process. It is thus possible that the electroweak phase transition may account for the observed baryon-antibaryon asymmetry in the universe [26-29]. Models invoking topological defects have also been proposed to account for the large scale inhomogeneity of luminous matter in the universe ]7, 23-25]. Topological defects, by perturbing space-time, could generate density fluctuations which would grow to allow hydrogen gas to be gathered in regions with sufficiently high density that stars and galaxies could form. The statistics of the distribution of galaxies created by such a process would depend on how the topological defects move after the phase quench
57
[30, 31]. It is thus important to understand the coarsening dynamics of defect tangles if one is to construct models that successfully "predict", for example, the galaxy-galaxy correlation function measured by observational astronomy. Considerable work has been done on models invoking ~-~ defects, cosmic strings [23]. More recently work on cosmological models employing % defects [7, 30, 31] has become of interest. Zurek [32] has suggested that the physics of cosmic string formation could be studied by investigating the formation of vortices in liquid 4He when the liquid is subjected to a rapid pressure quench from the normal phase to the superfluid phase. This experiment still remains to be done. We have preferred to work with liquid crystals instead because of the advantages afforded by a material whose phase transition occurs close to room temperature and whose defects can be directly imaged through an optical microscope. It should be noted that the correspondence between the behavior of liquid crystal or liquid helium defects and cosmological defects is not perfect. Liquid crystal dynamics are dissipative with inertial effects being negligible. The dynamics of cosmic strings is likely to be highly dissipative only for a relatively short time after the phase transition when the temperature of the universe is still sufficiently high so that the strings have large numbers of particles from which to scatter. At late times, when the universe is cold, cosmic strings would experience little damping and would whip back and forth at velocities that are a sizeable fraction of the speed of light. Though the Hubble expansion also provides a damping term in the field equations governing the dynamics of cosmic defects, it is generally not large enough to dominate the dynamics. However, when the universe is undergoing inflationary expansion this damping term would dominate. In this case the dynamics of disclination lines in liquid crystals could also closely mimic the dynamics of cosmic strings when expressed in comoving coordinates. Although the dynamics of defects in condensed matter systems may not always correspond closely to the dynamics of the analogous cosmological defects, the study of condensed matter systems
58
B. Yurke et al. / Coarsening dynamics in nematic liquid crystals
could significantly enhance our understanding of both.
2. Defects in uniaxial nematic liquid crystals
Liquid crystals exhibit a variety of mesophases [33-35] between the liquid phase and the solid phase and thus exhibit a variety of symmetry breaking phase transitions. Here we report on studies of the phase transition from the isotropic phase (the ordinary liquid phase) to the uniaxial nematic phase. The material we chose to work with, 4-cyano-4'-pentylbiphenyl (commercially known as K15 or 5CB) consists of long rodshaped molecules. In the isotropic phase the molecules are oriented randomly and are free to diffuse among themselves. In the nematic phase the molecules line up parallel to each other, that is, the molecules give up rotational entropy in order to maintain high translational entropy. Let n denote a vector pointed along a molecule. The liquid crystal does not distinguish between n and - n , that is, the material has inversion symmetry. The symmetry of the isotropic phase is the full rotation group 0(3). In the transition to the nematic phase, this symmetry is spontaneously broken to the symmetry of a cylinder, D~ h. Because of the inversion symmetry it is only necessary, however, to consider the symmetry breaking of the SO(3) subgroup of 0 ( 3 ) to 0 ( 2 ) . The vacuum manifold, which is the manifold of all possible ground states of the nematic phase, is the projective two-sphere, that is, a sphere with antipodal points identified with each other [8, 36]. According to homotopy theory for a projective two-sphere, 7rl, ~'2, and % defects may exist in a uniaxial nematic liquid crystal. The defects that are created in greatest abundance in a rapid pressure or temperature quench from the isotropic phase to the nematic phase are 7r~ defects in which the director field winds by + 7r or - 7r as one traverses a loop around the disclination line. These defects are called type + 1/2 and type - 1/2 disclination lines respectively. Figure 1 shows the director field configuration surrounding a + 1/2 and a - 1/2 disclination line. In the case shown the director field winds by +'rr
,/
+1/2
f
,y
-1/2
X
Fig. 1. An example of a type +1/2 and type - 1 / 2 disclination line.
and --rr, respectively, around the z-axis. Other types of winding by +Tr are possible. Disclination lines with a winding of -+27r are possible, but they are not topological singularities since Z
S
z/
,y
Fig. 2. A n example of a type +1 disclination line which is escaped along the z-axis.
"Y
Fig. 3. A monopole-antimonopole pair on a type +1 disclination line.
B. Yurke et al. +1/2
Coarsening dynamics in nematic liquid crystals
59
+1/2
+1
+1 Fig. 4. A type + 1 disclination line being pulled apart into two +1/2 disclination lines.
xS Fig. 6. The director field configuration for a texture.
Fig. 5. Defect tangle produced by a thermal quench of a freestanding film of nematic liquid crystal approximately 500 p~m thick. The picture is approximately 790 p.m wide. Type 1/2 defects are visible as dark sharp strings. Diffuse type 1 defects and monopoles are also visible in this photograph.
60
B. Yurke et al.
Coarsening dynamics in nematic liquid crystals
Fig. 7. Sequence of photographs showing texture decay in a freely suspended film of nematic liquid crystal.
B. Yurke et al, / Coarsening dynamics in nematic liquid crystals
®®
61
" ? [
,!
t=O.O sec
t=O
t=8.6 sec
t=60
~ii!iiiii!iiii!!!!:!!ii!~! ~ ~iiiiii~iiiiiii!!!iiiiii!!i!i~i~ iiiiiiiiiiii!iiiiiiiiii!iiiiiii!!!iiii!i!~=
t = 1 4 . 0 sec
t=80
.....
iiiiiii~i ~¸¸
~iiil;iiiii!~ili!!!~i~! iiiiiiiiiiiiiiiiiiiiii!i!~
t=27.5 sec
t=lO0
Fig. 8. Comparison of observed texture decay (left column) with numerical simulation (right column). Each video frame is 260 ~m wide.
62
B. Yurke et al. / Coarsening dynamics in nematic liquid crystals
the singularity at the core can be removed by suitably reorienting the molecules near the core [37-39]. This is referred to as escape in the third dimension. Figure 2 shows a type + 1 disclination line having a winding of +2~r about the z-axis. The singularity at the core is avoided by having the molecules tilt upwards and become parallel to the z-axis as one approaches the core. Figure 3 shows a type + 1 disclination line in which the direction of escape is downward both at the upper and lower end of the disclination line. However, in between there is a segment along which the direction of escape is upward. Where the segments with different directions of escape meet one has a pointlike singularity, known as a ~'2 defect or monopole [38]. At the upper point singularity the molecules all point radially inward, i.e., one has a "hedgehog" monopole. The lower monopole is a hyperbolic "hedgehog" and has the opposite monopole charge. Figure 4 shows that a type +1 disclination line can be pulled apart into two + 1 / 2 disclination lines. Such junctions of two type + 1 / 2 disclination lines with a type + 1 disclination line are commonly seen in defect tangles produced by rapid phase quenches. For comparison, the photograph in fig. 5 shows a defect tangle produced by a thermal quench of a freely suspended film of K15. The crisp narrow black lines are type 1/2 defects. The more diffuse bright lines are the type 1 disclination lines. A number of terminations of type 1 disclination lines on type 1/2 disclination lines can be seen. Also some of the type 1 disclination lines appear to be pinched into a dark spot at points along their lengths. These points are monopoles. Although % defects can in principle be created, they are rarely created in phase quenches from the isotropic phase to the nematic phase. Figure 6 shows a possible texture configuration. It consists of two concentric type 1 disclination lines of opposite winding number. At large distances from the defect the director field becomes uniform, in particular, the molecules orient their long axis along the z-axis. We have found that by inserting the tip of a glass needle into a freestanding film of nematic liquid crystal,
--500 ~m thick, type 1 disclination lines can be created as the needle is dragged through the fluid. In this manner we were able to produce defect tangles that were rich in type 1 disclination lines and that thus optimize the chances of creating a % defect configuration. The photographs of fig. 7 show what we believe to be a % defect in the process of unwinding. The first photograph in the sequence has no singularity present and can be interpreted as two loops of type 1 disclination lines. This texture unwinds by first having the two loops merge at one point to form a singularity which then decays to form a monopole-antimonopole pair. In successive frames of the figure, the monopoleantimonopole pair unwind the loop and then annihilate each other. During the run in which these photographs were taken, the microscope was switched back and forth between a camera and a video tape recorder. Figure 8 shows video frames of the texture unwinding. The first frame shows the moment at which the texture becomes singular at one point. Successive frames again show the unwinding of the texture. Alongside each video frame is shown the director field energy density of a computer simulation of texture decay in which the initial configuration consisted of a type - 1 disclination line inside a +1 disclination line in a configuration similar to that of fig. 6, but the two loops were placed offcenter with respect to each other. The outer loop overtakes the inner loop at one point creating a singularity which decays into a monopoleantimonopole pair. The singularity and the monopole-antimonopole pair show up in the energy density plots as pointlike concentrations of the strain energy. We note that ~'3 defects have been reported for cholesteric liquid crystals [40].
3. Equations of motion Strictly speaking, the uniaxial nematic liquid crystal in the nematic phase is characterized by a traceless symmetric tensor of second rank. However, it is more convenient for most purposes to work with n, the vector pointing along the long
B. Yurke et al. I Coarsening dynamics in nematic liquid crystals
axis of the molecule. In terms of the n vector field the elastic free energy F has the form
63
string as having a constant line tension T and a constant mobility F. Letting r denote the local radius of curvature, the dynamics are given by
F= ½{K,(~7-n) 2 + K2(n.V x n) 2 + K3ln x V x nl 2)
(1)
and is referred to as the Frank free energy [33, 34]. The constants K1, K2, and K 3 are respectively the splay, twist, and bend elastic constants. Similar terms arise in the free energy of superfluid 3He where the orbital angular momentum vector now plays the role of n. Since the reorientation viscosity of liquid crystals is generally sufficiently strong to make inertial terms negligible, the equation of motion for a liquid crystal is given by
On Y at
_
~F
8n~'
(2)
where y is the damping constant and the functional derivative is taken under the constraint that In[ 2 = 1. The equations of motion generated in this manner are fairly complicated and difficult to deal with. One often makes the equal constant approximation in which K 1 = K 2 = K 3 = K. In this case the field equations take the simplified form Ol'l a
Y Ot- = K[V2n~ + (7n~).(Vn~)n~].
(3)
This is the damped nonlinear sigma model. It is similar to the nonlinear sigma model used by cosmologists to study the evolution of a universe populated by global defects [41]. Because of the identification of n with - n when applied to the nematic liquid crystal eq. (3) admits stringlike solutions. The string line tension is proportional to In(R/ Re) where R is the radius of the disclination line (essentially the distance from the string to the nearest neighboring string) and Rc is the core radius. Similarly, the dissipation is proportional to ln(R/Rc). Because these logarithmic dependences of the string tension and the viscous dissipation are weak functions of R, to a good approximation one can model the dynamics of a
dr F dt
T r
(4)
For a type 1/2 disclination loop of radius r, this equation, upon integrating, yields r =
•/2T T-
(to - t ) ,
(5)
where t o is the time of collapse. The radius of the loop is thus expected to decrease as the square root of the time to collapse.
4. Coarsening dynamics The defects that are generated in the greatest abundance during a temperature or pressure quench from the isotropic to the nematic phase are type 1/2 disclination lines. Monopoles and texture defects tend to be rare. Also, since the type 1 defects are escaped in the third dimension, they are much less energetic than the type 1/2 defects. The interaction of type 1/2 defects thus dominate the coarsening dynamics. A scaling argument of the Lifshitz-Slyosov type [42] can be made to determine the type 1/2 string density p (line length per unit volume) as a function of time. One assumes that there is only one characteristic length ~. The string density is then proportional to ~-2 p ~ 1/~2.
(6)
The characteristic radius of curvature is also proportional to ~, so the characteristic line tension force per unit length fT is given by
ZT
(7)
The characteristic friction force per unit length fr is given by
fr = r v ,
(8)
64
B. Yurke et al. / Coarsening dynamics in nematic liquid crystals
where v is the characteristic velocity of the string. Equating the line tension force to the friction force one thus finds that the characteristic velocity is proportional to 1/~ v ~1/~.
(9)
The elastic energy per unit volume, E, stored in the disclination lines is given by E= T~/~ 3 ~ T p .
(10)
The rate W at which energy is being dissipated by friction is given by W = - - f F V ~ / ~ 3 oc I ' p 2 '
(11)
where eqs. (8) and (9) were used to obtain the proportionality. Equating the rate of change of the elastic energy dE/dt with the rate W at which energy is being dissipated yields the differential equation dp dt
F 2 c ~ p ,
(12)
where c is a constant of proportionality. This equation has the solution o
-
T c r ( t - to) '
a crystalline solid phase to nematic phase transition at 23°C and a nematic phase to isotropic phase transition at 35°C. We measured the slope Ap/AT of the coexistence curve to be 2.47 MPa/ K, between 0.7 and 17 MPa. A schematic of the apparatus used in these studies is depicted in fig. 9. The sample cell had two sapphire windows which, for most of the runs reported, were spaced 234-+ 23 pom apart. The view port to the cell was 3 mm in diameter. To prevent the pinning of defects to the windows, the windows were treated with the homeotropic alignment material n,n-dimethyl-noctadecyl-3-aminopropyltrimethoxysilyl chloride (otherwise known as D M O A P ) , using standard procedures [45]. Pressure was applied to the liquid crystal material in the sample cell via a diaphragm separating the liquid crystal from the pressurizing fluid which was water. Pressure jumps of less than 35 ms duration were applied to the sample cell by manually opening a valve. Transmission microscopy in which the cell was illuminated from the rear with unpolarized light was employed. A 10x objective having a depth of field large compared to the separation between the windows was used so that the disclina-
ILLUMINATION
(13) DIAPHRAGM
where t o is an integration constant. At late times the string density p thus scales as t -1. Our experimental measurements of the decay of the string density with time are described in section 6.
5. Apparatus In the following sections we describe experimental data on the dynamics of defects produced in rapid pressure quenches from the isotropic phase to the nematic phase. The nematic liquid crystal, 4-cyano-4'-n-pentylbiphenyl [43, 44], used in our studies was obtained from B D H Chemicals and used without further purification. At atmospheric pressure this material has
,I, SAMPLE CELL
~
VALVE
PRESSURE GAUGE
[ ~ MICROSCOPE PUMP
E ~]VALVE
L
I FLUID RESERVOIR
Fig. 9. Schematic of pressure quench apparatus.
B. Yurke et al. / Coarsening dynamics in nematic liquid crystals
tion lines throughout the cell would all appear in focus. No polarization filters were used. The microscope images were recorded on a high speed video recorder at a rate of 200 frames per second. For the study of the decay of the type
65
1/2 string density, video frames were digitized and image processed to determine the projected line length. A description of the image processing techniques used can be found in Chuang's thesis [46].
t = l . O sec
t = l . 7 sec
t = 2 . 9 sec
t=4.8 sec
Fig. 10. A coarsening sequence of a uniaxial nematic liquid crystal defect tangle. The dark threadlike defects are type 1/2 disclination lines. The tangle is 230 ixm thick and 360/xm wide.
B. Yurke et al. / Coarsening dynamics in nematic liquid crystals
66
6. Coarsening data A nematic liquid crystal subjected to a rapid pressure quench from the isotropic phase to the nematic phase generates a dense tangle of defects. The defect t~ingle consists predominately of type 1/2 disclination lines which undergo heavily damped motion in a direction to reduce the total strain energy of the liquid crystal. Figure 10 shows video images of the coarsening of a defect tangle at 1.0, 1.7, 2.9, and 4.8s after a pressure jump Ap of 4.69 MPa from an initially isotropic state in equilibrium at approximately 35°C and 3.6 MPa. Each picture is 360 txm wide. Figure 11 shows the coarsening data obtained from video images such as those shown in fig. 10. Data for the type 1/2 string density p as a function of time for four different pressure jumps each starting in the isotropic state at approximately 35°C and 3.6 MPa are shown. The projected string length was measured. The threedimensional string density (line length per unit volume) was obtained by assuming that the defect tangle is three-dimensional with the probability distribution of orientation of each line segment being uniform over the sphere. Each data point represents an average of 10 coarsening sequences taken at a given Ap. The data closely follow lines depicting the t-1 scaling ob-
256
'._:1.. 'ai~:~.~el . . . .
~-- 128 •~
IAi~.'a.b0' 'L
....
~_ ...... ........ + + : ~ .... -~ ~ zx.,. ........... +.+ %~
64
~'.......
I ....
-
4
~>i .... ...........
4 ........
e-
~
32
" <
16 12 -
I
'9,".
%! "-..
..... ~ , ~ . ~ 2 ' ~
1.0
2.0
4.0 8.0 Time [sec]
Fig. 11. T y p e 1/2 disclination time. D a t a were accumulated j u m p s AP: + , A P = 2 . 0 0 M P a ; 2.62MPa; ~, AP = 4.69MPa. scaling of t ~.
16.0
32.0
line density as a function of for four different pressure A, A P = 2 . 2 8 M P a ; *, A P = The dashed lines depict a
tained in section 4. A scaling relationship of p = t -~ with u = 1.02 _+ 0.04 is obtained from a least squares fit to this data. Finite size effects limit the reliability of the data for late times t. In particular, since the defects are not pinned to the sapphire windows of the sample cell, the defect tangle eventually collapses to a two-dimensional tangle and the type 1 / 2 disclination line density crosses over from a scaling of t 1 to the t 1/2 scaling studied by Orihara, Ishibashi, and Nagaya [18, 19] for a two-dimensional network. To take into account the uncertainties introduced by these finite size effects we quote our measured bulk coarsening exponent with greater error u = 1.02 _+ 0.09.
7. Loop data As the defect tangle coarsens and type 1/2 lines intersect each other and interconnect (intercommute), occasionally loops are formed. An example of this is shown in fig. 12. To check the model used for the coarsening argument of section 4, the collapse of type 1/2 loops was measured as a function of time. Figure 13 shows the time dependence of the loop radius for a type 1/2 disclination loop. By parameterizing the loop collapse as r = ( t - to) s, a least squares fit to the data gave a loop collapse exponent a = 0.49 _+ 0.002 for this data set. The measurement of seven such loop collapses gave an average loop collapse exponent a = 0.50 -+ 0.03 in good agreement with eq. (5). We also recorded the time of ,birth (the time of the intercommutation event that gave rise to the closed loop) and the time of death of all the type 1/2 disclination loops of the Ap = 4.69 MPa run. We included every type 1/2 loop that died after 1.00 s regardless of the number of type 1 disclination lines attached to the loop. From this data the loop density (the number of loops per unit volume) as a function of time could be determined. One expects the loop density to scale as the inverse of the correlation volume ~:3. The loop density is thus expected to scale as t -3/2 with time. Figure 14 shows our measured loop density as a function of time. At late times the loop density appears to scale with
B. Yurke et al. / Coarsening dynamics in nematic liquid crystals
t = l . 2 sec
67
t = 2 . 3 sec
t = l . 7 sec
I _ _ t=3.0 sec
t = 4 . 0 sec
t = 4 . 6 sec
Fig. 12. Sequences of video frames showing the production and collapse of a type 1/2 loop.
the expected power law of t 3/2 indicated by the solid black line. At early times (between 1 and 2s) the loop density does not appear to scale
~
I
' ' ' ' l
'
' ' ' 1
. . . .
I'
' ' ' l
'
properly. In fact, the density increases and builds up to a maximum at 2 s. There thus appears to be nonscaling behavior in the loop density. Cumulative Loop Density
' ' ' 1 ' ' " 1
48
4..]
.~ 32
-
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
2 E 1
16 o
0.5 ,-1
8 6
0.25 0.125
0.1
0.2
0.4 1.0 2.0 Time [sec]
4.06.0
Fig. 13. Loop radius as a function of time to collapse.
0.5
1.0
2.0 4.0 Time [sec]
8
Fig. 14. Loop density as a function of time.
16
B. Yurke et al. / Coarsening dynamics in nematic liquid crystals
68
8. Monopole data Figure 15 shows a sequence of video frames depicting the birth and death of a monopole. Although other mechanisms exist for forming monopoles, the most prevalent monopole production mechanism observed after a rapid pressure quench from the isotropic to the nematic phase consists of the collapse of a type 1/2 disclination loop having two type 1 disclination lines attached to it, as seen in the sequence of video frames. The figure also shows the most c o m m o n means by which monopoles are destroyed. The monopoles follow the type 1 disclination lines on which they are attached to the nearest termination of the type 1 on a type 1/2 string. The m o n o p o l e is destroyed when it
I
reaches the termination as can be seen by the absence of the m o n o p o l e in the last frame of the sequence. Again, if the coarsening is characterized by a single length scale sc, one would expect the m o n o p o l e density to scale as ~ 3 or as t 3/2. We have recorded the birth and death time of each m o n o p o l e contained in the sequence of runs A p = 2.28 MPa, 2.62 MPa, and 4.69 MPa. Monopoles are rare. So, in order to improve our statistics, we rescaled the time axis of the Ap = 2.28 MPa and the 2.62 MPa data run so that data of these runs could be c o m p a r e d with that of the 4.69 MPa run for equal string density. The resulting composite data is depicted in fig. 16. As with the loop density, the monopole density does not scale as expected. There is a sudden onset of m o n o p o l e production at early times. The mono-
/
t=5.9 see
t=6.8 see-
t=7.1 see
t=8.3 see
t=9.1 see
t--lO.6 s e c
Fig. 15. Monopole creation and annihilation sequence.
B. Yurke et al. / Coarsening dynamics in nematic liquid crystals
69
Cumulative Monopdle Density
_=
(a)
(b)
0.5 r
0.5
I ~-i
q I I I
1.0
2.0 4.0 Time [sec]
P I I i
8.0
16.0
Fig. 16. Monopole density as a function of time.
pole density reaches a maximum and then decays away more quickly than expected. Our observations of the loop density and the monopole density show that, strictly speaking, a nematic defect tangle does not scale with time. That is, the defect tangle at later times is not simply a magnified version of the defect tangle at earlier times. It may thus be likely that whenever more than one type of topological defect is present a defect tangle does not scale. One could, of course, argue that we have not reached the scaling regime. However, the system has had more than one decade in time (35 ms to 1 s) to reach the scaling regime before data is taken. Also type 1/2 disclination lines which dominate the coarsening dynamics exhibit the scaling expected for stringlike defects. Here we offer a partial explanation for what may be responsible for this nonscaling behavior. Depicted in fig. 17 is a type 1/2 disclination line that has been contorted so as to produce a loop. In order for the loop to generate a monopole upon collapse, the loop must have the same winding number along its entire length. Hence, the two sections of the disclination line near each other repel each other with a force F R which is proportional to the inverse of the distance d between them, F R o: K / d .
(14)
This repulsive force must be overcome by the curvature force F c which pulls the two disclina-
d~!
(c) Fig. 17. Loop production.
tion lines toward each other. This force is given by Fc ~ Kln(d/Rc)/R
,
(15)
where R c is the core radius and R is the radius of curvature. The factor l n ( d / R c ) accounts for dependence of the line tension on the defect line size. The radial extent of the defect is of the order of the distance of the defect to the next nearest defect d. If one takes d ~ R ~ ~:, then by equating F R with F c one finds a critical length scale ~:c which characterizes a string density threshold for the production of type 1/2 loops capable of producing monopoles A
~c = e R c ,
(16)
where A contains the geometrical factors that make the proportionalities, eqs. (14) and (15), equalities. The geometrical factor A is expected to be of order unity. If, however, it is 10, then e A is of order 2 × 1 0 4 . An A of 10 would thus provide a mechanism of amplifying a small length scale, the core radius R e which is of the order of 3 0 A , to a length scale ~:c of 6 0 ~ m which can easily be seen with optical microscopy.
70
B. Yurke et al. / Coarsening dynamics in nematic liquid crystals
9. Event frequencies As a defect tangle coarsens a number of processes happen that can be characterized by a well-defined event in time. For example, strings can intercommute, type 1 disclination lines can be pulled apart into type 1/2 disclination lines, monopoles are created and annihilated, loops collapse, and so forth. We have constructed a classification scheme for these events. For the case of the A P = 4 . 6 9 M P a data set we have systematically recorded all events occurring between 2 s and 5 s in the 400 ixm × 312 txm field of view of the video. For times between 1 s and 2 s we recorded all the events in one quarter of the video screen. For this data set we distinguished between the 16 types of events listed in table 1. As a listing of possible types of events, this table
is incomplete, since we have seen other types of events in other runs, such as monopoleantimonopole annihilation. We also observed 19 out of a total of 485 events (3.9%) which we were not able to classify because of obscuration. Table 2 lists the event frequencies for the various types of events. In the time interval 2.0 s ~< t < 5.0 s a total of 354 events were observed. In the time interval from 1.0 s < t < 2.0 s a total of 131 events were found. Also listed in table 2 are the classification numbers for the events. These numbers are defined as follows. An T is the difference in the n u m b e r of terminations of type 1 defects on type 1/2 defects before and after the event takes place. An M is the change in the number of monopoles after the event takes place. An L is the change in the number of links. Anls and An½s are
Table 1 Defect event classifications. Code
Event pictograph
a
X-)(
e
]E-)(
3
]lq[-)(
4
. ~ - -Itlr
6
"]1"][-_
7
-]fqF-
Description Intercommutation of two ± 1/2 strings (the two initial strings are separated in space) Decay of a ± 1 across two -+1/2's into two ± 1/2's Decay of two ± l's across two - 1/2's into two ± 1/2's lntercommutation of a ± 1/2 and a ± 1 resulting in a ± 1/2 with two T-intersections Unlinking of two ± l's from a ± 1/2
11
T--][-
12
C)--
Unlinking of two -+l's from a -+1/2 resulting in a single -+1/2 and a monopole carrying -+1 Decay of a ± 1 connecting two parts of the same -+1/2 into a single -+1/2 Decay of a ± 1 connected at both ends to a straight segment of -+1/2 Decay of a monopole carrying -+1 connected at both ends to a straight segment of -+1/2 Decay of a monopole sitting on a -+1 by absorption into a -+1/2 Collapse of a -+1/2 loop (no end products)
13
(~_,
Collapse of a -+1/2 loop with a -+1 across it
8 9
3D-D ""..Z.."-
10
16
~5----~-
17
X-X
Collapse of a -+ 1/2 loop with two ± l's coming out at either end, resulting in a single -+1 Collapse of a ± 1/2 loop with two attached -+l's, resulting in a monopole carrying -+1 Collapse of a -4-1/2 loop with four attached -+l's, resulting in a four--+ 1 vertex which escapes Intercommutation of two ± 1/2's resulting in two ± 1/2's connected by a -4-1
71
B. Yurke et al. / Coarsening dynamics in nematic liquid crystals
Table 2 Defect events, their probabilities, and associated observables. Probabilities were measured from the observation of 485 events from the AP = 4.69 MPa data. Code
Event
% Prob. <2s
% Prob. >2s
An v
An M
An L
Anls
1
X- )(
31.9
23.7
2
7[-)(
13.8
8.8
-2
-1
-1
3
]I-IF---)(
0.7
1.1
-4
-1
-2
4
-auf"- -/ql-
0
0.3
+2
+1 -1
6
71-][- ~
5.8
10.2
-2
7
71-][-- ~
0.7
0.6
-2
8
-]D-~
23.9
27.1
-2
-1
0
2.3
-2
-1
1.5
0.6
-2
9
""~-..--""-
10
"~".~-~-"
11
T-]I-
1.5
2.5
12
O-"
7.8
9.3
13
(~- ~
0
1.7
-2
14
-,,~,---~
5.1
5.9
-2
15
-,,~--- ~
1.5
1.4
-2
16
"<~>-@
0
0.6
-4
17
X-X
0
0.3
+2
t h e c h a n g e in t h e n u m b e r o f t y p e 1 a n d t y p e l i n e s in t h e l o c a l v i c i n i t y o f t h e e v e n t . N o t e events 8 and 9 are not topologically distinct a r e g e n e r a l l y e a s y to d i s t i n g u i s h d e p e n d i n g whether the type 1/2 or type 1 disclination exhibits greater curvature.
1/2 that but on line
Acknowledgement I s a a c C h u a n g w o u l d l i k e to t h a n k t h e M I T V I A f o r t h e i r s u p p o r t in c o o r d i n a t i n g this p r o j e c t .
+1
An1~s
-1
-1
-1
-1 -1 - 1
-1 -1
+1
-1 -1 +1
+1
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B. Yurke et al. / Coarsening dynamics in nematic liquid crystals
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