Orientational Ordering and Dynamics of Rodlike Polyelectrolytes

0.82 MB·English

by  Hossein Fazli

Orientational Ordering and Dynamics of Rodlike Polyelectrolytes
Hossein Fazli,1 Ramin Golestanian,1 and Mohammad R. Kolahchi1
1Institute for Advanced Studies in Basic Sciences, Zanjan 45195-159, Iran
(Dated: February 6, 2008)
The interplay between electrostatic interactions and orientational correlations is studied for a
modelsystemofchargedrodspositioned onachain,usingMonteCarlosimulation techniques. Itis
7 shownthatthecouplingbringsaboutthenotionofelectrostatic frustration,whichinturnresultsin:
0 (i) a rich variety of novel orientational orderings such as chiral phases, and (ii) an inherently slow
0 dynamicscharacterizedbystretched-exponentialbehaviorintherelaxationfunctionsofthesystem.
2
n PACSnumbers: 82.35.Rs,83.80.Xz,61.20.Lc
a
J
I. INTRODUCTION
1
1 x

Solutionsofhighlychargedpolymersareknownto de- f
]
t velopnovelstructuralpropertiesduetothe interplaybe-
f
o tween electrostatic interactions and entropic effects [1]. q
s For example, it has been shown that for most charged - L
t. biopolymers such as DNA, filamentous (F-)actin, and y -
a various viruses [2], electrostatic correlationsin the vicin- -
m -
ityofthemacroionscausedbymultivalentcounterions—
-
d- ions of opposite charge—canlead to like-chargedattrac- z
n tion[3]. Thisattractionmostoftendestabilizesthepoly- a
o electrolyte solution and leads to the formation of col-
c lapsed bundles.
[
In a recent experiment with F-actin, Wong et al. ob- FIG. 1: The schematic configuration of charged rods. The
1 served that when the density of multivalent counterions
centers of the rods of equal length L are fixed on a chain of
v is not yet sufficient to trigger complete collapse,the uni-
spacing a. Each rod has two rotational degrees of freedom,
9 form solution can become unstable and phase separate and a uniform distribution of point charges.
2
intocoexistinglow-andhigh-densityphases[4]. Thisex-
2
periment is perhaps a most direct manifestation of the
1
0 peculiar behavior of polyelectrolytes in high density so- long range nature of the interaction presents them with
7 lutions: the dense phase is characterized by multi-axial amultitudeofconfigurationsthataregoodminimumen-
0 liquidcrystallinebehavior[5],aswellasexceedinglyslow ergycandidatesandyetarefarapartintheconfiguration
t/ dynamics that leads to the formation of an actin gel [4]. space. This picture is partly supported by the fact that
a A similar self-assembly pattern has been observedin the there are so many of them. This picture is supported by
m
structure of the nuclear lamina—a charged filamentous the fact that in the presence of (a considerable amount
- integral membrane protein network that provides a cy- of)salttheaboveeffectsarewashedout,asscreeningcan
d
toskeletal support for the nuclear membrane [6]. eliminate the frustration by reducing the effective range
n
o Another interestingexample is the observationofslow of the mutual repulsion.
c modes in the dynamics of various high-density polyelec- Here, we set out to study the collective behavior of
v: trolyte solutions in the absence of salt, where it is sug- polyelectrolytes at close separations using Monte Carlo
i gested that correlations cause the formation of starlike (MC) simulation, with the primary goal of understand-
X
complexeswithconsiderablyslowdynamics[7,8]. These ingthecomplexitythatelectrostaticfrustrationcanbring
r examplesraisetheissuethatstrongmany-bodyCoulom- about. We considera mostsimple model systemofsimi-
a
bic correlationsfor extended (non-pointlike) chargedob- larlychargedrigidrods of equallength L with their cen-
jects could lead to novel features that are poorly under- ters forming a regular lattice of spacing a on a chain, as
stood. showninFig.1. Wefindavarietyofcollectivebehaviors,
Basedonsuchexperimentalevidence,onecanbroadly andmapoutthephasediagramforthesystemasafunc-
identify two characteristic features in this class of prob- tionoftemperatureandlatticespacing(inversedensityof
lems: the novel orientational ordering, and the slow dy- the rods), as shown in Fig. 2. In the low-density regime,
namics. Asafirststeptowardsunderstandingsuchprop- the rods are found to order in a staggered way upon de-
erties, one should note the frustration that is inherent creasingthetemperature,suchthattheyareperpendicu-
structurally in the collection of like-charged rodlike ob- lar both to their neighborsand to the axis that connects
jects: As they come together, they reorient themselves their centers, through a two-stage crossovertransition.
to minimize the total energy of the system. Here, the For intermediate densities, we find that a low-
2
inSec. IV. Somediscussionsandconcludingremarksare
presentedin Sec. V. Finally, a variantof the model that
Disordered
1.04 corresponds to polyelectrolyte combs is discussed in the
Appendix.
0NTL
qB2 p
q =
Partially ordered
II. THE MODEL
2
(d) (a)
0.25 We use the Metropolis algorithmto simulate a system
(c) D (a)„ 0 D =0 (b) tohfeNirrrrootdatsiowniathl dtehgerireecsenotfefrrseefidxoemd o(nθ,aφ)chleafitn,toanedxpolnorlye
aL* aL1 aLc La [9]. We assume that Nq (odd) point-charges of the same
sign are distributed symmetrically on each rod (Fig. 1).
(d) (a)
The screened Coulomb interaction energy of the system
of rods can be written as:
q2 Nr Nq e−κ|rmi−rnj|
z z U = , (1)
ε X X |r −r |
(c) (b) m>ni,j=1 mi nj
2D where ε is the dielectric constant of the medium, q =
Q/N where Q is the overall charge of the rod, r (r )
q mi nj
D isthepositionofchargei(j)onrodm(n),andκ−1 isthe
z z
Debye screeninglength. Note thatthe energyexpression
isinvariantunderthelocaltransformationθ →π−θ and
φ→φ+π foreachrod. Weusefreeboundaryconditions
FIG. 2: Phase diagram of the system in the a−T plane. to avoidcomplexities arisingfromthe Ewald-summation
Dashed line: The crossover transition line from a disordered of extended charge distributions [10].
regime to partially ordered one [region (a)]. Dash-dotted For each value of the lattice constant a, we start the
line: The crossover transition line from a partially ordered system from a random configuration of the rods at a
regimetothestaggeredground-stateofthelow-densityregime high temperature (where the system is in the disordered
[region (b)]. Solid line: First order transition line from a
phase) and gradually cool down toward lower tempera-
partially ordered regime to the chiral ordered phase of the
tures. At a given temperature, we start the calculation
intermediate-density case [region (c)]. Hashed region: The
of various thermal averages after equilibration and take
domain in which a multitude of transitions occur to phases
the last configuration at each temperature as the initial
withvariouskindsoforderings. Region(d): The“seaurchin”
configuration for a slightly lower temperature. We have
phase. Schematic configuration of rods corresponding to the
different parts of the phase diagram is indicated in thelower run a wide variety of simulations for different values of
panel. The values of the various transition points indicated Nr, Nq, and a, and found that all the qualitative results
in the phase diagram are ac = 0.40L, a1 = 0.10L, and are robust.
a∗ ≃0.02L.
III. THE PHASE DIAGRAM
temperature orderedphase appearsthroughafirst-order
phase transition, in which neighboring rods have a twist To capture the essence of the electrostatic frustration,
angleinadditiontotheninetydegreesofthelow-density we first consider the salt-free case that corresponds to
phase. In the high-density regime, a hierarchy of differ- κ = 0, and comment on the effect of screening later. In
ent phases is observed depending on the lattice spacing, Fig. 2, we show the phase diagram of the system in the
where there is a multitude of twist angles, a periodic plane of temperature and lattice constant for N = 100
r
structure with a basis, and out-of-plane arrangements and N =3.
q
of the rods. We also study the equilibrium relaxation In the low-density regime, where a is greater than a
properties of the system and find that the appropriate criticalvaluea =0.40L,therodsappeartoorderonde-
c
order parameter in the system has an anomalous relax- creasing the temperature through a two-stage crossover
ation characterized by a stretched-exponential behavior, transition. The first crossover (denoted by the dashed
which is a feature also seen in systems with frustration. line in Fig. 2) takes the system to a partially ordered
The rest of the paper is organized as follows. In Sec. phase in which the angular degree of freedom, θ, freezes
II, we introduce the model and outline the simulation to π,andtherodsfluctuatefreelyonlywithintheresult-
2
technique used. Section III is devoted to the description ing parallel plates [region (a) in Fig. 2]. The ordering in
of various parts of the phase diagram, which is followed theφdegreeoffreedomoccursthroughanothercrossover
by discussions on the relaxation dynamics of the system at a lower temperature (denoted by the dash-dotted line
3
1.2 ZONE001 0.15
ZONE001
ZONE001
1 NNr==210000 0.1
c/k a=0.35L Nr=50
r
B
0.8 E+ 0.05
- cooling
>
E
a=L
0.6 < 0
heating
0.4 -0.05
0 0.2 0.4 0.6 0.8 1 1.2 1.4
T 0.18 0.19 0.2 0.21 0.22
T
FIG. 3: The specific heat as a function of temperature T [in FIG. 4: The thermal average of energy per site [in units
units of q2/(εkBL)] for (a) a = L and (b) a = 0.35L, with of q2/(εL)] as a function of the temperature [in units of
threedifferent sizes Nr =50, 100, 200 and Nq =3. q2/(εkBL)] upon successive cooling and heating of the sys-
tem, for a = 0.35L, Nr = 100, Nq = 3. The self-energy of
thestaggered ground-stateconfigurationE+ issubtractedoff
in Fig.2). The crossovertransitiontemperaturesare de- from the energy. The hysteresis loop can be observed, which
termined from the specific heat of the system as a func- indicates a first order transition from the partially ordered
tion of temperature, which develops two rounded peaks. regime to the chiral phase.
In Fig. 3a, the specific heat of the system is plotted
as a function of the temperature T, which is made di-
mensionless by the combination q2/(εkBL), at a = L points are the averagesof five independent runs starting
from different initial random configurations. For each
for three different sizes N = 50, 100, 200 and N = 3.
r q temperature 2×106 MCSare used, with 106 MCS being
Eachdatapointistheaverageoverfiveindependentruns
for equilibration. It can again be seen that the specific
startingfromdifferentinitialrandomconfigurations. For
each temperature, 1.5×106 MCS are used with the first heat is not sensitive to the size of the system.
7.5×105 MCS being for equilibration. As can be seen To understand the nature of the phase transition, we
examinedthe energyofthe systemupon successivecool-
from Fig. 3a, the height of the peaks in the specific heat
ingandheating. InFig. 4,thethermalaverageofenergy
do not change appreciably for various values of the sys-
per site as a function of the temperature [that is made
tem size N =50, 100, 200.
The grournd-state configuration corresponding to this dimensionless using q2/(εkBL)] is plotted for a = 0.35L
corresponding to successive cooling and heating of the
regime is given as [see Fig. 2b]:
system. In this curve, 7.5×105 MCS are used for equili-
π π brationand 7.5×105 MCS for ensemble averages,andit
θ = and φ =m× , (2)
m m
2 2 corresponds to N = 100, N = 3. As can be seen from
r q
Fig. 4,theaverageenergyofthesystemshowshysteresis
where m indicates the position of the centers of rods on
thez axis(z =ma). Thisisinagreementwiththetwo- around the transition temperature, which is a signature
m
body minimum energy configuration of the rods [11]. A of a first order phase transition. One can also define
multipoleexpansionoftheelectrostaticenergyexpression an order parameter that could distinguish between the
in Eq. (1), which is presumably a good approximation two phases and thus monitor the phase transition as a
in the limit of a ≫ L, reveals that the system of rods function of temperature. A suitable choice for the order
effectively behaves as a collection of quadrupoles in an parameter could be
externalstaggeredelectric field; hence the perpendicular
ordering described by Eq. (2). This picture also agrees 1 Nr−1
M = hsin∆ i, (3)
well with the lack of genuine phase transitions in this N −1 X n
r n=1
regime,asitisexpectedfromquadrupolesinasymmetry
breaking external field. where∆n(t)=φn+1(t)−φn(t)−π2. Figure5showsaplot
In the intermediate-density regime, corresponding to of this order parameter versus temperature [in units of
a∗ < a < ac where a∗ ≃ 0.025L, lowering the tem- q2/(εkBL)] for a system of Nr =200 and Nq =3 at a=
perature for each value of the lattice constant causes a 0.35L. Inthisplot,106MCSareusedforequilibrationof
crossover transition from disordered to partially ordered the systemand 106 MCS forthermalaverages. It canbe
regime,whichisfollowedbyaphasetransition(solidline seen that the order parameter develops a discontinuous
inthephasediagraminFig.2)tolow-temperaturephases jump across the transition temperature. This is another
having chiral order. A plot for the specific heat in this indication that the transition across the solid line in the
regime, corresponding to a=0.35L,is shown in Fig. 3b phasediagramofFig.2isfirstorder. Theseobservations,
forsystemsizesN =50, 100, 200andN =3. Thedata however,should be complemented with more systematic
r q
4
1.02 a=0.6L
a=0.4L
0.12 a=0.3L
1.015 10-1
0.08
M E(0)1.01 D10-2
0.04 /
)
DE( 10-5 10-4 10-3 10-2 10-1
a-a
0 1.005 c
-0.04 1
0.19 0.2 0.21 0.22
T -0.4 -0.2 /Dp0 0.2 0.4
FIG. 5: The order parameter M [defined in Eq. (3)] versus FIG. 6: A plot of the ground-state energy per site as a func-
temperature[inunitsofq2/(εkBL)]forasystemofNr =200, tion of ∆ for a = 0.3L, a = ac = 0.4L, and a = 0.6L, with
Nq =3 at a=0.35L. Nr = 40 and Nq = 3. The value of ∆ which minimizes the
energy of the system vanishes continuously at a = ac. In-
set: The log-log plot of ∆g versus (ac−a) gives a slope of
investigations using energy histogram methods [12, 13]. β=0.500.
Below this transition, the system of rods appears to
develop a rich variety of orderings. For a1 < a < ac
where a1 = 0.1L [region (c) in the phase diagram in rods still respect the planar ordering, a multitude of
Fig. 2], the array of rods orders in a chiral phase with transitions occur taking the system through a variety of
the corresponding ground-state configuration given as: minimum energy configurations. For example, upon in-
creasingthe density, aphase appearsinwhichsuccessive
π π cross-like structures formed by two neighboring rods are
θ = and φ =m× +∆ (a) . (4)
m 2 m (cid:16)2 g (cid:17) rotated relative to each other around the z-axis by an
angle ∆′, which corresponds to a chiral ordering with a
Sincethetwistangle∆issmallinthevicinityofthetran-
basis. Finally, in the high-density region where a < a∗,
sition line, we can simplify our description by assuming
the rods are no longer confined in the planes. While
that∆itselfservesasthecorrespondingorderparameter
in this regime one also finds various kinds of orderings,
for the phase transition from (a) and (b) to (c). While
eventuallytherodstendtoapproachasphericalconfigu-
its value is observed to undergo a finite jump just below
ration that resembles a “sea urchin” [see Fig. 2(d)], as a
thetransitionline,theground-statevalue∆ (a)isfound
g approachesitssmallestpossiblevaluea0thatissetbythe
to vanish continuously at a , implying that the termi-
c thickness of the rods. Note that only a very smallchoice
nation point of the transition line at zero temperature of the cutoff length, such as our choice of a0 = 10−6L,
corresponding to a=a is a critical point.
c canallowforthesecomplicatedhigh-densityphasestoex-
Foramoredirectstudyofthetransitionintheground-
ist. Moreover,unlikethe lowerdensitycases,thispartof
state configuration at a , we calculate the exact energy
c thephasediagramdependsverysensitivelyonthechoice
per site for the system using the energy expressiongiven
ofchargedistributionontherods,namelythevalueofN
q
in Eq. (1) as a function of ∆ for the configuration given
as well as whether the charges are smeared or discrete.
in Eq. (4). This means that we freeze the charged rods
intotheconfigurationsdefinedbyEq. (4)andcalculated
in a point by point double summation the electrostatic
IV. RELAXATION DYNAMICS
energy of the configuration for each value of ∆. This
energy function, which is plotted in Fig. 6 for different
values of a, admits a Landau form and the correspond- The low-temperature chiral ordering (in either of its
ing minimum energy solution ∆ (a) vanishes at a as variousforms)isnoteasilyachievedinitscomplete form
g c
∆ = |a −a|β with β = 0.500 (see inset of Fig. 6). We when the system is cooled down from the disordered
g c
have checked that the value of β does not depend on N phase to temperatures below the transition line. This
q
(evenuptoN =1001),confirmingtheexpectationfrom means that the equilibrium dynamics of the system in-
q
Landau theory that β =1/2. volvesaratherslowprocess,whichwetrytoidentifyand
In the intermediate-density regime when the rods are study here.
broughtclosertoeachother,differentconfigurationsthat For a1 <a<ac, the array of rods is decomposed into
are close in energy appear. This changes the behavior of severaldomainsofthechiralstructurewithbothpositive
the system at a1 where the curve of energy versus ∆ and negative values of ∆, corresponding to the degener-
starts to have four degenerate minima. For a∗ <a < a1 ate minima of energy shown in Fig. 6. The neighboring
with a∗ ≃ 0.025L (hashed region in Fig. 2), while the left-handed and right-handed domains are regions that
ekBTL k =0 5
q2 k -1=L
Partially orderedq =p k -1=0.3L
2
1 TT==10..080 0.25
T=0.62
T=0.45
TT==00..3257 0.15
0.8 T=0.23
C(t) 1 ZZOONNEE000011
0.9 ZZOONNEE000011 0.04
0.6 0.8 ZONE001
g0.7 ZONE001 0.35 0.40 a
0.6 0.3 0.6 0.9 1.2 L
0.4 T
0.2 FIG. 8: First order transition line of the phase diagram for
different values of thescreening parameter.
0
100 101 102 103 104
t(MCS)
ture in the case of a = 0.25L with N = 100 and
r
N =3. Using nonlinear fits with three free parameters,
q
FIG. 7: Equilibrium relaxation function of the system for
wefindthatthelong-timepartoftherelaxationfunction
a = 0.25L and various temperatures, reported in units of
is well represented by the stretched-exponential form,
tqh2/e(εcukBrvLe)s.aTreheofntohnelinsteraertcfihtesd-teoxpthoenelnotniga-ltifmoremb.ehInavseiot:r oAf C0 exp[−(t/τ)γ]. Theexponentγ saturatestoaconstant
ofthe orderof 0.9 in the hightemperature regime,while
plot of γ as a function of temperature.
it drops at lower temperatures where the relaxation is
exceedingly slower. The stretched-exponential form for
typicallycontainjustafewrods,andarelinkedviawhat the relaxation is a feature that is often encountered in
canbethoughtofaskinks. Bydefinition,thekinksanni- the dynamics of frustrated systems [14].
hilatewhentheymeeteachotherorwhentheyreachthe
boundaries of the system. We observe that the density
of the kinks is more or less fixed to about one kink for V. DISCUSSION AND CONCLUSION
every40rodsorso,irrespectiveofthesizeofthe system.
These kinks are observed to have very slow dynamics, In a most simple realization of the system, we have
presumablyduetotheelectrostaticfrustration,sothata shownthatchargedlinearobjectsincloseseparationcan
considerablylongMC time is neededbeforeallthe kinks developnovelcollective orientationalcorrelationsas well
are annihilated and one of the domains spans the entire as an unusually slow dynamics, due to the frustration in
system. The MC time needed for the annihilation of the theorientationaldegreesoffreedom. Whileinoursimple
kinks increases with increasing the system size. We also modelsystemweassumethattherodsarepositionedina
note that when the energy has four minima (for a just lattice,weexpectthatremovingthisconstraintwillcause
belowa1)andtwokindsofchiralstructuresarepossible, the orientational correlations to manifest themselves as
different kinds of kinks are observed depending on the liquidcrystallinestructuressuchasthoseobservedinRef.
type of the two neighboring domains that are linked. [4]. Geometrical frustration induced by the dimension-
To put the study of the slow dynamics in a more ality of the arrangement could also affect the orienta-
quantitative framework, we probe the equilibrium relax- tional ordering, and it remains to be seen what kind of
ation properties of the system by measuring the auto- arrangement is favored by the liquid system itself, i.e.,
correlation function for ∆n(t), which is defined as: whatisthe coupling betweenthe positional-andthe ori-
entational correlations in this frustrated system.
C(t)= PnN=r−11R dt′∆n(t+t′)∆n(t′). (5) Letus examinethe effect ofscreeningby added-saltin
Nr−1 dt′∆ (t′)∆ (t′) the above results. For a low concentrationof added salt,
Pn=1 R n n the screening is weak and we expect that the results re-
To obtain C(t) at each temperature T, we start the sys- portedherewillholdtrue solongasthe screeninglength
tem from a random configuration at high temperatures is the largest length in the system. As the salt concen-
andquenchit toT. After runningthe systemfor await- tration is increased, however, the Debye length begins
ing time, t , we use the system configuration φ (t) in to compete with the other length scales and this could
w n
all time steps from t to t +t to calculate C(t) using change the phase behavior of the system.
w w av
Eq. (5). For long enough waiting times (t ≃ 2×106 We have performed simulations with the screened
w
MC steps for N =100)),the relaxationfunction C(t) is Coulomb interaction of Eq. (1) using different values
r
independentoft aswellasthesysteminitialconditions, for the Debye screeninglength. We find that the general
w
indicating that there is no difference between averaging structure of the phase diagram of Fig. 2 persists, while
over time [Eq. (5)] and averaging over different initial thesizeofthechiraldomaininthephasediagramshrinks
conditions of the system. Note that the waiting time is as κ is increased. In Fig. 8, the first order phase tran-
typically the time needed for the kinks to annihilate. sition line is shown for three values of the Debye length
InFig.7,weshowthesystemrelaxationfunctionwith for N = 100 and N = 3. It can be seen that even for
r q
logarithmic time scale for different values of tempera- κ−1 =Lthevalueofa barelychangesascomparedtoits
c
6
κ = 0 value of 0.4L. Our results suggest that κ−1 ∼ a sequence of the non-vanishing dipole moment of rods
c
setsthethresholdwherescreeningcanwipeoutthechiral that are grafted at one end as opposed to those that
ordering altogether. aregraftedfromthemid-point. Consequently,region(b)
We have neglected the effect of counterions in the in this case corresponds to dipoles in a staggered field.
above study. This can be justified by noting that the Similarly, in the chiral phase the neighboring rods are
polyelectrolytesareveryshortrodsandunlesswechoose out of phase by π (instead of π/2) plus a residual twist
extremely high values for q, it is unlikely that they con- angle. We also note that the transitions are occurring
dense on the rods. This means that they will move rel-
atively freely in the vicinity of the linear polyelectrolyte
array, and merely participate in the Debye screening of
the interaction. This effect can be taken into accountby Disordered
considering a local effective value for the Debye length
that incorporates the density of the counterions as well 2.1
0NTL p
aosnsthinataonfesffaeltc.tivMeo“rceeolvl”er,artohuenpdretsheencaerroafytphreovcioduenstethrie- qB2 (d)Partially ordered q =2 (a)
1.3
necessaryneutralizationofthesystem. Inourtreatment,
however, we did not need to care about the neutraliza- (c) D (a)„ 0
D =0 (b)
tion criterion because the centers of the charged rods
a1 ac a
are fixed in space and in effect it is only multipoles (not L L L
monopoles) that are interacting. (d) (a)
Finally, we note that the results obtained here are
related to the studies of the equilibrium configuration
of ions in a confined plasma, where even similar chiral
phasesareobservedastheconfinementpotentialbecomes
z z
anisotropic [17].
(c) (b)
D
Acknowledgments
2D z z
WearegratefultoM.Deserno,R.R.Netz,D.Stauffer,
T.A.Waigh,andG.C.L.Wongforinterestingdiscussions
and comments.
FIG. 9: Phase diagram of the polyelectrolyte comb model
in the a−T plane. Dashed line: The crossover transition
APPENDIX A: END-GRAFTED CHARGED
linefrom adisorderedregimetopartially orderedone[region
RODS
(a)]. Dash-dotted line: The crossover transition line from
a partially ordered regime to the staggered ground-state of
In this Appendix, we consider the case where the the low-density regime [region (b)]. Solid line: First order
charged rods are actually grafted at their ends instead transition line from a partially ordered regime to the chiral
of the mid-points. This will make a model for a poly- ordered phase of the intermediate-density case [region (c)].
Hashed region: The domain in which a multitude of transi-
electrolyte comb, i.e. a backbone with a linear array of
tionsoccurtophaseswithvariouskindsoforderings. Region
charged side chains, which has in fact been synthesized
(d): The“seaurchin”phase. Schematicconfigurationofrods
and studied [15, 16].
corresponding to the different parts of the phase diagram is
We have performed simulations on such a system and
indicated in the lower panel. The values of the various tran-
have found the same general behavior as the previous sition points indicated in the phase diagram are ac = 1.40L
case. The corresponding phase diagram for the case of and a1 =0.3L.
N = 7 is shown in Fig. 9, where the transition points
q
are found to be ac = 1.40L, and a1 = 0.3L. Note that
the region (b) in Fig. 9 corresponds to rods that are at larger values of the reduced temperature and lattice
anti-parallel to their neighbors, which is a direct con- spacing, because of the stronger dipolar interactions.
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1994). work in progress.
[7] For a review, see e.g.: D.F. Hodgson and E.J. Amis, in [14] Seee.g.: B.KimandS.J.Lee,Phys.Rev.Lett.78,3709
Polyelectrolytes: Science and Technology edited by M. (1997).Notethat∆worksasaninherentfrustrationpa-
Hara (Marcel Dekker,New York,1992). rameter f, and characterizes region (c) of thephasedia-
[8] M. Muthukumar,J. Chem. Phys. 107, 2619 (1997). gram (Fig. 2) as thefrustrated phase.
[9] Insamplingtheunitspheredescribedby(θ,φ)uniformly, [15] C.P. Whitby, P.J. Scales, F. Grieser, T.W. Healy, G.
one should be careful to include the sinθ factor in the Kirby, J.A. Lewis, and C.F. Zukoski, Journal of Colloid
measure. In most interesting domains in the phase dia- and Interface Science 262, 274 (2003).
gramofFig.2,however,thetendencytokeeptherodsin- [16] T.A. Waigh, privatecommunication.
planemakesthisfactordispensible,asourexplicitchecks [17] See: J.P. Schiffer, in Strongly Coupled Coulomb Systems
haveshown. edited by G.J. Kalman et al. (Plenum Press, New York,
[10] M. Deserno and C. Holm, J. Chem. Phys. 109, 7678 1998); and references therein.
(1998).
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