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Topologically Induced Glass Transition in Freely Rotating Rods
Sergei Obukhov, Dmitry Kobzev, Dennis Perchak, Michael Rubinstein
To cite this version:
Sergei Obukhov, Dmitry Kobzev, Dennis Perchak, Michael Rubinstein. Topologically Induced Glass Transition in Freely Rotating Rods. Journal de Physique I, EDP Sciences, 1997, 7 (4), pp.563-568.
�10.1051/jp1:1997175�. �jpa-00247344�
Topologically Induced Glass Transition in Freely Rotating Rods Sergei 0bukhov (~i*), Dmitry Kobzev (~), Dennis Perchak (~)
and Michael Rubinstein (~)
(~) Department of Physics, University of Florida, Gainesville, FL 32611-8440, USA (~) Research Laboratories, Eastman Kodak Company. Rochester, N~' 14650-2116, USA (~) Department of Chemistry, University of North Carolina, Chapel Hill, NC 27599-3290, US-~
(Recei;ed 21 June 1996. reiised 30 October 199G, accepted 30 December1996)
PACS.64.70.Pf Glass transitiom
PACS.64.60.Ht Dynamic critical phenomena
PACS.61.20.Lc Time-dependent properties; relaxation
Abstract. We present a simple minimal model which allows numerical and analytical study
of a glass transition. This is
a model of rigid rods with fixed centers of rotation. The rods can
rotate freely but cannot cross each other. The ratio L of the length of the rods to the distance
between the centers of rotation is the only parameter of this model. With increasing L we
observed a sharp crossover to practically infinite relaxation times in 2D arrays of rods. In 3D
we found a real transition to a completely frozen random state at Lc it 4.5.
One of the major problems of glass transition (for review see [1,2j) is that of the underlying
structural transition. Glass transition is traditionally viewed as a dynamical phenomena with temperature Tg close to the temperature Tc of some thermodynamic structural transition.
This creates a mutual interference bet,veen the two phenomena. In particular, the transition
temperature Tg usually depends on the cooling rate and the history of the sample, limiting the hopes to find the universal description of the glass transition.
Another major problem of glass transition is that of construction of a justifiable analytical theory. Any such theory should relate the mobility of a molecule with that of the surrounding molecules. These mobilities of surrounding molecules are coupled with those of their neighbors
and so on. The decoupling of the corresponding relations requires some approximations. There is typically no small parameter that can justify the decoupling procedure. This strong coupling
makes the theory unsuitable for quantitati,>e analysis. There is a clear need for a minimal model that would contain all the rele,>ant features of the phenomena and that is simple enough to allow numerical and analytical study. (The best kno,vn example of such a model is the Ising
model in magnetics.)
An important step towards creating a minimal model for glass the transition was recently
made by Edwards and E,>ans [3j, (see also Edwards and Vilgis [4]). They proposed a model of
randomly placed and randomly oriented rigid rods. This model is simple enough to be treated
analytically. The rotational mobility of the rods is blocked by the presence of other rods. The
only ,vay rods are allowed to move is along their longest dimension (along the direction of their
orientation). The mobility- of each rod depends on the mobility of the rods that can block
(*) Author for correspondence (e-mail: sergeitlphys.ufl.edu)
© Les #ditions de Physique 1997
564 JOURNAL DE PHYSIQUE I N°4
this motion. The small width to the length ratio [3,4j and the large number of rods interacting
with any single one justify the decoupling of the equations of motion. The authors derive the
Vogel-Fulcher law describing the divergence of the relaxation time, and the Williams-Watts stretched exponential relaxation in a glassy state.
Despite this remarkable progress in understanding of the analytical structure of glass tran- sition [3,4j, its physical observation is very difficult because of the effects of mutual alignment
of rigid rods leading to the formation of nematic phase. This ordering occurs in the same concentration range as the glass transition.
In our previous papers [5j we studied the relaxation time of flexible polymers with addi- tional constraints that their endpoints could only move along parallel planes. ive found that relaxation time in this case can be exponentially large in the number of entanglements of the
polymer with its neighbors. Thus the relaxation time has a very strong dependence on both
poly.mer length and concentration. This strong dependence is a remarkable change from the
po~v.er law dependence of the relaxation time on the molecular weight r oc N~.~ observed for a
linear polymer melt. This difference in the relaxation processes is due to additional constraints
imposed on the motion of chain ends.
Similar idea can be applied to the model of rigid rods, discussed above. Almost any additional constraint imposed on the motion of rigid rods makes the glass transition easier to achieve. For
example, we can connect the randomly oriented thin rods in pairs or in triplets by either rigid
or flexible bonds. In this case we will have a solution of almost massless objects. Each of these
objects restricts the mobility of approximately L~ other objects, where L is the length of rods measured in units of lattice spacing. The number density at which the glass transition can
occur is now I/L~, and volume fraction is (d/L)~, where d is the rod diameter. If the aspect
ratio d/L is small, this glass transition volume fraction (d/L)~ can be made much lower than the volume fraction d/L at which nematic ordering occurs.
The model [6j described below combines the ideas of the Edwards-Evans-Vilgis model ,vith the absence of any tendency towards structural ordering. Instead of connecting rods to each other, we will assume that positions of one end of each rod are fixed and that rods can freely
rotate around these points of attachment. Rods are not allo,,~ed to cross each other, otherwise
we treat them as having negligible thickness (zero excluded volume). Below we describe both
2D and 3D versions of this model. In 2D ~v.e find strong dependence of the relaxation time
r on the ratio L of rod length to the spacing between the rods, consistent with our analytical prediction r oc exp(L/Lo)~. In 3D we find the transition to a frozen configuration at Lc it 4.5.
In a recent paper by Renner, L6wen and Barrat [7j, the model [6j was studied on 3D fcc lattice
tinder different boundary conditions. The authors used the method of molecular dynamics and observed dramatic dynamical slowing down which was interpreted as the glass transition. We used Monte Carlo technique and a method of "frozen rods" (see below), which allows pinpoint
the transition with more certainty.
We have performed Monte Carlo simulations of the topologically restricted freely rotating rigid rod model. Each simulation run starts from a random initial configuration. There is no energy constant in our model, therefore all configurations have the same statistical weight. We generate a small random rotation for each rod around its point of attachment. Then ,ve check whether during this rotation the rod would intersect any other rod. If such crossing occurs, the move is rejected. We have considered both 2D and 3D models. In the 2D model the rods
are attached to the sites of the square lattice and point into the upper half-space, see Figure la
(to represent adsorption to a surface). In 3D model one end of each rod is attached to the node of a cubic lattice and the rods can point in any direction (Fig. lb). In both cases the only
parameter is the ratio L of rod length to the lattice spacing. All simulations were performed
in 9 x 9 x 9 box with periodic boundarv conditions in all directions. We have checked for size
fg
Q) ~
2~Q
~ ~
£ o zD
& 3D
. A estimates
from below
~ qi
~ ~ ~ ~ ~
~~ ln L
Fig. I. Fig. 2.
Fig. I. Rods attached to the sites of the a) square; and b) cubic lattice. The only parameter in
the problem is L. the ratio of the rod length to the lattice spacing.
Fig. 2. The relaxation time for a given length for both 2D and 3D models. For 2D model I < L < 6.5, and for 3D model I < L < 4. The slopes of the lines in the figure are equal to the corresponding
dimensionality (D
= 2,3). All times are normalized by the relaxation time To of the non-interacting system L < 0.5.
effects in the box 14 x 14 x 14. In the present paper we focus mainly on our results for the 3D model. The results for the 2D model and more details for the 3D model will be reported
elsewhere.
We study the relaxation of orientational correlation function
91t) = init)trio))
= icesflit)),
where nit) is a unit vector in the direction of the rod, flit) is the angle between the direction of the rod at time t and its initial direction and indicates the ensemble average.
Let us make a mean field estimate of the relaxation time for the array of freely rotating rods.
In order to rotate any rod by the angle of order unity, we should move about L~ other rods away from its path in a particular set of directions. Here D is the dimensionality of the array of rods. The total number of random attempts needed for the realizations of these particular rearrangements is of order of exp (L/Lo)~, where Lo is a constant of the order of unity-. The relaxation time of this entangled system of rods is r m To exp (L/Lo)~ where To a relaxation time of a single isolated rod.
The rilaxation time is plotted
as a function of the rod length in Figure 2. In the case of rods attached to the surface (2D), the relaxation time diverges with length L of rods as
r m Toexp(L/Lo)~. We observe that the relaxation time changes by at least four orders of
magnitude while the length of the rod changes from 2 to 10.
In case of the rods attached to the surface (2D model) and pointing into half-space, no glass
transition with divergent relaxation time is possible. At final L in 2D case any entangled configuration of rods can be continuously transformed without intersections into a completely aligned configuration. This transformation can be thought as a series of small (infinitely small)
566 JOURNAL DE PHYSIQUE I N°4
transformations. Each transformation consists of two steps. During the first step the whole
picture Figure la is stretched slightly in z direction such that position of any point with coordinates x, y, z is changed to x, y, (I + e)z. After such transformation the length of each rod becomes slightly larger than L, therefore at the next step,ve cut the extra length from the free end of each rod to reduce its length back to L. The result of these two transformations
(stretching and cutting) is equivalent to simultaneous rotation of all rods by small angles in such a way that they- do not cross each other during this rotation. This process can be repeated
many times until all the rods disentangle from each other and orient vertically (perpendicular
to the plane attachment).
The result of this thought experiment pro,>es that any configuration of rods attached to a 2D
plane can be continuously transformed without self intersections into the ordered configuration
with all rods perpendicular to the plane and then into any other configuration. In other ~v.ords there is no equilibrium glass transition for a 2D array of rods. But there is a dynamic transition,
~v.hich is a very- sharp crossover to extremely long relaxation times.
In three-dimensional case, when the rods are attached to the sites of a cubic lattice. ,ve observed a phase transition at L Gt 4.5 (see below). The divergence of the relaxation time at finite L due to this thermodynamic transition is con,>oluted with the exponential growth of the
relaxation time
r oc exp(L/Lo)~ due to the mean field effects discussed above. This exponential growth makes it very difficult to make reliable estimates of the relaxation time divergence at
finite L.
In order to determine the nature of the transition and to better define its position we have studied rod dy.namics at intermediate times. It is useful to represent the motion of a given
rod as diffusion in its angular space. A set of snapshots of the angular space of a typical rod at different values of parameter L = 2, 3, 4, and 5 are shown in Figure 3. The lines in the
figure are the projections of rods that are close enough to block the rotation of a chosen one.
Thus we show the projections of only those rods that can be "felt"' (can obstruct the motion of a given one). During the course of simulation all rods are moving and their projections (the
lines in Fig. 3) can move and e,wn disappear when a corresponding rod turns away from the test one.
At L
= 2 the density of the rods is low and they do not completely block the motion of a
test one. At L
= 3 the density of constraints is higher and at L
= 4 and 5 it is not ob,~ious whether the rotation of a test rod is possible at all. The rod may be localized within a certain window of mesh forever being blocked by given set of rods ("the dead lock" ).
For small angles 6 the motion of the rod is a simple diffusion in its angular space (6~) oc t as
long as the rod stays inside one window and doesn't interact with its neighbors. The average size of this ,vindow in angular space is 60(L) (mesh size in angular space) and the diffusion time needed for a rod to cross this average window is toIL). Numerically we found the average window size by- choosing such elementary- step in angular space that the attempted move in 50% of all cases was rejected (the rods would intersect ). The size of such step was taken as 60.
As long as the rod stays inside the ,vindo,v 6/60 < 1, ,ve have a universal line, representing a
simple diffusion (this part of the curve is not shown in Fig. 4). At longer times the surrounding rods slow the motiun of a given one. The surrounding rods mo,~e inside their respective windows at shorter time scales and participate in the motion of other rods at longer time scales. The
number of other rods that become involved in the motion of a gi,;en one grows expunentially
,vith flit) /60, and diffusion becomes exponentially slow (see Fig. 4).
These new ;ariables allo~v. us to compare the relaxation cur,;es for different lengths in the
same figure by plotting 6/60 as a function of logt/to. In the liquid state the displacement flit)
should asymptotically gro~v. linearly with t, whereas in glassy state it approaches a finit~ value.
In Figure 4 we identify the concave cur;es for L < 4 with a liquid state, and con,>ex cur,;es
~ 3
~ ~
L"2 ~~~ ~.~~
,
~
~ ~ 4.5
~ i
i
~
~ ~~
~
d ~
~ ~ ~
~
~
~
~- ~ ~~~
/
q
normal
~
~d~~~~~~ ~~~~~~
i IO 1°° ~~~ ~~~
~ ~
t/to
~~~ ~
Fig. 3. A part of the angular space (9,~i) of a typical rod for different L, -0.5 < cos9 < 0.5
(;ertical axis), -0.5 < ~i < 0.5 (horizontal axis). Lines on the picture are projections of the rods that
can block the rotation of a given one. The orientation of this typical rod can be represented by a
dot placed somewhere on the graph (not shown). cos9 is chosen instead of 9 in order to ensure the
homogeneity of the line distribution throughout the graph.
Fig. 4. Average root mean square angular displacement flit) of a rod as a function of time for different rod lengths: for L > 5 we are in glassy state, for L < 4 we are in liquid state. In the insert the relaxation at a given rod length is compared to the relaxation of the same system with 10 out of 729 rods held fixed (frozen). The difference between the two curves, the "susceptibility", is absent on the observable times at the transition point L m 4.5.
for L > 5 with a glass. The glass transition takes place for 4 < L < 5. At L
= Lc we expect to see a universal line separating relaxation in the liquid from relaxation in the glass.
In order to pinpoint the exact location of this line separating liquid- and glass-like relaxation and to determine Lc in our numerical experiment we used a method, which appears to be ,>ery
sensitive. lve freeze a small fraction (10 out of 729) of rods for each length. The relaxation must be slower due to additional constraints, since frozen rod cannot move and effectively
slows down relaxation of other rods (Fig. 4, insert). This procedure is analogous to a small increase in rod length, L ~ L*
= L(I + fi) except that in our case the lengths of the rods and the geometry of windows remain unaffected and we can compare the two relaxation curves
directly because the normalized units 60IL), to(L) are the same. The fraction of frozen rods
was determined empirically, so that the perturbation is still small, but the difference between the relaxation curves is sufficiently large for most lengths L.
If at some L the difference bet,veen the rela~~ation of the constrained and the original system disappears, then L must be very close to the transition L m Lc, because of the following
consideration: all relaxation curves follo,v the universal path for times shorter than their relaxation time r. At times shorter than the relaxation time of the system it is impossible to determine whether the system is a glass. or a liquid. As L approaches Lc, the time at which
corresponding relaxation cur;es branch out from the line corresponding to Lc becomes longer
and longer, since the relaxation time diverges at the transition point. Therefore, if L is close to
