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Conformation of liquid crystal side chain polymers in the smectic phase : models for anisotropic random walks
Jens Rieger
To cite this version:
Jens Rieger. Conformation of liquid crystal side chain polymers in the smectic phase : models for anisotropic random walks. Journal de Physique, 1988, 49 (9), pp.1615-1625.
�10.1051/jphys:019880049090161500�. �jpa-00210842�
Conformation of liquid crystal side chain polymers in the smectic phase :
models for anisotropic random walks
Jens Rieger
Theoretische Physik, Universität des Saarlandes, 6600 Saarbrücken, F.R.G.
(Reçu le 3 mars 1988, accepté sous forme définitive le 26 mai 1988)
Résumé.
2014On propose trois modèles pour expliquer la dépendance en température de l’anisotropie du rayon de giration de polymères cristaux liquides à chaîne latérale dans leur phase smectique. Dans ces modèles, on suppose que la conformation du squelette dans les plans entre lamelles smectiques peut être décrite
respectivement comme une marche aléatoire, comme une marche sans recoupement, ou comme une
trajectoire directe. Dans chaque modèle, on permet des passages du polymère à travers les lamelles, en les
affectant d’un facteur de Boltzmann. Par rapport aux degrés de liberté du squelette dans les plans, les
situations extrêmes correspondent au premier et au troisième modèle. Tout autre modèle plus réaliste donne
une anisotropie intermédiaire entre celles de ces deux modèles. On montre que les résultats obtenus suivant
ces modèles diffèrent assez peu selon les propriétés à grande échelle des systèmes qui ont été étudiés en
diffusion aux petits angles. On discute également les effets de volume exclu, qui jettent un doute sur les
théories existantes, et les situations hors de l’équilibre thermodynamique, qui rendent difficile l’interprétation
des expériences réalisées jusqu’ici.
Abstract.
2014Three models are proposed in order to explain the observed temperature dependent anisotropy of
the radii of gyration of liquid crystal side chain polymers in the smectic phase. In these models it is assumed that the conformation of the backbone in the planes between the smectic lamellae can be described by a
random walk, by a self-avoiding walk, or by a straight path respectively. In each model crossings of the side chain polymer through the lamellae are allowed and weighted by a suitable Boltzmann-factor. The first and the third model are the extremes which may occur with respect to the degrees of freedom the backbone has in the
planes. Every other, probably more realistic, model will show an anisotropy intermediate between these two models. It is shown that the results derived with the use of these models differ little with regard to large scale properties of the systems which are examined in small angle scattering experiments. Excluded volume effects the consideration of which casts some doubt on existing theories and non-equilibrium effects which make the
interpretation of existing experiments difficult are discussed.
Classification
Physics Abstracts
61.30
-05.40
-36.20
1. Introduction.
During the last decade there has been growing
interest in liquid crystal side chain polymers (LSP).
On the one hand these materials are interesting from
the technological point of view because of possible applications, e.g., in optical storage devices [1], on
the other hand LSPs are fascinating for physicists in general as one is confronted with phenomena whose explanation involves both polymer and liquid crystal (LC) physics. A LSP can be defined as being built by
means of a long polymer chain functioning as a
backbone (BB) with mesogenic moieties being at-
tached to it via flexible spacers. The rigid side chains
are responsible for the occurrence of typical LC
mesophases. For a comprehensive review on differ-
ent LSP systems see the recent articles by Finkel-
mann [2] and Shibaev and Plate [3]. The interesting
aspect of LSPs is for a theoretical physicist the following: because of the many conformational
degrees of freedom of the BB there is a tendency for
the LSP to assume a random coil conformation. This conflicts with the side chains tendency to gain a partially ordered mesophase (in the suitable tem-
perature range).
In this paper we are interested in the construction of models for the conformation of the BB in the smectic (SA) phase. Concerning this problem the following experiments are relevant: (a) small angle
neutron scattering (SANS) experiments [4-7] showed
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:019880049090161500
that on length scales being large compared with the
monomer-monomer distance the BB might be con-
sidered as an anisotropic random coil with Rz R 1- ,
where R, and R 1- are the radii of gyration of the BB parallel and perpendicular to the director of the side chains respectively. Both radii of gyration are tem- perature dependent [5-7] ; (b) it was found by small angle X-ray scattering (SAXS) experiments that the
smectic order is affected by the constraint that the side chains are attached to the BBs [8, 9]. A further point must be mentioned here : Boeffel et al. [10, 11]
found by 2H-NMR methods that the local orientation of the BB depends on the specific chemical structure
of the BB. Assuming that the results by Boeffel et al.
are not misleading because they might be a conse-
quence of a superstructure of the BB it is clear from their results that the conformation problem is hard
to attack from a pure microscopical point of view
since a small change in the structure of the BB seems
to lead to a totally different conformational be- haviour [10, 11].
In the following we confine ourselves to the case
where the BBs are mainly located in the planes
between the smectic lamellae which are built by the
side chains. It is an observational fact [4-7] that Rz =F 0. For that reason it will be assumed that the BB runs through one or several lamellae on the
average. Such a crossing is energetically unfavour-
able for the BB with its appending side chains locally destroys the smectic order, because along a crossing
of the BB the side chains can not be arranged in such
a way that they fit into the smectic lamellae. Note that the distance between two adjacent side chains is of the order of 5 A whereas the thickness of one
lamella is of the order of 30 A. That implies that
about five side chains can not match the lamellar order during one crossing. In our theory the crossings
will be taken into account by weighting the appear-
ance of one of them by a suitable Boltzmann-factor
exp (- E/kT ) where E is the free energy of one
crossing.
The issue of this paper is threefold : in the next section the packing (or excluded volume) problem of
the BBs in the interlamellar planes will be discussed
and two models will be proposed which take into account the excluded volume constraints. Further-
more it will be shown that the assumption that the
BB behaves like an ordinary two-dimensional ran-
dom walk in these planes [12, 13] is unrealistic. In sections 3 and 4 a theory for several types of random walks will be developed in order to understand the influence of the choice of a specific model on the
i predictions which are made for the experiment.
Section 5 provides a short discussion of the results and of the implications of our models for neutron
scattering experiments in the intermediate q-range.
Finally, in section 6 the influence of non-equilibrium
effects will be considered and the experiments on the
conformation problem of the BB will be discussed in the light of our results. Besides the possible applica- bility the presented models are interesting from the
basic research point of view because we provide in
section 3 the means to treat self-correlated random walks on non-cubic lattices with two different weights given to the random walk steps along different
directions.
2. The excluded volume problem.
The systems we consider have a well defined smectic
structure as can be seen in SAXS experiments [8, 14- 16]. So we are confronted with the following packing (or excluded volume) problem : each of the side chains by which the smectic lamellae are built is attached to one BB. From this it follows when
considering one interlamellar plane that on the
average half of the side chains adjoining this plane
have to be connected to one of the BBs which lie in this plane. That implies that there are on the average
no
«holes » in the planes : the planes are filled in a compact manner with BBs. On the other hand since each BB carries its own side chains with itself which have to be embedded in the lamellae there is for steric reasons not place enough for packing more
than ko
=2 or 3 BBs one on top of the other when
looking along the director of the side chains, see figure 1. The precise value of ko which may in some extreme situations even exceed the value of three
depends on the specific chemical structure of the
LSP under consideration. It is influenced, e.g., by
the respective effective diameters of the side chains and the BB, by the length of the spacer, and by the
curvilinear length of the BB between the points
where two adjacent side chains are attached.
There are two models which fulfill the above mentioned conditions. In the first model the BBs are
arranged in a sheaf-like structure [8], see figure 2. In
this model the BBs have the tendency to lie parallel
to each other in a plane. Their directional fields in two different planes are assumed to be uncorrelated.
Recent 2H-NMR experiments by B0153ffel et al. [11]
indicate that the BB is found in an elongated state in
the interlamellar planes. This speaks in favour of this
model. The second model maintains the idea of local random walk (RW) configurations. It can be ex- plained in terms of the k-tolerant compact RW in
two dimensions. The k-tolerant RW was introduced
by Malakis [17] and has recently found renewed
interest [18-20]. A k-tolerant RW is defined as a RW which may visit each point of a lattice at most k times (for k = 1 the case of the self-avoiding walk is regained). In our case we have the additional constraint that the lattice has to be completely
covered by several RWs in such a way that no point
is visited more than k
=ko times by the same or by
two different RWs because of the packing con-
Fig. 1.
-Schematic representation of a possible local configuration of two backbones (the thick lines) with the
side chains (the cylinders in the Fig.) being embedded in the smectic lamellae which are not shown for the sake of clearness. It shall be illustrated : (1) that it is difficult (if
not impossible) to find more than two backbones crossing
each other when projecting their respective paths onto the
x-y plane and (2) that with the aid of the appending side
chains very effective entanglements preventing the
backbone from longitudinal diffusion might be formed in smectic LSP systems.
straints. ko is a small integer larger or equal to two.
The precise value of ko depends on the specific
chemical structure of the LSP as was mentioned above. Within the coarse-grained point of view
which is mostly used in polymer physics and also
here several bonds of the real polymer or BB are
taken to define one statistical segment (Kuhn step)
of the RW in order to circumvent the consideration of the specific chemical structure of the polymer
chain [21]. That implies that k
=ko is changed to
some effective k:> ko. Unfortunately the solution of the two-dimensional k-tolerant compact RW prob-
lem for some fixed k is presently far out of reach.
That is the reason why we must confine ourselves in the following to some more simple models for the conformational behaviour of the BB in the inter- lamellar planes.
In the next section we present a hierarchy of RW
models which begins with the unrealistic case of the
ordinary. two-dimensional RW. On the next level the two-dimensional self-avoiding walk (SAW) and the
two-dimensional compact self-avoiding walk ard con-
sidered. The use of single chain statistics in these three models is justified for the following reasons :
the ordinary RW model is discussed in order to obtain insight into the properties of the most simple
Fig. 2.
-Typical conformation of one backbone (the
thick line) in the straight walk model. To make clear the sheaf-like structure also parts of other backbones are
depicted in the interlamellar planes. The lamellar thickness
(in the z-direction) is highly exaggerated.
model where the configurations the BB can exhibit
are only restricted by the lamellar structure of the system. Self- and mutual interactions (excluded
volume constraints) are neglected. This neglect also
underlies the existing theories [12, 13]. When discus- sing the SAW self-interactions are taken into ac- count : the fact that the BB must not pass through
itself is respected. The consideration of this effect is
important even for short sequences, i.e. in the case where the BB spends a small amount of arc-length in
the interlamellar plane between two consecutive
crossings. Recall that a SAW on the square lattice needs but four steps to form a forbidden configur-
ation. Turning to the many chain problem it must be
said that it seems presently not to be possible to treat
its influence on the present model since relatively
little is known of the statistics of a dense system of self- and mutual avoiding walks in two dimensions.
We end the hierarchy with the case where the BB has lost all its degrees of freedom in the planes namely the case where the BB is assumed to describe
a straight path between two crossings. Here the
mutual interactions of the BBs as regards the
excluded volume problem are taken into account
implicitely since the BBs are assumed to lie parallel
to each other in the respective planes. This last
model which is termed the straight walk model in the
following provides the theoretical treatment of the
picture of the sheaf-like structure which was pro-
posed above.
There are several reasons listed below from (i) to (iii) why we present our results despite the fact that
one of the two realistic models namely the k-tolerant
compact RW model presently can not be used for
the derivation of an expression for the radii of
gyration :
(i) it will be shown that all models yield similar expressions for the anisotropy factor a
:=R -; / R I.
Because of that it seems to be impossible to decide
from the knowledge of SANS data which kind of local structure is realised in the systems under consideration. This statement is of general nature
because the mentioned hierarchy contains the two
extremes of conceivable models (the simple RW and
the straight walk) ;
(ii) because the k-tolerant compact RW (kc-RW)
is located between the simple RW and the straight
walk (SW) with respect to this hierarchy the respec- tive anisotropy factors will obey the following re-
lation : a Rw - a kc-RW - aSW ;
(iii) concerning the existing theoretical ap-
proaches [12, 13] to the conformational problem of
LSP in the smectic phase the following point has to
be stressed : the description of the conformational behaviour of the BB in the interlamellar planes by a simple RW is highly unrealistic for a RW has fractal
dimension df
=2, i.e. even one RW will eventually
cover the whole plane. This problem was addressed by Wang and Warner [22] in their theory for the
nematic phases of LSP but not in the paper of Renz and Warner [13] in their approach to the BB
conformation problem in the smectic phase. The
latter author argues [23] that the leading term in the expression for the entropy of the system (see next section) is in any case proportional to the length of
the part of the BB which is considered. This point of
view is useful when one attempts to describe models where the number of degrees of freedom increases
exponentially with the chain length. As will be
shown below the straight walk model does not
belong to this class of models. Furthermore, the parameters which appear in the simple RW approach
can not be interpreted in the microscopic sense as
was done in reference [5] because the consideration of restrictions on the number of BB conformations
as proposed in this paper affects the parameters which appear in the simple RW approach.
Before turning to the theory a comparison will be
made between the assumptions we make and the
ones used by Renz and Warner in their theory [13]
for the conformation problem of LSP in the smectics
phase. In the RW models which will be discussed in the subsequent sections the RW is strictly confined
to a plane between the smectic lamellae, i.e. except for the jumps through the lamellae the BB is assumed to be well localized in the interlamellar
planes. This assumption is of course only justified at
temperatures where the lamellae are well defined,
i.e. at temperatures sufficiently below the smectic- nematic transition. Even in that case the BB exhibits fluctuations in the z-direction which are neglected in
the present theory. As regards this point the Renz- Warner theory might be closer to reality than ours
since these authors allow for these fluctuations by introducing a sinusoidal potential [13] instead of the sharp step potential we use. The different assump- tions made in the respective theories lead to a slight
difference in the final results for the anisotropy of
the radii of gyration as will be shown below. But it
seems to be unlikely that this difference will play a significant role when interpreting the experimental
results. Another difference between the present and the Renz-Warner theory is the following : we argue that the BB crosses the lamella as « fast » as
possible, i.e. the arc-length consumed by a jump through a lamella is assumed to be equal to d, the
lamellar thickness. As was stated above the reason
for this assumption is that the crossing energy is assumed to be large compared with kT. If the BB
would give more arc-length for a crossing the smectic order would be destroyed further with the resulting large energy penalty. Renz and Warner allow for
arbitrary arc-length of the part of the BB being
involved in a lamellar crossing. The fact that the BB is repelled from the core of the lamellae is taken into account by the above mentioned sinusoidal potential.
Renz and Warner postulate a second energetic
contribution namely a
«nematic
»energy which is the larger the smaller the angle is which the BB includes with the z-axis. They argue that the segment of the BB which points into the z-direction during a crossing forces the appending side chains to lie in the
x-y-plane within the lamella perpendicular to the
other side chains which form the respective lamella,
compare figure 2b in reference [13]. In my opinion
this effect plays no role in the real physical system since the side chains are decoupled from the BB orientation via alkane spacers. It is thus possible for
the side chains to align during a crossing, provided of
course the spacers are not too short. It seems to be safe to assume that the energy needed for the
bending of the spacers is smaller than the energy
required for the disturbance of the smectic order by
side chains being oriented perpendicular to the side
chains which form the lamellae. In conclusion we
would like to state that it seems to be unlikely that a
BB spends much more arc-length than d for a crossing of a lamella but we admit that there might
be a class of LSP [10, 11] to which our theories do not apply in the present form.
3. The theory for lattice RW models.
The conformation of the BB is described by several
types of RWs. Such a RW is placed on a lattice with
lattice spacing a and coordination number qo in the
interlamellar x-y-plane and step length Da
=d in the
z-direction, see figure 3 where the case qo = 4 (the
tetragonal case) is depkcted. One step of the RW
comprises several bonds of the real BB in order to receive RW steps which are statistically independent [21]. This means that the real BB with N bonds is
replaced by an equivalent chain with n N steps with the total length L of the BB not being affected,
i.e. Nb
=na
=L, where b is the monomer-mono- mer distance. This equivalent chain is assumed to show on a coarse-grained level the same behaviour
with respect to large scale properties as, e.g., the
mean squared end-to-end distance as the real BB
[21].
Fig. 3.
-Typical conformation of one backbone on the
tetragonal lattice, as represented by a simple random
walk. As in figure (2) the lamellar thickness is highly exaggerated.
Fig. 4
-An example of a typical entanglement. Some of
the side chains are depicted to show the distortion of the smectic lamellae by the crossings.
A crossing of the BB through a lamella is rep- resented by a step of length Da of the RW in the z-
direction. Each crossing costs the free energy E. We
assume that this crossing energy is considerably larger than kTsN, where TSN is the smectic to nematic transition temperature. This assumption seems to be justified for it was found by Volino et al. [24] that
even one mesogenic moiety in a classical LC which is displaced along its director in such a way that its center of mass is located between the lamellae contributes to the free energy with an amount of the
order of kTsN. Further contributions to the crossing
energy come from the fact that the smectic order is
locally destroyed by the crossing BB and from the bending of the spacers, see figure 4. In all models no
back tracing of the BB during lamellar crossing is
allowed since such a reversal is very improbable (in
the statistical sense) because of the high energy
being involved in such a process. Three models will
now be presented for the statistics of the RW on the interlamellar x-y-planes :
(1) the simple two-dimensional RW, i.e. a RW
which can move without constraint on the plane.
The number of configurations of a simple RW with k steps between two crossings is given by
(2) the two-dimensional self-avoiding walk (SAW). This is a first order approach to the packing problem. Vk is given by the following asymptotic
form
with y
=43/22 [25] (we use in the following
y = 4/3 for simplicity) and ql being an effective
coordination number with 1 q1 qo [21] ;
(3) the two-dimensional compact self-avoiding
walk. This is the collapsed version of the SAW. This walk covers a certain region completely without leaving any place unvisited except the places beyond
the boundary of this region. There is evidence [26, 27] that the asymptotic form of vk is given by
where q2 is another coordination number with 1 q2 qo. The exponent y’ unfortunately depends
on the specific form of the perimeter of the collapsed
SAW [27]. Present results indicate 1 y’ 2 [27, 28]. The use of relations (2) and (3) requires the assumption that the crossing energy is large com- pared with kT since the segment of the walk between two crossings must comprise enough steps in order
to allow for the application of these asymptotic
relations. This problem does not occur when using
relation (1) which is exact.
This hierarchy with respect to the restrictions which are imposed on possible configurations of the
RW is bounded by the straight walk. In this case the
walk has no degrees of freedom between two
crossings except the choice of the direction in the x-
y-plane. The straight walk model can not be treated with the methods proposed in this section and will therefore be discussed seperately in the next section.
Because we are interested in the qualitative be-
haviour of the system we limit ourselves to the
asymptotic case of long BBs i. e. n, No >> 1. But the method proposed can be extended in principle to the
case of finite chain length [29].
Presently our aim is to calculate the canonical
partition function Qn of a RW with n steps. With the
aid of Qn the mean squared end-to-end distances in the x-y-plane and in the z-direction will be derived.
Qn will be derived from the generating function or
isobaric partition function G (x ), see equation (8)
below. G (x ) is on the other hand computed from the
sequence generating functions (or partial isobaric partition functions) U(x ) and V (x ). The method of
generating functions (GF) is explained in more
detail in references [30, 31]. The GF for one crossing
of a lamella is given by
with uk = 2 Sk,D exp(- (3E) where P
=(kT)-1, 5 i, i is the Kronecker symbol. x acts as a counting
variable for the number of steps a sequence com-
prises (alternatively, x might be viewed as the activity). For the GF applying to the part of the RW being located in the interlamellar plane between two crossings we use
where Vk is given for the different models in
equations ((1)-(3)) respectively. The sum in equation (5) starts with k
=0 in order to include the
possibility of crossing one or more lamellae without lateral excursions between such crossings. For con-
venience we consider a RW starting and ending in
the interlamellar planes. The influence of this choice is negligible in the asymptotic limit of long chains.
The GF for the starting sequence reads
The sum starts with k = 1 since the first step is assumed to lie in a interlamellar plane. The GF containing complete information about all configur-
ations for all walk lengths is given by
G (x ) is related to the partition function Qn by
Because the free energy F
= -kT In Q n has to be proportional to n for n > 1 Qn must be of the form,
see reference [31]
Comparing equations (8) and (9) it can be seen that
xo is the pole of least modulus of G (x ). The average number of crossings is
The average number of steps in the x-y-plane is
We begin with the simple RW where uk
=0 G(x)
is then given by
xo is given by the solution of the following equation
For D = 1, 2, 3 xo can be calculated in a straightfor-
ward (but tedious) manner. In the following we
consider the case where D > 1 because it can be shown [32] that the final results for the anisotropy
factors which are derived for D > 1 are close to the exact solutions for the cases where D = 1, 2, 3 respectively. The respective values differ by some percent if /3E$:2. It can be assumed that the
crossing energy satisfies this relation [24]. Because
qo
>1 it is possible to expand the left side of
equation (13) about xo
=qo 1. Retaining the first two
terms of this expansion and using relation (9) we
obtain
Note that for D
=1 and E
=0 the exact result for the isotropic RW is regained. From equation (14)
nz and n-L are derived using equations ((10), (11)).
Omitting terms which are quadratic in qo D e- RE we
find
Because the successive crossings are not correlated
with respect to their directions the mean squared
end-to-end distance ri of the projection of the RW
onto the z-axis can be computed by regarding the
resulting one-dimensional RW with nz steps of
length Da :
In equation (17) we have again used the assumption
that qo De- PE 1. For the interlamellar RW se-
quences are also uncorrelated the mean squared
end-to-end distance rl of the projection of the RW
onto the x-y-plane is given by
The squared radii of gyration Rz2 and Rl which are
determined in small angle scattering experiments are proportional to r2z and ri respectively [33]. The anisotropy factor a is in the asymptotic limit there- fore given by
Now we proceed to the second case namely the
SAW model. The GF G (x ) is computed using the
coefficients Vk given by equation (2). Because we
know that the pole of G (x ) is located near x
=q 11 1 *e can use the following asymptotic be-
haviour of the GF V (x ) [34]
T (y ) is the gamma function. xo is given by the
solution of the following equation
The partition sum is derived in the same way as before and we find
Comparison of this expression with the one given for
the model of the simple RW, see equation (14),
shows that the introduction of excluded volume constraints in the way done here essentially replaces
qo by ql and changes the lamellar thickness Da into
an effective thickness Da/y. This change is due to
the loss of degrees of freedom in the interlamellar
planes. The mean squared end-to-end distance in the z-direction is derived as described above and is given by equation (16) inserting
The asterisk labels in the following the SAW results.
When considering the mean squared end-to-end
distance of the projection of the BB onto the x-y-
plane we use a model which is known in polymer physics (in a different context) under the name blob picture [21] : over short distances, i.e. between two crossings, excluded volume constraints are taken into account while correlations between SAW se-
quences which are separated by one more or more crossings are disregarded. This point of view is of
course problematic since the excluced volume prob-
lem is much more severe in two-dimensional than in three-dimensional systems for the screening effects
which simplify the theory in dense systems of the latter type are less effective in two dimensions. We propose the SAW model nevertheless for it is a first step into the direction of the k-tolerant RW model which is on the one hand thought to be in the same universality class as the SAW [18]. On the other
hand screening effects are more effective in the k-
tolerant RW case for each point of the lattice is
allowed to be visited a finite number of times and not only once as in the SAW case.
The projection of the whole BB onto the x-y-plane
looks in our model like a succession of [nz + 1 ]
blobs. The mean squared end-to-end distances of the parts of the walk which form the blobs are now taken
as the elementary steps of a simple RW since it was
assumed that the different blobs are uncorrelated because of screening effects. The mean squared end-
to-end distance of the part of the BB in the i-th blob is given by
where mi is the number of RW steps inside this blob and v
* =3/4 [25]. For the mean squared end-to-end
distance of the whole system in the x-y-plane we
obtain for a fixed set of {mi} with Y mi
=n - Dn z *
where the subscript denotes the fact that no averag-
ing with respect to the possible sets {mi} has been performed. The procedure of averaging is simplified by proceeding to the continuous case. Using the
method which is described in reference [35] for a
somewhat different system we finally obtain up to a constant of the order of unity
From equations ((24), (27)) the anisotropy factor
can be computed
where we have retained again only the leading
terms. Inserting nz* from equation (24) and using
y
=4/3 we obtain
where cp
=3 / (2 y )
=9/8. Comparing this ex- pression with the one derived for the simple RW we
find the somewhat surprising result that the apparent crossing energy is only weakly affected by the change to the SAW model. The main difference between the two respective anisotropy factors lies in
the fact that two different coordination numbers qo and ql appear which are at present not accessible by experiment, unfortunately. Accordingly the con-
clusion of these considerations is that it is not
possible to discriminate between the simple RW and
the SAW model by performing a small angle scat- tering experiment, even if the crossing energy were known. In principle it is of course possible to consider rz and r..L separately. But in that case two new problems occur when trying to compare the present theory with the experiments. First the value of the length of a statistical segment which appears
explicitly in the respective expressions via n and a must be known. Secondly it must be noted that the apparent crossing energy in the SAW case differs from the « true
»crossing energy (the simple RW case) only by a factor of the order of unity. As there
is great uncertainty about the value of this energy and as there are problems of determining this value by scattering experiments (see also below) it seems
to be hopeless to attack the problem from this side,
at least presently.
The case of the compact self-avoiding walk will
not be considered further for two reasons. First, as
was mentioned above, the exponent y’ which ap- pears in equation (3) depends on the specific form of
the two-dimensional globule. We do not dare to
make ad hoc assumptions as regards the statistics of these globules. Secondly, and this point is more important, with regard to the results obtained for the
simple RW and the SAW we do not expect results which differ drastically from the ones presented.
4. The theory for the straight walk model.
The method of GF does not apply here since all the approximations we made are not allowed for the straight walk model. Therefore a different approach
is presented. First note that the only entropic
contribution to the free energy comes from the fact that the crossings can be located anywhere along the
BB. It is assumed that the direction of the straight path the BB describes is randomly oriented in one plane with respect to the orientation in the adjacent plane. But all BBs in one plane are assumed to have
the same direction and therefore the entropic contri-
bution from this orientational disordering process to the free energy of one BB approaches zero when
considering macroscopic systems. From this it fol- lows that the free energy of one BB in the straight
walk approach is given by
N and E are defined above. D is the thickness of one
lamella measured in units of b (note that we use the
same symbol D as a measure of the lamellar thickness as in Sect. 3 though the respective values
of D differ). ni denotes the number of crossings and
is provisionally kept constant. f2 (N, D, n’) is the
number of different configurations of nz sequences
of D adjacent particles in excited states, i.e. D
monomers of the BB forming a crossing, on a linear
chain consisting of N particles. Equation (30) gives just the free energy of Tonk’s gas. (Note that for
D = 1 the problem is reduced to the consideration of Boltzmann statistics on a two level system.)
n (N, D, n’) is given by
The average number of excited sequences (crossings)
nz is found by differentiating F (N, D, n’) with
respect to ni :
for N >> 1 and N > Dnz. These two conditions imply
that systems with long BBs and with a high crossing
energy are described. With the aid of equations (16)
and (32) r,2 can be immediately given. The present model has the advantage that there are no
«
phenomenological » parameters. b is the mono- mer-monomer distance of the real BB and D is given by D
=d/b where d is the thickness of one lamella.
The derivation of rl is rather lengthy and will be
given elsewhere [35]. The result for nz being fixed is
found to be :
Neglecting the shortening of the part of the BB which lies in the interlamellar planes due to se-
quences which are involved in crossings and con- sidering the same limit as before we obtain after
averaging rl with respect to the probability density
function of nz [35]
with the weights p being given in (33). From equations (32) and (35) the anisotropy factor can
now be given
Note that for E/kT
=0 one obtains the result of the
isotropic RW « = 1/2 if N > 1. Next we consider the behaviour of
ain the high and low temperature limit in more detail :
where the second case is probably not accessible by experiment. Since the crossing energy is assumed to be large compared to KTSN we make in the following
a further approximation by setting the bracket term in (37) equal to unity. When comparing the straight
walk result for a with the previous results for the RW and the SAW one sees that the main difference lies in the different apparent activation energies.
The loss of all interlamellar degrees of freedom in the straight walk case leads to an effective crossing
energy Eeff
=2 E whereas for the SAW model only
a small difference between the apparent and the
« true » crossing energy was found.
5. Discussion.
In this section we return to the above proposed hierarchy of models. The results for the anisotropy
factors can be depicted in the following illustrative way if we write a in the general form
a - D2 q- D exp (- (3 Eeff) where D is the number of statistical segments (simple RW, SAW) or BB
monomers (straight walk) needed for a crossing of
one lamella
Since the simple RW and the straight walk are the
most extreme cases with respect to the number of
degrees of freedom the LSP has we suppose that a
more realistic model will lie in this table somewhere between the simple RW and the straight walk.
Refering to the comparison of the present theories with the Renz-Warner theory [13], see at the end of
section 2, it is interesting to compare the D-depen-
dence of r,2. We find for all cases the relation
r2z ~ D 2 which derives from the assumption that the crossing of one lamella consumes an arc-length of
d - D. Renz and Warner obtain r; -- D3/2 [13] using a
different approach as regards the possible crossing configurations. Unfortunately it seems to be unlikely
that this difference can be examined experimentally.
Returning to the presented hierarchy of models
we would like to note that it might be possible to
discern between the RW-like models (which are in
the following represented for simplicity by the simple RW model) and the straight walk case by performing a neutron scattering experiment in the
intermediate q-range (ql Rl > 1). If the BB
assumes a RW-like conformation in the planes
between the lamellae one should observe when
choosing the Kratky representation of the scattering
data (i.e. I (q) q2 vs. q, where I (q ) is the scattered intensity) the typical plateau over a wide q-range. If,
on the contrary, the straight walk model applies this
should show in a crossover from the plateau to a straight line with finite slope reflecting the rod-like structure of the BB configurations. In this latter case one should find I (qi ) - q- 1 for q J > (1) - 1 where (1) is the average length of the straight path the BB
describes between two crossings which is approxima- tively given by (I) = L/(Np + 1) with p being
defined in equation (33). Another difference be- tween the predictions of the two models as regards
the form of the scattering curve concerns its tempera-
ture dependence. When using the RW approach
there should be essentially no change in the
I (ql )-curve when changing the temperature since the local structure of the projection of the RW onto the x-y-plane is not affected. On the contrary the straight walk model predicts a decrease of the
ql -regime where I (qJ.. ) ’" q -2 (the plateau in the Kratky representation) when lowering the tempera-
ture. This prediction stems from the fact that the average length of the respective rod-like sequences of the BB increases with decreasing temperature which leads to rod-like scattering for qJ.. > (/) * B as
was stated above.
6. Non-equilibrium effects.
The experiments concerning the conformation of the BB of the LSP were done on LSPs with a degree of polymerisation from about 60 to several hundreds.
As is well known from polymer physics the ordering
of systems, e.g. the crystallisation of polyethylene, is
the more dominated by non-equilibrium effects the longer the polymers are. Constraints such as the
incomplete disentangling of the chains prevent the systems from reaching their ground state when lowering the temperature. The same non-equilib-
rium effects will probably affect the statistics of BB conformations during the nematic to smectic tran- sition if the experiments are done on the usual time
scales on LSP systems with a high degree of polymerisation. In that case most of the crossings
will not be thermally activated states but will be
induced and fixed by entanglements which can not
be resolved on the time scale of the experiment, see
figure 4. This problem is even more severe witch
LSPs than with classical linear polymers since the
side chains of different BBs may form very effective
topological barriers to relaxational motions, see figure 1. In reference [36] we explain briefly how the presented theories can also be applied to the case of entanglement induced crossings. We do not go into detail here because these considerations are still very
speculative. Especially it is not known what the critical BB length is at which the topological effects
which hinder the mutual rearrangement of the LSP molecules become dominant. In the case of ordinary
linear polymers these topological effects determine the properties of the systems from a polymerisation degree of a few hundreds on. For the hindering
mechanismes are probably much more effective in LSP systems we conjecture that this critical chain
length is of the order of or less than one hundred. If this is so it is safe to say that the reported exper- iments have been performed on LSP samples which
lie just in this critical range ; and indeed there is a recent experiment by Moussa et al. [5] which seems
to confirm this hypothesis. They report that for their PMA-CN sample in the smectic phase both Rz and Rl decrease with decreasing temperature. The de-
crease of R, which was also found for different other
samples [7] implies that at least a part of the crossings are thermally activated states. If the system has not the time to reach the equilibrium state when varying the temperature i.e. if some of the crossings
can not disappear when lowering the temperature due to entanglements the value which is derived for the apparent activation energy via an Arrhenius plot
will be too high with respect to the
«true » crossing
energy. But this point is difficult to pursue further for presently almost nothing is known about the value of the crossing energy in LSP systems. The decrease of R 1. with decreasing temperature men- tioned above seems to be a typical non-equilibrium
effect for the disappearance of crossings (i.e. the
decrease of R,) should naturally yield an increase in
R1. because the segments which formerly built the crossings are now part of the BB which lies in the x-
y-planes and this should lead to an increase in the overall size of the BB projected onto the x-y-plane.
That this in fact is not the case might be explained as
follows : if a BB runs somewhere along its extension from one plane to an adjacent one and returns from
that plane to the original one two crossings appear.
When lowering the temperature there is a certain probability that this sequence which lay in one of the adjacent planes collapses together with the part of the BB which was
«stored
»in the crossings to the original plane. This collapse produces a segment density in the x-y-plane which is larger than that
found in the equilibrium state. If the system has time
to react this surplus mass will diffuse to the ends of the BB. But if this diffusional process is hindered by
a special kind of entanglement which is produced by
side chains, see figures (1, 4), there will be an
increased BB density for the overall size can not
change in the x-y-plane. This leads by definition of the radius of gyration to the observed decrease of
Ri
In conclusion we claim that non-equilibrium ef-
fects are likely to occur when varying the tempera-
ture since it might take a very long time until the system has found its new equilibrium state. For it is
not known how large the influence of these effects is it is difficult to state to which extent it is possible to
compare the theories which all have been developed
for systems in equilibrium state to experiments.
What is needed is a theory which describes the
dynamics of LSPs in the smectic phase. But the
construction of such a theory seems presently to be a
very difficult task for up till now it has not been
possible to present a theory describing in a satisfying
manner the related problem of crystallisation in
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