HAL Id: jpa-00247781
https://hal.archives-ouvertes.fr/jpa-00247781
Submitted on 1 Jan 1992
HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
Scaling theory for the size of crumbled membranes in presence of linear polymers and other objects
T. Vilgis
To cite this version:
T. Vilgis. Scaling theory for the size of crumbled membranes in presence of linear polymers and other objects. Journal de Physique II, EDP Sciences, 1992, 2 (11), pp.1961-1972. �10.1051/jp2:1992106�.
�jpa-00247781�
Classification
Physics Abstracts
05.40 87.20 36.20
Scaling theory for the size of crumbled membranes in presence of linear polymers and other objects
T. A. Vilgis
Max-Planck-Institut fur Polymerforschung, Postfach 3148, 6500 Mainz, Germany
Institut Charles Sadron, CRM, 6 rue Boussingault, 67083 Strasbourg Cedex, France (Received12 June 1992, accepted in final form 12 August 1992)
Abstract. Condensed systems of polymers and membranes are studied. First it is shown that melts of membranes behave quite differently to melts of linear polymers. Whereas polymer chains in melts exhibit random walk behaviour, surfaces are always saturated. Most striking are results where surfaces are dissolved in linear polymers, where the size of the membrane is not changed,
I.e. v
=
4/5 as in the crumbled phase. The general value of v for D-dimensional manifolds is given by v = 2D/(2 +d~ where in the crumbled phase v
=
~ ~~.
Both values are identical for 2 + d
D
=
2. It is further shown that even for mixtures of membranes and stiff objects (Tobacco mosaic virusses) the crumbling exponent is also v = 4/5. This is a speciality for D
=
2 membranes. For
general values of D new exponents are predicted.
1. Introduction.
There is considerable interest in the statistical mechanics of surfaces as a model for
polymerized membranes [I]. Their conforrnational phase behaviour is very rich. Unlike linear flexible polymers, which can be treated as the special case of a one dimensional surface, surfaces show a flat phase and a crumbled phase [2] depending on intemal constraints, rigidity
etc. such polymeric membranes appear naturally in biology [3] and by polymerization of
Langmuir films or liquid bilayers [4, 5]. Such surfaces can be separated from their natural environment and solved in various solvents. The behaviour of such surfaces in ordinary low
molecular weight solvent has been studied extensively, both from the experimental and
theoretical side. As known from linear polymers, the most convenient way to characterize the statistics of the surface is to determine a size exponent v, R ~N~, or a fractal dimension
R~ N~. R is the dimension of the membrane and N~ is the total number of
monomers. v and
dr are thus related by di= DIV. D is the intemal dimension of the surface, I-e-
D
=
2 for two dimensional membranes and D
=
I for linear polymers and plays thus the role of the connectivity or spectral dimension. The different phases of such membranes can be summarized by a Flory estimate of various contributions to the total free energy. This has been nicely demonstrated [2], where amongst other results it has been shown that surfaces show a
rigid (flat) phase under certain circumstances, whereas linear polymers do not show a « flat »
1962 JOURNAL DE PHYSIQUE II N° 11
or rigid phase, I-e- R ~N. Although the importance of the flat phase in D
~ l surfaces is obvious, we restrict ourselves on the behaviour of crumpled surfaces. This does not seem to be useful since there are some doubts that crumpled phases do exist (see Eq, of [6] and references
therein). There is a large number of numerical simulations which indicates that two
dimensional membranes are always flat. This situation seems to be ruled by the strong
restriction of the size of the membranes in comparison to the local « persistence area ». For linear polymers (D
=
I local stiffness is not a dramatic restriction if the polymer is long enough. For two dimensional membranes this effect becomes more dramatic and local stiffness
might become long ranged which can be shown be purely mechanical arguments. Nevertheless,
a new model has been introduced by Renz and Baumgfirtner [6, 7] to overcome this stiffness effect. These authors simulate « real » flexible membranes with no local stiffness induced by
chemical bonds but the membrane is built up by connected spheres which are able to roll upon each other. These objects are among those to be considered in the following. However, recent
experiments seem to suggest the existence of crumbled membranes [9].
Let us, therefore, summarize the behaviour of the crumbled membranes. The statics of
crumpled phases in solvent have been studied by a Flory approach [10] and by various renormalization group techniques [I Il. Flory approximation predicts for v
v =
(~ ~, di
=
~ (l.1)
for two body interactions. For n-body interactions equation (I.I) can be generalized to
v =
2+ (n I)D
2 + (n I d (1,2)
and an upper and lower critical dimension
din) 2 nD
~~ (n )(2 D) (1,3a)
~~~ ~ (i,3b)
These results are all well known for D
= I, I.e. for linear polymers. It has been notified that for surfaces (D
= 2) the upper critical dimension is infinite for all values of n and polymer- specific properties such as a theta regime does not exist for surfaces.
In this paper condensed systems of membranes are studied. A natural question is how the statistics of a « melt » of membranes or as another typical example a single membrane in an environment of linear polymers can be described. Such cases have not been studied yet and are
of natural importance in biological systems. These considerations are of significance,
whenever the membrane concentration is larger than an overlap concentration for solutions, I-e-
~ 2(D-d)
~*~~ ~N 2+d (1.4a)
~d
for concentrated membrane solutions. Throughout the paper notation M
w N ~ is chosen, where
M is the total number of monomers, and that c* can also be represented in terms of the
corresponding fractal dimension
c*~M~~~~ (1.4b)
It will be shown in a more general context that in the concentrated membrane solution with
cm c* the membranes are saturated, I.e. they will show a statistic R~~N~ This becomes
immediately obvious since two membranes are always non intersecting in the crumbled phase,
I-e- the number
fi~2D Q
= j (1.5)
R is for D
=
2 always increasing with N even for d - oo, since the fractal dimension of the ideal (Gaussian) surface is infinite and the radius of gyration grows logarithically, I-e- R (log N )~'~
A more nontrivial case is the computation of the size of a surface in a polymeric
environment. Here one of the « solvents » is of one dimensional connectivity and nontrivial exponents can be expected. It is shown later that v for the D-dimensional manifold is given by
v'= ~~
(1.6)
For D
=
2 this is very special, since this exponent is identical to that of the elementary case,
I-e- membrane in solvent, although the D-dependence in equation (1.6) is very different from that in equation (1.1). Hence the fractal dimension for the case of surfaces in polymers is
different. It becomes
d)
=
~ ~ ~
(1.7)
independent of D, for all cases.
Another important situation is the behaviour of a membrane in the presence of rigid rod like molecules, such as tobacco mosaic virus molecules which are extremely rigid. Here the size exponent becomes
,,
2+3D (1.8)
" ~2(d+2)
This equation recovers the well known values for D
= (polymers in rods) which had been
given earlier [12]. Moreover, the upper critical dimension of d~~ = 3 can be recovered from (1.8). Another more general possibility is to dissolve a D-dimensional flexible manifold in
membranes. This can be realised by having one flexible membrane within others of different stiffness. The approach in this paper predicts for this case
~ ~ ~2
"~
2(d+ 21' ~~'~~
Most remarkably it is found that polymerized membranes D
=
2 are very special indeed. In all dimensions d they obey
v =
v'= v"
=
v"' (1,10)
Hence their size is not sensitive to the structure of the solvent and their environment. This is very unusual and rather different to the case of polymeric fractals [13] or linear polymers [12]
of different stiffness [13].
In the remainder of this paper the free energies leading to the exponents given in the
introduction will be derived from a systematic basis. It will be shown that the case
D
= 2 will be special, amongst cases, I-e- surfaces with a fractal connectivity. The different
expressions of equations (1.6-1.9) which are given the first time here will be unified by a more
rigorous approach.
1964 JOURNAL DE PHYSIQUE II N"
2. Formulation of the problem.
In order to derive a Flory free energy for a D-dimensional membrane in arbitrary environment
one can start from the path-integral representation for such objects. The theory will be
formulated for arbitrary objects, I-e- one membrane with Dj, solvent with in D~ connected manifolds. As mentioned earlier it will be assumed that the two species are of the same chemical nature and no thermodynamic interaction, apart from excluded volume interaction, will be present (no X-Parameter). The Hamiltonian for a Di-dimensional membrane is given by [1]
~ ja#jj2 ~~ ~~
Hi = d 'x + u jj d x d x' 8 lRj(x) Rj (x')] (2.1)
AX
where x is a Di-dimensional vector and I has d (embedding) dimensions. vi
j is the excluded volume interaction parameter. Extemal vector variables such as it will be denoted by
an arrow, whereas internal vectors will be in bold face representation. From dimensional analysis of the
Harniltonian (2.1) the usual excluded volume exponent can be found immediately, since the first term scales as N~ ~ ~ + ~ and the second one as N~~ ~~ Matching these leads to the result
given in equation (1.1).
If another species (« solvent ») is present it can be mathematically described by a similar many particle Hamiltonian, I-e-
n a§& 2
n
_ _
H~
= £ d~~x ~ + u~~ £ d~~xd~~x' lR((x) RI (x'ii (2.2)
AX
where n is the number of such «solvent» manifolds and is assumed to be large.
u~~ is the excluded volume interaction amongst the solvent molecules. For reasons which will become obvious later it is assumed that D~ w Dj. Equation (2.2) holds for flexible objects,
such as polymeric membranes or flexible polymers. A generalization to rigid objects can be
given on a scaling level which will become exact in the sense of later use. For rigid rod
polymers, the first term in equation (2.2) is of the order R~/N~ instead of R~/N, I-e-
v~ =
instead of v~ = 1/2. Thus resistivity scaling exponent f will enter, and on a scaling
level the first term of equation (2.2) has to be replaced by
Id/ ~i~ ~ R'~~(x) (2.3)
f
=
2 corresponds to flexible (Brownian) objects whereas ( = 2 D represents rigid objects.
It will be obvious later how this will work in general.
If the Dj-membrane is put into a solvent of D~-objects it will experience an additional interaction of the form
Vj~ = u i~ £ ld~'x d~~x' 8 ii (x) I( (x')] (2.4)
a j
The partition function of the system can then be written symbolically
Z
=
5~Ri
S~R~e~~' ~~~~ ~'~ (2.5)
To find an effective Hamiltonian for the membrane 1, the species 2 have to be integrated out.
This procedure yields
~fI ~D,~ ~~I ~
~ ~ ~D~ ~Di~> ~ ~~2 ~f~l~i(X)- I,IX)) (~ ~)
~~ ax
~
~~ i
+ u~~
$~(i)
here I is the d-dimensional wave vector conjugated to I (x ), and S((I) is the scattering function
or two point correlation function of the D~-dimensional object, I-e-
S((I)
=
~ d~~x d~~x'e~~~~'~~~~~'~~~~~ (2.7)
N ~
The mean field approximation used in the derivation of equation (2.6) will become appropriate
if the u,~ are identical and no thermodynamic phase separation will occur. This point will be discussed separately in a subsequent paper. For the first case equation (2.6) can be rewritten as
H(it
=
d~'x ~~~ ~
+ £ d~'x d~'x' ~ e'~~'~~~ ~'~~'~~ (2.8)
~~
j I + S°(I)
u
Dimensional analysis of equation (2.8) yields N~~ ~~~~' for the first term and
N~~' ~
~
N[~~ (2.9)
+ uN~~
for the second term. M~mN~~ are the total monomers of 2. Altematively equation (2.8)
corresponds to a Flory free energy
~ ~2D~
F
=
~
+ ~ (2.10)
N~ ~~~ + uN)~ R~
which is exactly (2.8) at I
= 0. Under the assumption uN~~ » I which is to incompressibility
limit [14] the interaction term can be reduced to
I Nl '
fi~D2 Rd
2
For N~
= (pointlike particles) equation (1.1) can be found immediately from (2.10).
IfN~~ and N~' are of the same order new exponents can be found. Therefore, a relationship
between Nj and N~ is needed, which can be found in natural cases. Thus we are going to
discuss different regimes for Ni and N~. It is desirable to change the size of membranes in a
controlled way, I.e. if the actual configuration is crumbled (v =4/5) it could become
important to tum them into a flat membrane R
~
N
i as one extreme case, or to tum them into
dense balls Ri ~Nf'~ Both configurations may contain for example different biological
information. Another application of such a controlled change of the membrane configuration in drug release. If, e.g. any drug can be released from a membrane it will be released faster from the flat phase as for example from the saturated bell. Thus we need exterial probes to control the size of membranes by their environment.
1966 JOURNAL DE PHYSIQUE II N° i1
3. Different regimes for Ni and N2.
In this paragraph three different regimes for N
j and N~ or altematively for the total number of
monomers in each species are discussed.
A. The first regime is whenever Nj »N~ or in terms of the total number of monomers
Mj
=
N~' » N~~
= M~. In this case the general answer for the exponent is known as it is given
in equation (1.1). Nevertheless these are some important correlations to be discussed. This can be done along the methods provided in reference [15].
B. Another important regime is N
j = N~. In this case it will be shown that the membranes
are always saturated, I-e- they satisfy R(
~ Mj. The case of N
j = N~ can be translated into the actual size of the ideal membrane, I-e- their natural (ideal) radius of gyration is given by
I I
(R()~~
= (R~)~~ (3.1)
or
~o ~~o~i ~~ ~~
~ ~
2 Dj
This ratio is vi/v~
=
If I is a surface and 2 is a flexible polymer this yields
2 D~
R( log R( (3.3)
which means that the ideal size of polymer is exponentially larger than the size of the
membrane. Therefore, one expects the membrane to form a dense ball of size Ri
M('~ Nf'~ A similar relation holds for the case if 2 are randomly oriented rigid rods, I-e-
R( log ~ R( (3.4)
where (
=
2 for flexible polymers and ( = I for rigid rod polymers.
C. Another important regime is given by R( R(, I-e- the actual (ideal) size of the objects
are comparable. For the numbers NJ and N~ this means
~o ~o
NJ' N~~ (3.5)
or
~ ~ , 2 Di
Therefore, we have vi/v~ = and if I is a D
=
2 membrane and D~ = I, I-e- linear
2 2 D~
object equation (3.5a) leads to
j
N~ (log N~)~ (3.6)
Thus, if the linear size of a membrane contains Nj 10~ monomers (Mj 10~~) which is a very large number, only chains of N~ 20 are needed to satisfy the condition of regime C.
These are very short chains indeed. It is the regime C which will turn out to be the most
interesting. It is also interesting from a methodic point of view. Flory type theories are a very delicate estimate of exponents, since they are only of « good quality » if only the ideal
quantities are used in the free energy. The reason has been discussed by de Gennes [10]. If the condition R( ~R( is used, this condition is satisfied at its best and we expect reasonable meanfield values for v.
There are more regimes for example N~'
~ N~~ which will be treated below along the same lines discussed above. All other cases can be treated the same way, as can be easily verified.
Notice that all conditions given in this paragraph can be translated into conditions for the
monomer number of the individual species, I-e- equation (3.5) can be written as starting from
i i
dj~ dj~
Rj
~ R~ leading to Mj M~ or
dj~
M~ M/~' (3.7)
is of course identical to (3.5a).
4. The exponents of the different regimes.
4.I REGIME A, N~'» N~~,
or N~'MN
~~. This regime can be treated with the approach
introduced above, and the results are obvious, since in the case N~' » N)~ ordinary swelling
will occur, whereas in the case N~~ » N)' saturation will take place and the surfaces will be
compact. This can be made more precise with the scaling concept used in reference [1II for the mixture of branched polymers and linear polymers, and is generalized to manifolds surfaces in this paper. The free energy of the surface surrounded by other objects can be written as (k~ T
= 1)
~2 F
= ~ ~
+R~F~~~ (4.1)
~~
where F~~~ is given by
F~~~
=
~ log (1 C (4.2)
M2
and the concentration C ~Mi/R~. These expressions for F~,~ are well known in polymer
physics.
The first term in equation (4.1) is the usual elastic free energy R~/M~'~~ for any fractal with a
measure introduced in (2.2). As in reference [15] (4.1) can be rewritten to a more appropriate
form.
F ='~j
~
j~ j C~'~ + i~ log (i C (4.3)
~ ~ ~
i
It is the first term in equation (4.3) which determines the size of the surface as a function of the
JOURNAL DE PHYSIQUE II T 2, N' II, NOVEMBER 1992 74
1968 JOURNAL DE PHYSIQUE II N" 11
solvent. The important parameter analogous to that defined in reference [11] is
M~ M~
a = =
~ ~
(4.4)
M~16~2 M~ Ii
As long as a » and no specific relation between Mj and M~ is present, F takes a minimum at
C 1. Therefore
~ ~flld /qD/d (~5)
l l
which is the saturated regime.
For chains and surfaces, this regime defines to a relation between Rj and R~
Ri « logR~ (4.6)
which corresponds to (3.3). Therefore the chains are much larger than the surface and the surface will collapse.
As long as a « the second term in (4.3) can be expanded and the minimum of F occurs at the concentration
d
M~ ~~~
~~ ~jI 1) ~~'~~
~f D ~d
I
Therefore, the size of the surface is given by R (Mi/C )~'~, therefore
_1 2+D
~ ~f d+2 ~fD(2+d) (~~)
2
I-e- ordinary swelling for M2
=
and « renormalized swelling » for M~ ~ 1. Notice that for
Mj = M~ (melt) one has
2
R Mfl~+~~ (4.9)
which will be recovered and discussed shortly below, by means of altemative methods.
4.2 THE MELT cAsE, Ni = N~, Dj = D~ = D. This is apart from the solution the other
elementary case. Here the mean field approximation which leads to equation (2.8) is most valid. The exponent for the size is predicted to be
v~ =
~ (4.10)
which generalizes the well known case for a melt of linear polymers (D = 1) derived first by
de Gennes [16a] and Daoud [16b]. Notice that the upper critical dimension is d$
=
2 D/(2 D) still infinity for D = 2 and d$
= 2 for D
= (which « explains » that molten chains are ideal). For D
= 2, equation (4.10) predicts v~ =
~ in d
= 3. To conclude that 5
membranes are swollen is erroneous since the fractal dimension Df of melt surfaces is given by Df
=
D/v
=
~ ~ ~
D which is for D
= 2 always larger than d. Thus the melt is saturated 2
