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Size and Scaling in Ideal Polymer Networks. Exact Results
Michael Solf, Thomas Vilgis
To cite this version:
Michael Solf, Thomas Vilgis. Size and Scaling in Ideal Polymer Networks. Exact Results. Journal de Physique I, EDP Sciences, 1996, 6 (11), pp.1451-1460. �10.1051/jp1:1996157�. �jpa-00247257�
J. Phys. I éYance 6 (1996) 1451-1460 NOVEMBER1996, PAGE1461
Size and Scaling in Ideal Polymer Networks. Exact Results
Michael P. Soif and Thomas A. Vilgis (*)
Max-Planck-Institut für Polymerforschung, Postfach 3148, 55021 Mainz, Germany
(Received 29 May1996, received in final form 12 July1996, accepted 26 July 1996)
PACS.61.41.+e Polymers, elastomers, and plastics
PACS.64.60.Cn Order-disorder transformations; statistical mechanics of model systems PACS.87.16.By Structure, bonding, conformation, configuration, and isomerism
of biomolecules
Abstract. The scattering function and radius of gyration of an ideal polymer network are calculated depending on the strength of the bonds that form the crosslinks. Our calculations
are based
on an exact theorem for the characteristic function of a polydisperse phantom net-
work that aliows for treating the crosshnks between pairs of randomly selected monomers as
quenched variables without resorting to replica methods. From this new approach it is found that the scattering function of an ideal network obeys a master curve which depends on one single parameter x = (ak)~N/M, where ak is trie product of the persistence length times the
scattering wavevector, N the total number of monomers and M the crosslinks in the system. By varying the crosslinking potential from infinity (hard à-constraints) to zero (free chain), we have also studied the crossover of the radius of gyration from the collapsed regime where Rg m O(1)
to the extended regime Rg ci O(@). In the crossover regime the network size Rg is found to be proportional to (N/M)~/~. The latter result can be understood in terms of a simple Flory
argument.
1. Introduction
Sulliciently crosslinked macromolecules form solid-like rubber networks whose spectacular elas- tic properties are commonly believed to be of entropic origin. As a consequence polymeric
networks are often successfully modeled as a set of independent random walks with the only restriction that the end points deform allinely under extemal stress. In a real network, how-
ever, permanent junctions between the macromolecules are randomly formed upon fabrication
leading to a high degree of polydispersity. To meet with this complicated physical situation
more sopbisticated theoretical models are required.
Recently vanous analytical studies il-?] bave aimed at a statistical description of polymer
networks taking randomness of the crosslinking positions into account. The matbematical challenge that arises in any of these approaches is that the permanent junction-points between
macromolecules are frozen and cannot be treated within the framework of Gibbsian statistical mecbanics. This bas first been realized by Edwards [1] and his replica formalism has by now
become tbe standard approacb in tbe field of polymer networks. Unfortunately tbe replica
(*) Author for correspondence (e-mail: vilgis©mpip-mainz.mpg.de)
@ Les Éditions de Physique 1996
metbod forces strong approximations to remain analytically tractable. A recent work by Pa- nyukov and Rabin [4] seems to overcome some of tbe dilliculties and derives promising results for tbe elasticity.
In tbis study we use a diflerent route of tbougbt. It bas recently been sbown [8] tbat for
randomly crosslinked Gaussian structures substantial progress
can be made by invoking quite diflerent mathematical tools than replica field theory. The purpose of the paper is to report
on further progress in this direction. Dur working model is an ideal polymer chain of N
monomers with M randomly selected pairs of monomers constrained to be in close neighbor-
hood. The distance constraints are modelled by harmonic potentials. By varying the strength of the crosshnking potential we cari continuously switch from a network situation with hard
à-constraints to tbe case of a free cbain.
Altbougb tbe above model is bigbly idealized, tbe system is most interesting in its own
rigbt, since it is considered a first step towards a systematic tbeory of polymer networks and gels. Moreover, the statistics of a huge macromolecule crosslinked to itself (randomly or not) has attracted a lot of attention recently because of its possible implications to protein
structure reconstruction from NMR data [9-11]. Despite of the recent interest to the best of
our knowledge tbese are tbe first exact results for ideal networks that have been reported.
2. Trie Ideal Network
We adopt the minimal model of a huge Gaussian chain that is M times crosslinked to it- self. In the Hamiltonian only terms that model chain connectedness and contributions due to
crosslinking are retained. Complicating factors such as entanglements, excluded volume are
deliberately neglected from the start. An appropriate Hamiltonian to begin with is
~~° ~2
(~~~ ~~~~~~ ~ ô ~Î(~~e
RJe)~ Il
~ ~
We bave assumed N +1 monomers wbose locations
in space are given by d-dimensional vectors IL~ Ii
= 0,1,..., N). Distance constraints exist between pairs of monomers labeled by i~ and
je. For further use we intriduce the
inverse strength of tbe crosslinking potential
~ j£j2
a
(~)
as tbe mean squared distance between monomers tbat form tbe crosslinks measured in units of tbe persistence lengtb a of tbe cbain (Fig. l). Limiting cases are given by z
= 0 (hard à-constraints) and z - cc (free cbain). Tbe wbole crosslinking topology is specified by a set of 2M integers C= (ie, je)$~. It bas been sbown
[8] tbat tbe model in (1) is equivalent to tbe Deam-Edwards model [1] without excluded-volume interaction if averages are understood in the following sense
jjdR~ ~-P7io
Î
~_~
= lim ~~ j3j
o z-o
fl dR~ ~-P7io
~=o
It is interesting to note that the entire range 0 > z < cc can be treated with tbe same formalism.
In [8j we bave explicitly sbown tbat ail limits are matbematically well defined. Tbus tbe bere
N°11 SCALING IN IDEAL NETWORKS 1453
~/l
Rie Rj~
Fig. 1. Modelling of a soft crosshnk between two monomers ie and je. A spring constant e - 0 leads to hard à-constraints, e - cc to free chain behavior.
presented metbod is indeed able to cover tbe range of all crosslink strengtbs, 1-e-, from very soft to tbe quencbed case.
To model M uncorrelated crosslinks tbe distribution of frozen variables C is assumed to be uniform
~~ ÎÎ2 ~ ~~~
e=1 0<~e <Je <N
Otber distributions are in principle possible but not considered in tbis investigation. As usual for systems witb permanent constraints care must be taken in evaluating averages of pbysical
quantities. Tbe strategy bere is flot to start witb tbe quencbed average over tbe frozen variables by employing for instance tbe replica trick, but to keep explicitly all crosslink coordinates C
during the calculation. Only at the very end tbe pbysical observable of interest is evaluated
for a particular realization of C which is generated by the distribution in (4). Clearly both
approaches will give the same results if only self-averaging quantities are considered.
The Hamiltonian in equation (1) together with the uniform distribution of crosslinks (4)
defines our working model for the ideal network.
3. Characteristic Function
In this section a brief review of the central mathematical theorem for the characteristic function of a Gaussian structure with internal à-constraints is given, together with its extension to
arbitrary crosslinking potential z. The characteristic function for the problem is introduced as
z~(E( cj
" (e~~'~) (à)
0
from which all expectation values can be obtained via diflerentiation. Simplifying notation has been adopted, where rj e ILj ILj-i Ii " 1,.
,
N) denote bond vectors along the backbone
E
Ei
,'
,' f~
Fig. 2. Externat field vector E projected parallel and perpendicular to U. The vector space U is spanned by the vectors pc defined in equation (7). The sketch is for two crosslinks
e = 1,2.
of the chain, and E
= (El, ...,EN), r
= (ri
;
.,rN) are N-dimensional super-vectors with d-dimensional vector components. Thus 20(E; C) is also the partition function of an ideal network in the presence of external fields Ej. Note that 20(E; C) depends explicitly on all external fields contained in the vector E, as well as on ail crosshnk positions C.
Without going into mathematical details, it is now possible to proof the following analytically
exact projection theorem [8j
~°l~i C) ~2
" ~xP (-pEl)
, 16)
wbere Ei is tbe lengtb of tbe external field vector E projected perpendicular to tbe vector space spanned by "crosslink vectors"
Pe " (Ù,. ,Ù,1,1,...,1,1,Ù,. ,Ù) (i)
~e+1to je
Tbe above statement
can be pictured in the following intuitive manner (Fig. 2). For each crosslink specified by the pair of integers 1 < ie < je < N, form the corresponding N- dimensional vector pe, equation (7), where the l's are assumed to run from the (ie +1)th
to the jeth position. The rest of the N components are filled with 0's. The whole set of M vectors pi,
..., PM defines a characteristic vector space for the problem, say U. There is
an unique decomposition of any field vector E parallel and perpendicular to U (Fig. 2), uiz., E = Ejj +Ei. The operator tbat pro jects E on U can be constructed from pe for any realization of crosslinks C. It is given by PP+, wbere P is tbe N x M rectangular matrix associated witb tbe crosshnk vectors pe (e = 1,.
,
M),
P + (Pi,
,
PM)
, (8j
wbile P+ is a generalized inverse of P [12j.
Tbe projection tbeorem (6) is only valid for hard crosslinking constraints z
= 0. A gen-
erahzation to arbitrary crosslinking potential z is, bowever, possible using tbe metbodh of
N°11 SCALING IN IDEAL NETWORKS 1455
reference [8j. Only tbe final result for tbe cbaractenstic function is quoted which reads
~°~~'~'~~ ~~~ lÎ~ ~~ j~
1ÎÎÎ~])
' ~~~
Here xe denotes any orthonormal basis associated with pe je
= 1,..., M), and we are corre-
sponding singular values [12j. For tetrafunctional crosslinks it was shown that we is always positive. From (9) it is straigbtforward to obtain expressions for tbe radius of gyration and
structure factor.
By tbe scattering function (form factor) we mean density fluctuations [13j normalized to one
So(k, z; C)
= ()pkl~)~ (10)
~Î2 ~~ ~~~ ~~~~~ ~J ) )o '
,Î~
wbere k is tbe scattenng wavevector, and tbe average is over tbe measure in (3) witbout baving
taken tbe z - 0 limit. In tbe following no distinction between N and N +1 will be made since N is assumed to be large. From equation (9) an exact expression for So can be obtained by
introducing tbe externat field vector E
= k c~j, where
Cq " (Ù,...,Ù,1,1,...,1,1,Ù,...,Ù)
,
Ill)
~+l to j
and k
= )k). From (9) and (10) it is found
~ N ~2~2 M j~
~ j2
~~~~ ~ ~~ ~ ~ ~ i ~~~ ~ ~
l
~ ii~l~Il ~~~~
Similarly for the radius of gyration [13j
~~~~~~~ ~ Î ~Î2 j~ 1ÎÎÎÎÎÎ)) ~~~~
~ J e=
It is worthwhile to note that the applicability of tbese results is not limited to random networks since all crosslinking coordinates are still imphcit in tbe formulas tbrougb xe and we.
By selecting diflerent ensembles for C= (ie,je)$
~ (random or not) any generalized Gaussian
structure witb internal crosslinking constrained can be treated by the same method. Further
generalizations of tbe working model (1) to structures built from more tban one cbain are feasible as long as tbe objects under investigation are simply connected. Otberwise tbe phantom cbaracter of tbe cbains and tbe neglect of entanglements leads to delocalization of those clusters of tbe network wbicb are not connected via crosslinks. A similar percolation problem anses at tbe vulcanization transition [14j wbicb, bowever, is not an objective of tbis study.
4. Discussion of ILesults
In applying equations (12, 13) to random networks, one bas to deal witb a sullicient number of
monomers N and crosslinks M. Moreover, tbe positions of crosslinks C= (ie,je)$ bave to be
cbosen at random. Tbe simplest scenario is to pick 2M integers ie, je (e = 1,.
,
&I) from tbe
2.0
/
''
l'
/
1.5 /
/ / / / '5H 1-O
ÉÇ/
H
o.5
o-o
~(i/2)
Fig. 3. Kratky plot of xso(x) for the ideal network (z
= 0, solid fine) and linear chain (z - cc, Debye function, dashed fine) as a function of @
= kRg in d
= 3. The open circles show a second realization of C to demonstrate self-averaging. Note that
we measure length scales in terms of Rg and
use a normalization So(0)
= 1 different to that in the expenmental literature. '
uniform distribution, equation (4), defined on tbe interval [0,Ni. For a given realization C we compute an ortbonormal basis x~ and singular values we of pe je
= 1,..., M). Any standard
technique like singular value decomposition [15] will sullice, for tbe ortbonormalization process presents only a mmor numerical task. We find tbat for number of monomers N > 10000 and
crosslinks M > 200 fluctuations between diflerent realizations of C difler by less tban 1 percent.
Tbis also presents an estimate for tbe numerical uncertainty of tbe calculation.
4.1. SCALING BEHAVIOR OF SCATTERING FUNCTION. Within trie framework of trie Deam- Edwards model (z
= 0) it was demonstrated [8] tbat tbe scattering function So of an ideal network is an universal function of wavevector k and mean crosslink density MIN as long as
N and M are sufficiently large to ensure self-averaging. A scaling form for So is motivated by tbe following argument. For a linear polymer witbout crosslinks the scattering intensity So(x) depends only on the product x
= k~R(, where R( = a~N/6 is the radius of gyration of the chain and So(x)
= 2(e~~ 1+ x)/x~ tbe Debye function [13]. In close analogy it was shown that for hard constraints (z
= 0) the radius of gyration of an ideal network is given by
R( ce 0.26a~N/M which suggests a scaling behavior, similar to that of linear chains without crosslinks. This scaling hypothesis for ideal networks was confirmed by our calculation based
on the expressions in equations (12, 13).
The numencally exact result for the scattenng function is presented in Figure 3. In the Kratky plot of Figure 3, xso(x) has been evaluated as a function of @ e kRg for z
= 0.
Independent of details of crosshnking topology C, all networks investigated fall on the same
N°11 SCALING IN IDEAL NETWORKS 1457
O.i
-~
fitoW O.Oi
ll~
O.ooi
O.Oi o-i i
fl
Fig. 4. Structure function Sa for different crosslinking potentials z. From right to left
z = 0,
10, 10~, 10~, 10~, 10~, cc for a network size of N
= 10000 and M
= 200 plotted in dimensionless units q =
kallù. The left curve (z - cc) is the Debye function of a linear chain.
master curve (solid line). Statistical fluctuations between diflerent networks were too small in the self-averaging regime to be seen on the scales used in Figure 3. No indication of a power- law decay for intermediate wavevectors k or other simplifying feature was detected, besides the
pronounced maximum in the Kratky plot at @ ce 2 which reflects the strong correlations of
the monomers due to crosslinking. For comparison, the case of a linear polymer with z - cc
(Debye function, dashed line) was also computed from (12).
The results of So for diflerent values of crosslinking potential z are illustrated in Figure 4 for
a network with N
= 10000 and M
= 200. By increasing the strength of the constraint from left
to right, large deviations of So from ideal chain behavior (Debye function, left curve) anse on
smaller and smaller length scales. For z
= 0 the network character persists down to even the shortest length scale k m 1 as a consequence of the high degree of crosslinking in the system
(MIN = 0.02). For sufficiently large wavevectors all curves decay as k~~ as expected when tbe scanning wavelength becoInes sInall coInpared to the Inesh size of the network. Again, the
Inaster curve for So can be obtained by plotting tbe X.axis in units of Rg.
4.2. THE COLLAPSE TRANSITION. In Figure 5 we bave calculated trie radius of gyration of
a network of N Inonolners and M crosslinks as a function of fi
= ela by use of equation (13).
From this investigation we can clearly distinguisb tbree diflerent scaling regiInes
~ 2
o.26 N/M
,
if e < ei
~ Cf 0.34 (ela) (N/M)~~~
,
if El < e < e2 (14j
~ N/6
,
if e m e2
,
__----
r'
,
M/6 ,
(1
o~
QÎ
o.26
O.I
O.l 10 100 1000 10000
(1/2)
Fig. 5. Radius of gyration of a polymer chain as a function of @
= ela. For the three solid curves the number of monomers
was varied from top to bottom N
= 5000, 10000, 20000; M was kept constant at 200. The short dashed fine shows
a network with N
= 20000 and M = 400.
with crossovers at ei CÎ afi@ and e2
ce afiÙ. The plateau values in Figure 5'correspond
to the two extremes R(la~
= 0.26 N/M (z - 0) to the left and R(la~
= N/6 (z - cc) to the
right.
In particular our investigation showed that the cases z
= 0 (hard constraints) and z
= 1
(constraints of the order of tbe persistence lengtb a) only difler by a numerical prefactor which varies from 0.26 for
z = 0 to about 0.27 for z
= 1. From this we conclude that an ideal network subject to uncorreiated crosslinking constraints is collapsed in a sense that its size is proportional to the square root of N/M. Therefore Rgla ci Cil since MIN is the mean
crosslink density in the network and of order unity in the thermodynamic limit N, M - oc.
Tbis finding seems to be at variance witb current speculations regarding tbe collapse transition of macromolecules [loi, wbere it was argued tbat a critical number of crosslinks M > M~ ci NI logN will force tbe system to collapse. Dur result for
z = 0 is in agreement witb recent Monte Carlo simulations by Kantor and Kardar Ill] wbo found for tbe mean squared end-
to-end distance R~la~ ci 1-à N/M. Tbis suggests tbe same one to six ratio for (Rg/R)~
in ideal networks as for linear polymers witbout crosslinking constraints and excluded-volume interaction [13j. We believe that the discrepancy between our exact results and these suggested
in [loi are due to the approximations used there.
Conversely, a free chain (z - cc) is an extended object with Rgla ce tJ(/V). Between the
collapsed and the extended regime we find a smooth crossover with (Rgla)~ being proportional
to (ela)@@. Remarkably this is the
same scahng as for randomly branched polymers
without excluded volume interaction. In the following we discuss the diflerent conformational states of the system in terms of simple scaling arguments.
