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Ind. Eng. Chem. Res. 2009, 48, 9514–9522
Self-Similar Dynamics of a Flexible Ring Polymer in a Fixed Obstacle Environment: A Coarse-Grained Molecular Model Balaji V. S. Iyer,†,‡ Ashish K. Lele,*,† Vinay A. Juvekar,† and Raghunath A. Mashelkar† Polymer Science and Engineering DiVision, National Chemical Laboratory, Pune, and Department of Chemical Engineering, Indian Institute of Technology, Mumbai
In this contribution we concern ourselves with an interesting problem, namely, the dynamics of ideal flexible ring polymers constrained in an array of fixed obstacles. The fundamental issue in this problem is to understand how a topologically constrained polymer chain is able to relax its conformation in the absence of chain ends. The key physics was provided in an elegant scaling theory by Rubinstein and co-workers (Obukhov, S. P.; Rubinstein, M.; Duke, T. Dynamics of a Ring Polymer in a Gel. Phys. ReV. Lett. 1994, 73, 1263-1267). In this work we develop a coarse-grained mean-field model based on the physical arguments of the scaling theory and derive constitutive relations for rings in fixed obstacle and melt environments. The model is composed of three distinct steps. In the first step the dynamics of an arbitrary section of a ring chain is worked out based on fractal Blob-Spring (BS) dynamics, and the center of mass diffusion and the relaxation spectrum of this section are determined. In the second step the center of mass diffusion obtained using the BS dynamics is used to model the one-dimensional diffusion of the section in a topologically constrained environment. In the final step we invoke the idea of dynamic self-similarity and argue that the dynamics described in the first and the second step, for any arbitrary section of the chain, applies to all sections of the chain. The constitutive relation is obtained consequently as the superposition of dynamic response of all sections of the ring chain. 1. Introduction Dr. B. D. Kulkarni, or BDK as we affectionately call him, has made exceptional contributions to chemical engineering science by ingeniously applying the rather exotic principles of nonlinear dynamics to process design and optimization. At the National Chemical Laboratory, where he built his illustrious career, he brought a unique theoretical flavor into a chemical engineering program that was otherwise steeped in the best traditions of combining deterministic modeling of transport phenomena and reaction engineering with objective empiricism and experimental validation. It is indeed our privilege to contribute this paper to the special issue of IEC&R brought out in BDK’s honor. The rheological response of a polymeric fluid is intimately connected to the dynamics of macromolecules that make up the fluid. Polymer dynamics in turn is influenced by their macromolecular architecture and the topological features of the environment. For example the rheology of a melt of entangled linear polymers is qualitatively different from the rheology of a melt of chemically similar unentangled polymers, although the static structure of the polymer chains in the two fluids is nearly identical.2 Entanglements create a topologically connected network that influences the dynamics of any individual polymer chain of the network. The network effect was imaginatively captured by the tube model,2 which proposes that any given polymer chain in the network is confined inside an effective tube created by the entanglements with neighboring chains. The tube prevents large scale lateral motions of the confined chain but allows for a one-dimensional diffusive motion along the contour of the chain.3 Entangled chains are thus assumed to revise their conformations via this snake like “reptation” motion. * To whom correspondence should be addressed. E-mail: ak.lele@ ncl.res.in. † National Chemical Laboratory. ‡ Indian Institute of Technology.
A rigorous formulation of the reptation dynamics has resulted in the development of a constitutive relation for entangled polymeric fluids.2 Subsequent improvements of the reptation model by the incorporation of other relaxation modes have resulted in sophisticated versions of the constitutive model.4,5 These have been shown to be highly successful in predicting linear and nonlinear viscometric flows as well as more complex flows.6 Macromolecular architecture also plays a significant role in governing the dynamics of polymers especially in an entangled state. For example, it is well-known that entangled branched polymers exhibit substantially different rheology than their linear counterparts. Fundamentally this difference arises from the fact that a branch is covalently pinned at one end to the rest of the macromolecule and hence reptation of the branch is prohibited. It has been proposed that branches renew their conformations by the so-called arm retraction process which involves onedimensional diffusive motion of the center of mass of the branch against a free energy barrier.7 Dynamics of more complex architectures such as H-, comb-, and dendritic-polymers have been modeled as a hierarchical relaxation of the branches by arm retraction followed by reptational relaxation of the backbone to which they are attached.8 Relaxation of polymers having an undefined branch on branch architecture has been modeled as relaxation of equivalent pom-pom polymers.9 These models have also been highly successful in predicting the viscometric response of nonlinear chain architectures in shear and extensional flows for both small and large deformations. All of the above models rely on the availability of at least one free end for the polymer chain through which tube renewal and the concomitant conformational rearrangement can occur. A topologically constrained cyclic or a ring polymer, on the other hand, has no free ends and hence the application of tube models to explain the dynamics of this peculiar chain architecture is far from being straightforward. Indeed, modeling the
10.1021/ie900535v CCC: $40.75 2009 American Chemical Society Published on Web 08/10/2009
Ind. Eng. Chem. Res., Vol. 48, No. 21, 2009 15
Figure 1. Lattice tree mapping of the structure of a ring chain in an array of fixed obstacles (unfilled squares). The pair of filled squares indicates a “gate”. The double-folded ring structure when collapsed to lines resembles a hyperbranched polymer. The ring structure can be mapped on to a Cayley tree as a closed random walk. The numbers assigned to the nodes of the tree indicate the smallest number of steps required to return to the origin.
dynamics of a topologically constrained ring chain has largely remained an unsolved problem up until recently.6 The differences in dynamics of topologically constrained linears and rings originate from the differences in their static structures. An ideal flexible ring polymer trapped in an array of fixed obstacles meanders through them when b , a , R, where b is the statistical segment or Kuhn length, a is the mean spacing between obstacles, and R is the size of the chain. The ring chain has a structure that is equivalent to a closed random walk on a Cayley tree of step size a (Figure 1).10 Each node of the Cayley tree is attributed a coordinate i, which is the smallest number of steps required to return from a given node to the origin. A closed random walk on a Cayley tree starting from the origin has to have equal distances moved from and to the origin; hence for a N step closed random walk the average coordinate i is of the order N1/2 steps from the origin. It follows that the spatial size of an ideal ring chain in an array of fixed obstacles corresponds to a random walk of i steps on the Cayley tree and hence is of the order R ≈ ai1/2 ≈ aN1/4. The mean square radius of gyration of an ideal ring chain was shown to be11 〈Rg2〉 )
z √2π 2 a √N z-2 8
(1)
where z is the coordination number of the Cayley tree lattice. Thus, an ideal ring chain is predicted to take up a highly collapsed conformation with a fractal dimension of 4. Indeed the size of the ring chain has the same molecular weight scaling as that of randomly branched polymers.12 In contrast, a linear chain in the same array of fixed obstacles follows Gaussian statistics. Note however that in an unconstrained state such as in a theta-solvent, both rings and linears are Gaussian coils12 and indeed show similar Zimm dynamics with relatively small differences.13 The time average structure of a ring chain in an array of fixed obstacles can be thought of as being composed of smaller substructures (branches and leaves, or loops) attached to a larger structure (trunk), and that this hierarchy repeats at all length scales in a self-similar manner.1 For the ideal ring chain, the trunk is on an average N1/2 steps long and there are an equal number N1/2 of loops attached to it. The trunk is in fact a hypothetical structure and the entire mass of the ring is located in the loops. Rigorous prediction of the static structure of rings in melt state has not been possible. However, an elegant scaling argument by Cates and Deutsch14 suggests that large flexible rings in melt state will have self-similar lattice tree structures with a fractal dimension of 5/2. Several Monte Carlo simulations
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agree with this predicted scaling, while others have suggested a more general fractal dimension in the range 2-4.16 The formulation of reptation dynamics for entangled linear polymers was based on the kink diffusion picture in which length defects, or kinks, stored in the chain diffuse along its contour resulting in center of mass motion and orientation relaxation.3 The kink diffusion concept has also been used to model dynamics of topologically entangled rings. In earlier models kinks were assumed to diffuse along the contour of the lattice tree structure, and all kinks were assumed to contribute to the center of mass motion of the ring chain. This resulted in the longest relaxation time of the ring14 to scale as τ ∼ N3 and the diffusion coefficient17 to scale as D ∼ R2/τ ∼ N-5/2. In an insightful paper Obukhov et al.1 proposed that the absence of chain ends would cause local accumulation of kinks in the loops which relax rapidly, while only those kinks that diffuse along the trunk would cause motion of the center of mass. In other words at any given time not all kinks contribute to the center of mass motion of the ring. The ideal ring chain in an array of fixed obstacles was coarse grained into blobs of size a and mass Ne. The authors showed that for any section comprising m blobs of the ring chain the curvilinear diffusion coefficient of the center of mass of the particular section along the trunk of the section scaled as Dc ∼ m-3/2. The longest relaxation time for the ring chain scaled as τ ∼ N5/2 and the self-diffusion coefficient of the ring scaled as D ∼ N-2. We will refer to this scaling model as the Duke-ObukhovRubinstein (DOR) model. Self-similarity in the static structure can be expected to result in dynamic self-similarity, and indeed the simulations of Obukhov et al.1 suggested such a behavior. The authors observed in their simulations that the time taken for any section of the ring chain of m blobs to pass through a pore in the array of fixed obstacles scaled as t ∼ 〈m2〉5/4. This implies that different sections of a topologically constrained ring chain will relax simultaneously in a self-similar manner following the DOR scaling model. On the basis of this hypothesis, Rubinstein and Colby18 proposed a constitutive relation for the linear viscoelastic behavior of a melt of rings. The constitutive law predicted a power law decay for the relaxation modulus G(t) ∼ t-2/5 at early times followed by an exponential decay at longer times. Recent experimental data by Kapnistos et al.19 supported these predictions. In the same paper the authors also suggested that for an ideal ring constrained in an array of fixed obstacles the relaxation modulus would decay as G(t) ∼ t-1/5. Their argument was based on the concept of “gates” which are pairs of obstacles that define the relative positions of loops on a trunk (Figure 1). Indeed it is the rearrangement of the relative positions of the loops that causes the center of mass diffusion of parts of the ring chain. In this paper we outline a formal derivation of the relaxation modulus of a ring chain in an array of fixed obstacles. Our approach is based on the physical ideas proposed in Kapnistos et al.,19 however, the derivation is different in some ways and more rigorous. Following Kapnistos et al., we assume that the relaxation of a nonconcatenated flexible ring chain in an array of fixed obstacles occurs by the simultaneous relaxation of all possible sections of a ring chain in a self-similar manner. Relaxation of any given section is modeled as a one-dimensional diffusion of its center of mass along the contour of the trunk of that section; the curvilinear diffusion coefficient is derived in a very specific and formal way in this paper. Relaxation of the section is deemed complete when the center of mass diffuses over the entire length of the trunk whereby it escapes from the
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solvent in which the ring and the array of obstacles are submerged, and ζeff is an effective friction coefficient on each blob. Equation 2 represents a series of coupled equations which can be decoupled by normal-mode analysis in the spectral dimension (see Appendix A.I.) to give dX0 ) f0 dt
(3)
dXp ) -kpXp + fp dt
(4)
ζ0
ζp
Here, Xp and fp are respectively the position vector and stochastic force on the pth mode (p ) 1, 2, ...) which represents a m/p subsection of the chain. The coefficients in eq (3-4) are defined as
Figure 2. A ring chain in an array of fixed obstacles represented as a fractal blob-spring chain. Each filled circle on the ring indicates a blob of mass Ne and size a. The blobs are connected by massless Hookean springs.
gate where it is attached to the rest of the ring chain. The net relaxation modulus is therefore written as the superposition of relaxations of all sections weighted by the number fraction of each section. There are thus three key steps in the derivation: (a) derivation of the curvilinear diffusion coefficient, (b) derivation of the confinement memory of the trunk, and (c) estimation of the number fraction of any given section of the ring chain. The next three sections provide the necessary derivations of quantities indicated above. We end this contribution with a brief mention about extending the model to the case of melt of rings. 2. Blob-Spring Model We begin by coarse graining a section of the ring chain of mK Kuhn segments into m ) mK/Ne blobs each of size a and mass Ne. Here, a is the average spacing between obstacles, and Ne is the average number of Kuhn segments between consecutive obstacles. At length scales less than a the ring chain does not see topological constraints, and therefore it may be assumed that the internal structure of each blob is described by Gaussian statistics, that is, Ne ) a2/b2, where b is the statistical length of a Kuhn segment. Indeed at a length scale of the order of the size of the ring chain R the presence of obstacles forces the polymer to follow non-Gaussian statistics given by fractal dimensionality of df ) 4 as described earlier. Further, we consider the blobs to be connected by massless Hookean springs of elastic constant k ) 3kBT/a2 to form a fractal blob-spring chain (Figure 2). The fast dynamics of such a chain can be modeled using the theory developed by Muthukumar20 for unentangled fractal polymers. There is however an important difference between the Rouse dynamics of a fractal ring chain and that of a fractal branched polymer as will be mentioned in the following discussion. The generalized Rouse equation for a fractal polymer can be written as ζeff
∂Rn ) k∇s2Rn + fn ∂t
(2)
In eq 2 ∇s2 denotes the Laplacian in spectral dimension, Rn is the position vector of the nth blob, fn is the stochastic force acting on the blob presumably by the molecules of the theta-
0 ζ0 ) mζeff
(5)
p ζp ) 2mζeff
(6)
p kp ) 2mkeff
(7)
Here, ζ0eff is the friction coefficient of a blob in the m section, p is the friction coefficient of a blob in the m/p subsection of ζeff the chain, and kp is the effective entropic spring constant derived from the fractal structure of the ring chain. For a polymer of p can be written following the fractal dimension df ) 4, keff Muthukumar20 argument as p )k keff
( ) ( ) psπ m
2
)k
pπ m
3/2
(8)
where ps is the normal coordinate transform variable in the spectral dimension (see Appendix A.I.). The difference between Rouse dynamics of a fractal branched polymer and a fractal ring chain arises in the way the frictional contribution is calculated. In case of the fractal branched polymer ζ0eff ) ζpeff ) ζblob, where ζblob is the friction coefficient of a single blob. Whereas in the case of the fractal ring chain we estimate ζpeff on the basis of the physical picture of ring dynamics proposed by the DOR model. As discussed earlier, a m/p subsection of the ring chain in a fixed obstacle environment has a fractal structure described by a hypothetical trunk with loops attached to it. The trunk, on an average, is made of (m/p)1/2 hypothetical blobs with each blob having a loop comprising (m/p)1/2 blobs attached to it. The loops relax their internal conformations rapidly relative to the trunk and therefore only one kink per loop per unit time contributes to the center of mass motion of the m/p subsection. This argument implies equivalently that the loops contribute an additional friction which is equal to the number of blobs in the loop. Thus we have: p ζeff )
( mp )
1/2
ζblob
(9)
and for the entire m section we have: 0 ) m1/2ζblob ζeff
(10)
The friction coefficient of a blob ζblob is simply obtained from the fact that a blob is made of Ne unentangled Kuhn segments following Gaussian statistics and undergoing Rouse dynamics so that
Ind. Eng. Chem. Res., Vol. 48, No. 21, 2009
ζblob ) Neζ
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(11)
where ζ is the segmental friction coefficient. The stochastic forces and the friction are related through the fluctuation-dissipation theorem: 〈(f0R(t1) f0β(t2))〉 ) 2ζ0kBTδRβδ(t1 - t2)
(12)
〈(fpR(t1) fqβ(t2))〉 ) 2ζpkBTδpqδRβδ(t1 - t2)
(13)
where δ is the Kronecker delta. We can now determine the relaxation spectrum and the center of mass diffusion coefficient of the m section of the ring chain as follows. The solution of eq 3 is given by X0 )
1 ζ0
∫
t
-∞
dt1 f0(t1)
(14)
The center of mass diffusion coefficient of the m section of the ring is given by 1 Dm ) lim 〈(X0(t) - X0(0))2〉 tf∞ 6t
(15)
The right-hand side of eq 15 can be obtained from eq 14 and eq 12 and we get 〈(X0(t) - X0(0))2〉 )
kBT t ζ0
(16)
Using eq 5 and eq 11 we get kBT 3/2 m ζblob
Dm )
)
a kBT b m 3/2ζ
1 ζp
∫
t
-∞
(
)
(t - t1) fp(t1) τp
ζp m 2 1 b2 ζ 2 1 1 a4 ζ 2 m ) N ) 2 3/2 2 kp p 3π3/2 kBT e p 3π b kBT
()
(18)
(19)
The right side of eq 19 is derived by making use of eqs 6-9. The longest relaxation time for the section is obtained for p ) 1 and given by τ1 )
1 a4 ζ 2 m 3π3/2 b2 kBT
(20)
The shortest relaxation time corresponding to p ) m scales as ∼Ne2, which is equal to the longest Rouse relaxation of an unentangled blob of Ne Kuhn segments and is given by τs )
1 b2ζ 2 1 a4 ζ ) Ne 3π3/2 b2 kBT 3π3/2 kBT
k BT 3/2 N ζblob
τN )
)
a k BT b N 3/2ζ
(22)
K
1 a4 ζ 2 N 3π3/2 b2 kBT
(23)
3. One Dimensional Diffusion
Here, τp is the relaxation time of the m/p section and is given by τp )
DN )
(17)
K
dt1 exp -
relaxation time of the entire ring chain containing N blobs (or equivalently, Nk Kuhn segments) are given by
Thus, the longest relaxation time of the ring chain scales as N2 which is similar to that for a linear Gaussian chain.
It may be noted that the molecular weight scaling of the diffusion coefficient in eq 17 is different from the scaling for a linear Gaussian chain. The dynamics of the m/p sections is governed by eq 4, the solution of which is given as Xp )
Figure 3. A m section of a ring chain is connected to the rest of the ring through a pair of gates (black filled squares), which define its position relative to the other parts of the ring chain. The m section is characterized by its trunk (blue line) and attached loops (green lines). The center of mass of the m section (filled blue circle) undergoes one-dimensional diffusion along the contour of its trunk due to the confining effect of the tube (dotted lines).
(21)
It is straightforward to show that because of dynamic selfsimilarity the center of mass diffusion coefficient and the longest
So far we have considered the influence of the array of fixed obstacles on the structure of the ring chain and thereby on its dynamics. However, we can also expect the topological constraints provided by the fixed obstacles to restrict the center of mass motion of the ring chain. We consider again a section of the ring chain made of m blobs which is connected to the rest of the chain through a pair of gates (Figure 3). At a given time the gates define the relative position of the m section with respect to the rest of the ring. The m section is characterized by its fractal structure, namely, a trunk and attached loops. Because of the topological constraints we may expect that the center of mass of the m section is not free to undergo random fluctuations, but is confined within a tube of length equal to the length of the trunk. The center of mass diffuses within the tube with a diffusion coefficient given by eq 17 and when it travels a mean square distance equal to the square of the length of the tube it “escapes” the confinement and also the pair of gates thereby changing its relative position with respect to the other sections of the ring chain. The diffusion coefficient given by eq 17 thus becomes the curvilinear diffusion coefficient and has the same molecular weight dependence as that of DOR model. While the trunk of the m section in Figure 3 might appear at first instance like the branch of a branched polymer, it is actually different in two aspects. First, it is not covalently pinned to the ring and second, the diffusive motion of its center of mass within the tube does not occur against a free energy barrier since the ring has already paid an entropic penalty when forced to double fold and meander through the array of obstacles. Thus, we may formulate the one-dimensional diffusion of the trunk along its contour akin to the reptation of a linear chain. Let Ψ(ξ,t;s) be
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the probability that the trunk moves a distance ξ while its ends have not reached the segment s of the original trunk contour point. The probability Ψ(ξ,t;s) satisfies the one-dimensional diffusion equation given by 2
∂Ψ ∂Ψ ) Dm 2 ∂t ∂ξ
τe )
3 π9/2 Γ 4 4 3×4
-4 a4
( [ ])
G(τe) )
cb k T m B
When ξ ) s, the segment s is reached by the trunk and Ψ(ξ,t;s) vanishes. Similarly when ξ ) s - Lm, the tube segment s is reached by the other end of the trunk and Ψ(ξ,t;s) vanishes. Here, Lm ) am1/2 is the length of the trunk of the m section. This gives the boundary conditions Ψ(ξ, t;s) ) 0
ξ)s
at
and
ξ ) s - Lm
Ψ(ξ, t;s) )
( )
2 lπs lπ(s - ξ) t sin sin exp -l2 m L L L τd
∑
( ) (
l
)
(27)
where the relaxation time of the m section, which is the time taken by the center of mass to escape the confining tube, is given by Lm2
τm d )
Dmπ2
)
1 a4 ζ 5/2 m π2 b2 kBT
(28)
3 1 a4 ζ ) 1/2 τs π2 b2 kBT π
(29)
where, the last equality is derived from eq 21. For m ) N τNd ) τring )
1 a4 ζ 5/2 N π2 b2 kBT
(30)
where, τdN ) τring is the longest relaxation time for the entire ring and its scaling with molecular weight is in agreement with the DOR model. The fraction of the confinement memory is then given by 1 Lm
∫
Lm
0
ds
τe
(34)
p
p
cb k T m B
∫
∞
0
( )
dp exp -2
cb τe ) kBT τp m
π8 τ
τ1 e
(35)
Using eq 33 and eq 20 in eq 35 and noting that cb ) c/Ne, we obtain G0 ) G(τe) )
8
(Γ[ 43 ]) Nc k T ≈ 0.5 Nc k T 2
√2π
5/2
B
e
B
(36)
e
Dynamic self-similarity, as indicated by the simulations of Obukhov et al.,1 suggests that all sections of the ring chain m ) 1, 2, ..., N relax in a manner described in section III. Unlike covalently connected branched polymers in which different sections of the polymer relax in a hierarchical manner with the relaxation of larger structures occurring only after smaller substructures attached to them have relaxed, the different sections of a ring chain relax simultaneously since they are not pinned. Thus, the overall relaxation modulus of the ring chain is given by the superposition of the relaxation moduli of all sections:
∑ n G (t) m
m
(37)
m
τd1 ) τ0 )
ψm(t) )
G(τe) )
G(t) )
Note that for m ) 1
( )
Moving from discrete to continuous expression:
(26)
The solution to the one-dimensional diffusion eq 24 subject to conditions 25 and 26 is:
(33)
∑ exp -2 τ
with the initial condition (25)
≈ 1.7τs
The relaxation modulus expression of fractal Rouse chain at τe can be shown (see eq 81) to be
(24)
Ψ(ξ, 0;s) ) δ(ξ)
ζ
b2 kBT
∫
s
s-Lm
dξΨ(ξ, t;s) )
∑ l 8π
2 2
l;odd
( )
exp -l2
t τm d (31)
nm in eq 37 is the number fraction of m sections of the ring chain each trapped in a pair of gates. Another interpretation of eq 37 is that the stress in the system comprising a ring in fixed obstacles is supported by the gates. Withdrawal of sections of the ring from their gates causes relaxation of the stress. We now calculate the fraction of such m sections, or equivalently, the fraction of gates that trap m sections of the ring chain. The fraction of such gates is equivalent to the probability of finding bonds in a hyperbranched polymer that divides the polymer into two parts of m and N - m segments (Figure 4). The equivalent hyperbranched polymer can be thought of as formed by the condensation polymerization of Az monomers where z is the coordination number. In this case the
4. Constitutive Relation The contribution of a m section to the relaxation modulus of the ring chain is related to the fraction of confinement memory eq 31 by Gm(t) ) G0ψm(t)
(32)
In eq 32 G0 is the modulus at time t ) τe (τe < τ1), which is the time required for the blob to move a distance of order of its size a, and at which time ψm(τe) ) 1. τe is obtained from the average mean-square displacement of a blob in an m section of order a2 and is given by (see Appendix A.I):
Figure 4. A m section of the ring chain is connected to the rest of the ring chain. The gate (filled squares) is equivalent to a bond of the hyperbranched polymer that divides it into two parts of m and N - m segments.
Ind. Eng. Chem. Res., Vol. 48, No. 21, 2009
probability of finding an unreacted Az monomer as part of an m section (m . 1) is given by18 um(ε) )
1 √2π
zz -- 12 m
-3/2
exp(-ε2m)
(38)
ε in eq 38 is the relative extent of reaction given by ε ) (φ φc)/φc, where φ and φc are, respectively, the conversion and critical conversion for gelation; ε takes values between 0 and -1. The required probability is then given by the product of the probability of finding an unreacted A monomer as part of the m section and the probability of finding an unreacted A monomer as part of the N - m section: um|N-m(ε) ) um(ε) × uN-m(ε) 1 z - 1 -3/2 m (N - m)-3/2 exp(-ε2N) ) 2π z - 2
(39)
1. While large rings are expected to retain a lattice animal configuration in melt, their fractal dimension can be anywhere between 2 < df < 4; the Cates and Deutsch argument gives df ) 5 /2 and the same has been indicated by various simulations as described in section 1. Thus, the equations derived in sections 1-4 will have to be derived for a more general fractal structure. 2. The value of Ne will be governed by the flexibility of the polymer, and need not be the same as that for a linear chain. 3. Constraint release, which was irrelevant for the case of fixed obstacles, will become an important relaxation mechanism since faster modes of neighboring chains will dynamically relax the topological constraints of slower modes of any given chain. There are various theoretical treatments to account for constraint release in the case of entangled linear chains.21-24 Following the tube dilation approach of Marrucci21 the constitutive relation for a melt of ring may be written as
for m . 1 and N - m . 1. Thus, the fraction of gates associated with m sections of a ring chain is given by normalizing the probability: nm )
um|N-m
∑u
1 ) m-3/2 4
m|N-m
(40)
The constitutive equation for rings in an array of fixed obstacles is given by the combination of eqs. 32-37 and eq 40 and may be written as G(t) )
1 4
∑m
-3/2
Gm(t) )
m
16
√2π
∫
1
dmm
( )
( [ ])
()
τ0 cbkBT t
1/5
∫
t/τ0
t/τring
dxx
(1/5)-1
(41)
exp(-x)
The integral is an incomplete gamma function -Γ[1/5, x] and can be approximated by 0.1 exp(-t/τring) for long times t/τring > 10 which shows that the relaxation modulus is dictated by exponential decay of the stress memory of the largest length scale (primary trunk) of the ring chain. The relaxation modulus expression can be approximated as
() ( ) τ0 t
1/5
exp -
t
τring
Lm2 Dmπ2
)
1 a4 ζ 2ν+2 m π2 b2 kBT
(46)
In eq 46 ν ) 1/df. Substituting in eq 44 and solving yields
(42)
G(t) ≈ 0.004cbkBT
(44)
1 1 1 ( ) (Γ[1 - ν])1/ 2ν cbkBT 1/(2ν) (ν+1)/(2ν) ν π √ 2 2 (45)
τm d )
In writing eq 41 we have assumed that since the right-hand side of eq 31 is a rapidly converging series, only the first dominant term may be considered. Also, the summation over m has been replaced by an integral over m and can be justified for large N using the Taylor-Maclaurin approximation. It is worth noting that although the summation over m is assumed to be from m ) 1 to N, the expression for nm given in eq 40 cannot really be used over the entire range m ) 1, N. Considering τdm ) τ0m5/2 and substituting x ) (t/τ0)m-5/2, eq 41 can be written as 2
2
and τdm given by
b B
t exp - m τd
)
nmψm(t)
In eq 44 the expression for nm is given by eq 40 and the expression for Gm(t) is given by eqs 31-36 with
(Γ[ 43 ]) c k T × -3/2
(∑ m
2
9/2
N
32 1 3 G(t) ) Γ 9/2 4 5√2 π
G(t) ) G0
G(τe) ) G0 )
m
9519
(43)
This expression is in agreement with that proposed by Kapnistos et al.19 except for the prefactor. 5. Comments on Melt of Rings The following things are expected to change in going from rings in an array of fixed obstacles to a melt of rings:
G(t) )
() τ0 t
1/(2ν+2)
[
∫
(t/τ0)
(t/τring)
x(1/2(2ν+2))-1 exp(-x)]2 (47)
where is given by 4 1 4 2ν + 2 π
(
) G0
)
2
(48)
The integral in the eq 47 is an incomplete gamma function -Γ[1/(4ν + 4), x] and can be approximated by an exponential decay 0.1 exp(-x) for large x. The long time response can thus be shown to be exponential decay so that an approximate constitutive equation can be written in an asymptotic form as G(t) ≈
()
τ0 10 t
( )
(49)
() ( )
(50)
1/(2ν+2)
exp -
t τring
For df ) 4 we obtain G(t) ≈ 3.2 × 10-5cbkBT
τ0 t
2/5
exp -
t τring
This equation is in agreement with that proposed by Kapnistos et al.19 except for the prefactor. The gain and loss modulus can be obtained from expression 49 through the transform G'(ω) ) ω
∫
∞
0
dt G(t) sin(ωt)
(51)
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Ind. Eng. Chem. Res., Vol. 48, No. 21, 2009
G''(ω) ) ω
∫
∞
0
dtG(t) cos(ωt)
(52)
The exponential decay in the relaxation modulus is expected to yield G′(ω) ≈ ω2 and G′′(ω) ≈ ω at low frequencies ω , 1/τring. At small times, that is, frequencies above the crossover frequency ωc ) 1/τring the power-law decay is expected to be stronger than the exponential decay and hence we consider only the transform of the power-law decay given by G'(ω) ≈ G''(ω) ≈ (ωτ0)1/(2ν+2)
(53)
The eqs 53 suggest that for higher than the crossover frequency the gain and loss moduli follow a power law frequency dependence. For df ) 4 we have G′,G′′ ≈ ω2/5. Thus, the high frequency behavior of rings is unlike linear entangled chains which show a constant gain modulus (the so-called plateau modulus) and a decreasing loss modulus after the crossover frequency. 6. Conclusions On the basis of the physical picture of the DOR model1 for dynamics of rings in an array of fixed obstacles we have developed a rigorous formalism to derive a constitutive relation for this system and shown that it can be extended to the case of melt of rings. Our formalism is based on Rouse dynamics of a fractal blob-spring chain and curvilinear diffusion of the center of mass over the longest length scale of the fractal structure. The constitutive equation is given as a mixing rule in which contributions to stress relaxation from different parts of the chain relaxing in a self-similar manner are superposed. Except for the prefactors the constitutive relations are in agreement with those proposed by Rubinstein and Colby,18 and Kapnistos et al.19 based on scaling concepts. The relaxation modulus shows a power law decay at short times and an exponential decay at long times. This corresponds to a power-law frequency dependence of the storage and loss moduli after the cross over point in dynamic oscillatory tests. Although we report concurrence of our formulation with the scaling ideas of Kapnistos et al.19 there exist certain issues that need careful consideration when comparing model prediction with experimental data of rings: (i) There is an absence of a consensus on the effect of synthesis route of rings on their rheological behavior. McKenna et al.25 had earlier pointed out that the synthesis under poor solvent conditions might lead to knotted structure which might lead to change in the rheological response, and it is important to note that the Kapnistos et al.19 synthesis is under such conditions. (ii) Earlier work of McKenna et al.26 on ring polystyrene indicated that entanglement molecular weight of the ring chain was approximately twice that of linears. Roovers27 estimated that the entanglement molecular weight of ring polybutadiene (synthesized under poor solvent conditions) was five times that of linear polybutadiene. This inference was based on the estimated value of plateau modulus of ring PBD melt. In fact Roovers oscillatory shear data do not show a clear indication of plateau in G′, which was attributed to an overlap of the rubbery and transition regimes. Models based on the hypothesis of dynamic self-similarity suggest that one should not expect to see a plateau in G′; indeed such models predict power law behavior of G′ after terminal regime. (iii) In the present model the parameter a refers to the average spacing between obstacles. For linear polymers, a is related to
the entanglement molecular weight. It is not clear as to what should be the value of a for a ring melt. (iv) In the case of extension of the model developed for the fixed obstacle environment to the case of melts we have considered that the Cayley tree structure remains intact in melt based on the results of simulations of Mu¨ller et al.16 The change in structure of the ring chain as the obstacle environment is changed from that of a fixed obstacle to an obstacle with a specific lifetime is still an open problem and is likely to yield a better understanding of the retention of the Cayley tree structure of a ring chain in a melt of rings. (v) The zero shear viscosity of the melt of rings according to the current model shows a molecular weight scaling of M1.5 while McKenna et al.26 reported a M3.8 scaling for rings. There exists no other experimental data on dependence of zero shear viscosity on molecular weight for rings and this discrepancy remains unexplained. (vi) It is now well accepted that the presence of even small amounts of linears in a ring melt has profound effect on the rheology. Therefore purification of rings is deemed critical for comparison with model predictions. Acknowledgment Balaji Iyer thanks CSIR for the Senior Research Fellowship. Appendix A.I. Mean Square Displacement of the Blob The normal coordinates are given by Rn ) X0 + 2
∑X
p
( ) psπn m
(54)
( )
(55)
cos
ps
fn )
f0 1 + m m
∑f
cos
p
ps
( )
(56)
( )
(57)
1 m
∫
dn cos
fp ) 2
∫
dn cos
Xp )
m
0
m
0
psπn m
psπn Rn m
psπn f m n
The mean-square displacement of a blob in terms of the normal coordinates (54-57) is given by 〈(Rn(t) - Rn(0))2〉 ) 〈(X0(t) - X0(0))2〉 + psπn 2 (Xp(t) - Xp(0)) cos m p
〈
〈[
∑ s
4 (X0(t) - X0(0))
( )] 〉 ) ( )〉 2
∑ (X (t) - X (0) cos p
ps
p
-
psπn m
(58)
The underlined term in expression 58 corresponds to the cross correlations between different modes in the fractal blob chain and vanish due to the linear independence of the normal coordinates. Consequently, the mean square displacement of the fractal Rouse blob is composed of two contributions, viz., the center of mass contribution and the internal modes contribution. Substituting eq 17 for the center of mass contribution in eq 58 we obtain
Ind. Eng. Chem. Res., Vol. 48, No. 21, 2009
〈(Rn(t) - Rn(0)) 〉 ) 6Dmt + 2
4
∑ p
∑ p1 [1 - exp(-p τt )] ) ∫
∞
2
f(p)πn 〈(Xp(t) - Xp(0))2〉cos2 m
(
3/2
)
(59)
0
1
p
dp
9521
( )]
[
t 1 1 - exp -p2 τ1 p3/2 (66)
The average mean square displacement can be shown to be where we have transformed variables from ps to p as per expression 8 so that f(p) ) ps ) p3/4(m/π)1/4 is a function of p which always takes integer values. We expect that the time taken by a blob to undergo a mean square displacement of order a2, is much smaller than the longest fractal Rouse relaxation time τ1. For t , τ1 the contribution of the internal modes for the displacement of the blob dominates over the center of mass contribution. The underlined term of expression 59 corresponds to the internal modes contribution and can be expanded to obtain 4
∑ (X (t) - X (0))
) 4 ∑ (〈X (t) 〉 + ( f(p)πn m ) f(p)πn 〈X (0) 〉-2〈X (t) · X (0)〉) cos ( m ) 2
p
p
p
cos2
2
p
2
p
p
(60)
p
From eq 18 we have 1 ζp2
∫
t
dt -∞ 1
∫
(
t
dt exp -∞ 2
(
)
(t - t1) τp
τe )
3 π9/2 Γ 4 4 3×4
)
ζ
b2 kBT
(68)
A similar derivation of mean-square displacement can be done for an arbitrary fractal dimension df ) 1/ν and it can be shown that a4 ζ 1 τe ) π(2ν+1)(1-ν)/νν1/ν(Γ[1 - ν])-1/ν 2 3 b kBT
cb m
σRβ )
(t - t2) 〈fpR(t1) fqβ(t2)〉 exp τq
∫
(
(61)
(69)
) (
a2 〈(Rn(t) - Rn(0)) 〉 ) 4m π3/2 2
1/2
∑ p
∫
)
(62)
( )]
[
1 t 1 - exp -p2 3/2 τ p 1 2 f(p)πn cos m
(
)
× (63)
m
∑
1 ((Rn(t) - Rn(0))2) m n)1
(
[
1 4 a2 t ) 1/2 3/2 1 - exp -p2 3/2 τ m π 1 p p 2 f(p)πn cos m
(
)
)] ∑ m
×
n)1
(64)
Moving from a discrete to continuous framework in n and p the summations are converted to integrals: ) ∫ ∑ cos ( f(p)πn m )
m
0
nβ
nR
n
〉
(70)
m
U)
To obtain the average mean-square displacement of any segment of the chain we sum the displacement of all the blobs and divide it by the number of blobs:
∑
∑ 〈 ∂R∂U R
U is the harmonic spring potential given by
Expression 62 yields ((Xp(t))2) ) 3kBT/kp for all t and 〈Xp(t) · Xp(0)〉 ) (3kBT/kp) exp(-t/τp). Substituting in eq 60 we obtain for t , τ1:
n)1
-4 a4
( [ ])
The microscopic expression for the stress tensor is given by2
×
kBT t t δRβδpq -∞ dt1 -∞ dt2 × ζp (t - t1) (t - t2) exp δ(t1 - t2) exp τp τq
2
(67)
A.II. Relaxation Modulus of the Fractal Rouse Chain
(XpR(t) Xqβ(t)) ) 2
m
1/4
From eq 67 and 20 the time taken to undergo an average mean square displacement of a2 is given by
Using expression 13 in eq 61 we obtain
average MSD )
[ ]( )
a2 3 t Γ π3/2 4 τ1
P
2
〈XpR(t) Xqβ(t)〉 )
average MSD ) 4m1/2
( f(p)πn m )
dn cos2
(65)
∑
k (R - Rn-1)2 2 n)2 n
(71)
in the spectral dimension. Combining eqs 70 and 71 the microscopic expression for the stress tensor for a fractal polymer chain can be written as σRβ )
cb k m
∫
N
0
dn〈∇sRnR∇sRnβ〉
(72)
where ∇s denotes gradient in the spectral dimension of the fractal network and cb is the number density of the blobs that make up the ring polymer and is given by cb ) c/Ne. Using the normal coordinate transformation 54-57, the stress tensor expression is given by σRβ )
cb m
∑k
ps〈XpRXpβ〉
(73)
ps
where, kps ) 2mk(psπ/m)2. Equation 73 is appropriately modified in the conjugate space, and the stress tensor expression is given by σRβ )
cb m
∑ k 〈X p
pR
Xpβ〉
(74)
p
We now impose a homogeneous deformation gradient υ j (r,t) ) κc(t) · r. During such a deformation the Langevin equation for the pth normal coordinate, Xp, becomes kp ∂Xp 1 ) - Xp + fp + κc(t) · Xp ∂t ζp ζp
(75)
From the Langevin eq 75 we obtain the equation for the correlation (XpR Xpβ) as
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Ind. Eng. Chem. Res., Vol. 48, No. 21, 2009
kp kBT ∂ 〈X X 〉 ) -2 〈XpR Xpβ〉 + 4 δ + κRµ〈Xpµ Xpβ〉 + ∂t pR pβ ζp ζp Rβ κβµ〈XpR Xpµ〉 (76) Equation 76 can be solved to obtain 〈XpR Xpβ〉 for any given homogeneous deformation gradient. For homogeneous shear where κc(t) is given by
[ ] 0 κ(t) 0 0 0 0 0 0 0
we have the equation for the xy component of the correlation given by kp ∂ 〈X X 〉 ) -2 〈Xpx Xpy〉 + κ(t)〈Xpy2〉 ∂t px py ζp
(77)
Considering the system to be close to equilibrium we have 〈Xpy2〉 ) kBT/kp, for which the solution to eq 77 is now obtained as 〈Xpx Xpy〉 )
kBT kp
∫
t
-∞
(
dt1 exp -2
)
(78)
)
(79)
(t - t1) κ(t1) τp
Substituting eq 78 in eq 74 we obtain σxy )
cb k T m B
∑∫
t
-∞
(
dt1 exp -2
p
(t - t1) κ(t1) τp
The phenomenological expression for stress tensor in terms of the relaxation modulus is given by σxy(t) )
∫
t
-∞
dt1G(t - t1) κ(t1)
(80)
Comparing eq 79 with eq 80 we obtain the contribution of the m section of the ring chain to the Blob-Spring relaxation modulus as G(t) )
cb k T m B
∑ exp(-2 τt ) p
(81)
p
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(6) McLeish, T. C. B. Present Puzzles of Entangled Polymeric Systems. Rheol. ReV. 2003, 197–232. (7) Pearson, D. S.; Helfland, E. Viscoelastic Properties of Star-Shaped Polymers. Macromolecules 1984, 17, 888–895. (8) McLeish, T. C. B.; Larson, R. G. Molecular Constitutive Equations for a Class of Branched Polymers: The Pom-Pom Polymer. J. Rheol. 1998, 42, 81–110. (9) McLeish, T. C. B.; Milner, S. T. Entangled Dynamics and Melt Flow of Branched Polymers. AdV. Polym. Sci. 1999, 143, 195–256. (10) Khokhlov, A. R.; Nechaev, S. K. Polymer Chain in an Array of Obstacles. Phys. Lett. 1985, 112A, 156–160. (11) Nechaev, S. Statistics of Knots and Entangled Random Walks. 2005, arXiv:cond-mat/9812205, arXiv.org e-Print archive. http://arxiv.org/abs/ cond-mat/9812205. (12) Zimm, B. H.; Stockmayer, W. The Dimension of Chain Molecules Containing Branches and Rings. J. Chem. Phys. 1949, 17, 1301–1314. ¨ ttinger, H. C. Bead-Spring Rings with Hydrodynamic (13) Liu, T. W.; O Interactions. J. Chem. Phys. 1987, 87, 3131–3136. (14) Cates, M. E.; Deutsch, J. M. Conjectures on the Statistics of Ring Polymers. J. Phys. (Paris) 1986, 47, 2121–2128. (15) Mu¨ller, M.; Wittmer, J. P.; Cates, M. E. Topological Effects in Ring Polymers: A Computer Simulation Study. Phys. ReV. E. 1996, 53, 5063–5074. (16) Mu¨ller, M.; Wittmer, J. P.; Cates, M. E. Topological Effects in Ring Polymers: Influence of Persistence Length. Phys. ReV. E. 2000, 61, 4078–4089. (17) Nechaev, S. K.; Semenov, A. N.; Koleva, M. K. Dynamics of a Polymer Chain in an Array of Obstacles. Physica 1987, 140A, 506–520. (18) Rubinstein, M.; Colby, R. H. Polymer Physics; Oxford University Press: New York, 2003. (19) Kapnistos, M.; Lang, M.; Vlassopoulous, D.; Pyckour-Hintzen, W.; Hitcher, D.; Cho, D.; Rubinstein, M. Unexpected Power-Law Stress Relaxation of Entangled Ring Polymers. Nat. Mater. 2008, 7, 997–1002. (20) Muthukumar, M. Dynamics of Polymeric Fractals. J. Chem. Phys. 1985, 83, 3161–3168. (21) Marrucci, G. Relaxation by Reptation and Tube Enlargement: A Model for Polydisperse Polymers. J. Poly. Sci. Polym. Phys. 1985, 23, 159– 177. (22) Tsenoglou, C. Molecular Weight Polydispersity Effects on the Viscoelasticity of Entangled Linear Polymers. Macromolecules 1991, 24, 1762–1767. (23) des Cloizeaux, J. Relaxation of Entangled Polymer Melts. Macromolecules 1990, 23, 3992–4006. (24) des Cloizeaux, J. Relaxation and Viscosity Anomaly of Melts Made of Long Entangled Polymers. Time-Dependent Reptation. Macromolecules 1990, 23, 4678–4687. (25) McKenna, G. B.; Hadziioannou, G.; Lutz, P.; Hild, G.; Strazielle, C.; Straupe, C.; Rempp, P.; Kovacs, A. J. Dilute Solution Characteristics of Cyclic Polystyrene Molecules and Their Zeroshear Viscosity in the Melt. Macromolecules 1987, 20, 498–512. (26) McKenna, G. B.; Hostetter, B. J.; Hadjichristidia, N.; Fetters, L. J.; Plazek, D. J. Dilute Solution Characteristics of Cyclic Polystyrene Molecules and Their Zero-Shear Viscosity in the Melt. Macromolecules 1989, 22, 1834–1852. (27) Roovers, J. Viscoelastic Properties of Polybutadiene Rings. Macromolecules 1988, 21, 1517–1521.
ReceiVed for reView April 2, 2009 ReVised manuscript receiVed July 10, 2009 Accepted July 15, 2009 IE900535V
