ma034808q_swelling.pdf

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Swelling Dynamics of Collapsed Polymers

Nam-Kyung Lee,*,,, Cameron F. Abrams,| A. J ohner,, and S. Obukhov,,

Laborat oi re E urope en Associe , I nst i t ut Charl es S adron, 67083 S t r asbourg Cedex, F r ance,Laborat oi re E urope an Associe , M a x -P l a n ck - I n s t i t u t f u r P ol ymerf orschung, A ckerm annw eg 10,55128 M ain z, Germ any, Departm ent of Physics, Sejong U ni versit y, Seoul 143-747, Sout h K orea,Depart m ent of Chemi cal E n gi neeri ng, Dr exel Un i versi t y, P hi l adel phi a, P ennsyl vani a 19104, andDepart m ent of P hysi cs, U ni versi t y of F l or i da, Gai nesvi l l e, F l or i da 32601

Received J un e 13, 2003; Revised M anu scri pt Received N ovember 10, 2003

ABS TRACT: We study different regimes in the dy nam ics of sw elling of a long polymer chain from acompact globule. Depending on the age or degree of self-entanglement inside of the globule, we considerthe limiting cases of freshly formed globule and old globule. Considering th e density of polymer in aglobule, which influences the mobility of solvent inside of the globule, we also distinguish wet and dryinitial configurations. For each of these regimes, we discuss the leading dissipation mechanism governingthe dyna mics of swelling. Our theoretical predictions are nicely corroborated by la rge-scale moleculardynamics simulations.

I. Introduction

H is t or ica l ly , t h e p roce ss of p oly mer c ol la p s e h asattracted more attention than the process of swelling. 1-11

A ma jor motiva tion of the colla pse studies is the a na logywit h p rot e in foldin g . O n t h e t h e ore t ica l s ide man ymodels were proposed based on de Gennes seminalexpanding sausage model.1 Nowadays it is possible toset up la rge-scale simulat ions w ith explicit solventpart icles that can discriminate between theories. Fore xamp le, Abrams e t a l .12 proposed tha t collapse isdominat ed by a st reamlining process where larger a ndlarger chain sect ions with a pearl-necklace structurealign, while the overall end-to-end distance remainsconstant , until , in a last step, the result ing dumbbellcollapses. This theory is supported by large-scale simu-lat ions.12

Relatively fewer theoretical efforts have been invested

in the reverse problem of homopolymer globule swell-ing.7,8,13,14 Sw elling of homopolymer globules bears somesimilarity with protein denaturation. This analogy is inma ny cases less deep than tha t betw een homopolymercollapse and protein folding, as early stages of swellingar e more sensitive to the details of the intera ctions a ndhence less universal. Most available theories describethe sw elling of a h omopolymer globule in term s similarto th ose depicting collapse.7,8,13 This implies at lea st tw oma jor simplifications: (1) The overall sha pe of expand-ing polymer is assumed to remain homogeneous uponswelling, and (2) a phantom chain model is used. Wewill make clear that both these assumptions may fail .The importance of topological constraints was recog-n iz ed b y R a b in , G r os be rg , a n d Ta n a k a , w h o f ir s tproposed a description of a knotted swollen globule. 14

We have suggested very recently that there exists anarr ested, intermediate sta te during swelling of a highlyentangled globule. This conjecture was confirmed bycomputer simulat ion. In a short note, we focused onthe dist inct monomer -monomer and entanglement -

entanglement correlat ions tha t develop in t he a rresteds t a t e .15 Eventua lly, the globule swells back to t he coilstate. This is corroborated by light scat tering experi-

ments (both sta tic and dyna mic) performed on very longp oly -NI PAM ch ain s in t h e g rou p of Wu .16,17 Theseexperiments are not time-resolved but allow measure-ment of the compactness of the polymer, through ther a t i o o f t h e h y d r od y n a m ic r a d i us t o t h e r a d i us ofg y ra t ion , in t h e in it ia l a n d f in a l s t a t e s . I n t h is p a p e r ,we discuss in s ome detail t he intr icacies of homopolymerglobule swelling.

Only very naively might the swelling from a compactglobule to an extended conformation be thought as thet ime re ve rsa l of col lap s e. Th e se t wo p roce ss es a redifferent both in their geometrical appearance and intheir scaling regimes. For example, during collapse,parts of a polymer a re brought together by contra ct ingt h e p ol ym er ch a i n, a n d a s a r es ul t t h e t r a n si en tcon f ig u rat ion s are ax ia l ly s y mmet r ic. O n t h e ot h e rhand, during swelling, the dynamics is often controlledby the rate of penetrat ion of solvent into the globule,and the result ing configurat ions are spherically sym-metric. The initial statistical fluctuations, like fluctua-tions of the end-to-end vector, do not play any role int h e s we l lin g p roce s s w h e reas t h e y do in col la p s e.12

Another difference between th e t wo processes is t hat ,for collapse, the initial coil configuration is rather wellchara cterized and universal (though strongly f luctua t-i ng ), w h i le t h e g lob ul a r s t a t e , w h i ch i s t h e i n it i a lconfigurat ion for sw elling, requires a ddit ional chara c-terizat ion, which w e now discuss.

Characterization of a Globule. Depending upon

the method of preparat ion of the init ial configurat ion(i.e., the history of the globule) and upon its compact-ness, we consider two ma jor par am eters related to thestructure of a globule and t o its density: (i) i ts a ge18

an d (ii) its solvent content. These span a phase dia gra mdepicted in Figure 1. We consider the a ge to ra nge fromfresh for newly collapsed globules to old for fullyequilibrated globules. S welling of a fresh globule isquite different from swelling of an old globule. Rightafter a polymer chain is collapsed into the sphericalcomp act g lobu le , t h e s eg me n t of a ch ain bet we e nmonomers i, k i s c o m p a c t l y f o l d e d a n d i s r e a d y t o

I ns t i t u t C ha rles S a d ron. Max-Pla nck-Inst itut fu r P olymerforschung. Sejong University.| Drexel University. University of Florida.

651M acromolecul es2004, 37 , 651-661

10.1021/ma 034808q CC C: $27.50 2004 America n Chemica l SocietyP ubl ish ed on Web 12/24/2003


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expa nd. The density of this globule is well equilibrat ed,but th ere are no new topological constr a ints preventingthe globule from expan ding. We expect t ha t t he a veragegeometric distance ri k between tw o monomers i, k in

a fr esh globule scales as ri k |i - k|1/3, for some ran geof values of |i - k|, where |i - k| is the distance betweenthe m onomers a long the contour of the chain.20 I n a nold globule, the complete equilibration of the configu-rat ion results in a Ga ussian ri k |i - k|1/2 dependenceof distance r i k for the close monomers. The price of thisequilibrat ion is mult iple topological constraints, i .e . ,entanglements. I f equilibrated, the whole globule canbe t h o u gh t as a s in gle la rg e k n o t . Th e s w e ll in g ofunentangled (fresh) and entangled (old) globules shouldthus be quite different (compar e Figure 2a an d F igure3b). Local correlations in both freshly prepared globulesan d old-enta ngled globules estimat ed from direct simu-lations will be presented in section 4.

Th e s econ d p arame t e r , me as u rin g t h e de ns i t y ofsolvent inside the globule, a llows us to distinguish drya nd w et globules. In a dry globule, the monomers of apolymer a re packed so tight ly th at molecules of solventc an n o t p e n e t ra t e amo n g t h e m. I n a we t g lo bu le , t h espace between monomers is large enough for solventmolecules to penetrate. Therefore, in marked contrastto the case of a wet globule, swelling of a dry globulemu s t be do min at e d by t h e mo n o me r s o lv at io n a t t h einterface.

In t he next sections w e consider dyn a mics of sw ellingfor each region, I , I I , I I I and IV, of the phase diagram(Figure 1). In each case we consider the leading dis-sipat ion mecha nism or other bot t leneck constra intswhich control the dynamics. Then we discuss the resultsof comp u t er s imula t ion s in l ig h t of ou r t h e ore t ica lpredictions.

II . Theory of Swelling of a Fresh(Entanglement-free) Globule

A. Very Fresh and Not so FreshWet Globules.We will represent t he str ucture of the expandin g globuleas a c lo s e ly p ac k e d ag g re g at e o f n blobs of size a.

I n s id e e a c h b lob t h e c h a i n h a s t h e s t r u ct u r e o f aG a u s s i a n r a n d o m w a l k c o n t a i n i n g (a/b)2 monomers,wherebis t he size of a monomer. The size a is controlledby the number density of the globule: a ) 1/(Fb2). Int h e p roce s s o f s we ll in g of a g lobu le , t h e s ize of t h eG au s s ian blo bs a(t) is in c re as e d wi t h t ime , u n t i l i tr ea c he s t h e s iz e o f t h e w h ol e c h a in . L a t e r w e w i llconsider the special case of a dry globule, when a(0) b. The entropic price of close packing and folding isabout kB T per blob.

The number of blobs

decreases with t ime, and the packing energy n kB T isdissipated through viscous friction. We find the rate ofswelling from the condition tha t t he decrease in pa ckingenergy is ba lan ced by the viscous dissipat ion. The la tteris dominated by the flow of solvent inside the globule.The st ructure of th e globule imposes the typical h ydro-dynamic scale a:

an d u s in g V ) R/t, wh e re R i s t h e s ize of a s we l lin gpolymer

we obtain

where initial conditions correspond to t0 ) a05/3N2/3 )F0

-5/3N2/3. We use dimensionless units ) 1, kB T )1 ,a nd b ) 1. To come back to actual units , we shouldreplace afa/b, RfR/b, a n d tft/0, wh e re 0 ) b3/kB T. The time needed for a swelling chain to reach the

Figure 1. Diagram of states for a collapsed globule.

Figure 2. S welling of a dry -fresh globule: monomer solvat ionand peeling of loops from the core.

Figure3. Swelling of an entangled globule. a fc depicts th eprocess of un iform expan sion of an entan gled wet globule; afc depicts self-disentanglement.

n(t) ) N

(a/b)2 (1)

kB Tdnd t

) (Va)2

n a3 (2)

R ) (N b2/a2)1/3a (3)

R(t) ) t1/5N1/5

a(t) ) t3/5N-2/5 (4)

652 Lee et al . M acromolecul es, Vol. 37, N o. 2, 2004


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G au s s ian s ize R N1/2 is about the Zimm relaxat iontime:

The time dependence R(t) t1/5 wa s a ls o fou n d in re f19. H o we v e r t h e p h y s ic s t h e re is q u i t e d i f fe re n t ashydrodynamics is negelected. H ydrodynamics changesthe frict ion law and sets a relevant pore sizeasma ller

tha n the ra dius. These two effects ha ppen to compensa teat the scaling level.In t his estimat e, we used only the most robust featur e

of the structure of the expanding globule: the density-controlled correlat ion distance a. Th is i s t h e on lyhydrodynamic scale for solvent penetrat ing inside theexpanding globule. Nevertheless, we should note heret h at t h e s t ru c t u re o f t h e e x p an din g g lo bu le is mo recomplicated and is characterized by one more lengthscale a.

The ideal freshly folded globule may exist only as atheoretical model. Any globule needs some time to beprepar ed. During tha t t ime the folds inside the globulemig h t t ry t o re arra n g e wi t h o u t c ro s s in g e ac h o t h e r .These rearrangements result in local Gaussian correla-

tionsrik |i - k|1/2 a long the cha in beyond the densitycontrolled scale a0 u p t o a0 ) a0Ne1/2. The maximumchain length at which the local Gaussian correlat ionsmight be observed is of the order of enta nglement lengt h|i - k| Ne(a0)2. The t ime needed for this rearra nge-ment is about Rouse-Zimm relaxa tion timet Ne2(a0)3.On shorter t ime scales, the rea rra ngement is possibleonly up to the scale

This type of local rearrangement is fast compared toswelling process considered a bove.20 Simple est imatesshow that , during the expansion process, the relat ionbetween a and a, even if it begins w ith a0 ) a0 for freshglobule, saturates at

and holds until a reaches a ) N1/2/Ne3/4 a nd a ) N1/2/Ne

1/4. At that moment the structure of the expandingc h ain c an be t h o u g h t o f as a G a u s s ian wa lk made o fN/Ne1/2 monomers folded Ne1/2 times. The fina l processof unfolding these Ne1/2 subcha ins into an equilibratedchain is st ill described by scaling formulas of eq 4. Thereason the structural scale a does not affect dynamicsof s we l lin g is s imple. Th e re la t iv e v eloci t ies of re -arr an gements of monomers w ith respect to each otherneeded for formation of scale a a r e s m a l l e r t h a n t h evelocity of solvent penetrating the globule, which pro-duces the dyn a mic bott leneck for the expan sion process.

B. Dry-Fresh globule. The above model of expansionin viscous media fails w hen th e globule is so dense tha tthe monomers of solvent cannot penetrat e. Instea d theycan only solvate monomers of the chain one by one,peeling t hem from t he core. In the picture il lustra t ingthis process, Figur e 2a, only one end of a polymer chainis shown, but configura tions with ma ny expanding loopsan d ma ny w etting centers ar e also possible, e.g., Figure2b. We can est imate t he wett ing ra te for one center a sdNw et /d t 0-1, wh e re 0-1 is the molecular diffusion-controlled wetting rate for a single wetting center (i.e.,

one stra nd peeling off from t he globule) and Nw et is thenumber of wet monomers. In dimensionless units d Nw et /d t 1.

A single w ett ing center is expected w hen th e forma-t ion of a wett ing center is a n a ct ivat ed process with anact ivat ion t ime ta c l a rg e r t h an t h e we t t in g t ime . T h ewett ing center shown on Figure 2b is the source of ach a i n i n a con f ig u ra t i on cl os e t o G a u s si a n . I n t h efollowing, it is assumed that the wet loop is larger thanthe dense core ra dius.

To obta in q ualita t ive scaling relat ions for chain sizeR(t) and blob sizea(t), we simply r eplace the number ofmonomers N i n eq s 1 a n d 3 b y t h e n u mb er of w etmonomers, Nw et t(dNw et /d t) t, an d as s u me 0 ) 1. Asingle wetting center is thus expected for an activationtime ta c l a rg er t h a n N; hence, this is not for very largeglobules. After simple estimates, we obtain (in dimen-sionless units)

D u rin g t h e we t t in g p ro c e s s , t h e lo o p is fo u n d t o bes l ig h t ly c o mp re s s e d as i t s radiu s is s mal le r t h an t h eG au s s ian s ize t1/2. A concentration profile is antici-

pated for the wet monomers stored in the loops, and aa s given by eq 7 corresponds t o the typical dista ncer R from the center. The power-law relations between Ra nd t, n, a n d t s u g g es t t h a t t h e s t a t ion ary p a r t o f t h eprofile (r < t2/5) is a ls o a p o wer law F(r)b3 ) (r/b). Atthe outer edge we assume a front , steep enough to beunessential in the integral of the concentration profile.Conserva tion of polymer t hen determ ines the exponentof the concentra tion profile ) -1/2, which leads to theprofile

No t e t h at re lax e d G au s s ian lo o p s ( is o la t e d o r in t h e

Da oud a nd Cotton sta r-regime with three body interac-tions) build a profile decaying as r-1. We est imate t het ime t n e e de d fo r t h e c o mp le t e we t t in g o f t h e dryglobule to be t N. At this t ime, a ll monomers of theinitially dry globule become wet, the globule has the sizeR ) N2/5, with blobs of size a ) N1/5. Further swellingof this wet globule proceeds as described by eqs 4 forwet globules.

I n t h e op pos i t e l imit , wh e re ma n y we t t in g ce n t ersform without act ivat ion, peeling proceeds all over th esurface of the globule. The peeled monomers are poorlywet and form a corona with a pore size of a few solventdiam eters only. The penetra tion of solvent thr ough thiscorona is ra te l imit ing an d the w hole globule becomespenetrable at a time of order N2/3. Though loop conden-

sat ion ta kes place, that results in only a few la rge loops,so this process is irrelevan t for th e kinetics. The globulesubsequently expands over a t ime N3/2 as described byeq 4 for initially wet globules.

III. Theory of Swelling of an Old (Entangled)Globule

At thermodynamic equilibrium, the compacted poly-mer chain is strongly entangled, and it can be thoughta s a single larg e knot. The swelling process is th ereforesubject to mult iple topological constra ints. We willestima te below t he swelling rat es for tw o most probablemechanism s of swelling: (1) uniform expansion of a nentangled wet globule, depicted in Figure 3a fFigure

t N3/2 (5)

a0 ) (t/a03)1/4a0

a ) aNe1/2 (6)

R(t) t2/5, a(t) t1/5 (7)

F(r)b3 ) (r/b)-1/2 (8)

M acromolecul es, Vol. 37, N o. 2, 2004 Sw elling Dyna mics of Collapsed P olymers 653


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3c, and (2) reptat ion of the chain out of the knottedconfigurat ion, as shown in Figure 3a fFigure 3c.

The initial density of the equilibrium globule deter-mines the correlation length r0* c-1. The presence ofentanglements brings the distance between entangle-ments along the chain se Ner* as a n addit ional lengthp a r a m et e r . Th e p re se nce of t w o s ca l es m a k es t h edyna mics more complicated: The process of swelling is

a two-stage process. In the init ial ( transit ional) stageof swelling, the chain expands a s far as i t is permit tedby the topological constraints. Then, in the second stageof expansion, the slow process of removal of topologicalconstraints governs the dynamics. As the t ime scalesch ara ct e r izin g t h e s e t w o s t e ps are v ery di ffere n t , w eterm t he fairly long-lived intermediate sta te the ar r esteds t a t e .

A. Initial Expansion. During this stage, the globules we lls f rom e q u il ibr ium t o t h e a rre s t e d, s t re t ch e dconfigurat ion, while the number of topological con-stra ints is conserved.

Characterization of Equilibrium and ArrestedConfigurations. We introduce the notion of an en-tangled cell in the init ial equilibrium configurat ion.This cell conta ins Ne1/2 subcha ins, each subcha in con-sists of Ne correlation blobs of size r0*. Each blob has(r0*)2 monomers, and the total number of monomers inan ent a ngled cell is m ) Ne3/2(r0*)2 (see Figure 4). Thetota l energy stored in the int eraction of the blobs insidean equilibrium cell is about Ne3/2. B efore th e swelling,the size of the entangled cell is r Ne1/2r0*. When thee n t an g le d c e l l is a l lo we d t o e x p an d, t h e n u mbe r o fmonomers and subchains in a cell remains the same,but the structure of the subchains is changed. This leadsto a n ew cell sizera in th e arr ested sta te (see Figure 4).In the stretched structure, the chain minimizes its freee n e rg y u n de r t h e c o n s t ra in t t h a t t h e n u mbe r o f e n -t an g le men t s an d t h e n u mbe r o f e n t an g le d s u bch ain s

Ne1/2 is conserved. This can be rationalized by a Flory

argument balan cing the elast ic energy a nd the intera c-t ion energy. As in the standard Flory argument for achain swollen in good solvent, 21 Gaussian elast icity isassumed, and the interaction energy is estimated in themean-field approximation:

The estimate is similar to the Flory estimate for singleswollen chain, with addit ional fa ctors Ne1/2 a n d (Ne1/2)2

in elast ic and interact ion energy accounting for Ne1/2

chains in th e same volume. This leads to th e est imateof the st retched cell size ra Ne7/10r0*6/5. The sa me ty peof deriva tion w as used by P u tz for sw elling of a w eaklycross-linked melt.22

R e mark ably , t h is arg u me n t c an be c a rr ie d o u t fo rscales L smaller than the size of the entangled cell. Weshould a ssume tha t inside a box of size L t h e re a re (L /r0*)1/2 different chains. The num ber of th ese cha ins isconserved during swelling. Under these conditions, onscales smaller than the size of the entangled cell , thestructure of the sw ollen polymer chain is a fracta l withexponent a ) 7/10 inst ead of 6/10 a s in the originalFlory a rgument for a swollen chain.

Another predict ion from the above Flory argumentcan be made abo ut t h e mon ome r/mon ome r de n si t ycorrelat ion exponent on scales sma ller tha n ra . I n s idethe sw ollen cell , a test sphere of radius F correspondsto an unswollen sphere of radius F5/7 init ially sharedby F5/7 strands that did not separate upon swelling.The t otal number of monomers inside the t est sphereis thus proport ional t o m F15 /7. This f ixes t he H aus-dorff dimension of the a rrested polymer t o Da ) 15/7.At larger scales the cells are arranged in a dense way

Figure 4. E xplana tion of the entangled cell . Snapshot (left) from a n MD simulation of a sw elling globule (N ) 50 000) in th earr ested sta te, showing a hemispherical section of th e globule. The enta ngled cell size is indicated a s ra . The gray scale depictsthe degree of constraint on thermal fluctuations of the monomers. 15 Dark regions indicate the accumulation of entanglements.The a ccompanying schemat ic (right) depicts the packing of expan ded enta ngled cells a nd t heir interna l stru cture.

F ) Fel + Fi

Ne1/2

ra2

Ner0*2 +

(Ne1/2)2Ne

2r0*

4

ra3 (9)



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and the dimension is d )3 . I n s u mma ry , t h e a rre st e dg lo bu le is de s c r ibe d a s a n arra y o f e n t an g le d c e l ls chara cterized by tw o exponents: An exponenta ) 10/7describing the internal strand structure inside a cell ,and an exponent Da ) 15/7 describin g th e (collective)structure inside a cell. (As mentioned in the Introduc-t ion , a s h ort ac cou n t of t h e s t ru c t u re o f t h e a rre s t e dglobule including ent a nglement/enta nglement correla -t ions was already given elsewhere. 15)

From a thermodynamical point of view, the chemicalpotentia l of the solvent in the globule ha s a ga in reachedi t s bu lk v a lu e in t h e a rre s t e d s t a t e . T h e s u bs e q u e n tstages will proceed under vanishing osmotic pressure,t h e g lobu le bein g e q u il ibra t e d u n der t h e re main in gtopological constraints. In the arrested state, the freeenergy has not yet completely relaxed. Following theFlory est ima te, the energy per cell amounts t o

as c o mp are d t o Fa (t ) 0) Ne3/2 in t h e in i t ia l s t a t e .Knowing t he structure of the qua si-equilibrat ed stretchedcon f ig u rat ion , we can e st ima t e t h e h y drody n a mic re -laxat ion t ime needed to reach this stat e: In the spirit

o f e q 2 , we balan c e t h e v is c o u s dis s ip at io n a n d t h edecrease of the free energy. The dissipat ion is domina tedby solvent penetrat ion and hence by the largest scalewith velocity

The incoming solvent sees the globule a s a porousmedium of pore size the cell size r. The rate equat ionfinally reads

The arrested sta te is thus approached wit h the termina lt ime

This rela xat ion is governed by st ress diffusion over theglobule radius at arrest with the collect ive diffusionconstant

Insert ing the est imate of Fa from eq 10 we obtain ta )N2/3Ne

1/5.B. Swelling of the Arrested Globule. To describe

the further sw elling from th e arrested sta te, we shouldlook for the dominant energy dissipat ion mechanism.When the enta ngled globule is expanding, t here ar e tw ovelocities w hich cha ra cterize this motion: the velocityof the ra dial expan sion of the globule in the solvent: VR) dR(t)/d t ) n1/3(t) dr(t)/d t, a n d t h e v e l o c i t y VL ofdifferent parts of the polymer chain relat ive to eachother, when the chain is threaded through itself . Toe st imat e t h is v eloci t y , c on s ider s we l lin g of a wh o leglobule wh en every elementa ry cell expands from r t o r+ r. The tota l tube length of the polymer cha in insideof the entangled cells within a globule was L pol ) n r Ne1/2.After the r t o r + rexpansion t he length of a tube isincreased t o L t u be ) n(r + r)Ne1/2. But the length of a

polymer rema ins a bout the sam e. The difference L tube

- L pol ) nr Ne1/2 gives the distance which the end ofthe chain should tra vel inside the globule. During thisthreading, some knots disappear (as depicted in Figure3c), and the number of cells is reduced. For the velocityof threading we obtain

wh ic h is a lway s larg e r t h an VR.I n t h e fo l lo win g , we dis c u s s t h e l i fe t ime td o f t h e

ar rested stat e for t wo different dissipat ion mecha nism.First , we assume that the dissipat ion is dominated bydirect conta cts betw een stra nds. Assuming th at the cellis p rev en t e d f rom e xp an s ion by Ne entanglementsb e t w e e n a n y t w o o f Ne1/2 s u bc h ain s in t h is c e l l , weconclude that energy stored in each entanglement isabout Ne9/10/Ne 1 in kB T u n i t s . I n t h e fo llowin g a l lestimates will be based on the Flory approach; thus thenumber of contacts is a ssumed to scale w ith t he Florye n erg y of a ce ll , Fa , for con s ist e n cy . Th e min ima ldistance between monomers of tw o chains forming a nentanglement is comparable to their size. This leads to

estimat e for th e rat e of energy dissipat ion, (VL2

/b2

)b3

Faper cell. A bala nce of dissipat ing energy simila r t o eq 2determines the rate equat ion:

The lifet ime of the a rrested sta te td is relevant to theearly stages of threading:

We now a ssume tha t t he dissipat ion is dominated byhydrodynamic f lows induced by t he motion of stra ndsrelat ive to each other. The typical velocity gradients

d u r in g t h r ea d i n g a r e s e t b y t h e d is t a n ce b et w e enmoving threads in a cell . As dissipat ion is dominatedby the la rger scale, the relevant distance is given by

A balance of dissipat ing energy similar to eq 2 deter-mines the rate equat ion:

The characteristic time td can be inferred d irectly fromeq 17:

U s in g t h e v a lu e s o f ds a n d Fa b y e q s 1 0 a n d 16, w eobtain

With the overall accuracy of Flory est imates used inderivation of the exponents, this estimate gives the sameresult as the est ima te based on direct frict ion betweent h e c h ain s, by n a t u re Ne is f inite and rather small . Asshown previously 15 and further discussed in section 4,

Fa Ne9/10 (10)

VR )d Rd t

(VR2/r2)r3 ) -

dFad t

(11)

ta N2/3

Ne11/10/Fa . (12)

Dc Fa

ra

VL ) n(t)Ne1/2 dr(t)

d t (13)

(VLb)

2

b3

n Fa ) - Fadn(t)

d t (14)

td ) N2

Ne-3/5 (15)

ds r/Fa1/3

Ne2/5 (16)

(VL

ds)

2

n r3

) - Fadn(t)

d t (17)

td N2Ne

3/2

ds2

Fa

(18)

td N2

Ne-1/5 (19)



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Ne ) 30 , N, the dependence on Ne could be dis-regarded. For long chains the lifet ime in the arresteds t a t e is mu c h larg e r t h a n t h e t ime fo r t h e g lo bu le t oget t ra pped. The lifetime tda s given a bove does not crossover smoothly t o the termina l t ime ta as given by eq 12when the globule contains only one cell. This suggestsa s h arp t ran s i t ion wh e n s el f-e n t an g leme n t s becomerelevant (N Ne3/2). We should remember tha t t he Floryargum ent gives inaccurat e est imates for the energy ina good solvent. A more detailed geometrical argumentbased on the fractal structure does not alter the resultssignificantly.

Inside knots th e concentra t ion ma y be become quitehigh and local chain structure may matter. Knots canbecome glassy a nd tra p the globule in the a rrested sta te.This is not accounted for by our theory.

C. Swelling of an Entangled Dry Globule. I n t h eprevious sect ion, we assumed that the penetrat ion ofsolvent into the globule does not control swelling. Wehad shown tha t for the wet globule the energy dissipa-tion from th e flow of solvent inside t he globule is sma llcompared t o the energy dissipat ion from t he threa dingpolymer chain. I n th e ca se of a d ry globule, the dista ncebetween monomers of a polymer chain is comparable

t o t h e g eome t r ica l s ize of s olve n t molecu les . Th isk in e mat ic c on s t ra in t on t h e p en e t ra t ion of s olv en t smig h t con t ro l t h e dy n a mics . We con s ide r f i rst t h epossibility of the init ia l sta ge of swelling, a s t he resultof which the compact globule becomes stretched by afactor R ) Ne1/5 2 . T h is is t h e s ame p ro ble m ass we l l in g o f a g e l wi t h max imu m p o re s ize big g e r o rsmaller than the size of solvent molecule. Whether ornot solvent molecules penetrate depends on their sizerelat ive to the ty pica l dista nce betw een cha ins (i.e., thepore size of a gel).

Consider f irst the case when penetrat ion of solventinto th e globule is possible. The tim e needed for wett ingcan be anomalously long, especially in the vicinity ofthe critical size.24 When t he globule is w et, th e swelling

dynamics will be the same as described above for thewet entan gled globule.

If solvent cannot penetra te into th e globule, or if thisprocess is too slow, we should consider the possibilityof the self-reptat ion of the chain out of the enta ngledconfigura tion. The ma in question here is whether t hereis a n e t e x t ern al force p u l lin g t h e ch ain ou t of t h eglobule. Naively one might assume that if one end of apolymer chain is outside the globule an d a nother endis inside, then there is a force pulling the external endout of the globule. More ca reful considera tion suggest st h at t h e p o ly me r c h ain c a n fo rm s e v e ra l p ro t ru din gloops pulled out of globule, as depicted in Figur e 2b wit han equivalent entanglement scheme in Figure 5a. One

o r t wo e n ds o f t h e c h ain c o u ld be h idde n in s ide t h eglobule; in this case there would be a net force pullingthem out the globule into the nearest loop, and aftershort a time both ends will be outside, as in Figure 5c.In this case, the sum of all forces acting on the chain iszero and t he time needed for th e cha in to reptat e out ofentanglements is a bout t he typical r epta t ion t ime.

When comparing this t ime with swelling t ime of wetentangled globule given in eq 19, the relaxat ion t imefor the dr y globule is significa ntly longer. In r ef 19, 1-Dpolymer loops were considered where reptat ion is ir-

relevant and exponentially slow dynamics was found.It should be noted that swelling very dense globules asconsidered in th is section is less universal th a n for wetones an d ma y depend on the very deta ils of the intera c-

t ion and local chain str ucture.

IV. MD Simulation Results with Comparison toTheory

Here we present MD simulation results on the swell-ing of globules in different regions of the phase diagram.Our simulation results discussed in section 4.1 coversregions II and IV in the phase diagram shown in Figure1. To some extent, region III is covered by the Rouse-type simulations presented in section 4.2.

In the following discussion of the simulation methods,sta nda rd L ennard-J ones units ar e employed, w ith en-e rg y in u n i t s o f kB T. L e n g t h is me as u re d in p ar t ic le

diameters, , t ime in )m2/kB T, a n d m a s s m ) 1.Par t ic le p airs wh ic h are bo n de d n e ig h bo rs a lo n g t h ech ain in t erac t t h ro ug h a h armo n ic p ot e n t ia l w i t h aspring constant of 50 kB T/2.

A. MD Simulation with Explicit Solvent Par-ticles. 1. Creation of the Initial Dry-Fresh Globule.D ry -fres h g lobu le s we re cre at e d by s imu la t in g t h ecollapse of solvated N ) 512 chains. The collapsesimulat ions w ere discussed in ref 12. B riefly, 40 indi-vidual NVE molecular dynamics (MD) simulations of asingle cha in in a box of 80 000 solvent part icles wereperformed, with potential parameters chosen such thatthe equilibrium state is understood to be a collapsedglobule. The chains were created as nonreversal randomwalk s wi t h a s t e p le n g t h o f 0 .97. All part icle pairs(except bonded neighbors) intera ct via 12-6 L e n n a rd-J ones potentia ls, which a re used to enforce an a ttr activeinteract ion with a 1kB Tw ell depth for solvent -solventan d monomer-monomer par ticle pa irs, a nd pure repul-sion for solvent-monomer pairs. For ea ch initia l config-u rat io n , an M D s imu lat io n is ru n u n t i l t h e radiu s o fgyra t ion of the chain stops changing, indicat ing a fullycollapsed state has been reached. The time step of thesimulat ion is 0.001, and no thermostat wa s used. Outof these 40, five were chosen at random as dry-freshglobules. As size of solvent part icle is as large as thesize of the monomers, few solvent pa rticles ar e tra ppedinside of the collapsed globule; it is therefore dry. Theinit ial states for the swelling simulat ions (describedbelow) are f iv e dis t in ct bu t s t a t is t ica l ly e q u iv a len t

t ) N3

Ne-1 (20)

Figure 5. S chemat ic of the sw elling of a dry globule: (a) Aninitially dry swelling globule consists of a dry core and wetloops. (b) There are forces on a polymer chain pulling towardoutside. (c) After some time, both ends of a polymer w ill beoutside and the t otal force will be zero. The chain then sw ellsaccording to t he self-reptation mecha nism.



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globule conforma tions immersed in explicit solventparticles.

2. Creation of the Initial Dry-Old Globule. Theinit ial conformation for a dry-old globule wa s createdby growing a ra ndom wa lk of length N ) 512 in a cubicbox that is 30.67on a side with periodic boundar ies.The wa lk is grown within a G aussia n confining poten-tial, in w hich the par ameter R0represents the strength

wh e re r i s t h e dis t an c e of t h e cu rre nt p ar t icle in t h ewa lk from the initia l particle. In the growth of the wa lk,tria l steps ar e accepted or rejected based on a Metropoliscriterion. First , the potentia l a t st ep position ri, Uconf(ri),is evalua ted. Then, a uniform ra ndom variat e Xover 0,1

is chosen. I f X > Uconf(ri), then the st ep is rejected, a ndnew trial steps are attempted (with correspondingly newran do m v ar ia t e s X) until X


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at tributed t o topological constra ints rea ched a fter sw ell-ing of the enta ngled globule is best seen from the Rouse-type simulations (i.e., without explicitly moving solventparticles) considered in this section, which allow us tos imu lat e c h ain s a s larg e as N ) 100 000. No solventpart icles were included in the simulat ions discussedhere, which neglects hydrodynamic interact ions. Thelack of h y drody n amics wi l l obv iou s ly in flu en ce t h edisentanglement process. However, the arrested struc-ture itself is not expected to depend on the details ofthe solvent -polymer dynamics.

We begin with a cubic box with dimension 171p erside, with periodic boundaries in all directions. We thenp la c e t h e f i r st m on om er of t h e c h a in a t a r a n d omlocat ion in t his box, a nd define th is a s t he origin. Wethen grow the random walk from this origin with stepsize of 0.97, with the condit ion that no step may take

the w alk beyond a st ipulat ed distance from t he origin,R0. We c h os e t o g row a wa lk o f 100 000 s t e p s in aspherical volume such tha t t he overa ll part icle numberdensity i nside the confi ni ng spherewa s 0.85-3. Hence,R0 is 30.4. D u rin g g rowt h of t h e wa lk , if a t r ia l s t e pwould lie more than R0from the origin, it is thrown outan d n e w t r ia l s t e p s are a t t e mpt e d u n t i l o n e is fo un dt h at l ie s wi t h in R0. Aft e r t h e g ro wt h of t h e w a lk , t h epush-off stage MD simulation is performed as describedpreviously in section 4.1.2. We examined several push-off stage durat ions; the one used most commonly was5000 steps in length , each step 0.01 for a total t ime tf- ti of 50.

In t he swelling sta ge, after t he push-off, the full WCAexcluded volume potentia l is u sed (i.e., b(t) ) 1.0 in eq4.1.2). The equa tions of motion a re integra ted using thevelocity-Verlet algorithm,26 with a t ime step tof 0.01.

Figure 7. Snapshots of a swelling fresh globule with t ime interval 100. Dark spheres are wet monomers, as defined in thetext.



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I n b ot h t h e p us h -of f a n d t h e s w e ll in g s t a g es , aLangevin-type thermostat is used to maintain a con-stant temperature, with frict ion ) 0 .5-1, as before.The set point temperature of the thermostat is 1.0.

Figure 10b shows the sq uared r adius of gyra t ion Rg2

for this chain (N ) 100 000) a s a funct ion of time. Afteran init ially ra pid expansion, the globule size sat urat esa t abo u t t ) 4000, demonstrat ing the concept of anarre s t ed s t a t e .

In the In troduction, and in our previous work,15 w epointed out that the structure of the arrested state ischaracterized by distinct monomer -monomer correla-t ion s an d e n t an g leme n t-entanglement correlat ions.The exponent a ) 7/10 describes t he int erna l st ra ndstructure via ri j |i - j|a . We observe excellentagreement w ith ri j2computed from simulation, plottedin Figu re 11b, which displa ys a best-fit exponent of 1.44,

giving a computed a of 0.72. The average number ofmonomers m(F) inside the test sphere of ra dius F d r a w naround a randomly chosen monomer is related to theHa usdorff dimension by m(F) FD a . T h e H au s do rf fdimension of the enta ngled cell should be Da ) 15/7 u pto scale ra . The structure factor S(q) o f t h e arre s t e dstructure is shown in Figure 11a. The Ha usdorff dimen-sion is seen only at the intermediate length scale whilet h e larg e q regime is dominated by surface effects. Toavoid the influence of the surface, we compute the realspace correlat ion funct ions w here an avera ge is ta kenover many test spheres (inside the globule) and arrestedglobule configura tions. At lar ger scales t he globule is adense packing of cells and the dimension is d ) 3. Thecrossover from fractal dimension Da )15/7 t o d )3 isperfectly ca ptured by th e simulat ion results as depictedin F igure 11c.

Figure 8. Snapshots of a swelling entangled globule with t ime interval 100. Dark spheres are wet monomers, as defined int he t ext .



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This scaling exponent is in excellent a greement w ithour computer simula tion result s. This sca ling providesa new insight on the nonlocal nature of entanglements.Indeed, i f the entanglement could be thought as localconstra int , similar to a chemical cross-link, we woulde xp ect s ep ara t io n of d i ffere n t ch a in s in s ide t h e e n -t an g le d c e l l du rin g s we l l in g . S t ran ds wo u ld t h e n bestretched ( ) 1) bu t a t a lo c a l s c a le ( ) 0.6). Thes t ra ig h t l in e s t a r t in g f ro m t h e mo n o me r s ize in t h es ca l i n g d a t a s h ow n i n F i gu r e 1 1 s t r on g ly s u gg es t stha t there is a self-similar picture on a ll scales belowthe enta ngled cell size. This mean s th at during sw ell-ing in the presence of topological constraints, sepa-

ra tion of different chain st ra nds inside the cell does notoccur.

Our data indicate that the number of strands in anygiven volume is a sta tistically conserved quan tity un dert h e s t re t c h in g de fo rmat io n . I t i s a ls o c le ar t h a t t h epictur e incorporat ed in Flory description is oversimpli-f ie d; s t r uct u r a l d is or d er s ee n i n F i gu r es 4 a n d 10in dica t e s t h a t s ome re dis t r ibu t ion of e n t a n g leme n t sta kes place wh en they slide and a ggregat e under stress,as discussed in ref 15.

For such a long chain, th e topological relaxa tion timetd (eq 18) is by far larger than the computat ion t ime,an d the cha in is never seen to esca pe from the a rresteds t a t e .

V. Conclusion

We s h owe d t h at s we ll in g of a col lap s ed p oly merglobule is a very intricate problem w here aging in thecol lap s ed s t a t e an d in it ia l comp act n e s s p la y a ma jorrole.

A freshly collapsed (fresh) globule has a crumpledstructure with internal fractal dimension three and isessentially rea dy to reexpan d. In t he simplest fresh-wetcase the swelling is homogeneous and the dissipat iondominated by the solvent penetration. The swelling timeis a bout th e coil relaxa tion tim e. A fresh-dry globule istoo compa ct for th e solvent to penetra te a nd only surfa cemonomers ar e solvated. Sw elling proceeds by peelingmonomers into loops. The w ett ing of the init ially dr yg lo bu le is a ra t h e r fas t p ro c e s s . T h e we t t in g s t e p isfollowed by an expansion step similar to the expansionof a n in it ia l ly w e t g lobu le w i t h ch ara ct e r is t ic t ime N3/2.

The configur a tion of a polymer chain in a n old globuleis locally Gaussian with an internal fractal dimensionof two. During a ging, the density of the globule does notevolve, but t he int erna l str ucture does. While evolvingfrom crumpled to Gaussian, the globule self-entangles.An old globule can be thought of a s a large knot . Thea d d i t ion a l e nt a n g l em en t s f or m ed d u r in g a g i n g a r etopological constra ints t ha t ar e to be released in ordert o s we l l t h e g lo bu le bac k t o t h e c o i l s t a t e . Swe l l in goccu r s i n t w o s t ep s. F i r st t h e g lob ul e s w el ls w i t hpreserved topology and rea ches an ar rested sta te, a st ep

Figure 9. Averaged profile of wet monomers during swellingof freshly prepared globules (N ) 512, t ) 400). The s emiloginset captures t he exponential cutoff.

Figure 10. S qua red radius of gyra tion of a swelling cha in asa funct ion of t im e. (a) I nit ia l ly fr esh (O) a n d o l d (0) w i t hexplicit solvent part icles, a veraged over five init ial globules.The chain length is N ) 512. Rg2 for the single old globulewhich most quickly escaped the arrested state is also shownas the da shed curve. (b) The Rouse-type sw elling of a cha in of

length N ) 100 000.

Figure 11.(a) Structure factor S(q) of a n N ) 100 000 swollenchain at t ime t) 50 000computed from MD simula tions. Thestraight line corresponds to the power law S(q) qD a (Da )15/7). (b, c) Rea l spa ce correla tion fu ncti ons of a sw elling N )100 000 globule: (b) single-str a nd correlat ion functions a t t )50 000(arrested), where best-fit value of exponent a fromr2ij |i - j|2a is 0.72 and the Flory a rgument gives a ) 0.7at arrest; (c) number of monomers mvs s ize of probe sphere Fshowing the collective Hausdorff dimension of Da ) 15/7 a ndthe crossover to Da ) 3 for t he ar r est ed st at e .



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w h i ch i s r e l a t iv el y f a s t w i t h a ch a r a c t er i st i c t i m ein cre as in g as N2/3. Su bs eq u e n t ly t h e g lobu le s lowlyd is en t a n g le s. We f ou n d t h a t t h e a r r e st e d s t a t e i scharacterized by distinct monomer -monomer correla-t ions (with Ha usdorff dimension 15/7) and entangle-ment -entanglement correlations that reflect entangle-ment sliding.15 We propose a t hread ing mechanism t ha tallows the wet globule to relax topological constra intsin a t ime proportiona l to N2. Finally, the old-dry globulehas to disentangle through self-reptat ion but the dy-namics may be dramatically slowed (as it is in glasses).

This work offers several prospects. Structures similarto those in the arrested state exist in swollen, weaklycross-links melts. The behavior of cross-linked entangledg el s u n d e r t e ns ion i s a l so s o me w h a t r el a t ed a s i tinvolves sliding knots. More loosely related problemsare healing and coalescence in polymeric systems.

Acknowledgment.The aut hors are gra teful for thesupport from LEA. N.-K.L appreciated the f inancials u p p o rt o f t h e B M B F -G e rman y v ia t h e Na n oce n t erM a i nz a n d t h e g e n er ou s s u pp or t of M P I P f or t h i scollaborat ion. C.F.A. gratefully acknowledges DrexelUniversity for startup funds.

References and Notes

(1) de Gennes, P. G. J. Phys. Lett. 1985, 46, L-639.(2) G rosberg, A. Yu; Kuznetsov, D. V. Ma cromolecul es1993, 2 6,

4249.(3) Timoshenko, E. G. ; Kuznetsov, Yu. A.; Dawson, K. A. J .

Chem. Phys1995, 102, 1816.(4) Kuznetsov, Yu. A.; Timoshenko, E. G. ; Dawson, K. A. J .

Chem. Phys. 1995, 10 3, 4807.(5) Bu guin, A.; Brochard-Wya rt, F.; de Gennes, P. G. C. R. Acad.

S ci . P a r i s 1996, 322, 741.(6) B y rne , A. ; K ie rna n, P . ; G re en, D. ; D a w s on, K . A. J . Ch e m.

Phys.1995, 10 2, 573.(7) P i t a rd , E . ; O rla nd , H . Europhys. Lett. 1998, 41, 467.(8) P i t a rd , E . E u r . P h y s . J . B 1998, 7, 665.

(9) Cha ng, R. ; Yethiraj , A. J. Chem. Phys. 2001, 11 4, 7688.(10) Milchev, A.; Binder, K. Europhys. Lett. 1994, 26, 671.(11) Chu, B .; Ying, Q.; Grosberg, A. Yu. M acromolecul es1995, 2 8,

180.(12) Abrams, C . F.; Lee, N.-K.; Obukhov, S. Eu rophys. Lett. 2002,

59, 391.(13) Rostia shivili, V.; Lee, N.-K.; Vilgis, T. A.J . Chem. Ph ys. 2003,

118, 937-951.(14) Ra bin, Y.; G rosberg, A.-Y.; Tan aka , T. Eur ophys. Lett.1995,

32, 505.(15) Lee, N.-K.; Abrams, C. F.; J ohner, A.; Obukhov, S. P. Phys.

Rev. Lett. 2003, 90, 225504.(16) Wu, C.; Zhou, S. Phys. Rev. Lett. 1996, 77, 3053.(17) Wan g, X.; Qui, X.; Wu, Chi. M acromolecul es1998, 3 1, 2972.(18) Here a nd below w e use the t erminology coined by G rosberg

in th e late 1980s (J. Phys. (Paris)1988, 49, 2095).(19) Pit ard, E . ; Bouchaud J .-P . E u r . P h y s . J . E 2001, 5, 133.(20) On la rger scales, such rear ra ngement s are also possible; this

s i t u a t ion is s imila r t o t he p rob le m of in t e rp e ne t ra t ion ofnonentan gled ring polymers. Extensive computer simula tionsof a melt of nonentangled ring polymers by Wittmer indicatethat a lthough there is part ia l interpenetrat ion of differentrings, the resulting configurations are compact and displaya R N1/3 leng t h d e pe nd enc e ( Wit t me r, J . P . P r iva t ecommunicat ion). I f we adopt corresponding numerical est i-mat es for th e t ime needed for this reara ngements, it appearstha t th is t ime is a lw ays close to the t ime needed for th e endof a p oly mer c ha in t o re pt a t e t rou gh t he g lob u le . Thisreptat ion creates mult iple topological constra ints , a nd, a s a

result , the globule is not fresh any more.(21) Flory, P . J .Pr incipl es of Polym er Chemi stry; Cornell Univer-

sity Press: I tha ca, NY, 1953.(22) P utz, M.; Krem er, K.; Evera ers, R. Phys. Rev. Lett .2000, 8 4,

298.(23) P utz, M.; K remer, K . Europhys. Lett. 2000, 49, 735.(24) We would expect th e same slow ma ny par ticle rear ra ngement

dynamics similar to that in structural glasses.(25) Weeks, J . D.; Chandler, D.; Andersen, H. C. J. Chem. Phys.

1971, 54, 5237.(26) Swope, W. C.; Andersen, H. C.; Berens, P. H.; Wilson, K. R.

J. Chem. Phys. 1982, 76, 637.(27) Kremer, K. ; Grest , G. S . J. Chem. Phys. 1990, 92, 5057.

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