5
Snaky Conformations of Strongly Adsorbed (2D) Comblike Macromolecules Igor I. Potemkin ² Physics Department, Moscow State UniVersity, Moscow, 119992 Russia, and Department of Polymer Science, UniVersity of Ulm, D-89069, Germany ReceiVed September 9, 2006; ReVised Manuscript ReceiVed NoVember 5, 2006 ABSTRACT: We study the decay of orientational correlations between segments, the gyration radius, and the Kuhn segment length of a 2D comblike macromolecule (brush) with random distribution of the side chains with respect to the backbone. This distribution is assumed to be fixed and controlled by an initial conformation of the molecule in 3D. Both “frozen’’ (quenched) fluctuations of left-right positions of the side chains and the thermodynamic fluctuations of the whole molecule in 2D are taken into account. We show that extra bending of the brush due to the locally uneven number of left and right side chains promotes the decay of the orientational correlation function, and reduces the gyration radius. The correlation function has a nonmonotonous decay with one minimum. 1. Introduction It is well-known that the Gaussian chain with a persistent flexibility mechanism has an exponential decay of orientational correlation function, i.e., a thermodynamic average of the cosine of the angle θ(s) between two tangents to the chain separated by contour length s is cos θ(s)) exp(-s/λ p ). 1 Here λ p is the persistence length which characterizes the length scale of the decay. Recently, it was shown that long range correlations in semidilute solutions and melts of linear chains are responsible for an unexpected, scale free power-law regime of the orien- tational correlation function, cos θ(s)s -ω , ω > 1. 2 Furthermore, the nonexponential decay is predicted for linear chain in a Θ-solvent. 3 In the present paper, we report about the unusual form of the correlation function of single comblike macromolecule strongly adsorbed on a plane surface (2D chain). This function has a nonmonotonous decay with one minimum corresponding to the negative value of the correlation function. Such behavior is a result of extra curvature of the macromolecule induced by its adsorption. The gyration radius and the Kuhn segment length are analyzed. Comblike macromolecules, sometimes called bottle-brushes, are polymers consisting of a long flexible main chain (backbone) and densely grafted side chains. 4,5 The peculiarity of such molecules is an enhanced stiffness induced by interactions between the side chains. 6-9 Relying on the strong dependence of the induced stiffness on the length of the side chains and on their grafting density, conformation of the brush molecules in a good solvent can vary from a rodlike to the coiled one. Strong adsorption of the brushes on a flat surface, when most of the side chains attain two-dimensional conformations, diversifies behavior of the molecules. In particular, stiffness of the brushes becomes more pronounced due to the stretching of the adsorbed side chains, 10-12 and the break of the local cylindrical symmetry of the macromolecule, i.e., formation of “left” and “right” (with respect to the backbone) side chains, is responsible for a number of effects. If the adsorbed side chains are allowed to be redistributed between left and right sides of the brush (usually this process requires strong external ac- tion 13,14 ), spontaneous curvature of the molecule occurs. 15-19 This curvature is driven by locally uneven distribution of the side chains that reduces the total free energy of the molecule. If an excess of the side chains is formed only in one side of the brush and this asymmetry is fixed along the whole molecule (the so-called segregating side chains), various conformations, such as spirallike 10 or horseshoe, 20 can be stable. It has been shown recently that conformations of strongly adsorbed brush molecules consisting of a poly(2-hydroxyethyl methacrylate) backbone and poly(n-butyl acrylate) (pBA) side chains strongly depend on the conditions of adsorption. 21 In particular, the molecules which are adsorbed on mica from a good solvent (chloroform), reveal elongated conformations whereas the spreading of the molecules on the surface from a 3D globular state gives much more compact conformations. In the present paper, we propose a theory which quantifies possible conformations of strongly adsorbed (2D) comblike macromol- ecules. For this purpose we use a model of the brush with random distribution of the side chains with respect to the backbone. This model is able to describe the observed variety of the molecular conformations. 2. Model Once a comblike macromolecule is adsorbed on a flat surface, the thermodynamic fluctuations are not able to change the distribution of the side chains with respect to the backbone, if attraction of the chains to the surface is very high. Indeed, a flip of one side chain requires simultaneous desorption and further readsorption of all monomer units of the chain. The thermal energy is too small to overcome the high attraction energy. To do it, some external action is needed. For example, lateral compression of a monolayer of the brush molecules on a liquid surface 13 or treatment of the molecules on a solid surface by various vapors 14 are able to change the position of the side chains. On the other hand, the thermodynamic fluctuations of the side chains in 2D are not suppressed and provide an equilibrium conformation of the molecule (at the given left- right distribution of the side chains). It is this regime of adsorption that will be discussed below. Let us assume that the local distribution of the side chains with respect to the backbone of strongly adsorbed brush mole- ² E-mail: [email protected]. 1238 Macromolecules 2007, 40, 1238-1242 10.1021/ma062099l CCC: $37.00 © 2007 American Chemical Society Published on Web 01/27/2007

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Snaky Conformations of Strongly Adsorbed (2D) ComblikeMacromolecules

Igor I. Potemkin †

Physics Department, Moscow State UniVersity, Moscow, 119992 Russia, and Department of PolymerScience, UniVersity of Ulm, D-89069, Germany

ReceiVed September 9, 2006; ReVised Manuscript ReceiVed NoVember 5, 2006

ABSTRACT: We study the decay of orientational correlations between segments, the gyration radius, and theKuhn segment length of a 2D comblike macromolecule (brush) with random distribution of the side chains withrespect to the backbone. This distribution is assumed to be fixed and controlled by an initial conformation of themolecule in 3D. Both “frozen’’ (quenched) fluctuations of left-right positions of the side chains and thethermodynamic fluctuations of the whole molecule in 2D are taken into account. We show that extra bending ofthe brush due to the locally uneven number of left and right side chains promotes the decay of the orientationalcorrelation function, and reduces the gyration radius. The correlation function has a nonmonotonous decay withone minimum.

1. Introduction

It is well-known that the Gaussian chain with a persistentflexibility mechanism has an exponential decay of orientationalcorrelation function, i.e., a thermodynamic average of the cosineof the angleθ(s) between two tangents to the chain separatedby contour lengths is ⟨cosθ(s)⟩ ) exp(-s/λp).1 Hereλp is thepersistence length which characterizes the length scale of thedecay. Recently, it was shown that long range correlations insemidilute solutions and melts of linear chains are responsiblefor an unexpected, scale free power-law regime of the orien-tational correlation function,⟨cos θ(s)⟩ ∼ s-ω, ω > 1.2

Furthermore, the nonexponential decay is predicted for linearchain in aΘ-solvent.3

In the present paper, we report about the unusual form of thecorrelation function of single comblike macromolecule stronglyadsorbed on a plane surface (2D chain). This function has anonmonotonous decay with one minimum corresponding to thenegative value of the correlation function. Such behavior is aresult of extra curvature of the macromolecule induced by itsadsorption. The gyration radius and the Kuhn segment lengthare analyzed.

Comblike macromolecules, sometimes called bottle-brushes,are polymers consisting of a long flexible main chain (backbone)and densely grafted side chains.4,5 The peculiarity of suchmolecules is an enhanced stiffness induced by interactionsbetween the side chains.6-9 Relying on the strong dependenceof the induced stiffness on the length of the side chains and ontheir grafting density, conformation of the brush molecules ina good solvent can vary from a rodlike to the coiled one.

Strong adsorption of the brushes on a flat surface, when mostof the side chains attain two-dimensional conformations,diversifies behavior of the molecules. In particular, stiffness ofthe brushes becomes more pronounced due to the stretching ofthe adsorbed side chains,10-12 and the break of the localcylindrical symmetry of the macromolecule, i.e., formation of“left” and “right” (with respect to the backbone) side chains, isresponsible for a number of effects. If the adsorbed side chainsare allowed to be redistributed between left and right sides ofthe brush (usually this process requires strong external ac-

tion13,14), spontaneous curvature of the molecule occurs.15-19

This curvature is driven by locally uneven distribution of theside chains that reduces the total free energy of the molecule.If an excess of the side chains is formed only in one side of thebrush and this asymmetry is fixed along the whole molecule(the so-called segregating side chains), various conformations,such as spirallike10 or horseshoe,20 can be stable.

It has been shown recently that conformations of stronglyadsorbed brush molecules consisting of a poly(2-hydroxyethylmethacrylate) backbone and poly(n-butyl acrylate) (pBA) sidechains strongly depend on the conditions of adsorption.21 Inparticular, the molecules which are adsorbed on mica from agood solvent (chloroform), reveal elongated conformationswhereas the spreading of the molecules on the surface from a3D globular state gives much more compact conformations. Inthe present paper, we propose a theory which quantifies possibleconformations of strongly adsorbed (2D) comblike macromol-ecules. For this purpose we use a model of the brush withrandom distribution of the side chains with respect to thebackbone. This model is able to describe the observed varietyof the molecular conformations.

2. Model

Once a comblike macromolecule is adsorbed on a flat surface,the thermodynamic fluctuations are not able to change thedistribution of the side chains with respect to the backbone, ifattraction of the chains to the surface is very high. Indeed, aflip of one side chain requires simultaneous desorption andfurther readsorption of all monomer units of the chain. Thethermal energy is too small to overcome the high attractionenergy. To do it, some external action is needed. For example,lateral compression of a monolayer of the brush molecules ona liquid surface13 or treatment of the molecules on a solid surfaceby various vapors14 are able to change the position of the sidechains. On the other hand, the thermodynamic fluctuations ofthe side chains in 2D are not suppressed and provide anequilibrium conformation of the molecule (at the given left-right distribution of the side chains). It is this regime ofadsorption that will be discussed below.

Let us assume that the local distribution of the side chainswith respect to the backbone of strongly adsorbed brush mole-† E-mail: [email protected].

1238 Macromolecules2007,40, 1238-1242

10.1021/ma062099l CCC: $37.00 © 2007 American Chemical SocietyPublished on Web 01/27/2007

cule and, consequently, its 2D conformation is determined byinitial conformation of the brush in solution (3D). If the moleculeis adsorbed from a good solvent, where it possesses elongatedrodlike or coiled conformation, one can expect that the distribu-tion is close to the symmetric one, and local deviations of thefraction of the side chains from the symmetric case are small.If the molecule has a globular conformation in the solution, thestrong adsorption may lead to the formation of pancakelikeglobule with a curved conformation of the molecule. Thisconformation is expected to have locally asymmetric distributionof the side chains, i.e., the fraction of left (right) side chains ishighly fluctuating parameter. Below we will analyze the caseof weakly asymmetric distribution of the side chains.

The 2D brush molecule is considered to be incompatible withthe surrounding medium (solvent or air), i.e. attractive forcesdominate between monomer units of the side chains and thebackbone. Minimization of the area of polymer-solvent contactsresults in a dense packing of monomer units of the brush (2Dmelt). N and M denote the numbers of the segments in thebackbone and the side chain, respectively,N . M . 1; a isthe size of the segment. We consider densely grafted polymerbrushes, i.e., the side chains are attached to each segment ofthe backbone and their number is equal toN.

Local excess of the side chains in one side of the backboneinduces bending of the brush molecule.10 The free energy ofthe bent brush per unit length comprises the elastic free energyof the side chains,17 whereas the bending energy of flexiblebackbone and the mixing entropy of the side chains can beneglected:17

Here the first and the second terms correspond to the elasticfree energy of the side chains forming convex and concave sidesof the brush, respectively (kBT ≡ 1). â, 1/2 e â e 1, is thefraction of the side chains of the convex side, andR is the radiusof the curvature of the brush section. Denote byε ) â - 1/2the small excess of the side chains,ε , 1. Then the bending ofthe brush segment induced by this asymmetry is also small,i.e., the curvature radius has to be much larger than the widthof the brush (the end-to-end distance of the side chains),17 aM/R, 1. Expansion of the free energyf in a series in powers ofεandaM/R gives

where the first term corresponds to the elastic free energy ofthe side chains of rectilinear segment of the brush, and thesecond term is the energy of small bending. The multiplierbefore the brackets has a meaning of the bending modulus10 κ:

The optimum curvature of the brush segment, 1/R0, is given byminimization of eq 2 with respectR, 1/R0 ) 6ε/aM.

In order to find the bending energy of the whole macromol-ecule, we need to summarize the contributions from all thesegments. In doing so, we have to take into account that eachsegment has (in general) a different curvature radius and thatthe position of the contour of the brush can fluctuate around a“trajectory” which is determined by the optimum radii. Denote

by x(s) a spatial coordinate of the brush contour located atcontour distances from the end of the chain, 0< s < L, L )aN. Then the curvature is defined as∂2x(s)/∂s2 and the totalbending energy takes the form

where the vectorn(s) is directed along the radius of the optimumcurvature at the points and has the valueε(s). n(s) is a randomparameter which describes “frozen” (quenched) fluctuations ofthe fraction of the side chains. To find correlation functions ofthis parameter, let us consider the discrete description of themolecule consisting of the finite number of segments, each ofsome given curvature. The signs of the curvatures of twodifferent segments (ith andjth) are statistically independent (eachsegment has equal probability to be bent to left or to right),therefore the averaging givesni ) 0 andninj ) ∆2δij. Hereδij

) 1 if i ) j, andδij ) 0 if i * j. The dimensionless parameter∆ , 1 is a measure of the local asymmetry of the distributionof the side chains: the higher∆, the higher the bending of thesegments. Passing to the continual description of the molecule,the averages can be rewritten as follows:

whereδ(s- s′) is the Diracδ function and the upper line meansthe average with respect to the quenched fluctuations (in contrastto the thermodynamic fluctuations).

For calculation of the orientational correlation function ofthe brush molecule, let us use the Gaussian model which isrestricted by the bending elasticity, eq 4, and by the conditionthat the average of the square of tangential unit vector of the

bush contour,u(s), has to be equal to unity at any point,⟨u(s)2⟩) 1, u(s) ) ∂x(s)/∂s. Here the angle brackets mean thethermodynamic average. Then the total free energy of themolecule takes the form:

whereλ is the Kuhn segment length of the brush and calculationof the orientational correlation function

is done in the Appendix, eq 19:

and the Kuhn segment length is determined by eq 18:

Taking into account that the end-to-end vector of the brush,r ,can be presented in the formr ) x(L) - x(0) ) ∫0

Ldsu(s), wefind

f ) â2R

2a2ln (1 + 2aMâ

R ) -(1 - â)2R

2a2ln (1 -

2aM(1 - â)R )

(1)

f ) M4a

+ aM3

12 (1R - 6ε

aM)2+ ... (2)

κ ) aM3

6(3)

Fbend) κ

2∫0

Lds (∂2x(s)

∂s2-

6n(s)aM )2

(4)

n(s) ) 0, n(s)n(s′) ) ∆2aδ(s - s′) (5)

F ) 1λ ∫0

Lds (∂x(s)

∂s )2

+ κ

2∫0

Lds (∂2x(s)

∂s2-

6n(s)aM )2

(6)

⟨cosθ (s, s′)⟩ ) ⟨u(s)u(s′)⟩ )∫Dx(s)

∂x(s)∂s

∂x(s′)∂(s′)

exp(-F)

∫Dx(s) exp(- F)(7)

⟨cosθ (s, s′)⟩ ) exp{-τ (1κ + 9∆2

aM2)}(1 - τ9∆2

aM2),τ ) |s - s′| (8)

λ ) 2κ

(1 + 9∆2κ

aM2 )2(9)

Macromolecules, Vol. 40, No. 4, 2007 Snaky Conformations 1239

by integration of eq 8.

3. Discussions

The decay of the average cosine of the angle between twotangential vectors of the brush molecule, eq 8, is presented inFigure 1. If we deal with perfectly symmetric distribution ofthe side chains with respect to the backbone,∆ ) 0, the brushreveals exponential decay (solid curve) and we can define thepersistence length,λp, which coincides with the bendingmodulus,λp ) κ, and is twice less than the Kuhn segment length,λ ) 2λp. The presence of local asymmetry of the distributionenhances the decay of the correlation function (dashed anddotted curves): the higher∆ is, the stronger the decay. Indeed,the local quenched bending of the brush to the left and to theright promotes the loss of the orientational correlations.Furthermore, the correlation function can be equal to zero atτ/κ ) 1/R and can have a negative minimal value,-exp(-2 -1/R) R/(1 + R), atτ/κ ) (1 + 2R)/R(1 + R), R ) 3∆2M/2. TheparameterR g 0 can attain high enough values,R . 1, if thelocal asymmetry of the distribution of the side chains is not sosmall, ∆ e 1. Thus, in contrast to chains with persistentflexibility mechanism (κ ) λp), the loss of the orientational cor-relations of the adsorbed brush,⟨cosθ⟩ ) 0, can be achieved atthe contour lengthsτ smaller than the bending modulus,τ )aM2/9∆2 ) κ/R , κ, i.e., the quenched fluctuations of the sidechains are responsible for this effect. In other words, anelementary length of the brush, where the thermodynamicfluctuations are suppressed, can be curved. If the contour lengthis slightly above the value where⟨cosθ⟩ ) 0, the correlationfunction has a minimum: the higherR, the deeper the minimum,Figure 1. Calculation of the characteristic size of the brush,

x⟨r2⟩ , eq 10, whose contour length corresponds to theminimum of the correlation function,L/κ ) (1 + 2R)/R(1 +

R), leads to the valuex⟨r2⟩ ≈ κ/R, R . 1, i.e. x⟨r2⟩ ≈ L.Therefore, despite of the local bending of a characteristic sizeaM/∆ (the curvature radius), the conformation of the brush iselongated and the average angle between the ends of themolecule belongs to the range 90° < θ < 270°.

The characteristic size of the molecule,x⟨r2⟩/κ, as afunction ofL/κ is shown in Figure 2. One can see that the higherasymmetry of the local distribution of the side chains∆ (thehigherR), the more compact conformation of the brush. If thecontour lengthL is smaller than both length scales,κ andaM2/9∆2, L(1/κ + 9∆2/aM2) , 1, the brush has a rodlike conforma-

tion, ⟨r2⟩ ≈ L2. Unusual behavior, comprising independence ofthe gyration radius of the brush on the contour length, is

predicted in the rangeaM2/9∆2 , L , κ, R . 1. Here⟨r2⟩ ≈2(aM2/9∆2)2. The physical picture of such behavior is thefollowing. As we showed above, the brush of the contour lengthτ ∼ aM2/∆2 has a linear dependence of the end-to-end distance

on τ, x⟨r2⟩ ∼ τ. It means that if we circumscribe such brush

by a circle of the radius∼aM2/∆2, there is a lot of “empty”(free of polymer) space within the circle, Figure 3a. Theconformation of longer brush can be imagined as a “filling” ofthe circle, Figure 3b, that is why the size of the molecule isdetermined by the radius of the circle rather than the contourlength of the molecule. Finally, if the length of the brush exceedsthe bending modulusκ, L . κ, the thermodynamic fluctuations

Figure 1. Average cosine of the angle between two tangential vectorsof the brush separated by the contour lengthτ as a function of thedimensionless ratioτ/κ at different values of the parameterR ) 9∆2κ/aM2 ) 3∆2M/2: R ) 0 (solid), 3 (dashed), 10 (dotted).

Figure 2. Ratio of the characteristic size of the moleculex⟨r2⟩ to thebending modulusκ as a function of the dimensionless contour lengthL/κ at different values of the parameterR ) 3∆2M/2: R ) 0 (solid),3 (dashed), 10 (dotted).

Figure 3. Schematic representation of 2D conformations of the brushmolecule of the contour lengthsτ ∼ aM2/∆2 (a) andaM2/∆2 , τ , κ(b). The radius of the circle is∼ aM2/∆2.

⟨r2⟩ ) ∫0

Lds∫0

Lds′⟨cosθ (s, s′)⟩

) 2

A2[AL - 1 + exp{-AL}] +

2B

A3[2(1 - exp{-AL}) - AL (1 + exp{-AL})]

A ) 1κ

+ 9∆2

aM2, B ) 9∆2

aM2(10)

1240 Potemkin Macromolecules, Vol. 40, No. 4, 2007

are responsible for conformation of the brush. The average of

the squared end-to-end distance takes the Gaussian form,⟨r2⟩≈ λL. Here the Kuhn segment lengthλ is done by eq 9. Relyingon attraction between uncorrelated (distant) segments of thebrush because of the line tension (the monomer units of theside chains attract each other), the conformation can be imaginedas a dense packing ofL/λ blobs, each of the sizeλ. The Kuhnsegment length has a very strong dependence on the distributionof the side chains. It takes the maximum value,λ ) aM3/3, at∆ ) 0, and the minimum value,λ ≈ 4aM/27∆4, at R . 1. Inthe latter case, our theory can formally be applied to the regimeof maximum bending of the segments,∆ ∼ 1, that corresponds

to the densely packed 2D globule,⟨r2⟩ ∼ a2MN.

4. Concluding Remarks

We propose a theory of strongly adsorbed comblike macro-molecules (brushes) with random distribution of the side chainswith respect to the backbone. The decay of the orientationalcorrelation function, the gyration radius, and the Kuhn segmentlength are analyzed. We show that local asymmetry in the left-right distribution of the side chains, which is controlled by theregime of adsorption (initial 3D conformation of the brush), (i)promotes the decay of the orientational correlation function and(ii) decreases the gyration radius. The correlation function isshown to have a nonmonotonous behavior with one minimum.In contrast to linear chain, the gyration radius of the brush canbe independent of the contour lengthL in certain range ofLvalues.

Acknowledgment. The author thanks M. Mo¨ller for stimu-lating discussions. Financial support of the DFG within the SFB569, the VolkswagenStiftung (Germany), and the RussianAgency for Sciences and Innovations is gratefully acknowl-edged.

5. Appendix

Calculation of the orientational correlation function of eq 7can be done using Fourier transforms for the vectorx(s):

The thermodynamic average of the left-hand side of eq 12 canbe presented as

where the symbolδ denotes variation derivative,δhω′/δhω ≡δ(ω - ω′); Dx ≡ ∏ω dxω is a product of differentials and thefree energy of the brush molecule is rewritten using the Fourier

transforms, eq 14. The Gaussian integrals of eq 13 are calculatedby the following substitution of variables

The result is

where we used the condition 2πδ(ω ) 0) ) L. The factor 2 inthe first term of eq 16 comes from the dimension of the system.Taking into account thatnωn-ω ) ∆2aL (see eq 5), we get

by substitution of eq 16 into eq 12 and integration with respect

to ω. The condition⟨u(s)2⟩ ) 1 determines the Kuhn segmentlengthλ by means of equating of the right-hand side of eq 17to unity atτ ) 0:

and ultimate expression of the correlation function takes thefollowing form:

References and Notes

(1) Flory, P. J. Statistical Mechanics of Chain Molecules; OxfordUniversity Press: New York, 1988.

(2) Wittmer, J. P.; Meyer, H.; Baschnagel, J.; Johner, A.; Obukhov, S.;Mattioni, L.; Muller, M.; Semenov, A. N.Phys. ReV. Lett. 2005, 93,147801.

(3) Panyukov, S. V. Private communication.(4) Matyjaszewski, K.; Xia, J.Chem. ReV. 2001, 101, 2921.(5) Sheiko, S. S.; Mo¨ller, M. Chem. ReV. 2001, 101, 4099.(6) Birshtein, T. M.; Borisov, O. V.; Zhulina, E. B.; Khokhlov, A. R.;

Yurasova, T. A.Polym. Sci. USSR1987, 29, 1293.(7) Fredrickson, G. H.Macromolecules1993, 26, 2825.(8) Rouault, Y.; Borisov, O. V.Macromolecules1996, 29, 2605.(9) Subbotin, A.; Saariaho, M.; Ikkala, O.; ten Brinke, G.Macromolecules

2000, 33, 3447.(10) Potemkin, I. I.; Khokhlov, A. R.; Reineker, P.Eur. Phys. J. E2001,

4, 93.(11) Sheiko, S. S.; Sun, F. C.; Randall, A.; Shirvanyants, D.; Rubinstein,

M.; Lee, H.; Matyjaszewski, K.Nature (London)2006, 440, 191.(12) Potemkin, I. I.Macromolecules2006, 39, 7178.(13) Sheiko, S. S.; Prokhorova, S. A.; Beers, K. L.; Matyjaszewski, K.;

Potemkin, I. I.; Khokhlov, A. R.; Mo¨ller, M. Macromolecules2001,34, 8354.

(14) Gallyamov, M. O.; Tartsch, B.; Khokhlov, A. R.; Sheiko, S. S.; Bo¨rner,H. G.; Matyjaszewski, K.; Mo¨ller, M. Macromol. Rapid Commun.2004, 25, 1703.

x(s) ) ∫-∞

∞ dω2π

xω exp (iωs),

xω ) ∫-L/2

L/2ds x(s) exp (-iωs), Lf∞ (11)

⟨u(s)u(s′)⟩ )1L∫-∞

∞dω2π

ω2⟨xωx-ω⟩exp{iω(s - s′)} (12)

⟨xωx-ω⟩ ) δ2

δhωδh-ω

∫Dx exp(-F + ∫-∞

∞dωhωx-ω)

∫Dx exp(-F)|h)0

(13)

F ) ∫-∞

∞ dω2π [ω2xωx-ω

λ+

κ

2 (ω4xωx-ω + 12ω2

aMxωn-ω +

36nωn-ω

a2M2 )] (14)

xω f xω +2πhω - 6κ

aMω2nω

κω4 + 2ω2/λ(15)

⟨xωx-ω⟩ )

δ2

δhωδh-ωexp( ∫-∞

∞dω′

πhω′h-ω′ - 6κω2hωn-ω/aM

κω4 + 2ω2/λ )|h)0

)

2L

κω4 + 2ω2/λ+

36κ2nωn-ω

a2M2 (κω2 + 2/λ)2(16)

⟨u(s)u(s′)⟩ ) x λ2κ

exp(-τx 2λκ) +

9∆2

aM2 xλκ

2exp(-τx 2

λκ) (1 - τx 2λκ)

τ ) |s - s′| (17)

λ ) 2κ

(1 + 9∆2κ

aM2 )2(18)

⟨u(s)u(s′)⟩ ) exp{-τ (1κ + 9∆2

aM2)} (1 - τ 9∆2

aM2) (19)

Macromolecules, Vol. 40, No. 4, 2007 Snaky Conformations 1241

(15) Potemkin, I. I.Eur. Phys. J. E2003, 12, 207.(16) Sheiko, S. S.; da Silva, M.; Shirvaniants, D. G.; Rodrigues, C. A.;

Beers, K.; Matyjaszewski, K.; Potemkin, I. I.; Mo¨ller, M. Polym. Prepr.(Am. Chem. Soc., DiV. Polym. Chem.) 2003, 44, 544.

(17) Potemkin, I. I.; Khokhlov, A. R.; Prokhorova, S.; Sheiko, S. S.; Mo¨ller,M.; Beers, K. L.; Matyjaszewski, K.Macromolecules2004, 37, 3918.

(18) de Jong, J. R.; Subbotin, A.; ten Brinke, G.Macromolecules2005,38, 6718.

(19) Subbotin, A.; de Jong, J.; ten Brinke, G.Eur. Phys. J. E2006, 20, 99.(20) Stephan, T.; Muth, S.; Schmidt, M.Macromolecules2002, 35,

9857.(21) Gallyamov, M.; Tartsch, B.; Mela, P.; Potemkin, I. I.; Sheiko, S.;

Borner, H.; Matyjaszewski, K.; Khokhlov, A. R.; Mo¨ller, J. Polym.Sci. B, to be submitted for publication.

MA062099L

1242 Potemkin Macromolecules, Vol. 40, No. 4, 2007