Living Polymer Systems at a Solid Substrate: Computer Simulation of

Article pubs.acs.org/Macromolecules

Living Polymer Systems at a Solid Substrate: Computer Simulation of a Soft, Coarse-Grained Model and Self-Consistent Field Theory De-Wen Sun* and Marcus Müller* Institut für Theoretische Physik, Georg-August-Universität Göttingen, Friedrich-Hund-Platz 1, D 37077 Göttingen, Germany ABSTRACT: Using particle-based simulation of a soft, coarsegrained model and self-consistent field theory (SCFT), we investigate the properties of dense living polymer systems with ring formation both in the bulk and in thin films. In the bulk, our results confirm that the molecular weight distribution of ring polymers exhibits a combination of an exponential decay and a power law. The exponential molecular weight distribution of linear chains is hardly affected by ring formation, only the corresponding mean molecular weight is slightly reduced. At lower segment density, the fraction of monomers of ring polymers is increased. In thin films, ring formation does not influence the width of the narrow interface, where the segment density rises to the bulk value. Since the molecular extension of ring polymers is smaller than that of linear chains, the spatial extent of the wide interphase is, however, reduced. In the narrow interface, we find that the local ring formation is affected by five aspects: (1) the mirroring of chain conformations and back-folding by the solid substrate (Silberberg argument) and other four aspects at the solid substrate, i.e., (2) the reduced segment density, (3) the enrichment of chain ends, (4) the pronounced segregation of nonbonded monomers, and (5) the reduced dimensionality of ring polymers. In the wide interphase, the local ring formation is enhanced mainly by the first aspect. By comparing the results from the particle-based simulation and SCFT, we observe good agreement in the wide interphase, but the difference in the local structure leads to differences in the narrow interface. Additionally, the SCFT results show that a decrease of the film thickness increases the global ring formation within the thin film.

I. INTRODUCTION A living polymer system is a polydisperse mixture composed of nonbonded monomers, linear chains and ring polymers of different molecular weights, where the reversible bond forming and bond breaking are controlled by a bonding free energy, Eb.1,2 As illustrated in Figure 1, in living polymer systems, each living monomer is assumed to have two bonding sites as dictated by the chemical structure of monomers. Thereby the formation of branched structures and networks is prohibited. Each linear chain has two free bonding sites at the opposite chain ends, whereas every ring polymer has no free bonding site but it can be converted into a linear chain by bond breaking. The molecular weight distribution is not determined by the initial synthetic conditions but it is controlled by the bonding free energy, Eb, and it can adapt to the external perturbations like, e.g., the presence of a solid substrate. Disallowing the formation of ring polymers, previous analytical theories and computer simulations1 demonstrated that the equilibrium molecular weight distribution in the bulk follows an exponential decay and the corresponding mean molecular weight, ⟨N⟩, is governed by the density, ρ0, of segments (both nonbonded and bonded monomers) and the bonding free energy, Eb, i.e., ⟨N ⟩ ∼

(

E

systems without ring formation, we also observed an exponential decay of molecular weight distribution for all but the nonbonded monomers. The differences at the smallest molecular weight come from the indistinguishability of the two free bonding sites of each nonbonded monomer in the particlebased simulation and the indistinguishability of the two chain ends belonging to the same linear chain in the self-consistent field theory (SCFT).3 If a living polymer melt is brought into contact with a solid substrate, both linear chains and ring polymers will change their conformations on various length scales in order to pack at the confining wall. In analogy to monodisperse homopolymer melts,4,5 we distinguish between the narrow interface, which is the region in the ultimate vicinity of the solid substrate where the density of segments rises to its bulk value, and the wide interphase, where the large-scale conformations of polymers are perturbed by the presence of the solid substrate. The width of the narrow interface is set by the layering of the fluid of segments at the solid substrate or the correlation length, ξ = b0 / 12κ0 , of density fluctuations, where b0 denotes the statistical segment length and κ0 represents the inverse

)

ρ0 b0 3 exp − 2k bT , B

Received: May 16, 2015 Revised: July 17, 2015

where b0 denotes the statistical segment length of long flexible linear polymer chains. In our former study of living polymer © XXXX American Chemical Society

A

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There are, however, some conditions in the bulk where ring formation becomes important. For instance, if the density of segments is low, the probability that the two chain ends belonging to the same linear chain meet with each other is relatively higher than that in a dense melt, and the formation of ring polymers is enhanced. Ring formation has also attracted attention in the context of phase transitions.10 In the present manuscript, we highlight the differences between linear chains and ring polymers both in the narrow interface and in the wide interphase by confining living polymer systems between two parallel solid substrates. Since the chain ends are enriched in the narrow interface of a homopolymer melt and the corresponding surface tension (slightly) decreases with N, the species with low molecular weights will preferentially segregate at the sharp segment-density gradient in this region of a living polymer melt.3 Since ring polymers do not possess chain ends, their behavior differs from that of linear chains in the narrow interface. Additionally, in the narrow interface, the polymers can be conceived as quasi-twodimensional and the reduced dimensionality affects the molecular weight distribution of ring polymers. The deepening of the correlation hole both in the narrow interface and in the wide interphase increases the probability of the two chain ends belonging to the same linear chain to meet with each other and therefore enhances the formation of ring polymers compared to that of linear chains. These properties of living polymer systems at solid substrates may find applications in tailoring the interaction between surfaces or nanoparticles in these polydisperse melts.2 Our paper is arranged as follows: In section II, we formulate the field-theoretical model and the self-consistent field theory (SCFT) for ring-forming living polymer systems and describe the corresponding particle-based simulation of our soft, coarsegrained model. Subsequently, we present our results both in the bulk and in thin films and quantitatively compare the results from the particle-based simulation and SCFT calculations in section III. In particular, we focus on the influences of a solid substrate on ring formation as a function of the bonding free energy, Eb. Finally, a brief summary concludes the manuscript.

Figure 1. Schematic drawing of living polymer systems composed of nonbonded monomers, linear chains, and ring polymers. Each living monomer (large cyan circle) has two bonding sites (small red dots), and the reversible bond forming and bond breaking give rise to polydisperse species with different molecular weights.

isothermal compressibility. The latter quantity stems from the repulsive interactions between segments that restrain the fluctuations of local segment density. Note that the width of the narrow interface is to leading order independent from the molecular weight. The width of the wide interphase is dictated by the spatial extension, Re, of macromolecules, and for polydisperse living polymer systems we make use of the spatial extension of a macromolecule with the mean molecular weight, Re2 = b02(⟨N⟩ − 1), of all species,3 including nonbonded monomers, linear chains, and ring polymers. Thus, in the limit of large ⟨N⟩, there is a clear scale separation between the narrow interface and the wide interphase. According to the Silberberg argument,6 the effects of a solid substrate in the wide interphase can be regarded as mirroring the chain conformations at the polymer-solid contact plane. This back-folding of chain confirmations onto themselves results in a deepening of the correlation hole in the intermolecular pair correlation function.7 In living polymer systems, different species, distinguished by their molecular weights, can segregate to the solid substrate such that there is a “fractionation” between the narrow interface, the wide interphase, and the bulk. In the present manuscript, we allow for the reversible formation of ring polymers, and ring polymers compete with linear chains for available monomers.8 In the bulk, the molecular weight distribution of linear chains and ring polymers qualitatively differs.8,9 Let pL(N) and pR(N) denote the molecular weight distributions of linear chains and ring polymers, respectively, then the ratio, pL(N)/pR(N), is proportional to the Boltzmann

II. MODEL AND COMPUTATIONAL TECHNIQUES A. Field-Theoretical Model and Self-Consistent Field Theory (SCFT) for Ring-Forming Living Polymer Systems. Self-consistent field theory (SCFT) has been utilized to study living polymer systems both in the bulk and in thin films,2,11,12 however, the formation of ring polymers has been ignored. In our former work,3 we followed the field-theoretical model proposed by Feng and Fredrickson2 for living polymer systems without ring formation. Here, we consider the reversible formation of ring polymers in the SCFT calculations and obtain the corresponding mean-field solutions. It is important to note that the field-theoretical model that ignores the noncrossability of chain molecules is adequate here because the reversible bond forming and bond breaking do not fix the topology which, e.g., in a dense melt of nonliving nonconcatenated ring polymers, results in a collapse of ring polymers in a melt of other ring polymers.13,14 Since the reversible bond forming and bond breaking can alter the topology, there is no such effect in ring-forming living polymer systems. Therefore, by virtue of the screening of excluded volume in a dense melt, the chains adopt Gaussian conformations on large length scales. Additionally, we will employ the particle-based simulation of a soft, coarse-grained

( ), of the additional bond in the ring polymer,

weight, exp

Eb kBT

the number of positions where a ring polymer can be broken, N, and the volume, ∼ Nd/2, that the two neighboring monomers can explore after their bond is broken (where d denotes the spatial dimension).9 Therefore, pR(N) ∼ N−τ exp(−N/N0) with τ = d/2 + 1. For a three-dimensional melt, τ = 5/29 and these predictions in the bulk have been confirmed by computer simulations.8,9 Due to this pre-exponential power law in the molecular weight distribution of ring polymers, the population of monomers belonging to ring polymers is usually very small and the reversible formation of ring polymers is often ignored. B

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Macromolecules model15 and compare the results obtained from these two different methods. Based on Feng and Fredrickson’s model2 for living polymer systems without ring formation and our modified treatment of the indistinguishability of the two chain ends belonging to the same linear chain,3 we formulate our field-theoretical model of living polymer systems, in which the formation of ring polymers is allowed. The starting point is the discrete Gaussian chain Ξ(μM , V , T ) =

∞

∞

∞

∑

∑

∑

∞

⋯

nL(N ) = 0

∞

∞

∑

∑

∞

∑

⋯

nR (3) = 0 nR (4) = 0

⋯

nR (N ) = 0

e μM Ntot λT 3Ntot

1 1 1 n(1)! nL(2) ! nL(3) !⋯nL(N ) !⋯ nR (3) ! nR (4) !⋯nR (N ) !⋯ ⎛ ⎞nL(2) 1 ∏ dR σ⎜⎜ ⎟⎟ σ=1 ⎝2⎠ n(1)

×

∑

⋯

n(1) = 0 nL(2) = 0 nL(3) = 0

×

model in the grand canonical ensemble. We prevent the formation of branched and network architectures by considering that each living monomer has two bonding sites and can form at most two bonds. Thus, the system is comprised of nonbonded monomers, linear chains and ring polymers. We control the chemical potential, μM, of all segments, volume, V, and temperature, T. The grand canonical partition function takes the form:

∫

⎛ ⎞nL(N ) 1 ×⎜ ⎟ ⎜2⎟ ⎝ ⎠

∫ ∏ dR α

L

αL = 1

nL(N )

∫ ∏ dR γ

N

L

γL = 1

⎛ ⎞nR (4)⎛ ⎞nR (4) 1 ⎜1⎟ ×⎜ ⎟ ⎜4⎟ ⎜2⎟ ⎝ ⎠ ⎝ ⎠

∫

⎛ ⎞nL(3)

nL(2)

2⎜ 1 ⎟

nL(3)

∫ ∏ dR β 3⋯

⎜2⎟ ⎝ ⎠

L

βL = 1

⎛ ⎞nR (3)⎛ ⎞nR (3) 1 ⎜1⎟ ⋯⎜ ⎟ ⎜3⎟ ⎜2⎟ ⎝ ⎠ ⎝ ⎠

nR (3)

∫ ∏ dR α 3 R

αR = 1

⎛ ⎞nR (N )⎛ ⎞nR (N ) 1 1 ∏ dR βR 4⋯⎜⎜ ⎟⎟ ⎜⎜ ⎟⎟ β =1 ⎝N⎠ ⎝2⎠

nR (4)

nR (N )

∫ ∏ dR γ N ⋯

R

R

γR = 1

⎛ /[R , R 2, R 3, ⋯ , R N , ⋯ , R 3, R 4, ⋯ , R N , ⋯] ⎞ σ αL βL γL αR βR γR ⎟. × exp⎜⎜ − ⎟ k T B ⎝ ⎠

kB is the Boltzmann constant. λT denotes the thermal de Broglie wavelength arising from integrating out the momenta. n(1) is the number of nonbonded monomers. nL(N) is the number of linear chains composed of N (N ≥ 2) monomers, whereas nR(N) is the number of ring polymers comprised of N(N ≥ 3) monomers. The formation of ring polymers with N < 3 is forbidden. The prefactor, [n(1)!] −1 , accounts for the indistinguishability of all nonbonded monomers. Likewise, the prefactor, [nL(2)!nL(3)!···nL(N)!···]−1, represents the indistinguishability of linear chains with the same number of monomers and the prefactor, [nR(3)!nR(4)!···nR(N)!···]−1, accounts for the indistinguishability of ring polymers with the same number of monomers. Rσ denotes the spatial position of nonbonded monomer σ. R γL N = (R γL,1, R γL,2, ···, R γL,N) are the

∞

Ntot = n(1) +

(1)

∞

∑ nL(N )N + ∑ nR (N )N . N =2

(2)

N =3

This formalism allows us to adjust the value of μM either to achieve a given segment density in the bulk (grand canonical ensemble) or to match the average density of segments in a film investigated by the particle-based simulation of a soft, coarsegrained model (canonical ensemble). The total Hamiltonian, / = /b + /nb, is composed of a bonded part, /b, and a nonbonded contribution, /nb. The nonbonded Hamiltonian, /nb, is a functional of the local microscopic volume fraction, ϕ(̂ r), κ0ρ0 /nb = [ϕ(̂ r) − 1]2 d3r, kBT 2 (3)

∫

spatial positions of N (N ≥ 2) monomers belonging to linear chain γL, while R γR N = (R γR,1, R γR,2, ···, R γR,N) are the spatial

positions of N (N ≥ 3) monomers belonging to ring polymer γR. The prefactors, (1/2)nL(2), (1/2)nL(3), ···, (1/2)nL(N), ···, arise from the treatment of the indistinguishability of the two chain ends belonging to the same linear chain. The prefactors, (1/ 3)nR(3), (1/4)nR(4), ···, (1/N)nR(N), ···, stem from the fact that we can arbitrarily select one monomer of a ring polymer to be the starting point to label all the monomers of this ring polymer, and there are additional prefactors, (1/2)nR(3), (1/2)nR(4), ···, (1/ 2)nR(N), ···, that result from the choice of labeling the monomers of a ring polymer in a clockwise or counterclockwise fashion. Ntot is the number of all segments (both nonbonded and bonded living monomers) conjugated to the chemical potential, μM, and takes the expression

ϕ(̂ r) =

+

1 ρ0

1 ρ0 ∞

n(1)

∑ δ(r − R σ) + σ=1

∞

1 ρ0

nL(N ) N

∑ ∑ ∑ δ(r − R γ N = 2 γL = 1 i = 1

)

L, i

nR (N ) N

∑ ∑ ∑ δ(r − R γ N = 3 γR = 1 j = 1

),

R, j

(4)

where κ0 characterizes the inverse isothermal compressibility of a dense melt and ρ0 is the reference density of segments used in the particle-based simulation of our soft, coarse-grained model (see below). The bonded Hamiltonian, /b, consists of the bonding energy associated with the reversible bond forming and bond breaking and the energy of harmonic springs that connect the bonded monomers, i.e., C

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Macromolecules /b = 2hnlinear + kBT ∞

+

nR (N )

∑ ∑ N = 3 γR = 1

nL(N ) N

∞

3 |R γ − R γL,i−1|2 2b0 2 L,i

∑ ∑∑ N = 2 γL = 1 i = 2

⎡N ⎤ ⎢∑ 3 |R − R |2 + 3 |R − R |2 ⎥ , γ γR, j − 1 γ γR,1 ⎢⎣ ⎥⎦ 2b0 2 R,j 2b0 2 R,N j=2

microscopic volume fraction, ϕ(̂ r), and the bonded Hamiltonian, /b, can be rewritten as

∞

nlinear = n(1) +

∑ nL(N ),

(6)

N =2

Ξ(μM , V , T ) =

∞

∞

∑

∑

∑

n(1) = 0 nL(2) = 0 nL(3) = 0

×

+

1 ρ0 +

∑ δ(r − R σ) σ=1 nL(N )

∞

∫0

∑ ∑ N = 2 γL = 1

nR (N )

∞

1 ρ0

∑ ∑ N = 3 γR = 1

/b = 2hnlinear + kBT +

N

δ[r − R γL(s)] ds

∫0

⎛ ⎞ 1 ×⎜ ⎟ ⎜2⎟ ⎝ ⎠

∑ ∑

3 2b02

∑

⋯

nL(N ) = 0

∞

∞

∑

∑

∞

∑

⋯

nR (3) = 0 nR (4) = 0

nL(3)

⎛ ⎞

nL(3)

L

⎝ ⎠

βL = 1

⋯

nR (N ) = 0

nL(N )

∫ ∏ +R β ⋯⎜⎜ 12 ⎟⎟

L

⎛ ⎞nR (3)⎛ ⎞nR (3) 1 ⎜1⎟ × ⎜⎜ ⎟⎟ ⎜2⎟ 3 ⎝ ⎠ ⎝ ⎠

∫0

3 2b0 2 N

∫0

N

∂R γR(s) ∂s

∂R γL(s) ∂s

2

ds

2

ds , (8)

and the corresponding grand canonical partition function is

∫ ∏ +R α ⎜⎜ 12 ⎟⎟ αL = 1

(7)

∑ ∑

N = 3 γR = 1

⎛ ⎞

nL(2)

δ[r − R γR(s)] ds ,

N = 2 γL = 1

1 1 1 n(1)! nL(2) ! nL(3) !⋯nL(N ) !⋯ nR (3) ! nR (4) !⋯nR (N ) !⋯ nL(2)

N

nL(N )

∞

nR (N )

∞

∞

⋯

n(1)

1 ϕ(̂ r) = ρ0

where b0 is the root of the mean-squared distance between two bonded monomers in the absence of nonbonded interactions and, in the following, all length scales will be expressed in units of b0. nlinear denotes the number of nonbonded monomers and linear chains. In the treatment of the bonding energy, when two potential bonding sites are connected by a bond, the corresponding energy is zero, otherwise, the energy of a broken bond is 2hkBT.2,16 This scheme is equivalent to assigning the bonding free energy Eb = −2hkBT3 to the formation of a bond. Thereby nlinear also represents the number of broken bonds. Each nonbonded monomer and each linear chain contribute one broken bond each. Since all potential bonding sites of a ring polymer are connected, ring polymers do not contribute to the number of broken bonds. In the continuum limit of the discrete Gaussian chain model, linear chains and ring polymers can be described as continuous open and closed space curves, respectively. The local ∞

(5)

⎝ ⎠

e μM Ntot λT 3Ntot n(1)

∫ ∏ dR σ σ=1 nL(N )

∫ ∏ +R γ ⋯ L

γL = 1

nR (3)

∫ ∏ +R α δ[R α (3) − R α (0)]

⎛ ⎞nR (4)⎛ ⎞nR (4) 1 ⎜1⎟ ×⎜ ⎟ ⎜4⎟ ⎜2⎟ ⎝ ⎠ ⎝ ⎠ ⎛ ⎞nR (N )⎛ ⎞nR (N ) 1 ⎜1⎟ ×⎜ ⎟ ⎜N⎟ ⎜2⎟ ⎝ ⎠ ⎝ ⎠

R

R

R

αR = 1 nR (4)

∫ ∏ +R β δ[R β (4) − R β (0)]⋯ R

R

R

βR = 1 nR (N )

∫ ∏ +R γ δ[R γ (N ) − R γ (0)]⋯ R

R

R

γR = 1

⎛ /[R σ , R α , R β , ⋯ , R γ , ⋯ , R α , R β , ⋯ , R γ , ⋯] ⎞ L R L L R R ⎟⎟ . × exp⎜⎜ − k T ⎝ ⎠ B

We conceive the ring polymers as linear chains with an

(9)

same linear chain reside at the same location.17,18 This condition gives rise to the δ−functions, δ[R αR(3) − R αR(0)],

additional constraint that the two chain ends belonging to the D

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Macromolecules δ[R βR(4) − R βR(0)], ···, δ[R γR(N ) − R γR(0)], ···, in the

Decoupling the nonbonded interactions via a Hubbard− Stratonovich transformation,19 we obtain the grand canonical partition function as

equation above. Ξ(μM , V , T ) ∼

⎛

⎞

∫ +ϕ(r) ∫ +ω(r) exp⎜⎝− /[ϕ(kr),T ω(r)] ⎟⎠,

(10)

B

with the effective Hamiltonian, /[ϕ(r), ω(r)], being κ0ρ0 /[ϕ(r), ω(r)] = kBT 2

∫ [ϕ(r) − 1]2 d3r − ρ0 ∫ [ω(r)ϕ(r)] d3r − ρ0 e−2h ρVg

1 V − ρ0 e−2h ρ0 gM 2

∞

∑ z MNQ L(N ) − ρ0 N =2

over the auxiliary fields, ϕ(r) and ω(r), respectively. zM = (gM/ λT3) eμM denotes the dimensionless activity of segments, where gM = (2πb02/3)3/2 is the volume of a living bond and it can be computed via the discrete Gaussian chain model.2 Q(1), QL(N), and QR(N) are the normalized partition functions of a

{

Q R (N ) =

N ∂R γL(s) 2 | ∂s

{

N

N ∂R γL(s) 2 | ∂s | 0

0

N ∂R γR (s) 2 | ∂s

∫ +R γRδ[R γR(N ) − R γR(0)] exp − 2b3 2 ∫0 | 0

{

ds

(11)

for N ≥ 2, (13)

N

ds − ∫ ω[R γR(s)] ds 0 N ∂R γR (s) 2 | ∂s

0

}

}

∫ +R γRδ[R γR(N ) − R γR(0)] exp − 2b3 2 ∫0 |

ds

}

}

for N ≥ 3, (14)

ω(r) = κ0[ϕ(r) − 1],

where

∫

N =3

1 N z M Q R (N ). N 5/2

ds − ∫ ω[R γL(s)] ds 0

∫ +R γL exp − 2b3 2 ∫

{

∞

∑

∫

∫ +R γL exp − 2b3 2 ∫0 | 0

1 V 2 ρ0 gM

nonbond monomer, of a linear chain with degree of polymerization N (N ≥ 2), and of a ring polymer with degree of polymerization N (N ≥ 3) in the external potential, ω(r)), respectively, i.e. 1 dR σ exp{−ω(R σ)}, Q (1) = (12) V

∫ +ϕ(r) and ∫ +ω(r) represent the functional integrations

Q L(N ) =

z MQ (1)

0 M

⎧ ⎪ 3 +R γL exp⎨− 2 ⎪ 2b0 ⎩

∫0

N

∂R γL(s) ∂s

2

∞

⎫ ⎪ ds⎬ = VgM N − 1 ⎪ ⎭

for N ≥ 2,

ϕ(r) = ϕ(r, 1) +

∫ +R γ δ[R γ (N ) − R γ (0)] ⎧ ⎪ 3 exp⎨− 2 ⎪ 2b0 ⎩ for N ≥ 3.

∑ ϕR (r, N ) N =3

(18)

with (15)

R

∑

∞

ϕL(r, N ) +

N =2

ϕ(r, 1) = R

(17)

e − 2h z M exp[−ω(r)], ρ0 gM

(19)

R

∫0

N

∂R γR(s) ∂s

2

⎫ ⎪ 1 ds⎬ = 3/2 V gM N − 1 ⎪ N ⎭

ϕL(r, N ) =

1 e − 2h N zM 2 ρ0 gM

∫0

N

qL(r, s)qL(r, N − s) ds

for N ≥ 2,

(16)

We obtain the self-consistent field equations by minimizing the effective Hamiltonian, /[ϕ(r), ω(r)], with respect to ϕ(r) and ω(r). Invoking the mean-field approximation, we find that the saddle-point value of ϕ(r) is the dimensionless local volume fraction and ω(r) is the corresponding potential field conjugate to ϕ(r). The self-consistent field equations are

ϕR (r, N ) =

∫0

(20) 2 3/2 1 1 ⎛ 2πb0 ⎞ 1 N ⎜ ⎟ zM 2 ρ0 gM ⎝ 3 ⎠ N

∫ dr0

N

qR (r0, 0; r, s)qR (r0, N ; r, N − s) ds

for N ≥ 3, (21)

E

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is ntot=nlinear+nring and the corresponding mean molecular weight is defined by ⟨N⟩ = Ntot/ntot. Likewise, ⟨N⟩linear = Nlinear/nlinear is the mean molecular weight of nonbonded monomers and linear chains, whereas ⟨N⟩ring = Nring/nring is the mean molecular weight of ring polymers. We consider a thin film with thickness D ≈ 22.63b0 and spatial discretization ΔL = D/256 ≈ 0.088b0, set κ0 = 1.5625 to restrain the fluctuations of local segment density, and vary the bonding free energy, Eb, in the range from −1kBT to −7kBT. Within the mean-field approximation, all quantities such as ϕ(r) and ω(r) only vary in the direction x perpendicular to the solid substrate, and one-dimensional calculations can be performed. The one-dimensional modified diffusion equations for the chain propagators are solved by a combination of the pseudospectral method20,21 and Richardson extrapolation22 under Dirichlet boundary conditions.16 We provide further details of these calculations in Appendix A. B. Particle-Based Simulation of a Soft, Coarse-Grained Model for Ring-Forming Living Polymer Systems. We complement the results of SCFT calculations by the particlebased simulation of a soft, coarse-grained model. We largely follow the algorithm in our previous work3,15 for living polymer systems without ring formation, but in the present work the reversible formation of ring polymers is allowed. We consider Ntot living monomers in a volume V with linear dimension D ≈ 22.27b0 and spatial discretization ΔL = D/32 ≈ 0.696b0 in all three Cartesian directions. The bonded Hamiltonian, /b, takes the form

where qL(r,s) is the chain propagator of linear chains, giving the probability of finding segment s at spatial position r, and satisfies the following modified diffusion equation, i.e., b2 ∂ qL(r, s) = 0 ∇2 qL(r, s) − ω(r)qL(r, s), ∂s 6

(22)

with the initial condition qL(r,0) = 1, whereas qR(r0,0;r,s) and qR(r0,N;r,s) are the chain propagators of ring polymers, which represent the probabilities of finding segment s at spatial position r from the starting point fixed at position r0 clockwise and counterclockwise, respectively, and satisfy the modified diffusion equations in the following, i.e., b2 ∂ qR (r0, 0; r, s) = 0 ∇2 qR (r0, 0; r, s) ∂s 6 − ω(r)qR (r0, 0; r, s),

(23)

b2 ∂ qR (r0, N ; r, s) = 0 ∇2 qR (r0, N ; r, s) ∂s 6 − ω(r)qR (r0, N ; r, s),

(24)

with the initial conditions qR(r0,0;r,0) = δ(r − r0) and qR(r0,N;r,N) = δ(r − r0), respectively. In terms of the chain propagators, the single-molecule and single-chain partition functions can be expressed as Q (1) =

1 V

∫ exp[−ω(r)] d3r,

1 Q L(N ) = V

(25)

∫ qL(r, s)qL(r, N − s) d r 3

/b 1 = 2 kBT

for N ≥ 2, (26)

⎛ 2πb 2 ⎞3/2 1 0 ⎟ Q R (N ) = N3/2⎜ ⎝ 3 ⎠ V qR (r0, N ; r, N − s) d3r

(27)

The factor, (2πb02/3)3/2, in the expressions of ϕR(r,N) and QR(N) come from the additional bond in a ring polymer. Integrating the dimensionless local volume fractions of nonbonded monomers, linear chains, and ring polymers, respectively, we obtain the corresponding global molecular weight distributions as pL (1) =

∫ [ϕ(r, 1)ρ0 ] d3r = e−2h gV zMQ (1), M

pL (N ) =

1 N

pR (N ) = N

1 N

ϕ(̂ r) =

(28)

(29)

z M Q R (N )

M

for N ≥ 3.

(31)

1 ρ0

∫ [ϕ(̂ r) − 1]2 d3r,

(32)

Ntot

∑ δ(r − R i),

(33)

i=1

where the local microscopic volume fraction, ϕ(̂ r), depends on the spatial positions, {Ri}, explicitly in the particle-based simulation. ρ0=Ntot/V is the average density of segments. The local segment density and the nonbonded interactions are evaluated on a cubic collocation lattice with a spatial resolution of ΔL. At each grid point, c, the local microscopic volume fraction can be evaluated from the spatial positions of living monomers on the basis of

M

∫ [ϕR (r, N )ρ0 ] d3r = 12 gV

i=1 j=1

κ0ρ0 /nb = kBT 2

∫ [ϕL(r, N )ρ0 ] d3r = 12 e−2h gV zM NQ L(N )

for N ≥ 2,

⎛ E 3[R i − R j]2 ⎞ b ⎟. + ⎟ 2 k T 2 b B ⎝ ⎠ 0

Ri and Rj are the spatial positions of monomers i and j, respectively. The matrix C[i,j] is employed to represent the connectivity of monomers in each system configuration. If monomers i and j are bonded with each other, C[i,j] = 1, and C[i,j] = 0 otherwise. The factor 1/2 compensates for double counting because the sum is taken over all pairs of monomers. The nonbonded Hamiltonian, /nb, is also expressed via a functional of the local microscopic volume fraction, ϕ(̂ r),

∫ dr0 ∫ qR (r0, 0; r, s) for N ≥ 3.

Ntot Ntot

∑ ∑ C[i , j]⎜⎜

1 N 5/2 (30)

In the following, we utilize Nlinear to denote the number of nonbonded monomers and bonded monomers belonging to linear chains, and use Nring to represent the number of bonded monomers belonging to ring polymers. nlinear is the number of nonbonded monomers and linear chains, while nring is the number of ring polymers. Thus, the total number of molecules

ϕ(̂ c|{R i}) =

1 (ΔL)3 ρ0

Ntot

∑ Π(c|{R i}), i=1

(34)

where Π(c|{Ri}) describes how to assign the position of a living monomer to the grid points on the collocation lattice. In the present work, we use a linear assignment function as we did F

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Figure 2. Normalized bulk molecular weight distribution, cL(N), of nonbonded monomers and linear chains in ring-forming living polymer systems for different bonding free energies, Eb, obtained by the particle-based simulation (part a on the left) and SCFT calculations (part b on the right), respectively. The inset shows how the bonding free energy, Eb, affects the mean molecular weight, ⟨N⟩linear, of nonbonded monomers and linear chains.

Figure 3. Normalized bulk molecular weight distribution, cR(N), of ring polymers for different bonding free energies, Eb, obtained by the particlebased simulation ((a) in the left) and SCFT calculations ((b) in the right), respectively. The large inset shows the ratio

( ) E

[pR (N )/pL (N )]/exp − k Tb , for different bonding free energies, Eb. The small inset displays how the bonding free energy, Eb, influences the B

mean molecular weight, ⟨N⟩ring, of ring polymers.

previously,3,15 i.e., each living monomer contributes to the 8 corners of the cubic grid cell it is located in. Then the nonbonded Hamiltonian, /nb, can be rewritten as κ0ρ0 /nb({R i}) = (ΔL)3 ∑ [ϕ(̂ c|{R i}) − 1]2 . kBT 2 c

We apply periodic boundary conditions in all directions for the systems in the bulk, while for the systems in thin films between two parallel solid substrates, periodic boundary conditions are only performed in the directions parallel to the solid substrates. The bonding free energy, Eb, ranges from −1kBT to −9kBT. We use 20 000 MC steps to equilibrate each system, and after this equilibration period, the measurements are performed every 10 MC steps. The total simulation for each system lasts 1 220 000 MC steps in order to obtain sufficient statistics of investigated properties, especially for that of ring polymers. κ0 is set to 1.5625 as in the SCFT calculations. We also provide further details of these calculations in Appendix A.

(35)

We utilize Monte Carlo (MC) simulations23 to investigate this soft, coarse-grained, particle-based model for ring-forming living polymer systems. Smart Monte Carlo (SMC) moves24 biasing the proposed trial displacement of a segment along the direction of strong bonded forces that give rise to Rouse-like dynamics25,26 are employed in order to benefit from the separation between the stronger bonded interactions and the soft nonbonded interactions.27 We define one MC step as the number of SMC trial displacements, during which−on average−each living monomer has the opportunity to move once. Additionally, during each MC step, each living monomer has a chance to change its connectivity. We employ reversible bond-forming and bond-breaking moves to equilibrate the connectivity between monomers,1,23 and these connectivity altering moves fulfill the detailed balance by accounting for the instantaneous number of possible bonding partners in the vicinity of a living monomer.23

III. RESULTS AND DISCUSSION A. Bulk Systems. For ring-forming living polymer systems in the bulk, the self-consistent field theory (SCFT) can be solved analytically (see Appendix B) and predicts the molecular

( ) exp( )

weight distributions to be pL (1) = exp nonbonded monomers, pL (N ) =

1 2

linear chains with N ≥ 2, and pR (N ) =

Eb kBT

V gM

Eb kBT

V gM

1 V 1 2 gM N 5/2

( ) for exp(− ) for exp(− ) for 1

exp − N

0

N N0

N N0

ring polymers with N ≥ 3, respectively, in three-dimensional G

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increases with −Eb monotonously and attains its maximum at Eb → −∞, when Eb → −∞, we obtain zM → 1 and N0 = (−ln zM)−1 → ∞. Therefore, the asymptotic molecular weight 1 distribution of ring polymers takes the form pR (N ) ∼ 5/2 , and N the corresponding limiting value of ⟨N⟩ring can be obtained by

space. Here N0 ≡ ( −ln z M)−1 ≈ ⟨N ⟩linear depends on the value of Eb. This analytical prediction demonstrates that the presence of ring polymers does not alter the exponential decay of molecular weight distribution of linear chains. In accord with previous results without ring formation, we find pL(N = 1) ≈ 2pL(N = 2)3 as well. Figure 2 presents the scaled molecular weight distribution, c L(N )⟨N ⟩linear = (pL (N )/nlinear)⟨N ⟩linear , and the corresponding mean molecular weight, ⟨N⟩linear, of nonbonded monomers and linear chains for different bonding free energies, Eb. Panel (a) depicts the results from the particle-based simulation, whereas panel (b) displays the SCFT predictions. Similar to the previous results for melts without ring formation8,9 and our analytical prediction, we observe that cL(N)·⟨N⟩linear collapses on a single exponential master distribution when plotted against

(

E

N/⟨N⟩linear. The average scales like ⟨N ⟩linear ∼ exp − 2k bT B

∞

numerically evaluating ⟨N ⟩ring =

∑N = 3 NpR (N ) ∞

∑N = 3 pR (N )

≈ 7.65 for dis-

crete molecular weight distribution. Additionally, for continuous molecular weight distribution, since ⟨N⟩ringcan be defined ∞

∫3 NpR (N ) dN

as ⟨N ⟩ring = be

∞

∫3 pR (N ) dN

, the corresponding limiting value can

obtained via analytical

calculations

of

∞

⟨N ⟩ring =

∫3 NpR (N ) dN ∞

∫3 pR (N ) dN

= 9 when Eb → −∞. Since the smallest

limiting value of ⟨N⟩ring is 3, the mean molecular weight of ring polymers only varies in a very small range. Figure 4 presents the fraction, Nring/Ntot, of the number, Nring, of monomers belonging to all ring polymers and the

)

apart from very minor corrections that arise from the indistinguishability of the two free bonding sites of each nonbonded monomer in the particle-based simulation and the indistinguishability of the two chain ends belonging to the same linear chain in the SCFT calculations, respectively, and the small fraction of monomers of ring polymers. Figure 3 presents the normalized molecular weight distribution, cR(N) = pR(N) /nring, of ring polymers and the corresponding mean molecular weight, ⟨N⟩ring, obtained by the particle-based simulationFigure 3aand SCFT calculationsFigure 3brespectively. The data for different bonding free energies, Eb, clearly show the deviations from a power-law distribution. To verify the analytical prediction, we plot the

( ) in the large insets of these E

ratio [pR (N )/pL (N )]/exp − k Tb B

two panels. Both the particle-based simulation and SCFT calculations corroborate the power-law behavior N −5/2 of this ratio and quantitatively agree. This power law suggests that the density, ρ0, of segments and the strength of soft repulsion, κ0, are appropriate to describe a dense melt; a different behavior,

( )∼N E

[pR (N )/pL (N )]/exp − k Tb

−2.92

Figure 4. Fraction, Nring/Ntot, of the number, Nring, of monomers belonging to all ring polymers and the number, Ntot = ρ0V, of all segments (both nonbonded and bonded monomers), as a function of the bonding free energy, Eb, for three different segment densities and two computational techniques in the bulk. Curves marked by open symbols and solid symbols correspond to the particle-based simulation (PBS) and SCFT calculations, respectively.

, would be expected in

B

8,9

the dilute limit. The small insets of these two panels show that the mean molecular weight, ⟨N⟩ring, of ring polymers increases when the value of −Eb becomes larger but the dependence is much weaker than that for linear chains. We also note that the formation of ring polymers slightly decreases the mean molecular weight, ⟨N⟩linear, of nonbonded monomers and linear chains compared to the case without ring formation because the value of zM is slightly smaller. The observation that ⟨N⟩ring only weakly depends on the value of −Eb is mainly due to the pre-exponential power law in the molecular weight distribution of ring polymers. This powerlaw dictates the value of ⟨N⟩ring for large, negative values of Eb and makes ⟨N⟩ring remain finite as Eb → −∞. According to our analytical solutions of SCFT for living polymer systems with ring formation in the bulk, the molecular weight distribution of ring polymers is pR (N ) ∼

1 N 5/2

( ) with N = (−ln z N

exp − N

0

0

−1 M)

number, Ntot=ρ0V, of all segments (both nonbonded and bonded monomers), as a function of the bonding free energy, Eb. The different data sets correspond to three different segment densities and two computational techniques, i.e., open symbols for the particle-based simulation and solid symbols for the SCFT calculations, respectively. Both techniques predict that the fraction, Nring/Ntot, increases when the bonding free energy, Eb, becomes more negative (at fixed segment density, ρ0). This effect arises from the additional bond whose Boltzmann factor favors the formation of ring polymers at large negative values of Eb. At fixed value of Eb, in turn, we observe that Nring/Ntot becomes larger when the density of segments decreases. This effect partially stems from the deepening of the correlation hole upon decreasing the segment density, i.e., a chain end that is available for bonding in the surrounding of an end segment is more likely to be the other end segment belonging to the same linear chain than the chain end of other linear chain as we decrease the segment density. Additionally, the local structure

and 0 < zM

< 1 for N ≥ 3, and ⟨N⟩ring is defined as ⟨N ⟩ring =

∞

∑N = 3 NpR (N ) ∞

∑N = 3 pR (N )

because in this work we employ the discrete molecular weight distribution (see Appendix B). Since the value of ⟨N⟩ring H

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Figure 5. (a) Normalized distance, R(N )/ N − 1 , along a linear chain as a function of N for the highest segment density and various values of Eb. (N)/pPBS (b) Ratio, pSCFT R R (N), as a function of the molecular weight, N, for the highest segment density and various values of Eb.

of the fluid of segments depends on the density of segments in the particle-based simulation. Qualitatively, the results obtained by the particle-based simulation and SCFT calculations agree, however, we find that the value of Nring/Ntot in the particle-based simulation is smaller than that in the SCFT calculations, especially when the density of segments is low and the value of −Eb is large. For the most negative bonding free energy, Eb = −9kBT, and the lowest segment density, Ntot = 8192, the difference can be nearly 10%. Partially these deviations between the particle-based simulation and SCFT calculations can be rationalized by the effect of segmental repulsion which, albeit very soft, gives rise to some packing effects in the fluid of segments and a swelling of chain conformations due to the excluded-volume effect in the particle-based simulation. The latter effect reduces the probability that the two chain ends belonging to the same linear chain meet in the particle-based simulation, and thereby it decreases the formation of ring polymers. This explanation is corroborated by the following consideration. In the SCFT calculations, the excluded-volume effect is screened down to the smallest length scale because the correlations are ignored or the invariant degree of polymerization, 5̅ , is infinitely large. Therefore, the chain conformations are Gaussian on all length scales and the excluded-volume interaction is replaced by the incompressibility constraint. However, in the particle-based simulation, since the correlations are duly accounted for and the corresponding density of chains is finite, the excluded-volume effects are important up to

At the lowest density of segments, we obtain for the overlap ρR

the screening length, ξ ∼ R e0/ 5̅ , of excluded-volume interactions. In the blob picture, on length scales smaller than ξ, i.e., inside the excluded-volume blob, the macromolecular conformations resemble that of isolated chain, whereas they adopt Gaussian statistics outside of the excluded-volume blob (albeit with the renormalized statistical segment length R

2

R

3

(N )

≈ 0.7 N , i.e., linear chains or ring parameter, 5 = 0 e0N polymers with low molecular weights do not strongly overlap and the local excluded-volume effects below the screening length of excluded-volume interactions are not screened. The strength of segmental repulsion, however, is weak and the deviations from the Gaussian chain behavior of the SCFT increases upon increasing N (weak-coupling branch). We observe in Figure 5a that R(N )/ N − 1 increases with N in the particle-based simulation because the Fixman parameter, vN 2/R e0 3(N ) = v N /b0 3, that describes the strength of the segmental repulsion increases, where v is the excluded volume of a segment. Additionally, in Figure 5a, we observe that the deviations from the Gaussian statistics due to the local excluded-volume interactions arise on small length scales, i.e., inside the excluded-volume blob, whereas on large length scales, we observe that the internal distances obey Gaussian statistics albeit with the slightly increased (renormalized) statistical segment length br. Thus, we expect the consequences of swelling in the particle-based simulation to be more pronounced for larger N. Indeed, if we plot the ratio, pSCFT (N)/pPBS R R (N), of the probabilities of a ring polymer with molecular weight N obtained by SCFT calculations and particle-based simulations, respectively, in Figure 5b, we observe that this ratio is larger than unity and it increases with N. For larger N, the linear chains are swollen more in the particle-based simulation, and their chain ends are less likely to meet than that in the SCFT calculations. Therefore, ring polymers form less frequently for large N in the particle-based simulation than that in the SCFT calculations. We note that, (N)/pPBS although pSCFT R R (N) is larger when N is large, the difference of Nring/Ntot between these two computational methods is still dominated by short ring polymers because of the pre-exponential power law in the molecular weight distribution of ring polymers. When the density of segments decreases, the excludedvolume effects on ring formation become stronger in the particle-based simulation because the polymers overlap less and therefore the difference of Nring/Ntot between these two computational methods increases. B. Film Systems. We confine the living polymer system with ring formation into a thin film with a constant thickness D (cf. section II) and discuss how the properties of ring-forming living polymer systems are influenced by the presence of a solid substrate. The number of all segments (both nonbonded and

2

br 2 ≡ N −e 1 ≥ N −e0 1 ≡ b0 2 ). Note here that Re0 is the root of the mean-squared end-to-end distance of a chain with N segments without excluded-volume interactions while Re is the root of the mean-squared end-to-end distance of a chain with N segments in a homogeneous melt. In the particle-based simulation of our soft, coarse-grained model, since the segmental excluded volume is small, its effect inside the excluded-volume blob can be treated as a perturbation, i.e., an expansion in powers of the Fixman parameter. I

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Macromolecules bonded monomers) within the film is Ntot = 32768 in the particle-based simulation and in the SCFT calculations we adjust the activity, zM, accordingly. We study the local properties as a function of the distance, x, from the solid substrate averaging over slabs of thickness ΔL. The spatial resolutions are ΔL = D/32 and ΔL = D/256 in the particlebased simulation and SCFT calculations, respectively. This discretization can be compared with the width of the narrow interface, ξ = b0 / 12κ0 ≈ 0.23b0 ≈ 0.01D. Within the spatial discretization, ΔL, which also sets an effective interaction range in the particle-based simulation (cf. Section IIB), the narrow interface at the solid substrate can not be resolved in the particle-based simulation. This is a quite common situation in highly coarse-grained polymer models where one coarsegrained segment corresponds to many chemical repeating units.28,29 In the SCFT calculations, the profiles have a finer spatial discretization than ξ, allowing us to study the local properties within the narrow interface. First, we present in Figure 6 the normalized profiles, Ntot(x)/ (NtotΔL/D), of the segment density for different bonding free

of Re(⟨N⟩) is significantly smaller compared with the corresponding results without ring formation,3 especially when the value of −Eb is large. For instance, Re(⟨N⟩) can be reduced by about 60% compared to the case without ring formation when Eb = −9kBT. The segment density profiles are almost independent from the value of Eb. Both calculations by the particle-based simulation and SCFT reveal a minor broadening of the narrow interface for large mean molecular weight (or more negative Eb). For small values of −Eb, however, the fraction of nonbonded monomers is significantly larger, and they segregate into the narrow interface (cf. Figure 8) by virtue of their higher translational entropy and the absence of conformationalentropy loss at the impenetrable boundary. Additionally, chain ends of linear chains are enriched in the steep segment-density gradient of the narrow interface.4 This enrichment of chain ends at the narrow interface facilitates the two chain ends belonging to the same linear chain to meet and form a ring polymer. Furthermore, as in the bulk, we expect that the lower density of segments in the narrow interface enhances the fraction of monomers belonging to ring polymers. For the investigated film thickness, the overall increase of the fraction of monomers belonging to ring polymers only increases by about 4−9% in the particle-based simulation and 7−11% in the SCFT calculations, respectively, when the bonding free energy, Eb, varies from −1kBT to −7kBT. Moreover, the overall population of monomers belonging to ring polymers remains much smaller than that of nonbonded monomers and linear chains, i.e., the enhancement of ring formation is a surface effect. Figure 7 presents the profiles of the fraction, Nend(x)/Ntot(x), of chain ends of linear chains for different bonding free

Figure 6. Segment density profiles, Ntot(x) /(NtotΔ L/D), for different bonding free energies, Eb. The arrow on the top marks the spatial extension, ξ, of the narrow interface. The main panel and large inset show the results from the particle-based simulation and SCFT calculations, respectively. The small inset displays how the bonding free energy, Eb, affects the root of the mean-squared end-to-end distance, Re(⟨N⟩), of a linear chain with the mean molecular weight, ⟨N⟩.

energies, Eb. The main panel and large inset correspond to the particle-based simulation and SCFT calculations, respectively. Similar to monodisperse melts of linear chains4,28 or living polymers without ring formation,3 there is a depletion zone− the narrow interface of width ξ − at the solid substrate. Since we fix the number of all segments within the film, the value of Ntot(x) /(NtotΔL/D) in the middle of the film exceeds unity by an amount of order 2ξ/D. The small inset compares the root of the mean-squared end-to-end distance of a linear chain with the mean molecular weight, ⟨N⟩, with the film thickness, D. We have chosen the range of Eb so that D > 2R e(⟨N ⟩) ≫ ξ . Thus, we expect that in the middle of the film the behavior is similar to the bulk behavior, and there is a scale separation between the narrow interface and the wide interphase with spatial extension Re(⟨N ⟩) . The mean molecular weight, ⟨N⟩, of all species, including nonbonded monomers, linear chains, and ring polymers, is significantly reduced by ring formation, and therefore the value

Figure 7. Profiles of the fraction, Nend(x)/Ntot(x), of chain ends of linear chains for various bonding free energies, Eb. The main panel shows the results from the particle-based simulation, whereas the large inset presents the results from SCFT calculations. The profiles are normalized by the corresponding thin-film average, Nend/Ntot, depicted in the small inset for different values of Eb.

energies, Eb. These profiles are normalized by the correspond∞ ing thin-film average, Nend/Ntot (with Nend = 2 ∑N = 2 nL(N ) being the number of chain ends of linear chains and Ntot=ρ0V), in order to highlight the effects of the solid substrate. This thinfilm average value, Nend/Ntot, is shown in the small inset as a function of the bonding free energy, Eb. J

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film. Thus, the nonmonotonous behavior arises due to the presence of nonbonded monomers that are numerous for small value of −Eb and segregate at the solid substrate but do not contribute to the chain-end fraction of linear chains. In order to corroborate this explanation, we present in Figure 8 the profiles of the fraction, Nnb(x)/Ntot(x), of nonbonded monomers. Indeed, the SCFT calculations in the large inset of Figure 8 reveal a significant fraction of nonbonded monomers of nearly 80% when Eb = −1kBT in the ultimate vicinity of the impenetrable boundary. This increase stems from entropic reasons−high translational entropy and no loss of conformational entropy−that apply to “non-living” polymers and, additionally, the solid substrate reduces the number of potential bonding partners (“missing-neighbor” effect). When the value of −Eb is large, however, the increase of nonbonded monomers in the narrow interface is much less pronounced, leaving room for chain ends of linear chains. There is no such pronounced segregation of nonbonded monomers observed in the particle-based simulation, cf. main panel of Figure 8. This is in accord with the monotonous decay of the chain-end fraction in the main panel of Figure 7. The reason for both effects is the difference of spatial resolution of the profiles and the different range of interactions between the particle-based simulation and SCFT results. We have verified that both of them give rise to qualitative differences between the particle-based simulation and SCFT calculations in the narrow interface. Farther than the interaction range or the statistical segment length away from the solid substrate−in the wide interphase and the middle of the filmboth models yield very similar results. The entropic reasons that give rise to the interface excess of nonbonded monomers and the enrichment of chain ends of linear chains also affect the formation of ring polymers. The reduced segment density and the pronounced enrichment of chain ends in the narrow interface favors the formation of ring polymers in the narrow interface. Additionally, the reduced dimensionality of ring polymers with their center of mass in the narrow interfaceeffectively two-dimensional ring polymers increases the probability of the two chain ends belonging to the same linear chain to meet and consequently increases the formation of ring polymers in the ultimate vicinity of the solid substrate. In addition to these four effects that only operate in the narrow interfaces, however, there is one important effect−the mirroring of chain conformations and back-folding by the solid

The particle-based simulation, shown in the main panel of Figure 7, exhibits an enrichment of chain ends at the narrow interface. Note that range of the enrichment zone in the vicinity of the solid substrate does not depend on the value of Eb or the mean molecular weight, i.e., the chain-end enrichment is characteristic for the narrow interface but does not extend to the wide interphase. This independence of the profiles in the wide interphase and the middle of the film from the bonding free energy, Eb, is also observed in the SCFT calculations, which are presented in the large inset of Figure 7. In fact, apart from a very minor depletion of chain ends in the wide interphase, the chain-end profiles (as well as the profiles of nonbonded monomers in Figure 8) do not reveal significant changes in the wide interphase as compared to the middle of the film or the bulk.

Figure 8. Fraction, Nnb(x) /Ntot(x), of nonbonded monomers as a function of the distance, x, from the solid substrate for different bonding free energies, Eb: Particle-based simulation in the main panel and SCFT calculations in the large inset. The small inset displays how the value of Eb influences the thin-film average, Nnb/Ntot.

The SCFT calculations indicate, however, that the fraction of chain ends exhibits a nonmonotonous behavior in the narrow interface when −Eb ≤ 3kBT. In the ultimate vicinity of the solid substrate, the chain-end fraction is reduced below the bulk value. In the narrow interface, the fraction of chain ends is increased and exhibits a maximum, whereas it decreases toward a constant value in the wide interphase and the middle of the

Figure 9. (a) Schematic drawing of chain conformations of a linear living polymer chain at the solid substrate illustrating the mirroring effect of the impenetrable boundary. (b) Modified probability distribution function, P0(N,x), of the two chain ends belonging to the same linear polymer with N monomers to meet with each other as a function of the distance, x, from the solid substrate according to the Silberberg argument. K

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Figure 10. Profiles of the fraction, Nring(x)/Ntot(x), of monomers belonging to all ring polymers for various bonding free energies, Eb. (a) Results from the SCFT calculations in the grand canonical ensemble at constant activity, zM, using the normalization by the bulk data (shown in the inset). (b) Main panel and large inset correspond to the particle-based simulation and SCFT calculations in the canonical ensemble, and utilize the normalization by the thin-film average, Nring/Ntot (shown in the small inset).

substratethat also significantly increases the formation of ring polymers in the wide interphase. According to the Silberberg argument,6 the solid substrate can be conceived as a reflecting boundary condition in the wide interphase. The mirroring of chain conformations and back-folding increases the self-density of extended macromolecules. Therefore, the two chain ends belonging to the same linear chain are more likely to be bonded together and form a ring polymer. Figure 9a presents the schematic drawing of chain conformations of a linear living polymer chain with and without the impenetrable boundary. Without the boundary, the probability distribution function of the end-to-end vector, R, of a linear Gaussian chain comprised of N monomers takes the standard three-dimensional form ⎛ 3R2 ⎞ ⎛ 3 ⎞3/2 ⎟. ⎟ exp⎜ − Pbulk(N , R) = ⎜ 2 2 ⎝ 2Nb0 ⎠ ⎝ 2πNb0 ⎠

width R e = b0 ⟨N ⟩ . The corresponding enhancement factor is plotted in Figure 9b. In particular, the Silberberg argument predicts that the formation of ring polymers at the solid substrate can be enhanced by two times. Figure 10 presents the profiles of the fraction, Nring(x)/ Ntot(x), of monomers belonging to all ring polymers. Figure 10a presents the SCFT results in the grand canonical ensemble normalized by the bulk data at the same activity, zM (shown in the inset of Figure 10a). The profiles in the main panel and large inset of Figure 10b correspond to the particle-based simulation and SCFT calculations, respectively, and are normalized by the thin-film average, Nring/Ntot, which is shown in the small inset of Figure 10b as a function of the bonding free energy, Eb. In accord with the Silberberg argument, the enhancement of ring formation occurs not only in the narrow interface but also additionally in the wide interphase. We can clearly observe in Figure 10a that the region of ring-forming enhancement extends farther away from the solid substrate as we increase the value of −Eb and the molecular weight of linear chains and ring polymers increases. Unlike the profiles of segment densities, chain ends of linear chains, and nonbonded monomers, the profiles of monomers belonging to all ring polymers are different in the wide interphase compared to the bulk (or the middle of the film). These findings qualitatively agree with the particle-based simulation in the main panel of Figure 10b and SCFT calculations in the large inset of Figure 10b in the canonical ensemble. The details of the profiles of monomers belonging to all ring polymers are dictated by the interplay of the different changes that occur in the narrow interface and in the wide interphase. The SCFT results in Figure 10a show that the fraction of monomers of all ring polymers exhibits a maximum in the narrow interface. This maximum behavior is observed for all values of Eb and the smaller the value of −Eb is, the farther away from the solid substrate it shifts. When Eb = −1kBT, the maximum occurs at x = 6D/256 ≈ 0.52b0, i.e., the position of the maximum remains in the narrow interface for all values of Eb investigated. The occurrence of maximum fraction of monomers of all ring polymers correlates with the maximum behavior of chain-end fraction and the segregation of nonbonded monomers at the solid substrate. In the grand canonical SCFT calculations, when the value of −Eb is large, the fraction of monomers of all ring polymers is

(36)

According to the Silberberg argument, the presence of a single impenetrable boundary modifies this expression to P(N , R) =

×

3 2πNb0 2

⎡ ⎛ 3R 2 ⎞ ⎛ 3R ′ 2 ⎞⎤ x ⎢exp⎜ − ⎜ − x 2 ⎟⎥ ⎟ exp + ⎢⎣ ⎝ 2Nb0 2 ⎠ ⎝ 2Nb0 ⎠⎥⎦

⎛ 3R 2 ⎞ 3 ⎜⎜ − ⎟, exp 2⎟ 2πNb0 2 ⎝ 2Nb0 ⎠

(37)

where R∥2 denotes the squared end-to-end distance parallel to the boundary. Since the boundary breaks the translational invariance in the x-direction perpendicular to the boundary, P(N,R) depends explicitly on the distances, x and x′, of the two chain ends from the boundary via Rx2 = |x − x′|2 and Rx′2 = |x + x′|2. The probability of the two chain ends belonging to the same linear chain to meet at a distance x=x′ away from the solid substrate is ⎛ 6x 2 ⎞⎤ ⎛ 3 ⎞3/2 ⎡ ⎢ ⎟⎥ , ⎜− ⎟ P0(N , x) = ⎜ 1 exp + 2 2 ⎝ Nb0 ⎠⎥⎦ ⎝ 2πNb0 ⎠ ⎢⎣

(38)

where the last factor quantifies the increase of the probability due to the presence of the solid substrate. This mirroring effect increases the probability of ring formation not only in the narrow interface but also throughout the wide interphase of L

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Figure 11. Profiles of the fraction, Nring(x)/Ntot(x) (N < 6), of monomers belonging to short ring polymers for different bonding free energies, Eb. (a) Grand canonical SCFT calculations using the normalization by the bulk value, Nring/Ntot (N < 6) (shown in the inset), at the same activity, zM. (b) Results from the particle-based simulation and SCFT calculations in the canonical ensemble normalized by the thin-film average (shown in the small inset) are presented in the main panel and large inset, respectively.

Figure 12. Profiles of the fraction, Nring(x)/Ntot(x) (N > 20), of monomers belonging to long ring polymers for various values of Eb. (a) Results from the grand canonical SCFT calculations normalized by the bulk value, Nring/Ntot (N > 20) (shown in the inset), at the same activity, zM. (b) Particlebased simulation and SCFT calculations in the canonical ensemble using the normalization by the thin-film average (shown in the small inset) are presented in the main panel and large inset, respectively.

enhanced by more than a factor of 2, i.e., the Silberberg argument and the additional effects in the narrow interface segment-density reduction, enrichment of chain ends of linear chains, segregation of nonbonded monomers, and reduced dimensionality of ring polymerscontribute to the enhancement of ring formation. For smaller mean molecular weights (or less negative Eb), however, the segregation of nonbonded monomers in the narrow interface reduces the fraction of monomers of all ring polymers and the corresponding enhancement of ring formation is less than that predicted by the Silberberg argument. The differences between the canonical and grand canonical SCFT calculations chiefly arise from the different types of normalization. The canonical SCFT calculations allow for a quantitative comparison with the data from the particle-based simulation. In contrast to the SCFT calculations, we find that the fraction of monomers of all ring polymers only exhibits a monotonous decay away from the solid substrate in the particle-based simulation (cf. main panel of Figure 10b) because of their larger range of nonbonded interactions. According to Figure 10, small value of −Eb highlights the effects of the enrichment of chain ends and the segregation of nonbonded monomers, whereas the effects of the reduced segment density and the reduced dimensionality of ring

polymers in the narrow interface and the mirroring effect will contribute more if the mean molecular weight is larger (or more negative Eb). In the following, we investigate the local formation of short ring polymers (N < 6) in the narrow interface and explore the local formation of long ring polymers (N > 20) in the wide interphase. Figure 11 presents the profiles of the fraction, Nring(x)/ Ntot(x) (N < 6), of monomers belonging to short ring polymers with less than N < 6 segments. In analogy to Figure 10, the main panel and large inset of Figure 11b correspond to canonical particle-based simulations and SCFT calculations, whereas Figure 11a presents the grand canonical SCFT results. Again we find good agreement between the particle-based simulation and SCFT calculations in the canonical ensemble, except for the maximum behavior in the SCFT calculations, which is even observed for short ring polymers. The grand canonical SCFT data in Figure 11a show that the range of enhancement of short ring polymers is smaller than the corresponding range for all ring polymers (cf. Figure 10a), and it is nearly independent from the value of Eb. This is expected because the Silberberg argument suggests that the range is set by the molecular size. Whereas the average ring-polymer size increases with the value of −Eb, the size of short ring polymers M

DOI: 10.1021/acs.macromol.5b01055 Macromolecules XXXX, XXX, XXX−XXX

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Figure 13. Profiles of the fraction, Nring(x,N)/Ntot(x), of monomers belonging to N-member ring polymers obtained by means of the grand canonical SCFT calculations for various values of N as indicated in the key. The different panels correspond to the results of different bonding free energies, Eb. The normalization factor as a function of N is shown in the inset.

Figure 14. (a) Fraction, Nring(D)/Ntot(D), of monomers belonging to all ring polymers and normalized by the corresponding bulk value at the same activity, zM, for three different values of Eb as indicated in the key. (b-d) Profiles of the fraction, Nring(x,N)/Ntot(x), of monomers belonging to all ring polymers as a function of the distance, x, from the solid substrate. Different symbols correspond to different film thicknesses, D, as given in the E legend and the different panels show the results for k Tb = −1, −3, and −5, respectively. The insets highlight the behavior of ultrathin films. B

N

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Macromolecules with N < 6 is largely independent from the bonding free energy, Eb. For all values of Eb, the maximum normalized fraction of monomers of short ring polymers, (Nring(x)/Ntot(x))/(Nring/ Ntot) (N < 6), is larger than the corresponding value averaged over all ring polymers. For small values of −Eb, we obtain an excess factor of 1.8, and for the most negative value Eb = −7kBT, we obtain nearly 2.4. This effect can be rationalized by the segregation of nonbonded monomers and short linear chains and the enrichment of chain ends of linear chains in the narrow interface that reduces the mean molecular weight of nonbonded monomers and linear chains in the narrow interface3 and preferentially results in the formation of short ring polymers. The same analysis for long ring polymers with N > 20 is presented in Figure 12. First, we note that the enhanced formation of long ring polymers extends much farther away from the solid substrate and the enhancement zone extends as we increase the value of −Eb and the mean molecular weight of ring polymers increases. Thus, the profiles of long ring polymers highlight the Silberberg effect in the wide interphase. Due to the spatial extend of the wide interphase, we have verified that this effect will enhance the high-molecular-weight tail of the ring-size distribution in thin films. Second, the enhancement of long ring polymers in the narrow interface is smaller than that of short ring polymers. The reason for the suppression of long ring polymers in the narrow interface is the overall reduction of the molecular weight in the narrow interface and the pronounced segregation of nonbonded monomers for small values of −Eb. In this case, even the particle-based simulation exhibits a nonmonotonous profile of the fraction of monomers of long ring polymers with a maximum that shifts farther away from the boundary upon decrease of −Eb. The reduced enhancement of long ring polymers in the narrow interface also gives rise to excess factors smaller than 2. Finally, in order to summarize how the mirroring of chain conformations and back-folding by the solid substrate and the effects in the narrow interface influence the local ring formation as a function of the molecular weight, Figure 13 presents the profiles of the fraction, Nring(x,N)/Ntot(x), of monomers belonging to ring polymers with N segments for four different E bonding free energies, i.e., − k Tb = 1, 3, 5, and 7, obtained by the

different number of grids in the grand canonical SCFT calculations. We utilize the same value of activity, zM, as in the previous investigations (Ntot = 32768 and D ≈ 22.63b0). On the one hand, this study demonstrates that the properties in the middle of the film with D ≈ 22.63b0 are not significantly affected by film-thickness effects and are characteristic for a semi-infinite system. On the other hand, we expect rather pronounced film-thickness effects for the strongest confinement. Figure 14a presents the fraction, Nring(D)/Ntot(D), of the number, Nring(D), of monomers belonging to all ring polymers and the number, Ntot(D), of all segments (both nonbonded and bonded monomers) as a function of the film thickness, D, for E three different bonding free energies, k Tb = −1, −3, and −5. B

This global fraction is normalized by the corresponding bulk value at the same value of zM. The limiting value 1 of the normalized fraction, (Nring(D)/Ntot(D))/(Nring/Ntot), is approached with a 1/D correction for D → ∞. In the opposite limit, the fraction of monomers belonging to ring polymers can be significantly increased. For ultrathin films, D ∼ 2b0, the fraction of monomers belonging to ring polymers is more than 5-fold, and the more negative the bonding free energy, Eb, is, the larger the increase of Nring(D)/Ntot(D) becomes. This pronounced increase of the fraction of monomers belonging to ring polymers stems from the depletion of local segment density, the reduced dimensionality of ring polymers in the ultrathin film, and the mirroring of chain conformations and concomitant back-folding because the ultrathin films only consist of the narrow interface region. The other panels of Figure 14 present the profiles of the fraction, Nring(x,D)/Ntot(x,D), of monomers belonging to ring polymers as a function of the distance, x, from the solid substrate for three different bonding free energies, Eb, and eight different film thicknesses, D. All the profiles are normalized by the corresponding bulk value, Nring/Ntot, at the same value of z M. For D ≤ 3b0, the profiles of Nring(x,D)/Ntot(x,D) significantly deviate from the profiles of thicker films. For these ultrathin films, the local fraction of monomers belonging to ring polymers is much increased and also the shape of the profile deviates from the thicker-film examples. For thicker films, the profiles at the solid substrate are rather independent from the film thickness, D, i.e., Nring(x,D)/Ntot(x,D) increases, passes through a maximum and decreases as a function of x. The film thickness, D, merely controls the cutoff of the decrease toward the bulk value. The crossover between the two behaviors occurs around D ≈ 3b0 independent of Eb or the mean molecular weight of ring polymers. Therefore, we conclude that the crossover thickness is set by the width of the narrow interface. These findings suggest that ring formation in living polymer system in ultrathin films or highly confined porous structures may be significantly enhanced.

B

grand canonical SCFT calculations. The highest local excess factor is obtained for the shortest ring polymer N = 3 and the most negative bonding free energy Eb=−7kBT. The largest integral excess of ring polymers is found for the longest ring polymers and the largest value of −Eb. Small values of −Eb corresponding to low molecular weight give rise to weaker excess factors, i.e., an nonmonotonous dependence on x and even a reduction of ring formation at the substrate compared with the bulk. In the previous investigations, we have chosen the film thickness D ≈ 22.63b0 large enough as to obtain nearly bulk behavior in the middle of the film. In Figure 14, we explore the effect of the film thickness using the grand canonical SCFT calculations. In this investigation, we employ the self-consistent field theory (SCFT) in the grand canonical ensemble to study the effect of film thickness, 1.41b0 ≤ D ≤ 26.87b0, on the global formation of ring polymers. We fix the spatial discretization as ΔL ≈ 0.088b0, and different film thicknesses correspond to

IV. CONCLUSIONS We have extended the self-consistent field theory (SCFT) of Feng and Fredrickson2 for living polymer systems to include ring formation and studied the effect of impenetrable boundaries on the formation of ring polymers. The SCFT calculations are quantitatively compared to the particle-based simulation of a soft, coarse-grained particle model.3,15 Comparing the results from the particle-based simulation and SCFT calculations, respectively, we find quantitative agreement O

DOI: 10.1021/acs.macromol.5b01055 Macromolecules XXXX, XXX, XXX−XXX

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compute-intensive solution of the modified diffusion equations, eq 23 and eq 24, for ring polymers. Since the computational costs increase with the molecular weights of ring polymers, we have restricted the range of Eb in the SCFT calculations. However, since the computational cost of particle-based simulations does not depend on the value of Eb, we set the range of Eb in the particle-based simulation same as that in our previous work.3 Additionally, we technically utilize the root of the meansquared end-to-end distance of a typical chain, Re0(Nref) with Nref = 32 segments in the absence of excluded-volume interactions as a unit of length. Since the value of Nref is arbitrary (although it is close to the mean molecular weight, ⟨N⟩), we express all length scales in units of b0. “Different” film thicknesses in the two different methods arise from different definitions of the statistical segment length in the discrete Edwards Hamiltonian (used in the particle-based simulation) and the continuous Gaussian chain model (employed in the calculations based on the SCFT). In the particle-based simulation, R e0(Nref ) = b0 Nref − 1 and D = 4Re0(Nref) ≈ 22.27b0, while in the SCFT calculations, R e0(Nref ) = b0 Nref and D = 4Re0(Nref) ≈ 22.63b0. However, we expect that this small difference arising from the use of different chain models will not give rise to a significant difference in properties. Moreover, it should be noted that there is also a difference between the statistical segment length, br, and the quantity, b0, for the chain in the absence of excluded-volume interactions. Using the SCFT calculations, we can express the average quantities by means of the molecular weight distributions of nonbonded monomers, linear chains, and ring polymers, i.e.,

for various properties in the wide interphase, however, there are some deviations in the narrow interface where the local structure of the fluid of segments and details of the interaction matter. In the bulk, we find that the molecular weight distribution of ring polymers exhibits a combination of an exponential decay and a power law and the formation of ring polymers does not alter the exponential decay of molecular weight distribution of linear chains. When the bonding free energy, Eb, becomes more negative, the mean molecular weight of ring polymers slightly increases, and the presence of ring polymers slightly decreases the mean molecular weight of nonbonded monomers and linear chains compared with the case without ring formation.3 These observations agree with the previous results in the bulk.8,9 Since the two chain ends belonging to the same linear chain have a higher probability to meet with each other, the formation of ring polymers is increased when the density of all segments (both nonbonded and bonded monomers) becomes lower. In thin films between two parallel solid substrates, we distinguish between a narrow interface whose spatial extend is independent of the polymer size and a wide interphase of width Re(⟨N⟩). For large negative bonding free energies, Eb, the mean molecular weight is large and the two regions are clearly separated. Ring formation decreases the mean molecular weight of all species and therefore slightly reduces the width of the wide interphase. In the narrow interface, we find that the local formation of ring polymers is affected by five aspects: (1) reduction of the segment density in the narrow interface, (2) enrichment of chain ends of linear chains at the solid substrate, (3) pronounced segregation of nonbonded monomers at the solid substrate, (4) reduced dimensionality of ring polymers at the solid substrate, and (5) the mirroring of chain conformations and back-folding by the solid substrate that can be described by the Silberberg argument. The last effect operates both in the narrow interface and in the wide interphase whereas the former four effects are limited to the narrow interface. These results suggest that the presence of a solid substrate can increase ring formation in living polymer systems−in particular for long ring polymers. Additionally, the SCFT results show that a decrease of the film thickness increases the global ring formation within the thin film, especially for the ultrathin films.

∞

∑ pL (N )N = e−2h V

Nlinear = pL (1) +

gM

N =2

+

1 − 2h V e 2 gM

∞

∑ z M NQ L(N )N ,

(39)

N=2

∞

Nring =

z MQ (1)

∑ pR (N )N = N =3

∞

1 V 2 gM

∑ N =3

1 z M N Q R (N ), N3/2 (40)

∞

■

nlinear = pL (1) +

APPENDIX A: FURTHER DETAILS OF COMPUTATIONAL TECHNIQUES In this appendix, we provide further details of our calculations using the particle-based simulation and self-consistent field theory (SCFT). In comparison to the particle-based simulation, we use a finer spatial discretization, ΔL = D/256 ≈ 0.088b0, in the SCFT calculations, which allows us to study the local properties within the narrow interface. The spatial discretization is chosen as to match the discretization of the chain contour. In the SCFT calculations, the finer spatial discretization corresponds to the finer chain-contour discretization, whereas the worse spatial discretization matches the worse chain-contour discretization in the particle-based simulation. The bonding free energy, Eb, varies from −1kBT to −9kBT in the particle-based simulation while it ranges from −1kBT to −7kBT in the SCFT calculations. The difficulty in the SCFT calculations when the value of −Eb is large arises from the occurrence of long ring polymers which requires a rather

+

1 − 2h V e 2 gM

∑ pL (N ) = e−2h V

∑

∑ z M NQ L(N ),

pR (N ) =

N =3

⟨N ⟩linear =

⟨N ⟩ring =

⟨N ⟩ =

1 V 2 gM

∞

∑ N =3

Nlinear = c L(1) + nlinear

Nring nring

z MQ (1)

(41)

N =2

∞

nring =

gM

N =2 ∞

1 z N Q R (N ), 5/2 M N

(42)

∞

∑ c L(N )N , N =2

(43)

∞

=

∑ c R (N )N , N =3

Nlinear + Nring Ntot . = ntot nlinear + nring

(44)

(45)

Then if we define P

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Macromolecules pL (r, 1) = [ϕ(r, 1)ρ0 ], 1 [ϕ (r, N )ρ0 ] N L

pL (r, N ) =

∞

(46)

nring (r) =

N =3

for N ≥ 2,

(47)

⟨N ⟩linear (r) =

1 [ϕ (r, N )ρ0 ] for N ≥ 3, (48) N R as the local molecular weight distributions at spatial position r of nonbonded monomers, linear chains, and ring polymers, respectively, the corresponding local quantities at spatial position r can be represented as pR (r, N ) =

1 [ϕ (r, N )ρ0 ], N R

N=3

Nlinear(r) = c L(r, 1) + nlinear(r)

(52)

∞

∑ c L(r, N )N , N=2

(53)

Nring(r)

⟨N ⟩ring (r) =

∞

∑ pL (r, N )N = [ϕ(r, 1)ρ0 ]

Nlinear(r) = pL (r, 1) +

∞

∑ pR (r, N ) = ∑

⟨N ⟩(r) =

N=2

nring (r)

∞

=

∑ c R (r , N )N , N =3

(54)

Nlinear(r) + Nring(r) Ntot(r) = , ntot(r) nlinear(r) + nring (r)

(55)

∞

+

∑ [ϕL(r, N )ρ0 ],

(49)

N=2 ∞

Nring(r) =

where cL(r, 1) = pL(r, 1)/nlinear(r), cL(r, N) = pL(r, N)/nlinear(r) with N ≥ 2, and cR(r, N) = pR(r, N)/nring(r) with N ≥ 3 are the local normalized molecular weight distributions at spatial position r of nonbonded monomers, linear chains, and ring polymers, respectively. Specifically, when the ring-forming living polymer systems are confined between two parallel solid substrates, all quantities such as ϕ(r) and ω(r) vary only in the direction x perpendicular to the solid substrates within the mean-field approximation and the corresponding calculations can be reduced to one-dimensional ones. Thereby the effective Hamiltonian, /[ϕ(x), ω(x)], becomes

∞

∑ pR (r, N )N = ∑ [ϕR (r, N )ρ0 ], N =3

(50)

N =3 ∞

nlinear(r) = pL (r, 1) +

∑ pL (r, N ) = [ϕ(r, 1)ρ0 ] N =2

∞

+

1 [ϕ (r, N )ρ0 ], N L

∑ N=2

(51)

κ0 ϱ0 /[ϕ(x), ω(x)] = 2 kBT

∫ [ϕ(x) − 1]2 dx − ϱ0 ∫ [ω(x)ϕ(x)] dx − ϱ0e−2h ρDg

1 D − ϱ0 e−2h 2 ρ0 gM

∞

∑

z M N Q L(N ) − ϱ0

N =2

, implies that average density of segments and the factor, N the investigated problems actually still remain three-dimensional. Based on this one-dimensional effective Hamiltonian, by invoking the saddle-point approximation, we can also obtain the corresponding self-consistent field equations and other related expressions, which are very similar to the threedimensional ones in section II. The differences come from that the spatial integrations are taken only in the direction perpendicular to the solid substrates, e.g., ϱD

z MQ (1),

0 M

1 pL (N ) = N

∫

z M N Q L(N ) pR (N ) =

1 N

ϱD 0 M

for N ≥ 3,

N =3

1 z N Q R (N ), 5/2 M N

(56)

1 zM N N2

∫ dx 0

N

qR (x0 , 0; x , s)qR (x0 , N ; x , N − s) ds

for N ≥ 3,

1/2

Q R (N ) = N

(60)

⎛ 2πb 2 ⎞1/2 1 0 ⎜ ⎟ ⎝ 3 ⎠ D

(x 0 , N ; x , N − s ) d x

respectively. The factors,

∫ dx0 ∫ qR (x0 , 0; x , s)qR

for N ≥ 3,

(57) 2πb0 2 3

1/2

( )

1 N2

(61) 2πb02 3

1/2

( )

and N1/2

, in

the expressions of ϕR(x,N) and QR(N) with N ≥ 3 stem from that the statistics of chain conformations of ring polymers in the two dimensions parallel to the solid substrates are canceled with each other.

(58)

∫ [ϕR (x , N )ϱ0] dx = 12 ρ 0g

z M N Q R (N )

∫0

ϱD 1 [ϕL(x , N )ϱ0] dx = e−2h 0 2 ρ0 gM for N ≥ 2,

∑

1/2

−5/2

∫ [ϕ(x , 1)ϱ0] dx = e−2h ρ 0g

∞

1 D 2 ρ0 gM

2 1 1 ⎛ 2πb0 ⎞ ⎜ ⎟ ϕR (x , N ) = 2 ρ0 gM ⎝ 3 ⎠

where ϱ0 = Ntot /D = ∫ ρ0 dy dz denotes the one-dimensional

pL (1) =

z MQ (1)

0 M

■

1 N 5/2

APPENDIX B: ANALYTICAL SOLUTIONS FOR BULK RING-FORMING LIVING POLYMER SYSTEMS In this appendix, we present the analytical solutions for ringforming living polymer systems in the bulk based on the selfconsistent field theory (SCFT). Integrating both sides of the expression for ϕ(r) in section II, we obtain

(59)

where we should note the representations of ϕR(x,N) and QR(N) with N ≥ 3, which are Q

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∫ ϕ(r) d3r = 1 1 + 2 ρ0 gM

e−2h 1 e−2h z MQ (1) + 2 ρ0 gM ρ0 gM

∞

∞

∑ z M NQ L(N )N N =2

1 z M N Q R (N ). N3/2

∑ N =3

⟨N ⟩linear =

∞

∑ N=3

+

1 1 2 ρ0 gM

∑ N=3

∑

∑ [z M exp(−ω homo)]N N N=2

+

1 [z M exp( − ω homo)]N , N3/2

pL (N ) =

for N ≥ 2,

nlinear

Nring =

1 − 2h V 1 V e z M exp( −ω homo) + e−2h 2 gM 2 gM

∞

∑ N =3

nlinear =

,

1 [z M exp( −ω homo)]N , N3/2

(67)

nring =

(68)

z M exp( −ω homo) , 1 − z M exp( −ω homo) 1 V 2 gM

⎛ N⎞ 1 V 1 1 V 1 exp⎜ − ⎟ z N= 5/2 M 5/2 2 gM N 2 gM N ⎝ N0 ⎠ (76)

∞

∑ N =3

1 [z M exp( −ω homo)]N , N 5/2

∞

1 V 2 gM

∑ N =3

1 V 2 gM

∞

∑ N =3

1 + (1 − z M)−1

∑ N =3

(77)

(78)

1 zM N , N 5/2

1 + (1 − z M)−2 ∞

⟨N ⟩ring = (69)

1 zM N , N3/2

1 − 2h V 1 V zM e z M + e − 2h , 2 gM 2 gM 1 − z M

⟨N ⟩linear =

V V 1 1 = e−2h z M exp( −ω homo) + e−2h gM gM 2 2

nring =

(75)

zM 1 − 2h V 1 V e z M + e − 2h , 2 gM 2 gM (1 − z M)2

Nlinear =

(66)

[1 − z M exp(−ω homo)]2

(74)

by defining N0 ≡ ( −ln z M)−1, and

by defining N1 ≡ ( −ln z M + ω homo) , and

Nring

⎛ 1 ⎞ V V z M = e − 2h exp⎜ − ⎟ , gM gM ⎝ N0 ⎠

for N ≥ 3,

z M exp( −ω homo)

(73)

for N ≥ 3,

(65)

−1

1 V = 2 gM

N=3

1 zM N = 1 N3/2

⎛ N⎞ 1 − 2h V 1 V e z M N = e − 2h exp⎜ − ⎟ 2 gM 2 gM ⎝ N0 ⎠

pR (N ) =

1 V 1 1 V 1 pR (N ) = [z M exp( −ω homo)]N = 2 gM N 5/2 2 gM N 5/2

Nlinear =

∞

∑

for N ≥ 2,

1 − 2h V 1 V e [z M exp(−ω homo)]N = e−2h 2 2 gM gM

⎛ N⎞ exp⎜ − ⎟ ⎝ N1 ⎠

1 1 2 ρ0 gM

pL (1) = e−2h

⎛ 1 ⎞ V V z M exp( −ω homo) = e−2h exp⎜ − ⎟ , gM gM ⎝ N1 ⎠

⎛ N⎞ exp⎜ − ⎟ ⎝ N1 ⎠

(72)

under the condition 0 < zM < 1. This equation can also be solved numerically, and the corresponding global quantities can be represented as

(64)

pL (N ) =

1 [z M exp( −ω homo)]N / N3/2

zM 1 e − 2h 1 e − 2h zM + 2 ρ0 gM 2 ρ0 gM (1 − z M)2

ϕhomo =

1 [z M exp(− ω homo)]N N3/2

(63)

pL (1) = e

(71)

Specifically, when ϕ homo = 1, i.e., the investigated homogeneous segment density is equal to the given density of segments, ρ0, ωhomo = 0, and the expression for ϕhomo is

under the condition 0