50th Anniversary Perspective: Polymer Conformation—A Pedagogical

Perspective Cite This: Macromolecules XXXX, XXX, XXX-XXX

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50th Anniversary Perspective: Polymer ConformationA Pedagogical Review Zhen-Gang Wang* Division of Chemistry and Chemical Engineering, California Institute of Technology, Pasadena, California 91125, United States ABSTRACT: The study of the conformation properties of macromolecules is at the heart of polymer science. Essentially all physical properties of polymers are manifestations of the underlying polymer conformations or otherwise significantly impacted by the conformation properties. In this Perspective, we review some of the key concepts that we have learned over nearly eight decades of the subject and outline some open questions. The topics include both familiar subjects in polymer physics textbooks and more recent results or not-so-familiar subjects, such as non-Gaussian chain behavior in polymer melts and topological effects in ring polymers. The emphasis is on understanding the key concepts, with both physical reasoning and mathematical analysis, and on the interconnection between the different results and concepts.

I. INTRODUCTION Polymer conformation refers to the spatial configuration of the constitutive atoms or atomic groups for a given, fixed connectivity. For most polymers, the ability to adopt different conformations arises from the rotational degree of freedom around the single chemical bonds connecting the backbone atoms. In the case of the simplest synthetic polymer, polyethylene, for example, there are three preferred rotational angles between the two C−C bonds around the axis of the intervening C−C bonds: one trans, having the lowest energy, and two gauche, having a somewhat higher energy.1,2 The availability of these different isomeric states, and the facile transitions between them, results in polymer chain flexibility, which has a wide range of thermodynamic and dynamic consequences. For some polymers, especially of filamental types, such as dsDNA and actin filaments, the flexibility is due to thermal bending fluctuations of the filaments,3,4 which arise from the small thermal fluctuations of the constituting atoms. In this case, the chain conformation space comprises the ensemble of space curves traced by the filaments. In both cases, chain flexibility on large scales is a consequence of the linear structure of the polymers. Polymer conformation is the most important concept in the study of polymer physics, as it is the molecular basis underlying essentially all the physical properties of polymers. All the salient features of macromolecules, such as its responsiveness, its large spatial extent, entanglement, and multiplicity of interactions, either are rooted in chain conformation or are significantly affected by it. Interestingly, it is also the large number of conformations and multiplicity of interactions that make it possible for some simplifying features to emerge in the theoretical description and properties of polymers, such as the applicability of mean-field description, scaling, and universal behaviors. © XXXX American Chemical Society

To see this last point, we start with an estimate of the number of conformations a linear polymer of moderate degree of polymerization can take, using the isomeric state representation.2 For our present purpose, we ignore the excluded volume interactions and the energetic differences between the trans and gauche statesaccounting for these effects will not alter the qualitative conclusions. If we take N = 1000, then the total number of internal isomeric states is Ω = 3(1000−2) ≈ 10474. (For comparison, the total number of baryons, including protons and neutrons, in the observable universe, is estimated to be only 1080.5) This is practically infinite. This realization has two implications. First, it is hopeless to enumerate all the conformation states; we must resort to a statistical description. Second, the behavior of a polymer chain will not be governed by any finite number of conformations but rather by the statistical ensemble of all conformations. The application of statistical description is what enables some simple results to emerge in seemingly very complex systems. The immensity of the number of conformations together with the large spatial extent of a typical polymer molecule is also responsible for the applicability of mean-field theory in describing dense polymer systems, such as polymer blends and block copolymer melts. The spatial size of a tagged chain in the melt is R ≃ N1/2b, where b is the Kuhn length (to be defined later). In the space pervaded by this chain, there are a total number of R3/(Nv) ≃ N1/2b3/v chains, where v is the volume per monomer unit. Thus, a polymer is interacting simultaneously with O(N1/2) other chains, whose interactions with the tagged chain provide an effective mean field for the tagged chain. Received: July 17, 2017 Revised: October 20, 2017

A

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concepts, with the help of both physical argument and some moderate amount of mathematical analysis. While polymer conformations are physically rather intuitive and pictorialthe random walk being an apt geometric representationit is useful to take the point of view where polymer conformations are considered to be the phase space for the statistical mechanical description of macromolecules. Although such a perspective is neither new nor unique, it affords certain conceptual clarity and unity and is in line with the ensemble view of modern statistical mechanics; the connection with entropy is immediately obvious. Thus, for example, the effect of force or geometric confinement can be viewed as biasing the conformation distributions or eliminating some conformations, whereas polymer dynamics can be considered as transitions between neighboring conformational states. Just as the subject itself is vast, there is a vast literature on polymer conformation, starting from the many foundational papers by the pioneers of polymer physics. A comprehensive reference list for the topics covered in this Perspective would add significantly to the length of this paper. Because the nature of this Perspective is more pedagogical, for the sake of keeping a relatively concise list of references, I will forego the usual rigor in referencing as in research articles where crediting and precedence are important considerations. Thus, I will include more references to the more recent and less familiar results, while for the more established results, I will cite fewer references and will sometimes refer to a textbook. Finally, a few words about notations. This Perspective discusses many different conformation properties. To denote each different property with a new symbol would introduce many different symbols. In addition, some symbols are the preferred notations by convention, e.g., N for the degree of polymerization (or chain length), F for the Helmholtz free energy, etc. To avoid introducing unduly large number of symbols, a symbol may represent multiple properties, depending on the context, as long as it is clear what the symbol denotes in its proper context. Thus, for example, f can variously denote the force, the Helmholtz free energy density, or some generic function. Another remark concerns the use of equal sign “=” and its variants. We use x = y to denote an exact equality, x ≈ y to indicate the value of y is approximately that of x, x ≃ y to mean y equals x (and have the same units) up to a numerical prefactor of order 1, and x ∼ y to indicate that x scales as y without regard to units. The rest of the Perspective is organized as follows. We start with the ideal chain model in section II and use it to illustrate a number of concepts and properties, such as chain stretching, confinement, structure factor, the concept of blob, etc. We then introduce interactions in real polymers in section III, where we discuss the excluded volume effects, the globule state, and the theta point for a single chain. Section IV is on concentration effects, where we start with a simplified derivation of the Flory− Huggins theory and then discuss several concentration effects, such as screening of excluded volume interactions, the behavior of chain size in different concentration regimes, the correlation hole effects, and the effects of the solvent molecular weight. Section V is devoted to an introduction of the topological interactions in polymers where we discuss the behavior of the simplest problem involving topological constraintsnonconcatenated and unknotted rings in dilute solutions and in melts. In section VI, we consider the coupling between chain conformation and phase transitions using polymer solutions

The conformation degrees of freedom also enables the description of interactions in terms of pseudopotentials. For example, in the theoretical study of inhomogeneous polymers such as interfaces in polymer blends and block copolymers, we often write the interaction between two polymer species or blocks in the form6,7 Uint = χ

∫ ϕA(r)ϕB(r) dr

(1.1)

While this form is quite familiar from the well-known Flory− Huggins theory,1,8,9 where the volume fractions refer to the fixed bulk composition, writing the interaction using the spatially dependent volume fractions of the two species at the same location implicitly requires some local averaging (coarse graining) over some length scales larger than the interaction range of the monomers, because for any polymers with monomers that take volume, the microscopic instantaneous densities (or volume fractions) cannot be both finite at the same spatial point. This coarse-graining is possible due to the multiplicity of local conformations. The use of such simple pseudopotentials involving only a small number of parameters allows certain universality in the description of polymeric systems; i.e., the larger length scale behavior is independent of the detailed microscopic models for the polymers. Polymer conformation is equally fundamental for dynamics.10,11 For example, viscoelasticity, one of the hallmarks of polymeric liquids, results from motion of the internal (i.e., conformational) degrees of freedom. Similarly, the unique manner in which polymers diffuse is a direct consequence of the conformational changes of both the tagged polymer itself and the conformational changes of the surrounding chains. The purpose of this Perspective is to review some of the most essential aspects of polymer conformations, highlight their interconnections, and point out some open questions. Because of the central importance of polymer conformation in macromolecular science, many excellent textbooks and monographs exist1,10,12−19 that treat the subject to varying degrees of breadth, depth, and rigor. Polymer conformation constitutes the bulk of polymer physics. Indeed, in the monograph by Grosberg and Khokhlov,16 the authors considered “the subject of the statistical physics of macromolecules” to be “the conformations and conformational motion of polymer chains...”. Given the vastness of the subject, it is clearly not possible to cover all the important aspects of polymer conformations. I have chosen to focus primarily on flexible homopolymers since understanding their properties is a prerequisite for studying more complex polymers. Readers will notice the omission of some obviously important and interesting topics, such as helix−coil transition in natural and some synthetic polymers, liquid-crystalline polymers, and block copolymers. Even within the confines of flexible homopolymers, I have left out some major topics, such as polyelectrolytes, polymers at surfaces and interfaces, and dynamics. Such omissions are inevitable owing to space limitation. Some of these subjects are also more specialized/ advanced. This Perspective is intended to primarily serve the new comers to this fieldgraduate students and postdocs who have had a first course in polymer physics but wish to have a more in-depth exposure and understanding of polymer conformations; I envision it to be an intermediary between introductory textbooks and research literature. With this objective in mind, the emphasis in this Perspective is on understanding the key B

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a random walk (with constant step length). This random walk analogy has been particularly productive in lattice model studies of a variety of chain conformation behaviors. The mapping to the continuous random walk forms the basis for the powerful path integral representation of polymers. Another useful quantification of the chain size is the square radius of gyration, defined as the square of the distance of a monomer from the center of mass of the chain, averaged over all monomers:

and binary polymer blends as examples. We include a short discussion on “molecular individualism” to remind us that polymer conformation is not all about universality but also about individuality. We conclude with some general perspectives and open questions.

II. IDEAL CHAIN The concept of an ideal chain plays a pivotal role in laying the foundation of polymer physics. An ideal chain is a polymer model in which no interactions exist apart from that responsible for chain connectivity. Depending on how the connectivity is modeled, there can be different versions of the ideal chain model. The simplest and perhaps the best known is the freely jointed chain (FJC) model due to Kuhn.20 Accounting for the fixed bond angle between consecutive bonds, we have the freely rotating model. Taking the continuum limit of the freely rotating chain model, we obtain the Kratky−Porod model or the worm-like chain model, a versatile model for describing semiflexible polymers. For theoretical treatment of long length scale properties of polymers, the most important model is the Gaussian chain model, which is a universal representation for all ideal chain models at sufficiently large length scales and for sufficiently long chains. The study of ideal chain models is important for several reasons. (1) It captures the most salient feature of a polymer its connectivityand can be used to illustrate many of the essential concepts concerning the statistical behavior of polymers. The model serves as a useful reference for including additional effects. (2) Polymers in dilute solution at the theta temperature and polymers in melt condition can be approximately described by the ideal chain model. In addition, for polymers with very large thermal blob size (such as dsDNA), the long length scale behavior is well described by the ideal chain model. (3) An ideal chain in free space defines the chain conformation space available to the polymer for a given specification of chain connectivity.21 Any interactions added to the ideal chain model, including external forces and geometric confinement, will bias and reduce the chain conformations. II.A. Freely Jointed Chain and Random Walk. The freely jointed chain (FJC) consists of N links or bonds, each of length b0, freely jointed consecutively by their ends. Let the unit vector of the ith bond be ui; the instantaneous end-to-end distance vector is

R g2 =

i=1

Rg =

(

1 2f

−

1 3f 2

1/2

)

N1/2b0 . Clearly, the larger the arm number,

Ω tot = (4π )N

(2.5)

On a 3-d lattice with coordination number z, the total number of conformations is clearly just zN. A quantity of paramount importance in the study of polymer conformations is the end-to-end distance vector distribution function p(R). In free space, owing to the translational invariance, we may consider one of the chain ends fixed at the origin. Thus, we seek the probability that the other end is located within some small volume element dR about R. (We use the shorthand dR to denote the volume element dRx dRy dRz.) Clearly

(2.1)

p(R) dR =

ω(R)dR Ω tot

(2.6)

where ω (R) is the density of states with one end at the origin and the other end at R. For arbitrary R, the result involves an integral of the inverse Langevin function.17 For |R| ≪ Nb0, i.e., the end-to-end distance is much less than the fully extended length of the polymer chain, and the distribution function becomes a simple Gaussian

(2.2)

The square root of ⟨R2⟩, called the root-mean-square end-toend distance, or just end-to-end distance for short, hereafter often denoted by Ree, is a linear measure of the spatial extent of the polymer. Ree is obviously given by R ee = N1/2b0

(2.4)

i=0

the smaller the Rg. Thus, for a fixed total N, Rg gives a measure of the compactness of the polymer. Since the link has a continuous rotational degree of freedom, the total number of conformations is not a countable integer. Nevertheless, we may evaluate the total configuration partition function Ωtot associated with the rotational degrees of freedom of the links and equate it with the total number of conformations. Obviously

where we use Ri to denote the position of the ith joint (including the end point of the chain). Obviously, by symmetry, ⟨R⟩ = 0, so the first nonvanishing moment is ⟨R2⟩ = ⟨R ·R⟩. Because of the statistical independence between the different bond orientation, we readily have ⟨R2⟩ = Nb02

N

∑ (R i − R CM)2

where RCM = ∑Ni=0Ri/(N + 1) is the instantaneous center of mass, and we have tacitly assigned the mass of the monomer to be at the joints. Further averaging over chain conformation, we obtain the mean-square radius of gyration ⟨Rg2⟩; its square root, commonly denoted as simply Rg, then provides a linear measure of the spatial extent of the polymer. Unlike the end-toend distance Ree, which is only meaningful for linear chains, Rg is well-defined for any chain architecture, such as stars, combs, etc. For linear polymers, we have Rg = 6−1/2N1/2b0 for large N. For an f armed star, each of N/f segments, we have

N

R ≡ R N − R 0 = b0 ∑ u i

1 N+1

⎛ 3R2 ⎞ ⎛ 3 ⎞3/2 ⎜− ⎟ ⎟ p(R) = ⎜ exp 2 2 ⎝ 2Nb0 ⎠ ⎝ 2πNb0 ⎠

(2.3)

(2.7)

This result can be understood using the central limit theorem: for |R| ≪ Nb0, the different components of the vector ui are independent, so each of the component of the vector R is a sum

This is the first nontrivial result in polymer physics. The visual representation of the freely jointed chain clearly evokes that of C

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Macromolecules of N independent stochastic variables. This also results in the factorization of the probability distribution into its Cartesian coordinate components. The probability distribution is directly related to the entropy decrease due to constraining the chain ends at R. To proceed, we integrate eq 2.6 over a small volume a3 where a ≃ b0 is a microscopic lengththis is necessary to obtain a finite probability in continuous space. Making use of eq 2.7, we get the entropy decrease due to constraining the free chain end within a3 of R: S(R) − Stot = −

where C∞ = limN→∞ CN. From eq 2.13, we see that CN approaches C∞ as 1/N. This result will be referred to and contrasted with in sections III.E and IV.F. For large N, insofar as long length scale properties, such as the mean-square end-to-end distance, are concerned, one could represent the freely rotating chain with an equivalent freely jointed chain of link length (the Kuhn length) b with the mapping

b2 = C∞b02

⎡⎛ ⎞3/2 ⎛ ⎤ 3 ⎟ a 3 ⎞⎥ ⎢ ⎜ ⎜ ⎟ + k ln ⎜ ⎟ B ⎢⎣⎝ 2π ⎠ ⎝ N3/2b03 ⎠⎥⎦ 2Nb02

Since C∞ > 1, each link in the FJC model can be considered to contain more than one bond of the freely rotating chain. This is an example of coarse graininga simplified representation of the original model with some finer details (in this case the bond angle) lumped into a newly defined parameter (the Kuhn length in the FJC model). One can include more realism by accounting for the steric hindrance in the torsional rotation. For relatively high barriers between the trans and gauche states, one may assume that the polymer exists in discrete torsional states corresponding to the energy minima. The chain conformation statistics for this isomeric state model2 can be calculated using the method of transfer matrix. We will not go into the specifics here; suffice it to say that the result of the calculation for the mean-square endto-end distance can again be written as ⟨R2⟩ = C∞Nb20, with a different C∞. The typical values of C∞ for most flexible synthetic polymers range between 4 and 12.17,19 An interesting limit of the freely rotating chain model is the Kratky−Porod model or wormlike chain (WLC) model.28 This limit is obtained by taking θ → 0, b0 → 0, and N → ∞ in such a way that Nb0 ≡ L and 2b0/θ2 ≡ lp are finite, where L is the contour length of the polymer whereas lp is defined as the persistence length. The persistence length measures the decay length of the correlation function for the bond vectors at positions s and s′ along the chain backbone

3kBR2

(2.8)

where Stot = kB ln Ωtot is the entropy of the unconstrained chain. The first term on the right-hand side is the familiar entropic elasticity, while the second term is the entropy reduction due to localization of the chain end within a volume a3. An interesting special case is R = 0, corresponding to the formation of a ring. The ring closure probabilitythe probability of bringing two ends to within a small volume a3is P(0) = a3p(0) =

⎛ 3 ⎞3/2 a3 ⎜ ⎟ ⎝ 2π ⎠ N3/2b 3 0

(2.9)

Equation 2.9 has the simple interpretation of the ratio of the microscopic volume a3 to the volume spanned by the polymer chain (N1/2b0)3. The ring closure probability, called the Jacobson−Stockmayer factor,22 plays an important role in ring−chain polymerization equilibrium,23,24 DNA cyclization,25 RNA folding,26 and network formation.27 The entropy decrease due to ring closure is, from eq 2.8 ⎡⎛ ⎞3/2 ⎛ ⎤ 3 a3 ⎞ S(0) − Stot = kB ln⎢⎜ ⎟ ⎜⎜ 3/2 3 ⎟⎟⎥ ⎢⎣⎝ 2π ⎠ ⎝ N b0 ⎠⎥⎦

⎛ |s − s′| ⎞ ⎟ ⟨u(s) ·u(s′)⟩ = exp⎜⎜ − lp ⎟⎠ ⎝

(2.10)

Taking the difference between eqs 2.8 and 2.10, we obtain the familiar expression for the entropy change of separating the two ends to R S(R) − S(0) = −

(2.11)

⟨R2⟩ =

II.B. Freely Rotating Chain and Wormlike Chain. The freely rotating chain model accounts for the fixed bond angles but assumes the torsional angle to be completely free. It is easy to see that if the fixed bond angle is θ, the correlation between unit bond vectors i and j is ⟨u i ·u j⟩ = (cos θ )|j − i|

L

∫0 ∫0

L

ds ds′ ⟨u(s) ·u(s′)⟩

= 2Llp − 2lp2(1 − e−L / l p)

(2.17)

For very large persistence length lp ≫ L, one can easily check that

(2.12)

⎛ L ⎞⎟ ⟨R2⟩ ≈ L2⎜⎜1 − 3lp ⎟⎠ ⎝

The mean-square end-to-end distance is ⎡ 1 + cos θ 1 − (cos θ )N ⎤ 2 ⟨R2⟩ = Nb02⎢ − cos θ ⎥ N (1 − cos θ )2 ⎦ ⎣ 1 − cos θ

(2.18)

The first term clearly corresponds to rigid-rod behavior, while the correction term is a result of chain retraction due to undulation, which can be used to experimentally determine the persistence length using microscopic visualization techniques.4 In the opposite limit, L ≫ lp, we have

(2.13)

The terms in the square brackets define the Flory characteristic ratio CN. For large N, the mean-square end-to-end distance is

⟨R2⟩ = C∞Nb02

(2.16)

This correlation function can be obtained directly from highresolution microscopic images in ds DNA29 and DNA bundles.30 The mean-square end-to-end distance is now

3kBR2 2Nb02

(2.15)

⟨R2⟩ ≈ 2Llp

(2.14) D

(2.19) DOI: 10.1021/acs.macromol.7b01518 Macromolecules XXXX, XXX, XXX−XXX

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Macromolecules This result can be mapped to that of eq 2.2 upon defining b = 2lp and N ≡ L/2lp. While the wormlike chain model was introduced as a limiting case of the freely rotating chain model, where the rigidity is due to geometric constraint, more recent applications of the WLC model is based on the energetic formulation by Saitô et al.,31 although this energetic description had been briefly discussed in the classic textbook by Landau and Lifshitz.32 In this formulation, one writes the bending energy due to curvature deformation of an elastic filament H=

κ 2

∫0

L

ds

⎛ ∂u ⎞2 ⎜ ⎟ ⎝ ∂s ⎠

From the full probability distribution, one can obtain any reduced probability distribution by integrating the appropriate degrees of freedom. For example, the end-to-end vector distribution is obtained by integrating all the positions of the internal beads. The number of beads in the Gaussian model is not unique and is usually chosen according to the level of details with which we wish to model the internal degrees of freedom. Clearly, more beads allow a description down to smaller length scales. For large N, it is mathematically convenient to consider it as a continuous variable and to replace the difference in eq 2.22 by the derivative and the summation by an integral. Doing so gives

(2.20)

where κ is the bending modulus having dimension [energy × length]. κ/(kBT) thus has the unit of length and can be shown to be just the persistence length appearing in eq 2.16. This energetic formulation is more natural for describing filamentary polymers such as ds DNA and actin filaments. II.C. The Universal Gaussian Chain. As should be clear from the discussions in the previous subsection, for polymers much longer than their persistence length, the different ideal chain models, the freely rotating chain, isomeric state chain, and the WLC can all be mapped to an equivalent FJC. Furthermore, the end-to-end distance distribution for the FJC is Gaussian for a sufficiently long chain. The Gaussian distribution is a result of the central limit theorem whereby the end-to-end vector is the sum of statistically independent vectors. Although real polymers have bending rigidity, as long as the rigidity persists only over short distance, it is possible to group the links in the polymer into “superlinks”. When the number of links in the superlink exceeds the persistence length, these superlinks become orientational uncoupled. Thus, distribution for the vector sum of these superlinks becomes Gaussian. When the superlinks contain sufficiently large number of links, such that they are each the vector sum of superlinks, then the distribution of these superlinks themselves become Gaussian. Therefore, for a sufficiently long chain, one can always represent it as a succession of links whose distribution is Gaussian. This presentation of an ideal polymer with Gaussian links is the Gaussian chain model and is the universal representation for any polymer when the chain is much longer than the persistence length. The great advantage of this Gaussian chain model is that the full probability distribution of its conformation can be written down exactly as the product for each of the Gaussian links: ⎛ 3 ⎞3N /2 ⎜ ⎟ ⎝ 2πb2 ⎠ ⎡ 3 N ⎤ × exp⎢ − 2 ∑ (R i − R i − 1)2 ⎥ ⎢⎣ 2b i = 1 ⎥⎦

H=

3kBT 2b2

2b2

i=1

⎛ ∂R ⎞2 ⎜ ⎟ ⎝ ∂s ⎠

(2.23)

(2.24)

We can easily check that the distribution eq 2.7 (which is the Green’s function with R′ = 0) is the solution of eq 2.24. Because of the Markovian nature of random walk, the Green’s function satisfies the following recursion relation: G(R, R′; N ) =

∫ dR1 G(R, R1; N − s)G(R1, R′; s) (2.25)

II.D. Segment Density Distribution. For a chain in free space, the average density is uniform because of translational invariance. However, it is of interest to inquire about the density distribution with respect to a fixed chain end or with respect to fixed center of mass. II.D.1. Fixed Chain End. The density distribution with one end fixed at the origin is given by c1(r) =

∫0

N

ds

∫ d R G (R , r ; N − s )G (r , 0 ; s )

(2.26)

This has a clear physical interpretation: the density is contributed by the sum (integral) of the internal segment s, which acts as the end point of the first s segments and the starting point for the next N − s segments. Making use of the Gaussian form of the end-to-end distance distribution function eq 2.7 and the normalization condition, we easily find

(2.21)

N

∑ (R i − R i− 1)2

ds

⎛ ∂ b2 ∂ 2 ⎞ − ⎟G(R, R′; N ) = δ(R − R′)δ(N ) ⎜ 6 ∂R2 ⎠ ⎝ ∂N

Interpreting the exponential factor as the Boltzmann weight in statistical mechanics, one can define an energy (Hamiltonian) as 3kBT

N

This continuous representation allows the mapping of the statistics of polymers onto the path integral formulation of quantum mechanics,33 thus unleashing the power of the manybody theory techniques developed in quantum field theory. In this continuum description, the end-to-end distance vector distribution, or more precisely the two-point correlation function (Green’s function) G(R,R′;N), which is the conditional probability that a chain of length N has its end at position R given that another end is at R′, can be shown to satisfy the following diffusion equation for a random walk in continuous space:

f (R 0, R1 ,..., R i ,..., R N ) =

H=

∫0

c1(r) =

(2.22)

which has a natural mechanical analogue as N Hookean springs with spring constant 3kBT/b2 connected in series. The junction points of the springs will be called beads. The appearance of kBT signals the entropic origin of these springs.

⎛ 3 r ⎞⎤ 3 ⎡ ⎢ − 1 erf ⎜ ⎟⎥ ⎝ 2N b ⎠⎦ 2πb2r ⎣

(2.27)

The 1/r behavior of the density near the origin reflects the twodimensional nature of the random walk: the number of monomers within a local volume r3 scales as r2/b2. E

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1 in 3-dimensions. Using eq 2.30, we find δ = 1/5, which is rather modest. Therefore, the Gaussian approximation using the mean-square radius of gyration as the variance is a reasonably good approximation for the density distribution about the center of mass. We should mention that although the average density distribution eq 2.30 is spherically symmetric, the instantaneous shape of a polymer is highly anisotropic, which was recognized long ago by Kuhn.20 The anisotropy can be characterized by decomposing R2g in eq 2.4 into components along the three principal axes, R21, R22, and R23, for any given chain conformation. For a random walk on lattice, Šolc found the ratio between the three principal components of the square radius of gyration tensor to be approximately ⟨R21⟩:⟨R23⟩:⟨R23⟩ ≈11.7:2.7:1.38 A similar result was found for the off-lattice discrete Gaussian chain, with ⟨R12⟩:⟨R32⟩:⟨R32⟩ ≈11.79:2.53:1.39 An average asphericity parameter A was introduced by Rudnick and Gaspari40 as a global measure of shape anisotropy (A = 0 for perfect spherical symmetry); it is given by

II.D.2. Fixed Center of Mass. Of greater interest is the segmental distribution about the center of mass. The distribution for a specific segment from the center of mass was worked out by Ishihara34 and by Debye and Bueche,35 and the result for the generic density was given in the book by Yamakawa.13 Here we follow a simpler derivation given in the Ph.D. thesis of A. Cohen.36 The derivation involves the following steps. First, it can be shown that the mean-square distance of the center of mass of a Gaussian chain from either end is Nb2/3. Thus, the distribution of an end relative to the center of mass of the chain is a Gaussian with variance Nb2/3: P(r; N ) =

⎛ 9r 2 ⎞ ⎛ 9 ⎞3/2 ⎜ ⎟ exp ⎜− ⎟ ⎝ 2πNb2 ⎠ ⎝ 2Nb2 ⎠

(2.28)

Next, an internal segment can be considered as the ends of two conjoining Gaussian chains of length ϵN and (1 − ϵ)N. The center of mass of each of these two chains ri (i = 1, 2) is distributed independently according to eq 2.28. The center of mass of the full chain is a weighted average: r = ϵr1 + (1 − ϵ)r2. The distribution of the segment at contour length ϵN is then

A=

pϵ (r) = ∫ dr1dr2δ[r − ϵr1 − (1 − ϵ)r2]

(2.31)

× P(r; ϵN )P[r; (1 − ϵ)N ]

These authors derived an exact result for random walks in any space dimension d: A = 2(d + 2)/(5d + 4), so in 3-dimensions, A = 10/19. II.E. Distribution for the Radius of Gyration. There is no simple closed form expression for the distribution of the radius of gyration P(Q). (Here we use Q to denote the fluctuating radius of gyration, while reserving Rg as the conventional notation for the root-mean-square radius of gyration ⟨Rg2⟩1/2.) Fixman41 first derived an integral representation for the distribution of the square of the radius of gyration P(Q2) using the discrete Gaussian chain model with n beads. For n ≫ 1, the expression is (using our notation for the continuous Gaussian chain)

(2.29)

Performing the spatial integration over r1 and r2, and integrating over ϵ (since the density can be contributed from any internal segments), we obtain ⎤3/2 ⎡ 9 dϵ⎢ ⎥ 0 ⎣ 2πN (3ϵ2 − 3ϵ + 1)b2 ⎦ ⎤ ⎡ 9r 2 × exp⎢ − 2 2⎥ ⎣ 2N (3ϵ − 3ϵ + 1)b ⎦

c(r ) = N

∫

2 2 2 2 2 2 2 2 2 1 ⟨(R1 − R 2 ) + (R1 − R3 ) + (R 2 − R3 ) ⟩ 2 ⟨(R g2)2 ⟩

1

(2.30)

In the literature, the segment distribution about the center of mass has often been approximated as a Gaussian with the variance given by the mean-square radius of gyration ⟨Rg2⟩ = (1/6)Nb2. It can be easily checked that eq 2.30 yields the correct ⟨Rg2⟩. Figure 1 shows a comparison between the exact result eq 2.30 and the Gaussian approximation. The deviation from the Gaussian distribution can be quantified by the nonGaussian parameter,37 which is defined as δ ≡ 3⟨r4⟩/(5⟨r2⟩2) −

P(Q 2) =

3 πNb2

∞

⎛

⎞−3/2

∫−∞ dρ⎝ sinXX ⎠ ⎜

⎟

⎛ 1 ⎞ exp⎜ − tX2⎟ ⎝ 4 ⎠

(2.32)

where t = Q2/R2g = 6Q2/(Nb2) and X2 = 4iρ. From P(Q2), we can obtain the distribution P(Q) using P(Q2) dQ2 = P(Q) dQ, i.e., P(Q) = 2QP(Q2). The integral eq 2.32 cannot be performed analytically but was evaluated numerically by Koyama.42 The limiting behaviors were explored by Fixman,41 by Forsman and Hughs,43,44 and by Fujita and Norisuye,45 who corrected an error in Fixman’s calculation. For a more complete discussion on the subject, we refer interested readers to the book by Yamakawa.13 Here we merely write the leading-order asymptotic behaviors. For t ≪ 1, the limiting behavior is P(Q ) ≈

⎛ 9⎞ 108 t −5/2 exp⎜ − ⎟ ⎝ 4t ⎠ π N b 1/2 1/2

(2.33)

and for t ≫ 1, the result is P(Q ) ≈

⎛ π2 ⎞ 121/2π 5/2 t exp ⎜− t ⎟ ⎝ 4 ⎠ N1/2b

(2.34)

We will make use of these asymptotic results when we discuss the Flory theory for excluded volume. II.F. Correlations and Structure Factor. Chain connectivity gives rise to correlations between monomers. A useful

Figure 1. Normalized radial density distribution of monomers around the center-of-mass (blue curve) p(r) = c(r)/N. For comparison, the Gaussian distribution with variance Rg2 is also given (red curve). F

DOI: 10.1021/acs.macromol.7b01518 Macromolecules XXXX, XXX, XXX−XXX

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Macromolecules

called the Kratky plateau and is a signature for Gaussian behavior from scattering experiments.46 The structure factor also plays a key role in many theoretical treatments involving monomer−monomer interactions and the interaction of a polymer with an external field (see sections II.H, IV.C, and IV.E). II.G. Deforming an Ideal Chain. Except when a polymer chain is in free space, any other conditions will lead to some deformation of polymer. We leave the deformation of polymer by continuous potential field to the next subsection. In this subsection, we consider two common types of deformation: geometric confinement and stretching a polymer by their ends. II.G.1. Ideal Chain in Confinement. Polymer confinement is a common theme in biology and nanoscience. A prominent example in biology is the packaging of viruses in which DNA or RNA with lengths many orders of magnitude larger than the viral capsidthe shell that contains the viral genomehas to be encapsidated into a small space.48 A polymer can be confined in any of the three space dimensions. When the confining length D is larger than the end-to-end distance of the polymer Ree, the effect on the chain conformation is minimal, although the overall translational degrees of freedom will still be affected. When D < Ree, the chain as a whole will be deformed because all the conformations that involve chain segments outside of the confinement are excluded. If D ≫ b, the confinement will not affect the orientation of the segments, so we can use the Gaussian model (or the lattice random walk model) to study the conformation properties. In the continuous Gaussian model, the confinement is enforced through boundary conditions on the diffusion equation (2.24). In the simplest case of hard-wall confinement, the boundary condition is G(R,R′;s) = 0 if either R or R′ is on the wall for any s. As an example, we calculate the entropy loss and corresponding free energy cost for confining a polymer chain in a slit of separation D. Let the coordinate of the confinement direction be z. By the factorization property of the Green’s function, the total Green’s function is G(R,R′;N) = G0(x,x′;N) G0(y,y′;N) G(z,z′;N) where the subscript 0 denotes the Green’s function in free space. The partition function of the polymer in the slit is

measure is the pair distribution functionthe conditional probability distribution of finding a monomer at distance r from any tagged monomer. The pair distribution can be obtained using the density distribution for the fixed end in section II.D.1. As in the derivation of the density distribution from the center of mass, a particular tagged monomer can be considered as the ends of two conjoining Gaussian chains of length ϵN and (1 − ϵ)N. The density around this particular monomer is the sum of the density from the two subchains. Since the tagged monomer can be at any position along the chain contour, we integrate over ϵ to arrive at c 2(r) =

3 ⎡ ⎢1 − 2 π b2 r ⎣

∫0

1

⎛ 3 r ⎞⎤ dϵ erf⎜ ⎟⎥ ⎝ 2ϵN b ⎠⎦

(2.35)

For r ≪ Nb , the second term in the bracket can be neglected, and we have 2

c 2(r) ≈

2

3 π b2 r

(2.36)

The 1/r behavior reflects the fact that random walk has a fractal dimension of 2: integrating eq 2.36 over a volume of radius r gives the number of monomers in the volume that goes as (r/ b)2. Experimentally, the correlated structure of a polymer is studied by scattering techniques through a measurement of the structure factor15,46,47 defined as g (k) =

1 N

N

∑ ⟨exp[i k·(R l − R m]⟩ l ,m

(2.37)

The structure factor measures the interference between two plane waves each being scattered by a monomer. Making use of the Gaussian nature of the internal distance distribution and approximating the sum by an integral, we have g (k) = ND(k 2R g2)

(2.38)

where D(x) is the Debye function given by D(x) =

2 −x (e − 1 + x) x2

(2.39)

It can be shown that the structure factor is simply the Fourier transform of the pair distribution function eq 2.35. The limiting behavior of g(k) is ⎧ N (1 − k 2R 2/3), kR ≪ 1 g g ⎪ g (k ) = ⎨ ⎪12/(k 2b2), kR g ≫ 1 ⎩

(2.40)

N 1 + Nb2k 2/12

1 Λ3

∫ dR ∫ dR′ G(R, R′; N )

=

A Λ3

∫0

D

dz

∫0

D

d z′ G (z , z′ ; N )

(2.42)

where A is the area of the slit and Λ is some length scale for the translational degrees of freedom, which has no thermodynamic consequence. The partition function of the polymer in free space is simply Z0 = (V/Λ3), where V is the system volume. To compute the conformation entropy change, we take the ratio of the partition function in confinement to that in free space for the same translational degrees of freedom, the latter being given by Z0′ = (AD/Λ3). The reduced partition function is

A good interpolating formula is g (k ) =

Z=

(2.41)

The limiting behaviors are to be physically expected: g(k) is proportional to the number of scattering objects within the wavelength of the scattering wave vector. When the wavelength is longer than the entire chain size, it encompasses nearly all chain segments, so g(k) ≃ N and decreases with increasing k. When the wavelength is much shorter than the radius of the gyration, it can only probe portions of the chain without being aware of the size of the chain. The 1/k2 behavior in this large-k limit is a hallmark of the Gaussian chain (random walk) statistics. One then expects a plateau in k2g(k) for large k; this is

q(D) ≡

Z 1 = Z0′ D

∫0

D

dz

∫0

D

d z′ G (z , z′ ; N )

(2.43)

which yields a loss of conformation entropy

ΔS = kB ln q(D) G

(2.44) DOI: 10.1021/acs.macromol.7b01518 Macromolecules XXXX, XXX, XXX−XXX

Perspective

Macromolecules G(z,z′;N) can be obtained by an eigenfunction expansion49,50 with the result 2 D

G ( z , z′ ; N ) =

⎛ pπz ⎞ ⎛ pπz′ ⎞ −p2 π 2Nb2 /6D2 ⎟ sin⎜ ⎟e D ⎠ ⎝ D ⎠

∞

∑ sin⎜⎝ p=1

(2.45)

and eq 2.43 becomes q(D) =

8 π2

∞

∑ p = 1,3 ,...

⎛ p2 π 2Nb2 ⎞ 1 ⎜− ⎟ exp p2 6D2 ⎠ ⎝

(2.46)

We can easily see that for D ≫ N b, q ≈ 1, while in the 1/2

opposite limit q ≈

8 π2

(

π 2Nb2 6D 2

2

2

exp −

), leading to an entropy of

confinement in this limit ΔS ≈ kB ln

2

k π Nb k π Nb 8 − B 2 ≈− B 2 2 π 6D 6D

Figure 2. Scaled disjoining pressure vs dimensionless plate separation. The vertical dashed line indicates the end-to-end distance Ree = N1/2b.

2

(2.47)

whose amplitudes exceed the tube diameter. The transverse mean-square fluctuation of a WLC grows with chain length s as 2s3/(3lp),58 from which Odijk54 introduced a new length scalethe deflection lengthdefined as

and a free energy of confinement ΔF = −T ΔS ≈

π 2Nb2 kBT 6D2

(2.48)

λ = D2/3lp1/3

The forgoing result for the confinement of a single ideal chain between two parallel plates can be used to compute the depletion force between the two surfaces in a solution reservoir of ideal chains. To this end, we calculate the grand partition function with chemical potential μ corresponding to a bulk concentration ρ0. Since the chains are noninteracting, the partition function is the same as for an ideal gas,51 and we obtain ⎡ AD ⎤ Ξ(D) = exp⎢ 3 q(D)e βμ⎥ ⎣Λ ⎦

which is the average chain length for the transverse fluctuation to become the size of the tube diameter D, i.e., the average distance for the chain to collide with the tube wall. Thus, the dominant conformation of the chain is undulation within a confining tube with a “period” on the order of λ. Drawing analogy with the Helfrich undulation force due to membrane fluctuation,59 Odijk showed the confinement free energy is given by

(2.49)

L ⎛ lp ⎞ ΔF ≈ A ⎜ ⎟ lp ⎝ D ⎠

from which we obtain the average concentration of chains inside the slit ρ(D) = ρ0 q(D)

⎛ ∂q ⎞ kBT ∂ ln Ξ = ρ0 kBT ⎜q + D ⎟ ⎝ A ∂D ∂D ⎠

The disjoining pressure

52

(2.51)

is Π = P(D) − P(∞), so

⎛ ∂q ⎞ Π = ρ0 kBT ⎜q − 1 + D ⎟ ⎝ ∂D ⎠

2/3

(2.54)

The most accurate determination of the numerical coefficient A gives a value of 2.3565 ± 0.0004 and 2.3552 ± 0.0006 respectively from Monte Carlo simulation60 and numerical solution of the eigenvalues of the differential equation describing a WLC.61 While the structure of a flexible polymer in confinement is rather indistinctivethe conformation space essentially forms a continuuma confined semiflexible polymer can exhibit some rather distinctive structural features. For example, a long WLC confined in a spherical cavity in the strong confinement limit takes predominantly the conformation of the chain wrapping around the interior surface of the cavity62 in order to minimize the bending energy cost. This result has important implications in DNA packaging in bacterial phages.63−65 II.G.2. Stretching an Ideal Chain. Polymer stretching is one of the most basic modes of deformation governing the mechanical and rheological properties. For example, the deformation of rubbers or polymer gels involves the stretching of strands between the cross-links. A pure stretching deformation of a linear chain involves separating the two chain ends. In order to isolate the stretching from translation, we fix one end of the polymer at the origin and pull the other end. Depending on whether we fix the pulling force, or control the end distance, the thermodynamic formulation is different. We use the Gaussian chain model to illustrate the main concepts.

(2.50)

with ρ0 = eβμ/Λ3, and the pressure P(D) =

(2.53)

(2.52) 53

This is the classic Asakura−Oozawa result. Clearly, from the behavior of q (cf. eq 2.46), ρ ≈ ρ0 for D ≫ N1/2b, while significant chain depletion sets in when D ≃ N1/2b. So the range of the disjoining pressure is N1/2b, and its depth is set by the bulk osmotic pressure ρ0kBT (see Figure 2). If the confining length is less than the persistence length, the Gaussian chain model is no longer valid, and we must account for the stiffness of the polymer explicitly. The confinement of stiff and semiflexible chains is a very rich problem. Many of the key concepts were introduced by Odijk,54−56 and a recent review article by Chen covers many new recent results.57 Here we briefly discuss confinement of a long wormlike chain (i.e., L ≫ lp) in a tube of diameter D under the condition of D ≪ lp. The conformation loss in this case is twofold: the loss of the overall rotation of the chain and the cutting off of undulations H

DOI: 10.1021/acs.macromol.7b01518 Macromolecules XXXX, XXX, XXX−XXX

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Macromolecules ⎡ 3R2 ⎤ ⎛ k T ⎞3/2 Q e = ⎜ B 2 ⎟ exp⎢ − ⎥ ⎝ KNb ⎠ ⎣ 2Nb2 ⎦

II.G.2.a. Pulling with Constant Force. With a constant applied force, the end vector is fluctuating subject to the Gaussian distribution and the additional Boltzmann weight due to the force. The partition function of a polymer chain with one end fixed at the origin and another end pulled by a constant force f is ⎛ 3 ⎞3/2 ⎟ Qf = ⎜ ⎝ 2πNb2 ⎠

∫

The free energy for the composite system is thus F = −kBT ln Q e =

⎞ ⎛ 3R2 dR exp⎜ − + β f·R⎟ 2 ⎠ ⎝ 2Nb

⎡ Nb2f 2 ⎤ ⎥ = exp⎢ ⎣ 6(kBT )2 ⎦

from which we obtain the (Gibbs) free energy

⟨f⟩ = 3K ⟨ΔR⟩ =

The average end-to-end distance vector is

⟨(ΔR)2 ⟩ = (2.57)

Nb2

S=−

Nb2

(2.58)

3kB⟨R⟩2 2Nb2

2Nb2

⎛ ek T ⎞ 3 kB ln⎜ B 2 ⎟ ⎝ KNb ⎠ 2

(2.67)

3kBR2 2Nb2

⟨(Δf)2 ⟩1/2 ⟨(ΔR)2 ⟩1/2 = 3kBT

(2.68)

(2.69)

(2.70)

Comparing eq 2.58 with eq 2.65 and eq 2.59 with eq 2.68, we see that for both the fixed force and fixed extension ensemble, we obtain the same form for the force−extension curve and stretching entropy; this is a manifestation of the equivalence of ensembles in statistical mechanics.51 However, in the general case, this equivalence only holds in the thermodynamic limit when the degrees of freedom are infinite. That this equivalence holds for any N in the case of Gaussian chain is due to the statistical independence of the different degrees of freedom. In general, the two ensembles are not equivalent for finite chain lengths.71,72 The Gaussian model is valid when f ≪ kBT/b, or R ≪ Nb. When the force is sufficiently large, the finite extensibility of the chain must be taken into account. It is straightforward to work out the force extension relation for a FJC in the fixed force ensemble; the result is17

(2.61)

II.G.2.b. Pulling with Constant Extension. We now consider the case where the other end is fixed at R. Experimentally, the position of the end is usually localized by an optical trap66−68 or by attaching to the tip of a cantilever in atomic force microscopy.69,70 It is therefore instructive to couple the free end of the chain to a harmonic potential located at R with a very large spring constant K. The deviation of this end from the center of the potential is ΔR. The partition function for this composite system is 2

+

so that we have

(The normalization condition for the end-to-end vector distribution in essence defines the reference state, such that both the entropy and Helmholtz free energy are zero when f = 0. Therefore, S and F in the above equations should be interpreted respectively as the entropy change and Helmholtz free energy change due to stretching.) Note that F can also be obtained by a Legendre transform of G

2

2Nb

2

⟨(Δf)2 ⟩ = 9KkBT

(2.60)

F = G + f·⟨R⟩

3kBR2

From eq 2.66, we can decrease the uncertainty in localizing the end by increasing the stiffness of the spring. However, the trade-off is that the fluctuation in the force increases. It is easy to show that

(2.59)

3kBT ⟨R⟩2

⎛ 3 ⎞3/2 ⎜ ⎟ ⎝ 2πNb2 ⎠

(2.66)

S(R) − S(0) = −

with the associated Helmholtz free energy cost

Qe =

kBT K

The entropy change due to stretching is then

The entropy of the system with respect to the undeformed state ∂G is S = − ∂T . Using the force−extension relation, we have

F = −TS =

(2.65)

∂F

3kBT ⟨R⟩

S(⟨R⟩) = −

(2.64)

The entropy of the composite system, from S = − ∂T , is

Thus, we obtain the force−extension relation

f=

⎛ KNb2 ⎞ 3 kBT ln⎜ ⎟ 2 ⎝ kBT ⎠

This has the same form as eq 2.58, but here the force is the average force felt by the spring. The average uncertainty of the end position ⟨(ΔR)2 ⟩is

(2.56)

∂G Nb2 f = ∂f 3kBT

2Nb2

+

3kBT R

2 2

⟨R⟩ = −

3kBT R2

The instantaneous force transduced to the spring is read off from the deviation vector ΔR as f = 3KΔR. The average of ΔR is calculated from the partition function eq 2.62, which gives the average force

(2.55)

Nb f G = −kBT ln Q f = − 6kBT

(2.63)

⎛ fb ⎞ ⟨R z⟩ = Nb 3⎜ ⎟ ⎝ kBT ⎠

2

∫ dΔR e−( 3/2Nb )(R−ΔR) −(3/2)βK(ΔR)

(2.62)

(2.71)

where 3(x) ≡ coth(x) − x−1 is the Langevin function. We can easily check that for small force we recover the linear force− extension behavior of the Gaussian chain, whereas for large

where the factor of 3 in front of K is introduced for convenience. For βK ≫ 1/(Nb2), the partition function is I

DOI: 10.1021/acs.macromol.7b01518 Macromolecules XXXX, XXX, XXX−XXX

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Macromolecules

spectrum. For large N, the expansion will be dominated by the ground state termthis leads to the so-called ground-state dominance approximation.14,75 In the ground-state dominance approximation, it is possible to write the entropy of the chain as an explicit functional of the segmental density. This was first worked out by Lifshitz,76 and the result is75

force, near full extension, the force−extension behavior is given by

fb L ≈ kBT L − ⟨R z⟩

(2.72)

where L = Nb is the maximum length of the chain at full extension. For the WLC, as the chain is being straightened out, there is additional free energy cost for ironing out the small bending undulation modes. This results in a stronger divergence of the force that goes as (L− ⟨Rz⟩)−2 near ⟨Rz⟩ ≈ L. Combining the behavior for both small and large deformations, Marko and Siggia proposed the following interpolating formula:73,74 ⎞2 2⟨R z⟩ fb 1⎛ L 1 ≈ + ⎜ ⎟ − kBT L 2 ⎝ L − ⟨R z⟩ ⎠ 2

S[c(r)] = −

kBb2 6

∫ dr [∇c1/2(r)]2

(2.75)

This result can be used to obtain the density profile and the free energy of a polymer chain in an attractive well by minimization of the free energy functional F[c(r)] = E[c(r)] − TS[c(r)] ⎧ ⎫ k Tb2 [∇c1/2(r)]2 ⎬ = dr⎨c(r)U (r) + B 6 ⎩ ⎭

∫

(2.73)

This formula yielded remarkable agreement with experimental force−extension data on λ-phage DNA73 (see Figure 3).

(2.76)

subject to the normalization condition ∫ dr c(r) = N. Another useful result concerns the free energy of a polymer chain in a weak, spatially varying potential. Since a constant shift in the potential by U0 has the trivial effect of simply adding a term NU0 to the total energy, we only need to consider potentials such that ∫ dr U(r) = 0, or in terms of the Fourier mode Ũ 0 = 0, where Ũ k is the Fourier transform of U(r): Ũk =

1 (2π )3

∫ dr U(r)eik·r

(2.77)

Solving eq 2.74 by perturbation to second order, and integrating over the two end positions, we get the partition of the polymer as Z= Figure 3. Extension vs force for 97 kb λ-phase DNA (squares). The solid line is the fit to the interpolating WLC formula eq 2.73 with L = 32.80 ± 0.10 μm and lp = 53.4 ± 2.3 nm. The dashed line is the fit to the FJC model with L = 32.7 μm and b = 200 nm. Reproduced with permission from ref 73. Copyright 1994 AAAS.

=

∫ dr ∫ dr′ G(r, r′; N )

⎡ Nβ 2 V exp ⎢ ⎣ 2(2π )3 V Λ3

⎤

∫ dk ŨkŨ −kg(k)⎥⎦ (2.78)

where g(k) is the structure factor of a Gaussian chain in free space given by eq 2.38. The free energy change due to the external potential is then

II.H. Gaussian Chain in External Field. The problem of an ideal chain in an external potential is important for two reasons. First, it is a physically interesting question how polymer conformation responds to external field. Second, the theoretical treatment of many chain systems eventually involves solving the problem of a single ideal chain in a field, either a fluctuating field or a self-consistent field.18 Here we consider the simplest case of a continuous Gaussian chain in an external potential. In the presence of an external potential, the Green’s function satisfies the modified diffusion equation

ΔF = −

Nβ 2(2π )3 V

∫ dk ŨkŨ −kg(k)

(2.79)

These results are used in the RPA (random phase approximations) theory for many-chain systems.10,18 Note that this free energy change is always negative, indicating that in a spatially varying potential with ∫ dr U(r) = 0, the chain will always be able to find favorable locations to lower its free energy. One can also check that the average energy change is ΔE = 2ΔF and the entropy change is ΔS = ΔF/T < 0. Thus, once again, we see that the conformation entropy is reduced due to interactions. For a Gaussian chain in a localized repulsive potential (a problem that arises in the self-consistent field theory treatment of excluded volume or electrostatic repulsion), there needs to be a constraint to keep the chain within the range of the potential. This can be done by fixing one of the chain ends or an internal monomer. A more natural constraint is to fix the center of mass of the chain. However, no simple solution exists for fixed center of mass in a potential.

⎞ ⎛ ∂ b2 ∂ 2 U ( R ) − + β ⎟G(R, R′; N ) ⎜ 6 ∂R2 ⎠ ⎝ ∂N = δ(R − R′)δ(N )

1 Λ3

(2.74)

Clearly U < 0 generates attraction to the polymer chain, whereas U > 0 generates repulsion. For most potentials, the equation needs to be solved numerically. Formally, the Green’s function can also be solved using eigenfunction expansion. For localized, attractive potential, the eigenvalues have a discrete J

DOI: 10.1021/acs.macromol.7b01518 Macromolecules XXXX, XXX, XXX−XXX

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Macromolecules II.I. The Blob Concept. Blob is a simple and intuitive concept for understanding the scaling behavior of many polymer properties.14 Scaling refers to the special property of how a physical quantity changes under a change in length or time scales. In a more narrow (but common) usage, scaling often refers to power-law dependence of a physical property. A blob is a subchain unit whose length scale corresponds to the scale below which the chain behavior is unchanged by other forces or interactions. Energetically, the blob size corresponds to a length scale at which the energy of interaction is roughly the thermal energy kBT. We illustrate this with the stretching of an ideal chain by tension under the condition f Ree ≫ kBT and f b ≪ kBT. The first inequality means that the elastic energy stored in the chain is much larger than the thermal energy, so on the length scale of the entire chain, the chain will be distorted (elongated along the direction of the force); the second inequality means that the force is weak enough as not to distort the chain on the scale of the Kuhn length, so that the Gaussian model is still a reasonable description.

again in agreement with the exact result eq 2.60 up to a numerical factor of order 1. Of course, for the Gaussian chain all these results can be obtained exactly. However, the blob scaling argument provides a physically more intuitive understanding. Its real power becomes more evident when we study chains with excluded volume effects and at finite concentrations.

III. REAL CHAIN The ideal chain models account only for the bonding interactions that make up the polymer while ignoring all other interactions. However, monomers on a real polymer take volumeno two monomer units can occupy the same space. Furthermore, in most cases, a polymer does not exist in a vacuum; it either is dissolved in solvents, including polymeric solvents, or is surrounded by other polymers of the same type as in polymer melts or glasses. We will consider the concentration effects in later sections. Here we focus on a single polymer in a small-molecule solvent. In the presence of solvent, the relevant interaction is the potential of mean force. For a polymer in a single solvent, within the framework of the McMillan−Mayer theory,51 the solvent degrees of freedom can be integrated out, and we are left with effective interactions between the polymer segments. While in a liquid the net bare interactions between all the species are attractive, the effective segment−segment interaction can be either attractive or repulsive. Furthermore, even if all the bare interactions are pairwise, the effective interactions can become nonpairwise. The exact form of these effective interactions are difficult to calculate from first principles.13 However, at the coarse-grained level, they can be described by local pseudopotentials of the following form:

Figure 4. Illustration of the force blob for a chain under tension.

Let ξ denote the length scale at which the elastic energy is just comparable to kBT, i.e. fξ ≃ kBT

U = kBT

(2.80)

Therefore, ξ ≃ kBT/f. Since the chain is undeformed up to length scale ξ, we have random walk statistics within the blob, so the blob size is related to the number of segments in the blob m by ξ ≃ m1/2b

where ĉ(r) =

Thus, m ≃ (ξ/b)2 and the number of blobs in the chain is (2.82)

We envision the blobs to be arranged sequentially along the direction of the force (taken to be along z-axis), so the extension is then

Rz ≃

Nb2f N ξ≃ m kBT

(2.83)

or f≃

kBT Nb2

Rz

(2.84)

Up to a numerical prefecture of order 1, this is the same as eq 2.58 or 2.65. The total free energy of deformation is F ≃ nbkBT ≃

k TR 2 N (fb)2 ≃ B 2z kBT Nb

⎤ (3.1)

N

∫0 ds δ[r − R(s)] is the microscopic density of

the segments, and v and w are respectively the second and third virial coefficients. Unlike the short-ranged correlations for bonding interactions that can be renormalized into an effective Kuhn length for sufficiently long chains, these interactions are always operative as long as two or three monomers are in close spatial proximity, regardless of how far apart they are along the chain backbone. For this reason, such interactions require different theoretical treatment. The third virial coefficient is usually positive. However, the second virial coefficient can be either positive or negative. For v > 0, the effective interaction between monomers is repulsive; i.e., it is more favorable for a monomer to be surrounded by solvent molecules than by other monomers, and a polymer chain will expand (swell) relative to its ideal size. This swelling, however, not only results in a numerically larger mean-square end-to-end distance Ree or radius of gyration Rg but also changes their scaling with respective to the chain length. For sufficiently long chains, Ree ≃ Rg ∼ N0.588. For v < 0, the effective two-body interaction is attractive. This attraction can cause the chain to collapse into a globular statethe analogue of a liquid drop. Since the volume of the globule is proportional to the total number of monomers, the size of the polymer scales as Ree ≃ Rg ∼ N1/3.

(2.81)

nb = N /m ≃ N (b/ξ)2 ≃ N (fb/kBT )2

⎡

∫ dr⎣⎢ 12 vc(̂ r)2 + 16 wc(̂ r)3 + ...⎦⎥

(2.85) K

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Macromolecules The second virial coefficient has dimensions of volume and in general consists of both entropic and energetic contributions. Therefore, we can write v as ⎛ Θ⎞ v = v0⎜1 − ⎟ ⎝ T⎠

In Flory’s uniform expansion model, the end-to-end vector distribution function for the expanded chain pα(R) is assumed to be related in an affine manner to that for the unperturbed Gaussian chain p(R), i.e., pα(R) = α3p(αR), where the factor α3 is required for normalization. The entropy change upon uniform expansion is calculated using the ideal chain expression eq 2.8, with a further averaging over R (in the unperturbed Gaussian chain state), which yields

(3.2)

where v0 is a volume factor, and by the Flory−Huggins theory it is given by the ratio of the square of the monomer volume to the volume of a solvent molecule. This turns out to be important when we discuss the solvent molecular weight effects in section IV.G. Θ defines a special temperature at which v = 0. It is the analogue of the Boyle temperature51 for low-density gases. At T = Θ, the so-called theta condition, the effective twobody interaction vanishes, and the chain becomes approximately ideal (see discussions in section III.E). For all solvent conditions, we may write R ≃ Rg ∼ Nν, where the Flory exponent ν = 0.588, 1/3, and 1/2 for good, poor, and theta solvents, respectively, for sufficiently long chains. We now examine these different solvent conditions separately. III.A. Good Solvent: Excluded Volume Effect. In this section, we discuss the various theoretical treatments of the excluded volume effects for a polymer chain in free space under good solvent conditions. III.A.1. Flory Theory of Excluded Volume. The first systematic theory for the excluded volume effect was due to Flory.1,77 The key idea in the Flory theory is that the size of a polymer is determined by the balance between two opposing forces: the excluded volume interaction which causes the chain to swell and the elastic entropy which opposes it. Roughly speaking, the excluded volume interaction from eq 3.1 is

ΔS = kB⟨ln[pα (R)/p(R)]⟩ 3 = kB⎡⎣ − 2 (α 2 − 1) + 3 ln α ⎤⎦

(3.8)

The last term can be interpreted as due to the increase in the phase volume for the chain end.16 The total free energy change for the uniformly expanded chain due to excluded volume interactions is then β ΔF =

∫ dr 12 vc(r)2 ≃ kBTv NR3

33/2 z (3.10) 2 Clearly, for z ≪ 1, α ≈ 1, so the chain behaves ideal. For z ≫ 1, α5 − α3 =

α5 ≫ α3 and we have α ≈ R ee ≈ R 0

3R2 2Nb2

(3.4)

Minimizing the total free energy yields R ee ≃ N3/5b(v /b3)1/5

(3.5)

Flory’s original treatment involves a more detailed calculation of each of these terms by introducing an expansion factor α under the assumption of uniform expansion. The expansion factor is defined as Ree = αR0 or Rg = αRg0 where the subscript 0 refers to the ideal chain. (In general, the expansion factor is different for the end-to-end distance than for the radius of gyration.13 They are identical in the uniform-expansion approximation.) Approximating the density profile as Gaussian around the center of mass with the variance Rg2 = α2Rg02, the excluded volume energy is calculated to be Uex =

1 ⎛⎜ 3 ⎞⎟ 5/2 ⎝ 2π ⎠ 2

3/2

2

Z=

3/2

Nv 3 kT= zkBT 3 B Rg 2α 3

⎛ 3 ⎞3/2 ⎛ v ⎞ 1/2 ⎜ ⎟ ⎜ ⎟N ⎝ 2π ⎠ ⎝ b3 ⎠

so

1/5 33/10 1/5 v⎞ 3/5 ⎛ ⎜ ⎟ z ≈ 0.97 N b ⎝ b3 ⎠ 21/5

(3.11)

∫0

∞

dα P0(α)e−βVeff (α) ≡

∫0

∞

dα e−βF(α)

(3.12)

where P0(α) is the distribution function for the dimensionless radius of gyration, with the subscript 0 denoting the Gaussian chain, and Veff(α) is the effective excluded volume potential def ined as

(3.6)

where we have defined the dimensionless excluded volume interaction parameter z=

33/10 1/5 z , 21/5

While Flory’s treatment yields a good estimate for the exponent ν and allows for a continuous crossover between the Gaussian chain behavior and swollen chain behavior in the overall chain size, the assumption of affine expansion is physically flawed: the transformation of the polymer from one fractal dimension (with fractal dimension df = 2) to another (with df = 5/3) cannot be described by a uniform expansion on all length scales. Furthermore, the affine deformation model does not provide sufficient entropic penalty for chain contraction. An alternative, physically more appealing approach is to formulate a variational free energy78 in terms of a macrostate variable of the chain, such as the end-to-end distance or radius of gyration. The latter is the preferred macrostate variable, since it is more easily measurable (by scattering techniques, for example), and it more naturally connects to the mean-field picture of the excluded volume interaction as a ball of gas within some spherical volume. The use of the radius of gyration has the additional advantage of being able to describe polymer architectures beyond a linear chain. Introducing the fluctuating radius of gyration Q, and defining the (fluctuating) expansion factor α = Q/Rg0,79 we may write the partition function relative to that of the Gaussian chain as

(3.3)

where we have ignored the contributions from the third virial coefficient [see point (3) below]. The main entropic contribution is the elastic stretching free energy when the average end-to-end distance is R, given by Fel = kBT

(3.9)

Minimizing eq 3.9 with respect to α, we obtain

2

Uex = kBT

3 2 33/2 (α − 1) − 3 ln α + z 2 2α 3

exp[−βVeff (α)] ≡

(3.7) L

⎡ 1 exp⎢ − v ⎣ 2

⎤ ′

∫ dr c(̂ r)2 ⎥⎦

0

(3.13)

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volume effects to become appreciable; it is determined by the condition v 1/2 g ≃1 (3.19) b3

where the prime on the average indicates a conditional average for a Gaussian chain with fixed radius of gyration Q. Note that eq 3.12 is formally exact. The most probable value for α is obtained by maximizing the integrand or minimizing F(α) in eq 3.12. Since a closed form expression for P0(α) is not available, we construct an interpolating formula by keeping the leading order exponential terms in the asymptotic expressions for the distribution of the radius of gyration, eqs 2.33 and 2.34. Further ignoring the small difference between π2 and 9, we get ⎡ 9 ⎤ P0(α) ≃ exp⎢ − (α 2 + α −2)⎥ ⎣ 4 ⎦

Thus, g ≃ b6/v2, giving rise to a spatial length scale ξT ≃ g 1/2b ≃

9 2 33/2 (α + α − 2 ) + z 4 2α 3

(3.14)

(3.15)

Minimizing the free energy eq 3.15, we obtain α 5 − α = 31/2 z

(3.16)

For large z, the average end-to-end distance is ⎛ v ⎞1/5 R ee ≈ R 031/10 z1/5 ≈ 0.89N3/5b⎜ 3 ⎟ ⎝b ⎠

(3.20)

This length defines a thermal blob (hence the subscript T).83,84 Physically, the thermal blob is a subchain in which the excluded volume interaction is of order kBT. Thus, only when the chain size is much larger than the thermal blob size will excluded volume interactions be significant. Since on physical grounds b3/v > 1, the thermal blob size is always larger than the Kuhn length, and a thermal blob will contain several Kuhn units. The ratio b3/v is related to the stiffness parameter p ≡ b2/(v2/3).16 For ds DNA, the stiffness parameter p ≈ 50,16 so g ≈ 105. Since each Kuhn unit is twice the persistence length, and the persistence length for ds DNA is about 150 base pairs, this gives g ≈ 3 × 107 base pairs and ξT ≈ 35 μm. Therefore, excluded volume effects usually do not need to be considered for ds DNA in good solvents. (3) Under good solvent conditions when v > 1, the effect of the third virial coefficient can be neglected. By an analysis similar to the second virial interaction term, we can show that in 3-dimensions the third virial coefficient interactions scales as (w/b6)N0 which will be completely overshadowed by the second virial interaction term eq 3.6. Physically, this is due to the low monomer density inside a polymer coil, which makes three-body contacts much rarer than two-body contacts. (4) The Flory theory suggests, and more fundamental analysis confirms,13,85 that the long length scale statistical behavior of a polymer chain in good solvent is determined only by R02 = Nb2 and the dimensionless parameter z and not by the individual parameters N, b, and v. If we write z in terms of R0 and the thermal blob size ξT

which suggests an elastic free energy of the form βFel ≃ (9/ 4)(α2 + α−2). The second term has the clear physical interpretation as the confinement free energy (see eq 2.48). This form of free energy was used in ref 16. For Veff, we make the mean-field approximation by using eq 3.6; the total free energy is then βF =

b3 b v

(3.17)

where we have used the same expansion factor for Ree. Although eq 3.15 was not proposed by Flory, we shall also term this treatment the Flory theory, as the essential physics in the Flory theory is balancing entropic elasticity with excluded volume. Several remarks are worth making. (1) It is straightforward to generalize the Flory treatment to arbitrary space dimension. Equation 3.10 now becomes v α d + 2 − α d = Cd d N 2 − d /2 (3.18) b

z=

⎛ 3 ⎞3/2 R 0 ⎜ ⎟ ⎝ 2π ⎠ ξT

(3.21)

Then we may treat R0 and ξT as the two independent parameters. However, since N, b, and v are the more familiar and more commonly used notations for describing a polymer chain, we will continue to use these notations. (5) Although the Flory theory yields exact results for ν for d = 1, d = 2, and d ≥ 4, and a good approximate value for ν in d = 3, it does not give the correct free energy. As we will see in section III.A.5, the universal part of the partition function for a chain with fully developed excluded volume, i.e., N ≫ (ξT/b)2, behaves as Nγ−1 with a universal exponent γ ≈ 7/6. Therefore, the free energy must behave as −(γ − 1) ln N for large N; this form of the free energy cannot be obtained in any mean-field theory. The exact reason why such a simple (and obviously approximate) theory was able to predict the exponent ν so accurately remains unknown.15 However, because of its simplicity, and the wealth of useful insight it yields, the Flory theory is often the first theory to go to in treating many new problems in polymer conformations. III.A.2. Self-Consistent Field Theory. An improvement of the Flory theory can be made by a self-consistent field (SCF) treatment first proposed by Edwards86 and reformulated and generalized by Freed.33,87 The physical idea is to replace the

where Cd = (1/2)(3d/2π)d/2. (Equation 3.16 can similarly be generalized.) From this, we can easily obtain that ν = 3/(d + 2) for d < 4, and ν = 1/2 for d ≥ 4. Remarkably, the exponent is exact for d = 1, 2 and d ≥ 4 and is very close to the best numerical value for d = 3. (The exactness of ν = 3/4 in d = 2 was conjectured by Cardy and Hamber80 and confirmed by Nienhuis.81) For d > 4, the right-hand side of eq 3.18 becomes vanishingly small with increasing N, so α = 1, i.e., no chain swelling. The case of d = 4 thus marks a special dimension, called the upper critical dimension. Below the upper critical dimension, the excluded volume effects are always important (relevant in the terminology of critical phenomena82), and above this dimension, the excluded volume effects becomes unimportant (irrelevant). Exactly at d = 4, the situation is more subtlethere are logarithmic corrections to the leading N1/2 scaling15 (see eq 3.33). This is quite general at the upper critical dimension. (2) Coming back to three-dimensions, the scaling behavior eq 3.11 only holds for z ≫ 1. For z ≪ 1, the chain behaves nearly ideal. Excluded volume becomes important when z ≃ 1. Thus, there exists a characteristic chain length g for excluded M

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III.A.4. Renormalization Group Theory. Since mean-field theories, including self-consistent field theory, do not properly account for the long-range fluctuation effects in a polymer chain with excluded volume, and simple perturbation expansion has limited convergence radius, a different framework is required. That many properties show power-law behavior with universal exponents is reminiscent of the critical phenomena in magnetic systems.95 Indeed, de Gennes14 demonstrated that the polymer excluded volume problem can be mathematically mapped to the n = 0 limit of a magnetic system near its critical point, where n is the number of components in the spin. Des Cloizeaux96 further extended this mapping to finite polymer concentrations. This polymer-magnet analogy enables the use of the powerful renormalization group theory to be applied to the study of polymers. Application of RG to polymers with excluded volume using the polymer-magnet analogy allows all the critical exponents to be straightforwardly calculated. However, this analogy applies to a polymer system with a polydispersed distribution of chain lengths determined by a chemical potential.14,15 Therefore, it does not directly yield the information on the conformation of a single polymer chain with fixed length. To focus directly on the chain conformation properties, de Gennes developed a physically more intuitive implementation of RG based on the decimation procedure (a type of coarse-graining).14,97 Here we briefly explain de Gennes’s idea. Readers who are interested in a more in-depth study of renormalization group theory are referred to several monographs devoted to the subject.82,98 The key idea in the renormalization group is to study how physical properties change as the observational length scale changes (by zooming in or zooming out). As we have mentioned, the representation of a polymer with excluded volume by a Gaussian chain with second virial interactions already assumed some coarse-graining; i.e., the representation is only valid beyond some length scales. Taking this representation as the starting point for a microscopic model with the parameters N, b2, and v, we inquire how the model will be altered if we further coarse-grain the system by integrating some intermediate degrees of freedom. For example, we can group g monomers into a new effective monomer. Thus, the new chain length becomes N′ = N/g. The Kuhn length b and the excluded volume parameter v will also be modified in some complicated way. However, we expect that with sufficiently large g or with repeated coarse-graining the dimensionless excluded volume parameter u ≡ v/b3 will reach a constanta fixed pointindependent of its initial value. This is because if we consider the polymer to be made up of coarse-grained units each comprising a sufficiently large number of chain segments, then the effective Kuhn length is the spatial size of the unit, ξ, whereas the second virial coefficient is proportional to the volume of the unit ξ3. The approach to a fixed point in u at sufficiently large length scales is the origin of the universal behavior as well as the power-law exponents. We now examine how b2 and u behave under a renormalization group transformation. Borrowing some earlier results from the analysis in the Flory theory, after a decimation by grouping g monomers, we expect

instantaneous two-body interaction by the interaction of a chain segment with the average local density, which is then calculated self-consistently. For a systematic introduction of the self-consistent field theory using field-theoretical techniques, we refer the readers to the excellent monograph by Fredrickson.18 Here, we will only provide the key results. We consider a polymer with one end fixed at the origin and the other end at position R. Under the self-consistent field approximation, the effective interaction felt by a chain segment is the excluded volume parameter times the local average segmental density c(r): ueff (r) = vkBTc(r)

(3.22)

The Green function G(r,r′;s) is, from eq 2.74 ⎡∂ ⎤ b2 ∂ 2 + vc(r)⎥G(r, r′; s) = δ(r − r′)δ(s) ⎢ − 2 6 ∂r ⎣ ∂s ⎦ (3.23)

Given G(r,r′;s), the average density is calculated by c=

1 G (R , 0 ; N )

∫0

N

ds G(R, r; N − s)G(r, 0; s) (3.24)

Equations 3.23 and 3.24 form a self-consistent set of equations for G and c. The mean-square end-to-end distance is then calculated as ⟨R2⟩ =

1 Z

∫ dR R2G(R, 0; N )

(3.25)

where Z = ∫ dR G(R,0;N). The SCF theory predicts the same value for the exponent ν as the Flory theory. For the root-mean-square end-to-end distance, Edwards’s calculation gives Ree ≈ 0.87N3/5b(v/b3)1/5; this is close to the results in eqs 3.11 and 3.17. A more rigorous formulation of the SCF was provided by Kosmas and Freed,88 who avoided some of the approximations in Edwards’s calculation and included some fluctuation effects. The same value for the exponent ν was obtained, but the numerical factor in Ree was altered slightly, with Ree ≈ 0.79N3/5b(v/b3)1/5. III.A.3. Perturbation Expansion. Using the excluded volume parameter z, it is possible to perform a systematic perturbation expansion for the expansion coefficient α. We refer to the monograph by Yamakawa for a more in-depth discussion of the various perturbation approaches.13 Here we quote the result for the polynomial expansion of α to the power z6 obtained by Muthukumar and Nickel89 4

α 2 = 1 + 3 z − 2.075z 2 + 6.297z 3 − 25.057z4 + 116.135z 5 − 594.717z 6 + ... (3.26)

The coefficients obviously increase with ascending powers, making the series expansion useless for finite z. However, using appropriate resummation techniques, useful information can be obtained about the asymptotic behavior for large z. Several different resummation methods90−92 all yielded the result ν = 0.588 ± 0.001. More refined analysis using higher order calculations and alternative series expansions yielded essentially the same value.93 Therefore, ν = 0.588 is the commonly accepted numerical value. Computer simulation using lattice self-avoiding walks on a cubic lattice with 33 × 106 steps94 found ν = 0.5876.

b′2 = b2gh1(g , u)

(3.27)

and u′ = ug 1/2h2(g , u) N

(3.28) DOI: 10.1021/acs.macromol.7b01518 Macromolecules XXXX, XXX, XXX−XXX

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exponent γ and the related ring-closure exponent θ. The partition function of a polymer with excluded volume is most aptly understood as the total number of self-avoiding walks (SAWs) on a lattice. Lattice model studies suggested101 that the total number of SAWs is given by

where the b′2 = b2g and u′ = ug1/2 results from the Gaussian chain behavior whereas h1 and h2 are correction factors due to excluded volume interactions. In general, these functions are complicated unknown functions of their arguments, which can only be obtained in a perturbative form. But we have already concluded that a simple perturbation expansion fails. However, recall in the Flory analysis, in d = 4 the excluded volume becomes unimportant. Thus, if we consider 4 − d ≡ ϵ as a smallness variable, the expansion can be controlled near ϵ → 0. Then a low-order expansion is valid. This is the origin of the ϵexpansion. Extending the above analysis to general ddimension, assuming ϵ to be small, keeping the leading order in ϵ and further letting g be a continuous variable g = s′/s with s′ = s + Δs, the above recursive relations become differential flow equations in the variable s: 2 ∂ ln b2 = 1 + 2u ∂ ln s π

ZNopen ≃ μ N N γ− 1

where μ depends on the lattice coordination number q (obviously, μ < q − 1) while γ is a universal exponent dependent only on spatial dimension. This form is preserved in the continuous limit with a proper interpretation of the effective coordination number μ. The best numerical estimate yields15,93 γ = 1.160 ± 0.003 ≈ 7/6

(3.29)

(3.30)

Equation 3.30 obviously possesses two fixed point, u* = 0 and π2

u* = 16 ϵ. The u* = 0 fixed point is the Gaussian fixed point. It is unstable, since any initial nonzero u0 will flow away from 0 in either the positive or negative direction depending on the sign of u0. (The flow in the negative direction will become bounded by the inclusion of the positive third virial coefficient.14) The

ZNclosed ≃

π2

fixed point at u* = 16 ϵ is stable since for any initial u0 > 0, the flow will always tend toward this fixed point. Note that the value of this fixed point only depends on the spatial dimension and does not depend on any microscopic parameters. This is the origin of universality. From eq 3.29 we can obtain the exponent ν. Note that for s → N and sufficiently large N, we may replace u by its fixed π

−d p(r ) = R ee f (r /R ee)

1 1 15 2 + ϵ+ ϵ 2 16 512

p(r ) ≃

(3.31)

⎛ r ⎞θ ⎟ ⎝ Re ⎠

−d R ee ⎜

(3.38)

Now the partition function for the ring is the partition function for the open chain multiplied by the probability that the two ends are at distance a within volume ad, i.e. ZNclosed ≃ ZNopenadp(a)

(3.32)

(3.39)

Since the overall exponents for N on the two sides must be the same, we are led to

which gives ν = 0.592 in d = 3, very close to the best available numerical value ν = 0.588. The renormalization group equations (3.29) and (3.30) also allow us to obtain the logarithmic correction in d = 4 (i.e., ϵ = 0). From eq 3.30 we find, for large s, u ≈ (π2/8)(ln s)−1. Substituting this result into eqs 3.29 and taking s → N, we obtain R ee2 ≃ Nb2(ln N )1/4

(3.37)

For r/Ree ≪ 1, we expect f(r/Ree) to be a power-law function in its argument, i.e.

This is the first-order result in the ϵ expansion. A more accurate result is obtained by going to second order; the result is ν=

(3.36)

where Ree ∼ N b is the root-mean-square end-to-end distance. On the other hand, the probability distribution of finding the two chain ends at separation r in the ensemble of all chain conformations must be of the form102

2

1 1 + ϵ 2 16

ad N μ R eed ν

point value u* = 16 ϵ. At s = N, b2 is essentially the meansquare end-to-end distance R2ee. Thus, from R2ee ∼ N2ν, we have ∂lnR2ee/∂lnN = 2ν, from which we identify

ν=

(3.35)

Because γ ≠ 1, the SAW and hence a polymer with excluded volume have a universal logarithmic dependence on the chain length; this dependence is not captured by the Flory theory or any mean-field theory. The origin of this exponent is the inequivalence of the monomers along the chain due to the open ends.14,16 Indeed, for a polymer that closes on itself, all the bonds are equivalent, so we expect the number of configurations to be simply exponential in N (or rather N + 1 if we consider the newly formed bond), multiplied by the geometric probability of finding the chain ends within some microscopic volume of order ad in a volume Rdee, i.e.

and 1 ⎛ 16 ⎞ ∂u = ϵu⎜1 − 2 u⎟ 2 ⎝ ∂ ln s π ϵ ⎠

(3.34)

θ = (γ − 1)/ν

(3.40)

In 3-dimensions, θ = 0.275 ≈ 5/18. From eq 3.38, we see that the probability distribution decreases as the two ends approach each other. This behavior is a manifestation of the excluded volume interactions which prevent the two ends from being close to each other and is to be contrasted with that of the Gaussian chain for which the probability distribution becomes nearly constant for r/R ≪ 1 (γ = 1, θ = 0) (see eq 2.7). The behavior of the end-to-end distance distribution at large separation r/Ree > 1 is also of interest. Not surprisingly, it is related to the free energy of stretching an excluded volume chain. Using eq 3.60, we obtain

(3.33)

This result has been confirmed by lattice cluster expansion99 and Monte Carlo simulation.100 III.A.5. Other Exponents. In addition to the Flory exponent ν, a number of other critical exponents can be defined. Here we discuss two of the most useful exponents: the partition function O

DOI: 10.1021/acs.macromol.7b01518 Macromolecules XXXX, XXX, XXX−XXX

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Macromolecules −d p(r ) ≃ R ee exp[−C(r /R ee)δ ]

⎛ 2 ⎞1/2 R 0 ⎡ ϵ⎛ 2π ⎞⎤ β ΔF = 2⎜ ⎟ ⎢1 + ⎜3 ln 2 + π − 1 − 1/2 ⎟⎥ ⎝ 3π ⎠ D ⎣ 8⎝ 3 ⎠⎦

(3.41)

where δ = (1 − ν)−1 and C is a numerical constant of order 1. Taking ν = 3/5 in 3-dimensions, δ = 5/2. This exponent was first conjectured by Domb et al.103 from exact enumeration of lattice self-avoiding walks and was proven theoretically by Fisher.102 Edwards and Singh obtained the distribution eq 3.41 using a variational calculation.104 However, from our analysis in section III.C, for a very long chain with large thermal blob size ξT ≫ b, the distribution should become Gaussian again at sufficiently large distances. III.B. Confining an Excluded Volume Chain. There has been intensive renewed interest in polymers under confinement in recent years. Much of this interest has been driven both by the fundamental physics of long ds DNA molecules in confinement105−107 and by the related practical applications.108−110 Since ds DNA is more appropriately described by a wormlike chain or helical wormlike chain,111 the more interesting regime of confining ds DNA is when the confinement length is comparable to or less than the persistence length, in which case the excluded volume has little effects. We refer the readers to a recent review article by Dai et al.112 on the subject of DNA confinement and to the review article by Chen57 on the theory of WLC confinement. Here we will focus on regimes of confinement when the confinement length is larger than the persistence length. In addition to the interests involving DNA, the study of confined geometry is of fundamental interest for understanding dimensional crossover in the system behavior as the space dimension is reduced. For example, approximate relationships between several critical exponents in d dimension and their counterparts in d − 1 dimension have been derived by studying confined systems using interdimensional scaling.113,114 In particular, the Flory result ν = 3/(d + 2) was derived in ref 114 using interdimensional scaling. Confinement of a polymer with excluded volume introduces two effects. First, the presence of the confining walls reduces the number of conformations available to the polymer, as for an ideal chain. Second, unlike an ideal chain in confinement, confinement of a polymer with excluded volume increases the excluded volume repulsion of the monomers, which leads to expansion of the chain in the unconfined direction(s). Unfortunately, no exact theory is available for a polymer with excluded volume in confined geometry. We must therefore resort to approximate theoretical argument. We consider the different confinement regimes separately, defined by the relative magnitude of the confinement length D to the length scales in the polymer. III.B.a. Weak Confinement Regime (D ≫ R0). In this regime, the chain conformation is essentially unaffected, aside from some small corrections, which can be calculated using perturbation theory. For example, for a polymer between two parallel hard plates, a first-order ϵ-expansion (ϵ = 4 − d) finds the relative change in end-to-end distance in the parallel direction to be115 2

2

⟨R ⟩ − ⟨R ⟩0 2

⟨R ⟩0

ϵ⎛ 2 ⎞ ⎜ ⎟ 4 ⎝ 3π ⎠

1/2

=

(3.43)

In these expressions, the subscript 0 denotes a chain property in free space and R0 = ⟨R2⟩01/2. Note that terms that are O(ϵ) are due to excluded volume effects. Thus, confinement increases the free energy even for an ideal Gaussian chain; excluded volume further adds to this increase. On the other hand, the chain size in the parallel direction can only increase because of excluded volume. We now consider the case of b ≪ D ≪ R0, which can be further divided into two different regimes. III.B.b. De Gennes Regime (ξT ≪ D ≪ R0). In this regime, the confinement breaks the chain into many blobs whose size is set by the confinement size D. For concreteness, we consider cylindrical geometry, but the analysis can be easily extended to the slit geometry. Because the conf inement blob size is larger than the thermal (excluded volume) blob size, the monomers inside a confinement blob obeys excluded-volume chain statistics. The number of monomers in a confinement blob of size D is (cf. eq 3.11) ⎛ b3 ⎞1/3⎛ D ⎞5/3 m≃⎜ ⎟ ⎜ ⎟ ⎝ v ⎠ ⎝b⎠

(3.44)

The chain inside the cylinder can be considered a train of N/m blobs, so the chain extension along the tube R∥ is R ≃

⎛ v ⎞1/3⎛ b ⎞2/3 ⎛ R ⎞2/3 N D ≃ Nb⎜ 3 ⎟ ⎜ ⎟ ≃ R 0⎜ 0 ⎟ ⎝D⎠ ⎝b ⎠ ⎝D⎠ m

(3.45)

Since each blob has a confinement free energy of roughly one kBT, the total free energy cost is F≃

⎛ v ⎞1/3⎛ b ⎞5/3 ⎛ R ⎞5/3 N kBT ≃ NkBT ⎜ 3 ⎟ ⎜ ⎟ ≃ kBT ⎜ 0 ⎟ ⎝D⎠ ⎝b ⎠ ⎝D⎠ m (3.46)

These results were first obtained by Daoud and de Gennes116 using a mathematical scaling argument. That F ∼ D−1/ν is a rather general result for a polymer in confinement (with at least one unconfined dimension). We now show this with the mathematical scaling argument used in ref 116. We start by writing the free energy in a general scaling form

F = kBTf (R 0/D)

(3.47)

For R0 ≫ D, we expect all the chain portions to feel the confinement equally; thus F ∝N. This is only possible if f (R 0/D) ≃ (R 0/D)1/ ν

(3.48)

To gain further insight, we repeat the calculation using a simple Flory-type argument. Minimizing the sum of the stretching energy and the excluded volume interaction F /kBT ≃

⎞ R0 ⎛ 4 20 1/2 ⎜ + 3 − 2 ln 2⎟ ⎠ D ⎝9 9

R2 Nb

2

+

vN 2 R D2

(3.49)

we obtain ⎛ v ⎞1/3⎛ b ⎞2/3 R ≃ Nb⎜ 3 ⎟ ⎜ ⎟ ⎝b ⎠ ⎝D⎠

(3.42)

(3.50)

Substituting this back to the free energy, we find

and the free energy change P

DOI: 10.1021/acs.macromol.7b01518 Macromolecules XXXX, XXX, XXX−XXX

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Macromolecules ⎛ v ⎞2/3⎛ b ⎞4/3 F ≃ NkBT ⎜ 3 ⎟ ⎜ ⎟ ⎝b ⎠ ⎝D⎠

according to the relative magnitude of the force blob size to end-to-end distance of the chain and the thermal blob size. III.C.a. Weak Stretching Regime (f ≤ kBT/R0). In this regime, linear response holds, so we must have

(3.51)

which differs from eq 3.46 in the exponents for both the (dimensionless) excluded volume and the confinement length. This observation is general: while the Flory argument yields good results for the chain size, it usually gives the incorrect scaling for the free energy. III.B.c. Extended de Gennes Regime (b ≪ D < ξT). For a chain with a large thermal blob size, ξT ≫ b, another regime exists. When D is less than the size of the thermal blob ξT ≃ b4/v, a subchain inside a blob experiences little excluded volume effect, so the statistics inside the blob becomes essentially Gaussian. The number of monomers in a blob thus becomes m ≃ (D/b)2, and so the confinement free energy is ⎛ b ⎞2 F ≃ NkBT ⎜ ⎟ ⎝D⎠

⎛ ∂⟨R z⟩ ⎞ ⟨R z⟩ = ⎜ ⎟f ⎝ ∂f ⎠0

where the subscript 0 refers to the zero-force limit. From statistical mechanics of fluctuations,51 we have 2 σRz ≡ ⟨R z2⟩ − ⟨R z⟩2 =

∂⟨R z⟩ ∂βf

(3.55)

At zero force, the first half of the equation gives us the zcomponent of the mean-square end-to-end distance, so ⎛ ∂⟨R z⟩ ⎞ 1 ⎜ ⎟ = R 02 3 ⎝ ∂βf ⎠0

(3.52)

(3.56)

Therefore, the extension−force relation is

However, the chain dimension along the cylinder is not simply (N/m)D because the excluded volume between the blobs is quite weak, so that there is significant overlap between consecutive blobs. (An alternative view is to consider the blobs as of an elongated shape.117) Using eq 3.3, the excluded volume interaction energy in a blob of size D with m ≃ (D/b)2 monomers in it is kBTvm2/D3 ≃ kBTvD/b4 ≃ kBTD/ξT; this is also the same energy scale for the interaction between two blobs. (Note that this energy is less than kBT by a factor of D/ ξT, so its contribution to the free energy is negligible.) The polymer can then be thought of as effective 1-dimensional chain with n = (N/m) effective monomers having an effective Kuhn length D, with dimensionless excluded volume interaction vD/ b4 between blobs. Applying the Flory theory for 1-dimension, with the right-hand side of eq 3.18 replaced by (vD/b4)n3/2, we find (for (vD/b4)n3/2 ≫ 1) the expansion factor α ≃ n1/2(vD/ b4)1/3, and so ⎛ v ⎞1/3⎛ b ⎞2/3 ⎛ R ⎞2/3 R = αn1/2D ≃ Nb⎜ 3 ⎟ ⎜ ⎟ ≃ R 0⎜ 0 ⎟ ⎝D⎠ ⎝b ⎠ ⎝D⎠

(3.54)

⟨R z⟩ =

⎛ b ⎞2/5 fb R 02 f ≃ N 6/5b⎜ ⎟ 3kBT ⎝ ξT ⎠ kBT

(3.57)

For f R0 ≃ kBT, the magnitude of ⟨Rz⟩ is on the same order as R0 . III.C.b. Pincus Regime (kBT/R0 < f < kBT/ξT). In this regime, the chain breaks into blobs of size ξ ≃ kBT/f. Since ξ > ξT, the blob behaves as a subchain with excluded volume, so the number of monomers in the blob is ⎛ kBT ⎞5/3⎛ b3 ⎞1/3 m≃⎜ ⎟ ⎜ ⎟ ⎝ fb ⎠ ⎝ v ⎠

(3.58)

and the chain extension is ⎛ b ⎞1/3⎛ fb ⎞2/3 N ⟨R z⟩ ≃ ξ ≃ Nb⎜ ⎟ ⎜ ⎟ m ⎝ ξT ⎠ ⎝ kBT ⎠

(3.53)

(3.59)

The deformation free energythe elastic free energy stored in the chaincan be obtained by integrating the force over the distance or, equivalently and more simply, by using F ≃ (N/m) kBT. From the extension−force curve and expressing the result in terms of the extension, we have

This is the same expression as we have obtained in the de Gennes regime from both the blob scaling and the simple Flory argument. So for this regime, we have the peculiar result that the chain behaves as an excluded volume chain for the chain size R∥, but the confinement free energy scales the same as for a Gaussian chain. We note that the extended de Gennes regime, and the extended Pincus regime in the next subsection, are the analogues of the mean-field regime of overlapping but essentially noninteracting Gaussian chains identified in the study of the solution behavior of semiflexible polymers.16,118,119 III.C. Stretching an Excluded-Volume Chain. In parallel to confinement in a tube, the behavior of an excluded-volume chain stretched by their ends can be classified according to the magnitude of the force. For f ≤ kBT/R0, the chain is only weakly deformed, and we expect linear force−extension curve. For f ≥ kBT/b, the finite extensibility and bending stiffness become important; the exact behavior depends on the chain model and has been briefly discussed in the context of stretching a wormlike chain. Since excluded volume effects become unimportant in the latter regime, we focus only on forces that are much less than kBT/b. In analogy to confinement in a tube, we can further divide this regime

⎛ ⟨R ⟩ ⎞5/2 F ≃ kBT ⎜ z ⎟ ⎝ R0 ⎠

(3.60)

In our analysis, we have implicitly assumed a constant-force ensemble. Since for a long chain, the extension−force relation is independent of ensembles, we may interpret eqs 3.59 and 3.60 at a fixed end-to-end distance R = ⟨Rz⟩. Then eq 3.60 determines the end-to-end distance distribution for R/R0 ≫ 1 (see section III.A.5). III.C.c. Extended Pincus Regime120,121 (kBT/ξT < f < kBT/b). In this regime, the force blob size is less than the thermal blob size, so that within a blob the subchain behaves as Gaussian. The number of monomers within a blob is ⎛ kBT ⎞2 m≃⎜ ⎟ ⎝ fb ⎠

(3.61)

so the extension is Q

DOI: 10.1021/acs.macromol.7b01518 Macromolecules XXXX, XXX, XXX−XXX

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Macromolecules ⟨R z⟩ ≃

fb N ξ ≃ Nb m kBT

(3.62)

As in the case of confinement, to observe this regime, we need ξT ≫ b. This can be realized by using either a long chain with enough stiffness, such as ds DNA, or a long chain not too far above its theta temperature, or a long chain in a polymeric solvent (see section IV.G). This regime has been observed in Monte Carlo simulations.121 III.D. The Globular State and the Coil−Globule Transition. Below the Θ temperature, the effective two-body interaction is attractive, causing the chain to contract relative to their ideal size. With sufficient attraction, the polymer forms a dense globule of nearly uniform density c0, and the size of polymer scales as R ≃ (N/c0)1/3. We now examine the structure and the thermodynamics of this state. To illustrate the key concepts, we only consider a large globule; there are subtleties when a globule is formed from a short chain.75 For in-depth discussion of the globular state and the coil−globule transition, including comparison with experiments, we refer to the series of articles by Grosberg and Kuznetsov.122−125 Within the framework of the Flory−Huggins theory (see section IV.A), for a single chain in a solvent with Flory− Huggins interaction parameter χ, we may integrate the solvent degrees of freedom subject to the overall incompressibility. This leads to the following free energy density for the interaction pseudopotential126 f=

kBT ̂ 2 ̂ [ϕ − χϕ ̂ + (1 − ϕ)̂ ln(1 − ϕ)] vm

Figure 5. Density profile for a globule for three values of χ with N = 104 and vm = 4πb3/3. Reproduced with permission from ref 126.

which is zero; the Laplace pressure due to curvature is negligible for large globules. Using eq 3.63 in eq 3.65, the equilibrium density of the globule inside the core is given by the solution of vmc0 + ln(1 − vmc0) + χvm2c02 = 0

For χ close to χΘ ≡ 1/2, the density is approximately 3 c0 ≈ (χ − χΘ ) vm

(χ − χΘ )N1/2b3/vm > 1

(3.68)

This condition defines the Ginzburg criterion the region of validity of mean-field theory. Since the core density is uniform for a large globule and is determined by the energy of a uniform bulk (without the translational entropy term), there is no role for the Kuhn length in the core. This means there is no conformation entropy contribution to the volume part of the free energy of a globule; conformation entropy becomes important only in the interfacial region where there is density gradient (see eq 2.75). This important insight was due to Lifshitz.76 The free energy of a globule of finite size can be written as a sum of a volume term and an interfacial term 82

Fg = Nf (c)c −1 + γ(6π 1/2)2/3 N 2/3c −2/3

(3.69)

where γ is the interfacial tension (which generally depends on the core density c) and (6π1/2N/c)2/3 is the surface area of a spherical core of volume N/c. The location of the surface is assumed to be defined by the Gibbs construction. The full free energy can be obtained by a self-consistent field calculation. By casting the free energy in the form of eq 3.69, we can obtain the interfacial tension. The interfacial tension is larger for smaller globules due to curvature correction; the core density is also slightly higher for smaller globules due to the Laplace pressure.127 For large N, the core density c differs from c0 (that obtained under the volume approximation) by a curvature correction of order N−1/3. This leads to a free energy difference of order N1/3. Thus, to order N2/3 (the order of the interfacial term), we may replace c by c0 and write

(3.64)

with respect to c, i.e. ∂[f (c)c −1] 1 = 0 = 2 [cf ′ − f (c)] ∂c c

(3.67)

Clearly, to have a well-defined globule, the core density must be larger than the self-density in an ideal chain N/(N1/2b)3. Therefore, we must have

(3.63)

where vm is the monomer volume and ϕ̂ (r) ≡ vmĉ (r) is the instantaneous local volume fraction. (Note that we have subtracted the inconsequential linear term (χ − 1)ϕ̂ .) It is easily seen by expanding the above free energy density that the second virial coefficient is simply vm(1 − 2χ), which yields the well-known value of χ at the Θ point as χΘ = 1/2. As first recognized by Lifshitz,76 density fluctuation in the core of the globule is short-rangedthe inside of the core behaves like a concentrated polymer solution, so self-consistent mean-field theory is applicable, and one may replace the instantaneous density in eq 3.63 with the average density. A self-consistent field calculation based on the ground-state dominance approximation was performed many decades ago by Lifshitz and co-workers using a virial expansion form of the interaction free energy.75,76 Figure 5 shows the density (volume fraction) profile from a more recent full self-consistent field calculation126 using the full interaction potential eq 3.63. For a very large globule, the free energy of the globule is dominated by the volume term. In this so-called volume approximation, the core density is obtained by minimizing the volume part of the free energy Fg = Nf (c)c −1

(3.66)

(3.65)

The f ′int term is obviously the chemical potential of a monomer inside the globule, so the term in the brackets on the right-hand side is the (osmotic) pressure of the monomers. This equation is simply the equality of the internal osmotic pressure with the osmotic pressure of the pure solvent outside of the globule, R

DOI: 10.1021/acs.macromol.7b01518 Macromolecules XXXX, XXX, XXX−XXX

Perspective

Macromolecules Fg ≈ Nf (c0)c0−1 + γ(6π 1/2)2/3 N2/3c0−2/3

Fint c ≃ kBTb2 0 + |f0 |Δ A Δ

(3.70)

Using self-consistent field theory in the Flory−Huggins framework, the following limiting behavior for the interfacial tension is obtained:126 2 ⎧ ⎪ 0.84(χ − 0.5) (b / vm)kBT , (χ − 0.5) ≪ 1 γ=⎨ ⎪ ⎩(0.18χ − 0.11)(b/vm)kBT , χ ≫ 1

≃ kBT

Δ≃b

(3.71)

Θ T ≈ −1 T Θ

γ ≃ kBTb

3vm |τ | 2w

(3.80)

⎡ bvm4/3 4/3 ⎤ v2 F ≃ kBT ⎢ −N m |τ |2 + N 2/3 5/6 |τ | ⎥ ⎥⎦ ⎢⎣ w w

(3.72)

(3.81)

(3.82)

The transition from globule to coil occurs when F(τtr) = 0, which yields |τtr| ≃ N −1/2

(3.73)

w1/4b3/2 vm

(3.83)

Thus, the globule−coil transition takes place slightly below the Θ temperature. In the infinite chain limit, the Θ point coincides with the globule−coil transition. It can be easily verified that, not surprisingly, at and near the transition, the width of the interface becomes comparable to the size of the core, i.e., Δ/R ≃ 1. While eq 3.83 defines the transition temperature from globule to coil, a globule is no longer a well-defined state when its free energy of formation (−F) becomes on the order of the thermal energy kBT. This occurs at a temperature slightly less than τtr by an amount of δτtr, by the condition

(3.75)

F(τtr − δτtr) ≃ −kBT

2

⎛ 3 N ⎞1/3 ⎛ w ⎞1/3 1/3 −1/3 R=⎜ ⎟ =⎜ ⎟ N |τ | ⎝ 2πvm ⎠ ⎝ 4π c0 ⎠

τ2

Noting that w ≃ eq 3.80 is the same as the first line in eq 3.71 aside from the numerical prefactor. The total free energy of the globule is then (ignoring numerical prefactors in the terms)

The volume of the globule is N/c0, and so the total free energy in the core and the radius of the globule are respectively v 3 Fcore = − NkBT m |τ |2 8 w

w 3/2

v2m,

(3.74)

v3 9 kBT m2 |τ |3 16 w

vm2

⎛ N ⎞2/3 bvm4/3 4/3 |τ | Fint = γA ≃ γ ⎜ ⎟ ≃ kBTN 2/3 5/6 ⎝ c0 ⎠ w

giving a free energy density f0 = −

(3.79)

and the total interfacial free energy is

Using eq 3.65, the core density is

c0 =

w1/2 −1 |τ | vm

The interfacial tension is then

where we have defined the scaled distance from the Θ temperature

τ=1−

(3.78)

Minimizing the total interfacial energy, we obtain

These two regimes correspond to the two states of globules the molten globule and the fully collapsed dense globule identified in the experimental study by Wu and Wang.128 Selfconsistent field calculation reveals a broad crossover between these two regimes, which was termed the swollen regime in ref 126. The structure and thermodynamics of the dense globule regime depend on the particular free energy model. However, in the swollen globule and molten globule regimes, the density is low and the free energy is dominated by the second and third virial terms. These regimes therefore afford a universal description which we briefly sketch here, following the work of Lifshitz and co-workers.16,75,76 We start with the virial expansion of the free energy, eq 3.1, with the second virial coefficient given by eq 3.2, so the free energy density is kT kT f = B vmτc 2 + B wc 3 2 6

b2vm v3 |τ | + kBT m2 |τ |3 Δ wΔ w

(3.76)

(3.84)

which yields |δτtr| ≃ N −1/2

(3.77)

w 3/4b−3/2 w1/2 ≃ 3 |τtr| vm b

(3.85)

|δτtr| characterizes the width or the sharpness of the transition. This width scales with the separation between the globule−coil transition temperature and the Θ temperature by the factor w1/2 /b3, which, as we will see, is usually less than 1. Specifying to the Flory−Huggins free energy, and assuming the solvent molecule has the same volume as the monomer, we have w = v2m. The scaled transition temperature and the width of the transition are then

The interfacial structure and free energy can be obtained by performing a full self-consistent field calculation as in ref 126 or by using the ground-state dominance approximation.75 These calculations are rather involved. Here we use a simple scaling argument to work out the interfacial thickness Δ and the interfacial tension γ of a large globule. In the interfacial region, there are two free energy contributions: the loss in conformation entropy due to density gradient (see eq 2.75) and a free energy cost relative to the interior because of reduced density. The free energy per unit area in the interfacial region for an interfacial width Δ is

⎛ b3 ⎞1/2 |τtr| ≃ N −1/2⎜ ⎟ = N −1/2p3/4 ⎝ vm ⎠ S

(3.86) DOI: 10.1021/acs.macromol.7b01518 Macromolecules XXXX, XXX, XXX−XXX

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Macromolecules

dimension for the tricritcal point is dc = 3. Therefore, in 3dimensions, we expect logarithmic corrections to the ideal chain behavior in the chain sizes in the long chain limit. Such logarithmic corrections have been computed using renormalization group theory by several authors.14,146−150 For the meansquare end-to-end distance, the result has the form

and |δτtr| ≃ N −1/2p−3/4

(3.87)

respectively, where p ≡ b is the stiffness parameter. Thus, for larger chain stiffness p, the globule−coil transition takes place deeper below the Θ temperature, and the width of the transition is narrower. Since the further below Θ temperature, the higher the density of the globule, there is a larger density drop across the transition. This, together with the narrower temperature range, makes the transition more firstorder-like for stiffer chains. The change from a continuous to a discontinuous transition as the stiffness increases has been demonstrated more explicitly using the Flory expansion factor by Yang and Wang.129 However, that argument does not yield information on the structure of the globule, and the free energy is problematic as in any Flory-type analysis. III.E. The Theta Point. The Θ point for a polymer is defined as the vanishing of the second osmotic virial coefficient A2.1 By analogy with the Boyle temperature for dilute gases, this condition signals the vanishing of effective two-body interchain interactions. However, just as the third virial coefficient of gases generally does not vanish at the Boyle temperature, the third osmotic virial coefficient of a dilute polymer solution remains finite and positive at the Θ point, suggesting effects of residual higher-order interactions at the (coarse-grained) segmental level. The three-body interaction was introduced by Zimm.130 Its importance was analyzed by Yamakawa131 following earlier work by Orofino and Flory132 and by Fixman.133 Because of three-body and other residual interactions, the temperature at which A2 = 0 generally depends on the chain length, leading to a chain-length-dependent ΘN. The Θ point is defined by the limit Θ = limN→∞ ΘN. Theoretical consideration suggested122,134,135 and computer simulations found136−142 2

/v2/3 m

ΘN − Θ = αN −1/2

⎡ ⎤ B R2 ≃ Nb2⎢1 − ⎥ ln(N /s0) ⎦ ⎣

where B is a numerical factor. The value of B is −37/363 from refs 146 and 14, 37/363 from refs 147−149, and 1.9/(44π) from ref 150. The interpretation of b in eq 3.91 is also slightly different between the different authors. To date, there have been no definitive evidence from either experiments or simulations to confirm the logarithmic corrections given above. Extensive Monte Carlo simulation of lattice self-avoiding walks with nearest-neighbor attractions was performed by Grassberger and Hegger141 and systematically compared with the predictions of Duplantier.148,149 The comparison shows much larger deviation from ideality than the type of logarithmic correction suggested in eq 3.91. Large deviations from Gaussian behavior in several chain size properties (end-to-end distance, radius of gyration, and hydrodynamic radius) have been reported by many computer simulation studies.137−139,141,151−153 Several of these studies reported that the various radii approach the asymptotic N1/2 scaling as N−1/2 rather than the expected N−1 as the chain length increases. Experimentally, marked deviations from Gaussian chain behavior at the Θ were reported in the ratio of the hydrodynamic radius to the radius of gyration.154 Also, the characteristic ratio CN for polystyrene in cyclohexane (a Θ solvent) was shown to approach C∞ slower than N−1.155 Khokhlov pointed out some time ago134 that the approach to ideal chain behavior depends on the property examined. Thus, while the two-body and three-body interactions exactly compensate each other for A2 at ΘN, the cancellation is incomplete for other properties, and the degree of compensation, as well as the sign of the residual effects, varies from property to property. Within the three-parameter model framework, Cherayil et al.150 performed first-order perturbation calculations using a continuous chain model, both with a shortrange cutoff and with dimensional regularization. They argued that the cutoff is required in order to capture certain effects due to the discreteness of the chain first discussed by Yamakawa131 and that only the cutoff scheme produces physically meaningful results. These authors were able to qualitatively describe a number of the trends observed in experiments and simulations.156 The results of Cherayil et al. were disputed by Duplantier,149 whose own calculations produced identical results from the cutoff and dimensional regularization schemes. Qualitatively, Duplantier’s result for Ree agrees with that of Cherayil et al.both predict chain contraction (relative to the ideal chain) at ΘN, but they disagree on the behavior in Rgref 149 predicts contraction while ref 150 predicts expansion. Furthermore, ref 149 predicts ΘN > Θ, in agreement with the earlier argument by Khokhlov,134 while ref 150 predicts the opposite. Subsequently, Cherayil et al.157 showed that differences in the cut-off schemes can lead to different results. On the other hand, Allegra158 emphasized the backbone distancedependent two-body interactions and predicted chain expansion at the Θ point, in agreement with the Monte Carlo simulation data of Webman et al.159 However, his theory also

(3.88)

(The scaling with N is to be expected by analogy with eqs 3.83 and 3.85.) The coefficient α was argued to be positive based on the quasi-monomer (renormalization) concept due to the effect of three-body interaction.134,135 Most computer simulation data also showed α > 0, but some studies reported α < 0.138,142 For T near ΘN, the second virial coefficient scales as ⎛ Θ ⎞ A 2 ∼ ⎜1 − N ⎟ ⎝ T ⎠

(3.89)

Therefore, at the infinite-chain Θ point, T = Θ A 2 ∼ −αN −1/2

(3.91)

(3.90)

The sign of A2 is determined by the sign of α. The temperature width between ΘN and Θ defines the Θ region of a polymer. The chain behavior in this region has been an issue of considerable controversy and disagreement. Drawing on the polymer-magnet analogy, de Gennes pointed out143,144 that the Θ point is a tricritical point in the terminology of critical phenomena.145 In the language of the three-parameter model, the theta point corresponds to the vanishing of the effective two-body interaction in the double limit of infinite chain length and infinite dilution. Because of the three-body repulsion, this occurs at a temperature where the bare two-body excluded volume parameter is slightly negative. Following similar analysis as presented in the Flory theory for excluded volume, one can show that the upper critical T

DOI: 10.1021/acs.macromol.7b01518 Macromolecules XXXX, XXX, XXX−XXX

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Macromolecules

N−1/2),98 so we expect the tricritical behavior with logarithmic corrections. Comparing eq 3.94 with eq 3.91 and taking κ ≈ bR5, we find that the crossover occurs roughly around Nc ≈ 105−107. This may explain the difficulty in observing the logarithmic corrections from experiments or computer simulations.

implies that ΘN < Θ, i.e., α < 0, which agrees with some simulation data but disagrees with the majority of the cases. More recently, Rubinstein and co-workers160 showed that the finite range of the two-body interaction results in long-range orientational correlations along the chain backbone in the close vicinity of the theta point, which in turn leads to power-law corrections to the ideal chain behavior. (The effect is quite similar to the non-Gaussian behavior for polymers in melt and concentrated solutions, and we defer a more detailed analysis to section IV.F.) In essence, the finite range of the two-body interaction adds a nonlocal term of the form κ U2′ = dr [∇c(r)]2 (3.92) 2

IV. CONCENTRATION EFFECTS IV.A. A Simplified Derivation of the Flory−Huggins Theory. The Flory−Huggins theory is the most commonly used theoretical framework for studying the thermodynamics of polymer solutions and blends. While our interest in this Perspective is on polymer conformations, rather than the phase behavior, the use of Flory−Huggins theory is convenient, for example, in discussing the solvent quality, or the second and third virial coefficient for the globular state, or the molecular weight effects of the solvent on the chain conformation. The original derivation,8,9 as well as most textbook presentation, of the Flory−Huggins theory uses a rather nontrivial combinatorial argument based on a lattice model for a polymeric species in a monomeric solvent. The generalization to multiple species of polymers in solution was made independently by Flory162 and by Scott and Magat163 using similar combinatorial arguments, and the more familiar form of the Flory−Huggins theory for polymer mixtures is taken as the limit of zero solvent content (though it is possible to directly derive the entropy of mixing between two polymers using the lattice model164). These derivations are cumbersome, obscure the physical assumptions, and are limiting in its interpretation, for example, of the Flory−Huggins parameter, which in these lattice models is associated with differences in the nearest-neighbor interactions between unlike and like monomer pairs. Here we present a simplified, yet more general derivation, without resorting to the construct of a lattice. For simplicity, we consider the case of incompressible mixtures, for which there is no volume of mixing and the Gibbs free energy mixing is the same as the Helmholtz free energy of mixing. We consider nA polymers of type A with degree of polymerization NA and nB polymers of type B with degree of polymerization NB. The volumes of the two polymers in their pure state are respectively VA and VB. After mixing, the volume is obviously V = VA + VB. We write the Helmholtz free energy of species i in its pure state as a sum of an ideal gas contribution and an excess part

∫

to the usual local two-body pseudopotential. In this equation, κ is given by κ=

1 6

∫ dr r 2f (r)

( >0)

(3.93)

where f = exp[−βV(r)] − 1 is the Mayer function. The second virial coefficient, as usual, is given by v = −4π∫ r2dr f(r). The form of the nonlocal potential eq 3.92 was first proposed by Fixman and Mansfield,153 who used it to calculate the nonGaussian correction to the hydrodynamic radius at the Θ point. When the second virial term is large (positive or negative), this nonlocal term has negligible effectit scales as κN2/R5g ≃ κN−1/2b−5 and becomes dominated by the second and third virial terms. However, when the second virial coefficient is zero, this term has a non-negligible effect on the chain size and other properties for finite chains. A complete study of the polymer in the Θ region would require supplementing the three-parameter model with the additional interaction term eq 3.92 to systematically examine the behavior of various measurable properties in this four-parameter model. Here we focus on the condition of vanishing effective second virial coefficient (which implicitly includes renormalization due to the three-body term122,135). Since the nonlocal term is relatively weak, a firstorder perturbation theory suffices, and we obtain ⟨R2⟩ = NbR2 −

3/2 60κ ⎛⎜ 3 ⎞⎟ N1/2 3 ⎝ ⎠ π 2 bR

(3.94)

for the mean-square end-to-end distance, where bR is a renormalized Kuhn length incorporating contributions from the nonlocal interaction, and in a complete theory, also from the third virial coefficient. Note that we have replaced the Kuhn length b in the correction term with the renormalized bR, since the difference due to this replacement is of higher order. For comparison, we also write the mean-square internal distances between two point s and s′ in a long chain far away from the ends161 ⟨[R(s) − R(s′)]2 ⟩ = |s − s′|bR2 −

⎤ ⎡ nΛ3 βFi = ni⎢ln i i − 1⎥ − ni ln Zi(Ni , bi) Vi ⎦ ⎣ ex (0) + Vi f ̃ (Ni , bi , ρ ) i

i

(4.1)

In this expression, the first term is due to the translational entropy of the polymer chain as a whole, the second term is the conformation entropy of a reference ideal chain, and f ex i is the excess free energy density due to all interaction effectsthe hardcore repulsion and van der Waals attraction between all monomers in the system, thus including both intrachain and interchain effects. ρi(0) = niNi/Vi is the monomer density in the pure state. Λi can be interpreted as the thermal de Broglie wavelength for the center of mass degrees of freedom, although its exact meaning is immaterial since it will be canceled out upon taken the free energy of mixing. bi can be interpreted as the Kuhn length, although we may use any ideal chain model as

3/2 32κ ⎛⎜ 3 ⎟⎞ |s − s′|1/2 3 ⎝ bR 2π ⎠

(3.95)

The different coefficient for the correction term reflects the end effects in the overall mean-square end-to-end distance that are absent for the internal mean-square distance. As pointed out in ref 160, these results imply that the Flory characteristic ratio CN approaches C∞ as N−1/2, which is in agreement with experimental data.155 For suf f iciently long chains, the square-gradient term must become irrelevant by power counting (it scales with N as U

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Macromolecules a reference. The appearance of bi in an expression signifies contribution due to chain conformation. In the mixture, we can similarly write the free energy as

and chain conformation by considering only the two-body energetic interactions. For the more general off-lattice formulation, in the spirit of the van der Waals treatment of dispersion interactions, we may write the energetic part of the excess free energy as

⎤ ⎡ n Λ 3 βFmix = nA ⎢ln A A − 1⎥ − nA ln ZA(NA , bA ) V ⎦ ⎣ ⎤ ⎡ nΛ 3 + nB⎢ln B B − 1⎥ − nB ln Z B(NB , bB) V ⎦ ⎣ ex ̃ + Vf (NA , bA , NB , bB , ρ , ρ ) A

mix

fĩ

ex,en

̃ f AB (4.2)

(4.8)

(4.9)

(4.10)

if the excess free energy is purely enthalpic in origin. We emphasize that, aside from the assumption of no volume change upon mixing, eq 4.7 is exact as long as χ includes all excess free energy contributions. Approximations are introduced only when a particular form for χ is assumed. In particular, the absence of the Kuhn length in χ assumes no change in the polymer conformation upon mixing. Both theory166−169 and computer simulation170 have shown that conformation difference results in entropic contributions to the Flory−Huggins parameter and can give rise to unusual miscibility behavior in binary polymer blends.169 Two special cases of eq 4.7 are worth mentioning. For NB = 1, and vA = vB = v0, we recover the original Flory−Huggins free energy of a polymer solution derived from the lattice model. The χ = 0 case is often termed the athermal limit, only because χ is taken to be energetic in origin in the common interpretation of this parameter following the original derivation of Flory and Huggins. However, it should be clear from our derivation that for a true athermal mixture χ generally is nonzero because of conformation change of the polymer with concentration. Thus, the Flory−Huggins theory is a poor description of dilute polymer solutions because of the large conformation change of the polymer upon mixing, but the approximation becomes better in the concentrated regime where the conformation change from the pure melt to the solution becomes less significant. Another special case concerns the mixing of two polymers that differ in their labels. Experimentally this is achieved by isotopic labeling, for example, by replacing the hydrogen atoms on a hydrocarbon polymer with deuterium. Even such small changes result in a significantly nonvanishing χ.171,172 However, conceptually, it is useful to consider two polymers that are completely identical except for their labels. In this case, χ = 0 and the Flory−Huggins theory is exact. We shall make use of this fact in later sections. IV.B. Overlap Concentration. Our description of the single-chain behavior is applicable to dilute solutions when the polymers are far apart. In this regime, occasional binary polymer−polymer interactions lead to slight modification of the polymer conformation and thermodynamics. The effects are in the form of perturbation expansions in powers of the concentration. The behavior qualitatively changes when the average spacing between the polymers becomes comparable to the radius of gyration of a polymer chainthis occurs when the concentration reaches c*, the overlap concentration. At c*, the

ex ex − ϕA f à (NA , bA , ρA(0)) − ϕBfB̃ (NB , bB , ρB(0))

(4.3)

where we have introduced the volume fraction ϕi = Vi/V and made use of the obvious relation between the monomer density ρi in the mixture and the monomer density ρ(0) in the pure i component, ρi = ϕiρ(0) i . Clearly, the first line of eq 4.3 is just the ideal part of the Flory−Huggins theory upon identifying the monomer volume vi = 1/ρ(0) i . From the derivation, the ideal part is due simply to the increased entropy associated with the increased volume accessible to the polymers of both species upon mixing. We now examine the remaining three terms which define the excess part of the free energy of mixing. Since any mixing property must vanish when ϕi = 0 or 1, the excess part of the mixing free energy can be written as an expansion in the form of ϕmA ϕnB, with m, n ≥ 1. The leading order term is therefore χ ex Δf ̃ = ϕA ϕB v0 (4.4) where we have introduced a reference volume v0 in order to make the Flory−Huggins parameter dimensionless in keeping with the conventional definition of χ.165 Including higher-order terms amounts to allowing composition dependence in χ. It is instructive (though not unique) to write f ̃ ex mix as (4.5)

f ̃ ex AB,

(which can be considered a definition of the excess free energy due to cross-interactions). Then, the parameter χ is ex

= aABρA(0)ρB(0)

2 ⎛ a1/2 a1/2 ⎞ χ = v0⎜⎜ AA − BB ⎟⎟ vB ⎠ ⎝ vA

ρ (0)ϕ ρ(0)ϕ β ΔF = A A ln ϕA + B B ln ϕB V NA NB ex ̃ + fmix (NA , bA , NB , bB , ϕA ρA(0) , ϕBρB(0))

ex

i = A, B

where aij is the interaction parameter in a generalized van der Waals equation of state for mixtures. Assuming Berthelot’s rule aAB = aAA aBB , we find

Note that the conformation part of the reference free energy remain unchanged, since by definition ideal chains are noninteracting. The free energy of mixing per unit volume is then

ex

̃ − f̃ − f̃ ) χ = v0(2f AB A B

= aii(ρi(0))2

and

B

̃ex = ϕ2f ̃ex + ϕ 2f ̃ex + 2ϕ ϕ f ̃ex fmix A A B B A B AB

ex,en

(4.6)

The final form of the free energy of mixing is then β ΔF ϕ 1−ϕ χ = ln ϕ + ln(1 − ϕ) + ϕ(1 − ϕ) V NAvA NBvB v0 (4.7)

From the derivation, it should be clear that the χ parameter includes changes in all types of interactions effectsexcluded volume, dispersion interactions, and chain conformation changes. The original lattice model interpretation of χ ignores contributions from changes in excluded volume interactions V

DOI: 10.1021/acs.macromol.7b01518 Macromolecules XXXX, XXX, XXX−XXX

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where we have replaced the density operator ĉ in eq 4.14 by the collective variable c because of the δ-function (eq 4.15). The problem of a Gaussian chain interacting with an external potential has been discussed in section II.H. Since the fluctuations are weak at high concentrations, it is a reasonable approximation to do a perturbation expansion to quadratic order in η, resulting in an expression similar to eq 2.79. Performing the integration over η and the subsequent integration over c, we obtain the renormalized two-body interaction potential. In reciprocal space, the potential (in units of kBT) takes the form

overall concentration equals the self-concentration within a polymer; this gives c* ≃

N R g3

(4.11)

For good solvent, using Rg ≃ N3/5b(b/ξT)1/5, we obtain

⎛ ξ ⎞3/5 c* ≃ N −4/5b−3⎜ T ⎟ ⎝b⎠

(4.12)

For c > c*, there is significant interchain penetration. The alternative scenario wherein the coils retract due to the squeezing of other chains would be unfavorable entropically. This would lead to chain size that scales as R g ≃ N1/3c −1/3

Vk̃ = v − v 2⟨c kc −k ⟩

where ⟨ck c−k⟩ is the density−density correlation function of the solvent chains and is given by

(4.13)

⟨c kc −k ⟩ =

which may hold transiently, for example, during rapid solvent evaporation. The onset of chain overlap results in new scaling behavior in many conformation, thermodynamic, and dynamic properties for polymer solutions.14,15,17 Therefore, c* is one of the most important concepts in the study of polymer solutions.173 IV.C. Screening. Because of the interchain penetration beyond the overlap concentration, the excluded volume interaction that causes chain swelling in dilute solution becomes screened beyond some correlation length. Such screening can be intuitively understood as arising from a cancelation between interchain and intrachain repulsions, as first pointed out by Flory.1 The concept was formalized by Edwards.174,175 More fundamentally, screening arises from the coupling of a tagged chain to the fluctuating medium of other chains. This can be explicitly demonstrated in the concentrated regime (whose criterion will be given later) where the fluctuations are relatively weak. Consider a tagged polymer of length N1 in the matrix of n chains of length N. We make N1 different from N to highlight the chain length dependence. The Hamiltonian of the system is 3 βH = 2 2b +

1 v 2

∫0

N1

⎛ ∂R ⎞2 3 ds ⎜ ⎟ + 2 ⎝ ∂s ⎠ 2b

n

∑∫ j=1

N

0

ξ −2 =

⎛ ∂R j ⎞2 dt ⎜ ⎟ ⎝ ∂t ⎠

1 + v 2

∫ dr[c1̂

2

2

n

∑∫ j=1

+ 2c1̂ c + c ] + i

0

N

(4.20)

12v 2cξ 2 1 ≈ cN b2

−1

(4.21) −1

for vc ≫ N . At melt density c ≈ v , so the effective excluded volume at long length scales is v/N; this result can be obtained from a virial expansion in the concentration of the tracer chains using the Flory−Huggins free energy. From eq 3.7, the dimensionless excluded volume parameter for a test chain of length N1 in a matrix of chains of length N is z=

⎛ 3 ⎞3/2 N11/2 ⎜ ⎟ ⎝ 2π ⎠ Nb3c

(4.22)

For a polymer in its own concentrated solution or melt, N1 = N, so the effect of excluded volume becomes negligiblethe chains behave nearly ideal on length scales larger than the correlation length; see, however, important non-Gaussian corrections in section IV.F. Since the screened interaction is fairly weak, we may apply perturbation theory. To first order, the mean-square end-to-end distance is10

turns the Hamiltonian eq 4.14 into an effective Hamiltonian of solvent chains interacting with the fluctuating field −iη,10,18

∫0

⎞ 12v 2c ⎛ 1 −2 ⎟ 2 ⎜ 2 b ⎝k + ξ ⎠

Vk̃ = 0 = v −

∫ dη exp{i ∫ dr η(r)[c(r) − c(̂ r)]}

⎛ ∂R ⎞2 3 ds ⎜ ⎟ + 2 ⎝ ∂s ⎠ 2b

(4.19)

The correlation length decreases with increasing concentration, and at the melt density cv ≈ 1 it becomes comparable to the Kuhn length. Two points are worth noting. First, the interaction of the tagged chain with the matrix chains generates an effective attraction. In real space this interaction takes the form of a screened Coulomb potential with the screening length ξ. Second, for kξ ≪ 1, i.e., on length scales larger than the screening length, this attraction all but cancels the bare excluded volume interaction. The residual net interaction is

(4.15)

N1

12 (vc + N −1) b2

Vk̃ = v −

where the terms in the first line are the ideal contributions due to chain elasticity (cf. eq 2.23). Introducing the collective concentration variable c(r) for the solvent chains using the Fourier representation of the identity operator

3 2b2

(4.18)

we can write eq 4.17 as

(4.14)

βH =

1 v + c g (k ) −1 −1

where g(k) is the single-chain structure factor given in eq 2.38. Using the interpolating formula eq 2.41 and defining the correlation length

∫ dr[c1̂ 2 + 2c1̂ c ̂ + c 2̂ ]

δ[c(r) − c (̂ r)] =

(4.17)

⎛ ∂R j ⎞2 dt ⎜ ⎟ ⎝ ∂t ⎠

⟨R2⟩ = Nb2[1 + ⟨U ⟩0 ] − ⟨(R N − R 0)2 U ⟩0

∫ dr η(c − c )̂

(4.23)

where U is the pair interaction energy with the screened potential eq 4.20 and the average ⟨...⟩0 is taken in the ideal

(4.16) W

DOI: 10.1021/acs.macromol.7b01518 Macromolecules XXXX, XXX, XXX−XXX

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Macromolecules

where the factor ckBT/N is the van’t Hoff limit for an ideal solution and f(c/c*) is a dimensionless scaling function. For c/c* ≪ 1, the Taylor expansion of f(c/c*) yields the second virial coefficient A2 ≃ (Nc*)−1 for a dilute polymer solution. In the semidilute regime c/c* ≫ 1 for long polymers, we expect the end effects to become unimportant, so that the osmotic pressure should only depend on the monomer concentration and not on the chain length. This dictates that the function f(c/c*) must be a power law of the form (c/c*)x with the exponent such that the N-dependence in c* exactly cancels the 1/N factor in the van’t Hoff law. Using eq 4.12, we obtain x = 5/4, and so

(unperturbed) state. A straightforward, though somewhat involved calculation using the potential eq 4.20 yields,175 for large N ⎛ 12vξ ⎞ ⎟ ⟨R2⟩ = Nb2⎜1 + ⎝ πb 4 ⎠

(4.24)

For consistency, the correction term needs to be small, leading to a condition for the concentration v c > c ** ≃ 6 (4.25) b c** defines the concentrated regime. IV.D. Semidilute and Concentrated Solutions. In the semidilute regime defined by c* < c < c**, the large concentration fluctuations cannot be accurately described by an RPA-type weak fluctuation theory. A more systematic treatment of the correlation effects will require the renormalization group theory.15,98,176−178 Here we present a simple scaling analysis.14 In the semidilute regime, excluded volume is screened out above the correlation length but is still operative below the correlation length. This allows us to identify the correlation length with the blob size, below which the chain conformation is unaffected by the interpenetration by other chains. The physical picture of the semidilute solution is that of a collection of space filling, nonoverlapping blobs of size ξ, from which it follows that the self-concentration in a blob is the same as the overall concentration. Thus m c≃ 3 ξ (4.26)

5/4

Π ≃ kBTc(cv)

ξ ≃ b(vc)

⎛ b3 ⎞1/2 ⎜ ⎟ ⎝v⎠

⎡c 1 31/2 (cv)3/2 ⎤ ⎥ Π = kBT ⎢ + vc 2 − π 2 b3 ⎦ ⎣N

⎛ N ⎞1/2 ⎜ ⎟ ξ ⎝m⎠

(4.27)

(4.28)

Using eq 4.27 and the relation between m and ξ, we get ⎛v⎞ R ≃ N1/2b⎜ 3 ⎟ ⎝b ⎠

(4.32)

the last term reflecting correction to mean-field behavior due to conformation change. A theory that interpolates between the semidilute and concentrated regimes has been worked out by Muthukumar and Edwards.179 IV.E. Correlation Hole and Effective Interaction between Chains. Consider a tagged chain in a concentrated solution or melt. Since the density fluctuation is small, the concentration of other chains near the tagged chain must be reduced due to the volume taken by the tagged chain. This slight depletion of chain segments near a tagged chain creates what is termed a correlation hole,14 which results from the combination of chain connectivity and near incompressibility of the solution or melt. Correlation hole is a key concept in the study of dense polymer systems. Experimentally, the correlation hole can be measured by neutron scattering of deuterium-labeled chains.46 If ϕ1 is the fraction of labeled chains in a melt of otherwise identical chains, the scattering function is given by

The end-to-end distance of a polymer can be obtained by noting that the chain forms a random walk of effective Kuhn length ξ, so R≃

(4.31)

In the concentrated regime, the concentration fluctuation around the uniform state is weak, so we expect mean-field behavior with Π = (1/2)vkBTc2. (Note the absence of b, indicating no change in the chain conformation entropy from the ideal state in this mean-field expression.) The crossover concentration is just c** given by eq 4.25 More refined calculations by Edwards174 gives for c > c**

where m is the number of monomers in a blob. Since within the blob the chain does not feel the presence of other chains, the Flory scaling holds, i.e., ξ ∼ m3/5 b (v/b3)1/5. The blob size is then −3/4

⎛ b3 ⎞1/2 ⎜ ⎟ ⎝v⎠

S(k) = ϕ1(1 − ϕ1)g (k)

(4.33)

1/4

(vc)−1/8

within the RPA approximation, where g(k) is the single chain structure factor of an ideal chain. The correlation hole creates an effective interaction between two tagged chains in a melt or concentrated solution. To demonstrate this, we consider the Flory−Huggins free energy of mixing labeled chains with unlabeled chains (in a gedanken experiment) in the limit of small fraction of labeled chains ϕ1. Since there is no essential difference between melts and concentrated solutions, henceforth we specify to melts and use the more conventional density ρ rather than concentration. Expanding the Flory−Huggins free energy density to quadratic order in ϕ1, we obtain

(4.29)

It is instructive to check that at c = c* the correlation length becomes the size of an isolated excluded volume chain, while at c = c** the correlation length becomes the thermal blob size ξT ≡ b4/v. Correspondingly, the size of the chain recovers respectively the Flory excluded volume scaling and the quasirandom walk behavior in these two cases. Another important property is the osmotic pressure of a polymer solution. Des Cloizeaux96 first demonstrated using the polymer-magnet analogy, that the osmotic pressure in the semidilute regime should obey the following scaling form

Π=

ckBT ⎛ c ⎞ f⎜ ⎟ N ⎝ c* ⎠

⎛ϕ ϕ 1 2⎞ βf = ρ⎜ 1 ln ϕ1 − 1 + ϕ ⎟ N 2N 1 ⎠ ⎝N

(4.30) X

(4.34)

DOI: 10.1021/acs.macromol.7b01518 Macromolecules XXXX, XXX, XXX−XXX

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Macromolecules where ρ is the total monomer density in the melt. It is instructive to write the above expression in terms of the density of the labeled chains ρ1p ≡ ρϕ1/N, and the free energy now becomes ⎛ ρ1p N ⎞ ⎛N⎞ ⎟ − ρ1p + ⎜ ⎟ρ12p βf = ρ1p ln⎜ ⎝ 2ρ ⎠ ⎝ ρ ⎠

(4.35)

The osmotic pressure due to the labeled chains is then ⎛ ⎞ ⎛N⎞ ∂f β Π1 = β ⎜⎜ρ1p − f ⎟⎟ = ρ1p + ⎜ ⎟ρ12p ⎝ 2ρ ⎠ ⎝ ∂ρ1p ⎠

(4.36)

from which we identify the second virial coefficient between two chains

N A 2′ = 2ρ

Figure 6. (a) Potential of mean force between two polymer chains in the melt for different chain lengths, density, and temperatures. There is a shallow attraction for r/Rg > 1.5, which is nearly imperceptible on the scale of the plot. Data obtained from the integral equation coarsegraining method, courtesy of M. Dinpajooh and M. G. Guenza.

(4.37)

On the other hand, the second virial coefficient is related to the interaction potential U(r) by51 A 2′ = 2π

∫0

∞

r 2 dr(1 − e−βU (r))

solutions, the orientation correlation on the coarse-grained level should decay exponentially beyond some correlation length. This turns out not to be the case. It should be clear that the excluded volume interaction should give rise to long-range correlations between the orientation of different segments on the chain. Since the orientational correlation is different near the chain ends that in the middle section of the chain, we focus on the internal segments in a very long chain far away from the ends. Let b(s) denote the tangent vector at contour position s, i.e., b(s) = ∂ R(s)/∂s; then the coarse-grained tangent−tangent correlation function between two monomers at position s and s′ on the chain is

(4.38)

Nondimensionalizing the distance by the radius of gyration of the chain (r̃ = r/Rg), we arrive at the following useful expression: 2π

∫0

∞

r 2̃ dr (1 ̃ − e−βU (r )̃ ) =

N 2ρR g3

(4.39) −1/2

Since the right-hand side of the expression scales as N , the potential is rather weak. Therefore, we may expand the exponential on the left-hand side; thus 2π

∫0

∞

r 2̃ dr ̃ βU (r )̃ =

N 2ρR g3

(4.40)

⟨b(s) ·b(s′)⟩ = −

The scale of the potential is thus U ≃ kBTN/(ρR3g), so we may write the potential as U (r )̃ =

N kBTh(r )̃ ρR g3

1 ∂ 2⟨[R(s) − R(s′)]2 ⟩ 2 ∂s ∂s′

(4.42)

Assuming the scaling for the internal mean-square distance to be the same as the mean-square end-to-end distance for an excluded volume chain, we then obtain the correlation function for the unit tangent vector u ≡ b/b to be

(4.41)

⎛ v ⎞2/5 ⟨u(s) ·u(s′)⟩ = a |s − s′|−4/5 ⎜ 3 ⎟ ⎝b ⎠

where h is a dimensionless function of the scaled distance variable r/Rg. This effective potential has been obtained from both integral equation theory180−182 and computer simulation.183 Figure 6 shows the scaled plot of the interaction potential computed from the integral equation coarse-graining method181,182 for different temperatures, chain length, and density. The data collapse nicely to a universal curve. (The small discrepancy for r/Rg ≪ 1 is likely due to the insufficiently long chain lengths.) IV.F. Nonideality Effects in Melts and Concentrated Solutions. As we have mentioned at the beginning of section IV.C, excluded volume interactions become screened beyond the screening length (the concentration blob size) due to cancellation between interchain and intrachain repulsions. At melt density, or for concentrations exceeding c**, the screening length decreases to the monomer size, and ideal chain behavior is expected.1 However, more careful studies in recent years reveal important nonideality effects in the chain structure. We have argued that in an ideal chain the intrachain correlations due to stiffness decay exponentially beyond the persistence length. If the chains were ideal in melts or concentrated

(4.43)

where a is a constant depending on the Flory exponent ν, a ∼ ν(2ν − 1). For simplicity, we take ν = 3/5 in the above expression. If we use the more accurate ν = 0.588, the exponent is −0.824. Importantly, for Gaussian chains, ν = 1/2, so the amplitude a = 0, and hence there is no long-range correlation. In concentrated solutions and melts, the excluded volume interactions are screened beyond the correlation length. However, as we have seen in the last section, correlation hole effects due to chain connectivity and near incompressibility give rise to effective interactions between two tagged chains. The same residual interactions are operative between monomers on the same chain. This leads to nontrivial corrections in the chain size and long-range intrachain correlations in the segmental orientation. These effects were discovered and discussed in a series papers by Wittmer et al.183−186 We start with the screened potential between monomers, eq 4.20. (For a more quantitative theory, we should use eq 4.17, but the key physics can be explained more simply using eq Y

DOI: 10.1021/acs.macromol.7b01518 Macromolecules XXXX, XXX, XXX−XXX

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Macromolecules 4.20.) For high density, the screening length is short, and we may perform an expansion to order k2 to obtain 1 b2 k 2 + ρN 12ρ

Vk̃ ≈

It is of interest to examine the mean-square distance between two points s and s′ along a very long chain far from the ends. Since we have stipulated that N11/2/N < 1, we may simply take N → ∞. A slightly more involved calculation gives

(4.44)

⟨[R(s) − R(s′)]2 ⟩ = |s − s′|bR2 −

This leads to an intrachain interaction of the form βU =

1 2ρN

∫ dr [ρ (̂ r)]2 +

b2 24ρ

∫ dr [∇ρ (̂ r)]2

8 ⎛⎜ 3 ⎞⎟ 3bR ρ ⎝ 2π ⎠

3/2

|s − s′|1/2 (4.50)

Because of the square-root correction, the ratio of ⟨[R(s) − R(s′) ]2⟩/|s − s′| approaches the effective Kuhn length as |s − s′|−1/2. This seems to be the case from computer simulation of coarse-grained polymer melt.187,188 Aside from the numerical factors, the square-root correction terms can be obtained heuristically from the following argument given by Wittmer et al.184 If the chain were Gaussian, then the mean-square distance of a section of the chain of length 2s would be equal to twice the mean-square distance of two conjoining subsections each of length s. We thus define a normalized deviation

(4.45)

Remarkably, the excluded volume interaction parameter v does not appear in the above expression, indicating that both terms in the interaction are completely entropic. Indeed, the second term is quite reminiscent of the Lifshitz entropy, eq 2.75, with the factor b2 signifying the conformation origin of this term. Note that the interaction potential eq 4.45 has the same form as the two-body part of the interaction for a chain near the Θ point, i.e., the second virial term and square gradient term (eq 3.92). Although the origin of eq 3.92 is due to the finite range of the potential rather than the correlation hole effects, the effects of these square gradient terms are the sameto penalize spatial gradient in the single-chain density profile, that is, to make the density profile of an ideal chain (see Figure 1) flatter; this increases the size of the chain. Using eq 4.45, it is straightforward to use the first-order perturbation theory eq 4.23 to calculate the mean-square endto-end distance of a test chain of length N1. We skip the algebra and simply give the result here:

K (s ) =

⟨R2(2s)⟩ − 2⟨R2(s)⟩ 2⟨R2(s)⟩

(4.51)

The same correlation hole repulsion between two chains (see section IV.E) also acts between the two conjoining subsections, thus leading to K(s) > 0. To leading order, this deviation should be proportional to the effective interaction potential between the two subsections, i.e. 1 K (s) ≃ βU ≃ 1/2 3 s ρbR (4.52)

3/2 3/2 4 ⎛⎜ 3 ⎞⎟ 1 ⎛⎜ 3 ⎞⎟ N13/2 + 3bρN ⎝ 2π ⎠ bρ ⎝ 2π ⎠ ⎤ ⎡5 × ⎢ N1s0−1/2 − 5N11/2 ⎥ ⎦ ⎣2

⟨R2⟩ = N1b2 +

Writing ⟨R2(s)⟩ = sb2R − D(s), we can easily verify that to leading order D(s) ≃ s1/2/(ρbR). Figure 7 shows the normalized mean-square distances as a function of the separation along the chain contour. The

(4.46)

where s0 is a cutoff, arising from our use of the small-k expansion of the potential. If we used the full expression (eq 4.20), there would be no need for such a cutoffξ would serve the role of a cutoff length. Therefore, we expect ξ ≃ s01/2b, so s0 ≃ (ξ/b)2. In other words, s0 is the number of monomers in a correlation blob. (For melt density, ξ ≃ b so s0 ≃ 1.) We choose the numerical factor such that this cutoff term, when combined with the ideal term, reproduces eq 4.24. Using the renormalized Kuhn length, we can write eq 4.46 as ⟨R2⟩ = N1bR2 +

3/2 3/2 4 ⎛⎜ 3 ⎞⎟ 5 ⎛⎜ 3 ⎞⎟ N13/2 − N11/2 3bR ρN ⎝ 2π ⎠ bR ρ ⎝ 2π ⎠

(4.47)

where we have replaced b in the perturbation terms by bR, the difference being of higher order. For the perturbation result to be valid, we require N11/2/N < 1. Two special cases are of interest: N1 = N, i.e., a chain in its own melt, and N → ∞, i.e, a chain in a matrix of infinitely long chains. For the first case, we have ⟨R2⟩ = N1bR2 −

3/2 11 ⎛⎜ 3 ⎟⎞ N11/2 3bR ρ ⎝ 2π ⎠

Figure 7. Ratio of the mean-square distance for a polymer chain in the melt to that of the ideal chain with the same renormalized Kuhn length, for b3Rρ = 8/3. The blue curve is for the end-to-end distance (eq 4.48), and the green curve is for the internal distance (eq 4.50).

(4.48)

logarithmic scale on the horizontal axis highlights the slow approach to the ideal value. While the non-Gaussian corrections are subdominant for mean-square distances, they result in nontrivial, long-range orientation correlations. Using eq 4.42, we find for the unit tangent vector u

while for the second case, we have ⟨R2⟩ = N1bR2 −

5 ⎛⎜ 3 ⎞⎟ bR ρ ⎝ 2π ⎠

3/2

N11/2

(4.49) Z

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3/2 1 ⎛⎜ 3 ⎟⎞ |s − s′|−3/2 3 ⎝ 3bR ρ 2π ⎠

v= (4.53)

⎛ 3 ⎞3/2 N11/2v ⎜ ⎟ ⎝ 2π ⎠ Nb3

w=

(4.58)

|τtr| ≃ N1−1/2N3/4p3/4

(4.59)

and the width of the transition |δτtr| ≃ N1−1/2N1/4p−3/4

(4.60)

So we have the interesting result that the transition occurs at a lower relative temperature and the transition width is broader for larger molecular weight of the solvent. (The ratio |δτtr|/|τtr| ≃ N−1/2p−3/2 decreases with increasing N.) We must keep in mind that these results are obtained by a direct translation of those for the description of a globule in monomeric solvent. Taken on their face value, eqs 4.59 and 4.60 predict that for N1 = N the distance from the Θ point increases with the chain length as N1/2 while the width of the transition decreases with N−1/2. However, this approach is likely to break down when the size of the solvent chain becomes larger than the interfacial width of the globule. Furthermore, for any finite chain lengths satisfying N1 ≤ N2, the tagged polymer should behave quasiideal in the coil state, while for monomeric solvent, the coil state is a swollen chain. On the other hand, if we define the Θ point to correspond to the N1 → ∞ limit at fixed N, where the tagged chain will be swollen in the coil state, the Θ point so defined may not be directly related to the coil−globule transition for chain lengths N1 ≤ N2. Therefore, the issue of coil−globule transition for the case of polymeric solvent is rather subtle. A useful first step would be to perform a selfconsistent field theory calculation to study the structure of the globule and its transition to the coil.

(4.54)

(4.55)

From the analysis in section III.A.1, we see N has the effect of increasing the thermal blob size to ξT ≃ Nb4 /v

v0 2 N

Thus, we see that the mean-field Θ condition now corresponds to χΘN = 1/2. Following the analysis in section III.D, we find that the scaled transition temperature is

Therefore, the long polymer will swell if N1 > N2 and will be quasi-ideal if N1 < N2. In the swelling regime, the size of the polymer behaves as ⎛ v ⎞1/5 R ≃ N13/5N −1/5b⎜ 3 ⎟ ⎝b ⎠

(4.57)

and

for |s − s′| ≪ N. We emphasize that this long-range orientation correlation is simply a consequence of the correlation hole due to chain connectivity and nearly incompressible nature of the polymers at high concentration or melt conditions. That this correlation exists for a wide range of internal distances reflects the fact that the correlation hole acts on all scales up to the chain size. One of the experimentally measurable consequences of this longrange orientation correlation is the behavior of the scattering function of labeled chains. For Gaussian chains, the Kratky plot k2S(k) vs k has a plateau for kRg ≫ 1 (cf. eq 2.40). The longrange orientation correlation introduces a nonanalytic k3 term in the inverse structure factor which destroys the plateau.186 IV.G. Solvent Molecular Weight Effects. The chain conformation has interesting dependence on the size of the solvent molecules. To illustrate this, we consider a single long polymer of N1 segments in an athermal solvent of shorter polymers of N segments. From eq 3.7 and using eq 4.21, the excluded volume parameter is

z=

v0 (1 − 2χN ) N

(4.56)

Therefore, changing the molecular weight of the solvent provides a simple and systematic means for tuning the thermal blob size, which is relevant for observing the extended de Gennes and extended Pincus regimes in chain confinement and stretching, discussed in sections III.B and III.C, respectively. The simple Flory-type and scaling theories suggest a monotonic increase in the chain size as the degree of polymerization of the solvent decreases with the crossover from quasi-ideal to swollen behavior occurring at N1 ≃ N2. However, lattice Monte Carlo simulation by Li et al.189 of an athermal mixture of long and shorter chains showed that the number of contacts between the long-chain and short-chain segments goes through a pronounced maximum at N1 ≃ N2, implying some degree of segregation between the two components. This behavior is unexpected from the simple Flory-type or scaling theories. However, the effective interaction between two polymers in a melt possesses a shallow attractive part, so in principle the asymmetry in the depth and range for the two polymer species could lead to some type of segregation. Further studies are required to understand this behavior. The molecular weight of the solvent has considerable effects on the coil−globule transition. Using the Flory−Huggins theory, we can show that the second and third virial coefficients for the long polymer are given respectively by

V. TOPOLOGICAL INTERACTIONS The connectivity in a polymer backbone means that two polymer strands (whether on the same chain or from different chains) cannot cross each other. For linear chains with free ends, this uncrossability affects the dynamics of the polymer motion and is ultimately responsible for the chain entanglement behavior for sufficiently long chains,10 but it does not affect the equilibrium properties: while two strands cannot pass each other by crossing, they can go around each other to access the same configuration space. However, for ring polymers, or polymers with fixed ends, the topological state becomes fixed due to uncrossability; the topological constraint thus reduces the conformation space a polymer can access. In this section, we present some basic concepts for the simplest ring polymers, i.e., nonconcatenated trivial knots (rings that are not linked and do not contain internal knots; see Figure 8). The topic of polymer topology is a rich and fascinating one and is relevant to many biological processes involving DNA. We refer to the monograph by Frank-Kamenetskii190 for a delightful introduction to some elementary topology and its relevance to DNA. For readers who desire a more rigorous exposition of the topology of closed curves and their connections to polymers, AA

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N < N0, topological constraint has little effect, and we have the freely jointed ring scaling. For N ≫ N0, the topological excluded volume leads to swelling of the ring. A simple Flory argument then yields

R g ≃ ξtop(N /N0)3/5

For more accuracy we may take ν = 0.588. This scaling has been confirmed by computer simulation.202−204 V.B. Interaction between Two Nonconcatenated Trivial Knots: Topological Excluded Volume. When two nonconcatenated rings approach each other to within their radius of gyration, the uncrossability between the two rings results in an entropic repulsion.206 Let Q̃ 2(R − R′) be the restricted partition function for two polymers with their center of mass respectively at R and R′, the interaction between the two polymers is then207

Figure 8. (a) An unknotted ring (trivial knot). (b) Two nonconcatenated (unlinked) trivial knots.

we refer to the review articles by Orlandini and Whittington191 and by Micheletti et al.192 V.A. A Single Trivial-Knot Ring in Free Space. The constraint of keeping the topology fixed results in interactions between different parts of the ring polymer, even in the absence of excluded volume.193,194 To highlight the topological effect, we consider a single trivial-knot ring made of freely jointed segments that are infinitely thin (thus having no volume) yet cannot cross each other. It is instructive to contrast this topologically constrained ring with a ring made of a freely jointed chain by connecting the two ends. We refer to the latter simply as a freely jointed ring. Clearly the freely jointed ring has many topological states, starting with the trivial knot and increasing in the complexity of the knots as the size of the ring increases. It was proposed theoretically195−198 and confirmed by computer simulation199,200 that the probability that a freely jointed ring remains a trivial knot is

Ptk ≃ exp( −N /N0)

U (r ) = −kBT ln

Q̃ 2(r ) Q̃ (∞)

(5.5)

2

where we have made use of the central symmetry of the interaction to write r = |R − R′|. At infinite separation, the rings are independent of each other, so Q̃ 2(∞) =

Q 12 V2

≡ q12

(5.6)

where Q1 is the single-ring partition which includes the overall translation of the center of mass and V is the volume of the system. On the other hand, if not for the nonconcatenation constraint, the two rings would be able to penetrate each other freely and generate many topologically linked states between the two rings, in which case we would have Q̃ 2(r) = q21 for any separation r. Therefore, the ratio of the two partition functions in eq 5.5 is just the probability that two trivial knots form a trivial link (i.e., remain nonconcatenated) at separation r, thus

(5.1)

where N0 = 243 ± 8 for the freely jointed chain model. The exponential decay in the trivial-knot probability has important implications for the size of a trivial-knot ring. The mean-square radius of gyration of a freely jointed phantom ring

U (r ) = −kBT ln P000(r )

(5.7)

This topological interaction was first calculated by Vologodskii et al.206,208 using lattice Monte Carlo simulation. Their numerical results suggested the following form for the unlinking probability

Nb2 (5.2) 12 is the ensemble average of all conformations having different topologies. Since more complex knots have smaller sizes,201 excluding all knots beyond the trivial knot should result in an increase in the ring size. Physically, the increased size in a trivial knot relative to a freely jointed ring arises from the effective repulsion between parts of the ring as a result of avoiding the formation of the more complex knots. This is closely related to the interaction between two nonconcatenated rings to be discussed in the next subsectionthe interaction between two internal sections of the ring due to the unknotting constraint is similar to the interaction between two small rings due to nonconcatenation. Using the result in that subsection, the excluded volume between two sections of size N′ in the ring is R g2 =

vtop ≃ R g3(N ′)[1 − P000(N ′)]

(5.4)

P000(r )̃ = 1 − B exp( −αr 3̃ )

(5.8)

where r̃ = r/Rg, and B and α are in general N-dependent parameters. This form was justified on heuristic ground for large r̃ by Deguchi and Tsurusaki.209 These authors performed off-lattice Gaussian polygon simulations and fitted the parameters B and α with the following N dependence: B = 1 − d0N −δ ,

α = α0N −μ

(5.9)

with δ = 0.54 ± 0.02, d0 = 2.1 ± 0.2, α0 = 0.55 ± 0.03, and μ = 0.16 ± 0.01. Figure 9 shows the interaction potential eq 5.7 recently calculated by Li et al.205 using Monte Carlo simulation. Using eq 4.38, we can calculate the topological second virial coefficient209 (based on the number concentration of rings) as

(5.3)

P00 0 (N′)

A 2′ = 2π

where is the probability that two nonconcatenated (i.e, unlinked) trivial-knot rings (indicated by the superscript 00) remain nonconcatenated (indicated by the subscript 0). Since the probability of knotting (the complement of eq 5.1) becomes significant only for N ≥ N0, we expect N′ ≃ N0. This defines the topological blob size ξtop = Rg(N0) ≃ N01/2b.194 For

∫0

∞

r 2dr[1 − P000(r )]

(5.10)

The terms in the brackets 1 − obviously has the interpretation as the probability of forming all nontrivial links between two trivial knots. A2′ in eq 5.10 is the quantification of the topological excluded volume between two trivial knots. P00 0 (r)

AB

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pair level), with the confinement free energy for compressing a ring βF ∼

ρR g3 N

+

Nb2 R g2

(5.11)

where the first term is just the number of rings within the pervaded volume of a test ring and the second term is the confinement free energy discussed in section II.G.1. The assumption that the topological interaction is proportional to the number of neighboring rings is a strong one. Several other proposals for the exponent ν have been put forth by different authors based on different assumptions. Khokhlov and Nechaev211 as well as Obukhov et al.212 suggested ν = 1/4 by assuming the ring to be an ideal randomly branched structure of folded loops (lattice animals). Brereton and Vilgis213 employed the linking number as the only topological invariant and obtained ν = 1/2 − 1/(6π) ≈ 0.45. Grosberg214 argued that a ring should assume a collapsed state if penetration by other rings due to threading (reptation) is suppressed, similar to the structure formed by a linear chain in its early stage of collapsing;215 thus ν = 1/3. The ν = 1/3 exponent has also been obtained by Sakaue216 using a Flory-type theory in which the topological interaction is treated following the van der Waals theory for classical fluid (thus including virial interactions of all orders). Computer simulation by Müller et al.217 using bondfluctuation model with N up to 512 yielded ν = 0.39 ± 0.03, in agreement with the Cates−Deutsch conjecture. Reigh and Yoon218 found ν = 0.36 ± 0.02 for N up to 5000 using a simple cubic lattice model. More recent simulations using longer chains219−222 increasingly point to ν = 1/3; the larger exponents are considered crossover effects.216,220,223 Increasing the persistence length has also been shown to decrease the apparent ν closer to 1/3.224 Although there is no rigorous theory for predicting the exponent ν for nonconcatenated unknotted rings in melts and solutions, accepting ν = 1/3, Obukhov et al.223 proposed a decorated loop model for the conformation of rings. In this model, each ring is envisioned to be made up of a self-similar hierarchy of loops (sections of the ring), with the smaller loops decorating the larger ones. The length scale of the smallest loops is on the order of the entanglement length of the linear polymer Ne. Loops of all sizes are assumed to behave ideally. With this geometric picture, these authors showed that the mean-square radius of gyration of a polymer ring in melts or concentrated solutions is given by

Figure 9. Potential of mean force between two unknotted, unlinked volumeless rings. Reproduced with permission from ref 205. Copyright 2016 Springer.

Using eq 5.8, the second virial coefficient is then A′2 = (2π/3) (BRg3/α) ∼ N3ν+μ, which means that the monomer second virial coefficient is A2 ≡ A′2/N2 ∼ N3ν+μ−2. Deguchi and Tsurusaki took ν = 0.5 and obtained A2 ∼ N−0.34, which appears to be consistent with their own simulation data. More recent simulation by Li et al.205 found A2 ∼ N−0.327, which is close to the Deguchi−Tsurusaki prediction. The ring size in these simulations was up to N ≈ 500, which barely exceeds the topological blob size. For sufficiently large rings, however, we must have ν = 0.588, which would give A2 ∼ N−0.076 if we took μ = 0.16. On the other hand, if the topological excluded volume scales the same as the pervaded volume by a trivial-knot ring, we must have μ = 0; then A2 ∼ N−0.236. The large-N behavior of P00 0 (r) warrants further study. V.C. Melts and Solutions of Rings. Just as for linear chains, we may define an overlap concentration for nonconcatenated unknotted rings to be when the monomer concentration in the solution becomes comparable to the selfconcentration in a ring c* ≃ N/R3g ∼ N−4/5 for large N. For concentrations exceeding c*, the rings interact with each other both through the usual excluded volume and through the topological excluded volume discussed above. However, while the excluded volume interactions become screened beyond the correlation length of the solutions, topological constraints remain in effect. Therefore, rings in solutions and melts do not generally behave ideally. Physically, since rings do not thread each other easily, we expect the interchain topological excluded volume to dominate over the intrachain topological excluded volume. Therefore, rings in melts or solutions are expected to be more compact than topologically unconstrained Gaussian rings. However, a first-principles theory for describing ring conformations in the melts and solutions has so far remained elusive because of the difficulty in the mathematical formulation of topological constraints.191 Since the topological constraint is only important for ring sizes larger than the topological blob size, small rings in melts should behave nearly ideal with Rg ∼ N1/2. However, the scaling behavior for large rings has been an issue of much debate until recently. Cates and Deutsch210 first conjectured that the radius of gyration for nonconcatenated, unknotted rings in melt should scale as Rg ∼ N2/5. They obtained this result by balancing the free energy due to topological interaction of a ring with other rings in its pervaded volume (considered at a

⎧ Nb2 ⎪ , N≤ 12 ⎪ ⎪ ⟨R g 2⟩ = ⎨ 2/3⎡ ⎛ N ⎞−1/3⎤ 2 ⎪ ⎛N⎞ ⎢ ⎥ξ , N > ⎪1.6⎜ Ñ ⎟ ⎢1 − 0.38⎜ Ñ ⎟ ⎥⎦ e ⎝ e⎠ ⎪ ⎝ e⎠ ⎣ ⎩

Nẽ Nẽ (5.12)

Nẽ b2 . 12

where Ñ e = 2Ne and ξe2 = The authors of ref 223 fitted eq 5.12 to a diverse set of simulation data from several different groups,217,219,225−228 with the fitted Ñ e ranging from 56 to 350 depending on the models. Note that although the size of the ring scales as N1/3, it does not form a dense globulethe number of other rings in the pervaded volume of a test ring is on the order of Ñ e1/2, which is significantly larger than 1. AC

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VI. CONFORMATION CHANGE IN PHASE TRANSITIONS Since polymer conformation depends on both the concentration and temperature (through enthalpic interactions), polymer conformation has a strong effect on, and is strongly affected by, a phase change. When one of the phases involved is spatially inhomogeneous, such as in the case of microphase transition in block copolymers, the conformation change in the ordered phase is captured even at the mean-field level.231−238 For macroscopic phase separation, mean-field theories (such as Flory−Huggins) contain no information about the chain conformation. Coupling between the chain conformation and phase behavior in this case is due entirely to fluctuation effect. It is useful to broadly distinguish between two kinds of fluctuations, long-wavelength fluctuations (such as those near a critical point or spinodal) and localized fluctuations (such as the formation of small clusters). For the long-wavelength fluctuations, we may further distinguish between strong fluctuations and weak fluctuations by invoking the Ginzburg criterion82a measure of the proximity to the critical point or spinodal. One definition of the Ginzburg criterion is based on the consideration of the relative importance of the leading fluctuation correction to the inverse susceptibility to its meanfield value:82

A picture similar in spirit to the decorated loops model is the fractal loopy globules model proposed by Ge et al.229 The fractal loopy globules model assumes a self-similar structure as in the decorated loops model, but instead of the detailed quantitative geometric construction in the decorated loops model, Ge et al. conjectured that the overlap parameter of the subsections (the loops) of the rings on all size scales Ne < N′ ≤ N O=

ρR3(N ′) N′

(5.13)

is the same and equals the Kavassalis−Noolandi magic value230 OKN ≈ 10−20, which is the condition for the onset of entanglement effects in linear polymers (and gives the condition for determining Ne). This conjecture is equivalent to stating that the loops in a ring of all sizes N′ have comparable degrees of topological interactions and are always “living on the edge” of being entangled with other loops of the same size. This conjecture is consistent with the structures revealed in the computer simulation findings by Rosa and Everaes.222 The dynamical consequences of the fractal loopy globules model seem to be in broad agreement with available computer simulation data.229 Because fewer rings can pervade the volume of any given ring in the melt in comparison to linear chains, the correlation hole for a ring polymer is expected to be deeper. Following the argument used in section IV.E, we conceptually label a small fraction of rings in the melt. Using the Flory−Huggins theory (which is exact since the labeled and unlabeled rings are otherwise identical), we obtain the same second virial coefficient for the labeled rings as for the case of linear chain melts. Thus, eq 4.37 remains true. Using Rg ≃ N1/3Ñ e1/6b for N ≫ Ñ e, we then have 2π

∫0

∞

r 2̃ dr (1 ̃ − e−βU (r )̃ ) ≃

κ −1 = κ0−1 − Δκ −1

−1 where κ−1 are respectively the mean-field value and 0 and Δκ leading-order correction of the inverse susceptibility. The system can be considered weakly fluctuating when |Δκ−1| < |κ−1 0 | and is strongly fluctuating if |Δκ−1| ≥ |κ−1 0 |. Renormalization group theory is required for strongly fluctuating systems near the critical point. For strongly fluctuating systems near offcritical spinodal, the lifetime of the metastable state becomes too short for the system to be treated with equilibrium statistical mechanics. In this section, we limit our discussion to long-wavelength weak fluctuations and localized strong fluctuations. VI.A. Polymer Solutions. Using the polymer-magnet analogy, Daoud and Jannink239 constructed a comprehensive temperature−concentration state diagram. An expanded state diagram was constructed to include chain stiffness by Schaefer et al.118 Using the Lifshitz theory, Grosberg and Kutznetsov124 systematically examined the effect of globules on the phase boundary and the interaction between two globules, from which the second virial coefficient for a dilute solution of globules was obtained. In this section, we focus on the effect of the formation of single-chain globules and multichain clusters on the phase behavior. The Flory−Huggins theory predicts the critical χ and critical composition (in terms of the polymer volume fraction) to be located at

1 1/2 3

ρNẽ

b

(5.14)

Note that the right-hand side of the equation is independent of N! Therefore, the effective interaction between two test rings in the melt is a universal function of the scaled distance r̃ = r/Rg. In particular, the free energy cost for any two rings to have their center of mass at the same location is identical for all ring sizes with N ≫ Ñ e. We can again estimate the magnitude of this interaction by noting that the right-hand side of eq 5.12 is usually less than 1. Thus, expanding the exponential, we find, to leading order U (0) ≃

kBT 1/2 ρNẽ b3

(5.15)

χc =

Therefore, the full interaction potential should be of the form U (r )̃ =

kBT 1/2 ρNẽ b3

(6.1)

1 (1 + N −1/2)2 ≈ 1/2 + N −1/2 2

(6.2)

and

hr (r )̃ (5.16)

ϕc = (N1/2 + 1)−1

where hr(r̃) is a universal dimensionless function of the scaled distance between two rings. (Note that eq 5.16 has the same general form as eq 4.41, with the replacement of N there by Ñ e.) It will be interesting to verify this result using computer simulation and explore its consequences.

(6.3)

For infinitely long chains the critical temperature coincides with the Θ point, both at χ = 1/2. For finite N, the critical χc exceeds χΘ; i.e., the critical temperature is lower than the Θ temperature. In terms of the distance from the theta point τ = χΘ − χ ( 1 for real polymers, we have |τtr| > |τc|

ΔGm = ΔFm − mkBT ln ϕ1 ≈ m2/3γA − m[kBT ln ϕ1 + γA]

(6.4)

(6.10)

Thus, the coil−globule transition takes place at a temperature slightly below the mean-field critical temperature for finite chain lengths. Well below the coil−globule transition |τ|N1/2 ≫ 1, the polymer-rich phase is a concentrated solution which is well described by the Flory−Huggins theory. The dilute phase consists of single-chain globules and multichain clusters. We now work out the general conditions for the concentration and size distribution of the clusters in the dilute phase. Let Fm denote the free energy of a cluster consisting of m chains and Cm its concentration. Because the polymer concentration is low in the polymer-poor phase well below the critical point, we ignore the interaction between different clusters. The free energy density, including the translational entropy of the clusters, can be written as

which can be used to estimate the globule concentration at the coexistence (the solubility limit) and the nucleation barrier when the concentration exceeds the solubility limit. The cluster size distribution in the dilute solution must be a decaying function of m at large m for the system to be stable. This is possible only if the net coefficient of the linear term in eq 6.10 is negative. When the coefficient is positive, the distribution becomes nonmonotonicfirst decreasing and then increasing without bound. The system is then metastable. Thus, vanishing of the linear coefficient yields the coexistence condition ⎛ γA ⎞ ϕ1co = exp⎜ − ⎟ ⎝ kBT ⎠

A more careful derivation using equality of chemical potential and osmotic pressure by including the translational entropy of the polymer in the concentrated phase gives

∞

F /V =

∑ {CmFm + kBTCm[ln(Cmam) − 1]} m=1

(6.5)

⎛ γA ⎞ ϕ1co = ϕH exp⎜ − ⎟ ⎝ kBT ⎠

In this equation, am is a reference volume arising from evaluating the partition function for internal degrees of freedom of the cluster, which cannot be determined from mean-field theories such as the self-consistent field theory. However, aside from setting a volume scale, am has little effect on the thermodynamics of the system; for simplicity, we take it to be the volume of a cluster. Then Cmam is just the volume fraction ϕm. Minimization of the free energy subject to a total fixed concentration of polymers yields ⎛ ΔF ⎞ ϕm = (ϕ1)m exp⎜ − m ⎟ ⎝ kBT ⎠

(6.6)

⎡ ⎤ b ϕ1co ≈ 3(χ − 0.5) exp⎢ −1.93 1/3 (χ − 0.5)4/3 N2/3⎥ ⎣ ⎦ v

(6.7)

where (36π)1/3N2/3c0−2/3 = A is the area of a single globule. Therefore, to a good approximation, the standard free energy of formation of the m-sized cluster is simply the reduction in the interfacial free energy ΔFm ≈ (m

2/3

− m)γA

(6.13)

for χ − χΘ ≪ 1 and ⎡ b ⎤ ϕ1co ≈ exp⎢ − 1/3 (0.89χ − 0.53)N 2/3⎥ ⎣ v ⎦

(6.8)

(6.14)

for χ ≫ 1 where we have taken c0v ≈ ϕH to be 1. Aside from the numerical prefactors, eq 6.13 was given earlier in ref 16. Because of the lower penalty for the chains in the dilute phase (proportional to N2/3 rather than N), the polymer concentration on the dilute side of the binodal (the solubility limit) is considerably higher than predicted by the FH theory (or any mean-field theory that ignore the formation of globules); see Figure 10.

As the concentration increases, the cluster size distribution shifts toward larger cluster sizes dictated by eq 6.6. It is convenient to write eq 6.6 as

⎛ ΔGm ⎞ ϕm = exp⎜ − ⎟ ⎝ kBT ⎠

(6.12)

where ϕH is the volume fraction in the concentrated phase (the subscript H standing for high concentration). This simple conclusion follows from the fact that the concentrated phase at coexistence is at nearly zero osmotic pressure due to the extreme dilution of the dilute phase for long chainsthis is the same condition that determines the density of a globule core and of the multichain clusters. Thus, the main difference between the chains in the dilute phase and the concentrated phase is the interfacial free energy cost of the globules and clusters in the dilute phase, which is compensated by the gain in translational entropy. At the coexistence, far away from the critical point (|τ|N1/2 ≫ 1), the concentration of dimers and higher-order clusters is much lower than that of the single-chain globule, so the total volume fraction in the dilute phase is mostly due to single-chain globules, i.e., ϕL ≈ ϕco 1 . The situation is similar to the vapor− liquid coexistence in simple fluids at low temperatures where the vapor pressure is low. Using the asymptotic behavior of the interfacial tension and the density near the Θ point, we have

where ΔFm ≡ Fm − mF1 can be considered the standard free energy of formation of the m-sized cluster. Note that minimization of the free energy eq 6.5 automatically results in equality of chemical potential for chains in the different-size clusters. At a given temperature, the density inside the single-chain globules and multichain clusters is determined by eq 3.66 and is independent of the total chain concentration. Similar to the free energy of a globule, we may write the free energy of an mcluster as Fm ≈ mNf (c0)c0−1 + γ(36π )1/3 m2/3N2/3c0−2/3

(6.11)

(6.9)

where AE

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Comparing with eq 6.11, we get the supersaturation at the pseudospinodal: ⎡ γA ⎤ S sp = ϕ1sp/ϕ1co ≃ exp⎢0.587 ⎥ kBT ⎦ ⎣

(6.20)

124

Grosberg and Kutznetsov derived the spinodal condition from consideration of the second virial coefficient of the osmotic pressure in the dilute phase. The general expression for the second virial coefficient is given by eq 4.38, where the interaction potential U(r) now is that between two globules with center of mass separation r. U(r) was calculated in ref 124 using self-consistent field theory in the ground state dominance approximation. Here we obtain the second virial coefficient using a simple scaling analysis. The interaction potential U(r) is zero when two globules are separated well beyond their radius of gyration and is the lowest when the two globules are fused into one. The fusion energy is

Figure 10. Modified phase diagram (on the dilute side) for polymer solution for N = 100. The pseudospinodal and cluster-modified binodal were calculated using the analogues to eqs 6.10 and 6.10, with quantities calculated using self-consistent field theory. Reproduced and adapted from ref 127.

U (0) = F2 − 2F1 = (22/3 − 2)γA = −0.413γA

(6.21)

so the potential must be of the following scaling form U (r ) = −0.413γAh(r /R g)

Equation 6.10 can be used to estimate the nucleation barrier in the metastable state when ϕ1 exceeds ϕco 1 . To be consistent, we use the approximate form of the coexistence concentration eq 6.11 and define the supersaturation S ≡ ϕ1/ϕco 1 . For S > 1, the free energy eq 6.10 has a maximum at m* =

γ 3A3 8 27 (kBT ln S)3

Since the range of the integral in eq 4.38 is Rg, and within the range exp (−βU) ≫ 1, the second virial coefficient is approximately ⎛ γA ⎞ γA ⎞ N ⎛ A 2′ ≃ −R g3 exp⎜0.413 ⎟ ≃ − exp⎜0.413 ⎟ kBT ⎠ c0 ⎝ kBT ⎠ ⎝

(6.15)

(6.23)

with a free energy barrier of ΔG* =

γ 3A3 4 27 (kBT ln S)2

Including the second virial term, the osmotic pressure of the dilute phase in terms of the polymer concentration ρ1p is β Π = ρ1p + A 2′ ρ12p

(6.16)

As expected, both the size of the critical nucleus m* and the nucleation barrier ΔG* diverge at the binodal S = 1. The metastable state is kinetically viable only if there is a significant free energy barrier to cluster growth. Metastability can no longer be maintained when the barrier becomes of order kBT.240 A pseudo-spinodal can be defined when the free energy barrier is some multiples of kBT. Choosing 10kBT as the criterion, ref 127 located the pseudospinodal using ΔG calculated from self-consistent field theory (see Figure 10). Here we use a more heuristic argument based on eq 6.10.241 We consider nucleation of the concentrated phase from the supersaturated dilute phase as a stepwise growth of the clusters, from single-chain globules to dimers, trimers, etc. In order for nucleation to be probable, the free energy increase must not be much larger than kBT. (Note the increment decreases as m increases.) We thus define the pseudospinodal by the condition that the free energy of formation of the dimer be of order kBT, i.e.

ΔG2 ≃ kBT

ρ1spp ≃

⎛ c0 γA ⎞ exp⎜ −0.413 ⎟ N kBT ⎠ ⎝

(6.25)

This is essentially eq 6.19 upon converting to volume fraction and noting ϕH = c0vm. VI.B. Binary Polymer Blends. The same issue of the coupling between the chain conformation change and phase separation exists for polymer blends. Using the same volume approximation for the globules as in in the case of monomeric solvent and approximating the interfacial tension using that for the flat interface,242 it is possible to determine the shift in the phase boundary due to formation of globules and clusters well below the critical temperature. Because of the limitation of space, we shall not pursue such a calculation. We leave this together with the issue of coil−globule transition in polymeric solvent to future work. Here we will briefly discuss the leading order correction to the chain size due to the unfavorable interaction between the two chain species for the special case of a symmetric A/B binary blend with the same monomer volume v0, Kuhn length b, and chain length N. The Flory−Huggins theory predicts a universal phase diagram in terms of the combination χN and the volume fraction of one of the species ϕA, with a critical point located at

(6.17)

(6.18)

from which we obtain ⎡ γA ⎤ ϕ1sp ≃ exp⎢ −0.413 ⎥ kBT ⎦ ⎣

(6.24)

Spinodal is located at ∂Π/∂ρ = 0, so we obtain the chain concentration at the spinodal to be

Since all the terms in eq 6.10 are large, the above condition is approximately ΔG2 ≃ 0

(6.22)

(χN )c = 2,

(6.19) AF

ϕAc = 1/2

(6.26) DOI: 10.1021/acs.macromol.7b01518 Macromolecules XXXX, XXX, XXX−XXX

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where D(k2R2g) is the Debye function given in eq 2.39. In ref 261, the corrections in the mean-square end-to-end distance and mean-square radius of gyration are given in terms of integrals over k, which in general can only be evaluated numerically. Here we obtain an approximate analytical expression for the mean-square end-to-end distance. Making use of the interpolation formula for the Debye function, we can write the screened potential as

and a spinodal curve given by the relation χN =

1 2ϕA ϕB

(6.27)

The absence of the Kuhn length in these expressions and the phase diagram signifies the absence of conformation entropy effects in the Flory−Huggins theory, a consequence of the mean-field (random mixing) approximation. Physically, however, because of the unfavorable interactions between the A and B chains, we expect a given A chain is more likely to be surrounded by other A chains than by B chains. This concentration correlation will in turn affect the conformation of the polymers. The parameter that characterizes the importance of fluctuation/correlation effects is the so-called invariant degree of polymerization defined as N̅ = N(ρ2b6). We have seen its square root N̅ 1/2 ≃ ρRg3/N many times throughout this Perspective; it is the number of chains in the pervaded volume of a test chain. The larger N̅ is, the more mean-field-like the system behaves as each chain is simultaneously interacting with a large number of chains. Therefore, for large N̅ , we expect the Flory−Huggins theory to provide a good description for the phase behavior except in a narrow region near the critical point where fluctuations become large. This region can be quantified by the Ginzburg number Gi ≡ |1 − χc/χ|, and theoretical analysis243 yields Gi ≡ |1 − χ c /χ | = αN̅ −1

1 b2 k 2 2 48 1 G̃ k ≈ + − χNϕB2 − (χN )2 ϕA ϕB3 2 ρN 12ρ ρN ρN 2b2 k + λ −2 (6.31)

where λ is the correlation length of the polymer blend, defined by λ −2 =

1 2

∫ dr[K(c + cB̂ )2 + 2χρ−1c cB̂ ]

(6.28)

3/2 3/2 ⎡ 11 ⎛⎜ 3 ⎞⎟ 8⎛ 3 ⎞ R ee2 = NbR2 ⎢1 − N̅ −1/2 − ⎜ ⎟ N̅ −1/2χN ⎢⎣ 3 ⎝ 2π ⎠ 3 ⎝ 2π ⎠ ⎛ ⎞⎤ 8 χNϕA ϕB⎟⎥ × ϕB2⎜1 + ⎝ ⎠⎦ 15

(6.33)

The first two terms are just the mean-square end-to-end distance in the homopolymer melt, eq 4.48, which we will denote as R2ee,0. Note that both the correlation hole correction to the chain size and enthalpic correction have the same dependence on N̅ , since both effects are governed by this parameter, as pointed out by Qin and Morse.261 To order N̅ −1/2, the relative change in R2ee due to enthalpic interaction is δR ee2 2 R ee,0

⎡ 2χNϕB2D(k 2R g2) ⎤ 1 ⎢1 − ⎥ ρND(k 2R g2) ⎢⎣ 1 − 2χNϕA ϕBD(k 2R g2) ⎥⎦

=−

3/2 ⎛ ⎞ 8 ⎛⎜ 3 ⎞⎟ 8 χNϕA ϕB⎟ N̅ −1/2χNϕB2⎜1 + ⎝ ⎠ 3 ⎝ 2π ⎠ 15

(6.34)

For χN less than the critical value 2, the correction is maximum in the tracer limit ϕA ≪ 1 and vanishes when ϕA = 1. For χN > 2, at a given ϕA, the magnitude of the correction increases with increasing χ up to the spinodal. Close to the spinodal, however, the theory is no longer applicable as the system loses its metastability due to nucleation.240,244,262 We note that the theory of Morse and co-workers treats the single-chain structure factor in a renormalized one-loop perturbation. A self-consistent procedure for determining the screened intrachain interaction and the single-chain structure has been developed by Schweizer, Yethiraj, and co-workers within the polymer reference interaction site model (PRISM) theory for polymer solutions, mixtures, block copolymers, and

(6.29)

where ρ = 1/v0 is the overall monomer density of the blend, tagging a test A chain, integrating over the concentration fluctuation of the remaining chains, and finally taking the incompressibility limit K → ∞, we obtain the following screened potential G̃ k =

(6.32)

Note that the first two terms are just the interaction due to correlation hole in a polymer melt (eq 4.44), while the third and fourth terms are due to the enthalpic interactions. In keeping with the self-consistency of the one-loop renormalization, χ in these questions should be understood as the effective χ.244,260,261 Interestingly, the last term produces no divergence at the spinodal when the correlation length diverges, reflecting the weak coupling between the chain conformation and the long wavelength concentration fluctuationa point noted by Qin and Morse.261 (However, other aspects of the theory break down near the spinodal.244,261) We can thus approximate the last term by setting λ−2 = 0, and a first-order perturbation calculation then yields, after renormalizing the Kuhn length as in section IV.F

where α is a numerical coefficient which can be much large than 1.244 Fluctuations in polymer blends and block copolymers have been studied by many authors using liquid-state theories,245−247 coarse-grained field theoretical approaches,166,167,244,248−252 and computer simulation.253−259 A systematic and rigorous theory was developed by Morse and co-workers.260,261 based on a renormalized one-loop perturbation expansion. Their work addresses a wealth of thermodynamic and structural issues related to fluctuations in binary polymer blends in the singlephase state and diblock copolymers in the disordered phase. In particular, it provides a consistent treatment of the coupling between the thermodynamics and the chain conformation (within the framework of the one-loop theory). It is beyond the scope of this Perspective to summarize the main results of this work. The most relevant result for our purpose here is change in the chain size due to the unfavorable enthalpic interaction. For the case of symmetric binary polymer blend, one can follow similar steps as in section IV.C to obtain the screened interaction potential for a test chain of type A. By taking a total interaction potential of the form Uint =

12 (1 − 2χNϕA ϕB) Nb2

(6.30) AG

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Macromolecules polyelectrolyte solutions.246,263−268 A recent field-theoretical variational method by Shen and Wang269 on the conformation and thermodynamics of polyelectrolyte solutions also features a similar self-consistent procedure. It will be interesting to explore the possibility of extending these nonperturbative approaches to examine the coupling between chain conformation and phase transitions.

different base-pairing combinations lead to different secondary structures, which in turn result in different three-dimensional structures. However, heterogeneity of the molecule or the interactions are not necessary for the system to exhibit molecular individualism, as evidenced by the example shown in ref 270. For example, the hairpin turns formed by a wormlike polymer in a confining tube272 is a distinct structure that must be treated separately from an undulating chain without such turns. A ring polymer in the matrix of long linear chains can be threaded by one chain, two chains, three chains, etc.,273 and these different states have different structural (e.g., radius of gyration) and dynamic (e.g., diffusion) properties. The material properties are often a result of the interplay between the distinct polymer conformations and the conformation continuum. For example, the unique rheological properties of end-associating polymer solutions274 reflects both the chain elasticity (which is due to entropy change in the conformation continuum) and the dynamic transitions between distinct conformation states such as loops, bridges, and dangling chains.275−277 Identifying and enumerating the distinct polymer conformations is in general a complex problem. For example, sophisticated algorithms have been developed to predict the secondary structure formation in RNA.278−280 Another wellknown example is protein folding281,282 where the challenge is the prediction of both the native structure and the folding pathway which involves a plethora of distinct intermediate states. While for simple systems such as the example given in Figure 11 where the structures can be envisioned from physical intuition, for more complex problems other methods will be needed. From a dynamical point of view, the existence of molecular individualism arises from separation of time scalestransitions between the different conformations within the same individuality take place on time scales that are much shorter than transitions between conformations in different individualities. The different individualities may correspond to different local free energy basins or may be separated by dynamical constraints, such as topological entanglements. From a statistical mechanical point of view, the essence of identifying dynamically and thermodynamically meaningful sets of distinct molecular conformations is the reduction from the fully microscopic coordinates to a low dimensional manifolds of coarse-grained states. Such dimensional reduction is in general a complex and nonlinear process and usually cannot be accomplished analytically. In recent years, several studies have used machine learning techniques to discover low-dimensional coarse-grained states that are essential in folding of biomolecules from molecular simulation trajectories.283−288 In soft matter and polymers, machine learning has been shown to be a promising strategy for identifying polymorphism in simple Lennard-Jonesian and model water systems,289 for categorizing the complex self-assembled patchy colloidal assemblies,290 and for classifying different states in the polymer coil−globule transition.291 Since the existing dimensional reduction methods use the particle coordinates as the degrees of freedom, it may be challenging to distinguish between topologically different states (such as the two conformations shown on the right in Figure 11). Nevertheless, machine learning can be a useful tool for learning and identifying complex molecular conformation states in polymers.

VII. MOLECULAR INDIVIDUALISM In most of our discussions in this Perspective, the polymer conformations form a continuum in the high dimensional conformation space, with constant transitions between the different conformations. In this case, the properties of a polymer chain are determined by the ensemble of chain conformations; no particular chain conformation stands out as being dominant over other chain conformations. However, in some cases, certain chain conformations stand out as distinctively different from others; these particular chain conformations are well separated from the others, and the transitions among these conformations are infrequent. De Gennes, in a commentary on the chain conformations revealed by elongational flow deformation of λ-phage DNA,270 coined the term “molecular individualism”.271 He invoked this term to refer to the phenomenon of distinct chain conformations that are observed under the same experimental conditions. However, we can generalize this concept to any chain conformations that are separated from the conformation continuum. Consider the simple case of a linear polymer with four evenly spaced bifunctional association groups. It is straightforward to enumerate all the possible pairing configurations formed by these four associating groups (see Figure 11). For Gaussian

Figure 11. (a) Different paring conformations for an associating polymer. (b) The Feynman diagram representation corresponding to the different conformations, with the orange dashed lines representing the pairing interactions. Note that the topological difference between the two conformations in case iii cannot be distinguished in the Feynman diagram.

strands, the statistical weight of these configurations can be easily calculated. If the pairing interaction is weak, all the configurations shown are part of the total conformation space for the open chain. Yet, for strong interactions, these conformations assume their distinctive identities and must be considered individually in the computation of thermodynamic and dynamic properties. As the polymer becomes more heterogeneous in composition and/or interactions, the diversity of these discrete conformations increases. A well-known example is the formation of secondary structures in RNA molecules, where AH

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VIII. OUTLOOK In this Perspective, I have given a “guided tour” to some essential concepts in polymer conformation. The emphasis, as I explained in the Introduction, is on understanding. Thus, as much as possible I have tried to sketch the steps that lead to a certain result and to combine mathematical analyses with physical arguments. I have also tried to highlight the connections between different results, for example, between the confinement free energy of a single polymer and the depletion force between two plates in a polymer solution or between the Pincus stretching regime and the Domb−Fisher exponent in the end-to-end distribution for an excluded volume chain, or between the nonideality of the chain conformation and the Lifshitz entropy, or between the Flory−Huggins theory and the effective interaction between two chains in the melt or concentrated solutions. Such connections may be obvious to the experts, but they are rarely explained in introductory textbooks. Thus, the analogy of this pedagogical review to a “guided tour”. Like in any guided tour, it is not possible to visit all the attractions, and so I had to pick and choose from a vast body of knowledge a subset to be presented within a reasonable length. The topics covered include both what would be considered familiar subjects in standard polymer physics textbooksthis is foundational material and is prerequisite for the more advanced topicsand some more recent results or not-so-standard subjects such as topological effects and the theta point. In spite of some obvious omissions, I hope this Perspective provides a coherent view of the essential concepts in polymer conformation. Another parallel for the sightseeing analogy is the beauty and grandeur of the subject matter. Polymers are wonderful not only because they are useful and essential for life but also because they are beautiful to behold (in our mind or with real microscopes). That a seemingly hopelessly large molecule can be reasonably represented as beads connected by harmonic springs, that this model is related to both Brownian diffusion studied by Einstein and the Schrödinger equation for a quantum particle, that such a simple model can actually describe the real molecule (at the coarse-grained level), and that the simple equations describing their behaviors can be used to interpret and guide experiment and can actually lead to better materialsall this is simply fantastic and awe-inspiring. I have enjoyed learning and relearning all the topics I have written about in this paper, and I hope the readers have likewise enjoyed reading about them and are motivated to learn more. I have alluded to some open/unsolved (unresolved) problems in several places in the article where they naturally present themselves. There are several obvious ones. Even for an ideal chain, the problem of a polymer with fixed center of mass in an external field does not seem to have a simple solution (except in a low-order perturbation). The problem is important because often the interacting chain problem is reduced to an ideal chain in some effective field. Therefore, the solution of this problem will enable a host of problems to be studied, for example, a self-consistent field calculation of the potential of mean force between two chains. Another obvious problem is the nature of the Θ point and the chain behavior in the Θ region. Although this region occupies only a small portion in the parameter space, conceptually and historically, the issue of the Θ point is an important one. Our knowledge of polymer physics would not

be complete without a clear and satisfactory resolution of the conflicting theories,14,15,98,146−150,158 discrepancy between theories and simulation data,141 and differences between the different simulation results.136−142 The work by Shirvanyants et al.160 is an important and promising step. The issue of topological interactions is another area where more work is needed. Because of the difficulties in analytically describing topological constraints,10 computer simulation has been the main tool for studying topological effects in polymers. However, the topological blob size is typically quite large,204,228 making it challenging to reach clear scaling regimes for long chains. In the Perspective, I mentioned the unlinking probability between two unknotted rings as a problem which requires further study to understand its scaling behavior for large ring sizes. A prediction (or rather a deduction) was also made about the effective interaction between two rings in the melts with a strength independent of the ring sizes. These results can only be tested with computer simulation of very large ring sizes. Finally, a consistent theory for phase transitions in polymeric systems that can describe both the chain conformation and thermodynamics remains elusive. In this Perspective, I discussed how the formation of globules can drastically affect the phase boundary and spinodal, but in the existing treatments, the globules are put in by hand124,127 and do not arise naturally from the theory as a localized form of fluctuation. On the other hand, the weak fluctuation theories244,248,260,261 are unable to describe the large-amplitude, localized fluctuations such as globules or micelles. Although not discussed in this paper, localized fluctuations in the form of micelles have qualitative effects in the microphase separation of highly asymmetric diblock copolymers.292,293 Recent work by Bates and co-workers highlights the role of these localized fluctuations in giving rise to complex free energy landscapes.294 In addition to these fundamental problems in polymer conformation, biology and new materials provide fertile grounds for studying many issues related to polymer conformation. The packaging of a bacteriophage involves crowding the virial capsid with the genome ds DNAa long semiflexible chain63,64,295and the lac repressor functions by clamping two operator sites along the ds DNA to form a loop.296−299 Understanding the DNA conformation in these systems provides the essential mechanistic insight into their functions. In the materials arena, the unique mechanical behavior and unusually high toughness in double-network gels300,301 result from the very different degrees of stretching between the polymer strands on the first, brittle network and on the second, more flexible network.302 In semiconducting polymers, chain conformation plays an essential role in affecting the energy and charge transport.303,304 Clearly, theoretical understanding of polymer conformation in these contexts can have great impact for the optimal design of the material properties. To illustrate this last point, I would like to end with the recent example of the work by Kornfield and co-workers305 on using end-associating, long telechelic polymers for safe jet fuel applications. The design of these polymers that self-assemble into “megasupramolecules” (MW > 5000 kg/mol) at low concentrations (