Equilibrium and Dynamics of Dilute Polymer Solutions

Equilibrium and Dynamics of DilutePolymer Solutions

Arti Dua

Abstract Polymers show universal behavior at long length and time scales.In this chapter, the equilibrium and dynamics of dilute polymer solutionsis presented in terms of theoretical models that form the basis of polymerphysics. The size of an ideal polymer is calculated from the freely jointed chainmodel, the freely rotating chain model and the Gaussian equivalent chainmodel. The Edwards Hamiltonian for a continuous ideal chain is obtainedfrom a continuous limit of the Gaussian equivalent chains. The size of areal chain in good and poor solvents is estimated from the free energy thatincludes entropic and enthalpic contributions.The temperature dependence ofchain size is discussed using the concept of thermal blobs. The force-extensionrelations for ideal and real polymers are illustrated using the tension blobs. Abrief introduction to stiff chains is presented in terms of the wormlike chainmodel. Starting from the Langevin equation for a Brownian particle, thesalient features of the Rouse and Zimm models are presented. The behaviorof polymers under extensional, rotational, simple shear and linear-mixed flowsis discussed using the finitely extensible Rouse chain.

1 Introduction

Polymers are large molecules composed of many smaller repeating units(monomers) bonded together. Solvent induced thermal fluctuations bringabout the internal rotation of bonds generating large numbers of confor-mational states [1-4]. On time scales much larger than the rotational rates,experimental measurements typically capture the average effects of the rapidfluctuations in the conformations. Even instantaneous measurements involveaverages over large ensembles of identical molecules. Given that the system

Department of Chemistry, Indian Institute of Technology, Madras, Chennai-600036, India

1

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has a large number of degrees of freedom, there is no general methodology,either classical or quantum mechanical, that can treat the system exactly.Apart from the enormously large number of conformations, each polymericsystem is characterized by its own unique chemistry at the microscopic levelbecause of the different chemical structures of the monomers. Nevertheless,at one level, polymeric systems can be quite simple, as the following suggests:the macroscopic properties of a collection of small molecules strongly dependon the intrinsic chemical details of the molecules, but when the same set ofchemical species are bonded to form a long molecule, they exhibit universalbehavior. In other words, at length scales much larger than the monomer size,the properties are insensitive to the structure of the individual molecules ofthe chain, but depends on the universal chain-like nature of the polymer.Thus, long wavelength properties can be described using general statisticalmechanical methods that depend on a few system dependent phenomenolog-ical parameters [1-11]. In what follows, we describe the theoretical methodsthat form the basis of polymer physics. Minimal models that depends ona few phenomenological parameters to describe long length-scale chain-likeproperties of the polymers is presented in the next section.

2 Ideal Polymers

An equilibrium property that governs the behavior of a polymer solution isthe chain size and its dependence on temperature. The understanding of thechain structure is key to unraveling the thermodynamical, mechanical anddynamical properties of polymer solutions. The size of a polymer depends onthe quality of solvent in which it is dissolved. In a real polymer, the relativestrengths of the monomer-monomer and monomer-solvent interactions deter-mine if the chain is swollen or collapsed. At high temperatures, the effectivemonomer-monomer interactions are repulsive and polymers are found in aswollen state. At low temperature, polymers form globules due to dominanceof attractive interactions. At a special temperature, called the θ-temperature,the attractive and repulsive parts of the effective monomer-monomer inter-actions cancel each other resulting in nearly ideal chain conformations. Itis worth mentioning that concentrated polymer solutions and linear poly-mer melt show near ideal chain behavior since surrounding polymers screenthe interactions between monomers. The conformational behavior of an idealpolymer forms the basis of most models in polymer physics [1-11]. In whatfollows, we describe several different models that estimate the size of an idealpolymer.

The end-to-end vector, R, of a flexible polymer consisting on n bond vec-tors is given by

R =n∑

i=1

ri, (1)

Equilibrium and Dynamics of Dilute Polymer Solutions 3

Fig. 1 Freely jointed chain

where ri is the ith bond vector of magnitude |ri| = l. The mean squareend-to-end distance of this chain is given by

⟨R2

⟩=

n∑

i=1

n∑

j=1

〈ri · rj〉 = l2n∑

i=1

n∑

j=1

〈cos θij〉 , (2)

where θij represents the angle between the bond vectors ri and rj and 〈· · ·〉corresponds to the ensemble average. Figure 1 represents a freely jointedchain. In a freely jointed chain model, all bond vectors are completely uncor-related with each other, that is, 〈cos θij〉 = 0 for i #= j. This leads to a verysimple expression for the mean square distance,

⟨R2

⟩= nl2. In general, re-

stricted bond-angle rotations and steric hinderance lead to local correlationsbetween bond vectors along the chain backbone. These local correlations areaccounted for by defining Flory’s characteristic ratio,

Cn =1n

n∑

i=1

n∑

j=1

〈cos θij〉 . (3)

In terms of Flory’s characteristic ratio, the mean square distance is given by⟨R2

⟩= Cnnl2. Flory’s characteristic ratio, Cn, increases with the increase

in the number of monomers. For n $ 1, it saturates to a finite value C∞yielding

⟨R2

⟩≈ C∞nl2. For many flexible polymers, C∞ is typically 7-9.

A simple estimate of C∞ is provided by the freely rotating chain model.The free rotating model assumes all torsional angles to be equally probable.It accounts for the local correlation by estimating 〈ri · rj〉 = l2(cos θ)|i−j|.In terms of the exponential function, (cos θ)|i−j| = exp(−|i − j|l/lp), wherelp = −l/ ln(cos θ) defines the length scale at which the correlations betweenthe bond vectors decay. Thus, the mean-square end-to-end distance of a freelyrotating chain is given by

⟨R2

⟩= l2

n∑

i=1

n∑

j=1

exp(−|i− j|l/lp), (4)

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which simplifies to yield

⟨R2

⟩= nl2

1 + cos θ

1− cos θ, (5)

where C∞ = (1 + cos θ)/(1− cos θ). For flexible polymers, the value of C∞calculated from the freely rotating chain model is less than expected. How-ever, the freely jointed chain model provides a rough estimate of the localcorrelations between the bond vectors. In the next section, a unified descrip-tion of ideal flexible polymers is provided in terms of the Gaussian equivalentchains.

3 Gaussian Equivalent Chains

As a rough approximation, the conformation of a flexible chain can be viewedas the trajectory of a random walker that makes n independent steps of fixedstep length l to cover a certain distance R [12, 13]. What is the probability forR? Since the steps are completely uncorrelated with respect to each other, theoverall probability of a given conformation is the product of the probabilitiesof the individual steps, that is,

P =n∏

0

p(ri), (6)

where ri is the bond vector of the ith step. Since the magnitude of the bond(step) length is fixed, the simplest way to define the bond probability is

p(ri) =1

4πl2δ(|ri|− l). (7)

The bond probability defined above accounts for the constraint of a constantbond length, but it leads to a rather complicated distribution function forR, the vectorial distance from one end of the chain to the other. At thesame time, since R is the sum of a large number of random variables (bondvectors), the probability function can be described by a Gaussian functionaccording to the central limit theorem. For large enough n and distance muchless than nl, the distribution is given by

P (R, n) =(

32πnl2

)3/2

exp(−3R2

2nl2

). (8)

The above distribution can be exactly produced by a bond probability thatis given, not by Eq. (1), but by


p(ri) =(

32πl2

)3/2

exp(−3r2

i

2l2

). (9)

A chain described by the above bond distribution function is referred to asa Gaussian chain. Since the Gaussian chain is made up of several Gaussianlinks, each effective step of the random walk is itself a random walk. TheGaussian chain is, therefore, a self-similar object, or fractal [14]. Its struc-ture looks the same on any length scale. If we group n0 monomers togetherto form a single unit, the polymer chain is again a random walk of largerstep length. To be specific, there are N = n/n0 effective steps of step lengthb = n1/2

0 l, with⟨R2

⟩= Nb2 = nl2. In other words, averaging over cer-

tain degrees of freedom (coarse-graining) does not change the general scalingbehavior of the long wavelength properties of the chain. In particular, thedistribution function depends on the ratio R2/

⟨R2

⟩, which is scale invari-

ant. The step length defined above is a coarse-grained length, which is theaverage over a large number of successive bond vectors. The chain thus ob-tained is a coarse-grained description of a real chain, and is equivalent toit at this level of description. The chain can, therefore, be called an equiva-lent Gaussian chain. The notion of equivalent chains was first introduced byWerner Kuhn [8, 9]. The coarse-grained length b is hence commonly referredto as the Kuhn length. In terms of the bond length l and the the number ofmonomers n, the Kuhn length and the effective number of monomers are givenby b = C∞l/ cos θ/2 and N = n cos2 θ/2/C∞. As long as the Kuhn length ismuch smaller than the contour length, an equivalent chain describes a flex-ible polymer. An equivalent chain is a good approximation to a real chainfor long wavelength properties. Indeed, most theoretical approaches modelflexible chains in terms of the equivalent chain description.

The equivalent Gaussian chain has the unphysical feature that it can ex-tend more than its contour length. In fact, the chain has an infinite storedlength. This feature is the reflection of the underlying Gaussian nature ofthe distribution function, which has a long length tail, and hence a finiteprobability for extension beyond its contour length. The infinite extensibilityis often a serious limitation when describing chain extension under variouskinds of force fields. Some of these issues are discussed in Sect. 9, wherea finitely extensible Gaussian model is presented for chain extension undershear flow.

4 Continuous Chains

The equivalent Gaussian chain can be subjected to a further level of coarse-graining by letting b → 0, N → ∞ and Rmax = Nb constant. This leads toa representation of the polymer as a continuous path with fixed ends. Theprobability that such a path begins at R0 and ends at R is given by a path

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integral representation:

G(R,R0;N) =∫ rN=R

r0=R0

D[rn] exp

[− 3

2b2

∫ N

0dn|rn|2

], (10)

where rn is the position vector of the nth monomer. N is the effective num-ber of monomers in the polymer. The integral over D[rn] is a continuousrepresentation of the discrete sum over all possible chain conformations, andit contains factors that ensure that G(R,R0;N) is normalized. The advan-tages of going over to a continuum chain description lie in the fact thatpath integrals can be evaluated using established techniques in quantum andstatistical mechanics [15, 16]. Moreover, the end-vector distribution functiondefined above satisfies a diffusion equation that is formally equivalent to theSchrodinger equation in quantum mechanics. The formal analogy is useful intreating many of the problems concerning a polymer in an external field.

Eq. (10) can be recognized as the Weiner probability distribution. Al-though the chain conformation is considered to be a continuous curve, thecontinuous limit is mathematically subtle since the curve is not differentiablein the classical sense [17, 18]. In other words, the curve does not have awell defined tangent at any point. This happens to be an inherent propertyof a self-similar object. However, Eq. (10) leads to a physically meaningfulprobability distribution that accounts for connectivity of the chain, which iscompletely entropic in origin. The “Hamiltonian” for the flexible polymer isof the form:

βH =3

2b2

∫ N

0dn |rn|2, (11)

where β = 1/kBT . Since the above Hamiltonian only accounts for the chainconnectivity, it represents an ideal chain Hamiltonian. If the polymer interactswith the external field, the end-to-end vector distribution function of Eq. (10)is modified to

G(R,R0;N) =∫ rN=R

r0=R0

D[rn] exp

[−

∫ N

0dn

(3

2b2|rn|2 + V [rn]

)], (12)

where V [rn] may be a solvent field, a gravitational field, or a field due tointramolecular excluded volume interactions between different parts of thesame chain. Non-local excluded volume and three-body interactions accountfor the quality of solvents in which polymers are dissolved. These interactionsare often described by a two-body and three-body pseudo-potentials [8-11]given by

βH =3

2b2

∫ N

0dn

(∂rn

∂n

)2

− v

2

∫ N

0dn

∫ N

0dmδ[rn − rm]


+w

6

∫ N

0dn

∫ N

0dm

∫ N

0dlδ[rn − rm]δ[rm − rl], (13)

where the first term describes the entropic elasticity of the chain; the sec-ond and the third terms represent the two-body repulsive excluded volumeand three-body repulsive interactions respectively; v and w represent thestrengths of the two-body and three-body interactions respectively. The sizeof a flexible polymer is characterized by its mean square radius of gyration,⟨R2

g

⟩. For a linear and flexible polymer,

⟨R2

g

⟩=

⟨R2

⟩/6, where

⟨R2

⟩is the

mean square end-to-end distance. The mean square end-to-end distance canbe evaluated from the following expression:

⟨R2

⟩=

∫∞−∞ dR R2 exp(−βH)∫∞−∞ dR exp(−βH)

. (14)

An ideal chain Hamiltonian is given by Eq.(11). The evaluation of the chainsize involves a simple Gaussian integration, which yields

⟨R2

⟩= Nb2. A

real chain Hamiltonian, described by Eq. (13), involves both entropic andenthalpic contributions. The evaluation of the chain size is, therefore, a diffi-cult task. In the next section, we provide a brief overview of the Flory theorywhich provides an estimate of the chain size in the presence of enthalpicinteractions [1, 2].

5 Temperature Dependence of Polymer Size

The size of a polymer depends on the effective interactions between monomers,which are characterized by excluded volume (second virial coefficient) andthree-body interaction parameters. The excluded volume parameter v esti-mates the effective interaction between the monomers, v = −

∫d3rf , where

the Mayer f -function is f = exp(−βHc)−1, and Hc represents the interactionbetween different monomers. A similar treatment for three-body potentialsyields an expression for w. Typical average values of these parameters arev ∼ b3(1 − θ/T ) and w ∼ b6, where θ is the temperature at which thechain is ideal [6-11]. Away from the θ temperature, the sign of the excludedvolume parameter v determines whether the effective two-body interactionsare repulsive or attractive. For T > θ, the effective interactions betweenmonomers are repulsive. At temperatures higher than the θ temperature,therefore, monomers like the solvent in which they are dissolved and the sol-vent is termed as a good solvent. In the opposite limit of T < θ, the effectivetwo-body interaction is attractive and the solvent is termed as a poor solvent.

The size of a real polymer was first estimated by Flory by identifying theentropic and energetic contributions to the the total free energy F (R, N),

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F (R, N) = Fent(R, N) + Fint(R, N), (15)

where N is the effective number of monomers and R is the end-to-end vector.The entropic contribution to the free energy can be calculated from the

probability distribution for end-to-end vector of an ideal polymer which fol-lows Gaussian statistics,

P (R, N) =(

32πNb2

)3/2

exp(−3R2/2Nb2), (16)

resulting in the following entropic contributions to the free energy:

S(R, N) = kB lnP (R, N) (17)

= S(0, N)− 3kBTR2

2Nb2. (18)

The free energy of a polymer is given by F (R,N) = U(R,N)−TS(R,N). Foran ideal polymer there are no energetic contribution, and the entropic freeenergy is given by Fent(R,N) = 3kBTR2

2Nb2 . For a real polymer, the energeticcontribution to the free energy per unit volume can be calculated by carryingout a virial expansion in powers of the monomer number density cn. Thecoefficient of the c2

n term is proportional to the excluded volume v and thecoefficient of the c3

n is related to the three body interaction coefficient w ≈ b6.

Fint

V=

kBT

2(vc2

n + wc3n + · · ·) ≈ kBT

2(v

N2

R6+ w

N3

R9+ · · ·) (19)

The total free energy, with contributions from entropic and enthalpic terms,is given by

F (R,N)kBT

≈ R2

Nb2+ v

N2

R3+ w

N3

R6(20)

From the above equation, the size of a real polymer in a good and poor canbe evaluated.

In a good solvent, the repulsive nature of the effective monomer-monomerinteractions swell the chain. Since under these conditions, the three bodyinteraction term is not important, the size of the polymer can be evaluatedby just retaining the entropic and the repulsive excluded volume interactionterms,

F (R,N)kBT

≈ R2

Nb2+ v

N2

R3. (21)

By minimizing the above free energy with respect to R, i.e. ∂F (R,N)∂R = 0,

the size of the polymer in a good solvent can be estimated. The equilibriumsize is given by R ≈ N3/5v1/5b3/5. The latter is a universal expression forthe chain size in a good solvent. In contrast to the exponent of 0.6, the moresophisticated renormalization group theories estimate this exponent as 0.588


[8, 9]. The Flory theory, inspite of its simplicity, predicts the right scaling forthe chain size.

In a poor solvent, the effective attractive monomer-monomer interactionslead to a collapsed globular structure. Since the resulting structure is highlycompact, the entropic contributions are not important and the size of thepolymer can be evaluated by just retaining the three body repulsive and thetwo-body attractive excluded volume interaction terms,

F (R,N)kBT

≈ −|v|N2

R3+ w

N3

R6. (22)

By minimizing the above free energy with respect to R, the equilibrium sizein a poor solvent can be estimated as R ≈ (N/|v|)1/3b. Therefore, a globule-to-coil conformational transition can be brought about by changing the signof the excluded volume interaction term, which amounts to changing thesolvent quality or temperature.

To understand the temperature dependence of a real polymer, it is custom-ary in polymer physics to introduce the concept of blobs [6]. Since polymersare large molecules, there exist a separation of length scales within the poly-mer structure. Irrespective of the nature of the interaction energy, the size ofa blob corresponds to the length scale at which the interaction energy is ofthe order of the thermal energy, kBT . On length scales smaller than the blobsize, the chain follows the unperturbed statistics. On length scales larger thanthe blob size, the chain statistics is determined by the interaction energy.

The free energy of a real polymer has entropic and enthalpic contributions.The length scale at which the free energy is of the order of kBT determinesthe size of a thermal blob, ξT . The entropic contribution to the free energy isβFent = R2/Nb2. Since length scales below the blob size are dominated by en-tropy, we have ξT ∼ n1/2

T b where nT is the number of monomers in a thermalblob. To determine nT , we set βFint ∼ 1, which yields nT ∼ b6/v2. On lengthscales larger than ξT , the thermal blobs are self-avoiding in a good solvent andspace-filling in a poor solvent. In terms of the thermal blob, therefore, the sizein a good and poor solvents are given by R ∼ (N/nT )3/5ξT ∼ N3/5v1/5b3/5

and R ∼ (N/nT )1/3ξT ∼ (N/|v|)1/3b. The latter expressions are the same asobtained from the Flory theory.

6 Polymers under tension

The free energy of an ideal polymer only has entropic contribution given byF (R, N) = 3kBTR2/2Nb2. In order to hold the chain at a fixed end-to-endvector R, one needs to apply equal and opposite force at the chain ends,

f =∂F (R, N)

∂R=

3kBT

Nb2R. (23)

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The force-extension relation is linear and follows Hooke’s law for small defor-mation. This is the result of the Gaussian approximation for the end-to-endvector distribution. The above relation suggests that the spring constant isk = 3kBT/Nb2 . The dependence of the spring constant on temperature is areflection of the entropic elasticity.

The linear relation for ideal chains can also be obtained from the conceptof blobs [6, 7]. The length scale at which the free energy due to the appliedforce is of the order of kBT defines the tension blob, ξf ∼ kBT/f . For lengthscales shorter than the tension blob, the chain statistics is unperturbed and isgoverned by entropy, that is, ξf ∼ n1/2

f b, where nf is the number of monomersin the tension blob. For length scales larger than the tension blob, the chainstatistics is determined by the applied force yielding a stretched state R ∼(N/nf )ξf ∼ Nb2f/kBT . The latter suggests that for an ideal chain f ∼(kBT/Nb2)R. This recovers the linear relation between force and extensionin agreement with Eq. (23).

For real chains at high temperatures, the unperturbed statistics is that of aself-avoiding walk. For length scales shorter than the tension blob, therefore,ξf ∼ n3/5

f b. At large length scales, the chain is in a stretched state givenby R ∼ (N/nf )ξf ∼ Nb(fb/kBT )2/3 implying f = kBT/b(R/Nb)3/2. Incontrast to ideal chains the force-extension relation for real chains is non-linear. The scaling arguments discussed in this section are only valid forsmall deformations. For large deformations, the force-extension relation canbe derived from the wormlike chain model. In the next section the chainstatistics of stiff polymers is presented in terms of the wormlike chain model.

7 Stiff Chains

The discussion so far has been limited to a class of polymers that are com-pletely flexible. The flexibility in such molecules arises because of the unhin-dered rotation of bond vectors through a fixed bond angle. In stiff polymers,like double stranded DNA, the flexibility is due to the fluctuations of thecontour length about a straight line. The wormlike chain (WLC) is a contin-uous chain description of stiff polymers. The WLC model is a special case ofthe freely rotating model for small bond angles. The concept of the worm-like chain was first introduced by Kratky and Porod. The wormlike chain isalso referred to as the Kratky-Porod Chain. The mean-square distance of thewormlike chain can be calculated from Eq. (4). Since the WLC is a continuousdescription, the summations in Eq. (4) can be expressed as integrals,

⟨R2

⟩=

∫ L

0ds′

∫ L

0ds exp(−|s− s′|/lp), (24)


where L = nl = Nb, lp = 2l/θ2 and b = 2lp for θ * 1. The resultantexpression is given by

⟨R2

⟩= 2Llp − 2l2p(1− exp(−L/lp)). (25)

The persistence length, lp, is a measure of the distance over which the bondvectors are correlated, and hence detrmines the effective stiffness of the chain.That Eq. (25) is a valid description in both stiff and flexible limits can beseen as follows: when Llp * 1,

⟨R2

⟩= L2 (the rod limit); when Llp $ 1,⟨

R2⟩

= 2Llp = Nb2 (the flexible limit).For a chain with a constant contour length, there exist a unit tangent

vector given by u(s) = ∂r(s)/∂s, where r(s) is the position of a point at arclength s from one end of the chain. For a stiff chain, the orientation of theunit vector u(s) changes slowly with respect to the arc length variable s. Ingeneral, for chains with variable stiffness, the bending energy is proportionalto the change in the orientation of the unit vector with respect to the arclength variable s, ∂u(s)/∂s. Thus the lowest order scalar function of ∂u(s)/∂sthat can contribute to the bending energy must be quadratic in this variable.Since u(s) · ∂u(s)/∂s = 0, the only quadratic form is

H =ε

2

∫ L

0ds

(∂u(s)

∂s

)2

, (26)

where ε is the elastic energy for bending, which is equal to kBT lp. The con-nectivity term in the above Hamiltonian is absent because of the constraint|u(s)| = 1, which follows from simple differential geometry. Since the orien-tation of the unit tangent vector varies slowly along the stiff chain backbone,the unit tangent vectors are correlated over longer distances compared to aflexible chain. In general, the correlations between the unit tangent vectorsat two different points along the chain backbone decay exponentially, and isof the form

〈u(s) · u(s′)〉 = exp(−L/lp) (27)

Saito, Takahashi and Yunoki (STY) have estimated the tangent vectordistribution function in terms of an infinite series, with the above constrainton the unit tangent vector [19]. Their distribution provides a correct estimatefor the second and fourth moments of the end-to-end distance, R. However,the probability distribution for the end-to-end distance cannot be calculatedwithin their formalism, even in terms of an infinite series. In order to sim-plify the analysis, Harris and Hearst relaxed the local constraint of constantunit tangent vectors to a global constraint of constant contour length byintroducing a Lagrange multiplier, and treating it as a free parameter [20].Although their approach reproduces the mean-square end-to-end distance byfixing the free parameter, it fails to yield other second moment quantitiesexcept in the rod and flexible limits. In a different approach, Freed relaxedthe local constraint |u(s)| = 1 to statistical constraint

⟨u(s)2

⟩= 1, produc-

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ing a rather simple model in which the chain Hamiltonian has a connectivityterm along with a bending energy term [21]. The coefficient in front of thebending energy term is kept as a free parameter and is fixed by requiringthat

⟨u(s)2

⟩= 1. For this model, the probability distribution function that

the chain ends are at 0 and R with tangent vectors U′ and U is of the form

G(R, 0;U,U′;N, 0) =∫ u(L)=U

u(0)=U′D[u(s)]δ

[R−

∫ L

0dsu(s)

]

exp

[−

(3

4lp

∫ L

0ds|u(s)|2 +

lp2

∫ L

0ds|u(s)|2

)].(28)

However, the distribution function estimates⟨R2

⟩= Llp, which is just the

flexible chain result. There is no dependence on the bending energy ε. Thisspurious behavior is because of the inhomogeneous nature of the chain definedby Eq. (28). The inhomogeneity becomes more explicit when the mean squarefluctuations,

⟨u(s)2

⟩, of the tangent vectors are calculated at different points

along the chain backbone. It turns out that there is as much as a factor of2 increase in

⟨u(s)2

⟩for s at the ends of the chain as compared to s at the

middle [22]. Therefore, the excess fluctuations at the chain ends need to besuppressed by incorporating an energy penalty for fluctuations at the ends.A Hamiltonian that does that is given by

βH =3

4lp

∫ L

0ds|u(s)|2 +

3lp4

∫ L

0ds|u(s)|2 +

34(|u0|2 + |uN |2). (29)

The probability distribution for the end-to-end distance derived from thisHamiltonian provides correct predictions for various second moments. Inparticular, it reproduces the predictions of the Kratky-Porod model whileremaining analytically tractable.

8 Rouse and Zimm Models

The diffusion of a large molecule in a solvent of small molecules generallyfollows Brownian motion. Since a Brownian particle is much heavier thanthe surrounding fluid particles, there exist a wide separation of time scalesin their respective microscopic dynamics. The Brownian particle changes itsvelocity on a much slower time scale; the fluid particles, on the other hand,change their velocities on a much faster time scale. As a result, the Brownianparticle suffers a large number of collisions with the fluid particles in itspath through the fluid. It is, therefore, possible to identify the velocity ofthe Brownian particle as a slow mode of the system, and the velocities ofthe fluid particles as fast modes of the system. On time scales much longer


Fig. 2 Bead and springRouse chain.

than the frequency and duration of individual collisions, the dynamics of theBrownian particle can be expressed in terms of an equation in which thedetails of the collision process are contained in the statistical properties ofa stochastic variable (the fast modes). Such an equation was suggested byLangevin [23], and is given by

mr(t) = −ζ r(t) + f(t) (30)

The first term in this equation represents the inertial force. The second termis the systematic force with a mean −ζv, where ζ is the friction coefficient.The systematic force is the average frictional force, and is proportional tothe velocity of the particle, but acts opposite to its direction of motion. Thethird term represents the random force f(t) whose time average is zero. Therandom force contains details of the “fast” dynamics. It is the resultant of alarge number of microscopic collisions, and so follows a Gaussian distributionby virtue of the central limit theorem [24-26]. These collisions can generallybe assumed to be uncorrelated on time scales much longer than the collisiontime, and can be regarded as a sequence of independent impulses. Since theBrownian particle attains thermal equilibrium with the fluid particles over asufficiently long time, these considerations dictate the second moment of therandom force given by

〈fn(t)fm(t′)〉 = 2kBT ζδ(t− t′)δnm. (31)

The delta function indicates that the force at t is uncorrelated with the forceat t′. Since the process is also assumed to be Gaussian, all cumulants higherthan the second are zero.

Eq. (31) is the simplest form of the fluctuation dissipation theorem [27]. Itstates the the dissipation due to frictional forces and the fluctuations due torandom forces are related to one another since they have the same microscopicorigin.

14 Arti Dua

For long enough length and time scales, the Langevin equation describesthe Brownian motion of the centre of mass of the polymer. In contrast, thechain-like dynamical properties are characterized by length and time-scalesthat are longer than the dynamics characteristic of the monomers, but thatare shorter than the Brownian dynamics characteristic of the centre of mass.The dynamical behavior of the polymer in this intermediate regime is inde-pendent of the chemistry of the chain. The intermediate time regime, there-fore, describes the dynamical aspects of the universal chain-like properties[28-31]. The simplest model that describes low frequency global features ofthe polymer dynamics through a few phenomenological parameters is referredto as the Rouse model [28]. This model considers the chain to be a Gaussianequivalent chain with N beads and Kuhn length b. In the Rouse model, apolymer is represented as a bead and spring chain as shown in Figure 2. Apartfrom the inertial force, the frictional force and the random, the ith bead alsoexperiences an elastic force from its immediate neighbors. The equation ofmotion for the ith bead is given by

mrn(t) = −ζbrn(t) + fn(t)− 3kBT

b2

N∑

n=1

Anmrm(t), (32)

where ζb is the bead friction coefficient, which is related to the random forcefn, as in the case of the Brownian motion. The last term of the above equationaccounts for the elastic force, with Anm being the Rouse matrix, defined as

Anm = 2δnm − δn,m+1 − δn,m−1 (33)

Since polymers are large molecules, the dynamics of polymers in solutions isdominated by viscous effects. In the over-damped limit, the inertial term inEq. (32) can be ignored. Therefore, the equation of motion in the continuumlimit is given by

ζ rn(t) =3kBT

b2

∂2r(n, t)∂n2

+ fn(t) (34)

In general, force acting on any point along the chain backbone can causemotion of the fluid around it, which in turn can affect the velocity of the othersegments. This interaction, which is mediated by the motion of the solventfluid, is called the hydrodynamic interaction. A more general expression forthe dynamics of polymers is given by

ζ rn(t) = −∫ N

0dn′D[rn − rn′ ]

δH

δr′n+ fn(t), (35)

where H is the flexible chain Hamiltonian given by Eq. (11). The nonlocalhydrodynamic interaction is accounted for by kernel D[rn − rn′ ]. The Rousemodel neglects the hydrodynamic interactions. It assumes that the velocityat any point along the chain backbone is determined only by the force acting


on it. The kernel is, therefore, given by δ[rn − rn′ ]. The model that accountsfor nonlocal hydrodynamic interactions in pre-averaged fashion is referred toas the Zimm model [31].

A standard way of treating the Rouse model is to decompose the motionof the polymer into the normal coordinates such that each mode is capableof independent motion. The decomposition into independent modes is doneby identifying the normal mode variable q = 2πp/N , where p = 0, 1, 2, 3 · · ·are the Rouse mode variables. A small q explores large length scales insidethe chain; the p = 0 mode, therefore, represents the motion of the center ofmass of the chain. Several dynamical quantities of interest can be expressedin terms of these normal modes [11]. However, a simple estimate of the dif-fusion coefficients and the relaxation times can be made from the followingconsiderations: according to the Einstein’s relation, the diffusion coefficientof a Brownian particle is given by D = kBT/ζ; the relaxation time can beestimated by determining the time required for the particle to move a dis-tance of its size, τ ∼ R2/D. In the Rouse model, each bead has the frictioncoefficient of ζ. So the total friction coefficient of the chain is ζR = Nζ.Therefore, the diffusion coefficient of the Rouse chain is DR = kBT/Nζ andthe relaxation time is τR ∼ N2b2ζ/kBT . In the Zimm model, the hydrody-namic interactions couple the dynamics of different parts of the chain. As aresult, the Zimm chain moves like a solid object with the friction coefficientgoverned by the Stoke’s law, ζZ ∼ ηR, where η is the solvent viscosity. Thediffusion coefficient and the relaxation time of the Zimm chain are given byDZ = kBT/ηR and τZ ∼ N3/2b3η/kBT respectively.

In the next section, the estimation of a few dynamical quantities under flowfields is discussed by modifying the Rouse model to account for the effects offlow.

9 Polymers Under Flow

Polymers are known to deform under various kinds of flow, the extent ofdeformation depending strongly on the nature of flow [32-39]. A pure elon-gational flow produces large changes in the size of the polymer coil. A purerotational flow, on the other hand, rotates the chain without inducing de-formation. Most practical flows consist of a mixture of elongational and ro-tational components, so the dynamics of a chain in such flows depends onthe relative magnitudes of these two components. In a simple shear flow,the magnitude of the rotational and elongational components are equal, thestretched state produced by the elongational component is destabilized bythe rotational component. Therefore, the extended state of polymers in shearflow is inherently unstable and undergoes large fluctuations [32]. The varia-tion of the mean polymer size as a function of the the flow rate and decayof the time correlation function for various values of the flow rate are two

16 Arti Dua

quantities that can be studied from statistical mechanical analysis [32-39]. Ifthe hydrodynamic interactions are ignored, the dynamics can be describedby the Rouse model, supplemented with a term that accounts for the forcefield due to flow

ζ rn(t) = k∂2r(n, t)

∂n2+ γA · rn(t) + fn(t), (36)

where k = 3kBT/b2 is the spring constant, γ is the flow rate, and A is thevelocity gradient tensor, which contains the details of the applied flow. Thevelocity gradient tensor for linear-mixed flow is given by

A =

0 1 0α 0 00 0 0

where α = 0 corresponds to simple shear, while α = 1 and α = −1 correspondto pure elongational and rotational flows respectively. Eq. (36) can be solvedfor the chain extension as a function of the flow rate. However, as discussedearlier, the Rouse chain is a bead and spring chain, consisting of Hookeansprings that can be extended indefinitely. Real polymers show Hookean be-havior only for very small extensions. The Hooke’s law is expected to breakdown at large values of the applied flow. Eq. (36) needs to account for theconstraint of inextensibility. For a discrete polymer model, FENE (finitelyextensible nonlinear elasticity) ansatz is often used by writing the springconstant as k = k′/

(1− l2i /l2m

), where k′ = 3kBT/b2, li is the instanta-

neous bond extension, and lm is the maximum extension of the bond. Thus,the chain pays a high penalty for extension beyond its maximum length. Amore practical ansatz preaverages the instantaneous bond extension to yieldk = k′/

(1−

⟨l2i

⟩/l2m

). In the same vein, the following ansatz serves well for

continuous chains [33-35]:

k = k′(

1−⟨R2

⟩0/

⟨R2

⟩m

1− 〈R2〉 / 〈R2〉m

), (37)

where⟨R2

⟩and

⟨R2

⟩0

represent the mean square end-to-end distances inthe presence and absence of flow.

⟨R2

⟩m

is the maximum observed meansquare end-to-end distance under flow.

In terms of the normal modes, rp(t) = 1N

∫ N0 dnrn(t) cos(2pπn/N), the

above equations can be rewritten as

ζp∂rp(t)

∂t= kprp(t) + γrp(t) + fp(t), (38)

where kp = p2π2k/N , ζ0 = Nζ and ζp$=0 = 2Nζ. The above expression canbe solved for the chain extension under steady state to yield


Fig. 3 Mean fractionalextension, x versus theWeissenberg number, Wifor different fixed values ofα. The full lines are thetheoretical curves calcu-lated from Eqs. (40) and(41). The symbols repre-sent data from simulationsof Ref.[39] with the follow-ing meanings: open squares,100%E; plus signs, 51%E;crosses, 50%E, open circles,49.5%E; astrisks, 0%E.Thisfigure has been reproducedfrom Ref. [35]

⟨R2

⟩=

8N

∑

p=1,odd

[1kp

+γ2ζ2(1 + α)2

2kp(k2p − γ2ζ2α)

]. (39)

If τ = ζN2b2/3π2kBT is defined as the longest relaxation time and Wi = γτas the Weissenberg number, the above equation can be rewritten as

z = β

[(1− z)(1− β)

+4

3π2

(1− z

1− β

)3

Wi2(1 + α)2S

], (40)

where z =⟨R2

⟩/

⟨R2

⟩m

, β =⟨R2

⟩/

⟨R2

⟩m

, and S is given by

S = − π2

8αWi2

(1− β

1− z

)2

+π

8(αWi2)5/4

(1− β

1− z

)5/2

(tan(aπ/2)+tanh(aπ/2)),

(41)where a = (αWi2)1/4((1− β)/(1− z))1/2.

If the mean fractional extension of the chain is defined as x =√

z, theresults can be presented in terms of Figs. 3 and 4, which show the variationof the mean fractional extension as a function of the Weissenberg number Wi[35].

Figure 3 compares the theoretical result for the mean fraction extension asa function of Wi with results from the Brownian dynamics simulations of Chuet al. [39] for six different values of α represented in terms of %E = 50(1+α).In a pure extensional flow (100%E), the chain undergoes a sudden increasein its size at very small value of Wi. In a pure rotational flow (0%E), thechain only rotates and does not extend. In the case of simple shear, thepresence of a large number of fluctuations result in 50% full extension. In thelatter case, our theoretical curves reproduce the trend seen in single moleculeexperiments on DNA under shear [37, 38]. Figure (4) represents the variationof mean extension as a function of Wieff =

√αWi. A significant degree

18 Arti Dua

Fig. 4 Mean fractionalextention, x versus theeffective Weissenberg num-ber, Wieff =

√αWi for

α > 0. The full lines arethe theoretical curves cal-culated from Eqs. (40) and(41). The symbols repre-sent data from simulationsof Ref.[39]. This figure hasbeen reproduced from Ref.[35]

of data collapse suggest Wieff is a reasonable scaling variable for α > 0.Our theoretical predictions compare well with simulation and experimentaldata on polymers under flow. The neglect of the hydrodynamic interactionsleads to deviations at low Weissenberg numbers. The finitely extensible Zimmmodel can provide a better comparison at low Weissenberg numbers.

10 Conclusions

In this chapter, a unified description of ideal polymers is presented in terms ofthe Gaussian equivalent chains. The static properties of the Gaussian equiv-alent chains, like the size and its dependence on solvent quality and temper-ature, are discussed in terms of simple scaling arguments and theories basedon statistical mechanics. The non-linearity in the force-extension relationfor real polymers is obtained from the concept of tension blobs. The diffu-sion coefficient and the relaxation time of a polymer are calculated from theRouse model. In the presence of the hydrodynamic interactions, the dynamicquantities of interest are estimated from the Zimm model. The behavior ofpolymers under flow is presented in terms of the finitely extensible Rousemodel, which is shown to provide a reasonable accurate description of poly-mers in extensional, rotational, simple shear and linear-mixed flows for weakflows.

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Equilibrium and Dynamics of Dilute Polymer Solutions