Kinetics of Swelling and Collapse of a Single Polyelectrolyte Chain

Article pubs.acs.org/Macromolecules

Kinetics of Swelling and Collapse of a Single Polyelectrolyte Chain Soumik Mitra† and Arindam Kundagrami*,†,‡ †

Department of Physical Sciences and ‡Centre for Advanced Functional Materials, Indian Institute of Science Education and Research Kolkata, Mohanpur 741246, India ABSTRACT: Kinetics of swelling and collapse of a single, isolated, flexible polyelectrolyte (PE) is described by a theoretical model considering uniform spherical expansion along with charge regularization. The equation of motion resulting from the osmotic and viscous forces is first solved numerically to obtain the temporal profiles for size and charge of the PE for swelling and deswelling in good solvent and collapse in poor solvent. Further, simpler analytical expressions for the equation of motion are obtained in the high- and lowsalt limits. Finally, asymptotic analytical expressions for the time evolution of the size of the PE are derived. Of three major effects, like-charge repulsion of monomers, entropic force, and two-body attraction dominate respectively the processes of swelling, deswelling, and collapse. In general, the profiles show that the chain swells faster and farther for higher temperatures, lower dielectric mismatch, and lower concentrations of monovalent salt and deswells faster to smaller sizes for reverse trends of the same parameters. The kinetics is faster for lower molecular weights. Deswelling, followed by collapse, along with condensation of counterions, occurs in sufficiently poor solvents. For all configurational changes, charge of the PE chain decreases with decreasing size. Our results are in qualitative agreement with those available from simulations and experiments.

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INTRODUCTION The conformational behavior of polymers has been a topic of interest through several past decades. The equilibrium size and configurational properties of isolated polymer molecules in solutions depend sensitively on the solvent quality and vary from expanded conformations in good solvents to random walk conformations (Gaussian) in theta (Θ) solvents and collapsed conformations in poor solvents. Since the first discovery of the coil−globule transition in polymers,1 a considerable number of theoretical2−13 and simulation studies14−25 and experiments26−30 have been performed to investigate conformational properties, mainly collapse dynamics, of single polymer chains in solutions. This has been a major topic of interest owing to its intricate features as well as deep influences to many biological phenomena such as protein folding31−35 or polymer translocations.36−38 Among the early theoretical approaches, notable are from De Gennes3,4 and Grosberg6 which proposed two-stage kinetic models for collapse dynamics of a single, uncharged, homopolymer chain, which responds to changes in temperature or solvent quality by crumpling, blob formation, and finally through anisotropic configurational changes to form compact globules under possible topological constraints. Later, Dawson and co-workers7,8 proposed through the Gaussian selfconsistent method using Langevin equations a three-stage kinetics of collapse: first, a rapid, exponential decay in size of the polymer chain, then a coarsening stage of the Gaussian coil, followed by a slow relaxation of the compact globule attaining the equilibrium state. Allegra and Ganazzoli9 considered the intrinsic viscosity and hydrodynamic interactions within the © XXXX American Chemical Society

Rouse−Zimm diffusion formalism to obtain the time dependence of chain contraction in response to solvent quality. Pitard and Orland10 used the Edwards Hamiltonian in the Langevin dynamics formalism to study the swelling or collapse from a theta (Θ) to respectively a good or bad solvent and arrived at the time dependency of the radius of gyration in the form [Rg2(t) − Rg2(0)] ∼ ±t3/4 for polymer chains. Concurrently, Klushin described11 the self-similar kinetic coarsening of the collapsing chain composed of clusters, a very similar picture based on which a phenomenological model was proposed immediately after by Halperin and Goldbart,12 using the Flory− Huggins mixing entropy and elastic free energy of the chain. Notable is that all of the above analysis are for uncharged polymer chains. Molecular, Brownian, and Langevin dynamics simulations explored two major issues of the kinetics of polymer chain collapse: the time dependence of size (Rg or Rg2 vs t) reveals that collapse is faster, in line with De Gennes’ prediction of two-stage kinetics, with poorer solvent quality14,18−20,23 and lower molecular weight.15,20,22 Simulations in details investigated two- and three-stage collapses involving cluster formation and coarsening15,16,20 as well. Later, hydrodynamic interactions were found to accelerate the collapse process in recent works.21,23,24 In contrast, simulation studies on expansions of polymer chains in good solvents are rare.22 Received: October 18, 2016 Revised: February 8, 2017

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DOI: 10.1021/acs.macromol.6b02267 Macromolecules XXXX, XXX, XXX−XXX

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deswelling of a single, isolated, and flexible polyelectrolyte (PE) chain in solution. Considering uniform spherical expansion or contraction of the chain, we derive the equation of motion of a hypothetical spherical surface, subject to osmotic pressure, encapsulating the chain. The osmotic pressure, in turn, is derived from a free energy as a function of the size and charge of the chain. The system is assumed to be stable with respect to charge variations during configurational changes. The quasistatic nature of the dynamics allows for the charge to selfregulate inside the system and, therefore, makes it a timedependent quantity as well. The model accounts for the interplay between chain conformations and counterion adsorption, a crucial feature of charge-regularized multicomponent systems. The configurational changes occur through a slow and overdamped process with no hydrodynamic interactions, laying the foundation to use an equilibrium free energy to formulate the dynamic equation of motion. By solving the equation of motion, we find the time dependencies of size and charge of the chain as functions of temperature, dielectric properties, and monovalent salt concentrations of the solution, and the monomer density and the molecular weight of the polymer, for swelling and deswelling of the polyelectrolyte in good solvents, and collapse in poor solvents. The free energy is further approximated to provide simple differential equations for high- and low-salt limits. In addition, analytical expressions for the time dependency of the size of the PE chain are also obtained in certain limits and compared with the full numerical solution. In the rest of the paper, we first develop the analytical theory for the simple model and derive the equation of motion of the PE chain. Then we incorporate the free energy and find the limiting cases. The numerical results are discussed in detail thereafter. Finally, analytical expressions in terms of power laws are obtained, with a summary of the key results mentioned in the conclusions.

Light-scattering studies initially helped26 identify the role of temperature and solvent quality in collapse from Θ-temperatures of polystyrenes in cyclohexane. Further light-scattering experiments28,29 identified the roles of the molecular weight and solvent quality in determining the characteristic time of chain collapse and found results closely in line with theoretical predictions and molecular simulations mentioned above. Both poly(methyl methacrylate) (PMMA) in isoamyl acetate28 and PMMA in pure acetonitrile(AcN) and in the mixed solvent of AcN + water(10% of volume)29 found the coil−globule transition to be slower for higher molecular weight. In a later work, fluorescence experiments measured the ratio of excimer to monomer emissions intensities for single, synthetic poly(Nisopropylacrylamide) (PNIPAM) chains, suggesting30 a twostage collapse kinetics through a double-exponential decay (with two relaxation times). Understanding the collapse dynamics of homopolymers has found relevance in the recent interest in the biological problem of protein folding.31,32 As a first approximation, simulations on protein like polymers18,35 and copolymers24 with both hydrophilic and hydrophobic blocks on the backbone of the uncharged chains provide the temporal profiles of Rg2 versus t as a function of hydrophobicity. It is evident that theoretical and simulatiom models, as well as a few experiments, have investigated the collapse dynamics of uncharged polymer chains and protein-like molecules with mixed hydrophobicity in reasonable detail for the past 40 years. However, studies on time-dependent configurational changes of charged polymers, let alone of charge-regularized polymers for which the degree of ionization gets adjusted self-consistently in response to the solvent conditions or proximity among monomers in the system, are scarce to the best of our literature survey. Lee and Thirumalai, extending the formalism of Pitard and Orland,10 explained the collapse of PE and polyampholyte (PA) chains,13 respectively, in terms of counterion-mediated attractions and charge fluctuations. Their model allowed for collapse of PE chains from rod-like shapes to intermediate “pearl-necklace” structures, which merge at the late stages to form the compact globule. Further, a rare Langevin dynamics simulation by the same authors39 found rapid condensation of explicit counterions alongside the collapse of strongly charged flexible polyelectrolytes (also seen in experiments40), in addition to the traditional dependence of collapse dynamics on solvent quality. A multistage collapse process is suggested by this work as well, which predicts rapid counterion condensation first, followed by formation of globules in the poor solvent. Being interested in the kinetic conformational changes of charged polymeric systems, we noticed that the swelling dynamics for charged polymer (polyelectrolyte) gels, in contrast to single polyelectrolyte chains, are studied in detail.41−45 The spatiotemporal profiles of inhomogeneous density, stress, and charge, and the overall size of neutral or charged polymer gels, obtained from solving the diffusion-like constitutive equations originating from Newton’s law, indicate43−45 that the swelling of gels is a damped, diffusive process. Inspired by the above formalism, in this study we have applied very similar concepts to compute the swelling and collapse profiles for size and charge of a single polyelectrolyte chain in a dilute solvent. We propose a kinetic model to calculate the temporal variations of size (in terms of the radius of gyration) and charge (in terms of the degree of ionization) during swelling and

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THEORY The physical process of swelling of a single, isolated polyelectrolyte (PE) chain is conceived as follows. The PE chain is assumed to undergo uniform spherical expansion46 and is released (t = 0) from a squeezed state of radius R1 (Figure 1a). As the chain is under positive osmotic pressure, Π, it swells

Figure 1. Schematic representation of a single polyelectrolyte chain swelling due to osmotic pressure acting on the surface of the hypothetical sphere encapsulating the chain.

until it reaches equilibrium of radius R2, at which Π becomes zero (Figure 1b). It is further assumed that as the chain is isolated (in infinite dilution), it goes through quasi-equilibrium configurations during the entire swelling process. This allows one to apply the equilibrium properties of the uniform spherical expansion model for the kinetics of the swelling process along B


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Macromolecules with the equilibration of charge species at all times during configurational changes of the chain. The deswelling process is just the reverse of the swelling process, except for the fact that here, in the initial state (and for the entire swelling process), Π is negative. The chain deswells from the forcefully expanded state to the equilibrium state of smaller size with zero osmotic pressure. The collapse process is very similar to the deswelling process, except that the degree of deswelling is much higher with the chain collapsing to an uniformly dense globule in which the radius of gyration of the chain, Rg, goes as N1/3, N being the number of monomers. To set up a minimal model following the above description, we consider the chain to be within a spherical cavity of radius R and the osmotic pressure of the solution within the cavity to act on its surface. The pressure will be positive and outward for swelling and negative and inward for deswelling and collapse. We aim to study the motion of the surface of the spherical cavity, i.e., the change in the radius R, which in turn represents the conformational changes of the chain. Our model is inspired by the ideas of the kinetics of swelling of polymer gels where an equation of motion is written for an elemental volume of the gel following Newton’s law.42−45 Solution of the equation of motion in terms of the displacement vector for the cross-link points in the gel provides the inhomogeneous spatiotemporal profiles for density, charge, osmotic pressure of the gels, and the time evolution of its size.44,45 In the same spirit, we consider the swelling of the spherical surface (like a spherical balloon) as mentioned above for the PE chain (Figure 1), driven by the osmotic pressure. For an elemental surface ΔS of the sphere, the equation of motion can be written as σsΔS

d2R dR = −ζ ΔS + ΠΔS dt dt 2

σs

d2R dR = −ζ +Π dt dt 2

ζ

l1̃ =

=−

ζ

(1)

(2)

(7)

2 dl1̃ 1 ⎛ 6 ⎞ ∂F + ⎜ 2⎟ =0 π ⎝ Nl ⎠ ∂l1̃ dt

∂F ∂l1̃

(3)

N ,T

⎛ 6 ⎞ 2 ⎜ ⎟R ⎝ Nl 2 ⎠ g

(8)

where the partial derivative is taken keeping N and T constant. The functional dependency of the free energy F(l1̃ ,N,T) on l1̃ comes through relations between V, Rg, and l1̃ as mentioned above. Equation 8 is the basic constitutive equation of motion in our study. We may note that for a charged polymer (PE) chain the free energy will be a function of the degree of ionization, f, as well. f is defined as the fraction of total monomers which do not form ion pairs (i.e., for which the counterions are uncondensed and hence mobile in the solution). Here, both the monomer and the counterion are taken to be monovalent. Ideally, f depends on the size of the chain, given by l1̃ ,47,49 through the interaction energy of ion-pair formation, electrostatic repulsion of uncharged monomers, and the entropy of the free ions. Therefore, F is understood as F(l1̃ ,f,N,T) and = N ,T

∂F ∂l1̃

+ f ,N ,T

∂F ∂f

⎛ ∂f ⎞ ⎜ ⎟ ⎝ ∂l1̃ ⎠ l ̃ ,N ,T 1

(9)

Now, we make the key assumption that the time scale of diffusion of free counterions is much shorter to that of the monomers. Therefore, the free ions diffuse to equilibrium with an effectively frozen configuration of the PE chain, still away from equilibrium, in the background at each time step. This assumption is strongly supported by the simulation of Lee and Thirumalai.13,39 Therefore, at equilibrium (of free ions), the chemical potential of the free ions in the solvent and the condensed ions on the backbone of the chain must be equal. This condition renders the second term in eq 9 vanish with

∂F ⎛⎜ ∂R ⎞⎟ ∂R ⎝ ∂V ⎠ N , T 1 ∂F 4πR2 ∂R

(6)

where l is the Kuhn length of the chain and a constant of the problem. In terms of the expansion factor l1̃ , eq 6 reads

relating it to the phenomenological free energy that is a function of volume (V), number of monomers (N), and temperature (T) of this system of single PE chain in high dilution. In our model, V = (4/3)πR3 corresponds to the volume enclosed by the sphere, leading to an expression of the osmotic pressure in terms of the radius R as Π =−

dR 1 ∂F =− dt 4πR2 ∂R

It is obvious that the radius R of this hypothetical sphere depends on the number of monomers of the PE chain. In this uniform spherical expansion model, we choose the radius of gyration, Rg, of the PE chain to be equal to the radius, R, of the hypothetical sphere. Further, we convert the above equation in terms of the expansion factor l1̃ for the size of the chain in the uniform spherical expansion model defined as47

The osmotic pressure Π in the canonical ensemble is defined as

⎛ ∂F ⎞ Π = −⎜ ⎟ ⎝ ∂V ⎠ N , T

(5)

Generically, the friction between the polymer and the solvent is high enough to ensure an overdamped conformational kinetics for the swelling or deswelling process, which allows one to ignore the inertial term, leading to

where the left-hand side denotes the inertial term, σs being the uniform surface mass density. The first term on the right-hand side gives the force due to friction which is proportional to the velocity of the moving surface, ζ being the coefficient of friction for a unit surface. This leads to σs

d2R dR 1 ∂F = −ζ − dt dt 2 4πR2 ∂R

∂F ∂f

(4)

=0 l1̃ , N , T

(10)

which allows one to use the differential expression for the fixed charge case

Using this form of the osmotic pressure the equation of motion, eq 2 takes the form C


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Macromolecules ∂F ∂l1̃

= N ,T

∂F ∂l1̃

F2 = (fρ ̃ + cs̃ ) log(fρ ̃ + cs̃ ) + cs̃ log cs̃ − (fρ ̃ + 2cs̃ ) f ,N ,T

(11)

F3 = −

with the variable charge being derived from minimizing the total free energy with respect to f at each time step. A schematic explanation of the above procedure is presented in Figure 2, in

1 3/2 4π lB̃ (fρ ̃ + 2cs̃ )3/2 3

F4 = −(1 − f )δ(lB/l) F5 =

3/2 w w 1 3 ̃ 4⎛ 3 ⎞ + 33 [l1 − 1 − log l1̃ ] + ⎜ ⎟ 2N 3 ⎝ 2π ⎠ N l ̃ 3/2 Nl1̃ 1

+2

6 2 ̃ N1/2 f lB 1/2 Θ0(a) π l̃ 1

(14)

where ⎛ 2 1 ⎞ 1 ⎜ 5/2 − 3/2 ⎟ exp(a) erfc( a ) + ⎝a ⎠ a 3 a π π 2 + 2 − 5/2 − 3/2 (15) a a 2a 2 2 Here a = κ̃ Nl1̃ /6 and κ̃ = 4πlB̃ ( fρ̃ + 2c̃s), with the following dimensionless quantities: lB̃ = e2/4πϵϵ0lkBT, the scaled Bjerrum length which sets the length scale for the electrostatic interaction between the charged species in the solution, ρ̃ = ρl3, the monomer density of the system, κ̃ = κl, the inverse of the Debye screening length, and c̃s = csl3, the number concentration of added salt. The contributions to the total free energy are described as follows: (i) F1, the entropy of mobility of condensed counterions on the polymer backbone, (ii) F2, the translational entropy of the uncondensed (mobile) counterions and all other free charge species, (iii) F3, the part arising from the fluctuations in densities of mobile ions as obtained from the Debye−Hückel theory,53 (iv) F4, the adsorption energy of the condensed counterion−monomer pairs, and (v) F5, the configurational free energy of the flexible PE chain, obtained variationally from an excluded-volume Edwards Hamiltonian,46 with w and w3 being the two-body and three-body interaction parameters, respectively. The above expression of F(l1̃ ,f,N,T) can be inserted in the equation of motion of swelling/deswelling (eq 8) to provide Θ0(a) =

Figure 2. Schematic representation of the free energy profile as a function of size and charge.

which the free energy profile is drawn. The system comes down in free energy with size variation (swelling or deswelling) along the hinge line, on which the free energy is at the minimum with respect to charge variation. We re-emphasize that the above simplification is valid for this special case of charge-regularized swelling/deswelling in which the dynamics of the counterions are orders of magnitude faster than that of the monomers. Numerically, the full derivative with respect to variations of both size (l1̃ ) and charge ( f) of the PE chain (eq 9) will be ∂F ∂l1̃

= N ,T

F(l1̃ + h , f (l1̃ + h), N , T ) − F(l1̃ , f (l1̃ ), N , T ) (l1̃ + h) − l1̃

(12)

where h → 0. However, the approximation mentioned above simplifies this to (eq 11) ∂F ∂l1̃

f ,N ,T

ζ′

F(l1̃ + h , f (l1̃ ), N , T ) − F(l1̃ , f (l1̃ ), N , T ) = (l1̃ + h) − l1̃

π 2

dl1̃ T ∂ + (∑ Fi ) = 0 dt N ∂l1̃ i

(16)

where Fi’s are the dimensionless expressions of the components of the free energy in eq 14, with ζ rescaled to ζ′ = ζl4π/(62kB), a constant of the problem. Equation 16 is solved using the numerical expression, eq 13, for general values of a, which corresponds to the concentration of mobile ions (or, mainly, the salt concentration). The kinetic model provides the timedependent conformational dynamics (swelling or deswelling) of the PE chain. A major assumption in constructing the free energy F5 is that of a uniform expansion or contraction with spherical symmetry. Worthy of note is that the partial derivative of F with respect to l1̃ (keeping f constant) contains only the F5 term as the rest of the free energy contributions are not explicit functions of the variable l1̃ . However, the full free energy, F, is required to be minimized to the get the degree of ionization all along the quasi-static kinetic process and also at equilibrium. A brief comment on the applicability of the above model is in order. The free energy used in the current model is in conformity with uniform spherical expansion and can thus be used for a spherical chain only. The formulation will not be

(13)

Both the derivatives (eqs 12 and 13) give the same result as the variation of the free energy with respect to f at the value it is minimized through f is zero, i.e., F (l1̃ + h, f(l1̃ + h), N, T) = F(l1̃ + h, f(l1̃ ), N, T) (Figure 2). Free Energy. It is to be emphasized that any free energy expression of a PE chain as a function of its size and charge47,50−52 may be chosen to solve eq 8 as long as the chain swells or shrinks spherically and the free energy contains the radius of gyration. To proceed further, we choose a model proposed by Muthukumar47 for the free energy(in units of NkBT) of the system consisting of an isolated flexible PE chain with ionizable monomers, counterions, and the added salt ions (monovalent and of the same species as that of the counterions) in high dilution, for which F(l1̃ ,f,N,T) = NkBTΣiFi where F1 = f log f + (1 − f ) log(1 − f ) D


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Macromolecules immediately applicable for anisotropic swelling, for example, for rod-like structures.20,25 One may need to construct the free energy as a function of size (and possibly shape) of the PE chain, whereas addressing the time-dependent boundary conditions of such anisotropic swelling might remain a challenge. The immediate generalization of this formalism may consider a long, cylindrical PE chain expanding in the radial direction, for which a suitable free energy must be constructed as a function of its radial size. Unlike in PE gels, where one uses the free energy density (which is independent of geometry), here the equation of motion deals with the (geometry-dependent) free energy of the entire system. Other examples include an extended and rod-like chain collapsing to a spherical (compact) globule through various intermediate stages, including formation of structures like “pearl necklace”,20 in which the phenomena like coalescence and Ostwald ripening should be taken into account. Treatments of such anisotropic swelling are beyond the scope of the current work. Limiting Cases. It is apparent from eq 15 that for a generic value of a (= κ̃2Nl1̃ /6) or salt concentration one may not escape a full numerical solution of eq 16. However, the limiting cases for low and high salt concentrations, with low and high values of a, respectively, might be explored to yield easier expressions for Θ0(a) (eq 15), which may lead to analytical solutions to some extent. Low-Salt Limit. In this limit the function Θ0(a) (eq 15) is expanded in Taylor series around small values of the argument a and is given by Θ0(a) =

2 − 15

limit to hold numerically, we need 2/15 > πa /12 , leading to a < 0.8. Lower values of a will give better results from the analytical expression. High Salt Limit. In the limit of high salt (high values of a), the function Θ0(a) eq 15 can be expanded as Θ0(a) =

F5 =

1

ζ′

2 15

⎫ 6 2 ̃ N1/2 ⎪ ⎬=0 f lB 3/2 ⎪ π l̃ ⎭ 1

3 ⎜⎛ 6 ⎟⎞ 2⎝N⎠

1/2

(22) 2

1 π 3/2

f2 1 5/2 (fρ ̃ + 2cs̃ ) l ̃ 1

(23)

1 π 3/2

⎫ f2 1 ⎪ ⎬=0 (fρ ̃ + 2cs̃ ) l ̃ 5/2 ⎪ ⎭ 1

(24)

The above expression implies that for large salt concentrations the electrostatic interactions become short-ranged, and the twobody, nonelectrostatic interaction parameter w gets modified to

w′ = w +

(18)

f2 fρ ̃ + 2cs̃

(25)

Hence, the additional component of two-body (short-ranged) interactions originating from the Coulombic repulsion among monomers is independent of molecular weight, depends mostly on degree of ionization (f) and salt concentration (c̃s), and becomes insignificant when c̃s approaches f. Effectively, very high salt removes electrostatics from the problem. The solution to eq 24 gives the kinetic behavior in the limit of high salt. One notes from eq 21 that for the high-salt limit to hold numerically, we need 1/3a > √π/2a3/2, leading to a > 7.0. Higher values of a will give better results from the analytical expression. Therefore, both for low- and high-salt limits, the numerical derivative can be replaced by analytical expressions in terms of l1̃ , f, and c̃s (for high salt only), which immediately delivers a simpler differential equation of the form dl1̃ /dt = g(l1̃ ).

(19)

⎧ dl1̃ ⎛ 3 ⎞3/2 w 1 T⎪ 3 ⎡ 1⎤ 3 w3 ⎜ ⎟ ⎢ ⎥ 1 2 + ⎨ − − − 5/2 ⎪ ⎝ 2π ⎠ N ⎩ 2N ⎣ N l1̃ 4 dt N l̃ l1̃ ⎦ 1 −

w3 3 Nl1̃

⎧ dl1̃ ⎛ 3 ⎞3/2 w 1 T⎪ 3 ⎡ 1⎤ 3 w3 ⎜ ⎟ ⎢ ⎥ + ⎨ − − − 1 2 4 5/2 ̃1 ⎦ ⎝ ⎠ π N⎪ N N dt 2 2 N l ̃ ⎣ l1̃ l1 ⎩ −

Inserting the above in the equation of motion (eq 16) gives ζ′

+

that leads to the equation of motion in terms of salt concentration

∂F5 ⎛ 3 ⎞3/2 w 1 1⎤ 3 w3 3 ⎡ ⎢1 − ⎥ − 2⎜ ⎟ = − ⎝ 2π ⎠ N l1̃ 4 2N ⎣ N l ̃ 5/2 ∂l1̃ l1̃ ⎦ 1 6 2 ̃ N1/2 f lB 3/2 π l̃

3 ⎜⎛ 6 ⎟⎞ 2⎝N⎠

1/2

−

leading to an analytical expression for the derivative of F5 with respect to l1̃

2 15

3/2 N l1̃

∂F5 ⎛ 3 ⎞3/2 w 1 3 ⎡ 1⎤ 3 w3 ⎢1 − ⎥ − 2⎜ ⎟ = − 5/2 ⎝ 2π ⎠ 2N ⎣ N l1̃ 4 N l̃ ∂l1̃ l1̃ ⎦ 1

3/2 w w 3 ̃ 4⎛ 3 ⎞ + 33 F5 = [l1 − 1 − log l1̃ ] + ⎜ ⎟ 3/2 2N 3 ⎝ 2π ⎠ ̃ Nl1̃ N l1

−

w

Using the relations a = κ̃ Nl1̃ /6 and κ̃ = 4πlB̃ ( fρ̃ + 2c̃s), the analytical expression for the derivative of free energy for high salt is obtained as

π a 4a 1 8a + − π a3/2 + − ... 12 35 24 189

1

3/2

6 2 ̃ N1/2 1 f lB 1/2 π 3a l1̃ 2

2

6 2 ̃ N1/2 f lB 1/2 π l̃

3 ̃ 4⎛ 3 ⎞ [l1 − 1 − log l1̃ ] + ⎜ ⎟ 2N 3 ⎝ 2π ⎠ +2

For very small a (i.e., a → 0) we get Θ0(a) = 2/15 for the lowest order in a, and all a-dependent terms vanish. In this case the chain free energy (eq 14) becomes

4 15

(21)

Truncating this to the lowest order, we get Θ0(a) = 1/3a. Thus, the PE chain free energy becomes

(17)

+

1 π 3 π 1 − 3/2 + 2 − 5/2 + 3 ... 3a 2a a 2a a

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RESULTS AND DISCUSSION The aim of this study is to find the kinetic behavior, i.e., the temporal profiles of size (l1̃ vs t) and degree of ionization (charge) (f vs t), of a single, isolated polyelectrolyte (PE) chain, by solving the constitutive equation (eq 16) through the

(20)

The solution to this equation, which is free from the derivative of the free energy, gives the kinetic behavior of a PE chain in the low-salt limit. One notes from eq 17 that for the low-salt E


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temperature is inherently slower which is taken into account in our solutions. Now we shift to the actual dynamic process which is slow and nearly quasi-static compared to the motion of the counterions. This allows the free ions to equilibrate at each infinitesimal dynamic step, in turn allowing the charge to selfregulate within the system. This is achieved through the minimization of free energy with respect to charge (eq 10), leading to a dynamic regularization consistent with the evolution of the size of the chain. The charge-regularized l1̃ vs t curves (Figure 4a) show trends quite opposite to the fixed-charge ones. We remember47,54,55

numerical scheme of eq 12 or 13, under the condition of charge regularization (eq 10). From many free energy expressions [F(l1̃ , f)] of the PE chain available in the literature,47,50−52 we choose the one proposed by Muthukumar47 (eq 14), in the form of F(l1̃ ,f,lB̃ ,δ,c̃s,ρ̃,N), where lB̃ , δ, c̃s, ρ̃, and N are parameters. Three types of conformational changes are looked into: (i) swelling in good solvent, in which the PE chain typically starts from the nonequilibrium configuration of Gaussian size (l1̃ = 1) to expand to the equilibrium co nfiguration of higher sizes; (ii) deswelling in good solvent, in which the chain starts from an unstable configuration of large size (l1̃ ≫ 1) and shrinks to the equilibrium size, which is still greater than the Gaussian size; and (iii) collapse, in which the chain shrinks below the Gaussian size of l1̃ = 1 in a poor solvent. Swelling. The parameters for the swelling kinetics are chosen such that the equilibrium size of the PE chain is reasonably larger (l1̃ ∼ 10−50) than the Gaussian size of l1̃ = 1. Once l1̃ (t = 0) = 1 is chosen as the starting size, the positive osmotic pressure (eq 16) for sizes lower than the equilibrium size swells the chain spherically toward equilibrium. All swelling processes are observed in a good solvent for which w, the twobody interaction parameter, is taken to be zero (the solvent still behaves well for the presence of charge in the system). Effect of Temperature, lB̃ . lB̃ , the reduced Bjerrum length, is a measure of temperature of the solution with fixed dilectric constant. Equivalently, lB̃ sets the relative strength between thermal energy and electrostatic interactions in the system. First, we work with the simplest case of fixed charge for which f, the degree of ionization, is chosen to be fixed despite change in configuration of the chain during swelling. Consequently, in eq 14, the terms F1 to F4 are rendered irrelevant, and F5 confirms a higher equilibrium size with a higher lB̃ or lower temperature. Notable is that for fixed charge entropic considerations are insignificant, and both the equilibrium and the approach to it are majorly determined by electrostatics. The chain entropy (first term in F5) progressively loses its relevance with higher charge. A higher lB̃ ensures higher electrostatic repulsion among charged monomers, resulting in a higher degree of swelling, which is evident from Figure 3. The parameters chosen are47 N

Figure 4. Swelling: time evolution of size (a) and charge (b) for different values of temperature (T = 900/lB̃ ).

that size (l1̃ ) of a polyelectrolyte chain with regularized charge varies nonmonotonically with temperature (lB̃ ), as a higher lB̃ induces more condensation of counterions, hence less repulsion among monomers, leading to a smaller size. In general, the final size decreases with higher lB̃ (Figure 4a), except for high temperatures for which we observe nonmonotinicity and a reverse trend (curve for lB̃ = 1.0 is higher than that for 0.5). The temporal evolutions still display diffusive behavior. One may propose that by observing the trends for l1̃ vs t curves, it is possible to ascertain whether the PE system has fixed or variable charge. The profiles for the degree of ionization (Figure 4b) do not show nonmonotonicity with lB̃ and show a sharp increase in initial times followed by quick saturation. To understand the features of charge profiles, we recall the analytical expression49 of charge as a function of the Coulomb strength, lB̃ δ. For the extended chain configurations in low or high salt, one may apply the adiabatic approximation, in which the variation in f is determined by F1, F2, and F4 in eq 14 only (as, for extended chains, F5 and its variation are negligible compared to the first three terms), leading to ̃

f= Figure 3. Swelling: time evolution of size, measured by l1̃ (t), for different values of temperature, measured by lB̃ , for fixed charge (degree of ionization): (a) f = 0.1, (b) f = 0.9.

−(cs̃ + e−δlB) +

̃

̃

(cs̃ + e−δlB)2 + 4ρ ̃e−δlB 2ρ ̃

(26)

which indicates no dependency on the size, l1̃ , for extended chains. The expression further suggests no dependency of charge either on shape and size or on molecular weight (N) of the polymer molecule. Thus, the kinetic profiles show a steady value of charge, which decreses with lB̃ , for most part of the swelling for which the chain remains extended. The charge is low at the initial stages when the chain configuration is close to a relatively dense Gaussian size (l1̃ = 1). These basic results on swelling of PE chains from this study show a striking similarity in nature to that of swelling of PE gels.43−45 The qualitative similarity with the experimental results43 for the swelling of polyacrylamide gels in water clearly indicates that both the PE gel and an isolated PE chain show a

= 1000, ρ̃ = 0.0005, c̃s = 0, w = 0.0, w3 = 0.0, and δ = 3.5. The degree of swelling is significantly higher for a higher charge ( f = 0.9 in Figure 3b compared to f = 0.1 in Figure 3a), expectedly, as a higher charge will push the chain to a higher size to minimize the penalty of electrostatic energy due to repulsion. The diffusive features of the l1̃ vs t curve is visible, although the origin is not understood yet. We note that the factor of T in front of the derivative in the kinetic equation (eq 16) rescales the friction factor ζ′ → ζ′/T. Therefore, the dynamics for lower F


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length (higher temperatures), lower dielectric mismatch, and lower concentrations of monovalent salt. The swelling curves look diffusive-like, with the swelling saturating at long times. The kinetics of configurational changes are faster during swelling for shorter chains with lower molecular weight. For all configurational changes, charge of the PE chain decreases with decreasing size, especially when the size is in the vicinity of the Gaussian value. This is in conformity with mutual dependency of size and charge in equilibrium.47,49 Deswelling. The deswelling of the PE chain is determined from the inward motion of the hypothetical spherical surface encapsulating the chain consistent with this uniform spherical expansion model. The initial size is chosen to be higher than the equilibrium size, and hence the osmtic pressure remains negative throughout the deswelling process. The parameters are chosen such that the equilibrium size of the chain is above the Gaussian value and in the range of l1̃ ∼ 1−50. Once l1̃ (t = 0) = 100 is chosen as the starting size, the negative osmotic pressure makes the chain deswell spherically toward equilibrium. All deswelling processes are still observed in a good solvent for which w, the two-body interaction parameter, remains zero. Effect of Temperature, lB̃ . The temporal profiles of the size [l1̃ (t)] for various temperatures (lB̃ ∼ 1/T) are plotted in Figure 7 for two fixed charges, f = 0.1 and 0.9. As explained for

universal diffusive behavior during swelling toward equilibrium configurations. As evident from eq 26, δ affects most processes similar to lB̃ , and hence the profiles for different δ should resemble those of lB̃ , which is indeed confirmed in the Appendix. Effect of Salt, c̃s. Addition of salt decreases the chain size in two ways. Salt screens Coulomb repulsion among monomers and decreases the charge of the chain (average effective charge of a monomer, i.e., f) inducing more condensation of counterions. In Figure 5, we show the temporal profiles for

Figure 5. Swelling: time evolution of size (a) and charge (b) for different values of salt concentration, c̃s.

swelling at modest concentrations of monovalent salt (c̃s) for which the sizes are reasonably close to the Gaussian configuration. A small amount of salt drastically reduces the charge, hence the size, which is reflected both in the small value and low variation of charge with size. The equilibrium size and charge expectedly decrease with c̃s, the profiles being diffusive and qualitatively similar to those for lB̃ and δ. Effect of Chain Length, N. The chain length (equivalently, the number of monomers of the chain, N, which is proportional to the molecular weight) does not affect the charge in the expanded state significantly (eq 26). However, it trivially increases the factor l1̃ (eq 7). The profiles plotted in Figure 6

Figure 7. Deswelling: time evolution of size, measured by l1̃ (t), for different values of temperature, measured by lB̃ , for fixed charge (degree of ionization): (a) f = 0.1, (b) f = 0.9.

swelling, the majority of the kinetics is controlled by electrostatic interactions, and entropy plays negligible role when charge is fixed. As expected, the equilibrium sizes are higher for lower temperatures for fixed charge, which induces sharper deswelling at higher temperatures. No nonmonotonicity in equilibrium size as a function of temperature is observed. As the equilibrium sizes will be larger for higher values of charge, the deswelling is sharper for lower charge if the starting size is the same. The parameters chosen are the same as in swelling: N = 1000, ρ̃ = 0.0005, c̃s = 0, w = 0.0, w3 = 0.0, and δ = 3.5. For the charge regularized case of deswelling, similar to swelling, the charge gets self-regulated in a quasi-static manner commensurate to the dynamics of the chain size. The chargeregularized l1̃ vs t curves (Figure 8) show, again similar to swelling, trends quite opposite to the fixed-charge ones, as the size (l1̃ ) varies nonmonotonically with temperature. Features of deswelling curves too may indicate whether the PE system has fixed or variable charge. As predicted by eq 26, no dependency on the size for extended chains is observed for deswelling, too. For both cases of fixed and variable charge, a higher equilibrium size is reached faster through deswelling. Effect of Salt, c̃s. As explained before, additional salt decreases the equilibrium size by reducing the average charge ( f) of monomers through condensation and screening the

Figure 6. Swelling: time evolution of size (a) and charge (b) for different values of chain length (molecular weight), N.

confirm the above. In this case, too, variation of f with time is short-lived compared to l1̃ and reaches saturation as soon as the chain is reasonably extended. Most notable is that the shorter chains (or polyelectrolytes of lower molecular weight) swell faster to equilibrium, which is justified if one considers longer relaxation times related to the time scale of configurational changes for longer chains. One may note that the factor of 1/N in front of the derivative in the kinetic equation (eq 16) rescales the friction factor ζ′ → Nζ′. Therefore, the dynamics for a higher chain length is inherently slower which is taken into account in our solutions. The effect of monomer density, ρ̃, on the swelling kinetics is discussed in the Appendix. The key results obtained from the numerical solution of the equation of motion (eq 16) of the PE chain swelling in a good solvent are as follows. It is observed that the chain swells in a good solvent faster and farther for lower values of Bjerrum G


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The key results obtained from the numerical solution of the equation of motion (eq 16) of the PE chain deswelling in a good solvent are as follows. The chain deswells faster to smaller sizes for higher values of Bjerrum length (lower temperatures), higher dielectric mismatch, and higher concentrations of monovalent salt. Similar to swelling, the kinetics of configurational changes are faster for shorter chains with lower molecular weight. Keeping further similarity to swelling, for all configurational changes, charge of the PE chain decreases with decreasing size, especially when the size is in the vicinity of the Gaussian value. The availability of studies on kinetics of configurational changes for charged polymer systems in the literature, to the best of our knowledge, is extremely rare. A Langevin dynamics simulation by Lee and Thirumalai39 observed rapid condensation of counterions for collapse of strongly charged flexible polyelectrolytes, which falls in line with our findings. The temporal profiles for size (l1̃ ) are qualitatively supported by other simulation results15,20,22,23 for the deswelling of uncharged single polymer chains. Further, similar deswelling curves (Rg2 vs t) are observed for simulations on other uncharged polymeric systems such as copolymers with hydrophobic−hydrophilic blocks (protein-like)18,24,35 as functions of hydrophobic number fractions of the chains. Flourescence experiments on coil−globule transition in PNIPAM chain (N = 3100)30 through the ratio of excimer to monomer emissions intensities have identified a doubleexponential decay during deswelling with two relaxation times. The rapid initial decay of the size gradually ending with the chain slowly reaching a steady equilibrium state, i.e., a compact globule, in this experiment, is reminiscent of the deswelling curves obtained in our numerical analysis. Further, the swelling curves obtained in simulations22 qualitatively resemble the results of our study. Therefore, the essence of the results obtained in this study under approximation of spherical expansion showing the deswelling dynamics for different chain lengths of the polyelectrolyte is in good qualitative agreement with the simulation18−20,22−24,35 and experimental28−30 results available. Also, Guy Ziv et al.35 showed using lattice Monte Carlo simulations that collapse of proteins, considered as heteropolymers made of blocks of hydrophobic and hydrophilic regions follow a similar trend (although Rg vs t data is available) as obtained from our model. Though our system of study is a homopolymer chain whereas proteins are complicated heteropolymers, still one could aspire to modify a homopolymer free energy and similarly the dynamics to extrapolate it to study protein collapse. Collapse. In this section, we study the conformational changes of the PE chain in a poor solvent. Both the swelling and deswelling processes considered in this work occurred in a good solvent in which the size of the chain never goes lower than Gaussian (l1̃ ≥ 1). In a poor solvent, the two-body interaction parameter w is negative (we need to add a positive term w3 (eq 14) to stabilize the collapse47,49) for which the size may decrease to values lower than Gaussian (l1̃ < 1). In addition to the same set of parameters used for swelling and deswelling, w = −1.6 and w3 = 0.3 are chosen. In Figures 11a and 11b, the equilibrium size and charge are plotted against temperature (lB̃ ), respectively. We note that for lB̃ ∼ 2.85 there is a first-order transition from the expanded to collapsed state accompanied by an abrupt reduction in charge. The mechanism for this transition is well understood49,50 in terms of the balance

Figure 8. Deswelling: time evolution of size (a) and charge (b) for different values of temperature (T = 900/lB̃ ).

electrostatic repulsion among the remaining charged monomers. However, f does not vary much and the screening effect does not bring in much changes compared to unscreened Coulomb repulsion at extended chain configurations. These effects for a monovalent salt are visible for the deswelling profiles in the presence of salt (Figure 9), where the l1̃ versus t

Figure 9. Deswelling: time evolution of size (a) and charge (b) for different values of salt concentrations, c̃s.

curves coinside for most part of deswelling, for which the chains remain extended, for all salts. The variation is visible, for both charge and size, when the chain reaches close to the Gaussian size. The results indicate that addition of salt with chemically similar counterions to that of the chain−solvent system induces a stronger and more rapid collapse of the PE chain in solution. Effect of Chain Length, N. Figure 10 shows that smaller chains have faster dynamics leading to a lower equilibrium size.

Figure 10. Deswelling: time evolution of size (a) and charge (b) for different values of chain length (molecular weight), N.

This is an expected result, supported by simulations15,20,22 for uncharged polymers, especially with the factor of 1/N in eq 16 which rescales the friction factor ζ′ → Nζ′. Further, relaxation times are longer for longer chains related to the time scale of their configurational changes. We have previously noticed that charge is not much affected by N for extended sizes. The profiles plotted in Figure 10 confirm the above. Experiments with PMMA (uncharged) in isoamyl acetate28 and in acetonitrile (AcN) and AcN + water29 identify slower kinetics with higher molecular weight as well. The effect of the dielectric mismatch parameter δ and monomer density ρ̃ on the deswelling kinetics is discussed in the Appendix. H


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along with a drop in charge. Although the swelling curve for l1̃ > 0.67 takes the usual shape of concave downward (not shown in the figure), the collapse curve for l1̃ < 0.67 has this distinct convex upward shape which is very different from the deswelling curves we found earlier in good solvents. Similar features had been noticed before in theoretical studies9 for the collapse of uncharged chains below the Gaussian size in poor solvents. We further note that the scheme of kinetics proposed in this study may be used as a tool to explore and identify the local minima and maxima in the free energy landscape of the system. With decreasing temperatures (increasing lB̃ ), the local minimum for the larger size (Figure 12c) will gradually disappear, and only one minimum, the global minimum, will survive (see Figure 1 of ref 50). A similar effect can be induced using a substantially large two-body attraction parameter (−w) to ensure a collapse of the chain to a sub-Gaussian size. In those situations, one may also aim to study the collapse transition to a sub-Gaussian size (l1̃ < 1), starting from an expanded chain of initial size substantially larger than the Gaussian value (l1̃ ≫ 1). The kinetics of collapse from an expanded chain configuration shows two distinct regimes (Figure 13a) with

Figure 11. Collapse: equilibrium values of size (a) and charge (b) as functions of temperature (lB̃ ). Time evolution of size (c) and charge (d) around the maximum of the free energy [l1̃ = 0.67 and f = 0.053] which falls between two minima.

between electrostatic repulsion of like-charged monomers and hydrophobicity due to the poorness of the solvent. One may identify two minima49,50 associated with such firstorder transitions, the one with smaller equilibrium size being the global minimum representing the collapse transition. In Figures 12a and 12b we show such a minimum in terms of l1̃

Figure 13. Kinetics of collapse depicting (a) size and (b) charge, starting from an extended chain configuration with w = −85.0 and w3 = 3.85.

competing trends. In the initial part the process follows the trend similar to that of deswelling, showing a distinctly convexdownward curve. However, as the chain size approaches the Gaussian value, the monomers become progressively closer, and the short-ranged two-body attraction comes into play. Eventually, the two-body attraction wins over the Gaussian entropy to enforce a rapid, avalanche-like decrease in size. The convex-upward trend of sub-Gaussian collapse is clearly visible in this regime before the collapse gets arrested by the compensating repulsive three-body interaction. One may note that the effective, short-ranged attraction among the monomers can be induced through hydrophobicity (solvent poorness) as well as other effects such as electrostatic ion-bridging due to divalent salt ions48 or dipolar attraction of bound ion pairs.47 The divalent cation will competetively displace the monovalent counterion and hence will form an effective, positive monovalent charge at the location of the monomer. This effective positive charge will attract another negatively charged monomer from anywhere on the contour of the PE chain, forming the ion bridge. This interaction can be modeled through a short-ranged two-body attraction 49 potential quite similar to that described by the negative w mentioned in this subsection. Therefore, one may explore the feasibility of applying the current model to divalent salts as well. Limiting Cases. Previously, we observed that one may find analytical expressions for ∂F/∂l1̃ for high- and low-salt limits. This reduces the numerical analysis to that of solving a

Figure 12. Collapse: (a, b) identify the global free energy minimum; (c, d) identify the local free energy maximum.

and f, whereas the minimum is located at l1̃ (global minimum) = 0.091, f (minimum) = 0.0226 for lB̃ = 3.0. The upper minimum is similarly found to be located (not shown in any figure) at l1̃ (local minimum) = 22.9, f (minimum) = 0.20. Between these two minima, expectedly, a local maximum in l1̃ (but still a minimum in f, as the free energy is minimized with f at all times no matter what) lies at l1̃ (local maximum) = 0.67, f (minimum) = 0.053. This local maximum is evident from Figure 12c and (d), as well as from the kinetics described in what follows. We show in Figure 11c,d that if one starts from a size above the local maximum (l1̃ > 0.67), the chain swells toward the local minimum with higher size, accompanied by a rise in charge as well. However, if one starts from a size below the local maximum (l1̃ < 0.67), the chain slides down the free energy landscape from the other side of the maximum and collapses toward the global minimum with the smaller size, I


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Macromolecules differential equation of the form dl1̃ /dt = g(l1̃ ), instead of calculating the derivative ∂F/∂l1̃ numerically. We plotted the size and charge obtained both from numerical and analytical results in Figure 14. In Figure 14a,b we find reasonable

̃ is the initial size of the chain, which, in this work, is where l10 ̃ = 1. taken to be equal to the Gaussian value, l10 In the high-salt limit from eq 24 one obtains ζ′

dl1̃ T⎛ 3 + ⎜⎜ − dt N⎝ 2

f2 6 1 1 ⎞ ⎟=0 3/2 ̃ ⎟⎠ N π (fρ ̃ + 2cs̃ ) l5/2

(29)

leading to the power law dependency in time as l1̃

7/2

3/2 Tf 2 7 ⎛⎜ 6 ⎞⎟ 7/2 − l10̃ = t 8ζ′ ⎝ Nπ ⎠ fρ ̃ + 2cs̃

(30)

The full-numerical and analytical solutions for lB̃ = 3.0, δ = 3.5, N = 1000, c̃s = 0.0, ρ̃ = 0.0005 (parameters same as before), and f = 0.209 obtained from the analytical expression for charge (eq 26) using the above-mentioned values of the parameters are shown in Figure 15a. Further, it is possible to use the power law

Figure 14. Limiting cases: comparison of numerical and analytical results for high salt: temporal profiles for size (a) and charge (b), and for low salt: temporal profiles for size (c) and charge (d).

matching between respectively the size and charge, obtained from the full numerical solution (eq 16, dotted lines) and the analytical expression (eq 24, solid lines) for high salt (high values of a). The parameters chosen for high a are same as in swelling or deswelling, except that the salt is chosen as c̃s = 0.1. The range of a explored for high salt is 28.9−126.5. Similarly, in Figure 14c,d we match the size and charge for low salt (analytical expression from eq 20). The parameters chosen for low a are same as in swelling or deswelling, except that the dielectric mismatch is chosen as δ = 5.0. The range of a explored for low salt is 0.43−0.054. Analytical Expressions. The numerical solution of the kinetic equation (eq 16) with the full free energy, or even for the limiting cases of high and low salts as described above, includes contribution of all effects (entropic, two-body, Coulombic) but fails to provide deeper insight. One may further explore effects of dominant contributions applicable to respective kinetic processes, and try to obtain analytical expressions, at least in the asymptotic limits. Swelling. In the case of swelling above the Gaussian size, the two-body interaction parameter is taken to be zero. Further, the conformational entropy is always maximum for the Gaussian chain. Hence, the major or dominant driving force for swelling must come from the electrostatic repulsion of monomers, allowing us to retain only the fourth term in F5 of the free energy (eqs 14 and 15, with asymptotic forms for Θ0 as derived in eqs 20 and 24, for low- and high-salt limits, respectively) as the leading one. The behavior of this term is different in the two asymptotes, i.e., the low- and high-salt cases, respectively. In the low-salt limit from eq 20 one obtains ζ′

dl1̃ T⎛ 2 + ⎜⎜ − dt N ⎝ 15

6 2 ̃ N1/2 ⎞ ⎟=0 f lB ̃ ⎟⎠ π l3/2

Figure 15. Comparison of full numerical solution and analytical solution in the asymptotic limits: (a) swelling in low salt, (b) deswelling in no salt, and (c) collapse in no salt.

expressions with the coefficients as arbitrary parameters to fit the full numerical solution and hence extract useful information about the system from the fit parameters. For example, in the case of swelling in low-salt conditions, one can easily take the above power law (eq 28) with α=

5/2

5 T 2 5/2 − l10̃ = 2 Nζ′ 15

6 2 ̃ 1/2 f lBN t π

6 2 ̃ 1/2 f lBN π

(31)

and fit this to the full numerical solution. The parametric fit is shown in Figure 15a, which gives the value for the fit parameter α = 0.848. Knowing the temperature and chain length (molecular weight), one can determine the degree of ionization of the system from this fit parameter. The swelling curves obtained from the full numerical solution, analytical expression, and parametric fit in the low-salt limit show (Figure 15a) similar trends and hence confirm the like-charge repulsion indeed to be the dominant effect for swelling. Deswelling. In the case of deswelling, the two-body interaction parameter is still taken to be zero. As one starts from a size far above the Gaussian value, the effect due to likecharge repulsion of monomers will be progressively lesser with the degree of chain expansion, but its significance will still depend on the degree of ionization. For sufficiently low charge, the stable, deswelled state will be Gaussian, and the entire deswelling will be dominated by the entropic force, which

(27)

leading to the power law dependency in time as l1̃

5 T 2 2 Nζ′ 15

(28) J


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Macromolecules becomes zero for the Gaussian size. This allows us to retain only the first term in F5 of the free energy (eqs 14 and 15) as the leading one. The higher is the charge content, the more is the deviation expected from the full-numerical result. Applying the above in eq 16, one gets ζ′

dl1̃ 3T ⎛ 1⎞ = ⎜1 − ⎟ 2 dt l1̃ ⎠ 2N ⎝

l1̃

γ= (32)

(33)

⎛ 3T ⎞ (l1̃ − l1̃ f ) exp(l1̃ ) = exp(l10̃ )(l10̃ − l1̃ f ) exp⎜ − 2 t ⎟ ⎝ 2N ζ′ ⎠ (34)

which can further be plotted with 3T 2N 2ζ′

(35)

as the fit parameter. Equation 34 now applies to deswelling for any salt conditions and any amount of charge of the PE chain. The full-numerical solution and the solution for the transcentental equations with lB̃ = 3.0, δ = 3.5, N = 1000, c̃s = 0.0, and ρ̃ = 0.0005 (parameters same as before) are shown in Figure 15b. β = 0.000 45, which sets the deswelling rate of the PE chain and retains the information on the molecular weight of the polymer. The comparative deswelling curves show (Figure 15b) similar trends, confirming the configurational entropy to be the major driving force as predicted. Collapse. In the collapse process, the charge is minimal in a PE chain (and hence the effect of salt is minimal as well), the variation of entropy is small for sub-Gaussian sizes, and the major driving force comes from the attractive two-body interaction. Thus, retaining only the corresponding term in the free energy (eqs 14 and 15), the equation of motion for any salt conditions reduces to ζ′

3/2 dl1̃ w 1 2T ⎛⎜ 3 ⎞⎟ = N ⎝ 2π ⎠ dt N l ̃ 5/2 1

7/2

3/2 7 2T ⎛⎜ 3 ⎟⎞ wt 7/2 − l10̃ = ⎝ ⎠ 2 Nζ′ 2π N

= γt

(38)

3/2 7 2T ⎛⎜ 3 ⎞⎟ w 2 Nζ′ ⎝ 2π ⎠ N

(39)

■

CONCLUSION In this study we propose a kinetic model to calculate the time variations of size (in terms of the radius of gyration) and charge (in terms of the degree of ionization of monomers) during swelling and deswelling of a single, isolated, and flexible polyelectrolyte (PE) chain in solution within the approximation of uniform spherical expansion. We start from applying Newton’s law to an elementary, hypothetical surface, subject to osmotic pressure and viscous damping, encapsulating the PE chain, and derive the equation of motion (eq 8) in terms of size and free energy of the chain. We choose a free energy47 (eq 14) as a function of size and charge available in the literature and solve the equation motion with the condition of charge regularization; i.e., the free energy is minimized through charge variation at each time instant (eq 10). The full, numerical solution provides temporal profiles for size and charge of the PE chain for its configurational changes during three processes observed: (a) swelling in good solvent, (b) deswelling in good solvent, and (c) collapse in poor solvent. We further derive asymptotic analytical expressions in terms of power-law dependencies of size on time, identifying the key physical effects for all three processes. The general features observed for these kinetic processes are as follows: 1. Of three major driving forces leading to configurational changes of the PE chain, the like-charge repulsion, the entropic force, and the two-body attraction dominate respectively the swelling, deswelling, and collapse processes. 2. The chain swells faster and farther for higher temperatures, lower dielectric mismatch, and lower concentrations of monovalent salt and deswells faster to smaller sizes for lower

(36)

leading to the power law behavior l1̃

7/2

fitting which to the full numerical solution yields γ = −0.0162. Further, γ can be employed to estimate w and, consequently, the chemical mismatch parameter χ, defined as41 w = 1 − 2χ. Therefore, one can obtain very reliable analytical solutions, as simple power laws or algebraic transcendental equations, to the equation of motion for all three processes, viz. swelling, deswelling, and collapse. Such solutions providing time dependencies of the chain size help us estimate the charge content, molecular weight, and hydrophobicity of a PE chain as quantitites hidden in coefficients which can be calculated as fittable parameters. Experimental verification of such power laws may be an interesting step ahead in this subject. One may note the difference in the proposed mechanism of collapse between our work and that of Lee and Thirumalai.13 In our model, the nonelectrostatic hydrophobicity remains the principal driving force (seen in some experiments40) of attraction between the monomers which leads to collapse. The latter model predicts that the origin of such attractions lies in ion-mediation in PE or charge fluctuations in PA chains. Further, the time scales and power laws pertinent to formation of intermediate structures such as the “pearl-necklace” phases, are beyond the scope of our model of uniform spherical configurational changes.

̃ is the initial, expanded value of l1̃ . One notes that at where l10 long times or when the size is equal to the equlilibrium value at the end of deswelling, l1̃ = l1f̃ , which allows one to replace the above by

β=

− l10̃

where

leading to ⎛ 3T ⎞ (l1̃ − 1) exp(l1̃ ) = exp(l10̃ )(l10̃ − 1) exp⎜ − 2 t ⎟ ⎝ 2N ζ′ ⎠

7/2

(37)

̃ = 0.67 is the initial sub-Gaussian size of the PE chain where l10 as mentioned in the full numerical example. The full-numerical solution and the solutions for the aymptotic equations with lB̃ = 3.0, δ = 3.5, N = 1000, c̃s = 0.0, ρ̃ = 0.0005, w = −1.6, and w3 = 0.6 (parameters same as before) are shown in Figure 15c. This confirms the unusual temporal profile for collapse as the effect coming purely from the negative two-body interaction. The collapse is arrested, of course, by w3, which is insignificant other than that and, hence, ignored to facilitate the analytical calculation. From the asymptotic solution one can again take up a general parametric form K


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Macromolecules temperatures, higher dielectric mismatch, and higher concentrations of monovalent salt. 3. The kinetics of configurational changes are faster, for both swelling and deswelling for shorter chains. 4. For collapse in poor solvent, up to a certain degree of poorness, two free energy minima are formed, and the pathway to swelling or collapse is determined by the initial size which can be on either side of the local free energy maximum. The collapse in poor solvent occurs with a concomitant condensation of counterions. Beyond a certain poorness, only one free energy minimum is formed for the collapsed size, and one may get a deswelling, starting from a size far above Gaussian, followed by collapse to sub-Gaussian sizes. 5. For all configurational changes, charge of the PE chain decreases with decreasing size, especially when the size is in the vicinity of the Gaussian value or below it. In addition to the numerical analysis, the free energy is expanded in the high- and low-salt limit to obtain analytical expressions which simplify the numerical analysis and yield results close to the full numerical solution. Especially, the highsalt limit suggests the long-ranged electrostatic interaction to be replaced by a short-ranged interaction that modifies the twobody, nonelectrostatic, chemical mismatch parameter and vanishes for very high salt. It is encouraging to note that a simple mean-field model that treats the conformational dynamics in a bulk fashion is able to provide results qualitatively matching with those obtained from experiments and extensive simulations probing the microscopic details of the system. One may aspire to extend this model to polymers with coexisting hydrophobic and hydrophilic blocks such as in proteins.18,24,35 More importantly, this model shows

Figure 18. Deswelling: time evolution of size (a) and charge (b) for different values of dielectric mismatch parameter, δ.

quality may pose the next level of challenges. In comparison to the previous works, this model may aim to further explore the exponent of time dependence of swelling and collapse,10 effective relaxation times and possible stages of collapse,30 dependency of the kinetics on viscosity,8 chain stiffness,19 and hydrophobicity.20

Figure 19. Deswelling: time evolution of size (a) and charge (b) for different values of monomer density, ρ̃.

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APPENDIX In the Appendix, we present additional results on swelling and deswelling profiles as functions of the dielectric mismatch parameter, δ, and the monomer density, ρ̃. Swelling

Effect of Dielectric Mismatch, δ. The time evolution profiles for l1̃ and f with the dielectric mismatch parameter δ are very similar (Figure 16) to those for lB̃ , except that the nonmonotonicity for high tempeatures (low values of lB̃ ) is absent for δ. δ regulates only f (see F4 in eq 14 and eq 26) but not l1̃ as is evident from its absence in F5. Charge of the chain monotonically decreases with δ, resulting in its smaller size due to decreased electrostatic repulsion of monomers. lB̃ = 3.0, corresponding to room temperature T = 300 K is chosen along with the same values of other parameters mentioned above. Effect of Monomer Density, ρ̃. The leading term in monomer density, ρ̃, goes inversely proportional to the charge, f (eq 26). Therefore, a higher ρ̃ reduces f, which in turn reduces the equilibrium size. The kinetic profiles (Figure 17) are similar to those for the Coulomb strength (lB̃ and δ), as for smaller sizes close to Gaussian, a dense system will attract counterions to the chain backbone.

Figure 16. Swelling: time evolution of size (a) and charge (b) for different values of the dielectric mismatch paremeter, δ.

Figure 17. Swelling: time evolution of size (a) and charge (b) for different values of monomer density, ρ.

promise to address the inhomogeneous charge distribution on heteropolymers which include proteins to explore their collapse kinetics. In addition to equilibrium phase behaviors, this meanfield model serves to be a powerful tool in qailitatively describing to a good extent a quasi-static dynamic process, which may encourage future research on applying a suitable heteropolymer free energy to this model. Incorporation of hydrodynamic interactions and microscopic control on solvent

Deswelling

Effect of Dielectric Mismatch, δ. The time evolution profiles for l1̃ and f with the dielectric mismatch parameter δ are very similar (Figure 18) with same parameter values of swelling) to those for lB̃ . As mentioned for swelling, δ regulates only f, in the same footing as lB̃ for most parameter range (eq 26). Therefore, the nonmonotonicity for high tempeatures (low values of lB̃ ) is absent for δ. L


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Macromolecules Effect of Monomer Density, ρ̃. As mentioned before, monomer density, ρ̃, goes inversely proportional to the charge, f (eq 26), in the first approximation. A higher ρ̃ reduces f, in turn reducing the equilibrium size. This leads to the counterintutive result that a denser system will have a sharper deswelling to a smaller size (Figure 19). Similar to swelling, charge variation with ρ̃, and even with size for fixed ρ̃ for reasonably extended chains, is relatively more pronounced for this set of parameters.

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AUTHOR INFORMATION

Corresponding Author

*E-mail [email protected]; phone +91 98302 22545 (A.K.). ORCID

Arindam Kundagrami: 0000-0002-1469-7123 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The funding of this work is entirely supported by Ministry of Human Resource and Development (MHRD), India, through IISER Kolkata. The authors thank Swati Sen for stimulating discussions and help in preparing figures and the reviewers for their encouraging comments and critical analysis which helped obtain key analytical results.

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Kinetics of Swelling and Collapse of a Single Polyelectrolyte Chain

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