Multibody Interactions, Phase Behavior, and Clustering in

Subscriber access provided by NEW YORK MED COLL

Article

Multibody Interactions, Phase Behavior and Clustering in Nanoparticle-Polyelectrolyte Mixtures Gunja Pandav, Victor A Pryamitsyn, Jeffrey R Errington, and Venkat Ganesan J. Phys. Chem. B, Just Accepted Manuscript • DOI: 10.1021/acs.jpcb.5b07905 • Publication Date (Web): 16 Oct 2015 Downloaded from http://pubs.acs.org on October 22, 2015

Just Accepted “Just Accepted” manuscripts have been peer-reviewed and accepted for publication. They are posted online prior to technical editing, formatting for publication and author proofing. The American Chemical Society provides “Just Accepted” as a free service to the research community to expedite the dissemination of scientific material as soon as possible after acceptance. “Just Accepted” manuscripts appear in full in PDF format accompanied by an HTML abstract. “Just Accepted” manuscripts have been fully peer reviewed, but should not be considered the official version of record. They are accessible to all readers and citable by the Digital Object Identifier (DOI®). “Just Accepted” is an optional service offered to authors. Therefore, the “Just Accepted” Web site may not include all articles that will be published in the journal. After a manuscript is technically edited and formatted, it will be removed from the “Just Accepted” Web site and published as an ASAP article. Note that technical editing may introduce minor changes to the manuscript text and/or graphics which could affect content, and all legal disclaimers and ethical guidelines that apply to the journal pertain. ACS cannot be held responsible for errors or consequences arising from the use of information contained in these “Just Accepted” manuscripts.

The Journal of Physical Chemistry B is published by the American Chemical Society. 1155 Sixteenth Street N.W., Washington, DC 20036 Published by American Chemical Society. Copyright © American Chemical Society. However, no copyright claim is made to original U.S. Government works, or works produced by employees of any Commonwealth realm Crown government in the course of their duties.

Page 1 of 45

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

The Journal of Physical Chemistry

Multibody Interactions, Phase Behavior and Clustering in Nanoparticle-Polyelectrolyte Mixtures Gunja Pandav,† Victor Pryamitsyn,† Jeffrey Errington,‡ and Venkat Ganesan∗,† The University of Texas at Austin, McKetta Department of Chemical Engineering, Austin, TX 78712., and Department of Chemical and Biological Engineering, University at Buffalo, The State University of New York, Buffalo, New York 14260-4200. E-mail: [email protected]

∗ To

whom correspondence should be addressed University of Texas at Austin, McKetta Department of Chemical Engineering, Austin, TX 78712. ‡ Department of Chemical and Biological Engineering, University at Buffalo, The State University of New York, Buffalo, New York 14260-4200. † The

1 ACS Paragon Plus Environment


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Abstract We present the results of a computational study of the interactions, phase-behavior and aggregation characteristics of charged nanoparticles (CNPs) suspended in solution of oppositely charged polyelectrolytes (PEs). We used an extension of the mean-field polymer selfconsistent field theory (SCFT) model presented in our earlier work (Macromolecules, 47, 6095 (2015)) to explicitly characterize the multibody interactions in such systems. For dilutemoderate particle volume fractions, the magnitudes of three and higher multibody interactions were seen to be weak relative to the contributions from pair interactions. Based on such results, we embeded the pair-interaction potentials within a thermodynamic perturbation theory approach to identify the phase behavior of such systems. The results of such a framework suggested that the gas and FCC crystal phases were thermodynamically stable, whereas the fluid-like phase was metastable in such systems. To complement the parameters studied using SCFT, we used a recently developed multibody simulation approach to study the aggregation and cluster morphologies in CNP-PE mixtures. For low particle charges, such systems mainly exhibited clusters arising from direct contact aggregation between CNPs. However, for higher particle and PE charges and low PE concentrations, large regions of PE-bridged clusters were seen to form. We present a morphological phase diagram summarizing such results.


Page 2 of 45

Page 3 of 45

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


1

Introduction

Mixtures of charged nanoparticles (CNP) and oppositely charged polyelectrolytes (PE) form an important class of materials widely encountered in food formulations and biological systems. 1–9 Since the CNP and PE can electrostatically bind with each other, such mixtures can exhibit rich phase behavior and complexation characteristics which in turn exert a significant influence on the structural and rheological properties of the mixture. 10,11 Not surprisingly, experimental characterization of the phase behavior of CNP-PE mixtures has attracted considerable attention. 5,11–20 In many situations, such mixtures have been observed to be immiscible, typically phase separating by one of two distinct means: (i) In situations where the net interactions between the particles and polyelectrolytes are weak or repulsive in nature, the system undergoes macrophase separation in which two phases respectively enriched in the CNP and the PE components are formed; (ii) In mixtures characterized by strongly favorable interactions between the CNP and PE, complex coacervation occurs in which a two-phase region is formed where both the polymer and particles segregate to one of the phases, while the other phase is depleted of both components. 1,3,5,11,12,14,19–22 Complex coacervation is frequently accompanied by the formation of PE-bridged particle aggregates, which have been demonstrated to exhibit non-trivial response to temperature, pH, shear and other ambient conditions. 1,2,4,23 Not surprisingly, characterization of such phases and the parametric dependence of their occurrence has also become of significant interest for applications. Although the CNP-PE systems have been widely studied in many earlier experimental works, a complete theoretical understanding of such systems is still lacking. Full-scale simulations in which the charged particles, polymers and ions are treated on an equal rigorous footing are computationally expensive. 24,25 Hence, interparticle interactions are routinely approximated by the effective, polymer-mediated two-body potentials deduced at the dilute limit of CNP concentrations, and are used in simulations involving just the particle system. 26–29 Motivated by such ideas, in recent works, 30,31 we used a numerical implementation of polymer self-consistent field theory to study the effective interactions between two charged spherical particles in polyelectrolyte solutions. We refer the reader to Refs. 30,31, for an elaborate discussion of other earlier work 25,32–61 pertain3 ACS Paragon Plus Environment


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

ing to these general class of problems and refrain from repeating the discussion here. Explicitly, in Ref. 30 we considered a model in which the particles possessed a specified uniform surface charge density, and the polymers contained a prespecified amount of dissociated charges which was assumed to be smeared along the chain. We characterized the polymer-mediated interactions between the particles as a function of the particle charge, polymer concentrations and particle sizes. In Ref. 31 we extended such a model to study situations in which either or both the particle and the polymers possessed partially dissociable groups. Additionally, we also considered the case when the dielectric constant of the solution depends on the local concentration of the polymers and when the particle’s dielectric constant was lower than that of the solvent. In both Refs. 30 and 31, we demonstrated that the polymer-mediated interactions consisted of a short range attraction arising from polymer-mediated depletion interaction and a long-range electrostatic repulsion. A longer range, albeit much weaker, bridging attraction 35,37,54,62 was also evident for some parametric regimes. Allowing for partial dissociation of the polymer and particle was seen to have a strong influence on the strength of the repulsive portion of the interactions. Rendering the dielectric permittivity to be inhomogeneous had an even stronger effect on the repulsive interactions. 31 While an understanding of the pair-interactions between particles proves extremely useful for a qualitative understanding of the physics of CNP-PE mixtures, most often, interactions deduced at infinite dilution of particles are by themselves not useful unless they are relevant for the prediction of the structure and phase behavior at finite particle concentrations. Motivated by such considerations, in a recent work, we developed a full multibody simulation framework (referred to henceforth as SCMF framework) to study mixtures of charged particles and uncharged polymers. 63,64 Such a framework treated the polymer segments, ions and the particles on an equal footing and enabled the study of multiparticle systems. In such a context, we examined the influence of multibody effects on the structure of particle-polymer mixtures. Our results indicated that the pair-interactions deduced at infinite dilute concentration of particles overpredicted the aggregation between particles relative to the results of the full multibody simulations. Such effects were shown to be a consequence of the interplay between the respective multibody effects on the


Page 4 of 45

Page 5 of 45

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


depletion and electrostatic interactions. We note that there are complementary advantages underlying the SCFT methodology and the SCMF framework developed in Ref. 64. On the one hand, the SCFT methodology is attractive in its ability to yield results for the effective, polymer-mediated interparticle potentials for fixed configurations of arbitrary number of particles of specified shapes and distribution of charges. Since such quantities rely on free energy calculations, the multibody simulation approach of Ref. 64 is not convenient for such a purpose. Moreover, a mean-field framework is especially applicable for denser polymer solutions, 65 which are expensive to study in a simulation approach which tracks all the polymer degrees of freedom. On the other hand, the multibody framework of Ref. 64 proves especially useful for analysis of the structure of the particle-polymer mixture, and for characterizing the cluster/aggregation characteristics, especially those resulting from polymer-bridging effects. In contrast, studying such features prove cumbersome within the SCFT methodology. 55,57 Moreover, mean-field theories are much less accurate in regimes of dilute polymer concentrations and high charges, which are characterized by strong fluctuation effects. 65 In contrast, our earlier studies demonstrated that the SCMF approach can semiquantitatively capture the influence of such fluctuation effects. 63 Finally, the numerical framework underlying SCFT multibody simulation framework is less successful in its ability to deal with systems characterized by strong polymer charges. However, our studies indicated no comparable issues for the SCMF framework for such parametric regimes. Based on the above reasoning, in this work we use the SCFT methodology of Refs. 30,31 to explicitly study the relative strengths of two- , three- and multi-body interactions to examine the range, if any, of parameters for which the pair-interactions provide a quantitative approximation to the effective polymer-mediated free energies. We then embed the pair-interaction potentials within a simple thermodynamic perturbation theory framework to develop the phase diagram for such systems. 66–77 Subsequently, we use the framework of Ref. 64 to study regimes of high polymer and/or particle charges and dilute polymer concentrations, which are not accessible to our numerical SCFT framework. Such regimes are expected to be characterized by aggregation and



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

clustering mediated by polymer bridges, which are more complicated to study within a framework which integrates out the polymer degrees of freedom. Our SCMF results help identify a morphological phase diagram and do indicate the occurrence of large regions of PE-bridged clusters for appropriate conditions of polymer, particle charges and concentrations. The rest of the article is organized as follows. In Section 2, we present a brief discussion of the SCFT and SCMF framework. Subsequently in Section 3.1, we present a short review of our results reported in Ref. 30 for the pair interactions at infinite dilution. In Section 3.2.1, we present the PE-mediated interparticle potentials for three-body interactions and compute the contribution of multibody effects to the total free energy of CNP-PE system. In Section 3.3, we present the phase diagram for CNP-PE mixtures deduced using thermodynamic perturbation theory in conjunction with the pair interaction potentials. In Section 4, we present results from SCMF simulations which characterize the influence of CNP, PE charges and concentrations on the structure of aggregates. We conclude with a brief summary of our results in Section 5.

2

Simulation Method

2.1

Multibody Interactions and McMillan-Mayer Approach

We considered a system of positively charged spherical particles of radius Rc in a solvent containing a mixture of negatively charged polyelectrolytes, negatively charged coions and positively charged counterions (see schematic in Fig. 1). The total charge of the particles is denoted as QC , and is assumed to be distributed in a spherically symmetric manner over the entire volume of the particle. The charge on the PE is denoted as Q pol . Since the electrostatic field resulting outside a particle with spherically symmetric volumetric charge distribution with total charge QC is identical to that arising for a spherically symmetric surface charge with total charge QC , we adopted the former model as it provided better numerical performance. In our previous articles, 30,57,78 we have used the Mayer’s cluster expansion technique as a framework to identify the effective interactions between particles in complex fluids. Within such a 6 ACS Paragon Plus Environment

Page 6 of 45

Page 7 of 45

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


-

-

ܳ௖ ܳ௖

-

ܳ௖

ܳ௖

-

ܳ௖

Figure 1: A schematic of the model under consideration in this work. We consider positively charged spherical particles of radius Rc immersed in a solution of negatively charged polyelectrolytes, counterions. The total charge of the particle is QC and is assumed to be distributed in a spherically symmetric and uniform manner. The charge on the PE is denoted as Q pol and was assumed to be smeared along the chain.

framework, discerning the polymer-mediated interactions between the particles in the PE solution requires the calculation of the grand canonical free energies for the mediating fluid in the presence of a fixed configuration of particles. 57,79–81 In the following, we use the notation ΞI (ri , r j , · · · ; z p ) to denote the grand canonical partition function 57,78–81 of the PE solution whose activity coefficient is fixed as z p and which contains I particles at positions ri , r j · · · . The I body interaction potential corresponding to such a situation is denoted as UI (ri , r j · · · ). In such a notation, the polymermediated interaction potentials can be written as:

U1 (z p ) = log [Ξ1 (z p )] − log [Ξ0 (z p )] + Qc ΨG

(1)

U2 (|r1 − r2 |; z p ) = log [Ξ2 (r1 , r2 ; z p )] − 2U1 (z p ) − log [Ξ0 (z p )] + 2Qc ΨG 3

U3 (r1 , r2 , r3 ; z p ) = log [Ξ3 (r1 , r2 , r3 ; z p )] − ∑ U2 (ri , r j ; z p ) − 3U1 (z p ) − log [Ξ0 (z p )] + 3Qc ΨG i> j

In the above representation, we have accounted for the fact that the one-body potential is independent of the position of the particle and the two-body interactions are only dependent on the distance between the particles. The term ΨG in the above equation denotes a “gauge” potential which arises 7 ACS Paragon Plus Environment


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 8 of 45

from the need to enforce an electroneutrality condition. In this work, we were specifically interested in quantifying the magnitude of multibody interactions relative to the pair-interaction potentials U2 (|r1 − r2 |; z p ). Towards this objective, we effected two classes of numerical computations: 1. Three body interactions for specified geometric configurations: We considered representative geometric configurations (see Figure 2), and explicitly computed the three-body potentials U3 (r1 , r2 , r3 ; z p ). For the cases L and IT shown below, we fixed the distance denoted as r1 and choose to vary the edge indicated as r in Figure 2. For the equilateral triangle arrangement (ET), we varied the distance r which results in simultaneous displacements of all three particles. 1

1

r1 1

r

r

r

r1 r

3

2

r

3

2

r

2

Linear (L)

Isosceles triangle(IT)

3

Equilateral triangle(ET)

Figure 2: Particle configurations used to examine three-body effects.

2. Multibody interactions in randomly generated particle configurations: We considered randomly generated particle configurations {ri } and obtained the “excess” free energy, Fmb , of such configurations compared to the free energy estimated through pair interactions: , j=Nc i= j−1

Fmb (ri , z p ) = log(Ξ[ri , z p ]) −

∑ ∑

j=2

U2 ( ri − r j , z p ) − NCU1 (z p ) − log [Ξ0 (z p )] .

(2)

i=1

We used four randomly generated configurations for different particle volume fraction to calculate Fmb (ri , z p ).


Page 9 of 45

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


2.2

Self-Consistent Field Theory of PE-CNP system

Equation (1) suggests that discerning the PE-mediated two-body, three-body (and multibody) interactions between the particles requires the calculation of the grand canonical free energies ln Ξ0 , ln Ξ1 , ln Ξ2 · · · for the PE solution in the presence of a fixed configuration of particles. In our work, we used a mean-field framework for the PE solution to determine the free energies ln ΞI as function of the polymer solution chemical potential z p , or equivalently, the bulk polymer concentrations c. 65,82 The model used in this work is identical to the framework presented in Ref. 30. Hence, to maintain brevity we do not repeat the details here and instead refer the reader to the original article. In brief, we model the polyelectrolyte chains as continuous, flexible Gaussian chains with a prespecified overall charge which is independent of solvent or pH conditions (strong acids). 82 We neglected all nonelectrostatic interactions (enthalpic and excluded volume) between the polymer monomers and ions. We also ignore all enthalpic interactions between the polyelectrolyte and particles and retain only electrostatic and excluded volume interactions between the particle and the polymer segments. The electrostatic interactions were themselves modeled using a classical Coulomb potential with a spatially constant dielectric value. The presence of the particles and their interactions with the polymers were modeled explicitly by incorporating a steeply repulsive potential wcp (r) which ensures the impenetrability of polymer monomers, co- and counterions into the particle core: wcp (r) =

σ r − Rc

12 (3)

where r denotes the distance between the center of the particle and the monomer or counterion component, σ is a parameter chosen to minimize numerical issues arising from sharp interfaces. Using such a framework and the equations detailed in Ref. 30 we obtained the requisite grand canonical free energies, ΞI (ri , r j , · · · ; z p ).



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

2.3

Single Chain in Mean Field Simulation Framework

We complement our SCFT results by using the multibody approach based on Single Chain in Mean Field simulation (SCMF) methodology 83–89 developed in our previous article. 64 The details of SCMF simulations were presented in our previous article for the mixtures of charged particles and neutral polymers. 64 To maintain brevity of this article, the modifications and the details pertinent to the present work are reported in the Supporting Information. We mainly used the SCMF framework to study the structure of aggregates and clusters formed in the multiparticle systems. An especially advantageous aspect of the SCMF simulations is its ability to also probe aggregates resulting from polymer bridging effects. Towards these objectives, we used a procedure proposed by Sevick et al. 90 in which the clusters are identified through a connectivity matrix based on physical contact between particles. Such a matrix is then further modified to account for indirect contact among particles and thereby identify unique clusters. Particles were considered in physical contact with each other when the distance between particles was less than one grid spacing i.e. the center-to-center distance was less than 2Rc + 4x. We expect this to be an appropriate measure since resolutions below 4x are inherently coarse-grained in our simulations. To characterize PE-bridged clusters, the connectivity matrix is modified to account for electrostatic binding between CNP and PE. The PE is assumed to be connected to a particle if the distance between a monomer on the polymer chain and the particle was less than Rc + 4x. Using the cluster size distributions thus calculated, we classify the aggregates into different states. The details of the classification are discussed in Section 4.1.

2.4

Parameters

For our studies, we worked in units in which the Bjerrum length was 0.7 nm corresponding to the Bjerrum length of water at room temperature. The polymer concentration is reported as C/C∗ , where C denotes polymer concentration and C∗ is the overlap concentration. Within SCFT framework, the PEs were assumed to be ideal chains, i. e. the excluded volume interactions were neglected. However, in case of SCMF simulations, the model included excluded volume interactions 10 ACS Paragon Plus Environment

Page 10 of 45

Page 11 of 45

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(Equation 3 in Supplementary information) u0 = 10 nm3 . However, independent simulations (not reported) confirmed that the excluded volume interactions had only a small influence and that the properties of a neutral polymer solution with such interactions were practically identical to an ideal solution. All simulations are carried out at small screening length conditions for which only particle and polyelectrolyte counterions are included. For both SCFT and SCMF simulation frameworks, we used a periodic simulation box 200 × 200 × 200 nm3 (the screening length for our lowest concentrations was ≈ 24nm). To solve the diffusion equation resulting in the SCFT, we have used a pseudo-spectral method. 65 We used a 3D spatial grid of size 1283 and chain discretizations of s = 32; 64; 128; 256 with an extrapolation the results to s → ∞. We fixed the size of the polymer as Rg = 24 nm (where Rg is the unperturbed radius of gyration of polymer chain) and particle radius Rc = 10 nm (for SCFT results, we also studied particles of size Rc = 20 nm and report the corresponding results). The SCFT results were for polymer charge Q pol = 20 and the SCMF results correspond to Q pol = 60 and Q pol = 120. For both frameworks, the charge on each particle is assumed to be equally distributed throughout the particle. We note that such an approach does not possess the ability to model strong ion correlation effects. However, such effects are more relevant for systems of multivalent ions and/or strongly charged systems. Our parameters were not in such regimes and hence the referee is correct in his observation that such characteristics are not relevant for our parameters. The SCMF simulations were initiated with particles placed in cubic lattice conformation and random configurations of polymer and ions. We carried out pre-equilibration Monte Carlo (MC) only on polymer and ions to generate initial configurations which avoid particle-polymer and particle-ion overlaps. The monomer, particle and ion positions were evolved in three dimensions using random displacement MC moves. In addition to the monomer displacement moves, slithering snake moves wherein the polymer chain is regrown by one segment in either direction were used. Each Monte Carlo step (MCS) consisted of 100 MC random displacement attempts per monomer, one MC random displacement attempt per particle and one slithering snake move per chain. The



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 12 of 45

potential fields are updated based on the inhomogeneous density of monomers, particles and ions after one MC step per monomer and one MC step per particle. After particle displacements, we effect MC steps on polymer segments and ions to ensure sufficient relaxation and minimal overlap between particle and monomers, ions. Additional details of the algorithm are similar to those described elsewhere. 64,84 The system was equilibrated for 5 × 104 MCS after which the properties were averaged and clusters were analyzed.

3

SCFT Results and Thermodynamic Perturbation Theory

3.1

Polymer-mediated pair-interaction potentials in CNP-PE mixtures

In our previous article, 30 we examined the interactions between CNP-PE mixture for the two charged nanoparticles in PE solutions. The resulting interaction potentials consisted of a shortrange attraction and a long-range repulsion. An even longer-range attraction attributed to polymerbridging effects was also evident for some parameters, but the latter was weaker relative to the short-range attraction and the electrostatic repulsions. The short-range attraction was demonstrated to arise from the polymer depletion near the surface of the particles, and as a consequence, was stronger for higher polymer concentrations. The long-range repulsion arose from the electrostatic interactions accompanying the overall charge density profiles resulting from the particle, counterion and polymer charges. Figure 3(a) presents representative potentials corresponding to conditions of high particle charge and low polymer concentrations. At such low polymer concentrations, the interparticle interactions are purely repulsive, mirroring the fact that the depletion interactions are much weaker relative to the electrostatic repulsions, and hence, the interaction potentials display only a repulsion and the bridging-induced long range attraction (magnified in the inset). In contrast, for higher polymer concentrations and for uncharged or weakly charged particles (cf. Figure 3(b)), the interaction potentials developed a short-range attraction in addition to the long-range repulsion. An interesting outcome of the analysis presented in Ref. 30 was the demonstration that the range of the deple12 ACS Paragon Plus Environment

Page 13 of 45

tion potential was much smaller than the correlation length of the neutral polymer solution (which corresponds to Rg for the ideal polymers considered in SCFT), and was determined by adsorption of the polymer segments on the oppositely charged particle-counterion cloud. We invoke this fact later in this article to justify the relative magnitudes of multibody interaction effects. (a)

(b)

3

0.3 Rg=24 nm

Rg=12 nm

0.2

2.5

1.5 1 0.5

0.5 0.4 0.3 0.2 0.1 0 -0.1 20

0.1 0

U2(r)

U2(r)

2

U2(r)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


40

-0.1

Rg=24 nm, Qc=0

-0.2

Rg=12 nm, Qc=0

-0.3

Rg=24 nm, Qc=5

-0.4

Rg=12 nm, Qc=5

60

r-2Rc 0 -0.5

-0.5 0

20

40

60

80

0

20

r-2Rc

40

60

80

r-2Rc

Figure 3: (a) Two-body interactions for Qc = 36 and Rg = 12nm (C/C∗ = 0.01), Rg = 24nm, (C/C∗ = 0.02). The inset shows a magnification of the region exhibiting long-range attraction. (b) Two-body interactions for Qc = 0.0, 5.0 and Rg = 12nm (C/C∗ = 0.01), Rg = 24nm, (C/C∗ = 0.02).

3.2 3.2.1

Multibody Effects Three Body Interactions

In this section, we present results examining the contribution of three and multibody interactions. In Figures 4 and 5, we present results for the case in which the particle radius was fixed at Rc = 20 and the polymer concentration was varied (in supporting information Figure 2, we display similar results for Rc = 10). Consistent with the discussion of the preceding section, we observe that the two-body potential exhibit a short-range attraction and long-range repulsion, and the strength of short-ranged attraction increases with PE concentration. In Figures 4(b-d) and 5(b-d), we present results for three body interactions for the different geometrical arrangements discussed earlier.



0.7

(a)

0.7

(b) C/C*=0.07 C/C*=0.14 C/C*=0.25 C/C*=0.42 C/C*=0.7 C/C*=1.15

0.6

0

10

20

30

50

C/C*=0.07 C/C*=0.14 C/C*=0.25 C/C*=0.42 C/C*=0.7 C/C*=1.15

-0.8 -1.3 -1.8 0.7

40

0.4

U3(r)

-0.3

0.5

0 -0.1 0.7

C/C*=0.07 C/C*=0.14 C/C*=0.25 C/C*=0.42 C/C*=0.7 C/C*=1.15

0.6 0.5

0.4

L

0.3

0

10

20

30

0.5

0.1

0

IT

0.3

0.1

50

C/C*=0.07 C/C*=0.14 C/C*=0.25 C/C*=0.42 C/C*=0.7 C/C*=1.15 C/C*=1.86

r

0.4

0.2

40

r-2Rc

(d)

0.6

0.2

-0.1

ET

0.1

r-2Rc

(c)

0.3 0.2

U3(r)

U2(r)

0.2

U3(r)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 14 of 45

0 0

10

20

30

40

50

-0.1

0

10

r-2Rc

20

30

40

50

r-2Rc

Figure 4: (a) Two-body interactions, three-body interactions for (b) ET, (c) L, and (d) IT arrangements for Rc = 20nm, Qc = 10 and Rg = 24nm. For the L and IT arrangements, the two particles were in contact, and the distance of third particle is denoted as r. Qp=10 Rc=20nm The three-body interactions are seen to be repulsive for the triplet arrangements, and moreover, the strength of repulsion is seen to grow with PE concentration. The equilateral triangle arrangement is seen to exhibit the strongest three-body interactions. Moreover, with increasing particle charge, we observe an increase in the magnitude of the three-body interactions. The above results can be physically rationalized based on the polymer density characteristics. Indeed, when a third particle is brought in the vicinity of two fixed particles, the polymer density experienced by the third particle is expected to be lower than that experienced in the situation when the particle is brought in the vicinity of a single isolated particle. As a consequence, depletion attractions in three body configurations are weaker than what may be expected based on the sum of two body potentials. Such an effect manifests as the repulsive three body interactions seen in


Page 15 of 45

(a)

(b) 0.7 C/C*=0.07 C/C*=0.14 C/C*=0.25 C/C*=0.42 C/C*=0.7 C/C*=1.15

U2(r)

1 0.5

C/C*=0.07 C/C*=0.14 C/C*=0.25 C/C*=0.42 C/C*=0.7 C/C*=1.15

0.6 0.5 0.4

U3(r)

1.5

0

0.3

ET

0.2 0

10

20

30

40

50 0.1

-0.5

0 -1

-0.1

r-2Rc

(c)

10

20

30

40

50

r-2Rc

0.7 C/C*=0.07 C/C*=0.14 C/C*=0.25 C/C*=0.42 C/C*=0.7 C/C*=1.15

0.5

0.4 0.3

L

C/C*=0.07 C/C*=0.14 C/C*=0.25 C/C*=0.42 C/C*=0.7 C/C*=1.15 C/C*=1.86

0.6 0.5 r

0.4

U3(r)

0.6

0.2

0.1

0.1

0

IT

0.3

0.2

-0.1

0

(d)

0.7

U3(r)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


0 0

10

20

30

40

50

-0.1

0

10

r-2Rc

20

30

40

50

r-2Rc

Figure 5: (a) Two-body interactions, three-body interactions for (b) ET, (c) L, and (d) IT arrangements for Rc = 20nm, Qc = 20 and Rg = 24nm. For the L and IT arrangements, the two particles were in contact, and the distance of thirdQp=20 particle Rc=20nm is denoted as r. Figures 4 and 5. Increasing the particle charge increases the influence of the electrostatic interactions relative to the depletion interactions and leads to stronger three body interactions. Since the linear and isosceles triangle arrangements consider particles in contact, the range of influence of the particle pair upon the third body is more limited, and hence results in weaker three body interactions compared to the equilateral triangle arrangement. An interesting auxiliary consequence of the such results is that in situations where the interparticle potentials are attractive, such interactions are expected to be strongest for the case of the linear arrangements of the particles. Such considerations suggest direct particle clustering in such systems are more likely to occur into linear anisotropic arrangements rather than the isotropic configurations promoted by the equilateral triangle arrangements. 91,92



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

More importantly, we observe that for most of the parameters and arrangements considered in this article, the three body interactions are relatively weak when compared to the pair interactions (the results for smaller particles, presented in supporting information Figure 2, show similar trends). This result may be surprising since in a previous article 64 we demonstrated that multibody effects on polymer depletion interactions could be significant even at dilute particle concentrations for neutral polymer solutions. The qualitatively different behavior exhibited by the system considered in the present work can be rationalized by noting that, in CNP-PE systems the range of depletion of the polymer density profiles do not correspond to the polymer correlation length (in this case Rg ). Instead, in our earlier article 30 we demonstrated that the polymer depletion characteristics for our systems were determined by the interplay between polymer “adsorption” on the oppositely charged counterion and particle cloud and the polymer exclusion from the particles. Such physics led to much narrower depletion thickness relative to the neutral polymer system. As a consequence, the range and magnitude of the three-body interactions are seen to be considerably weaker for the CNP-PE systems.

3.2.2

Multibody free energy of randomly distributed particles

The preceding section demonstrated that the magnitude of three-body interactions in CNP-PE systems are weak relative to the pair-interaction potentials. To complement such results, we present a direct estimation of the overall contribution of the multibody interactions for finite particle volume fractions. Towards this objective, we calculated the excess multibody free energy (total free energy excluding one and two-body interactions) represented by Fmb (r1 . . . rn ; z p ) (see Eq. 2). Since such computations are expensive, we performed such calculations only for four randomly placed particle configurations at four distinct particle volume fractions. In Figure 6, we display the results as a histogram of the excess free energies (per-particle) in the different configurations (labelled 1 4). Overall, we observe that the multibody free energy contributions are small (less than 0.03kB T per particle) for lower particle volume fractions φ p = 0.15. Even for the highest particle volume fraction φ p = 0.27, we observe that the contribution of the excess multibody free energy is only of


Page 16 of 45

Page 17 of 45

the order 0.25kB T . Together, the results presented in Sections 3.2.1 and 3.2.2 suggest that for particle charge, polymer concentration regimes examined within the SCFT framework, the magnitude of multibody interactions effects are relatively weak, and that the pair-interaction potentials deduced at infinite dilution can be used as a reasonable approximation to study the thermodynamics of CNPPE mixtures. We note that systems characterized by strong depletion interactions are likely to form aggregated structures which are not represented by random configurations of particles. However, through the combination of three body interactions in specified configurations and the free energies of the random configurations, we expect that the estimates provided for the multibody interactions are still reasonable. 0.350 0.300

φp=.03

φp=.07

φp=.13

φp=.27

0.250

F୫ୠ ܑ

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


0.200 0.150 0.100 0.050 0.000 -0.050

1

2

3

4

Figure 6: Multibody free energy of randomly distributed particles for four different particle configuarations, at Qc = 20, C/C∗ = 1.87, Q pol = 20, Rg = 24 nm.

3.3

Phase Diagram based on Two-Body Potentials

The results presented in the previous two sections suggest that the use of the pair-interaction potentials deduced at infinite dilution concentration of particles is expected to be a reasonable approximation for studying the thermodynamics and phase behavior of CNP-PE mixtures. Motivated by such findings, in this section we present phase diagram results based on thermodynamic perturbation theory (TPT) calculations, which has been shown to be a reasonable framework for qualitative, and in many cases, semiquantitative characterization of the boundaries of the different 17 ACS Paragon Plus Environment


phases which emerge out of the interaction potentials. 66–77,93,94 3

(a) 2.5

Gas-solid binodal Gas-fluid binodal

2

C/C*

Gas-fluid spinodal 1.5

1

0.5

PE-bridged clustering zone 0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

ϕp 3

(b) 2.5

2

C/C*

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 18 of 45

1.5

1

0.5

0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

φp Figure 7: Phase diagram from Thermodynamic Perturbation Theory for (a) Qc = 10 and (b) Qc = 20 for Rg = 24nm and Rc = 20nm.

In Appendix, we describe the details of the TPT we used in this work. The framework we employ takes advantage of the fact that since our interaction potentials were determined in a semigrand canonical ensemble, free energy calculations based on the interparticle potentials automatically ensure equality of the polymer chemical potentials in the two phases. Subsequently, the polymer concentrations corresponding to the respective equilibrium phases and particle concentra18 ACS Paragon Plus Environment

Page 19 of 45

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


tions (i. e. the polymer uptake in the respective phases) can be deduced by using an equilibrium SCFT simulation for the corresponding volume fraction of particles. In Supporting Information (Figure 3) we present representative uptake plots depicting the polymer concentrations as a function of the chemical potential z p and the particle volume fraction φ p for the disordered particulate phase. Such uptake information can be used to recast the phase diagrams emerging from TPT into the polymer concentration-particle volume fraction planes. Representative phase diagrams derived from the TPT are displayed in Figures 7(a) and (b) (In Supporting Information Figure 4 we present the phase diagram in the z p − φ p plane). The lower boundaries of these phase diagram are characterisitic of the gas-solid boundaries of hard spheres except for the renormalization of the particle size and volume fractions arising from the electrostatic repulsions. The fluid-solid boundaries seen for higher polymer concentrations are characteristic of systems possessing the short-range attraction, and have also been noted in other colloids characterized depletion interactions in which the “depletants” are much smaller than the particles themselves. 95 The occurrence of such features for our system may be surprising since the polymer sizes (Rg = 24) are comparable to particle sizes (Rc = 20). However, as discussed earlier, the depletion interactions in our system have a range which is considerably smaller the polymer sizes and are relatively insensitive to the polymer concentration. From Figures 7a and b, we observe that with increasing particle charge the phase boundaries shift upward in polymer concentrations. Such a result can be justified by noting that increasing the particle charge leads to an increased electrostatic repulsion between the particles and correspondingly necessitates a higher polymer concentration (attraction) to create a similar driving force for phase separation. Many earlier studies of colloidal system possessing phase diagrams of the kind shown in Figure 7 have discussed the formation of dynamically arrested glassy phases which occur in the region indicated as metastable clustering zone. 96–101 Interestingly, the critical point of the fluid-gas phase transition occurs around φ p = 0.5 for our systems, which contrasts with typical values (φ p . 0.25) noted in contexts such as for mixtures of neutral polymers and colloids. 57,94 Such results suggest



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

that in CNP-PE systems, the region charactering metastable clusters and gelation can potentially occupy a large region of particle volume fractions. 102 In Supplementary Information, we present results which demonstrate the formation of such clusters. 103 In summary, in this section we presented results from polymer self-consistent field theory to demonstrate that for the range of parameters considered in the present work, the multibody interactions are weak relative to the pair interaction potentials in CNP-PE mixtures. Based on such considerations, we used a simple thermodynamic perturbation theory to construct phase diagrams for such systems. Such phase diagrams resembled those predicted for neutral polymer-particle systems and exhibits large regions of parameters characterized by aggregation and gelation. In the next section, we extend the range of particle charges and polymer charges considered and turn our attention to the regions of low polymer and particle concentrations where other kinds of aggregates form.

4

Polymer Bridging Interactions and Structural Consequences

A surprising feature of the results presented in the previous section was that the phase behavior of the systems considered in this article resembled the characteristics seen in the context of systems characterized by hard core and polymer depletion interactions. Moreover, the binodals depicted in Figure 7 indicate that the phase separation is representative of a segregative phase separation in which the resulting phases are enriched in polymers and particles respectively. Such results appear to contradict some experimental observations which have demonstrated the occurrance of a “complex-coacervation” like behavior in which resulting phases are such that one of the phases contains a concentrated mixture of particles and polymers whereas the other phase is depleted of both particles and polymers. 1,23 We rationalize such differences by pointing out that the parametric regimes explored in the previous section primarily pertained to polymer concentration and charge regimes wherein the most relevant interactions were the depletion attraction and electrostatic repulsion. In contrast, for higher polymer and particle charges, the long-range bridging attraction


Page 20 of 45

Page 21 of 45

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


U (r )

C / C*  0.095; QC  20 C / C*  0.190; QC  20 C / C*  0.095; QC  36 C / C*  0.190; QC  36 C / C*  0.045; QC  20 C / C*  0.090; QC  20 C / C*  0.191; QC  20 C / C*  0.381; QC  20

r  2 RC

Figure 8: (Reprinted with permission from Ref. 30) Interaction potentials U(r) for different bulk polymer concentrations C/C∗ and particle charges QC for Rg = 12 (closed symbols) and Rg = 24 (open symbols). Particle radius Rc = 20nm. Only the long-range attractive portion of the interactions are displayed and the lines are meant to be a guide to the eye.

evident in Figure 3 becomes much more significant (see Figure 8, which is reproduced from Ref. 30). We hypothesize that the regimes which are characterized by strong long-ranged bridging attractions are more likely to lead complex coacervation like phase behavior in the regimes indicated schematically in Figure 7. Unfortunately, the numerical framework we had used for the SCFT methodology was unable to access the regimes of high polymer charges due to issues with convergence for the Poisson Boltzmann equation. Moreover, regimes of high polymer charges and dilute polymer concentrations have also been suggested to be significantly influenced by thermal and electrostatic fluctuation effects 104,105 which are absent in the mean-field framework. 65 Hence, the regime of validity of results derived from a successful numerical implementation of SCFT would have been even otherwise limited. Furthermore, we anticipate that such regimes are likely to be characterized by long-range attractions of the order of polymer Rg , and are likely to suffer from strong multibody interaction effects. Motivated by the above considerations, we adopt the recently developed SCMF framework which is capable of overcoming many of the issues discussed above. 64 Explicitly, such a simulation method allows us to access regimes of high polymer and/or particle charges (as we demonstrate below). Moreover, our earlier studies revealed that the SCMF method is capable of semiquantita21 ACS Paragon Plus Environment


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

tively capturing the effects arising from the polymer concentration and electrostatic fluctuations. 63 Moreover, by virtue of its multibody nature, the SCMF methodology accommmodates interactions to all orders in particle concentrations. An added advantage of the SCMF method is its ability to directly characterize the polymer bridging characteristics. However, a disadvantange of the SCMF framework is that effecting phase equilbrium calculations are significantly more complex and hence we restrict ourselves only to a classification of the phases based on the structural characteristics. Due to the vastness of the parameter space, we do not embark on a comprehensive study, and instead, we only consider a few selected parameters representative of higher particle and polymer charges and classify the structural characteristics resulting as a function of particle and polymer concentrations. In this section, we present results which are complementary to the regimes to studied in the previous section.

4.1

Classification of Aggregates

As described in Section 2.3, we used the SCMF methodology in conjunction with a cluster identification algorithm 90 to identify both aggregates formed from direct particle contacts and through polymer bridging. In Figure 9 we display representative snapshots displaying such aggregates obtained from SCMF simulations. To characterize the different kinds of CNP clusters and CNP-PE clusters, the primary measure we used was the particle-particle radial distribution function g(r). States which were characterized by a peak in g(r) at particle contact were classified as particle aggregates. In contrast, states which were characterized by a peak in g(r) at an interparticle distance much larger than the particle contact (at distances of the order of polymer Rg ) were classified as polymer-bridged aggregates (PB). Some of our simulations resulted in aggregates which exhibited peaks at both particle contact and at a distance of the order of polymer Rg . Such states were identified as being a combination of both particle and polymer-bridged aggregates. To quantify the extent of PE-bridging of the particles, we used a parameter known as bridging fraction, B f , defined as the fraction of PE chains binding with more than one particle. 22 ACS Paragon Plus Environment

Page 22 of 45

Page 23 of 45

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


To further classify the clusters (both direct particle and polymer-bridged aggregates) resulting in our simulations, we used an approach recently proposed by Godfrin et. al. 106 For the latter, we computed N(s), the average fraction of particles contained in a cluster of size s:

N(s) = (s/N p )n(s),

(4)

with N p denoting total number of particles, n(s) representing the average number of clusters and s denoting the size of cluster i.e. number of particles in a cluster. 107 Subsequently, we classified the aggregates as monomer dominated states (M) containing small clusters, aggregated states (A) containing large clusters and percolated (P) state containing a system spanning cluster. 106,108,109 States which exhibited a monotonically decaying N(s), and which decayed to zero by s ≈ 10, were classified as M. 106 States which exhibited a monotonically decaying N(s), but containing larger clusters where termed as aggregates (A). If N(s) exhibits a peak for s ' N p , where N p denotes the number of particles in the system, we classify such states as a percolated state P. For the finite size systems examined in this study, such percolated states could be a manifestation of macrophase separation.



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

PE-bridged aggregate Qp=40, C/C*=0.04 Direct contact particle aggregates

Qp=10, C/C*=0.16

Qp=10, C/C*=0.75

Figure 9: Representative snapshots displaying PE-bridged and direct contact particle aggregates for φ p = 0.025, Rg = 24 nm, Rc = 10 nm, and Q pol = 60. The particles are shown in red and PE monomers are shown in blue. The size of particle and PE monomers is only representative and not to scale.

4.2

Effect of Particle Charge

We present the SCMF results by first illustrating the influence of particle charge on the structure of aggregates. In Figure 10(a) we display the particle-particle radial distribution functions as a function of particle charge at fixed polymer charge, Q pol = 60 and concentration C/C∗ = 0.16. At low particle charges, the PE-induced depletion interactions are seen to dominate, resulting in direct aggregation between particles which manifests as a sharp peak in g(r) at particle contact. With increasing particle charge, the propensity to aggregate is seen to reduce as a result of the increased electrostatic repulsion between particles. Such effects are also seen to be reflected in the cluster size distributions based on direct CNP contacts (shown in Figure 10(b)). For instance, at Qc = 5, 10, monomer dominated clusters are seen to form. In contrast, at Qc = 30 only clusters


Page 24 of 45

Page 25 of 45

comprising of two-particles are formed. The direct clustering between CNPs is seen to vanish for the particle charge of Qc = 40. Interestingly, with increase in particle charge, a second peak at higher interparticle distance is seen to appear in Figure 10(a). The length scale of this peak is seen to be of the order of 2Rc + Rg , which is a signature that such states are particle clusters mediated by PE bridging. In the inset to Figure 10(a), we display bridging fraction B f . Consistent with such a speculation, we observe that the bridging fraction of the polymers increases with increase in particle charge. Together, the above results indicate that at a fixed dilute polymer concentration, an increase in the particle charge results in a transformation from direct particle aggregates to polymer-bridged clusters. (a)

(b)

3

2

0.14

Qc=5

0.12

Qc=10

0.1

Qc=20

0.08

Qc=30

0.06

Qc=40

0.1 0.01

N(s)

Bf

4

g(r)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


0 10 20 30 40

Qc

Qc=10 Qc=20

1

0

Qc=5

0.001

Qc=30

0.0001 0

20

40

60

80

1e-05

100

r (nm)

1

2

4

8

s

Figure 10: Effect of particle charge on (a) particle-particle radial distribution function and (b) cluster size distribution based on direct clutering between CNPs for C/C∗ = 0.16, φ p = 0.025, Rg = 24 nm, Rc = 10 nm, Q pol = 60. The inset in (a) displays the bridging fraction as a function of particle charge.

4.3

Effect of Polymer Concentration

In Figure 11, we display the particle-particle radial distribution function, g(r), as a function of polymer concentration for fixed particle charges Qc = 10 and Qc = 40. From the peak in g(r) at contact seen in Figure 11(a), we conclude that at low particle charges (Qc = 10) direct particle aggregation manifests. With increasing polymer concentration, the aggregation propensity, as re25 ACS Paragon Plus Environment


flected in the magnitude of the peak in g(r), is seen to increase. Such trends are consistent with the presence of a short-range depletion interactions whose strength increases with increasing polymer concentration (cf. discussion in Section 3). In contrast, at high particle charges (Qc = 40, Figure 11b), the peak in the g(r) at contact is seen to vanish. In such a case, the electrostatic repulsion between particles dominate the depletion attraction induced by polymers and overcomes the tendency towards contact aggregation. However, a secondary peak at much larger distances (of the order of 2Rc + Rg ) is seen to emerge. Concomitant with the emergence of such characteristics, we observe that the bridging fraction also increases. Such characteristics again reflect a bridging-induced particle aggregation. (a)

1.5 Bf=0.18 Bf=0.11

10 8 6 4

(b)

*

* C/C =0.04 Bf=0.52

*

C/C =0.08 Bf=0.24

*

* C/C =0.12 Bf=0.14

C/C =0.04

*

C/C =0.08

Bf=0.08 Bf=0.06 Bf=0.04

C/C =0.12

Bf=0.02 Bf=0.01

C/C =0.5

1

*

C/C =0.16

*

C/C =0.16 Bf=0.1 * C/C =0.25 B =0.06

*

C/C =0.25

g(r)

12

g(r)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 26 of 45

*

Bf=0.02 * C/C =0.75 Bf=0.01

*

C/C =0.75

f

*

C/C =0.5

0.5

2 0

0

20

40

60

80

0

0

20

40

60

r (nm)

r (nm)

Figure 11: Effect of polymer concentration on the particle-particle radial distribution function at φ p = 0.025, Rg = 24 nm, Rc = 10 nm, Q pol = 60 for (a) Qc = 10 and (b) Qc = 40.

4.4

Effect of Polymer Charge

In the introduction to this section we speculated that the propensity to form polymer-bridged clusters to be strongly correlated to the charge on the polymer molecules themselves. In Figure 12(a) we validate such a hypothesis by presenting results for two different overall polymer charges Q pol . The peaks in the radial distribution function g(r) confirm that polymer-bridging mediated aggregation between particles is significantly stronger for the higher PE charge. In Figure 12(b) we


Page 27 of 45

compare the cluster size distribution based on PE-bridged aggregation. Consistent with the trends observed on g(r), PEs having higher charge are indeed seen to form larger aggregates involving a greater fraction of the particles. 2

(a) Qpol=120

1

(b)

Qpol=60

1.5 0.1 *

C/C =0.04

1

N(s)

g(r)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


0.5

0

*

C/C =0.16

0.01

*

C/C =0.75

0.001

0

20

40

60

80

r (nm)

0.0001

1

2

4

8

16

s

Figure 12: (a) Particle-particle radial distribution function for different polymer charge and (b) Cluster size distribution depicting PE-bridging aggreagtion between CNPs for Q pol = 60 (dashed lines) and Q pol = 120 (solid lines). Other parameters are fixed at φ p = 0.025, Rg = 24 nm, Rc = 10 nm and Qc = 40.

4.5

Morphological Phase Diagrams

In this final section, we use the classification discussed in Section 4.1 to summarize the results of our SCMF simulations as morphological phase diagrams in Figure 13 (the results for Qc = 5 are shown in Figure 8 of Supplementary Information). Overall, we observe that for low particle charge and moderate to high polymer concentrations, much of the structures result from direct aggregation between the particles. Such results can be justified by recalling that such states were characterized by strong short-range attractions from polymer depletion effects. In contrast, at dilute polymer concentrations, for which such attractions are expected to be weaker, we observe a stronger likelihood of forming polymer-bridged aggregates. With increasing particle charge, we observe a considerable expansion of the polymer concentration regime in which polymer-bridged aggregates result. These trends can be understood to be a consequence of the increased electrostatic repulsion between the particles and (as a result) the reduced importance of polymer depletion 27 ACS Paragon Plus Environment


effects. PB

M

A

Qc=20

P

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

C/C*

C/C*

Qc=10

0.4

PB

M

A

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0

0

ϕp

0.01

Qc=30

PB

M

0.1

0.01

A

Qc=40

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

C/C*

C/C*

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 28 of 45

0.4

0.1

ϕp PB

M+PB

0.4

0.3

0.3

0.2

0.2

0.1

0.1 0

0 0.01

ϕp

0.01

0.1

ϕp

0.1

Figure 13: Aggregate states as a function of particle volumee fraction, particle charge and polymer concentration. Direct aggregation between particles classified as no clustering, monomer dominated clusters, aggregates, and percolated states as a function of PE concetration and particle charge for Rc = 10 nm and Q pol = 60 respectively.

Together, the above results serve to identify the regimes of formation of polymer-bridged aggregate structures which resemble the complex coacervates seen in experiments. Explicitly, regimes characterized by high polymer charges are seen to be necessary to facilitate sufficient and substantial polymer bridging. Moreover, the polymer concentration/particle charges should be in such parametric conditions wherein the electrostatic repulsions between the particles dominate the short-range polymer depletion attractions. If such conditions are met, there is seen to be a 28 ACS Paragon Plus Environment

Page 29 of 45

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


preponderance of polymer bridged aggregates. An increase in polymer concentration or a decrease in particle and/or polymer charge is seen to result in particle aggregate structures representative of systems characterized by a short-range attractive interactions.

5

Summary

We examined the multibody interactions in charged nanoparticle (CNP)-polyelectrolyte (PE) mixture using self-consistent field theory methodology. The effect of interactions on the structure of the system was characterized using single chain in field simulations. We used a model wherein the charges on the CNPs and PEs were assumed to be completely dissociated and the charge does not vary as a function of solution conditions. Our results based on SCFT indicated that for regimes characterized by low particle and polymer charges, the three and higher-body interactions were weak relative to the two-body interactions. Based on such considerations, we used the thermodynamic perturbation theory to generate the phase diagram for CNP-PE mixtures. Such an analysis indicated that for such systems only gas and crystal phases are thermodynamically stable, but however contains a hidden/metastable liquidgas phase envelope. We then turned our attention to the regimes not amenable to study within our SCFT framework, which were characterized by high particle charges, low polymer concentrations and high polymer charges. The characterization of the structure of aggregates using single chain in mean field simulations showed evidence of PE-bridged clusters at low PE concentrations. The depletion effects were seen to dominate at higher PE concentrations a resulting in propensity for contact aggregation between particles. In future studies, we plan to build on the SCMF methodology presented in this article to examine similar effects in weakly charged systems and in presence of polarization effects resulting from differences in dielectric permittivities.



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 30 of 45

Acknowledgement This work was supported in part by grants from Robert A. Welch Foundation (Grant F1599), National Science Foundation (DMR-1306844) and the US Army Research Office under grant W911NF-13-1-0396. The authors acknowledge the Texas Advanced Computing Center (TACC) at The University of Texas at Austin for providing computing resources that have contributed to the research results reported within this paper.

Supporting Information Available Details of SCMF methodology and cluster characterization using MC method are discussed. The three-body interactions for Rc = 10 nm, PE uptake data, phase diagram based on two-body potentials in z p − φ p plane, and SCMF morphological phase diagram for Qc = 5 are displayed. This material is available free of charge via the Internet at http://pubs.acs.org/.

A

Thermodynamic Perturbation Theory

The thermodynamic perturbation theory 66–77 is based on the phenomenological observation 110 that the pair correlation function g(r) of a pairwise interaction system of the density ρ with a hard core pairwise interaction (for example Lenard-Jones interaction) can be approximated very closely by pair correlation functions of the hard spheres of appropriate diameter of one and volume fraction φp : g(r, ρ) ≈ gHS (r Re f f , φ p )

(5)

where Re f f is an effective hard-core range and φ p = π6 R3e f f ρ. The Helmholtz energy per particle A of the fluid or crystal can then be expressed in the form: 66–68,72,74,76,111–114 Z ∞

A = AHS [φ p ] + 12φ p

1

gHS (r, φ p )U2 (Rc r) r2 dr

(6)

To effect phase diagram calculations, we compare the free energies of different phases within 30 ACS Paragon Plus Environment

Page 31 of 45

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


the Equation (6). For a gas/fluid-like system, we use the Carnahan-Starling equation of state for estimating the free-energy A: 115

AHS [φ p ] = log(φ p ) − 1 +

φ p (4 − 3φ p ) (φ p − 1)2

(7)

and a Percus-Yevick approximation for the radial distribution function: 116

gHS (r, φ p ) = gPY (r, φ p ).

(8)

In such a representation, the free energy of the gas/fluid phase AGF can be written as:

AGF

φ p (4 − 3φ p ) = log(φ p ) − 1 + + 12φ p (φ p − 1)2

Z ∞ 1

gPY (r, φ p )U2 (Rc r) r2 dr

(9)

For the solid phase, we assumed that the resulting crystal possessed FCC structure and used a simple estimate for the free energy AHSFCC and the radial distribution functions gFCC (r, φ p ) of such a phase: 74  √ = − log 4 2  

AHSFCC

π √ 3 2φ p

!1/3

 − 1

and (

ncut

)

gFCC (r, φ p ) = Θ[r − 1] g1 (r, φ p ) + ∑ gi (r, φ p ) i=2

where A − α1 (R1 −r)2 e 2 r √ ni α − α (r−Ri )2 √ gi (r, φ p ) = e 2 i≥2 24 2πφ p rRi

g1 (r, φ p ) =



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

We set ncut = 5, ni = {12, 6, 24, 12, 24, . . .} (for FCC lattice) and !1/3 √ o n√ √ √ 3 2φ p 2, 3, 2, 5, . . . (i ≥ 2), Ri = π  −2 !1/3 π 3 π 2/3  √ − 1 . α= 2 3 2φ p Values of A, R1 and α1 are calculated from the following equations: 66,67 1 1−

√

23φ p π

1/3

=

1 − 4φ p g1 (r, φ p ) Z ∞

n1 π √ 3 2φ p

= 24φ p

!1/3

1

Z ∞

n1 = 24φ p

1

g1 (r, φ p )r2 dr g1 (r, φ p )r3 dr


Page 32 of 45

Page 33 of 45

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


References (1) Dubin, P. L.; Ross, T. D.; I., S.; Yegerlehner, B. E. In Ordered Media in Chemical Separations; Hinze, W. L., Armstrong, D. W., Eds.; American Chemical Society, 1987; Chapter 9, Coacervation of Polyelectrolyte-Protein Complexes, pp 162–169. (2) Strege, M.; Dubin, P.; West, J.; Flinta, C. Protein Separation via Polyelectrolyte Complexation. ACS Symposium Series 1990, 427, 66–79. (3) Turgeon, S.; Beaulieu, M.; Schmitt, C.; Sanchez, C. Protein-polysaccharide interactions: phase-ordering kinetics, thermodynamic and structural aspects. Current Opinion in Colloid & Interface Science 2003, 8, 401 – 414. (4) Cooper, C.; Dubin, P.; Kayitmazer, A.; Turksen, S. Polyelectrolyte-protein complexes. Current Opinion in Colloid & Interface Science 2005, 10, 52–78. (5) Turgeon, S. L.; Schmitt, C.; Sanchez, C. Protein-polysaccharide complexes and coacervates. Current Opinion in Colloid & Interface Science 2007, 12, 166–178. (6) Boddohi, S.; Kipper, M. J. Engineering Nanoassemblies of Polysaccharides. Advanced Materials 2010, 22, 2998–3016. (7) Palivan, C. G.; Fischer-Onaca, O.; Delcea, M.; Itel, F.; Meier, W. Protein-polymer nanoreactors for medical applications. Chemical Society Reviews 2012, 41, 2800–2823. (8) Becker, A. L.; Henzler, K.; Welsch, N.; Ballauff, M.; Borisov, O. Proteins and polyelectrolytes: A charged relationship. Current Opinion in Colloid & Interface Science 2012, 17, 90 – 96. (9) van Rijn, P. Polymer Directed Protein Assemblies. Polymers 2013, 5, 576–599. (10) Dickinson, E.; Eriksson, L. Particle Flocculation by Adsorbing Polymers. Advances in Colloid and Interface Science 1991, 34, 1–29.



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

(11) Dickinson, E. Colloid science of mixed ingredients. Soft Matter 2006, 2, 642–652. (12) Tolstoguzov, V. Phase behaviour of macromolecular components in biological and food systems. Nahrung-Food 2000, 44, 299–308. (13) Sennato, S.; Truzzolillo, D.; Bordi, F. Aggregation and stability of polyelectrolyte-decorated liposome complexes in water-salt media. Soft Matter 2012, 8, 9384–9395. (14) Chollakup, R.; Beck, J. B.; Dirnberger, K.; Tirrell, M.; Eisenbach, C. D. Polyelectrolyte Molecular Weight and Salt Effects on the Phase Behavior and Coacervation of Aqueous Solutions of Poly(acrylic acid) Sodium Salt and Poly(allylamine) Hydrochloride. Macromolecules 2013, 46, 2376–2390. (15) Antonov, Y. A.; Moldenaers, P. Structure formation and phase-separation behaviour of aqueous casein-alginate emulsions in the presence of strong polyelectrolyte. Food Hydrocolloids 2011, 25, 350–360. (16) Pelaez-Fernandez, M.; Moncho-Jorda, A.; Callejas-Fernandez, J. Structure of charged colloid-polymer mixtures. EPL 2010, 90, 46005. (17) Spruijt, E.; Westphal, A. H.; Borst, J. W.; Stuart, M. A. C.; van der Gucht, J. Binodal Compositions of Polyeleetrolyte Complexes. Macromolecules 2010, 43, 6476–6484. (18) Tolstoguzov, V. Some thermodynamic considerations in food formulation. Food Hydrocolloids 2003, 17, 1 – 23. (19) Weinbreck, F.; de Vries, R.; Schrooyen, P.; de Kruif, C. G. Complex coacervation of whey proteins and gum arabic. Biomacromolecules 2003, 4, 293–303. (20) Du, X.; Dubin, P. L.; Hoagland, D. A.; Sun, L. Protein-Selective Coacervation with Hyaluronic Acid. Biomacromolecules 2014, 15, 726–734.


Page 34 of 45

Page 35 of 45

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(21) Kizilay, E.; Kayitmazer, A. B.; Dubin, P. L. Complexation and coacervation of polyelectrolytes with oppositely charged colloids. Advances in Colloid and Interface Science 2011, 167, 24 – 37, Polyelectrolyte-Macroion Coacervation. (22) Kizilay, E.; Maccarrone, S.; Foun, E.; Dinsmore, A. D.; Dubin, P. L. Cluster Formation in Polyelectrolyte-Micelle Complex Coacervation. The Journal of Physical Chemistry B 2011, 115, 7256–7263. (23) Cabane, C.; Wong, K.; Duplessix, R. Macromolecules For Control Of Distances In Colloidal Dispersions. ACS Symposium Series 1989, 384, 312–327. (24) Skepo, M.; Linse, P. Dissolution of a polyelectrolyte-macroion complex by addition of salt. Physical Review E 2002, 66, 051807. (25) Carlsson, F.; Linse, P.; Malmsten, M. Monte Carlo Simulations of Polyelectrolyte-Protein Complexation. The Journal of Physical Chemistry B 2001, 105, 9040–9049. (26) Bolhuis, P. G.; Louis, A. A. How To Derive and Parameterize Effective Potentials in ColloidPolymer Mixtures. Macromolecules 2002, 35, 1860–1869. (27) Kung, W.; Gonzalez-Mozuelos, P.; de la Cruz, M. O. A minimal model of nanoparticle crystallization in polar solvents via steric effects. Journal of Chemical Physics 2010, 133, 074704. (28) Zhang, D.; Gonzalez-Mozuelos, P.; de la Cruz, M. O. Cluster Formation by Charged Nanoparticles on a Surface in Aqueous Solution. Journal of Physical Chemistry C 2010, 114, 3754–3762. (29) Gnan, N.; Zaccarelli, E.; Sciortino, F. Tuning effective interactions close to the critical point in colloidal suspensions. Journal of Chemical Physics 2012, 137, 084903. (30) Pryamitsyn, V.; Ganesan, V. Interplay between Depletion and Electrostatic Interactions in Polyelectrolyte-Nanoparticle Systems. Macromolecules 2014, 47, 6095–6112. 35 ACS Paragon Plus Environment


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

(31) Pryamitsyn, V.; Ganesan, V. Pair Interactions in Polyelectrolyte-Nanoparticle Systems: Influence of Dielectric Inhomogeneities and Partial Dissociation of Polymer and Nanoparticles. Submitted 2015, (32) Muthukumar, M. Adsorption of a Polyelectrolyte Chain to a Charged Surface. Journal of Chemical Physics 1987, 86, 7230–7235. (33) Ellis, M.; Kong, C. Y.; Muthukumar, M. Polyelectrolyte adsorption on heterogeneously charged surfaces. Journal of Chemical Physics 2000, 112, 8723–8729. (34) Joanny, J. F. Polyelectrolyte adsorption and charge inversion. European Physical Journal B 1999, 9, 117–122. (35) Podgornik, R. Polyelectrolyte-mediated bridging interactions. Journal of Polymer Science Part B-polymer Physics 2004, 42, 3539–3556. (36) Borukhov, I.; Andelman, D.; Orland, H. Scaling laws of polyelectrolyte adsorption. Macromolecules 1998, 31, 1665–1671. (37) Borukhov, I.; Andelman, D.; Orland, H. Effect of polyelectrolyte adsorption on intercolloidal forces. Journal of Physical Chemistry B 1999, 103, 5042–5057. (38) Wang, Q. Charge inversion by flexible polyelectrolytes on flat surfaces from self-consistent field calculations. Macromolecules 2005, 38, 8911–8922. (39) Chatellier, X.; Joanny, J. F. Adsorption of polyelectrolyte solutions on surfaces: A DebyeHuckel theory. Journal De Physique Ii 1996, 6, 1669–1686. (40) Acres, R. G.; Feyer, V.; Tsud, N.; Cadino, E.; Prince, K. C. Mechanisms of Aggregation of Cysteine Functionalized Gold Nanoparticles. Journal of Physical Chemistry C 2014, 118, 10481–10487.


Page 36 of 45

Page 37 of 45

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(41) Murthy, A. K.; Stover, R. J.; Borwankar, A. U.; Nie, G. D.; Gourisankar, S.; Truskett, T. M.; Sokolov, K. V.; Johnston, K. P. Equilibrium Gold Nanoclusters Quenched with Biodegradable Polymers. ACS Nano 2013, 7, 239–251. (42) Tombacz, E. pH-dependent surface charging of metal oxides. Periodica Polytechnicachemical Engineering 2009, 53, 77–86. (43) Zhang, F.; Ali, Z.; Amin, F.; Feltz, A.; Oheim, M.; Parak, W. J. Ion and pH Sensing with Colloidal Nanoparticles: Influence of Surface Charge on Sensing and Colloidal Properties. Chemphyschem 2010, 11, 730–735. (44) Borukhov, I.; Andelman, D.; Orland, H. Random polyelectrolytes and polyampholytes in solution. European Physical Journal B 1998, 5, 869–880. (45) Messina, R.; Holm, C.; Kremer, K. Polyelectrolyte adsorption and multilayering on charged colloidal particles. Journal of Polymer Science Part B-polymer Physics 2004, 42, 3557– 3570. (46) Dickinson, E. Stability and rheological implications of electrostatic milk proteinpolysaccharide interactions. Trends in Food Science & Technology 1998, 9, 347–354. (47) Ryden, J.; Ullner, M.; Linse, P. Monte Carlo simulations of oppositely charged macroions in solution. Journal of Chemical Physics 2005, 123, 034909. (48) Jonsson, M.; Linse, P. Polyelectrolyte-macroion complexation. I. Effect of linear charge density, chain length, and macroion charge. The Journal of Chemical Physics 2001, 115, 3406. (49) Podgornik, R.; Parsegian, V. A. Charge-fluctuation forces between rodlike polyelectrolytes: Pairwise summability reexamined. Physical Review Letters 1998, 80, 1560–1563. (50) Rouzina, I.; Bloomfield, V. A. Macroion attraction due to electrostatic correlation between



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

screening counterions .1. Mobile surface-adsorbed ions and diffuse ion cloud. Journal of Physical Chemistry 1996, 100, 9977–9989. (51) Carrillo, J.-M. Y.; Dobrynin, A. V. Molecular dynamics simulations of polyelectrolyte adsorption. Langmuir 2007, 23, 2472–2482. (52) Turesson, M.; Woodward, C. E.; Akesson, T.; Forsman, J. Simulations of surface forces in polyelectrolyte solutions. Journal of Physical Chemistry B 2008, 112, 5116–5125. (53) Shi, A. C.; Noolandi, J. Theory of inhomogeneous weakly charged polyelectrolytes. Macromolecular Theory and Simulations 1999, 8, 214–229. (54) Podgornik, R. Two-body polyelectrolyte-mediated bridging interactions. Journal of Chemical Physics 2003, 118, 11286–11296. (55) Surve, M.; Pryamitsyn, V.; Ganesan, V. Universality in structure and elasticity of polymernanoparticle gels. Physical Review Letters 2006, 96, 177805. (56) Podgornik, R.; Saslow, W. M. Long-range many-body polyelectrolyte bridging interactions. Journal of Chemical Physics 2005, 122, 204902. (57) Surve, M.; Pryamitsyn, V.; Ganesan, V. Depletion and pair interactions of proteins in polymer solutions. The Journal of Chemical Physics 2005, 122, 154901. (58) Belloni, L.; Ferreira, P. G. Interactions between charged colloids in semi-dilute polyelectrolyte. Philosophical Transactions of the Royal Society of London Series A-mathematical Physical and Engineering Sciences 2001, 359, 867–877. (59) Forsman, J. Density functional theories of surface interactions in salt solutions. Journal of Chemical Physics 2009, 130, 064901. (60) Forsman, J.; Nordholm, S. Polyelectrolyte Mediated Interactions in Colloidal Dispersions: Hierarchical Screening, Simulations, and a New Classical Density Functional Theory. Langmuir 2012, 28, 4069–4079. 38 ACS Paragon Plus Environment

Page 38 of 45

Page 39 of 45

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(61) Sing, C. E.; Zwanilden, J. W.; de la Cruz, M. O. Ion Correlation-Induced Phase Separation in Polyelectrolyte Blends. Acs Macro Letters 2013, 2, 1042–1046. (62) Dahlgren, M. A. G.; Waltermo, A.; BLlomberg, E.; Claesson, P. M.; Sjostrom, L.; Akesson, T.; Jonsson, B. Salt Effects On the Interaction Between Adsorbed Cationic Polyelectrolyte Layers - Theory and Experiment. Journal of Physical Chemistry 1993, 97, 11769–11775. (63) Pandav, G.; Ganesan, V. Computer Simulations of Dendrimer-Polyelectrolyte Complexes. Journal Of Polymer Science Part B-Polymer Physics 2014, 118, 10297–10310. (64) Pandav, G.; Pryamitsyn, V.; Ganesan, V. Interactions and Aggregation of Charged Nanoparticles in Uncharged Polymer Solutions. Submitted 2015, (65) Fredrickson, G. H. The Equilibrium Theory of Inhomogenous Polymers; CLARENDON PRESS, OXFORD, 2005. (66) Barker, J. A.; Henderson, D. Perturbation Theory and Equation of State for Fluids: The Square - Well Potential. The Journal of Chemical Physics 1967, 47, 2856–2861. (67) Barker, J. A.; Henderson, D. Perturbation Theory and Equation of State For Fluids .2. A Successful T heory of Liquids. Journal of Chemical Physics 1967, 47, 4714–&. (68) Rascon, C.; Mederos, L.; Navascues, G. Theoretical approach to the correlations of a classical crystal. Physical Review E 1996, 54, 1261–1264. (69) Velasco, E.; Mederos, L.; Navascues, G. Analytical approach to the thermodynamics and density distribution of crystalline phases of hard spheres. Molecular Physics 1999, 97, 1273–1277. (70) Rascon, C.; Velasco, E.; Mederos, L.; Navascues, G. Phase diagrams of systems of particles interacting via repulsive potentials. Journal of Chemical Physics 1997, 106, 6689–6697.



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

(71) Velasco, E.; Mederos, L.; Navascues, G.; Hemmer, P. C.; Stell, G. Complex phase behavior induced by repulsive interactions. Physical Review Letters 2000, 85, 122–125. (72) Mederos, L.; Navascues, G.; Velasco, E. Self-consistent nonperturbative theory for classical systems. Physical Review E 2002, 65, 016131. (73) Yu, Y.-X.; Wu, J. A fundamental-measure theory for inhomogeneous associating fluids. Journal of Chemical Physics 2002, 116, 7094. (74) Navascues, G.; Velasco, E.; Mederos, L. Theories of phase behavior of colloidal suspensions. International Journal of Modern Physics B 2004, 18, 143–169. (75) Paricaud, P.; Varga, S.; Jackson, G. Study of the demixing transition in model athermal mixtures of colloids and flexible self-excluding polymers using the thermodynamic perturbation theory of Wertheim. Journal Of Chemical Physics 2003, 118, 8525–8536. (76) Serrano-Illan, J.; Navascues, G.; Velasco, E.; Mederos, L. Self-consistent nonperturbative theory: Treatment of colloidal-type interactions. Journal of Chemical Physics 2003, 119, 1510–1516. (77) Surve, M.; Pryamitsyn, V.; Ganesan, V. Nanoparticles in solutions of adsorbing polymers: Pair interactions, percolation, and phase behavior. Langmuir 2006, 22, 969–981. (78) Pryamitsyn, V.; Ganesan, V. Effect of confinement on polymer-induced depletion interactions between nanoparticles. Journal of Chemical Physics 2013, 138, 234905. (79) William G. McMillan, J.; Mayer, J. E. The Statistical Thermodynamics of Multicomponent Systems. The Journal of Chemical Physics 1945, 13, 276–305. (80) Dijkstra, M.; van Roij, R.; Evans, R. Phase behavior and structure of binary hard-sphere mixtures. Physical Review Letters 1998, 81, 2268–2271. (81) Surve, M.; Pryamitsyn, V.; Ganesan, V. Dispersion and percolation transitions of nanorods in polymer solutions. Macromolecules 2007, 40, 344–354. 40 ACS Paragon Plus Environment

Page 40 of 45

Page 41 of 45

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(82) Wang, Q.; Taniguchi, T.; Fredrickson, G. H. Self-consistent field theory of polyelectrolyte systems. Journal of Physical Chemistry B 2004, 108, 6733–6744. (83) Muller, M.; Smith, G. D. Phase separation in binary mixtures containing polymers: A quantitative comparison of single-chain-in-mean-field simulations and computer simulations of the corresponding multichain systems. Journal Of Polymer Science Part B-Polymer Physics 2005, 43, 934–958. (84) Daoulas, K. C.; Muller, M. Single chain in mean field simulations: Quasi-instantaneous field approximation and quantitative comparison with Monte Carlo simulations. Journal Of Chemical Physics 2006, 125, 184904. (85) Daoulas, K. C.; Muller, M.; Stoykovich, M. P.; Kang, H.; de Pablo, J. J.; Nealey, P. F. Directed copolymer assembly on chemical substrate patterns: A phenomenological and singlechain-in-mean-field simulations study of the influence of roughness in the substrate pattern. Langmuir 2008, 24, 1284–1295. (86) Wang, L.; Broadbelt, L. J. Explicit Sequence of Styrene/Methyl Methacrylate Gradient Copolymers Synthesized by Forced Gradient Copolymerization with Nitroxide-Mediated Controlled Radical Polymerization. Macromolecules 2009, 42, 7961–7968. (87) Wang, L.; Broadbelt, L. J. Factors Affecting the Formation of the Monomer Sequence along Styrene/Methyl Methacrylate Gradient Copolymer Chains. Macromolecules 2009, 42, 8118–8128. (88) Wang, J. F.; Muller, M. Memory Effects of Diblock Copolymer Brushes and Mixed Brushes. Langmuir 2010, 26, 1291–1303. (89) Steinmuller, B.; Muller, M.; Hambrecht, K. R.; Smith, G. D.; Bedrov, D. Properties of Random Block Copolymer Morphologies: Molecular Dynamics and Single-Chain-in-MeanField Simulations. Macromolecules 2012, 45, 1107–1117.



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

(90) Sevick, E. M.; Monson, P. A.; Ottino, J. M. Monte-Carlo Calculations Of Cluster Statistics In Continuum Models Of Composite Morphology Rid B-6682-2009. Journal Of Chemical Physics 1988, 88, 1198–1206. (91) Mossa, S.; Sciortino, F.; Tartaglia, P.; Zaccarelli, E. Ground-State Clusters for Short-Range Attractive and Long-Range Repulsive Potentials. Langmuir 2004, 20, 10756–10763. (92) Campbell, A. I.; Anderson, V. J.; van Duijneveldt, J. S.; Bartlett, P. Dynamical Arrest in Attractive Colloids: The Effect of Long-Range Repulsion. Phys. Rev. Lett. 2005, 94, 208301. ´ nÂRcore ˇ (93) Hagen, M.; Frenkel, D. Determination of phase diagrams for the hardÃ´câCˇ attractive Yukawa system. Journal of Chemical Physics 1994, 101, 4093–4097. (94) Rotenberg, B.; Dzubiella, J.; Hansen, J.; Louis, A. Thermodynamic perturbation theory of the phase behaviour of colloid/interacting polymer mixtures. Molecular Physics 2004, 102, 1–11. (95) Dijkstra, M.; van Roij, R.; Evans, R. Phase diagram of highly asymmetric binary hardsphere mixtures. Physical Review E 1999, 59, 5744–5771. (96) Poon, W. C. K.; Haw, M. D. Mesoscopic structure formation in colloidal aggregation and gelation. Adv. Colloid Interface Sci. 1997, 73, 71–126. (97) Donati, C.; Glotzer, S. C.; Poole, P. H.; Kob, W.; Plimpton, S. J. Spatial correlations of mobility and immobility in a glass-forming Lennard-Jones liquid. Physical Review E 1999, 60, 3107–3119. (98) Bergenholtz, J.; Poon, W. C. K.; Fuchs, M. Gelation in model colloid-polymer mixtures. Langmuir 2003, 19, 4493–4503. (99) Puertas, A. M.; Fuchs, M.; Cates, M. E. Competition between glass transition and liquid-gas phase separation in attracting colloids. J. Phys. Condens. Matter 2007, 19, 205140–.


Page 42 of 45

Page 43 of 45

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(100) Kob, W.; Roldan-Vargas, S.; Berthier, L. Non-monotonic temperature evolution of dynamic correlations in glass-forming liquids. Nature Physics 2012, 8, 164–167. (101) Berthier, L.; Kob, W. Static point-to-set correlations in glass-forming liquids. Physical Review E 2012, 85, 011102. (102) Matia-Merino, L.; Lau, K.; Dickinson, E. Effects of low-methoxyl amidated pectin and ionic calcium on rheology and microstructure of acid-induced sodium caseinate gels. Food Hydrocolloids 2004, 18, 271–281. (103) Errington, J. Direct calculation of liquid-vapor phase equilibria from transition matrix Monte Carlo simulation. Journal of Chemical Physics 2003, 118, 9915–9925. (104) Lee, J.; Popov, Y.; Fredrickson, G. Complex coacervation: a field theoretic simulation study of polyelectrolyte complexation. The Journal of chemical physics 2008, 128. (105) Riggleman, R. A.; Kumar, R.; Fredrickson, G. H. Investigation of the interfacial tension of complex coacervates using field-theoretic simulations. Journal Of Chemical Physics 2012, 136, 024903. (106) Godfrin, P. D.; Castaneda-Priego, R.; Liu, Y.; Wagner, N. J. Intermediate range order and structure in colloidal dispersions with competing interactions. Journal of Chemical Physics 2013, 139, 154904. (107) Stauffer, D. Scaling Theory of Percolation Clusters. Physics Reports-review Section of Physics Letters 1979, 54, 1–74. (108) Chen, S. H.; Rouch, J.; Sciortino, F.; Tartaglia, P. Static and dynamic properties of water-inoil microemulsions near the critical and percolation points. Journal of Physics: Condensed Matter 1994, 6, 10855. (109) Jadrich, R. B.; Bollinger, J. A.; Johnston, K. P.; Truskett, T. M. Origin and detection of



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

microstructural clustering in fluids with spatial-range competitive interactions. Physical Review E 2015, 91. (110) Verlet, L. Computer Experiments On Classical Fluids .i. Thermodynamical Properties of Lennard-jones Molecules. Physical Review 1967, 159, 98–&. (111) Choi, Y. M.; Ree, T.; Ree, F. H. Hard-sphere Radial-distribution Functions For Facecentered Cubic and Hexagonal Close-packed Phases - Representation and Use In A Solidstate Perturbation-theory. Journal of Chemical Physics 1991, 95, 7548–7561. (112) Hemmer, P. C.; Velasco, E.; Mederos, L.; Navascues, G.; Stell, G. Solid-solid transitions induced by repulsive interactions. Journal of Chemical Physics 2001, 114, 2268–2275. (113) Jackson, G.; Swol, F. V. Perturbation-theory of A Model Hcp Solid. Molecular Physics 1988, 65, 161–173. (114) Kyrlidis, A.; Brown, R. A. Model For Density-functional Thermodynamic Perturbation Analysis of Lennard-jones Solids. Physical Review E 1993, 47, 427–438. (115) Carnahan, N. F.; Starling, K. E. Equation of State For Nonattracting Rigid Spheres. Journal of Chemical Physics 1969, 51, 635–656. (116) Trokhymchuk, A.; Nezbeda, I.; Jirsak, J.; Henderson, D. Hard-sphere radial distribution function again. The Journal of Chemical Physics 2005, 123, 024501.


Page 44 of 45

Page 45 of 45

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


PE-bridged aggregates Direct contact particle aggregates

For Table of Contents use only. “Multibody Interactions, Phase Behavior and Clustering in Nanoparticle-Polyelectrolyte Mixtures” Gunja Pandav, Victor Pryamitsyn, Jeffrey Errington and Venkat Ganesan


Multibody Interactions, Phase Behavior, and Clustering in

Recommend Documents