Elucidating the Effects of Metal Complexation on...

July 24, 2018Volume 51

Number 14

MacromoleculesMacromolecules

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Elucidating the Effects of Metal Complexation on Morphological andRheological Properties of Polymer Solutions by a Dissipative ParticleDynamics ModelKolattukudy P. Santo, Aleksey Vishnyakov, Ravish Kumar, and Alexander V. Neimark*

Department of Chemical and Biochemical Engineering, Rutgers, The State University of New Jersey, Piscataway, New Jersey 08854,United States

*S Supporting Information

ABSTRACT: When a salt is added to a polymer solution, metalcations may coordinate with polymer ligands forming interchainand intrachain links. Metal coordination leads to drastic changesof polymer morphology, formation of clusters, and, ultimately, asol−gel transition that affect the solution rheology. Althoughmetal coordination is ubiquitous in polymeric systems, thephysical mechanisms of coordination-induced morphologicaland rheological changes are still poorly understood due to themultiscale nature of this phenomenon. Here, we propose acoarse-grained dissipative particle dynamics (DPD) model tostudy morphological and rheological properties of concentrated solutions of polymers in the presence of multivalent cations thatcan coordinate the polymer ligands. The coordinating metal is introduced as a 3D complex of planar, tetrahedral, or octahedralgeometry with the central DPD bead representing the metal cation surrounded at the vertices by either four or six dummy beadsrepresenting coordination sites, some of which are occupied by counterions to provide electroneutrality of the complex.Coordination is modeled as the dynamic formation and dissociation of a reversible link between the vacant coordination siteand a ligand described by the Morse potential. The proposed model is applied to study the specifics of the equilibriummorphology and shearing flow in polyvinylpyrrolidone−dimethylformamide solutions in the presence of metal chlorides.Coordination leads to interchain and intrachain cross-links as well as to metal cations grafted onto polymer chains by a singlelink. The interchain cross-links induce a sol−gel transition to a weak gel phase as the metal concentration increases. Because ofthe reversible nature of interchain cross-links, the weak gel phase behaves as a viscoelastic fluid, the viscosity of which graduallyincreases with the metal concentration and decreases as the shear rate increases. The change of viscosity due to interchaincoordination cross-links scales with the interchain cross-link density and the metal concentration according to the power lawwith the exponent ν ≈ 1.15. The influence of the grafted metal atoms on the viscosity is found to be much weaker, while theeffect of the intrachain cross-links is found to be negligible. The simulation results are in qualitative agreement with availableliterature data. The proposed DPD model provides a physical insight into the morphological features of polymer solutions in thepresence of multivalent slats and can be extended to other coordinating systems such as metal-substituted polyelectrolytes.

I. INTRODUCTIONMaterials containing metal−polymer complexes possessintriguing mechanical and catalytic properties. Their potentialapplications include ion-conducting solid membranes used inbatteries,1 semiconducting materials,2 pharmacological appli-cations such as antibacterial3 and anti-inflammatory drugs,4

membranes for artificial organs,5 polymer catalysts,6 materialswith tunable mechanical strength,7 and wastewater manage-ment.8 These materials contain metal atoms, which formreversible coordination bonds with polymer ligands ( e.g.carboxyl, hydroxyl or amide groups on polymer ), cross-linkingthe polymer chains and affecting the the system morphologyand rheology. The qualitative difference between cross-linkingby covalent and coordination bonds is that the latter areweaker and, in many cases, dissociate spontaneously or underexternal factors like shear stress. In this work, we suggest acoarse-grained simulation model of the dynamics of reversible

complexation in polymer solutions aiming at predicting thesystem morphological and rheological properties drawing onan example of concentrated polyvinylpyrrolidone (PVP)−dimethylformamide (DMF) solutions doped with nondisso-ciated metal chlorides. This system was studied experimen-tally,15 which provides an opportunity to correlate thesimulation and experimental data.The influence of coordination on the structure and rheology

of polymer solutions has been extensively studied in theliterature.9−20 If a metal atom makes only single coordinationbond with one chain, the charge on the ion grafted to the chaincan transform a neutral polymer into a polyelectrolyte. Theelectrostatic/steric repulsion between the chains caused by the

Received: March 8, 2018Revised: May 19, 2018

Article

Cite This: Macromolecules XXXX, XXX, XXX−XXX

© XXXX American Chemical Society A DOI: 10.1021/acs.macromol.8b00493Macromolecules XXXX, XXX, XXX−XXX

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grafted coordination links (G-links) induces conformationalchanges leading to chain expansion. When a single metal atomcoordinates several polymer ligands, either intrachain (chela-tion) or interchain cross-links occur. Intrachain cross-links (C-links), which connect different fragments of the same chain,cause its contraction, while interchain cross-links (X-links)connecting different polymer chains lead to clustering andgelation, as the polymer and/or metal concentration increases.The specifics of the resulting sol−gel transition were widelyreported in the literature.9,11−13,21,22 Rheological properties ofpolymer solutions, such as viscosity, are strongly influenced bycoordination between the metal atoms and the monomers ofthe polymer chains that serve as ligands.9,10,13,14,19,23−26 Thechain expansion enhances entanglement effects in semidiluteand concentrated polymer solutions and contribute to viscosityincrease. A dramatic increase in viscosity may be caused by X-links that effectively increase the molecular weight and lead tothe formation of elastic networks of dissociable bonds referredto as weak gels.27−29 On the other hand, chain contractioncaused by C-links reduces entanglements and decreases theviscosity, especially at lower polymer concentrations.A number of experimental studies15,19,26,30 specifically

targeted PVP, or Povidone, a water-soluble polymer, themonomer of which consists of a γ-lactam ring (Figure 1) thatincludes carboxyl and amide groups. PVP is extensively used inpharmaceutical and cosmetic products31 and as a modelcompound for proteins, especially in studies of metal-loenzymes. Metal atoms coordinate with carboxyl oxygens ofthe monomers, and this causes change in material propertiessuch as viscosity that varies with the metal concentration. Haoet al.15 performed viscosimetric and spectroscopic studies ofPVP solution in DMF with dissolved metal chlorides (MCln),such as chlorides of calcium, cobalt, and lithium, at differentconcentrations. They found that the viscosity of PVP−DMF−MCln solutions depends heavily on the concentration of themetal chloride as well as on the type of metal chloride. Theauthors demonstrated using NMR and FTIR measurementsthat metal atom interacts with the carbonyl oxygen of thepolymer and, based on the data, hypothesized possiblecoordination with the nitrogen, confirming the role ofcomplexation.

Simulation techniques available for modeling polymer−metal complexation are insufficiently developed. Generally,such dynamic simulation should account for dissociation andformation of coordination bonds between metal atoms andappropriate ligands. Treating chemical reactions in the courseof molecular dynamics simulation is computationally ex-pensive, even when a classical reactive force field is available.At the same time, predicting dynamic properties and,especially, rheology of nanostructured polymeric systemstypically requires spatial and temporal scales not accessiblewith reactive force fields. This gap can be bridged withsimplified modeling approaches that mimic chemical reactionsin atomistic32 or coarse-grained simulations.33,34 Existingtheoretical models are mostly limited to dilute solutions.10

The simplest model of complexation used in the literature isthe nonbonded model,35 in which the metal atom interacts withthe ligands simply by van der Waals and Coulomb interactions.This model cannot take into account the number and spatialarrangement of coordinating bonds specific to the particularmetal, as the ligands tend to close-pack around the metal atom.In the bonded model,36 the metal atom is covalently bonded toligands, and this preserves the coordination number andcomplex geometry. This model is applicable to stable metal−ligand complexes but cannot be used in the systems whereligand dissociation and exchange may occur. The semibondedmodel introduced by Pang,37,38 called the cationic dummy atomapproach, incorporates advantages of both bonded andnonbonded model. It has dummy atoms placed around thecentral metal atom that are covalently bonded to it and allowsfor the dummy atoms to interact with the ligands throughpurely electrostatic potentials. The ion charge is equallydistributed on the dummy atoms, the van der Waalsparameters of which are set to zero, and the central atomhas only van der Waals interactions with other atoms. It isworth noting that the bonded approach is applied to themodels of associating polymers that do not specifically targetcomplexation.39

However, even this level of simplification may not besufficient for atomistic simulations of rheological properties,which require spatial and temporal scales that are onlyaccessible on a coarse-grained level, where individual atomsand molecules are lumped together and are presented by

Figure 1. Coarse-grained models for (a) PVP monomer and (b) DMF solvent. (c−e) Models for metal chlorides (MClpSq) with tetrahedral (upperrow), octahedral (middle row), and planar (bottom row) geometries. The metal atom (blue) is in the center. The chlorine atoms (C), shown ingreen, occupy some of the coordination sites (S). The sites available for coordination are shown in white. The geometry is kept rigid by strong M−S and S−S bonds (not shown).

Macromolecules Article

DOI: 10.1021/acs.macromol.8b00493Macromolecules XXXX, XXX, XXX−XXX

B



quasiparticles or beads. In particular, dissipative particledynamics (DPD), a coarse-grained method of superbcomputational efficiency,40,41 was shown to be practical insimulations of dissociation reactions in polymers. Lisal etal.33,34 considered polymerization in polymer melts using DPDsimulations. Protonation was included via Monte Carlo (MC)-like steps that involve bond formation and dissociation.Formation and breakup of temporary bonds were consideredalso by Karimi-Varzaneh et al.42 with a purpose of mimickingchain entanglements in polymer solutions and melts. In ourrecent work,43 protonation−deprotonation equilibrium wasmodeled by introducing proton as a special “P bead” that couldform dissociable Morse bonds with bases (proton-acceptingbeads). This model was applied to dilute acid solutions,sulfonated polystyrene, and Nafion.43−45

In this work, we incorporate coordination reactions intoDPD by presenting the coordinating metal ion as a semirigid3D complex that consists of a central “metal cation” beadsurrounded by the dummy beads representing coordinationsites, some of which are occupied by counterions (Figure 1).By applying bond and angle potentials between the centralbead and coordination dummy beads, the complex is kept in aparticular spatial arrangement that reproduces the geometryplanar, tetrahedral, or octahedralof the given metal complex.Unlike other beads, the vacant dummy beads are not involvedin any nonbonded interactions but are able to form dissociableMorse bonds with ligand beads, which represent functionalgroups of polymer or solvent that are able to coordinate withthe metal atom. To demonstrate the capabilities of theproposed model, we attempt to simulate morphological andrheological properties of concentrated polyvinylpyrrolidone(PVP)−dimethylformamide (DMF) solutions doped withnondissociated metal chlorides that can form reversiblecoordination bonds with the chain monomers and solventmolecules that both serve as coordinating ligands. The systemchoice is dictated by availability of experimental data15 forqualitative comparison with the simulation results.The paper is structured as follows: In section II, we describe

the computational methodology, details of the DPD simulationscheme, parametrization strategy, models for metal complexesand coordination interactions, and the Lees−Edwardsapproach to calculate viscosity in shear flow. In section III,we discuss the simulation results that includes analysis ofequilibrium morphological properties of metal-complexedPVP−DMF solutions, distinguishing different types of intra-chain and interchain cross-links, analyzing sol−gel transitiondue to interchain reversible cross-linking. Special attention ispaid to the rheological behavior of polymer solutions withdifferent degrees of coordination and the relationships betweenthe viscosity, the shear rate, salt concentration, and the amountof interchain cross-links. Conclusions are summarized andcritically discussed in section IV.

II. MODELS AND METHODSDPD Simulations. In DPD simulations,40,41 groups of atoms are

represented as beads that follow Newton’s equations of motion,interacting through different pairwise forces:

∑= + + + + +≠

F F F F F Ffii j

ij ij ij ij ij ijC B A D R M

(1)

FijC is the short-range conservative repulsion force between beads i and

j traditional in DPD simulations which has the form

= − ( )r aF r( ) 1ij ij ijr

R ijC ij

c, where Rc is the effective bead diameter and

aij the repulsion parameter. FijC(rij) = 0 for rij ≥ Rc. Fij

B(rij) and FijA(rij)

are the bond and the angular forces, respectively. All permanentbonds and angles between the beads are harmonic. Bonds and anglesmaintain the connectivity of the polymer chain within the segmentsand between the segments and also maintain the geometry of metalcomplexes. The drag force Fij

D and the random force FijR constitute a

Langevin thermostat and are given in terms of friction coefficient γand a Gaussian random noise θij as

γ σ θ= − · = r w wF r v r F r( ) ( ) ;ij ij ij ij ij ij ij ijD D R R

(2)

wD and wR are weight functions related by wD = (wR)2 with

= −( )w r( ) 1ijr

RR ij

cfor rij ≤ Rc and zero for rij > Rc. σ

2 = 2γkBT. The

drag and the random forces are applied to all beads except thevacancies (for which γ is set zero) with the dissipation parameter γ =4.5, previously applied in simulations with similar intracomponentrepulsion parameter aii.

40 Finally, FijMis the Morse force responsible for

the coordination interactions described below.Polymer and Solvent Models. We apply the most common

implementation of DPD, where the effective bead diameter Rc andintracomponent short-range repulsion parameter aii are equal for allbead types representing polymer (PVP) and solvent (DMF)fragments. The dissection of PVP and DMF into beads is shown inFigure 1a,b. The beads represent the fragments of approximatelyequal volumes, which are calculated from effective Bondi volumes offunctional groups used in group-contribution activity coefficientmodels.46,47 PVP monomer is described by a coarse-grained structureconsisting of the three beads P, V, and N (Figure 1a), representing theatom groups −CH2−CNH−, −CH2−CO−, and −CH2−CH2−,respectively. N beads are linked to each other to form the polymerchain while the P beads contain the carbonyl oxygen that makescoordination with the metal. The solvent DMF (Figure 1b) isdissected into two beads D and E representing atom groups (H)N−CHO and CH2−CH3, respectively; the D beads also have the oxygensthat can form coordination bonds. Assuming the ratio vk = 41.56 Å3

between the actual molar volume and the effective group volumeparameter, we obtain the average volume of the N, P, V, D, and Ebeads of 61.0 Å3, which corresponds to Rc = 0.568 nm. Theintracomponent repulsion parameter aii = 78.5kBT/Rc which matchesthe compressibility of DMF, is taken equal for all bead types. SincePVP is soluble in DMF, the intercomponent parameters aij would becloser, and as we here intend only to construct a generic model, wetake aij = 78.5kBT/Rc for the polymer and solvent beads. The bondlengths of N−N, N−P, V−P, and N−V bonds are taken to be 0.51Rc,and that of the D−E bond is 0.39Rc, estimated from the radialdistribution functions of the atom groups obtained from atomisticMD simulations. The details of the calculation of the DPD parametersare given in the Supporting Information, section I.

Coordination Metal Complexes. The metal atoms that arecapable of making coordination bonds are modeled in the spirit of thecationic dummy atom model.37,38As shown in Figure 1c, the metalcomplex is modeled as the central metal cation presented by M bead(blue) surrounded by coordination sites, which are either occupied bycounterions (C beads drawn in green in Figure 1) or vacant andavailable for coordination with ligands. The coordination vacanciesare represented as “dummy” S beads while the complex beads areconnected by harmonic bonds and angles to preserve coordinationgeometry specific to the particular metal salt. The coordinationvacancies (S beads) interact with ligands, which are either the beadsthat represent polymer ligands (for example, P beads) or the solventD beads through a particular coordination potential discussed below.We consider three different geometries of coordinating metalcomplexes: tetrahedral, octahedral, and planar as shown in Figure1c, with the metal atom at the center and the coordination sites at thevertices. An obvious limitation of the model is its inability to changethe geometry of the complex depending on the ligands and theenvironment. In reality, the same metal ions are sometimes capable offorming complexes of different morphology and different coordina-tion number (for example, Cu2+ may form tetrahedral or octahedral



C

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complexes depending on the environment). In this work, we limitourselves to modeling neutral metal complexes like chlorides, whichdo not dissociate in DMFan aprotic solvent that interacts weaklywith anions. We assume that in the metal salt complex a certainnumber (denoted as “p”) of vacancies are permanently occupied bychloride ion ligands (C beads) which neutralized the metal cation andcannot be replaced by other ligands (Figure 1). The complexstructures are denoted here as MClpSq with p being the number ofcounterions and q the number of vacancies (vacancy symbols S maybe replaced by ligand symbols upon coordination). For example, if themetal atom has a coordination number of 6 (octahedral geometry)and charge of +3 (this is typical, for example, for aluminum cation, p =3), three positions in its octahedral coordination shell will bepermanently occupied by chloride anion ligands, and three other (q =3) vacancies may be filled with either polymer or solvent ligands; theformula is therefore MCl3S3. Altogether, we consider nine differenttypes of metal complexes which differ by geometry and the number ofavailable coordination vacancies shown in Figure 1: metal complexeswith four coordinating sites of planar and tetrahedral geometry andmetals complexes with six coordinating sites of octahedral geometry,which contain from one to three chlorine ions depending on the metalvalency to secure electroneutrality. If all positions in the coordinationshell but one are permanently occupied by counterions, the metalcomplex may be grafted to polymer forming a side fragment; if two ormore vacancies are available, intrachain (chelation) or interchaincross-linking may occur.The parameters for interactions of metal M and counterion C

beads are chosen somewhat arbitrarily, since we mostly target thequalitative influence of coordination on polymer solution properties.Because the salts do not dissociate in DMF and the metal bead ispermanently linked to the chloride beads with harmonic bonds, eachcomplex is always neutral and only possesses quadrupole and dipolemoments; these interactions may influence the conformations.However, the exact conformations of molecules at scales below theeffective bead diameter are well beyond the capability of coarse-grained simulations, and it would require the entire toolset of classicalMD to reproduce the conformation on the molecular level. Toestimate the general magnitude of electrostatic forces in the system atlonger distances, we calculated the Boltzmann-weighted angle averageof interactions between two quadrupoles Q in a dielectric solvent:48

π=

ϵ ϵE r

Qk T r

( )14

5 (4 )QQ(eff)

4

B 02 10

Assuming that the chlorine beads bear −e charge, the metal−liganddistance of 0.18 nm typical for such compounds, and ϵ = 4.33 (DMFat room temperature), we obtain EQQ

(eff)(r) = 3.4 × 10−113/r10. At r ≈0.55−0.6 nm, EQQ

(eff) becomes comparable with the energy changesassociated with typical torsion angle rotations, which are ignored incoarse-grained simulations. Although it is possible to treat the

electrostatic interactions explicitly, we assume that due to their short-range nature, they can be effectively included in the short-rangerepulsion. Metal atoms are modeled as small beads with diameterabout 0.22 nm (0.4Rc) that create strong repulsion with all otherbeads to avoid overlap between the complexes. The repulsionparameters of C beads are also set larger (Table 1). The interactionsof M and C with the ligands are cut off at distances shorter than M/C−S bond to ensure that the short-range repulsion does not preventcoordination. The M−S bond length is 0.4Rc in tetrahedral andoctahedral geometries, but it is 0.5Rc in the planar ones. The M/S−S/C bond length and angle parameters corresponding to each geometryare given in Table 1. The vacant coordination sites S do not interactwith other beads, aSj = 0; however, the sites interact with the ligandbeads (P or D) via the coordination potential described below.

Coordination Potential. Coordination between metal complexvacancies and ligand beads is modeled as a truncated, smoothed, andshifted Morse potential, similar to the model for dissociable bonds.43

The Morse potential of well depth K has the following form:

ϕ = −α α− − − −r r K( , ) (e 2e )r r r r0

2 ( ) ( )0 0 (3)

where r0 corresponds to the location of the minimum of ϕ and αdetermines the curvature of the well. The coordination interaction isvery short-ranged and therefore has to be truncated at thecoordination cutoff distance rc. To avoid discontinuity of the forceat r = rc, the truncated potential ϕ(r,r0) − ϕ(rc,r0) is complementedwith a linear term, −(r − rc)ϕ′(rc,r0). This alternation, however,affects the equilibrium position of the potential. To restore thelocation of the potential minimum to r0, we shift the Morse potentialby a constant value −rs. The final expression for the Morse-typecoordination potential is

ϕ ϕ ϕ= − − − − − ′ −U r r r r r r r r r r r r( ) ( , ) ( , ) ( ) ( , )c 0 s c 0 s c c 0 s

(4)

From the condition of the minimum, Uc′(r0) = 0, we can determine rs:

α= −

−

α

α

− −

− −r1

log1 e1 e

r r

r rs

2 ( )

( )

c 0

c 0

i

kjjjjj

y

{zzzzz

(5)

In our model, the potential minimum is at the coordination site, r0 =0, so that the coordinated ligand bead occupies exactly thecoordination vacancy (the energy minimum is reached when thecenters of S bead and ligand bead coincide).

In all systems studied, the coordination potential is appliedbetween vacant coordination site S beads and polymer ligand P beadsof PVP oligomers. In select systems, we also applied the coordinationpotential between S beads and solvent ligand D beads of DMF, thusmaking the polymer and solvent compete for the vacancies in thecoordination shell. The value of the Morse parameter is chosen to be

Table 1. DPD Parameters for PVP, DMF, and Metal Complexes

short-range repulsion, aij(kBT/Rc)

P V N D E M C S

M 200 200 200 200 400 400 200 0.0C 200 200 200 200 200 200 200 0.0S 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

bonds

bond r0(Rc) Kb(kBT/Rc2)

N−N, N−P, N−V, P−V 0.51 320D−E 0.39 440M−S, M−C 0.4 (tetrahedral and octahedral), 0.5 (planar) 320S/C−S/C 0.66 (tetrahedral), 0.707 (planar), 0.56 (octahedral)

angles

angle θ0 (deg) Kθ(kBT)

S/C−M−S/C (S/C adjacent) 90 (planar and octahedral), 109 (tetrahedral) 20S/C−M−S/C (S/C diagonal) 180 (planar and octahedral) 20



D


K = 60kBT, the value of rc is set to be 0.2Rc, and α = 10, which gives rs= 0.0127Rc and a well depth of Uw = −20.05kBT.Simulations. We consider PVP−DMF−MClpSq systems with box

dimensions of approximately 30 × 30 × 30 Rc3 containing 81 000

particles at a density of ρ = 3Rc−3. The polymer chain length is nseg =

60 and the total number of polymers, Npol = 270, giving the totalnumber of segments to be Nseg = 16 200. The mass fraction of thepolymer is 0.6 with the number of DMF solvent molecules also equalto nsol = 16 200. The metal chloride is added to the system replacingequal number of solvent beads, at different concentrations with thetotal numbers varying in the range NM = 0−2000. Defining theconcentration of the metal atoms as CM = NM/Nseg, CM ∈ [0; 0.1235].Noteworthy, because of the use of different bead sizes, the

simulation is performed at the constant pressure conditions with theapplied pressure corresponding to that of PVP−DMF system withoutions at density ρ = 3Rc

−3. This approach was employed in simulationsof Kacar et al.49,50 and our previous simulations.43−45

The viscosity of the polymer−solvent−metal chloride system iscalculated according to the method of Lees and Edwards51 by creatingthe Couette flow in X-direction with linear velocity profile at a givenshear rate γ = dv/dy in the Y-direction (see Supporting Information,section III). The stress component in the XY plane is calculated bysumming over the per atom virial stress due to the “real” beads(excluding S beads)

∑ ∑σ = − +>V

mv v fr r

r1

xyi

ix iyj i

ijijx ijy

ij

i

k

jjjjjjjy

{

zzzzzzz (6)

where vix and viy are respectively are the x and y components of thevelocities of the ith atom while rijx and rijy are the x and y componentsof the vector rij = ri − rj and f ij is the total force between the atoms iand j. m is the mass of the beads, and V is the total volume of thesystem. The viscosity of the fluid is then given by

ησγ

=

xy

(7)

Simulations are performed using LAMMPS software.52 A typicallength of simulation is 1.8 million time steps with a time step value ofΔt = 0.01τ, including 1 million steps of equilibration, followed by800 000 steps of shear flow to measure viscosity. Structural andrheological properties are determined at varying the salt concentrationand shear rate for PVP−DMF solutions doped with nine types of saltcomplexes MClpSq of different coordination geometry and number ofvacancies in the coordination shell shown in Figure 1.

III. SIMULATION RESULTSSystem Morphology. Figure 2a shows a snapshot of the

equilibrated PVP−DMF solution containing dichloride of ametal with octahedral coordination MCl2S4 (see Figure 1).Two positions in the coordination shell are permanentlyoccupied with chloride anions, and four are available to formcoordination bonds with ligand beads P in polymer. Themorphological details of the equilibrated systems are analyzedbelow. The degree of coordination χ is defined as the fractionof coordination vacancies filled by P or D ligands. (A vacancyis considered as filled if the corresponding S bead interactswith at least one ligand bead via the coordination potential;although coordination with more than one ligand is notforbidden, such occasions are practically never observed due tothe short range of the coordination potential and conservativerepulsion between the ligands.) The fraction of filled vacanciesis defined as

χ =*N

N qL

M (8)

where NL* is total number of coordinated ligands and NMq =NS, the total number of available coordination sites in thesystem. χ depends on the depth of the coordination potentialUw, complex geometry, repulsion between the coordinated

Figure 2. Specifics of coordination in PVP−DMF−MClpSq solutions. (a) Typical snapshot of the equilibrated solution: (blue) polymer beads,(yellow) solvent beads, (red) M and Cl complex beads, example of octahedral coordination complex MCl2S4. (b−d) Dependence of the fraction ofcoordination sites occupied by the ligands in metal chloride solutions with tetrahedral (b), planar (c), and octahedral (d) coordination geometriesin the inert solvent (e) Average number of occupied coordination sites per metal complex nl as a function of the total number of available sites q. (f)Competitive coordination of tetrahedral MCl2S2 (dashed lines) and octahedral MCl3S3 (solid lines) complexes with polymer and solvent ligands.Red lines represent fraction of sites coordinated with polymer, green lines show fraction of sites coordinated with solvent, and blue lines representthe overall fraction of occupied vacancies.



E




ligands, chain flexibility, and overlap between the chains. InFigures 2b−d, χ is plotted against the salt concentrationexpressed as the number of metal atoms per monomer, CM =NM/Nseg, in MClpSq systems of different geometries, in whichmetal atoms coordinate with only polymer ligands (P beads),while the solvent is chemically inert. In general, χ decreasesvery slowly with CM, showing that in the concentration regimeconsidered the number of ions barely affects coordination. Atthe same time, χ decreases with the number of availablevacancies q, partly due to the repulsion between thecoordinated ligands which increases as the neighboring vacantsites gets filled and partly due to entropic spatial restrictionsthat the chains coordinating with the same metal impose oneach other. For systems with p + q = 4 (tetrahedral and square)χ generally remains high, and the influence of q remains weak.It becomes much more pronounced for the octahedralgeometry with q > 3 because of larger number of coordinatedligands causing more interligand repulsion and stronger spatialrestrictions associated with the polymer chains. This fact canalso be inferred from the variation of the average number ofligands per metal bead, nl = ⟨NL*⟩/NM, with q, where ⟨ ⟩represents the average over different salt concentrations,described in Figure 2e. nl is nearly proportional to q at q < 4and is independent of the coordination geometry (nodifference between planar and tetrahedral complexes isobserved). Octahedral complexes with higher q exhibitsaturation regime with a slower growths of occupied sitefraction due to spatial restrictions. However, it should be notedthat in the case of octahedral MCl2S4 and planar MCl2S2,placing the vacant sites at diagonally opposite vertices wouldlead to reduced interligand repulsion and may lead to increasein fraction of coordinated sites.Figure 2f illustrates the specifics of complexation in the

systems, where the metal coordinates with both solvent andpolymer ligands, showing the fraction of vacancies filled bypolymer P ligand beads of PVP (χP = NP*/NS) and solvent Dligand beads of DMF (χD = ND*/NS) as a function of the saltconcentration. Although the total fraction of filled vacancies (χ= χP + χD) remains constant and close to 1, the fraction of Pligands in the coordination shell increases monotonically withthe salt concentration at the expense of D ligands. In oursimulations the increase of salt concentration occurs at theexpense of the solvent molecules (the total volume of thesystem and the polymer fraction remains constant as CMincreases). That means that the solvent fraction somewhatdecreases with CM but the combinatorial factor cannot explainthe increase of the fraction of coordination sites filled by thepolymer as CM increases. For example, in the two systemsshown in Figure 2f, the octahedral MCl3S3 (solid lines) and thetetrahedral MCl2S2 (dashed lines), the fraction of D beads inthe entire pool of ligands decreases by 23.5% and 17.6%,respectively, as NM increases from 100 to 2000, while thefraction of D beads linked to the metal atoms decreases by 11%in both cases. We believe that the increase of χP at the expenseof χD is caused by the certain level of segregation that ispresent in our system (Figure 2a): an increase in the localpolymer concentration around a metal ion due to coordinationleads to solvent expulsion from the metal vicinity. Still theability of the solvent to be coordinated with the metal increasesthe overall coordination, which is higher in comparison withthe systems shown in Figures 2b and 2d where the solvent ischemically inert. The D-coordination enhances the total degreeof coordination χ by 5% in octahedral MCl3S3 and by 1% in

tetrahedral MCl2S2 systems (blue lines). Despite the shallowerdepth of the coordination potential, the entropic penalty forsolvent coordination is not as high as for the polymercoordination, since polymer chains are more difficult to bearranged around the metal atom and their coordination leadsto an additional entropy loss.Complexation affects morphological features of polymer

solutions such as chain conformations and clustering due toentanglement and cross-linking. To quantify the changes inpolymer morphology, we adopt a simplified classificationconsidering only three major types of complexes presented inFigure 3. The metal complexes that are linked to polymer

chains through just one coordination site are called graf tedcomplexes, or G-links, as shown in Figure 3c, where snapshotsof polymer chains with grafted metal complexes in tetrahedralMCl3S1 systems at different salt concentrations are given. If ametal atom is connected to multiple points on the same chain,making an intrachain cross-link (C-link) or chelation, thensuch complexes are described as chelate complexes or C-complexes, and if a metal coordinates with more than onechain, forming an interchain cross-link or X-link, then we denotesuch complexes as X-complexes. Representative snapshots of Xand chelate links in octahedral MCl2S4 systems are shownFigures 3a and 3b, respectively, X-links connecting threedifferent chains (colored differently) while C-links connect 3−4 points on the same chain.Now we define the number nij as the total number of metal

atoms in the system that are linked to i polymer ligandsbelonging to j connecting chains. Obviously, i ≥ j and both i, j≤ q. If j = 0, then i = 0, and n00 represents the number of metalatoms that are either uncoordinated (free) or coordinated onlyto solvent beads. i = j = 1 describes G-links, and n11 gives thetotal number of G-complexes in the system. If j = 1, then 2 ≤ i

Figure 3. Examples of different types of links formed by metalcomplexes. (a) X-links: interchain cross-links formed by octahedralMCl2S4 complexes connecting three chains shown in different colors.(b) C-links: intrachain or chelate cross-links formed by octahedralMCl2S4 complexes at CM = 0.006. (c) G-links: grafted tetrahedralMCl3S1 complexes at CM = 0.012 (left) and CM = 0.123 (right).Colors: the coordination sites, S beads colored in yellow and shownlarger for visibility since they are occupied by P beads, the chlorideions are shown in purple, and the central metal atom in pink. Green,red, blue, and gray are used for different polymers.



F



≤ q represents pure intrachain C-links. On the other hand, 2 ≤j ≤ q represents interchain X-links involving j chains. Ingeneral, if j > 1, then there are j X-linked sites and (i−j) C-linked sites per complex among the nij metal ions.We can define the fraction of X-linked sites per complex in

the system as

∑ ∑χ =≥ =qN

jn1

i j

q

j

q

ijXM 2 (9)

and the fraction of C-linked sites in the same fashion

∑χ = −= ≥qN

i j n1

( )i j i j

q

ijCM , 1, (10)

The fraction of G-links per ion is simply given as χG = n11/qNM, while the fraction of free ions is χ0 = n00/qNM (note thatχ0 is not the fraction of total uncoordinated sites as it does notinclude the sites of the linked metal beads that are free). Thesum χX + χC + χG = χ. Additionally, one can define the averagenumber of X-linked sites per complex as nX = qχX.Figure 4 presents the specifics of complexation and cross-

linking of PVP chains as a function of salt concentration.Figures 4a−e delineate cross-linking in PVP−DMFin thesystems where the metal atoms only coordinate with thepolymer while the solvent is inert. The average fractions of X-linked and G-linked sites per metal atom as a function of saltconcentration are shown in Figures 4a,b for planar andtetrahedral and octahedral geometries, respectively. Generally,cross-linking increases, and the fraction of G-complexesdecreases with q, simply because more vacancies becomeavailable to different chains. The trend is, however, differentfor q > 3 octahedral geometry because χ decreases with q due

to short-range conservative and entropic repulsion betweencoordinated ligands (discussed with Figure 2). The respectivefractions of intrachain C-links are illustrated in Figures 4c,d. χCdecreases with q in tetrahedral and planar systems (Figure 4c).In general, chelation is very pronounced: for example, for q =2, χC is substantially higher than χX, especially for planarMCl2S2, where the number of X-links is reduced at the expenseof chelation. This shows that the repulsion between thecoordinated ligands is not the major factor affecting χX, as itshould reduce C-links as well. That is, if a metal complexcoordinates with a ligand belonging to, say, polymer 1 via oneof its vacancies, the neighboring vacancy is likely to becoordinated with another monomer of the same chain ratherthan with a different chain. Chelation of a metal atom with thesame chain through more than two vacancies requires severespatial restrictions on the chain and almost never occurs. Forexample, for octahedral complexes χC remains almost constantas q increases from 3 to 5. The extra available vacanciescoordinate with other chains, and X-linking becomes more andmore pronounced.Additional factors affecting cross-linking can be construed

from Figures 4a−d. Both χX and χC are sensitive to the metalcomplex geometry; for example, χX is considerably lower (andχC is higher) in planar MCl2S2 systems compared to thetetrahedral ones (Figures 4a,c) because of a different geometryof the complexes: the distance between the adjacent vacanciesis somewhat longer in planar complexes (0.7Rc vs 0.66Rc,respectively), which are greater than the polymer segmentlength, 0.51Rc, and therefore the second site S2 of the ions canlink only with the second (or farther) nearest neighbors of theP bead that is linked to S1, which are 21/2 × 0.51Rc = 0.72Rcaway, assuming that the chain is flexible. Thus, the planar oneshave higher probability to reach out the neighboring P beads

Figure 4. Specifics of cross-linking in PVP−DMF−MClpSq solutions. (a−e) Systems with a chemically inert solvent. (a) The fractions of interchaincross-linked sites, χX (filled symbols), and grafted sites, χG (open symbols), per complex in systems having tetrahedral (solid lines with diamonds)and planar (dashed lines with squares) metal complexes as a function of salt concentration. (b) χX (filled) and χG (open) in the octahedral ionssystems. Colors: magenta, green, blue, and red respectively represent systems with q = 2, 3, 4, and 5. (c, d) The fraction of intrachain linked sitesper ion χC in different systems (same representation scheme). (e) Average number of interchain cross-linked sites per metal as a function of q. (f)Effects of competitive solvent coordination. Solid lines represent octahedral MCl3S3 systems and dashed lines, the tetrahedral MCl2S2 systems.Colors: green = χX, blue = χC, and red = χG.



G



than the tetrahedral ones having shorter S−S bond lengths.Note also that with q = 3 the octahedral metal atom has muchhigher fraction of X-links compared to other ions. This may bedue to the fact that apart from having a still smaller S−S bondlength of 0.56Rc, the larger number of chlorides in the ioninduces more local steric repulsion, leading to expansion of theattached polymer chain, increasing the interchain contacts andthereby the probability of making X-links.The average number of X-linked sites per metal complex, nX

= q⟨χX⟩, where ⟨ ⟩ denotes averaging over different saltconcentrations, is plotted in Figure 4e. Like nl, ⟨nX⟩ increaseswith q at q ≤ 3 and then saturates. Figure 4e demonstrates that⟨nX⟩ depends of the complex geometry. Experiments show thatadding metal ions with larger coordination number leads tolarger increase in viscosity of polymer solutions15 due to thegreater tendency to make interchain cross-links. Oursimulations show that the amount of cross-links increaseswith the coordination number of the complexes up to a certainvalue, beyond which it is practically independent of thenumber of vacancies q.Figure 4f characterizes X-linking in solutions of octahedral

MCl3S3 and tetrahedral MCl2S2 systems, in which the metalatoms coordinate with both polymer and solvent ligands.Competition between polymer and solvent ligands consid-erably reduces both fractions of X-links and C-links by factorsof 2−3. At the same time, the tendencies observed for thesystems with the inert solvent remain the same: X-linkingincreases with q, while chelation decreases. Another significantchange is the presence of a substantial amount of G-links,which decreases with q, yet constitutes the largest fraction intetrahedral MCl2S2 systems. These results show that both X-links and C-links are less probable compared to coordinationwith free solvent molecules while G-links are equally asprobable as the solvent coordination. This also demonstratesthe effect of the polymer entropy loss associated withinterchain and intrachain cross-links, which constrict polymerconfigurations.Specifics of Chain Clustering by Reversible Cross-

Linking.We define a cluster (to differentiate it from a polymeraggregate) as a set of polymer chains connected by any numberof X-link bonds. That is, if two polymer chains are coordinatedby the same metal complex, they belong to the same cluster. Acluster therefore can be viewed as larger branched polymer

molecule with increased effective molecular weight equal toMcnseg, where Mc is the number of chains in the cluster,described as the size of the linked cluster. The cross-linkersattach to the polymers randomly, unlike in the case ofassociating polymers where cross-linking points are distributeduniformly on the chains, and an isolated linked cluster is like abranched polymer with random branch points and loops. Theaverage cluster size increases with the metal concentration. Atsome point, the system of X-links percolates; that is, a clusterturns into a continuing network of connected chains. Thismoment should generally correspond to gelation (that is, thesystem turns into a gel acquiring some elastic properties),discussed in classical bond percolation models.27,29,53,54

However, in the system with reversible coordination that weconsider here, coordination bonds are not permanent; theyform and break and can be of different strengths. Some ofcoordination bonds are in fact very short-lived and weak. Thus,the cluster size fluctuates in time. The lifetime of the linksdepends on the activation energy as τlink ∼ eEa/kBT, where Ea isdetermined by the well depth Uw of the coordination potential,the repulsion between the neighboring coordinated ligands,and other entropic factors. Note that since Ea depends on thenumber of neighboring coordinated ligands and the chainconfigurations around the metal complex, it is expected tochange with additional coordination bond that the clusterforms. That is, the percolation scenario of clustering andgelation in polymer solutions with reversible cross-links isdynamic which provides the solution fluidity even beyond thegelation threshold observed on static snapshots of the polymernetwork. In such “weak” or “annealed” gels,28,29,55 materialproperties such as viscosity do not change drastically at the gelpoint, although can be high, in contrast to strong gels wherebonds are permanent (covalent) and viscosity diverges at thepercolation threshold. Figures 5a,b show static snapshots of thelargest linked cluster at a particular instant of time, in systemsof octahedral MCl3S3 at two different metal chlorideconcentrations. With CM = 0.006 (NM = 100), the size ofthe largest clusterMc

max = 10, which is an isolated cluster. WhenCM is increased to 0.012 (NM = 200), Mc

max is increased to 57,forming a gel network of linked polymers spanning the wholesimulation box. Such a sol−gel transition happens in a narrowrange of metal concentration as is depicted in Figures 5c,d,consistent with the power law divergence of cluster size in

Figure 5. (a, b) Snapshots of the largest cross-linked polymer cluster in MCl3S3 systems at different salt concentrations. Colors: P, V, and N beadsare colored green, red, and yellow, respectively. Solvent and metal beads are not shown. (c, d) Variation of the average size of the largest clusterwith salt concentration in different MClSq (c) and MCl2Sq (d) systems.



H



percolating systems at gel point. Figure 5c describes thevariation of average Mc

max/Npol (averaged over 200 000 timessteps = 2000τ after equilibration) with CM in MClSq systems ofdifferent geometry, which shows that all polymer chains areclustered into a single cluster, Mc

max reaches Npol, within anarrow range of metal concentration. All systems in this plothave q ≥ 3 and therefore have a similar high degree of cross-linking and exhibit characteristic clustering behavior with sharpgelation transition. On the other hand, in Figure 5d, whichdepicts clustering in MCl2Sq, the planar and tetrahedralsystems with q = 2 show a slower transition from sol to gel,while the transition in the octahedral system that has q = 4 issimilar to the MClSq systems shown in Figure 5c. This effect isbecause, as shown in Figure 4, the maximum fractions of X-links are observed for q ≥ 3 and the slower clustering is due tolower X-link fraction in the tetrahedral and planar MCl2S2,with planar system having the lowest.Rheological Properties. The major interest in studying

materials containing metal complexes of polymers stems fromthe fact that the rheological and other material properties ofpolymeric systems are strongly affected by complexation withmetals. Hao et al.15 showed that the critical shear-thinning rate,γc (the shear rate at which the viscosity starts to decrease withincrease in shear rate after a Newtonian shear-independentplateau), decreases with the increase in metal chlorideconcentration. This indicates that the terminal relaxationtime of the polymers,56 which is the reciprocal of γc, increases,and therefore the effective molecular weights of the polymersare increased by interchain cross-linking. In Figure 6a, we plotthe variation of relative viscosity defined as the ratio of theviscosity of the PVP−DMF−MClpSq solution to the viscosityof the pure solvent DMF, ηrel = η/ηDMF, obtained from theshear flow simulations, against the shear rate, γ, at different saltconcentrations. η denotes the apparent viscosity of thesolution, and ηDMF is the viscosity of the pure DMF solvent,

which is found to be shear-independent and equal to 7.075 τk TR

B

c3

in our simulations (see the Supporting Information, section IV,and Figure S2). The plots are qualitatively similar to theexperimental data of Hao et al.,15 as viscosity decreases withthe shear rate nearly in the entire range of γ values considered,with the exception of low shear rates at high metal chlorideconcentrations. In our simulations, shear-independent plateauis not reached, as the systems are highly viscoelastic. It is worthnoticing that shear-independent Newtonian regimes in DPDsimulations are observed within the range of shear rates studied

here for pure solvents, as shown in the SupportingInformation, section IV, for pure DMF fluid, or less viscoelasticsystems.57 At the same time, the shear rates in DPDsimulations are generally much higher compared to those inexperimental studies. For this comparison, one has to convertthe DPD time unit τ to real time units in seconds, which is byno means unequivocal. We estimated the conversion factorassuming the equality of the viscosity of DMF fluid found (seethe Supporting Information, section IV) equal to be

η τ= 7.075 k TRDMF

B

c3 to the experimental the viscosity of pure

DMF at ambient conditions ηDMF,exp = 0.802 mPa s.58 Thiscomparison yields τ = 5 ps. Additionally, comparison of thesimulated DMF diffusion coefficient (see the SupportingInformation, section V) with the actual diffusion coefficient ofDMF, Dexp = 1.6 × 10−9 m2/s,59 yields τ ∼ 6 ps. Note that thevalues of τ estimated by these two different methods are veryclose. Assuming the thus estimated value of τ, the lowest γconsidered in this study corresponds to 2 × 107 s−1, which ismuch higher compared to the experimental range of shear ratesstudied in ref 15.It is also possible that a shear thickening regime as observed

by Xu et al.60 may exist just before the critical shear thinningrate, indicated by the apparent increase of viscosity with shearrate in the red curve (CM = 0.093) in Figure 6a. Unfortunately,finding such a regime in DPD simulations would be reallyproblematic, since lowering the shear rate decreases the shearstress and thus increases the statistical error. The Newtonianregime may be observed in DPD simulations for sufficientlydilute systems; however, from our experience, we find that forthe limited chain lengths used in the simulations the effects ofcomplexation would be too small to identify. Studies of dilutesystems containing longer chains require significantly largersimulation box and longer time than those used in this workand can be hardly performed with necessary accuracy in thesimulations with explicit solvent. Yet, we can conclude that theresults shown in Figure 6a are in qualitative agreement with theexperiments.15

Figure 6b describes variation of viscosity ηrel at a given shearrate, γ = 0.01τ−1, with CM for octahedral MCl3S3 (circles) andtetrahedral MCl2S2 (triangles) systems. The figure shows thatthe increase in ηrel is most pronounced in systems where metalcan coordinate only with polymer beads with the solventunable to participate in coordination (red lines). The viscosityreduces as the solvent coordination is introduced (green lines)and is the lowest when there is no complexation (blue line).

Figure 6. (a) Relative viscosity ηrel vs the shear rate γ for solutions of tetrahedral MCl2S2 complexes different metal chloride concentrations. (b)Variation of viscosity with ion concentration at different coordination conditions in octahedral MCl3S3 (circles) and tetrahedral MCl2S2 (triangles)systems. Red lines represent viscosity of the system in which only metal−polymer coordination is present, the green curves show viscosities whenthe metal ion coordinate with both polymer and solvent, and the blue curves represent viscosity in the presence of ions without coordination (ηrel* ).



I










Note that the octahedral systems in Figure 6 are more viscousthan tetrahedral ones that as shown in Figure 4 may beattributed to the increased cross-linking in the octahedralsystems compared to the tetrahedral ones. Note also that thatηrel increases with CM even when the complexation is absent.This is because the fraction metal chloride is not small, and itsmaterial characteristics are different from the solvent. In oursimulations, as the chlorides have larger repulsion parametersthan the solvent, their presence leads to expansion of thepolymers and therefore to more entanglement effects.Noteworthy is the obtained viscosity−concentration de-

pendencies do not exhibit any stepwise behavior that onewould expect for the systems with sol−gel transition shown inFigure 5. That is, the formation of dynamic gel network withreversible cross-links does not change dramatically the systemfluidity. This effect deserves to be explored further. Theconnection between viscosity and morphological character-istics such as the fractions of interchain and intrachain cross-links is analyzed below.Relationship between Morphology and Viscosity.

The interrelation between viscosity and structural features ofmetal-complexed polymer systems has not been analyzed indetail before, although similar systems of associating polymerssuch as ionomer gels with reversible cross-linking (dynamicgels) have been theoretical studied. Associative thickeners(ATs) such as telechelic polymers have been studied usingtransient network theory61,62 formulated initially by Green andTobolsky63 and developed further by Tanaka and Ed-wards,64−66 Annable and co-workers,67 and others. Telechelelicpolymers are, however, much simpler systems than the onesstudied here since they have associating groups only at theends, and consequently, the chain relaxation time τr isindependent of the cross-link density and is given by a singleprocess, that is, the dissociation of one of the chain ends fromthe cross-linking junctions. Here the viscosity is given by theGreen−Tobolsky (GT) formula η = Gτr, where the shearmodulus G = nelkBT, where nel is the number of elasticallyeffective chains (chains that cross-links). With τr independentof the cross-link density, the Green−Tobolsky formula impliesa linear scaling with respect to the cross-link density.Unlike telechelic polymers, metal complexed polymers have

associating stickers randomly distributed on the chains. ATswith multiple stickers on the chains also have been consideredin the literature.61,68−71 Here, the chain relaxation depends onthe cross-link density, and a linear dependence of viscosity oncross-link density does not necessarily exist. Gonzalez68,69

suggested that the viscosity of ionomer gels containingreversible interchain cross-links increases exponentially withthe average cross-links per chain. Later, Rubinstein and co-workers70,71 noted that this is an overestimate, and theviscosity dependence in reversible gels should be much weaker,and suggested several power law dependences of the viscosityon polymer concentration at different regimes using the stickyreptation model. These studies considered polymers withequally spaced along the chains sticky points, where cross-linksbetween the chain segments can be formed, and found that thecross-link density varied with the polymer concentration in acomplicated fashion. In metal-complexed polymer systems,however, the cross-link density is determined by theconcentration of coordinating metal complexes, while thepolymer concentration remains constant with cross-link pointsrandomly distributed on the chains. Leibler et al.10 consideredviscosity of ion-complexed dilute polymer solutions andderived an expression for intrinsic viscosity in terms of thefraction of G-linked sites using the Flory model of polymersthat takes into account of the excluded volume effects inducedby the attached metal ions, where the intrinsic viscositydepends on the fraction of G-linked segments per polymerunits f as [(1 − f)2 + const·f 2]3/5. Earlier, Kuhn and Majer72

suggested that in very dilute solutions the intrinsic viscosityreduces with intrachain links per chain, nC, by a factor (1 −const·nC). However, none of these works analyzed the effectsof interchain links on viscosity. Many experimental groups inthe past9,11,12,73 studied sol−gel transition in polymer solutionsoccurring as the metal concentration increases. Shibayama etal.11,12 investigated sol−gel behavior in PVA−Congo-Red (CR)systems and observed gel melting and “re-entrant” sol−geltransition, which essentially attributed to weak cross-linkingbetween the polymers and the electrostatic repulsion betweenthe polymer and the metal ions. Such behaviors exhibited byweak gels cannot be explained by conventional site-bondpercolation models28,29 that apply to strong gels.In order to analyze the effects of complexation on the

viscosity of the PVP−DMF−MClpSq solutions, we consider ηrelL

= ηrel − ηrel* , where ηrel* is the relative viscosity of the systemswhere the ions make no complexation (like the blue line inFigure 6). With the effect of mere physical presence of ions onviscosity being removed by subtracting ηrel* , ηrel

L represents theincrease in viscosity due to coordination, which includes theformation of G-, C-, and X-links. Figures 7a−c show thedependence of ηrel

L on CM in double-logarithmic coordinates forcomplexes with different geometries and coordination number

Figure 7. Change in viscosity due to metal links ηrelL = ηrel − ηrel* at different salt concentration in (a) MClSq, (b)MCl2Sq, and (c) MCl3Sq systems

with different coordination geometries. Colors: red = octahedral, green = tetrahedral, and blue = planar. The dashed lines represent thecorresponding linear fits.



J



q at γ = 0.01τ−1. In all cases, the ηrelL obeys a power law

dependence on CM:

η η− * = νBCrel rel M (11)

where ν is the exponent and B is the value of ηrelL when CM = 1.

In Figure 7a, which depicts viscosities of MClSq systems ofdifferent geometries, the values of ηrel

L are similar for allgeometries, all of which have q ≥ 3 and similar average X-linked sites per ion. The values of the exponent in octahedral,planar, and tetrahedral systems respectively are νo = 1.24, νp =1.2, and νt = 1.09, whereas the corresponding B values are104.52, 93.14, and 54.04, respectively. The B values herecannot correspond to any physical variable, as for q > 1, CM = 1is well beyond the saturation point, where the complexationreaches a maximum and eq 11 is expected to hold only formetal concentrations well below the saturation point. Figure 7bpresents the viscosities in MCl2Sq systems which shows thatviscosity is higher in octahedral systems which has q = 4compared to other two geometries each having q = 2, withplanar systems being the least viscous. Again, this can becorrelated to the fraction of X-links in the three systemsdepicted in Figure 4e, which shows that the octahedral systemshave the highest X-linked sites per metal atom while planarsystems have the lowest. The values of ν and B for differentgeometries are found to be νo = 1.09, νp = 1.11, and νt = 1.16and Bo = 69.5, Bp = 33.03, and Bt = 51.6.Figure 7c shows viscosities of MCl3Sq systems compared to

X-linked and G-linked systems, as the octahedral MCl3S3system has the highest X-linking (see Figure 4), while intetrahedral and planar systems with q = 1, only G-links arepossible. As shown in the figure, the viscosities of purely G-linked systems are much lower than the viscosity of the X-linked octahedral systems and also lower than the other X-linked systems described in Figures 7a,b, despite the fact thatthe fraction of G-links is nearly 100%. The power lawexponents in this case are νt = 1.46, νp = 1.32, and νo = 1.18.The systems with solely G-links have larger exponents with theaverage value ν ∼ 1.4 than the average value, ν ∼ 1.15, in theX-linked systems. B values are found to be Bt = 52.7, Bp = 49.6,and Bo = 139.1.The above analysis shows that the major factor that

determines the viscosity of the metal-complexed polymersystems is the fraction of interchain cross-links. In systems forwhich q > 1, the amounts of grafted links are small ornegligible, the viscosity is found to be controlled solely by theX-link fraction, the effect of intrachain C-links is weak ornegligible. Note that C-links causes contraction of chains whenthey connect segments that are far, however, most of them mayconnect nearest segments along the chain and in this case, theyare more like G-links rather than C-links. Especially when theinterchain links are present, the viscosity is dominated byclustering, as it would cause more resistance to the flow. InFigure 8, we plot the change in relative viscosity against the X-link density, defined as the ratio of the number of X-linkedsites to the total number of polymer segments

ρχ

=N q

NXX M

seg (12)

across systems with different geometries at different saltconcentrations. It is interesting that regardless of the saltconcentration or the geometry, ηrel

L is found to obey a powerlaw dependence on the interchain X-link density in the form

given by eq 11, with ν = 1.18 and B = 59.7. This demonstratesa very strong connection between viscosity and interchaincross-links in metal-complexed polymer systems. The scalingexponent in eq 11 indicates an approximately linear depend-ence as in transient network theory of telechelic polymers,although at this point it is not obvious how to correlate theshear modulus and relaxation time to the cross-link density.Measuring G would require simulation of the systems underoscillatory shear. Kurokawa et al.9 fitted the experimentalviscosity values below the gel point with an exponent value of0.7 consistent with the irreversible percolation models, inwhich the viscosity diverges at the gel point. Our systems aremostly beyond the gel point, and as the weak gel systemsappear more viscous compared to sols, observation of largerexponents in the simulations of X-linked and G-linked systemsis sensible.Although we found different power law exponents ν ∼ 1.15

in X-linked systems and ν ∼ 1.4 in purely G-linked systems,their values may depend on several factors such as length of thechains and strength and range of the coordination potentialthat were not explored in our simulations. In addition, theymay also depend on the shear rate, as the shear rate at whichviscosities in Figures 7 and 8 are calculated, γ = 0.01τ−1,corresponds to a strong shear-thinning regime (Figure 6a).The chain length nseg = 60 is very small compared to that inreal polymer systems which scales to thousands. ν may increasefor longer chains and stronger coordination interactionbetween the metal and ligands. As the viscosity diverges instrong gels, the viscous nature of weak gels should depend onthe bond strength. Analysis of these effects is beyond the scopeof this paper and will be studied in a future work.

IV. CONCLUSIONSWe suggested a coarse-grained model of metal complexation inpolymer solutions implemented into the dissipative particledynamics (DPD) framework. As a characteristic example, themodel is parametrized to mimic polyvinylpyrrolidone−dimethylformamide−metal chloride solutions. The systemcomponents are modeled as soft beads representing character-istic fragments of polymer and solvent, metal ions, and

Figure 8. Dependence of the relative viscosity on the X-link densityfor complexes of different geometries at varying salt concentration.Triangles, squares, and circles respectively represent planar,tetrahedral, and octahedral complexes. Black, green, yellow, magenta,blue, cyan, and red points respectively represent the saltconcentrations CM = 0.012, 0.025, 0.037, 0.05, 0.062, 0.093, and0.124. Linear fitting shown by red line with the slope of ν = 1.18.



K



counterions. The beads interact through conservative pairwiserepulsion potentials and are connected by harmonic bonds tosecure proper conformations of the compounds considered.The metal complex denoted as MCpSq is modeled as thecentral metal cation M bead surrounded by n = p + qcoordination sites, p of which are occupied by counterion Cbeads and q other are vacant and available for coordinationwith ligands; the number of vacancies, q = n − p. The vacanciesare represented as “dummy” S beads capable of formingreversible coordination bonds with polymer and solventligands through dissociable cut and shifted Morse potentials.The complex beads are connected by harmonic bond and anglepotentials to preserve the geometry specific to the particularmetal salt: planar (n = 4), tetrahedral (n = 4), and octahedral(n = 6). We consider nine types of metal salt complexes MCpSqof different valency p = 1, 2, and 3, geometry and coordinationnumber n = 4 and 6.The proposed DPD model captures essential features of

metal−polymer coordination such as complex spatial geometryand coordination bond reversibility, while allowing forsimulations of large systems and calculations of macroscopicproperties such as viscosity. The model capabilities aredemonstrated with drawing on the example of morphologicaland rheological properties of concentrated PVP−DMFsolutions depending on the concentration of complex-formingmetal chlorides. While modeling metal salt complexes, weassume that the chloride anions are permanently bonded to themetal cation at the coordination sites and remaining vacantsites are available for coordination with polymer and/orsolvent ligands.While analyzing the specifics of complexation morphology,

we considered three types of metal−polymer coordinationbonds: interchain cross-links or X-links, intrachain chelatecross-links or C-links, and singly bound complexes or graftedG-links. Although in most of the simulations the coordinationinteraction was turned on only with polymer ligands, selectivesimulations were performed to show the effect of competitivecoordination with solvent ligands.It is found that in all systems considered the fraction of

coordinated vacancies (disregarding of their type) has a weakdependence on the salt concentration that is characteristic forweakly interacting complexes. The total number of metal−polymer links linearly increases with the salt concentration.The fraction of coordinated vacancies depends on the complexgeometry; it is significantly smaller for octahedral complexescompared to planar and tetrahedral. The number ofcoordinated vacancies in octahedral complexes varied weaklyaround 50% of the number of available vacancies. This effectmay be explained by the restrictions on the conformations ofpolymer chains around the complex and interligand repulsionthat hinder formation of multiple (>3) coordination bondswithin one complex. Competitive coordination with thesolvent ligands is more favorable due to reduced geometricalrestrictions than with the polymer ligands, especially of low saltconcentrations.The number of interchain X-links increases with the number

of vacant sites for planar and tetrahedral complexes and isalmost constant for octahedral complexes of different valency.Intrachain C-links are more prominent at low q. In addition,the cross-links are also affected by the geometrical factors suchas distances between the vacant sites within the complex.In concentrated polymer solutions, the interchain cross-

linking results in clustering of polymer chain connected by

reversible coordination bonds and in a transition from sol togel phase as the metal concentration is increased. The clustersize distribution was calculated from the momentary snapshotstaken along the simulation trajectory and thus reflects the staticpolymer morphology. This static analysis shows that the sol−gel transition happens at relatively low salt concentrations, asthe polymer concentration is high, with the cluster sizediverging within a narrow range of salt concentration aroundthe gel point. While this picture is consistent with the classicalpercolation theory, the reversible nature of coordination bondsmakes the cluster distribution dynamic with their continuouscluster breakup and rearrangement. The dynamic effects causea weak or annealed gel phase without drastic changes inmaterial properties such as viscosity at the percolationthreshold.The viscosity of the metal-complexed PVP−DMF solutions

was calculated by modeling the shear flow with Lees−Edwardsboundary conditions at different shear rates. For all systemsconsidered, the viscosity at given shear rate gradually increaseswith the salt concentration without any specific features in theregion of the sol phase transformation into a weak gel withreversible cross-links. The viscosity decreased with the increasein the shear rate above a certain critical shear thinning rate, asexpected due to the viscoelastic nature of the polymericsolutions, in qualitative agreement with experiments.15

Noteworthy is the salt interaction with solvent affects thesolution viscosity in different ways. Competing coordinationwith solvent ligands reduces the number of interchain cross-links and thus decreases the polymer solution viscositycompared to the solution without metal−solvent coordination.Another important factor is that even in the absence ofpolymer and coordination effects, the solution viscositymoderately increases with the salt concentration. Since ourgoal is to analyze the relationship between the viscosity and thedegree of cross-linking, we construct the concentrationdependence of the relative viscosity taking as a baseline theviscosity of the solution without coordination. We find that therelative viscosity obeys a power law dependence with respectto both metal concentration and the X-link density with theexponent ν ∼ 1.15 in both cases. At the same time, forcomplexes with q = 1, which allow for purely G-linkcoordination, the exponent is larger, ν ∼ 1.4. This differencearises from the difference in morphological modificationscaused by X-links and G-links; while X-links bind togetherpolymer chains with reversible bonds, G-links induce chainconformational expansion due to the grafted metal complexes,increasing the chain overlap and possibly entanglementsbetween the chains. At the same time, the increase in viscositydue to G-links is found to be much smaller than that caused bya comparable amount of X-links. The exponents determinedhere for weak gel phases of polymer solutions with reversiblemetal−ligand complexation are close to the values observed forsol viscosities by Kurokawa et al.9 in dilute poly(vinylalcohol)−borate solutions, calculated using irreversible site-bond percolation theory and the exponents in the theoreticalmodels10 that take into account of the excluded volume effectsinduced by the attached metal ions in dilute polymer−metalcomplex solutions.The DPD model proposed here can be adopted and

extended for other systems with polymer−metal complexationlike metal-substituted polyelectrolytes.



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■ ASSOCIATED CONTENT*S Supporting InformationThe Supporting Information is available free of charge on theACS Publications website at DOI: 10.1021/acs.macro-mol.8b00493.

Details of the DPD parametrization of polymer, solventand metal and coordination sites, the Lees−Edwardsmethod, and the viscosity of solvent DMF (PDF)

■ AUTHOR INFORMATIONCorresponding Author*E-mail [email protected] (A.V.N.).ORCIDAlexander V. Neimark: 0000-0002-3443-0389NotesThe authors declare no competing financial interest.

■ ACKNOWLEDGMENTSThis work was supported by the NSF CBET Grant No.510993 “GOALI: Theoretical Foundations of InteractionNanoparticle Chromatography” and DTRA Grant No.HDTRA1-14-1- 0015 “Mass Transport, Kinetics, and CatalyticActivities of Multicatalyst Polyelectrolyte Membrane”. Calcu-lations were performed using the Extreme Science andEngineering Discovery Environment (XSEDE)79 (ProjectNo. DMR-160117), which is supported by NSF Grant No.ACI-1053575 and Grant ACI-1548562.

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