Coarse-Grained Simulations of Polymer-Grafted Nanoparticles

Article pubs.acs.org/JPCB

Coarse-Grained Simulations of Polymer-Grafted Nanoparticles: Structural Stability and Interfacial Behavior Nitish Nair,*,† Michelle Park,‡,⊥ Jan-Willem Handgraaf,§ and Flavia M. Cassiola*,∥ †

Shell India Markets Private Limited, Bangalore 560048, India Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, United States § Culgi B.V., P.O. Box 252, Leiden 2300 AG, The Netherlands ∥ Shell International Exploration and Production, Westhollow, Houston, Texas 77082-3101, United States ‡

ABSTRACT: In the tertiary oil recovery method known as “polymer flooding”, the viscosity of the injected water is increased by dissolving partially hydrolyzed polyacrylamide so as to lower the mobility ratio and raise the vertical and areal sweep efficiencies. However, its drawbacks include the degradation of the polymer in the reservoir due to (1) shear while passing through chokes, perforations, and pore throats, (2) morphological changes induced by divalent ions, and (3) complete hydrolysis of the polymer at high temperatures. These factors adversely affect the viscosity of the polymer flood. Past experimental research showed that polymer-grafted nanoparticles (PNPs) could achieve the same viscosity enhancement at lower quantities than traditional linear polymers. The PNPs have the putative advantage of greater stability when confronted with the aforementioned reservoir conditions. In this work, we use dissipative particle dynamics (DPD) to simulate the oil−PNP−water system at the mesoscale and estimate its sensitivity to brine in ways that could serve as guidelines to experiments. We study the effect of salinity on the structure of linear and branched polyelectrolytes before extending the DPD model to PNPs at the oil−water interface. To this end, we parameterize the interactions of the polymer with the oil and water phases, and broadly map out solvent conditions that change the graft’s morphology and affect the interfacial behavior of the grafted particle. We find that the equilibrium location of the grafted nanoparticle in an oil−brine system depends on its grafting density and the salinity. its neighbors in solution.14 However, its drawbacks include the reduction of the polymer’s efficacy in the reservoir due to (1) shear while passing through chokes, perforations, and pore throats, (2) morphological changes induced by divalent ions, and (3) complete hydrolysis of the polymer at high temperatures. The presence of divalent salts poses a danger because they screen the negatively charged monomers in HPAM and cause the polymer to coil up−such collapsed states are less likely to form entangled networks, thus adversely impacting the solution viscosity.15 All of these factors, salinity being the most important, adversely affect the viscosity of the polymer flood. Novel structures such as branched PAMs,16 bare silica nanoparticles,17 nanofluids,18 and polymer-grafted/coated nanoparticles19 have been tested with respect to long-term stability, susceptibility to divalent attack, oil recovery efficiency, and adsorption tendencies. Ponnapati et al.20 and Ye et al.21 showed that polymer-grafted nanoparticles (PNPs) could achieve the same viscosity enhancement at lower quantities than those of traditional linear polymers.20 The PNPs have the putative advantage of greater stability when confronted with conditions in the reservoir22 and can even be used as surfaceactive agents.23 Silica nanoparticles coupled with anionic

1. INTRODUCTION In the tertiary oil recovery method known as “polymer flooding”, the viscosity of the injected water is increased by dissolving polyacrylamide (PAM) or partially hydrolyzed PAM (HPAM) or biopolymers such as xanthan gum to lower the mobility ratio and raise the vertical sweep efficiency.1−8 Under unfavorable mobility conditions in waterflooding, flow instabilities at the oil−water interface, arising out of viscosity differences, result in fingers of water piercing through the oil phase, thus lowering the efficiency of recovery. The development of these narrow corridors of high permeability is termed “viscous fingering”. Mobility control9,10 refers to the set of techniques that remedy viscous fingering by lowering the mobility of water with respect to oil. This is achieved by increasing the viscosity of the injected water phase, as mobility (kr/μ) is inversely proportional to the solution viscosity, hence the utility of water-soluble polymers such as PAM and HPAM. The presence of polymer solution leads to a much more stable water−oil interface. In addition to mobility control, HPAM is also being investigated as a conformance control agent, wherein polymers are used to block high-permeability zones in the reservoir and force fluids for enhanced oil recovery (EOR) through low-permeability regions.11,12 This selective blockage of large pores occurs as a result of polymer adsorption on the rock or the formation of gels in the pores.13 In general, viscosity enhancement requires that the polymer exists in an uncoiled state, which increases the probability that it will entangle with © 2016 American Chemical Society

Received: June 20, 2016 Revised: August 11, 2016 Published: August 11, 2016 9523

DOI: 10.1021/acs.jpcb.6b06199 J. Phys. Chem. B 2016, 120, 9523−9539

Article

The Journal of Physical Chemistry B

The list of characteristics of linear HPAM − molecular weight, extent of hydrolysis, monomer arrangement − and HPAM-grafted nanoparticles − grafting density, chain length, polydispersity − demands a slew of experiments to measure polymeric behavior under various solution conditions. All of these parameters affect the viscosity of the solution in intricate ways via the structural stability of the polymers. Past work shows the relation between the hydrodynamic volume of HPAM and the solution viscosity:65 polymers that occupy larger volumes in solution have a greater chance of interacting with each other and forming interpolymer linkages that ultimately boost the viscosity. Salinity hinders the viscosity enhancement because ions screen electrostatic repulsions along the polymer chain, thereby causing them to coil up and reduce the hydrodynamic volume. It is important to note our premise at the outset: coarse-grained techniques such as DPD can simulate polymer behavior under different solvent conditions and identify chemistries, configurations, and structures that are most likely to resist hostile environments, for example, divalent ions in brine. We use the radius of gyration to quantify the span of the polymer in solution. Before reaching this predictive stage, however, we need to characterize the molecular models for PAM and HPAM; specifically, the intermonomer, monomer− water and monomer−oil interactions. The goal of this article is to study the effect of salinity on the morphology of free and grafted PAM/HPAM-like polymers and the interfacial activity of the latter in oil−water systems. This article is arranged as follows: Section 2 lays out the details of the coarse-grained simulation method and the associated parameterization protocols that will be used for free polymers and PNPs. The results for free linear and branched polymers are presented in Section 3, whereas Section 4 contains those for the PNPs.

surfactants have been shown to stabilize foams for oil recovery.24 From the computational standpoint, standard Molecular Dynamics and Monte Carlo simulations have been used to study the behavior of polyelectrolytes25−27 and surfacefunctionalized nanoparticles.28−33 This work uses a less intensive method, dissipative particle dynamics (DPD),34−37 to simulate the oil−PNP−water system at the mesoscale after first characterizing the interactions of a polymer chain with aqueous and oleic phases. DPD is a popular tool for coarsegrained simulations: it has been used to analyze the structural and mechanical properties of polymer-grafted biomolecules,38−40 the self-assembly of NPs in polymer brushes,41 and, more significantly, the behavior of functionalized NPs42,43 and PNPs at the oil−water interface.44−46 The common thread underlying these papers on PNPs at the oil−water interface is the influence of the polymer grafts in manipulating the position of the nanoparticle. For example, if a PNP with hydrophobic grafts is initially in the water phase, the grafts will drag the PNP to the oil−water interface; if the attraction to the oil is strong enough, the PNP will be transported completely to that phase. The coarse-grained potentials in DPD enable us to view this sequence of events on reasonable time scales. Although approximate, these force fields capture the essential behavior of polymers and yield insights into the overall physical and chemical processes at work. Electrostatic interactions in DPD were later added by Groot,47 thus enabling its use in modeling polyelectrolyte brushes,48−51 adsorption on surfaces52 and liquid−liquid interfaces,53 and comblike polymers.54 The philosophy of DPD is to reduce complex molecular structures to coarse-grain representations and has therefore generated interest in the polymer physics community. Typical EOR polymers have molecular weights of the order of 106−107 Da, which translates to 104−105 monomers in the case of PAM. If one assumes a sphere circumscribing the average configuration of an EOR polymer, the diameter of such a sphere would be ⌀(102) nm, a length scale that lies within the purview of DPD. Polymers with molecular weights in the 106 Da range may be treated as a chain of beads linked by harmonic springs. After its inception, many authors successfully tested the efficacy of DPD in reproducing well-established static and dynamic properties of polymers.55−60 The rheological behavior of polymer solutions and melts has received a fair degree of attention in the computation domain.55,57,59,61 Linear polymers in DPD generally have chain lengths in the 10−100-bead range, and these models have been able to reproduce traditional scaling laws for static and dynamic properties.55−63 DPD has recently been used to characterize the structural properties of copolymer-grafted gold nanoparticles in a good solvent.64 We also stick to the 10−100-bead range of chain lengths in this work. The end goal is to verify whether the DPD mechanism and our parameterization scheme can describe (1) structural features of polymers and (2) the effect of monovalent and divalent salts on the morphology of the chain. Once this has been accomplished, it is trivial to extend the formalism to larger systems, given the availability of computing power. The soft interbead repulsions in DPD do not make it an attractive tool for quantitative predictions of polymer viscosity. However, it may be used to develop qualitative scaling relations between the zero-shear viscosity and molecular weight.

2. SIMULATION METHOD DPD is a coarse-grained technique that can access length and time scales that lie beyond the range of more detailed atomistic approaches such as Molecular Dynamics. Section 2.1 outlines the forces involved in DPD and focuses on the “repulsion parameter”, which is the handle we use to tweak interactions between beads representing different chemical groups. The chemical entities in our systems are (1) acrylate and (2) acrylamide to represent the HPAM, (3) silica for the nanoparticle, (4) water or (5) oil as the solvent. Each chemical entity is denoted by a “bead” in DPD parlance. In Sections 2.2−2.3, we describe the method by which we identified bead types and assigned repulsion parameters to each interbead interaction present in our simulations. All of the simulations in this article use the DPD module in the Chemistry Unified Language Interface (Culgi version 8.0) software from Culgi B.V., The Netherlands (2004−2016). 2.1. Forces in DPD. The basis of DPD lies in developing coarse-grained versions of molecules and their interaction potentials. During DPD simulations, the nonbonded force acting on a bead is calculated by considering the net effect of three different forces: (1) a conservative force, (2) a random dissipative force, and (3) a frictional force. Both the random dissipative force and the frictional force are pairwise forces that act as the thermostat in DPD. Conservative forces distinguish the chemical nature of every bead. Our flavor of DPD uses three different conservative forces. 9524


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chemical moieties listed in Table 1, all of which are components of hydrophilic polymers.

1. Electrostatic force: This term allows us to model ionic species such as salt in brine and the charged head groups of surfactant molecules. 2. Bead−bead spring interactions: These represent the bonded interactions between connected beads. Our simulations use the harmonic form for the bead−bead spring interaction. 3. Bead repulsion: Beads interact with each other through a soft repulsive force in DPD simulations. Our implementation of DPD uses the Hoogerbrugge and Koelman form.34 The bead repulsion between beads is “soft” compared to force fields in molecular dynamics. This implies that the beads are allowed to overlap up to a certain extent. The mathematical form of the Hoogerbrugge−Koelman interaction is given in eq 2.1. rij ⎞ ⎛ Fir = ∑ aij⎜1 − ⎟ ⎝ d⎠ j

Table 1. Chemical Structures of Monomers Used in the Molecular Simulation along with Their Respective Molecular Volumes and Bead Representations

(2.1)

In eq 2.1, d forms the cutoff for the repulsive force between beads. This is set to 1 DPD length unit in our work. We discuss this length scale in greater detail in Section 2.2. The term aij determines the strength of the interaction in terms of kBT. This captures the chemical interaction between beads. Groot and Warren developed a formalism relating the bulk compressibility of systems to the aij term.66 They showed that soft-core interaction with aii = 78 for water beads reproduced the bulk compressibility. In the literature, the DPD repulsion parameter between similar, or “like”, beads, aii, is set to 78. We use this value as a reference interaction between bead pairs that are chemically identical. We retain the same like−like repulsion between oil beads as well. This assumption makes the oil and water phases equally compressible though this may not be the case in reality. For the first-cut approach adopted in this article, we simply seek a phase that is (1) chemically different from water and (2) an unfavorable solvent for hydrophilic EOR polymers. The interaction between two unlike beads is estimated through their enthalpy of mixing. Past work in the literature has used the Flory−Huggins interaction parameter, χij, to estimate aij.66 The χ-parameter between two monomers, i and j, dictates the enthalpy of mixing when they are blended. A highly positive χij (>1) implies that a mixture of two homopolymers composed of monomers i and j, respectively, will separate into two distinct domains, given the right temperature. The most widely used estimate for the DPD repulsion parameter, aij , is shown in eq 2.266 χ aij = aii + (2.2) 0.689

Cations and anions belonging to the added salt are treated like water beads with associated positive and negative charges, respectively. When a part of the PAM chain undergoes hydrolysis, the acrylamide monomer is converted to acrylate, which is an ionic species. The remaining three functional groups in Table 1 are constituents of the poly(AMPS-co-AA) copolymer having acrylamido methylpropane sulfonic acid (AMPS) and acrylic acid (AA) as its monomers. Data from the literature suggest that AMPS is more tolerant than PAM toward salinity,68 so we use it as a test case for our coarse-grained model, specifically to assess whether the combination of force field and chain architecture will reproduce the experimental observation. The second column in Table 1 shows the chemical structures of these groups, and the third column contains their molecular volumes. The first three and last entries have volumes similar to the water bead (90 Å3), so we represent each one with a single bead. AA and acrylamide are neutral groups, whereas sulfonic acid and acrylate are negatively charged. We denote neutral beads by P and negatively charged beads by Q. The fourth column contains the bead nomenclature in the DPD setup. The AMPS-co-AA monomer has a molecular volume of 314 Å3, that is, ∼3 times the volume of a water bead. Hence, we assume that the AMPS-co-AA monomer is composed of three beads, two of the P type for the neutral part of the molecule and one of the Q type for the negatively charged head group, as we assume that the sulfonic acid is deprotonated to the sulfonate form. The bead for the latter ion is the last entry in Table 1 with the approximate volume of 113 Å3 with CH3 as the alkyl fragment. In addition to neutral and charged monomers, we have a bead for water, W, that conceptually contains three water molecules. Figure 1a shows this bead along with the discretization of the PAM chain in Figure 1b. As mentioned

In the literature, some of the ways to estimate the χ parameter include empirical methods1,8 and techniques such as COSMORS.67 2.2. DPD Interactions: Setting Bead Types. DPD groups a collection of atoms into a bead and treats the latter as a single entity that interacts with its neighbors according to the specified force field. DPD simulations traditionally lump three water molecules into a single bead and set the resulting molecular volume, 90 Å3, as the volumetric basis.66 As the governing equations in DPD are nondimensionalized, one DPD length unit corresponds to 6.46 Å.66 Harking back to eq 2.1, we mandate that the cutoff for the conservative force, d, equals 6.46 Å. Apart from water, we are primarily interested in the 9525


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Figure 1. Representation of compounds as beads in the DPD method: (a) water bead, (b) acrylate and acrylamide beads constituting HPAM, (c) relevant interbead interactions in the simulation.

Figure 2. Representation of branched polyelectrolyte in the DPD formalism: (a) chemical structure of AA and AMPS reduced to the P−Q form used in the simulation, (b) structure of a linear polyelectrolyte with the contour of the chain traced by the white arrows, (c) structure of a branched polyelectrolyte with the red arrows showing the branch points

repulsion parameter, aij, in eqs 2.1−2.2 is the handle we use to tweak the interactions between beads i and j. In a sense, aij determines the chemical nature of the beads involved in the simulation. In addition to PAM and HPAM, which are linear polymers, we also model poly(AMPS-co-AA), a random copolymer of 2acrylamido-3-methylpropane sulfonic acid (AMPS) and AA. Figure 2a shows the molecular structure of the copolymer with m repeating units of its AA backbone and n repeating units of the dangling AMPS branch. We use the bead types specified in Table 1 to represent this copolymer. We use a chain of P beads for the AA backbone, which we assume to be electrically neutral. The branch is represented by two P beads for the neutral segment of AMPS and one charged Q bead as its head group. The logical progression from the molecular structure to the DPD form is depicted in Figure 2a. The structural difference between DPD versions of the linear and branched polyelectrolytes is shown in Figure 2b,c. Figure 2b displays a linear polyelectrolyte, for example, HPAM, wherein the negatively charged monomers (purple beads) are interspersed among the neutral ones (green beads). The white arrows trace

previously, Q and P stand for the charged acrylate and neutral acrylamide, respectively. Any polymer is then a chain of these beads that are joined together by harmonic springs, which represent the bonds between the functional groups. The polymer chain can either be linear, for example, PAM, or branched, for example, poly(AMPS-co-AA) with the sulfonic acid group dangling from the backbone. In the case of the latter, we randomly position branching points along the linear backbone and grow the branches at those locations. We will return to the branched configurations later, but they share the same interaction parameters as the linear chains, as the same DPD beads are involved. Figure 1c shows the various interactions at play in the polymer−water system. When we add salt (monovalent or divalent) to mimic brine, we assume that the ion is solvated and hence treat it like a water bead except for the presence of a positive or negative charge on it. If the polymer contains only one monomer, acrylamide on the left-hand side of Figure 1c, the relevant interactions are P−W, P−P, and W−W. In the case of the random copolymer, HPAM, on the right-hand side of Figure 1c, three additional pairwise interactions enter the picture: Q−W, Q−Q, and Q−P. The 9526


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basic trends observed in experiments, for example, the effect of solvent quality on polymer morphology. Rather than dwelling on the internal arrangement of monomers, we focus on the effect of the solvent on the global structure of the free polymer or the grafted chains in the case of a polymer-grafted nanoparticle. Section 3.1 sets the values of aij between a neutral homopolymer, water, and oil. In Section 3.2, we extend the parameterization to linear polyelectrolytes and study their structural behavior in saline solutions. Finally, Section 3.3 looks at the stability of branched polyelectrolytes, for example, poly(AMPS-co-AA), in saline media and compares it to that of its linear counterpart as a test of the repulsion parameters derived in Sections 3.1−3.2. 3.1. Parameterizing Polymer−Water−Oil Interactions. We first study an infinitely dilute solution of a neutral homopolymer, for example, PAM, in a solvent such as deionized water or oil. We define the oil as a four-bead chain that repels the polymer and is therefore a poor solvent. This is a reasonable assumption for the hydrophilic polymers that are used in EOR. Therefore, the polymer chains in our model remain in the water phase and do not enter the oil phase. With a base repulsion of aij = 78 between chemically identical beads (i.e., W−W or P−P), we vary the P−W interaction in the following way: aij = 70 for a good solvent, aij = 80 for an ideal solvent, and aij = 100 for a poor solvent. The DPD simulation followed the polymer’s motion through the solvent for each set of repulsion parameters and computed Rg during the production phase. This process was repeated for three different chain lengths, 10, 50, and 100 beads, with three independent trials for each. The final Rg was an average of the values obtained from each trial. The final plot of ln(Rg) versus ln(Nmon − 1) is shown in Figure 3 for the three settings of aij. The linear

the contour of the chain. On the other hand, Figure 2c shows a branched random copolymer where the backbone (green beads) is neutral but the dangling branches (purple beads) are charged. The white arrows once again show the contour of the linear backbone, and the red arrows show the branches leading off specific points on the backbone. 2.3. DPD Interactions: Parameterization Protocol. We now know that three values of aij have to be estimated for the PAM−water system. The radius of gyration (Rg) quantifies the span of a polymer and hence is a useful quantity to describe its structural behavior in solution. The volume occupied by the polymer is a function of intramolecular and intermolecular interactions−a high (low) Rg indicates that the chain is stretched (coiled). The squared radius of gyration is a thermodynamic quantity defined as N

R g2(t ) =

1 mon ⇀ ⇀ ⇀ ⇀ ∑ ( ri − rCT)·( ri − rCT) Nmon i = 1

(2.3a)

N

⟨R g2⟩ =

1 f 2 ∑ R g (t j) Nf j = 1

(2.3b)

where Nmon is the number of monomers in the polymer chain, Nf is the number of frames or snapshots of the chain recorded r i is the position vector of the ith during the simulation, ⇀ r CT is the position vector of the monomer in the polymer, and ⇀ centroid of the polymer chain. Equation 2.3a describes the variation of R2g with time, whereas eq 2.3b gives the timeaveraged value of R2g at the end of the simulation. We get the radius of gyration of the chain from R g = ⟨R g2⟩ . We use the following scaling relation69,70 between Rg and the number of monomers to measure the effect of the solvent on the polymer’s morphology.

R g ∼ (Nmon − 1)ν

(2.4)

The value of ν in eq 2.4 varies according to whether a solvent is “good” (ν = 0.588), “ideal” (ν = 0.5), or “poor” (ν = 0.33). As PAM and HPAM are soluble in water, it is a good solvent. Conversely, both polymers are insoluble in oil, which makes it a poor solvent. We therefore seek to parameterize the PAM− water interactions in DPD in such a way as to obtain the Rg− Nmon scaling law with the exponent ν = 0.6 for water and ν = 0.33 for oil. We use this scaling relationship to establish a link between a simple bead−spring chain and basic PAM behavior. We will not further refine the force field in this article by matching the density and Tg of the polymer. The only requirements of our coarse-grained force field are (1) reproduce the essentials of PAM behavior in water and (2) capture the effect of monovalent/divalent cations on the morphology of HPAM. The set of parameters generated in Section 3.1 will be used for the HPAM chain as well with one modification−the acrylate beads will have a charge of −1.

Figure 3. Scaling relationships between Rg and the number of monomers, N, for a neutral homopolymer immersed in solvents of different qualities.

variation of ln(Rg) with ln(Nmon − 1) is evident in the figure. The slopes for the good (green triangles), ideal (red squares), and poor (blue diamonds) solvents are respectively 0.6, 0.51, and 0.31. Equation 2.4 tells us that the slope is the exponent, ν, of the Rg−Nmon scaling law. The values of ν thus calculated match our expectations for the different types of solvents. In the case of polymer−water repulsion, we decided to use aij = 80 for the following reason: the Flory−Huggins χ-parameters for PAM−water and polyacrylate−water have been measured as 0.49 and 0.5, respectively.71 We can convert this to aij using eq 2.2 with aii set to 78. This gives us aij for PAM−water as 80. Subsequent calculations will use this value.

3. SIMULATIONS OF FREE POLYMERS IN SOLUTION Before simulating a polymer-grafted nanoparticle, we need to capture the interactions between a single polymer chain and the solvent (water or oil) to a reasonable extent. The goal is to find a set of repulsion parameters (aij) that will produce an Rg−Nmon scaling relationship (eq 2.4) commensurate with the desired solvent property. Additionally, our coarse-grained force field links the behavior of PAM/HPAM-like hydrophilic polymers to 9527


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Figure 4. Variation of the scaling exponent, ν, as a function of the oil−polymer repulsion factor, aij, which is used to set the oil−polymer interaction.

We performed a finer scan of aij to fit the polymer−oil repulsion factor. Figure 4 shows the calculated scaling exponent, ν, as a function of aij, which was varied from 70 to 100. We see that ν reaches a value of 0.31 at aij = 83 and varies negligibly beyond that point. The blue arrow in Figure 4 indicates the aij that we chose for polymer−oil repulsion, aij = 85. 3.2. Effect of Salinity on Linear Polyelectrolytes. Now that we have a set of repulsion parameters to describe the neutral homopolymer−water/oil interactions, we shift our attention to polyelectrolytes. We construct a polyelectrolyte chain by replacing P-type beads in a homopolymer chain with Q bearing a negative charge. The Q beads can be placed either periodically or randomly along the length of the chain. The entire system is kept electrically neutral by adding positively charged counterions to the simulation box, one for each Q in the chain. The presence of Q monomers causes the chain to expand due to electrostatic repulsions between the negatively charged beads. A steady increase in the number of Q monomers should produce a concomitant increase in Rg. In reality, counterion condensation takes place above a critical population of negatively charged monomers72,73 wherein counterions adsorb onto the polymer chain and screen the negatively charged Q monomers. The screening effect reduces the electrostatic repulsions that stretch the chain, and Rg therefore drops. Once the number of counterions increases above a limit, charge inversion occurs and Coulombic repulsion enters the fray once again except that it is now between positively charged ions. The Bjerrum length, lB, is the distance from an ion at which its electrostatic interaction with another ion exactly matches thermal fluctuations. Above lB, thermal fluctuations dominate over electrostatic interactions. The value of lB for water is 7 Å and plays a significant role in determining whether counterion condensation occurs or not. In Figure 5, we test the effect of the inclusion of Q monomers (pink beads) among P (purple beads) in polymer chains that are 10 (blue diamonds) and 50 beads (red squares) long. We vary the periodicity of the charged monomer along the x axis. The notation is as follows: a charge periodicity of four means that one in every four beads in the chain is Q, and

Figure 5. Effect of placement of the anionic monomer (pink beads) in the polyelectrolyte on Rg for two chain lengths, 10 beads (blue diamonds) and 50 beads (red squares).

the structure is PPPQPPPQ. Similarly, a charge periodicity of 1 denotes a PQPQPQ structure. The y axis shows Rg of the chain in Å. For both chain lengths, as we go from right to left, the number of Q beads increases and so does Rg due to Coulombic repulsion. The rise in Rg for the 50-bead-long chain (red squares) is greater than the 10-bead chain (blue diamonds) because the former can accommodate more Q beads at the same charge periodicity. Even at the maximum Q loading for a PQ copolymer, we do not see any hint of counterion condensation, which would manifest itself via a drop in Rg. Beads in our simulation have diameters in the vicinity of 6.46 Å (i.e., the DPD length scale), and the Bjerrum length of the solvent, water, is 7 Å. In the polymer chain with the highest number of charged Q beads (i.e., a periodicity of one), each neutral P bead is followed by a negatively charged Q. The distance between two successive Q beads along the chain is ∼2 × 0.7 = 1.4 DPD units, or 9 Å, because the bond length is 0.7 DPD units. This means that the average distance between charged monomers is greater than lB, that is, where thermal fluctuations prevail over electrostatic interactions. This is why we do not see counterion condensation in any of our simulations. 9528


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DPD model of AMPS has a two-bead-long branch whose structure is shown in Figure 2a. To compare the effect of a divalent salt on Rg of equivalent linear and branched polyelectrolytes, we have to first identify a pair of these structures that have comparable radii of gyration at zero salinity. We construct the branched polyelectrolytes by fixing the length of the neutral backbone (PPPP...) and then randomly assigning branch points according to a “branch fraction”. For instance, if the backbone has 10 beads and the branch fraction is 0.3, the final polyelectrolyte will have three branches (i.e., three Q beads) at random locations along its length. The total number of beads in the polymer in this example is 13. Let us suppose that its radius of gyration in deionized water is Rg,b. We need to construct a linear polyelectrolyte that will have a similar radius of gyration, Rg,l, under the same conditions. The reason is that changes in either polymer due to the addition of divalents must be compared on a one-to-one basis. Hence, the algorithm is as follows: 1. Fix the structure of the branched polyelectrolyte (backbone length, branch fraction) 2. Find its zero-salinity radius of gyration, Rg,b, via DPD 3. Vary chain length and fraction of Q of the linear polyelectrolyte 4. Find zero-salinity Rg,l for each setting via DPD 5. Choose the combination that satisfies Rg,l ≈ Rg,b Table 2 contains the results of this exercise for a branched polymer with a one-bead-long branch. Table 2a(b) contains the

We move the polyelectrolyte model one step ahead by adding a monovalent salt in a simulation box containing a 10bead-long chain and water. Figure 6 shows Rg as a function of

Figure 6. Effect of placement of the anionic monomer (pink beads) in the polyelectrolyte on Rg for three salinities, 0 wt % (blue diamonds), 0.36 wt % beads (red squares), and 3.8 wt % (green triangles) of monovalent salt.

the charge periodicity for three different salt concentrations, 0 (blue diamonds), 0.36 (red squares), and 3.8 wt % (green triangles). The addition of salt hastens the charge-screening process, which leads to the collapse of the polyelectrolyte. We do not see a sizeable drop in Rg of the chain when the salt concentration is 0.36 wt %. At the higher salinity of 3.8 wt % (comparable to seawater), Rg drops by 9.5% from the zerosalinity value at the highest Q loading (charge periodicity = 1, encircled in red). The drop at the next charge periodicity of two is much lower, 3%. Our polyelectrolyte model captures the adverse effect of salinity on morphology, albeit at a high fraction of the negatively charged monomer. The model is on the right track, as the intrinsic viscosity of HPAM (alternatively, Rg) decreases in the presence of NaCl.74 Even so, it must be kept in mind that the results in Figure 6 are for a chain length of 10 beads; the number of Q monomers at charge periodicities >1 is too low to be affected by the presence of salinity. We expect more significant drops in Rg of longer polyelectrolytes, as they will contain more Q monomers at comparable charge periodicities and will therefore be more sensitive to salt in the solution. 3.3. Effect of Salinity on Branched Polyelectrolytes. The previous section focused on the effect of a monovalent salt on the radius of gyration of linear polyelectrolytes. We now study the structural stability of linear and branched polyelectrolytes when exposed to a divalent salt solution, for example, CaCl2. This work was inspired by Xue et al. who showed that increasing the fraction of AMPS in a poly(AMPSco-AA) random copolymer increased its stability in high-salinity brine.68 AMPS contains a sulfonate group that is inherently more stable toward Ca2+ but we chose to investigate whether separating the charged monomer from the polymer backbone would reduce its susceptibility to divalent attack. We also want to check whether our DPD force field for polyelectrolytes can reproduce the AMPS-related result from Xue et al. The resulting copolymer has a neutral backbone and pendant charged monomers. We tested two lengths of the dangling branch, one and two beads. In the former case, the single bead was charged, whereas in the latter, the first bead (attached to the backbone) was neutral and the second was charged. The

Table 2. Identification of a Linear Polyelectrolyte That Is Equivalent to a Branched Electrolyte with Branch That Is One Bead Longa backbone length

total #monomers

Q fraction

Rg (Å)

0.3 0.5 0.3 0.5

0.23 0.33 0.23 0.33

14.66 17.58 23.5 29.7

0 0 0 0

0.21 0.38 0.3 0.5

14.65 17.18 24 30.48

branch fraction (a)

30 30 50 50

39 45 65 75

34 32 50 50

34 32 50 50

(b)

a

Each row in table (b) denotes a linear polyelectrolyte that has a radius of gyration comparable to the branched version in table (a) at zero salinity.

data for the branched (linear) polyelectrolyte. Each row in Table 2a is paired with the corresponding row in Table 2b. In other words, each branched polymer in Table 2a has a radius of gyration that is comparable to its linear counterpart in Table 2b. Table 2a shows that we tested four types of branched electrolytes, two backbone lengths coupled with two branch fractions. The second column is the total number of beads comprising the polymer, that is, the sum of the number of P and Q beads. In the case of the linear polyelectrolyte (Table 2b), the first two columns are the same. The third and fourth columns in each table show the fractions of branches and Q monomers. The branch fraction is zero for the linear polymer, whereas the Q fraction is adjusted along with the backbone length so as to have matching Rg columns in both tables. We see that the Rg values of the branched polymers (Table 2a) are numerically close to their linear counterparts (Table 2b) with a 9529


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Figure 7. Effect of divalent salt concentration on Rg of linear and branched polyelectrolytes for backbones of length (a) 30 beads and (b) 50 beads. The branch is one bead long.

Figure 8. Effect of divalent salt concentration on Rg of linear and branched polyelectrolytes for backbones of length (a) 30 beads and (b) 50 beads. The branch is two beads long.

from 23.5 to 19 Å, a 25% drop in Rg for the linear polymer versus a 19% drop for the branched equivalent. A similar trend is observed for the higher branch fraction of 0.5 (orange triangles, green triangles), but the difference between Rg at 3.8 wt % salt is less than 1 Å. Taking these results together, we can make two inferences for structures with one-bead-long branches: (1) The difference between the branched and linear polyelectrolytes with respect to stability in a saline medium becomes more apparent as the backbone length increases; (2) Increasing negative charge on the polyelectrolyte at the same backbone length, 50 beads in this example, increases its exposure to divalent attack. We repeated the calculations presented above with two-beadlong branches as an ansatz for poly(AMPS-co-AA)−the exact structure is shown in Figure 2a. The equivalent linear polyelectrolyte was identified in a similar manner as previously described. We show the effect of salinity on these linear and branched polymers in Figure 8a,b. In both figures, only one branch fraction, 0.3, is considered. The linear polyelectrolyte in

maximum deviation of 3%. At this point, we have an equivalent linear polyelectrolyte for each branched species studied. In Figure 7a,b, we vary the concentration of the divalent salt from 0 to 3.8 wt % and compute Rg of the branched and equivalent linear polyelectrolytes. The backbone is 30 beads long in Figure 7a and 50 beads long in Figure 7b. The branch is one bead in length in both figures. The lines for “Branched (X)” and “Linear (X)” must be compared against each other, where “X” is the branch fraction. Figure 7a for the 30-bead backbone shows that increasing the salinity decreases Rg to the same extent in the linear and branched chains. That is, moving the charged monomer away from the backbone does not improve its tolerance to divalents. Both branch fractions behave identically in this regard. Figure 7b for the 50-bead backbone is more interesting. Let us first consider the plots for a branch fraction of 0.3 (blue diamonds, red squares). As the salinity increases from 0 to 3.8 wt %, Rg of the linear chain (red squares) drops from 24 to 18 Å, but Rg of its branched counterpart (blue diamonds) drops 9530


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Figure 9. Structure of PNPs: (a) discretization of the HPAM chain into the bead−spring form that is grafted onto a nanoparticle core with randomly selected grafting sites (brown beads). (b) PNPs with neutral, 25% acrylate, and 50% acrylate grafts immersed in a solvent.

panel of Figure 9a: (1) an all-acrylamide chain that is electrically neutral, (2) a polyelectrolyte with 25% acrylate, and (3) a polyelectrolyte with 50% acrylate. HPAM used in the polymer flood has an average acrylate content (alternatively, the degree of hydrolysis) of 25%. The 50% acrylate chain gauges the effect of the extent of hydrolysis on polymeric stability in saline environments. The model nanoparticle is shown on the extreme right of Figure 9a. It is not a monolithic structure but is composed of a conglomeration of smaller beads. We construct the polymer-grafted nanoparticle through a custom-made script in the CULGI environment. The radius of the particle is three DPD length units or ∼2 nm. In this snapshot, the brown beads denote grafting sites that are randomly assigned. The first bead of the grafted chain is bonded to the grafting site on the nanoparticle through a highly negative repulsion factor, that is, a strong attraction, between these two beads. The pink beads denote the surface of the nanoparticle that will be exposed to the solvent. Interfacial activity will be possible only if the nanoparticle surface has been chemically functionalized.75 Similarly, the dispersibility of the bare nanoparticle in water, which is essential for EOR, may be increased by functionalizing its surface with hydrophilic polymers or surfactants.22 The bare nanoparticle in our simulation has an aij of 150 with water and oil. It is therefore equally (un)comfortable in either phase. All of the polymer chains grafted onto the nanoparticle surface have the same chemical nature, for example, they are all neutral or all with 25% acrylate. In a chain that has Nmon + 1 beads, the first bead (colored gray in Figure 9a) is the one that is grafted to the nanoparticle, that is, it is the anchor for the remaining Nmon monomers. The algorithm picks a grafting site on the nanoparticle surface at random and attaches the polymer chain to it. This attachment is accomplished by setting the repulsion parameter between the grafted bead and the grafting site to an extremely negative value. A highly negative repulsion parameter between two beads is just a strong attractive force. We use this technique to ensure that the graft does not desorb from the particle during the simulation. This procedure is repeated until the desired grafting density (number of chains/

Figure 8a for the 30-monomer backbone suffers an 18% decrease in Rg when the divalent salt concentration is increased from 0 to 3.8 wt %. Rg of its branched counterpart drops by 10%. The adverse effect of salinity is even more apparent in the case of the 50-bead backbone plots in Figure 8b. Rg of the linear (branched) polymer decreases by 30% (17%) when exposed to the divalent salt. When we compare the plots in Figure 8 with those in Figure 7, it is clear that increasing the length of the branch by one unit increases the stability of the polyelectrolyte in the presence of divalent ions. Therefore, despite our simple force field and structural representation of poly(AMPS-co-AA), our model is able to discern its stability, relative to that of an equivalent linear polyelectrolyte, in a saline medium. The investigations into the behaviors of free linear and branched polymers in this section have given us confidence in the basic polyelectrolyte model. We increase its complexity in the next section by grafting such polymers onto a nanoparticle. The aim in simulating this system is to characterize the morphologies of the grafted chains and interfacial activity of the grafted nanoparticle in the presence of salt.

4. SIMULATION OF PNPS The purpose of this section is to study the behavior of PNPs embedded in a single brine phase or at the interface of brine and oil. We are interested in two aspects of this problem: (1) the effect of salinity on the conformations of grafted polymers, both neutral and charged, and (2) the role played by polymer− water and polymer−oil interactions on the motion of the nanoparticle near the oil−water interface. With this in mind, Section 4.1 first focuses on grafting polymer chains onto spherical nanoparticles. Section 4.2 investigates the effect of salinity, monovalent and divalent salts, on the radii of gyration of grafted neutral homopolymers and polyelectrolytes. Finally, Section 4.3 looks at PNPs at the oil−brine interface. 4.1. Construction of Grafted Nanoparticles. Figure 9a shows the HPAM chain with the negatively charged acrylate bead (Q) and the neutral acrylamide bead (P). We use three types of polymer chains in this study, as depicted in the central 9531


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Figure 10. Effect of grafting density and salinity on the average Rg of PNPs with (a) neutral grafts, (b) 25% acrylate grafts, and (c) 50% acrylate grafts.

Figure 11. Effect of grafting density on the distribution of cations around PNPs with (a) neutral grafts, (b) 25% acrylate grafts, and (c) 50% acrylate grafts. The salinity in all cases is 3.65 wt % of the monovalent salt.

average Rg of the grafts in the absence of added salt. As the graft density increases, so does Rg−the higher population density on the nanoparticle surface forces the chains to stretch out into the solution instead of forming mushroom-like structures. An example of the latter conformation is shown in the inset for ρ = 0.24 chains/nm2 in Figure 10a. Steric hindrance between the chains kicks in as ρ increases. The introduction of monovalent salt does not affect Rg (red squares), as both plots coincide with each other. The absence of negatively charged monomers does not provide the cations from the salt with a hook to latch onto the graft and induce collapse, as in Figure 6. Rg for grafts with 25% acrylate are presented in Figure 10b. Here, we vary the concentration of the monovalent (i.e., 1:1) salt from 0 to 3.65 wt % and use one formulation with just a divalent salt at a concentration of 3.66 wt %. In the absence of any salt, the average graft Rg varies between 7.5 and 8.5 Å with increasing ρ. The equivalent range for the neutral grafts in Figure 10a was 6.8−7.4 Å. This upward shift in the Rg plot is due to the addition of negatively charged acrylate beads; electrostatic repulsions between these beads cause the chain to expand more than a neutral polymer would under the same circumstances. However, unlike the neutral grafts, the 25% acrylate chains suffer a loss in Rg as the concentration of the monovalent salt is increased. Although 0.04 wt % of the salt has no effect (red squares), 0.39 wt % (black crosses) and 3.65 wt % (green triangles) definitely lower Rg for each value of ρ. The 3.66 wt % divalent salt solution (pink inverted triangles) decreases Rg even more than its monovalent counterpart (3.65 wt %, green triangles). We see a similar progression of the Rg plots in Figure 10c for the 50% acrylate case. The spread in the Rg plots as a function of salinity is higher in Figure 10c than in Figure 10b, that is, when there are more acrylate beads in the

surface area of nanoparticle) has been achieved. The grafted nanoparticle is introduced into the simulation box along with the solvent, counterions (if the graft is a polyelectrolyte), and monovalent/divalent salt (if brine is used). Figure 9b shows three examples of the polymer-grafted particle in a simulation box. From left to right, we see grafts that are neutral (green beads), 25% acrylate (orange beads), and 50% acrylate. Each graft in these snapshots contains 10 beads. The grafting density is 0.5 grafts/DPD area (1.2 chains/nm2), which translates to 55 chains attached to the nanoparticle surface. The solvent beads and ions are also present in the box but are not shown for the sake of clarity. A typical simulation with a single solvent takes one of these boxes as the input and lets it evolve according to the DPD force field. We record Rg of each graft during the production phase and report the average at the end. Therefore, the simulation of a single polymergrafted nanoparticle will produce a single value of Rg that has been averaged over all of the grafts. 4.2. Effect of Salinity on Graft Conformation. Figure 10a−c shows the average Rg of 10-bead-long grafted polymers as a function of the graft density, ρ. Note that we compute the average Rg of the grafted polymers and not of the nanoparticlegraft assembly. We vary the salt concentration and observe its influence on Rg. In the case of the neutral homopolymer grafts in Figure 10a, we only use a monovalent salt; the plots for the polyelectrolytes in Figure 10b,c use both monovalent and divalent salts. Each plot shows Rg for three graft densities: 0.24, 1.2, and 2.4 chains/nm2 corresponding to 11, 55, and 113 grafts on the nanoparticle surface, respectively. Figure 10a verifies whether the neutral PAM-like grafts are affected by salinity. We do not expect it to be so and that is exactly what the plots tell us. The blue circles denote the 9532


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Figure 12. Snapshots from the simulation of a polymer-grafted particle with grafting density = 0.24 chains/nm2 and an acrylate content of (a)−(c) 25% and (d)−(f) 50%. The salt concentration is (a) and (d) 0 wt %, (b) and (e) 3.65 wt % monovalent salt, (c) and (f) 3.65 wt % divalent salt.

remaining in solution rather than penetrating the graft layer. The peak intensities in Figure 11a for the neutral grafts are lower than those in Figure 11b,c for the polyelectrolytes with 25% acrylate and 50% acrylate, respectively. In both Figure 11b,c, we see that the cation density in the polymer shell (0 < r < 20 Å) is the highest for ρ = 2.4 chains/nm2 and the lowest for ρ = 0.24 chains/nm2, that is, contrary to the trend in Figure 11a. For a certain extent of hydrolysis, a colloid with a higher grafting density has a higher acrylate content. The electrostatic attractions between the negatively charged acrylate monomers and the cations pull the latter into the polymer corona. This explains the order of the density curves in Figure 11b,c. A divalent salt would lead to similar trends in the cation distribution near the nanoparticle surface. The cations in the graft layer screen the Coulombic repulsions between acrylate monomers and therefore reduce Rg of the grafted nanoparticle below the value at zero salinity (Figure 10b,c). However, Figure 10b,c also tells us that at the same salt concentration, higher grafting densities lead to higher Rg. For instance, at a salinity of 3.66 wt % of divalent salt (pink triangles, Figure 10b), the PNP with ρ = 0.24 chains/nm2 has Rg = 7 Å, whereas the PNP with ρ = 2.4 chains/nm2 has Rg = 7.9 Å. Higher grafting densities lead to higher radii of gyration even in strong brines due to (1) steric hindrance between the grafts that prevents them from collapsing onto the PNP surface; (2) Coulombic repulsions between acrylates that are not totally screened by cations in the graft layer; (3) Coulombic repulsions between the electrostatically trapped cations in the graft layer. The last point arises from the cation density profiles in Figure 11−higher graft densities have higher cation concentrations near the PNP surface. Given the brushlike environment at high ρ, it is possible that a charge inversion occurs wherein cations “adsorbed” to the polyelectrolyte grafts start to repel each other. The tables and plots presented thus far paint a rather dry numerical picture of phenomena in the simulation box. More graphic descriptions are shown in Figure 12a−f for a nanoparticle with 10-bead-long grafts and in Figure 13a−d for a nanoparticle with 50-bead-long grafts. The grafting density in both cases is 0.24 chains/nm2. Figures 12a−c and 13a,b show

grafted chains. Consider the comparable cases of 3.66 wt % divalent salt (purple inverted triangles) in Figure 10b,c: at ρ = 0.24, 1.2, and 2.4 chains/nm2, Rg drops respectively by (5.3, 8.8, 7.1%) for the 25% acrylate case (Figure 10b) and (8.2, 12.6, 8.3%) for the 50% acrylate case (Figure 10c). Cations from the salt diffuse through the graft layer and screen the interacrylate Coulombic repulsions leading to lower radii of gyration, though higher graft densities are preferable to lower ones. We also performed such simulations for 50-bead-long grafts (data not shown) and observed similar trends. An increase in the extent of hydrolysis makes the grafted chains more susceptible to saltinduced collapse. The intermonomer repulsion (aij = 78) is lower than monomer−water (aij = 80) repulsion, so there is an enthalpic gain associated with intrachain interactions. Therefore, Rg of the screened polyelectrolyte decreases from its original value at zero salinity. Figure 11a−c shows the density of cations from the monovalent salt in the vicinity of the nanoparticle. The salt concentration is 3.65 wt %. The y axis in each of these plots is the density of cations (number of cation beads/box volume) at a distance r from the surface of the nanoparticle. The x axis is the distance of the ion from the surface of the nanoparticle (referred to as “colloid” in these figures). The curves in Figure 11a−c were obtained for 10-bead-long grafts. In the case of the neutral grafts, the graft layer extends until a distance of 20 Å from the colloid surface. The region beyond that point is the bulk solution as labeled in Figure 11a. The three curves in each plot refer to the cation distribution for the three polymer grafting densities: 0.24 chains/nm2 (blue circles), 1.2 chains/ nm2 (red squares), and 2.4 chains/nm2 (black crosses). As the grafting density, ρ, increases in Figure 11a, the cation concentration near the nanoparticle surface decreases. The peak in the density profile at r ∼ 2.5 Å has an intensity of 0.25 for ρ = 0.24 chains/nm2 and 0.1 for ρ = 2.4 chains/nm2. The subsequent peaks in each curve decay thenceforth. The colloid with the highest grafting density of neutral chains (black crosses, ρ = 2.4 chains/nm2) prohibits cations from entering the polymer shell to the greatest extent. The cations are not electrostatically attracted to the neutral monomers in the grafts. Hence they choose the energetically favorable route of 9533


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In Figure 13, the systems in the first column had zero salinity, and those in the second column had 3.65 wt % of divalent salt. For both graft lengths and extents of hydrolysis, the grafts exist in an extended state at zero salinity and start to contract as salinity increases, and it is more prominent for divalent cations. Whereas this effect is somewhat noticeable in Figure 12 for the 10-bead graft, it is much more evident for the longer grafts in Figure 13. The star-shaped grafted nanoparticle is reduced to a relatively amorphous lump in the presence of divalent ions. The influence of the divalents is all the more potent when the degree of hydrolysis is 50% (Figures 12d−f and 13c,d). This is because multivalent ions can incapacitate multiple acrylates along the chain by screening their negative charges, leading to collapsed polymer conformations. 4.3. Grafted Nanoparticles at the Oil−Water Interface. In this final section, we shall simulate the behavior of the polymer-grafted nanoparticle in a box containing two phases: water and oil. The grafts are 10 beads in length and have a degree of hydrolysis of 50%. We qualify the oil domain as one in which the polymer is insoluble. Each oil molecule is a fourbead-long chain representative of dodecane. We construct a simulation box with equal volumes of oil and water. The oil is represented as yellow beads in subsequent figures, whereas the water region has intentionally been left empty so that the embedded nanoparticle may be visible. We implement periodic boundary conditions in all directions. The oil−water interface is a distinct boundary at the center of the simulation box. The grafted nanoparticle is initially placed at the center of the box. Once the simulation begins, the motion of the nanoparticle is governed by the polymer−water and polymer−oil interactions. As mentioned before, the bare nanoparticle itself has no

Figure 13. Snapshots from the simulation of a polymer-grafted particle with grafting density = 0.24 chains/nm2 and an acrylate content of (a)−(b) 25% and (c)−(d) 50%. The salt concentration is (a) and (c) 0 wt %, (b) and (d) 3.65 wt % divalent salt.

simulation snapshots for grafted particles with 25% acrylate; Figures 12d−f and 13c,d show configurations for grafted particles with 50% acrylate. The panels are arranged in ascending order of cation valency. Pictures in the first column of Figure 12 are from simulations that had zero salinity, whereas those in the second and third columns had 3.65 wt % of monovalent and divalent salts, respectively.

Figure 14. Effect of (a)−(c) grafting density at zero salinity and (d)−(f) salinity at constant grafting density on the equilibrium position of the grafted nanoparticle in a brine−oil system. The grafting density of chains varies as (a) 0.24 chains/nm2, (b) 1.2 chains/nm2, and (c) 2.4 chains/nm2. The valency of the salt varies as (d) 0 (0 wt %), (e) 1 (3.65 wt %), and (f) 2 (3.65 wt %). 9534


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Figure 15. Effect of grafting density and salinity on the distribution of oil molecules near the nanoparticle surface. The acrylate content of the graft varies as (a), (c), (e) 25% and (b), (d), (f) 50%. The valency of the salt varies as (a), (b) 0 (0 wt %), (c), (d) 1 (3.65 wt %) and (e), (f) 2 (3.65 wt %).

ρ = 0.24 chains/nm2) to the water phase (Figure 14c, ρ = 2.4 chains/nm2). This is because an increase in ρ increases the number of grafted polymers and hence the magnitude of polymer−oil repulsions. Additionally, Figure 10 tells us that PNPs with higher graft densities have higher radii of gyration. At zero salinity in Figure 14a−c, the grafts are fully extended, thereby maximizing attractive interactions with the water beads. The grafted nanoparticle therefore finds it energetically more favorable to move completely into the water phase at high ρ and minimize contact with oil. This result agrees with Yong who found that highly hydrophilic grafts force the nanoparticle to stay in the water phase, as that is more energetically beneficial than staying at the oil−water interface.46 Both sets of

preference for either phase. The water region also contains counterions and salt with either monovalent or divalent cations. As we go from top to bottom in Figure 14a−c, the graft density increases from 0.24 (Figure 14a) to 1.2 (Figure 14b) and 2.4 chains/nm2 (Figure 14c). The salinity is zero throughout this sequence. Similarly, as we go from top to bottom in Figure 14d−f, the cation valency increases from 0 (i.e., zero salinity, Figure 14d) to 1 (Figure 14e) and 2 (Figure 14f). The grafting density is uniform, 1.2 chains/nm2, throughout this set. Each panel in Figure 14 shows the position of the grafted particle at equilibrium. In Figure 14a−c, we see that increasing the grafting density alone makes the nanoparticle migrate from the oil−water interface (Figure 14a, 9535


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acrylate and ρ = 0.24 and 1.2 chains/nm2, we see that the peaks at r ∼ 2.5 Å increase as the cation valency increases. The oil density for ρ = 2.4 chains/nm2 (black crosses) becomes nonzero at r ∼ 15 Å in the absence of salinity (Figure 15a), r ∼ 12 Å with monovalent salt (Figure 15c), and r ∼ 10 Å with divalent salt. The increase in peak intensity for ρ = 0.24 and 1.2 chains/nm2 combined with the recession in the oil density for ρ = 0.24 and 1.2 chains/nm2 indicates that oil molecules are able to creep closer to the nanoparticle surface as the concentration of multivalent cations in the brine increases. Hence, the density of oil molecules in the vicinity of the particle surface increases as we go through Figure 15a−c−e. The same trend is observed for grafted particles with 50% acrylate (Figure 15b,d,f): as the cation valency increases from top to bottom, the oil density also increases in the range 0 < r < 20 Å, implying that oil molecules are able to penetrate the polymer shell. These findings agree with the equilibrium positions of the grafted particle in Figure 14d−f. The monovalent (Figure 14e) or divalent cations (Figure 14f) cause the polyelectrolyte grafts to collapse, thereby reducing the overall polymer−oil repulsion. The concentration of oil in the particle’s immediate surroundings within the polymer shell consequently increases and this is reflected in the oil density plots of Figure 15. A lateral comparison of the density profiles in Figure 15 shows that for the same amount of added salt, grafts with 50% acrylate are more effective than those with 25% acrylate in preventing the oil from approaching the nanoparticle’s surface. This is understandable, as the interacrylate electrostatic repulsions will be greater for the graft with more acrylate monomers. The difference between the oil density curves for the two degrees of hydrolysis increases with the grafting density. A nanoparticle that is densely grafted with grafts having a high acrylate content bars the oil from wetting its surface. It will therefore be less likely to stay near the oil−water interface, preferring the bulk water phase instead. Thus, the proportion of negatively charged monomers (or the extent of hydrolysis) will affect the equilibrium position of the grafted nanoparticle. This behavior has implications for the viscosity-enhancement property of the PNP. The effective viscosity of the water phase is directly proportional to the bulk concentration of suspended PNPs. If the PNP is interfacially active, it will not contribute to altering the viscosity of the bulk water phase; alternatively, if the PNP prefers to dissolve in the bulk water phase, it will contribute to increasing its viscosity. The viscosityenhancement capability of the PNP is highly sensitive to the functionalization of its surface.

simulations were conducted at zero salinity, so the findings are comparable. A slightly hydrophilic graft will maintain the PNP at the interface while simultaneously fanning out to decrease the oil−water IFT.46 At the other extreme, a highly hydrophobic graft that prefers the oil to water will transport the entire PNP from the water domain to the oil domain.44−46 The converse is observed in Figure 14d−f when cation valency is varied while keeping ρ fixed at 1.2 chains/nm2. We know from Figure 14b that a grafted particle with ρ = 1.2 chains/nm2 prefers the water phase to the interface at zero salinity. That is the starting point shown in Figure 14d. As the cation valency increases, the equilibrium position of the nanoparticle shifts from the water phase (Figure 14d, zero salinity) to the oil−water interface (Figure 14f, divalent salt). We have already proved in Section 4.2 that the introduction of salt, monovalent or divalent, causes the polyelectrolyte grafts to contract. The grafts are in an extended configuration in Figure 14d due to the absence of any salt. The nanoparticle as a whole, with ρ = 1.2 chains/nm2, prefers the aqueous phase to maximize attractive interactions with water and minimize repulsions with oil. As the cation valency increases, the chains start to contract, minimizing their surface area that is exposed to the solvent. Figure 13a,b displays the contraction of grafts with 25% acrylate (ρ = 0.24 chains/nm2) in brine. In addition, Figure 10c shows the decrease in Rg of a PNP with ρ = 1.2 chains/nm2. More significantly, at the same concentration, a divalent salt (Figure 10c, pink inverted triangles) reduces Rg to a greater extent than a monovalent salt (Figure 10c, green triangles). A divalent cation can screen two acrylate monomers simultaneously and is more potent than a monovalent cation in causing a polyelectrolyte to collapse. The contraction of the polymers in brine reduces the surface area of the graft layer exposed to the oil. This reduction in exposed surface area is directly proportional to the valency of the cation. The overall polymer−oil repulsion therefore decreases as we travel down from Figure 14d to 14f. This decrease in the polymer−oil repulsion lowers the force that pushes the PNP away from the interface into the water phase. The equilibrium position of the nanoparticle consequently shifts to the interface as the cation valency increases. The same trend in Figure 14d−f can be obtained by simply increasing the concentration of the monovalent salt, as the grafts will contract at high salinity. The trends in Figure 14 also hold for grafts with an acrylate content of 25%. We expect the grafting density, not the salinity, to affect the equilibrium position of a nanoparticle with neutral grafts. Although our model polymer is not amphiphilic, the amphiphilicity of the PNP in these situations may be tuned through the polymer−oil interaction, via the graft architecture, grafting density, extent of hydrolysis, and concentration of monovalent/divalent cations in brine. The contraction of the grafts as a function of cation valency is reinforced on viewing the density of oil molecules near the nanoparticle surface (Figure 15a−f). The oil density profiles (number of oil beads/box volume) graphically quantify the wetting of the nanoparticle surface by the oil. The first column, Figure 15a,c,e, contains the oil density profiles for grafted particles with 25% acrylate. The second column, Figure 15b,d,f, is for grafted particles with 50% acrylate. Each panel has three curves for the three grafting densities: 0.24 chains/nm2 (blue circles), 1.2 chains/nm2 (red squares), and 2.4 chains/nm2 (black crosses). In the first row of panels, salinity is zero; the second row has 3.65 wt % of monovalent salt, and the third row has 3.65 wt % of divalent salt. In Figure 15a,c,e with 25%

5. CONCLUSIONS In this article, we have disclosed a coarse-grained molecular simulation-based approach to analyze the structural stability of polymers used in EOR. The model polymers were based on neutral PAM and partially hydrolyzed PAM. We studied their behavior under two conditions: (1) free polymer in a solvent and (2) multiple polymers grafted to a nanoparticle and the whole assembly embedded in a solvent. We used the first case to calibrate the force field that governed the polymer−water and polymer−oil interactions. In both cases, we quantified the effect of salinity, monovalent and divalent salts, on the radius of gyration, Rg, of the polyelectrolyte. The presence of salt in water is expected to reduce the hydrodynamic volume of a polyelectrolyte. The cations are electrostatically attracted to the anionic monomers and screen their charges. This causes the polymer to lose its extended 9536


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beneficial combination of system parameters. We note that the actual computation of viscosity is an involved problem that requires more accurate force fields. Hence, it lies in the domain of atomistic simulations. We can extend these methods to studying novel polymeric structures for EOR, such as comb polymers and Linked Polymer Systems (LPS), under saline conditions. The former is a highly branched structure, typically with hydrophilic branches and a hydrophobic backbone, and the latter is a cross-linked network of linear polymers. Comb PAMs are being investigated as viscosity-enhancement agents, whereas the LPS are used as conformance control agents. In the case of LPS, it is important to monitor the extent of cross-linking (alternatively, the activity of the cross-linking agent) so as not to form a viscous gel with near-zero mobility in reservoir pores. A coarse-grained polyelectrolyte model could be used to track the formation of cross-links as a function of cross-linker concentration and salinity. Ideally, a phase diagram mapping out different configurations − free polymer, colloidal aggregates and gel − would be useful in identifying polymer and cross-linker concentrations that yield the desired cross-linked states. Such a phase diagram, if predicted through simulations, would guide or minimize experimental efforts in achieving the target product. Our model can also be used to study the solvent conditions under which two or more PNPs would aggregate or adsorb onto a surface. This is important in the context of using PNPs as interfacially active species−aggregation with each other or adsorption on rock would reduce their interfacial activity. It is therefore clear that coarse-grained simulations have a significant role to play in the EOR polymer domain.

conformation and form a coil. Our simulations of linear polyelectrolytes in water reproduced this phenomenon. The drop in Rg is steeper for polyelectrolytes with a higher content of negatively charged monomers. Therefore, as the degree of hydrolysis of PAM increases, so does its vulnerability to cations. Although this is a well-established experimental observation, its replication in the simulation gives us confidence in the polymer−water−ion interactions that we have imposed. We performed a second test with branched polyelectrolytes where the negatively charged monomers dangled from the linear backbone. The branch was either one or two beads long in our computations. The latter setup represented the poly(AMPS-coAA) polymer, which is more stable toward divalent attack according to the literature. Our simple representation of such a branched polyelectrolyte indeed suffered a smaller drop in Rg in the presence of divalent ions when compared to its linear counterpart. This suggests that moving the anionic monomer away from the backbone should improve the polyelectrolyte’s tolerance to salinity. The final investigation performed in this report looked at the conformations of polymers grafted on the surface of a nanoparticle in brine and oil−brine systems. We characterized the average Rg of the grafts as a function of the salinity and the grafting density. Increasing the latter increases Rg due to steric hindrance between neighboring grafts on the crowded nanoparticle surface. Increasing the salinity lowers Rg as observed in the free polymer simulation. This decrease in Rg occurs almost uniformly across the chosen grafting densities. These simulations used grafts that were 10 beads in length, but we expect the trend to hold, and be more prominent, for longer grafts. The study culminated in a system where the grafted nanoparticle was positioned at the interface of oil and water domains. We simulated the motion of the nanoparticle under different salinities in the aqueous phase and for various grafting densities as well. The polymer was designed to be hydrophilic and oleophobic, therefore increasing the grafting density at constant salinity moved the grafted particle away from the interface into the water domain. The converse occurs when the salinity is increased (zero to monovalent and divalent salt) at a constant grafting density. In this way, the interfacial activity of the polymer-grafted nanoparticle can be tuned by varying at least two parameters: salinity and grafting density. Both sets of results were reported for a single degree of hydrolysis, so that is one more handle that can significantly affect the particle’s behavior in a two-phase system. The inspiration for this work was the connection between viscosity enhancement and the morphology of the polyelectrolyte in solution. The various simulations in this article have shown the tight link between the salinity and polymer conformation, especially at high degrees of hydrolysis. The zeroth-order assumption is that the system handles listed in the previous paragraph − grafting density, salinity, and extent of hydrolysis − will simultaneously affect the graft conformation and the solution viscosity in the same direction. For example, if a polyelectrolyte graft is not susceptible to salt-induced collapse due to a combination of graft density and extent of hydrolysis, it will lead to viscosity enhancement. Additionally, the interfacial activity of the PNP detracts from its viscosity-boosting function because the effective viscosity of the water phase depends on the concentration of PNPs suspended in the bulk. The surface functionalization of the nanoparticle in terms of the parameters listed above governs its efficiency. The ultimate purpose of the model is to screen various polymer chemistries to identify this

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AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected]. Tel: +91-725-902-7943 (N.N.). *E-mail: fl[email protected]. Tel: +1-281-658-7484 (F.M.C.). Present Address ⊥

Amgen Inc., Cambridge, Massachusetts, United States (M.P.).

Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We thank Sudip Roy and Foram Thakkar from Shell Technology Centre, Bangalore, and Cesar Mantilla from Shell Technology Centre, Houston, for going through the article and providing valuable suggestions. We are also grateful to Shell for funding this work and providing permission to publish this article.

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REFERENCES

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Coarse-Grained Simulations of Polymer-Grafted Nanoparticles

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