Self-Consistent Field Modelling of Microplastic Particle Formation and Adsorption of Macromolecular Pollutants

Abstract

Accumulation of microplastics (MPs) in aqueous environments poses a serious ecological problem nowadays. MP particles are able to adsorb pollutants of different kinds and to transport them to living organisms, leading to biotoxicity. Hence, investigation of the adsorption of pollutants of different molecular weights onto MP particles is an important task. We employed the numerical Scheutjens–Fleer self-consistent field method to study (i) the formation of MP particles consisting of homopolymer macromolecules and (ii) the adsorption of pollutant homopolymer chains onto the MP particles. Under poor solvent conditions, the polymer macromolecules were shown to form MPs with a constant density inside the particle and with an interfacial layer at its periphery. The size of the MP particles and the thickness of the interfacial layer were controlled by the solvent quality. MP particles were shown to adsorb pollutant polymer chains from the surrounding liquid due to higher compatibility of the MP particle with the pollutant polymer chains as compared to the solvent. The amount of adsorbed polymer pollutant increased with the increase of its concentration in solution. Softer MP particles were shown to adsorb larger amounts of pollutants due to a broader interfacial layer. The conformational characteristics of the adsorbed polymer chains (trains, loops, and tails) were studied in detail.

1. Introduction

Microplastics (MPs) are plastic particles smaller than 5 mm. MPs in the environment are classified into two major categories: primary and secondary MPs [1,2]. Primary MPs are generated due to commercial and industrial processes. Examples include microbeads in facial scrubs, shampoos, body washes, toothpaste, disinfectants, air-cleaning media, etc. Secondary MPs evolve from the degradation of large plastic products promoted by weathering, ultraviolet radiation, and physical crushing. Both primary and secondary MPs originate from the most commonly used plastics, in particular, polyethylene terephthalate, polyethylene, polyvinylchloride, polyurethane, polystyrene, and polypropylene.

According to the literature, MPs have unique physicochemical properties. In addition to their own detrimental effects, MPs possess a large specific surface area, hydrophobic properties, and minuscule size, which allow them to adsorb various environmental chemicals and to transport them from the environment to living organisms [3,4]. Particularly notable is the extremely large specific surface area. This characteristic makes MPs highly reactive toward various pollutants and leads to the high ability of MP particles to adsorb various contaminants [5,6,7,8,9]. Adsorption and desorption are two key factors regulating the vector potential of MPs, resulting in the accumulation of chemicals in organisms. This effect has been actively studied in the literature. Primary research has focused on the adsorption processes of heavy metals and pesticides [10,11,12,13,14,15,16,17], organic compounds such as polycyclic aromatic hydrocarbons [18,19,20,21,22,23,24,25] and pharmaceutical substances, notably antibiotics [26,27,28,29,30,31,32].

While most studies have focused, for example, on the adsorption of low-molecular weight pollutants (i.e., hydrophobic organic chemicals), recent research has also highlighted the interaction of MPs with macromolecular contaminants. Numerous studies have confirmed that MPs can enhance the bioaccumulation of hydrophobic pollutants in organisms, mediated by their role as pollutant vectors exceeding the uptake from contaminated water alone. Indeed, the concentration of the contaminant at the MP surface could be much higher than in the environment, thus creating an additional risk for organisms [6]. It should also be mentioned that hydrophobic interactions are highly dependent on the environmental temperature, which is extremely important for adsorption–desorption processes.

Meanwhile, electrostatic interactions play a significant role in the adsorption process as well. In the case of negatively charged MPs, they can accumulate cationic polymers, which are widely used in water treatment, the food industry, and cosmetics and can act as environmental toxins. The study by Yuzhanin et al. demonstrated that in aqueous solutions, cationic polymers, such as poly(diallyldimethylammonium chloride) and poly-L-lysine hydrobromide, are electrostatically adsorbed onto anionic polystyrene microspheres—a model system for MPs. Notably, the adsorbed polymers can migrate between particles, leading to a uniform distribution of macromolecules across all microspheres in the system [33]. In [34], two types of micro-sized polymer particles were described, mimicking the behavior of real microplastics. Both types of anionic species electrostatically adsorb cationic polymers. The adsorption is accompanied by neutralization of the particle charge and aggregation of the species at mutual neutralization of the particles and polycation charges. Polycations pass from their complexes with microgels to free microgels, which results in dissolution of the aggregates and formation of homogeneous solutions; however, the same polycations are not desorbed from microspheres when free microspheres are added, and the aggregates are preserved. No redistribution or dissolution is observed in the microgel–polycation–microsphere ternary systems.

Despite the active development of experimental studies on the adsorption of various macromolecules on the surface of MPs, which is associated with their widespread distribution in the environment and potential impact on ecosystems and living organisms, theoretical studies of the patterns of adsorption of homo- and hetero-polymers on MP particles, relevant in this field, have not been sufficiently reported.

In this paper, we focused on the study of hydrophobic effects in the organization of MPs and their interactions with polymers. It is well-known that the most common industrial polymers are highly hydrophobic and thus can accumulate different polymers from the surroundings due to hydrophobic interactions [35,36,37]. Theoretical studies and computer simulations provide a unique opportunity to assess the impact of key system parameters over a wide range, which is often difficult to achieve in experiments. In particular, it is possible to vary the structural characteristics of the polymer (chain length, flexibility, chemical heterogeneity, volume fraction in the solution); the properties of the solution (solvent quality, ionic strength, pH, temperature); the morphology of the MP surface (particle size, hydrophobicity, charge, degree of roughness). Typically, studies of this kind use generalized and simplified polymer models, and their relation to the specific chemical structure of macromolecules is determined through the parameters of interaction of monomer units with each other and with the low-molecular solvent molecules and the intrinsic characteristics of the polymer chain, such as its thermodynamic stiffness. Such studies not only provide a deeper understanding of the mechanisms of adsorption but also predict the behavior of polymers under real conditions, which is important for assessing environmental risks and developing strategies to minimize the negative impact of MPs.

This paper presents the results of research on the adsorption of high-molecular-weight pollutants—homopolymers—on model MP particles. Hydrophobic effects are described by the Flory parameter [38]. The study was carried out by using the Scheutjens–Fleer self-consistent field method [39]. The features of the formation of model MP particles in a solution of linear polymers and during the collapse of a single star-shaped macromolecule with a large number of arms of equal length were studied. We investigated the effects of the homopolymer chain length, the volume fraction of the homopolymer in the solution, and the interaction parameters of the homopolymer–solvent, homopolymer–MP particle, and MP particle–solvent on the adsorption due to the hydrophobic interactions.

The paper is organized as follows. In Section 2, the model of the system under study is presented and the application of the numerical self-consistent field (SCF) method by Scheutjens and Fleer to study the adsorption of homopolymers on model MP particles is described. Section 3 is devoted to the presentation and discussion of the results obtained. Section 4 reports the conclusions and final comments.

2. Model and Method

2.1. Model

In the present work we study an MP particle immersed in a polymer solution. Both the particle and the solution contain homopolymers, which we denote as polymer A and polymer B, respectively. The MP particle is formed by n_A molecules of polymer A with a degree of polymerization N_A under thermodynamically poor solvent conditions. The MP particle contains n_A N_A units of polymer A and has a density close to unity. The polymer solution contains soluble chains of polymer B, each consisting of N_B monomer units; the bulk volume fraction of polymer B in the solution is ${\phi }_B^b$. The polymer volume fraction φ is a convenient dimensionless measure of the polymer concentration c in the solution. These two quantities are directly related to each other as φ = ca³, where a is the monomer unit size equal to the lattice cell size. In what follows, we will use the terms “polymer volume fraction” and “polymer concentration” as synonyms.

Polymer–solvent and polymer–polymer interactions are described in terms of the Flory–Huggins theory [38] by using the Flory interaction parameters. The choice of these parameters (χ_AS and χ_BS for polymer—solvent interactions and χ_AB for polymer A—polymer B interactions) plays a crucial role. These are determined by ensuring that the polymer adsorption on the particle does not disrupt its integrity. For polymer A, positive values of the Flory parameter corresponding to strong precipitant conditions, χ_AS ≥ 2, were chosen. For polymer B, the athermal solvent conditions were taken: χ_BS = 0. Finally, for the interaction of the dissolved polymer B with polymer A, forming the MP particle, positive values χ_AB ≥ 0 of the Flory parameter corresponding to a weak incompatibility between polymers A and B were chosen. In this case, adsorption of polymer B onto the MP particle occurs due to the screening of unfavorable polymer A—solvent interactions by more favorable polymer A—polymer B interactions. The parameter determining the adsorption is the difference Δχ_A = χ_AS − χ_AB: the greater this value, the stronger is the effective attraction of B units to the MP particle. The Flory interaction parameter is defined as [38]

$${\chi }_{ij}={\displaystyle \frac{1}{z}}·{\displaystyle \frac{w_{ij}-(1/2)(w_{ii}+w_{jj})}{k_{B}T}},$$

(1)

where w_ii, w_jj, and w_ij are the energies associated with the pair contacts (ii, jj, and ij), z is the lattice coordination number (or number of lattice cells that are the nearest neighbors to a given cell), k_B is the Boltzmann constant, and T is the temperature. Hence, all three interaction parameters (χ_AS, χ_BS, and χ_AB) can be controlled by varying the temperature. According to the Flory–Huggins theory, the values of χ_AS and χ_BS define the quality of the solvent S for polymers A and B, respectively, namely, good solvent conditions when a polymer swells in the solvent for χ_AS and χ_BS < 1/2, and poor solvent conditions when the polymer collapses for χ_AS and χ_BS > 1/2. The specific case χ_AS, χ_BS = 1/2 corresponds to theta solvent conditions when the net two-body polymer–solvent interaction becomes zero.

We also consider, as a reference system, the MP particle formation under poor solvent conditions by a star-shaped A-polymer (polymer star) having n_A homopolymer arms, each consisting of N_A monomer units.

2.2. Method

The system was studied by using the Scheutjens–Fleer self-consistent field (SF-SCF) method [39]. This approach has been widely used to model polymer adsorption for more than 40 years [40]. In the SCF approach, various interactions between particles in the system are replaced by average effective interactions. The SF-SCF method is based on the mean-field approximation and a discrete lattice representation of the underlying space. Its advantage lies in its computational efficiency with a high convergence rate combined with reproducible results. Due to the symmetry of the problem, a one-gradient spherical lattice was chosen for the calculations [39]. Calculations were performed by using the sfbox program, which implements the SF-SCF method, developed in the Laboratory of Physical and Colloid Chemistry at the University of Wageningen, the Netherlands.

The calculations were performed in two stages (Figure 1): first, the solvent and the polymer A were present in the simulation cell, and an MP particle was formed. Here, an additional “seed” monomer unit of type A was placed in the first layer of the system—the center of the spherical lattice. This seed unit facilitates the assembly of the units into a spherical particle but has no actual effect on its size and the polymer A density distribution. In the second stage, polymer B with the bulk volume fraction ${\phi }_B^b$ was added to the system. For each set of system parameters, the equilibrium distributions of the volume fractions of its components were calculated. In the case of a star-shaped polymer forming the MP particle, the center of the star was placed in the center of the spherical lattice.

3. Results and Discussion

3.1. MP Particle Formation

First, we discuss the formation of MP particles under poor solvent conditions. For all considered systems, polymer volume fraction profiles were calculated, which show that under thermodynamically poor solvent conditions, a dense spherical particle with a constant-density core and a thin transition interfacial layer (“fringe”) is formed (Figure 2a). In the case of the star-shaped polymer (Figure 2b), the core of the MP particle is denser due to the chain connectivity at the central point (branch point) of the polymer star.

The particle size of the MP can be determined by using the average density of the polymer in it:

$$\overline{{\phi }_A}={\displaystyle \frac{{\sum }_r{\left[{\phi }_A\left(r\right)\right]}^{2}L(r)}{{\sum }_r{\phi }_A(r)L(r)}}={\displaystyle \frac{{\sum }_r{\left[{\phi }_A\left(r\right)\right]}^{2}L(r)}{N_A{n}_A}},$$

(2)

where $L\left(r\right)=4\pi \left(r^2-r+1/3\right)$ is the number of lattice cells in the layer with number r; therefore the product φ_A(r)L(r) is the number of polymer units in the layer with number r, and summation over all layers gives the total number of units in the system N_tot = N_A n_A. Then, if we approximate the MP particle by a sphere with a constant density $\overline{{\phi }_A}$, the radius of the sphere is

$$R_0={\left({\displaystyle \frac{3N_A{n}_A}{4\pi \overline{{\phi }_A}}}\right)}^{1/3}.$$

(3)

Figure 3 shows the dependencies of the MP particle radius and its average density on the solvent quality—the Flory parameter χ_AS—for particles formed in a solution of chains with the degree of polymerization N_A = 10, 20, 50, 100 and a particle formed by a collapsed star of 20 arms of length N_A = 500, with a fixed total number of monomer units in the system N_tot = N_A n_A = 10⁴. It is evident that in the strong precipitant regime, the particles’ size and their average density are the same for different N_A. Therefore, the R₀(χ_AS) and $\overline{{\phi }_A}\left({\chi }_{AS}\right)$ dependencies for the star play the role of the master curves, and the dependencies for the solution coincide with those at large χ_AS but show an increasing difference with decreasing χ_AS. Moreover, there is a threshold value of χ_AS below which the formation of a stable MP particle by linear polymer chains is not possible, whereas the MP particle formed by a star macromolecule is stable at any χ_AS value.

In order to quantify the thickness of the outer interfacial layer, we analyzed the decay of the polymer volume fraction profile, which is defined by the absolute value of the first derivative of the function $\phi \left(r\right)$: $W\left(r\right)=\left|{\phi }^{\prime }\left(r\right)\right|$. In the framework of the lattice model this function is calculated as a difference between the values of $\phi \left(r\right)$ for neighboring spherical layers: $W\left(r\right)=|\phi \left(r+1\right)-\phi \left(r\right)|$. Figure 4a shows the function $W\left(r\right)$ for the MP particle formed by linear polymer chains at different values of χ_AS and at N_tot = N_A n_A = 10⁴. We found that the maximum of the function $W\left(r\right)$ (i.e., the maximal decay of the density profile) is very close to the radius of the MP particle R₀ introduced according to Equation (2). As an example, the vertical dashed line in Figure 4a shows the value of R₀ at χ_AS = 3. One can see that the value of R₀ is very close to the maximum of the function $W\left(r\right)$. Since the position of the interfacial layer is determined by the maximum of the function $W\left(r\right)$, the thickness of the interfacial layer $∆R$ can be related to the width of the function $W\left(r\right)$ in the vicinity of its maximum. Therefore, we define $∆R$ using the following mean-square quantity:

$${∆R}^2={\displaystyle \frac{{\sum }_r{\left[r-R_0\right]}^{2}W\left(r\right)L\left(r\right)}{{\sum }_{r}W\left(r\right)L\left(r\right)}},$$

(4)

The dependency of the interfacial layer thickness on the solvent quality (Figure 4b) shows that the thickness is about the monomer unit size and increases with amelioration of the solvent quality. Differences in the layer thickness for MP particles assembled by short or long chains, including stars, are virtually indistinguishable. In the poor solvent regime, the MP particles formed by linear macromolecules and by the star have identical interfacial layer thickness. Since the position of the interfacial layer can be identified from the steepest decay of the polymer volume fraction profile below, for analyzing the adsorption of pollutants on MP particles we will calculate the size of the particle R₀ from the position of the maximum of $W\left(r\right)=\left|{\phi }^{\prime }\left(r\right)\right|$ function.

3.2. Polymer Adsorption onto MP Particle

3.2.1. Density Profiles

Figure 5a shows density profiles of polymer A forming MPs under poor solvent conditions at χ_AS = 2 and 2.5. In addition to polymer A and solvent S, polymer B was also present in the system at certain concentration ${\phi }_B^b$, which was varied over a wide range from ${\phi }_B^b={10}^{-6}$ to ${\phi }_B^b=0.9$. The interaction parameters χ_AS, χ_BS, and χ_AB were chosen in such a way that the presence of the polymer B, taken even at a very high concentration, did not affect the φ_A(r) profile in any way, so the profiles shown in Figure 5a are universal in the sense that they are determined by the χ_AS value only. It can be seen that at higher values of χ_AS, a denser particle is formed.

As is well known, a typical conformation of an adsorbed chain consists of alternating trains (adsorbed sequences) and loops flanked at the ends by two tails (Figure 6).

To distinguish the adsorbed and non-adsorbed parts in the chain conformations, it is necessary to define the boundary of the “soft” MP particle. Indeed, the particle density profile consists of an extended high density “plateau” in the center and a thin “fringe” at the periphery; in the fringe region, a rather sharp drop to almost zero density is observed (Figure 5a). The boundary of the dense part can be defined as the point (i.e., the layer number) at which the drop in the density profile is maximal, that is, it is the maximum point for the derivative dφ_A/dr (and the minimum for its absolute value). For the lattice model we employ, the “point” means the layer number, and for the cases shown in Figure 4, the value of R₀ is the same and equals R₀ = 11 due to the discreteness of the lattice. It does not change when a polymer is added to the system in the specified range of Δχ_A values. Only in the case χ_AS = 2, χ_AB = 0, and ${\phi }_B^b=0.9$ does the position of the layer boundary change and become equal to R₀ = 10, but since this corresponds to a single point and an extremely high polymer concentration, we can simply exclude this point in the parameter space from further consideration.

Having defined the position of the boundary (or the cutoff radius) R₀, we can provide a rigorous definition for the adsorbed sequences, or trains, loops, and tails in our model:

• A train is a continuous section of the polymer chain whose monomer units are located in layers R ≤ R₀ + 1
• A loop is a continuous section of a polymer chain whose end units are located in the layer R = R₀ + 2, and all units of the loop have coordinates r ≥ R₀ + 2. The loop connects two trains
• A tail is a continuous sequence of units at the end of the polymer chain, where its first unit is located in the layer r = R₀ + 2, and all tail units, including the free end, have coordinates r ≥ R₀ + 2. The tail is connected to a train.

Furthermore, the definition of the R₀ boundary allows us to divide all B-type chains into adsorbed and free (non-adsorbed) ones:

• A chain is considered as adsorbed if it has at least one monomer unit with the coordinate r ≤ R₀ + 1
• A free chain has no contact with the particle, and all its links are located in layers r ≥ R₀ + 2.

Hence, the contributions of the adsorbed (${\phi }_B^a$) and free (${\phi }_B^f$) polymer to the density profile ${\phi }_B$ can be separated:

$${\phi }_B(r)={\phi }_B^a(r)+{\phi }_B^f(r),$$

(5)

How does the adsorption of polymer B occur on a particle assembled from polymer A? Figure 5b shows the density profiles of polymers A and B at different concentrations of polymer B on the particles assembled at χ_AS = 2 and 2.5. We see that the B-polymer is indeed adsorbed on the particle being localized at the particle/solvent interface; the higher the polymer concentration in the solution, the more the polymer is adsorbed on the surface. At the same time, a softer particle with a looser surface layer adsorbs a larger amount of polymer than a denser particle with the same value of the effective adsorption parameter Δχ_A. This is in full accordance with the picture of polymer adsorption in wide potentials: wider potentials adsorb more polymer [41].

3.2.2. Adsorption Isotherms

Let us turn to the integral characteristics of adsorption. To quantify the amount of adsorbed polymer, we use the criterion introduced above which allows the separation of “adsorbed” chains from “non-adsorbed” ones. Based on this criterion, the amount of polymer adsorbed on the particle can be calculated:

$$\Gamma =\sum _r{\phi }_A^a\left(r\right)L(r)=\sum _r\left[{\phi }_A\left(r\right)-{\phi }_A^f\left(r\right)\right]L(r).$$

(6)

Figure 7 shows isotherms of polymer adsorption onto an MP particle formed by n_A = 50 macromolecules of polymer A consisting of N_A = 100 monomer units at different values of the adsorption parameter Δχ_A. We see that the amount of adsorbed polymer increases with increasing concentration, and a softer particle (formed at lower values of χ_AS) adsorbs a larger amount of polymer.

Another integral characteristic that is ideologically similar to Γ, but not directly proportional to it, is the number of units immediately connected to the particle surface (i.e., polymer B units that fall in the range r ≤ R₀ + 1):

$$\theta =\sum _{r⩽R_0+1}{\phi }_A\left(r\right)L\left(r\right).$$

(7)

For θ (Figure 8a), the same trends are observed as for Γ: an increase with increasing polymer concentration in solution and higher values for softer (less dense) particles.

The root-mean-square thickness of the adsorbed layer is defined as

$$t_{rms}^2={\displaystyle \frac{{\sum }_{r\ge R_0+1}{\left(r-R_0\right)}^2{\phi }_A^a\left(r\right)L\left(r\right)}{{\sum }_{r\ge R_0+1}{\phi }_A^a\left(r\right)L\left(r\right)}},$$

(8)

where the particle boundary R₀ is determined by the maximum decay in the density of polymer A at the periphery (Figure 5a). It is clear that the thickness of the adsorbed layer is determined by the amount of polymer adsorbed on the particle, so it is also an increasing function of the polymer concentration (Figure 8b). On the other hand, we see that the dependence of thickness on the adsorption parameter Δχ_A (effective adsorption energy) is opposite: the higher Δχ_A, the thinner is the adsorption layer. This fact is well known, starting from the pioneering works by de Gennes [42,43]. The higher the adsorption energy, the greater is the number of units that bind directly to the surface and the shorter are the loops and tails. It is also noteworthy that the dependence of the layer thickness on the polymer concentration is rather weak for moderately strong adsorption (for example, at Δχ_A = 1.5) and its variation with increasing concentration is of the order of the monomer unit size. At high concentrations the thickness of the adsorbed layer practically does not depend on the adsorption strength.

3.2.3. Individual Chain Characteristics: Trains, Loops, and Tails

So far, we have considered “multi-chain” characteristics of polymer adsorption on the surface of the MP particle. At the same time, it is interesting to look at the average characteristics of a single chain in the ensemble of all adsorbed chains.

These are, first of all, the quantities associated with the elements of the adsorbed chain-conformation (Figure 6): adsorbed sequences, or trains (tr), loops (lp), and tails (tl) and their distributions in the chain which give access to their average values—the average fraction of chain links contained in elements of type i in the chain (ν_i), the average number of such elements in the macromolecule (n_i), and the average length of a sequence of monomer units of this type (ℓ_i).

Figure 9 shows the dependencies of these characteristics for trains on the polymer concentration in solution. Note that the fraction of monomer units in trains (ν_tr) is equivalent to the fraction of adsorbed units in a single chain, this being of primary importance when a single-chain adsorption is studied. Here, we see a very interesting and even unexpected effect: with an increase in the polymer concentration in solution, the fraction of adsorbed units in an individual macromolecule decreases, in contrast to the fact that the total number of adsorbed units in all adsorbed macromolecules increases (as one can see in Figure 8a). The number of trains in the chain (n_tr) and their average length (ℓ_i) also decrease—in most of the polymer concentration range. This suggests that an increase in the polymer concentration enhances competition between the dissolved macromolecules for binding sites on the finite-sized MP particle surface, and the “anchoring” of new chains on the surface occurs due to the desorption of some units in the already adsorbed chains (it should be noted that the equilibrium state of the system is a dynamic equilibrium). In the low polymer concentration regime, all three characteristics (ν_tr, n_tr, and ℓ_tr) are weakly dependent on the concentration ${\phi }_B^b$ and have the form of a plateau, which is more extended in the case of relatively weak adsorption (lower Δχ_A values). In the high concentration regime, the concentration dependencies of ν_tr, and ℓ_tr are non-monotonic and pass through the minimum; the $n_{tr}\left({\phi }_B^b\right)$ dependence monotonically decreases. To understand the reason for the change in the behavior of ν_tr, and ℓ_tr at high ${\phi }_B^b,$ we calculated the concentration dependencies of the number of solvent molecules in the adsorption region as

$${\theta }_s=\sum _{r⩽R_0+1}{\phi }_s\left(r\right)L\left(r\right)=\sum _{r⩽R_0+1}[1-{\phi }_A\left(r\right)-{\phi }_B\left(r\right)]L\left(r\right)$$

(9)

which is the number of “vacancies” that can be filled by polymer B. It can be seen that in the region of low polymer concentrations (${\phi }_B^b<{10}^{-1}$) θ_s changes slightly, i.e., the number of filled vacancies remains almost constant, which explains the decrease of ν_tr and ℓ_tr upon redistribution of monomer units. On the other hand, at high concentrations (${\phi }_B^b>{10}^{-1}$), a notable (more than six times) decrease in θ_s is observed, which indicates that the polymer forming the concentrated solution begins to fill these “vacancies”, replacing the solvent (which amount in the system decreases with increasing ${\phi }_B^b$) in the interfacial layer of the MP particles; this causes the growth of ν_tr and ℓ_tr with increasing concentration.

It is also interesting to note that the fraction of adsorbed units (ν_tr), the number of trains (n_tr), and their average length (ℓ_tr) are slightly larger for adsorption on particles formed under poorer solvent conditions. If we were discussing the adsorption of a single macromolecule, we would expect a completely opposite trend: a softer particle has a wider adsorption layer and can therefore adsorb the polymer more easily. In the case of a polymer solution, we actually observe an increase in the integral number of adsorbed macromolecules as well as in the number of monomer units bound to the particle’s surface, as the particle softens. This suggests once again that an increase in the amount of adsorbed polymer during adsorption from solution is achieved by increasing the number of adsorbed macromolecules, while each of the macromolecules individually uses only a part of its “capabilities” (i.e., monomer units) to bind to the particle.

As the polymer concentration increases, the average fraction of units in the loops (ν_lp) decreases too (and the average number of loops and trains behaves similarly because of the obvious ratio n_lp = n_tr − 1), while the average loop length (ℓ_lp) increases (see Figure 10). The stronger the effective attraction of polymer B to the particle, the shorter are the loops. The deterioration of the solvent quality for polymer A causing the particle compaction and narrowing of its interfacial layer leads to the decrease in both the average loop length and the number of units in the loops. It is interesting to note that the concentration dependencies for loops are monotonic and have neither minima nor maxima. The loops are not very long; their average size is 5 units ± 2 units depending on the polymer concentration and the adsorption strength.

Finally, Figure 11 shows the dependencies (ν_tl, n_tl,, and ℓ_tl) for tails flanking the adsorbed macromolecule. We see that an increase in the polymer concentration leads to an increase in the number of units in tails and the average tail length. The average number of tails also increases; this value generally varies in a narrow range because it can take a value from 0 to 2.

Hence, based on the obtained results, we can draw some general conclusions concerning the adsorption isotherms and concentration dependencies of the adsorption characteristics:

• As the polymer concentration increases, the amount of polymer adsorbed on the MP particle increases
• The number of units in polymer chains that are directly bound to the particle (i.e., belong to trains, but not to loops or tails) behaves similarly
• The increase in the amount of adsorbed polymer upon increasing polymer concentration leads to an obvious increase in the root-mean-square thickness of the adsorbed layer. At the same time, with a stronger effective attraction of polymer B to the particle, the thickness of the adsorbed layer is smaller
• Softer MP particles adsorb a larger amount of polymer, forming a thinner adsorbed layer
• The characteristics of an individual adsorbed polymer chain exhibit a non-trivial and counterintuitive behavior—the fraction of adsorbed units in a single chain decreases with increasing polymer concentration in solution. An increase in the concentration also leads to a decrease in the fraction of units in loops, with a certain increase in the average loop length, and to an increase in the average tail length and the fraction of tail-forming units in the chain.

3.2.4. Effect of the Length of Dissolved Chains on the Adsorption

Let us see how the length of dissolved polymer B chains affects the adsorption isotherms. Figure 12a shows the adsorption isotherms obtained for chains of three different lengths (N_B = 20, 100, and 500); the values of the adsorption parameter Δχ_A fall in the range from 1.5 to 2. We see that the curves are divided into three families, each of them having a common point at ${\phi }_B^b\to 1$. The top family of curves is for N_B = 500, and the lowest family is for N_B = 20. Hence, longer chains are adsorbed to a greater extent on the MP particle and as a result form a thicker layer, as shown in Figure 12b.

The same trends can be seen in the direct dependencies of these characteristics on the chain length in the solutions of different concentrations shown in Figure 13. We observe a rapid growth for short chains and a much slower growth at large N_B. At the same time, the layer thickness monotonically increases with the increasing length of macromolecules.

Another interesting characteristic is the number of adsorbed macromolecules given by the ratio Γ/N_B. These dependencies are shown in Figure 14. We see that the number of adsorbed macromolecules is a decreasing function of the chain length at high polymer concentrations, while at moderate and low concentrations it behaves nonmonotonically. The decrease in the number of adsorbed macromolecules is obviously related to increasing steric interactions that do not allow several macromolecules to be adsorbed on a particle due to their mutual repulsion, which obviously increases with increasing molecular weight. The range of N_B where Γ/N_B grows extends with increasing chain length (correspondingly, the position of the maximum in the Γ/N_B vs. N_B dependence shifts to the right). Figure 15 shows that this non-monotonicity weakens (disappears) with a decrease in the effective energy of attraction of the B units to the particle. In the case of a non-monotonic dependence (at high concentrations of polymer B in solution), a decrease in the effective attraction energy leads to a less pronounced Γ/N_B vs. N_B dependence. Note that at low polymer concentrations, the number of macromolecules adsorbed on the particle is small and may even be less than one (see the shadowed area in Figure 15a). This means that despite the monomer units’ attraction to the surface, adsorption is weak or even absent (in fact, if the system contained several MP particles, only some of them would adsorb one macromolecule).

3.2.5. Adsorption onto a “Soft” MP Particle vs. Adsorption onto the “Equivalent” Hard Sphere

As indicated in the Introduction, the main feature of the MP particle is its “softness”, i.e., the presence of the transition layer on the periphery in contrast to a solid particle that has a sharp boundary with the surrounding solvent. The presence of a “soft” blurry boundary means an increase of the adsorption potential width, which, as it is known, enhances adsorption [41]. To illustrate this effect, we performed calculations for the adsorption of polymer B from solution onto a solid particle of spherical shape—a hard ball—formed by polymer A units and having a radius equal to the “cutoff radius” R₀ of the MP particle. All interaction parameters (χ_AS, χ_BS, and χ_AB) for the new system remained the same as in our original system.

A comparison of the dependencies of the number of units in the adsorbed chains (Γ), the number of immediately adsorbed units (θ), and the root-mean-square thickness of the adsorbed layer (t_rms) for the two systems shows a better adsorption capacity of a “soft” MP particle compared to a hard spherical particle, see Figure 16. These differences are better illustrated in the dependencies for θ, where the two families of adsorption isotherms are well separated. This separation is due to a significant difference in the adsorption capacity of hard and soft particles. The thicknesses of the adsorbed layers differ in the opposite direction—a slightly thicker adsorption layer forms on the hard particle than on the soft one. This is also due to the fact that the units adsorbed on the solid particle are located strictly on the surface, in the nearest-neighbor spherical lattice layer, while in the case of a soft particle they also penetrate deeper into the particle. In addition, an increase in the width of the adsorption potential “draws” the polymer into this potential well and contributes to its localization within this well.

3.2.6. Effects of the Adsorption Characteristics on the Effective Adsorption Energy

The dependencies shown in Figure 7, Figure 8, Figure 9, Figure 10 and Figure 11 can be re-plotted at fixed bulk concentration ${\phi }_B^b$ of the polymer in solution by choosing the effective adsorption energy Δχ_A = χ_AS − χ_AB as the variable parameter. Thus, we obtain the dependencies of the collective and individual characteristics of the adsorbed chains on the effective attraction of the dissolved polymer B to polymer A forming the MP particle.

Figure 17 shows the dependencies of the number of units of adsorbed chains (Γ) and the root-mean-square thickness of the adsorbed layer (t_rms) on the effective adsorption parameter Δχ_A for adsorption on MP particles formed at χ_AS = 2 (solid lines) and 2.5 (dashed lines). Increasing Δχ_A results in the monotonic increase of all three characteristics. The root-mean-square thickness of the layer shows a very weak dependence on Δχ_A (in particular, we note that the variation of the value of t_rms on these dependencies is of the order of the monomer unit size), which weakens further with a decrease in the polymer concentration in solution. We also note that the solvent quality for polymer A (which determines the polymer density within the particle density) practically does not affect the adsorption characteristics if we take Δχ_A as the control parameter—this is clearly visible for the behavior of Γ; for t_rms the difference is more visible on the plot, but the magnitude of this difference is small, of the order of magnitude 1/10 of the monomer unit size.

3.2.7. Effect of the MP Particle Size on Adsorption

Finally, we consider the effect of the size of the MP particle, which is determined by the number of particle-forming macromolecules of polymer A. It is clear that an increase in the particle size will correspondingly increase the surface available for the adsorbing polymer B. Namely, with an increase in the number of macromolecules n_A by α times, its radius will increase by α^1/3 times, and its area by α^2/3 times. Therefore, in our calculations, we increased the number of macromolecules n_A by 3 times; then the surface area of the particle increased by 2 times (more precisely, by 3^2/3 ≈ 2.08 times). Figure 18 shows the adsorption isotherms and the layer thickness. For a particle formed by 150 chains, the integer “cutoff radius” R₀ naturally increases R₀—it becomes equal to 16, and we have: (16/11)³ ≈ 3.08, which corresponds to a threefold difference in the number of chains forming large and small particles, respectively. It should come as no surprise that the absolute number of adsorbed macromolecules and adsorbed units increases with increasing particle size, and the thickness of the adsorbed layer decreases. It is interesting to see the ratios for Γ and θ for a large and small particle (we denote them with the indices “1” and “2” for a small and large particle, respectively). Moreover, this comparison should be made in two ways: we compare not only the direct ratios Γ₂/Γ₁, θ₂/θ₁ but also the ratios Γ₂/(αΓ₁) and θ₂/(αθ₁) because one large particle can be split into α small ones. In this case, the area ratio will be α^2/3/α = α^−1/3 (when a large particle is assembled from several small ones, the total area decreases, while when a large particle disintegrates into several small ones, it increases).

These dependencies are shown in Figure 19, with two ordinate axes. Two auxiliary horizontal lines are drawn on the graphs—the upper one corresponds to the boundaries Γ₂/(αΓ₁) = 1 and θ₂/(αθ₁) = 1, that is, in the case when the same amount of polymer is adsorbed on one large particle of MP and on those that are split α times, and the lower limit of the corresponds to Γ₂/Γ₁ = 2 and θ₂/θ₁ = 2, when the ratio of the amounts of the adsorbed polymer corresponds to a twofold ratio of the areas. Using these auxiliary curves, we can see that the amount of polymer adsorbed on a large particle exceeds the amount of polymer adsorbed on a small particle by more than two times—this is a consequence of the fact that the polymer is adsorbed not only on the surface, but also in the thickness of the transition (adsorption) layer. On the other hand, when the particle is divided into smaller ones, the amount of adsorbed polymer increases and the ratio Γ₂/(αΓ₁) < 1 (it slightly exceeds 1 only at very low concentrations ${\phi }_B^b$ at Δχ_A corresponding to strong adsorption).

4. Conclusions

We employed the Scheutjens–Fleer self-consistent field (SF-SCF) method to study the formation of MP particles consisting of monodisperse homopolymer macromolecules under thermodynamically poor solvent conditions and to investigate the adsorption of pollutant homopolymer chains onto the MP particles. Since the problem has obvious spherical symmetry, we used the one-gradient version of the SF-SCF method on a spherical lattice. It was shown that the polymer macromolecules form MP particles with a constant polymer density inside the particle and with an interfacial boundary layer at its periphery. The size of the MP particle and the thickness of the interfacial layer both decrease with a deterioration of the solvent quality. On the other hand, in moderately poor solvent, the MP particle becomes unstable and disintegrates into separated macromolecules. In order to suppress and even exclude this effect, we also considered MP particle formation by a single multi-arm star polymer, which keeps its integrity at any solvent quality. In the poor solvent limit, the MP particles formed by linear macromolecules and by the star molecule of the same total molecular weight have identical size and interfacial layer thickness. In a real situation (i.e., under experimental conditions), the adsorption of the pollutant polymer chains onto the model MP particle should not break the particle’s integrity. To take this into account, the polymer–polymer and polymer–solvent interaction parameters were chosen appropriately: the solvent was poor for the particle-forming polymer A and good for soluble polymer B; a slight incompatibility between polymers A and B prevents the MP particle disintegration. The driving force of polymer adsorption on the particle is thus the screening of unfavorable particle–solvent interactions by the adsorbed polymer.

The effects of the polymer chain length, its concentration in the solution, and the interaction parameters on the adsorption were investigated. It was shown that the amount of the adsorbed polymer pollutant increases with increasing bulk concentration of the pollutant in solution. Softer MP particles were shown to adsorb a larger amount of the polymer than solid ones. Responsible for this effect is the interfacial layer of the MP particles composed of macromolecules.

Note that the increased adsorption capacity of “softer” MP particles was observed by J. Li et al. [32] who investigated the adsorption of five (low-molecular) antibiotics—sulfadiazine, amoxicillin, tetracycline, ciprofloxacin, and trimethoprim—on five types of microplastics (polyethylene, polystyrene, polypropylene, polyamide, and polyvinyl chloride) in freshwater and marine systems. It was shown that polyamide had the highest adsorption capacity in freshwater, which is attributed to its porous structure.

With an increase in the length of the dissolved polymer chains, both the adsorbed amount and the number of monomer units in direct contact with the particle increase. At the same time the number of adsorbed polymer chains (that is, the adsorbed amount divided by the chain length) as a function of the chain length shows a decrease at high polymer concentrations in the solution and non-monotonic behavior at moderate and low polymer concentrations.

The analysis of individual chain characteristics shows that the fraction of adsorbed monomer units in each polymer chain decreases, in contrast to the increase in the total amount of adsorbed polymer pollutant. This indicates that increasing polymer concentration enhances competition for adsorption sites on the MP particle surface, and the adsorption of new chains is implemented via partial desorption (that is, the decrease of the fraction of adsorbed units) of already adsorbed chains.

The results obtained in the present study by using the numerical SF-SCF method are quite general and provide a deeper insight into the features of the adsorption of macromolecular pollutants onto MP particles, including details of the conformations of the pollutant macromolecules onto the surface of MP particles, the effects of the interfacial layer, the adsorption capacity, etc. The proposed method can be extended in future work to study the adsorption of macromolecular pollutants of more complex structure, such as block copolymers or heteropolymers, onto MP particles. The results obtained in this research are relevant to support the solving of ecological problems associated with aqueous environmental pollution.