Polymer Wrapping onto Nanoparticles Induces the Formation of Hybrid Colloids

Abstract

Polymers stabilize the nanoparticles onto which they wrap, avoiding coagulation and undesired phase separation processes. Wrapping gives rise to hybrid colloids, and is useful in bio-intended applications. In non-covalent interaction modes, polymers physically adsorb onto the nanoparticles’ surface, NPs, and some of their portions protrude outside. Both their non-interacting parts and the free polymers are in contact with the solvent, and/or are dispersed in it. Wrapping/protruding ratios were forecast with a simple statistical thermodynamic model, and the related energy calculated. The wrapping efficiency is controlled by different contributions, which stabilize polymer/NP adducts. The most relevant ones are ascribed to the NP-polymer, polymer–polymer, and polymer–solvent interaction modes; the related energies are quite different from each other. Changes in the degrees of freedom for surface-bound polymer portions control the stability of adducts they form with the NPs. The links between wrapped, free, and protruding states also account for depletion, and control the system’s properties when the surface adsorption of hosts is undesired. Calculations based on the proposed approach were applied to PEO wrapping onto SiO₂, silica, and nanoparticles. The interaction energy, W, and the changes in osmotic pressure associated with PEO binding onto the NPs have been evaluated according to the proposed model.

1. A Brief Historical Background

A historical view of the concept of the hybrid indicates that it had origin in the very early stages of civilization, when humans started breeding animals and growing comestible plants [1]. Mythology surrounding the concept emerged from the human imagination. Accordingly, our ancestors described sphinxes, minotaurs, medusae, chimeras, centaurs, and so forth [2]. Many diverse morphological features imparted to hybrids demonstrate the contemporary attitudes to running, swimming, and flying, and these mythological creatures also combined conflicting characters—such as wings and horns, or beaks and hoofs—in the same animal. Our ancestors believed that occurrence of so many imaginary animal types was an evident proof of the power, or wrath, of gods.

The concept of the hybrid endured for a long time, has been continually revised, and is still used today. For instance, emperors, kings, and noble families of the 16th to 18th centuries collected stuffed exotic animals, existing in nature or reconstructed from taxidermists, and put them in “wunder-kammers” (wonder cameras) [3]. The concept was used in botany, zoology [4], and genetics [5]. In chemistry, the resonance of benzene [6], and the LCAO-MO theory of atomic/molecular orbitals [7] were defined on similar grounds. In colloid sciences, the term actually applies to entities whose domains differ in physical nature, packing modes, composition, and so forth. This is the reason why the term is still used.

2. Introduction

There are clear distinctions between nanoparticles, association colloids, hybrids, polymers, etc., but their physical foundations are related. What we know about hybrid colloids is that: (a) their subdomains show diverse, and very peculiar, features. They can be fully organic, or organic–inorganic, etc. (b) The domains self-organize, self-sustain, and self-help, provide composites, follow the constraints dictated by stability conditions (thermodynamic or kinetic in origin). (c) The links between the subdomains can be covalent or not. Links form entities retaining some properties of the original species and completely new ones. Hybrid colloids include bio-based and soft matter-based entities. These systems are part of important research fields forecast by P.G. de Gennes [8], the founder of Soft Matter.

We focus on the subset of hybrid colloids, made up of polymers and nanoparticles. The forces responsible for their formation span from vdW (van der Waals) to electrostatic, including steric forces. Combinations of these forces may give rise to a wide diversity of entities. The resulting hybrid nanoparticles find use in electronics, surface coating, materials sciences, gene therapy, food, bio-analytical methods, etc. [9,10,11,12,13]. The reasons for using functionalized entities as hybrids is because uncoated NPs adsorb ions, solvents, other species, and interact each other, with hosts [14], self-associate [15], and may phase-separate [16]. These processes are detrimental for practical purposes and must be avoided.

Both covalent and non-covalent functionalization give rise to hybrids. Non-covalent functionalization occurs through the physical adsorption of surfactants, or polymers, onto NPs. These adsorbents share some commonalities, and substantial differences too. Surfactants adsorb quite easily, but tend to be released; polymers are preferred (if adsorption kinetics is immaterial) because of this. Polymer adsorption occurs by NP dipping in the presence of solvent, sometimes with cosolvents. Thereafter, the adducts are manufactured for the required purposes. Surface coverage stabilizes NPs [17], thus preventing coalescence. Polymer-wrapped NPs—PW-NPs—are stable and may remain as such for a long time. They are obtained by different pathways [18,19,20]. Information on their preparation is outside the scope of this study, and is reported in selected articles and reviews [21,22,23].

Physical wrapping is concomitant to polymer partition with the bulk, and depletion may occur [24,25,26,27]. The phenomenon is due to unbalanced osmotic effects, with dispersant flow from regions between PW-NPs. It is dependent on the polymer volume fraction, φ, and occurs in dispersions of homopolymers, block copolymers, polyelectrolytes, surfactants, their mixtures, and polymer–surfactant systems. Polymers are partitioned with the medium; the process is controlled by the affinity toward NP’s, polymer/NP ratios, T, etc.

Wrapping is controlled by electrostatic, sterical, hydrophobic, vdW forces, and combinations thereof [28]. At the molecular level, wrapping gives rough surfaces. Some polymer portions surface-adsorb, while others protrude to the bulk. The protrusion probability is high: cases are known where wide coronas self-assemble around NPs [29,30]. The wrapped layer can be densely packed and many nms thick. Wrapping is useful if the coagulation of the biopolymers onto the NPs is not desired [31]. Wrapping modes are dense or loose, depending on polymer affinity towards NPs, on the strength of the interaction between polymers and surfaces, and many other factors. Concentrated regimes can be met on NP surfaces. The projection in Figure 1 indicates that some polymer portions pack in compact domains; loose packing is also possible. The transitions between these regimes depend on polymer content, on its affinity with the NP surface, and on the amount of formerly adsorbed polymer. Wrapping proceeds until a dense packing is attained. The conformation of the protruding groups depends on both polymer–solvent and polymer–polymer interaction energies. Protruding portions repel each other, giving rise to different structures, as was observed in PEO-based [32] and PPO-based systems. (PEO stands for poly-ethylene-oxide, PPO for poly-propylene-oxide.)

The statistical thermodynamic model we introduced here was never applied to polymer-NP systems—in this sense it is unique and original. It is assumed that polymers physically wrap and protrude outside NP’s as decorated coronas, as shown in Figure 2, which displays polymer protrusion outside the particle (Medusa’s head). Wrapping/protruding ratios were obtained assuming that the two states are different in nature. Surface-bound segments are termed fractional loops, α (≤1), while protruding ones, termed trains, are defined as ε [33]. (Note that α + ε = X_b, the latter being the amount of polymer moving as a whole kinetic entity with NP’s.) Multilayer adsorption is possible. Wrapping is the result of some energy terms, the effects of which extend in many concentric layers around each NP.

The quintessential aspects of wrapping are reported on below. We first describe the model and provide an approximate analytical solution to the problem. The manuscript reports some calculations, which show how the effect can be significant. Next, we discuss the case of mixtures made of silica NPs and a homopolymer, PEO, to which the model was applied. Small modifications of the theory are required if it should be applied to soft-matter matrices.

3. The Model

At saturation, surface coverage is retained. Wrapping efficiency is quantified by Langmuir-like, or other isotherms [34]: despite the complexity of hybrid systems, simple isotherms such as those described above give satisfactory results. The model applies to hard (SiO₂, TiO₂, CNTs, graphenes, etc.), or soft NPs (latexes, droplets, vesicles). It is assumed below that: (a) the NPs’ curvature radii are constant; (b) polymer length is ≤NPs’ diameter; (c) wrapping length is ≤the full polymer one; (d) polymers and NPs are size monodisperse; (e) different parts of the same polymer chain may wrap; (f) the polymer partition between bulk and surface states is possible; (g) the aforementioned X_b term is ≤X_tot.

A polymer chain consists of loop terms, α, representing the number of all wrapping portions. α parts and protruding ones, ε, differ in physico-chemical properties. T is the segments number in a chain (thus, αT is a number). NP-polymer interaction modes belong to a category, polymer–polymer and polymer–solvent ones to others. Solvent is present in the coronas around NPs in an undefined way, and its contribution to the system stability is unknown. X_f is the mole fraction of free polymer in the medium. In addition, (X_tot = X_f + X_b). α and ε (=1 − α) are defined as

MαT = (1 − X_f)N°P^α(1 − P)²

(1a)

MεT = M(1 − α)T = (1 − X_f)N°P²(1 − P)^ε

(1b)

M is the number of monolayers lying on the NP surface, N° the number concentration of polymers, P the adhesion probability. If multi-layer adsorption holds true, the amount of wrapping polymer is calculated by summing Equation (1a,b) over all possible M layers, which are assembled in concentric sheets around NPs. The equilibrium between α and ε states, K_α,ε, is defined as

K_α,ε = α/ε = α/(1 − α)

(2)

Combining Equation (1a,b) with Equation (2) results in

K_α,ε = α/(1 − α) = [P^α(1 − P)²/P²(1 − P)^(1−α)]

(3)

Equation (3) is always ≤0; its value depends on the polymer affinity towards NPs and on the repulsive forces between the protruding chains. Packing can be dense or loose, and the solvation substantial. The loop length is related to probability. [It is possible to write explicit forms of α and ε as a function of X_b]. Thus,

αP/(1 − P) = [P^α(1 − P)²/P²(1 − P)^((1−P)/P)]

(4)

αP³/(1 − P)^((1−P)/P) = P^α(1 − P)³

(5)

P^α/α = [P³(1 − P)^((1−P)/P)/(1 − P)³]

(6)

In Equation (6), the probability is P^α. The partition between α and ε states depends on polymer affinity toward NPs, and is directly proportional to the number of adsorbed segments. Wrapping polymers are ribbons of uniform cross section, a. They can be twisted around their major axis to form a sort of “tortiglioni”, a type of rolled pasta.

The adhesion energy, E_ad, depends on wrapping length. It is the sum of different terms, and is defined as

E_ad = (W + π)al_if_i + η

(7)

The subscript I (>1) in Equation (7) indicates the number of binding entities; l_i, the segment length in extended conformation; f_i, a shrinking parameter, ≤1, which depends on segments conformation; η is a shrinking energy, related to polymer conformation [35]; π is a pressure acting normally to the NP’s surface; W an adhesion work per unit volume.

The probability of the ith state, P_i*, is controlled by the adsorption energy; E_ad: is expressed as

P_i* = k_i exp ^{−[(π +W)afili/kT]} exp ^−[η/kT]

(8)

k_i’s are stiffness factors, peculiar to each ith state; k_i and f_i terms are interrelated; f_i’s values refer to shrinking—if they approach unity, the wrapping energy is the maximum compatible with a given length.

Summing P_i*’s over all possible layers gives the overall amount of the wrapping part, P_a, defined as

P_α = P_tot = Σ_i=1P_i* = Σ_i=1 k_i exp ^{−[(π +W)afili/kT]} exp ^−[η/kT]

(9)

Equation (9) relates the probability to the statistical weight of a given state; note that the term P_i* is a distribution function, controlled by k_i. The summation in Equation (9) contains many terms that are not energetically equivalent. Wrapping occurs if the lengths, l_i ’s, and W terms are large. Accordingly, a maximum in P_i* vs. l_i plots occurs. Figure 3, which deals with the binding probability calculated assuming different exponents.

According to Figure 3, P_i* reaches a maximum in a certain l_i range, depending on f_i; the same conclusions hold for the distribution width. The P_i* function is not symmetrical with respect to l_i, and decreases with increases in the number of bound segments. In other words, not all segments may wrap. Both Figure 3 and, more precisely, Figure 4 indicate the location of P_i*. It was normalized in such conditions to provide evidence regarding the location of the maximum.

The value of the normalized P_i,norm scales with the number of bound segments, and the resulting function is steep. A comparison between Figure 3 and Figure 4 indicates that P_i* depends on a, l_i, and f_i.

A derivation of Equation (9) provides information on the system stability. In each layer we get

ln(P_i/k_i) = −[(π + W)af_il_i/kT] − [η/kT]

(10)

Accordingly, a and η are implicit form of f_il_i. Derivation with respect to l_i and f_i gives:

(dln[P_i/k_i]/df_i)_li = −(π + W)al_i − (dη/df_i)_li = 0

(11a)

(dln[P_i/k_i]/dl_i)_fi = −(π + W)af_i − (dη/dl_i)_fi = 0

(11b)

when Equation (11a,b) balance, we get the equalities:

(π + W)al_i = (dη/df_i)_li

(12a)

(π + W)af_i = (dη/dl_i)_fi

(12b)

Note that, if Equation (12a,b) are zero, the term W = −π. The segment’s length and the shrinking term determine the wrapping efficiency, which is controlled by conformational constraints and by sterical hindrance.

Polymer wrapping may be a strongly cooperative process [36], as in ss-DNA adsorption onto CNTs [37], or when polyelectrolytes adsorb onto oppositely charged NPs [38]. The processes occur if the attractive forces at work between polymers and NPs are higher than polymer–polymer repulsive ones; that is, if |E_i(P-P)|_att > |E_i(P-NP) + E_i(P-P)|. The first term refers to polymer-NP interactions, whereas the P-P one to polymer-polymer interactions. The difference between such terms is easily derived with respect to l_i, and represents the energy.

The π and W terms depend on wrapping length. W is proportional to the polymer affinity towards NPs, whereas π refers to P-P interactions. The balance of two such contributions can be expressed in terms of chemical potentials. For instance, the wrapping of sodium polyacrylate, PAANa, onto SiO₂ is driven by different forces with respect to that of the conjugated acid, PAA. A critical analysis of these facts, and of many similar ones, clarifies many puzzling phenomena met in polymer-NP systems [39,40,41].

Despite the significant contributions that have been issued to date, many wrapping details are lacking. For instance, information from ζ-potential does not give complete information on how many, and which, polymer segments effectively interact. Calorimetric approaches, conversely, may be helpful. The above quantity, in fact, depends on the polymer partition with the bulk, and on the number of segments that are surface bound. In a two-site approximation, the interaction heat of polymer-NP systems, ΔH_i,sol, depends on wrapping, at constant T. We suppose below that polymers are monodisperse and partly dehydrate upon wrapping. The process is determined if ΔH_dehyd, the dehydration enthalpy, is large. ΔH_tot is the result of terms that depend on surface bound and free states. Enthalpic contributions due to ε (i.e., 1 – α) units are the same as those due to dissolution of the free polymer. The difference between the systems with and without NPs is proportional to α, provided ΔH_wrap ≠ ΔH_i,sol. In systems containing both NPs and polymer, the term ΔH_dil,mist is due to the free polymer, X_f (= N_f/N°), and a second one to (1 − α)(1 − X_f). The polymer part interacting with NPs is (1 − α)(1 − X_f). Heat effects ascribed to the dissolution of the protruding polymer parts are the same as the free ones. The only terms responsible for substantial changes in ΔH values are related to α.

Therefore,

(ΔH_sol,mixt/N°) = ΔH_sol[X_f + (1 − X_f)(1 − α)T] + ΔH_wrap[(1 − X_f)αT]

(13a)

(ΔH_sol,mixt/N°) = n_f(ΔH_sol/N°) + (N° − N_f)[(αTΔH_wrap/N°) + ((1 − α)TΔH_sol/N°]

(13b)

The difference between Equation (13a,b) is ΔH_wrap. The (1 − α) contribution is due to non-wrapping units, having the same ΔH_sol as bulk polymer. In this approach, we supposed that conformational changes are athermal. Thus, the difference between ΔH_sol and ΔH_dil,mixt is directly proportional to α. We are convinced that isothermal titration calorimetry, ITC, may give substantial information in this regard. Recent experiments showed that protein adsorption onto NPs covered by linear polymer brushes is hindered, when the reverse holds true for cyclic polymer brushes [42].

4. Results

Data relative to a model [PEO-SiO₂] system are reported below. No measurements were run; a model was introduced and calculations were made based on it. The uncertainties in energy terms are reported here in the Results section. PEO adsorbing onto aqueous SiO₂ is under consideration. Both components are monodisperse: the diameter of SiO₂ is 200 nm. The number of EO units in the polymer is 50–60 [43], so = its length is grossly the same as the diameter of NP diameter. (A range is considered because its conformation is unknown.) Its conformational energy is hardly quantified, when hydration energy is nearly equivalent to the transfer energy from water to the NP surface at about 5.5 kJ per EO unit [44]. The location of PEO contact points on the NP surface can be continuous (many points in a series) or random. It is immaterial whether or not the adsorbed ribbon length, l_i, is made of i subsequent units.

Upon adsorption, PEO releases water, which is the reason for the analogy between wrapping and micelle formation. In addition, the adsorbed PEO layer is thinner than the fully extended polymer chain. The value refers to units located in the corona inner core: for units facing the bulk, the transfer energy can be lower. Combining these hypotheses with the number of oxyethylene units provides the required solution. The number of surface bound units (8–10) was inferred from the distribution functions reported in Figure 3 and Figure 4. The number of wrapping units described above provides an interaction energy, W, in the range 45 ± 5.0 kJ mol⁻¹.

The presence of PEO increases the medium osmotic pressure. The whole effect is modulated by the interactions among the polymer and NPs. If the amount of SiO₂ is constant, and a two-site approximation for the PEO state holds true, the overall osmotic pressure, π, is due to free PEO, free NPs, and adducts. Since the osmotic coefficients of such mixtures are known [45], the Gibbs energy can be inferred. π gives the polymer chemical potential, μ*. The latter progressively decreases due to the combined effect of composition and surface adsorption. At saturation, it reaches a minimum at a given PEO/NP ratio, as indicated in Figure 5. The shape of the curve depends on a progressive polymer uptake onto NPs until surface saturation is attained. In the calculations for osmotic pressure, we supposed that PEO is linear, monodisperse, and water-soluble. It has many configurations, and the adsorption kinetics onto NPs is presumably slow.

PEO units exert a pressure in a direction normal to the NP surface. The effect is proportional to concentration, that is to the thickness of the adsorbed layer. Integration with respect to [PEO] volume fraction gives the expected π value, and μ*. A similar behavior is expected if we assume that many EO units progressive overlap each on the other until a certain layer thickness is attained [46]. We calculated that value imposing the coverage to be 15 nm thick. The energy minimum ascribed to overlapping units is 25 ± 6.0 kJ mol⁻¹ at a volume fraction close to 0.10–0.12.

The considerations of the η values are more complex. It is known that aqueous PEO undergoes dehydration, conformational changes, and phase transitions by raising T. The process is controlled by terms which favor the onset of a compact structure, concomitant to phase separation [47]. Presumably, a similar behavior holds when PEO adsorbs and is compacted onto surfaces [48,49]. The possible conformations of the PEO segments control binding onto SiO₂ and desolvation. The two processes are likely interrelated and control the polymer packing density onto NPs. Estimates of the energy concomitant to that process have been made, but the results are merely tentative and strongly model dependent.

5. Discussion

Wrapping efficiency is relevant for practical purposes. Many aspects of wrapping are known, mostly with regard to applications. There are still obscure aspects to be carefully considered. The forces that control wrapping depend on NPs and polymeric species. Different stabilizers have advantages and drawbacks for a given NP type. Uncharged polymers modulate adsorption and are used in the preparation of biointended materials.

According to the model presented here, the wrapping polymer units, l_i, are ribbons of constant width tilted around their major axis, in such a way that they are compacted on the surface. The wrapping energy for the system that has been discussed in more detail depends on polymer length. Wrapping domains are continuous, although this statement is not generally true. The π and W terms in Equation (7) are diagnostic of the process efficiency, and indicate that repulsive or attractive forces occur at saturated surfaces.

The term W depends on the interaction energy between the NPs, whereas π depends on the polymer surface concentration. At saturation, the Gibbs energy, modulated by the balance of two such contributions, is minimum, and is proportional to the number of interacting segments. Affinity towards selected NPs depends on the interaction energy, E_i,(P-NP), on the number of bound segments, and on packing density as well. Wrapping ends at surface saturation, and multilayer adsorption is formally possible. Polymers may change their rotational free energy upon dehydration, even though this point is still matter of controversy [50]. Equilibria between wrapping and protrusion indicate that α depends on the polymer.

Different binding units may coexist in block copolymers used in wrapping technologies. The case of Pluronics F 127 (an alternating PEO–PPO–PEO sequence) [51], is diagnostic. It contains separate blocks, each of its own length, and subunits are located in different parts of the main chain. Pluronics has a sequence-sensitive hydrophilic–lipophilic balance, HLB, which depends on its molecular details. Wrapping is proportional to HLB, T, and PPO/PEO is proportional to the number ratio. The binding of poly(L-lysine)-PEO block copolymers onto SiO₂ has a similar behavior [50]; it is possible that wrapping depends on the unit’s length and the net charge of polypeptide moieties.

Ancillary aspects of PW-NPs systems imply depletion, and hosts adsorption into wrapped segments. The former process occurs when the volume fraction of free polymer, N°X_f/ρ_polym = φ_f, is higher than a critical value, φ_cr. φ_f depends on polymer content and wrapping. Depletion of the aforementioned systems is obtained by deriving φ_f with respect to φ_tot, while still keeping the other control variables fixed.

Adsorption of host species in the corona may be forecast too. The partition therein is obtained by calculating the chemical potential of the host in the environment in question. Host-containing PW-NPs apply in biomedicine. Hosts bound to protruding segments are released first, before the wrapped ones.

6. Conclusions

Hybrids obtained by the physical adsorption of polymers onto NPs have a three-dimensional arrangement of macromolecular components, as shown in Figure 1 and Figure 2. A range of plausible scenarios for the wrapping, depletion, binding of hosts, and for the fate of PW-NPs adducts interacting with cellular systems, are reported in selected articles [52,53]. Considerations based on a critical analysis of such systems are in good agreement with the conclusions drawn here.

A simple statistical thermodynamic approach to wrapping and developing a user-friendly model has been introduced here. The statistical procedures developed take in due account several fundamental aspects of wrapping. They show that the polymer affinity towards NPs is mainly controlled by the interaction energies. These control the number of wrapping segments, the packing density, and end at saturation. Experiments were carried out on such systems some years ago, and it is difficult to gather detailed information on wrapping density from experiments alone. Studies reported to date suffer from the fact that the forces at work in the wrapped layer are badly defined. More finely developed and refined statistical approaches to wrapping may help in obtaining more details on the process efficiency.

The approach reported here is unique in many respects, despite its simplicity. The approach introduced here could benefit from calorimetric experiments, which are easily rationalized on thermodynamic grounds. Calorimetric experiments could be used to assess the validity of this treatment on solid grounds.