Polydisperse Telechelic Polymers at Interfaces: Analytic Results and

Article pubs.acs.org/Langmuir

Polydisperse Telechelic Polymers at Interfaces: Analytic Results and Density Functional Theory Jan Forsman* Theoretical Chemistry, Chemical Centre, Lund University, P.O.Box 124, S-221 00 Lund, Sweden

Clifford E. Woodward† School of Physical, Environmental and Mathematical Sciences, University College, University of New South Wales, ADFA, Canberra ACT 2600, Australia ABSTRACT: We use a recently developed continuum theory to expand on an exact treatment of the interfacial properties of telechelic polymers displaying Schulz−Flory polydispersity. Our results are remarkably compact and can be derived from the properties of equilibrium, ideal polymers at interfaces. A new surface adsorption transition is identified for ideal telechelic chains, wherein the central block is an equilibrium polymer. This transition occurs in the limit of strong end adsorption. Additionally, closed expressions are derived for the ideal continuum telechelic chain in contact with two large spheres, using the Derjaguin approximation. We analyze the interactions between colloids as a function of polydispersity and molecular weight, and the results are compared with polymer density functional theory in the dilute limit. Significant variations in polymer mediated forces are observed as a function of polydispersity, molecuar weight, and chain stiffness.

I. INTRODUCTION Polymers tethered onto particle surfaces have proved a very useful way to alter and control particle interactions. For example, the tethering of polymers onto nanoparticles may enhance their solubility in the corresponding melt.1,2 Interesting superstructures may also form in these systems when phase separation is frustrated by connectivity constraints.3 Biocompatible polymers such as polyethylene glycol (PEG) can be used to protect the surfaces of implants from nonspecific adsorption.4 Heterotelechelic polymers (of the type ABC) have been purposefully employed to adsorb specific target proteins to surfaces.5 More recent experimental work on the grafting of telechelic polymers onto surfaces and nanoparticles has also appeared,.6−9 Telechelic polymers are polymers that contain end-functionalized groups. The simplest are triblock polymers of the type ABA or ABC. Here the end blocks strongly adsorb or chemically react with particle surfaces, while the central blocks are repelled by the particles. When the end groups do not react chemically with their host surfaces, telechelic polymers adsorb reversibly. This means that the adsorbed surface density will vary, according to the adsorption strength as well as other ambient (thermodynamic) conditions. This will be the case whether polymer molecules remain ”trapped” between particle surfaces or easily exchange with the surrounding solution. The former situation is realized when polymer molecules diffuse slowly from the region between particles over the time scale of particle motions. This is likely when the adsorption of the ends is very large. On the other hand, the latter case is most relevant for investigations of thermodynamic equilibrium in such systems. © 2012 American Chemical Society

Current methods of telechlic polymer synthesis can also control degrees of polydispersity.4 This adds another degree of freedom with which to alter interactions. The role played by polydispersity in determining the thermodynamic behavior of mixtures of particles and telechelic polymers has not been thoroughly investigated, either theoretically or experimentally.10 A recent interesting example of a polydisperse telechelic polymer system consists of surfactants aggregated to form chain-like micelles (CLM) of varying flexibility and length, which can end-associate with colloidal particles.11 The CLM are assumed to be living polymers which have an exponential molecular weight distribution. A simple theoretical treatment, assuming ideal polydisperse telechelic polymers, was able to broadly account for the experimental phase diagram. This is likely a reasonable approximation below the semidilute concentration regime for polymers For many years, self-consistent field (SCF) theories12,13 for polymers have provided a useful (albeit approximate) approach to treat nonuniform polymer fluids, especially in the concentrated regime. A generalization of the SCF approach to polymer brushes was carried out by Milner et al.14 some time ago. Those authors assumed a degree of correlation, extraneous to the usual SCF approach, by noting that tethered chains at high surface coverage are expected to be strongly stretched. By cleverly exploiting an analogy with classical dynamics, one Received: November 21, 2011 Revised: January 19, 2012 Published: January 24, 2012 4223

dx.doi.org/10.1021/la204576q | Langmuir 2012, 28, 4223−4232

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where the molecular weight distribution is of the Schulz−Flory form.33,34 The PDFT becomes essentially identical with the ideal polymer model in the limit of very dilute solutions. We also developed an analytical version of the theory for ideal living polymers35 at surfaces. In the absence of a self-consistent field, the continuum theory can also be solved exactly for general Schulz−Flory forms. Remarkably, these are derivable from the already reported solutions to the ideal equilibrium polymer system.36 These two levels of theory will provide the tools with which we will investigate the polymer particle mixture. Here we shall focus on the colloid limit, where the particle radius is large compared with the average radius of gyration of the polymers. This simplifies the treatment, as the thermodynamics of the mixture can be determined by an effective pair potential, which is the potential of mean force between a pair of colloidal particles immersed in the polymer solution. Furthermore, this potential can be obtained by considering the planar limit for particle surfaces and the Dejaguin approximation, thus significantly reducing the complexity of the theoretical analysis. Some recent simulation work has also explored the pair interaction between colloids induced by telechelic polymers.37 In the next section we shall give a brief outline of the derivation for ideal polydispersed telechelic polymers in the presence of particle surfaces. We first consider the interesting case of living telechelic polymers. These polymers display an exponential molecular weight profile, which can be considered the first in a hierarchy of Schulz−Flory forms. Analytical results are obtained for planar surfaces and for spherical surfaces, within the Derjaguin approximation. It is important to note that some of the results for the planar surfaces were also previously obtained by van der Gucht et al.36 Additionally, inversion of the Laplace transform allows us to extract the asymptotic form of the interaction between two spheres in monodisperse telechelic polymers. The problem of nonideal telechelic chains is then addressed using polymer density functional theory (PDFT). We make comparisons with the ideal chain results. Some final remarks conclude the paper.

can then conclude the residual mean-field has a quadratic form. That work was, in turn, applied to telechelic chains by Milner and Witten,15 who assumed that bridging chains were a small perturbation of loops. This gave a fairly weak attraction at separations corresponding to the point where the effective brushes on each surface first come into contact. Other workers have also performed theoretical studies of telechelic polymers.16−20 A study of triblock polymers with adsorbing ends was also carried out by Wijmans et al.21 using a lattice version of the SCF approximation, and more recently, Cao and Wu used polymer density functional theory (PDFT) to show that the source of of this attraction was due to bridging chains.22 In these works, it was assumed that the adsorption of chain ends was truly reversible and the strong stretching assumption was not in play, instead the polymer molecules responded only to the mean-field arising from the excess free energy functional. Earlier simulation work23 showed that reversible adsorption gave rise to significant exchange between populations of chains with different modes of adsorption, depending upon factors such as polymer length, surface separation, and adsorption strength. We will denote these populations as bridges and loops (both ends adsorbed), tails (one end adsorbed), and free (no adsorption). An important outcome of this work was that adsorption strength could be used as a valuable means for controlling the thermodynamics of mixtures of particles and telechelic polymers. It was clear from that earlier work that adsorption strength and polymer molecular weight must be considered together. For example, in a triblock copolymer of the type ABA, the length of the adsorbing A blocks as well as the adsorption energy per A monomer will determine the strength with which the polymer will adsorb. This provides a useful way to modify the adsorption properties of polymers without modifying chemical functionality. On the other hand, as the A blocks will adsorb in a generally flat configuration, surface saturation may become a problem, especially for particle sizes near the protein limit. Thus, it is also useful to consider the effect of the length of the nonadsorbing B block. Configurational entropy will generally counter adsorption of chains. It is well-known that polymer depletion can provide a thermodynamic driving force that destabilizes mixtures of particles and nonadsorbing polymers. In this case, the onset of phase separation can be controlled through varying the relative sizes of polymers and particles. The addition of adsorbing ends (as in telechelic polymers) adds another dimension to the themodynamic parameter space that controls the interplay between adsorption and depletion. These types of interactions give rise to frustrated correlations, which often lead to interesting self-assembled structures.3 Thus, the adsorption modes described above will be strongly affected by the molecular weight of the nonadsorbing block. Furthermore, this dependence can be tuned to some extent in the case of polydisperse polymers, wherein one may expect to see quite different adsorption profiles compared with the monodisperse case, depending upon the degree of polydispersity. This will have a profound effect on interparticle forces. To the best of our knowledge, a detailed theoretical examination of these mechanisms has not yet been reported in the literature. The purpose of this article is to provide such a study. In particular, we will investigate the role played by polymer polydispersity and molecular weight in determining the interactions between particles in telechelic polymers. We shall use a PDFT, similar to that employed by Cao and Wu22 and others.24−32 We recently showed how the PDFT could be easily generalized to include polydispersity, in the case

II. IDEAL TELECHELIC POLYMERS: CONTINUOUS CHAIN THEORY A. Telechelic Chains with a Living Central Block. In the continuous chain theory, the polymer is modeled as an extensible continuous thread. This is in contrast to the discrete interacting chain model, treated with polymer density functional theory, as will be described below. While somewhat idealized, the continuous model allows one to obtain analytical results, especially in the ideal chain limit, wherein all monomer− monomer interactions are assumed negligible, i.e., under Θ conditions. So treated, the end−end distribution of an ideal polymer chain in the presence of an applied external field, ψ(r), is described by the function G(r,r′;s), which gives the statistical weight of a chain segment of length s, with ends at r and r′. In the continuum limit, this quantity satisfies the following ”diffusion” equation, ∂G(r, r′; s) σ2 2 = ∇ G(r, r′; s) − ψ(r) G(r, r′; s) 6 ∂s

(1)

with boundary condition G(r, r′; 0) = δ(r − r′)

(2)

Here σ is the Kuhn length. The potential ψ(r) will depend upon the particular problem being treated. As with refs 10 and 36, 4224


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where ϕ(z) = √κ cosh(√κz/Rg) − λ sinh(√κz/Rg) and λ = ϵRg. The functions in the denominator are given by

we shall consider an ideal polymer solution between hard planar surfaces, separated by a distance D. We assume an implicit Θ solvent, or at least a dilute polymer solution, so that monomer− monomer interactions can be neglected. The ends of the polymers contain strongly adsorbing sites that bracket a central block of length s (telechelic polymers). While this central block is relatively ”nonadsorbing” compared with the ends, we shall assume for the moment that a weak short-ranged adsorption potential acts between the central monomers and the surfaces. It is well-known that the action of such a potential can be realized by application of the following homogeneous boundary conditions,38 1 1 ∇z G(z , z′; s)|z = 0 = − ∇z G(z , z′; s)|z = D = −ϵ G G

A(κ, D) =

B(κ, D) =

∞

d s F (s ) G ( z , z′ ; s )

(3)

= bi(z)

∫

z=0

floop = φpR g κ

The solution of eq 6, with boundary conditions from eq 7, can be obtained via standard methods. The result is

κ φ(z′) φ(D − z) = 2R g A(κ, D) B(κ, D)

z′ < z ≤ D

(12)

(13)

Defining the sticking parameter, χi = Qi/Rg, we obtain the following for adsorbing chains: Ploop = χ1χ2 f loop, Pbrdg = χ1χ2 f brdg, and Ptail = (χ1 + χ2)f tail. Explicit expressions for the f x are as follows:

(7)

0 ≤ z < z′

D

λ sinh(ξ) + Dφp Pfree = 2φpR g κ A(κ, D)

(6)

κ φ(z) φ(D − z′) Ĝ(z , z′; κ) = 2R g A(κ, D) B(κ, D)

∫

where φp is the polymer density, measured in units of σ−3. Substituting eq 8 into eq 12 leads to the following decomposition of the population into f ree, loop, bridge, and tail contributions: Ptot = Pfree + Ploop + Pbrdg + Ptail. Free polymers are not adsorbed via any end to any surface, bridges have both ends adsorbed but on different surfaces, loops have both ends adsorbed on the same surface, and tails have only one end adsorbed and the other dangling. For free polymers,

= −ϵ z=D

D

Ptot = φp dz dz′ Ĝ(z , z′; κ) b1(z) b2(z′) 0 0

where Rg2 = sσ2̅ /6. In all that follows, we shall set the Kuhn length, σ = 1. Therefore, all distances are measured in units of σ. Note that the boundary conditions are easily obtained from eq 3, ∇ Ĝ (z , z′; κ) =− z Ĝ

(11)

Here the strength of the surface interaction with the terminal monomers is quantified by the parameters Qi. The end−end distribution for the telechelic polymers is Ĝ (z, z′; k) b1(z) b2(z′). B. Adsorption Modes: Population Explosion of Short Loops. The total surface densities (number of polymers per unit area) between the surfaces is given by

A differential equation for Ĝ can be obtained by averaging the left- and right-hand sides of eq 1,

∇z Ĝ(z , z′; κ) Ĝ

(10)

exp( −βVi(z)) = Q i δ(z − 0) + Q i δ(z − D) + 1

(5)

R g2 2 ∇z Ĝ (z , z′; κ) − Ĝ (z , z′; κ) = −δ(z − z′) κ

κ cosh(ξ) − λ sinh(ξ)

where ξ = √κD/2Rg. The distribution Ĝ (z, z′; k) is for ideal chains weakly adsorbing onto the surfaces. It does not yet account for the strong attraction between the end monomers and the surfaces. We can describe this interaction using a shortranged surface attraction, Vi(z). The end monomers are labeled by the index i = 1,2, respectively, allowing for asymmetry. The surface Boltzmann factor can be represented by a sum of Dirac delta functions, which reflects the short-range of the surface adsorption, i.e.,

κ determines the average polymer length, which is given by s/κ. ̅ In our evaluations, κ is ultimately set to unity. However, it is convenient to maintain it as an explicit variable in the expressions below, as it will be used to develop results for more general molecular weight distributions. Using F(s), we define the average end−end distribution,

∫0

(9)

and

where ϵ is a measure of the adsorption energy and we have assumed planar symmetry. Here the z-coordinate is perpendicular to the two parallel planar surfaces, situated at z = 0 and z = D, respectively. The strong surface adsorption of the end monomers is assumed to act on a very short section of chain (relative to Rg). Hence, it is highly nonlinear and is treated in a different way, as we shall show later. Polydispersity enters the model by assuming that the length, s, of the central block has an associated distribution F(s). For the analysis in this section we shall use an exponential distribution. An exponential distribution is also consistent with a living central polymer block, which is the system studied in recent experiments on chainlike micelles.11 We describe the distribution using an auxiliary parameter, κ. Thus, κ F(s) = exp( −κs / s ̅ ) (4) s̅

Ĝ (z , z′; κ) =

κ sinh(ξ) − λ cosh(ξ)

κ cosh(2ξ) − λ sinh(2ξ) A(κ, D) B(κ, D)

(14)

κ3/2 fbrdg = φpR g A(κ, D) B(κ, D)

(15)

sinh(ξ) ftail = 2φpR g κ A(κ, D)

(16)

Similar results for this system were also obtained by van der Gucht et al.36 Minor differences occur, as those authors defined a sticking potential for the end monomers over a small but finite range, while we use a delta function adsorption, eq 11. However, the qualitative conclusions remain similar.

(8) 4225


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the two surfaces become distinct. The populations of tails and free chains are not shown, as they are insignificant on the scale of this figure. In real chains, the length scale for cooperative binding will be of the order of the Kuhn length, or the persistence length for stiff polymers. The bridging chains cannot take advantage of cooperative binding when the surface separation is much larger than this length. On the other hand, short loops bind very strongly to create a divergent population. However, these loops do not directly influence particle interactions beyond short surface separations. Upon further increase in separation, D/Rg ∼ O(1), the large population of short loops remains essentially constant while a small (finite) increase in loops occurs due to the noncooperative binding of longer polymers. The population of the latter is of order η2. This increase occurs over a length scale (b) of Rg and is described by the function Ploop (D/Rg). Clearly, loop explosion will only occur when a significant population of short chains exist, as is the case with living chains. As we shall show below, it does not occur for the more general Schulz−Flory molecular weight distributions, wherein the number of short chains approaches zero. Real discrete telechelic chains, where the central block has a size less than or of the order of the Kuhn length, will exhibit a significant loop population due to this phenomenon. One is likely to obtain this situation when the central polymer block is somewhat stiff. While we have argued that short chains will not directly influence surface forces at large separation, we need be cognizant of the fact that, for real polymers, a large surface population will lead to crowding, a mechanism not at play in the ideal model considered here. That is, the surface adsorption of end monomers of short loops will likely dominate those of other modes of binding and indeed may prevent the formation of tails and bridges. This conjecture will be tested in future work. When monomers exclude each other, the large number of adsorbed short loops may well interfere with the adsorption of longer polymers, significantly reducing the latter’s influence on particle interactions. C. Schulz−Flory Distribution. As we have shown in previous work,10,39 the living chain model presented above can be straightforwardly extended to the case of more general polydispersity by noting that the exponential distribution can be considered as a zeroeth-order Schulz−Flory (S−F) distribution.40 Using the auxiliary parameter κ, the nth order S−F function is given by

Interestingly, one can combine the loop and bridge contributions to obtain the following succinct result cosh(ξ) floop + fbrdg = 2φpR g κ A(κ, D)

(17)

In the rest of this paper, we shall focus on surfaces which are purely repulsive to the central block, i.e., λ → −∞. For convenience, we shall assume that χ1 = χ2 = χ. To obtain sufficient populations of adsorbed species, the adsorption strength of terminal blocks to the surfaces, as measured by χ, must be sufficient to overcome the entropic repulsion, determined by λ. For separations, D/Rg ∼ 1, both f loop and f tail ∼ |λ|−1, while f bridge ∼ |λ|−2. Thus, at this separation, finite populations of tails and bridges will be obtained by the choice χ/λ → η as λ → −∞, where η ∼ O(1). We denote this the sticking limit. This limit has drastic implications on the loop population. In particular, we note that (close to the sticking limit) when the surfaces are in ”contact”, D = 0, Ploop(D = 0) = φpR g κηχ ∼ O(χ)

(18)

Thus, Ploop(D=0) diverges in the sticking limit, undergoing something of a loop ”population explosion”. When the separation increases slightly so that D ≳ |λ|−1, the loop population can be written as (a) (b) Ploop ≈ Ploop + Ploop (D / R g )

(19)

where, in the sticking limit, → 2φpRgκηχ. That is, as D increases from contact, the large loop population increases to twice its value quite suddenly, over a distance ∼|λ|−1. A concomitant decrease in the number of bridges also occurs over this range. This behavior indicates that the explosion in the population of loops in the sticking limit is due to very short polymers that bind both ends cooperatively to a given surface. Thus, the binding of one end pays all or most of the entropic cost of surface binding for the whole molecule. Figure 1 shows (a) Ploop

F (n)(κ, s) =

κn + 1 Γ(n + 1)

sn s ̅ n+1

exp( −κs / s ̅ )

(20)

where Γ(x) is the gamma function. A S−F distribution is obtained by choosing the value κ = n + 1, where n is the polydispersity index and is conveniently treated as an integer. As already noted, n = 0 correponds to the exponential distribution. The distribution narrows as n increases, and the polymer sample eventually becomes monodisperse in the limit n → ∞. Generalizing eq 5, we obtain

Figure 1. Plot of surface densities, Ploop, and Pbrdg versus D/Rg for ideal telechelic chains with living polymer central blocks (n = 0). We have used an average polymer length of s ̅ = 200 and adsorption strength χ = 100000/Rg. The surface repulsion was chosen, so that η = −1 and the polymer density φp = 0.0005. Note that while the adorption strength, χ, is high, it is conteracted by a large entropic repulsive contribution λ, so that the quantity η is a modest value.

(n) Ĝ (z , z′; κ) =

∫0

∞

ds F (n)(κ, s) G(z , z′; s)

(21)

This distribution satisfies the following recursive differential equation,

this behavior for a strongly repulsive surface, reasonably close to the sticking limit. At contact (D = 0), there is no distinction between short loops and bridges and the populations of both are equal. However, upon a small increase in separation, 0 < D/Rg ≪ 1, the number of short bridges drops dramatically (to essentially zero) and the number of short loops doubles, as

R g2 2 (n) (n) ∇z Ĝ (z , z′; κ) − Ĝ (z , z′; κ) κ (n − 1) = −Ĝ (z , z′; κ) 4226

(22)


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where Ĝ (0) is identical to Ĝ , described in the previous section. This provides a way of generating results for higher order S−F distributions. However, we can also use the variable κ to obtain an alternative recursion formula, (n + 1) (n) Ĝ (z , z′; κ) = Ĝ (z , z′; κ) −

We begin by evaluating the sum of loop and bridge contributions to the free energy. From eq 17, we obtain (0) 2 βΩ(0) loop + βΩbrdg = − 2φpR g κχ

κ n+1

(n) ∂Ĝ (z , z′; κ) × ∂κ

= (23)

2φpR g κχ 2 λ

1−

1 κ /λ tanh(ξ)

κ ≈ 2φpR g κηχ 1 + tanh(ξ) λ

{

which follows from the specific form of the S−F distribution. We found this approach the most convenient to use in this work. Other thermodynamic properties of the system, which are linear functionals of the end−end distribution, obey this relation. For example, the population of loops for higher order S−F distributions is given by (n) κ ∂Ploop (n + 1) (n) Ploop = Ploop − n + 1 ∂κ

cosh(ξ) A(κ, D)

} (25)

Recall that ξ = √κD/2Rg. Here the approximation in the last line of eq 25 becomes essentially exact in the sticking limit. As discussed earlier, short loop adsorption becomes infinite in this limit. It contributes to the surface tension of two isolated surfaces, as is indicated in eq 25. We denote this surface tension (0) term as 2ΔΓloop , where

(24)

ΔΓ(0) loop = φpR g κηχ

where we have again used the superscript to denote the use of the nth order S−F distribution. From eq 18, we see that the term due to the loop explosion in living chains (n = 0) is linear in κ. Substituting it into eq 24 immediately shows that it does not contribute for higher order S−F distributions. Figure 2

(26)

The tail contribution to the free energy is given by βΩ(0) tail = − 4φpR g χ κ 4φpR g η

≈

κ

sinh(ξ) A(κ, D)

tanh(ξ)

(27)

Finally, the f ree chain contribution to the free energy is βΩ(0) free = − 2φpR g λ ≈

2φpR g κ

sinh(ξ) κ A(κ, D)

tanh(ξ)

(28)

The surface tension of an isolated surface is given by Γ = (0) ΔΓloop + ΔΓ(0), where the first term is given by eq 26. The second term is obtained by noting tanh(ξ) → 1 as ξ → ∞. (0)

Figure 2. Plot of surface densities versus D/Rg for ideal telechelic chains containing a central block with Schulz−Flory polydispersity (n = 1). We have used an average polymer length of s ̅ = 200 and adsorption strength χ = 100000/Rg. The surface repulsion was chosen so that η = −1 and polymer density φp = 0.0005.

ΔΓ(0) =

φpR g κ

(κη + 1)2

(29)

Recall that, for a living polymer, κ = 1. The net interaction between the surfaces can then be expressed as

shows the population of all adsorption modes for the case n = 1. In this case, the molecular weight distribution goes to zero in the limit of short chains. The lack of short chains means that loop explosion no longer occurs and the populations of the different adsorption modes become comparable. D. Free Energy. The free energy is also a linear functional of Ĝ (n)(z,z′;k). Indeed, the total Gibbs free energy per unit (n) , can also be partitioned into free, loop, bridge, surface area, Ωtot (n) (n) (n) (n) (n) = Ωfree + Ωloop + Ωbrdg + Ωtail . These and tail contributions: Ωtot terms are closely related to the adsorption densities described (n) (n) = −Pfree + Dϕp, and βΩx(n) = above. In particular, we have βΩfree (n) −Px , where x = loop, bridge, and tail. Here β−1 (=kBT) is the (n) is the free energy per unit of thermal energy. Note that Ωtot unit area in excess of the bulk. Thus, the bulk term is subtracted from the f ree component above. The net interaction between (n) (n) = Ωtot − 2Γ(n), where the particle surfaces is given by ΔΩtot (n) Γ is the surface tension for a single surface. For the moment, we will only consider living polymers, n = 0, and obtain expressions for the free energy in the sticking limit.

(0) βΔΩ(0) tot = 2ΔΓ {tanh(ξ) − 1}

(30)

Thus, short loops play no role in the net free energy, even for separation distances below the Kuhn length of the polymer. The reason for this is a cancellation between loop and bridge terms. As D approaches contact (below the Kuhn length), the number of short loops will decrease (by about a half). This desorption process contributes a strong repulsive term to the free energy. However, it is accompanied by a growth in short bridges and a strong attraction, which negates the loop contribution. In earlier work, we presented similar (albeit more complex) expressions. Equation 30 represents the much simplified result that follows for the situation where the surfaces are repulsive to the central block. This is a ubiquitous case in experimental scenarios, which makes our result particularly useful. A number of conclusions follow immediately from this expression. For example, the role of the adsorption strength, χ, is completely embodied in the 4227


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separations. Figure 5 shows the separate (net) free energy contributions for the case n = 2. We note that, while the maximum

quadratic form, eq 29. Remarkably, this expression predicts that the interaction free energy between particles varies nonmonotonically with respect to the adsorption strength η. The minimum interaction strength occurs at η = −1, where it disappears altogether! Whether the adsorption strength is weakened or strengthened from this value, a stronger attraction is obtained. We should note that this is only the case for living polymers. However, with the methods described above, the free energy can be relatively easily generalized to other degrees of polydispersity, as described by S−F distributions of higher order. An example of this is given in Figure 3, which shows the

Figure 5. Free energy components (units of β−1) of ideal telechelic chains containing a central block with a Schulz−Flory polydispersity of order n = 2. We have used an average polymer length of s ̅ = 200 and an adsorption strength χ = 100000/Rg; the surface repulsion was chosen so that η = χ/λ = −2 and polymer density ϕp = 0.0005.

in the bridging attraction occurs at D/Rg ≈ 2, the long-range of the repulsive contributions (especially the tail term) means that the minimum in the total free energy occurs at D/Rg ≈ 3. The qualitative picture is totally different for equilibrium polymers, which display attractive interactions down to very short separations. These behaviors could be tested experimentally, and they indicate that some modulation of particle interactions is possible, by varying the degree of polydispersity. E. Interaction between Spheres: Analytic Expressions in the Colloid Limit. In many cases experimentalists deal with polymer/particle mixtures, where the particles are considered to be essentially spherical. The interaction between two spherical particles, with radius RS and surface separation D, can be estimated with the Derjaguin approximation (DA). This becomes a better approximation when RS ≫ Rg, i.e., toward the so-called colloidal limit. The DA for living telecheic polymers is given by the following relation,

(n) βΔΩtot

Figure 3. Total free energy of ideal telechelic chains containing a central block with a Schulz−Flory polydispersity of order n. We have used an average polymer length of s ̅ = 200 and an adsorption strength χ = 100000/Rg; the surface repulsion was chosen so that η = χ/λ = −2 and polymer density φp = 0.0005. The arrow indicates the order of increasing n, with n = 0, 1, 2.

net free energy for varying degrees of polydispersity n = 0−2. As suggested earlier, the interactions can be modified in an interesting way, through appropriate choices of the adsorption strength, η. Here we have chosen, η = −2, which, from eq 29, means that the sum of loop, bridge, and tail contributions to the free energy add to zero (for the case n = 0). That is, the total free energy is in this case independent of adsorbed polymer contributions and is equal to the contribution due to free polymers alone. The behavior of the free energy for increasing n can be correlated with the populations of adsorbing polymers. Figure 4 shows the number of adsorbing polymers per unit area,

ΔΩ(0) coll(κ ; R ) ≈ πRS

∞

∫D

dz ΔΩ(0) tot (z)

(31)

where D = R − 2RS, with R the distance between the centers of (0) the colloidal-sized particles. The quantity ΔΩcoll (R) acts as an effective pair interaction between the spherical particles. This simple model precludes explicit consideration of the polymer. Strictly speaking, one would also need to include the effect of many-body interactions as well, in order to obtain an accurate description of the polymer mediated forces. However, within the colloid limit, one can ignore the higher-order interactions (beyond the pair interaction), due to the relatively small size of the polymer molecules. Substituting the expression for (0) ΔΩtot (z) into eq 31, we get Figure 4. Plot of surface densities versus D/Rg for ideal telechelic chains containing a central block with Schulz−Flory polydispersity (n = 2). We have used an average polymer length of s ̅ = 200 and adsorption strength χ = 100000/Rg. The surface repulsion was chosen so that η = χ/λ = −1 and polymer density φp = 0.0005.

(0) βΔΩ(0) coll(κ ; R ) = 4πΔΓ

=−

for S−F polydispersity n = 2. This should be compared with the cases of Figures 1 and 2. We note that as the value of n increases, so does the short ranged repulsion, as well as the weak attraction at a separation D/Rg ≈ 3. This is due mainly to bridging, which dominates the (depletion) attraction at these

RSR g κ

4πφpR g2RS κ

∫ξ

∞

dx {tanh(x) − 1}

(κη + 1)2 ln{1 + e−2ξ} (32)

While this result holds for living chains, it is easily generalized to other S−F descriptions of polydispersity. That is, the free energy is a linear functional of the end−end distribution. 4228


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ignore intermolecular interactions to some extent, while intramolecular interactions can be modeled with chain stiffness. This also makes it possible to make some contact with the ideal, continuous chain model, solved above. It should be noted, however, that chain stiffness is not easily incorporated in the continuous chain model. On the other hand, we have already developed an exact PDFT for semiflexible chains that is relatively easy to solve numerically. Including stiffness in the polymer modeling generally causes the behaviors in flexible polymers to be enhanced, especially in dilute solutions, due to the larger radius of gyration. The PDFT for semiflexible chains has been described in detail elsewhere, so we will only provide a bief description in this work. The monomers are modeled as hardspheres with diameter, σ, connected to form a necklace-like polymer; that is, the bond length and the diameter are equal to σ. Excluded volume effects are handled via an excess functional obtained by integrating the “generalized Flory-dimer” equation of state.26 Furthermore, intrinsic chain stiffness is introduced via a bond angle potential EB, with

Furthermore, one is able to use inverse Laplace transforms to generate a series representation for the corresponding interaction in monodisperse telechelic chains. We show this as follows: ΔΩ(0) coll(κ ; R ) =

∫0

∞

ds

κ mon) exp( −κs / s ̅ )ΔΩ(coll (s ; R ) s̅ (33)

(mon) where ΔΩcoll (s;R) is the free energy for monodisperse polymers (0) of length s. It is equal to the Laplace inverse of ΔΩcoll (κ; R)/κ, evaluated at s = s.̅ In order to perform the inversion, we note that, for D > 0, we obtain the following:

∞

ln{1 + e−2ξ} =

∑ n=1

( −1)n + 1 −2nξ e n

(34)

Substituting this series into eqs 32 and 33, to obtain (0) ΔΩcoll (κ;R)/κ, we find terms of the type exp(−√κD/Rg), exp(−√κD/Rg)/κ, and exp(−√κD/Rg)/κ2. These can be inverted to give the final result ∞ mon) (R g ; R ) = −4πφpR g2RS ΔΩ(coll

∑ n=1

⎛ s s 1⎞ βEB(si , si + 1) = ϵ⎜1 − i i + ⎟ ⎝ σ2 ⎠

( −1)n + 1 n

⎧⎛ 2 2 ⎞ ⎛ ⎞ ⎪ n D ⎟ erfc⎜ nD ⎟ 2 1 × ⎨⎜ + η − ⎜ 2R ⎟ ⎟ ⎪⎜⎝ 2R g2 ⎝ g⎠ ⎠ ⎩

(36)

Here, si denotes the bond vector between monomers i and i + 1 and ϵ is the strength of the bending potential. Forsman and Woodward41 presented a PDFT for semiflexible monodisperse polymers. The corresponding PDFT for polydisperse chains was formulated by Woodward and Forsman, first for flexible42 and subsequently for semiflexible43 polymers. Unlike the continuum theory, the surface potential enters via an explicit functional form, rather than as a boundary condition. In order to model telechelic behavior, we adopt two kinds of surface potentials, Ve(z) and Vc(z), acting on end and central block monomers, respectively. The surface potentials have a truncated form and are each zero beyond a range of 4σ,

⎛ ⎞ −n2D2 /4R g2 ⎫ ⎪ e 2 nD ⎜ ⎬ + ⎜η − 1⎟⎟ π ⎪ ⎝ 2R g ⎠ ⎭ (35)

As far as we are aware, eqs 32 and 35 are the first reported analytic results for the interaction between two colloidal spheres immersed in a telechelic polymer model, albeit under ideal (theta) conditions. Using the method described above, these results can be used to describe the effective pair interaction for colloidal particles in ideal telechelic polymers over a spectrum of polydispersities, ranging from living to monodisperse chains. As such, it constitutes one of the main results of this paper.

⎛ 2 aα ⎞ βVα(z) = 2π⎜ − ⎟ − C if x ≤ 4σ ⎝ 45z9 3z3 ⎠ =0

otherwise

(37)

Here, z denotes the distance perpendicular to the given surface. The parameter C is chosen such that Vα(4σ) = 0. The index α denotes either “c” or “e” monomers. In all our calculations below, we chose ae = 1 and ac = 0, which were fixed. Furthermore, the bulk density of monomers, nmb , was fixed at nmb σ3 = 0.01, which places us in the dilute regime for all the polymer lengths considered. The stiffness of the semif lexible polymers was characterized by ϵ = 6 (for flexible chains, this parameter is of course zero). A. Polydispersity and the Onset of Bridging Attraction vs Loop Repulsions PDFT. The resulting interaction free energy for telechelic polymers, where the degree of polydispersity is varied, is shown in Figure 6. For equilibrium chains, the interaction is monotonically attractive and seems to display the behavior predicted by the ideal chain system, eq 30. In the ideal system, all contributions to the free energy are monotonically varying and the bridging attraction will compensate for the strong repulsions of the tail and loop contributions at all separations. As the system becomes more monodispersed, the various contributions to the free energy become non-monotonic. In the PDFT results, we note the onset of a short-ranged repulsion and weak attraction at intermediate separations, as the polydispersity decreases. This is also reminiscent of the ideal

III. DENSITY FUNCTIONAL THEORY FOR POLYDISPERSE TELECHELIC POLYMERS While the ideal model presented above allows one to obtain a number of useful analytic results, it is strictly only valid when the chains are under Θ conditions or else are sufficiently dilute so that intermolecular interactions can be ignored. Furthermore, use of the continuum limit allows one to obtain the chain distributions by solving a linear second-order differential equation. To extend our theoretical treatment to more general systems, one needs to use a more complex model of a discrete chain, with intermolecular interactions and a higher level of theory. The polymer density functional theory (PDFT) has proved a very useful approach for treating interacting polymer fluids, especially when the density of polymers is high and the interactions are dominated by intermolecular interactions. Here it has proved to be quite accurate when compared with computer simulations. Recently, Cao and Wu used PDFT to show a weak attraction due to bridging at the point where the loop brushes from each surface first come into contact.22 Here, we shall use PDFT to explore the role played by adsorption strength and polymer moleclar weight in determining the overall particle interaction. We shall work in the dilute regime, where one can 4229


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Figure 6. (a) Total free energy per unit area in units of kBT versus D/σ for telechelic flexible polymers, obtained from the polymer density functional theory. All polymers have average length s ̅ = 100, and the bulk monomer density is given by nmb σ3 = 0.01. The degree of polydispersity for the central block is varied according to the following values of the S−F order: n = 0 (blue diamonds); n = 1 (red circles); n = 3 (green triangles); n = 19 (purple squares); and monodisperse (blue crosses). (b) As for part a, but for semiflexible polymers with ϵ = 6.

Figure 7. Total free energy per unit area in units of kBT versus D/σ for semiflexible telechelic flexible polymers, obtained from the polymer density functional theory. In all cases the bulk monomer density is given by nmb σ3 = 0.01 and the stiffness parameter is ϵ = 6. (a) Equilibrium polymers (n = 0) with the following average molecular weights: s ̅ = 100 (blue diamonds); s ̅ = 200 (red circles); s ̅ = 400 (green triangles). (b) As for part a with n = 3. (c) as for part b but for monodisperse polymers.

adsorption of chain ends. That is, as the molecular weight increases, the entropic cost for adsorption becomes greater. This effect is due to the confinement of the chain close to the surface. For ideal chains the correlation length is of the order of the radius of gyration and so the entropic cost grows with chain length. For excluding chains this also applies in the dilute regime. However, crossing into the concentrated regime, excluded volume effects will counter the entropic repulsion due to the surfaces.38 Thus, if the bulk density is increased, the effect of molecular weight on the surface forces would be diminished. This effect is also seen in the ideal model via eq 30. For a large and fixed adsorption strength, the interaction free energy (0) βΔΩtot ∼ 1/s ̅2. Results for equilibrium chains in the ideal model are shown in Figure 8. This kind of behavior is hidden in models which consider the strong adsorption limit, wherein loops and bridges are assumed to dominate the free energy. It is clear that it is the so-called telechelic, bridge, loop, and tail contributions to the interactions which decrease as s ̅ increases, so that eventually the free term dominates.

chain behavior shown in Figure 3. The role played by polymer stiffness is to push these behaviors out to larger separations. In this context, we should point out the scale difference between (a) and (b) in Figure 6. The confirmation of this effect in real experimental systems would appear straightforward in principle. The qualitative similarity between the ideal continuum chain and the PDFT results leads us to conclude similar mechanisms are at play in both models. In particular, Figure 5 indicates that the bridge contribution becomes significant at separations where the loop and tail terms are diminishing. This behavior is quite different from what is observed in equilibrium polymers. B. Molecular Weight Dependence. In Figure 7 we highlight the role played by molecular weight in determining particle interactions in polymers with varying degree of polydispersity. We note that, in general, attractions becomes weaker as s ̅ increases. A similar result is also seen in the flexible chain model. For equilibrium chains we have the remarkable result that the attraction appears more short-ranged as the average molecular weight increases! This is due to the diminishing 4230


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AUTHOR INFORMATION

Corresponding Authors

*Electronic address: [email protected]. †Electronic address: [email protected]. Notes

The authors declare no competing financial interest.

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REFERENCES

(1) Wool, R. Polymer interfaces: structure and strength; Hanser/ Gardner: Munich, 1995. (2) Lger, L.; Raphal, E.; Hervet, H. In Polymers in Confined Environments; Granick, S., Binder, K., de Gennes, P.-G., Giannelis, E., Grest, G., Hervet, H., Krishnamoorti, R., Lger, L., Manias, E., Raphal, E., Wang, S.-Q., Eds.; Advances in Polymer Science; Springer: Berlin/ Heidelberg, 1999; Vol. 138; pp 185−225. (3) Akcora, P.; Liu, H.; Kumar, S. K.; Moll, J.; Li, Y.; Benicewicz, B. C.; Schadler, L. S.; Acehan, D.; Panagiotopoulos, A. Z.; Pryamitsyn, V.; Ganesan, V.; Ilavsky, J.; Thiyagarajan, P.; Colby, R. H.; Douglas, J. F. Nat. Mater. 2009, 8, 354. (4) Amado, E.; Kressler, J. Soft Matter 2011, 7, 7144. (5) Heredia, K. L.; Tao, L.; Grover, G. N.; Maynard, H. D. Polym. Chem. 2010, 1, 168. (6) Ashcraft, E.; Ji, H.; Mays, J.; Dadmun, M. Macromol. Chem. Phys. 2011, 212, 465. (7) Haung, J. H. Z.; Mays, J.; Dadmun, M.; Smith, G.; Bedrov, D.; Zhang, Y. Langmuir 2010, 26, 202. (8) Haung, J. H. Z.; Mays, J.; Dadmun, M. Macromolecules 2008, 41, 1009. (9) Patton, D.; Knoll, W.; Advincula, R. Macromol. Chem. Phys. 2011, 212, 485. (10) Woodward, C. E.; Forsman, J. Phys. Chem. Chem. Phys. 2011, 13, 5746. (11) Helgeson, M. E.; Wagner, N. J. J. Chem. Phys. 2011, 135, 084901. (12) Edwards, S. Proc. Phys. Soc. 1965, 85, 613. (13) Helfand, E. J. Chem. Phys. 1975, 62, 999. (14) Milner, S. T.; Witten, T. A.; Cates, M. E. Macromolecules 1988, 21, 2610. (15) Milner, S. T.; Witten, T. A. Macromolecules 1992, 25, 5495. (16) Johner, A.; Joanny, J. F. J. Chem. Phys. 1992, 96, 6257. (17) Tang, W. H.; Witten, T. A. Macromolecules 1996, 29, 4412. (18) Bjorling, M.; Stilbs, M. Macromolecules 1998, 31, 9033. (19) Bhatia, S. R.; Russel, W. B. Macromolecules 2000, 33, 5713. (20) Meng, X.-X.; Russel, W. B. Macromolecules 2003, 36, 10112. (21) Wijmans, C. M.; Leermakers, F. A. M.; Fleer, G. J. J. Colloid Interface Sci. 1994, 167, 124. (22) Cao, D.; Wu, J. Langmuir 2006, 22, 2712. (23) Misra, S.; Mattice, W. L. Macromolecules 1994, 27, 2058. (24) Woodward, C. J. Chem. Phys. 1991, 94, 3183. (25) Woodward, C. J. Chem. Phys. 1992, 97, 695. (26) Woodward, C. E.; Yethiraj, A. J. Chem. Phys. 1994, 100, 3181. (27) Yethiraj, A. J. Chem. Phys. 1998, 109, 3269. (28) Forsman, J.; Woodward, C. E.; Freasier, B. C. J. Chem. Phys. 2002, 117, 1915. (29) Yu, Y.-X.; Wu, J. J. Chem. Phys. 2002, 117, 2368. (30) Chen, H.; Ye, Z.; Peng, C.; Liu, H.; Hu, Y. J. Chem. Phys. 2006, 125, 204708. (31) Chen, H.; Ye, Z.; Cai, J.; Liu, H.; Hu, Y.; Jiang, J. J. Phys. Chem. B 2007, 111, 5927. (32) Chen, H.; Cai, J.; Ye, Z.; Peng, C.; Liu, H.; Hu, Y.; Jiang, J. J. Phys. Chem. B 2008, 112, 9568. (33) Woodward, C. E.; Forsman, J. Phys. Rev. Lett. 2008, 100, 098301. (34) Woodward, C.; Forsman, J. Macromolecules 2009, 42, 7563. (35) Woodward, C.; Forsman, J. J. Chem. Phys. 2010, 130, 244703. (36) van der Gucht, J.; Besseling, N. A. M.; Fleer, G. J. J. Chem. Phys. 2003, 119, 8175. (37) Testard, V.; Oberdisse, J. C. L. Macromolecules 2008, 215, 41.

Figure 8. Total free energy per unit area in units of kBT versus D/σ for equilibrium (n = 0) telechelic flexible polymers, obtained from the ideal continuum model for varying molecular weights. In the direction of the arrow we have s ̅ = 100, 200, and 300, respectively. The adsorption strength of end monomers to the walls was characterized by χ = 1000/Rg, and the repulsion of the central monomers was characterized by λ = −10Rg. The bulk monomer density is given by nmb σ3 = 0.01.

IV. CONCLUSION In this paper we have shown how a model of polydisperse ideal telechelic polymers between planar surfaces can be solved analytically in the continuum chain limit, when the molecular weight distribution of the central block has the general Schulz− Flory form with integral polydispersity index, n. In the case of surfaces, which are purely repulsive, with respect to the central monomers, closed expressions are also available for the interaction between large colloidal particles within the Derjaguin approximation. For equilibrium polymers, we have identified a loop explosion, which occurs due to cooperative adsorption of short loops. While this does not anomalously affect particle interactions for ideal polymers, it is clear that surface crowding will likely have a significant role to play for monomers which repel one another. Our results show that changing the degree of polydispersity (varying the S−F order) has a drastic effect on surface forces, due mainly to changing the relative magnitudes of bridge, loop, tail, and free contributions. This is observed for both the ideal continuum model, as well as a dilute interacting monomer model, solved with the PDFT. With the PDFT we also showed that, for a fixed adsorption strength, the interactions between particles was significantly modulated by varying the average polymer molecular weight, due to the changing entropic cost of adsorbing a polymer end. In equilibrium chains, we observed an attraction of shorter range when the polymers had a larger molecular weight (on average). This seemingly anomalous result still awaits experimental confirmation. Similar effects were seen in the ideal model, which indicated that the bridge, loop. and tail contributions became weaker as the molecular weight increased. The consequences of these results on experimental systems are potentially important. For example, in a mixture of polydisperse telechelic polymers and particles, it becomes clear that shorter polymers will be more likely adsorbed on particles, giving rise to stronger interparticle forces, than may be expected from more monodispersed samples at the same average molecular weight. Additionally, a large population of very short telechelic polymers may well lead to crowding on particle surfaces, which will have the effect of reducing telechelic contributions to the interactions, so that depletion forces become dominant. At the moment we have studied central monomers which are repulsive to surfaces. On the other hand, entropic repulsion can be moderated by increasing central monomer attractions, which makes a range of new effects become possible. 4231


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(38) deGennes, P. G. Macromolecules 1981, 14, 1637. (39) Woodward, C. E.; Forsman, J. J. Chem. Phys. 2010, 133, 154902. (40) Schulz, G. Z. Phys. Chem. 1939, B43, 25. (41) Forsman, J.; Woodward, C. E. J. Chem. Phys. 2003, 119, 1889. (42) Woodward, C. E.; Forsman, J. Phys. Rev. Lett. 2008, 100, 098301−1. (43) Woodward, C. E.; Forsman, J. Macromolecules 2009, 42, 7563.

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Polydisperse Telechelic Polymers at Interfaces: Analytic Results and

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