Research Letters in Physical Chemistry

Volume 2008 (2008), Article ID 504917, 4 pages

http://dx.doi.org/10.1155/2008/504917

## Homogenization of Mutually Immiscible Polymers Using Nanoscale Effects: A Theoretical Study

^{1}New Materials Department, CIDETEC, Paseo Miramón 196, 20009 Donostia-San Sebastián, Spain^{2}Departament of Polymer Science and Technology, Faculty of Chemistry and Polymat, UPV/EHU, P.O. Box 1072, 20018 Donostia-San Sebastián, Spain

Received 21 January 2008; Accepted 9 May 2008

Academic Editor: Liming Dai

Copyright © 2008 Sarah Montes et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A theoretical study to investigate homogenization of mutually immiscible polymers using nanoscale effects has been performed. Specifically, the miscibility behavior of all-polymer nanocomposites composed of linear-polystyrene (PS) chains and individual cross-linked poly(methyl methacrylate)-nanoparticles (PMMA-NPs) has been predicted. By using a mean field theory accounting for combinatorial interaction energy and nanoparticle-driven effects, phase diagrams were constructed as a function of PMMA-NP size, PS molecular weight, and temperature. Interestingly, complete miscibility (i.e., homogeneity) was predicted from room temperature to 675 K for PMMA-nanoparticles with radius less than ~7 nm blended with PS chains (molecular weight 150 kDa, nanoparticle volume fraction 20%) in spite of the well-known immiscibility between PS and PMMA. Several nanoscale effects affecting miscibility in PMMA-NP/PS nanocomposites involving small PMMA-nanoparticles are discussed.

#### 1. Introduction

Nanoscale effects are responsible
for the growing scientific interest in both nanomaterials and nanocomposites [1]. Often, nanoobjects display interesting physical
phenomena such as surface plasmon resonance in metallic nanoparticles (NPs), ballistic
transport in carbon nanotubes (CNTs), or fluorescence emission in quantum dots
(QDs), among others. Most of them are useful for a huge of potential applications
covering from nanoelectronic to improved sensors devices [2]. Also, nanocomposites composed of well-dispersed
nanoobjects into a polymer matrix lead to unexpected mechanical [3] and/or rheological properties [4]. A representative example is the large decrease (up
to 80%) in melt viscosity upon nanoparticle addition first observed in polystyrene-nanoparticle
(PS-NP)/linear-polystyrene (PS) blends by Mackay et al. [5]. The underlying physics of this unexpected behavior
is being currently explored [6] and should be rationalized
in terms of scaling concepts at the nanoscale [7].
Blend miscibility (homogeneity) is a necessary condition to observe such a viscosity
drop in *all-polymer nanocomposites* (i.e.,
polymer-nanoparticle/linear-polymer blends).

Recently, we have introduced an entropic model for predicting the miscibility behavior of PS-NP/PS nanocomposites with very good agreement between theory and experiment [8]. Additionally, the theory has been employed for the prediction of the interaction parameter, the miscibility behavior, and the melting point depression of athermal poly(ethylene) (PE)-nanoparticle/linear-PE nanocomposites using chain dimensions data from Monte-Carlo (MC) simulations [9]. Our main findings indicate that dilution of contact, hard sphere-like, nanoparticle-nanoparticle interactions plays a key role in explaining the miscibility behavior of polymer-nanoparticles dispersed in a chemically identical linear-polymer matrix [8, 9].

In a recent work, the athermal model has been extended to calculate the phase diagram of weakly interacting all-polymer nanocomposites by accounting for combinatorial interaction energy and nanoparticle-driven effects [10]. Complete miscibility was predicted for PS-nanoparticles with radius < 6 nm blended with poly(vinyl methyl ether) (PVME) at low concentrations. When compared to linear-PS/PVME blends displaying phase splitting at K, the miscibility improving effect of sub-10 nm PS-nanoparticles was clearly highlighted [10].

In this letter, we
explore theoretically the conditions for homogenization of two mutually
immiscible polymers by changing all the linear-polymer chains of one of the
components by cross-linked polymer nanoparticles. Specifically, we consider the
PS/poly(methyl methacrylate) (PMMA) pair as a model system. Immiscibility
between PS and PMMA is well known in the literature as a result of unfavorable
interactions between styrene (S) and methyl methacrylate (MMA) repeat units
[11–13]. Here, miscibility diagrams for PMMA-NP/PS nanocomposites are reported
as a function of PMMA-NP size, PS molecular weight (*M _{n}*) and
temperature. Finally, several nanoscale effects affecting the miscibility
behavior of PMMA-NP/PS nanocomposites are also discussed.

#### 2. Theoretical Approach

For a binary blend to be
thermodynamically stable against phase separation, the following well-known conditions
must be fulfilled: In a mean field, theoretical framework, , the free energy of mixing (per
unit volume) for an all-polymer nanocomposite composed of spherical
polymer-nanoparticles (component 1) of volume fraction , radius *R _{p}*, nanoparticle volume , and linear-polymer chains of
degree of polymerization and monomer volume is given by [10]
where

*k*is the Boltzmann constant,

*T*is the absolute temperature, is the radius of gyration of linear-polymer 2,

*r*is the radius of a repeat unit 2, and and are the entropic and enthalpic components of the blend interaction parameter (), respectively.

_{2}Equation (1*a*) provides
the contribution to the total free energy of mixing due to the *combinatorial entropy* of mixing, whereas (1*b*)
gives the contribution to the free energy of mixing due to *nanoparticle-driven*
effects. Equation (1*b*) takes into account the dilution of hard sphere-like nanoparticle-nanoparticle interactions upon mixing (first RHS term) [8, 9] and the stretching of the linear-polymer chains due to the presence of the nanoparticles (second RHS term). Referring to this latter term, a Ginzburg-type expansion term is adopted to account for the fact that the
polymer-nanoparticles cause stretching of the polymer chains in their vicinity [14].

Equation (1*c*) accounts
for the temperature-dependent
and nanoparticle size-dependent *interaction* effects in the all-polymer nanocomposite. A prefactor
is introduced in (1*c*) since the number of surface contacts with monomers 2 for
each nanoparticle becomes smaller as one increases the nanoparticle radius [14]. This ratio tends toward unity inasmuch as it should, *r _{1}* being the radius of a repeat unit 1.

For the sake of simplicity, in (1) we have omitted a term (~constant) arising from the (non-zero) reference free energy of the pure component 1 in the disordered molten state at (maximum packaging of monodisperse spheres) as dicussed in [9] which has no effect in the resulting spinodal equation (second derivative of the free energy of mixing with respect to ).

The condition just determines the spinodal miscibility boundary in the phase diagram, which is given by where we have employed .

Equations (1)–(2) are presumably valid only for (the theoretical maximum nanoparticle packaging volume fraction without freezing) and for (it should be noted that is typically around 0.3 nm) [10].

#### 3. Results and Discussion

It is well known that PS and PMMA lead to heterogeneous (phase separated) blends due to the immiscibility between components at high molecular weights [11–13]. The S/MMA temperature-dependent interaction parameter () has been determined by several experimental techniques such as small angle neutron scattering (SANS) (using deuterated block copolymers) [11, 12] and cloud-point (CP) measurements (using oligomer mixtures and taking into account end-group effects) [13] with good agreement between them. As an example, by SANS measurements [11] and by CP experiments [13]. Since the entropic contribution, , is much greater than the () term, the temperature dependence of was found to be relatively weak.

We have employed the temperature-dependent interaction parameter determined by CP measurements to calculate the spinodal miscibility boundary of PMMA-NP/PS composites as a function of nanoparticle radius at constant blend composition (see Figure 1). Complete miscibility across the 275–675 K temperature range was predicted for PMMA-nanoparticles with radius less than 6.8 nm in spite of the well-known immiscibility between PMMA and PS homopolymers. Conversely, for PMMA-nanoparticles with radius higher than 7.2 nm, complete immiscibility (phase separation) is expected. For PMMA-NP of radius in between 6.8 and 7.2 nm, partial miscibility was predicted as a function of temperature (the blends displaying upper critical solution temperature (UCST)-type behavior). No significant changes was observed when values from SANS experiments [11, 12] were employed in the calculations.

In order to rationalize
these nanoscale-driven results, we have examined the values of the different
factors governing (2) (second derivative of the energy of mixing). (arising from *combinatorial* effects) was found to be positive (favorable to
mixing) and increased linearly with temperature. Conversely, (arising from *interactions*) was
negative and increased in absolute value with *T*. In general, so miscibility was conditioned by favorable values of the term. At constant , the size of the PMMA-nanoparticles
was the main factor affecting the term (see (2*b*)). At a given temperature, was found to decrease upon increasing the PMMA-NP size leading, respectively, to
partial and complete immiscibility at PMMA-NP radius of 7 and 7.5 nm. As a
result, miscibility in PMMA-NP/PS nanocomposites can be mainly attributed to two
combined effects: (1) reduced (unfavorable) PS chain stretching by smaller PMMA-nanoparticles
and (2) favorable dilution of (hard sphere-like) nanoparticle-nanoparticle
interactions upon mixing.

The effect of PS molecular weight on the predicted phase diagram for PMMA-NP/PS nanocomposites is illustrated in Figure 2. As expected, a reduction in PS molecular weight shifts the miscibility boundary toward smaller nanoparticle sizes due to the large entropy penalty paid for nanoparticle inclusion into short linear-polymer chains. Hence, the critical PMMA-NP radius for complete immiscibility () changes from 5.4 to 7.2 and 7.8 nm upon changing the PS molecular weight from 50 to 150 and 500 kDa, respectively.

Concerning the effect of blend composition on for PMMA-NP/PS ( kDa) nanocomposites, a linear increase in (up to ~29%) on going from ( nm) to ( nm) was observed.

#### 4. Conclusions

A mean field theoretical model,
accounting for combinatorial interaction energy and nanoparticle-driven
effects, has been employed to investigate homogenization of mutually immiscible
polymers using nanoscale effects. The PS/PMMA pair has been selected as a
model system since immiscibility between PS and PMMA is well documented in the
literature, and reliable values of the S/MMA interaction parameter are
available as a function of *T*. Specifically, we have investigated the effect on blend
miscibility (homogeneity) of the replacement of linear-PMMA chains by
cross-linked individual PMMA-nanoparticles.

Hence, phase diagrams have been constructed for PMMA-NP/PS nanocomposites as a function of PMMA-NP size, PS molecular weight, blend composition, and temperature. Interestingly, complete miscibility across the 275–675 K temperature range was predicted for PMMA-nanoparticles with radius less than 6.8 nm blended with PS ( kDa, ). Increasing PS molecular weight and nanoparticle content was found to have a small positive effect on PMMA-NP/PS nanocomposite miscibility.

Homogenization of PMMA-NP/PS nanocomposites was mainly attributed to two combined nanoeffects: reduced PS chain stretching by the smaller PMMA-nanoparticles, and favorable dilution of contact (hard sphere-like) nanoparticle-nanoparticle interactions upon mixing.

#### Acknowledgments

Financial support by MEC (Grant no. CSD2006-53), Basque Government (Grupos Consolidados IT-274-07), and Diputación de Gipuzkoa through C. I. C. Nanogune—Consolider and Nanotron Project is gratefully acknowledged.

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