**UFRJ**

# Physical Chemistry of Polymers

(Parte **1** de 7)

Physical Chemistry of Polymers: Entropy, Interactions, and Dynamics

T. P. Lodge*

Department of Chemistry and Department of Chemical Engineering & Materials Science, UniVersity of Minnesota, Minneapolis, Minnesota 55455

M. Muthukumar

Department of Polymer Science & Engineering and Materials Research Science and Engineering Center, UniVersity of Massachusetts, Amherst, Massachusetts 01003

ReceiVed: January 23, 1996; In Final Form: May 14, 1996X

A brief examination of some issues of current interest in polymer physical chemistry is provided. Emphasis is placed on topics for which the interplay of theory and experiment has been particularly fruitful. The dominant theme is the competition between conformational entropy, which resists distortion of the average chain dimensions, and potential interactions between monomers, which can favor specific conformations or spatial arrangements of chains. Systems of interest include isolated chains, solutions, melts, mixtures, grafted layers, and copolymers. Notable features in the dynamics of polymer liquids are also identified. The article concludes with a summary and a discussion of future prospects.

1. Introduction

1.A. General Remarks. Macromolecules form the backbone of the U.S. chemical industry, and are essential functional and structural components of biological systems.1 Yet, the very existence of long, covalently bonded chains was in dispute only 70 years ago. The past 50 years has seen a steady growth in understanding of the physical properties of chain molecules, to the point that the field has achieved a certain maturity. Nonetheless, exciting and challenging problems remain. Polymer physical chemistry is a richly interdisciplinary field. Progress has relied on a combination of synthetic ingenuity, experimental precision, and deep physical insight. In this article, we present a brief glimpse at some interesting current issues and acknowledge some notable past achievements and future directions.

1.B. Basic Concepts. When N monomers join to form a polymer, the translational entropy is reduced. However, the entropy associated with a single molecule increases dramatically, due to the large number of different conformations the chain can assume.2-1 Conformational changes occur at both local and global levels. Local conformational states with differing energies depend on the chemical nature of substituent atoms or side groups, X, as sketched in Figure 1. Typically, there are three rotational conformers at every C-C bond. These states, and the torsional energy as a function of rotation about the middle C-C bond, are represented in Figure 2. If ∆² , kT, there exists complete static (i.e., equilibrium-averaged) flexibility. Even for higher values of ∆²/kT, where the trans conformation is preferred, the chain will still be flexible for large N. We can define a statistical segment length, b, over which the local stiffness persists; b depends on the value of

∆²/kT. 12 But, beyond this length, bond orientations are uncor- related. The parameter which determines the overall chain flexibility is b/L, where L, the chain contour length, is ∼N. If b/L , 1, the chain has complete static flexibility; for b/L . 1, the chain is a rigid rod.

Similarly, ∆E/kT determines the dynamical flexibility. If ∆E/ kT , 1, the time τseg ∼ exp(∆E/kT) required for trans T gauche isomerizations is short (i.e., picoseconds to nanoseconds in solution), and the chain is dynamically flexible. For higher values of ∆E/kT, dynamical stiffness arises locally. However, for large scale motions, involving times much greater than τseg, the chain can still be taken to be dynamically flexible. The chemical details of the monomers and solvent affect the local properties, b and τseg. Macroscopic, or global, properties doX Abstract published in AdVance ACS Abstracts, July 15, 1996.

Figure 1. N monomers combine to form one linear chain, here with an all-carbon backbone and a pendant group denoted X.

Figure 2. Schematic of the potential energy as a function of rotation about a single backbone bond and the corresponding trans and gauche conformers.

S022-3654(96)0244-4 C: $12.0 © 1996 American Chemical Society not depend directly on the local static and dynamic details and can be represented as universal functions (i.e., independent of chemical identity) by “coarse-graining” the local properties into phenomenological parameters. Polymer physical chemistry deals with both the macroscopic properties, dictated by global features of chain connectivity and interactions, and the phenomenological parameters, dictated by the local details.

Connectivity leads to long-range spatial correlations among the various monomers, irrespective of potential interactions between monomers.13,14 Such a topological connectivity leads naturally to statistical fractals,15 wherein the polymer structure is self-similar over length scales longer than b but shorter than the size of the polymer. For the ideal case of zero potential interactions, the monomer density F(r) at a distance r from the center of mass decays in three dimensions as 1/r,6,7 as shown in Figure 3. Consequently, the fractal dimension of the chain

Df ) 2. This result is entirely due to chain entropy, but the long-ranged correlation of monomer density can be modified by potential interactions. In general, F(r) decays as (1/r)d-D , where d is the space dimension. Equivalently, the scaling law between the average size of the polymer, e.g., the radius of gyration Rg, and N is Rg ∼ Nν, where ν ) 1/Df. The value of

Df is determined by the compromise between the entropy arising from topological connectivity and the energy arising from potential interactions between monomers. For example, most polymer coils with nonspecific short-ranged interactions undergo a coil-to-globule transition upon cooling in dilute solutions, such that the effective fractal dimension increases to about 3. Or, the chain backbone may be such that b increases at low T; in this case the chain can undergo a coil-to-rod transition, where

Df decreases monotonically to about 1. This competition between conformational entropy and monomer-monomer in- teractions represents a central theme of this article.

When specific, strong interactions such as hydrogen-bonding or electrostatic forces are present, chain conformations can suffer entropic frustration, as illustrated in Figure 4 for charge-bearing monomers.16,17 In the process of forming the fully registered state (a) between the oppositely charged groups, the chain, via random selection, can readily form a topological state such as (b). The chain is entropically frustrated in state (b) since the two registered pairs greatly reduce the entropic degrees of freedom of the chain. The chain needs to wait until the pairs dissociate, accompanied by a release of entropy, and the process of registry continues. This feature of entropic frustration is common in macromolecules containing chemically heterogeneous subunits and in polymers adsorbing to an interface. When a chain is frustrated by topological constraints, not all degrees of freedom are equally accessible, and standard arguments based on the hypothesis of ergodicity may not be applicable. The identification of the resulting equilibrium structure is a challenge to both experiment and theory. The kinetics of formation of such structures is also complicated by the diversely different free energy barriers separating the various topological states. In general, the distance between different trajectories of the system diverges with time, and the presence of free energy minima at intermediate stages of evolution delays the approach to the final optimal state.17 Several of these features are exemplified by biological macromolecules.

1.C. Recent Developments. Synthesis. Conventional polymerization methods, either of the step-growth (e.g., polycondensation) or chain-growth (e.g., free radical) class, produce broad molecular weight distributions and offer little control over long-chain architectural characteristics such as branching. Although of tremendous commercial importance, such approaches are inadequate for preparing model polymers, with tightly controlled molecular structures, that are essential for fundamental studies. For this reason, living polymerization has become the cornerstone of experimental polymer physical chemistry. In such a synthesis, conditions are set so that growing chain ends only react with monomers; no termination or chain transfer steps occur. If a fixed number of chains are initiated at t ) 0, random addition of monomers to the growing chains leads to a Poisson distribution of chain length, with a polydispersity (Mw/Mn) that approaches unity in the high N limit. Block copolymers can be made by sequential addition of different monomers, branched chains by addition of polyfunctional terminating agents, and end-functionalized polymers by suitable choice of initiator and terminator. Examples of chain structures realized in this manner are shown in Figure 5.

The most commonly used technique, living anionic polymerization, was introduced in the 1950s.18,19 It is well-suited to several vinyl monomers, principally styrenes, dienes, and methacrylates. However, it suffers from significant limitations, including the need for rigorously excluding oxygen and water, a restricted set of polymerizable monomers, and reactivity toward ancillary functionalities on monomers. Consequently, there is great interest in developing other living polymerization protocols.20 Over the past 15 years, group transfer,21 ring- opening and acyclic diene metathesis, 2-24 cationic, 25,26 and even free radical27 living polymerization methods have been demonstrated. Soon a much broader spectrum of chemical functionalities will become routine players in the synthesis of model polymers, and commercial products will rely increasingly on controlled polymerization techniques.

Theory. The genesis of polymer theory is the realization that a conformation of a polymer chain can be modeled as the

Figure 3. Schematic of a random coil polymer, and the density distribution F(r) for b , r , Rg.

Figure 4. Illustration of (a) complete registry and (b) entropic frustration for a chain bearing both positive and negative charges.

13276 J. Phys. Chem., Vol. 100, No. 31, 1996 Lodge and Muthukumar

trajectory of a random walker.2-1 In the simplest case, the apparent absence of any energy penalty for self-intersection, the statistics of random walks can be successfully applied. In the long-chain limit, the probability distribution G(RB,N) for the end-to-end vector of a chain, RB, is Gaussian, under these (experimentally realizable) “ideal” conditions. Now G satisfies a simple diffusion equation, analogous to Fick’s second law:

Thus, N corresponds to time, G to concentration, and b2 to a diffusivity. The right-hand side simply requires that the chain beginning at the origin arrive at RB after N steps. When there is a penalty for self-intersections, due to excluded volume interactions between monomers, it is possible to compose a pseudopotential for segmental interactions. The diffusion equation now contains an additional potential term which, in turn, depends on G:13,28

This requires a self-consistent procedure to determine G, from which various moments of experimental interest can be derived; in essence, this process amounts to making an initial guess for G, calculating the potential term, and numerically iterating until the chosen G satisfies eq 1.2.

For multicomponent polymer systems, the local chemical details and the various potential interactions between effective segments can be parametrized by writing an appropriate

“Edwards Hamiltonian” (see eq 2.1 and associated discussion). 7-9

Standard procedures of statistical mechanics (with varying levels of approximation) are then employed to obtain the free energy as a functional of macroscopic variables of experimental interest. 8,29-31 Such density functional approaches lead to liquid- state theories derived from coarse-grained first principles.8 The free energy so derived, reflecting a quasi-microscopic description of polymer chemistry, is also used to access dynamics.

Simulation.32,3 Lattice walks are used to determine G for a single chain with potential interactions, with some site potential energy to simulate chain contacts. A key feature of this approach is to use generating functions34-37 where pN are probability functions describing chains of N steps; this greatly reduces the computational complexity. For fully developed excluded volume, the exact method enumerates all possible nonintersecting random walks of N steps on a lattice; assuming all configurations are a priori equally probable, various averages are then constructed. In the alternative Monte Carlo method, a chain of successively connected beads and sticks is simulated on various lattices, or off-lattice, and statistical data describing the chain are accumulated. The stick can be either rigid or a spring with a prescribed force constant; the latter case is referred to as the bond-fluctuation algorithm. 38

As before, the beads interact through an appropriate potential interaction. Once an initial configuration is created, a randomly chosen bead is allowed to move to a new position without destroying the chain connectivity. The energy of the chain in its new configuration is computed, and the move is accepted or rejected using the Metropolis algorithm.39 Instead of making local moves, so-called pivot algorithms can be used to execute cooperative rearrangements.40,41

The use of molecular dynamics,42 in which Newton’s law is solved for the classical equation of motion of every monomer, has been restricted to rather short chains.43,4 Such atomistic simulations are difficult for polymers since even a single chain exhibits structure from a single chemical bond (ca. 1 Å) up to

Rg (ca. 10-103 Å), and the separation in time scale between segmental and global dynamics is huge. Brownian dynamics is an alternative method, 45 wherein Newton’s equation of motion is supplemented with a friction term and a random force, which satisfy a fluctuation-dissipation theorem at a given T. Since the friction coefficient is in general phenomenological, this Langevin equation is usually written for an effective segment. All of the above methodologies are in current use.

Experimental Techniques. Polymers require a variety of techniques to probe their multifarious structures, dynamics, and interactions. Polymer structure may be probed in real space, by microscopy, and in Fourier space, by scattering. Both approaches are important, but scattering has been more central to the testing of molecular theory. Classical light scattering (LS) and small-angle X-ray scattering (SAXS) have been used for over 50 years, but small-angle neutron scattering (SANS) has, in the past 25 years, become an essential tool.46-49 The key feature of SANS is the sharp difference in coherent scattering cross section between hydrogen and deuterium; isotopic substitution thus permits measurement of the properties of single chains, or parts of chains, even in the bulk state. All three experiments give information on the static structure factor, S(q):

(Parte **1** de 7)