Eugene Helfand

Fluctuation effects in the theory of microphase separation in block copolymers

The effect of composition fluctuations on the microphase separation transition in diblock copolymers is investigated. Such fluctuation corrections, which were neglected in the mean field treatment of Leibler, are found to be significant for the molecular weights usually encountered. The analysis is facilitated by reducing the block copolymer Hamiltonian to a form previously studied by Brazovskii. Our principal results are the following: (i) A symmetric diblock copolymer is predicted to undergo a first order phase transition at a larger value of χN than the second order transition found by Leibler. The Flory interaction parameter is denoted χ and N is the number of statistical segments per chain. The location of the transition is predicted to be at (χN)t=10.495+41.022 N−1/3, where the peak in the scattering function attains its maximum value of 0.12328 N1/3. (ii) We find windows in composition, with finite width, through which it is possible to pass from the disordered phase to each of the ordered micropha...

Theory of the Interface between Immiscible Polymers. II

A theory is developed to predict the interfacial properties between two immiscible polymers. A self‐consistent‐field approach is employed to describe the configurational statistics of polymer molecules in the interfacial region. The density profiles and surface tension are calculated, and compare well with experiments. The interfaces are found to be much broader than for ordinary liquids. Also discussed are the distribution of chain ends and block‐copolymer joints in the interface, and the effects of adding moderate amounts of solvent.

Scaled Particle Theory of Fluid Mixtures

An extension of a previous one‐component theory of hard‐sphere systems (in three, two, and one dimensions) and the surface tension of real systems is made to mixtures. The theory is based on consideration of an approximate expression for the work of adding an additional hard sphere to a mixture. Comparison between theory and molecular‐dynamics calculations of the various contributions to the virial pressure (related to contact distribution functions) of such hard‐sphere mixtures is excellent. Comparison of the theory with experimental surface tensions of mixtures of simple liquids is satisfactory.

Theory of inhomogeneous polymers: Fundamentals of the Gaussian random‐walk model

In earlier work a theory of inhomogeneous polymers was developed from a mean field theory point of view. Applications were made to polymer−polymer interfaces and to the microdomain structure of block copolymers. Here a connection is made between the intuitive mean field arguments for these problems and more fundamental statistical mechanics. Functional integral techniques are employed extensively.

Aspects of the Statistical Thermodynamics of Real Fluids

By extending the ideas previously applied to the statistical mechanical theory of hard sphere fluids of Reiss, Frisch, and Lebowitz, an approximate expression has been determined for the work of creating a spherical cavity in a real fluid. In turn the knowledge of this entity permits an evaluation of properties such as the surface tension and the normal heats of vaporization of fluids and the Henrys law constants of fluid mixtures. The agreement between the calculated and experimental properties is satisfactory.

Theory of unsymmetric polymer–polymer interfaces

Solutions have been obtained to equations which described the interface between two immiscible polymers and are more general than the equations first introduced by Helfand and Tagami. Gaussian random−walk statistics are assumed for the macromolecules. As a consequence of the present work, limitations of the earlier theory are removed, particularly the assumption that the properties of the two polymers when pure are identical. Calculations are performed for a variety of polymers and comparison with experiment is made.

Theory of the Kinetics of Conformational Transitions in Polymers

We develop a theory, valid for large friction, for the rate of conformational (rotational isomeric state) transitions in polymers. Such changes are classified according to the effect which the transformation in a finite portion of the molecule has on the attached polymeric chains. Major consideration is given to type 2* transitions, in which the attached chains are merely translated as the conformational alteration occurs. Study of a simplified model indicates that the rates of 2* transformations are not seriously inhibited by the attached polymer chains.

Conformational state relaxation in polymers: Time‐correlation functions

A study is made of relaxation processes in polymer molecules which proceed via conformational transitions of the chain backbone from one rotational isomeric state to another. The conformational time‐correlation functions are determined for several models of the process. Each model allows for the occurrence of both independent and cooperative conformational transitions. The resulting correlation functions contain a modified Bessel function which is associated with the diffusional nature of the process. This functional form has recently proven useful in fitting time‐correlation functions determined in polymer simulations, which indicates that it will be of value in fitting data obtained in relaxation experiments on polymers.

Kinetics of conformational transitions in chain molecules

A theory is developed for the rate of conformational transitions (trans→gauche) of bonds in chain molecules, such as alkanes and polymers. This is a multidimensional extension of Kramers’ reaction rate theory. Central to the understanding of how changes in the chain’s geometry affect transition rate is the determination and examination of the reaction coordinate. The reaction coordinate is a localized mode; i.e., the rotational motion of the transforming bond is accompanied by motion in neighboring bonds, but this motion diminishes with distance. Comparison is made between the calculated rates and those determined by Brownian molecular dynamics simulations.

Molecular dynamics of infrequent events: Thermal desorption of xenon from a platinum surface

Molecular dynamics techniques are employed to compute the rate of thermal desorption of Xe atoms from a platinum surface at temperatures between 100 and 2000 °K. A generalized Langevin approach is used to substantially reduce the number of degrees of freedom required to achieve an accurate description of the desorption process. A variety of techniques developed to improve the feasibility of simulating slow processes are explored. Using a ’’compensating potential’’ procedure, desorption rates can be computed efficiently even at temperatures for which the residence time of Xe on the surface is of the order of 1 sec (1014 vibrational periods).