Theory of Poly(ethylene glycol) in Solution - ACS Symposium Series

Chapter 2

Theory of Poly(ethylene glycol) in Solution

Downloaded by UNIV OF MISSOURI COLUMBIA on August 31, 2014 | http://pubs.acs.org Publication Date: August 5, 1997 | doi: 10.1021/bk-1997-0680.ch002

Gunnar Karlström and Ola Engkvist Department of Theoretical Chemistry, University of Lund, P.O. Box 124, S-221 00 Lund, Sweden

The behavior of the poly(ethylene glycol) molecule in aqueous solutions is reviewed, using a comparison of experimental and theoretical means. It is stressed that the understanding of the conformational equilibria for the poly(ethylene glycol) molecule is of large importance both for the description of the phase behavior of poly(ethylene glycol) - water system, as well as for the properties of poly(ethylene glycol) solutions. In particular the influence of additives to the solution is discussed, together with the properties of poly(ethylene glycol) coated surfaces and the use of poly(ethylene glycol) for purifying biochemical compounds.

Poly(ethylene glycol) (PEG) or as it is often called poly(ethylene oxide) (PEO) is a polymer of considerable technical importance. PEG has perhaps its largest use as an additive to control the viscosity of paint and as an additive when paper is produced. It is also used as a head group in nonionic surfactants and to cover surfaces in order to prevent other polymers (e.g. proteins) from binding to the surface. Naturally, a large amount of both physical and chemical information has been collected about PEG and PEG solutions in general and about the PEG-water system in particular (1,2). The purpose of this chapter is to try to establish a relationship between the macroscopically observed phase behavior and a microscopic description of a PEG molecule in a water solution. It has long been known that, while completely soluble in water at low temperature, PEG loses its solubility at elevated temperatures. This process is normally given the name clouding. Complete phase diagrams for the PEG-water system were first determined by Sakei and coworkers (3). A similar behavior has been observed for ethylhydroxyethylcellulose (EHEC) in both water and formamide by Samii et. al. (4) (EHEC is a cellulose molecule substituted with ethyl and ethylene oxide groups so that it is effectively covered with ethylene oxide (EO) groups). In a similar way surfactants with ethylene oxide head groups phase-separate from a water 16

© 1997 American Chemical Society

In Poly(ethylene glycol); Harris, J., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1997.


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K A R L S T R O M & ENGKVIST

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Theory of PEG in Solution

or formamide solution at elevated temperatures (5). These observations are consistent with a model suggesting that the EO segments become more hydrophobic at higher temperatures. Another line of research of relevance to this discussion starts with the investigations performed by Podo and coworkers (6,7), who began in the 1970s to investigate the conformational equilibrium of 1,2-dimethoxyethane (DME), which can be regarded as a very short PEG chain, as a function of the solvent. Their main conclusion was that the polarity of the PEG chain was changed with the solvent polarity. Thus in polar solvents the PEG chain preferred polar conformations and in non polar solvents non polar structures were favored. These observations have later been verified by other investigations (8,9). Recently Raman and IR investigations by Matsuura and coworkers (10,11) have indicated that for short PEG chains this trend does not hold in a very dilute water solution, where more non-polar conformations again start to be more populated. From these experiments it is also clear that this last trend is most pronounced for chains that contain 3 EO units and becomes weaker for longer chains. For a PEG molecule containing more than 10 EO units, this effect would hardly be observed. The outline of this chapter will be that first we will briefly introduce some concepts of general polymer solution theory, with focus on the PEG-water system. This will be followed by a section dealing with molecular modeling of the PEG-water system. The knowledge, that we have presented in these two sections, will then be used to discuss the effect of different additives to a water solution of PEG, the behavior of a surface coated with PEG and finally the interaction between a protein and a solution containing PEG. General Polymer Solution Theory. The objective of this chapter is to see what information about PEG in general and about the PEG - water system in particular can be obtained from comparison of results from theoretical modeling of PEG solutions (and similar systems) with experimental data. Despite the fact that the focus should be placed on the physical properties and behavior of the PEG molecules, it is necessary to start by a short introduction into general polymer - solution modeling. The purpose of this is not to give a full account of what has been done within the field but rather to introduce the reader to the benefits and shortcomings of the three main strategies used to study polymers in solution. Virial Expansion Techniques. The starting point for this type of modeling is to disregard the solvent and focus only on the effective interactions between the polymer molecules (12). The concept of virial coefficients was originally developed to describe the deviation from ideal behavior for gases. For one mole of an ideal gas we may write pV=RT

(1)

and for a real gas the following relation holds to a good approximation


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POLY(ETHYLENE GLYCOL)

pV=RTf 1 + - + - ^ +

...)

(2)


In these equations p is the pressure, V the volume, R the gas constant, and T the absolute temperature. The constants B and C are called the second and third virial coefficients and they measure the deviation from ideal behavior. As we will see below the constants B and C are only constants at a fixed temperature, and in reality they vary quite strongly with temperature. To get some idea about the physics behind these constants, it is convenient to start with van der Waal's gas law for one mole of gas. (3)

In this equation b is a correction to the ideal gas law due to the fact that each molecule has a volume and that consequently only the volume ( V - b ) is available for the molecules. Thus b measures the volume of one mole of molecules. The other correction term (a) originates from the facts that the molecules attract each other and that this reduces the pressure. In order to obtain the pressure of the ideal gas law this reduction of the pressure must be added to the experimentally observed pressure. Consequently a is a measure of the attraction between molecules. From what has been said above it is clear that both a and b are positive constants. If van der Waal's gas law is transformed to a form similar to the virial expansion one obtains (4)

In the second term in the parentheses p/RT can be substituted by V using the ideal gas law and one obtains (5) Comparing equations 2 and 5 we see that B = b - a/RT. Here we can identify two limiting cases. At sufficiently low temperatures a will dominate and the second virial coefficient will be negative, whereas at sufficiently high temperatures b will dominate, resulting in a positive second virial coefficient. A negative virial coefficient means that there is an effective attraction between the molecules and a positive virial coefficient that there is an effective repulsion between the molecules. In terms of the phase behavior this means that at low temperatures, where the second virial coefficient is negative one observes a liquid phase in equilibrium with a gas phase, and at high temperatures where the second virial coefficient is positive one observes only a gas phase. The link between virial expansions for gases and for polymers in solution is straight forward. In this type of model one can neglect the solvent, since it only is a medium in which the polymer molecules are moving, in a similar way to the movements of gas molecules in a vacuum. There are however also some differences.



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Theory of PEG in Solution

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For the gas molecules we know that there is always a long-range attraction between the molecules, i.e. a is positive. For the polymer solution this is not necessarily true. In principle a may be negative. In general it is true, however, that at sufficiently short distances polymer molecules will repel each other and this means that at sufficiently high temperatures the polymers will dissolve in the solvent, (provided that the solvent is still a liquid and the polymer does not disintegrate). From this we may conclude that one would expect that the solubility of polymers in a solvent should increase with increasing temperature and that the second virial coefficient should increase with increasing temperature. This type of analysis has been performed for the water dextran - PEG system by Edmond and Ogston (13,14), and an analysis of the PEG water system reveals that this system behaves abnormally. The effective interaction between PEG molecules, as measured by the second virial coefficient, becomes more attractive at higher temperatures, even when it is measured in units of RT. A similar type of thermodynamic analysis of the PEG -water system have been made by Kjellander and Florin (15,16). The important message from this type of analysis is that there are some anomalies in the PEG-water system. The analysis as such is based on macroscopic concepts and provides very little information about the mechanism on a molecular level. It was observed by Kjellander and Florin-Robertsson, however, that the thermodynamics of the PEG-water system was similar to that observed in water solutions of hydrocarbons and other non polar molecules. For systems of that type one normally observes a solubility minimum in water at temperatures close to 20 C. The phenomenon is normally referred to as the hydrophobic effect, and is well described in the book with the same name by Tanford (17), and more recently by Israelachvili and Wennerstrom (18). The basic idea is that at low temperatures, the solubility of non-polar substances is increased due to a formation of a relatively ordered water layer around the non polar substance. The structure is preserved in order to maintain as many hydrogen bonds between the water molecules as possible, despite the presence of the solute. This is energetically favorably compared to a non-structured solvation of the hydrophobic substance, but associated with an entropic cost. At higher temperatures where entropy becomes more important the solubility decreases, or increases less than could be expected from the temperature increase. There are two weak points when this model is used to explain the observed behavior of the PEG - water system. First of all the clouding occurs in a temperature range (100 - 200 °C) where an increased solubility is observed for ordinary non-polar substances. Second the concept of "structured water" is not uniquely defined on a molecular level, and consequently it is not possible to design microscopic molecular models that can reproduce what is experimentally observed, or one may say that the model can not be proved false since it does not allow for any predictions. 0

Microscopic Models of Clouding in the PEG - Water System. From the previous section we learned that the phase separation observed at higher temperatures is a manifestation of an increased effective attraction between the EO



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POLY(ETHYLENE GLYCOL)

segments. This feature must be handled by any model designed to describe the observed phase behavior. The ideal entropy of mixing always favors the formation of a one-phase system and this effect will always be larger when the temperature is increased. Thus there must be another effect, with entropic origin, that is stronger than the mixing entropy. In the literature there exist two different microscopical explanations that will be reviewed here. The older one is based on the work by Hirschfelder, Stevenson and Eyring from 1937 (19), where the authors show that if the intermolecular potential between two species has small, strongly attractive regions and large repulsive ones, then the effective interaction between two substances may change from being attractive to being repulsive at higher temperatures. It is rather easy to associate the attractive regions with hydrogen bonds and the other regions with non hydrogen bonded interaction. An explanation along these lines was first given, for the PEG - water system, by Lang (20) and later Goldstein (21) gave it a mathematical formulation. The other microscopic model for the phase separation at elevated temperatures focuses on the conformational degrees of freedom of the PEG chain (22). Based on the notation that some of the conformations are more polar than others, one realizes that the different conformations must interact with the solvent in different ways. (The notation more polar means here, that the local dipole density is higher than for a less polar conformation.) A weak point in this latter model is that it requires that there are more non-polar conformations than there are polar ones. Quantum chemical calculations on D M E show that most of the relevant conformations have a dipole moment of 1.6 to 1.8 D, and the only conformation with low energy that lacks dipole moment is the one which is anti around the two C - 0 bonds as well as around the C-C bond. Today there exist two distinct ways to model a polymer solution. The most straight forward and in some sense the most accurate of these is to perform all molecular Monte Carlo (MC) or Molecular Dynamics (MD) simulations of the liquid polymer solution (23). The advantage of this type of modeling is that it gives a full description of the probability distributions for the studied system, provided that the intermolecular potentials employed are sufficiently accurate enough. The drawback is that it is not possible at present to study systems that are so large that accurate thermodynamic properties can be determined and phase diagrams calculated. Since this is exactly what we initially are focused on in this chapter we are forced to turn to the other type of thermodynamical modeling based on lattice theory. This type of model is normally given the name Flory - Huggins theory after Flory and Huggins, who independently developed the model (24). The model gives an expression for the entropy and energy of mixing between a solvent and a solute and is derived by looking for possible conformations of a polymer chain on a lattice. For a full derivation see reference (25). The basic assumptions behind the theory are that a polymer segment and a solvent molecule have the same size, that they are randomly mixed, apart from the connectivity of the polymer chain, and that the enthalpic part of the polymer - solvent interaction depends only on the solvent and solute concentrations. The following expressions are obtained


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K A R L S T R O M & ENGKVIST

A x = U 5ni

-TS

a i x

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Theory of PEG in Solution (6)

t t i ] (

mix - " H ^ ( «Pi I"

Theory of Poly(ethylene glycol) in Solution - ACS Symposium Series

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