Theta States

Here, unperturbed state. Equation (4a) represents the probability distribution of the vectorial quantity r. A less detailed form of representation is the distribution zt (r) showing the probability that the magnitude r of r has a certain value irrespective of direction. Thus, the probability that the chain end-to-end length is in the range r to r+dr irrespective of its direction is... 

It is now recognized that a continuum of architecture and properties, which begins with the classical branched polymers, resides between these two classes. Typical branched structures such as starch or high pressures polyethylene are characterized by more than two terminal groups per molecule, possessing substantially smaller hydrodynamic volumes and different intrinsic viscosities compared to linear polymers, yet they often exhibit unexpected segmental expansion near the theta state . 

Simulation methods have been proved to be useful in the study of many different molecular systems, in particular in the case of flexible polymers chains [ 14]. According to the variety of structures and the theoretical difficulties inherent to branched structures, simulation work is a very powerful tool in the study of this type of polymer, and can be applied to the general problems outHned above. Sometimes, this utility is manifested even for behaviors which can be explained with simple theoretical treatments in the case of linear chains. Thus, the description of the theta state of a star chain cannot be performed through the use of the simple Gaussian model. The adequate simulation model and method depend strongly on the particular problem investigated. Some cases require a realistic representation of the atoms in the molecular models [10]. Other cases, however, only require simplified coarse-grained models, where some real mon-... 

It should also be noted that ternary and higher order polymer-polymer interactions persist in the theta condition. In fact, the three-parameter theoretical treatment of flexible chains in the theta state shows that in real polymers with finite units, the theta point corresponds to the cancellation of effective binary interactions which include both two body and fundamentally repulsive three body terms [26]. This causes a shift of the theta point and an increase of the chain mean size, with respect to Eq. (2). However, the power-law dependence, Eq. (3), is still valid. The RG calculations in the theta (tricritical) state [26] show that size effect deviations from this law are only manifested in linear chains through logarithmic corrections, in agreement with the previous arguments sketched by de Gennes [16]. The presence of these corrections in the macroscopic properties of experimental samples of linear chains is very difficult to detect. 

Fig. 2. Bead density profiles. Solid line Brushes, mean-field and scaling theory (step function) dashed-dotted line generalization of the Milner et al. theory for brushes in the theta state dashed-double dotted line Milner et al. theory for brushes (EV chains) dashed line EV stars dotted line EV combs. Variable r is scaled to give zero bead density for the smooth curves of brushes at r=l. The brush curves are normalized to show equal areas (same number of units). The comb and star densities are arbitrarily normalized to show similar bead density per volume unit as the step function and EV curves for brushes at the value ol r where these curves intercept...

An increase of g in the theta state with respect to the ideal values is similarly obtained by Ganazzoli et al. [52,53] through the use of a theoretical approach based on the self-consistent minimization of the intramolecular free energy. Their results indicate a significant expansion of the star arms due to the core effects. The same type of calculations have later been used to describe the star contraction in the sub-theta regime [54]. Guenza et al. [55] described a star chain at the 0 point as a semiflexible chain with partially stretched arms that take into account the star core effect. Their results are also consistent with experimental data. 

Fig. 16. Generalized Kratky plot of the form factor of a star from numerical data of a MG simulation [161]. x=q . e/kgT=0.1 corresponds to EV chains ande/kgT=0.3 is close to the theta state...

Dimensions of an actual polymer random coil in a theta state. 

Ratio of the mean-square end-to-end distance, r ), of a linear macromolecular chain in a theta state to NLY, where N is the number of rigid sections in the main chain, each of length L if all of the rigid sections are not of equal length, the mean-square value of L is used, i.e.,... 

Note If the solution of the chain molecules is not in a theta state, the segments change mutual orientation only approximately randomly. 

Hypothetical freely jointed chain with the same mean-square end-to-end distance and contour length as an actual macromolecular chain in a theta state. 

Note 1 In some respects, a polymer solution in the theta state resembles an ideal solution and the theta state may he referred to as a pseudo-ideal state. However, a solution in the theta state must not be identified with an ideal solution. 

Ratio of a dimensional characteristic of a macromolecule in a given solvent at a given temperature to the same dimensional characteristic in the theta state at the same temperature. The most frequently used expansion factors are expansion factor of the mean-square end-to-end distance, Ur = (/o) expansion factor of the radius oj gyration, as = (/0) relative viscosity, = ([ /]/[ /]o), where [ ] and [ /]o are the intrinsic viscosity in a given solvent and in the theta state at the same temperature, respectively. 

As explained earlier (Sect. 1.3.1), macromolecules in a low-molecular-weight solvent prefer a coiled chain conformation (random coil). Under special conditions (theta state) the macromolecule finds itself in a force-free state and its coil assumes the unpertubed dimensions. This is also exactly the case for polymers in an amorphous melt or in the glassy state their segments cannot decide whether neighboring chain segments (which replace all the solvent molecules in the bulk phase) belong to its own chain or to another macromolecule (having an identical constitution, of course). Therefore, here too, it assumes the unperturbed ) dimensions. 

The deterioration of the solvent qnality, that is, the weakening of the attractive interactions between the polymer segments and solvent molecules, brings about the reduction in the coil size down to the state when the interaction between polymer segments and solvent molecules is the same as the mutual interaction between the polymer segments. This situation is called the theta state. Under theta conditions, the Flory-Huggins parameter % assumes a value of 0.5, the virial coefficient A2 is 0, and exponent a in the viscosity law is 0.5. Further deterioration of solvent quality leads to the collapse of coiled structure of macromolecules, to their aggregation and eventually to their precipitation, the phase separation. 

Theta (6) solvents are solvents in which, at a given temperature, a polymer molecule is in the so-called theta-state. The temperature is known as the theta-temperature or the Flory temperature. (Since P. J. Flory was the first to show the importance of the theta-state for a better understanding of molecular and technological properties of polymers, theta temperatures are also called Flory temperatures. ) In the theta-state, as explained above, the solution behaves thermodynamically ideal at low concentrations. 

The chain conformation is determined by the interaction between neighboring segments and the interaction between distant segments along a polymer which, via chain flexibility, are located in each other s vicinity. The former effect determines the local chain stiffness. The latter is referred to as the excluded volume effect and influences the overall conformation. Both types of interaction can be of electrostatic and nonelectrostatic origin. In the absence of excluded volume effects (flexible polyions in a theta state or... 

Caroline and co-workers have recently reported measurements of translational diffusion coefficients in solutions of PS in two mixed-solvent systems at or near theta conditions. In the solvent CCb-methanol (85), they observed the diffusion theta state, defined when the coefficient y of Equation 41 equals 0.5, to occur at 25°C and a volume fraction of CCI4, (fyCCU = 0.8025. In this system there is strong preferential adsorption of the polymer for CCI4, and it is not possible to define a true theta state such that y = a = V2 and A2 = 0 simultaneously. Under diffusion theta conditions, the concentration dependence of Dt apparently is closely described by the Pyun-Fixman hard-sphere model. In the mixed solvent benzene—2 propanol, polystyrene exhibits a true theta condition at T = 25.5°C and (benzene) = 0.04. Frost and Caroline confirmed that y = 0.5 within experimental error in this system (86) and report that values of the parameter fcf are scattered between the extreme values corresponding to the predictions of Yamakawa (and Imai) and the soft-sphere model of Pyun-Fixman (or the Freed theory). 

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