Excluded volume integral

The theories of polymer solutions upon which steric-stability theories are based are usually formulated in terms of a portmanteau interaction parameter (for example Flory s X Parameter and the excluded volume integral) which does not preclude electrostatic interactions, particularly under conditions where these are short range. It is thus appropriate to consider whether polyelect-roly te-stabilisation can be understood in the same broad terms as stabilisation by non-ionic polymers. It was this together with the fact that polyelectrolyte solutions containing simple salts show phase-separation behaviour reminiscent of that of non-ionic... 

The final results of many perturbation theories (often called two-parameter theories) contain the combinations Nb and N" /3 as the only sample dependent quantities. Here N is the number of segments, each of length b, in an equivalent freely jointed polymer chain, related by (r )o = Nb" (see p. 64), and /3, the so-called excluded volume integral, is the volume excluded by one chain segment to another. 

The second virial coefficient B2 of the colloidal particles can be calculated in general from the potential of mean force via the excluded volume integral ... 

For the macTomolecule in solution, realization of the analogous condition requires a relatively poor solvent in which the polymer segments prefer self-contacts over contacts with the solvent. The incidence of self-contacts may then be adjusted by manipulating the temperature and/or the solvent composition until the required balance is established. Carrying the analogy to a real gas a step further, we require the excluded volume integral for the interaction between a pair of segments to vanish that is, we require that 3=0. This is the necessary and sufficient condition.6,13... 

The link between chain dimensions and the thermodynamic behaviour of dilute polymer solutions is of great importance and has its origin in the excluded volume integral fig- This arises because fie is the volume excluded by one segment to any other segment which therefore can be part of the same or a different molecule. Theories for fie lead to equations of the form... 

In a homopolymer, the chain segments are taken to be identical, and the x functions and excluded volume integrals j are the same for all ij. Since U(Rij) must in general comprise both attractive and repulsive terms, is expected to depend on temperature, and may be zero at a particular temperature or at more than one temperature (see Section 3.3,4). By contrast, is expected to depend only weakly on temperature and to be positive, albeit possibly very small. " ... 

The two-parameter formalism makes no stipulations concerning the temperature dependence of the radius of gyration and of the second virial coefficient, except that both quantities must reflect any temperature dependence of the excluded volume integral P through the variable z. Without an explicit theory of the behavior of j , we can only assert general thermodynamic requirements. We recall that the osmotic pressure is given by —(RT/Vi) In, with the activity of the solvent and Fj its molar volume. Then using equation (63), we can write the chemical potential of the solvent as a... 

Converting the probability function described above into an excluded volume is accomplished by integrating the probability 1 - 0(d) of exclusion over a spherical volume encompassing ah values of d ... 

The quantity b has the dimension of a volume and is known as the excluded volume or the binary cluster integral. The mean force potential is a function of temperature (principally as a result of the soft interactions). For a given solvent or mixture of solvents, there exists a temperature (called the 0-temperature or Te) where the solvent is just poor enough so that the polymer feels an effective repulsion toward the solvent molecules and yet, good enough to balance the expansion of the coil caused by the excluded volume of the polymer chain. Under this condition of perfect balance, all the binary cluster integrals are equal to zero and the chain behaves like an ideal chain. 

If there were no intramolecular interactions (such as bonding or excluded volume), then V(R) = 0, and the next guess for the density profile can be obtained directly from Eq. (75). The presence of V(R) necessitates either a multidimensional integration or (more conveniently) a single-chain simulation. 

Where (pm is the maximum concentration at which flow is possible -above this solid-like behaviour will occur. q>/(pm is the volume effectively occupied by particles in unit volume of the suspension and therefore is not just the geometric volume but is the excluded volume. This is an important point that will have increasing relevance later. Now integration of Equation (3.53) with the boundary condition that as... 

This is based on the belief that for an analysis of universal excluded volume effects the detailed. shape of p(r) is not relevant as long as it simulates a short range repulsive interaction. Indeed, the integral over all configurations averages over the factors of P(Vi—r j), and for a Gaussian chain the probability... 

This equation acknowledges that real molecules have size. They have an exclusion volume, defined as the region around the molecule from which the centre of any other molecule is excluded. This is allowed for by the constant b, which is usually taken as equal to half the molar exclusion volume. The equation also recognizes the existence of a sphere of influence around each molecule, an interaction volume within which any other molecule will experience a force of attraction. This force is usually represented by a Lennard-Jones 6-12 potential. The derivation below follows a simpler treatment (Flowers Mendoza 1970) in which the potential is taken as a square-well function as deep as the Lennard-Jones minimum (figure 2a). Its width x is chosen to give the same volume-integral, and defines an interaction volume Vx around the molecule, which will contain the centre of any molecule in the square well. This form of molecular pair potential then appears in the Van der Waals equation as the constant a, equal to half the product of the molar interaction volume and the molar interaction energy. 

Integral equations theories are another approach to incorporate higher order correlations, and consequently also lead to lowered osmotic coefficients. There are numerous variants of these theories around which differ in their used closure relations and accuracy of the treatment of correlations [36]. They work normally very well at high electrostatic coupling and high densities, and are able to account for overcharging, which was first predicted by Lozada-Cassou et al. [36] and also describe excluded volume effects very well, see Refs. [37] for recent comparisons to MD simulations. 

Meirovitch developed the scanning method to study a system of many chains with excluded volume contained in a box on a square lattice.With this method, an initially empty box is filled with the chain monomers step by step, with help of transition probabilities. The probability of construction of the whole system is the product of the transition probabilities selected, and therefore, the entropy of the system is known. Consequently standard thermodynamic relations can be used to make highly accurate calculations of pressure and chemical potential, directly from the entropy. In principle, all these quantities can be obtained from a single sample without the need to carry out any thermodynamic integration. 

The excluded volume v is defined as minus the integral of the Mayer /-function over the whole space ... 

The identification of fast coordinates can be difficult in simulations of condensed phases. Some of the fast modes are bond or angle vibrations that can be identified and integrated separately. However, other fast modes are transient. They are fast for a short duration of time and slow otherwise. The transient fast modes are collisions for example, two atoms that are close and feel strong repulsive forces due to excluded volume interactions. The relevant degree of freedom (the distance between the atoms) is a fast mode during the collision event and a slow mode before or after the short collision period. The fundamental complication in the treatment of these modes is the identity crisis of these fast/slow coordinates. 

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