Macromolecules in a Dilute Solution

The picture considered in the previous section is idealised one the macromolecule does not exist in isolation but in a certain environment, for example, in a solution, which is dilute or concentrated in relation to the macromolecules (Des Cloizeaux and Jannink 1990). The important characteristic for the case is the number of macromolecules per unit of volume n which can be written down through the weight concentration of polymer in the system c and the molecular weight (or length) of the macromolecule M as 

The mean distance between the centres of adjacent macromolecular coils d ss n /3 can be compared with the mean squared radius of gyration of the macromolecular coil (S2), which presents the mean dimension of the coil. Taking the definition (1.21) into account, one can see, that a non-dimensional parameter n R2)3/2 is important for characterisation of polymer solutions. The condition 

The curves illustrate two variants of the concentration dependence of the mean size of a macromolecular coil in solution. The example is taken of a macromolecule in a good solvent, so that at low concentrations the size of the macromolecular coil is larger than the size of ideal coil, R2)/ R2)o 1. 

Macromolecules in dilute solutions (c C 1) can be considered as not interacting with each other, though this is not always valid (Kalashnikov 1994 Polverary and de Ven 1996). 

To consider the behaviour of a single macromolecule in the solution, the interaction of the atoms of the macromolecule with the atoms of solvent molecules has to be taken into account, apart from the interactions between the different parts of the macromolecule. To find the distribution function for the chain co-ordinates, one ought to consider N+1 big particles of chain interacting with each other and each with small particles of solvent. One can anticipate that after eliminating the co-ordinates of the small particles in the 

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Volume from which a macromolecule in a dilute solution effectively excludes all other macromolecules. 

In addition to the thermodynamic quality of the solvent [34], the properties of a polymeric solution depend strongly on the concentration of the dissolved macromolecules. In a dilute solution the macromolecules are isolated from each other and consequently the interactions between the diluted polymer molecules play a minor role. Increasing the polymer content, a critical concentration is reached which gives rise to contact between different polymeric coils leading to a penetration at higher... 

Let us note that Eq. (41) is a generalisation of the equation for the dynamics of a macromolecule in a dilute solution (Sect. 3.1) the effective viscous liquid in which the Brownian particle moves has been replaced by an effective viscoelastic liquid in the case of a concentrated solution this introduces the concept of microviscoelasticity. Of course, if memory functions turn into -func-... 

Zhulina KB, Adam M (2005) Diblock copolymer micelles in a dilute solution. Macromolecules 38 5330-5351... 

Excluded volume determines space occupancy in biopolymer solutions. Competition between macromolecules for space in a mixed solution determines the phase separation threshold. In a dilute solution of biopolymers, macromolecules hardly interact with one another, individual macromolecules are independent of one another, and biopolymers are cosoluble. The effects of spatial limitations are enhanced by the transition from a dilute mixed solution, to a semi-dilute biopolymer solution where molecules come into contact with one another, interact, compete for the same space, and do not mix in all proportions. 

The main difficulty in the experimental investigations of EB in flexible-chain polymer solutions is due to the low value of the observed effect. Specific Kerr constants for a flexible-chain polymer bearing no charge are K 10 cm g (V/300) even for polar macromolecules and, hence, the difference between birefringence in a dilute solution and the Kerr effect in the solvent alone is very slight. 

The GPC-viscometry with universal calibration provides the unique opportunity to measure the intrinsic viscosity as a function of molecular weight (viscosity law, log [17] (it versus log M) across the polymer distribution (curves 3 and 4 in Fig. 1). This dependence is an important source of information about the macromolecule architecture and conformations in a dilute solution. Thus, the Mark-Houwink equation usually describes this law for linear polymers log[i7] = ogK+ a log M (see the entry Mark-Houwink Relationship). The value of the exponent a is affected by the macromolecule conformations Flexible coils have the values between 0.5 and 0.8, the higher values are typical for stiff anisotropic ( rod -like) molecules, and much lower (even negative) values are associated with dense spherical conformations. 

Flexible macromolecules. Calculations of the attractive potential energy according to equation (15.11) show that for spherical colloidal particles immersed in a dilute solution of rigid spheres, the attraction rarely exceeds k T. For articulated macromolecules, the configurational entropy of the chains is decreased in the neighbourhood of the interfaces and this provides a source of non-zero values for w(x,d). Asakura and Oosawa (1954) approximated this entropy decrement by analysing the problem in terms of the classical theory (Carslaw, 1921) for diffusion in a vessel with walls that absorb diffusing particles. The end result for parallel flat plates of area A is... 

A theoretical analysis of the effect of counterion localization in a dilute solution of weakly charged branched polyions of different topologies [31-33] and ionic microgels [34, 35], was performed on the basis of a cell model, similar to that used here for a star-like PE. The elastic term in the free energy that accounts for the conformational entropy of a uniformly swollen branched macromolecule, has to be specified depending on the polyion topology. The shape of the cell might also be modified. For example, in the case of a molecular PE brush, a cylindrical instead of spherical cell should be used. 

Mark-Houwink equation n. Also referred to as Kuhn-Mark-Houwink-Sakurada equation allows prediction of the viscosity average molecular weight M for a specific polymer in a dilute solution of solvent by [77] = KM, where K is a constant for the respective material and a is a branching coefficient K and a (sometimes a ) can be determined by a plot of log [77] versus logM" and the slope is a and intercept on the Y-axis is K. Kamide K, Dobashi T (2000) Physical chemistry of polymer solutions. Elsevier, New York. Mark JE (ed) (1996) Physical properties of polymers handbook. Springer-Verlag, New York. Ehas HG (1977) Macromolecules, vols 1-2. Plenum Press, New York. 

The power law exponent fell between the values of 0.5 observed for a 0 solvent and 0.6 for a thermodynamically good one, confirming the above description of tetrahydrofiiran as a moderately good solvent. Poly(3-hexylthiophene) macromolecules exist as isolated flexible-coil chains in a dilute solution with a persistent length of 2.4 0.3 nm. 

All measurements, of course, have to be made at a finite concentration. This implies that interparticle interactions cannot be fully neglected. However, in very dilute solutions we can safely assume that more than two particles have only an extremely small chance to meet [72]. Thus only the interaction between two particles has to be considered. There are two types of interaction between particles in solution. One results from thermodynamic interactions (repulsion or attraction), and the other is caused by the distortion of the laminar fiow due to the presence of the macromolecules. If the particles are isolated only the laminar flow field is perturbed, and this determines the intrinsic viscosity but when the particles come closer together the distorted flow fields start to overlap and cause a further increase of the viscosity. The latter is called the hydrodynamic interaction and was calculated by Oseen to various approximations [3,73]. Figure 7 elucidates the effect. 

In this form, van t Hoff s law of osmotic pressure is also used to determine the molar masses of biological and synthetic macromolecules. When the osmotic pressure is measured for a solution of macromolecules that contains more than one species of macromolecule (for example, a synthetic pol5mer with a distribution of molar masses or a protein molecule that undergoes association or dissociation), the osmotic pressures of the various solute species II, are additive. That is, in sufficiently dilute solution... 

The use of dilute polymer solutions for molecular weight measurements requires the macromolecules to be in a true solution, i.e., dispersed on a molecular level. This state may not be realized in certain instances because stable, multimolecular aggregates may persist under the conditions of "solution" preparation. In such cases, a dynamic equilibrium between clustered and isolated polymer molecules is not expected to be approached and the concentration and size of aggregates are little affected by the overall solute concentration. A pronounced effect of the thermal history of the solution is often noted under such conditions. 

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