Size exclusion distribution coefficient

A more useful and fundamental parameter than elution volume is the dimensionless size exclusion distribution coefficient (Kq) which is related to and Vg by the equation ... 

Figure 7 Dependences of the size exclusion distribution coefficient, K, for the small globular protein hen egg white lysozyme versus the ionic strength, /, of the mobile phase for several notional size exclusion sorbents of different average pore diameter, particle size, and surface chemistry characteristics. The sorbents employed in these investigations were 1, Synchropak GPC 100 2, Waters I-125 3, Shodex OH Pak B-804 4, Lichrosorb Diol 5, Tosoh TSK SW 3000 and 6, Tosoh TSK SW 2000. The Interplay of hydrophobic interaction and electrostatic phenomena, superimposed upon the size exclusion effect due to the differences in the pore sizes of the support materials, is particularly evident with these sorbents at high- and low-ionic-strength conditions. (Data ad ed from Ref. 98.)...

It is clear that the separation ratio is simply the ratio of the distribution coefficients of the two solutes, which only depend on the operating temperature and the nature of the two phases. More importantly, they are independent of the mobile phase flow rate and the phase ratio of the column. This means, for example, that the same separation ratios will be obtained for two solutes chromatographed on either a packed column or a capillary column, providing the temperature is the same and the same phase system is employed. This does, however, assume that there are no exclusion effects from the support or stationary phase. If the support or stationary phase is porous, as, for example, silica gel or silica gel based materials, and a pair of solutes differ in size, then the stationary phase available to one solute may not be available to the other. In which case, unless both stationary phases have exactly the same pore distribution, if separated on another column, the separation ratios may not be the same, even if the same phase system and temperature are employed. This will become more evident when the measurement of dead volume is discussed and the importance of pore distribution is considered. 

Quality assurance for size exclusion supports is based primarily on the reproducibility of molecular weight calibrations. Although the reproducibility of the exclusion and inclusion limits is important, the distribution coefficients (Ko) of included standards are a better indication of duplication. Table 10.3 (page 314) shows such data for the SynChropak GPC and CATSEC supports. 

When a dilute solution of a polymer (c << c ) is equilibrated with a porous medium, some polymer chains are partitioned to the pore channels. The partition coefficient K, defined as the ratio of the polymer concentration in the pore to the one in the exterior solution, decreases with increasing MW of the polymer (7). This size exclusion principle has been used successfully in SEC to characterize the MW distribution of polymer samples (8). 

In fundamental SEC studies retention is often described in terms of a distribution coefficient. The theoretical distribution coefficient Kg is defined as the ratio of solute concentration inside and outside of the packing pores under size exclusion conditions. The experimental distribution coefficient as defined in Equation 1, is a measurable quantity that can be used to check the theory. 

Size exclusion chromatography is a unique separation technique based on molecular size (hydrodynamic volume) differences among solutes. The distribution coefficient of an eluting solute is defined as... 

Theory. The most widely accepted mechanism of size separation is based on steric exclusion (1). In terms of thermodynamic properties, the distribution coefficient consists of enthalpic and entropic contributions ... 

Molecular exclusion chromatography. The stationary phase in molecular exclusion chromatography is a material containing pores, the dimensions of which are chosen to separate the solutes present in the sample based on their molecular size. This can be perceived as a molecular sieve allowing selective permeation. This technique is known as gel filtration or gel permeation, depending on the nature of the mobile phase, which is either aqueous or organic. The distribution coefficient in this technique is called the coefficient of diffusion. 

This is a well-known result obtained by Casassa46). In the exclusion mode, the longer the macromolecule the greater is the energy that has to be spent to place it in the pore. The distribution coefficient Kd exponentially decreases to zero with increasing size of the macromolecule. The larger the molecule, the smaller is VR in accordance with Eq. (3.1) in the exclusion mode 0VR/6N < 0. b. The case of attraction... 

Fig. 7. Interrelation between molecular weight and retention volume for macromolecules of different functionality at chromatography in the exclusion (1-3), the critical (4), and the adsorption (5) separation modes 59). In the general case, the distribution coefficient Kd is a function of the pore size D, the chain length N, the interaction energy with the pore wall of the backbone segments 0 and the terminal segment 0f containing the functional group (zones 1,2 and 3 correspond to the cases shown in Fig. 2)...

As it was shown in Section 3.2, close to the critical conditions the distribution coefficient Kd is a function of chain length, pore size D and the energy of interaction of units with pore walls, 0. For a chosen molecule and adsorbent, Kd = Kd(0), and, therefore, by changing 0 one can successively achieve the transition from the adsorption to the exclusion mode and vice versa, finding in this way the critical conditions necessary for separation according to the functionality. 

Figure 13.3. Appearance of non-size-exclusion effects on SEC-elution curves of polyelectrolytes and other charged analytes including low-molecular-weight organic acids. Kd is the distribution coefficient and Ve, V0, and Vt are elution volume of the analyte, column void volume, and total column volume, respectively.

In order to calculate the molecular weight M) or molecular-weight distribution (MWD) of the polymer, the dependence of the Soret coefficient on M must be known. Because is virtually independent of M, at least for random coil polymers, the dependence of retention on M reduces to the dependence of D on M. The separation of molecular-weight components by D (or hydrodynamic volume, which scales directly with D) is a feature that thermal FFF shares with size-exclusion chromatography (SEC). In the latter technique, the dependence of retention on D forms the basis for universal calibration, as D scales directly with the product [rjjM, where [17] is the intrinsic viscosity. Thus, a single calibration plot prepared in terms of log([i7]M) versus retention volume (F,) can be used to measure M for different polymer compositions, provided an independent measure of [17] is available. In thermal FFF, a single calibration plot can only be used for multiple polymers when the values of for each polymer-solvent system of interest are known. However, a single calibration plot can be used with multiple channels. In... 

The above simple model of a steric exclusion mechanism was considered by several authors attempting to describe quantitatively the gel chromatographic separation process. Distribution coefficients were expressed on the basis of the model considerations of the dimensions of both the separated molecules and the pores of gel, as well as of the stochastic model approaches (for reviews see e.g.. Refs. 1, 3-6), and also of the thermodynamic reasoning on the changes of conformational entropy of macromolecules due to their transfer from the interstitial volume into the pores in the course of separation [7]. However, besides the steric exclusion from the pores, at least two other size-based mechanisms are operative in the ideal gel chromatography ... 

Flow field-flow fractionation (flow FFF) was developed by Beckett and Hart (1988) to determine the MW distribution of humic and fulvic acid. This method determines the diffusion coefficient which is then correlated to the MW with standards of known solutions, such as polystyrenesulphonate. A different result was obtained, compared to size exclusion chromatography when using Aldrich humic acids. In Becketts study the complexity of this kind of analysis is demonstrated by Beckett and Hart (1988) and different methods are questioned. 

To understand how steric exclusion differs from the other forms of chromatography, refer to Equation 21.4. In this context, V a and Va are referred to as the void volume and the total pore volume, respectively. The distribution coefficient depends on the molecular weight of the sample and on the pore size of the packing. The equilibrium established in exclusion chromatography is described by Equation 21.1 ATx is defined by Equation 21.2. In a true permeation process, assuming all pores to be accessible to a small solute molecule, and = 1- If none of the pores... 

The theory of adsorption at porous adsorbents predicts the existence of a finite critical energy of adsorption e, where the macromolecule starts to adsorb at the stationary phase. Thus, at > the macromolecule is adsorbed, whereas at e < e the macromolecule remains unadsorbed. At e = Ec the transition from the unadsorbed to the adsorbed state takes place, corresponding to a transition from one to another separation mechanism. This transition is termed critical point of adsorption and relates to a situation, where the adsorption forces are exactly compensated by the entropy losses TAS = AH [2, 7]. Accordingly, at the critical point of adsorption the Gibbs free energy is constant (AG = 0) and the distribution coefficient is Kj = 1, irrespective of the molar mass of the macromolecules. The critical point of adsorption relates to a very narrow range between the size exclusion and adsorption modes of liquid chromatography. It is, therefore, very sensitive towards temperature and mobile phase composition. 

Fundamentally, V, is the sum of the void volume occupied by all solutes, a portion of the internal pore volume defined by the size exclusion differential equilibrium constant and a portion of the surface of the column packing defined by the distribution coefficient describing interactions between the column and solute K. This condition leads to the general equation... 

The enthalpic term in eqn [8] can be considered as a partition coefficient Kp, whose value is unity AH° = 0) if the size exclusion is the only interaction of the solute molecules with the column packing. If the adsorption and/or partition of the solute takes place in the chromatographic system, Kp > 1 AH° is negative), or if an incompatibility between the solute and the column packing exists, Kp lies between zero and unity AH° is positive). Consequently, the en-tropic term in eqn [8] represents the distribution coefficient Ka of pure size exclusion. Ka acquires values between zero and unity and, in agreement with the experimental findings, is independent of the temperature. Hence, the distribution coefficients can be redefined as... 

If Kp 1, the nonexclusion interactions take place in the separation mechanism. The absence of such interactions requires the SEC to be performed under conditions of only entropic, size exclusion interactions. In such a case, Kp = 1 and Ka = K ec- E follows from this thermodynamic approach that the distribution coefficient depends on the chemical character of the solute molecules and that of the solvent, as well as on the matrix constituting the porous particles. 

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