Strategies in Size Exclusion Chromatography - American Chemical

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Chapter 4

Modeling of Size Exclusion Chromatography by Monte Carlo Simulation

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Jean-Pierre Busnel, Christophe Degoulet, and Jean-François Tassin Laboratoire de Physico-Chimie Macromoleculaire, Université du Maine, Unité de Recherche Associée 509, BP 535, 72017 Le Mans Cedex, France

By using Monte Carlo simulation, polymer chains of various flexibility and thickness can be constructed inside pores of various geometries. This allows to evaluate the corresponding steric partition coefficient Κ and the chromatographic radius R defined as the radius of the rigid sphere with the same Κ value. When comparing with R defined as the radius of the sphere with the same product [η].Μ, R /R is clearly different from 1, and is not strictly independent of the flexibility and the relative thickness of the macromolecule. However, this ratio is generally almost stable and experimentally, for flexible polymers, the so-called universal calibration (UC) is often found to work well despite the lack of academic evidence. c

η

c

η

Size exclusion chromatography (SEC) is now the current denomination for the chromatographic process first introduced by Moore (/) as Gel Permeation Chromatography (GPC). SEC has rapidly become a powerful tool for characterising die molar mass distribution of polymers. Recent technological improvements have made this technique even more attractive. Unfortunately this method relies on calibration using standards and does not yield the absolute molar masses as soon as the studied macromolecule is different from the one used in the calibration. Universal calibration (2) is an elegant solution to that problem as is simply relates each macromolecule size (i.e. each elution volume) to a product [η]Μ. Although the on-line viscosity measurements (3-5) raise experimental problems (6,7) especially with high performance column sets, intrinsic viscosity ([η]) measurements can nowadays be reliably (8,9) performed.

0097-6156/96/0635-0052$15.00/0 © 1996 American Chemical Society

In Strategies in Size Exclusion Chromatography; Potschka, M., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1996.

Downloaded by NORTH CAROLINA STATE UNIV on October 10, 2012 | http://pubs.acs.org Publication Date: May 30, 1996 | doi: 10.1021/bk-1996-0635.ch004

4. BUSNEL ET AL.

53

Modeling of SEC by Monte Carlo Simulation

It has been claimed by many authors (first by Benoît (JO)) that experimentally the U C is valid whatever the chemical nature and the structure of the polymer (77). However from a theoretical point of view the situation is different. Casassa (72)and Giddings (73) assume that the elution process is governed by the equilibrium distribution of the solute between the mobile phase and the stagnant phase inside pores. The equilibrium is characterised by the distribution coefficient, K, defined as the ratio between the concentration inside the pores to the concentration outside the pores. Using statistical mechanics, they have calculated Κ for random flight linear and branched chains as well as for rods, confined in pores with simple geometries. Casassa showed that UC is valid for any linear or branched macromolecule containing a large number of statistical segments. The results of Giddings lead to different distribution coefficients for a thin rod and for a flexible chain with the same viscometric radius. These idealised theoretical calculations indicate that U C can be applied to polymers of different architecture, but that rigidity might be a limiting factor. The aim of this paper is to study the influence of the flexibility of linear chains through a Monte Carlo simulation of the size exclusion phenomenon. Definition of Different Sizes of a Macromolecule A single macromolecule has a complex temporal and spatial distribution of conformations and only average dimensions can be calculated or measured. A special mention is given to the radius of gyration which corresponds to a clear geometric definition. The simplest way to compare sizes as measured by different experimental techniques is by defining the corresponding size as the radius of a rigid sphere which has the same measured property as the macromolecule. The following sizes can be defined : - The Stokes radius obtained from measurements of the transiational diffusion coefficient: ( 1 )

^"oTcnoDt where k, Τ, η and D are the Boltzman constant, the absolute temperature, the solvent viscosity and the transiational diffusion coefficient. - The viscometric radius obtained from mass and intrinsic viscosity measurements: 3[η]Μ 1/3 ΙΙΟπΝ, (2) where N is the Avogadro number. - The chromatographic radius R obtained from SEC (R is the radius of the sphere that would have the same elution volume in a pure SEC experiment). Using these definitions, U C applies if the relation between and R doesn't depend on the nature of the sample (linear or branched, flexible or rigid). For linear chains, a good estimate of R^ can be obtained from theoretical work of Yamakawa et al. (14). R values have been calculated using a computer simulation in a wide range of realistic situations, especially for chains of variable stiffness ranging from freely rotating segments to rigid rods. ο

t

a

c

c

c

c

In Strategies in Size Exclusion Chromatography; Potschka, M., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1996.

54

STRATEGIES IN SIZE EXCLUSION CHROMATOGRAPHY

Principle of Simulation In a large variety of real situations SEC experiments appear as pure equilibrium chromatographic processes. Recent measurements in our laboratory show that there is no significant elution volume shift (less than 0.05%) when changing flow rate from 0.5 to 1 c ^ . m i n for monodisperse standards analysed on classical columns either in THF or in water (Polystyrene standards in THF in the M W range 2.10 to 2.10 g.mol" on columns P L GEL, mixed bead, particle diameter 10 μπι ; in water 0.1 M ammonium acetate, Pullulan standards in the M W range 2.10 to 10 on TSK PW5000, particle diameter 10 urn). So the key parameter for SEC is the partition coefficient Κ related to the elution volume : V =V +KV , where V is the solute elution volume, V the void volume and V the pore volume of the column. For a rigid sphere of radius Rc, Κ is simply the volume fraction of the pores accessible to the centre of mass and for a number of pore geometries, there is a simple relation between R and Κ : 1

4

6

1

4

Downloaded by NORTH CAROLINA STATE UNIV on October 10, 2012 | http://pubs.acs.org Publication Date: May 30, 1996 | doi: 10.1021/bk-1996-0635.ch004

6

e

D

0

p

e

p

c

- For a cylindrical pore of circular section with radius R^ or of square section with side 2Rp : 2

K =(l-R /R ) - For a spherical pore with radius Rp or a cubic pore with side 2R^ : c

p

(3)

3

K - ( l . R ^ ) (4) In the general case, Κ is best defined as the equilibrium constant for solute exchange from intrapore volume to bulk solvent as proposed by Casassa :

Solute in bulk solvent

>

Solute inside pore

Κ has been calculated as follows : For a given pore geometry, a macromolecule is created by randomly choosing a starting point inside the pore volume. Then segments are successively placed using a random walk taking into account conformational constraints. If a segment reaches a pore wall the macromolecule doesn't exist inside the pore. After a large number of trials, [S]b is proportional to the number of trials and [S] is proportional to the number of successful trials. Therefore, Κ appears as the value of the fraction of successful trials, for a large number of trials. Chain geometry. We have studied freely rotating chains (Figure 1) defined by : - The number of segments N . - The length 1 and the diameter d of each segment - A fixed value of α - A value of β at random between 0 and 2π. For highly flexible chains α = π/2 and Ν varies from 100 to 10 . For worm­ like chains N=100 and a varies from π/2 to 0. To take into account the effect of segment thickness it is sufficient to note that a thin molecule of chromatographic radius Rc in a pore of linear dimension Rp corresponds to a thick molecule (segment diameter d) with a chromatographic radius Rc+ d/2 in a pore of size Rp + p

4

In Strategies in Size Exclusion Chromatography; Potschka, M., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1996.

4. BUSNELETAL.

Modeling of SEC by Monte Carlo Simulation

55

Downloaded by NORTH CAROLINA STATE UNIV on October 10, 2012 | http://pubs.acs.org Publication Date: May 30, 1996 | doi: 10.1021/bk-1996-0635.ch004

d/2. Mean values of the radius of gyration are obtained directly after simulation, as the position of each segment is explicitly known. Pore Geometry. Observations by electron microscopy have been performed on various column packings : Controlled Porous glass (15-18) , Methacrylates gels (19), hydrophilic TSKPW gels (20), Styrene-Divinylbenzene gels (21). These packings are found to consist of beads obtained by partial fusion of small irregular particles. This produces very deep, tortuous channels whose cross sections are irregular but never with highly thin protuberances or sharp angles. To take into account the geometry encountered along a given pore, a reasonable model would be a combination of long cylinders with circular or square sections and of more compact closed volumes (spheres or cubes) and these four pore geometrical models were tested. Linear calibration curves are obtained by mixing several pore sizes, moreover due to the complex real geometry, each pore size corresponds to a distribution of sizes. So the elution volume for a given molecule in real situation is governed by the combination of different Κ values in different pore sizes and pore volumes and only an apparent Κ value is observed. For this reason we have tested situations with a unique pore size, for Κ values between 0.1 and 0.9, and in many cases it seems sufficient to check the situation for the central value K=0.5. Comparison with Theoretical Results The validity of the simulation procedure can be tested by comparison with explicit analytical results, available in limiting cases. The radius of gyration obtained by simulation using a pore of infinite size can be compared with theoretical values (22). For a chain with Νfreelyjointed segments with length 1, R is given by: g

N(N + 2)1

2 ( 5 )

φ -

6(N + 1) Small values of Ν allow accurate checking of the uniformity of orientation 4 randomness. After generation of 5.10 chains, deviation with theory was less than 0.1%. For a chain with a large number offreelyrotating segments : N(N + 2)1 1 + cosa ^ 6(N + 1) 1-cosa Using values of Ν from 100 to 500 and a from 0.2 to π/2, deviation with theory is less than 0.1% after generation of more than 10 chains. Pure helices are indeed obtained when β is fixed. Concerning theoretical values of the partition coefficient in simple pores, two special cases have been investigated in the literature. Casassa (12) predicted for random coils containing large numbers of segments in cylindrical pores : 2

CO

Λ

(7) i=i af Here