Dynamics of chromatography : (in three parts)

168 Pages·1965·464.603 MB·English

by  Giddings, John Calvin

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Principles and Theuru
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5 A Series of Monographs
i EDITORS
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3 J. Calvin Giddings Roy A. Keller
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‘ Stephen Dal Nogare Erich Heftmann
Rudolf Kaiser John H. Knox
Ivor Smith
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DYNAMICS OF
CHROMATOGRAPHY
PART 1
Principles and Theory
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DYNAMICS OF
HROMATOGRAPHY
A J. CALVIN GIDDINIGSP
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PART I
Principles and Theory
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CONTENTS
vii
6 IX
One. Introduction and Dynamics of Zone Migration 1-11
A Viewpoint 1
Terminology 1
Dynamics ofMigrationand Equilibrium 2 3
Forms oftheMigration Equation 5
Chromatographic Methods 8
References 11
Two. Dynamics of Zone Spreading 13-94
Importance ofZone Spreading 13
TheEvolution ofZone Spreading Concepts 14
RoleoftheTheoretical Plate Concept 20
Origin in Random Processes 26
TheRandom Walk Model 29
Longitudinal MolecularDiffusion 35
Adsorption-Desorption Kinetics 36
Diffusion-Controlled Kinetics 39
Flow Pattern andEddy Diffusion 47
The CouplingTheory ofEddyDiffusion 52
Summary ofApplications to Chromatography 61
Criticism ofthe Random Walk Model A 65
Rigorous StochasticTheory: KineticEffects 66
ZoneSpreadinginNonuniformandLinkedColumnsandinRegions
External to the Column 79
References 91
Nonequilibrium and the Simple Mass Transfer
a Term 95-118
Nlitureand Effect ofNonequilibrium 95
Plate Height for Simple One-Site Adsorption
of fer Mass Transfer through a Uniform Film
xii CONTENTS
Chapter Four. The Generalized Nonequilibrium Theory 119-193
4.1 Nature oftheProblem 119
4.2 Step-WiseKinetics: Adsorption Chromatography 120 . p Chapter One
4.3 Theory ofDiffusional Mass Transfer 131
4.4 Diffusion in the Stationary Phase 136
Introduction and Dynamics
4.5 Diffusion in the Mobile Phase 149
4.6 Some Mixed Mass Transfer Mechanisms 158
4.7 Extensions, Limits, and Assumptions 169 of Zone Migration
4.8 Summary ofPractical Results 177
4.9 References 188
Chapter Five. Packing Structure and Flow Dynamics 195-225
1.1 A Viewpoint
5.1 Introduction 195
omatography has been defined in numerous and sometimes contra-
5.2 Packing Structure 196
ctory ways We wish to avoid the tedious differentiation between the
5.3 The.Laminar Flow Process 203
omatographicandthenonchromatographic.Wepreferheretoapproach
5.4 Turbulent Flow 217
L€ word “chromatography” by association with the laboratory technique
5.5 References 224 .
rig that name. We should like to approach the laboratory technique
Chapter Six. Diffusion and Kinetics in Chromatography 227-264 would a surgeon, dissecting it to discover the underlying features and
e processes related thereto. We may look, first, at those events which
6.1 MolecularDisplacement and Chromatography 227 over the dimensions of a single granular particle; then, step by
6.2 Nature ofDiffusion 227 , change our magnification to the level ofthe small pores and finally
6.3 Diffusion Rates in Liquids and Gases and on Surfaces 230 he molecular world itself. We should then like to synthesize a whole
6.4 Diffusion through Porous Materials .243 ofchromatography from the discoveries thus made.
6.5 Kinetics of Single-Step Processes: Adsorption-Desorption
It is inevitable that we shall partially fail in this task, for the detailed
Phenomena 249
are too complex and varied, and empirical work has not yet
6.6 References 262
the type of precise, discriminatory data needed. This is merely
way of saying that the science of chromatography is not complete.
Chapter Seven. The Achievement of Separation .265-5-302
e must still invent models, average out details, compartmentalize the
7.1 Theory and Practice 265 lated, rely on inadequate data, ignore the tenuous, and otherwise
7.2 Elements ofResolution 266 plify our thinking to a comprehensible level. At the end ofall this we
7.3 Optimum Conditions Based on the Resolution Concept 275 I fortunately,thatourkaleidoscopicpictureisin accord withthemajor
7.4 Fast Separations 286 gs of experimental chromatography. We thus have at our disposal
7.5 Comparison ofGas and Liquid Chromatography 293 ns ofcorrelating, predicting, and, most importantly, understanding
7.6 References 30] phenomenon ofchromatography. This is the realjustification for the
t task.
Appendix A: Principal Symbols 303
1.2 Terminology
Appendix B: Significant Equations 310 the most part we have used terms and expressions commonly
ng in the cliromatogiupliic literature. In a few cases it has been
rytodrawgasandliquid chromatograpliytogetherbycompromise.
Author Iridr-.r 315
5&5 liquid clirornatograpliy. for instuiicc, it lius been conventional to
Subject Index 31.9
l
2 DYNAMICS OF CHROMATOGRAPHY DYNAMICS or ZONE MIGRATION 3
affix subscriptg to gas phase terms (e.g., Cg) and subscript lto the liquid ratio characterizing migration rates. The quantities R and Rf, while
stationary phase terms. While this distinction is appropriate for Part closely related, are sometimes mistakenly equated.“ The latter differs
II, here we must (except in specific cases) use subscript m for the mobile slightly (about 15%) from R by being the ratio ofthe zone velocity to the
phase—liquid or gas—-—and s for the stationary phase. velocity ofthe solvent(mobile phase)frontinpaper orthinlayer chroma-
Frequent use is made of expressions such as “column migration” or tography. Sincethevelocityofthefrontisgreaterthanthatofthefollowing
“processes within the column.” The use of the word “column” merely mobile phase, R > Rf.
identifies the location of the relevant process, and is not intended to In gas chromatography the retention parameter most commonly used
exclude noncolumn techniques such as paper and thin layer chroma- is the capacity factor k. This is related to R by R = l/(1 + k). It is
tography. We could usethe generalterm “chromatographicbed,” butthis entirely valid to characterize retention by either parameter. Despite the
hasbeendoneonlyonoccasion. Bythesametokentheword “particle” = discord with much gas chromatographic literature, the use ofR is more
“fiber” = “bead” in the relevant cases. desirablefrom several points ofview. First, it and closely related param-
The terms “mobile phase” and “stationary phase” are, of course, eters (e.g., Rf) have been used in liquid chromatography for years. It
standard in chromatography. We have been rather free with the former, is unfortunate that a departure from this sound terminology was ever
sometimes referring to it by the obvious alternates, “mobile fiuid” and made, as it has tended to widen the gulfbetween related techniques. (The
“moving fluid.” The stationary phase divides itself into two categories, use of R for “resolution” has compounded the difficulties; Rs is used
depending on whether molecular attachment occurs by adsorption here.) Second, R is proportional to the migration rate, and migration in
(adsorption chromatography) or absorption (partition chromatography). differential form is the basis of separation. Third, and perhaps most
The latter may be distinguished by the term “bulk stationary phase.” important, nearly every mathematical expression containing R is more
The substance being chromatographed is the “solute” or the “sample.” complex when written in terms of k. Of the more than 150 numbered
The source ofmigration in chromatographyis the downstream motion equations in this book containing R (excepting those characterizing R),
of the mobile phase. We must clearly understand the measure of this over 90% are simpler (fewer type characters) as a consequence of not
motion in order to make quantitative sense out of migration. The ap- using k. Most of the well-known equations of chromatography follow
propriate measure is the flow velocity, and it comes in several forms. trend. The simplicity associated with R is related to the fact that it is
At any particular microscopic point there is a local velocity v’. This direct migration velocity parameter. A
varies widely within and between the particles. Its average over 5-10 Numerous other terms and parameters appear throughout the volume.
particle diameters is the regional velocity, v. If, furthermore, there is a These are explained along the way. And a section, Symbols, attheend of
serious variation of this across the column (as in many preparative book summarizes the principal mathematical quantities.
columns), the average over the entire cross section is denoted by 5. The
lattervariationscanusuallybeignoredandthe observedvelocityidentified
1.3 Dynamics of Migration and Equilibrium
with v. i
Thebestmeasureofthe mobilephasevelocityvisthedistance/timeratio tionisachievedthroughdifferentialmigration. Hencethemigration
of some inert material which does not attach to the stationary phase. is a cornerstone of chromatography. This section introduces the
Such material will occasionally be swept at great speed through the open gration concept from a dynamic viewpoint.
flow passages between particles, and sometimes held back in stagnant As usually observed, a zone migrates along smoothly at some fraction
pockets and pores. As an average, however, its velocity best represents of the mobile phase velocity. At the molecular level, by contrast to
the overall motion of mobile phase. On occasion we are interested only appearances, migration occurs in a state ofchaos, each molecule
in the mean interparticle velocity (thus excluding mobile phase held ng in a stop-and-go sequence independent ofany othermolecule.
stagnant within pores that might exist), which can be denoted by time a molecule affixes itself to the stationary phase, its migration
v0. interrupted while the zone as a whole passes over. One molecule may
Due to retention, chromatographic zones move along at only a fraction ahead of another that is immobilized. only to be overtaken by the
R of the mobile phase velocity. Thus the zone velocity (i.e., the velocity molecule at a subsequent point. The process might best be described
ofthe zone’s ceiilcr ofgravity) is Rn. The quantity R is the éééiéffiiflfiiiml 9. game ofuncoordinated leapfrog. This analogy gives a picture ol‘how
4 DYNAMICS OF CHROMATOGRAPHY DYNAMICS or ZONE MIGRATION 5
the zone can move continually ahead although only a fraction of the the exponential is usually about equal to the time required for a single
zone’s molecules are moving at any one time. sorption step. Since in most runs a large number (about 103-104) of
Since each molecule follows a random, jerk-like path through the sorptions and desorptions take place, the probability rapidly approaches
column, there is a statistical dispersion of the zone’s molecules, that is, R with great precision. Thus the zone as a whole may be presumed at
the zone broadens out. The zone spreading which accompanies migration equilibrium. except perhaps for about one-thousandth ofits lifetime right
inallnormalformsofchromatographyisthusareflectionofthemolecular at the beginning. Hence the average distance migrated can usually be
basis ofmigration. Although any given molecule isjumping rapidly back based on equilibrium considerations to within one part in a thousand.
and forth between mobile and stationary phases, the probability at any Onoccasion, where thereisaheterogeneous orunevensorptivematerial
instant(after aninitialtransientperiod)thatitis inagivenphasebecomes withcertainpartsresistingfastequilibration,theprobability,afterdecaying
a rigidly fixed physical parameter. Let the probability that the molecule rapid-ly at first, may only slowly make its final approach to R. If this
is in the mobile phase be R. As more and more randomjumps are made slowness is enough to disturb the equilibrium approximation, it will
and the relative degree of statistical fluctuations restrained (as always) usually manifest itselfin nonsymmetrical zones, usually with tailing‘-‘* (see
by the large number of such transfers, the time fraction spent in the Sections 2.13 and 6.5). _
mobile phase will approachthis fraction R. Since net downstreammotion The above tells us that the migration of a zone as a whole is usually
is achieved solely during residence in the mobile phase? the molecule governed by equilibrium. But it does not tell us whether equilibrium is
moves only with a fraction ~R relative to the velocity v of the mobile validthroughouttheentire width ofthezone. Recallthat R is an average
phase. Thus the R value may be definedidentically as a relative migration timefraction and thatzone spreading occursmainly as a result ofindivid-
rate (as earlier), as the probability that a solute molecule is in the mobile ual molecules fluctuating around this value.
phase or as the limiting fraction oftime a molecule spends in the mobile If, after considerable migration, we select a given molecule near the
phase. Furthermore, if we have a large number of such molecules (e.g., front ofa zone, this molecule will naturally be one which has reached its
all those in a zone), the fraction in the mobile phase at equilibrium must advanced position by spending a fraction of time slightly larger than R
equal the same parameter R. This can be used essentially as a fourth say, R + AR) in the mobile phase. At the particular time t (or at any
definition ofR. ebefore 2‘)whenwelookatthismolecule, itsprobability ofbeinginthe
The latter definition leads us to the viewpoint that zone migration, obilephaseisjustthisvalueR + ARTHencethefraction(= probability)
despite its chaotic molecular origin, is related to equilibrium parameters. f molecules in the mobile phase near the front of a zone exceeds the
All equilibrium has a kinetic origin, so this dualviewpointis notunusual. uilibrium value R. This same argument shows that the probability-—
Sincetheequilibriumassumptionis abasicpartofmuchchromatographic nce the number fraction at the rear of a zone—-is less than R. In fact
work, particularly the identification of components through R values, it e precise value R is applicable only at the zone center; that is, true
merits further comment. , r ' uilibrium occurs atthezonecenterandnowhereelse. Thiswillbeshown
Ifwehavealargenumber ofmoleculesinachromatographiczone, then a differentkind ofargumentin Chapter 3. It sufiices here to say thatif
thefraction in the mobile phase at any point and time is identical to the aretorelatechromatographicmigrationtotheequilibriumdistribution
probability that any given molecule is in the mobile phase at the same molecules between phases, we must refer to the center ofthe zone, and
point and time. Let us follow this simple probability concept to establish must have a system with a fairly rapid molecular exchange between
whereandwhenequilibriumisvalid.AvalueofexactlyRdenotescomplete
equilibrium. i
1.4 Forms of the Migration Equation
In many systems the initial zone is carried onto the head ofa column
in a stream ofmobile fluid, and migration commences without hesitation. R value, being now established as an equilibrium parameter when
A given molecule, thrust suddenly onto a column, remains in the mobile uated at the center of normal zones, can be expressed in several
phase a finite time until its first capture by the stationary phase. The l ways.
probability that the molecule is in the mobile phase at the instant of
Thismakesuseoftheconceptofcoiuilitional"probabilities.Thegivenmoleculedeparts
contact with the column is thus one. The probuhilitty decays exponentially
thenorm only becaiise we have purposely selected a rapidly a.i.lv;i.ncing"molecule.
(in the simplest cases) to R as sorption eon'unei1ees. The time eonéiant of nonaver-ageprobability, R, isconditionalontheselection ofsucliamolecule.
6 DYNAMICS or CHROMATOGRAPHY
Z-DYNAMICS OF ZONE MIGRATION 7
If R is the equilibrium fraction of solute in the mobile phase, then The c terms as defined here are overall concentrations, taking no account
l -—- Ris the solute equilibrium fraction inthe stationaryphase. The term of the microscopic distribution of solute within the phases (the latter is
R/(l -—- R)is thus the ratio ofthe solute inthe mobile phase to thatin the measured by the local concentration units discussed above, always
stationary phase. Along with most equilibrium ratios, this one is the denoted by a prime, e.g., cs’).
concentration-volume product c.m’V,m, applicable to the numerator We have already seen that R is the fraction of solute in the mobile
(mobilephase)dividedbythecorrespondingtermcs’Vsofthedenominator phase at equilibrium. Thus R = (cm/c)eq. Henceforth equilibrium con-
(stationary phase); that is, ntrations will be denoted by an asterisk so that R is now given by the
ple expressioni
-.i‘ -.__ fella" (1.4-1)
1 —R c,’l/1., R = c,.,,*/c (1.4-5)
a like manner the solute content ofthe mobile phase over that ofthe
The equilibrium ratio ofconcentration in the two phases'(c,’/cm’)eq is the
tionary phase at equilibrium is
distribution coeflicienti or partition coefficient K (sometimes denoted
as Gt). Thus R c =l=
-———— = —'“— 1.4-6)
-3-=51 (mm 1 — R cf‘ (
1—R KK
though both c,,,* and c,* change from point to point in the zone, their
The quotient V.,,,/V, is simply the mobile phase to stationary phase tio is constant. Equations using overall concentration terms, like the
volume in the various forms ofpartition chromatography (cross-sectional bove two, will be employed extensively later.
areas, frequently used in place of volumes, are equally correct). For Taking another approach, we have already noted that a molecule left
adsorption chromatography V, can be replaced by the surface area. ng enough in the column will spend a fraction closely approaching R
OnsolvingthelastequationforR, wehave anequivalentoftheclassical its total time in the mobile phase. In any finite interval R will not
expression of Martin and Synge“ , yequalthisfractionduetothemolecule’sfluctuations, butifwetake
average time fraction for all solute molecules, some fluctuating in one
R=_li_- nan
m+xn ion and some in another, R may be identified precisely. Thus
I R = rm/(rm + rs) = rm/i (1.4-7)
If retention is due to several simultaneous mechanisms (e.g., partial
adsorption in partition chromatography), the above arguments can be here in total time 2‘ an average time rm is spent in the mobile phase and
easily extended to yield the generalization? in the stationaryphase. This equation is due to LeRosen.8 As with the
lier cases we may write this in other equivalent forms; for example,
R . V“ — -— ~ (1.4-4) R i
——-=fl 148
1—R i ( )
In any case R can always be related to the equilibrium term (or terms) K
and to the amount ofeach phase present. ‘The above equations can be related more directly to the dynamics of
In much ofchromatographic practice it is impossible and unnecessary tionand desorptionasfollows. Duringitsmigrationdownthecolumn
to deal withthelocalconcentrations c,’ and cm’, applicable tomicroscopic given molecule will sorb and desorb many times in rapid succession.
regions within the phases. Given a unit volume of the column material, each sorption step must be followed by a desorption step, the same
it is often only necessary to know the following; the total amount c of ber of each must occur during migration (give or take one for end
solute in that volume, the amount cs associated with the stationary phase, ). Let this number be r. Now if a molecule spends a total time is
andthe amount cm inthe mobile phase. We have, ofcourse, 0,. + cm = c. the stationary phase, and desorbs r times, the average stay must be
This is denoted by 1,, since it is the average time required to get the
T This ratio only approximates the true thermodynamic distributioncoelilelent,except
in ideal solutions and gases, since the latter is a ratio ofactivities (rather thar1eeneen- total concentration c is not starreIId because it does not chanI-r~e as solutetransfers
l|'nlions).“ phases. We have 0, 4- r... 2 i:',* + r,..* E C‘.
8 DYNAMICS or CHROMATOGRAPHY DYNAMICS or zone MIGRATION 9
molecule desorbed. Similarly, the average stay in the mobile phase before book. Equilibrium treatments of nonlinear chromatography, by contrast
an adsorption can occur is ta = rm/r. On writing these expressions as to dynamic treatments, abound in the classical paperst on the subject.
r, = tdr and rm = tar, equations (1.4-7) and (1.4-8) become (2) Liquid-Gas. As far as experimental work goes, there are two main
camps divided according to whether a liquid of a gas is used for the
R = ta/(ta + td) mobile phase. As shown in Chapter 7, there is little basic differencein the
natureofthedynamicsinvolved. Thus theworkofthisvolumeisapplicable
-—R- = 5* (1.4-10) almost in total to both. But the numerical differences in certain physical
1 R 1:,
properties (e.g., diffusivity, viscosity) are so enormous as to cause a very
real inequality in practical performance (speed, resolution, etc.). The
thus relating R to the dynamic parameters ta and rd. As will be shown in
specific and detailed nature of gas chromatographic processes will be
Section 2.7, z‘,, and rd are equal to the reciprocal of the (apparent) rate
discussed in Part II, while liquid chromatography will be reserved for
constants ofadsorption and desorption.
Part III.
In summarizing this section, we should draw attention to the fact that
(3)Partition-Adsorption. Thisisperhapsthemostbasicdivisionbetween
R is based on equilibrium considerations but that true equilibrium even
techniques because there is a fundamental difference in the retention
in the most ideal cases is applicable at only one point (the center) in the
mechanism. Each form is frequently used in both gas and liquid chroma-
zone. The reference ofRto strictly equilibrium conditions, out ofkeeping
tography. Adsorption chromatography is based on the retention ofsolute
with nearly all the solute in the zone, is reflected by certain (unrealistic)
by surface adsorption, whereas partition chromatography is based on the
restraints applicable in most ofthe equations for R in this section. Thus
adsorption ofsolute by bulkjt stationary phase usually held in place by an
eachtimeRiswrittenasaconcentrationratio,wemusttaketheequilibrium
inert scaffold ofsolidparticles. Both methods, alongwith certain “mixed”
ratio rather than the real value found at some random point in the zone.
cases, are considered at length in this volume.
And eachtimeRistakenas atimeratio (inmobileand stationaryphases),
(4) Zone-Fr0nral. Solute may be introduced as a discrete zone or as a
it must be the ratio applicable to the hypothetical average molecule or to
step function (i.e., as a continuous train of solute beginning with a sharp
amolecule spendinginfinitetimeinthe columnratherthanthe fluctuating
. front). Resolution is best when the discrete zone is introduced as narrow
value ofreal molecules.
'- as possible (approaching a 6-function) and, in the frontal case, when the
\ i front begins as a sharp, right-angle profile. A 6-function profile (infinitely
1.5 Chromatographic Methods
r thin zone) introduced into a linear column which has a rapid exchange
Itis unnecessaryto describeinfullthe numerousmodifications ofchroma- betweenphaseswilldevelop onthecolumninto a Gaussian(“normalerror
tography. These have been cataloged in many books on the subject. We curve”) zone. The sharp front will develop into an S-shaped “error
should like, however, to indicate very briefly the dynamicsimilarities and ‘ function.” The mathematical treatment of zone shape in linear chroma-
differences ofthe main categories. ' tography automatically describes fronts because the latter is the integral
Depending on the characteristics of interest, chromatography may be of the former (Section 2.14). Since zone chromatography is the common
splitinnumerous, arbitrary ways. Thelistbelowisby no means complete, departure pointinthelaboratory our discussion will largely center on this
but will illustrate the enormous.scope ofthe subject. Inethod. The terms 5-function and Gaussian will consequently appear
(1) Linear-Nonlinear. In linear chromatography all solute molecules r Often.
behave independently; in nonlinear chromatography there is some i (5) Elurion-Nonelution. Solute may be washed clearthroughthe column
measurabledegreeofinteractionbetweenthem.Nonlinearchromatography lforanalysis or itmaybestoppedatsomepointwithinthechromatographic
is usually thought of as involving only nonlinear sorption isotherms. It l-bed. The latter is especially useful in paper and thin layer techniques. The
may, however, also involve nonlinear dynamic processes such as diffusion lfiynamics ofthe two cases are essentially identical except that in one case
and kinetics. Nonlinearity always yields to linearity with sufficiently small ‘the process is terminated earlier. Both cases are covered in this volume.
samples of solute. The dynamics of interacting molecules, as l'on1-id in |
-ti Forexample, references 9 and 10.
nonlinear chromatography, is so complex that it lins never been treated
1 Theword“bulk” inthisbook lnenns threetli111e1'1sio11a_l, not !1te‘LZCS51:l!'ll_*y’ lliat it Ilns U16
with real sa.'l'isfac'tio1‘1, and ll1erel1,11';: corlstilfutes only it minor part gf 111$ properties oflarge rnasses.