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Anal. Chem. 1984, 56.2104-2109
the extraneous effects. The use of some form of sheath flow in which the solute is kept away from the edges by a stream of pure carrier, as also discussed above, is another possible solution. While the use of sheath flow was suggested in our earlier paper dealing with capillary liquid chromatography in FFF-type channels (16), the form of sheath flow shown in Figure 3 was actually used in a prior publication (15);this was done by injecting the solute into the channel via a separate port located downstream from the carrier inlet. A reduction in plate height of about 20% was observed. This approach is certainly viable; the important but unanswered question concerns the fraction of practical FFF experiments for which this solution is, or will be, worth the additional experimental effort.
ACKNOWLEDGMENT The authors wish to thank Marcia Hansen for obtaining the experimental diffusion coefficient for methylene blue in ethanol. LITERATURE CITED (1) Giddings, J. C. J . Chem. Phys. 1988, 4 9 , 81. (2) Hovingh, M. E.; Thompson, G. H.; Glddlngs, J. C. Anal. Chem. 1970, 42, 195.
(3) Giddings, J. C. Anal. Chem. 1981, 53, 1170A. (4) Giddings, J. C.; Myers, M. N.; Caldweil, K. D. Sep. Sci. Techno/. 1981, 16, 549. (5) Karalskakis, G.; Myers, M. N.; Caldweii, K. D.; Giddings, J. C. Anal. Chem. 1981, 53, 1314. (6) Giddings, J. C.; Yang, F. J.; Myers, M. N. Anal. Chem. 1978. 4 8 , 1126. (7) Myers, M. N.; Caldwell, K. D.; Glddings, J. C. Sep. Sci. 1974, 9 , 47. ( 8 ) Giddings, J. C.; Karaiskakis, G.; Caldwell, K. D.; Myers, M. N. J . Colloid Interface Sci. 1983, 92,66. (9) Giddings, J. C.; Yoon, Y. H.; Caldweil, K. D.; Myers, M. N.; Hovingh, M. E. Sep. Sci. 1975, 10, 447. (IO) Giddings, J. C.; Schure, M. R., submitted for publication in Chem. €ng. Sci . (11) Schure, M. R.; Myers, M. N.; Caldwell, K. D.; Byron, C.; Chan, K. P.; Giddings, J. C. submitted for publication in Environ. Sci. Technoi. (12) Giddings, J. C. J . Gas Chromatogr. 1963, 1 , 12. (13) Giddings, J. C. “Dynamlcs of Chromatography”; Marcel Dekker: New York, 1965; Part 1, Principles and Theory. (14) Giddings, J. C.; Seager, S. L. Ind. Eng. Chem. Fundam. 1962, 1 , 277. (15) Smith, L. K.; Myers, M. N.; Giddings, J. C. Anal. Chem. 1977, 4 9 , 1750. (16) Glddlngs, J. C.; Chang, J. P.; Myers, M. N.; Davis, J. M. J . Chromatogr. 1983, 225, 359.
RECEIVED for review April 12, 1984. Accepted May 25, 1984. This project was supported by Public Health Service Grant GM10851-26 from the Natiional Institutes of Health.
Chromatographic Band Broadening Theory Using a Random Walk with a Step-Length Distribution Stephen G . Weber
Department of Chemistry, University of Pittsburgh, Pittsburgh, Pennsyluania 15260
Chromatographlc band broadenlng Is studied theoretlcaily by using the random walk approach In which the step-length distribution Is Included In the variance calculation (random fly). The mass transfer term that Is derlved Is dependent on q , the average number of adsorptlon/desorptlon events occurrlng In a partlcle. The mass transfer term Is large If q Is large but dlffuslon In the particle Is slow, or H q Is small, elther because of a llmlted number of colllslons of a solute molecule wlth the statlonary phase/stagnant moblle phase Interface, or because of a rate llmltlng adsorption or desorptlon.
The advance of chromatography has been aided by theory over the past several decades. In general three different theoretical approaches to band broadening are used, the mass-balance approach, the nonequilibrium approach, and the stochastic approach (1). While the former two possess distinct advantages, most importantly the prediction of higher moments, the stochastic approach is interesting because of the analogy with microscopic chemical reaction processes, although the mathematical treatment is fairly difficult (2-5). One stochastic approach, the random walk, is simple and leads to correct results (6-10), but in the form usually applied the relationship between the dynamics of chemical reactions and chromatography has been lost. There are still advances being made that make the random walk theory applicable to many systems (11,12),even though the mathematical development of stochastic processes has been occurring for a century. Certainly chromatographers can benefit from the application of more detailed random walk approaches. The random walk approach as it has been applied to chromatography consists of the following model of the chro-
matographic process (ref 10 and references therein). The frame of reference for the process is moving with respect to the laboratory frame; the frame employed travels at a velocity a correpsonding to the average velocity of a particular peak or band. Individual molecules, however, do not actually travel at i7,they are either in the stationary zone (stationary phase or stagnant mobile phase) with a velocity of zero or in the mobile zone (flowing mobile phase) with a velocity u,. When a molecule moves from mobile zone to stationary zone it is said to have taken a step backward, and when it leaves the stationary zone it is said to have taken a step forward. The molecules execute a one-dimensional random walk. The variance of the distribution of molecules along the chromatographic axis ($) is related to the plate height, H = u:/L, where L is the column length. For a one-dimensional random walk the variance is given by u t = n12,where n is the number of steps taken and the step length is exactly 1. The values of n and 1 are determined for a particular process (e.g., stationary zone diffusion). The variances resulting from independent processes are added to calculate the overall variance. Of course, an individual molecule does not undergo steps of exactly length 1 each time, so it is more appropriate to view 1 as the average step length, (1). Then a? = n(1)2. It is possible to be more realistic and still use the simple mathematics of the random walk by using the step-length distribution P ( r ) . In such a treatment, not only the average length of a single step is used but also the distribution or variability of those step lengths is used (5,13-15). Figure 1 shows four step-length distributions that will be covered in this paper. Each of the distribution functions has a unit area indicating that a molecule always takes a step somewhere. Also, each distribution has two “halves”, one representing the forward step and one representing the backward step; each
0003-2700/84/0356-2104$01.50/00 1984 American Chemical Society
ANALYTICAL CHEMISTRY, VOL. 56, NO. 12, OCTOBER 1984
a
2105
t and the second moment is
The first moment is the retention time in chromatography. The second moment is related to the variance. The variance is actually the second moment of the peak taken in a coordinate system in which pl = 0. This is called the second central moment, second cumulant, or variance (9) K~
= 112 - 1112 -1
+I
0 X
C
1 -1
+I
0
d
-1
0
+I
X
Flgure 1. P ( x ) is the
We will rely heavily on these relationships below. The variability of the step lengths is incorporated by using a slightly more general form than that shown above for the variance of a population that has undergone a random walk of n steps (13) 612
X
probability distribution of step lengths, l Is the average step length, equal to k"v,( f,m)/(l k"): (a) conventional random walk assumption, P ( x ) = 1/2 at f l ; (b) axial diffusion causes a spread in the distribution of forward (mobile phase) steps; (c) other mobile phase mass transfer process have been included; (d) intraparticle processes have been included.
+
half of the distribution has an area of 1/2. Figure l a shows the distribtuion of step lengths that is used in the traditional random walk approach discussed earlier. The molecule has a probability of 1 / 2 of moving one step forward and a probability of 1 / 2 of moving one step backward. To see the physical significance of a distribution, consider Figure lb. Here the process of' axial or longitudinal diffusion in the mobile phase has been represented. When a molecule is taking a step of average length ( I ) in the flowing mobile phase it is all the while undergoing diffusion. This can either shorten or lengthen the actual step length in any particular case, leading to a distribution of possible forward step lengths. This paper will demonstrate the method for using a steplength distribution to calculate the plate height for distributions such as those illustrated in Figure 1. In particular a detailed description of intraparticulate mass transfer is given. The stochastic treatment is naturally quite useful for formulating the relationship between kinetics and diffusion since both are stochastic processes. This derivation is the only one of which I am aware that treats intraparticle diffusion and kinetics together.
RESULTS Use of a Step-Length Distribution in the Random Walk. Since this is a statistical treatment, statistical nomenclature abounds; thus a definition of some terms is required. The zeroth moment, wo, of a function is the area under the curve. The first moment of a function P(x),p1 is
(4)
(5)
= np2
Here p2 is the second moment (not the second central moment) of the distribution p ( ~ )It. is assumed that n 2 10 or so. The use of eq 5 also implies the assumption that the number of steps taken in the random walk is not a stochastic variable, that is, for a given set of parameters n is known. If the step length is only known through P ( x ) , then n must certainly not be known exactly. However, the distribution function describing the number of steps taken to reach the end of the column, given P(x), is much narrower than P(x). In fact, it can be shown for a Gaussian P ( x ) with a variance of K 2 that the square of the relative standard deviation in the distribution of the number of steps is l / n that of the distribution of step lengths. Thus even for n as low as 10 we can ignore the variability of n and assume it is known. Let us apply this formula to the distribution in Figure l a in an attempt to generate the formula uz = n(1)2, since Figure l a represents the case for which that formula applies, namely, an invariant step length ( I ) . We have
P(x) = p2
=
-
1/,6(2?
(1)2)
(6)
= J x 2 P ( x ) dx
(7)
+ (U2)
(8)
f/2((02
=
(1)2
(9)
and from eq 5 u1 =
n(1)2
(10)
which is the expected relationship. We will show below that eq 5 allows the incorporation of other sources of variance for a truer picture of the chromatographic process. Consider that the step lengths are not really accurately represented as a 6 function at x = ( l ) , but rather there is a spreading due to various physical processes. In only one case has the step length distribution been used in a stochastic approach to chromatography, Benyon et al. (5) assumed a Gaussian profile of "transit times" with a variance that is not calculable. This ends up as the eddy diffusion contribution to their final plate height. A general equation for pz in chromatography will now be established and then it will be applied to the remaining distributions in Figure 1. The equation will be set up for a one-site process only. The extension to multiple sites is straightforward; however, since it will not be treated here the increase in complexity is not warranted. The essential qualitative feature of this treatment is that the forward and
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ANALYTICAL CHEMISTRY, VOL. 56, NO. 12, OCTOBER 1984
Table I. Definition of Terms geometrical factors in intraparticulate diffusion mass transfer term in the equation for plate height diffusion coefficient in phase i: m, mobile; s, stationary; sm, stagnant mobile; fm, flowing mobile; sz, stationary zone (s + sm) effective diffusion coefficient in phase i pore diameter stationary phase film thickness particle diameter plate height reduced plate height Hld, phase capacity factor zone capacity factor first-order diffusion rate constant first-order association rate constant first-order dissociation rate constant step length in random walk column length number of collisions a molecule makes with the s/sm interface from the sm side total number of collisions a molecule makes with the slam interface step length distribution average number of times a molecule is adsorbed in one stationary phase particle average time a molecule spends in phase i during one random walk; see definition of Di for definition of i average velocity of a band mobile phase velocity in the fm zone intraparticle void fraction interparticle void fraction second central moment, variance distance parameter in Eyring equation for D i = 0, 1, 2; zeroth, first, and second moments (respectively)of P ( x ) VedplDrn variance of chromatoeraDhic band reverse steps can be treated independently, and that for each of these the distribution’s variance and position (K2 and pl, respectively) are treated individually. This leads to the solution of a number of small problems instead of one large problem, e.g., in the mass-balance approach. To obtain the overall variance,:u we must know n and p2 (see eq 5 ) for the chromatographic process. The value for n has been given by a random walk approach (7) as eq 11.
n = 2L/ue(tfrn)
(11)
Recall that we are assuming that n is not a stochastic variable. A table of variable definitions is given in Table I. (Throughout this paper a convention for diffusion coefficients and residence times will be used. The average time a molecule spends in a certain chromatographic zone, z, is given as ( t z ) ,where z is fm (flowing mobile phase) or sm (stagnant mobile phase), s is stationary phase, and sz is stationary zone (s sm). Diffusion coefficients also use the same symbology; in addition the superscript “eff’ indicates an effective diffusion coefficient, that is, including tortuosity factors.) The value for p2 comes from eq 2, 3 and 4. The function P ( x ) is broken into two halves
+
= Pfrn(x) + Psz(z)
(12)
Pfrn(r) and Ps,(x) represent the probability distributions for forward and backward steps, respectively. Equation 2, with po = 1 since there is unit probability that the molecule will take a step, becomes PZ
+ P,z(x)) dx
(13)
+ s x 2 P s z ( x dx )
(14)
= Jx2(Pfrn(x)
= s x 2 P f m ( x )dx
Table 11. Variances and First Moments for
ut
figure
Pl,frn2
lb Id
processes
K2,fm
KZ,0z
Computation
2Dfmeff(tfrn)0 (1)’ 2Dfmeff(tfrn) eq 35 [[u,l(l +
axial dif axial dif,
ILl,sz2
(1)2 same
k’?l ( t %l)2
stat zone
mass transf
These can be viewed as individual contributions of foreward and backward steps to p2. Furthermore, each of these two contributions is given by a variance and a first moment; see eq 15 and 16. There is an analogous equation for p2,sz.The ~ 2 , f r n=
2Sx2Pffm(x) dx
= 2(~2,fm+
pl,fm2)
(15) (16)
factor of 2 arises because of division of the integral in eq 15 by bo, which, for each half of the distribution, is 1/2. Then, CLZ
=
%(KZ,frn
=
%bz,fm
+ ~2,sz)
+ ~2,sz)+ %(P1,fm2 + ~1,sz’)
(17) (18)
Equation 18 shows that the overall chromatographic variance per step is given by four independently calculable terms. In particular it demonstrates the main theoretical point of this paper, that the chromatographic variance is governed not only by the step length but by the variability of the step lengths. Application to Various Distributions. Figures lb-d show distributions of increasing complexity that will be discussed below. In all cases the variances and first moments of each half of the distribution will be combined by using eq 18 to arrive at the contribution to :u resulting from the processes represented in Figure 1. Figure l b shows a normal distribution of step lengths for the forward mobile phase step. The distribution is due to axial diffusion. Table I1 gives the values of K2 and pL1for this process. K2,fm is calculated by using the Einstein equation k2(axial diffusion) = 2Dfrneff(tfm)
(19)
Then from eq 18 and 19
+ (1)2
uf = Dfmeff(t*,)
(20)
From a previous random walk treatment (7) the value of ( 1 ) has been found to be given by eq 21
k”
(1) = 1 k” ue (tfm)
+
(21)
Recall that k” is the zone capacity factor
While k’is the phase capacity factor k’ =
(t,) (tfrn) + (tsrn)
They are related by eq 24 in which ez is the interparticle void volume as a fraction of the total column volume and cp is the intraparticle (pore) volume as a fraction of the total column volume k” = ep/tz + (1 cp/cz)k’ (24)
+
Using equations 11,20, and 21, one can write an expression for u t
ANALYTICAL CHEMISTRY, VOL. 56, NO. 12, OCTOBER 1984
be given by the Poisson distribution.
Substituting (t,,)/k”for (tfm)(eq 2 2 ) one has
P(r) = q‘e-*/r! From the Einstein equation and a structure factor for spherical particles (8) one has
Here we view the particle as having an effective diffusion coefficient which is equal to the time-weighted average of solute diffusion coefficients in stationary and stagnant mobile phases (10). The ratio of times spent in these two phases is related to k‘, so for Dszeffone has
- k’(1 + ez/tp)Dseff+ Dameff 1
+ k’(1 + tz/tp)
2107
(29)
This formulation of can be used to experimentally determine D, (16,171.Using eq 27 in eq 26 and converting to reduced variables, one has
This h represents intraparticle diffusional resistance to mass transfer and extraparticle longitudinal diffusion. Thus we have very simply incorporated the mobile phase contribution to axial diffusion into the expression for h using a single random walk model. More complete (and therefore complex) step-length distributions are shown in Figure lc,d. Figure ICprimarily includes mobile mass transfer and eddy diffusion in addition to axial diffusion since these processes increase the variability of the times spent in the flowing mobile phase. It is important to note that mobile phase mass transfer is not reduced to a simple problem by this device, it is only true that if one knew the step-length distribution due to mobile phase mass transfer, and convolved that with the distribution due to axial diffusion, then the problem would be treatable by the random walk method. In Figure I d the distribution of backward steps has been modified to include intraparticle diffusion, adsorption/desorption kinetics, and it could also include particle size distribution. We will now deal with the variation of forward and backward steps, except for mobile phase mass transfer. To include the association-dissociation kinetics we must determine how kinetics alters the step-length distribution. This we will do in the following section. A molecule enters the stationary zone from the mobile zone and begins by diffusing in the stagnant mobile phase. If we assume that the bulk of the stagnant mobile phase is chemically homogeneous, then while the molecule is in the stagnant mobile phase the energy of the molecule will be the same between diffusional jumps. At some point the molecule will be near the wall of a pore and thus be poised to adsorb (or partition). It will then adsorb, or not. Here we can rather formally describe adsorption as a molecule’s move intoa region which involves a change in that molecule’s energy. If it adsorbs it can diffuse in or on the stationary phase, then after a while it may find itself poised to desorb. It then may or may not desorb. If it does, then the molecule is back in the stagnant mobile phase, having completed one adsorb/desorb event. The adsorption/desorption process is a stochastic one, and as long as the adsorption and desorption probabilities remain constant the distribution of the number of these events will
(31)
where r is the number of adsorption/desorbtion events that occur in a single stationary phase particle, q is the average value of r (0 < q < a),and P(r)is the probability that r events actually occur. The first moment of the distribution is q, and the variance is also q. Since the molecule diffuses in both phases (s and sm) there is a contribution to the variance from intraparticulate diffusion. The overall distribution of the step lengths resulting from variation in the time a molecule spends inside a particle is the convolution of the two independent distributions for diffusion and the kinetics. The variance of the convolute is the sum of the variances of the two independent distributions. However, to add the two variances they must have the same units; K2 for diffusion is in (length)2and for kinetics is dimensionless. To determine K2 for kinetics in units of (length)2one must convert from q, a unitless quantity, to one with units (length)2. This can be done with two conversion factors
The first term in parentheses is the average time per adsorption event, (t,)/ q , and the second is the average velocity of the band
The value of equation
K2
for diffusion is eq 34 from the Einstein
= 2DE$ff(t,z) (34) The variance for the backward step length distribution is now given by K2
(35) which represents the sum of the two variances. Using the values of K~ and bI2shown in Table I1 and eq 11, 21, and 27 in eq 18 for total band variance one has
2L(D,Zffk”+ D
eff) fm
(36)
Ve
In terms of reduced plate height and velocity this becomes
+
2(Da,”ffk ” D fmeff) / D m v (37) This represents the band broadening from both the B (intraand interparticle) and C (intraparticle) terms. The Evaluation of q . We have taken a statistical approach to q. It remains to determine q for a given set of chromatographic conditions. The number of adsorption events a molecule undergoes inside a support particle is equal to the number of times the molecule presents itself at the surface, E,and the probability that once at the surface the molecule will be adsorbed, Pa
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ANALYTICAL CHEMISTRY, VOL. 50, NO. 12, OCTOBER 1984
q = fiP, (38) First we will determine ii. The number of times a molecule “collides” with the interface from the solution side is m. This must be equal to the average time spent in the stagnant mobile phase divided by the average time a molecule spends in a pore between encounters, ( tp). Since (tsz)
=
(tsrn)
+ (ts)
q may also be given from the perspective of stationary phase processes. In this case d, represents the stationary phase thickness, 6 represents a geometrical factor analogous to a, and kD represents the diffusion rate in the stationary phase.
(39) 6 k‘(ep
lo
‘=
EZ)
tp
d,2 D, kd -d,2 D‘ kD
(51)
Evaluation of the C Term. The first term on the righthand side of eq 37 is the mass transfer, or C term. It is possible to use it to explore certain limiting cases. 1. Large q. For lare q equilibrium is reached in each particle of stationary phase and the C term reduces to
and from eq 22 and 24
then we have
dn2
1
To compute ( tP) we will resort to the Einstein equation once again, this time in three dimenions, with the distance traveled equal to the pore diameter, d.
(43) Here a is a geometrical factor which is a function of pore geometry and D,, is the stagnant mobile phase diffusion coefficient. Thus we have
in general agreement with Knox (18),and Horvath and Lin (19) and Chen and Weber (IO) except the more general D,, has been used instead of DSm. 2. Small q. For 2
