SEPARATION BY FLOW 681
lization t e m p e r a t ~ r e . ~ ,16,1i ~ ” , ~In , Figure 7 the apparent heat of fusion is plotted against ljl. For the infinitely thick crystals, a value of 68 =t 1 cal/g is estimated for the heat of fusion. This coincides with the value of 68.6 cal/g for A H , derived from polymer diluent measurements.la Heat of fusion data for crystals grown in xylene reported by Fischer and Hinrichsen, l 6 and Mandelkern, Allou, and Gopalan? (16) E. W . Fischer and G. Hinrichsen, Pol),mer, 7, 195 (1966). (17) D. A . Blackadder and T. L. Roberts, Makromol. Chem., 126,116(1969). (18) L. Mandelkern, Rubber Chem. Techml., 32, 1392 (1959).
fit the line in Figure 7 within experimental error. We emphasize that the apparent heat of fusion of crystals prepared in either hexadecane or xylene is the same for crystals of comparable thickness. Conclusion
In summary, it has been shown that crystals prepared in either xylene or hexadecane at equal undercoolings possess the same lamellar thickness, melting temperature, thickening characteristics, and degree of crystallinity although the macroscopic structure, as revealed by light and electron microscopy, is not identical.
Separation by Flow. 11. Application to Gel Permeation Chromatography Charles M. Guttman and Edmund A. DiMarzio InJtitute f i w Muterids Rewurcfi, Nutionul Bureuu of StundurdA, WuJhington, D. C. 20234. Receiced Mus 8 , 1970
ABSTRACT: Models of a gel permeation chromatography column are proposed in which there is flow through each of the beads as well as around them. Diffusion is allowed within and outside of the beads. By makiag general arguments on particle current flow. the volunie elution is computed as a function of solute particle size for a simplified view of the column. The equation for the location of volume elution peaks thus derived shows functional dependences on the particle radius and the column geometry very much like equations derived by previous workers for models in which there was no flow in the beads. The method of Hermans to calculate the peak broadening is extended to allow for flow within the beads. Two times characterize the system; the time for a particle to diffuse into and out of the bead and the time to flush the particle out of the bead. The width of the volume elution peak no longer becomes infinite as the diffusion coefficient goes to zero (in contradistinction to the work of Hermans) since the residence time within the bead is never larger than the flush time. Explicit formulas are given for the first three moments of the volume elution; it is shown that the elution volume for a monodisperse species is Gaussian. I n all cases systems with open pores which allow flow show better separation capabilities than those which do not allow flow.
I.
Introduction
In a previous paper’ we have shown that an isolated polymer molecule flowing down a thin capillary and undergoing Brownian motion will have an average velocity greater than that of the solvent. This is because the center of the particle (assumed to be a rigid sphere) cannot get any closer to the walls of the capillary than its radius. It, therefore, samples only those solvent velocities away from the walls; since the solvent velocity is larger the further the distance from the wall, large molecules will have larger average velocities than small molecules. In gel permeation chromatography (gpc) the large solute particles elute out earlier (have smaller elution volumes) than the small particles. This is also true of the above described separation by flow phenomenon (sbf). Furthermore, Benoit and coworkers2have pointed out that the important particle variable in gpc is the hydrodynamic volume. This is also true for sbf. It is, therefore. natural to inquire into the relationship between gpc and sbf.
In this paper we consider the relationship between gpc and sbf. In order to d o this we have attacked the more general problem of separation in a media all of which is flowing. Other workers3 have treated a gpc system as a two-phase system: the mobile region between the beads and the stationary region within the beads. In the mobile region they allowed both for diffusion and flow while in the beads only diffusion was operative. O n the other hand, we view the beads as being constructed of many fine pores in which both flow and diffusion take place. Thus polymer molecules in the beads can both diffuse out to the mobile phase or they can be flushed out by the flow of carrier fluid through the beads. Two new and apparently distinct results arise from considering the effect of flowing within the beads. First, for the average volume elution, we find that in the high flow rate and/or low solute diffusion constant limit, the sbf phenomena described above is operative; however, as one goes to the low flow rates and/or the
( I ) E. A . DiMwzio and C. M. Guttrnan, Mucromo[ecules, 3, 131 (1970). ( 2 ) 2 . Grubisic, P. Rempp, a n d H. Benoit, J . Poli,m. Sci., Port E , 5 , 753 (1967).
(3) ( a ) J. J . Herinatis, ibid., Par1 A , 2, 1217 (1968); (b) H. Drterrnann, “Gel Chromatography,” Springer-Verlag, New York, N . Y . , 1968.
682 GUTTMAN, DIMARZIO
Macromolecules
high solute diffusion constant region sbf does not seem to be an important effect. Second, we find that effect of flow within the beads is to give a peak width which is more narrow than that which would be obtained if there were no flow within the beads. This effect is large for large flow rates and/or low solute diffusion constants and is minimal for s b w flow rates andlor high diffusion constants. Presently existing gpc systems normally are operated at flow rates in which the effects of flowing within the bead appear to be minimal. However, Bombaugh and Levangie? have recently used high flow rates for gpc separation. At these flow rates, the effects of flow with the beads may well make substantial contributions to the separations of polymer molecules. In section I1 we describe the various geometric models for gpc columns on which we shall make our computations. In section I11 using a stochastic treatment we evaluate the first moment of the volume elution distribution for systems containing only flowing volume (no stagnant volume). In the two limiting cases we can evaluate it is shown that volume elution cs. particle size curves are virtually the same as those obtained from the mobile volume-stagnant volume approach previously proposed by others. 31) In section IV the second and third moments of the volume elution distribution are derived using an adaptation of a method of Hermans.3a Using these moments we show the elution volume is Gaussian for a monodisperse system. These moments numerically are the same as those of Hermans when the time to diffuse into and out of the bead is short compared to the time to flush the polymer from the bead. When the diffusion time is large compared to the flush time the second and
Figure 1. The bank model of a gpc column. Thz large tubes in a given bank (on right) represent the totality of interstitital regions at that same level in the column (on left). Thesmall tubes in this bank represent the totality of fine tubes within the beads at this same level. The space between banks serves as a mixing region and is not considered to have any volume. The available volume in each bead is viewed as consisting of a labryinth of fine capillaries through which fluid can both diffuse and flow. (4) K . J. Bombaugh and
1357 (1969).
R. F. Levangie,
Anal. Chem., 41,
third moments remain finite. This is in contradistinction to the results of Hermans. The flow within the beads prevents infinite tailing. In the final section we discuss the various implications of the derived formulas. 11. Description of the Models
In this section we shall describe some models for a gpc column. A gpc column is made up of fine gel beads (ca. 50 p in diameter) packed together. The beads are porous; in our models we assume that the pores go through the entire bead. Thus we assume the carrier fluid flows around, into and tlirougli the beads. The surface of the beads divides the system into two regions; that within the beads and that outside. In this model of the column the region within the beads is viewed, for the purpose of calculation and simplicity, as made up of small open cylinders, all of radius rs, and all of length I . We assume, naively, that each cylinder is of uniform radius and that no cylinder intersects another. These cylinders are assumed to be aligned in one direction and this direction is chosen as the direction of fluid flow. The number of small tubes per unit volume is chosen so that the total volume within small tubes is the same as that available to solvent within the beads. Furthermore these tubes must be bunched together so that the distance between bunches is comparable to the size of the interstitial region between beads in the real column. Thus both flow and diffusion are allowed within the beads as well as within the interstitial region between beads. In this paper none of the tubes will be closed, Thus there is no stationary or stagnant volume in our system. The above model does not yet specify a detailed geometry for the system. We have purposely maintained generality because the method used in section IV to compute the broadening and skewness does not require a detailed specification. However, in order to calculate average elution volumes as is done in section 111 we have need of a more specific model. This latter model is a series of banks of tubes separated by mixing regions (see Figure 1). We shall refer to this model as the bank model. Each bank is made up of parallel arrays right circular cylinders of different radii; the fluid flows through the cylinders (rather than around them). For concreteness one can view each bank as a membrane riddled with holes. For clarity of presentation we assume there are tubes of only two radii r i and r,; their numbers in each bank are N Land N,y and their lengths are all 1. The banks are thus all identical and there are n of them. This series of n banks is called a column. The path of the particle through the column is as follows. A pressure head Ap forces the particle (which is suspended in carrier fluid) through the column from top to bottom. The particle emerges at time t later after having traversed n p f large tubes and np, small ones. We assume that particles in the mixing region lose memory of the tubes they came out of. Accordingly the probability p i of a particle jumping into and through the tubes of the next bank (i = I , s) is independent of which tube it emerged from. This assumption will be more valid for larger diffusion coefficients
Vol. 3, N o . 5. September-October 1970
SEPARATION B Y FLOW 683
and less valid for smaller diffusion coefficients. Its validity is also a function of the actual geometry of the system. For example, if all the large tubes were bunched together at one end of the bank (membrane) and the small tubes at the other end, one would expect that by piling the banks in register one could have large tubes in one bank vertically above large tubes in the adjacent banks. Consequently, particles coming from large tubes would tend to go into large tubes and so on. One obviously minimizes this effect by mixing the tubes within each bank. The above bank model can be viewed in its own right as an object of study, or as a model for gpc. As a model for gpc, the large tubes represent the flow region outside of and around the beads; the small tubes represent pores in the beads. Any particle emerging from a small tube has the option of choosing to go into a small tube again (into a bead) or into a large tube (flow around a bead). Thus the totality of small tubes in one bank represents all the pores in all the beads at one level in the gpc column, and the totality of big tubes in this bank represents all the regions between beads at this same level. The length of the tubes is proportional to the diameter of the beads. 111. Evaluation of the Probabilities of Tube Choice. Calculation of Volume Elution Using Bank Model In this section we present a derivation of the average elution volume of a particle from a column as a function of the particle size and column geometry. The column is, as described before, viewed as a collection of banks and the path of each particle is viewed as a result of the stochastic process of choosing tubes within banks. Although the average elution volume is obtainable from this point of view with relative ease, broadening and skewness of the elution volume peak are much more difficult to calculate. For this reason in section IV we turn to an alternate procedure to obtain the broadening and skewness. A. Relation of Probabilities of Tube Choice, pi, to Steady-State Currents. Let us consider a steadystate situation in which we are introducing continuously at a rate, S,, particles of size N at the top of the column and eluting them at the same rate from the bottom of the column. In every bank the particle current in tubes of radius rz is Sai and the total current flowing through all the tubes of a bank is Sa = CNiSai
Note that since each bank is equivalent to any other we do not need to index the bank. It is required of course that the mixing region above the first bank is given a distribution of particles which is identical with that of the subsequent mixing regions. Now consider the net number of particles, d m ? , going through a tube of type i in a time, dt. It is din, = S a , & (3.2) Obviously of all the particles trucersing a particular bank, the fraction f , of particles going thiough type i capillaries is (3.3)
Since each of the n banks are equivalent, the total number of type i tubes that a particle travels through is
n,
=
fin
(3.4)
What we have shown t o be true for particles on the average in the steady state is statistically true for one particle. This is because in dilute solution each particle acts independently of the others. Thus the probability, p , , that a particle will travel through an i type tube in a given bank is
The usefulness of eq 3.5 derives from the fact that we have reduced the problem of evaluating a probability, p l , to that of evaluating a steady-state particle current, S a l . We have not been able t o evaluate Sai for the general case. However, there are two limiting cases which bracket the general case and we now proceed to their computation. B. The Case When p z is Proportional to Fluid Flow (Low Diffusion-High Velocity). Let us say we tag certain fluid particles radioactively. Then the flow properties of the tagged particles are identical with those of the untagged particles. We suppose a slug of solution of such tagged particles is put above the bank homogeneously; we ask, then, what fraction of the tagged molecules go through the large tubes and what fraction go through the small tubes. Clearly since these particles behave no differently than the fluid they are in, the fraction going through the large tubes is just the volume fraction of fluid going through the large tubes. But for the fluid we know that the volume fraction going through a given tube is just the fraction of flow of the fluid that goes through that tube. Thus, if q l f is the fluid flow through one tube of radius i, the normalized probability for the fluid, and therefore for radioactive particles to go through tubes of type i, is (3.6) Qi, of course, is the total fluid flow through the bank. Equation 3.6 is most useful because it is a rigorous result valid for all possible tube geometries, fluid velocities, and diffusion coefficients. However, the particle size must be the same as the fluid particle size, We might expect these probabilities to hold for particles larger than the fluid but still very small compared to the tube radius since these particles, in the main, act like fluid particles. Even for arbitrary size solute particles the above probabilities are expected to hold in the case of small diffusion coefficient D, of the solute or high fluid velocities. Consider the particles to be spread uniformly in the mixing region. If there is no diffusion (or little diffusion) during the time the particle is in the mixing region, then those particles which are in a volume of fluid will, by and large, be carried with that volume of fluid into the tube the fluid goes into. Thus the particles go into the tube with the same probability as the surrounding fluid, and we have p 1 as above. The only particles for which this is no: true are those which
684 GLTTMAW, DIMARZIO
Mucromolecules
are too large to fit into the small tubes. Since the large particles elute out first it is unreasonable to expect that they plug up the small tubes. Thus we may presume that such particles accumulate above the tube until diffusion takes over and drives the particles to the larger tubes. If this is done, the same probabilities hold except we need a cutoff for particles too large for small tubes. Thus we have for particles of arbitrary radius in the limits of small diffusion coefficient or high velocity
Equation 3.7 is restricted to two tube sizes, s (for small) and I (for large). p 1 can be calculated for other distributions of tube size but we will not need these results for the bank model because the essential physical content of the model in its application to gpc requires only two tube sizes. C. Estimation of p i in the Equilibrium Limit (High Diffusion-Low Velocity). Let us suppose that the fluid flow velocity is sufficiently small that the equilibrium distribution of particles is obtained. At equilibrium the concentration of particles is constant throughout the column. As shown in Appendix A this restriction is unnecessarily strong and can be relaxed. If we call this concentration C, then we obtain immediately for the current in a tube of radius i
In eq 3.8 T(r, - a)' is the effective area and (r,,)?is the average particle velocity in tube i (see eq 3.14). If the solute particles are impenetrable spheres, a is their radius; but if the particle is a polymer molecule, a is an effective radius. This point will be discussed in more detail in section V. In the above we defined concentration of particles per unit effective area, that is per K(r, - 0 ) ' . In order to make contact with previous work we will from here on use concentration defined as number per total crosssectional area, that is, per mi2. Thus we have
C, =
ri2 C, = k l C , (rl - a)*
volume, V,., for a particle from a column is simply the product of the average time the particle spends in the column, ( t ) ,and the carrier fluid flow, Q f (3.1 1 ) The carrier fluid flow is obtainable directly and we need only compute ( t ) . The average time a particle spends in the column is
(0= n,(t), + nr(t)r
(3.12)
where n , is the number of small tubes a particle traverses going through the column, n l is the number of large tubes a particle traverses going through the column, ( t i s is the time for a particle to flow through a small tube, and ( t ) ! is the time for a particle to go through a large tube. Now the time to traverse a tube of length lwith a velocity (r,,)?is simply
We have shown in previous papers',: that
where u,j2 is the average fluid velocity in the tube, a is the particle radius, ri is the tube radius, and y is a number near 0.1 which depends on the mass distribution in the polymer. Now n , is simply the product of n, the number of banks in the column, and the p f ' s derived previously. Thus we have for the volume elution of a particle of radius u
We notice that Q i , the fluid flow through a bank (and thus through a column), is simply Qi
=
N191'
+ N4vr
(3.16)
where r q/
7ru1r12 2
= --
(3.9)
for the equilibrium limit where i is s or 1. Equation 3.9 was derived for rigid psrticles of radius a. More generally, k l is the partition coefficient and is determined from statistical mechanics as a ratio of partition functions. Thus the above equation serves to define an effective a in these cases. From eq 3.5 we obtain immediately
that is, the fluid flow through a large tube and a smnll tube, respectively. Using the equilibrium limit of the pi's calculated in part C of this section we have for volume elution
where V , is the total flowing volume in the column, ;.e., V,. = anl[Nlr12 N , r s 2 ] ;@ i sthe fraction of flowing volume in large tubes; and
+
One observes that eq 3.10 and 3.7 are the same when n is small.
One can presume that the exact p i are bracketed by the two limiting forms given in eq 3.10 and 3.7. Fortunately the elution curves obtained from them are similar. D. Calculations of Volume Elution. The elution
( 5 ) E A. DiMarzio and C. M. Guttman, Polrrn. L e t t . , 7 , 267 (1967).
Vol. 3, N o . 5 , Seprember-October 1970
SEPARATION BY
FLOW 685
a parameter which is 1 when most of the fluid flow is in the large tubes. For the flow limit we have for V ,
+ + 0.5(1 - +) X for ci
< r,, and =
V&J
for a > r s . In both of the above equations we have assumed that the particle radius, a, is much less than the radius of the large tubes, that is air1
K,, which is consistent with the results of Boni, et a / . , but not with those of Yau, et a / . In this connection it should be observed that we are discussing the average elution volume (first moment of the elution volume distribution) and not the location of the peak in the elution volume. Thus in general one must allow for some (linear) combination of K, and Kc which reduces to Ki for high flow rates and/or low diffusion, and reduces to K, for high diffusion and/or slow flow. The dimensionless parameter controlling the admixture isloobviously (u,//2D,5). By eliminating a/r, from 5.2 and 5.3 (y = O), one obtains a relation between K, and K i
Kr
=
K,.
+ ( K2 p--K,
(5.7)
However, our formulas are valid only for small a/rs; we are unable to assess the effect of u/r, 'v 1. It is clear, however, that Kf in this range needs serious modification. (2) Peak Spreading and Skewness. The reader should notice that the effect of sbf in its modification of the solute velocity in the small tubes has a small effect on the broadening. This is seen by changing (ct,)$ to u, in eq 4.18. This change can modify the second moment by no more than a factor of 2. This is not to say that there is little effect of flow within the small bead. In fact, the expression we obtain for the second moment (4.18) is substantially different from that of Hermans (who considers only diffusion into the bead) because of the presence of the Langevin function. In the limit of small (cL,),, eq 4.18 goes over to the Hermans result which is applicable to blocked tubes
But for small D, we do not get an indefinitely large moment as Hermans does; rather, because of the bounded character (E(=)= 1) of the Langevin function, our second moment remains finite (as does also the third). The physical interpretation of this is as follows. There are two characteristic times involved; one is the time for diffusion into and out of the bead, I2/2Drand the other the time for the fluid to flush the bead, //(cl,)
