448
MICHAELJ. EITEL
Concentration Fluctuations in Dilute Polymer Solutions. I.
A Theoretical Treatment of Flow-Time Fluctuations in a Capillary Viscometer by Michael J. Eitel Department of Chemistry, Northeastern University, Boston, Masaachuaetts
0.9116
(Received March 17, 1967)
The present theoretical treatment of flow-time fluctuations in a capillary viscometer assumes Newtonian flow behavior for dilute polymer solutions. Temporary consideration was given to equivalent experiments, which display volume fluctuations instead of flow-time fluctuations. Gradual flow processes were simulated by a model, which is concerned with the stepwise movement of many precisely sized and positioned “lamellae,” which are annular subvolumes of solutions that reside temporarily within the capillary. On the basis of statistical principles, the solute molecular weight and the lamellar size were correlated with the expected fluctuations in solute concentration, and the familiar Huggins equation was applied to calculate the variance of the local viscosity in each subvolume. Iterative computer calculations were used to summarize the cumulative fluctuation effects from a multitude of lamellae, and the computed fluctuations in the extruded volume were correlated with the experimentally observable fluctuations in flow time. Sample calculations for a typical solute-solvent system predict that the contribution of solute-caused fluctuations in viscometric flow time are readily detectable in a suitably chosen viscometer. However, the magnitude of these fluctuation effects was found to be highly sensitive to the geometry of the viscometer under consideration. Introduction When dilute polymer solutions are subjected to moderately rapid flow, the flow-time data from a capillary viscometer may exhibit pronounced fluctuations. While flaws in experimental technique or dust contamination may often account for such imperfections in reproducibility, it is conceivable that under suitable conditions the solute itself may contribute to the fluctuation phenomena. Considerations of the state of disorder (in solution) demand that the solute distributes itself by forming minute clusters, and the combined behavior of the resulting regions of nonaverage local viscosity may affect the reproducibility of macroscopic flow-time observations. In this investigation, the predictable concentration fluctuations will be correlated with fluctuations in viscometric flow rate, and the anticipated magnitude of such fluctuation phenomena will be examined. The existence of concentration fluctuations in dilute colloidal systems has been confirmed convincingly by Westgren’s observations of particle populations within a minute volume in the field of vision of an ultramicroscope.’ The concentration fluctuations were found to be predictable, and previous theoretical work (due to Smoluchowski’) had laid the foundation for correlating the fluctuations of such microscopic particle counts with solute molecular weight. While these studies demonstrated directly the existence of temporal fluctuations The Journal
of
Physical Chemistry
in concentration, it is equally meaningful to describe fluctuation phenomena in terms of spatial particle clusters (mentioned previously) , as a distribution of particles applying at every instant. In the present correlation of concentration fluctuations with viscometric measurements, a reasonable model will be devised that simulates the gradual flow processes within a capillary viscometer. On the basis of this model, the definition of precisely sized and positioned subvolumes (inside the capillary) will become appropriate, and it is recognized that exceptional local viscosity can be attained within such subvolumes, since particle populations therein are determined by chance. From elementary considerations of flow within a capillary, it follows that concentration fluctuations do not affect discharge rates with uniform efficiency at all locations within the capillary, for it is most readily visualized that any exceptional local viscosity occurring near the capillary axis is ineffective in altering discharge rates. However, the flow rate is more sensitive to local viscosity at other loci within the capillary, and the combined statistical fluctuation effects from a multitude of subvohmes will be shown to be largely cumulative. Statistical Considerations In the quantitative treatment of concentration ( 1 ) Sees. Chandrasekhar, Rev. Mod. Phys., 15,44 (1943),Chapter 111.
CONCENTRATION FLUCTUATIONS IN DILUTEPOLYMER SOLUTIONS fluctuations, frequent use will be made of the fundamental error-propagation theorem. It applies to any function F(x,y, . . .), where x, y, . . , are quantities whose mechanisms of fluctuation must be completely independent of each other. Then the variances (Le., the squares of the standard deviation values) for F , z, y, . . , are interrelated by2
CP(F) = (dF/bz)2a2(x)
+ (dF/by)%2(y) + . . .
(1)
The partial derivatives occurring here are to be taken a t their average values. This statistical law will find immediate use in the computation of concentration fluctuations within small subvolumes inside the capillary. These subvolumes (lamellae) are defined in order to conserve the simplicity of cylindrical symmetry: lamella i is a thin annular volume of thickness Ar$ and length L, equal to the length of the capillary. If rt designates the radius of the annulus, then the lamellar volume is -2nrtLAri, where the minus sign arises from the desired use of Art as a negative quantity. If one had a way of counting the average number, nt, of polymer molecules contained in a lamella-sized volume of solution, then one could calculate the solute concentration ci’ (in grams per cubic centimeter) from the relation cz’ = -niM/2artLXArz
(2)
where M is the particle molecular weight, and 3l is Avogadro’s number. Application of eq 1 yields readily a2(ci’) = (M/2nriLXAr,)2c2((nz>)
(3)
Equation 3 can be developed further by applying additional statistical relationships. If follows from the binomial distribution that the precision in selecting nt particles (to be in lamella i) from a total of N particles is governed by the relation3 a2(nd = Np(1
- PI
(4)
p is the probability of having a particle enter into the p ) is the probability for the sample volume, while (1 particle to stay away from that particular lamella. It is important to consider N as the invariant number of particles within the whole viscometer, not as the variable number of particles remaining in the viscometer reservoir a t any particular time. Since p is small compared to unity, and since N p is equal to n(,eq 4 simplifies to a statement of the so-called central-limit theorem
-
u2(nt)= nt
-Mc‘/2nrlLXArt
cometric work is established by the application of a viscosity equation due to Kugginsq4 It is desirable to express the equation in the form rl =
rlO(1
+ [rl’lc’ + IC’[q’12c’2 + * ..)
(7) where k’ is the Huggins constant (in unaltered form), while 7 and ~0 are the viscosities of the solution and solvent, respectively. [q’ ] has been introduced here instead of the intrinsic viscosity, [ q ] ,in order to render the notation compatible with c’ (in grams per cubic centimeter). Since c is customarily expressed in grams per deciliter when [r’lc’ = [ q ] c . The Huggins equation will be assumed to describe also the concentration dependence of the local viscosity for the contents of each lamellar volume. Thus q$ = qo(1
+ [q’lct’ + k’[q’l%’2 + . . .)
(9)
Although it is not to be taken for granted that the same coefficients [q’] and IC’ apply in both eq 7 and 9, this condition is introduced here as a simplifying assumption. While it would be inappropriate to apply the errorpropagation theorem to macroscopic, averaged viscosity values (occurring in eq 7)) the subvolume viscosities (described by eq 9) can be validly subjected to this treatment. Thus, one can evaluate a2(q1); the contribution uc2(qi),exclusively due to fluctuations in solute concentration, is then given as
where
[”I
= [q’]
+ 2k’[q’]2ct’ + . . .
(10)
Since this result can be combined with eq 6, it will be readily seen that the fluctuations in local viscosity are inversely proportional to the size of the subvolume. Therefore, most appropriate values of Ar must be sought, and then any further subdivision of such Ar values must be avoided, for, otherwise, eq 10 would indicate inappropriately amplified fluctuations. Fundamental Viscometric Relations The dilute solutions and solvents under consideration will be assumed to conform with Newtonian flow characteristics. Then it follows from the definition of Newtonian viscosity that
(5)
Using eq 3 and 2, one obtains now u2(cl’) =
449
(6)
Because of the occurrence of Art in the denominator, we conclude that Ari must not be allowed to become an infinitesimal quantity. The relation of the foregoing developments to vis-
Aui/Ar = -Pr/2qtL (11) where - Aui/Ar is the velocity gradient across the thick(2) For example, see G. Friedlander, J. W. Kennedy, and J. M. Miller, “Nuclear and Radiochemistry,” 2nd ed, John Wiley and Sons, Inc., New York, N. Y., 1964,p 183. (3) See ref 2, p 174. (4) M. L. Huggins, J. Am. Chem. SOC.,64, 2716 (1942). Volume 72, Number 2 February 1968
MICHAEL J. EITEL
450
t = 4V7pJ?rR4pg COS
+
where pt =
1
+ [l + (p2gR4
COS
+/16L72)]”z
(16)
Although eq 12, 13, 14, and 16 have been expressed in terms of the solution viscosity, 71, they can be rewritten easily so as to involve the solvent viscosity, 70. In particular, the analog of eq 16 can be expressed in terms of a new quantity, po to = 4V7opo/~R~pg COS
+
where PO =
Figure 1. Pictorial explanation of symbols pertaining to movement of lamellae, in relation to the extruded volume and the contents of the capillary. For clarity, the upward direction has been chosen as the direction of flow.
ness of a lamella i (see Figure 1), and P is the “effective” drop in hydrostatic pressure (to be discussed) along the length of the capillary. Using eq 11, a relation is readily obtained for evaluating the average extrusion velocity u as a function of r
u
=
P(R2 - r2)/47L
(12)
where R is the radius of the capillary. Despite its own usefulness, eq 12 is commonly encountered merely as an intermediate in the derivation of Poiseuille’s law. The latter establishes a relation between the flow-time, t, and the volume, V, for the upper reservoir of the viscometer 7 =
P?rR4t/8VL
(13)
For flow under gravity, the pressure drop P across the length of the capillary is not always strictly proportional to L, but since the liquid becomes accelerated from rest to attain the appropriate rate of discharge, the imparted kinetic energy may have to be accounted for. This can be accomplished by applying a pressure correctionb
AP = pR4P2/64q2L2
(14)
where p is the density of the fluid. The remaining effective pressure, P , can be evaluated from the drop in hydrostatic pressure (across the length of the capillary) by subtracting the quantity in eq 14. For a capillary that is inclined from the vertical direction by an angle +, one obtains, using eq 13
P
= Lpg COS
+ - (V2p/n2R4t2)
(15)
where g is the constant of gravitational acceleration. Equations 13 and 15 can be utilized to solve the quadratic equation for 1, yielding The Journal of Phyeica 1 Chenaiatry
1
+ [ l + (p2gR4cos +/l6L70~)]’/~ (17)
Here to is the average flow time observable with solvent, and p is the solvent density. Since it appears to be appropriate to neglect the minute differences in solution and solvent density, it follows readily that 2 < p t < PO for each solute-solvent system under consideration. In an effort to complete the listing of viscometric relations to be cited in subsequent developments, a viscometric quantity is to be introduced, which depends neither on solute concentration nor on the viscosity of the fluid. The extrusion distance x has these properties, and an average value of x is obtained if the average extrusion velocity, u, is maintained during the flow time, t. Thus x = ut
(18)
Application of eq 12 and 13 yields z = 2V(R2 - r2)/nR4
(19)
Actually, it is remarkable that P, t, and 9 are absent from this formula simultaneously. In an implicit way, this coincidence will be exploited by the procedure described in the next paragraph.
A Model for Describing the Flow Processes In order to cope with fluctuation conditions effectively, it is expedient, to consider, temporarily, an equivalent experimental arrangement, in which the extrusion time is held constant, while the extrusion volume is allowed to fluctuate (the converse is true during actual experimental determinations). The proposed change affects neither the fluctuations of the residence time, T , nor its average value. The latter is characteristic of r for any given lamella, i.e. T
= tL/z
(20)
By application of eq 1, it follows, from eq 19 and 20, that u ~ ( T= )
[ & ~ 4 / 2 ( ~ 2- r2)>2d(t>/Vz
(21)
(6) See, for example: D. P. Shoemaker and C. W. Garland, “Experiments in Physical Chemistry,” McGraw-Hill Book Co., Inc., New York, N. Y., 1962,pp 78-82.
CONCENTRATION FLUCTUATIONS IN DILUTEPOLYMER SOLUTIONS for constant V , while u ~ ( T )=
[7rLR4/2(R2- r2)>12t2a2(V)/V4 (22)
for constant t. Then a comparison of the two experimental methods is afforded by a“V)/V2
=
.“(t)/P
(23)
For the alternative experimental arrangement, it is expedient also to add the contributions to the extrusion volume V in a, special way. For each Ar, at the lamellar radius, r , there will be a shift in the total extrusion length, Az, such that the contribution A V to the extrusion volume is A V = ar2Ax
(24)
In Figure 1, A V would be a horizontal slice of thickness Az. It is easily seen that the whole volume of discharge, V , is accounted for by the sum of all values of AV (for small increments Ar). According to the model, each value of AV depends solely on the viscosity conditions that prevailed (at several residence times, T ) in the particular “stratum” of the solution, which migrated through one (hypothetical) stationary, lamellashaped subvolume within the capillary. Then the Az values of the surrounding strata have no direct influence on the AT‘ value under consideration; and adjacent volume contributions A V can attain widely different magnitudes (for widely fluctuating viscosity conditions) independently of each other. The mutual independence of the individual terms in the summation
V
=
XAV, k
becomes of keen importance, when the error-propagation theorem is applied, later on. Az is also to be expressed as a sum of independently fluctuating terms. For the sake of simplicity, the gradual renewal of solution inside the capillary will be simulated by a stepwise process. Each lamella-sized sample i of solution is considered to make its contribution to the extrusion increment, Az, during the characteristic residence time, T ~ . The contribution is designated by axLl,where the subscript L has been used to emphasize the length of the lamella, which generates a contribution to the magnitude of Az. After the expiration of T $ , a new lamella-sized sample replaces the old one, and the new contribution to the magnitude of Ax will be virtually independent of the preceding value for axLz. Obviously m
Az = z A z L I i-1
where m = z/L; Le., m gives the total number of lamellae that the stratum of length x is comprised of. From the properties of the model, it follows that m must be a whole number, for each meaningful set of values for r and Ar. Therefore, it is seen that Ar must
45 1
necessarily conform to precisely defined conditions, The relation between x and r demands that the Ar values for correctly defined lamellae near the axis of the capillary should be larger than for lamellae near the wall of the capillary. Considerations of a more quantitative nature will be presented below. Before proceeding further, it is worthwhile to introduce further assumptions and to examine implied ones. In Figure 1, Ax is shown as a quantity that grows in magnitude, in conformance with eq 26, as additional lamellae (with their individual contributions AzLt) emerge from the capillary. The diagram must not be misunderstood to mean that shear displacements are assumed to persist in a lamella after its exit from the capillary, for according to the model, the shear effects are confined to regions within the capillary. However, the simplifying assumption will be made that we are justified in neglecting any effects of turbulence a t either end of the capillary and any effects resulting from the migration of solute particles (from one lamella to an adjacent one) will be neglected. Furthermore, it will be considered adequate to treat the solute as randomly distributed point particles (represented by their centers of gravity) in all statistical procedures. The latter expedient eliminates the possibility of having any particles belong (partially) to more than one subvolume. After successfully expressing V and Az as sums of independently varying terms, a relation must be sought to correlate AxLt with independently fluctuating variables. Qualitatively, it is seen that AxLt depends on two factors: the viscosity prevailing within the lamella i and the time allotted for shear displacement to take place. T { is actually a property of the adjacent lamella, located a t the next incremented value of r . r z fluctuates entirely independently of the concentration c ~ ’ ,which has a controlling influence on ?$ and on Aut for the interval Ar. The result with the desired properties is AZL$= T ~ A u ~
(27)
Using eq 11, eq 27 is readily put in the form A Z L= ~ -Prr4Ar/2qtL
(28)
AxLt is positive only if the incrementation, involving Ar, is carried out toward decreased values of r . Detailed analysis of the flow processes has shown that the residence times for all lamellae within an individual stratum are interrelated. During actual flow, the ith lamella is literally connected to the (i - 1)th lamella; hence, the residence times can differ only by a small increment, i.e. T(
=
74-1
+
6T$*
(29)
where 6rs* designates a contribution caused solely by the properties of lamella i. From this recursion forVolume 78, Number 8 February 1958
MICHAEL J. EITEL
452 mula, it is apparent that the deviation of ri from the average expectation value is i
87( =
carj* j=1
(30)
where the values of 6rj* are statistically independent measures of flow-rate fluctuations. For a set of m lamellae, the variations 8rs are added up, yielding for the deviation in the residence time of the composite
otherwise, when 6t < 0 (a condition of sizable probability). Some terms are conspicuously absent from eq 32. In view of eq 28, it is obvious that d2f&hi2 is zero. Moreover, many leading terms of the Taylor series cancelled mutually, because of the readily verifiable relation m
= -C(W/brs)6rTI
fm+lWT
i=l
m
6t =
-C(m -j j-1
+ 1)6Tj*
(31)
Of course, each factor 6rj* can be either positive or negative. Statistical Treatment of Fluctuations Despite the condition that we should consider only an integral average number of lamellae (m) within each stratum, part of an additional lamella may become extruded if 6t happens to be positive (when time is left over), indicating that the sum of residence times for m lamellae was exceptionally short. Thus it is meaningful 1)th lamella, to consider the contribution of a (m when the variation of Ax is computed from the variation of the constituent AxLt values (designated below by fs, for brevity). Since each AxLi depends on the variations of both v i and ri, one must resort to a Taylor-series expansion for a description of 6(AxLi). Then the variation of Ax is obtained as a composite of such series expansions around the average value for Ax. One retains as the important terms
The cancellation phenomenon is a consequence of the stipulation that t must remain constant, for the model under consideration. Examination of eq 32 shows that two types of terms occurring therein refuse to change their sign, regardless of the fluctuation conditions that may exist. Terms of this nature are the last member of eq 32, and terms containing the partial derivative b2fi/dvi2. These terms make the only nonzero contributions when we consider the average value of AX), a quantity to be equated with zero
+
where the first cluster of dots applies to sets of terms with subscripts i from 2 to m. Higher order terms in the expansion were deemed negligible, because of the small size of the anticipated fluctuations. I n eq 32, some subscripts were retained for the mere purpose of facilitating the identification of terms, for all occurring derivatives are set to be their average values. In the last two terms of eq 32, the ratio (6t/r) was used to limit the quantities (Srm+land 6qm+1, since the latter are the variations for a whole lamella, in conformance with the definition of the other quantities 6ri and 6qt. Moreover, it will be useful to let 67m+l
= 6%
and 8rm+1 =
6Tm
The Journal of Physical Chemistry
__
Wi
UC2(7J
hi
2
m-2-+---
bfm+l
Srm6t
brm+1
7
-0
(35)
Here a contraction of all variances u C 2 ( r iwas ) possible, since all quantities 11% have equal variance in lamellae of identical volume, and the subscript c serves to emphasize that concentration fluctuations are considered to be the sole cause of these variance values. Considerations based on eq 30 and 31 reveal that every value of (6rm6t)must be nonpositive. The terms of eq 35 can be introduced into eq 32, yielding
T
Here the first set of dots has similar meaning as in eq 32, except that the very last member (containing 6qm6rm) has been combined with the subsequent term in eq 36, in order to have 67, occur only once. This expedient prepares the latter equation for further development. The right-hand side of eq 36 must now be squared and averaged (in that order) in order to yield the statistically meaningful quantity a2(Ax). Since all 6qI and all 6r
