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Reconsidering the Analysis of Refractive-Index-Matched Polymer/ Polymer/Solvent Tracer Diffusion Experiments Brian F. Hanley*
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Department of Chemical Engineering, Louisiana State University, Baton Rouge, Louisiana 70803, United States ABSTRACT: A method for analyzing quasielastic light-scattering (QELS) data from compatible polymer−polymer−solvent systems, in which the refractive index increment for one polymer species relative to the solvent is close to zero while the other is present in dilute quantities, is explained. Specifically, the technique makes use of the fact that for such systems the scattered intensity at wave vector q⃗ can be equally well expressed in terms of the tracer polymer’s molecular-weight distribution or in terms of a distribution of decay rates obtained from the inverse Laplace transform of the autocorrelation function. Appropriate integrations over these two distributions result in one-to-one mappings of individual tracer molecular weights in the tracer’s molecular-weight distribution to individual decay rates (and thus, tracer diffusivities) in the decay rate/tracer diffusivity distribution. Thus polydispersity in the molecular-weight distribution of the tracer polymer becomes advantageous. Employing this technique, a single QELS tracer experiment can yield extensive data on the (Dtr, Mtr) relationship for a given background polymer concentration, cb, and molecular weight, Mb. We demonstrate the applicability of this method by extracting tracer diffusion coefficient/molecular weight curves from some autocorrelation functions for the systems PS/PVME/o-fluorotoluene and PSAN/PMMA/toluene reported in the literature.
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INTRODUCTION Polymer diffusion in dilute, semidilute, and concentrated solutions has been the subject of significant research over the past 70 years. In particular, microscopic models involving geometrically constrained chain motion above the overlap concentration, c*, coupled to techniques borrowed from the general theory of continuous phase transitions that identify c* as a type of critical point have enjoyed considerable success explaining the macroscopic dynamical behavior of these systems. The tube model of Edwards, the reptation model of de Gennes, and the power-law scaling conjectures of Hervet et al. are the starting points for almost all subsequent theories of polymer motion in semidilute and concentrated solution.1−3 There remains, however, a good deal of uncertainty about the specific details of polymer self-diffusion behavior. Most theories for the self-diffusion of linear polymers in solution or in the melt start with a study of monodisperse “test” chains of a given length moving in a matrix of monodisperse background chains (which might have a different chain length than that of the “test” chain) of varying concentration. The test chain and the background polymer are usually assumed to have the same monomeric composition. Even in this relatively simple case, several diffusive mechanisms have been proposed that can contribute to the overall motion of the test chain depending on the ratio of the length of the test chain to the length of the chains making up the matrix (or background) along with the concentration of the matrix polymer.4−6 Chain-length polydispersity in either the tracer or the matrix (or both) complicates the theory even further because the relative contributions of the available diffusion mechanisms can vary along the tracer’s chain-length distribution, even for a monodisperse matrix. Theoretical and © XXXX American Chemical Society
experimental cases can, of course, be designed to some limited degree so that one mechanism dominates. For example, reptation as a diffusive mechanism has been studied both theoretically and experimentally by immobilizing the background matrix, either by cross-linking the matrix polymer or by using a matrix polymer with a much higher molecular weight than the test chain. In its simplest incarnation, dynamic light scattering from refractive-index-matched ternary polymer/polymer/solvent systems has proven to be a relatively straightforward method for studying the diffusion of a dilute tracer species in a binary solution composed of a different background polymer and a small-molecule solvent chosen so that the refractive index increment of this pair, ∂n/∂cb, is close to zero. The method gives the experimenter control of the tracer polymer’s average molecular weight, molecular-weight distribution, and topology; in addition, the background polymer’s average molecular weight, molecular-weight distribution, topology, and concentration can also be controlled.7−9 In theory, the experimenter even has limited control over thermodynamic interactions among the polymers and the solvent as long as a window of thermodynamic compatibility exists and as long as ∂n/∂cb remains near zero for the polymer designated as background.10 More elaborate experiments are possible; Phillies reviewed much of the available tracer diffusion data as of 2011.11 Received: July 2, 2018 Revised: October 29, 2018
A
DOI: 10.1021/acs.macromol.8b01408 Macromolecules XXXX, XXX, XXX−XXX
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THEORY The autocorrelation function measured during a tracer diffusion experiment is given by12
molecular weight and of solvent quality has been extensively studied for both binary polymer/solvent systems as well as for the optically matched ternary system PS/PVME/o-ft that is investigated here.21,22 We will assume, based on the aforementioned measurements and predictions, that the polystyrene chains assume their random flight dimensions at the background polymer concentrations of interest in this paper.13,21 We will further assume that the molecular-weight dependence of the random flight dimensions of the high-polystyrene-content PSAN copolymer can be approximated by the dimensions of polystyrene homopolymer of the same weight-average molecular weight. These assumptions should not introduce any appreciable error. S[q , R G(M tr)] =
2 2 {e[−qR G(M tr)] − 1 4 [qR G(M tr)]
+ [qR G(M tr)]2 }
The equations above relate the electric-field autocorrelation function measured in the tracer diffusion experiment to the tracer’s weight fraction distribution. This distribution can be determined by a number of techniques, with gel permeation chromatography (and appropriate absolute molecular weight detectors) being the most common.23 For several of the experiments to be analyzed below, polystyrene standards (M̅ w/ M̅ n < 1.1 as determined by the manufacturer) were used as the tracer species. Except for the low- and high-molecular-weight tails of these standards, the weight fraction distribution of polymer can be well-represented either by a Schultz distribution (eq 6) or by a log-normal distribution (eq 7).24
Figure 1. Comparison of an experimentally measured molecular-weight distribution for a polystyrene standard (part: PSS-dps800k, lot: ps4087di, M̅ w = 800 K, M̅ n = 764 K) from PSS Polymer Standards Service with a Schultz distribution fit.
C(t ) = B(1 + β[g1(t )]2 )
(1)
with g1(t ) = A
∫0
∞
2
M′tr w(M′tr )S(q , M′tr )e−Dtr(M′tr )q t dM′tr (2)
A=
{
∫0
w(M tr) =
−1
∞
M′tr w(M′tr )S(q , M′tr ) dM′tr
}
y z + 1M trz e−yM tr Γ(z + 1)
(3)
S[q,RG(Mtr)] is the form factor function for the tracer. The form factor function is dependent on the spatial configuration of the scatterer;13,14 however, for dimensionless distances, qRG, less than ∼0.75, S[q,R] can be approximated well by the following expression for scatterers of any configuration15,16 ij (qR G)2 yzz zz S[q , R G] ≅ jjj1 + j 3 z{ k
(5)
w(M tr) =
e−[
(6)
(ln(M tr) − ln(M0))2 ] 2σ 2
(2π)1/2 σM 0e σ
2
/2
(7)
For the Schultz distribution M̅ n = z/y, M̅ w = (z + 1)/y, and thus M̅ w /M̅ n = (z + 1)/z; for the log-normal distribution 2 2 2 M̅ n = M 0e σ /2 , M̅ w = M 0e3σ /2 , and M̅ w /M̅ n = e σ . A GPC chromatogram for a polystyrene standard (part: PSSdps800k, lot: ps4087di, M̅ w = 820 K by light scattering, M̅ n = 764 K) from PSS Polymer Standards Service is shown in Figure 1 along with a fit of the chromatogram to a Schultz distribution.25 The electric-field autocorrelation function for the tracer, g1(t), can also be written quite generally in the form of a Laplace transform of a distribution of tracer diffusivities
−1
(4)
As long as experiments are performed at sufficiently small qRG, then measurements of the radius of gyration of the tracer chain as a function of the molecular weight for the tracer (at fixed background polymer concentration) by standard total lightscattering techniques suffice for the calculation of S[q,RG] if RG for the tracer chain cannot be approximated from models or measurements already in the literature. Models exist for the behavior of RG(c) for binary solutions17,18 and for RG(ctr,cB) for ternary solutions of two polymers and a small-molecule solvent that have been confirmed by experiment.19,20 These studies demonstrate that linear chains in good solvents contract as the overall polymer concentration is increased. For concentrations in excess of the chain overlap concentration, the dimensions of linear tracer chains approach those achieved in a θ solvent. In this paper we will discuss and analyze autocorrelation functions obtained from linear tracer chains of polystyrene and poly(styrene-co-acrylonitrile) (PSAN) copolymer with high styrene content in semidilute/concentrated solutions of background polymer. The radius of gyration for polystyrene as a function of
g1(t ) =
∫0
∫0
∞
2
G(D′tr )e−D ′tr q t dD′tr
(8)
∞
G(D′tr ) dD′tr = 1
(9)
or as a Laplace transform of a distribution of apparent Stokes− Einstein radii26 g1(t ) =
∫0 B
∫0
∞
P(R′)e
−(
kBT )q2t 6πηR′
dR ′
(10)
∞
P(R′) dR′ = 1
(11) DOI: 10.1021/acs.macromol.8b01408 Macromolecules XXXX, XXX, XXX−XXX
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Figure 2. (a) Typical scattered intensity distribution function versus molecular weight. The shaded area of the distribution would be included in the analysis. (b) Cumulative scattered intensity distribution generated from the integral of the distribution in panel a. (c) Typical apparent Stokes− Einstein radius distribution function. The shaded area of the distribution would be included in the analysis. (d) Cumulative distribution function of apparent Stokes−Einstein radii generated from the integral of the distribution in panel c.
diffusivities between Dtr and Dtr + dDtr, and P(R) dR is the fraction of the scattered intensity at wave vector q⃗ contributing to g1(t) by tracer molecules with apparent Stokes−Einstein radii between R and R + dR. After some manipulation, the relationships above can be rewritten in integral form as
These representations are essentially equivalent because by definition the apparent Stokes−Einstein radius for a given tracer diffusivity, even in semidilute and concentrated solutions, is given by Dtr =
kBT 6πηR
(12)
A
The choice of viscosity, η, in eq 12 is immaterial because it ultimately cancels out in the transformation of R values to Dtr values. In the analysis to be presented below, we have used the solvent viscosity for that reason.27 The distributions G(Dtr) and P(R) are obtained from numerical inversion of the Laplace transform represented by the autocorrelation function, g1(t). A number of algorithms have been developed for this purpose.28−31 The equivalence of eqs 2, 8, and 10 can be used to elucidate the following relationships AM trw(M tr)S(q , M tr) dM tr = G(Dtr ) dDtr
(13)
AM trw(M tr)S(q , M tr) dM tr = P(R ) dR
(14)
∫0
M tr
∞
M′tr w(M′tr )S(q , M′tr ) dM′tr =
∫D
G(D′tr ) dD′tr
tr
(15)
A
∫0
M tr
M′tr w(M′tr )S(q , M′tr ) dM′tr =
∫0
R
P(R′) dR′ (16)
Equation 15 relates a particular tracer molecular weight, Mtr, to a particular tracer diffusivity, Dtr; eq 16 relates a particular tracer molecular weight, Mtr, to a particular tracer with Stokes− Einstein radius, R. Note the limits of integration and their relative positions. The equations above give the experimenter a prescription for relating specific tracer molecular weights in the tracer’s molecular-weight distribution to specific tracer diffusion coefficients or to specific tracer Stokes−Einstein radii, simultaneously in the same sample: a) Determine the molecular-weight distribution of the tracer polymer independently. b) Measure the autocorrelation function for samples prepared with this tracer and a background polymer
AMtrw(Mtr)S(q,Mtr) dMtr represents the fraction of the scattered intensity at wave vector q⃗ contributing to g1(t) by tracer molecular weights between Mtr and Mtr + dMtr. Similarly, G(Dtr) dDtr can be thought of as the fraction of the scattered intensity at wave vector q⃗ contributing to g1(t) by tracer C
DOI: 10.1021/acs.macromol.8b01408 Macromolecules XXXX, XXX, XXX−XXX
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Figure 3. (a) Tracer autocorrelation function g1(t) generated from eq 21. (b) Comparison of the input apparent Stokes−Einstein radius distribution (eq 23) with numerically evaluated distribution functions obtained from the inverse Laplace transform of the data generated from eq 21. (c) Dtr versus Mtr determined from the area-matching technique described in the text. Note that the individual cases are almost indistinguishable. (d) Instantaneous Dtr versus Mtr power-law slope determined from the area-matching technique described in the text and from eq 16.
background polymer average molecular weight, M̅ b), via eq 17 or 1832
with an average molecular weight, M̅ b, and concentration, cb. c) Use one of the available numerical Laplace transform inversion routines to estimate a distribution of tracer diffusivities or a distribution of Stokes−Einstein radii. A preliminary approximation for either one of these distributions can be calculated ahead of time from the tracer’s molecular-weight distribution and some expected relationship between Mtr and Dtr. This approximation is useful for screening the results of any numerical Laplace transform inversion given the ill-posed nature of the inversion problem.
d[ln(Dtr )] d[ln(M tr)] d[ln(Dtr )] d[ln(M tr)]
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= cb , M̅ b
AM tr2w(M tr)S[q , R G(M tr)] Dtr G(Dtr )
=− cb , M̅ b
AM tr2w(M tr)S[q , R G(M tr)] RP(R )
(17)
(18)
SAMPLE CALCULATIONS To test that the algorithm described above functions as described, we have subjected two simulated tracer autocorrelation functions created with assumed tracer molecular-weight distributions and Dtr/Mtr relationships to Laplace transform inversion using the algorithms described and implemented by Hansen.30,33 Hansen’s routines are geared toward determining the distribution of Stokes−Einstein radii rather than decay rate (or diffusion coefficient) distributions, so the analyses below were carried out using eqs 14 and 16. In both examples, Schultz molecular-weight distributions were employed to construct correlation functions, g1(t), by numerical integration of eq 2 down to the level g1(tmax) ≈ 0.015. The laser wavelength was taken to be 488 nm and the scattering angle 90°.34 Then, these simulated autocorrelation functions were subjected to numerical
d) Relate specific tracer molecular weights to specific diffusion coefficients or Stokes−Einstein radii by repeated numerical integrations over the appropriate distribution functions. e) Use different concentrations of tracer polymer, ctr, and extrapolate tracer diffusion coefficient/tracer molecular weight results to ctr = 0 if necessary. Once the analysis above has been completed and associations have been made between particular tracer molecular weights and particular tracer diffusivities or radii, it is possible to calculate the instantaneous power-law slope for these particular values of Dtr and Mtr (for a fixed background polymer concentration, cb, and D
DOI: 10.1021/acs.macromol.8b01408 Macromolecules XXXX, XXX, XXX−XXX
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Macromolecules AM trw(M tr)S(q , R G) = AM trM tr2 e−yM tr
(19)
Let us also assume that the tracer molecules diffuse via reptation Dtr =
1 M tr2
(20)
The electric-field autocorrelation function, g1(t), for this case can be calculated from g1(t ) = A
∫0
∞
2
−2
M tr3 e−yM tr − q M tr t dM tr
(21) −2
−4
Let us take T = 300 K, q = 2.7312 × 10 m , and η = 5 × 10 Pa· s (Dtr has units of m2/s; R has units of meters); then, A = 2.6667 × 10−24. The autocorrelation function calculated from eq 21 with the parameters specified above is shown in Figure 3a. From the assumed Mtr−2 molecular-weight dependence of the tracer diffusivity, the apparent Stokes−Einstein relationship (eq 12) yields 7
R = 4.39474 × 10−19M tr2
(22)
And we find the distribution of superficial Stokes−Einstein radii given by P(R ) = 6.9035 × 1012Re−3016.92
R
(23)
Figure 3b is a comparison of the input apparent Stokes− Einstein radius distribution function computed from eq 23 with two approximate distributions based on the inverse Laplace transforms of the autocorrelation function data of Figure 3a, as returned from the numerical Laplace inversion code of Hansen.33 To construct the first approximation, we only used correlation function data where g1(tmax) ≥ 0.12. The value of 0.12 was chosen as a potential lower limit on g1(t) based on the expectation that most light-scattering experiments are performed in homodyne mode. Because g2(t) = g1(t)2, this sets the lower limit on the actually measured g2(t) at ∼0.015. The second inversion used correlation function data all the way down to g1(tmax) ≥ 0.01, which is not achievable in a typical homodyne experiment but could possibly be reached in a heterodyne experiment. The two approximate distributions differ only slightly from each other. The integral method for computing (Mtr, R) pairs was then applied to these two numerical approximations (see eq 16). Once the (Mtr, R) pairings had been made, values for the tracer diffusion coefficient, Dtr, were calculated from the apparent Stokes−Einstein radii using eq 12. Figure 3c is a plot of (Mtr, Dtr) points for the two approximate distributions shown in Figure 3b together with the Dtr = Mtr−2 assumed at the outset of these calculations. The curves are nearly indistinguishable from one another. Finally, the instantaneous Dtr/Mtr power-law exponent as a function of Mtr, calculated via eq 17, is displayed in Figure 3d. Example 2: Schultz MWD with M̅ w = 1.2 × 105; M̅ w/M̅ n = 1.5; Rouse/reptation crossover In this example, we also use a Schultz molecular-weight distribution for the tracer polymer
Figure 4. (a) Dtr versus Mtr for eq 25. (b) Instantaneous Dtr/Mtr powerlaw slope calculated from eq 25. Note that the instantaneous slope varies continuously as a function of the tracer’s molecular weight and that it assumes values typical of those observed via experiment.
Laplace transform inversion. The “[E]xtra smoothness” regularization method (see equation 10 in Hansen30) was selected for both inversions. Each P(R) distribution was returned with 150 points. The algorithm for creating (Mtr, R) pairings from the AMtrw(Mtr)S(q,Mtr) dMtr and P(R) dR distributions was the same for both simulations. First, normalized cumulative distributions were prepared from the AMtrw(Mtr)S(q,Mtr) dMtr and P(R) dR distributions by numerical integration using a trapezoidal method. A particular apparent Stokes−Einstein radius would be selected from the normalized P(R) distribution; then, the tracer molecular weight associated with this value of R would be determined via linear interpolation of the cumulative [Mtr, AMtrw(Mtr)S(q, RG)] curve so that the cumulative fractional areas for both distributions would be the same. (For simplicity, S(q, RG) was taken to be one in these calculations.) To avoid effects associated with the tails of these distributions, R values were only selected if the area fractions associated with them fell between 0.1 and 0.9 (see Figure 2). Once the pairings (Mtr, R) had been made, values of Dtr were determined from eq 12 to produce individual (Mtr, Dtr) points. Example 1: Schultz MWD with M̅ w = 1.5 × 106; M̅ w/M̅ n = 1.5; pure reptation Consider the following Schultz molecular-weight distribution for the tracer polymer
−5
AM trw(M tr) = 6.5104 × 10−20M tr3 e−2.5 × 10
M tr
(24)
M̅ w = 120 000 and M̅ n = 80 000 for this distribution function; once again, S(q, RG) has been taken to be one. Furthermore, we have assumed that the tracer diffusivity, Dtr, of a molecule with molecular weight Mtr is given by E
DOI: 10.1021/acs.macromol.8b01408 Macromolecules XXXX, XXX, XXX−XXX
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Figure 5. (a) Tracer autocorrelation function g1(t) generated from eq 21. (b) Comparison of the input apparent Stokes−Einstein radius distribution (eq 23) with numerically evaluated distribution functions obtained from the inverse Laplace transform of the data generated from eq 21. (c) Dtr versus Mtr determined from the area-matching technique described in the text. Note that the individual cases are almost indistinguishable. (d) Instantaneous Dtr versus Mtr power-law slope determined from the area-matching technique described in the text and from eq 18.
Table 1. Summary of Polymer-Related Data for Experiments PS/PVME/O-Fluorotoluenea name
M̅ w,tr
Polymer Sources tracer background
V−V V−W V−X PB
1 050 000 1 050 000 1 050 000 422 000
NBS 1479 synthesized NBS 1479 synthesized NBS 1479 synthesized Toyo Soda synthesized
name
M̅ w,tr
V−E V−F
120 000 120 000
(M̅ w/M̅ n)tr
ctr (mg/mL)
M̅ w,b
(M̅ w/M̅ n)b
cb (g/mL)
0.46 0.46 0.46 0.79
1 300 000 1 300 000 1 300 000 1 300 000
∼1.6 ∼1.6 ∼1.6 ∼1.6
0.0601 0.0793 0.0975 0.061
1.09 1.09 1.09 1.09 PSAN/PMMA/Tolueneb (M̅ w/M̅ n)tr
ωtr
M̅ w,b
(M̅ w/M̅ n)b
ωb
1.7 1.7
∼0.0003 ∼0.0003
117 000 410 000
∼2.5 ∼2.5
0.436 0.38
Monsanto Polysciences39 Monsanto Polysciences
T = 30 °C; scattering angle = 90°; γ = 514.5 nm. bT = 26.5 °C; scattering angle = 90°; γ = 488 nm.
a
log10(Dtr ) =
−2.8227 1+
7.5738 × 106 M tr1.3025
− 12.9585
P(R ) ≅ (25)
The specific constants in eq 25 come from the analysis of a limited set of Dtr/Mtr data for the system PSAN (∼76% styrene by weight)/PMMA/toluene with a PMMA weight fraction of 0.38. These data will be discussed in more detail below. A plot of Dtr versus Mtr as well as a plot of the instantaneous Dtr/Mtr power-law slope are shown in Figure 4. Obtaining the apparent Stokes−Einstein radius distribution function, P(R), from eq 25 is tedious, but straightforward
( −z − 12.9585)2.071004 R(z + 15.7812)4.071004 É ÅÄÅ 0.7677509Ñ ÑÑ ÅÅ ij −z − 12.9585 yz ÑÑ Å zz expÅÅ−4.781234jj ÑÑ ÅÅ ÑÑ k z + 15.7812 { ÅÇ ÑÖ
with
ij 3.897716 × 10−19 yz zz z = log10jjj z R k {
F
(26) DOI: 10.1021/acs.macromol.8b01408 Macromolecules XXXX, XXX, XXX−XXX
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Macromolecules As in the prior example, eq 2 was used to numerically compute the electric-field autocorrelation function g1(t), which is shown in Figure 5a. Figure 5b is a comparison of the input Stokes− Einstein radius distribution function computed from eq 26 with two approximate distributions based on the inverse Laplace transforms of the autocorrelation function data of Figure 5a using g1(t) data cutoffs of 0.12 and 0.01. The two numerically approximated distributions differ more noticeably here than in the previous example, which is expected given the relatively more complex Dtr/Mtr constitutive equation. Figure 5c is a plot of (Mtr, Dtr) points for the two approximate distributions shown in Figure 5b together with eq 25. The estimated curves are nearly indistinguishable from one another over a wide range of molecular weights. Finally, the instantaneous Dtr/Mtr power-law exponent as a function of Mtr, calculated via eq 18, is displayed in Figure 5d.
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EXPERIMENTAL RESULTS
Some tracer diffusion autocorrelation function data reported by Wheeler and also by Hanley for the system PS/PVME/o-fluorotoluene
Table 2. Summary of Inputs Supplied to the Numerical Laplace Transform Inversion Algorithm name
smoothing
tmin (s)
tmax (s)
Rmin (nm)
Rmax (nm)
V−V V−W V−X PB V−E V−F
E E E E E E
0 0 0 0 0 0
0.0278 0.0697 0.131 0.010862 1.37 0.35
100 100 1000 5 5000 500
100 000 500 000 5 × 107 50 000 5 × 106 5 × 106
will be examined here using the fractional area-matching technique discussed above. Further autocorrelation function data reported by Hanley for the system PSAN/PMMA/toluene will also be analyzed by the same method.35,36 Preliminaries. Details of the sample preparation and experimental protocols used in collecting C(t) data are reported elsewhere.35−37 Table 1 summarizes details related to the experiments to be examined here. The sample names refer to the figure numbers in Hanley’s Ph.D. thesis, except for the PS/PVME/o-fluorotoluene named PB, where the autocorrelation function appears in Lodge et al.12,36 Table 2 includes information used during the Laplace inversion process. The minimum and maximum apparent Stokes−Einstein radii were selected so that P(Rmin) and P(Rmax) were close to zero. The “E” option (for “E”xtra smoothness) was selected to minimize the appearance of potentially extraneous peaks. In the PS/PVME/o-fluorotoluene experiments, the tracers were linear polystyrene standards: NBS 1479 with a reported M̅ w = 1 050 000 and Toyo Soda F-40 with a reported M̅ w = 422 000. Both standards had a polydispersity index of
