THE SCALING OF DIFFUSION WITH MOLECULAR WEIGHT IN [PDF]

THE SCALING OF DIFFUSION WITH MOLECULAR WEIGHT IN ENTANGLED. POLYMER SOLUTIONS. A Thesis. Presented to. The Graduate Fac

0 downloads 3 Views 371KB Size

Recommend Stories


Determination of the Molecular Weight of Polyacrylamide Fractions by [PDF]
polyacrylamide, osmometry, fractionation, molecular weight distribution. INTRODUCTION .... The poly - acrylamide with low molecular weight is soluble in water .

the high molecular weight components of sputum
You have to expect things of yourself before you can do them. Michael Jordan

changes in molecular weight of dna accompanying
What we think, what we become. Buddha

The Molecular-Weight Distribution of Glycosaminoglycans
Forget safety. Live where you fear to live. Destroy your reputation. Be notorious. Rumi

Differentiation of the Low-Molecular-Weight Heparins
We must be willing to let go of the life we have planned, so as to have the life that is waiting for

Steady-State Molecular Diffusion
Your big opportunity may be right where you are now. Napoleon Hill

molecular-weight heparin thromboprophylaxis
When you talk, you are only repeating what you already know. But if you listen, you may learn something

Determination of the Molecular Weight of Low-Molecular-Weight Heparins by Using High-Pressure
And you? When will you begin that long journey into yourself? Rumi

Diffusion entropy analysis on the scaling behavior of financial markets
Learn to light a candle in the darkest moments of someone’s life. Be the light that helps others see; i

Idea Transcript


THE SCALING OF DIFFUSION WITH MOLECULAR WEIGHT IN ENTANGLED POLYMER SOLUTIONS

A Thesis Presented to The Graduate Faculty at the University of Akron

In Partial Fulfillment of the Requirements for the Degree Masters of Science

Jason Randall August, 2005

THE SCALING OF DIFFUSION WITH MOLECULAR WEIGHT IN ENTANGLED POLYMER SOLUTIONS

Jason Randall

Thesis

Approved:

Accepted:

_______________________________ Advisor Dr. Shi-Qing Wang

________________________________ Dean of the College Dr. Frank N. Kelley

_______________________________ Faculty Reader Dr. Ernst von Meerwall

________________________________ Dean of the Graduate School Dr. George R. Newkome

_______________________________ Department Chair Dr. Stephen Z. D. Cheng

________________________________ Date

ii

TABLE OF CONTENTS Page LIST OF TABLES ……...…………………………………………………………….…. v LIST OF FIGURES ……..…………………………………………………………....… vi CHAPTER I. INTRODUCTION ………………………………...………………………………... 1 II. BACKGROUND 2.1 Diffusion ……………….……………….……...………………………………..... 4 2.2 Reptation ………………………………………………………………………..... 9 2.3 Contour Length Fluctuation ………………………….…………….…………..... 11 2.4 Constraint Release ……………………………………………….………..…..… 14 2.5 Resolving Theory with the Scaling of Diffusion with Molecular Weight …….... 17 III. EXPERIMENTAL 3.1 Material Selection …...………………………………………………......…….... 20 3.2 Sample Preparation …….....………………………………………………..….... 21 3.3 Testing Methodology ……...…………...……………………………..……….... 22 IV. RESULTS AND DISCUSSION 4.1 The Scaling of Self-Diffusion with Molecular Weight ……………………..…… 24 4.2 The Scaling of Trace Diffusion with Molecular Weight 4.2.1 Samples Prepared at a Ratio of 1:5 (~16%) Short Chain to Long Chain …..... 28 iii

4.2.2 Samples Prepared at a Ratio of 1:10 (~8%) Short Chain to Long Chain …..... 30 4.2.3 Discussion of Unexpectedly Fast Trace Diffusion Values …………….…….. 32 V. SUMMARY …………………………………………………………………….… 36 REFERENCES ………………………………………………………………………… 38

iv

LIST OF TABLES Table

Page

3.1

Molecular Characteristics of PBD Samples …………...………………………. 23

3.2

Level of Entanglement for Samples at Expected Concentrations ……………... 23

4.1

Self-diffusion Measurements and Normalized Values ………………………... 25

4.2

Trace Diffusion Measurements and Normalized Values for 1:5 Short Chain/Long Chain Ratio …………………………………………………….… 29

4.3

Trace Diffusion Measurements for 1:10 Short Chain/Long Chain Ratio …..…. 31

v

LIST OF FIGURES Figure

Page

2.1

Evaluation of trace diffusion coefficient. The extrapolated trace diffusion coefficient values are shown in open symbols. (after Ref 12) …....….. 5

2.2

Sequences for both rf pulses (top) and gradient pulses (bottom) as a function of time. (after Ref. 23) ………………………...…………………...…. 6

2.3

Typical echo attenuation plot showing a single diffusion coefficient present in the sample. …………………………………………………………… 8

2.4

Typical echo attenuation plot showing two separate diffusion coefficients present in the sample. ……………………………………………… 8

2.5

Diagram of the reptation model. The polymer chain is represented by the solid line, while constraints are shown by X marks, and the envisioned tube is given by the dashed lines. …………………...……..…..….. 10

2.6

Graph of viscosity versus normalized chain length on logarithmic scale. The line with slope 3.0 does not accurately capture the data. (after Ref. 10) … 11

2.7

Illustration of contour length fluctuation. (a) Initial picture is the same as reptation. ……………………………….……. 13 (b) The polymer chain moves outside of the tube by a distance R’. ……….….. 13

2.8

Illustration of constraint release. (a) The polymer is initially confined as in the reptation model. ……...……….. 15 (b) One of the constraints is removed, allowing the polymer chain to move to a new location. ……………………………………………………………… 15 (c) The constraint moves back in, keeping the polymer in the new conformation. ………………………………………………………………….. 16

2.9

Comparison between normalized self and trace diffusion coefficients for h-PBD blend melts at 175 °C showing asymptotic scaling. (after Ref 12) …... 19

vi

4.1

The scaling of the self-diffusion coefficient Ds with molecular entanglement ratio M/Me(φ). …………………………………………………………………. 26

4.2

Typical echo attenuation plot for 44k PBD at φ = 0.51 at 70 °C. ………...…… 27

4.3

Typical echo attenuation plot for 100k PBD at φ = 0.42 at 70 °C. ……………. 27

4.4

The scaling of the trace diffusion coefficient Dtr with molecular entanglement ratio M/Me(φ). …………………………………………………………………. 29

4.5

Normalized self and trace diffusion coefficients for 1:5 short chain/long chain ratio. ………………………………………………………...…………… 30

4.6

Echo attenuation plot for 12k in 1.5M (1:5) at φ=0.35 at 70 °C. ……………… 31

4.7

Echo attenuation plot for 100k in 1.5M PBD (1:10) at φ = 0.35 at 70 °C. ….… 32

4.8

Echo attenuation plot for 1.5M PBD at φ=0.35 at 70 °C. ……………………... 35

vii

CHAPTER I INTRODUCTION

For years, the use of scaling laws to accurately predict the behavior of polymers has become an essential tool to the polymer community. Many of these scaling laws are based upon the relationship between the polymer’s molecular weight and a key physical property of interest such as viscosity, relaxation time, or diffusion constant. These scaling laws must work within the theoretical models they are based upon however, or they are useless to accurately describe the behavior they hope to capture. Of particular interest to the present work is the scaling of diffusion with molecular weight, and the physical theory used to explain that scaling. From using the basic reptation theory1-2, a projected scaling of diffusion with molecular weight is approximately -2.01, a factor confirmed by various trace diffusion measurements made in melts3-6. Trace diffusion measurements are a special case of self-diffusion measurements where a short chain polymer is present in a long chain matrix at very small concentrations. This is important to note because the self-diffusion coefficient scales with molecular weight to a larger negative exponent. Self-diffusion measurements on melts and solutions alike show a scaling of the diffusion coefficient with molecular weight to the -2.4 power4-8. To describe this theoretically, this phenomenon has been commonly attributed to the well known 1

modification of reptation referred to as contour length fluctuation9-10. Scaling in reptation with CLF of the reptation time with molecular weight is approximately 3.4 and not 3. This deviation translates to a -2.4 scaling in the diffusion coefficient. There is a problem with this line of reasoning, however. Contour length fluctuation should not be the physical reason why there is a departure of scaling of selfdiffusion with molecular weight from M-2.0 scaling, because this departure should also have been seen in trace diffusion, which is not the case. Because of this another reason was sought out, and eventually put forth by Wang11. In this study and later results12 it was suggested that constraint release was actually the physical effect governing the shift in scaling of self-diffusion. This idea was supported by the observation of asymptotic scaling of the self-diffusion coefficient in polymer melt systems, that is, that the selfdiffusion coefficient will eventually scale to -2.0 as in trace diffusion instead of -2.4 as seen elsewhere. As of yet these findings have not been confirmed for polymer solutions, which is an important final step as it would unify the experimental results and theoretical picture of what happens in diffusion. The present work was intended to answer the question of what happens to diffusion in polymer solutions, in particular, whether self-diffusion follows asymptotic scaling to -2.0 as was observed in melts, or whether trace diffusion actually shows the same -2.4 scaling as in self-diffusion as suggested by others13-15. Experiments were carried out using the pulse-gradient NMR technique, often employed in diffusion measurements. The previous work showing asymptotic scaling in melts12 also employed this method. All samples were 1,4-polybutadiene of various

2

molecular weights and at various concentrations in order to get reasonable molecular entanglement ratios (>2). The results of this work showed that trace diffusion data laid directly on top of the self-diffusion data, and both experienced the same -2.4±0.1 scaling with molecular weight. Though this agreed with previous research in the area, definitive conclusions as to the nature of diffusion in solutions had to be left for others to make, as this work was carried out near the limits of the technique and showed signs of uncertainty.

3

CHAPTER II BACKGROUND

2.1 Diffusion Before elaborating on the results of this study, it is important to develop the basic background knowledge necessary for understanding the importance of doing this work, beginning with diffusion. Diffusion is the spreading of molecules throughout a given system, in this case a single polymer chain moving through an entire polymer network. In polymeric systems, diffusion has been measured experimentally using a variety of techniques12,16-18, including infrared spectroscopy, forced Rayleigh scattering, forward recoil spectroscopy (FRES), and pulse-gradient NMR (pg-NMR) - the technique used in this work. One theoretical tool often employed in these studies is scaling law, in particular the scaling of the diffusion coefficient, D, with the relaxation time, τ, as shown in equation 2.119 below:

D~

Rg

2

τ

(2.1)

where Rg is the radius of gyration of the polymer. Here it is important to distinguish between two commonly referred to types of diffusion in the literature, the first being selfdiffusion, and the second tracer or trace diffusion. Self-diffusion is most accurately described as the Brownian motion of molecules through other molecules, that is, the random motion of molecules in the absence of any 4

gradient or other driving force. For the purposes of this research it is treated as one polymer chain of a particular molecular weight moving through other like chains of similar molecular weight. This does not necessitate that the overall system must consist of similar chains, as self-diffusion exists for each component in a multi-component system. The concept of trace diffusion is somewhat more involved, and is essentially a special case of self-diffusion. Trace diffusion is the self-diffusion of a minor component in the trace limit, commonly conceptualized in polymeric materials by envisioning a singular probe chain surrounded by a matrix of different chains. In Figure 2.1, a typical plot of self-diffusion with changing concentration is shown12. By extrapolating this graph to the φ = 100% limit, the minor component (now at 0%) is still represented by a self-diffusion value. This value then becomes known as the trace diffusion coefficient.

Figure 2.1 Evaluation of trace diffusion coefficient. The extrapolated trace diffusion coefficient values are shown in open symbols. (after ref. 12) 5

Of the various techniques used to study both self-diffusion and trace diffusion in polymers, pulse-gradient nuclear magnetic resonance is one of the most prolific8,12,20-22, and is the technique of choice in this research. The technique works by applying an rf stimulated-echo pulse sequence 90° – τ1 – 90° – (τ2 - τ1) − 90° - τ1 - echo, where τ1 and τ2 are the times separating pulses. At a fixed τ1 and τ2, the spin-echo attenuation, A, can be measured as a function of the pulsed gradient magnitude G, time ∆ separating gradient pulses, or gradient pulse time length δ23. A diagram of the pulse sequences for both gradient and rf sequences can be seen in Figure 2.223.

Figure 2.2 Sequences for both rf pulses (top) and gradient pulses (bottom) as a function of time. (after ref. 23)

The expression24 for diffusional echo attenuation created by Stejskal and Tanner is then applied: 6

A(2τ , G ) = exp − γ 2 DX A(2τ , G = 0 )

(

)

(2.2)

where g is the known gyromagnetic ratio of the nuclide at resonance and

δ  X = δ 2 G 2 ∆ −  3 

(2.3)

By plotting the natural log of A versus X, an echo attenuation plot is created, with a straight line having a slope –γ2D. The diffusion coefficient can then be extracted directly from these plots. Two such plots are shown in Figures 2.3 and 2.4. In the first plot, a single diffusion coefficient is shown, although a slight curvature to the plot exists. This departure from the straight line is due to polydispersity of the polymer measured, and can be accounted for by additional models25. In the second plot, two separate attenuation rates can be seen, with each rate representing a different component in the material tested. This resolving of two different diffusion coefficients using this technique is possible and seen in work done by von Meerwall25,26. Note that the only way to distinguish between components in wide-line pg-NMR is by the difference in their diffusion coefficients, as it is not a spectroscopic technique23. In order to study diffusion, according to equation 2.1, knowledge of the polymer’s relaxation time is necessary. In dealing with entangled polymers, as does the focus of this research, this knowledge can be acquired through the application of theoretical models, in particular the reptation model and its modifications, which are covered in the next sections.

7

Figure 2.3 Typical echo attenuation plot showing a single diffusion coefficient present in the sample.

Figure 2.4 Typical echo attenuation plot showing two separate diffusion coefficients present in the sample. 8

2.2 Reptation The theoretical picture of polymer chains has been refined a number of times through the years. The current, widely known portrayal of the movement of a linear polymer chain entangled in other polymer chains is the reptation model of de Gennes1, and Doi and Edwards2, of which a representation is shown in Figure 2.5. In this model, one polymer chain can be envisioned to exist inside a tube, created by a series of constraints produced by the entanglement with other, surrounding chains. In Figure 2.5 these constraints are represented by the X-marks, while the tube can be envisioned by following the dashed line through the X marks near the polymer chain. The chain’s movement through the original tube and all subsequent tubes it passes through is then dictated by the parameters of those tubes – their diameters a, and lengths L, as it travels along the contour of the tubes. These parameters are good for conceptualizing the problem, but are not readily described by knowledge of the polymer. Instead, the polymer chain and its movement are described by physically measurable quantities, with one of the most important being reptation time. The reptation time, τrep, is the time required for a polymer chain to diffuse out of the original tube and is given in the relation1,2:

τ rep ≈

ς b2 N 3 kTN e

∝M3

(2.4)

where ζ is the segmental friction coefficient, b is the Kuhn length, N the chain length and Ne the entanglement chain length. By substitution into equation 2.1, the following relation is achieved:

9

D∝

R2

τ



M = M −2 3 M

(2.5)

The reptation time can also be used in conjunction with the plateau modulus, Ge, for approximating the zero shear viscosity1,2,19, η, as:

η ≈ Geτ rep ∝ M 3

(2.6)

Figure 2.5 Diagram of the reptation model. The polymer chain is represented by the solid line, while constraints are shown by X marks, and the envisioned tube is given by the dashed lines.

Since the reptation time scales as the cube of molecular weight, the zero shear viscosity must also scale as molecular weight cubed. This result is not in complete agreement with actual data27, and lead to the proposal of a modification to the reptation theory discussed in the next section.

10

2.3 Contour Length Fluctuation Contour length fluctuation, or CLF, is one of the most important modifications of the original reptation theory27. The need for such modification arose when comparing data for moderately entangled polymers above the critical molecular weight to the reptation theory. The discrepancy between the actual data and that predicted by the theory is demonstrated in Figure 2.610. While the original theory predicts that the zero shear viscosity will scale with M3 in reality the scaling follows a 3.4±0.2 power law. This is noticeable by the departure of the data in Figure 2.6 from the line of slope 3 on the log-log plot.

Figure 2.6 Graph of viscosity versus normalized chain length on logarithmic scale. The line with slope 3.0 does not accurately capture the data. (after Ref. 10)

The mechanics behind contour length fluctuation are simple27. A polymer chain inside a tube can be pictured to be moving back and forth through the tube. At some

11

point in time, it may partially travel outside the tube entirely. When the chain does this it lowers the tube length to an effective tube length, Leff, represented simplistically as: Leff = L − 2R '

(2.7)

This process is demonstrated in Figure 2.7. In 2.7 (a), the polymer chain starts as in figure 2.5 as the standard picture of reptation. The polymer chain then moves along the contour and actually exits the tube in Figure 2.7 (b). This leads to the overall shortening of length as described by the theory by an amount equal to the length of tube the chain has departed. Since this can occur at both ends of the tube, a factor of two is necessary in the actual equation. This of course is an oversimplification, as R’ must be based on the parameters of the tube as well as the polymer reptating. According to Doi2 then, the effective length Leff is: X *a   Leff = L1 −  R  

(2.8)

where X is a numerical factor taken to be 1.45, a is the diameter of the tube, and R is the radius of gyration of the polymer. This then eventually leads to the equation for reptation time τrep2:

τ rep ≈

ς b2 N 3  kTN e

1 − X 

Ne N

  

2

(2.9)

In practice equation 2.9 leads to an effective scaling with molecular weight to the 3.4 power. Going back to equation 2.1, diffusion can be found proportional to M-2.4: Dself ∝

R2

τ



12

M = M − 2.4 M 3.4

(2.10)

Figure 2.7 Illustration of contour length fluctuation. (a) Initial picture is the same as reptation. (b) The polymer chain moves outside of the tube by a distance R’.

Using the same relation for zero shear viscosity as above:

η ≈ Geτ rep ∝ M 3.4

13

(2.11)

These theoretical predications for both zero shear viscosity and self-diffusion are considered to be in good agreement with actual data. Contour length fluctuation has since been widely referred to in order to describe both zero shear viscosity scaling2 as well as the scaling of the diffusion coefficient with molecular weight8, which will be more critically examined in section 2.4.

2.4 Constraint Release The first modification to the reptation model was the idea of constraint release28 (CR), proposed because pure reptation considers a tube comprised only of static obstacles. The principle behind constraint release is that the tube the polymer is reptating in is not a static entity, but rather is in a state of flux itself. Consider a system of polymer chains together, with one of those chains – a probe chain - reptating inside a tube consisting of the other chains. The tube as discussed in section 2.2 is merely putting constraints upon the probe chain, however, each of these chains is also reptating inside a tube. As such, it is possible at some point for the chains to move in such a way so as to release some of the constraints the probe chain feels. When this occurs, the probe chain is allowed additional conformation choices. If the probe chain moves into a different conformation, it is possible that another constraint may move back in, thus fixing the tube in the new conformation. In this manner the constraints fluctuate and thus the tube that the probe chain “sees”. This is the principle behind constraint release, and can be more readily visualized using Figure 2.8. In this diagram, once again constraints are pictured as Xmarks. As these marks are removed going from 2.8 (a) to 2.8 (b), the tube, represented as a dashed line, shifts. Once in this 14

Figure 2.8 Illustration of constraint release. (a) The polymer is initially confined as in the reptation model. (b) One of the constraints is removed, allowing the polymer chain to move to a new location.

15

Figure 2.8 Illustration of constraint release. (c) The constraint moves back in, keeping the polymer in the new conformation.

conformation, another constraint moves back in in Figure 2.8 (c), thus temporarily locking the polymer in its new conformation. The tube’s relaxation time can be modeled using reptation or reptation with contour length fluctuation included, with an additional correction, producing the equation19:

τ CR ≈ τ rep

 N Pr obe   Ne

  

2

(2.12)

where τrep is either equation 2.4 or 2.9. The Probe superscript here is to differentiate the length of the chain inside the tube NProbe with the lengths of the surrounding chains N used in finding τrep in this equation, since the focus here is on the reptation time of the

16

tube and not the probe chain. This approach stands in contrast to the way the previous reptation times have been presented, as they focused upon length of the probe chain solely and ignored the surroundings. Note that τCR is proportional to the square of the ratio of polymer chain length to entanglement polymer chain length. This ratio need not be large (~10-20) in order for the difference between τCR and τrep to be two orders of magnitude. With this difference in time scale, it could be argued that the tube was effectively static and not undergoing constraint release after reasonably low (again ~1020) entanglement ratios. In practice the constraint release correction is rarely used in monodisperse cases because of this very result. However, the model can be useful for examining polydisperse samples if that polydispersity is broad enough to include chains that could be considered unentangled. In the next section, it will also be argued to be responsible for scaling phenomenon in diffusion.

2.5 Resolving Theory with the Scaling of Diffusion with Molecular Weight Since contour length fluctuation is used to explain the discrepancy in the scaling law for the zero shear viscosity (3.4 vs. 3), it has also been used to explain why in many experiments, both in solution and in melts, the self-diffusion coefficient Dself scales as molecular weight to the -2.4±0.24-8. Fairly extensive research has been done on polymer solutions and self-diffusion, in particular a recent work by Tao, Lodge, and von Meerwall8 asserted that in polymer solutions the self-diffusion coefficient scaled as approximately -2.4 even to reasonably high (~100+) molecular weight entanglement ratios (M/Me). This example shows not only fairly high linearity in diffusion coefficient scaling, but shows it over a wide breadth of molecular weights explored. This is of 17

importance more so than the conclusion, which was in agreement with previous research that showed self-diffusion scaling as -2.4 with molecular weight, because there has been evidence to the contrary11. An examination of literature on polymer melts and selfdiffusion reveals findings4-7 similar to those in solutions. What would appear fairly clear cut is muddied, however, by the examination of research on trace diffusion data. There are a number of papers3-6 that point to the scaling of the trace diffusion coefficient, Dtrace, with molecular weight as -2.0. This of course directly contradicts the assertion that contour length fluctuation is responsible for the results in self-diffusion, for if CLF was the governing physics, the same result would necessarily be seen by both self-diffusion and trace diffusion. Clearly then evidence existed that contour length fluctuation could not be what was affecting the scaling of self-diffusion. Wang11 would revisit this problem recently, and suggested that CLF did not actually affect mass transport processes such as diffusion. His conclusion was that the deviation in self-diffusion was actually due to constraint release. This view was supported by evidence11-12 (Figure 2.912) that the -2.4 scaling in self-diffusion actually asymptotically approached the trace diffusion scaling of -2.0 at a critical entanglement ratio M/Me of approximately 15-20. This suggested that entanglement would at some point make the tube move so slowly relative to the movement of the chain inside that eventually constraints could not release, the CR effect would be eliminated, and self-diffusion samples would take on the ideal scaling predicted by the reptation model. Meanwhile, data taken on trace diffusion16-18 drew mostly on samples where the CR effect could be argued to have already been eliminated due to their level of entanglement, thus showing the -2.0 scaling. 18

Ds in Ref. 29

o

h-PBD at 175 C

1

10

Ds in Ref. 6 Ds in Ref. 5 Ds in Ref. 8 (1.3) Dtr in Ref. 3

DM

2

Dtr in Ref. 5

-0.5

0

10

-1

10

10

0

1

2

10

10

M/Me

Figure 2.9 Comparison between normalized self and trace diffusion coefficients for hPBD blend melts at 175 °C showing asymptotic scaling. (after Ref 12)

This problem was first elucidated using data present in the literature11 and then reconfirmed via independent measurements12. Their research was only performed on melts however, and was not confirmed for solutions. It is with this conflicting evidence and views in mind that the present work was done, in order to resolve this discrepancy in solutions. Scaling of the trace diffusion coefficient to -2.0 and self-diffusion coefficient to -2.4 initially and then -2.0 asymptotically with molecular weight would be in agreement with data on melts and hopefully achieve a more unified theory applicable to both melts and solutions.

19

CHAPTER III EXPERIMENTAL

3.1 Material Selection The polymers chosen for all experiments were linear 1,4-polybutadiene (PBD) of varying molecular weights. The entanglement molecular weight (Me) for polybutadiene is 1700 g/mol. All polymer samples were synthesized by the Goodyear Research Center with the exceptions of 18k and 22k PBD, which were purchased from Polymer Source. The 1.5M PBD will be referred to as the long chain component, or simply long chains, while all other samples (100k, 44k, 22k, 18k, and 12k) are considered to be the short chain component or short chains. Table 3.1 gives a more detailed listing of the physical properties of these polymers. Molecular weights of the short chain polymers were chosen upon their ability to be considered entangled even in solution. A factor of 2 M/Me(φ) or greater was calculated for each concentration and molecular weight combination, see table 3.2. Low polydispersity was desired for short chain samples in order to get a singular diffusion coefficient measurement as opposed to an entire spectrum of diffusion coefficients. The long chain matrix was chosen to have a high enough molecular weight so as to be significantly slower moving than any of the short chains so as not to cloud diffusion measurements, as well as being so highly entangled so as to exclude the CR effect.

20

Carbon tetrachloride (CCl4), 99%+ purity, was selected to be the final solvent for the solutions. Carbon tetrachloride is a non-protonated solvent, chosen for its transparency in proton pulse gradient NMR measurements. The CCl4 was purchased from Sigma-Aldrich.

3.2 Sample Preparation For the following, self-diffusion samples refer to short-chain samples that contain no long-chain matrix in solution or simply the long-chain matrix by itself. Trace diffusion samples denote short and long-chain blends in solution. This is done to prevent confusion, since all samples undergo self-diffusion measurements. See table 3.2 for a breakdown of the concentrations and molecular weights used. Self-diffusion samples were prepared by first dissolving short-chain polymer in excess CCl4. The mixture was mechanically agitated for at least 2-3 days until a homogeneous solution was created. The excess solvent was then evaporated slowly until the correct concentration was reached. Low molecular weight samples (44k and under) were then transferred to 7mm glass NMR tubes via pulling the solution into a transfer tube by syringe, then pushing the solution back out into the bottom of the tube. New transfer tubes were used for each new sample. Higher molecular weight samples were transferred in small solid pieces via tweezers and a loading rod which were cleaned between use. Each tube was sealed temporarily by a Teflon cap to prevent evaporation until they could be hermetically sealed. Samples were also prepared using the tracer diffusion method of preparation (see below) to determine the validity of the process.

21

Tracer diffusion samples were prepared first by blending the short-chain component with the long-chain component at an approximate ratio of 1:5 or 1:10 in excess toluene. Samples were magnetically stirred until uniformity was achieved (~4 days), after which the blends were cast as films and the excess toluene was allowed to evaporate off. These films were then divided into small pieces, and placed in the bottom of an NMR tube. The appropriate amount of CCl4 was then added to achieve the desired concentration, and the tube was then capped temporarily (about 5 minutes) until they could be sealed. Samples prepared in this manner were allowed to come to equilibrium over the course of several weeks.

3.3 Testing Methodology Pulse-gradient, spin-echo NMR measurements were carried out on a 33 MHz Spin-lock CPS-2 spectrometer with a wide gap electromagnet operating off-resonance by -3kHz with single-phase rf phase-sensitive detection. Samples were preheated and run at a temperature of 70.5 °C, with gradient coils supplying a calibrated horizontal gradient of 634 Gauss/cm. Experiments were performed at a fixed G by varying the gradient pulse length δ, though δ of 8ms was never exceeded to minimalize residual gradient effects. The rf pulse spacing between the first two 90° pulses was 15ms, while the spacing ∆ between gradient pulses was 150ms. A steady gradient of Go = 0.35 Gauss/cm parallel to G was used for convenience in data collection. At least ten values of δ were used to produce a maxmimum echo attenuation to below 3% of the original echo. To improve the signal to noise ratio, six or more passes for each delta were averaged together. The echo signal A 22

was measured as the magnitude of the Fourier transform of the beat between echo and reference, after a correction for rms baseline noise. Data analysis was conducted offline by the current version of a FORTRAN program30 written to account for two component diffusion and known polydispersity effects25 as well as residual gradient effects31. In the case of two diffusion component samples, the program reports back a fast component Dshort and a slow component Dlong. To account for polydispersity effects, the program assumes a log-normal molecular weight distribution. All resulting values in this work carry an error of 5% or more.

Table 3.1 Molecular Characteristics of PBD Samples Sample

1,2-PBD

1,4-PBD

12k 19k 22k 44k 100k 1.5M

8.6% 90% 91.7% 91.8% 92.0%

Mn

Mw

(kg/mol) 11.6 18.2 21.7 43.5 121 1310

(kg/mol) 11.8 21.0 22.6 43.9 129 1753

Mw / Mn

Source

1.03 1.15 1.05 1.01 1.07 1.34

Goodyear Polymer Source Polymer Source Goodyear Goodyear Goodyear

Table 3.2 Level of Entanglement for Samples at Expected Concentrations φ

M

Me(φ)

[M/Me(φ)]

0.35

6K

0.424

4.76K

0.514

3.8K

12K [2] 12K [2.5] 12K [3.2]

19 K [3.2] 19 K [4] 19 K [5]

23

22 K [3.6] 22 K [4.5] 22 K [5.7]

44 K [7.3] 44 K [9.2] 44 K [11.6]

100 K [17] 100 K [21] 100 K [26.5]

CHAPTER IV RESULTS AND DISCUSSION

4.1 The Scaling of Self-Diffusion with Molecular Weight Before making the more complicated trace diffusion measurements, a confirmation of previous results on polymer solutions was sought. The numerical results can be found in Table 4.1, along with the renormalized values used in later plots. Figure 4.1 shows the scaling of the self-diffusion coefficient D with entanglement molecular weight ratio M/Me(φ). Me(φ) is always calculated here with the following equation: M e (φ ) = M e * φ −1.2

4.1

where Me is 1700 g/mol for PBD. The predicted scaling of -2.4±0.1 is indeed present. With certain molecular weights and concentrations, multiple samples were run, with some variance showing up between samples. This is evident especially in the 100k samples, as the difference can be as large as a factor of 1.5. This is most likely attributed to a discrepancy between the recorded concentration, and the actual concentration, as CCl4 is a highly volatile solvent and prone to evaporating very quickly. For nearly all the samples, 100k being the exception, only a single diffusion coefficient was observed. The 100k samples showed the presence of some low molecular weight oligomer, though this more than likely made up as little as 1% of the total solution content and had very

24

Table 4.1 Self-diffusion Measurements and Normalized Values M

φ

1.2

2

Dφ M

M/Me

D

(cm*g) /mol s 2.91 2.85 2.26 2.58 2.64 2.66 2.35 2.28 1.69 2.08 2.04 1.98

2

2

(g/mol) 12000 12000 12000 12000 12000 12000 12000 12000 12000 12000 12000 12000

0.32 0.35 0.42 0.42 0.42 0.42 0.42 0.44 0.51 0.51 0.51 0.51

1.8 2.0 2.5 2.5 2.5 2.5 2.5 2.6 3.2 3.2 3.2 3.2

(cm /s) 7.94E-08 6.97E-08 4.39E-08 5.02E-08 5.14E-08 5.16E-08 4.57E-08 4.25E-08 2.61E-08 3.21E-08 3.16E-08 3.06E-08

19600 19600 19600

0.33 0.36 0.59

3.0 3.4 6.1

3.68E-08 3.03E-08 6.05E-09

3.74 3.41 1.23

21700 21700 21700

0.35 0.42 0.51

3.6 4.6 5.7

2.14E-08 1.24E-08 5.53E-09

2.86 2.08 1.17

44000 44000 44000 44000

0.35 0.42 0.51 0.56

7.3 9.2 12 13

1.78E-09 9.84E-10 5.05E-10 4.19E-10

0.98 0.68 0.44 0.40

120000 120000 120000 120000 120000 120000 120000 120000

0.35 0.39 0.42 0.42 0.45 0.52 0.54 0.56

20 23 25 25 27 32 34 35

2.60E-10 1.92E-10 1.74E-10 8.85E-11 1.10E-10 9.48E-11 5.85E-11 8.55E-11

1.06 0.89 0.89 0.46 0.61 0.62 0.40 0.61

2

little influence on the results due to being present in such small amounts. Upon first examination of the data this would not seem the case, as quite a number of points are involved in the initial faster slope. This is due however to the fact that the echo signal strength depends on spin-spin relaxation time. The spin-spin relaxation time is 25

significantly longer for low molecular weight components in blends and solutions, and thus simply appears to makeup more of the sample. An example of the echo attenuation plots for 44k can be seen in Figure 4.2, demonstrating a singular diffusion coefficient after polydispersity effects. Also shown are results for 100k in Figure 4.3, showing the trace oligomer as a fast-diffusing component (the steep slope) as well as some data scattering.

10

10

-7

-8

2

D (cm / s)

-2.4

10

-9

10

-10

10

-11

10

0

10

1

12k @ φ = 0.35 12k @ φ = 0.42 12k @ φ = 0.51 19k @ φ = 0.35 19k @ φ = 0.51 22k @ φ = 0.35 22k @ φ = 0.42 22k @ φ = 0.51 44k @ φ = 0.35 44k @ φ = 0.42 44k @ φ = 0.51 100k @ φ = 0.35 100k @ φ = 0.42 100k @ φ = 0.51

10

2

M / M (φ) e

Figure 4.1 The scaling of the self-diffusion coefficient Ds with molecular entanglement ratio M/Me(φ).

26

Figure 4.2 Typical echo attenuation plot for 44k PBD at φ = 0.51 at 70 °C.

Figure 4.3 Typical echo attenuation plot for 100k PBD at φ = 0.42 at 70 °C. 27

4.2 The Scaling of Trace Diffusion with Molecular Weight 4.2.1 Samples Prepared at a Ratio of 1:5 (~16%) Short Chain to Long Chain

Measurements were then conducted on trace diffusion samples having a short chain to long chain volume fraction ratio of approximately 1:5. Although still roughly 16% short chain then, these values were taken as being approximately the values expected from extrapolation to 100% long chain. This was done as there was experimental difficulty in observing diffusion at true trace concentrations in these samples. The numerical results for these experiments can be found in Table 4.2. A plot of the approximate trace diffusion coefficient versus the molecular entanglement ratio can be seen in Figure 4.4. The results do not follow the predicted -2.0±0.1 scaling, but rather fall within the boundary of the -2.4±0.1 scaling expected of the self-diffusion coefficient. Indeed, the suggested slope in Figure 4.4 is indicative that the systems studied actually have at least as fast diffusion as their self-diffusion counterparts, a phenomenon discussed more in section 4.2.3. It is now possible to create a plot similar to that shown in Figure 2.9, by not only normalizing the diffusion coefficients by the square of molecular weight, but also by accounting for the effect of concentration. In removing the effect of concentration a similar form to equation 4.1 is followed, but multiplying by φ1.2 rather than dividing by it. The results of such a normalization can be seen in Figure 4.5. Due to the normalization, any deviation from the predicted scaling for either self-diffusion or trace diffusion is magnified, even between batches of the same sample. Instead of getting the expected asymptotic scaling, both self and trace diffusion coefficients appear to have a scaling with

28

molecular weight beyond -2.0 for the entire range of molecular weights tested, as expected from their unnormalized plots.

Table 4.2 Trace Diffusion Measurements and Normalized Values for 1:5 Short Chain/Long Chain Ratio φ

M (g/mol) 12000 12000 44000 44000 120000

10

0.42 0.40 0.41 0.47 0.29

M/Me

D

Dφ1.2M2

2.49 2.35 8.88 10.46 15.98

(cm2/s) 4.35E-08 5.32E-08 9.73E-10 8.30E-10 3.73E-10

(cm*g)2/mol2 s 2.21 2.55 0.65 0.65 1.22

-7

12k in 1.5M @ φ = 0.42 44k in 1.5M @ φ = 0.42 44k in 1.5M @ φ = 0.51 100k in 1.5M @ φ = 0.35

10

-8

tr

2

D (cm / s)

-2.5

10

10

-9

-10

10

0

10

1

10

2

M/M

e

Figure 4.4 The scaling of the trace diffusion coefficient Dtr with molecular entanglement ratio M/Me(φ). 29

10

1

D measurements s

D measurements

~ -0.6

2

2

[(cm*g) / (mol s)]

tr

0

DM φ

2 1.2

10

10

-1

10

0

10

1

10

2

M/M

e

Figure 4.5 Normalized self and trace diffusion coefficients for 1:5 short chain/long chain ratio.

In many of the 1:5 ratio blends, an additional diffusion component faster than any of the values tabulated was recorded. Figure 4.6 is an example echo attenuation plot with the additional component shown by the initial, very steep sloped line, similar to that seen in Figure 4.3. In all likelihood this can be traced back to very small amounts of THF or toluene used in the initial blending step that was not completely removed even after the week in the vacuum oven. An order of magnitude faster than any component measured, and on the order of protonated solvents in solution, this likely had little or no effect on the results due to being present in small amounts relative to everything else but must be accounted for in the data reduction.

30

Figure 4.6 Echo attenuation plot for 44k in 1.5M (1:5) at φ=0.35 at 70 °C.

4.2.2 Samples Prepared at a Ratio of 1:10 (~8%) Short Chain to Long Chain

A few samples were prepared at a lower long chain to short chain ratio (1:10) to investigate the possibility that too much short chain was present in the previous samples, thus speeding up their diffusion coefficient values. Tabulated values for these results can be found in Table 4.3, however, no plots will be shown due to only having a relatively small amount of data.

Table 4.3 Trace Diffusion Measurements for 1:10 Short Chain/Long Chain Ratio M (g/mol) 44000 120000 120000 120000

φ 0.51 0.35 0.42 0.51

M/Me

D

7.85 13.62 16.95 21.40

(cm2/s) 1.58E-10 1.02E-10 9.77E-11 7.94E-11

31

In Figure 4.7, a typical plot for one of the 100k samples at this short-chain:longchain ratio is shown. Of note is the amount of data scatter present, for this scatter combined with seeing two diffusion components (a similar phenomenon was observed for samples at 1:5 short chain to long chain ratio) made determination of an exact diffusion coefficient almost impossible. As such, the tabulated values in Table 4.3 can only be considered approximations, and in that regard show no discernable or significant difference between the values achieved for samples at 1:5 ratio and these values.

Figure 4.7 Echo attenuation plot for 100k in 1.5M PBD (1:10) at φ = 0.35 at 70 °C.

32

4.2.3 Discussion of Unexpectedly Fast Trace Diffusion Values

As noted in the previous sections, the trace diffusion coefficient values uncovered in this work were unexpectedly fast. What follows is an examination of possible reasons for this discrepancy with the projected values as well as a look at the physical implication if these values are the true trace diffusion coefficient values. It is of great importance to note that in order to attempt to remove fast components possibly present in the 1.5M, analysis of a sample of only 1.5M PBD at volume fraction 0.30 in CCl4 created using the trace diffusion method described in section 3.2 was performed. Figure 4.8 is the resulting echo attenuation plot, which shows not only a fast component, but a slow component to it as well. The fast component is on the order of the diffusion coefficient found for 12k samples, while the slow component is on the order of the 100k samples. While the signal from the fast component could only be attributed to some trace level of oligomer in the samples, most likely in the base 1.5M stock, the slow diffusion coefficient is quite probably the very low end of the molecular weight distribution of the 1.5M. The question then becomes, particularly in the 1:10 samples since there is less short chain to acquire a signal from, what is actually being measured – the short chain or this contaminant? The actual answer is both signals are seen, and it is because of this unfortunate occurrence that all trace diffusion data has an additional level of uncertainty built into it that cannot be quantified. It could be argued that a concentration gradient was present inside the tube, and led to inaccurate measurements as the spot measured within the tube contained a low concentration of polymer relative to the expected homogeneous concentration. Visual inspection was the first method used to determine if a gradient was present, and there was 33

no outward visible sign of difference in refractive index throughout the samples. Samples were also measured again over a month after their initial measurements were taken, and the results achieved were within the error of the original measurements. There is also the matter of the limitation of the technique. Estimates23 have been made that the lower boundary for measurement of a diffusion coefficient by pg-NMR is approximately 10-11 cm2/s. This approximation is based on having the most ideal sample conditions. Some of the diffusion coefficients measured herein are less than an order of magnitude from this approximation, and have shown evidence of not exhibiting ideality. Certainly with measurements for 100k samples, the values achieved must be considered in context of their proximity to the lower limit. With this in mind, there are two possible physical explanations for what is actually occurring for these trace diffusion samples. Either trace diffusion must behave as self-diffusion for solutions, a conclusion that has been hinted at in other work16-18, or constraint release is still playing a role in these samples due to the amount of small chain and contaminant present. The former is self-explanatory, that trace diffusion in solutions actually scales as -2.4 with molecular weight. The latter relies on the short chains’ ability to diffuse through one another in between the long chains rather than being obstructed by them. Effectively, constraint release would still be occurring. It is important to mention that the previous work showing trace diffusion scaling as -2.4 was done at much lower volume fractions of short chain to long chain. Thus it could be argued that constraint release was still playing a large role in those samples. If the argument that this work showing -2.4 scaling in trace diffusion is attributable to constraint release is made, the same argument must then be applied to the previous work showing the same scaling. 34

However, to ascribe all the results to constraint release would seem unlikely given previous data on melts12 which suggests that the diffusion coefficient decreases linearly with concentration and not abruptly. If at roughly 85% long chain the solution is still truly showing -2.4 scaling, the final true trace diffusion coefficient scaling with molecular weight could not be much different and still hold linear scaling in concentration. Given the contamination found in the original 1.5M batch product used, which goes to the question of whether or not the concentrations are indeed correct and the constraint release effect is removed, as well as being near the boundary of the technique, no conclusive statement as to the behavior of diffusion in polymer solutions can be made, though this lends further credence to the possibility that trace diffusion and self-diffusion behave the same in polymer solutions.

Figure 4.8 Echo attenuation plot for 1.5M PBD at φ=0.35 at 70 °C. 35

CHAPTER V SUMMARY

In summation, diffusion experiments on linear PBD were carried out utilizing pulse-gradient NMR in order to elucidate the true dynamics behind diffusion in polymer solutions, be that the effect of contour length fluctuation or constraint release. The question behind this work first arose when examining the traditional scaling laws11 used in context with the theory used to describe the motion of the polymer chains, specifically the scaling of diffusion with molecular weight due to contour length fluctuation8 was in disagreement with trace diffusion data12. Work was first done on melts observing an asymptotic scaling behavior of self-diffusion, suggesting that constraint release was responsible for the scaling of diffusion with molecular weight12. Similar research on polymer solutions was then required to discover if the same mechanistic phenomenon governed both melts and solutions. The results of the experiments showed that trace diffusion and self-diffusion actually overlapped along the projected scaling for self-diffusion with molecular weight of approximately -2.4. There was a question about the validity of this result however, due to the presence of contamination in the stock 1.5M PBD used that had values on the order of the diffusion coefficients being measured. The possibility existed that constraint release had not been properly removed because of this, or perhaps even in spite 36

of this. Due to the uncertainty of the measurements an explicit conclusion based on the evidence was not possible, though the results were not out of line with older work done in the area16-18. Final conclusions then remain open for future work to refine and determine whether trace diffusion indeed scales like self-diffusion in polymer solutions, or whether similar scaling is seen in solutions as in melts and therefore constraint release is the governing principle behind the scaling of diffusion with molecular weight.

37

REFERENCES

1. de Gennes, P. G. Scaling Concepts in Polymer Physics; Cornell University Press: Ithaca, New York, 1979. 2. Doi, M.; Edwards, S. F. The Theory of Polymer Dynamics, 2nd ed.; Clarendon Press: Oxford, England, 1988. 3. Klein, J.; Fletcher, D.; Fetters, L. J. Nature (London) 1983, 304, 526. 4. Antonietti, M.; Coutandin, J.; Sillescu, H. Makromol. Chem. 1987, 188, 2317. 5. von Seggern, J.; Klotz, S.; Cantow, H.-J. Macromolecules 1991, 24, 3300. 6. von Meerwall, E.; Wang, S. F.; Wang, S. Q. Polym. Prepr. (Am. Chem. Soc. Div. Polym. Chem.) 2003, 44, 287. 7. Pearson, D. S.; Fetters, L. J.; Graessley, W. W.; Strate, G. V.; von Meerwall, E. Macromolecules 1994, 27, 711. 8. Tao, H.; Lodge, T. P.; von Meerwall, E. Macromolecules 2000, 33, 1747. 9. Doi, M. J. Polym. Sci. Polym. Phys. Ed. 1983, 21, 667. 10. Milner, S. T.; McLeish, T. C. B. Phys. Rev. Lett. 1998, 81, 725. 11. Wang, S. Q. J. Polym. Sci., Polym. Phys. Ed. 2003, 41, 1589. 12. Wang, S.; von Meerwall, E.; Wang, S. Q.; Halasa, A.; Hsu, W.-L.; Zhou, J. P.; Quirk, R. P. Macromolecules 2004, 37, 1641. 13. Kim, H.; Chang, T.; Yohanan, J. M.; Wang, L.; Yu, H. Macromolecules 1986, 19, 2737. 14. Nemoto, N.; Kojima, T.; Inoue, T.; Kishine, M.; Hirayama, T.; Murata, M. Macromolecules 1989, 22, 3793. 15. Remoto, N.; Kishine, M.; Inoue, T.; Osaka, K. Macrolmolecules 1990, 23, 659. 38

16. Tirrell, M. Rubber Chem. Technol. 1984, 57, 522. 17. Bachus, R.; Kimmich R. Polymer 1983, 24, 964. 18. Green, P. F.; Kramer, E. J. Macromolecules 1986, 19, 1108; Green, P. F.; Mills, P. J.; Palmstrom, C. J.; Mayer, J. W.; Kramer, E. J. Phys. Rev. Lett. 1984, 53, 2145. 19. Rubinstein, M.; Colby, R. Polymer Physics; Oxford University Press: Oxford, England, 2003. 20. Pace, R. J.; Datyner, A. J. Polym. Sci. Polym. Phys. Ed. 1979, 17, 1675. 21. Kulkarni, M. G.; Mashelkar, R. A. Polymer 1981, 22, 1665. 22. Berens, A. R.; Hopfenberg, H. B. J. Membrane Sci. 1982, 10, 283. 23. von Meerwall, E. J. Non-Crystalline Solids 1991, 131-133, 735. 24. Stejskal, E. O.; Tanner, J. E. J. Chem. Phys. 1965, 42, 288. 25. von Meerwall, E.; Palunas, P. J. Polym. Sci., Polym. Phys. Ed. 1987, 25, 1439. 26. Shim, S. E.; Parr, J. C.; von Meerwall, E. D.; Isayev, A. I. J. Phys. Chem. B 2002, 106, 12072. 27. Doi, M. J. Polym. Sci. Polym. Phys. Ed. 1983, 21, 667. 28. Klein, J., Macromolecules 1978, 11, 852. 29. Pearson, D. S.; Fetters, L.J.; Graessley, W. W.; Strate, G.V.; von Meerwall, E. Macromolecules 1994, 27, 711. 30. von Meerwall, E. D.; Ferguson, R. D. Comput. Phys. Commun. 1981, 21, 421. 31. von Meerwall, E.; Kamat, M. J. Magn. Reson. 1989, 83, 309.

39

Smile Life

When life gives you a hundred reasons to cry, show life that you have a thousand reasons to smile

Get in touch

© Copyright 2015 - 2024 PDFFOX.COM - All rights reserved.