Small Angle X-Ray Scattering

INTRODUCTION

CHAPTER 3 CHAPTER 3

3.3 Small Angle X-Ray Scattering

deduced. However, it is expected that the average value will be very close to the density of the dominant disordered phase.

of gyration can be determined from

dlnf = -R2 /

dh2 g 3. (7)

For large angles in SAXS, the so-called Porod region, and for infinitely long and narrow slits, h -+ oo and

(8) where S is the total surface area of the matter contained in the volume V. For this region, Porod's range of inhomogeneity, lm can be defined as:

(9)

Then the total surface area per unit mass becomes S = 4 X 10⁴<:(1-

t:)

limh-+oo[h²I(h)]

<7a

f

₀⁰⁰ hf ( h) dh ^m2

/g,

(10)

where ^E is volume fraction of the pores. Care must be exercized in evaluating the integral in the denominator of the above equation. The lower limit poses no problem, since at values of h close to zero, hf ( h) is very small regardless of the value of f(h). For large values of hf(h), however, errors can become significant since the intensity values, f(h), are small and may be influenced by other sources of extraneous and parasitic radiation (Compton, thermal, and atomic structure scattering). The area measurements have significance only if it is verified that the product h³ f(h) is constant over a large interval. In this case the above integral can be approximated[35] as:

f

⁰⁰ hf(h)dh = {h" hf(h)dh

+ f

⁰⁰ hk/ h3_dh,

lo lo 1

^{ho ( 11)}

where k is the limiting value of h³ f(h), and h₀ must be within the range of appli- cability of Porod's law where dlnf/dlnh = -3 ash-+ oo.

An alternative approach of determining x-ray parameters is due to Debye and coworkers[40] and applies to completely random two phase systems. The character- istic function is given as:

/o(r)

= exp(-r

/a),

(12)

and the intensity for point optics is given as:

I(h)=

A ₂ ,

[ 1

+ ( 4~;e)2]

⁽¹³⁾

where A is a constant and 20 is the scattering angle. The correlation distance, a, can be obtained from a plot of 1-¹!² against (20)², which should yield a straight line, if the material is indeed random. Then:

A ( slope ) ¹¹²

a = 27r intercept ' ( 14)

The correlation distance, a, has been shown to be similar to the range of inho- mogeneity, lm, and the total surface area of the voids per unit mass is defined as before. Other secondary parameters that can be calculated from lm or a, if the pore volume fraction E is known, are: dvoids which corresponds to the average dimension of segments in the voids and is given by:

dvoids =

lm/(1 - t:),

(15)

and dsolid is defined as the average distance between pore walls within the carbon matrix and is given by:

(16) All of the above parameters apply to both dense and dilute void concentrations with the exception of R₉^, which applies only to dilute systems.

3.3.2 Results and Di'scussion

Several experimental x-ray profiles are shown in Fig. 11. It can be observed that the intensity of scattered radiation exhibits a continous profile, decreasing with the angle

of scattering. The intensity increases with heat treatment temperature and with the degree of oxidation as indicated from the plain polymer profiles. It also varies with the nature of the copolymer, depending on the concentration, and, presumably, size and shape of the voids. The fact that the carbon formed from glycerol-Triton and PFA exhibits the highest scattering is due partly on its enhanced porosity and partly on the degree of conversion that is the highest of all carbons in this figure.

The experimental curves were corrected for smearing effects, using the method of Schmidt and Hight[41] but with a unity weighting function that approximates a negligible slit width-to-height ratio[42]. The resulting intensity curve approximates the scattering that would have been observed from point collimation under the same conditions. The Debye plots generated for various copolymers, shown in Fig.

12, are reasonably linear. Correlation length values, a, have been estimated and summarized in Table VI. Values of the mean cord intercept length for the pores and the solid were calculated from the values of a. Most of the void sizes are below lOA, being in good agreement with previously reported values[l8,36,38] for glassy carbons that have undergone similar heat treatment. The voids in the 18% PEG char are exceptionally large, indicating the presence of a number of pores large enough to be penetrated by nitrogen at 77 K. These results are in good agreement with the large BET area observed for this material (Table III). The fact that the slope in the Debye plot for this last material is steep only for a limited range of scattering angles and then becomes milder at higher angles shows the existence of smaller voids along with the large ones. The Debye plot for the 25% carbon black containing material shows two or three distinct void sizes; the largest one, indicated by scattering at very small angles, corresponds to 150A (diameter) pores in excellent agreement with the porosimetry data. The smallest pores correspond to the values measured for the plain polymer matrix. Guinier plots for the copolymers exhibit a narrow angular range of linearity, suggesting a rather broad pore size

distribution. Radii of gyration for the linear region are included in Table VI. Porod plots indicate significant deviation from Porod's Law ( d In I/ d In h oc -3) at the tail of the scattering curve, suggesting anisotropic density fiuctuations[18]. Furthermore the Porod variant plots exhibited no maximum in the h³ I(h) value suggestive of interfaces with sharp edges and corners[43,44]. Since the Porod Law does not hold, the range of inhomogeneity was not calculated. Total surface areas were computed based on Debye's parameter a.

The scattered radiation from the materials heat-treated at elevated temperatures increases with temperature and degree of oxidation. This intensity increase is due to both the increase of the amount of the inhomogeneity and the enhancement in contrast between the two regions that produce the heterogeneity. The amount of inhomogeneity increases because the voids augment in number and size, the contrast is due to the expulsion of volatiles from the pores. As shown in Fig. 13 Porod 's law holds well at large angles for all of the materials. especially the partially burned ones. This suggests that the materials become progressively more homogeneous with heat treatment,as reported previously[18,39]. The Porod Invariant, h³ I(h), tends to limiting values at large angles, indicating a smoothing of the interfaces with pyrolysis temperature and oxidation (Fig. 14). The resulting lm values are in good agreement with Debye's correlation lengths a for the various materials, as shown in Table VII. Also shown are values for the mean cord intercepts, d, for the voids and the solid, and surface areas calculated from the lm values. The average void size is seen to increase with heat treatment and, degree of oxidation at the expense of the solid matrix in between voids. As also suggested by Perret and Ruland[18], the average thickness of the carbon matrix between the pore walls, of the oxidized materials, dsolid, is comparable to the layer stack height, L0 obtained from wide-angle scattering measurements (Table V). This indicates that pores are separated by only one stack carbon layers and supports the hypothesis of Jenkins

and Perret and Ruland [19,18] regarding the structure of glassy carbons.

Guinier plots for the heat treated materials are shown in Fig. 15. Guinier's law holds well for a large range of scattering angles indicating a narrow pore size distribution. At very small angles, the scattered intensities increases, probably be- cause of enlargement of surface pores by the combustion process. Calculated radii of gyration are listed in Table VII. These values, however, are questionable since the void fraction in glassy carbons is more than a few percent and Guinier's Law is ap- plicable only to dilute systems. Surface area calculations indicate the development of a large amount of porosity upon heat treatment. If one compares the 616 m² /g measured by SAXS to the 9 m² /g measured by BET for the material pyrolyzed to 1500 K, it is apparent that these pores are closed to N_{2 •} Partial oxidation causes an initial increase in surface areas because of pore openning, followed by a continous drop in areas upon further oxidation because of pore enlargement. The value of surface area for the plain polymer that has been partially burned in air at 1500 K is in excelient agreement with the N ₂ BET surface area (see Fig. 7), suggesting the development of a well-interconnected and open network of micropores.

In summary, the results of SAXS indicate that the initial materials which were heat-treated to 800 K possess a vast network of micropores that have an average size of less than 10

A.

The only exeptions are the materials containing carbon black spheres and large amounts of PEG. The size distributions of pores are broad and the pores are anisotropically distributed within the particle. The voids are not very well developed, formed between ribbon like layers of glassy carbon, and contain sharp edges and corners. As the carbonizing temperature increases, the void size increases because of evolution of heavy volatile hydrocarbons, structural ordering and densification of the carbon matrix. The data suggests that the voids become more rounded but not necessarily spherical[37], and more uniform in size and spatial distribution. The pore openings are small, and most of this porosity is closed to

N₂ at 77 K and open only to C02 at 195 K[38]. Upon partial combustion, the pores enlarge because of the combined effect of heterogeneous reaction between the carbon at the pore walls and the oxidizer and enhanced structural rearangement caused by the catalytic effects of 0 2 • After this treatment, the pore openings are sufficiently large for N ₂ to penetrate in BET measurements.