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Conformation of polymer molecules at solid-liquid interfaces by small-angle neutron scattering.


As for interfacial phenomena involving small molecules species, the behavior of polymer molecules at interfaces is an essential aspect of a wide variety of physical-chemical phenomena. Some examples are stabilization of colloidal particles (1), flocculation and wetting (2), adhesion (3), chromatography (4), restricted capillary flow in tertiary oil recovery (5), and biocompatibility (6).

The feature that distinguishes polymers from their small-molecule counterparts is their long-chain structure at the molecular level, and it is this longchain structure that makes the study of macromolecules confined to interfaces totally unlike any other areas of surface science. Nonetheless, like other interesting and important cases of molecules confined at interfaces--say, physical adsorption of gases in separation processes--or chemisorption of reactants and products in heterogeneous catalysis--the specific coverage of the adsorbing species, [gamma], is a quantity of paramount importance. With modern microscopic and surface science techniques, it is now common to be able to express [gamma]in mass of adsorbate per unit area of adsorbent surface. In considering polymers, however, [gamma], is only the beginning. In addition, the conformation of the polymer molecules in the interface is every bit as important--do the chains lie flat, or are they highly extended, and so on?

In considering polymer adsorption--including conformation effects--one must recognize two quite different domains of coverage that, in most respects, must be treated by different molecular models. The two domains are sparse coverage and moderate to heavy coverage.

1) Sparse coverage can be defined as an assembly of polymer molecules in the interface with centers of gravity of nearest neighbors separated by distances that are greater than the component of the mean-square radius-of-gyration in the direction parallel to the surface. The analogous situation for solutions would be concentrations well below the critical concentration for chain overlap (7). It was demonstrated long ago that the conformational statistics of polymer molecules sparsely covering an adsorbing surface are greatly perturbed both parallel to and normal to the surface (8, 9). 2) For moderate to heavy coverage, the domains of neighboring adsorbed polymer molecules overlap significantly, and, at least to a good first approximation, the density of repeat units is a function only of Z, the distance from the solid surface out into the continuum liquid. The analogous situation in solutions would be concentrations well above the critical concentration for chain overlap--i e., moderately concentrated to concentrated solutions. When this model of interfacial polymer is appropriate, only conformational statistics normal to the surface need be considered. These conformational statistics are expressed in terms a function called the polymer density profile [phi](Z), the volume fraction of polymer as a function of Z. A criterion for guaranteeing moderate to heavy coverage is that the volume fraction of polymer is the same for all elements of volume a given distance from the surface that are large compared to the volume of a repeat unit but small compared to [<[S.sub.2]>.sub.3/2] where <[S.sub.2]> is the mean-square radius of gyration of the polymer molecules.

It is important to be careful in the interpretation of the term "moderate coverage" in the context used here. Just as solutions of sufficiently high molecular weight polymer can be in the "moderately concentrated" regime at concentrations as low as one percent or less, an analogous interface can be in the "moderate coverage" regime even when the maximum polymer concentration in the interface is only 1% or less.

Comparison of sparsely covered surfaces with truly dilute solution and moderately to heavily covered surfaces with semidilute solutions are shown in Figs. 1 and 2.

It should be noted that the concept of a polymer density profile will always lose its meaning for sufficiently large Z. For sufficiently large Z, the volume fraction of polymer will always decrease with increasing Z to a value too small for the criterion described above to be obeyed. In treating the results of experiment in terms of theory, it is expected (or at least hoped) that the breakdown of the concept of a polymer density profile in the dilute (large Z) "tail" will have little or no effect on data interpretation.

There are only two experimental techniques that purport to be able to determine polymer density profiles directly--small angle neutron scattering (SANS) for polymers confined to the interface between diluent and high surface-area substrates (10-18) and neutron reflectivity (NR) for polymers confined to the interface between solvent and flat, extended surfaces (19, 20).

To date, SANS and NR experiments have been restricted to cases of moderate to heavy coverage. Intermolecular interference between individual molecules for the sparse coverage case would greatly complicate the interpretation of experiments.

It should be mentioned that the specific surface coverage, [gamma] (in mass of polymer per unit area of surface), is a research topic of long-standing interest, and one that has been explored for decades by many experimental techniques (21). It can be calculated directly from the polymer density profile when [phi](Z) is available:

(1) [Mathematical Expression Omitted]

where [bar]V is the partial specific volume of the polymer in the diluent.


Experimental System

The experimental system considered here consists of spherical silica particles, bare and covered with tethered poly(n-butyl methacrylate) chains. The silica particles were prepared and characterized under the direction of D. T. Wu by the research staff of the Marshall Laboratory of the DuPont Company by the method developed by Stober et al. (22). Transmission electron micrographs indicated that the particles were nearly monodisperse with a number-average diameter of about 2070 A and a weight average diameter of 2090 A. In addition, Yokoyama (23) measured the sizes of about 300 particles using a Horiba particle size analyzer. By this technique, the average particle diameter is 2080 A with a standard deviation of 156 A. The size distribution is thus quite narrow, with approximately 90% of the particles having a diameter within 1.5 standard deviations of the mean. In addition, sedimentation field flow fractionation measurements gave number-average and weight-average diameters of 1940 A and 2110 A respectively. Finally, a particle size of 2200 A-was determined by dynamic light scattering.

Considering the results of the particle size measurements, we elected to take the particle diameter as 2150 A.

The polymer was poly(n-butyl methacrylate), prepared by the group transfer polymerization described by Sogah et al. (24), which gives a narrow molecular weight distributions product. Polymerization was carried out in tetrahydrofuran using dimethyl ketene methyl trimethylsilylacetal as the initiator and tetrabutyl ammonium m-chlorobenzoate as the catalyst. The polymer was prepared by Wu, and described previously by Wu et al. (25). They determined molecular weight distribution by gel permeation chromatography, and the results gave a value of [M.sub.W] of 45,000 and a heterodispersity index, [M.sub.W]/[M.sub.N], of 1.02.

The silica particles were coated with polymer by Wu and his associates by chemically grafting the polymer molecules to the silica surface. The grafting process required the addition of tri-ethoxy silyl propyl methacrylate as the terminal repeat unit of the growing polymer chains. This was effected by adding the reagent to the reacting mixture before termination of the reaction with alcohol. These terminal repeat units then react with the silanol surface to form a covalent bond.

The chemical grafting process is described in detail by Wu et al. (26). It must be performed in a "common" solvent--that is, one that provides sufficient electrostatic repulsion between bare silica particles to stabilize the dispersion, while at the same time solubilizing the butyl methacrylate polymer. To accomplish this, the particles, after preparation, are transferred to ethanol by repeated centrifugation. This concentrated silica dispersion is then diluted with ethylene diacetate forming a mixture, which is 20/80 by weight ethanol/ethylene diacetate. The grafting reaction is carried out by addition of a large excess of polymer and heating the solution to 65[degrees]C for 12 furs.

Knowing the particle diameter and the amount of polymer grafted to the surface, it is possible to determine the surface coverage of the particles. The amount of grafted polymer was determined by thermogravimetric analysis (TGA). In this procedure, polymer is burned off of the silica particles by heating in a stream of air. Experiments were performed as described by Charsley et al. (27). Approximately 30 mg of the dry polymer-coated silica was placed on the TGA balance. The thermobalance was heated from room temperature to 700[degrees]C at 20[degrees]C/min. The purge gas was air at a flow rate of 100 cc/min. Effluent from the TGA was directed to a mass spectrometer, where it was analyzed for water and carbon dioxide resulting from oxidation of the polymer. The sample weight and the derivative of the weight as a function of temperature were recorded. Output from the mass spectrometer and the mass balance gave the mass of polymer burned off of the silica particles. The result gave a surface coverage of 3.44 mg/[m.sup.2].

Small-Angle Neutron Scattering: Data Treatment

Geometry of Scattering. Before examining data treatment per se, it is useful to consider the geometry used to describe a scattering experiment. The essential features are shown in Fig. 3.

The intensity of the neutrons scattered by an assembly of particles in dispersion into any given infinitesimal solid angle, d[omega], is characterized by the differential scattering cross section, [(d[sigma] / d[omega]).sup.d], of the assembly of scattering particles. The differential scattering cross section is a function of direction defined by d[omega] and is defined by the following equation:

(2) [Mathematical Expression Omitted]

where [] is the neutron flux of the incident beam (in neutrons per unit time per unit area), [f.sub.scat] is the neutron flux of the scattered neutron beam pointing in the direction of d[omega], and R is the radius of a reference sphere. The superscript d in Eq 2 indicates that the differential scattering cross section applies to the scattering from the aggregate of all of the particles in dispersion that are intersected by the incident beam. Or, in other words, in the notation used here, (d[sigma] / d[omega]).sup.d] represents scattering of the dispersion--particles and dispersant--minus scattering from the pure dispersant. Since (for all systems considered in this work) [f.sub.scat] is inversely proportional to [R.sup.2], the differential scattering cross section gives complete information on scattering of the particles as a function of direction.

When there is random orientation of the scattering particles in space--as is the case here--scattering is independent of [phi], and [(d[sigma] / d[omega]).sup.d] is a function of [theta] only. Furthermore, for the size of the scattering particles discussed in the work, it can be shown that [(d[sigma] / d[omega]).sup.d] is given by

(3) [(d[sigma] / d[omega]).sup.d] = N [(d[sigma] / d[omega]).sup.p]

where [(d[sigma] / d[omega]).sup.p] is the differential scattering cross section of a single particle averaged over all orientations, and N is the total number of particles intersected by the incident beam.

The SANS experiments were done at Los Alamos National Laboratory. A schematic of the scattering experiment is shown in Fig. 4. Since scattering is only a function of [theta], that is the only angle shown in this Figure. The scattered neutron flux is measured as a function of the so-called "momentum transfer vector," Q, which is defined by the equation

(4) Q = (4[pi] / [lambda]) sin ([theta] / 2)

where [lambda] is the wavelength of the neutrons. It can be shown that for assemblies of randomly oriented particles, scattering must be a function of Q only.

Data from a scattering experiment are, however, often reported as a dimensionless intensity, which we will denote as I, defined by the equation

(5) I = 1 / [] [(d[sigma] / d[omega]).sup.d]

where [] is the cross-sectional area of the incident beam passing through the dispersion. The intensity, I, and the differential scattering cross section of a collection of scattering particles is given by

(6) I = lN [(d[sigma] / d[omega]).sup.p]

where l is the path-length of the neutrons through the dispersion and N is the number of scattering particles per unit volume of dispersion. Experimental data are modeled by comparing predicted to observed intensities using Eq 6.

Theoretical Scattering from Bare Silica Particles. Before examining the experimental data, it is useful to consider the theoretical prediction of scattering from bare silica particles dispersed in a liquid. The differential scattering cross section, [(d[sigma] / d[omega]).sup.d], for a particle of any shape averaged over all orientations is written

(7) [Mathematical Expression Omitted]

where B is the contrast of the particles (scattering length density of particle minus that of the dispersant) as a function of position vector r relative to the geometric center of the particle. Again, it should be remembered that Eq 7 refers to the differential scattering cross section of the dispersion minus that of the pure dispersant.

When the scattering objects have spherical symmetry, as is the case here, averaging is not necessary and the scattering equation is written

(8) [Mathematical Expression Omitted]

For an assembly of identical spherically symmetric particles, when integrated over the angular coordinates, Eq 8 becomes

(9) [Mathematical Expression Omitted]

where B(r) is the contrast as a function of the distance from the geometric center of the particle.

For homogeneous spheres, the contrast can be factored out of the integral and Eq 9 written

(10) [Mathematical Expression Omitted]

where [B.sub.s] is the (uniform) contrast of a sphere.

If we define a function we will denote as the scattering function of the spheres, [F.sub.s](Q), by the following equation

(11) [Mathematical Expression Omitted]

Equation 10 becomes

(12) [Mathematical Expression Omitted]

For an assembly of monodisperse spheres of radius and volume [V.sub.s], the scattering function is given by (28)

(13) [Mathematical Expression Omitted]

and the scattering cross section by

(14) [Mathematical Expression Omitted]

The theoretical scattering cross section as a function of Q for an assembly of monodisperse spheres show maxima and minima, which, for sufficiently small spheres, are well resolved experimentally by SANS. An example is shown in Fig. 5.

If, however, spheres are as large as 1000 A, the maxima and minima fall close together across the experimental range of Q and are difficult to resolve with the best SANS equipment available today. Even with infinite resolution, the SANS experiment is essentially a process of sampling the rapidly oscillating theoretical curve at a set of values of Q, as shown in Fig. 6.

However, no instrument has infinite resolution. An intensity is reported for a set of "nominal" values of that are, in fact, neutron fluxes integrated over a range of Q bracketing each "nominal" value. The prediction of the experimentally measured scattering cross section can thus be written as

(15) [Mathematical Expression Omitted]


(16) [Mathematical Expression Omitted]

where [delta][q.sub.2] and [delta][q.sub.1] are the upper and lower bounds of the range of momentum transfer about the stated value of Q, and G(Q') is a weighting function centered around Q, the nature of which is a function of the experimental setup.

In the limit as R goes to infinity (or in practical terms, for R >> 1 / Q for all experimental Q) the integral in Eq 16 is expressed in closed form:

(17) [Mathematical Expression Omitted]

or, since we are considering R >> 1 / Q

(18) [Mathematical Expression Omitted]

Equation 18 comes close to being obeyed for the spheres of 2150 A used in this study. Indeed, for the higher values of experimental Q, Eq 18 is accurate to within a few percent. In general, and for the lower values of Q considered here, the mean values given by Eq 16 must be determined by numerical integration.

Figure 7 compares the experimental scattering with the model described in Eqs 15 and 16 where <[F.sub.s][(Q).sup.2]>, is determined by numerical integration for the first 10 lowest values of Q and by Eq 18 for the rest.(*)

Agreement between experiment and the model predictions for uniform, monodisperse spheres is excellent up to values of Q equal to about 0.04 [A.sup.-1]. Except for about the first 10 lowest-Q data points, scattering values are described by Eq 18. The striking deviations between the simple model, Eq 18, and the experimental points for Q > 0.04 [A.sup.-1] is, however, well understood. The spheres are not homogeneous, but have random inhomogeneities, as described by Debye and Bueche long ago (29), and these inhomogeneities are responsible for scattering in the higher Q range. The Debye-Bueche analysis has proven to be successful in interpreting scattering from glassy materials (30, 31) and amorphous polymer systems (32, 33).

Spheres With Random Inhomogeneities. Glassy materials (such as the amorphous silica particles used in this research) can be characterized by random fluctuations in scattering length density. Particles in dispersion thus demonstrate random fluctuations in contrast. These random variation in contrast were written by Debye and Bueche as

(19) B(r) = [B.sub.0] + [eta](r)

The term [eta](r) is a random fluctuation above and below the mean value of the contrast, [B.sub.0]. The average value of the random fluctuation is equal to zero, i.e.,

(20 <[eta](r)> [equivalent] [bar][eta](r) = 0

Debye and Bueche showed that the presence of random inhomogeneities introduces a contribution to the differential scattering cross section of the particle that is written as

(21) [Mathematical Expression Omitted]

where b is a parameter with units of length that is a measure of the average distance between neighboring maxima and minima in the fluctuating scattering length density.

An important result of the above analysis is that there is no interaction between scattering from the (hypothetical) homogeneous spheres themselves--Eq 15, or, when applicable, Eq 18--and from the random inhomogeneities, Eq 21. The differential scattering cross section is thus predicted to be just the sum of the two. For the case of large R, the result for particles with random inhomogeneities is simply (for the higher Q)) written

(22) [Mathematical Expression Omitted]


(23) [Mathematical Expression Omitted]

or, alternatively,

(24) [Mathematical Expression Omitted]

Two points should be noted: 1) The contribution from the inhomogeneities will always dominate the scattering at sufficiently large Q. 2) The measure of the intensity of the fluctuation in scattering length density, [epsilon], should be a relatively small number; 20% variations above and below [[rho].sub.s] would give values of [epsilon] of the order of 0.15. It follows, therefore, that for R [nearly equal to] 1000 A and b [nearly equal to] 10 A, the switch-over would be in the range of Q [nearly equal to] 0.03 [A.sup.-1].

The effect of inhomogeneities on scattering in our experiments is illustrated in Fig. 8. The data are the same as in Fig. 7, except that scattering from inhomogeneities has been added. Scattering is associated with a value of [epsilon] of 0.14 and b of 10 A.

Figure 9 compares the model with experimental results when incorporating scattering both from the hypothetical homogeneous spheres and the random density fluctuations. Agreement between experiment and the theoretical sphere model is quite satisfactory.

Adsorbed Polymer: The Polymer Density Profile

When adsorbed polymer overlays the spheres the scattering function of the system is given by (28)

(25) [Mathematical Expression Omitted]

where [B.sub.p] is the contrast of the polymer. We take the first integral in the square brackets over the volume of the sphere and the second integral over all space. The function [phi]'(r) is related to the polymer density profile by the relationships

(26) [phi]'(r) = 0 if r < R where Z = r - R = [phi](Z) if r [greater than or equal to] R

The second integral in the square brackets in Eq 25 can be written as

(27) [Mathematical Expression Omitted]

It is convenient to recast Eq 27 in the following form

(28) [Mathematical Expression Omitted]

where [T.sub.s](Q) and [T.sub.c](Q) are referred to here as the sine and cosine transforms of the polymer density profile, and are defined by the equations

(29) [Mathematical Expression Omitted]


(30) [Mathematical Expression Omitted]

In most cases of interest when 2150 A diameter spheres are used, R >> Z up to values of Z for which [phi](Z) is negligible. In these most common of cases, the term in parentheses in Eqs 29 and 30 can be replaced with unity, making for simpler computation of the transforms.

It is important to note in Eq 28 that the scattering function of the adsorbed polymer has the same rapidly oscillating trigonometric functions as the sphere themselves, and thus require the same averaging process in modeling the scattering experiment.

Scattering of particles with random inhomogeneities and with adsorbed polymer is, for the larger values of Q, given by the equation

(31) [Mathematical Expression Omitted]

Determining the Polymer Density Profile

The polymer density profile can be determined as follows. Scattering from the dispersion of bare particles is subtracted from scattering from the dispersion of particles with adsorbed polymer. This difference scattering is given by the following equation:

(32) [Mathematical Expression Omitted]

In principle, the ideal experimental conditions are when the scattering lengths of the dispersant and spheres are identical, i.e., [B.sub.s] is equal to zero--the so-called contrast-matching condition. If that be the case, Eq 32 can be inverted directly to give [phi](Z). Unfortunately, in this work the total signal-to-noise ratio was too small under contrast-matching conditions for sufficiently precise data to be taken. Best results were obtained far from the match point.

Data analysis thus consists of comparing [Mathematical Expression Omitted] with predictions based of various proposed polymer density profiles, [phi](Z). The better the fit between the experimental and predicted difference scattering, the more probable that the proposed [phi](Z) represents a good representation of the true polymer density profile.

It must be recognized that it is impossible to assert that any proposed [phi](Z) is the correct polymer density profile--only that it is consistent with the experimental data. Furthermore, it could be possible that two or more analytical or numerical forms of [phi](Z) that give "similar looking" polymer density profiles could give equal agreement with experimental data when substituted into Eq 32.


Figure 10 shows the experimental difference scattering, [Mathematical Expression Omitted], for the polymer-substrate system studied in this research along with predicted values obtained by assuming the double exponential polymer density profile given in Eq 33

(33) [Mathematical Expression Omitted]

and shown in Fig. 11. The double exponential polymer density profile is not associated with any particular theory but was chosen to give the best agreement (given time and patience) between the prediction of the model and the experimental results. The area under this polymer density profile gives a value for the specific coverage of 3.42 mg/[m.sup.2], which is in excellent agreement with the value of 3.44 mg/[m.sup.2] determined by TGA. Clearly, this level of agreement is to a very large extent fortuitous. Our experience with TGA experiments indicates that they will give values of [gamma] only to within about [+ or -] 10%. Furthermore, there is no reason to believe that the area under the polymer density profile should give a value of [gamma] to any better accuracy than [+ or -] 10%. Indeed, we suggest that agreement between the values of [gamma] determined by these two experimental techniques to within [+ or -] 20% can be interpreted as consistent.

Experimental error, as reported by the Los Alamos SANS facility, is also shown in Fig. 10. Whereas the open circles indicate the most probable values of the difference scattering, the open triangles and squares represent the most probable values [+ or -] one standard deviation. We suggest that a reasonable minimum criterion for agreement between a proposed polymer density profile and the experimental difference scattering is that the computed difference scattering lie within the envelope defined by the experimental error curves.

One should note that Eq 33 gives a value of zero to the polymer density profile at Z = 0. This is, of course, not physically possible. But it is easy to show that, at a coverage of 3.4 mg/[m.sup.2], the volume fraction of polymer at the surface would be about 0.02-0.05. Inspection of Fig. 11 shows that little distortion of the displayed density profile at low-Z would be required for this function to demonstrate such a small value of [phi](0). The profile at the smallest values of Z affects the difference scattering most at the highest values of Q. If the profile were thus modified at low-Z to give a realistic volume fraction of polymer at Z = 0, the modification would only slightly affect the difference scattering (Fig. 10) in the "tail" where experimental error is the greatest. We have thus elected not to make such a refinement since that would be pushing the interpretation beyond the accuracy of the experiments. We would simply prefer to assert that the true polymer density profile has an intercept of 0.02-0.05 and not the 0.0 shown in Fig. 11.

Figure 11 shows that for 0 < Z < 40 A the volume fraction of polymer increases with increasing Z. The region between Z = 0 and Z equal to about 30 A is sometimes referred to as a depletion layer. This term is used since this "layer" that lies next to the surface has a volume fraction of polymer that is less than that near the maximum value of the profile. The existence of a depletion layer is of considerable interest, since such behavior is in sharp contrast to that expected for polymer density profiles associated with the physical adsorption of homopolymers or polymers grafted to surfaces that are also adsorbing surfaces. For these interfacial regions, the value of [phi](Z) is expected to be a maximum at the surface and decrease with increasing Z.

The striking contrast between the difference scattering by interfaces with a depletion layer and those without is shown in Figs. 10 and 11. In these Figures we show the polymer density profiles and predicted difference scattering for our "best" model and for a mean-field model developed by Milner, Witten, and Cates (34, 35) (which we will denote as the MWC theory). Although the MWC theory was formulated for polymer molecules tethered to otherwise nonadsorbing surfaces, their results predict a polymer density profile of the following form, which does not demonstrate a depletion layer:

(34) [phi](Z) = A([L.sup.2] - [Z.sup.2]) if 0 [greater than or equal to] Z [greater than or equal to] L = 0 otherwise

Parameter L in the MWC theory is the value of Z where the polymer density profile vanishes--a parameter that can be considered as a measure of the thickness of the polymer layer. The parameter A is related to the chain length, and the polymer-polymer and polymer-solvent interaction energies (which were lumped into an interaction strength [omega]). For comparison of the MWC theory with our "best" model, we scaled the parameters in Eq 34 to give the correct specific surface coverage and a reasonable value for L. The predicted difference scattering for the MWC theory is shown along with that of our "best" model in Fig. 10. The scaled MWC theory is shown along with our "best" model in Fig. 11.

Although the MWC theory was, in fact, designed for tethered polymer chains, it clearly does not apply to our experimental system. And although the MWC theory totally misses the mark in describing the physics of this interfacial system, it is reported to be in good agreement with numerical calculations (34, 35). It seems clear, therefore, that since the MWC analysis and the numerical work are in agreement, the problem lies in assumptions associated with this particular implementation of the mean-field theory that applies to both.

Milner, Witten, and Cates write, "If the interaction strength w is small and the average concentration <[phi]> is large, the mean-field approximation is valid" (35). They go on to say, "In this case, a chain emerging from the surface encounters many other chains before it noticeably avoids itself." For our experimental system we find that [phi](0) = 0.02-0.05--which may well be too small for the MWC theory to apply--and even at Z = 20 A, [phi] is equal to only about 0.2. Furthermore, although the picture that best describes our interfacial region is that shown in Fig. 2, it can also be determined that a segment of a chain emerging from the surface in our interfacial region does not encounter many segments from other chains before it encounters another segment from the same chain. It is unlikely, therefore, that the assumptions required for the MWC analysis to apply are obeyed for our experimental system.

We should like to add that, in our experience, it is very difficult to graft chain ends to a surface to a much higher level of coverage than was done in the research reported here. There is thus a real question as to whether or not the MWC theory would ever be applicable to chemically grafted polymer. The question can only be answered by more experimental work.

It is interesting, however, that rather old theoretical work based on the famous 1943 treatise on chain statistics by Chandrasekhar (36) does seem to describe the physics of our interfacial system rather well. In particular, theories by Meier (37) and Hesselink (38) for tethered chains immersed in theta solvents give good semiquantitative pictures of the profiles for the tethered chains studied in this work. This is illustrated in Fig. 12. Not only do the Meier and Hesselink profiles demonstrate the required depletion region, their predicted maxima have no arbitrary constants, but are determined entirely by the unperturbed mean-square radius of polymer molecules. The two theoretical curves in Fig. 12 are scaled to give the same maximum values as the "best" model. More on the Hesselink and Meier theories and associated Monte Carlo calculations will be presented in a subsequent paper.


This work was supported by the Marshall Laboratory of the DuPont Company and by the State of Pennsylvania through the Ben. Franklin Partnership. Small-angle neutron scattering measurements were done at the LANSCE facility of the Los Alamos National Laboratory. Indeed, we are especially grateful to the staff at LANSCE for their time and effort devoted to our experimental effort.


(In order of introduction in the text)

[gamma] = Specific coverage of surface with adsorbed or tethered polymer, commonly expressed in mg/m2.

Z = Distance from the adsorbing surface out into the continuum fluid. [phi](Z) = Polymer density profile, i.e., the volume fraction of polymer in the interface as a function of Z. [bar]V = Partial specific volume of the dispersant. [(d[sigma]/d[omega]).sup.d] = Differential scattering cross section of the assembly of scattering particles. The superscript implies that the particles are in dispersion. N = The total number of dispersed particles intersected by the incident beam. [([d[sigma]/d[omega]).sup.P] = The differential scattering cross-section of a single particle averaged over all orientations. The superscript reminds the reader that this terms applies to individual particles. [phi] = Angle between scattered and incident neutron beams. [lambda] = Wavelength of the neutrons. Q = The absolute value of the scattering vector (usually simply referred to as the scattering vector). Equal to (4[pi]/[lambda])sin([theta])/2). [] = The cross sectional area of the incident beam that passes through the dispersion. I = A dimensionless scattering intensity, equal to (1/[])(d[sigma]/d[omega])[sup.d] l = The path-length of the neutrons through the dispersion. N = The number of scattering particles per unit volume of dispersion. r = Position vector (usually taken with the origin at the center of mass of the particle). B(r) = Contrast of the particle as a function of position (contrast being defined as scattering length density of particle minus that of the dispersant). [B.sub.s] = The (constant) contrast of a uniform sphere. R = Radius of spherical particle. [V.sub.s] = Volume of spherical particle. [F.sub.s](Q) = Scattering function of homogeneous spheres, equal to [3[V.sub.s]/[(RQ).sup.3]][sin(RQ)-(RQ)cos(RQ)]. ([F.sub.s][(Q).sup.2])>[sub.l] = Average of [F.sub.s][(Q).sup.2] over a range of scattering vectors (denoted as Q') centered over some specified value of p. [B.sub.0](r) = Average value of the contrast for a particle with a contrast that fluctuates at random from place to place. n(r) = Fluctuation in contrast, above and below [B.sub.0](r), for a particle with a contrast that fluctuates at random from place to place. b = A parameter with units of length that is a measure of the average distance between neighboring maxima and minima in contrast for a particle with a contrast that fluctuates at random from place to place. [epsilon] = A measure of the relative magnitude of the contrast fluctuation, equal to ([n.sup.2])/[Mathematical Expression Omitted]. [B.sub.p] = Contrast of bulk polymer. [phi]'(r) = A function related to the polymer density profile by the relationships: [phi]' = 0 if r < R, and [phi]'(r) = [phi](Z) if r [greater than or equal to] R and where Z = r - R. [T.sub.s](Q) = By definition, the sine transform of the polymer density profile, given by Q[Mathematical Expression Omitted][1 + (Z/R)][phi](Z)sin(QZ) dZ. [T.sub.c](Q) = By definition, the cosine transform of the polymer density profile, given by Q [Mathematical Expression Omitted][1 + (Z/R)][phi](Z)cos(QZ) dZ.


(*) Equation 18 is quantitative over the complete range of experimental Q if R > 2000 A, but spheres that large were not available for this research. We expect, however, that they will be available for future experiments.


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W. C. FORSMAN and B. E. LATSHAW, Present address: Air Products and Chemicals, 7201 Hamilton Boulevard, Allentown PA 18195.
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Author:Forsman, W.C.; Latshaw, B.E.
Publication:Polymer Engineering and Science
Date:Apr 1, 1996
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