Structure of Physically Adsorbed Polymer Layers Measured by

Langmuir 2002, 18, 855-864

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Structure of Physically Adsorbed Polymer Layers Measured by Small-Angle Neutron Scattering Using Contrast Variation Methods John H. E. Hone,† Terence Cosgrove,*,† Marina Saphiannikova,† Timothy M. Obey,† John C. Marshall,† and Trevor L. Crowley‡ School of Chemistry, University of Bristol, Cantock’s Close, Bristol BS8 1TS, U.K., and Department of Chemistry, Salford University, Salford, Greater Manchester M5 4WT, U.K. Received May 14, 2001. In Final Form: October 23, 2001

The scattering from poly(ethylene oxide) adsorbed on polystyrene latex particles was measured as a function of the isotopic composition of the aqueous solvent. Oscillations in the data were observed even when the particle was matched in scattering length density to the solvent. These oscillations are a new and important characteristic of the scattering from an adsorbed polymer layer compared to the scattering from free chains. The contrast variation approach allowed the scattering from the interference between the layer and the particle to be measured explicitly. The data have been fitted using a model function to find the volume fraction profile of the adsorbed layer without the complications that can arise from densityfluctuation contributions to the scattering. It was therefore possible to obtain the fluctuation contribution itself by simple subtraction. The fluctuation scattering was found to be in good agreement with scaling predictions and with a new function derived for an exponential volume fraction profile.

Introduction It is well-known that the small-angle neutron scattering from a colloidal dispersion coated with a polymer layer contains information on the structure of the adsorbed layer.1-17 This structural information is normally presented as the volume fraction of the polymer at a distance z from the surface, known as a volume fraction profile φ(z). The explicit form of the volume fraction profile is important in colloid science in the steric stabilization of colloidal particles, and very few experimental methods apart from those based on neutrons or X-rays exist.18 The * Corresponding author. E-mail: [email protected]. † University of Bristol. ‡ Salford University. (1) Crowley, T. L. Ph.D. Thesis, University of Oxford, 1984. (2) Ottewill, R. H. In Colloidal Dispersions; Goodwin, J. W., Ed.; Special Publication 43; Royal Society of Chemistry: Cambridge, England, 1982. (3) Auvray, L.; de Gennes, P. G. Europhys. Lett. 1986, 2, 647. (4) Auvray, L. In Mecanique Physique Chimie Sciences De L’Univers Sciences De La Terre; Comptes Rendus De L’Academie Des Sciences Serie II: 1986; Vol. 302, p. 859. (5) Auroy, P.; Auvray, L.; Leger, L. J. Phys.: Condens. Matter 1990, 2, SA317. (6) Auvray, L.; Auroy, P. In Neutron, X-ray and Light Scattering; Linder, P., Zemb, T., Eds.; Elsevier Science Publishers: Amsterdam, 1991. (7) Auroy, P.; Auvray, L.; Leger, L. Physica A 1991, 172, 269. (8) Auroy, P.; Auvray, L.; Leger, L. Macromolecules 1991, 24, 2523. (9) Auvray, L.; Auroy, P.; Cruz, M. J. Phys. I 1992, 2, 943. (10) Cosgrove, T.; Crowley, T. L.; Vincent, B.; Barnett, K. G.; Tadros, T. F. Faraday Symp. Chem. Soc. 1981, 16, 101. (11) Cosgrove, T.; Obey, T. M.; Vincent, B. J. Colloid Interface Sci. 1986, 111, 409. (12) Cosgrove, T.; Crowley, T. L.; Ryan, K.; Webster, J. R. P. Colloids Surf. 1990, 51, 255. (13) Cosgrove, T.; Ryan, K. Langmuir 1990, 6, 136. (14) Cosgrove, T.; Heath, T. G.; Ryan, K. Langmuir 1994, 10, 3500. (15) Forsman, W. C.; Latshaw, B. E. Polym. Eng. Sci. 1996, 36, 1114. (16) Ye, X.; Tong, P.; Fetters, L. J. Macromolecules 1997, 30, 4103. (17) Caucheteux, I.; Hervet, H.; Rondelez, F.; Auvray, L.; Cotton, J. P. New Trends Phys. Phys. Chem. Polym., [Proc. Int. Symp.] 1989, 63. (18) Fleer, G. J.; Cohen Stuart, M. A.; Scheutjens, J. M. H. M.; Cosgrove, T.; Vincent, B. Polymers at Interfaces; Chapman & Hall: London, 1993.

volume fraction profile can be extracted from two different types of SANS experiments. First, and most straightforwardly, the scattering length density of the solvent can be matched to that of the substrate particles, a condition commonly referred to as “contrast match”. The measured scattering is only that produced by the adsorbed layer, with no contribution from the substrate particles. The second, more involved method requires the dispersion of both the coated and bare particles in a medium with a scattering length density different from that of the particles and the adsorbed layer. Manipulation of these data allows the scattering arising from the interference between the diffuse adsorbed layer and the colloidal substrate to be determined. The precise way in which this scattering may be measured is described in detail below. There are two salient differences between the two methods. The technique of contrast matching is extremely rapid, typically requiring a measuring time of a few hours per sample. However, local deviations (δc) from the average polymer concentration (c) in the layer give rise to a scattering contribution proportional to 〈δc2〉, which is greater than zero. As yet, the precise magnitude and form of this additional scattering for a physically adsorbed layer is generally unknown and difficult to quantify without very good data. Conversely, by measuring the interference scattering, the volume fraction profile can be found without the complications of additional unquantified scattering because 〈δc〉 ) 0, but the measurement time is considerably longer and the data manipulation quite convoluted. In this work, we have followed the scaling approach of Auvray and de Gennes3,4 to obtain an estimate of the magnitude and form of the fluctuation scattering contribution. We have also determined the interference scattering experimentally from one system at two different solvent compositions. This approach allows the volume fraction profile for the adsorbed layer to be determined with a much greater degree of certainty than previous methods allowed. Once known, the profile has been used

10.1021/la010709z CCC: $22.00 © 2002 American Chemical Society Published on Web 01/11/2002

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to calculate the scattering expected from an adsorbed layer at contrast match without any scattering from fluctuations. This calculated curve, when subtracted from the measured on-contrast scattering data, gives the additional scattering from adsorbed layer concentration fluctuations. We compare these data to theoretical predictions. Neutron Scattering Theory. In a SANS experiment, the scattering intensity is measured as a function of the momentum transfer, Q, which can be calculated from the wavelength of the incident radiation, λ, and the scattering angle, θ, thus:

Q)

4π θ sin λ 2

()

sity of the solvent must be different from that of the particle and the layer. Equation 4 applies to situations in which it is assumed that the particles are monodisperse and that the SANS instrument has infinite resolution. If ∆(Qro) is the spread in Qro arising from the polydispersity in the particle size (∆ro) and finite instrumental Q resolution (∆Q) such that

∆(Qro) ) Q∆ro + ro∆Q . 1,

then averaging over the polydispersity and instrument resolution gives

(1)

sin2(Qro) ) cos2(Qro) )

1

Here we use Crowley’s theory, which describes the scattering from a polymer-coated particle I(Q) as the sum of four parts: (1) scattering due to the particle Ipp(Q), (2) scattering due to the polymer layer Ill(Q), (3) interference between the particle and the adsorbed layer Ipl(Q), and (4) an incoherent background Iinc:

I(Q) ) Ipp(Q) + Ipl(Q) + Ill(Q) + Iinc

[

Ipp(Q) ) 9(Fp - Fs)2φpVp

]

sin(Qro) - Qro cos(Qro) 3

Q ro

3

2

(3)

where φp is the particle volume fraction, ro is the particle radius, Vp is the particle volume, and F is the scattering length density of the particle (p) and solvent (s), respectively. Interference Scattering. The interference scattering can be obtained by accurately measuring the scattering from three samples: (1) an off-contrast coated dispersion (i.e., Fp * Fs * Fl where Fl is the scattering length density of the polymer), (2) a dispersion of identical composition but without the adsorbing polymer, and (3) an on-contrast coated dispersion (i.e. Fp ) Fs * Fl). By subtracting the bare particle scattering and the scaled on-contrast scattering (i.e. the scattering from the layer alone as if it were in the off contrast solvent) from the total scattering, we obtain the interference between the layer and the substrate. We have assumed that the bare particle dispersion has the same volume fraction and polydispersity as the coated particles and that we can ignore scattering from free polymer in solution. The interference term (which has no fluctuation contribution1) is given by

Ipl(Q) ) 24πφp(Fp - Fs)(Fl - Fs) Q4ro2 [sin(Qro)

[sin(Qro) - Qro cos(Qro)]

∫0t φ(z) cos(Qz) dz + cos(Qro)

∫0t φ(z) sin(Qz) dz]

(4)

provided that ro . t, where t is the thickness of the layer and φ(z) is the volume fraction of polymer at a distance z from the surface. Assuming that the scattering length density of each component in the system and the particle size are known, it is possible to obtain φ(z) at any given solvent composition; however, the scattering length den-

1 2

(5b)

and

sin(2Qro) ) 0

(5c)

The interference scattering (eq 4) can be simplified to1

(2)

We start by studying each term in eq 2 in turn. Particle Scattering. The particle scattering is given by2

(5a)

Ipl(Q) )

12πφp(Fp - Fs)(Fl - Fs) 4

2

Q ro

[

∫0t φ(z) cos(Qz) dz -

∫0t φ(z) sin(Qz) dz]

Qro

(6)

In the experiments described here, the samples and the instrumentation used precluded the use of the simplification introduced in eq 6 because of the low polydispersity of the latex (ro ) 47.5 ( 4.6 nm) and the high Q resolution of the D22 camera. Under these circumstances, eq 4 cannot be used in its present form but must be modified to account for both of these features. The particle size distribution fits well to a log-normal distribution; therefore, the scattering is calculated for each radius and is weighted for the proportion of the population with that size.2 To overcome the finite Q resolution of the detector, the fit has been convoluted with a simple Gaussian function with a variable width. This fit is implemented by averaging the calculated intensity at each Q point; in other words,

Ires(Q) )

(

i)+n

∑

I(Q + iq)(Q + iq)-β exp -

i)-n i)+n

Q-β

∑

i)-n

(

exp -

1 (i - (n + 1)) 2

1 (i - (n + 1)) 2

4

)

2

4

)

2

(7)

where q is the (variable) step in Q that specifies the amount of smearing, -n to +n is the total number of steps (n ) 5 for all data fitting), and the exponential function is an appropriate Gaussian weighting. The index β is the power law of the underlying Q dependence of the scattering: for layer scattering, β ) -2; for the interference term, β ) -3; and for the particle scattering, β ) -4. To demonstrate the effects of polydispersity and instrumental resolution on the data, the interference scattering from a hypothetical adsorbed layer (arbitrarily chosen to have an exponential volume fraction profile) is plotted in Figure 1. The four curves have been calculated using different theoretical forms of the layer scattering equation, namely, scattering from (a) a layer adsorbed to monodisperse particles (ro ) 450 Å) at infinite resolution (eq 4), (b) a layer adsorbed to polydisperse particles (ro ) 450 ( 32 Å, σs ) 0.07; σs describes the width and skewness

Structure of Physically Adsorbed Polymer Layers

Langmuir, Vol. 18, No. 3, 2002 857

Previously we have shown that for smooth-layer scattering data20 neither eq 10 nor eq 9 completely describes the measured data, so another term to describe the fluctuations (I˜) has been used, thus

Ill(Q) ) I(Q) + I˜(Q,a)

Figure 1. Calculated interference scattering (with a constant 10 cm-1 added to remove the negative values) for (a) monodisperse particles (solid line, eq 4), (b) polydisperse particles (dotted line), (c) polydisperse particles with finite instrument resolution (dashed line), and (d) complete smoothing (dot-dash line, eq 6). All of the data were calculated with an exponential volume fraction profile.

of the log-normal particle size distribution) at infinite resolution, (c) a layer adsorbed to the same polydisperse particles, smeared with a resolution function q (q ) 5.0 × 10-4 Å-1) that quantifies the spread in Q at each data point, and (d) the scattering fully averaged for both resolution and polydispersity effects (eq 6). Layer Scattering. As described in the Introduction, it is possible to suppress the scattering from the particles selectively by adjusting the isotopic composition of the solvent in which the particles are dispersed (i.e., contrast match). At contrast match, the Ipp(Q) and the Ipl(Q) terms are both zero because Fp - Fs ) 0; therefore, the observed scattering Ill(Q) arises from the adsorbed layer only:1,3-9

Ill(Q) ) I(Q) + I˜(Q)

I(Q) )

12π∆Fls2φp Q2ro

|

∫0 φ(z) sin(Qro + Qz) dz| t

2

6π∆Fls2φp 2

Q ro

|

∫0t φ(z) exp(iQz) dz|2

φ(r b) ) φ(r) + δφ(r b)

(12)

where φ(r) is the average profile and δφ(r b) is the local deviation from the average concentration. The total polymer contribution to the scattering is given by

φp FF* Vp

(13)

∫ dV φ(rb) exp(-iQB br )

(14)

Ill(Q) ) where

F ) ∆Fls

is the scattering amplitude and F* is its complex conjugate. The integral ∫ dV indicates integration over all space in spherical coordinates and is given by

∫ dV ) ∫0π sin θ dθ ∫02π dφ ∫ r2 dr

(9)

(15)

Substitution of eq 12 into eq 13 gives the total scattering:

where ∆Fls ) Fl - Fs. Equation 9 describes the scattering from the layer if it is assumed that the adsorbing particle is monodisperse and that the scattering instrument has infinite Q resolution. In earlier work, it was assumed that neither of these criteria could be met,10-14,19,20 in which case eq 9 may be simplified using the assumptions of eq 5 to1

I(Q) )

where a is the monomer length. Using scaling arguments and assuming that the volume fraction profile decays as z-4/3, Auvray and de Gennes3 predicted that I˜ ∼ Q-4/3. Theoretical Scattering from Spatial Concentration Fluctuations. The scattering from an adsorbed polymer layer, Ill(Q), as discussed above, arises from two contributions: (1) the presence of the layer (the hI(Q) term) and (2) local spatial variations in the polarizability, δR (for light scattering) or variations in the scattering length density, δF, of the polymer (for neutron scattering). More precisely, I˜ ∝ δF2 (or δR2) ∝ δφ2, where φ is the polymer volume fraction. At first we shall consider both the scattering from the average profile hI(Q) and the fluctuation contribution I˜(Q) to the scattering resulting from spatial concentration heterogeneities in an adsorbed polymer layer. Next, we will focus on the I˜(Q) term. As the system of interest has spherical symmetry, the derivation will be performed using the spherical coordinate system b r ≡ (r, θ, φ) with the origin at the center of the particle. The complete polymer volume fraction profile φ(r b) is then given by

(8)

where hI(Q) is the scattering from the average structure of the layer and I˜(Q) is the scattering from spatial concentration fluctuations (described in more detail below). If the fluctuation contribution is ignored, the scattering can be described in terms of the scattering due to the average concentration in the layer (i.e., Ill(Q) ) hI(Q)) as follows:

(11)

(10)

φp Ill(Q) ) ∆Fls2 Vp

∫ dV1[φ(r1) + δφ(r b1)] exp(-iQ Bb r 1) ∫ dV2 [φ(r2) +

δφ(r b2)] exp(iQ Bb r 2) (16)

or equivalently

Ill(Q) )

∆Fls2φp [ Vp

∫ ∫ dV1 dV2 φ(r1) φ(r2) e-iQB (rb -rb ) + ∫ ∫ dV1 dV2 δφ(rb1) δφ(rb2) e-iQB (rb -rb )] (17) 1

1

(19) Washington, C.; King, S. M. Langmuir 1997, 13, 4545. (20) Cosgrove, T.; Hone, J. H. E.; Howe, A. M.; Heenan, R. K. Langmuir 1998, 14, 5376.

2

2

b2) and because the integrals with cross terms φ(r1) δφ(r

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φ(r2) δφ(r b1) are equal to zero. (Taking the first cross-term as an example, we have

from the particle surface for the case of an adsorbed polymer, we obtain

∫∫ φ(r1) δφ(rb2) e-iQB (rb -rb ) dV1dV2 1

2

g(z,r) )

that can be split into two independent integrals. Therefore, the cross-term can be rewritten as

∫φ(r1) e-iQB br dV1 ∫δφ(rb2) eiQB br 1

2

dV2.

The second integral has equal positive and negative contributions and is therefore equal to zero; hence, the whole term is also zero.) The first term in eq 17 represents the scattering from the average profile hI(Q), and the second term is the scattering from spatial concentration fluctuations I˜(Q) where r1 and r2 are two spatial positions within the adsorbed layer. At this stage, we shall drop the average profile term (discussed above) and introduce two physical restrictions to simplify the problem: (1) the system is spherically symmetric and (2) correlations between regions of different concentration die out within a so-called correlation length ξ. From restriction (1) it follows that the correlation term b2) depends only on the magnitudes of b r1 and b r δφ(r b1) δφ(r r2, not on their directions. Likewise, the phase )b r1 - b factor e-iQBbr can be replaced by its average sin(Qr)/Qr taken over all directions of b r.21 By means of these simplifications, the fluctuation term is reduced to the form

I˜(Q) ) ∆Fls2φpVp-1(4π)2

∫rr + tdr1 r12 ∫ dr r2〈δφ(r1) δφ(r1, r)〉 o

o

sin(Qr) (18) Qr

where ro is the particle radius and t is the thickness of the layer. Here the 〈 〉 brackets mean averaging over all r. directions of b r1 and b For the case of a thin adsorbed layer (i.e., t , ro), r1 ≈ ro, so I˜(Q) can be approximated by

o

o

I˜(Q) ) 12π∆Fls2φpυoro-1

By shifting to surface coordinates z ) r1 - ro and substituting Vp ) 4/3πro3, one obtains

I˜(Q) ) 12π∆Fls2 φpro-1

∫0t dz ∫ dr r2〈δφ(r1) δφ(r1, r)〉

sin(Qr) Qr (20)

Next we introduce the pair correlation function g(r) for a semidilute polymer solution derived by de Gennes:22

1 g(r) ) [〈c(0) c(r)〉 - c2] c

sin(Qr) Qr (24)

The scaling theory of de Gennes23 shows that at a distance z from the particle surface, a weakly adsorbed layer is “similar” to a semidilute polymer solution of correlation length ξ[φ(z)]. Hence, using a self-similar argument, ξ[φ(z)] is predicted to be equal to z. The relation ξ(φ) ) aφ-3/4, established for bulk polymer solutions, then leads to the average profile

φ(z) ≈

(az)

-4/3

for z ∈ D

(25)

where D is the size of the proximal region, which is approximately equal to the size of a monomer or to the thickness of the adsorbed polymer trains.23 Referring to Auvray and de Gennes,3 we are able to estimate the scattering due to spatial fluctuations in the adsorbed polymer layer if we divide the integral ∫t0 dz into two parts:

I˜(Q) ) I˜blob(Q) + I˜chain(Q)

(26)

For the case when the correlation length ξ ) z is smaller than Q-1 (z ≈ 2-250 Å, i.e., Q < 0.4), we see the scattering from blobs of size z. The pair correlation function for a semidilute solution is given by22

g(ξ,r) ∼ ξ

-1/3 -1 -5/3

r a

rξ

exp(-r/ξ),

(27)

One can note that no correlations exist between polymer regions separated by a distance exceeding ξ (restriction 2). In this case, the phase factor sin(Qr)/Qr is approximately equal to one, hence

I˜blob(Q) ≈ 12π∆Fls2φpυoro-1a-5/3

∫0Q

-1

dz φ(z)

∫0ξ(z) dr r2/3

(28)

and after integration with respect to dr

∫0Q

I˜blob(Q) ≈ (36/5)π∆Fls2φpυoro-1a-5/3

-1

(21)

dz φ(z) ξ(z)5/3 (29)

Substitution of eq 25 into eq 29 taking into account that ξ(z) ) z leads to

where c is the average monomer number density. By inserting

c(r) ) c + δc(r)

∫0t dz φ(z) ∫ dr r2 g(z, r)

g(ξ,r) ∼ r-4/3a-5/3, sin(Qr) (19) Qr

(23)

where c(z) ) φ(z)/υo is the average monomer number density at position z, and υo ) a3 is the monomer volume. Substitution of eq 23 into eq 20 gives

I˜(Q) ) ∆Fls2φpVp-1(4πro)2

∫rr + t dr1 ∫ dr r2〈δφ(r1) δφ(r1, r)〉

〈δc(z) δc(z,r)〉 c(z)

(22)

and introducing the dependence of g(z, r) on the distance (21) Debye, P. Ann. Phys. (Berlin) 1915, 46, 809. (22) de Gennes, P. G. Scaling Concepts in Polymer Physics; Cornell University Press: Ithaca, NY, 1979.

I˜blob(Q) ≈

27π∆Fls2φpa4 5(Qa)4/3ro

(30)

For the case when Q-1 < z < t (z ≈ 2-250 Å, i.e., Q > 0.009), we see the scattering from individual chains. (23) de Gennes, P. G. Macromolecules 1981, 14, 1637.


Langmuir, Vol. 18, No. 3, 2002 859

I˜chain(Q) ) 12π∆Fls2φpυoro-1

∫Qt

-1

dz φ(z)

sin(Qr) Qr (31)

∫0ξ(z) dr r2/3a-5/3

Here, the phase factor is not equal to one; however, Edwards24 showed that the Fourier transform of the second integral in eq 31 is equal to (1/(Qa)5/3). This term reduces eq 31 to

I˜chain(Q) ≈ 12π∆Fls2φpυoro-1(Qa)-5/3

∫Qt

-1

dz φ(z) (32)

which after inserting eq 25 and integrating with respect to z gives

I˜chain(Q) ≈

36π∆Fls2φpa4 (Qa)4/3ro

[

]

t -1/3 1 - 1/3 Q

(33)

Figure 2. Calculated fluctuation scattering (ro ) 450 Å, φ ) 0.047, ∆Fls ) 10-6 Å-2) for the case when the layer thickness t ) 200 (dashed line) and 300 Å (dotted line) using eqs 30 and 33, respectively, and when Qt . 1 (solid line) using eqs 30 and 34.

For the case where t -1/3 , Q1/3 or equivalently Qt . 1, eq 33 can be rewritten as

I˜chain(Q) ≈


(34)

This general result is the same as that given by Auvray and de Gennes3 but with scaling factors included. The assumption that Qt . 1, which leads to the result in eq 34, is only valid at high Q or for thick layers, as demonstrated by Figure 2. Equations 30 and 34 show that the fluctuation term has a Q-4/3 dependence and that the average layer scattering has a Q-2 dependence (eq 10). Therefore, the average layer scattering decays more rapidly and the fluctuation term is more important at high Q than at low Q. At low Q, the limit becomes invalid; however, as Q increases, Qt f 1 and I˜chain(Q) tends toward its maximum value given by eq 34. Hence we have shown that for a profile varying as z-4/3 in the regime a < z < Q-1, the spatial fluctuations in the adsorbed layer scale as Q-4/3. For the region where Q-1 < z < t, the fluctuation term varies as Q-4/3 at high Q, but the fluctuation scattering intensity decreases at low Q. The Q-Dependency of I˜ for an Exponential Profile. Because neutron scattering data from adsorbed homopolymer layers do not often fit well to a profile of the form z-4/3,25-27 the data are invariably fitted to an exponentially decaying volume fraction profile. This fitting alters the assumptions used in the previous section because the profile now has a different mathematical form. Instead of using eq 25, we shall replace it with

( )

φ(z) ) φs exp -

z zo

Figure 3. Calculated fluctuation intensity (∆Fls ) 10-6 Å-2, φp ) 0.047, ro ) 450 Å), I˜blob(Q), for a volume fraction profile scaling as z-4/3 (solid line, eq 30) and for a profile decaying exponentially according to φ(z) ≈ φs exp(-z/zo) (dashed line), where zo is taken to be 25 Å, φs ) 1, and the monomer length a ) 5 Å.

relation ξ(φ) ) aφ-3/4 (discussed above). Then, substitution of eq 35 into eq 29 leads to

I˜blob(Q) ≈ (36/5)π∆Fls2φpυoro-1

∫0Q

-1

dz φ(z)-1/4

(36)

We are now in a position to integrate the following equation along z:

I˜blob(Q) ≈ (36/5)π∆Fls2φpυoro-1φs-1/4

∫0Q

-1

( )

dz exp

z 4zo (37)

(35)

where φs is the fraction of the surface in direct contact with the polymer, and zo controls the span of the profile. In this case, ξ(z) * z and should be calculated using the (24) Edwards, S. F. Proc. Phys. Soc., London 1965, 85, 613. (25) Cosgrove, T.; Heath, T. G.; Ryan, K.; Crowley, T. L. Macromolecules 1987, 20, 2879. (26) Cosgrove, T. Chem. Ind. (London) 1988, 45. (27) Cosgrove, T. J. Chem. Soc., Faraday Trans. 1990, 86, 1323.

Integrating between the limits gives

[ ( ) ]

I˜blob(Q) ≈ 30π∆Fls2φpro-1a3φs-1/4zo exp

1 -1 4zoQ (38)

Figure 3 shows a log-log plot of the variation of the fluctuation intensity as a function of Q when calculated for a z-4/3 scaling profile (eq 30) and an exponentially decaying profile (eq 38).

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Although it is theoretically possible to derive two different forms of the fluctuation scattering, according to the shape of the volume fraction profile, it may not be possible to distinguish such subtle differences experimentally, particularly at high Q where the scattering intensity falls below 0.01 cm-1. We shall now determine the contribution I˜chain for an exponentially decaying profile. Substituting eq 35 into eq 32, we find

I˜chain(Q) ≈ 12π∆Fls2φpφsυoro-1(Qa)-5/3

∫Q t

-1

( )

z zo (39)

dz exp -

Upon integration with respect to z, this equation becomes

I˜chain(Q) ≈

12π∆Fls2φpa3φszo (Qa)5/3ro

[ ( ) ( )] exp -

1 t exp Qzo zo (40)

which is insignificant for small values of zo. Experimental Section Materials. A partially deuterated polystyrene latex (D-PSL) was prepared using a surfactant-free emulsion polymerization method.28 The glassware used for the polymerization reaction was cleaned in a concentrated base bath (ethanol saturated with NaOH) for at least 12 h, followed by 15 min in an acid bath (20% HNO3). Excess acid was removed with copious amounts of water. A three-necked round-bottomed flask was charged with 600 g of H2O (Milli Q, Millipore) and placed in a thermostatically controlled oil bath heated to 94 °C. The three necks of the flask were fitted respectively with a nitrogen line, a stirrer, and a reflux condenser. The H2O was degassed under nitrogen for ∼30 min. Some of the H2O was removed and used to dissolve 0.414 g of the initiator (ammonium persulfate, final concentration 2.5 × 10-3 mol‚dm-3, Aldrich 98+%). Approximately 0.9 mL of distilled H-styrene (BDH 99%) and ∼2.7 mL of freshly distilled D-styrene (Aldrich 98+ atom % D) were added and degassed for 10 min. Finally, the dissolved initiator was quickly added to the reaction mixture, and the reaction was then left to proceed under a nitrogen atmosphere for 18 h. The system was allowed to cool before the latex was poured through glass wool into dialysis tubing, which had been previously boiled several times in distilled water. The latex was purified by dialysis against 20 changes of water over a period of 2 weeks. The latex was concentrated by centrifugation and redispersed in D2O (Fluorochem, 99.9+ atom % D) to a solids concentration of 11.28 ( 0.04% w/w, equivalent to φp ≈ 11.25%. The H2O content of the resulting latex was determined using high-resolution NMR. The peak heights in the proton spectra of several H2O/D2O mixtures were measured and used to produce a calibration curve. From this curve the H2O content of the latex was found to be 13.1% w/w. Several electron micrographs of the dry particles were taken using a JEOL CX100 transmission electron microscope (TEM). The size and polydispersity of the latex were determined by analyzing the micrographs using a SeeScan image analysis system connected to a Sony CCD camera that was calibrated with micrographs of a 468 nm diffraction grating taken at the same magnification. Approximately 1000 particles were sized in this way. It was found that the average particle diameter was 95.0 ( 9.1 nm. The measured size distribution was also fitted with a log-normal distribution and found to give a modal particle diameter of 96.4 nm with a distribution skewness of 0.058. Poly(ethylene oxide) (PEO) was obtained from Polymer Laboratories Ltd. (batch no. 20829-4) and had a peak molecular weight (Mp) of 114 000 g‚mol-1 (≡ 114 kD) and a polydispersity index (Mw/Mn) of 1.04. (28) Juang, M. S.-D.; Krieger, I. M. J. Polym. Sci., Polym. Chem. Ed. 1976, 14, 2089.

Table 1. Instrument Configuration for D22 detector detector distance/ offset/ collimation/ wavelength/ m mm m Å 1.40 12.00

380 380

11.0 11.0

7.0 7.0

Q range/ Å-1 0.0049-0.0537 0.046-0.537

Coated Particle Preparation. The scattering length density of the particles was determined by means of a contrast-match plot,2 which involves dispersing the bare latex in various D2O/ H2O mixtures and recording the resulting scattering intensity. The square root of the initial intensity, xI(0) (or more practically, the intensity at the lowest Q value), is plotted against H2O concentration. When xI(0) ) 0, the scattering length density of the solvent and the particle are equal; therefore, the concentration of H2O required to contrast match the particles can be determined, and this value was used in subsequent experiments. The samples for the neutron scattering experiment were prepared by weighing out approximately 0.3 g of D-PSL. A calculated amount of stock polymer solution (∼0.2 mL, prepared in D2O) was then added to the D-PSL using a Gilson pipette. The amount of polymer added was sufficient to coat the sample29 and to give a continuous phase polymer concentration of ∼1 mg‚mL-1 in the bulk. To these samples, appropriate amounts of H2O and D2O were added such that the resulting dispersion consisted of 4.70% w/w D-PSL, a sufficient amount of polymer to give full surface coverage of PEO plus 1 mg‚mL-1 in the bulk and 12.35% w/w H2O dispersed in D2O. Measurements. The SANS measurements were performed on the D22 small-angle scattering instrument at the Institut Laue Langevin, Grenoble, France. The D22 instrument has a 2-D area detector consisting of 128 × 128 cells, each measuring 7.5 mm × 7.5 mm. Measurements were made at two sample-detector distances to give the full Q range (0.0049-0.537 Å-1). Table 1 shows the two instrument configurations used. The incoherent scattering from water was measured to remove perturbations in the data due to variations in the efficiency of each detector cell. These data, in combination with a knowledge of the solid angle and the primary beam intensity, were used to reduce the raw intensity data to absolute units (cm-1). Data Reduction. All of the samples were measured in 2 mm Hellma quartz cells, except water, which was measured in a 1 mm cell because of the very high scattering at high Q (1.4 m). The background noise (electronic and neutron scattering) was measured using the neutron absorber boron carbide (B4C) and subtracted from all of the data sets. The transmission of all of the samples and empty cells was measured and used to correct the data. Finally, because the scattering from empty cells can be significant at low Q, each sample was corrected for the scattering from its own cell. For the on-contrast samples with adsorbed polymer, the scattering from the bare particles dispersed in the same solvent as that of the coated latex was measured and subtracted from the layer scattering such that any residual scattering from the particles as a result of small deviations from contrast would be removed. Residual scattering from excess polymer in solution was negligible.

Results and Discussion The scattering from one of two different off-contrast coated and uncoated samples is shown in Figure 4. Both the coated and uncoated samples show periodic oscillations due to the relatively monodisperse spherical particles and indicate an approximate particle diameter of 900 Å. It is principally the numerical difference between these two data sets that introduces some errors into the interference data because both samples scatter very strongly but the difference between them is small, in comparison. It was therefore necessary to run each sample for up to 2 h to (29) Barnett, K. B. Ph.D. Thesis, University of Bristol, 1986.


Langmuir, Vol. 18, No. 3, 2002 861

Figure 4. Log-log plot of the scattering from PEO-coated PSLD9 in 6.2% v/v H2O (b) and bare PSL-D9 in 6.2% v/v H2O (0) fitted to eq 3 with polydispersity and Q-resolution corrections Table 2. Parameters Resulting from the Fits to Bare PSL-D9 Dispersed in 6.2% v/v H2O and 22% v/v H2O, with Op ) 0.047 parameter Fp - Fs/Å-2 Fs/Å-2 rp/Å σs resolution q/Å-1 (q increment for smoothing) background, Iinc/cm-1

PSL-D9 in 6.2% H2O

PSL-D9 in 22.0% H2O

-0.57 × 10-6 5.91 × 10-6 464 ( 2 0.058 7.7 ( 0.2 × 10-4

0.58 × 10-6 4.76 × 10-6 471 ( 2 0.058 7.7 ( 0.2 × 10-4

0.121 ( 0.004

0.346 ( 0.013

obtain extremely good statistics, even though the scattering from every off-contrast system was relatively large. The fits to the particle scattering data have been obtained by fixing the particle volume fraction at 0.047. The particle-solvent scattering length density difference, modal particle size and polydispersity, resolution, and incoherent background have been fitted. The parameters from the fits to the two different bare particle contrasts are given in Table 2. There is very good agreement between the fits for both the modal particle radius and the resolution smearing. The radius value also agrees well with the independent measurement of the modal particle radius by TEM (482 Å). The data manipulation required to obtain the interference function can be represented by

Ipl(Q) ) I(Q) - Ipp(Q) - Ill(Q)

∆Fls2 ∆Flp2

(41)

where I(Q) is the measured scattering from the coated sample dispersed in the off-contrast solvent, ∆Fls is the scattering length density difference between the layer and the off-contrast solvent (∆Fls ) -5.28 × 10-6 and -4.12 × 10-6 Å-2 for 6.2% and 22.0% H2O, respectively), and ∆Flp is the scattering length density difference between the layer and the particle, which is the same as the difference between the scattering length density of the layer and the contrast-match solvent (∆Flp ) -4.70 × 10-6 Å-2). Note that the incoherent background was subtracted before further corrections were made. The interference scattering for both of the off-contrast samples has been explicitly determined and is shown in Figure 5. The data from the on-contrast experiment is shown as a log-log plot in Figure 6. Rather than assuming that we

Figure 5. Interference term, Ipl(Q), for adsorbed 114 kD PEO on PSL-D9 (22.0% v/v H2O (0) and 6.2% v/v H2O (O)) fitted to eq 4 with polydispersity and resolution smearing. The fitting parameters to the two curves are identical except for the scattering length density differences.

Figure 6. Log-log plot of the on-contrast scattering from 114 kD PEO adsorbed to PSL-D9 (solid line fitted to eq 11 with polydispersity and resolution smearing) and calculated layer scattering using the volume fraction profile from a fit to the interference data, not including the fluctuation contribution (dashed line).

have obtained contrast exactly, the scattering from the same uncoated dispersion is measured and subtracted from the coated particle scattering. In this way, any residual contribution from the particle is removed. In actual fact, a fit to the on-contrast bare particles gave a scattering length density difference between the particles and the solvent of 0.05 × 10-6 Å-2, which is equivalent to an error of (0.7% w/w D2O that indicates that we are as close to the contrast-match point as inherent errors in the sample preparation and knowledge of the position of contrast match allow. Although the particles are effectively invisible, because there is a hollow sphere in the center of the adsorbed layer, there are oscillations in the data that provide a size equivalent to the particle diameter, as discussed above. Data Analysis Our intention is to fit the interference data to obtain the adsorbed layer volume fraction profile in the absence of scattering arising from concentration-fluctuation variations. From this profile we can calculate the scattering expected from the layer in the absence of fluctuations and subtract this calculated function from the measured data

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Figure 7. Volume fraction profile for 114 kD PEO adsorbed to PSL-D9 (solid line derived from the interference scattering; dashed line derived from the on-contrast scattering).

Figure 8. Scattering due to adsorbed layer fluctuations for 114 kD PEO on D9-PSL (O) as a log-log plot. The lines represent fits over the range 0.004 < Q/Å-1 < 0.1 to eqs 42 (solid line) and 38 (dashed line).

Table 3. Parameters Resulting from the Fit to the Layer Scattering from 114 kD PEO Adsorbed to PSL-D9 (Constrained Parameters in Parentheses)

Table 4. Parameters Derived from the On-Contrast Volume Fraction Profiles

parameter

value

preexponential factor, φs decay length, zo/Å particle size, rp/Å monomer length, a/Å particle distribution skewness, σs resolution smearing, q/Å-1 incoherent background, Iinc/cm-1

0.128 ( 0.005 31.86 ( 1.3 (482) 3.1 ( 0.04 (0.058) 6.34 ( 0.01 × 10-4 0.00513 ( 0.0002

to get the measured layer fluctuations. However, to start with, we shall fit the on-contrast scattering independently, including a fluctuation term in the fit, to obtain a volume fraction profile from the layer scattering (Figure 7). On-Contrast Data Fitting. The layer scattering data presented in Figure 6 has been fitted using eq 11, with smearing for polydispersity and resolution effects. In the polydispersity smearing, the skewness of the particle size distribution (σs ) 0.058) and the modal particle radius (rp ) 482 Å) was fixed to constrain the fit to known parameters. The scattering length density difference between the layer and the particle or the solvent was -4.70 × 10-6 Å-2. An attempt was also made to fit the on-contrast data in the absence of a fluctuation term. It was found that the data could be fitted, but to get a sufficiently high intensity at low Q, the volume fraction of polymer at the interface was unrealistically high and highly compressed. To fit the data, it is necessary to insert a variable model volume fraction profile into eq 11. The specific form of the volume fraction profile chosen was an exponentially decaying function given by eq 35. An exponential volumefraction profile was chosen because PEO physically adsorbed to polystyrene latex particles has been shown to have this shape from a direct inversion of SANS interference data30 and from the theoretical predictions of Scheutjens-Fleer mean-field theory.18 The parameters derived from the fitting process are shown in Table 3, and the volume fraction profile is shown in Figure 7 along with the profile obtained from the interference data (discussed below). The dependence of the fluctuation scattering on Q has been taken to be a -4/3 power law not only as a result of (30) Cosgrove, T.; Vincent, B.; Crowley, T. L.; Cohen Stuart, M. A. In Polymer Adsorption and Dispersion Stability; Goddard, E. D., Vincent, B., Eds.; ACS Symposium Series 147; American Chemical Society: Washington, DC, 1984.

parameter adsorbed amount, Γ/mg‚m-2 root-mean-square layer thickness, δrms/Å second moment of the layer thickness, σ/Å hydrodynamic layer thickness, δh/Å

on-contrast profile

independent measurement

0.48 ( 0.04 44.4 ( 1.8

0.4831

31.9 ( 1.2 24030

Table 5. Parameters Resulting from the Fit to the Interference Scattering from 114 kD PEO Adsorbed to PSL-D9 (Constrained Parameters in Parentheses) parameter preexponential factor, φs decay length, zo/Å particle size, rp/Å particle distribution skewness, σs resolution smearing, q/Å-1

value 0.139 ( 0.008 29.1 ( 1.8 (482) (0.058) 6.1 ( 0.3 × 10-4

the data presented later (Figure 8) but also because of the theoretical prediction of Auvray and de Gennes3 as well as our derivation in the Theory section. The parameters derived from the profile are shown in Table 4, and it can be seen that there is good agreement with an independent measurement of the adsorbed amount. Also, the monomer length obtained from fitting the total scattering with nonlinear least-squares analysis gives a reasonable value of ∼3 Å for PEO. Interference Data Fitting. The interference data presented in Figure 5 have been fitted using eq 4, including a baseline for the incoherent scattering with smearing for polydispersity and resolution effects. In the polydispersity smearing, the skewness of the particle size distribution was fixed at 0.058, as determined from a fit to the measured particle size distribution. The modal particle radius was allowed to vary so that maxima and minima in the data could be fitted accurately. As before, it was necessary to insert a variable model volume fraction profile in eq 4 and to use an exponential volume fraction profile to fit the data. The parameters arising from a fit to the two combined interference data sets (shown in Figure 5) are given in Table 5. The derived volume fraction profile is shown in Figure 7.

Structure of Physically Adsorbed Polymer Layers Table 6. Parameters Derived from the Two Volume Fraction Profiles parameter adsorbed amount, Γ/mg‚m-2 root-mean-square layer thickness, δrms/Å second moment of the layer thickness, σ/Å

interference profile

on-contrast profile

0.48 ( 0.06 40.4 ( 2.5

0.48 ( 0.04 44.4 ( 1.8

29.1 ( 1.8

31.9 ( 1.2

The two profiles are remarkably similar; consequently, the derived parameters in Table 6 are also similar. The agreement between the curves, which have been measured by two different and totally independent methods, is remarkable. This result in itself suggests that there are adsorbed layer fluctuations that must be accounted for when fitting on-contrast layer scattering data; however, from the interference profile, it is now possible to derive the absolute fluctuation scattering for this system. From the profiles, the adsorbed amount, Γ, the rootmean-square layer thickness, δrms, and the second moment of the layer have been determined20 and are shown in Table 6. The adsorbed amounts derived from the profiles are very similar to the values determined from the standard depletion method (0.48 mg‚m-2 31). The hydrodynamic thickness of a 114 kD PEO layer on PSL is approximately 240 Å by extrapolation.30 Although this value is considerably larger than the root-mean-square layer thickness derived from the profiles (see Table 6), similar root-meansquare thicknesses have been reported before for PEO on PSL12 and are thought to be a result of the insensitivity of the neutrons to the segments in the highly extended polymer tails.18,30 For comparison, we have marked the value of the radius of gyration for the same polymer in aqueous solution (Figure 7). Calculating the Average Layer Scattering. Now that we have a fluctuation intensity independent volume fraction profile for the adsorbed PEO derived from the interference data, we can insert the profile into eq 9 to obtain the scattering we would expect from the layer, excluding fluctuations. The data are then smeared with the particle polydispersity (σs ) 0.058) and the instrumental resolution (q ) 6.34 × 10-4Å-1 obtained from a fit to the on-contrast data, see Table 3). This calculation uses a particle size of 964 Å, which is the value used to fit the on-contrast data, and an added incoherent background of 0.0068 cm-1, which is the minimum value in the oncontrast data. This background value has profound implications on further data manipulation. The calculated scattering is that from the average profile of the layer only (i.e., hI(Q)). There is no fluctuation contribution. The calculated curve is shown in Figure 6 (dashed line) along with the measured on-contrast layer scattering (data points) (i.e., Ill(Q) ) hI(Q) + I˜(Q)). For all values of Q, the calculated scattering from the average profile is less than the measured scattering, which indicates that there is a measurable degree of scattering due to spatial concentration fluctuations. The maxima and minima are at approximately the same positions in Q because the substrate diameter chosen for the calculation was the same as the measured value taken from the particle size distribution. Fluctuation Contribution. The scattering due to concentration fluctuations in the layer, I˜(Q), can be explicitly obtained by subtracting the calculated scattering, hI(Q), from the measured scattering data, Ill(Q), shown (31) Shar, J. A.; Cosgrove, T. Private communication, 1998.

Langmuir, Vol. 18, No. 3, 2002 863

in Figure 6. The resulting fluctuation term is shown in Figure 8. The fluctuation scattering has an approximate power law dependence. The measured layer data decay into the incoherent background scattering, so the value of the measured incoherent level was added to the calculated data so that the backgrounds would cancel when the data sets were subtracted. The accuracy of the experiment is insufficient for these data to be relied upon with any degree of certainty; hence, all of the data with an intensity less than 0.01 cm-1 were disregarded for the purposes of data analysis. The solid line in Figure 8 is a fit with a Q-4/3 dependence given explicitly by

I˜(Q) )


(42)

which is the sum of eqs 30 and 34. All of the parameters except the monomer length are known, so the value for a can be deduced. Although the Q-4/3 fit is quite adequate, it is clear that this curve does not fit the data over the entire (acceptable) Q range. We have therefore introduced our prediction for the fluctuations derived from an exponentially decaying volume fraction profile, given by eq 38. It should be noted that this term is derived from just one of the regimes describing the fluctuations (see Neutron Scattering Theory section); therefore, a fit using eq 38 will underestimate the scattering intensity and overestimate the monomer length. The monomer-length values obtained by the two methods are 3.1 Å for the z-4/3 profile and 5.3 Å for the exponential profile. It was expected that the fit to eq 38, which is obtained from an exponential profile rather than from blob or whole-chain regimes, would give a larger monomer size. However, this fit is consistent with the idea that a represents the size of a statistical segment because PEO is a reasonably flexible polymer. Further theoretical work will be required before this idea can be stated with any certainty. However, the data show that the fluctuation scattering from an adsorbed layer can be reasonably well approximated by a -4/3 power-law Q dependence, allowing on-contrast SANS data to be fitted with relative certainty, at least for neutral homopolymers of intermediate length adsorbed to small particles. A better fit to the fluctuation scattering data is obtained by using the theoretical prediction derived in this work that assumes an exponentially decaying volume fraction profile. Conclusions In this work, we present the SANS data from PEO adsorbed to polystyrene latex particles at contrast match and the scattering from the interference between the adsorbed layer and the substrate particles. In both cases, the data show periodic oscillations due to the low particle polydispersity and the high instrumental resolution. New equations are presented to account for these previously unobserved oscillations. Analysis of the interference data allows the extraction of the volume fraction profile of the adsorbed polymer without the complication of having to allow for unquantified scattering arising from spatial density fluctuations within the layer. This profile was then used to generate the expected scattering from the same system dispersed in contrast-match solvent, and this data was subtracted from the measured on-contrast data. This technique provides a measure of the fluctuation scattering intensity,

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which was found to decay according to Q-4/3 as predicted by scaling theory. A new equation to predict the form of the fluctuation scattering more accurately on the basis of an exponentially decaying volume fraction profile, is also presented. This work shows the importance of including a parameter to account for adsorbed layer fluctuations when fitting SANS data from adsorbed polymer layers under conditions of contrast match.

Hone et al.

Acknowledgment. We are very grateful to Dr. S. Egelhaaf for his help as the D22 instrument scientist. We also thank the ILL, Grenoble, for the provision of beam time. Dr. S. King is thanked for reading the manuscript prior to submission. J.H. and J.M. acknowledge Kodak Limited and the EPSRC for funding. M.S. acknowledges the Royal Society for a fellowship. LA010709Z

Structure of Physically Adsorbed Polymer Layers Measured by

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